Communications In Mathematical Physics - Volume 287

Commun. Math. Phys. 287, 1–40 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0598-y Communications in Mathe...

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Commun. Math. Phys. 287, 1–40 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0598-y

Communications in

Mathematical Physics

Mating Non-Renormalizable Quadratic Polynomials Magnus Aspenberg1, , Michael Yampolsky2, 1 Mathematics Division, RTH-Royal Institute of Technology, st-10044 Stockholm, Sweden 2 Mathematics Department, University of Toronto, 40 St George Street, Toronto,

ON, M5S 2E4, Canada. E-mail: [email protected] Received: 27 January 2008 / Accepted: 14 April 2008 Published online: 16 August 2008 – © Springer-Verlag 2008

Abstract: In this paper we prove the existence and uniqueness of matings of the basilica with any quadratic polynomial which lies outside of the 1/2-limb of M, is nonrenormalizable, and does not have any non-repelling periodic orbits.

1. Introduction 1.1. Two definitions of mating. The idea of mating quadratic polynomials was introduced by Douady and Hubbard [Do2] as a way to dynamically parameterize parts of the parameter space of quadratic rational maps by pairs of quadratic polynomials. We will present several different ways of describing the construction, which lead to equivalent definitions in the case which is of interest to us. Consider two quadratic polynomials f 1 (z) = z 2 + c1 and f 2 (z) = z 2 + c2 whose Julia sets J1 and J2 are connected and locally connected. For i = 1, 2 denote i the Böttcher coordinate at infinity ˆ i → C\ ˆ D, ¯ i : C\K where K i is the filled Julia set of f i . It gives a conjugation i ◦ f i (z) = (i (z))2 , for i = 1, 2. Carathéodory’s Theorem implies that i−1 extends to a continuous parameterization ∂D → Ji . Setting γi : t → i−1 (e2πit ) ∈ Ji , The first author was partially supported by the Foundation Blanceflor Boncompagni-Ludovisi, née Bildt and by the Fields Institute. The second author was partially supported by an NSERC Discovery Grant.

2

M. Aspenberg, M. Yampolsky

we have f i (γi (t)) = γi (2t).

(1.1)

The topological space X = (K 1 K 2 )/(γ1 (t) ∼ γ2 (−t)) is obtained by glueing the two filled Julia sets along their boundaries in reverse order. Note that by (1.1) the dynamics of f 1 | K 1 and f 2 | K 2 correctly defines a dynamical system F : X → X, F = ( f 1 K 1 f 2 K )/(γ1 (t) ∼ γ2 (−t)). 2

If X is homeomoprhic to S 2 , then we say that f 1 and f 2 are topologically mateable. In this case, we call the mapping F the topological mating, and use the notation F = f 1 T f 2 . ˆ Assume further that there exists a homeomorphic change of coordinate ψ : X → C ◦

◦

which is conformal on K 1 ∪ K 2 and such that ˆ →C ˆ R = ψ ◦ F ◦ ψ −1 : C is a rational mapping. We then say that R is a conformal mating (or simply a mating) of f 1 and f 2 , and write R = f 1 f 2 . The pair of quadratics f 1 and f 2 is then called conformally mateable. Conformal mateability thus implies, in particular, topological mateability. Let us give another useful definition of mating. Let © be the complex plane compactified by adjoining the circle of directions at infinity {∞ · e2πiθ : θ ∈ S 1 }. Given two quadratic polynomials f 1 and f 2 as before, consider the extension of f i to the circle at infinity given by f i (∞ · e2πiθ ) = ∞ · e4πiθ . Glueing the two circles at infinity in reverse order, we obtain a 2-sphere = ©1 ∪ ©2/ ∼∞ , with the equivalence relation ∼∞ identifying (∞ · e2πiθ1 ) with (∞ · e2πiθ2 ) whenever θ1 = −θ2 , and a well defined map f 1 F f 2 equal to f i on ©i , i = 1, 2. The map f 1 F f 2 is called the formal mating between f 1 and f 2 . For each θ ∈ S 1 we denote Ri (θ ) the external ray of f i with angle θ given by i−1 ({r e2πiθ for r ≥ 1}). Label Rˆ i (t) the closure of Ri (t) in . We define the ray equivalence relation ∼r on in the following way: x ∼r y if and only if there exists a finite sequence of closed external rays { Rˆ i j (t j )} j=1,...,k with the property Rˆ i j (t j ) ∩ Rˆ i j+1 (t j+1 ) = ∅, for 1 ≤ j ≤ k − 1 and Rˆ i1 (t1 ) x, Rˆ ik (tk ) y. If f 1 and f 2 are topologically mateable then it follows from the definition that the topological space ©1 ©2/ ∼∞ modulo ∼r is again a 2-sphere and f 1 T f 2 = f 1 F f 2/ ∼r .

Mating Non-Renormalizable Quadratic Polynomials

3

We can now give another equivalent definition of conformal mating in terms of ray equivalence: f 1 and f 2 are conformally mateable if there exists a rational mapping ˆ →C ˆ and a pair of semiconjugacies φi : K i → C, ˆ i = 1, 2, R:C R ◦ φi = φi ◦ f i , ◦

such that the following holds: φi is conformal on K i , and φi (z) = φ j (w) if and only if z ∼r w. The map R is called a conformal mating between f 1 and f 2 . Recall that two branched coverings Fi : S 2 → S 2 , i = 1, 2 with finite postcritical sets Pi are equivalent in the sense of Thurston if there exist orientation preserving homeomorphisms of the sphere φ and ψ such that φ ◦ F1 = F2 ◦ ψ, and ψ is isotopic to φ rel P1 . Using Thurston’s characterization of postcritically finite rational mappings as branched coverings (see [DH2]), Tan Lei [Tan] and Rees [Re1] demonstrated that if f i (z) = z 2 + ci , i = 1, 2 is a pair of postcritically finite quadratics and the parameters c1 and c2 are not in conjugate limbs of the Mandelbrot set, then the formal mating f 1 F f 2 (or a certain degenerate form of it) is equivalent to a quadratic rational map R in the sense of Thurston. Further, Rees [Re2] and Shishikura [Sh1] showed that under the above assumptions, f 1 and f 2 are conformally mateable. Note that the condition that c1 and c2 are not in conjugate limbs is clearly necessary for topological mateability. Indeed, otherwise the cycles of external rays {R1 (t j )} and {R2 (s j )} landing at the dividing fixed points of the respective maps have opposite angles t j = −s j (see e.g. [Mi3]). Thus { Rˆ 1 (t j )} ∪ { Rˆ 2 (s j )} separates and therefore / ∼r is not homeomorphic to S 2 . It is remarkable that this condition is also sufficient when f 1 and f 2 have finite critical orbits, as this includes cases when both Julia sets are dendrites with empty interior. First examples of matings not based on Thurston’s characterization of rational maps appeared in the paper of Zakeri and the second author [YZ]. Before formulating it, recall that an irrational number θ ∈ (0, 1) is of bounded type if there exists B > 0 such that θ can be expressed as an infinite continued fraction with terms bounded by B. Theorem. Let θ1 and θ2 be two irrationals of bounded type, such that θ1 + θ2 = 1. Then the pair of quadratic polynomials f i = e2πiθ j z + z 2 , j = 1, 2 are conformally mateable. The mating R = f 1 f 2 is unique up to a Möbius change of coordinates, and is identified algebraically. However, it is very far from being postcritically finite. The postcritical sets of its two critical points are quasicircles, bounding a pair of Siegel disks. The approach taken in [YZ] consists in defining a dynamical puzzle partition of ˆ for the mapping R. The renormalization theory of critical circle the Riemann sphere C maps [Ya] can be used to show that nested sequences of puzzle pieces shrink to points. This provides a combinatorial description of the Julia set of R, sufficient to verify that it is a mating. The history of the problem we consider in this paper is as follows. In 1995 J. Luo [Luo] has proposed an approach to constructing a particular class of non postcritically finite matings of the following sort. A quadratic polynomial f c (z) = z 2 + c is called starlike if c is contained in one of the hyperbolic components attached to the main cardioid of the Mandelbrot set M. The name is due to the fact that Hubbard trees associated to such components have only one branching point. A Yoccoz’ quadratic polynomial has only repelling periodic cycles, and is renormalizable at most finitely many times. Yoccoz (see e.g. [Hub]) has proved that such

4


polynomials are combinatorially rigid, and have locally connected Julia sets. Luo has proposed mating starlike maps with Yoccoz’ ones, arguing that the Yoccoz’ puzzle partition for quadratics can be transplanted into the quadratic rational map. In this paper we carry this program out for a particular instance of a critically finite starlike polynomial f −1 (z) = z 2 − 1, whose Julia set is known as the basilica. We use the symbol ◦◦ as a graphical reference to this particular quadratic parameter, to avoid awkward notation. Thus f −1 becomes f ◦◦ , and its Julia set is denoted J◦◦ . We prove: Main Theorem. Suppose c is a non-renormalizable parameter value outside the 1/2-limb of M such that f c does not have a non-repelling periodic orbit. Then the quadratic polynomials f c and f ◦◦ are conformally mateable, and their mating is unique up to a Möbius coordinate change. It will be evident from the argument how to adapt it to work for an arbitrary starlike map; however, we decided to specialize to the case f◦◦ for the sake of clarity. Potentially, the methods of the proof should also work for the case of a general Yoccoz’ parameter c, or even an infinitely renormalizable parameter with good combinatorics. Since f ◦◦ has a superattracting orbit 0 → −1 → 0, any candidate mating R must exhibit a superattracting orbit of order 2. Let us place the critical point at ∞ and assume that R(∞) = 0, R 2 (∞) = ∞. The following family will serve as our candidate matings: Ra (z) =

z2

a . + 2z

The critical points of Ra are ∞ and −1. A crucial obstacle now (and a principal difference with [YZ]) is that there is no algebraic approach to specifying the candidate mating of f c and f ◦◦ . Instead, and similarly to Yoccoz’ rigidity result, we will define a puzzle partition in the parameter space of Ra , and select the mating as the unique intersection point of a specific sequence of puzzle-pieces. 2. Basic Properties for Ra and f◦◦ For ease of reference, we summarize in this section some of the basic properties of the mapping f ◦◦ (z) = z 2 − 1 and the quadratic rational maps in the family Ra . We refer the reader to [Mi1] for the discussion of the properties of Fatou and Julia sets, and to [Mi3] for the properties of external rays of polynomial maps. 2.1. Basic properties of f ◦◦ . Let us begin with the following general statement (cf. [Mi1]). Lemma 2.1. Let U be a simply-connected immediate basin of a superattracting perˆ → C ˆ of period q. Denote φ : U → D a iodic point of a rational mapping F : C Böttcher coordinate: φ(F q (z)) = (φ(z))d for some d > 1. An internal ray is a curve φ −1 ({r e2πit | r ∈ [0, 1)}. Then: • suppose, p is a repelling or parabolic periodic point on the boundary of U . Then p is the landing point of an internal ray whose period is divisible by the period of p; • conversely, every periodic internal ray lands at a repelling or parabolic periodic point in ∂U .


5

Let B0 , B−1 be the immediate basins of attraction of 0 and −1 respectively for f ◦◦ . Let B∞ be the basin of attraction at infinity. Note that f ◦◦ : B0 → B−1 is also a 2 → 1 covering branched at 0. Lemma 2.2. For any two Fatou components A and B of f ◦◦ , neither of which is the attracting basin of infinity, exactly one of the following holds: (1) A ∩ B = ∅. (2) A ∩ B is only one point, which is a pre-fixed point for f ◦◦ . (3) A = B. The statement of the lemma follows immediately from the Maximum Principle. Note that the boundaries of the Fatou components B0 and B−1 touch at the repelling fixed point α of f ◦◦ . Since the mapping f ◦◦ is hyperbolic, its Julia set is locally connected. In particular, ˆ if : C\K ( f ◦◦ ) → C\D denotes the Böttcher coordinate at ∞, the Carathéodory’s Theorem implies that −1 extends continuously to ∂D. Moreover, every external ray R(θ ) = −1 ({r e2πiθ | r > 1}) lands at a point of the Julia set. We denote γ (θ ) = lim+ (r e2πiθ ). r →1

Hyperbolicity of f ◦◦ also implies: Lemma 2.3. Let Fi be an arbitrary infinite sequence of distinct Fatou components of f ◦◦ . Then diam Fn → 0. We will also make use of the following lemma: Lemma 2.4. A point z ∈ J◦◦ is a landing point of precisely two external rays if and only if z is a preimage of the fixed point α. No other point z ∈ J◦◦ is biaccessible. The angles of the two external rays which land at α are easily identified as 1/3 and 2/3.

2.2. Properties of maps in the family Ra . In what follows, we will refer to the illustration of the parameter space for the family Ra pictured in Fig. 1. For Ra let A∞ be the immediate basin of attraction at infinity, and A0 the Fatou component containing 0. Let us note: Proposition 2.5. The Fatou components A0 and A∞ are distinct and simply-connected. The critical point −1 of Ra is never contained in A∞ . Proof. We have A0 = A∞ by Denjoy-Wolff Theorem. If A∞ is multiply-connected, then, necessarily, −1 ∈ A∞ , by the Riemann-Hurwitz formula. Thus A0 contains all critical values of Ra . In this case, it follows (see e.g. [Mi2], Lemma 8.1) that the Julia set of Ra is totally disconnected, and that every orbit in the Fatou set converges to an attracting fixed point, which is impossible.

6


Fig. 1. The parameter set for Ra

Note that whenever a is such that −1 ∈ A0 , the Fatou set of Ra is the union of A0 and A∞ . The Julia set J (Ra ) is the common boundary of the two Fatou components, and we have (see, for instance, [CG], Theorem 2.1 on p. 102): Proposition 2.6. If −1 ∈ A0 , then J (Ra ) is a quasicircle. In the parameter space (Fig. 1) the above values of a form the “exterior” hyperbolic component which we denote P∞ . More generally, a capture hyperbolic component for the family Ra contains maps for which there exists an iterate Ran (−1) ∈ A∞ . The smallest such n will be referred to as the generation of the capture component. For instance, a = 2 is the center of the biggest red “bubble” in Fig. 1, in which we have Ra2 (−1) ∈ A∞ . The corresponding Julia set is depicted in Fig. 2. Similarly to the statement of Lemma 2.2, we will show in Sect. 5: Lemma 2.7. Suppose that the parameter a is chosen outside of the closure P¯∞ . Then given any two Fatou components A and B in the basin of ∞ of Ra exactly one of the following holds: (1) A ∩ B = ∅, (2) A ∩ B is only one point, (3) A = B. Moreover, if Case (2) occurs, then A¯ ∩ B¯ is either a preimage of the fixed point xa ≡ A¯ 0 ∩ A¯ ∞


7

Fig. 2. A capture dynamics: dynamical plane of R2

or a pre-critical point. For the latter possibility to occur, the parameter a must belong to the boundary of a capture component. Denote Mat the set of parameter values a not contained in any of the capture components. This set is colored in black in Fig. 1. The interior of Mat contains matings with ◦

basilica, and thus should be naturally identified with M with the 1/2-limb removed. As an example of a mating in Mat, consider Fig. 3. This image was popularized on the cover of Stony Brook preprint series; it is the mating of Douady’s rabbit with basilica.

3. Orbit Portraits for Quadratic Polynomials In this section we provide a brief summary of several results on the combinatorics of external rays of quadratic polynomials following Milnor’s paper [Mi3]. All proofs are given in [Mi3]. Let the points {x1 , x2 = f (x1 ), . . . , x p = f (x p−1 )} form a periodic orbit of a quadratic polynomial f c (z) = z 2 + c with period p. Assume further, that this orbit is either repelling or parabolic, and hence the landing set of a finite collection of periodic external rays R(θi ) (see e.g. [Mi1]). Definition 3.1. For each 1 ≤ i ≤ p, let Ai = {θ1i , . . . , θki } denote the set of angles of the external rays landing at xi . The collection O = {A1 , . . . , A p } is called the orbit portrait of the cycle (x1 , . . . , x p ). According to the type of the cycle, the orbit portrait is either repelling or parabolic.

8


Fig. 3. Stony Brook preprint cover: the dynamical plane of the mating of basilica and Douady’s rabbit

Given the periodicity of xi , the iterate f ci permutes the rays with angles in Ai . The following is immediate: Lemma 3.1. Given an orbit portrait O = {A1 , . . . , A p } the size of Ai is the same for all i. Moreover, Ai+1 = 2 Ai mod Z, and if |Ai | ≥ 3, then the cyclic order of the angles θ ij ∈ Ai is the same as that of their images 2θ ij mod Z ∈ Ai+1 . Definition 3.2. For A = {θ1 , . . . , θk } ⊂ T, write exp(A) = {e2πiθ1 , . . . , e2πiθk } ⊂ S 1 . A formal orbit portrait is a collection {A1 , . . . , A p } of subsets of T for which the following properties hold: • each Ai is a finite subset of T; • for each j modulo p, the doubling map t → 2t mod Z carries A j bijectively onto A j+1 preserving the cyclic order around the circle; • all of the angles in A1 ∪ · · · ∪ A p are periodic under doubling with the same period r p; • for each i = j, the convex hulls of the sets exp(Ai ) and exp(A j ) are disjoint. The valence of an orbit portrait O is vO = |Ai |. Every angle in Ai is periodic of period pr . Since there are pvO angles in O, the quantity vO/r is the number of distinct cycles of external rays in the orbit portrait O. Lemma 3.2. Only two possibilities can occur: either vO = r or vO = 2 and r = 1. Assume that vO ≥ 2. For each Ai , the complement T\Ai consists of finitely many complementary arcs. Each such arc corresponds to a sector between two of the rays landing at xi .


9

Lemma 3.3. Let O = {A1 , . . . , A p } be a formal orbit portrait. Then every complementary arc for Ai , except for one is mapped one-to-one under z → 2z onto a complementary arc of Ai+1 . The exception is the critical arc of Ai , which has length greater than 1/2. The image of the critical arc wraps around the whole unit circle, covering one of the complementary arcs of Ai+1 twice. If the portrait O is realized by a quadratic polynomial, then for each i, the sector corresponding to the critical arc of Ai contains the critical point 0. Lemma 3.4. Assume that vO ≥ 2. There exists a unique shortest complementary arc in O. If the portrait is realized by a quadratic polynomial f c , then the sector corresponding to this arc can be characterized among the pv sectors formed by the rays landing at points xi as the one which contains the critical value c = f c (0) and no points of the orbit xi . Definition 3.3. The complementary arc in the previous lemma is referred to as the characteristic arc of the orbit portrait. 4. Bubble Rays To construct a Yoccoz puzzle partition for the quadratic rational maps in Mat, we will use chains of Fatou components in place of external rays. This method was employed in [YZ] and [Ro2], it was also suggested in [Luo]. We begin by describing such chains in the filled Julia set of f ◦◦ ; this discussion, while mostly trivial, will serve as a useful preparation for handling maps in the family Ra . 4.1. Bubble rays for f ◦◦ . Recall that B0 and B−1 denote the components of the immediate super-attracting basin of f ◦◦ , labelled according to the point in the critical orbit they surround. ◦

Definition 4.1. A bubble of K ◦◦ is a Fatou component F ⊂ K ◦◦ . The generation of n (F) = B . The a bubble F is the smallest non-negative n = Gen(F) for which f ◦◦ 0 − Gen(F)

center of a bubble F is the preimage f ◦◦ (0) ∩ F. If F = B0 , then let G be the bubble with the lowest value of Gen(G) for which G¯ ∩ F¯ = ∅. We will refer to G as the predecessor of F, and to the point x = root(F) ≡ G¯ ∩ F¯ as the root of F. A bubble ray B is a collection of bubbles ∪m≤∞ Fk such that for each k the intersection 0 Fk ∩ Fk+1 = {xk } is a single point, and Gen(Fk ) < Gen(Fk+1 ). Note that by Lemma 2.2, each of the points xk is a preimage of the α-fixed point of f ◦◦ . If m < ∞, we will refer to the component Fm as the last bubble of B. Hyperbolicity of f ◦◦ readily implies: ◦

Proposition 4.1. There exist s ∈ (0, 1), and C > 0 such that for a bubble F ⊂ K ◦◦ we have diam(F) ≤ Cs Gen(F) . In particular, for each infinite bubble ray B = ∪∞ 0 Fk there exists a unique point x ∈ J◦◦ such that Fk → x in Hausdorff sense.

10


We refer to x as the landing point of B. By Lemma 2.2 we have: Proposition 4.2. If two bubble rays B1 , B2 have the same landing point, then one of them is contained in the other one. By Lemma 2.1, each pre-periodic point on the boundary of a bubble is a landing point of an internal ray. We may therefore define: Definition 4.2. The axis of a bubble ray B = {Fk }m≤∞ is the closed union 0 γ (B) ≡ ∪m 0 γk , where γk for k ≥ 1 is the union of two internal rays of Fk connecting its center to the points xk−1 and xk , and γ0 is the internal ray of F0 terminating at x0 . Let x be the landing point of an infinite bubble ray B. As the Julia set J◦◦ is locally connected, Carathéodory’s Theorem implies that there exists at least one external ray R(θ ) landing there. By Lemma 2.4, such θ is unique. Let us refer to the number −θ as the angle of the bubble ray B and denote it (B) ≡ −θ. By Proposition 4.2, (B1 ) = (B2 ) implies that one of these rays is a subset of the other. We will call a bubble ray B periodic if the angle (B) is periodic under doubling; the period of the ray will refer to the period of its angle. Note that the angle of a bubble ray can be determined intrinsically, from the choice of the bubbles themselves. Indeed, consider the spine (K ◦◦ ) ≡ K ◦◦ ∩ R = [−β, β], where β is the non-dividing fixed point of f ◦◦ . The spine may also be seen as the union of the axes of the bubble rays B+ , B− starting with the bubble B0 and terminating at ±β respectively. Let B = ∪Fk be an infinite bubble ray, landing at x = β. Consider the forward k (x). Define a sequence s(B) = (s )∞ of 0’s and 1’s as follows. We set iterates xk = f ◦◦ i 1 • si = 0 if xi is above the spine, or equivalently, if there is a bubble Fk with k ≥ i which is above the spine; • si = 1 if xi is below the spine, or equivalently, if there is a bubble Fk with k ≥ i which is below the spine; • if i is the first instance when neither of these two possibilities holds, set si = 1, and s j = 0 for all j > i (note, that in this case we necessarily have xi = −β. For B ⊂ B+ we set s(B) = (0)∞ 0 . We will sometimes refer to the dyadic sequence s(B) as the intrinsic address of B. Noting that (β, +∞) = R(0), and (−∞, −β) = R(1/2), we immediately have Proposition 4.3. For each infinite bubble ray B we have (B) = −

∞ i=1

2−i si , where s(B) = (si )∞ 0 .


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4.2. Bubble rays for a map Ra . The definition of a bubble ray for a rational mapping Ra is completely analogous to Definition 4.1. Definition 4.3. A bubble of Ra is a Fatou component F ⊂ ∪Ra−k (A∞ ). The generation of a bubble F is the smallest non-negative n = Gen(F) for which Ran (F) = A∞ . The center of a bubble F is the preimage Ra− Gen(F) (∞) ∩ F. A bubble ray B is a collection of bubbles ∪m≤∞ Fk such that for each k the intersection 0 Fk ∩ Fk+1 = {xk } is a single point, and Gen(Fk ) < Gen(Fk+1 ). The structure of bubble rays for Ra is particularly easy to describe when a ∈ Mat, and somewhat more difficult in the capture case. We consider the simpler possibility first. The case a∈ Mat. Consider the Böttcher coordinates b1 : D → B0 , and b2 : D → A∞ . The identification φ ≡ b2 ◦ b1−1 : B0 → A∞ conjugates the dynamics of f ◦◦ and Ra . Note that by Lemmas 2.1 and 2.7 the components A∞ and A0 have a single common boundary point x = limr →1− b2 (r ) and is fixed by the dynamics of Ra . By Lemma 2.7 we have the following: Proposition 4.4. If two bubbles F1 and F2 of Ra touch at a boundary point z, then z is a preimage of x. By Lemma 2.1, the axis γ (B) of a bubble ray B of Ra can be defined as before. To ˆ which is the define the spine a begin by considering the union of internal rays l∞ ⊂ C image under b2 of the segment (−1, 1). Let l0 ⊂ A0 be its preimage, and set t1 = l¯∞ ∪ l¯0 ∪ {x}. We now inductively define tn = tn−1 ∪ h 1 ∪ h 2 , where h i are the two components of Ra−1 (tn−1 )\A∞ intersecting tn−1 . Definition 4.4. We set a = ∪tn , and endow this arc with positive orientation as induced by the orientation of (−1, 1) → l∞ . Further, for a bubble F of Ra with F ∩ a = ∅, we say that F is above the spine, if the unique finite bubble ray connecting it to the spine lies above a with respect to the orientation of a . In the complementary case, we say that the bubble F is below the spine. We define the intrinsic address s(B) of a bubble ray B in exactly the same fashion as before. −n The oriented spine allows us to extend inductively the conjugacy φ : f ◦◦ (B0 ) → −n Ra (A∞ ) so that: Proposition 4.5. Denote ◦

L = K ◦◦ ∪

∞

n=0

−n f ◦◦ (α)

.

12


Then φ extends as a conjugacy to the whole of L. Moreover, this conjugacy obeys the property: s(φ(B)) = s(B) for each bubble ray B in K ◦◦ . Definition 4.5. For an infinite bubble ray B of Ra we set the angle of B equal to (B) ≡ (φ −1 (B)). By construction, we have (B) =

∞

2−n sn , where s(B) = (sn )∞ 1

(4.1)

n=1

for each bubble ray B of Ra . The case when a belongs to a capture component. Let us exclude the trivial possibility when the critical value −a = Ra (−1) ∈ A∞ , and denote n > 1 the smallest natural number for which Ran (−1) ∈ A∞ holds. The conjugacy φ can still be extended consis−(n−1) tently with the orientation to f ◦◦ (B0 ). Denote F −a the bubble of Ra containing ◦

the critical value, and set H = φ −1 (F) ⊂ K ◦◦ . ◦

Definition 4.6. We define an equivalence relation ∼ on K ◦◦ as follows. Connect the two ˆ ◦◦ . The equivalence relation identifies preimages H1 , H2 of H by a simple arc h ⊂ C\K ◦

any two bubbles G 1 , G 2 ⊂ K ◦◦ if there exists l ≥ 0 such that G 1 is connected to G 2 −l by a component of f ◦◦ (h). For points xi ∈ G i we set x1 ∼ x2 if this happens, and if n+l n+l f ◦◦ (x1 ) = f ◦◦ (x2 ). One readily verifies: Lemma 4.6. In the capture case, the mapping φ extends as a surjective conjugacy from ∞ ∞ ◦ −i K ◦◦ ∪ f ◦◦ α −→ Ra−i (A∞ ∪ {x}). i=0

z 1 ∼z 2

i=0

5. Parabubble Rays Removing the α-fixed point from the basilica K ◦◦ separates it into two connected components. We will denote them L for “left”, and R for “right”. Put Re = R\B 0 , (the subscript e, standing for "exterior" of the right half of the basilica). As we will see below, there is a natural correspondence between the components of the interior of Re , and the capture hyperbolic components in the parameter plane of the family Ra . For the remainder of this section, let us fix the notation Ra (z) = R(z, a), Ran (z) = R n (z, a).


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Definition 5.1. Let a0 be such that Rak0 (−1) = ∞ for some k ∈ N, and let n be the smallest such value of k. Then a connected set P of parameters a containing a0 , such that R n (−1, a) ∈ A∞ is called a capture hyperbolic component or a parabubble. The point a0 is called a center of P. We will see further that it is unique. Finally, we say that the generation of P is n, and write Gen(P) = n. Set ξn (a) = R n (−1, a). Then we have ξn+1 (a) =

(ξn

a a = . + 2ξn (a) ξn (a)(ξn (a) + 2)

(a))2

(5.1)

From (5.1) it follows by a straightforward induction, that Lemma 5.1. The degree of ξn is the nearest integer value to 2n+1/3. We now state: Lemma 5.2. For n ≥ 2, the degree of ξn is equal to the number of bubbles of generation n in the basilica which are contained in R. Proof. To each bubble B ⊂ K ◦◦ we associate an interval (a, b) = I B ⊂ R/Z, where a, b are the angles of the external rays meeting at the root of B. It is easy to see that the centers of the intervals I B of all bubbles of generation n are symmetrically distributed around the unit circle and that each I B does not intersect 1/3 or −1/3. It is easy to verify that the closest integer to 2n × (2/3) is equal to the number of I B which are contained in the interval (−1/3, 1/3). The claim follows from Lemma 5.1. Denote Aa∞ the set A∞ for the map Ra . Let ˆ a : Aa∞ → C\D be the Böttcher coordinate for Ra normalized so that a (∞) > 0. Note that a is analytic in a. A direct calculation implies a (z) =

1 z + o(1), as z → ∞. 2

(5.2)

If −a ∈√A∞ then a can be extended around ∞ until we hit a critical point z = 1 ± 1 − a for Ra2 . However, the Green’s function g(z, a) = log |a (z)| is still well defined on A∞ and moves continuously with a, and g(z, a) → 0 as z → ∂ A∞ for all a ∈ C. Let P∞ be the open set of parameters where −a ∈ A∞ , that is, where J (Ra ) is a quasicircle. This capture component obviously contains an open neighborhood of ∞. By the λ-Lemma of [MSS] we have: Lemma 5.3. The Julia set J (Ra ) moves holomorphically for all a ∈ P∞ . Let us continuously extend the Green’s function g(z, a) on the whole sphere so g(z, a) = 0 outside A∞ . The proof of Theorems III.3.2 in [CG] can be easily adjusted to the family Ra2 : A∞ → A∞ , to show that the Green’s function g is uniformly Hölder α-continuous for |a| ≤ C, some α = α(C) ∈ (0, 1]. As a consequence, g(−a, a) → 0 as a → ∂ P∞ , (see Theorem III.3.3 [CG]). Moreover, by (5.2), the function a (−a) has

14


a simple pole at ∞. Since g(−a, a) → 0 as a → ∂ P∞ , the Argument Principle implies ˆ that a (−a) takes every value in C\D exactly once. We get the following: c has logarithmic capacity Lemma 5.4. The set P∞ ∪ {∞} is simply connected and P∞ equal to 1/2.

It is easy to verify that A∞ does not necessarily move continuously at ∂ P∞ if we step inside P∞ (e.g. at a = 3), but the following holds. Lemma 5.5. The set A∞ moves holomorphically for all parameters a ∈ (P ∞ )c . We have a ∈ ∂ P∞ if −a ∈ ∂ Aa∞ . Proof. Put ψa = a ◦ a−1 . Then ψa maps Aa∞0 onto Aa∞ . If a ∈ / P ∞ then −a ∈ / A∞ 0 c a0 by definition and we have that ψa (z) = ψ(z, a) is a holomorphic motion on A∞ × P ∞ . By the -Lemma, ∂ Aa∞ also moves holomorphically. If −a1 ∈ ∂ A∞ for some a1 ∈ / P ∞ then since A∞ moves holomorphically, the point −a1 is an image of some point z 1 ∈ ∂ Aa∞0 under ψ, i.e. ψ(z 1 , a1 ) = −a1 . The analytic function ψz 1 (a) satisfies ψz 1 (a1 ) + a1 = 0. Either ψz 1 (a) + a ≡ 0 or not. If so, then −a ∈ ∂ A∞ for all a ∈ (P ∞ )c , which is clearly false. If not so, then choose a small disk B(a1 , ε) ⊂ (P ∞ )c and some z 2 ∈ Aa∞0 , with z 2 sufficiently close to z 1 , such that |ψz 1 (a) − ψz 2 (a)| < |ψz 1 (a) + a| for a ∈ ∂ B(a1 , ε). By Roche’s Theorem, ψz 2 (a) + a = 0 must have a solution b ∈ B(a1 , ε), which means that −b ∈ Ab∞ , which is a contradiction. Corollary 5.6. The statement of Lemma 2.7 holds for a ∈ ( P¯∞ )c . Moreover, for every such a, the bubbles of Ra have locally connected boundaries. Proof. Consider a mapping Ra with the parameter a ∈ ( P¯∞ )c contained in a capture component. Since Ra is a hyperbolic mapping, the boundary of every Aa∞ is locally connected by the standard considerations. The second claim follows. The first claim is now immediate. 5.1. Internal parameter rays.. If P is a capture component of generation n ≥ 1, for t ∈ P let gn (t) = t (R n (−1, t))), so that gn maps a ∈ P to the Böttcher coordinate for Ran (−1) in A∞ . The function ξn is a rational function and has a pole of finite order at the center of every capture component (later we show that it is in fact a simple pole). We proceed with the following definition. Definition 5.2. An internal parameter ray of angle θ is a connected component of the set {gn−1 (r e2πiθ ) : r > 1}. Lemma 5.7. Let P be a parabubble with Gen(P) = n ≥ 2, and let θ ∈ T be periodic (pre-periodic) under doubling. Then an internal parameter ray of P with angle θ lands at a point a0 ∈ ∂ P. Moreover, the point p(a0 ) = Ran0 (−1) is a repelling periodic (pre-periodic) point on the boundary of A∞ .


15

Proof. To fix the ideas, we assume that θ = 0 so that p(a0 ) is the repelling fixed point where A∞ and A0 touch. Set γn (t) = gn−1 (te2πiθ ), for t > 1, where we assume that gn−1 (te2πiθ ) belongs to a chosen connected component of {gn−1 (r e2πiθ ) : r > 1}. We want to show that lim+ γn (r ) exists and is equal to a0 . First note that

r →1

|a−1 (r ) − p(a)| ≤ δ(r ),

(5.3)

where δ(r ) → 0 as r → 1, which follows by Lemma 2.1. Also, note that the left hand side of (5.3) is a continuous function of both a and r on P × (1, ∞). This implies that a−1 (r ) → p(a) uniformly as r → 1 on P. Therefore, for a = γn (r ), |−1 γn (r ) (r ) − p(γn (r ))| ≤ δ(r ),

(5.4)

where δ(r ) → 0 as r → 1. Now, for |a−a0 | ≤ ε, we have |Ran (−1)− p(a)| ≤ ε (ε) → 0, as ε → 0. On the other hand, since the zeros of |Ran (−1) − p(a)| are isolated, we can find a C > 0 such that if 0 < ε ≤ |a − a0 | ≤ C, then |Ran (−1) − p(a)| ≥ ε . If γn (r ) does not land at a0 , take an a ∈ γn (r )\B(a0 , ε), where r is sufficiently close to 1, so that (5.4) holds for δ(r ) ≤ ε/2. But since |a−a0 | ≥ ε we have |Ran (−1)− p(a)| ≥ ε , for a = γn (r ), which is a contradiction. Hence γn (t) must land at a0 . The landing property for periodic parameter rays in P∞ follows from the standard theory in e.g. [CG], Theorem 5.2: Proposition 5.8. If θ is rational then the internal parameter ray of angle θ in P∞ lands at a parameter a ∈ ∂ P∞ . Moreover, if θ = 0 is periodic then Ra has a parabolic cycle and if θ is strictly preperiodic then Ra is a postcritically finite map. Consider the conjugacy φ from Lemma 4.6. We have the following: Lemma 5.9. Let P be a parabubble of generation n ≥ 2 and address σ . (I) There exists a unique bubble W ∈ K ◦◦ such that the following holds. Let a ∈ P and denote Ba the bubble of Ra which contains the critical value −a. Then φ −1 (Ba ) = W . (II) On the other hand, for each bubble W ∈ R, there exists a unique parabubble P such that for any a ∈ P we have φ(Ba ) = W , where −a ∈ Ba . (III) Moreover, a ∈ ∂ P if and only if −a ∈ ∂ Ba . (IV) The parabubble P is an open set, has a unique center, and is simply connected. Proof. The first and third claim are immediate consequences of Lemma 5.5. The same lemma implies that P is an open set. We have ξn (a) = R n (−1, a) → ∂ A∞ as a → ∂ P by Lemma 5.9, so a ◦ ξn → ∂D as a → ∂ P. By the Argument Principle, this means that every capture component P is ˆ mapped by a ◦ ξn onto C\D as a d −→ 1 covering. We want to show that d = 1. Let P be a parabubble of generation n, and F the corresponding bubble for Ra in which −a lies. Note that the map φ in Lemma 4.6 is an injection of all bubbles of generation ≤ n. Hence we can define B = φ −1 (F). The root of B then is a landing point

16


x of an internal ray of B with angle θ = 0 (by Lemma 2.1). The predecessor C touches B at x. It follows from Lemma 5.7 that an internal parameter ray with angle θ = 0 in P will land at a parameter a such that Ran (−1) is the unique repelling fixed point on the boundary of A∞ . It follows that there is a corresponding parabubble Q to C (in the same way as P corresponds to B), such that P touches Q at a. Moreover, gen(Q) < gen(P), since gen(C) < gen(B). Proceeding in this way we see that for every parabubble P, there is a finite sequence of internal parameter rays connecting the center of P with a point on ∂ P∞ . Reversing this process we also see that for every bubble B in the right basilica R there is a corresponding parabubble P, in the sense that if F is the bubble for Ra in which −a lies, then B = φ −1 (F). We cannot have such correspondence to the left basilica simply because a = 0 is a singularity for the family Ra and no sequence of parabubble rays can end there. We have to prove that there is one and only one bubble in the right basilica corresponding to every parabubble. By Lemma 5.1 the only thing we have to show is that it is impossible to have one parabubble P corresponding to two different bubbles B1 and B2 in the right basilica. This would imply that the parabubble has two distinct centers. By the λ-lemma of [MSS], any two centers in the same parabubble P would correspond to quasi-conformally conjugate rational maps. Since these maps would also be postcritically finite, Thurston’s Theorem implies that a center is unique. Hence every parabubble corresponds to a unique bubble in the right basilica and (II) is proven. Now, since the degree of ξn coincides with the number of parabubbles of generation n, the Pigeonhole Principle implies that ξn has a simple pole at the center of each parabubble ˆ of generation n. By the Argument Principle, a ◦ ξn : P → C\D assumes every value ˆ in C\D exactly once, so indeed d = 1. It follows that every capture component is simply connected. By Lemma 5.9, the mapping ψ : a → −a → φ −1 (−a) ◦

is an injection from the capture locus of the family Ra to R. It is straighforward to extend this mapping to the roots of the (para)bubbles, except for the roots contained in the boundary of P∞ . Denote by C the union of capture components of the family Ra and Ce = C\P∞ . Since dynamical bubbles may only touch at a single point, which is a preimage of the c fixed point where A∞ and A0 as long as a ∈ P ∞ , our discussion implies: Proposition 5.10. If P and Q are two parabubbles not equal to P∞ , and P = ψ(P), Q = ψ(Q), then the following holds: (1) P ∩ Q ∩ (P ∞ )c = ∅ ⇔ P ∩ Q = ∅, (2) P ∩ Q ∩ (P ∞ )c is exactly one point ⇔ P ∩ Q is exactly one point, (3) P = Q ⇔ P = Q . Moreover, ◦

ψ(C) = R.


17

Similarly to the notation for dynamical bubbles, if the intersection of the closures of two parabubbles P ∩ Q = {a} is exactly one point and Gen(P) > Gen(Q), let us refer to Q as the predecessor of P and a as the root of P. Let {a j } be the set of all touching points between parabubbles not including those which lie on the boundary of P∞ . The above proposition implies that ψ continuously extends to a homeomorphism ⎞ ⎛ ∞ ◦ ⎝ ( f − j (α) ∩ Re )⎠ . ψ : Ce ∪ {a j } → Re j=1

Definition 5.3. Let B = {Fk }∞ 0 ⊂ K ◦◦ be an infinite bubble ray with angle (B) = θ ∈ (−1/3, 1/3). We call the corresponding sequence of capture components {Pk }∞ 0 , with ψ(Pk ) = Fk , a parabubble ray in C with angle θ , and write (P) = θ . Similarly to the definition for dynamical bubble rays, we define the axis for a parabubble ray P to be the union of the internal parameter rays γk , γk ⊂ Pk which land at the points P k ∩ P k−1 = xk and P k+1 ∩ P k = xk respectively, starting from ∞. In the next section we show that certain infinite bubble rays and parabubble rays land at a single point. 6. Landing Lemmas 6.1. Dynamical bubble rays. We begin with the following lemma. Lemma 6.1. Assume that B is a periodic infinite bubble ray B such that the axis is disjoint from the closure of the postcritical set. Then the axis γ for B lands at a single periodic point which is either repelling or parabolic. Proof. Let be the closure of the postcritical set and let S be the set of cluster points for γ . If the period of γ to itself is n then R n maps ∪ S into itself. Hence R −n can be lifted ˆ by the universal covering D of C\( ∪ S) to a map fˆ : D → D such that fˆ(D) ⊂ D is a strict inclusion. Hence R n is strictly expanding with respect to the Poincaré metric on ˆ C\( ∪ S). Since γ is invariant under f we can take a starting point x0 ∈ γ and set f (x0 ) = x1 , and xk = f (xk−1 ). Let γk be the part of γ between xk and xk+1 . The hyperbolic distance between xk and xk+1 decreases as k increases. Take a point p ∈ S. Then the hyperbolic distance from any point on γ to p is infinite, since S is contained in the boundary of the ˆ hyperbolic set C\( ∪ S). Since the hyperbolic length of γk decreases for increasing k, any neighbourhood N of p has the property that there is a smaller neighbourhood N ⊂ N such that if γk ∩ N = ∅ then γk ⊂ N . But this means that f (N ) ∩ N = ∅. So p has to be a fixed point. Since S is connected, S must contain only this point. By the Snail Lemma, p must be a parabolic or repelling point (cf. [Mi1], Lemma 16.2).

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We next prove that the axis of a periodic (or preperiodic) bubble ray cannot accumulate on some bubble. Lemma 6.2. Let B be a periodic infinite bubble ray for which the axis is disjoint from the closure of the postcritical set. Then the axis of B cannot accumulate at some bubble. Proof. Without loss of generality, in order to reach a contradiction, it suffices to suppose that the axis of B accumulates at ∂ A∞ . Since B is periodic, we know from Lemma 6.1 that the axis for the bubble ray B lands at a single periodic point p on the boundary of A∞ . The bubble ray B then encloses a domain D whose boundary is a connected part I = A∞ of ∂ A∞ and half of the boundary of all the other bubbles in B. Since p and B is fixed under some iterate n we have that D is invariant under R n . This means that any bubble B in D must never be mapped into A0 ∪ A∞ , since this set lies outside D (the fact that there exists some bubble in D is obvious). This is clearly impossible, since bubbles by definition are preimages of A∞ . We are now in position to prove a landing lemma for periodic or preperiodic bubble rays. Lemma 6.3. Assume that B is periodic infinite bubble ray, for which there exist an N such that all bubbles in B of generation at least N are disjoint from the closure of the postcitical set. Then B lands at a single point. Proof. Assume that B is periodic of period q. We have seen (Lemma 6.1 and Lemma 6.2) that the axis γ of the bubble ray must land on a periodic point x. Since the postcritical set is disjoint from any bubble B in B with Gen(B) ≥ N , we have an annulus R around this B of some definite modulus m > 0 such that there are −q −qn well defined inverse branches of Ra on R ∪ B, where Ra (B) ∈ B for all n ≥ 0. This means that the lengths of the γk in the proof of Lemma 6.1 are commensurable with the diameter of the corresponding bubbles Fk , by the Koebe Distortion Lemma. Hence the bubble ray B converges to the same periodic point as the axis γ lands on.

6.2. Orbit portraits for Ra . We have seen in Sect. 4 that bubble rays have angles inherited from the angles of external rays in the basilica (although these angles are not always well defined, as in the capture case for instance). With the theory about orbit portraits for quadratic polynomials in Sect. 3 in mind, it is now straightforward to define an orbit portrait for Ra . Definition 6.1. Let x1 , x2 , . . . , x p be a (repelling or parabolic) periodic orbit, where Ra (xi ) = xi+1 , Ra (x p ) = x1 . Assume that there are a finite number of periodic infinite bubble rays landing on xi , with well defined angles; let Ai be the corresponding angles for the bubble rays landing at xi . Then the orbit portrait for Ra is the set O = {A1 , A2 , . . . , A p }. Given two angles θ1 = θ2 we let [θ1 θ2 ] ⊂ T be the arc of the unit circle swept by going in counter-clockwise direction from θ1 to θ2 . We say that θ lies between θ1 and θ2 if θ ∈ [θ1 θ2 ]. Before we state the next lemma we make some more definitions.


19

Definition 6.2. Let B1 and B2 be two bubble rays starting from A∞ with well defined angles θ1 and θ2 and axes γ1 and γ2 . Assume that B1 and B2 land at a common point p. Denote D the domain bounded by the axes γ1 , γ2 which does not contain any bubble rays with angles in T\[θ1 θ2 ]. Define the outer boundary of the sector bounded by B1 , B2 as the union of the arcs of the boundaries of the bubbles in these two bubble rays lying outside D together with ˆ lies between their endpoints. Similarily, define the inner boundary. We say that z ∈ C B1 and B2 if z ∈ D and z ∈ / B1 ∪ B2 . This notion of being between two bubble rays also makes sense for bubble rays even if B1 and B2 do not land on a common point. Definition 6.3. Assume that the two bubble rays B1 and B2 have intrinsic addresses s(B1 ) = (x0 , x1 , . . . , ) and s(B2 ) = (y0 , y1 , . . .) respectively. We say that an infinite bubble ray B, with intrinsic address s(B) = (z 0 , z 1 , . . .), lies between B1 and B2 if yi ≤ z i ≤ xi for all i ≥ 0. Equivalently, the angle (B) ∈ [(B1 ) (B1 )]. This definition also makes sense for parabubble rays in an exactly analoguous way. Lemma 6.4. Let O = {A1 , . . . , A p } be a formal orbit portrait with vO ≥ 2 and let I = [t− t+ ] be its characteristic arc. If the formal orbit portrait O is realisable by some Ra then −a cannot lie on the outer boundary of a bubble ray with angle t− or t+ . Proof. Since a ∈ Mat, in the case when −a belongs to the boundary of some bubble, we have a conjugacy φ from Proposition 4.5 between the dynamics of f ◦◦ on the interior of K ◦◦ and that of Ra on its Fatou set. Now suppose O is realised and let Ai be the set of angles of the bubble rays landing at xi . Assume that A2 contains the characteristic arc. Let B− and B+ be the bubble rays corresponding to the angles t− , t+ ∈ A2 and let A+ , A− be the bubble rays corresponding to the critical arc in A1 , i.e. so that A− and A+ are mapped onto B− and B+ respectively. Also, let D be the domain enclosed by the axes of B− and B+ . There are two more preimages of B− and B+ , call them A− , A+ respectively. Also, = (A ), a = (A ). Since I = (a , a ) is let a− = (A− ), a+ = (A+ ) and a− c − + − + + , a lie entirely inside I , and thus the bubble the critical arc in A1 we have that both a− c + rays A− , A+ lie entirely inside the domain Dc enclosed by the axes for the bubble rays forming the critical arc. Now, assume that −a lies on an outer boundary of a bubble in B+ (the proof is the same if −a ∈ B− ). Then the critical point −1 must belong to a bubble in A+ . Since Ra is 2 − 1 in a neighbourhood of −1 and orientation preserving, we have that the bubble ray A+ must touch A+ at −1. Since −a is outside D, this implies that −1 must be outside Dc . Thus A+ must be outside Dc , which is a contradiction. The following lemma tells us when a specific orbit portrait is realised. Lemma 6.5 (Realization of orbit portraits). Let O = {A1 , . . . , A p } be a formal orbit portrait with a characteristic arc I = [t− t+ ]. Let Pt− , Pt+ be the corresponding parabubble bubble rays, with angles t− and t+ and assume that a belongs to a parabubble P between Pt− and Pt+ . Then the orbit portrait O is realised by Ra . The proof follows that of Lemma 2.9 in [Mi3].

20


Proof. Note that all infinite bubble rays with angles in any A j are well defined since their forward images do not intersect the critical value. Let be the closure of the postcritical set for Ra and let ρ(z) be the induced hyperbolic ˆ metric on C\. Let C be the critical bubble containing −1 and V the critical value bubble containing −a. There is a unique finite bubble ray ending at V . Its preimage is two finite bubble rays B1 and B2 both ending at C. Their axes γ1 and γ2 join in C and form a ˆ closed simple curve in C. Take a hyperbolic disk D C which covers the critical point −1 and let L=

∞

Rak (γ1 ∪ γ2 ∪ D).

k=0

It is easy to see that the complement of L is two topological disks U1 and U2 . We have R(L) ⊂ L and ⊂ L. Moreover dist(, L c ) ≥ ε > 0 for some definite ε > 0. It is easy to check that the n th preimages of U1 and U2 consist of 2n+1 topological disks. Moreover, all preimages of the U j will be on a definite distance ε > 0 from so we have a uniform constant c = c(ε) > 1 so that ρ(R(x), R(y)) ≥ cρ(x, y) for x, y lying in any of these preimages of Ui . It follows that the preimages of Ui shrink to points. Thus the symbol sequence of some point with respect to the initial partition L is unique. In particular the landing points of the periodic bubble rays in O will have the same symbol sequence if and only if they land at a common point. To show that O is indeed realised it now suffices to show that all the landing points of the bubble rays with angles in A j lie entirely in one of the components Ui . Since they are mapped onto each other they will have the same symbol sequence in that case. The preimages of the characteristic arc [t−1 t+ ] under the doubling map will be two smaller arcs I and I at the end of the critical arc. Since every A j ∈ O cannot have any element in I or I we have that all bubble rays corresponding to angles in A j are completely contained in U1 or completely contained in U2 . Thus all the angles in every A j have the same symbol sequence, so they land on a common point, and so O is a realised bubble portrait. Lemma 6.6. Assume that Ra has a parabolic fixed point z 0 , with Ra (z 0 ) = e2πi p/q , where p/q ∈ Q with ( p, q) = 1. Then there are precisely q periodic bubble rays B j , j = 1, . . . , q, landing at z 0 . These bubble rays are mapped onto each other under the action of Ra , with combinatorial rotation number p/q. Proof. For simplicity, consider the mapping Ra with a = 32/27 which has a simple parabolic with eigenvalue 1. After a suitable change of coordinates shifting the fixed point to the origin, this mapping takes the form ζ → ζ + ζ 2 + O(ζ 3 ) in a neighborhood of ζ = 0. Denote A and R the attracting and repelling petals of Ra correspondingly. Note that Montel’s Theorem guarantees that the repelling petal contains a bubble B.


21

Now, B is the end of some finite bubble ray C F . Taking the preimages of the bubble ray C F we get a sequence of bubble rays Ck = Ra−k (C F ), whose ends will converge to z0 . Since preimages will increase the generation and since there are finitely many finite bubble rays of any fixed generation, for any N there must be some bubble B0 of generation N contained in infinitely many Ck . Let Ck0 ⊃ B0 be the Ck containing B0 with lowest generation and Ck1 ⊃ B0 the second lowest. Then Ram (Ck1 ) = Ck0 for some m ≥ 1 and the preimage of B0 under Ram is a longer bubble ray B1 ⊃ B0 . Moreover, B1 ⊂ Ck1 . Taking further preimages of B1 under the same branch f = Ra−m we get a sequence Bn of nested finite bubble rays such that Bn ⊂ Ckn . Moreover, the “difference” between Bn and Ckn , i.e. the number of bubbles in Dn = Ckn \Bn is a fixed constant K for all n. The bubble Dn is also a preimage of the starting set D0 under f . Since the postcritical set accumulates on z 0 , it is disjoint from Dn . Thus there is a neighbourhood around all bubbles in D0 where f n is defined for all n ≥ 0. Now, the Koebe Distortion Lemma implies that all bubbles in Dn shrinks to points, namely the parabolic fixed point z 0 , since one of them, namely the end of Ckn ⊃ Dn , converges to z 0 . Hence there is a subsequence of bubbles in Bn which converge to z 0 (but we do not know a priori that the bubble ray itself will converge to z 0 ). However, by construction, the bubble ray B = ∪n Bn is periodic. We can now apply Lemma 6.3 to B, which shows that B lands at a single point, which must be equal to z 0 . Let us show that the period of B is 1. A priori, C is periodic with a period which divides m. Assume the period is p = 1 and that Ra (B j ) = B j+1 for 1 ≤ j ≤ p − 1, Ra (B p ) = B1 . By simple combinatorial considerations (see e.g. [Mi3]), these bubbles form their own orbit portrait. But this means that some point z ∈ R\A in the domain bounded by two consecutive bubble rays B j and B j+1 , will be mapped into A, which is impossible. Hence p = 1. 6.3. Parameter bubble rays. Let us first note the following evident statement: Lemma 6.7. Assume that an orbit portrait O is realized for some rational map Ra by bubble rays landing at a repelling orbit {xi }. Let at , t ∈ [0, 1] be a continuous path with a0 = a along which the corresponding periodic orbit {xit } remains repelling. Assume further that for every t no iterate of the critical value −at is contained in the boundary of a bubble ray with angle γ ∈ O. Then the orbit portrait O is realized for all Rat . The following proposition has an analogue in [Mi1], Theorem 4.1 (and Lemma 4.2). Since the proof is completely similar, we omit it. Proposition 6.8 (Milnor; Parameter Path). Given a parameter a0 such that Ra0 has a parabolic fixed point z 0 with combinatorial rotation number p/q and an orbit portrait O (from Lemma 6.6). Then there is a path γ emerging from a0 in parameter space so that a ∈ γ implies that Ra has a repelling fixed point z = z(a) with orbit portrait O and an attracting periodic orbit with period q, close to z(a). The set A of parameters where the attracting periodic orbit in the above lemma exists, is bounded by a finite number of analytic curves. Indeed, A = {a : |(R q ) (z i (a), a)| < 1}. The condition |(R q ) (z i (a), a)| = 1 represents an analytic curve with a finite number of singularities. We conclude that there is a “wedge” W˜ , that is, two analytic curves γ1

22


and γ2 which meet at a0 such that for a small neighbourhood B(a0 , ε), an open set E bounded by γ1 , γ2 and ∂ B(a0 , ε) has the property that inside E, we have O realised and z i (a) is an attracting periodic orbit of period q (as in the above lemma). By Lemma 6.7 and Lemma 6.4 the parabubble rays Pt + , Pt − lie outside of the wedge ˜ W. Lemma 6.9. Let a0 be as in the above lemma and assume that (t + , t − ) is the characteristic arc for O. Then for any ε > 0, we have B(a0 , ε) ∩ Pt = ∅, for at least one t = t + , t − , where Pt + , Pt − denote the parabubble rays with angles t + , t − respectively. Proof. Assume the contrary. Then there is an ε > 0 such that B(a0 , ε) is disjoint from the parabubble rays Pt for t = t + , t = t − . By the above argument, and Theorem 6.8, the orbit portrait O is realised in B(a0 , ε) ∩ N , where N = {a : |Ra (α(a))| > 1}, and α(a) is the (local) continuation of the parabolic fixed point z 0 (this is possible if the multiplier is = 1). Hence there is a parameter a1 ∈ B(a0 , ε), such that Ra1 also has a parabolic fixed point z 1 . But since the combinatorial rotation number is changed for a1 the new wedge W˜ 1 emerging from a1 has to exhibit a different orbit portrait O1 . But W˜ 1 must intersect B(a0 , ε) ∩ N , and so both orbit portraits O1 and O are realised, which is impossible. The lemma follows. Proposition 6.10 (Parabubble wakes I). Let a0 be such that Ra0 has a parabolic fixed point z 0 with eigenvalue Ra 0 (z 0 ) = e2πi p/q , ( p, q) = 1. Denote O = {{θ1 , . . . , θq }} the orbit portrait from Lemma 6.6, and let I = [t− t+ ] be its characteristic arc. Then the corresponding parabubble rays with angles t+ and t− land on a0 . Proof. The standard considerations of parabolic dynamics imply that q

Ra0 (z) = R(z) = (z − z 0 ) + b(z − z 0 )q+1 + O((z − z 0 )q+2 ), for some b = 0. For a close to a0 the fixed point z 0 will bifurcate into q + 1 fixed points q (for Ra ) z k (a), which are analytic in a neighbourhood of a0 , and where z k (a0 ) = z 0 , for k = 1, . . . , q + 1. One of these fixed points must be a fixed point for Ra as well if q q ≥ 2, while the other fixed points (for Ra ) are all repelling, indifferent or attracting. By Lemma 6.9 there must be a subsequence of parabubbles Pn k ⊂ Pt + (or Pt − ) such that Pn k ∩ B(a0 , ε) = ∅, for all k ≥ N (ε). Hence, for sufficiently large k, if a1 ∈ Pn k , then −a1 ∈ B n k , where B n k is the corresponding dynamical bubble in the bubble ray Bt + , i.e. with same address as P n k . Since a1 is a capture parameter the fixed points z i (a1 ) (under q Ra ) cannot be attracting. They cannot be neutral so they must be repelling. We now use the standard theory of parabolic bifurcation (see for ex [Sh2] Sect. 7, k [Sh3], [DH1]). For a suitable small perturbation, we get q fundamental domains S+,a k , 1 ≤ k ≤ q, for the repelling and attracting petals respectively for the perturbed and S−,a map Ra . They have the property that k k ∩ S−,a = {α(a), z k (a)}. S+,a

Moreover, there exist analytic functions k+,a , k−,a (the perturbed Fatou coordinates) k = S k \{α(a), z (a)} and which are defined and injective in a neighbourhood of S˜+,a k +,a q k k ˜S−,a = S−,a \{α(a), z k (a)} respectively, and conjugate the dynamics of Ra to that of the unit translation. With a choice of normalization, these coordinates will vary locally analytically with a.


n,

23

k , then there is an n ≥ 1 such that R qn (z) ∈ S˜ k , and for the smallest such If z ∈ S˜−,a a +,a

qn

k+,a (Ra (z)) = k−,a (z) −

1 + n + const, β(a)

where β(a) = βk (a) is an analytic function in a punctured neighbourhood of a0 , β(0) = 0, defined by (Ra ) (α(a)) = e2πiβ(a) . q

k the Écalle-Voronin cylinders, obtained as the quotients Denote C+k , C− q

q

k k k C+k = S+,a mod Ra C/Z, C− = S−,a mod Ra C/Z.

We get that for z ∈ C+k , k−,a ◦ Ra ◦ (k+,a )−1 (z) = z + qn

1 mod Z. β(a)

(6.1)

The function τa (z) =

1 + z mod Z β(a)

k is called the transit map. viewed as an isomorphism C+k → C− Now let us fix some k so that the critical point −1 belongs to the k th attracting petal. For simplicity let us drop the indices k in the above discussion and only focus on these particular Fatou coodinates. Then for any prescribed bubble Bl in Bt + we can find a qn parameter a ∈ B(a0 , ε) such that Ra (−1) ∈ Bl , for some n ≥ 1, n = n(l). Fix a = a1 as above. For this specific perturbation, we already have −a ∈ B n k , so we know that n = 1 in (6.1). The bubbles Bn move holomorphically and with uniformly bounded distortion in the Fatou coordinates for a in some disk B(a0 , ε) (the lifted dynamical bubbles in Bt + , in the Fatou coordinates, are all unit translates of each other). The function τa (−1) = 1/β(a) + z mod Z has derivative

∂a τa (z) ∼ D

1 −m = , (a − a0 )m (a − a0 )m+1

for some m ∈ Q, m > 0. This, and the distortion considerations, imply that Pn k converge to a0 . It remains to show that all parabubbles Pl ⊂ Pt + converge to a0 , instead of just a subsequence lk . This follows from the fact that n = n(a) in (6.1) is continuous function of a which only assumes integer values. Hence n = 1 for all l and Pt + lands on a0 . Of course a similar statement holds for Pt − . Let us write W = W (t + , t − ) for the parabubble wake being a set of points between the parabubble rays from the above lemma. Also, let O = O(t + , t − ) be the corresponding orbit portrait. Note that the characteristic arcs corrsponding to different orbit portraits around the fixed point are disjoint. Lemma 6.11 (Parabubble wakes II). The parabubble rays in the above lemma cut out an open set in the complex plane, called the bubble wake W = W (t + , t − ) such that a ∈ W if and only if Ra exhibits the repelling orbit portrait O = O(t + , t − ).

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Proof. By Lemma 5.9 the set A∞ moves holomorphically and the critical value −a belongs to the boundary of a bubble if and only if a belongs to the boundary of the corresponding parabubble. By Lemma 6.7 if for a single parameter a ∈ W = W (t + , t − ) the map Ra realises the orbit portrait O = O(t + , t − ), then the same is true for every parameter in W . On the other hand, O cannot be realised for any parameter value outside W . Indeed, O is not realised for a in any of the capture components outside W , since this would imply that the critical value is outside the characterisic arc. 7. A Puzzle Partition for Ra The idea of a puzzle partition for a Julia set originated in the work of Branner and Hubbard [BH]. It has been further developed by Yoccoz (see e.g. [Hub] and [Mi5]), to study the local connectedness of the Mandelbrot set at Yoccoz parameters, and the local connectedness of the corresponding Julia sets. We employ the Branner-Hubbard-Yoccoz approach to maps of the family Ra using partitions given by landing bubble rays.

7.1. The Yoccoz puzzle for quadratic polynomials. Let us recall the main steps of Yoccoz’ construction for a quadratic polynomial f c without non-repelling orbits with a connected Julia set. Let α stand for the dividing fixed point of f c . It is the landing point of a cycle of q > 1 external rays of f c . Denote these rays R1 , . . . , Rq . Recall that the Böttcher coordinate ˆ ˆ : C\K ( f c ) → C\D, conjugates f c to the dynamics of z → z 2 . Fix an arbitrary r > 1 and let Er be the equipotential curve Er = −1 ({r e2πiθ : θ ∈ [0, 1]}. Let U0 be the graph formed by U0 = R1 ∪ · · · ∪ Rq ∪ Er ∪ {α}. The puzzle pieces of depth 0 are the bounded components of C\U0 . Denote these q j topological disks P0 , j = 0, . . . q − 1. By definition, the Yoccoz’ puzzle pieces of depth d ≥ 1 are the first preimages of the puzzle pieces of depth d − 1 under f c . What makes puzzle partitions of Julia sets so useful in the study of local connectedness are the following two straightforward observations: Proposition 7.1. The following two properties hold: j

j

• (Markov property) any two puzzle pieces Pd and Pd are either disjoint, or one of them is contained in the other; j • the intersection Jc ∩ Pd is a connected set. The Markov property allows us to make the following definition for any point z ∈ Jc which is not a preimage of α.


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Definition 7.1. For any z ∈ Jc with α ∈ / ∪ f cn (z), let Pd (z) denote the puzzle piece of depth d which contains z. Let us also set Ad (z) = Pd (z)\Pd+1 (z). We will refer to Ad (z) as an annulus, even though it may be degenerate. The sequence of annuli Ad (0) will be called the critical annuli. The following is a consequence of the Grötzch Inequality (see e.g. [BH]): Lemma 7.2. Let Ai , i ∈ N be a sequence of bounded conformal annuli in the plane with simply-connected complementary components. Denote Wi the bounded component of C\ A¯ i . Assume that Ai+1 ⊂ Wi and mod Ai = ∞. Then diam

Wi = 0

Yoccoz has demonstrated, in particular: Lemma 7.3. Assume that f c is non-renormalizable. Then mod Ad (0) = ∞. His proof uses the concept of a tableau developed by Branner and Hubbard [BH]. Below we extract a definition suitable for a generalization from [Mi5]. To motivate some of the notation, fix a point z ∈ Jc , and consider its orbit under f c : z = z 0 → z 1 → z 2 → . . . . Note that the puzzle piece Pd (z j ) is mapped onto Pd−1 (z j+1 ), either as a conformal isomorphism or a branched double covering, depending on whether the piece Pd (z i ) contains the critical point or not. Definition 7.2. Let S(z) be the largest integer d ≥ 0, for which Pd (z) = Pd (0). If Pd (z) = Pd (0) for all d, put S(z) = ∞, and if Pd (z) = Pd (0) for all d, put S(z) = −1. We then distinguish the following three possibilities: • Critical case. d < S(z i ). Here the critical point lies in Pd (z i ) = Pd (0). Hence the annulus Ad (z i ) is mapped onto its image as an unbranched two-to-one covering. One easily deduces that mod Ad (z i ) =

1 mod Ad−1 (z i+1 ). 2

• Off-critical case. d > S(z i ). Here the critical point is outside Ad (z i ) so that Ad (z i ) is mapped conformally onto its image Ad−1 (z i+1 ). Indeed, mod Ad (z i ) = mod Ad−1 (z i+1 ).

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• Semi-critical case. d = S(z i ). This means that the critical point lies in the annulus Ad (z i ), and 1 mod Ad−1 (z i+1 ). 2 Definition 7.3 (A critical tableau). A critical tableau is a two-dimensional array of nonnegative real numbers (µd,n ), d, n ≥ 0 together with a marking, formed according to a set of rules given below. Each position of the tableau is marked as critical, semi-critical, or off-critical. An iterate I in the tableau is a move in the north-western direction in the array: mod Ad (z i ) >

µd,n −→ µd−1,n+1 . I

The rules of a critical tableau are as follows. • Every column of a tableau is either all critical; or all off-critical; or has exactly one semi-critical position (d0 , n) and is critical above (d > d0 ) and off-critical below. The 0th - column is all critical. • If µd,n > 0 then I(µd,n ) > 0. Moreover, if (d, n) is marked off-critical, then I(µd,n ) = µd,n ; if (d, n) is marked semi-critical, then I(µd,n ) < 2µd,n ; if (d, n) is marked critical, then I(µd,n ) = 2µd,n . • Let position (d0 , n) be marked as either critical or semi-critical. Draw a line northeast from this position, and do the same from the position (d0 , 0) in the tableau. Then the marking above the second line must be copied above the first one. • Suppose that (d, 0) is marked critical, (d − k, k) is also critical, and (d − i, i) is offcritical for i < k. Assume that (d, n) is semi-critical for some n. Then (d − k, n + k) is also semi-critical. Finally, we say that a tableau is recurrent if sup{d| (d, k) is critical for some k > 0} = ∞; we say that it is periodic if there exists k > 0 such that the k th column is entirely critical. The relevance to the quadratic Yoccoz’ puzzle should be evident from the above discussion: Definition 7.4 (The critical tableau of a Yoccoz’ puzzle). For f c as above, we let µd,n = mod Ad ( f cn (0)). We note: Proposition 7.4. The critical tableau of the Yoccoz’ puzzle of f c is periodic if and only if f c is renormalizable. The basis of the Yoccoz’ result is given by the following theorem: Theorem 7.1. Assume that (µd,n ) is a tableau, which is recurrent and not periodic. Assume further that there exists d such that µd,0 > 0. Then µd,0 = ∞. d


27

7.2. A puzzle partition for Ra . The puzzle pieces for Ra which we construct are similar to those just described but instead of external rays we use bubble rays. More specifically, choose a parameter a in a parabubble wake W (t + , t − ), and let the corresponding orbit portrait be O(t + , t − ) = {{θ1 , . . . , θq }}. Denote Bi = Bθi the bubble ray with angle θi starting with the bubble A∞ , and let αa be the common landing point of these rays. Another repelling fixed point of Ra , that in the intersection of A¯ 0 and A¯ ∞ will be denoted pa . Definition 7.5. The thin initial puzzle-pieces of Ra are the connected components of

ˆ (∪i Bi ) ∪ {αa } . C\ Similarly, a thick initial puzzle-piece of Ra corresponding to a thin puzzle-piece P is the set P¯ ∪ B 1 ∪ B 2 , where Bi are the two bubble rays which bound P. Finally, an initial puzzle-piece of Ra is a domain obtained as follows. Let γi be the axis of Bi terminating at α and ∞. Further, let ˆ : A∞ → C\D be the Böttcher coordinate, fix an arbitrary r > 1, and let D = −1 ({|z| > r }) and D = Ra−1 (Dr ) ∩ A0 . The initial puzzle-pieces are the connected components of ˆ (∪γi ) ∪ {αa } ∪ D¯ ∪ D¯ . C\ q

We denote the initial puzzle-pieces P01 , . . . , P0 . The puzzle pieces of depth n are the j n th preimages of P0i ; they will be denoted Pn . The basic properties being the same for all three kinds of puzzle-pieces, we will only formulate the results for the last kind. We begin by noting: j

Lemma 7.5 (Markov property). For any two puzzle pieces Pni , Pm one of the following two possibilities holds: they are disjoint, or one is a subset of the other. This allows us again to define for a point z ∈ J (Ra ) which is not a preimage of αa Pd (z) as the puzzle-piece of depth d which contains z. Further, set Ad (z) = Pd (z)\Pd+1 (z); we refer to this set as a complementary annulus, although it could be degenerate. We again label the annuli as critical, off-critical, and semi-critical depending on the position of the critical point −1. A critical annulus Ad+k (−1) will be called a child of the critical annulus Ad (−1) if Rak : Ad+k (−1) → Ad (−1) is an unramified double covering.

28


We define Ta to be a marked array Ta = (mod Ad (Ran (−1))), d, n ≥ 0, with the positions marked as critical, off-critical, or semi-critical if the respective annuli are. The following proposition is verified in a straightforward way, completely similarly to the quadratic case. We therefore omit the proof. Proposition 7.6. The marked array Ta is a critical tableau. However, it may happen that there is no non-degenerate annulus in the tableau Ta . We will need to modify the construction of the annuli slightly to guarantee the existence of one. 7.3. Non-degenerate annuli. The construction of a non-degenerate critical annulus for Ra is somewhat more delicate than that for a quadratic polynomial. We begin with the following: ˆ D. ¯ Lemma 7.7. We have P1 (−1) C\ Proof. There are q ≥ 3 infinite bubble rays Bk , k = 1, . . . q landing at α. First, let us argue that at least one bubble ray Bk contains A−2 (the Fatou component of Ra containing −2) and another contains A0 . Suppose this is not the case. Then all, but possibly one, external angles θk for Bk will belong to (1/6, 1/3) ∪ (2/3, 5/6). But then all, but possibly one, of the images of θk under doubling will belong to (1/3, 2/3), which is disjoint from (1/6, 1/3) ∪ (2/3, 5/6). Since q ≥ 3 this gives a contradiction. We want to show that the preimages Bk of Bk landing at the preimage of α have the same property, that is at least one bubble ray Bk contains A−2 and another contains A0 . If this is not the case then the images of all, but possibly one, Bk have angles in (1/3, 2/3), which is impossible. Hence the region P1 (−1) is bounded by four bubble rays which all emerge from A−2 or A0 . It is easy to see that this region is compactly contained in Ac∞ , and the lemma follows. 1 Now let us denote Z 11 , . . . , Z q−1 the puzzle-pieces of level 1 which are not adjacent to αa , but to its other preimage αa . It is easy to see that if Ta is not a periodic tableau, q j then some iterate of the critical point −1 under Ra will escape to one of the pieces Z 1 . The first time this happens, say after the n th iterate, we can pull back the degenerate j qn annulus P0 (−1)\P1 under Ra . See Fig. 8 for an illustration. This will give a degenerate critical annulus Am (−1). However, by Lemma 7.7, the only place where the boundaries of Pm (−1) and Pm+1 (−1) touch is a preimage of the segment l of two internal rays containing A∞ ∩ A0 which connects D and D . The invariance of A∞ ∪ A0 implies:

Lemma 7.8. The pinching of any child of Am (−1) is disjoint from Pm (−1) ∩ Pm+1 (−1). This means in particular the following: Corollary 7.9. Let Am (−1) be as above. Let Am j (−1) be any child of Am (−1). Then the critical puzzle pieces Pm j (−1) satisfy Pm j+1 (−1) Pm j (−1) and Pm j+1 +1 (−1) Pm j +1 (−1).


29

Fig. 4. Bubble rays for f ◦◦ and Ra . The picture below is a mating of f ◦◦ with a hyperbolic parameter in the 1/3-limb of M. Three periodic bubble rays land at a repelling fixed point of the rational map. The solid lines follow their axes. Their angles are 1/7, 2/7, and 4/7 respectively. The axes of the same bubbles are shown inside K ◦◦ in the above pictures. The broken lines show the position of the spines

Let us now thicken the arc l to a strip S which is invariant under the branch of Ra−1 fixing pa . Let us now replace Am (−1) with a non-degenerate annulus A˜ m (−1) obtained by thickening Am (−1) with the preimage of S. Definition 7.6. The thickened tableau Tã is obtained from Ta by preserving the same marking, and replacing the moduli of the images and pre-images of Am (−1) with those of A˜ m (−1). We derive: Theorem 7.2. Assume that the critical tableau Ta is recurrent and not periodic. Then Pd (−1) = {−1}.

30


Fig. 5. An example of a parameter wake, W (1/7, 2/7). The axes of the parabubble rays P1/7 , P2/7 which bound the wake are indicated. Their common landing point is the parameter value a0 for which Ra0 has a parabolic fixed point z 0 with eigenvalue e2πi/3 . The orbit portrait of z 0 is {1/7, 2/7, 4/7}

Fig. 6. The Yoccoz puzzles of depths 0 and 1, with q = 3 external rays landing at α

Proof. By Corollary 7.9, it is evident that Tã is a critical tableau, and Theorem 7.1 holds for it. By Lemma 7.2 this implies the result. We now handle the non-recurrent case: Lemma 7.10. If there is some N so that PN (−1) is disjoint from the orbit z 0 → z 1 → . . ., then ∩n Pn (z 0 ) = {z 0 }. Proof. Let us first thicken the puzzle pieces at the initial depth 0. Let B be a linearizing neighborhood of α, and let G be a linearizing neighborhood of p = A∞ ∩ A0 . Denote G 1 , . . . , G k the finitely many preimages of G which


31

j

Fig. 7. A bubble puzzle of depth 1. Note that the pieces P0i and P1 touch at an arc connecting A0 and A∞

/ /

/ / /

/ /

/ / / /

/

/ /

/

Fig. 8. A bubble-puzzle of depth 1 together with some preimages of the pieces Z 11 and Z 12 . The broken lines show the “equipotential” of depth 4. Note that Z 41 is degenerate in the sense that its boundary touches the boundary of P0 (−1), whereas Z 42 is not

• intersect one of the B1 , . . . , Bq , and j j • are not contained in B. Let P˜0 be a thick puzzle-piece of depth zero. We denote Pˆ0 j the union of P˜0 with B and with those of the domains G i which intersect with it. j j Thickened puzzle pieces Pˆd of depth d are the d th preimages of Pˆ0 . Note that the Markov property fails for these domains, however for each puzzle piece Pd (z) there exists a unique thickened puzzle piece Pˆd (z) which compactly contains it.

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Now the proof goes precisely as in [Mi5]. Set Ui = PˆNi −1 , numbered so that U0 = PˆN −1 (−a). Equip every Ui with the Poincaré distance ρi (x, y). Note that for each i > 0 there are exactly two univalent branches g1i and g2i of Ra−1 defined on Ui , each of which carries Ui into a proper subset of some U j . It follows that for each puzzle piece PNi −1 , i > 0, the branch gki shrinks the Poincaré distance by some definite factor λ < 1. Since the orbit z 0 , z 1 , z 2 , . . . avoids the critical puzzle piece we get that diam(PN +h (z 0 )) ≤ δλh , and the statement of the lemma follows.

7.4. Combinatorics of the puzzle. We make some definitions first. Let a1 , a2 be two parameters in the same wake W . We say that Ra1 and Ra2 have the same combinatorics of the puzzle up to depth d if there exists an orientation preserving homeomorphism ˆ →C ˆ such that the following holds: φ:C • φ homeomorphically maps distinct puzzle pieces Pki of depth k ≤ d of Ra1 to distinct j puzzle-pieces Q k of depth k of Ra2 ; • for all k ≤ d we have φ : Pk (−1) → Q k (−1); • finally, φ respects the dynamics, that is, j

j

Pki = Ra1 (Pk ) if and only if φ(Pki ) = Ra2 (φ(Pk )). Similarly, we will say that a quadratic polynomial f c and Ra have the same combinatorics of the puzzle up to depth d, if there exists an orientation-preserving continuous surjection φ which maps puzzle-pieces of f c to those of Ra up to depth d, sending critical pieces to critical ones, and respecting the dynamics. Proposition 7.11. Let f c be a quadratic polynomial without non-repelling fixed points. For every d there exists a parameter a such that Ra and f c have the same combinatorics of the puzzle down to depth d. Moreover, consider the puzzle-piece Pd (c) of f c , and let β1 , . . . , βk be the angles of external rays which bound it. Then bubble rays with the same angles bound the puzzle piece Q d (−a) of Ra . Finally, there exists an open set d in the a−plane, with d ⊂ d−1 , and 0 = W such that Rb has the same combinatorics of the puzzle to depth d and −b is contained in the particular puzzle piece of level d if and only if b ∈ d . Proof. The proposition follows by a straightforward induction on the depth d. The base of induction, with d = 0 is given by Lemma 6.11. Assuming the statement is true at depth d − 1, consider the pullback of the puzzle of level 1 inside the critical value piece Pd−1 (−a). By assumption, this picture has the same combinatorial structure as the similar one for f c . By Lemma 6.7, as the parameter a moves through d−1 , the critical value sweeps out Pd−1 (−a). We can hence select a parameter a to match the combinatorics of the puzzle of f c down to level d. The parameter plane statement follows from similarly obvious consideration and is left to the reader. Definition 7.7. We call a set d as above a parameter puzzle piece.


33

8. Existence of a Mating Fix a Yoccoz’ polynomial f c which is not critically finite, non-renormalizable, and such that c does not belong to the 1/2-limb of the Mandelbrot set. By Proposition 7.11, there exists a parameter value a such that Ra has the same combinatorics of the puzzle as f c for all d ∈ N. Lemma 8.1. Consider any z ∈ J (Ra ) which is not a preimage of αa or pa . Then the nested sequence of puzzle pieces Pd (z) shrinks to z : Pd (z) = {z}. Proof. Assume first that there exists some N > 0 such that the orbit of z is disjoint from PN (−1). In this case, the claim is implied by Lemma 7.10. In the opposite case, for each n ∈ N consider the first instance i such that Rai (z) ∈ Pn+1 (−1). Then the annulus A˜ n+i (z) is a conformal copy of A˜ n (−1). By construction, all these annuli around z are disjoint, and hence by Theorem 7.2, mod A˜ n (z) = ∞. By Lemma 7.2, we have the claim.

Lemma 8.2. Every bubble ray for Ra lands. Proof. This is obviously true for the preimages of the rays landing at the fixed point α. Let z be an accumulation point of any other ray B = ∪∞ 0 Fi . There is an infinite sequence of nested puzzle pieces Pd (z) containing z, and by the previous lemma, Pd (z) = {z}. Now by Lemma 2.7 the bubbles Fi do not cross the boundaries of Pd (z), and hence Fi → z. 8.1. Construction of semiconjugacies. Consider the conjugacy ◦

−n φ : K ◦◦ ∪n f ◦◦ (α) → ∪n Ra−n (A∞ ∪ { p})

defined in Proposition 4.5. By Lemma 8.2 and Lemma 5.6 it extends by continuity to a ˆ semi-conjugacy K ◦◦ → ∪Ra−n (A∞ ) = C: φ1 ◦ f ◦◦ (z) = Ra ◦ φ1 (z).

(8.1)

Let z ∈ Jc and not a preimage of α, and let Pd (z) be the sequence of Yoccoz’ puzzlepieces of depth d containing z. Let Q d (z) be the corresponding pieces in the puzzle of Ra and define φ2 (z) = ∩Q d (z). By construction, φ2 extends continuously to ∪n f c−n ({α}) and for the extended map φ2 ◦ f c = R a ◦ φ2 . Let ∼r denote the ray equivalence relation generated by the quadratics f ◦◦ and f c . We proceed to demonstrate:

34


Theorem 8.1. We have φi (z) = φ j (w) if and only if they are in the same ray equivalence class, z ∼r w. We begin with the following definition. Definition 8.1. For q > 1, let θ1 → θ2 → · · · → θq → θ1 be a period q orbit of the doubling map. The angles θi partition the circle into arcs Ai , i = 1, . . . , q, which we enumerate in the counter-clockwise order starting from the arc containing 0. For θ ∈ T which does not eventually fall into the orbit under doubling, we denote σθ1 ,...,θq (θ ) the itinerary of θ with respect to the partition Ai , viewed as an infinite string in {1, . . . , q}∞ . In the case when θ is a preimage of one of the θi the itinerary σθ1 ,...,θq (θ ) will be a finite string of digits between 1 and q – to avoid ambiguity, the last Ai will be chosen to the right of θi . In a very similar way, let us define a symbol sequence σ (z) ∈ {1, . . . , q}∞ with respect to the initial Yoccoz puzzle for f c or the initial Yoccoz bubble-puzzle for Ra as follows. Enumerate the initial puzzle-pieces of f c as P0k , k = 1, . . . , q in counterclockwise order around α, starting with P01 0. Set Q k0 to be the puzzle piece of Ra , which corresponds to P0k . Put j k if f c (z) ∈ P0k , for z ∈ J ( f c )\ ∪n f c−n (α), σ (z) = j k if Ra (z) ∈ Q k0 , for z ∈ J (Ra )\ ∪n Ra−n (αa ∪ pa ). Since φ1 is a semi-conjugacy the following lemma is immediate. Lemma 8.3. Assume that z ∈ K ◦◦ is uni-accessible and let φ1 (z) = ζ . Let −β be the angle of the external ray landing at z. If z is not a preimage of the α-fixed point, then σ (ζ ) = σ−θ1 ,...,−θq (−β). Recall now that a point in the Julia set J◦◦ is bi-accessible if and only if it is a preimage of α◦◦ . The latter is the landing point of two external rays, R1/3 , and R2/3 , forming a period 2 cycle. Let d be the function d : z → 2z mod Z. Lemma 8.4. Let Rθ be a ray landing at a bi-accessible point x ∈ J◦◦ . Then the landing point of R−θ in Jc is uni-accessible. Proof. The angle −θ has a finite orbit under the doubling, and hence the orbit of the landing point y of the ray R−θ is also finite. By assumption, f c is not critically finite, and hence the orbit of y does not include 0. Denote n the first iterate for which d n (−θ ) ∈ {1/3, 2/3}, and z = f n (y). Since f n is a local homeomorphism on a neighborhood of y, the number m of accesses is the same for y and z. Assume that m > 1. Note first that z cannot be a fixed point, as otherwise the ray portrait {{1/3, 2/3}} is realized for f c , and c is in the 1/2-limb. Hence z has period 2. By the properties of periodic external rays all rays landing at z have the same period, 2, and same for f (z). Hence, there are m × 2 ≥ 4 angles in T whose period under the doubling is equal to 2. By inspection, 1/3 and 2/3 are the only angles with this property, and we have arrived at a contradiction.


35

By assumption, there exists q > 2 such that there is a cycle of rays Rθ1 , . . . , Rθq landing at the dividing fixed point α of f c . By construction, a cycle of bubble rays Bθ1 , . . . , Bθq with the same angles lands at the fixed point αa . Lemma 8.5. We have φ1 (z) = φ1 (w) if and only if z ∼r w. Proof. By Lemma 2.7, only uni-accessible points can be identified. From Lemma 8.4 the lemma now follows if at least one of z and w is bi-accessible. Hence we can assume that both z and w are either landing points of infinite bubble rays B1 , B2 ⊂ K ◦◦ , or that one of z and w or both lies on a uni-accessible point on the boundary of a bubble. Denote −β1 , −β2 the angles of the external rays landing at z and w respectively. (In the case when z and w are landing points of infinite bubble rays Bi , note by definition, that the angles of these bubble rays are β1 and β2 respectively.) By Lemma 8.3, φ1 (z) = φ1 (w) if and only if σ−θ1 ,...,−θq (−β1 ) = σ−θ1 ,...,−θq (−β2 ).

(8.2)

Now, consider the external rays Rβi of f c . Since the combinatorics of the puzzles of f c and Ra is the same for every depth, these two rays have a common landing point if and only if (8.2) holds. The statement of the lemma now follows from Lemma 8.4. Lemma 8.6. We have φ2 (z) = φ2 (w) if and only if z ∼r w. Proof. Note that by Lemma 8.4, if z = w, then z ∼r w if and only if both of these points are uni-accessible, and denoting β1 , β2 their external angles, we have d n (β1 ) = 1/3, d n (β2 ) = 2/3 for some n. On the other hand, if ζ = φ2 (z) = φ2 (w), then ζ ∈ Ra−n ( pa ) for some n. It is thus enough to show that φ2 (z) = φ2 (w) = pa if and only if z, w are the landing points of the external rays R1/3 , R2/3 respectively. By construction, at most two points in Jc are mapped to pa by φ2 , so we only need to prove the second implication. The landing points z, w of rays R1/3 , R2/3 form a cycle of period 2, hence, the period of the cycle ζ1 = φ2 (z), ζ2 = φ2 (w) is at most 2. By Lemma 2.7, these points do not lie in the boundary of any of the bubbles. Assume that ζ1 = pa = ζ2 . Then there exists a bubble ray of angle θ landing at ζ1 . Since the combinatorics of the puzzle is the same for Ra and f c , σθ1 ,...,θq (θ ) = σθ1 ,...,θq (1/3). This bubble ray then lands at a point in J◦◦ with the external angle 2/3, which is a contradiction. We finish the proof of Theorem 8.1 with the following: Lemma 8.7. We have φ1 (z) = φ2 (w) if and only if z ∼r w.

36


Proof. If z ∈ K ◦◦ is uni-accessible then let −β be the angle of the external ray landing at z and put ζ = φ1 (z). By Lemma 8.3, σ−θ1 ,...,−θq (−β) = σ (ζ ). If ζ = φ2 (w), then w lies in the same puzzle-pieces as the point ζ , by definition. An external ray Rγ (there can be more than one) which lands at w must by Lemma 8.3 satisfy σθ1 ,...,θq (γ ) = σ (ζ ). Obviously, one solution is γ = −β, and therefore z ∼r w. Conversely, if z ∼r w, then φ1 (z) = φ2 (w) by construction. If z ∈ K ◦◦ is bi-accessible then the lemma follows from Lemma 8.4. We conclude: Main Theorem, the existence part. Suppose c is a non-renormalizable parameter value outside the 1/2-limb of M. Then the quadratic polynomials f c and f ◦◦ are conformally mateable.

9. Uniqueness of Mating To transfer the results of shrinking puzzle pieces in the dynamical plane to the parameter plane, we use a variation of the approach of Yoccoz (see [Hub]). Our arguments follow the presentation of [Ro1]. Let us recall the following definition. Definition 9.1. Let X be a connected complex mainfold. A holomorphic motion over a set E ⊂ C is a function ˆ ϕ : X × E → C, where ϕ(λ, z) is holomorphic in the variable λ ∈ X and injective in z ∈ E. We make use of a stronger version of the λ-lemma of Man˜e-Sud-Sullivan [MSS], due to Slodkowski [Slo]. The λ-Lemma. A holomorphic motion over a set E has a unique extension to a holoˆ morphic motion over E. The extended motion gives a continuous map ϕ : X × E → C. ˆ extends to a quasiconformal map of C ˆ to itself. For each λ ∈ X , the map ϕλ : E → C Let us fix a parameter c satisfying the conditions of the Main Theorem. Let n be the nested sequence of parameter puzzle-pieces of Proposition 7.11 in the a-plane. Our aim is to show: Theorem 9.1. We have diam(n ) → 0.


37

Let us fix a parameter a0 ∈ ∩n . Let P be a parabubble intersecting some n . Denote by Ba the bubble containing the critical value −a for Ra with a ∈ P. Let k ∈ N be the smallest such that for any a ∈ P, R k (−a) ∈ Aa∞ . Let ˆ a : Aa∞ → C\D denote the normalized Böttcher coordinate at infinity. By Lemma 5.6, it extends homeomorphically to the boundary. We then obtain a homeomorphism P → Ba0 by the formula F : a → Ra−k ◦ a−1 ◦ a ◦ Rak (−a). 0 0 Pasting these homeomorphisms together, we obtain Lemma 9.1. There is a homeomorphism from the closure of the boundary of the parameter puzzle piece of depth n into the closure of the boundary of the puzzle of depth n for Ra0 . We now construct a holomorphic motion on the boundary of the puzzle at an initial level. a0 ˆ where I a0 is the → C, Lemma 9.2. There is a holomorphic motion h n : n × In+1 n+1 a a . Moreover, 0 closure of the boundary of the puzzle of depth n + 1. We have h an (In+1 ) = In+1 a0 Ra ◦ h an (z) = h an0 ◦ Ra0 (z), for any z ∈ In+1 .

Proof. Indeed, as a varies throughout n , the critical value does not hit the bubble rays corresponding to the puzzle of depth n according to Lemma 5.5. We get from Lemma 5.9 that Aa∞ moves holomorphically on n . So do the preimages of Aa∞ as long as we do not hit the critical value. It follows that every bubble B in the boundary of the puzzle of depth n moves holomorphically according to the formula ηa (z) = Ra−k ◦ a−1 ◦ a0 ◦ Rak0 (z),

(9.1)

where k is smallest integer such that Rak (z) ∈ A∞ , for z ∈ B. Since the critical value does not intersect the puzzle of depth n, we can pull back this puzzle once so that the puzzle of depth n + 1 moves holomorphically as well. The λ-Lemma now extends the motion to its closure. It follows from (9.1) that a ) = I a0 and that the diagram h an (In+1 n+1 h an

a0 In+1 −−−−→ ⏐ ⏐ R a0

a In+1 ⏐ ⏐R a

h an−1

Ina0 −−−−→ Ina is commutative.

a be the puzzle piece bounded by h a (∂ P a0 ), where P a0 is the Definition 9.1. Let Dn+1 n n+1 n+1 puzzle piece Pn+1 surrounding the critical value −a0 at depth n + 1.

38


We have the following: Lemma 9.3. The parameter a ∈ m \m+1 if and only if the critical value −a ∈ a \D a . Dm m+1 Proof. Take a non self-intersecting path at from a0 to the boundary of m ,t ∈ [0, 1], crossing the boundary of m+1 exactly once. Then the critical value −at has to cross the a0 a ⊃ D a . Assume this happens at boundary of h amt (∂ Pm+1 ), since we always have Dm m+1 at t = t0 . Then for t > t0 we get that −at ∈ / Dm+1 , since we are outside m+1 . Similarly, a0 −at ∈ Dm+1 for t < t0 . Proof of Theorem 9.1. Let us first handle the harder case, when the critical tableau of f c is recurrent. Extend the holomorphic motion on m 0 at depth m 0 by the λ-Lemma, so that we get a holomorphic motion on m 0 with dilatation K = K (δ(a, ∂m 0 )), which depends on the conformal distance δ(a, ∂m 0 ) from a to the boundary of m 0 . Let us call this extended motion h˜ m 0 . m −m Now, lift the motion h˜ m 0 via the unbranched covering maps Ra j 0 for a ∈ m j . We get a holomorphic motion ˆ h˜ m j : m j × Aam0j −→ C, where Aam0j = Pm j (−a0 )\Pm j +1 (−a0 ) is an annulus surrounding the critical value (the Am j are children to Am 0 ). Since holomorphic composition does not change the dilatation, it follows that this lifted motion has the same dilatation K as h˜ m 0 . Moreover, the annuli a \D a Aam0j move holomorphically; set Aam j = h˜ m j (Aam0j ). In other words, Aam j = Dm m j +1 . j a By Lemma 9.3, we have that a ∈ m j \m j +1 if and only if −a ∈ Am j . Define the parameter annuli An = n \n+1 . Fix the number N = m j from now on and let N = . Define a map defined on , by H = H N : a → h˜ a−1 (−a). We see that H N : A N → AaN0 . On the boundary of it is injective, which follows directly from Lemma 5.9. The next issue is to show that the map H N is quasiconformal with a definite bound on the dilatation independent of N . Here the proof is again the same as in [Ro1]; let us differentiate the relation h˜ aN (H N (a)) = −a. Then we get ∂h aN (H N (a))∂ H N (a) + ∂h aN ∂ H N (a) = 0. This implies that the Beltrami coefficient µ(a) = ∂ H N/∂ H N satisfies |µ(a)| =

|∂h aN (H N (a))| KN − 1 = < 1, a |∂h N (H N (a))| KN + 1

where K N is the dilatation of h aN . However, if we consider the conformal representation χ : N → D, the λ-Lemma implies that KN =

1 + |χ (a)| . 1 − |χ (a)|

Since the set m j is compactly contained in m 0 for j ≥ 2, we get that K m j ≤ K , for all j ≥ 2.


39

We claim that the map H N is injective. First of all, it is injective on the boundary of A N . Moreover, if we solve the Beltrami equation for µ, then we get a quasiconformal map φ : A N → φ(A N ), so that ∂φ = µ∂φ. It follows that H N ◦ φ −1 is conformal. By the Riemann- Hurwitz formula, there can not be any branch points in A N . Since H N is injective on the boundary of A N , it follows that H N ◦ φ −1 maps φ(A N ) conformally onto AaN0 . It follows that H N must be a homeomorphism. Since the annulus Am 0 (−1) may be degenerate, we again consider the thickened annuli A˜ m (−1) from Definition 7.6 – we select the initial thickened piece so that it moves holomorphically with a. Consider the thickened annulus A˜ m 0 (−a0 ) for a = a0 . The map H N (N = m j ) defined above then easliy extends to the thickened non-degenerate parameter annulus A˜ N . It follows that 1 1 mod A˜ am0j ≤ mod A˜ m j ≤ mod A˜ am0j . K K Since mod A˜ N = ∞ we have mod A˜ N = ∞, and we conclude from Lemma 7.2 that the parameter pieces N shrink to a single point, which has to be a0 . In the non-recurrent case, consider the puzzle of depth N so that the critical puzzle piece PN (−1) is disjoint from the postcritical set. As the critical value −a varies through N the puzzle at depth N +1 moves holomorphically as in Lemma 9.2. Hence every annulus A N (z) moves holomorphically. Extend this holomorphic motion by Slodkowski’s Theorem and denote the extended motion by h˜ similar to the above argument. Since every annulus An (−a0 ), for n > N , is a univalent pullback of some A N (z) (since R − a0 is non-recurrent) we can lift the holomorphic motion h˜ to the parameter piece n over Pn (−a0 ). Define a map Hn : An → An (−a0 ) in exactly the same way as above. The proof of the fact that the parameter annuli shrink to a single point is now similar to the recurrent case and we leave the details to the reader. We conclude: Main Theorem, the uniqueness part. The mating in the Main Theorem is unique. Acknowledgements. We wish to thank Mitsu Shishikura for useful comments. We are grateful to Vladlen Timorin for dicussing his own results on the family Ra with us, and for a helpful remark on a preliminary version of the paper. We are grateful to Roland Roeder for help with computer graphics and stimulating conversations. We thank Tan Lei for useful comments on an earlier version. Finally, the first author thanks Rodrigo Perez for stimulating discussions. This work was carried out at the Fields Institute, during the Dynamics Thematic Program of 2005-6. We gratefully acknowledge the Institute’s support and hospitality.

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Douady, A.: Systéms dynamiques holomorphes. Seminar Bourbaki. Astérisque. 105-106, 39–64 (1983) Douady, A.: Disques de Siegel at aneaux de Herman. Seminar Bourbaki. Astérisque. 152-153, 151– 172 (1987) Douady, A., Earle, C.: Conformally natural extension of homeomorphisms of the circle. Acta Math. 157, 23–48 (1986) Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes I & II. Preprint, Pub. Math. d’Orsay 84-02, 85-05, 1984/1985 Douady, A., Hubbard, J.H.: A proof of Thurston’s topological characterization of rational functions. Acta Math. 171, 263–297 (1993) Epstein, A.: Counterexamples to the quadratic mating conjecture. Manuscript Hubbard, J.H.: Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, In: “Topological methods in modern mathematics” (Stony Brook, NY, 1991), Houston, TX: Publish or Perish, pp. 467–511, 1993 Luo, J.: Combinatorics and holomorphic dynamics: Captures, matings, Newton’s method. Thesis, Cornell University, 1995 Lyubich, M.Yu.: The dynamics of rational transforms: the topological picture. Russian Math. Surveys 41, 43–117 (1986) Lyubich, M.Yu.: Dynamics of quadratic polynomials i-ii. Acta Math. 178, 185–297 (1997) Lyubich, M.Yu.: Parapuzzles and SRB measures. Asterisque. 261, 173–200 (2000) Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. École Norm. Sup. (4) 16(2), 193–217 (1983) McMullen, C.: Complex Dynamics and Renormalization. Ann. Math Stud. Vol. 135, 1994 Milnor, J.: Dynamics in One Complex Variable: Introductory Lectures, Wiesbaden: Vieweg, 1999 Milnor, J.: Geometry and dynamics of quadratic rational maps. Experim. Math. 2, 37–83 (1993) Milnor, J.: Periodic orbits, external rays, and the Mandelbrot set: An expository account. Asterisque 261 (1999) Milnor, J.: Pasting together Julia sets - a worked out example of mating. Expe. Math. 13(1), 55–92 (2004) Milnor, J.: Local connectivity of Julia sets: expository lectures. In: The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., 274, Cambridge: Cambridge Univ. Press, pp. 67–116, 2000 Moore, R.L.: Concerning upper semi-continuous collection of continua. Trans. Amer. Math. Soc. 27, 416–428 (1925) Rees, M.: Realization of matings of polynomials as rational maps of degree two. Manuscript, 1986 Rees, M.: A partial description of parameter space of rational maps of degree two: part i. Acta Math. 168, 11–87 (1992) Roesch, P.: Holomorphic motions and parapuzzles, In: The Mandelbrot set, Theme and variations, London Math. Soc. Lecture Note Ser., 274, Cambridge: Cambridge Univ. Press, 2000 Roesch, P.: Topologie locale des méthodes de Newton cubiques. Thesis, E.N.S, Lyon, 1997 Shishikura, M.: On a theorem of M. Rees for matings of polynomials. In: The Mandelbrot set, Theme and variations, London Math. Soc. Lecture Note Ser., 274, Cambridge: Cambridge Univ. Press, 2000 Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. 147, 25–267 (1998) Shishikura M.: Bifurcation of parabolic fixed points. In: The Mandelbrot set, Theme and variations, London Math. Soc. 274, Cambridge: Cambridge Univ. Press, 2000 Shishikura, M., Tan, L.: A family of cubic rational maps and matings of cubic polynomials. Exp. Math. 9(1), 29–53 (2000) Slodkowski, Z.: Natural extensions of holomorphic motions. J. Geom. Anal. 7(4), 637–651 (1997) Tan, L.: Matings of quadratic polynomials. Erg. Th. and Dyn. Sys. 12, 589–620 (1992) Tan, L., Yin, Y.: Local connectivity of the Julia set for geometrically finite rational maps. Sci. China Ser. A 39(1), 39–47 (1996) Yampolsky, M.: Complex bounds for renormalization of critical circle maps. Erg. Th. Dyn. Sys. 18, 1–31 (1998) Yampolsky, M., Zakeri, S.: Mating Siegel quadratic polynomials. J. Amer. Math. Soc. 14(1), 25–78 (2001) Zakeri, S.: Biaccessibility in quadratic Julia sets. Erg. Th. Dyn. Sys. 20(6), 1859–1883 (2000)

Communicated by L. Takhtajan

Commun. Math. Phys. 287, 41–65 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0710-3

Communications in


The Power of Quantum Systems on a Line Dorit Aharonov1, , Daniel Gottesman2, , Sandy Irani3, , Julia Kempe4, 1 School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel.

E-mail: [email protected]

2 Perimeter Institute, Waterloo, Canada. E-mail: [email protected] 3 Computer Science Department, University of California, Irvine, CA 92697, USA.


4 School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel.

E-mail: [email protected] Received: 28 January 2008 / Revised: 2 October 2008 / Accepted: 8 October 2008 Published online: 13 January 2009 – © Springer-Verlag 2009

Abstract: We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses. Supported by Israel Science Foundation grant number 039-7549, Binational Science Foundation grant number 037-8404, and US Army Research Office grant number 030-7790. Supported by CIFAR, by the Government of Canada through NSERC, and by the Province of Ontario through MRI. Partially supported by NSF Grant CCR-0514082. This work was mainly done while the author was at CNRS and LRI, University of Paris-Sud, Orsay, France. Partially supported by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848, by an ANR AlgoQP grant of the French Research Ministry, by an Alon Fellowship of the Israeli Higher Council of Academic Research, by an Individual Research grant of the ISF, and by a European Research Council (ERC) Starting Grant.

42

D. Aharonov, D. Gottesman, S. Irani, J. Kempe

1. Introduction The behavior of classical or quantum spin systems frequently depends very heavily on the number of spatial dimensions available. In particular, there is often a striking difference between the behavior of one-dimensional systems and of otherwise similar twoor higher-dimensional systems. For instance, one-dimensional systems generally do not experience phase transitions except at zero temperature, whereas phase transitions are common in other dimensions. As another example, satisfiability with nearest neighbor constraints between constant-size variables set on a two- or larger-dimensional grid is a hard problem (NP-complete, in fact), whereas for a one-dimensional line, it can be solved in polynomial time. We thus ask: what is the computational power associated with a line of quantum particles? There are a number of ways to interpret the question. For instance, we can ask whether the evolution of such systems can be efficiently simulated by a classical computer. We can ask whether the system is powerful enough to create a universal quantum computer under various scenarios for how we may control the system — in which case we cannot hope to simulate this behavior efficiently on a classical computer, unless of course BQP = BPP. (BPP and BQP are the classes of problems efficiently solvable on a probabilistic classical computer and on a quantum computer, respectively.) We can also ask how difficult it is to calculate or approximate interesting properties of the system, such as its ground state energy (that is, the lowest eigenvalue of the Hamiltonian operator for the system). For many one-dimensional quantum systems, it is indeed possible to classically compute many properties of the system, including in some cases the dynamical behavior of the system. The method of the density matrix renormalization group (DMRG) [Whi92, Whi93,Sch05] has been particularly successful here, although there is no complete characterization of which one-dimensional systems it will work on. Indeed, DMRG provides a good example of the difference between one- and two-dimensional systems; there are only a few results applying DMRG techniques to simulate special two-dimensional systems. However, it has long been known that one-dimensional quantum systems can also, under the right circumstances, perform universal quantum computation. It is straightforward to create a quantum computer from a line of qubits with fully controllable nearest-neighbor couplings by using SWAP gates to move the qubits around. Even a one-dimensional quantum cellular automaton can perform universal quantum computation [Wat95,SFW06]; the smallest known construction has 12 states per cell. While many one-dimensional systems are relatively simple and can be simulated classically, the general one-dimensional quantum system must thus have complexities that are inaccessible classically. In two interesting (and closely related) subfields of the area of quantum computation, studied extensively over the past few years, it has been conjectured that one-dimensional systems are not as powerful as systems of higher dimensionality. These cases are adiabatic evolution, and the QMA-completeness of the local Hamiltonian problem.

1.1. Results related to adiabatic computation. In an adiabatic quantum computer, the Hamiltonian of the system is slowly shifted from a simple Hamiltonian, whose ground state is easy to calculate and to create, to a more complicated Hamiltonian, whose ground state encodes the solution to some computational problem. The quantum adiabatic theorem guarantees that if the Hamiltonian is changed slowly enough, the system stays close

The Power of Quantum Systems on a Line

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to its ground state; the time required to safely move from the initial Hamiltonian to the final Hamiltonian is polynomial in the minimal spectral gap between the ground state and the first excited state over the course of the computation. The first adiabatic quantum algorithm was introduced by Farhi et al. [FGGS00, FGG+ 01], though the idea of encoding the solution to an optimization problem in the ground state of a Hamiltonian appeared as early as 1988 [ACdf90,ACdf88]. The computational power of the model was clarified in [AvDK+ 04], where it was shown that adiabatic quantum computers could in fact perform any quantum computation. Adiabatic computation can thus be viewed as an alternative model to quantum circuits or quantum Turing machines. Adiabatic quantum computers might be more robust against certain kinds of noise [CFP02], and the slow change of parameters used in an adiabatic quantum computer might be more amenable to some kinds of experimental implementation. To this end, it is important to understand what sorts of physical systems can be used to build an adiabatic quantum computer, and in particular, we would like the interactions of the physical Hamiltonian to be as simple as possible. Several research groups have devoted effort to this question recently. To describe their work, let us make the following definition: Definition 1 (Local Hamiltonian). Let H be a Hermitian operator (interpreted as a Hamiltonian, giving the energy of some system). We say that H is an r -state Hamiltonion if it acts on r -state particles. When r = 2, namely, when the particles are qubits, we often omit mention of the number of states per particle. We say that H is k-local if it can be written as H = i Hi , where each Hi acts non-trivially on at most k particles. Note that this term does not assume anything about the physical location of the particles. We say that H is a d-dim Hamiltonian if the particles are arranged on a d-dimensional grid and the terms Hi interact only pairs of nearest neighbor particles. Note that the term d-dim implies that the Hamiltonian is 2-local. We will in this paper assume for simplicity that each local term Hi in the Hamiltonian is positive definite and of norm at most 1. It is always possible to ensure this by shifting and rescaling the Hamiltonian H → λ(H + E 0 ). Provided we also similarly shift and rescale all other energies in the problem (such as the spectral gap), this transformation will not at all alter the properties of the Hamiltonian. We will further assume, to avoid pathologies, that there are at most polynomially many terms Hi and that the coefficients in each Hi are specified to polynomially many bits of precision. Aharonov et al. [AvDK+ 04] made a first step towards a practical Hamiltonian, by showing that a 2-dim 6-state Hamiltonian suffices for universal adiabatic quantum computation. Kempe, Kitaev and Regev [KKR06], using perturbation-theory gadgets, improved the results and showed that qubits can be used instead of six-dimensional particles, namely, adiabatic evolution with 2-local Hamiltonians is quantum universal. Their interactions, however, were not restricted to a two-dimensional grid. Terhal and Oliveira [OT05] combined the two results, using other gadgets, to show that a similar result holds with 2-dim 2-state Hamiltonians. Naturally, the next question was whether a one-dimensional adiabatic quantum computer could be universal. Since one-dimensional systems are fairly simple, it was conjectured that the answer was “no.” However, in this paper, we prove Theorem 1. Adiabatic computation with 1-dim 9-state Hamiltonians is universal for quantum computation. As mentioned, this could be important experimentally, as one-dimensional systems are easier to build than two-dimensional ones, and adiabatic systems might be more

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robust than standard ones. Any simplification to the daunting task of building a quantum computer is potentially useful. However, a more systematic way of dealing with errors will be needed before it is possible to build a large adiabatic quantum computer. The results of [CFP02] only imply adiabatic quantum computation is robust against certain types of control errors, and it remains an interesting open question to show that adiabatic quantum computation can be made fault-tolerant in the face of general small errors. An important step in this direction is [JFS06], which introduces some quantum error-correcting codes in the adiabatic model. It is also worth noting that the adiabatic construction given in the proof of Theorem 1 actually has a degenerate ground state, which makes it less robust against noise than a non-degenerate adiabatic computer, although still resistant to timing errors, for instance. Using a 1-dim 13-state Hamiltonian similar to the QMA-complete 1-dim 12-state Hamiltonian family described below, it is possible to break the degeneracy. This construction is outlined in Sect. 4.1. It is natural to ask whether our result holds even when we restrict ourselves to translationally invariant Hamiltonians. We sketch one way to do this in Sect. 2. However, it does not appear to be possible to make the translational invariant adiabatic construction nondegenerate.

1.2. Results related to QMA-completeness. Theorem 1 means that efficient simulation of general one-dimensional adiabatic quantum systems is probably impossible. One might expect that calculating, or at least approximating, some specific property of a system, such as its ground state energy, would be more straightforward, as this does not require complete knowledge of the system. This intuition is misleading, and in fact it can be harder to find the ground state energy than to simulate the dynamics of a system. More concretely, it has long been known that it is NP-hard to find the ground state energy of some classical spin systems, such as a three-dimensional Ising model. Kitaev [KSV02] extended these results to the quantum realm, by defining the quantum analogue of NP, called QMA (for “quantum Merlin-Arthur”). QMA is thus, roughly speaking, the class of problems that can be efficiently checked on a quantum computer, provided we are given a “witness” quantum state related to the answer to the problem. QMA is believed to be strictly larger than both BQP and NP. That is, a problem which is QMA-complete can most likely not be solved efficiently with a quantum computer, and the answer cannot even be checked efficiently on a classical computer. Kitaev proved that the problem of approximating the ground state energy of a quantum system with a 5-local Hamiltonian is complete for this class. (The ability to solve a QMA-complete problem implies the ability to solve any problem in QMA.) The exact definition of the local Hamiltonian problem is this: Definition 2 (Local Hamiltonian problem). Let H = i Hi be an r -state k-local Hamiltonian. Then (H, E, ∆) is in r -STATE k-LOCAL HAMILTONIAN if the lowest eigenvalue E 0 of H is less than or equal to E. The system must satisfy the promises that ∆ = (1/poly(n)) and that either E 0 ≤ E or E 0 ≥ E + ∆. We can define a language d-DIM r -STATE HAMILTONIAN similarly. Kitaev’s proof uses what is called the circuit-to-Hamiltonian construction of Feynman [Fey85], which we sketch in Subsect. 1.5. The main result of [AvDK+ 04], namely the universality of adiabatic computation, was based on the observation that the circuit-to-Hamiltonian construction is useful also in the adiabatic context, where it is used


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to design the final Hamiltonian of the adiabatic evolution. It turns out that the same connection between adiabatic universality proofs and QMA-completeness proofs can also be made for many of the follow-up papers on the two topics. It is often the case that whenever adiabatic universality can be proven for some class of Hamiltonians, then the local Hamiltonian problem with the same class (roughly) can be shown to be QMA-complete — and vice versa. Note, however, that there is no formal implication from either of those problems to the other. On one hand, proving QMA-completeness is in general substantially harder than achieving universal adiabatic quantum computation: In an adiabatic quantum computer, we can choose the initial state of the adiabatic computation to be any easily-created state which will help us solve the problem, so we can choose to work on any convenient subspace which is invariant under the Hamiltonian. For QMA, the states we work with are chosen adversarially from the full Hilbert space, and we must be able to check ourselves, using only local Hamiltonian terms, that they are of the correct form. On the other hand, proving adiabatic universality involves analyzing the spectral gap of all of the Hamiltonians H (t) over the duration of the computation, whereas QMA-completeness proofs are only concerned with one Hamiltonian. Both [KKR06 and OT05] in fact prove QMA-completeness; universal adiabatic quantum computation follows with a little extra work. Thus, the 2-DIM HAMILTONIAN problem is QMA-complete. However, the 1-DIM r -STATE HAMILTONIAN problem remained open for all (constant) r . It was suspected that the problem was not QMA-complete, and in fact might be in BQP or even BPP. For one thing, ground state properties of one-dimensional quantum systems are generally considered particularly easy. For instance, Osborne has recently proven [Osb07] that there are efficient classical descriptions for a class of onedimensional quantum systems. DMRG techniques have been employed extensively to calculate ground state energies and other properties of a variety of one-dimensional quantum systems. Furthermore, the classical analogue of 1-DIM r -STATE HAMILTONIAN is easy: Take r -state variables arranged on a line with constraints restricting the values of neighboring pairs of variables. If we assign a constant energy penalty for violating a constraint, the lowest energy state satisfies as many constraints as possible. This problem, a one-dimensional restriction of MAX-2-SAT with r -state variables, is in fact in P; it can be solved with a recursive divide-and-conquer algorithm or by dynamic programming. For instance, we can divide the line in half, and run through all possible assignments of the two variables xi and xi+1 adjacent to the division. For each pair of values for xi and xi+1 , we can calculate the maximal number of satisfiable constraints by solving the sub-problems for the right and left halves of the line. Then we can compare across all r 2 assignments for the one that gives the largest number of satisfied constraints. Thus, to solve the problem for n variables, it is sufficient to solve 2r 2 similar problems for n/2 variables. By repeatedly dividing in half, we can thus solve the problem in 2 O((2r 2 )log n ) = O(n log(2r ) ) steps. MAX-k-SAT in one dimension is also in P, and can in fact be reduced to MAX-2-SAT with large enough variables. Despite the intuition that one-dimensional systems should not be too difficult, we prove: Theorem 2. 1-DIM 12-STATE HAMILTONIAN is QMA-complete. The theorem implies a striking qualitative difference between the quantum and the classical one-dimensional versions of the same problem — one is easy, whereas the other is complete for a class which seems to be strictly larger than NP. This might seem

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surprising, but in retrospect, we can provide an intuitive explanation. The reason is that the k-local Hamiltonian essentially allows us to encode an extra dimension, namely, time, by making the ground state a superposition of states corresponding to different times. In other words, the result implies that the correct analogue of one-dimensional local Hamiltonian is two-dimensional MAX-k-SAT, which is of course NP-complete. Indeed, there are many cases in physics where one-dimensional quantum systems turn out to be most closely analogous to two-dimensional classical systems. One such example is given by Suzuki who showed that the d-dimensional Ising model with a transverse field is equivalent to the (d + 1)-dimensional Ising model at finite temperatures [Suz76], although the extra dimension there is quite different from the dimension of time addressed in this paper.

1.3. Implications of our results. One consequence of Theorem 2 is that there exist onedimensional systems which take exponentially long to relax to the ground state, with any reasonable environment or cooling strategy. To see this, we invoke a quantum version of the modern Church-Turing thesis, 1 which would state that any reasonable physical system can be efficiently simulated on a quantum computer. As it invokes a notion of a “reasonable” physical system, this is of course not a provable statement, but one can be convinced of it by looking at examples. The details of doing so for any particular system may be complicated, but no counter-example is known, and barring undiscovered laws of physics modifying quantum mechanics, the quantum Church-Turing thesis is believed to be true. In this particular case, we could thus use a quantum computer to simulate the system plus the environment. If the system reliably relaxes to the ground state in a time T , then by causality, we need only simulate an environment of size O(T 3 ) (for a threedimensional environment); the quantum simulation can be performed using standard techniques and runs in polynomial time. We can then use the simulation to solve 1-DIM 12-STATE HAMILTONIAN problems. Since the latter problem is difficult, that implies the simulation must also take a long time to reach the ground state. This observation assumes, of course, that QMA is exponentially hard to solve by a quantum computer. Similarly, we can argue that the class of systems presented in this paper will take an exponentially long time to relax to the thermal equilibrium state at very low temperatures (again, assuming QMA is hard for a quantum computer). To see this, suppose that there is a state of the system with energy less than E (the bound on the ground state energy from Definition 2), and note that at a temperature less than O(∆/n) (with ∆ polynomially small in the system size n), the Gibbs distribution gives a constant probability of the system occupying the low-lying state, even if there are exponentially many eigenstates with energy just above E + ∆. Of course, if there is no state with energy less than E + ∆, then the system will never have energy less than E + ∆, no matter what the temperature. Therefore, if the system were able to reach its thermal equilibrium state, by observing it, we would be sampling from the Gibbs distribution, and would, with good probability, be able to answer the QMA-complete 1-DIM 12-STATE HAMILTONIAN problem. Exponentially long relaxation times are a characteristic of a spin glass [BY86], making this system a candidate to be a one-dimensional spin glass. Spin glasses are not completely understood, nor is there a precise definition of the term, but it has been unclear whether a true one-dimensional spin glass can exist. There are a number of properties relevant to being a spin glass, long relaxation times among them, and clas1 The modern Church-Turing thesis is discussed, for instance, in [vEB90].


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sical one-dimensional systems generally lack that property and most other typical spin glass properties. On the other hand, some other one-dimensional quantum systems are known [Fis95] that exhibit long relaxation times and some, but not all, other spin-glasslike properties. Our result may help shed further light on the situation. Unfortunately, we are unable to present a specific Hamiltonian with long relaxation times. As usual in complexity, it is difficult to identify specific hard instances of computationally hard problems. Similarly, we cannot say very much about other physically interesting properties of the system, such as the nature or existence of a liquid/glass phase transition. This paper is a merger of [Ira07] and version 1 of [AGK07], two similar papers originally written independently. The extended abstract of the merged version appeared in [AGIK07]. Following the initial version of our paper, various groups have improved and extended our result. For instance, [NW08] gives a 20-state translation-invariant modification of our construction (improving on a 56-state construction by [JWZ07]) that can be used for universal 1-dimensional adiabatic computation. Kay [Kay08] gives a QMA-complete 1-dim r -state Hamiltonian with all two-particle terms identical. Nagaj [Nag08] has improved our QMA construction to use 11-state particles. 1.4. Notation. A language is computer science terminology for a computational problem; it can be defined as a set of bit strings. Frequently we consider the language to be a proper subset of all possible bit strings, but here we will be more concerned with languages which satisfy a promise, meaning we consider the language as a subset of a smaller set of bit strings which satisfy some constraint. We will refer to the set of strings which satisfy the constraint as the instances of the problem. The instances then can be partitioned into two sets L yes , the set of instances which are in the language, and L no , the set of instances which are not in the language. The decision problem associated with the language is to decide if a given bit string is in the language or not. The instances represent different possible inputs about which we are attempting to reach a yes/no decision. For example, satisfiability (or SAT) is the decision problem for the language composed of Boolean formulas which have satisfying assignments of the variables; each instance encodes a possible formula, some of which can be satisfied and some of which cannot. A complexity class is a set of languages. The precise definition of the complexity class QMA is as follows: Definition 3. A language L is in QMA iff for each instance x there exists a uniform polynomial-size quantum circuit 2 C x such that (a) if x ∈ L yes , ∃|ψ (the “witness”, a polynomial-size quantum state) such that C x |ψ accepts with probability at least 2/3, and (b) if x ∈ L no , then ∀|ψ, C x |ψ accepts with probability at most 1/3. We only consider strings which are instances of L. We say we reduce language L to language M if we have a function f converting bit strings x to bit strings f (x) such that if x ∈ L yes then f (x) ∈ Myes and if x ∈ L no , then f (x) ∈ Mno . We do not put any requirement on the behavior of f when the input x does not satisfy the promise. f must be computable classically in polynomial time. The point is that if we can solve the decision problem for M then we can automatically also solve the decision problem for L by first converting x to f (x) and then checking if f (x) is in M. There are other notions of reduction, but this will be sufficient for our purposes. A language L is complete for a complexity class C if any language in C can be reduced to L. Thus, solving a C-complete language L implies the ability to solve any problem in 2 A uniform circuit is one whose description can be generated in polynomial time by a Turing machine.

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C with polynomial time overhead; L is thus one of the most computationally difficult problems in C. We will be interested in reducing an arbitrary QMA language L to variants of the language r -STATE k-LOCAL HAMILTONIAN. We will use x to refer to an instance of the original problem L, and C x to refer to the checking circuit associated with the instance x. That is, we assume that information about the instance is encoded in the structure of the checking circuit. The circuit C x acts on n qubits. Some of the qubits will be ancilla qubits which must be initialized to the state |0, whereas others are used for the potential witness for x ∈ L. In the context of adiabatic quantum computation, we will also use C x to refer to a more general quantum circuit, unrelated to a QMA problem. In this case, we consider C x to be in a form where all input qubits are |0. When the adiabatic quantum computation is intended to compute some classical function (such as factoring), x is the classical input to the function, which we hardwire into the structure of the circuit itself.

1.5. Outline of the approach. To explain the main idea behind the proofs of our two main theorems, we recall Kitaev’s proof that 5-LOCAL HAMILTONIAN is QMA-complete. The fact that r -STATE k-LOCAL HAMILTONIAN is in QMA is not difficult. The witness is the ground state of the Hamiltonian (or indeed any state with energy less than E), and a standard phase estimation technique can be used to measure its energy. The accuracy of the measurement is not perfect, of course, which is why we need the promise that the energy is either below E or above E + ∆. To prove completeness, we need to reduce an arbitrary QMA language L to 5-LOCAL HAMILTONIAN. The translation to local Hamiltonians is done by creating a Hamiltonian whose ground state is of the form t |φt |t (ignoring normalization), where the first register |φt is the state of the checking circuit C x at time t and the second register acts as a clock. We call this state the history state of the circuit. 3 The main term in Kitaev’s Hamiltonian sets up constraints ensuring that |φt+1 = Ut |φt , where Ut is the (t + 1)st gate in C x . This term, denoted Hprop , ensures that the ground state is indeed a history state, reflecting a correct propagation in time according to C x . The clock is used to associate the correct constraint with each branch of the superposition; any state which does not have the correct time evolution for the circuit will violate one or more constraints and will thus have a higher energy. Kitaev’s Hamiltonian includes more terms, which guarantee that the input to C x is correct, that the final state of the circuit is accepted, and that the state of the clock is a valid clock state and not some arbitrary state. In the context of adiabatic evolution, these additional terms are not needed, since we have control over the initial state, but they are needed to prove QMA-completeness, since we must be able to check (i) that the witness being tested for the QMA-complete problem has the correct structure, corresponding to a valid history state for a possible input to C x , and (ii) that C x accepts on that input. In order to prove the universality of adiabatic quantum computation, [AvDK+ 04] let Kitaev’s Hamiltonian Hprop be the final Hamiltonian in the adiabatic evolution, and set the initial Hamiltonian to be diagonal in the standard basis, forcing the initial ground state to be the correct input state |φ0 |0 of the circuit. At any point during the adiabatic quantum computation, the state of the system is then in a subspace I spanned by the states |φt |t, and the spectral gap of any convex combination of the initial and final Hamiltonians restricted to this subspace is at most polynomially small in the size of the 3 We call a state a history state if it has the above form for any input |φ to C , not just the correct input. x 0


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original circuit. Therefore, the adiabatic quantum computation can be performed with at most polynomial overhead over the original circuit, to reach a state which is very close to the history state, from which the result of the original circuit can be measured with reasonable probability. Our idea for proving universality of adiabatic evolution and QMA-completeness in one dimension is similar. One would consider a quantum circuit C x , and design a 1-dim r -state Hamiltonian which will verify correct propagation according to this circuit, and then use this Hamiltonian as the final Hamiltonian for the adiabatic evolution or as the instance of the QMA-complete problem corresponding to the instance x of the original problem. This idea, however, is not easy to realize when our Hamiltonian is restricted to work on a one- or two-dimensional lattice, since we cannot directly access a separate clock register, as only the subsystems nearest to the clock would be able to take advantage of it in order to check correct propagation in time. Instead, following the strategy of [AvDK+ 04], we must first modify the original circuit C x into a new circuit C˜ x . This allows us to distribute the clock, making the time implicit in the global structure of the state. When time is encoded somehow in the configuration, then it can be ensured that the transition rules allow only propagation to the correct next step. The construction of [AvDK+ 04] used the following arrangement: the qubits of the original circuit were put initially in the left-most column in the two-dimensional grid, and one set of gates was performed on them. To advance time and perform another set of gates, the qubits were moved over one column, leaving used-up “dead” states behind. Doing all this required going up to 6 states per particle instead of 2, with the qubits of the original circuit encoded in various two-dimensional subspaces. The time could therefore be read off by looking at the arrangement and location of the two-dimensional subspaces containing the data. This construction relies heavily on the ability to copy qubits to the next column in order to move to the next block of gates in the computation, so a new strategy is needed in one dimension. For the modified circuit C˜ x , we instead place the qubits in a block of n adjacent particles. We do one set of gates, and then move all of the qubits over n places to advance time in the original circuit C x . It is considerably more complicated to move qubits n places than to move them over one column in a two-dimensional arrangement, and we thus need extra states. The adiabatic construction can be done using 9 states per particle, which we increase to 12 for QMA-completeness. A straightforward application of existing techniques can then complete the proof of one-dimensional universal adiabatic quantum computation, but there is an additional wrinkle for proving QMA-completeness. In particular, previous results used local constraints to ensure that the state of the system had a valid structure; for instance, in [AvDK+ 04], terms in the Hamiltonian check that there are not two qubit states in adjacent columns. However, using only local constraints, there is no way to check that there are exactly n qubit data states in an unknown location in a one-dimensional system — there are only a constant number of local rules available, which are therefore unable to count to an arbitrarily large n. We instead resort to another approach, which might be useful elsewhere. While our modified circuit has invalid configurations (containing, for instance, too many qubit states) which cannot be locally checked, we ensure that, under the transition rules of the system, any invalid configurations will evolve in polynomial time into a configuration which can be detected as illegal by local rules. Thus, for every state which is not a valid history state, either the propagation is wrong, which implies an energy penalty due to the propagation Hamiltonian, or the state evolves to an illegal

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configuration which is locally detectable, which implies an energy penalty due to the local check of illegal configurations. We call this result the clairvoyance lemma. The structure of the paper is as follows: in Sect. 2 we describe how to map circuits to one-dimensional arrangements. Sect. 3 shows our result on the universality of adiabatic computation on the line, and Sect. 4 gives the result on QMA-completeness. We conclude in Sect. 5. 2. The Basic Construction In this section, we will describe our construction which maps a quantum circuit C x to a modified circuit C˜ x . We first present the larger 12-state QMA construction, and explain how to modify it to get the 9-state adiabatic construction at the end of this section. There are a number of properties which C˜ x must satisfy. It should perform the same computation as C x , of course. In addition, each gate in C˜ x must interact only nearest-neighbor particles in a line; on the other hand, we allow those particles to be 12-dimensional. The gate performed at any given time cannot depend explicitly on the time (the number of gates already performed), but can depend on location. We will define the gates in terms of a variety of possible transition rules, and to remove any ambiguity, we will ensure that for a legal state of the system, only one transition rule will apply at any given time. For QMA-completeness, we need some additional properties ensuring that enough of the constraints are locally checkable. The problem of moving to one dimension is somewhat similar to a quantum cellular automaton in that the transition rules need to depend only on the local environment and not on some external time coordinate, but differs from a cellular automaton in a number of ways. A cellular automaton acts on all locations simultaneously and in the same way, whereas our transition rules are required to only cause one pair of particles to change per time step (provided the system is in a state of the correct form), and the transition rules differ slightly from location to location. A better analogy is to a single-tape Turing machine. We will have a single “active site” in our computation, analogous to the head of a Turing machine, which moves around manipulating the computational qubits, changing states as it does so in order to perform different kinds of actions. The transition rules of a Turing machine are independent of the location. In the construction below, we have a few different kinds of locations, for instance for the different sorts of quantum gates used in C x . Each kind of location has a different set of transition rules that apply (although many of the rules are the same for all types of location), and the position of a site determines which set of rules applies to that site. For the adiabatic construction, we could instead proceed with translational-invariant rules as with a Turing machine by initializing each site with an additional marker indicating what type of location it is supposed to be. This would require a substantial increase in the number of states per site (perhaps to 100 or so), and we do not know how to use translational invariant nearest-neighbor rules for the QMA result. Instead we use position-dependent rules. This enables us to use essentially the same construction for both the adiabatic result and the QMA-completeness result, and reduces the required number of states in the adiabatic case. In the beginning we put the quantum circuit we wish to simulate into a canonical form. Let n be the number of qubits in the circuit, labeled 1 . . . n from left to right in a line. We assume that the circuit is initialized to the all |0 state and consists of R rounds. Each round is composed of n − 1 nearest-neighbor gates (some or all of which may be the identity gate); the first gate in a round acts on qubits 1 and 2, the second on qubits


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2 and 3, and so on. Any quantum circuit can be put into this form with the number of rounds at most proportional to the number of gates. We will assume that the first round and the last round contain only identity gates, as we will need those two rounds for additional checking in the QMA-completeness result. In the 1D arrangement, there will be a total of n R 12-state particles, arranged on a line in R blocks of n qubits, each block corresponding to one round in the circuit. Roughly speaking, we imagine that in the beginning the qubits are all in the first block. The first round of quantum gates is performed one by one on these qubits, after which they are moved to the next block, where the second round of gates is performed, and so on. Once the qubits reach the last block and undergo the last round of gates, their state will be the final state of the circuit. The main difficulty we will have to overcome is our inability to count. Since there are only a constant number of states per site, there is no way to directly keep track of how far we have moved. Our solution is to move the full set of qubits over only one space at a time. We keep moving over until the qubits reach the next block. We can tell that we have reached a new block, and are ready to perform a new set of gates, by making the transition rules different at the boundary between two blocks. When the qubits are correctly aligned with a block, a qubit state and a non-qubit state will be adjacent across a block boundary, whereas while the qubits are moving, adjacent pairs of particles which cross a block boundary will either have two qubit states or two non-qubit states. We denote a block boundary by · ·. The 12 states in each site consist of 2-state subsystems (different versions of a qubit holding data), represented by elongated shapes (e.g., ), and 1-state subspaces, represented by round shapes (e.g., ). Two of the 2-state systems and two of the 1-state site types will be “flags” or “active” sites, which will be represented by dark shapes and can be thought of as pointers on the line that carry out the computation. Light-colored shapes represent a site that is inactive, waiting for the active site to come nearby. There will only be one active site in any (valid) configuration. We have the following types of states: Inactive states : Qubits to right of active site : Qubits to left of active site : Unborn (to right of all qubits) : Dead (to left of all qubits)

Flags (active states) : Gate marker (moves right) : Right-moving flag : Left-moving flag, moves qubits right one space : Turning flag

The and active sites are qubit states. In the analogy to a Turing machine, they can be thought of as the head sitting on top of a qubit on the tape. The and flags are one-dimensional subspaces, which sit between or next to the particles which are in qubit states. Definition 4. We use the term configuration to refer to an arrangement of the above types of states without regard to the value of the data stored in the qubit subsystems. Valid (or legal) configurations of the chain have the following structure: · · · (qubits) · · · , where the · · · string or · · · string might not appear if the qubits are at one end of the computer. The (qubits) string consists of either n sites (for the first two possibilities) or n + 1 sites (for the last three choices) and is of one of the following forms:

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··· ··· ···

··· ···

··· ··· ···

In the first three cases, either the string or the string might be absent when the active site is at one end of the qubits. (The flag is only needed at the left or right end of a string of qubits.) When the flag is , the n qubit sites are lined up inside a single block; for all other values of the flag, the (qubits) string crosses a block boundary. ··· · · · · · · · · · where the The initial configuration is and the qubits are in the state corresponding to the input of the original circuit C x . Following the transition rules below, the sweeps to the right, performing gates as it goes. When it reaches the end of the qubit states at the border of the next block, it becomes a , which in turn creates a flag which sweeps left, moving each qubit one space to the right in the process. When the flag reaches the left end, it stops by turning into a which then creates a flag, which moves right through the qubits without disturbing them. The flag hits the right end of the set of qubits and becomes a , which begins a flag moving left again. The - - - cycle continues until the qubits have all been moved past the next block boundary. Then we get a arrangement, which spawns a new , beginning the gate cycle again. The evolution stops when the qubits reach the last block boundary and the gate flag reaches the end of the line, i.e., the final configuration ··· is · · · · · · · · · . We have the following transition rules. Something of the form XY is never across a block boundary, whereas X Y is. Y and X represent the left and right end of the chain, respectively. → (performing the appropriate gate between the two encoded 1. (Gate rule) qubits) 2. (Turning rules right side) → , → , → , → 3. (Sweeping left rules) → , → 4. (Turning rules left side) → , → , → , → → , → , → 5. (Sweeping right rules) 6. (Starting new round rule) → We can see by inspection that these rules create the cycle described above; Fig. 1 follows an example with n = 3 step-by-step through a full cycle. A single cycle takes n(2n + 3) moves, so the full computation from the initial configuration to the final one takes a total of K = n(2n + 3)(R − 1) + n − 1 steps. (The additional n − 1 steps are for the gate flag to move through the final block.) The only tricky part is to note the following fact: Fact 1. For any configuration containing only one active site, there is at most one transition rule that applies to move forward one time step and at most one transition rule to move backwards in time by one step. For valid configurations, there is always exactly one forwards transition and exactly one backwards transition, except for the final and initial states, which have no future and no past, respectively. Proof. We can see this by noting that the transition rules only refer to the type of active site present and to the state of the site to the right (for , , and ) or left (for ) of the active site (or left and right respectively for backwards transitions). For a configuration


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Fig. 1. A full cycle, with n = 3. Time goes down the first and then the second column

with only one active site, there is therefore at most one transition rule that applies. We can also see that for every legal configuration, there is indeed a transition rule that applies; the few cases that may appear to be missing cannot arise because of the constraints on alignment of the string of qubits with the blocks.

The illegal cases without forward transitions are cases without backwards transitions are , , play an important role in Sect. 4.

,

, , and

, and , and the illegal . These arrangements will

Remark. For adiabatic computation, we can simplify this construction a bit. For the proof of QMA-completeness, we will need to distinguish the particles to the left of the active site from the particles to its right, but this is not necessary for universal adiabatic computation. Therefore, we may combine the and states into a single qubit subspace , and we may combine and into a single state . That leaves us with essentially the same construction, but with only 9 states per particle instead of 12. 3. Universality of Adiabatic Evolution in One Dimension We first recall the definition of adiabatic computation [FGGS00,FGG+ 01]. A quantum system slowly evolves from an easy to prepare ground state of some initial Hamiltonian H0 to the ground state of a tailored final Hamiltonian H1 , which encodes the problem. The state evolves according to Schrödinger’s equation with a time-varying Hamiltonian H (s) = (1 − s)H0 + s H1 , with s = t/T , where t is the time and T is the total duration of the adiabatic computation. Measuring the final ground state in the standard basis gives the solution of the problem. The adiabatic theorem tells us that if T is chosen to be large enough, the final state will be close to the ground state of H1 . More specifically, we can define instantaneous Hamiltonians H (t/T ), and let gmin be the minimum of the spectral gaps over all the Hamiltonians H (t/T ). T is required to be polynomial in 1/gmin . To prove Theorem 1, given a general quantum circuit, we would like to design an efficient simulation of this circuit by adiabatic evolution on a one-dimensional line. The construction in Sect. 2 associates to any circuit C x with n qubits a modified circuit C˜ x using n R9-state particles in one dimension (with some states merged according to the remark in the end of Sect. 2), together with transition rules which evolve the system for

54


K = n(2n + 3)R + n − 1 steps through K + 1 configurations from initial to final. We denote the quantum state of the t th configuration by |γ (t) for t = 0, . . . , K . Observe that the |γ (t) are orthogonal to each other, since each has a different configuration. We note that the last state |γ (K ) contains the final state of the original circuit, encoded in the last block of particles. Next we construct a one-dimensional adiabatic evolution whose final Hamiltonian has as its ground state the history state of this evolution, namely, √

1 K +1

K

|γ (t).

t=0

To define H1 ≡ Hprop , we simply translate the transition rules of the previous section to 9-state 2-local Hamiltonians. In other words, any transition rule which takes a state |α to |β is translated to a Hamiltonian of the form 21 (|α α| + |β β| − |α β| − |β α|). E.g., the first rule becomes 21 (| | + U | |U † − U | | − | |U † ), summed over a basis of states for the encoded qubits and over all nearest neighbor pairs of particles (recall we have combined and ). Here U is the gate corresponding to this location and acts on the two qubits encoded as | . Restricted to the eight-dimensional subspace spanned by | and | (each for values of |00, |01, |10, and |11 for the two qubits) for a single pair of particles, that would give the matrix 1 I −U , 2 −U † I with I the 4 × 4 identity matrix. Our initial Hamiltonian has the initial configuration |γ (0) as its ground state. To define H0 , we penalize all configurations that do not have in its |0 state (| (0)) in the first position: H0 = I − | (0) (0)|1 This Hamiltonian of course has a highly degenerate ground state, but the important point is that |γ (0) is the only state which satisfies H0 in the invariant subspace spanned by the |γ (t). We assume we are able to select the correct ground state |γ (0) to be the actual initial state of the adiabatic computation. We could avoid the need for this by various methods, for instance by having an additional phase of adiabatic evolution from a simple non-degenerate Hamiltonian to H0 . Another option is to break the degeneracy of the ground state of H1 by using the full 12-state construction with the Hamiltonian described in Sect. 4. Even then, we either need to modify H0 to be non-degenerate (this approach requires a new analysis of the spectral gap, which we have not done), or to add an additional 13th start state. We outline the latter approach in Sect. 4.1. To analyze the spectral gap of any Hamiltonian in the convex combination of H0 and H1 , we essentially follow the proof of [AvDK+ 04], using an improvement due to [DRS07]. We first observe that Claim. The subspace K0 spanned by the states |γ (t) is invariant under any of the Hamiltonians H (s). This is easily seen to be true by the fact that it is invariant under the initial and final Hamiltonians. From this claim, it follows that throughout the evolution we are inside


55

K0 , so we only need to analyze the spectral gap in this subspace. We next move to the basis of K0 spanned by the |γ (t) states. In this basis H0 restricted to K0 looks like ⎛ ⎞ 0 0 ... 0 ⎜0 1 ... 0⎟ ⎟ H0 |K0 = ⎜ ⎝ ... ... . . . ... ⎠ , 0 0 ... 1

and H1 restricted to K0 becomes ⎛ 1 2 ⎜ 1 ⎜− ⎜ 2

⎜ ⎜ ⎜ H1 |K0 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

− 21

1 − 21

0 − 21 .. . .. . 0

0

··· . 0 ..

1 − 21 .. .. . .

0 .. .

0 − 21 1 0 − 21 ··· 0

⎞ 0 .. ⎟ .⎟ ⎟ .. ⎟ .. . .⎟ ⎟ ⎟. .. ⎟ . ⎟ ⎟ 1 −2 0⎟ ⎟ 1 − 21 ⎠ 1 − 21 2

It can easily be seen that the history state (which in this basis is simply the all ones vector) is a zero eigenstate of this Hamiltonian. Finally we need to analyze the spectral gaps of all convex combinations of these two. This has been done in Sect. 3.1.2 of [AvDK+ 04] and simplified with improved constants in [DRS07]; we refer the reader there for the details. The result is that the spectral gap is at least 1/[2(K + 1)2 ], an inverse polynomial in n and R, which itself is an inverse polynomial in the number of gates in the original circuit. This proves Theorem 1. 4. 1D QMA The propagation Hamiltonian Hprop introduced in Sect. 3 is insufficient for QMA-completeness. Now we have a circuit C x which checks the witness for a QMA problem, taking as input the witness and some ancilla qubits in the state |0. However, any correct history state for the circuit C x will have zero eigenvalue for Hprop , even if the t = 0 component of the history state is not correctly initialized, or if C x does not accept the witness. Even worse, Hprop also has zero eigenvalue for any other state which is a uniform superposition of states connected by the transition rules, even if the superposition includes illegal configurations. To solve these problems, we will introduce three new terms to the overall Hamiltonian H: H = Hprop + Hinit + Hfinal + Hpenalty . The initialization term Hinit will constrain the initial state of the modified checking circuit C˜ x so that all ancilla qubits are initialized to |0, and the final Hamiltonian Hfinal verifies that the checking circuit does accept the witness as input. Hpenalty will penalize illegal configurations using local constraints. As mentioned in the Introduction, not all illegal configurations can be penalized directly. Instead, some of them will only be penalized because they evolve into a locally checkable illegal configuration.

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To create the initialization term, we will assume without loss of generality that all the gates performed in the first block are the identity. We will use the gate flag to check that qubits are correctly initialized instead of using it to do gates in the first block. Then we get the following Hamiltonian term: | (1) (1)|i . Hinit = i

The sum is taken over ancilla qubits i in their starting positions in the first block of n sites. (The remaining qubits in the first block of n sites are used to encode the potential witness.) Hinit creates an energy penalty for any ancilla to be in the state |1 when the gate flag passes over it. Since the gate flag sweeps through the whole block, this ensures that the ancilla qubits must be correctly initialized to |0 or the state suffers an energy penalty. Similarly, for the final Hamiltonian, we assume no gates are performed in the final block, and use to check that the circuit accepts the output. That is, we get the following term in the Hamiltonian: Hfinal = | (0) (0)|out . This causes an energy penalty if the output qubit “out” is in the state |0 when the gate flag sweeps over it. In general, a correct history state for some potential input witness state will have, for the final block, a superposition of terms with the output qubit in the state |0 and terms with the output qubit in the state |1, and the energy penalty is thus proportional to the probability that the circuit rejects the potential witness. Now we move to Hpenalty , which is a bit more involved. We will describe a set of local penalties that will enforce that each illegal configuration will be penalized either directly or because it will evolve into a configuration that will be penalized. We forbid the following arrangements: 1. X, X (X is anything but ), X , X (X is anything but ). 2. In the first block on the left: , . In the last block on the right: , . 3. X, X (X is anything but or ), X , X (X is anything but or ). 4. , , , , , . 5. , . 6. Any adjacent pair of active sites (e.g., or ), with or without block boundaries. 7. , , , , , , , (but the first two are OK with a block boundary and the next four are OK without a block boundary). Note that the arrangements forbidden in group 7 are precisely those missing a forward or backwards transition rule in the note following Fact 1. We encode these rules into a penalty Hamiltonian in the straightforward way: Hpenalty = |X Y X Y |i,i+1 , XY

where the sum is taken over the forbidden arrangements XY listed above for all adjacent pairs of sites (tailored appropriately to the location of block boundaries and the first and last blocks). Claim. A configuration that satisfies the rules in groups 1 through 6 is of one of the legal forms described in Sect. 2, or it is in one of three cases: (i) a (qubits) string of the


57

· · · (such a configuration is allowed in Sect. 2 only when is all the form · · · way on one end of the (qubits) string), (ii) a (qubits) string that has incorrect length: different from n when the active site is or or different from n + 1 when the active site is or , or (iii) a (qubits) string of length n, but with either an active site and the (qubits) string aligned with a block boundary, or an active site and the (qubits) string not aligned with a block boundary. Note that we do not need the rules in group 7 for this claim; they are used later to deal with the illegal states that cannot be checked locally. Proof. The rules in group 1 enforce that the configuration is of the form · · · (qubits) · · · by ensuring that all unborn ( ) states are at the far right and all dead ( ) states are at the far left. Group 2 ensures that the particles are not all or all (and in fact ensures that there is at least one block of n composed of something else). Group 3 ensures that within the (qubits) string, any qubits are on the right end and any qubits are on the left end. Groups 2 and 4 guarantee that the (qubits) string is not empty and does not consist of only or qubits. Group 5 then checks that the (qubits) string does not directly jump from to , so there is at least one active site in between, and the rules in group 6 ensure that there cannot be more than one adjacent active site. Since, by groups 1 and 3, an active site can only occur to the right of all and sites and to the left of all and sites, all active sites are together in a group, so the rules in groups 5 and 6 ensure that there is exactly one active site. Comparing with Definition 4, we see that this leaves only the exceptions listed in the claim.

The three exceptions in the above claim cannot be ruled out directly via a local check, since counting cannot be done using local constraints. 4 We can only rule out the exceptions by considering both the penalty Hamiltonian Hpenalty and the propagation Hamiltonian Hprop . In order to do so, we first break down the full Hilbert space into subspaces which are invariant under both Hpenalty and Hprop . Let us consider the minimal sets of configurations such that the sets are invariant under the action of the transition rules. This defines a partition of the set of configurations. Given such a minimal set S, we consider the subspace K S spanned by all the configurations in S. The penalty Hamiltonian is diagonal in the basis of configurations, so K S is invariant under Hpenalty , and the set S is closed under the action of Hprop , so K S is invariant under Hprop as well. The space K S belongs to one of three types: 1. All states in S are legal. (We call this subspace K0 .) 2. All states in S are illegal, but are all locally detectable, namely, none of them belong to the exceptions of Claim 4. 3. All states in S are illegal, but at least one of them is not locally detectable. Note that the set S cannot contain both legal and illegal configurations, since legal configurations do not evolve to illegal ones, and vice-versa. We want to show that subspaces of type 2 or 3 have large energy. The main challenge will be to give a lower bound on the energy of a space K S of type 3, which we will do by lower bounding the fraction of configurations in S that violate one of the conditions 1-7 above. We want to prove: 4 If we are willing to add a 13th state, we could split the state into two states, one to turn around on the · · · with local rules. However, left and one to turn around on the right. Then we could rule out · · · this strategy will not work for strings with the wrong number of qubit states.

58


Lemma 1 (Clairvoyance lemma). The minimum eigenvalue of Hprop +Hpenalty , restricted to any K S of type 2 or 3, is (1/K 3 ), where K is the number of steps in C˜ x . Proof. To prove the theorem, we deal separately with each type of invariant subspace. Type 2 is straightforward: the full subspace K S has energy of at least 1, due to Hpenalty . We now focus on type 3. We observe several properties of type 3 subspaces. First, notice that the number of qubit states ( , , , and ) and the number of active sites ( , , , and ) are conserved under all transition rules. Since S is a minimal set preserved by the transition rules, all the configurations in S therefore contain the same number of qubit states. Since S is of type 3, it must contain an undetectable illegal configuration, such as a (qubits) string of the wrong length, and by Claim 4 has exactly one active site somewhere in the string. By Fact 1, there is at most one forwards transition rule and one backwards transition rule that applies to each configuration in S. Note also that configurations with one active site which is or always have both a forward and backwards transition rule available. Therefore, if S is of type 3 it must contain at least one configuration with a or active site. Going further, we claim that a fraction of at least (1/n 2 R) of the configurations in S violate one of the rules in groups 1 through 7 above. To see this, we divide into three cases: · · · : In this – S contains a configuration with a qubit string of the form · · · case, the forward transition rule that applies to the illegal configuration is either → or → . This gives us a configuration which can be locally seen as · · · . This violates group 1, as sites should be to the illegal, such as · · · left of all other kinds of sites. Therefore S contains at least 1/2 locally checkable illegal configurations. In fact, it is a much larger fraction, as all the backwards transition rules and further forward transition rules will also produce locally checkable illegal states. – S contains a configuration with a active site: Via forward transitions, the will move to the right until it hits either the right end of the (qubits) string or until it hits a block boundary. If it does not encounter both together, it has reached one of the illegal configurations forbidden in group 7. Via backwards transitions, the moves to the left until it hits either the left end of the (qubits) string or until it hits a block boundary. Again, if it does not encounter both together, we must now be in one of the configurations forbidden in group 7. If the right and left ends of the (qubits) string are both lined up with block boundaries, and there are no other block boundaries in between, then the (qubits) string contains exactly n sites and is correctly aligned with the block boundary. Therefore it is actually a legal state, which cannot occur in S of type 3. Thus, we must have one of the locally checkable illegal states, and it must occur after at most n transitions (in either direction). The configurations themselves, however, could be part of a larger cycle with transitions involving , , and active sites. However, the length of the (qubits) string is at most n R, and we can have at most n - - - cycles before getting a , so the locally checkable illegal states comprise a fraction at least (1/n 2 R) of all the states in S. – S contains no configuration with a active site: S still must contain configurations with . These configurations can form part of a normal cycle, but the normal cycle eventually leads to a transition to a configuration. For forward transitions, this occurs when the left end of the (qubits) string is aligned with a block boundary, and for backwards transitions, it occurs when the right end of the (qubits) string is aligned with a block boundary. The only way to avoid this happening is for S to


59

have a configuration with no forward transition and another with no backwards transition. Since the only such configurations contain one of the arrangements , , , or , all of which are forbidden by the rules in group 7, S must contain a locally checkable illegal configuration. Again, the length of the (qubits) string is at most n R, and we can have at most n - - - cycles in a row, so the locally checkable illegal states comprise a fraction at least (1/n 2 R) of all the states in S. We can now invoke Lemma 14.4 from [KSV02] to lower bound the energy of the overall Hamiltonian for a type 3 subspace: Lemma 2. Let A1 , A2 be nonnegative operators, and L 1 , L 2 their null subspaces, where L 1 ∩ L 2 = {0}. Suppose further that no nonzero eigenvalue of A1 or A2 is smaller than v. Then A1 + A2 ≥ v · 2 sin2 θ/2, where θ = θ (L 1 , L 2 ) is the angle between L 1 and L 2 . In our case, A1 is the propagation Hamiltonian Hprop , and its null subspace, restricted to K S , consists of equal superpositions over all configurations in the invariant subspace S. (There are multiple such states, with different values of the encoded qubit states.) A2 is the penalty Hamiltonian Hpenalty , diagonal in the basis of configurations. Then sin2 θ is the projection (squared) of the superposition of all shapes on the subspace of locally checkable illegal configurations; that is, it is the fraction of locally checkable illegal configurations in the invariant set. The minimum nonzero eigenvalue of Hpenalty is 1, but (as in [KSV02]) the minimum nonzero eigenvalue of Hprop is (1/K 2 ), where K = n(2n + 3)(R − 1) + n − 1 is the number of steps in C˜ x . Thus, if S is a set containing a configuration which is illegal but cannot be locally checked, all states in K S have an energy at least (1/K 3 ). We can now prove Theorem 2. We start by assuming the circuit C x accepts correct witnesses and rejects incorrect witnesses with a probability exponentially close to 1. This can be achieved, for instance, by checking multiple copies of the witness. Then we will show that if there exists a witness which is accepted by C x with probability at least 1 − O(1/K 3 ), then there is a state with energy at most O(1/K 4 ), whereas if all possible witnesses are only accepted by C x with probability at most 1/K 3 , then all states have energy at least (1/K 3 ). We already know that on subspaces of type 2 and 3, the minimum eigenvalue of Hprop + Hpenalty is (1/K 3 ). We therefore restrict attention to the subspace K0 of type 1, built from only legal configurations, and can hence ignore Hpenalty . From this point on, the proof follows [KSV02], but we include it for completeness. The type 1 space K0 contains only valid history states. However, some have incorrect ancilla inputs and are penalized by Hinit , while others include an incorrect witness input, and are penalized by Hfinal . Only history states corresponding to a correct witness have low energy. If there exists a witness which is accepted by C x with probability at least 1 − 1/K 3 , 1 then the history state | = √ t |φt for that witness will have small enough K +1 energy:

|Hprop + Hpenalty + Hinit | = 0,

60


so we only need to calculate |Hfinal |. The only term in | that contributes √ is the one with on the output qubit in the last block, which has a coefficient 1/ K + 1. Therefore, the energy due to Hfinal is 1/(K + 1) times the probability that the witness is rejected by C x . That is, it is O(1/K 4 ). Now let us consider what happens for a “no” instance. The following lemma will complete the proof of Theorem 2: Lemma 3. If there is no witness that causes the circuit to accept with probability larger than 1/K 3 , then every state in K0 has energy at least (1/K 3 ). Proof. We will invoke Lemma 2 once more, with A1 = Hprop , and A2 = Hinit + Hfinal . From now on, all the operators we discuss will be restricted to K0 . The null subspace of the propagation Hamiltonian Hprop contains states which correspond to a history ˜ |ψ = √ 1 t |φt of the circuit C x for some input state |φ0 , not necessarily correct. K +1 Fix such a history state |ψ. We will upper bound its projection squared on the null space of A2 . Let us first study the structure of this null space. The null space of Hinit is spanned by states for which either the is not on an ancilla qubit in the first block or the is on an ancilla qubit, but the ancilla qubit is a |0. Let init be the projection on the null space of Hinit . The null space of Hfinal contains states for which either the is not on the output qubit in the last block or the is on the output qubit and the output qubit is a |1. Let final be the projection on the null space of Hfinal . The two projectors commute, so the projector onto the null subspace of Hinit + Hfinal is init final . We would thus like to upper bound

ψ| init final |ψ. To this end, we expand the input state |φ0 for |ψ into |φ0 = α|φ0v + β|φ0i , where |φ0v is a state with all valid ancilla input qubits and |φ0i is an orthogonal state consisting of a superposition of states with one or more invalid ancilla input qubits. Note that |ψ = α|ψ v + β|ψ i , where |ψ v is the history state for input |φ0v and |ψ i is the history state for input |φ0i . We have:

ψ| init final |ψ = |α|2 ψ v | init final |ψ v + |β|2 ψ i | init final |ψ i +2Re α ∗ β ψ v | init final |ψ i , where we have used the fact that the two projectors commute, so init final is Hermitian. We now bound each of the three terms. For the first term, note that ψ v | init = ψ v |. We might as well assume the output qubit is the very last qubit, so the reaches it only at t = K , and we have that

ψ v | final |ψ v =

1 K+p v v K + φ K , | final |φ K = K +1 K +1

where p is the probability that the circuit C x accepts |φ0v . We thus find that the first term is at most |α|2 (K + O(1/K 3 ))/(K + 1). For the second term, it suffices to upper bound ψ i | init |ψ i since

ψ i | init |ψ i = init |ψ i 2 ≥ final init |ψ i 2 = ψ i | init final |ψ i ,


61

again using the fact that the two projectors commute. Since the computation C˜ x begins by sweeping a through the first n sites, and thereafter never has within the first n sites, only the first n states |φt can possibly be outside the null space of Hinit . Also, note that φt | init |φt = 0 unless t = t . We can thus write n−1 1 i i i i

ψ | init |ψ =

φt | init |φt . (K + 1 − n) + K +1 t=0

We have φt | init |φt = 0 if the t th qubit of the input is an ancilla qubit and is in the state |1, and φt | init |φt = 1 otherwise. Since the |φti states are superpositions of terms with at least one incorrect ancilla input qubit, we have

ψ i | init |ψ i ≤ K /(K + 1). For the third term, we have |Re α ∗ β ψ v | init final |ψ i | ≤ | ψ v | init final |ψ i | = | ψ v | final |ψ i |. Note, however, that φtv |φti = 0, since |φt is related to |φ0 by a unitary transformation and φ0v |φ0i = 0. This gives us 1 | φ v | final |φ iK | K +1 K 1 v ≤ final |φ K K +1 = O(K −5/2 ).

| ψ v | final |ψ i | =

The above expression might not be 0 since both |ψ v and |ψ i might have a non-zero component which is accepted by the circuit, but since |φ v is only accepted rarely (with probability O(1/K 3 )), it cannot be too large. Summing up all contributions, we have:

ψ| init final |ψ ≤ |α|2 =

K + O(1/K 3 ) K + |β 2 | + O(K −5/2 ) K +1 K +1

K + O(K −5/2 ). K +1

From this we can conclude that sin2 (θ ) is at least (1/K ), where θ is the angle between the null spaces of Hprop and Hinit + Hfinal . Lemma 2 then tells us that every state has energy (1/K 3 ). 4.1. Non-degenerate universal adiabatic hamiltonian. If we wish to create a universal adiabatic quantum computer with a non-degenerate ground state, the Hamiltonians from Sect. 3 do not suffice. In fact, even if we used a 12-state construction with the final Hamiltonian H1 equal to H from the QMA-completeness construction above, then H1 would be non-degenerate, but the initial Hamiltonian H0 in Sect. 3 still is not. We sketch

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here a construction that adds an additional 13th state to make sure that the Hamiltonian remains non-degenerate throughout the adiabatic evolution. The new state is a start state , which serves as the state for the input qubits in the initial configuration. We will add a set of terms to H0 and H1 (which also includes Hinit , Hfinal , and Hpenalty , in addition to Hprop ). These terms will guarantee that the unique ··· · · · · · · · · · . Furthermore, H0 will have the propground state of H0 is erty that H0 |γ (t) = 1 for any t > 0, which allows us to use the calculation of the gap given in Sect. 3. To do this, we add a term that enforces the condition that if the first particle is in state , then the second particle is in state . This is accomplished by forbidding any configuration that has in the first location and any state except in the second. For example, to forbid states in particles 1 and 2, we add the term | |12 . Similarly, for i = 2, . . . , n − 1, we enforce that if particle i is in state then particle i + 1 must be in state as well. Then we add a term that says if particle n is in state then particle n + 1 is in state . For i > n, add terms that enforce that if particle i is in state , then so is particle i + 1. Similar terms should also be added to H1 to assure that its ground state remains non-degenerate when the new start state is added. Finally, we need to slightly alter the transition rules used to build Hprop . For i = 1, . . . , n − 1, replace the rule → in locations i and i + 1 with → (0). With this change, Fact 1 still holds, with the small caveat that valid configurations now must have any qubits in the first block set to |0. Note that none of the configurations in the subspace spanned by the |γ (t) violate any of the extra conditions. Furthermore, the additional terms guarantee the uniqueness of the ground state of H0 given above. In order to establish the non-degeneracy of any of the Hamiltonians H (s), we need to prove that there are no low energy states outside of the subspace spanned by the |γ (t), which follows from the Clairvoyance lemma and a simplified version of Lemma 3. 5. Discussion and Open Problems We have shown that 1-dim 12-state Hamiltonians can be used for both universal adiabatic quantum computation and to produce very difficult, QMA-complete problems. A similar result holds also for a related quantum complexity class, called QCMA, which is the subclass of QMA where the witnesses are restricted to be classical. To see this, observe first that our reduction from an arbitrary QMA language to the 1-DIM 12-STATE HAMILTONIAN problem is witness-preserving, at least once the acceptance probability has been amplified. Given a witness for the original QMA language, we can, in fact, efficiently construct on a quantum computer a witness for the corresponding instance of 1-DIM 12-STATE HAMILTONIAN using the adiabatic algorithm. This implies that if the witness for the original problem is efficiently constructible (which means we may as well assume it is a classical bit string describing the circuit used to construct the quantum witness), then the witness for 1-DIM 12-STATE HAMILTONIAN is also efficiently constructible. Thus, we have also shown that the sub-language of 1-DIM 12-STATE HAMILTONIAN which has the additional promise of an efficiently-constructible ground state is complete for QCMA. There remain many interesting related open problems. Clearly, it is interesting to ask whether the size of the individual particles in the line can be further decreased, perhaps as far as qubits. We could, of course, interpret our 12-state particles as sets of 4 qubits, in which case our Hamiltonian becomes 8-local, with the sets of 8 interacting qubits arranged consecutively on a line. The perturbation-theory gadgets used to convert 8-local interactions to 2-local ones do not work in a one-dimensional system: The pairs


63

of interacting qubits form a graph, which needs to have degree at least 3 (or 4 for some of the gadgets used by [OT05]). If we do apply the approach of [OT05], however, to the system described here, we get a 2-dim Hamiltonian on a strip of qubits of constant width. The constant is rather large, but this still constitutes an improvement over the result of [OT05]. Another approach to decreasing the dimensionality of the individual particles is to find a new protocol for removing the explicit reference to time from the circuit C x . It seems likely that some further improvement is possible in this regard, but it is unlikely that an improved protocol by itself can take us all the way to qubits, as we need additional states to provide the control instructions. If it is possible to have universal adiabatic quantum computation and QMA-completeness with 1-dim 2-state Hamiltonians, we will probably need new techniques to prove it. It may also be that there is a transition at some intermediate number of states between 2 and 12 for which universality and QMAcompleteness become possible. This would be analogous to the classical 2-dimensional Ising spin problem without magnetic field, which is in P for a single plane of bits, but is NP-complete when we have two layers of bits [Bar82]. Another interesting line of open questions is to investigate the energy gap. There are two energy gaps of relevance, both interesting. One is the “promise gap” ∆ in the definition of QMA. We have shown that we have QMA-completeness when ∆ is polynomially small relative to the energy per term. This can easily be improved to a constant value of ∆ by amplification: t copies of the ground state will either have energy less than t E or above t (E + ∆), amplifying the promise gap to t∆. A more interesting question is whether ∆ can be made a constant fraction of the total energy available in the problem, the largest eigenvalue of H ; if so, that would constitute a quantum version of the PCP theorem. Hastings and Terhal [HT] have argued that it is not possible to do this in any constant number of dimensions unless QMA = P: To approximate the ground state energy, we can divide the system into blocks of a constant size, and diagonalize the Hamiltonian within each block. We can then consider the tensor product of the ground states for each separate block; since such a state ignores the energy due to Hamiltonian terms interacting different blocks, it will not be the true ground state, but its energy can differ from the ground state energy by at most the surface area of each block times the number of blocks (normalizing the maximum energy per term to be 1), whereas the maximum eigenvalue of H is roughly proportional to the volume of each block times the number of blocks. Thus, the tensor product state approximates the ground state energy up to a constant fraction of the total energy, and the fraction can be made arbitrarily small by increasing the size of each block. This gives a polynomial-time classical algorithm for approximating the ground state energy to this accuracy. When the interactions in H are not constrained by dimensionality, the argument breaks down, so this quantum version of PCP remains an interesting open problem for general r -state k-local Hamiltonians. We can also look at the spectral gap, the gap between the ground state and the first excited state of H . For adiabatic computation, we are interested in the minimal spectral gap over the course of the computation. We have shown that universal adiabatic quantum computation is possible if the spectral gap is polynomially small relative to the energy per term. What happens if it is bounded below by a constant? Hastings has recently shown [Has] that it is possible to efficiently classically simulate the adiabatic evolution of a 1-dim r -state Hamiltonian system with constant spectral gap. We should therefore not expect to be able to perform universal adiabatic quantum computation with such Hamiltonians, as that would imply BQP = P. Hastings’ argument builds on his earlier paper [Has07], which showed that the ground state of a gapped 1-dim r -state Hamiltonian

64


has an efficient classical description as a matrix product state. By repeatedly updating the matrix product state description, one can then keep track of the adiabatic evolution with resources polynomial in the path length in parameter space and the inverse error of the approximation. For d-dim r -state Hamiltonians with constant spectral gap and d ≥ 2, the question remains open in general, although Osborne has proven [Osb06] that such systems can be efficiently classically simulated for logarithmic time. For the QMA-completeness problem, we know very little about the possible values of the spectral gap. If the spectral gap is constant relative to the energy per term (or even (1/ ln ln n R)), the ground state has a matrix product state representation [Has07] and the problem is in NP. In our construction, there are enough low-energy states that the spectral gap cannot be much larger than the promise gap ∆, but it might be much smaller. In the “no” instances, there are likely very many states which violate only a small number of transition rules, penalty terms, or initial conditions, and these have energy just above E + ∆, so likely the spectral gap is exponentially small in the “no” instances. We know less about the size of spectral gap in the “yes” instances. States which do not correspond to valid histories have an energy at least ∆ larger than the ground state, but unfortunately, there may be different valid histories with energies less than E + ∆ but above the ground state energy. The difficulty is that the original problem might have many potential witnesses. Some may be good witnesses, accepted with high probability, whereas others may be mediocre witnesses, accepted with a probability near 1/2. There could, in fact, be a full spectrum of witnesses with only exponentially small gaps between their acceptance probabilities. In order to show that the “yes” instances can be taken to have a spectral gap which is at least inverse polynomial in the system size, we would need a quantum version of the Valiant-Vazirani theorem [VV86], which would say that we can always modify a QMA problem to have a unique witness accepted with high probability. Acknowledgement. This work was partly done while three of the authors (D. A., D. G., and J. K.) were visiting the Institute Henri Poincaré in Paris, and we want to thank the IHP for its hospitality. We wish to thank Daniel Fisher, Matt Hastings, Lev Ioffe, Tobias Osborne, Oded Regev, and Barbara Terhal for helpful discussions and comments and Oded for help with the design of the state icons.

References [ACdf88] [ACdf90] [AGIK07]

[AGK07] [AvDK+ 04] [Bar82] [BY86] [CFP02]

Apolloni, B., Carvalho, C., de Falco, D.: Quantum stochastic optimization. Stochastic Processes and Their Applications 33(5), 233–244 (1988) Apolloni, B., Cesa-Bianchi, N., de Falco, D.: A numerical implementation of “quantum annealing”. In: Stochastic Processes, Physics and Geometry: Proceedings of the Ascona-Locarno Conference. River Edge, NJ: World Scientific. 1990, pp. 97–111 Aharonov, D., Gottesman, D., Irani, S., Kempe, J.: The power of quantum systems on a line. In: FOCS. Proc. 48th Ann. IEEE, Symp on Foundations of Computer Science, Los Alamitos, CA: IEEE Comp. Soc., 2007, pp. 373–383 Aharonov, D., Gottesman, D., Kempe, J.: The power of quantum systems on a line. http://arXiv. org/abs/0705.4077v2 [quant-ph], 2007 Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. In: Proc. 45th FOCS, Los Alamitos, CA: IEEE Comp. Soc., 2004, pp. 42–51 Barahona, F.: On the computational complexity of ising spin glass models. J. Phys. A: Math. Gen. 15, 3241–3253 (1982) Binder, K., Young, A.P.: Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801–976 (1986) Childs, A., Farhi, E., Preskill, J.: Robustness of adiabatic quantum computation. Phys. Rev. A 65, 012322 (2002)


[DRS07] [Fey85] [FGG+ 01] [FGGS00] [Fis95] [Has] [Has07] [HT] [Ira07] [JFS06] [JWZ07] [Kay08] [KKR06] [KSV02] [Nag08] [NW08] [Osb06] [Osb07] [OT05] [Sch05] [SFW06] [Suz76] [vEB90] [VV86] [Wat95] [Whi92] [Whi93]

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Deift, P., Ruskai, M.B., Spitzer, W.: Improved gap estimates for simulating quantum circuits by adiabatic evolution. Quant Infor. Proc. 6(2), 121–125 (2007) Feynman, R.: Quantum mechanical computers. Optics News 11, 11–21 (1985) Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292(5516), 472–476 (2001) Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. http://arXiv.org/list/quant-ph/0001106, 2000 Fisher, D.S.: Critical behavior of random transverse-field ising spin chains. Phys. Rev. B 51(10), 6411–6461 (1995) Hastings, M.: Personal communication Hastings, M.: An area law for one dimensional quantum systems. JSTAT. P08024 (2007) Hastings, M., Terhal, B.: Personal communication Irani, S.: The complexity of quantum systems on a one-dimensional chain. http://arXiv.org/abs/ 0705.4067v2[quant-ph], 2007 Jordan, S.P., Farhi, E., Shor, P.W.: Error-correcting codes for adiabatic quantum computation. Phys. Rev. A 74, 052322 (2006) Janzing, D., Wocjan, P., Zhang, S.: A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation. http://arXiv.org/ abs/0710.1615v2[quant-ph], 2007 Kay, A.: The computational power of symmetric hamiltonians. Phys. Rev. A. 78, 012346 (2008) Kempe, J., Kitaev, A., Regev, O.: The complexity of the local hamiltonian problem. SIAM J. Comp. 35(5), 1070–1097 (2006) Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Providence, RI: Amer. Math. Soc., 2002 Nagaj, D.: Local Hamiltonians in Quantum Computation. PhD thesis, MIT. http://arXiv.org/ abs/0808.2117v1[quant-ph], 2008 Nagaj, D., Wocjan, P.: Hamiltonian quantum cellular automata in 1d. Phys. Rev. A 78, 032311 (2008) Osborne, T.: Efficient approximation of the dynamics of one-dimensional quantum spin systems. Phys. Rev. Lett. 97, 157202 (2006) Osborne, T.: Ground state of a class of noncritical one-dimensional quantum spin systems can be approximated efficiently. Phys. Rev. A 75, 042306 (2007) Oliveira, R., Terhal, B.: The complexity of quantum spin systems on a two-dimensional square lattice. Quant. Inf. Comp. 8(10), 0900–0924 (2008) Schollwöck, U.: The density-matrix renormalization group. Rev. Mod. Phys. 77, 259– 316 (2005) Shepherd, D.J., Franz, T., Werner, R.F.: Universally programmable quantum cellular automaton. Phys. Rev. Lett. 97, 020502 (2006) Suzuki, M.: Relationship between d-dimensional quantal spin systems and (d+1)-dimensional ising systems. Prog. Theor. Phys. 56(5), 1454–1469 (1976) van Emde Boas, P.: Handbook of Theoretical Computer Science. volume A, Chapter 1. Cambridge, MA: MIT Press, 1990, pp. 1–66 Valiant, L.G., Vazirani, V.V.: Np is as easy as detecting unique solutions. Theor. Comput. Sci. 47(3), 85–93 (1986) Watrous, J.: On one-dimensional quantum cellular automata. In: Proc. 36th Annual IEEE Symp. on Foundations of Computer Science (FOCS), Los Alamitos, CA: IEEE Comp. Sci, 1995, pp. 528–537 White, S.R.: Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992) White, S.R.: Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B 48, 10345 (1993)

Communicated by M. B. Ruskai


Communications in


Thermal Conductivity for a Momentum Conservative Model Giada Basile1 , Cédric Bernardin2 , Stefano Olla3 1 WIAS, Mohrenstr. 39, 10117, Berlin, Germany. E-mail: [email protected] 2 Université de Lyon, CNRS (UMPA), Ecole Normale Supérieure De Lyon,

46, Allée D’Italie, 69364, Lyon Cedex 07, France. E-mail: [email protected] 3 Ceremade, UMR CNRS 7534, Université de Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France. E-mail: [email protected] Received: 29 January 2008 / Accepted: 19 July 2008 Published online: 29 October 2008 – © Springer-Verlag 2008

Abstract: We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t −d/2 in the unpinned case and like t −d/2−1 if an on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases. 1. Introduction The mathematical deduction of Fourier’s law and heat equation for the diffusion of energy from a microscopic Hamiltonian deterministic dynamics is one of the major open problems in non-equilibrium statistical mechanics [6]. Even the existence of the thermal conductivity defined by the Green-Kubo formula, is a challenging mathematical problem and it may be infinite in some low dimensional cases [13]. Let us consider the problem in a generic lattice system where dynamics conserves energy (between other quantities like momentum, etc.). For x ∈ Zd , denote by Ex (t) the energy of atom x. To simplify notations let us consider the 1-dimensional case. Since the dynamics conserves the total energy, there exist energy currents jx,x+1 (local functions of the coordinates of the system), such that d Ex (t) = jx−1,x (t) − jx,x+1 (t). dt

(1)

Another consequence of the conservation of energy is that there exists a family of stationary equilibrium measures parametrized by temperature value T (between other possible parameters). Let us denote by < · > = < · >T the expectation of the

68

G. Basile, C. Bernardin, S. Olla

system starting from this equilibrium measure, and assume that parameters are set so that < jx,y > = 0 (for example if total momentum is fixed to be null). Typically these measures are Gibbs measure with sufficiently fast decay of space correlations so that energy has static fluctuation that are Gaussian distributed if properly rescaled in space. Let us define the space-time correlations of the energy as S(x, t) = < Ex (t)E0 (0) > − < E0 >2 . If thermal conductivity is finite, S(x, t) should be solution of the diffusion equation (in a proper large space-time scale) and thermal conductivity (TC) can be defined as 1 2 x S(x, t). t→∞ 2t T 2

κ(T ) = lim

(2)

x∈Z

By using the energy conservation law (1), time and space invariance (see Sect. 3), one can rewrite ⎞⎛ t ⎞ ⎛ t 1 ⎝ κ(T ) = lim jx,x+1 (s)ds ⎠ ⎝ j0,1 (s )ds ⎠ t→∞ 2t T 2 x∈Z

=

1 T2

∞

0

jx,x+1 (t) j0,1 (0) dt,

0

(3)

x∈Z 0

which is the celebrated Green-Kubo formula for the thermal conductivity (cf. [17]). One can see from (3) why the problem is so difficult for deterministic dynamics: one needs some control of time decay of the current-current correlations, a difficult problem even for finite dimensional dynamical systems. Furthermore in some one–dimensional systems, like the Fermi-Pasta-Ulam chain of unpinned oscillators, if total momentum is conserved by the dynamics, thermal conductivity is expected to be infinite (cf. [13] for a review of numerical results on this topic). Very few mathematically rigorous results exist for deterministic systems ([8,15]). In this paper we consider stochastic perturbations of a deterministic Hamiltonian dynamics on a multidimensional lattice and we study the corresponding thermal conductivity as defined by (3). The stochastic perturbations are such that they exchange momentum between particles with a local random mechanism that conserves total energy and total momentum. Thermal conductivity of Hamiltonian systems with stochastic dynamical perturbations have been studied for harmonic chains. In [5,7] the stochastic perturbation does not conserve energy, and in [3] only energy is conserved. The novelty of our work is that our stochastic perturbations conserve also momentum, with dramatic consequences in low dimensional systems. In fact we prove that for unpinned systems (where also the Hamiltonian dynamics conserve momentum, see the next section for a precise definition) with harmonic interactions, thermal conductivity is infinite in 1 and 2 dimensions, while it is finite for d ≥ 3 or for pinned systems. Notice that for stochastic perturbations of harmonic systems that do not conserve momentum, thermal conductivity is always finite [3,7]. This divergence of TC in dimension 1 and 2 is expected generically for a deterministic Hamiltonian non-linear system when unpinned. So TC in our model behaves qualitatively like in a deterministic non-linear system, i.e. these stochastic interactions reproduce some

Thermal Conductivity for a Momentum Conservative Model

69

of the features of the non-linear deterministic hamiltonian interactions. Also notice that because of the conservation laws, the noise that we introduce is of multiplicative type, i.e. intrinsically non-linear (cf. (6) and (7)). On the other hand, purely deterministic harmonic chains (pinned or unpinned and in any dimension) have always infinite conductivity [15]. In fact in these linear systems energy fluctuations are transported ballistically by waves that do not interact with each other. Consequently, in the harmonic case, our noise is entirely responsible for the finiteness of the TC in dimension 3 and for pinned systems. Also in dimension 1 and 2, the divergence of TC for unpinned harmonic systems is due to a superdiffusion of the energy fluctuations, not to ballistic transport (see [2,12] where this behavior is explained with a kinetic argument). For anharmonic systems, even with the stochastic noise we are not able to prove the existence of thermal conductivity (finite or infinite). If the dimension d is greater than 3 and the system is pinned, we get a uniform bound on the finite size system conductivity. For low dimensional pinned systems (d = 1, 2), we can show the conductivity is finite if the interaction potential is quadratic and the pinning is generic. For the unpinned system we have to assume that the interaction between nearest-neighbor particles is strictly convex and quadratically bounded at infinity. This is because we need some information on the spatial decay of correlations in the stationary equilibrium measure, that decay slow in the unpinned system [9]. In this case, we prove the conductivity is finite in dimension d ≥ 3 and we obtain upper bounds in the size N of the system of the form √ N in d = 1 and (log N )2 in d = 2 (see Theorem 3 for precise statements). The paper is organized as follows. Section 2 is devoted to the precise description of the dynamics. In Sects. 3, we present our results. The proofs of the harmonic case are in Sect. 4 and 5 while the proofs of the anharmonic case are stated in Sect. 6. The final section contains technical lemmas related to equivalence of ensembles. Notations. The canonical basis of Rd is noted (e1 , e2 , . . . , ed ) and the coordinates of a vectoru ∈ Rd are noted (u1 , . . . , ud ). Its Euclidean norm |u| is given by |u| = (u1 )2 + . . . + (ud )2 and the scalar product of u and v is u · v. If N is a positive integer, ZdN denotes the d-dimensional discrete torus of length N and we identify x = x + k N e j for any j = 1, . . . , d and k ∈ Z. If F is a function from Zd (or ZdN ) into R then the (discrete) gradient of F in the direction e j is defined by (∇e j F)(x) = F(x + e j ) − F(x) and the Laplacian of F is

given by (F)(x) = dj=1 F(x + e j ) + F(x − e j ) − 2F(x) . 2. The Dynamics In order to avoid difficulties with definitions of the dynamics and its stationary Gibbs measures, we start with a finite system and we will define thermal conductivity through an infinite volume limit procedure (see sect. 3). We consider the dynamics of the system of length N with periodic boundary conditions. The atoms are labeled by x ∈ ZdN . Momentum of atom x is px ∈ Rd and its displacement from its equilibrium position is qx ∈ Rd . The Hamiltonian is given by HN =

x∈ZdN

⎤ 2 1 |p | ⎣ x + W (qx ) + V (qx − qy )⎦ . 2 2 ⎡

|y−x|=1

70


We assume that V and W have the following form: V (qx − qy ) =

d

j

j

V j (qx − qy ),

W (qx ) =

d

j=1

j

W j (qx ),

j=1

and that V j , W j are smooth and even. We call V the interaction potential, and W the pinning potential. The case where W = 0 will be called unpinned. We consider the stochastic dynamics generated by the operator L = A+γS .

(4)

The operator A is the usual Hamiltonian vector field px · ∂qx − ∂qx H N · ∂px , A= x

while S is the generator of the stochastic perturbation and γ > 0 is a positive parameter that regulates its strength. The operator S acts only on the momentums {px } and generates a diffusion on the surface of constant kinetic energy and constant momentum. This is defined as follows. If d ≥ 2, for every nearest neighbor atoms x and z, consider the d − 1 dimensional surface of constant kinetic energy and momentum 1 |px |2 + |pz |2 = e ; px + pz = p . Se,p = (px , pz ) ∈ R2d : 2 The following vector fields are tangent to Se,p : i, j

j

j

X x,z = ( pz − px )(∂ pzi − ∂ pxi ) − ( pzi − pxi )(∂ p j − ∂ p j ), z

so

d

i, j 2 i, j=1 (X x,z )

x

generates a diffusion on Se,p (see [11]). In d ≥ 2 we define i, j 2 1 X x,x+ek 2(d − 1) x i, j,k i, j 2 1 X x,z , = 4(d − 1) d d

S=

x,z∈Z N |x−z|=1

i, j

where e1 , . . . , ed is canonical basis of Zd . Observe that this noise conserves the total momentum x px and energy H N , i.e. S px = 0 , S H N = 0. x

In dimension 1, in order to conserve total momentum and total kinetic energy, we have to consider a random exchange of momentum between three consecutive atoms (because if d = 1, Se,p has dimension 0), and we define S=

1 (Yx )2 , 6 d x∈Z N


71

where Yx = ( px − px+1 )∂ px−1 + ( px+1 − px−1 )∂ px + ( px−1 − px )∂ px+1 which is vector field tangent to the surface of constant energy and momentum of the three particles involved. The corresponding Fokker-Planck equation for the time evolution of the probability distribution P(q, p, t), given an initial distribution P(q, p, 0) is given by ∂P = (−A + γ S)P = L ∗ P, ∂t

(5)

where L ∗ is the adjoint of L with respect to the Lebesgue measure. i, j Let {wx,y ; x, y ∈ ZdN ; i, j = 1, . . . , d; |y − x| = 1} be independent standard i, j i, j Wiener processes, such that wx,y = wy,x . Equation (5) corresponds to the law at time t of the solution of the following stochastic differential equations: dqx = px dt, dpx = −∂qx H N dt + 2γ px dt √ d γ i, j i, j X x,z px dwx,z (t). + √ 2 d − 1 z:|z−x|=1 i, j=1

(6)

In d = 1 these are: γ dpx = −∂qx H N dt + (4 px + px−1 + px+1 )dt 6 γ + (Yx+k px ) dwx+k (t), 3

(7)

k=−1,0,1

where here {wx (t), x = 1, . . . , N } are independent standard Wiener processes. Defining the energy of the atom x as Ex =

1 1 2 p + W (qx ) + 2 x 2

V (qy − qx ),

y:|y−x|=1

the energy conservation law can be read locally as Ex (t) − Ex (0) =

d

Jx−ek ,x ([0, t]) − Jx,x+ek ([0, t]) ,

k=1

where Jx,x+ek ([0, t]) is the total energy current between x and x + ek up to time t. This can be written as t Jx,x+ek ([0, t]) =

jx,x+ek (s) ds + Mx,x+ek (t). 0

(8)

72


In the above Mx,x+ek (t) are martingales that can be written explicitly as Itô stochastic integrals Mx,x+ek (t) =

γ (d − 1)

t

i, j i, j X x,x+ek Ex (s) dwx,x+ek (s).

(9)

i, j 0

In d = 1 these martingales are written explicitly as Mx,x+1 (t) =

γ 3

t

(Yx+k Ex ) dwx+k (t).

(10)

0 k=−1,0,1

The instantaneous energy currents jx,x+ek satisfy the equation LEx =

d

jx−ek ,x − jx,x+ek ,

k=1

and it can be written as a s jx,x+ek = jx,x+e + γ jx,x+e . k k

(11)

The first term in (11) is the Hamiltonian contribution to the energy current 1 a jx,x+e = − (∇V )(qx+ek − qx ) · (px+ek + px ) k 2 d 1 j j j j V j (qx+ek − qx )( px+ek + px ) =− 2

(12)

j=1

while the noise contribution in d ≥ 2 is s = −γ (∇ek p2 )x γ jx,x+e k

(13)

and in d = 1 is s γ jx,x+1 = −γ ∇ϕ( px−1 , px , px+1 ), 1 2 2 + 4 px2 + px−1 + px+1 px−1 − 2 px+1 px − 2 px px−1 ]. ϕ( px−1 , px , px+1 ) = [ px+1 6

In the unpinned case (W = 0), given any values of E > 0, the uniform probability measure on the constant energy-momentum shell ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ px = 0, qx = 0 N ,E = (p, q) : H N = N E, ⎪ ⎪ ⎩ ⎭ x∈Zd x∈Zd N

N

is stationary for the dynamics, and A and S are respectively antisymmetric and symmetric with respect to this measure. For the stochastic dynamics, we believe that these measures are also ergodic, i.e. total energy, total momentum and center of mass are the only conserved quantities. Notice that because of the periodic boundary conditions, no other


73

conserved quantities associated to the distortion of the lattice exist. For example in d = 1 the total length of the chain x (qx+1 − qx ) is automatically null. In the pinned case, total momentum is not conserved, and the ergodic stationary measures are given by the uniform probability measures on the energy shells N ,E = {(p, q) : H N = N E } . In both cases we refer to these measures as microcanonical Gibbs measures. We denote by < · > N ,E the expectation with respect to these microcanonical measures. We will also consider the dynamics starting from the canonical Gibbs measure d < · > N ,T with temperature T > 0 defined on the phase space (R2d )Z N by < · > N ,T =

e−H N /T dq dp. Z N ,T

To avoid confusion between these measures we restrict the use of the subscript E for the microcanonical measure and the subscript T for the canonical measure. 3. Green-Kubo Formula and Statement of the Results In the physical literature several variations of the Green-Kubo formula (3) can be found ([13,7]). As in (3), one can start with the infinite system and sum over all x ∈ Zd . One can also start working with the finite system with periodic boundary conditions and sum over x ∈ dN , where dN is a finite box of size N and take the thermodynamic limit N → ∞ (before sending the time to infinity). In the finite case there is a choice of the equilibrium measure. If < · > is the canonical measure at temperature T , one refers to the derivation à la Kubo. If < · > is the microcanonical measure at energy E N d , one refers to the derivation à la Green. Because of the equivalence of ensembles one expects that these different definitions give all the same value of the conductivity, provided that temperature T and energy E are suitably related by the corresponding thermodynamis relation. Nevertheless a rigorous justification is absent in the literature. In the sequel we will consider the microcanonical Green-Kubo formula (noted κ) and the canonical Green-Kubo formula (noted κ) ˜ starting from our finite system. In the harmonic case we work out the microcanonical Green-Kubo version that we compute explicitly. Similar computations are valid (with less work) for the canonical version of the Green-Kubo formula and will give the same result. In the anharmonic case equivalence of ensembles is less developed and we deal only with the canonical version of the Green-Kubo formula. The microcanonical Green-Kubo formula for the conductivity in the direction e1 is defined as the limit (when it exists) ! " 1 E N ,E Jx,x+e1 ([0, t]) J0,e1 ([0, t]) , t→∞ N →∞ 2T 2 t d

κ 1,1 (T ) = lim lim

(14)

x∈Z N

where E N ,E is the expectation starting with the microcanonical distribution < · > N ,E , and the energy E = E(T ) is chosen such that it corresponds to the thermodynamic energy at temperature T (i.e. the average of the kinetic energy in the canonical measure). In the harmonic case T = E.

74


Similarly the canonical version of the Green-Kubo formula is given by ! " 1 E N ,T Jx,x+e1 ([0, t])J0,e1 ([0, t]) 2 t→∞ N →∞ 2T t d

κ˜ 1,1 (T ) = lim lim

(15)

x∈Z N

when this limit exists. Here E N ,T indicates the expectation with respect to the equilibrium dynamics starting with the canonical measure < · > N ,T at temperature T . These definitions are consistent with (2)–(3) as we show at the end of this section. Our first results concern the (α, ν)-harmonic case: V j (r ) = αr 2 , W j (q) = νq 2 , α > 0, ν ≥ 0.

(16)

Theorem 1. In the (α, ν)-harmonic case (16), the limits defining κ 1,1 and κ˜ 1,1 exist. They are finite if d ≥ 3 or if the on-site harmonic potential is present (ν > 0), and are infinite in the other cases. When finite, κ(T ) and κ(T ˜ ) are independent of T , coincide and the following formula holds: γ (∂k1 ω)2 (k) 1 κ˜ 1,1 (T ) = κ 1,1 (T ) = dk + , (17) 2 8π dγ ψ(k) d [0,1]d

where ω(k) is the dispertion relation ⎛ ω(k) = ⎝ν + 4α

d

⎞1/2 sin (π k )⎠ 2

j

(18)

j=1

and

# 8 dj=1 sin2 (π k j ), if d ≥ 2 ψ(k) = if d = 1. 4/3 sin2 (π k)(1 + 2 cos2 (π k)),

(19)

Consequently in the unpinned harmonic cases in dimension d = 1 and 2, the conductivity of our model diverges. In order to understand the nature of this divergence we define the (microcanonical) conductivity of the finite system of size N as ⎛⎡ ⎤2 ⎞ 1 1 ⎜⎢ ⎥ ⎟ κ N1,1 (T ) = (20) E N ,E ⎝⎣ Jx,x+e1 ([0, t N ])⎦ ⎠ , 2T 2 t N N d d x∈Z N

where t N = N /vs with vs = limk→0 |∂k1 ω(k)| = 2α 1/2 the sound velocity. This definition of the conductivity of the finite system is motivated by the following consideration: ∇k ω(k) is the group velocity of the k-mode waves, and typically vs is an upper bound for these velocities. Consequently t N is the typical time a low k (acoustic) mode takes to cross around the system once. One defines similarly κ˜ N1,1 (T ) by ⎛⎡ κ˜ N1,1 (T ) =

1 1 E N ,T 2 2T t N N d

⎤2 ⎞ ⎜⎢ ⎥ ⎟ Jx,x+e1 ([0, t N ])⎦ ⎠ . ⎝⎣ x∈ZdN

(21)


75

We conjecture that κ N (resp. κ˜ N ) has the same asymptotic behavior as the conductivity defined in the non-equilibrium stationary state on the open system with thermostats at the boundary at different temperature, as defined in eg. [3,6,15]. With these definitions we have the following theorem: Theorem 2. In the harmonic case, if W = 0: (1) κ N ∼ N 1/2 if d = 1, (2) κ N ∼ log N if d = 2. In all other cases κ N is bounded in N and converges to κ. Same results are valid for κ˜ N . In fact we show that, in the harmonic case, we have lim

N →∞

κ˜ N1,1 (T ) κ N1,1 (T )

= 1.

(22)

This is a consequence of Eq. (51) that one can easily check is also valid if the microcanonical measure is replaced by the canonical measure. In the anharmonic case we cannot prove the existence of either κ˜ 1,1 (T ) or κ 1,1 (T ), but we can establish upper bounds for the canonical version of the finite size GreenKubo formula (21). Extra assumptions on the potentials V and W assuring a uniform control on the canonical static correlations (see (86–89)) have to be done. In the unpinned case W = 0, (89) is valid as soon as V is strictly convex. In the pinned case W > 0, (86) is “morally” valid as soon as the infinite volume Gibbs measure is unique. Exact assumptions are given in [4], Theorem 3.1 and Theorem 3.2. In the sequel, “the general anharmonic case” will refer to potentials V and W such that (86) (or (89)) is valid. Theorem 3. Consider the general anharmonic case. There exists a constant C (depending on the temperature T ) such that • For d ≥ 3, (1) either W > 0 is general (2) or if W = 0 and 0 < c− ≤ V j ≤ C+ < ∞ for any j, then κ˜ N1,1 (T ) ≤ C. • For d = 2, if W = 0 and 0 < c− ≤ V j ≤ C+ < ∞ for any j, then κ˜ N1,1 (T ) ≤ C(log N )2 . • For d = 1, if W = 0 and 0 < c− ≤ V ≤ C+ < ∞, then κ˜ N1,1 (T ) ≤ C

√

N.

• Moreover, in any dimension, if V j are quadratic and W > 0 is general then κ˜ N1,1 (T ) ≤ C.

76


The proof of this statement is in Sect. 6. We now relate the definition of the Green-Kubo (14) and (15) to the variance of the energy-energy correlations function (2). Consider the infinite volume dynamics on Zd under the infinite volume canonical Gibbs measure with temperature T > 0. The expectation is denoted by ET . Fix t > 0 and assume that the following sum makes sense: i, j xi x j ET [ (Ex (t) − T )(E0 (0) − T ) ] = xi x j S(x, t). (23) DT (t) = x∈Zd

x∈Zd

If x = 0, by space and time invariance of the dynamics, we have 1 ET [(Ex (t) − T )(E0 (0) − T )] = − ET [ (Ex (t) − Ex (0)) (E0 (t) − E0 (0)) ] . (24) 2 By definition of the current, we have for any y ∈ Zd : Ey (t) − Ey (0) =

d

Jy−ek ,y ([0, t]) − Jy,y+ek ([0, t]) .

(25)

k=1

By two discrete integration by parts one obtains ! " i, j ET Jx,x+ei ([0, t]) J0,e j ([0, t]) DT (t) =

(26)

x∈Zd

so that the thermal conductivity is equal to the space-time correlations of the total current ! " 1 ET Jx,x+ei ([0, t]) J0,e j ([0, t]) . t→∞ 2T 2 t d

κ i, j (T ) = δ0 (i − j) lim

(27)

x∈Z

Of course this derivation is only formal even for fixed time t > 0. The problem is to define the infinite volume dynamics and to show S(x, t) has a sufficiently fast decay in x. For the purely Hamiltonian dynamics, it is a challenging problem. For the stochastic dynamics it seems less difficult but remains technical. To avoid these difficulties we adopt a finite volume limit procedure starting from (3). This explains the definitions (14) and (15). Consider now the closed dynamics on ZdN starting from the microcanonical state. The rest of the section is devoted to the proof of the following formula: ⎛⎡ ⎤2 ⎞ 1 1 ⎜⎢ ⎥ ⎟ E N ,E ⎝⎣ Jx,x+e1 ([0, t])⎦ ⎠ 2 d 2T t N d x∈Z N

⎛⎡

⎤2 ⎞ t ⎜⎢ ⎥ ⎟ γ ON a = (2T 2 t N d )−1 E N ,E ⎝⎣ jx,x+e (s)ds ⎦ ⎠ + + d , 1 d N d

(28)

x∈Z N 0

and an identical formula in the canonical case (with E N ,E substituted by E N ,T ). The term γ /d in (28) is the direct contribution of the stochastic dynamics to the thermal conductivity. In the microcaconical case we actually prove that is equal to


77

γ /d only for the harmonic case. A complete proof of (28) for anharmonic interaction demands an extension of the equivalence of ensembles estimates proven in Sect. 7. In the grancanonical case this problem does not appear. Starting in the microcanonical case, remark that the first term on the RHS of (28) can be written as ⎛⎡ ⎤2 ⎞ t ⎜⎢ ⎥ ⎟ a (2T 2 t N d )−1 E N ,E ⎝⎣ jx,x+e (s)ds ⎦ ⎠ 1 x∈ZdN 0

1 = 2 T

) ∞ ( a s + a 1− E N ,E jx,x+e (s) j0,e (0) ds. 1 1 t d

(29)

x∈Z N

0

If γ = 0, which corresponds to the purely Hamiltonian system, as N and then t goes to infinity, and if one can prove that the current-current correlation function has a sufficiently fast decay, then one recovers the usual Green-Kubo formula (3). To prove (29) one uses space and time translation invariance of the dynamics ⎛⎡ ⎤2 ⎞ t ⎜ ⎟ a (2T 2 t N d )−1 E N ,E ⎝⎣ jx,x+e (s)ds ⎦ ⎠ 1 x

d −1

= (2T t N ) 2

x,y

d −1

= (T t N ) 2

x,y

d −1

= (T t N ) 2

1 = 2 T

∞ ( 1− 0

s t

t

t

ds

0

0

t

s ds

0

x,y

0

a a du E N ,E jx,x+e (s) j (u) y,y+e1 1

a a du E N ,E jx,x+e (s) j (u) y,y+e 1 1

0

s

t

ds

0

a a du E N ,E jx−y,x−y+e (s − u) j (0) 0,e 1 1

0

)+

a a E N ,E jx,x+e (s) j0,e (0) ds. 1 1

x

We now give the proof of (28). Because of the periodic boundary conditions, since j s if a gradient (cf. (13)), the corresponding terms cancel, and we can write

Jx,x+e1 ([0, t]) =

x

t 0

a jx,x+e (s) ds + 1

x

Mx,x+e1 (t)

x

t Je1 (s) ds + Me1 (t)

= 0

(30)

78


so that

⎛* +2 ⎞ Jx,x+e1 ([0, t]) ⎠ (t N d )−1 E N ,E ⎝ x

⎛⎡ ⎤2 ⎞ t ⎜ ⎟ = (t N d )−1 E N ,E ⎝⎣ Je1 (s)ds ⎦ ⎠ + (t N d )−1 E N ,E M2e1 (t) 0

⎛⎡ d −1

+2(t N )

E N ,E ⎝⎣

t

⎤

⎞

Je1 (s) ds ⎦ Me1 (t)⎠ .

(31)

0

The third term on the RHS of (31) is shown to be zero by a time reversal argument and the second term on the RHS of (31) gives in the limit a contribution equal to γ /d. To see the first claim let us denote by {ω(s)}0≤s≤t the process {(px (s), qx (s)); x ∈ ZdN , 0 ≤ s ≤ t} arising in (6) or in (7) for the one-dimensional case. The reversed process {ωs∗ }0≤s≤t is defined as ωs∗ = ωt−s . Under the microcanonical measure, the time reversed process is still Markov with generator −A + γ S. The total current Jt (ω· ) = x Jx,x+e1 ([0, t]) is a functional of {ωs }0≤s≤t . By (6–7), we have in fact that Jt (·) is an anti-symmetric functional of {ωs }0≤s≤t , meaning Jt ({ωs∗ }0≤s≤t ) = −Jt ({ωs }0≤s≤t ).

(32)

In fact, similarly to (8), we have Js (ω·∗ )

s =

(Je1 )∗ (ω∗ (v))dv + M∗e1 (s), 0 ≤ s ≤ t,

(33)

0

where (M∗e1 (s)) is a martingale with respect to the natural filtration of (ωs∗ )0≤s≤t 0≤s≤t ∗ a ∗ a and (Je1 ) = x ( j )x,x+e1 is equal to −Je1 = − x jx,x+e . 1 We have then by time reversal E N ,E [Jt (ω· )Je1 (ω(t))] = −E N ,E [Jt (ω·∗ )Je1 (ω∗ (0))] ⎡⎛ t ⎞ ⎤ = −E N ,E ⎣⎝ (Je1 )∗ (ω∗ (s))ds + M∗ (t)⎠ Je1 (ω∗ (0))⎦ ⎡⎛ = −E N ,E ⎣⎝

0

t

⎞

⎤

Je1 ∗ (ω∗ (s))ds ⎠ Je1 (ω∗ (0))⎦ ,

(34)

0

where the last equality follows from the martingale property of M∗ . Recall now that (Je1 )∗ = −Je1 . By variables change s → t − s in the time integral, we get ⎡⎛ t ⎞ ⎤ E N ,E [Jt (ω· )Je1 (ω(t))] = E N ,E ⎣⎝ Je1 (ω(s))ds ⎠ Je1 (ω(t))⎦ . (35) 0


79

It follows that ⎞ ⎤ ⎤ ⎡⎛ t ⎡ t E N ,E ⎣⎝ Je1 (ω(s))ds ⎠ Me1 (t)⎦ = E N ,E ⎣ Je1 (ω(s))Me1 (s)ds ⎦ 0

t =

0

⎛

⎡

ds E N ,E ⎣Je1 (ω(s)) ⎝ Js (ω· ) −

0

s

⎞⎤

Je1 (ω(v))dv ⎠⎦ . = 0

(36)

0

For the second term on the RHS of (31) we have 2 , i, j γ 2 X (p /2) x,x+e1 x (d − 1)N d x N ,E i, j , 2 γ j i i j p = p − p p x x+e1 x x+e1 (d − 1)N d x N ,E i= j . / 2γ j i ( px px+e = )2 1 N ,E (d − 1)N d x i= j / . 2γ j j i i ( p − p p p ) . x x+e x x+e 1 1 N ,E (d − 1)N d x

(t N d )−1 E N ,E M2e1 (t) =

i= j

Thanks to the equivalence of ensembles (cf. Lemma 7), this last quantity is equal to 2γ

T2 + N −d O N , d

(37)

where O N remains bounded as N → ∞. The calculation in d = 1 is similar. The contribution of the martingale term for the conductivity is hence γ /d and we have shown (28). Notice this is the only point where we have used the equivalence of ensembles results of Sect. 7 that we have proven only in the harmonic case. We conjecture these are true also for the anharmonic cases. Observe that all the arguments above between (30) and (37) apply directly also to the canonical definition of the Green-Kubo but without the small error in N (because for the canonical measure momentums px are independently distributed and the equivalence of ensembles approximations are in fact equalities). Therefore we have the similar formula to (28): 1 E N ,T Jx,x+e1 ([0, t])J0,e1 ([0, t]) 2 2T t x ⎛⎡ ⎤2 ⎞ t ⎜ ⎟ γ a = (2T 2 N d t)−1 E N ,T ⎝⎣ jx,x+e (s)ds ⎦ ⎠ + . 1 d x

(38)

0

In the next sections we will consider the (α, ν)-harmonic case and we will compute explicitly the limit (as N → ∞ and then t → ∞) of the two first term on the RHS of (31).

80


4. Correlation Function of the Energy Current in the Harmonic Case We consider the (α, ν)-harmonic case (16). We recall that Je1 = x jx,x+e1 . Because s of the periodic boundary a conditions, and being jx,x+e1 a spatial gradient (cf. (13)), we have that Je1 = x jx,x+e1 . We are interested in the decay of the correlation function: C1,1 (t) = lim

N →∞

1 a a E N ,E (Je1 (t)Je1 (0)) = lim E N ,E ( j0,e (0) jx,x+e (t)), (39) 1 1 d N →∞ N x

where E N ,E is the expectation starting with the microcanonical distribution defined above. For λ > 0, let u λ,N be the solution of the Poisson equation a jx,x+e λu λ,N − Lu λ,N = − 1 x

given explicitly in Lemma 2 of Sect. 5. By Lemma 1, we can write the Laplace transform of C1,1 (t) as ∞

dte−λt C1,1 (t) dt = lim

N →∞

a j0,e u 1 λ,N

N ,E

.

(40)

0

Substituting in (40) the explicit form of u λ,N given in Lemma 2, we have: a

α2 − j0,e u = gλ,N (x − y) (qe1 − q0 ) · (pe1 + p0 )(px · qy ) N ,E λ,N 1 N ,E 2γ x,y =

α2 gλ,N (x − y) (qe1 · p0 − q0 · pe1 )(px · qy ) N ,E 2γ x,y +

α2 gλ,N (x − y) (qe1 · pe1 − q0 · p0 )(px · qy ) N ,E . 2γ x,y (41)

Observe that the last term on the RHS of (41) is null by the translation invariance property. So we have (using again the translation invariance and the antisymmetry of gλ,N ) a

α2 − j0,e u = gλ,N (x − y) (qe1 − q−e1 ) · p0 )(px · qy ) N ,e . 1 λ,N N ,e 2γ x,y Define K N (q) = N d E −

1 qx · (ν I − α)qx . 2 x

In the unpinned case ν = 0, conditionally to the positions configuration q, the law of p Nd is µq = µ√ (defined in Lemma 6), meaning the uniform measure on the surface 2K (q) N

# (px )x∈Zd ; N

0 1 2 p = K N (q); px = 0 . 2 x x x


81

By using properties (i),(ii) and (iii) of Lemma 6, one has for x = 0, / .

j j i µq p0i px (qei 1 − q−e )q ((qe1 − qe1 ) · p0 )(px · qy ) N ,E = y 1 i, j

=

.

i µq p0i pxi (qei 1 − q−e )qyi 1

i

=−

d , i=1

N ,E

/ N ,E

2K N (q) i (q i − q−e )qyi 1 d N d (N d − 1) e1

N ,E

/ 1 . i 2 i i i ( p ) (q − q )q . e −e y 0 1 1 N ,E Nd − 1 d

=−

(42)

i=1

For x = 0, one gets / .

j j i µq p0i p0 (qei 1 − q−e ((qe1 − qe1 ) · p0 )(p0 · qy ) N ,E = )q y 1 i, j

=

/ i i µq p0i p0i (qei 1 − q−e )q y 1

d . i=1

=

N ,E

d ,

p0i

2

N ,E

(qei 1

i=1

i − q−e )qyi 1

.

(43)

N ,E

In the pinned case ν > 0, conditionally to the positions configuration q, the law of p Nd is λq = λ√ (defined in Lemma 5), meaning the uniform measure on the surface 2K N (q) 0 # 1 2 p = K N (q) . (px )x∈Zd ; N 2 x x We proceed in a similar way and we observe that if x = 0, λq ( p0i pxi ) = 0 (cf. ii) of Lemma 5) Since gλ,N is antisymmetric (see (64–65)) and such that z gλ,N (z) = 0, one obtains easily in both cases (pinned and unpinned) −

a j0,e u 1 λ,N

N ,e

, 2 α2 i i i i p0 (qe1 − q−e1 )qy =− gλ,N (y) 2γ y N ,E i , 2 2 α 1ν=0 i i i i p0 (qe1 − q−e1 )qy + gλ,N (y − x) 2γ N d − 1 N ,E x=0,y i , ) 2 ( 2 1ν=0 α i p0i (qei 1 − q−e = − 1+ d gλ,N (y) )qyi . 1 N − 1 2γ y N ,E i

(44) Let N (x), x ∈ such that

ZdN ,

be the unique solution of (ν I − α) N = δe1 − δ−e1

x∈ZdN

N (x) = 0.

(45)

82


By (iii) of Lemma 7 and (77), we have 1 1 1 1 ) 2 2 ( 1 1 a

1 α E ν=0 1− j u λ,N − 1+ d gλ,N (y) N (y)11 0,e1 1 N ,E N − 1 2γ d 1 1 y 2 31/2 1 C log N C log N 11 C log N ≤ gλ,N (y)1 ≤ (gλ,N (x))2 ≤ . d d/2 N N λN d/2 y x

(46)

Hence the last term of (46) goes to 0. Taking the limit as N → ∞ we obtain (see (80)) ∞

e−λt C1,1 (t) dt =

0

α2 E 2 gλ (z)(z), 2dγ z

(47)

where gλ are solutions of the same equations as gλ,N but on Zd and is the solution of the same equation as N but on Zd . Using Parseval relation and the explicit form of the Fourier transform of gλ (cf. (74)) and , one gets the following formula for the Laplace transform of C1,1 (t) for d ≥ 2: 2 3 sin2 (2π k1 ) α2 E 2 1 . (48) dk d d 2 j d ν + 4α j=1 sin (π k ) λ + 8γ j=1 sin2 (π k j ) [0,1]d

By injectivity of Laplace tranform, C1,1 (t) is given by: 2 3 α2 E 2 sin2 (2π k1 ) dk C1,1 (t) = d ν + 4α dj=1 sin2 (π k j ) [0,1]d

⎧ ⎨ exp

⎩

−8γ t

d j=1

⎫ ⎬

sin2 (π k j ) . ⎭

(49)

For the one dimensional case, the equation for gλ,N (resp. gλ ) is different (see (75) ) and we get the following integral representation of the correlation function of the energy current: 1 C1,1 (t) = αE

2

4γ t 2 sin (π k)(1 + 2 cos2 (π k) . dk cos2 (π k) exp − 3

(50)

0

In any dimension, we have the following unified formula for C1,1 (t) E2 C1,1 (t) = 4π 2 d

[0,1]d

(∂k1 ω(k))2 e−tγ ψ(k) dk,

(51)


83

where ω(k) is defined by (18) and ψ(k) by (19). Observe that the same formula holds if we replace E N ,E by E N ,T . In this last case, the situation is simpler since we do not need equivalence of ensembles. Standard analysis shows the behavior of C1,1 (t) as t goes to infinity is governed by the behavior of the function (∂k1 ω(k))2 and ψ(k) around the minimal value of ψ which is 0. In fact, ψ(k) = 0 if and only if k = 0 or k = (1, . . . , 1). By symmetry, we can treat only the case k = 0. Around k = 0, ψ(k) ∼ a|k|2 and (∂κ 1 ω(k))2 ∼ b(ν + |k|2 )−1 (k1 )2 , where a and b are positive constants depending on ν and α. Essentially, C1,1 (t) has the same behavior as

(k 1 )2 e−aγ t|k| 1 = d/2+1 2 ν + |k| t 2

dk

√ [0, t]d

k∈[0,1]d

(k1 )2 e−aγ |k| . ν + t −1 |k|2 2

dk

(52)

Hence, we have proved the following theorem: Theorem 4. In the (α, ν)-harmonic case, the current-current time correlation function C1,1 (t) decays like • C1,1 (t) ∼ t −d/2 in the unpinned case (ν = 0) • C1,1 (t) ∼ t −d/2−1 in the pinned case (ν > 0) 5. Conductivity in the Harmonic Case Lemma 1. Consider the (α, ν)-harmonic case. For any time t, the following limit exists: C1,1 (t) = lim

N →∞

1 E N ,E (Je1 (t)Je1 (0)). Nd

(53)

and ∞

dte−λt C1,1 (t) dt = lim

N →∞

a j0,e u 1 λ,N

N ,E

.

(54)

0

The same result holds with E N ,E replaced by E N ,T . Proof. We only prove this lemma in the microcanonical setting. Let us define f N (t) =

1 E N ,E (Je1 (t)Je1 (0)). Nd

(55)

We first prove the sequence ( f N ) N is uniformly bounded. By Cauchy-Schwarz and stationarity, we have

1 4 2 4 2 Je1 (t) N ,E Je1 (0) N ,E d N 1 . / = d J2e1 . N ,E N

| f N (t)| ≤

(56)

84


We now use symmetry properties of the microcanonical ensemble to show this last term is bounded above by a constant independent of N , a N −d < J2e1 > N ,E = < j0,e ja > N ,E 1 x,x+e1 x

=

α2 4

d .

j

j

j

j

(qei 1 − q0i )(qx+e1 − qx )( pei 1 + p0i )( px+e1 + px

x i, j=1

/ N ,E

.

In the unpinned case ν = 0, conditionally to the positions configuration q, the law Nd of p is µq = µ√ (defined in Lemma 6). 2K N (q) By using properties (i), (ii) and (iii) of Lemma 6, one has / α2 . i i i i i i 2 (qe1 − q0i )(q2e − q − 3q + 3q )( p ) . −e1 e1 0 0 1 N ,E 4 d

N −d < J2e1 > N ,E =

(57)

i=1

By Cauchy-Schwarz inequality, the modulus of this last quantity is bounded above by 1 17 α[8 < E02 > N ,E + < Ee21 > N ,E ] = < E02 > N ,E , 2 2

(58)

where the last equality is a consequence of the invariance by translation of < · > N ,E . Let Nd (X 1 , . . . , X N d ) be a random vector with law λ√ , meaning the uniform measure on NdE √ the N d -dimensional sphere of radius N d E. The vector of energies (Ex , x ∈ ZdN ) has the same law as (X 12 , . . . , X 2N d ). By Lemma 4, E(X 14 ) = < E02 > N ,E is bounded above by a constant independent of N . Hence there exists a positive constant C such that | f N (t)| ≤ C.

(59)

Similarly, inequality (59) can be proved in the pinned case ν > 0. Let f (t) be any limit point of the sequence ( f N (t)) N ≥1 and choose a subsequence (Nk )k≥0 such that ( f Nk ) converges to f (for the pointwise convergence topology). By Lebesgue’s theorem, we have ∞

∞

e−λt f (t)dt.

(60)

e−λt f N (t)dt = − < j0,e1 , u λ,N > N ,E

(61)

lim

e

k→∞

−λt

f Nk (t)dt =

0

0

But we have that ∞ 0

and we have seen in Sect. 4 this last quantity converges as N goes to infinity to ∞ 0

e−λt f ∞ (t)dt,

(62)


85

where f ∞ is given by (see (18–19) for the notations) E2 (∂k1 ω(k))2 e−tγ ψ(k) dk. f ∞ (t) = 4π 2 d

(63)

[0,1]d

By injectivity of the Laplace transform, we get f (t) = f ∞ (t). Uniqueness of limit points implies ( f N (t)) N ≥1 converges to f ∞ (t) for any t. It follows also we can inverse time integral and infinite volume limit in the left hand side of (54) and the lemma is proved.

Lemma 2 (Resolvent equation). 2 −1

u λ,N = (λ − L)

−

3 a jx,x+e 1

x

where gλ,N (z) is the solution (such that

=

α gλ,N (x − y)px · qy , γ x,y

z gλ,N (z)

= 0) of the equation

2λ gλ,N (z) − 4gλ,N (z) = (δ(z + e1 ) − δ(z − e1 )) γ

(64)

for d ≥ 2, or " 2λ 1 ! gλ,N (z)− 4gλ,N (z) + gλ,N (z + 1) + gλ,N (z − 1) = (δ(z + 1)−δ(z − 1)) γ 3 (65) for d = 1. Moreover, Au λ,N = 0 and Lu λ,N = γ Su λ,N . Proof. We only give the proof for the dimension d ≥ 2 since the proof for the one dimensional case is similar. Let u λ,N = γα x,y gλ,N (x − y)px · qy . The generator L is equal to the sum of the Liouville operator A and of the noise operator γ S. The action of A on u λ,N is null. Indeed, we have: ⎛ ⎞ α α [(α−ν I )qx ] · ⎝ gλ,N (x − y)qy ⎠ + gλ,N (x−y)px · py . Au λ,N = γ x γ y,x y (66) Here, and in the sequel of the proof, sums indexed by x, y, z are indexed by Z N and sums indexed by i, j, k, are indexed by {1, . . . , d}. Summation by parts can be performed (without outcoming boundary terms since we are on the torus) and we get Au λ,N =

α α [(α − ν I )gλ,N ](x − y)qx qy + gλ,N (x − y)px · py . γ x γ y,x

(67)

Remark now that the function δ(· − e1 ) − δ(· + e1 ) is antisymmetric. Hence gλ,N , and consequently gλ,N , is still antisymmetric. We have therefore Au λ,N which is of the form: {a1 (x − y)px · py + a2 (x − y)qx · qy } (68) Au λ,N = x,y

86


with a1 , a2 antisymmetric. Using the antisymmetricity of a1 and a2 , it is easy to show that the last two sums are zero and hence Au λ,N = 0. A simple computation shows that if ∈ {1, . . . , d} then S( px ) =

i, j 1 (X y,y+ek )2 ( px ) 2(d − 1) y i= j,k

=

2 2(d − 1)

i, (X x,x+e )2 ( px ) + k

i=,k

i=,k 2 (X x−ek ,x )2 ( px ) 2(d − 1) i=,k

6 1 5 ( px+ek − px ) − ( px − px−e ) = k d −1 i=,k

=

2( px ).

Since the action of S is only on the p’s, we have γ Su λ,N = α gλ,N (x − y)S(px ) · qy x,y

= 2α

gλ,N (x − y)(px ) · qy

x,y

= 2α

(gλ,N )(x − y)px · qy ,

x,y

where in the last line, we performed a summation by parts. Since gλ,N is a solution of (64), we have λu λ,N − γ Su λ,N =

α a px · (qx+e1 − qx−e1 ) = − jx,x+e . 1 2 x x

(69)

of the function v on Let us define the Fourier transform v(ξ ˆ ), ξ ∈ v(ξ ˆ )= v(z) exp(2iπ ξ · z/N ). ZdN ,

ZdN

as (70)

z∈ZdN

The inverse transform is given by v(z) =

1 v(ξ ˆ ) exp(−2iπ ξ · z/N ) Nd d

(71)

ξ ∈Z N

On Zd we define similarly: v(k) ˆ =

v(z) exp(2iπ k · z), k ∈ [0, 1]d .

z∈Zd

and its inverse by

v(z) =

v(k) ˆ exp(−2iπ k · z). [0,1]d

(72)


87

For λ > 0, the function gλ : Zd → R is the solution on Zd of the equation 2λ gλ (z) − 4gλ (z) = δ0 (z + e1 ) − δ0 (z + e1 ), d ≥ 2, γ 2λ 1 gλ (z) − (4gλ (z) + gλ (z + 1) + gλ (z − 1)) = δ0 (z + 1) − δ0 (z − 1), d = 1. γ 3 (73) Then we have gˆ λ (k) =

−2iπ sin(2π k1 ) , if d ≥ 2 d 2λ 2 j + 16 j=1 sin (π k ) γ

(74)

and gˆ λ (k) =

−2iπ sin(2π k) , if d = 1. 2λ 8 2 2 + sin (π k) 1 + 2 cos (π k) γ 3

(75)

Since gλ,N is the solution of the same equation as gλ but on ZdN , we have the following formula for gˆ λ,N : gˆ λ,N (ξ ) = gˆ λ (ξ/N ).

(76)

The following bound follows easily from Parseval relation:

(gλ,N (x))2 ≤

x∈ZdN

γ2 λ2

(77)

Similarly, the function N defined in (45) has Fourier transform given by ˆ ), ˆ N (ξ ) = (ξ/N

(78)

where ˆ (k) =

−2i sin(2π k1 ) . ν + 4α dj=1 sin2 (π k j )

(79)

Let us denote by z ∗ the conjugate 7 8∗ of the complex number z and observe that the funcd ˆ tion k ∈ [0, 1] → gˆ λ (k) (k) ∈ R+ is continuous. Hence we have the following convergence of Riemann sums:

gλ,N (y) N (y) =

y∈ZdN

−−−−→ N →∞

[0,1]d

1 gˆ λ,N (ξ )[ˆ N (ξ )]∗ Nd d ξ ∈Z N

∗ ˆ dk gˆ λ (k)[(k)] =

gλ (y).(y).

(80)

y∈Zd

The limits as λ → 0 of the above expressions give the values for the conductivity (up to a multiplicative constant) when this is finite. If ν = 0 it diverges if d = 1 or 2.

88


6. Anharmonic Case: Bounds on the Thermal Conductivity We consider in this section the general anharmonic case and we prove Theorem 3. Recall (38), then all we need to estimate is ⎛⎡ ⎤2 ⎞ N ⎜ ⎟ a (81) (2T 2 N d+1 )−1 E N ,T ⎝⎣ jx,x+e (s)ds ⎦ ⎠ . 1 x

Let us define

E N ,T

0

= Je1 , then we have the general bound ([16], Lemma 3.9) ⎛⎡ ⎤2 ⎞ N / . ⎜⎣ ⎟ Je1 (s)ds ⎦ ⎠ ≤ 10N Je1 , (N −1 − L)−1 Je1 ⎝ a x jx,x+e1

N ,T

0

/ . ≤ 10N Je1 , (N −1 − γ S)−1 Je1

N ,T

.

(82)

1 Recall that S(px ) = 2(px ) if d ≥ 2 and S( px ) = (4 px + px+1 + px−1 ) if d = 1, 6 (N −1 − γ S)−1 Je1 =

d

G N (x − y) px V j (qy+e1 − qy ), j

j

j

(83)

j=1 y

where G N (z) is the solution of the resolvent equation ⎧ " 1! ⎪ ⎪ N −1 G N (z) − 2γ (G N )(z) = − δ0 (z) + δe1 (z) , d ≥ 2 ⎪ ⎪ ⎪ 2 ⎨ γ −1 G (z) − )(z) + (G N )(z + 1) + (G N )(z − 1)] N [4(G N N ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎩ = − 1 [δ0 (z) + δ1 (z)] , d = 1. 2

(84)

The left-hand side of (82) is equal to − 5T N d+1

d j=i

. / j j j j (G N (x) + G N (x + e1 )) V j (qx+e1 − qx )V j (qe1 − q0 )

x

N ,T

. (85)

• Pinned case.

/ . j j j j In the pinned case, the correlations V j (qx+e1 − qx )V j (qe1 − q0 )

nentially in x, 1. / 11 1 j −c|x| 1 V (qx+e − qxj )V (qej − q j ) 1 . j 1 0 N ,T 1 ≤ Ce 1 1 j It follows that the previous expression is bounded by |G N (x) + G N (x + e1 )|e−c|x| . C T 2t N d x

N ,T

decay expo-

(86)


89

Since G N is bounded in d ≥ 3, it follows that (81) is uniformly bounded in N . In low dimensions, our estimates are too rough and we obtain only diverging upper-bounds. Nevertheless, if V j (r ) = α j r 2 are quadratics and W j are general but strictly positive then . / j j j j V j (qx+e1 − qx )V j (qe1 − q0 ) N ,T 5 6 j j j j j j (87) = α j 2 < qx q0 > N ,T − < qx−e1 q0 > N ,T − < qx+e1 q0 > N ,T . As a function of x, this quantity is a Laplacian in the first direction and by integration by parts, the left-hand side of (81) is upper bounded by |(G N )(x) + (G N )(x + e1 )| e−c|x| . C (88) x

By Lemma 3, this quantity is uniformly bounded in N . • Unpinned case. In the unpinned case, we assume that 0 < c ≤ V j (q) ≤ C < +∞. We have (cf. [9], Theorem 6.2, that can be proved in finite volume uniformly) 1. / 11 1 j −d 1 V (qx+e − qxj )V (qej − q j ) 1 (89) j 1 0 N ,T 1 ≤ C|x| . 1 1 j In the one dimensional case, the random variables r x = qx+1 − qx are i.i.d. and < V (r x ) > N ,T = 0. Only the term corresponding to x = 0 remains in the sum of (85). By Lemma 3, we get the upper bound . / √ ≤ C N. (90) (G N (0) + G N (1)) V (r02 ) N ,T

For the unpinned two dimensional case, we obtain the upper bound |G N (x) + G N (x + e1 )||x|−d C x∈Z2N

≤ C log N

|x|−2

x∈Z2N

∼ C(log N )2 . For the case d ≥ 3, we use the first point of Lemma 3, (89) and the fact that |x|−d ∼ log N .

(91)

(92)

x∈ZdN

Lemma 3. Let G N be the solution of the discrete equation (84). There exists a constant C > 0 independent of N such that • • • •

G N (x) ≤ C(|x|d−2 + N −1/2 ), d ≥ 3 G N (x) ≤ C √ log N , d = 2 G N (x) ≤ C N , d = 1 |G N (x + e1 ) + G N (x − e1 ) − 2G N (x)| ≤ C, d ≥ 1.

90


Proof. In the proof, C is a constant independent of N but which can change from line to line. We first treat the case d ≥ 3. We use Fourier’s transform representation of G N : G N (x) = −

−2iπ k·x/N 1 2iπ k1 /N e , (1 + e ) 2N d θ N (k/N ) d

(93)

k∈Z N

where θ N (u) = N −1 + 8γ form:

d

j=1 sin

2 (π u j ).

G N can also be written in the following

1 G N (x) = − [FN (x) + FN (x − e1 )] , 2

(94)

where 1 e−2iπ k·x/N . Nd θ N (k/N ) d

FN (x) =

(95)

k∈Z N

Let us introduce the continuous Fourier’s transform representation of the Green function F∞ on Zd given by: exp(2iπ x · u) F∞ (x) = du, (96) θ (u) [0,1]d

where θ (u) = 8γ dj=1 sin2 (π u j ). Remark that F∞ is well defined because d ≥ 3. We have to prove there exists a constant C > 0 independent of N such that FN (x) ≤ C(|x|d−2 + N −1/2 ).

(97)

Observe that by symmetries of FN , we can restrict our study to the case x ∈ [0, N /2]d . We want to show that FN (x) is well approximated by F∞ (x). We have N N (x) + F∞ (x) − F∞ (x), FN (x) − F∞ (x) = FN (x) − F∞

(98)

where

exp(2iπ x · u) du. θ N (u)

N (x) = F∞ [0,1]d

(99)

9 For each k ∈ ZdN , we introduce the hypercube Q k = dj=1 [k j /N , (k j + 1)/N ) and we divide [0, 1]d following the partition ∪k∈Zd Q k . By using this partition, we get N

N FN (x) − F∞ (x) =

k∈ZdN Q k

du

e2iπ k·x/N − e2iπ u·x θ N (k/N )

+ Qk

due2iπ u·x

(

) 1 1 − . θ N (k/N ) θ N (u)

(100)


Remark that

due2iπ u·x =

91

e2iπ k·x/N ϕ(x/N ), Nd

(101)

Qk

where ϕ(u) =

d :

e2iπ u

d : sin(π u j ) . (π u j )

j

j=1

(102)

j=1

It follows that the first term on the right-hand side of (100) is equal to (1 − ϕ(x/N ))FN (x) so that FN (x) =

( ) N (x) 1 F∞ 1 1 . due2iπ u·x + − ϕ(x/N ) ϕ(x/N ) θ N (k/N ) θ N (u) d

(103)

(104)

k∈Z N Q k

The next step consists to show that the second term on the right-hand side of (104) is small. In the sequel, C is a positive constant independent of N but which can change from line to line. For each u ∈ Q k , we have sin2 (π u j ) − sin2 (π k j /N ) = π sin(2π c j )(u j − k j /N ).

(105)

for some c j ∈ [k j /N , (k j + 1)/N ). Consequently, we have | sin2 (π u j ) − sin2 (π k j /N )| ≤

C | sin(π k j /N )|. N

(106)

Moreover, there exists a positive constant C such that ∀k ∈ ZdN , ∀u ∈ Q k , θ N (u) ≥ Cθ N (k/N ).

(107)

It follows that the modulus of the second term on the right-hand side of (104) is bounded by d 1 N −1 | sin(π k j /N )| C . |ϕ(x/N )| Nd θ N (k/N )2 d

(108)

k∈Z N

j=1

Since the modulus of the function ϕ(u) is bounded below by a positive constant on [0, 1/2]d , this last term is of the same order as N

−1

d j=1

[0,1]d

| sin(π u j )| du. θ N (u)2

(109)

Elementary standard analysis shows that this term is of the same order as N

−1

1 0

rd dr. + r 2 )2

(N −1

(110)

92


For d ≥ 4, this term is clearly of order N −1 . For d = 3, the change of variables r = N −1/2 v gives an integral of order N −1 log N . In conclusion, we proved F N (x) FN (x) = ∞ +O ϕ(x/N )

(

) log N . N

(111)

Moreover, it is not difficult to show that N |F∞ (x) − F∞ (x)| ≤ C N −1/2 .

(112)

Since we have (cf. [14], Theorem 4.5) F∞ (x) ≤ C|x|2−d

(113)

we obtained the first point of the lemma. For the 1- and 2-dimensional estimates, we have that |G N (x)| ≤ G N (0) and by standard analysis, there exists a constant C > 0 independent of N such that G N (0) ≤ C

dk [0,1/2]d

N −1

+

d

1

j=1 sin

2 (π k)

.

(114)

By using the inequality sin2 (π u) ≥ 4u 2 , one gets G N (0) is of same order as dk [0,1/2]d

1 . N −1 + |k|2

(115)

√ This last quantity is of order N if d = 1 and log N if d = 2. Let us now prove the final statement. Assume d ≥ 2 (the case d = 1 can be proved in a similar way). We have |G N (x + e1 ) + G N (x − e1 ) − 2G N (x)| 1 1 1 1 1 2 1 −2iπ k·x/N e 1 1 2iπ k1 /N 2 (1 + e ) sin (π k1 /N ) =1 d 1 1N 1 θ (k/N ) N d 1 1 k∈Z N ≤

4 sin2 (π k1 /N ) Nd θ N (k/N ) d k∈Z N

≤ (2γ )−1 .

(116)


93

7. Appendix: Equivalence of Ensembles In this part, we establish a result of equivalence of ensembles for the microcanonical measure < · > N ,E , since it does not seem to appear in the literature. The decomposition in normal modes permits to obtain easily the results we need from the classical equivalence of the ensemble for the uniform measure on the sphere. This last result proved in [10] says that the expectation of a local √ function in the microcanonical ensemble (the uniform measure on the sphere of radius k in this context) is equal to the expectation of the same function in the canonical ensemble (the standard gaussian measure on R∞ ) with an error of order k −1 . In fact, the equivalence of ensembles of Diaconis and Freedman is expressed in terms of a very precise estimate of variation distance between the microcanonical ensemble and the canonical ensemble. In this paper, we need to consider equivalence of ensembles for unbounded functions and to be self-contained we prove in the following lemma a slight modification of estimates of [10]. Lemma 4. Let λrnn 1/2 be the uniform measure on the sphere # Srnn 1/2 = (x1 , . . . , xn ) ∈ Rn ;

n =1

0 x2 = nr 2

of radius r and dimension n − 1 and λr∞ the Gaussian product measure with mean 0 and variance r 2 . Let θ > 0 and φ be a function on Rk such that |φ(x1 , . . . , xk )| ≤ C

2 k =1

3θ x2

, C > 0.

There exists a constant C (depending on C, θ, k, r ) such that 1 1 lim sup n 1λrnn 1/2 (φ) − λr∞ (φ)1 ≤ C .

(117)

(118)

n→∞

Proof. This lemma is proved in [10] for φ positive bounded by 1. Without loss of generality, we can assume r = 1 and we simplify the notations by denoting λrnn 1/2 with λn and λr∞ with λ∞ . The law of (x1 + . . . + xk )2 under λn is n times a β[k/2, (n − k)/2] distribution and has density (cf. [10]) f (u) = 1{0≤u≤n} ·

( )(k/2)−1 ( ) u ((n−k)/2)−1 (n/2) u 1 1− . (119) n (k/2)[(n − k)/2] n n

On the other hand, the law of (x1 + . . . + xk )2 under λ∞ is χk2 with density (cf. [10]) g(u) =

1 2k/2 (k/2)

e−u/2 u (k/2)−1 .

(120)

With these notations, we have 1 1 n 1λ (φ) − λ∞ (φ)1 ≤ C

∞ 0

u θ | f (u) − g(u)|du.

(121)

94


The RHS of the inequality above is equal to ∞ 2C

u

θ

(

)+ ∞ f (u) − 1 g(u)du + C u θ (g(u) − f (u))du. g(u)

0

(122)

0

(

)+ f (u) In [10], it is proved 2 − 1 ≤ 2(k +3)/(n −k −3) as soon as k ∈ {1, . . . , n −4}. g(u) The second term of (122) can be computed explicitly and is equal to * + ((2θ + k)/2) θ n θ (n/2) 2 − . (123) (k/2) (θ + n/2) A Taylor expansion shows that this term is bounded by C /n for n large enough.

We recall here the following well known properties of the uniform measure on the sphere. Lemma 5 (Symmetry properties of the uniform measure on the sphere). Let λrk be the uniform measure on the sphere # 0 k Srk = (x1 , . . . , xk ) ∈ (Rd )k ; x2 = r 2 =1

of radius r and dimension dk − 1. i) λrk is invariant by any permutation of coordinates. ii) Conditionally to {x1 , . . . , xk }\{xi }, the law of xi has an even density w.r.t. the Lebesgue measure on Rd . In the same spirit, we have the following lemma. Lemma 6. Let µrk be the uniform measure on the surface defined by # 0 k k k d k 2 2 Mr = (x1 , . . . , xk ) ∈ (R ) ; x = r ; x = 0 . =1

=1

We have the following properties: i) µrk is invariant by any permutation of the coordinates. j

ii) If i = j ∈ {1, . . . , d} then for every h, ∈ {1, . . . , k} (distinct or not), µrk (xhi x ) = 0. iii) If h = ∈ {1, . . . , k} and i ∈ {1, . . . , d}, µrk (xhi xi ) = −

µk (x2 ) µk (x2 ) r2 =− r h =− r . dk(k − 1) k−1 k−1

(124)

Lemma 7. (Equivalence of ensembles.) Consider the (α, ν)-harmonic case. There exists a positive constant C = C(d, E) such that:


95

1 1, 1 C E 2 11 1 j i 2 i) If i = j, 1 p0 pe1 − 21 ≤ d . 1 d 1 N N ,E 1 1. 1 i i j j / 1 1≤ C . ii) If i = j, 11 ( p0 pe1 p0 pe1 ) 1 Nd N ,E iii) For any i and any y ∈ ZdN , we have 1 1 ( )2 1. 1 Clog N / E 1 j j 1 j j 2 − N (y)1 ≤ . 1 qy (q−e1 − qe1 )( p0 ) 1 1 N ,E d Nd Proof. Let us treat only the unpinned case ν = 0. The pinned case is similar. We take the Fourier transform of the positions and of the momentums (defined by (70)) and we define ˜ ) = (1 − δ(ξ ))ω(ξ )q(ξ ˆ ), p(ξ ˜ ) = N −d/2 (1 − δ(ξ ))p(ξ ˆ ), ξ ∈ ZdN , q(ξ

(125)

4 where ω(ξ ) = 2N −d/2 α dk=1 sin2 (π ξ k /N ) is the normalized dispersion relation. The factor 1 − δ in the definition above is due to the condition x px = x qx = 0 assumed in the microcanonical state. Then the energy can be written as 6 1 5 ˜ )|2 + |q(ξ ˜ )|2 |p(ξ 2 ξ =0 6 1 5 2 ˜ )) + Im 2 (p(ξ ˜ )) + Re2 (q(ξ ˜ )) + Im 2 (q(ξ ˜ ) . = Re (p(ξ 2

HN =

ξ =0

˜ Re(q) ˜ are even and Im(p), ˜ Im(q) ˜ are odd: Since px , qx are real, Re(p), ˜ ) = Re(p)(−ξ ˜ ˜ ) = Re(q)(−ξ ˜ Re(p)(ξ ), Re(q)(ξ ), ˜ ) = −Im(p)(−ξ ˜ ˜ ) = −Im(q)(−ξ ˜ Im(p)(ξ ), Im(q)(ξ ).

(126)

On ZdN \{0}, we define the relation ξ ∼ ξ if and only if ξ = −ξ . Let UdN be a class of representants for ∼ (UdN is of cardinal (N d − 1)/2). With these notations and by using (126), we have 6 5 ˜ )) + Im 2 (p(ξ ˜ )) + Re2 (q(ξ ˜ )) + Im 2 (q(ξ ˜ ) . Re2 (p(ξ HN = (127) ξ ∈UdN

It follows that in the microcanonical state, the random variables ˜ ), (Im p)(ξ ˜ ), (Req)(ξ ˜ ), Im q)(ξ ˜ ))ξ ∈Ud ((Rep)(ξ

N

√ are distributed according to the uniform measure on the sphere of radius N d E (which is not true without the restriction on the set UdN ). The classical results of equivalence of ensembles for the uniform measure on the sphere ([10]) can be applied for these random variables.

96


i) By using inverse Fourier transform and (126), we have ,

j

p0 pei 1

2 -

1 = 2d N N ,E

.

p˜ j (ξ ) p˜ j (ξ ) p˜ i (η) p˜ i (η )

ξ,ξ ,η,η =0

/

2iπ e1 · (η + η ) N e . −

N ,E

(128) It is easy to check by using (ii) of Lemma 5 that the only terms in this sum which are nonzero are only for ξ = −ξ and η = −η . One gets hence ,

2 j p0 pei 1

N ,E

, 1 11 j 112 11 i 112 = 2d . 1 p˜ (ξ )1 1 p˜ (η)1 N N ,E

(129)

ξ,η=0

Classical equivalence of ensembles estimates of [10] show that this last sum is equal to (E/d)2 + O(N −d ). ii) Similarly, one has .

j

j

( p0i pei 1 p0 pe1 )

/ N ,E

=

1 N 2d

.

p˜ i (ξ ) p˜ i (ξ ) p˜ j (η) p˜ j (η )

/

ξ,xi ,η,η =0

( ) 2iπ e1 · (ξ + η ) . × exp − N

N ,E

(130)

It is easy to check by using (ii) of Lemma 5 that the only terms in this sum which are nonzero are for ξ = −ξ and η = −η. One gets hence .

j

j

( p0i pei 1 p0 pe1 )

/ N ,E

=

, 1 11 i 112 11 j 112 p ˜ p ˜ (ξ ) (η) 1 1 1 1 N 2d N ,E ξ,η=0 ) ( 2iπ e1 · (ξ +η) . × exp N

(131)

Using classical equivalence of ensembles estimates ([10]), one obtains .

/

⎛

E2 ⎜ 1 j j ( p0i pei 1 p0 pe1 ) = 2⎝ d e N ,E d N

⎞2 2iπ e1 ·ξ ⎟ −d −d N ⎠ + O(N ) = O(N ). (132)

ξ =0

iii) By using the symmetry properties, we have . / q˜ j (ξ )q˜ j (ξ ) p˜ j (η) p˜ j (η )

N ,E

=0


97

for ξ = −ξ or η = −η . Hence one has / . j j j q x q z ( p 0 )2 1 = 3d N

N ,E

. / q˜ j (ξ )q˜ j (ξ ) p˜ j (η) p˜ j (η )

ξ,ξ ,η,η =0

N ,E

exp (−2iπ(ξ · z + ξ · y)/N ) ω(ξ )ω(ξ )

, 12 - exp (−2iπ ξ · (z − y)/N ) 1 11 j 1 i q ˜ (ξ ) p ˜ (η) 1 1 N 3d ω(ξ )2 N ,E ξ,η=0 , 12 - exp (−2iπ ξ · (z − y)/N ) 1 11 j 1 j q ˜ = 2d (ξ ) p ˜ (e ) , 1 1 1 N ω(ξ )2 N ,E =

ξ =0

Estimates of [10] give 1 ( )2 11 1. / C E 1 1 j 2 j 2 − 1 (q˜ (ξ )) ( p˜ (e1 )) 1≤ d. 1 N ,E d 1 N It follows that . / j j j j qy (q−e1 − qe1 )( p0 )2

N ,E

=

E 2 e−2iπ ξ ·(−e1 −y)/N − e−2iπ ξ ·(e1 −y)/N + RN , d N 2d ω(ξ )2 ξ =0

where |R N | ≤ C N −2d

| sin(2π ξ 1 /N )| . d 2 k k=1 sin (π ξ /N ) ξ =0 4α

To obtain iii) observe that 1 e−2iπ ξ ·(−e1 −y)/N − e−2iπ ξ ·(e1 −y)/N = N (y) N 2d ω(ξ )2 ξ =0

and ⎧ ⎪ ⎨log N /N , d = 1 N −2d ∼ 1/N , d = 2 d 2 k ⎪ ⎩1/N d , d ≥ 3. k=1 sin (π ξ /N ) ξ =0 4α | sin(2π ξ 1 /N )|

Acknowledgements. We acknowledge the support of the ANR LHMSHE n.BLAN07-2184264 and of the accord GREFI-MEFI.

98


References 1. Basile, G., Bernardin, C., Olla, S.: A momentum conserving model with anomalous thermal conductivity in low dimension. Phys. Rev. Lett. 96, 204303 (2006) 2. Basile, G., Olla, S., Spohn, H.: Energy transport in stochastically perturbed lattice dynamics. http:// arxiv.org/abs/0805.3012, 2008 3. Bernardin, C., Olla, S.: Fourier’s law for a microscopic model of heat conduction. J. Stat. Phys. 121(3/4), 271–289 (2005) 4. Bodineau, T., Helffer, B.: Correlations, Spectral gap and Log-Sobolev inequalities for unbounded spins systems. In: Differential equations and mathematical physics (Birmingham, AL, 1999), AMS/IP Stud. Adv. Math., 16, Providence, RI: Amer. Math. Soc., 2000, pp 51–66 5. Bolsterli, M., Rich, M., Visscher, W.M.: Simulation of nonharmonic interactions in a crystal by selfconsistent reservoirs. Phys. Rev. A 4, 1086–1088 (1970) 6. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: A challenge to theorists, In: Mathematical Physics 2000, Fokas, A. et al. (eds.), London, Imperial College Press, 2000, pp. 128–150 7. Bonetto, F., Lebowitz, J.L., Lukkarinen, J.: Fourier’s Law for a harmonic Crystal with Self-Consistent Stochastic Reservoirs. J. Stat. Phys. 116, 783–813 (2004) 8. Bricmont, J., Kupianen, A.: Towards a Derivation of Fouriers Law for Coupled Anharmonic Oscillators. Comm. Math. Phys. 274(3), 555–626 (2007) 9. Deuschel, J.D., Delmotte, T.: On estimating the derivative of symmetric diffusions in stationary random environment, with pplications to ∇φ interface model. Prob. Theory Relat. Fields 133, 358–390 (2005) 10. Diaconis, P., Freedman, D.: A dozen de Finetti style results in search of a theory. Ann. Inst. H. Poincaré (Probabilités Et Statistiques), 23, 397–423 (1987) 11. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24. Amsterdam: North-Holland Publishing Co., 1989 12. Jara, M., Komorowski, T., Olla, S.: Limit theorems for additive functionals of a Markov chain. http:// arxiv.org/abs/0809.0177, 2008 13. Lepri, S., Livi, R., Politi, A.: Thermal Conduction in classical low-dimensional lattices. Phys Rep. 377, 1– 80 (2003) 14. Mangad, M.: Asymptotic expansions of Fourier transforms and discrete polyharmonic Green’s functions. Pacific J. Math. 20, 85–98 (1967) 15. Lebowitz, J.L., Lieb, E., Rieder, Z.: Properties of harmonic crystal in a stationary non-equilibrium state. J. Math. Phys. 8, 1073–1078 (1967) 16. Sethuraman, S.: Central limit theorems for additive functionals of the simple exclusion process. Ann. Probab. 28(1), 277–302 (2000) 17. Spohn, H.: Large Scale Dynamics of interacting Particles. Berlin-Heidelberg-New York: Springer, 1991 Communicated by A. Kupiainen


Communications in


Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit Drago¸s Iftimie1 , Milton C. Lopes Filho2 , Helena J. Nussenzveig Lopes2 1 Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan,

Bâtiment du Doyen Jean Braconnier, 43, Blvd du 11 Novembre 1918, F–69622 Villeurbanne Cedex, France. E-mail: [email protected]

2 Departamento de Matemática, IMECC, Universidade Estadual de Campinas - UNICAMP,

Caixa Postal 6065, Campinas, SP 13083-970, Brazil. E-mail: [email protected]; [email protected] Received: 31 January 2008 / Accepted: 8 April 2008 Published online: 11 October 2008 – © Springer-Verlag 2008

Abstract: In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypotheses: first, that the initial exterior domain velocity converges strongly in L 2 to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [4–6].

1. Introduction The purpose of the present work is to study the asymptotic behavior of families of solutions of the incompressible Navier-Stokes equations, in two and three space dimensions, in the exterior of a single smooth obstacle, when both viscosity and the size of the obstacle become small. More precisely, let be a smooth and bounded domain in Rn , n = 2, 3, such that is connected and simply connected if n = 2 and R3 \ is connected and simply connected if n = 3. Let ε > 0 and set ε = Rn \ε. Let u 0 be a smooth, divergence-free vector field in Rn , which gives rise to a smooth solution u of the Euler equations, defined on an interval [0, T ]. Let u ν,ε ∈ L ∞ ((0, T ); L 2 (ε )) ∩ Cw0 ([0, T ); L 2 (ε )) ∩ L 2 ((0, T ); H01 (ε )) be a weak Leray solution of the incompressible Navier-Stokes equations, with viscosity ν, in ε , satisfying the no-slip boundary condition at ∂ε . We prove that there exists a constant C = C(u 0 , , T ) > 0 such that, if the following hypothesis holds:

100

D. Iftimie, M. C. Lopes Filho, H. J. Nussenzveig Lopes

[H] We have that sup u ν,ε (·, 0) − u 0 L 2 (ε ) → 0,

ε≤Cν

as ν → 0, then we have that supε≤Cν u ν,ε − u L ∞ ((0,T );L 2 (ε )) → 0, as ν → 0. Furthermore, if we assume that [H] occurs at a certain rate in ν we can obtain an explicit error estimate in L 2 . In addition, we prove that if we fix an initial vorticity ω0 in R2 , smooth and compactly supported in R2 \{0} and consider u ν,ε (·, 0) = K ε [ω0 ] + m H ε , where K ε denotes the Biot-Savart operator in ε , while H ε is the normalized generator of the harmonic vector fields in ε , and m = ω0 , then Hypothesis [H] is satisfied. In the case of dimension three, if we fix an initial vorticity ω0 in R3 , smooth, divergence-free and compactly supported in R3 \{0} and consider u ν,ε (·, 0) = K ε [ω0 ], where K ε again denotes the Biot-Savart operator in ε , then it is proved in [4] that Hypothesis [H] is satisfied. In both cases we have rates √ for the convergence of the initial data in such a way that u ν,ε − u L 2 (ε ) = O( ν) when ν → 0, uniformly in time. A central theme in incompressible hydrodynamics is the vanishing viscosity limit, something naturally associated with the physical phenomena of turbulence and of boundary layers. In particular, a natural question to ask is whether the limiting flow associated with the limit of vanishing viscosity satisfies the incompressible Euler equations. This is known to be true in the absence of material boundaries, see [1,18] for the two dimensional case and [9,24] for the three dimensional case. Also, if the boundary conditions are of Navier type, see [2,8,17,28], noncharacteristic, see [26] or for certain symmetric 2D flows, see [15,16,19], convergence to an Euler solution remains valid. The most relevant case from the physical point of view corresponds to no slip boundary conditions. In this case, we have results on criteria for convergence to solutions of the Euler system, see [10,11,25,27], and also results when the data for the boundary layer equations are analytic, see [13,21–23], but the general problem remains wide open. To be a bit more precise, let us assume that u 0 is a solution of the Euler equations in the exterior domain and that u ν is a solution of the Navier-Stokes equations with viscosity ν, with no-slip boundary condition in . Suppose further that u ν and u 0 have the same initial velocity v0 and that both u 0 and the family {u ν } are smooth, defined on a fixed time interval [0, T ]. It is easy to see that δ E(ν, t) ≡ u 0 (·, t) − u ν (·, t) L 2 () is uniformly bounded in ν and t ∈ [0, T ] (by 2v0 L 2 if v0 has finite energy), but it is not known whether δ E → 0 when ν → 0. In fact, given the experimentally and numerically observed behavior of high Reynolds number flows in the presence of boundaries, it is reasonable to conjecture that δ E does not, in general, vanish as ν → 0; see, for instance, Sect. 15.6 of [20] for illustrations of high Reynolds number flow past a cylinder. Of course, this leaves open the possibility that u ν might approach another solution of the Euler equations, different from u 0 . This article contains an answer to the following question: can we make δ E small, by making both the viscosity and the obstacle small? This problem was one of the main motivations underlying the authors’ research on incompressible flow around small obstacles. Our previous results include the small obstacle limit for the 2D inviscid equations, see [5,14] and for the viscous equations, see [4,6]. The work we present here is a natural outgrowth of this research effort.

Vanishing Viscosity Limit Around a Small Obstacle

101

The remainder of this article is divided into three sections. In Sect. 2, we state and prove our main result, namely the convergence in the small viscosity and small obstacle limit, assuming convergence of the initial data. In Sect. 3 we study the problem of convergence of the initial data: for two space dimensions, we adapt techniques developed in our previous work, while in the three dimensional case we refer the reader to recent work by D. Iftimie and J. Kelliher, see [4]. In Sect. 4, we interpret the smallness condition on the obstacle as the condition that the local Reynolds number stays below a certain (small) constant. In addition, still in Sect. 4, we obtain an enstrophy estimate for the wake generated by the small obstacle and we list some open problems. 2. Main Theorem We use the notation from the Introduction to state and prove our main result. We consider the initial-value problem for the Navier-Stokes equations in ε , with no-slip boundary condition, given by: ⎧ ν,ε ∂ u + u ν,ε · ∇u ν,ε = −∇ p ν,ε + νu ν,ε , in ε × (0, ∞), ⎪ ⎨ t ν,ε div u = 0, in ε × [0, ∞), (1) ν,ε (x, t) = 0, u for x ∈ ∂ε , t > 0, ⎪ ⎩ ν,ε ε u (t = 0) = ϑ (x), for x ∈ ε , t = 0. We assume that the initial velocity ϑ ε ∈ L 2loc (ε ) is divergence-free and tangent to ∂ε , but we do not assume that it satisfies the no-slip boundary condition. In three dimensions we assume further that ϑ ε ∈ L 2 (ε ). Under these hypotheses it was shown by H. Kozono and M. Yamazaki, see [12], that, in two dimensions, there is a unique global strong solution to (1) with initial velocity ϑ ε , while, in three dimensions, there is a global Leray weak solution u ν,ε of (1), see [3]. More precisely, in three dimensions there exists u ν,ε ∈ L ∞ ([0, ∞); L 2 (ε )) ∩ Cw0 ([0, ∞); L 2 (ε )) ∩ L 2loc ([0, ∞); H01 (ε )) such that u ν,ε is a distributional solution of (1) and the following energy inequality holds true: t u ν,ε (t) L 2 (ε ) + 2ν ∇u ν,ε (s) L 2 (ε ) ds ≤ u ε0 L 2 (ε ) ∀t ≥ 0. (2) 0

ϑε

u ν,ε (·, t)

Both and are defined only in ε , but we will consider them as defined on the whole space by extending them to be identically zero inside ε. Let u 0 be a smooth, divergence-free vector field defined in all Rn , and let u be the corresponding smooth solution of the Euler equations; in two dimensions u is globally defined while in three dimensions it is defined, at least, on an interval [0, T ]. We are now ready to state our main result. Theorem 1. Assume that ϑ ε − u 0 L 2 (Rn ) → 0 as ε → 0. Fix T > 0, arbitrary if n = 2, and smaller than the time of existence of the smooth Euler solution if n = 3. Then there exists a constant C1 = C1 (, u 0 , T ) such that, if ε ≤ C1 ν, then u ν,ε (·, t) − u(·, t) L 2 (Rn ) → 0 as ν → 0.

102


√ Moreover, if we assume that ϑ ε − u 0 L 2 (Rn ) = O( ν), then there exists also C2 = C2 (T, u 0 , ) such that √ u ν,ε (·, t) − u(·, t) L 2 (Rn ) ≤ C2 ν, for all 0 < ε < C1 ν and all 0 ≤ t ≤ T . Before we proceed with the proof, we require two technical lemmas. To state the first lemma we must introduce some notation. As in the statement of the theorem, u denotes the smooth Euler solution in Rn . In dimension two, we denote by ψ = ψ(x, t) the stream function for the velocity field u, chosen so that ψ(0, t) = 0. In dimension three, ψ denotes the unique divergence free vector field vanishing for x = 0 and whose curl is u. In other words, we set (x − y)⊥ · u(y, t) y ⊥ · u(y, t) ψ(x, t) = dy + dy 2π |x − y|2 2π |y|2 R2 R2 in dimension two so that u = ∇ ⊥ ψ and x−y y × u(y, t)dy − × u(y, t)dy ψ(x, t) = − 3 3 R3 4π |x − y| R3 4π |y| in dimension three so that u = curl ψ. In both two and three dimensions one has that ψ and ∇ψ are uniformly bounded on the time interval [0, T ]. Let R > 0 be such that the ball of radius R, centered at the origin, contains . Let ϕ = ϕ(r ) be a smooth function on R+ such that ϕ(r ) ≡ 0 if 0 ≤ r ≤ R + 1, ϕ ≥ 0 and ϕ(r ) ≡ 1 if r ≥ R + 2. Set ϕ ε = ϕ ε (x) = ϕ(|x|/ε) and u ε = ∇ ⊥ (ϕ ε ψ) in dimension two and u ε = curl(ϕ ε ψ) in dimension three. In both dimensions two and three, the vector field u ε is divergence free and vanishes in a neighborhood of the boundary. We also re-define the pressure p = p(x, t) from the Euler equation in Rn with data u 0 so that p(0, t) = 0. Lemma 2. Fix T > 0. There exist constants K i > 0, i = 1, . . . , 5 such that, for any 0 < ε < ε0 and any 0 ≤ t < T we have: (1) (2) (3) (4) (5)

∇u ε 2L 2 ≤ K 1 , u ε L ∞ ≤ K 2 , u ε − u L 2 + u ε − ϕ ε u L 2 ≤ K 3 ε, ∇ψ∇ϕ ε L ∞ + ψ∇ 2 ϕ ε L ∞ ≤ K 4 /ε, p∇ϕ ε L 2 + ∂t ψ∇ϕ ε L 2 ≤ K 5 ε.

Above, we used the notation ∇ψ∇ϕ ε L ∞ = i, j ∂i ψ∂ j ϕ ε L ∞ in dimension two and ∇ψ∇ϕ ε L ∞ = i, j,k ∂i ψk ∂ j ϕ ε L ∞ in dimension three. Similar notations were used for the other terms.


103

Proof. Some of the inequalities above can be improved in dimension three. However, it turns out that these improvements have no effect on the statement of Theorem 1. Therefore, to avoid giving separate proofs in dimension three we choose to state these weaker estimates. Recall that both u and ∇u are uniformly bounded. First we write ∂i u ε = ∂i ∇ ⊥ (ϕ ε ψ) = u∂i ϕ ε + ∂i ψ∇ ⊥ ϕ ε + ψ∂i ∇ ⊥ ϕ ε + ϕ ε ∂i u in dimension two and ∂i u ε = ∂i curl(ϕ ε ψ) = u∂i ϕ ε + ∇ϕ ε × ∂i ψ + ∂i ∇ϕ ε × ψ + ϕ ε ∂i u in dimension three. The supports of the first three terms of the right-hand sides of the relations above are contained in the annulus ε(R + 1) < |x| < ε(R + 2), whose Lebesgue measure is O(ε2 ). Furthermore, |∇ϕ ε | = O(1/ε), |∇ 2 ϕ ε | = O(1/ε2 ) and |ψ(x, t)| = O(ε) for |x| < ε(R + 2), since ψ(0, t) = 0. Taking L 2 norms in the expressions above gives the first estimate. Next we observe that u ε = ϕ ε u + ψ∇ ⊥ ϕ ε or u ε = ϕ ε u + ∇ϕ ε × ψ. Clearly ϕ ε u is bounded and to bound the second term, we use again that ψ(0, t) = 0, which proves the second estimate. For the third estimate, observe that u ε − u and u ε − ϕ ε u are bounded, as we have just proved, and have support in the ball |x| < ε(R + 2). For the fourth estimate, we use again that ψ(0, t) = 0. The last estimate follows from two facts: that the functions whose L 2 -norm we are estimating have support on the ball |x| < ε(R + 2) and that they are both bounded, since p(0, t) = 0 and ψt (0, t) = 0. We also require a modified Poincaré inequality, stated below. Lemma 3. Let be the obstacle under consideration and let R be such that ⊂ B R . Consider the scaled obstacles ε and the exterior domains ε . Then, if W ∈ H01 (ε ) we have W L 2 (ε ∩B(R+2)ε ) ≤ K 6 ε ∇W L 2 (ε ∩B(R+2)ε ) . Proof. The case ε = 1 is standard. The remainder of the proof requires a scaling argument. Let W ∈ H01 (ε ) and set Y = Y (x) = W (εx). Then Y ∈ H01 (1 ). Using the case ε = 1 there exists a constant K 6 such that Y L 2 (1 ∩B R+2 ) ≤ K 6 ∇Y L 2 (1 ∩B R+2 ) . Undoing the scaling we find: Y 2L 2 ( ∩B ) 1 R+2 ∇Y 2L 2 (

1 ∩B R+2 )

= =

|W (εx)| d x = 2

1 ∩B R+2 1 ∩B R+2

W 2L 2 (

ε ∩B(R+2)ε )

εn

;

ε2 |∇W (εx)|2 d x = ε2−n ∇W 2L 2 (

ε ∩B(R+2)ε )

.

The desired result follows immediately. We are now ready to prove Theorem 1. Proof of Theorem 1. We begin by noting that, since u is a smooth solution of the Euler equations in Rn × [0, T ], it follows that √ u ν,ε (·, t) − u(·, t) L 2 (ε) ≡ u(·, t) L 2 (ε) ≤ Cε ≤ C ν, (3) if ε < Cν. Hence it remains only to estimate the L 2 -norm of the difference in ε , which we do below.

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We first give the proof in two dimensions and then we indicate how the proof should be adapted to three dimensions.

2.1. Case n = 2. The vector field u ε is divergence free and satisfies the equation u εt = −ϕ ε u · ∇u − ϕ ε ∇ p + ∂t ψ∇ ⊥ ϕ ε . We set W ν,ε ≡ u ν,ε − u ε . The vector field W ν,ε is divergence free, vanishes on the boundary and satisfies: ∂t W ν,ε − νW ν,ε = − u ν,ε · ∇u ν,ε − ∇ p ν,ε + νu ε + ϕ ε u · ∇u + ϕ ε ∇ p − ∂t ψ∇ ⊥ ϕ ε . We perform an energy estimate, multiplying this equation by W ν,ε and integrating over ε . We obtain 1 d W ν,ε 2L 2 + ν∇W ν,ε 2L 2 2 dt = −ν ∇W ν,ε · ∇u ε d x − W ν,ε · [(u ν,ε · ∇)u ν,ε ] d x ε ε ν,ε ε ν,ε ε + W · [(ϕ u · ∇)u] d x + W · ϕ ∇ p dx − W ν,ε · ∂t ψ∇ ⊥ ϕ ε d x. ε

ε

ε

(4) We will examine each one of the five terms on the right-hand-side of identity (4). We look at the first term. We use Cauchy-Schwarz and Young’s inequalities followed by Lemma 2, item (1), to obtain ν

ε

∇W

ν,ε

ν ∇W ν,ε 2L 2 + K 1 . · ∇u d x ≤ 2 ε

(5)

Next we look at the second and third terms together. We write |I| ≡ − W ν,ε · [(u ν,ε · ∇)u ν,ε ] d x + W ν,ε · [(ϕ ε u · ∇)u] d x ε ε ν,ε ν,ε ε ν,ε ε ν,ε ε = − W · [((W + u ) · ∇)(W + u )] d x + W · [(ϕ u · ∇)u] d x ε ε = − W ν,ε · [(W ν,ε · ∇)u ε ] d x − W ν,ε · [(u ε · ∇)u ε ] d x ε ε ν,ε ε + W · [(ϕ u · ∇)u] d x , ε


105

where we used the fact that W ν,ε · [((W ν,ε + u ε ) · ∇)W ν,ε ] = 0. Finally, we add and subtract W ν,ε · [(u ε · ∇)u] to obtain −W ν,ε · (W ν,ε · ∇u ε ) + W ν,ε · [u ε · ∇(u − u ε )] |I| = ε ν,ε ε ε + W · [((ϕ u − u ) · ∇)u] d x ν,ε ν,ε ε ν,ε ε ε ≤ W · (W · ∇u ) d x + W · [u · ∇(u − u )] d x ε ε + W ν,ε · [((ϕ ε u − u ε ) · ∇)u] d x ε = W ν,ε · (W ν,ε · ∇u ε ) d x + (u − u ε ) · [(u ε · ∇)W ν,ε ] d x ε ε ν,ε ε ε + W · [((ϕ u − u ) · ∇)u] d x ε ν,ε ε ν,ε ≤ (W · ∇u ) · W d x + u − u ε L 2 u ε L ∞ ∇W ν,ε L 2 ε

+W ν,ε L 2 ϕ ε u − u ε L 2 ∇u L ∞ . For each i = 1, 2 we have that ∂i u ε = ∂i ψ∇ ⊥ ϕ ε + ψ∂i ∇ ⊥ ϕ ε + ∂i ϕ ε u + ϕ ε ∂i u.

(6)

Therefore, |I| ≤ (∇ψ∇ϕ ε L ∞ + ψ∇ 2 ϕ ε L ∞ )W ν,ε 2L 2 (A ) + ϕ ε ∇u L ∞ W ν,ε 2L 2 ε

+u − u ε L 2 u ε L ∞ ∇W ν,ε L 2 + W ν,ε L 2 ϕ ε u − u ε L 2 ∇u L ∞ ,

(7)

where Aε is the set ε ∩ B(R+2)ε , which contains the support of ∇ϕ ε . Hence, using Lemma 2, items (2), (3) and (4), together with Lemma 3, in the inequality (7), we find |I| ≤

K4 2 2 ε K 6 ∇W ν,ε 2L 2 + ϕ ε ∇u L ∞ W ν,ε 2L 2 ε +K 2 K 3 ε∇W ν,ε 2L 2 + K 3 ε∇u L ∞ W ν,ε L 2 .

(8)

Next we look at the fourth and fifth terms in (4). Recall that we chose the pressure p in such a way that p(0, t) = 0. We find |J | ≡ W ν,ε · ϕ ε ∇ p d x − W ν,ε · ∂t ψ∇ ⊥ ϕ ε d x ε ε ≤ W ν,ε · p∇ϕ ε d x + W ν,ε · ∂t ψ∇ ⊥ ϕ ε d x . ε

ε

We estimate each term above to obtain, using Lemma 2 item (5), |J | ≤ ( p∇ϕ ε L 2 + ∂t ψ∇ ⊥ ϕ ε L 2 )W ν,ε L 2 ≤ K 5 εW ν,ε L 2 .

(9)

106


We use estimates (5), (8) and (9) in the energy identity (4) to deduce that 1 d ν ν W ν,ε 2L 2 + ν∇W ν,ε 2L 2 ≤ ∇W ν,ε 2L 2 + K 1 + K 2 K 3 ε∇W ν,ε L 2 2 dt 2 2 2 ν,ε 2 ε ν,ε 2 ∞ +K 4 K 6 ε∇W L 2 + ϕ ∇u L W L 2 + ε(K 5 + K 3 ∇u L ∞ )W ν,ε L 2 ≤

ε2 ν ν ν ∇W ν,ε 2L 2 + K 1 + ∇W ν,ε 2L 2 + K 22 K 32 + K 4 K 62 ε∇W ν,ε 2L 2 2 2 4 ν ν,ε 2 2 2

W K ε L2 + 5 . +K 0 W ν,ε 2L 2 + 2 2

(10)

5 = K 5 + K 3 ∇u L ∞ . Above we have used the notation K 0 = supε ϕ ε ∇u L ∞ and K At this point we choose ε so that ν 0 < ε < min ε0 , . (11) 8K 4 K 62 With this choice, letting y = y(t) = W ν,ε 2L 2 , we obtain dy ε2 2 2 ≤ ν K 1 + 2K 22 K 32 + K 5 ε + (2K 0 + 1)y ≤ C 1 ν + C 2 y. dt ν

(12)

From Gronwall’s inequality it follows that

u ν,ε − u ε 2L 2 ( ) ≤ C(T, u 0 , ) ν + ϑ ε − u ε0 2L 2 ( ) . ε

ε

(13)

If ϑ ε − u 0 L 2 → 0 then it follows from (13) together with Lemma 2, item (3), and from (3), that u ν,ε − u L 2 → 0, as desired, where the constant √ C1 can be chosen to be (8K 4 K 62 )−1 . If we assume further that ϑ ε − u 0 L 2 = O( ν), then the second part of the statement of Theorem 1 easily follows. This concludes the proof in the two dimensional case. 2.2. Case n = 3. The proof in dimension three is similar to the previous one. There are two differences: notation and the justification that we can multiply the equation of W ν,ε by W ν,ε . First, about notation. One has to replace everywhere the term ∂t ψ∇ ⊥ ϕ ε by ∇ϕ ε ×∂t ψ and also relation (6) becomes ∂i u ε = ∇ϕ ε × ∂i ψ + ∂i ∇ϕ ε × ψ + ∂i ϕ ε u + ϕ ε ∂i u. These two modifications are just changes of notations. These new terms are of the same type as the old ones, so the estimates that follow are not affected. Second, we multiplied the equation of W ν,ε by W ν,ε . The solution u ν,ε , and therefore ν,ε W too, is not better than L ∞ (0, T ; L 2 (ε ))∩ L 2 (0, T ; H 1 (ε )). But it is well-known that some of the trilinear terms that appear when multiplying the equation of W ν,ε by W ν,ε are not well defined in dimension three with this regularity only. In other words, one cannot multiply directly the equation of W ν,ε by W ν,ε . Nevertheless, there is a classical trick that allows us to perform this multiplication if the weak solution u ν,ε verifies the energy inequality. What we are trying to do, is to subtract the equation of u ε from the


107

equation of u ν,ε and to multiply the result by u ν,ε − u ε . This is the same as multiplying the equation of u ν,ε by u ν,ε , adding the equation of u ε times u ε and subtracting the equation of u ν,ε times u ε and the equation of u ε times u ν,ε . Since u ε is smooth, all these operations are legitimate except for the multiplication of the equation of u ν,ε by u ν,ε . Formally, multiplying the equation of u ν,ε by u ν,ε and integrating in space and time from 0 to t yields the energy equality, i.e. relation (2), where the sign ≤ is replaced by =. Since we assumed that the energy inequality holds true, the above operations are justified provided that the relation we get at the end is an inequality instead of an equality. But an inequality is, of course, sufficient for our purpose. Finally, to be completely rigorous, one has to integrate in time from the beginning. That is, we would obtain at the end relation (12) integrated in time. Clearly, the result of the application of the Gronwall lemma in (12) is the same as in (12) integrated in time. This completes the proof in dimension three. Remark. The proof above is closely related to the proof of Kato’s criterion for the vanishing viscosity limit in bounded domains, see [10]. Both results are based on estimating the difference between Navier-Stokes solutions and Euler solutions by means of energy methods. In Kato’s argument, the difference is estimated in terms of the Navier-Stokes solution, on which Kato’s criterion was imposed. In contrast, our proof estimates the difference in terms of the full-space Euler solution, which is smooth in the context of interest. 3. Compactly Supported Initial Vorticity Now that we are in possession of Theorem 1 we will examine two asymptotic problems for which we can prove the convergence condition on the initial velocity. We focus on flows with compactly supported vorticity, and the diameter of the support of vorticity becomes the order one length scale, relative to which the obstacle is small. Let us begin with the three dimensional case. We consider an initial vorticity ω0 which is assumed to be smooth, compactly supported in R3 \{0}, and divergence-free. Let ε > 0 be sufficiently small so that the support of ω0 is contained in ε . The domain ε is assumed simply connected so that there exists a unique divergence-free vector field, tangent to ∂ε , in L 2 (ε ), whose curl is ω0 , see, for example, [4]. We take ϑ ε to be this unique vector field. We take u 0 to be the unique divergence-free vector field in L 2 (R3 ) whose curl is ω0 , given by the full space Biot-Savart law. In [4], D. Iftimie and J. Kelliher studied the small obstacle asymptotics for viscous flow in ε , for fixed viscosity, in three dimensions. They proved that the small obstacle limit converges to the appropriate Leray solution of the Navier-Stokes equations in the full space. One important ingredient in their proof was precisely to verify strong convergence of the initial data; in our notation, Iftimie and Kelliher proved that 3

ϑ ε − u 0 L 2 (R3 ) = O(ε 2 ), as ε → 0. We may hence apply Theorem 1 to obtain the following corollary. Corollary 4. Let ω0 ∈ Cc∞ (R3 \{0}) and consider u 0 and ϑ ε defined as above. Fix T > 0 and assume that the solution u = u(x, t) of the incompressible Euler equations in R3 , with initial velocity u 0 , exists up to time T . Let u ν,ε be a Leray solution of

108


(1) with initial velocity ϑ ε . Then there exist constants C1 = C1 (, ω0 , T ) > 0 and C2 = C2 (, ω0 , T ) > 0 such that √ u ν,ε (·, t) − u(·, t) L 2 (R3 ) ≤ C2 ν, for all 0 < ε < C1 ν and all 0 ≤ t ≤ T . Next we discuss at length the case n = 2. In dimension two the exterior domain is no longer simply connected. This means that the vorticity formulation of the Euler equations is incomplete, and we must specify the harmonic part of the initial velocity as well as the initial vorticity, see [5] for a thorough discussion of this issue. To specify the asymptotic problem we wish to consider, we must choose the initial data for (1). Let ω0 be smooth and compactly supported in R2 \{0}. Let K denote the operator associated with the Biot-Savart law in the full plane and set H = x ⊥ /(2π |x|2 ), to be its kernel. Let K ε be the operator associated with the Biot-Savart in ε , i.e., K ε = ∇ ⊥ −1 ε , where ε is the Dirichlet Laplacian in ε . Let H ε be the generator of the harmonic vector fields in ε , normalized so that its circulation around ∂ε is one. The divergencefree vector fields in ε with curl equal to ω0 are of the form K ε [ω0 ] + α H ε , with α ∈ R, see [5]. In [6] the authors studied the asymptotic behavior, as ε → 0, of solutions of (1) with ν fixed and initial velocity K ε [ω0 ] + α H ε . It was shown in [6] that u ν,ε converges to a solution of the Navier-Stokes equations in the full plane with initial data K [ω0 ] + (α − m)H , where m = ω0 , as long as |α − m| is sufficiently small. For the vanishing viscosity limit, we must consider only the case α = m. There are two reasons for this. First, K ε [ω0 ] + α H ε converges weakly to K [ω0 ] + (α − m)H in distributions, see Lemma 10 in [6], but, as we shall see, this convergence is not strong in L 2 (see Remark 1 following the proof of the next lemma). Second, one cannot expect solutions of the Euler equations in the full plane with initial velocity K [ω0 ] + (α − m)H to be smooth (even existence is not clear) unless α = m. In view of this discussion, set u 0 = K [ω0 ] and ϑ ε = K ε [ω0 ] + m H ε . With this notation, we can prove strong convergence in L 2 of the initial data, as follows. Lemma 5. Fix ε0 such that the support of ω0 does not intersect ε for any ε < ε0 . There exists a constant C > 0, depending on and ω0 such that ϑ ε − u 0 L 2 (R2 ) ≤ Cε. Proof. We begin the proof with a construction whose details can be found in [5]. In Sect. 2 of [5], an explicit formula for both K ε and H ε can be found in terms of a conformal map T , which takes into the exterior of the unit disk centered at zero. The construction of T and its behavior near infinity are contained in Lemma 2.1 of [5]. Using identities (3.5) and (3.6) in [5], we have that the vector field H ε can be written explictly as H ε = H ε (x) =

1 (T (x/ε))⊥ DT t (x/ε) , 2π ε |T (x/ε)|2

and the operator K ε can be written as an integral operator with kernel Kε , given by Kε =

1 (T (x/ε) − T (y/ε))⊥ (T (x/ε) − (T (y/ε))∗ )⊥ , DT t (x/ε) − 2π ε |T (x/ε) − T (y/ε)|2 |T (x/ε) − (T (y/ε))∗ |2


109

where x ∗ = x/|x|2 denotes the inversion with respect to the unit circle. Furthermore, we recall Theorem 4.1 of [5], from which we obtain ϑ ε L ∞ (ε ) ≤ Cω0 L ∞ ω0 L 1 , 1/2

1/2

for some constant C > 0. To understand the behavior for ε small in the expressions above, we need to understand the behavior of T (x) for large x. We use Lemma 1 in [7], which is a more detailed version of Lemma 2.1 in [5], to find that there exists a constant β > 0 such that T (x/ε) = βxε−1 + h(x/ε),

(14)

with h = h(x) a bounded, holomorphic function on 1 satisfying |Dh(x)| ≤ C/|x|2 . Therefore, |DT (x/ε) − βI| ≤ C

ε2 . |x|2

(15)

We will need a further estimate on the bounded holomorphic function h = h(z) = T (z) − βz, namely that |h(z 1 ) − h(z 2 )| ≤ C

|z 1 − z 2 | , |z 1 ||z 2 |

(16)

for some constant C > 0 independent of z 1 , z 2 . This estimate holds since, by construction (see Lemma 2.1 in [5]), we have that h(z) = g(1/z) with g a holomorphic function on (1 )∗ , whose derivatives are bounded in the closure of (1 )∗ . Here, (1 )∗ denotes the image of 1 through the mapping x → x ∗ = x/|x|2 to which we add {0}. Here, we are using the following fact: If D is a bounded domain with ∂ D a C 1 Jordan curve then any bounded function f ∈ C 1 (D) with bounded derivatives is globally Lipschitz in D. This fact is a nice exercise in basic analysis, which we leave to the reader. Therefore we have 1 |z 1 − z 2 | 1 1 1 ≤C − =C . (17) |h(z 1 ) − h(z 2 )| = g −g z z z z |z ||z | 1

2

1

2

1

2

In order to estimate ϑ ε −u

0 L 2 (ε ) we use the fact that the support of ω0 is contained in ε for ε sufficiently small to write 1 (T (x/ε) − T (y/ε))⊥ (x − y)⊥ DT t (x/ε) ω0 (y) dy − 2π [ϑ ε (x) − u 0 (x)] = |T (x/ε) − T (y/ε)|2 |x − y|2 ε ε 1 (T (x/ε))⊥ t DT (x/ε) + |T (x/ε)|2 ε ε (T (x/ε) − (T (y/ε))∗ )⊥ ω0 (y) dy ≡ Aε + Bε . − |T (x/ε) − (T (y/ε))∗ |2

Let us begin by estimating Bε . We make the change of variables η = εT (y/ε), whose Jacobian is J = | det(DT −1 )(η/ε)|, a bounded function. Additionally, we set z = εT (x/ε). With this we find: ⊥ z (z − ε2 η∗ )⊥ ε t B = DT (x/ε) ω0 (εT −1 (η/ε)) J dη. − 2 |z − ε2 η∗ |2 {|η|>ε} |z|

110


We observe now that there exists ρ independently of ε such that the support of ω0 (εT −1 (η/ε)) is contained in the set {|η| > ρ}. Moreover, one can bound |z − ε2 η∗ | ≥ |z| − ε2 |η∗ | ≥ |z| − ε2 /ρ ≥ |z|/2 provided that ε2 ≤ ρ/2. Therefore we can write ε2 |η∗ | ε2 |ω0 (εT −1 (η/ε))| J dη ≤ C 2 , |Bε | ≤ C 2 ∗ |z| {|η|>ρ} |z||z − ε η | where C depends on the support of ω0 , on the L 1 -norm of ω0 and on the domain through the bounds on the conformal map T and its derivatives. Finally, we use this estimate in the integral of the square of Bε : 1 |Bε |2 d x ≤ Cε4 dz ≤ Cε2 , 4 ε {|z|>ε} |z| as desired. Next we treat Aε . First we re-write Aε in a more convenient form: 1 β (T (x/ε) − T (y/ε))⊥ (x − y)⊥ ε t ω0 (y) dy DT (x/ε) A = − ε |T (x/ε) − T (y/ε)|2 |x − y|2 ε β (x − y)⊥ 1 DT t (x/ε) − I ω0 (y) dy + |x − y|2 ε β ≡ Aε 1 + Aε 2 . By (15), the term Aε 2 can be easily estimated: 1 ε2 ε2 |ω |Aε 2 | ≤ (y)| dy ≤ C , 0 |x|2 ε |x − y| |x|2 so this reduces to an estimate similar to the one we found for Bε . Next we examine Aε 1 . We use the expression for T given in (14) to write (x − y + ( ε )[h( x ) − h( y )])⊥ DT t ( xε ) (x − y)⊥ β ε ε ε A 1= − ω0 (y) dy. β |x − y|2 |x − y + ( βε )[h( xε ) − h( εy )]|2 ε With this we have: |Aε 1 | ≤ C

| βε [h( xε ) − h( εy )]|

ε

|x − y + ( βε )[h( xε ) − h( εy )]||x − y|

|ω0 (y)| dy.

We will make use several times of the estimate we obtained for h given in (17). First |

y ε2 |x − y| ε x

−h . h ≤C β ε ε |β||x||y|

(18)

Using (18) gives |Aε 1 | ≤ C

ε

ε2

|x − y

+ ( βε )[h( xε ) −

h( εy )]||β||x||y|

|ω0 (y)| dy.

Let R, r > 0 be such that the support of ω0 is contained in the disk of radius R and outside the disk of radius r . We will estimate Aε 1 in two regions: |x| ≥ 2R and |x| < 2R.


111

Also, recall that the estimate of the L 2 -norm of Aε 1 is to be performed in ε so we may assume throughout that |x| ≥ Cε. Suppose first that |x| ≥ 2R. Then we find: |Aε 1 | ≤ C

ε2 . |x|2

Above we used that r < |y| ≤ |x|/2 and hence |x − y + ( βε )[h( xε ) − h( εy )]| ≥ C|x| if ε is sufficiently small, since h is bounded. Finally, in the region Cε ≤ |x| < 2R we use (16) and the fact that |y| is of order 1 to bound

y x |x − y| ε h −h ≥ ||x − y| − ε2 (|x − y|/|x||y|)| ≥ |x − y + β ε ε 2 for ε small enough. Therefore ε2 |A 1 | ≤ C |x| ε

Clearly this last portion has

ε

L 2 -norm

ε2 |ω0 (y)| dy ≤ C ≤ Cε. |x − y| |x|

in the region |x| < 2R bounded by Cε.

Remark 1. Let ω0 ∈ Cc∞ (R2 \{0}) does not converge strongly in L 2

and α ∈ R, α = m. We observe that K ε [ω0 ] + α H ε to K [ω0 ] + (α − m)H . We argue by contradiction, assuming this convergence holds. In view of the lemma above, K ε [ω0 ] + m H ε converges strongly to K [ω0 ] so we must have (α − m)H ε → (α − m)H.

This does not hold, as it can be easily seen in the case of the exterior of the disk. In this case, H ε = H outside the disk of radius ε, but H ε vanishes for |x| < ε. Since H L 2 ({|x| ε}. Then the conformal map T is the identity, so Aε ≡ 0 and all that is needed is the easier estimate for Bε . Remark 3. The constant α − m is precisely the circulation of ϑ ε around the boundary of ε . The condition α − m = 0 is physically reasonable, in particular because viscous flows vanish at the boundary, and therefore, so does their circulation. This is the condition for the small obstacle limit of ideal flow to satisfy Euler equations in the full plane, see [5] and also for the small obstacle limit of viscous flows to satisfy the full plane NavierStokes equations for all viscosities, see [6]. The argument in [6] required sufficiently small α − m to obtain the appropriate limit when ε → 0, and the smallness condition was actually α − m = O(ν) as ν → 0. We conclude this section with the formal statement of a corollary which encompasses Theorem 1 and Lemma 5. Corollary 6. Let ω0 ∈ Cc∞ (R2 \{0}) and consider u 0 and ϑ ε defined as in Lemma 5. Let u = u(x, t) be the global smooth solution of the incompressible Euler equations in R2 , with initial velocity u 0 . Let u ν,ε be the solution of (1) with initial velocity ϑ ε . Fix T > 0. There exist constants C1 = C1 (, ω0 , T ) > 0 and C2 = C2 (, ω0 , T ) > 0 such that √ u ν,ε (·, t) − u(·, t) L 2 (R2 ) ≤ C2 ν, for all 0 < ε < C1 ν and all 0 ≤ t ≤ T .

112


4. Physical Interpretation and Conclusions The behavior of incompressible viscous flow past a bluff body, such as a long cylinder or a sphere, is a classical problem in fluid dynamics, to the extent of having a conference series devoted to it, see http://www.mae.cornell.edu/bbviv5/. Let us consider the simplest situation, two-dimensional flow of a viscous fluid with kinematic viscosity µ, filling the whole plane minus a disk of diameter L, with constant driving velocity U at infinity. The disturbance caused by the disk, known as its wake, depends only on the Reynolds number associated with the flow, given by Re ≡

LU . µ

The observed behavior of the wake begins, for small Re, as a steady solution of the Navier-Stokes equations, but the wake undergoes a series of bifurcations as Re grows, progressively developing steady recirculation zones (4 < Re < 40, periodic recirculation and a Von Karman street (40 < Re < 200), nonperiodic vortex shedding (200 < Re < 400), leading to turbulence (Re > 400). See [20], Sect. 15.6, for details and illustrations. In our problem, which involves nearly inviscid flow past a small bluff body, the qualitative behavior of the wake of the small obstacle is determined by the local Reynolds number, which encodes the way in which an observer at the scale of the obstacle experiences the flow. Basically, by making our obstacle small, we are making the flow more viscous at its scale. Roughly speaking, a change of variables x = εy leaves the obsε tacle fixed while the Reynolds number behaves like . Thus, the assumption ε ≤ C1 ν ν corresponds to the bounded Reynolds number. Let us make this argument more precise. We assume that the Navier-Stokes system under consideration, (1), is dimensional, i.e. has time and space measured in seconds and meters, and mass normalized so that fluid density is one. In these units, the kinematic viscosity for air is µ = 14.5 × 10−6 m 2 /sec, and for water it is µ = 1.138 × 10−6 m 2 /sec, both at 15o C. Let us restrict our discussion to the two-dimensional case. The smallness condition in Theorem 1, (11), reads ε < C1 µ, and the dimensional constant C1 , requires closer scrutiny. Actually, the constant C1 is given by: C1 =

1 , 8K 4 K 62

where K 4 appears in Lemma 2, item (4), and K 6 is from Lemma 3. K 6 is a nondimensional constant that depends on the shape of the obstacle . The constant K 4 can be chosen as K4 =

sup

x∈Rn , t∈[0,T ]

ε(|∇ψ∇ϕ ε |(x, t) + |ψ∇ 2 ϕ ε |(x, t)).

The function ψ above is the stream function of the full-plane Euler flow, adjusted so that ψ(0, t) = 0. Also, ε∇ϕ ε is O(1), localized near the obstacle and ε∇ 2 ϕ ε is O(1/ε), also localized near the obstacle. Therefore, both terms included in K 4 are associated with first derivatives of the stream function at the obstacle, i.e. with the local velocity

4 supt |u(0, t)| (we can also assume that the u(0, t). Therefore, we can write K 4 = K limiting Euler flow is stationary, to avoid the time dependence).


113

From the point of view of the obstacle, the inviscid velocity u(0, t) acts as a constant (in space) forcing velocity imposed at infinity, and therefore, the qualitative behavior of the wake of the obstacle is determined by the local Reynolds number Reloc ≡ u(0, t)ε/µ. Clearly, condition (11) can be rewritten as Reloc
0 there exists a constant C > 0, independent of ν such that

T

ν,ε (t) dt ≤ C.

0

Proof. We go back to relations (10) and (12) and include the viscosity term which had been ignored. We find: dy ν + ∇W ν,ε 2L 2 ≤ C1 ν + C2 y. dt 4 We next integrate in time to obtain W ν,ε (·, T )2L 2 − W ν,ε (·, 0)2L 2 + ≤ C1 T ν + C2

T 0

ν 4

0

T

∇W ν,ε (·, t)2L 2 dt

W ν,ε (·, t)2L 2 dt.

Now we use Theorem 1 and ignore a term with good sign to obtain ν 4

T 0

∇W ν,ε (·, t)2L 2 dt ≤ C T ν + W ν,ε (·, 0)2L 2 ≤ C T ν,

114


where we used Lemma 5 together with item (3) from Lemma 2 to estimate the initial data term. From this we conclude that T ∇W ν,ε (·, t)2L 2 dt ≤ C. 0

Finally, we observe that ν,ε ≤ C∇W ν,ε 2L 2 + C∇u ε 2L 2 ≤ C∇W ν,ε 2L 2 + C K 1 , by item (1) in Lemma 2. This concludes the proof. Finally, let us consider some open questions naturally associated with the research presented here. First, one would like to weaken, and ultimately remove, the smallness condition on the size of the obstacle; this is the most physically interesting follow-up problem. Second, one would also like to consider two dimensional flows with nonzero initial circulation at the obstacle, in order to study the interaction of the vanishing viscosity and vanishing obstacle limits in more detail. This would improve the connection of the present work with the authors’ previous results in [5,6]. An easier version of this second problem would be to consider an initial circulation of the form γ = γ (ν) and find out how fast γ has to vanish as ν → 0 in order to retain our result. A third problem is to describe more precisely the asymptotic structure of the difference between the full-space Euler flow and the approximating small viscosity, small obstacle flows. Acknowledgements. This research is supported in part by FAPESP Grant # 2007/51490-7. The second author was supported in part by CNPq Grant #303301/2007-4. The third author was supported in part by CNPq Grant #302214/2004-6 and FAPESP Grant #2006/04861-7. This work was done during the Special Semester in Fluid Mechanics at the Centre Interfacultaire Bernoulli, EPFL; the authors wish to express their gratitude for the hospitality received. The authors would like to thank Peter Constantin, Jim Kelliher and Franck Sueur for many helpful comments.

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Communications in


Remodeling the B-Model Vincent Bouchard1 , Albrecht Klemm2 , Marcos Mariño3 , Sara Pasquetti4 1 Jefferson Physical Laboratory, Harvard University, 17 Oxford St., Cambridge, MA 02138, USA.


2 Physikalisches Institut der Universität Bonn, Nußallee 12, D-53115 Bonn, Germany.


3 Department of Physics, CERN, 1211 Geneva 23, Switzerland.

E-mail: [email protected]; [email protected]

4 Institut de Physique, Université de Neuchâtel, Rue A. L. Breguet 1, CH-2000 Neuchâtel,

Switzerland. E-mail: [email protected] Received: 31 January 2008 / Accepted: 25 April 2008 Published online: 10 September 2008 – © Springer-Verlag 2008

Abstract: We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi–Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A p fibrations, where the amplitudes compute the ’t Hooft expansion of vevs of Wilson loops in Chern-Simons theory on lens spaces. We also use our formalism to predict the disk amplitude for the orbifold C3 /Z3 . Contents 1.

2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toric Calabi-Yau Threefolds with Branes . . . . . . . . . . . . . . . . . . . 2.1 Mirror symmetry and topological strings on toric Calabi-Yau threefolds 2.1.1 A-model geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 B-model mirror geometry. . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Open string mirror symmetry. . . . . . . . . . . . . . . . . . . . . 2.1.4 Topological open string amplitudes. . . . . . . . . . . . . . . . . . 2.2 Moduli spaces, periods and flat coordinates . . . . . . . . . . . . . . . 2.2.1 Moving in the moduli space. . . . . . . . . . . . . . . . . . . . . . 2.2.2 Open string phase structure. . . . . . . . . . . . . . . . . . . . . . 2.3 The open and closed mirror maps . . . . . . . . . . . . . . . . . . . . .

119 119 120 120 120 121 121 122 123 124 125 125 127 129

118

V. Bouchard, A. Klemm, M. Mariño, S. Pasquetti

2.3.1 Closed flat coordinates. . . . . . . . . . . . . . . . . 2.3.2 Open flat coordinates. . . . . . . . . . . . . . . . . 2.3.3 Picard-Fuchs equations. . . . . . . . . . . . . . . . 2.3.4 Open phase transitions. . . . . . . . . . . . . . . . . 2.3.5 Small radius regions. . . . . . . . . . . . . . . . . . 2.3.6 The O(−3) → P2 geometry. . . . . . . . . . . . . . 3. A New B-Model Formalism . . . . . . . . . . . . . . . . . . 3.1 The formalism of Eynard and Orantin for matrix models 3.1.1 Ingredients. . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Recursion. . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Symplectic transformations. . . . . . . . . . . . . . 3.1.4 Interpretation. . . . . . . . . . . . . . . . . . . . . . 3.2 Our formalism . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Ingredients. . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Recursion. . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Symplectic transformations. . . . . . . . . . . . . . 3.2.4 Interpretation. . . . . . . . . . . . . . . . . . . . . . 3.3 Computations . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Disk amplitude. . . . . . . . . . . . . . . . . . . . . 3.3.2 Annulus amplitude. . . . . . . . . . . . . . . . . . . 3.3.3 Genus 0, three-hole amplitude. . . . . . . . . . . . . 3.3.4 Genus 1, one-hole amplitude. . . . . . . . . . . . . 3.3.5 Higher amplitudes. . . . . . . . . . . . . . . . . . . 3.4 Moving in the moduli space . . . . . . . . . . . . . . . . 4. Genus 0 Examples . . . . . . . . . . . . . . . . . . . . . . . 4.1 The vertex . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Framing. . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Disk amplitude. . . . . . . . . . . . . . . . . . . . . 4.1.3 Annulus amplitude. . . . . . . . . . . . . . . . . . . 4.1.4 Three-hole amplitude. . . . . . . . . . . . . . . . . 4.1.5 The genus one, one hole amplitude. . . . . . . . . . 4.1.6 Framed vertex in two legs. . . . . . . . . . . . . . . 4.2 The resolved conifold . . . . . . . . . . . . . . . . . . . 5. Genus 1 Examples . . . . . . . . . . . . . . . . . . . . . . . 5.1 Local P2 . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Framed outer amplitudes. . . . . . . . . . . . . . . . 5.1.2 Framed inner amplitudes. . . . . . . . . . . . . . . . 5.2 Local Fn , n = 0, 1, 2 . . . . . . . . . . . . . . . . . . . 6. Orbifold Points . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Chern–Simons theory and knots in lens spaces . . . . . . 6.2 The orbifold point and a large N duality . . . . . . . . . 6.3 Orbifold amplitudes . . . . . . . . . . . . . . . . . . . . 6.3.1 Orbifold flat coordinates. . . . . . . . . . . . . . . . 6.3.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The C3 /Z3 orbifold . . . . . . . . . . . . . . . . . . . . 7. Conclusion and Future Directions . . . . . . . . . . . . . . . A. Useful Conventions . . . . . . . . . . . . . . . . . . . . . .

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130 131 132 133 134 134 137 138 138 139 140 140 140 141 141 141 142 143 143 143 144 146 146 148 150 150 150 151 151 152 153 154 156 156 156 157 159 160 162 163 168 169 170 172 173 175 176

Remodeling the B-Model

119

1. Introduction 1.1. Motivation. Topological string theory is an important subsector of string theory with various physical and mathematical applications, which has been extensively investigated since it was first formulated. This has led to many different ways of computing topological string amplitudes, based very often on string dualities. Topological strings come in two types, the A-model and the B-model, which are related by mirror symmetry. The A-model provides a physical formulation of Gromov–Witten theory, while the B-model is deeply related to the theory of deformation of complex structures. Both models have an open sector whose boundary conditions are set by topological D-branes. The main advantage of the B-model is that its results are exact in the complex moduli (i.e. they include all α corrections), which makes it possible to study various aspects of stringy geometry not easily accessible in the A-model. One can in particular obtain results for the amplitudes far from the large radius limit, around non-geometric phases such as orbifold or conifold points. Sphere and disk amplitudes are given by holomophic integrals in the B-model geometry. In particular, sphere amplitudes are determined by period integrals over cycles; those were first calculated for the quintic Calabi–Yau threefold in [16]. For non-compact Calabi-Yau threefolds, the mirror geometry basically reduces to a Riemann surface, and the disk amplitudes are given by chain integrals directly related to the Abel-Jacobi map [5,6]. Note that disk amplitudes ending on the real quintic inside the quintic threefold have also been calculated [43], using a generalization of the Abel-Jacobi map [26]. (g) In contrast, B-model amplitudes Ah at genus g and with h holes, on worldsheets with χ < 1, have an anomalous, non-holomorphic dependence on the complex moduli which is captured by the holomorphic anomaly equations. These were first formulated in the closed sector in [11], and have been recently extended to the open sector in various circumstances [13,23,44]. The holomorphic anomaly equations can be solved to determine the amplitudes, up to an a priori unknown holomorphic section over the moduli space — the so called holomorphic ambiguity — which puts severe restrictions on their effectiveness. Modular invariance of the amplitudes completely governs the non-holomorphic terms in the amplitudes [1,23,31,32,48] and reduces the problem of fixing the holomorphic ambiguity to a finite set of data for a given g and h. Recently, boundary conditions have been found in the closed sector [31] (the so-called conifold gap condition and regularity at the orbifold point) which fully fix these data in many local geometries (like the Seiberg–Witten geometry or local P2 ) [27,31]. In the compact case they allow to calculate closed string amplitudes to high, but finite genus (for example, g = 51 for the quintic) [32]. For open string amplitudes the situation is worse: appropriate boundary conditions are not known, and the constraints coming from modularity are much weaker. In fact, it is not known how to supplement the holomorphic anomaly equations with sufficient conditions in order to fix the open string amplitudes.1 In view of this, it is very important to have an approach to the B-model that goes beyond the framework of the holomorphic anomaly equations. In the local case (toric or not), such an approach was proposed in [2], which interpreted the string field theory of the B-model (the Kodaira–Spencer theory) in terms of a chiral boson living on a “quantum” Riemann surface. However, the computational framework of [2] is only effective in very simple geometries, and in practice it is not easy to apply it even to backgrounds like local P2 . 1 This applies also to the case in which it has been argued that there is no open string moduli [43].

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In [39] it was argued that all closed and open topological string amplitudes on local geometries (including the mirrors of toric backgrounds) could be computed by adapting the recursive method of Eynard and Orantin [22] to the Calabi–Yau case. This method was obtained originally as a solution to the loop equations of matrix models, giving an explicit form for its open and closed amplitudes in terms of residue calculus on the spectral curve of the matrix model. The recursion solution obtained in this way can then be defined formally for any algebraic curve embedded in C2 . In [39] it was argued that this more general construction attached to an arbitrary Riemann surface computes the amplitudes of the chiral boson theory described in [2], and in particular that the formalism of [22] should give the solution to the B-model for mirrors of toric geometries, providing in this way an effective computational approach to the Kodaira–Spencer theory in the local case. Various nontrivial examples were tested in [39] successfully. However, many important aspects of the B-model, like the phase structure of D-branes, as well as the framing phenomenon discovered in [5], were not incorporated in the formalism of [39]. 1.2. Summary of the results. In this paper we build on [39] and develop a complete theory of the B-model for local Calabi-Yau geometries in the presence of toric D-branes. Our formalism is based on a modification of [22] appropriate for the toric case, and it leads to a framework where one can compute recursively and unequivocally all the open and closed B-model amplitudes in closed form, non-perturbatively in the complex moduli, albeit perturbatively in the string coupling constant. In particular, our formalism incorporates in a natural way the more subtle aspects of D-branes (like framing) which were not available in [39]. Moreover, the proposed formalism is valid at any point in the moduli space. Basically, once one knows the disk and the annulus amplitudes at a given point, one can generate recursively all the other open and closed amplitudes unambiguously. Thus this formalism goes beyond known approaches in open topological string theory, such as the topological vertex, as it allows to study closed and open amplitudes not only in the large radius phase but also in non-geometric phases such as conifolds and orbifolds phases. 1.3. Outline. In Sect. 2 we review relevant features of mirror symmetry as well as open and closed topological string theory. We put a special emphasis on phase transitions in the open/closed moduli spaces, and on the determination of the corresponding open and closed mirror maps. Section 3 is the core of the paper, where we propose our formalism. We also explain how it can be used to compute amplitudes explicitly at various points in the moduli space. In Sects. 4 and 5 we get our hands dirty and do various checks of our formalism, for local geometries. In Sect. 6 we study open and closed amplitudes in a non-geometric phase corresponding to the blow-down of local P1 × P1 . We check our results against expectation values of the framed unknot in Chern-Simons theory on lens spaces. We also propose a prediction for the disk amplitude of C3 /Z3 , which corresponds to the orbifold phase in the moduli space of local P2 . Finally, we summarize and propose various avenues of research in Sect. 7. 2. Toric Calabi-Yau Threefolds with Branes In this section we introduce basic concepts of mirror symmetry and topological string theory for non-compact toric Calabi-Yau threefolds with Harvey-Lawson type special


121

Lagrangians. In particular, we discuss the target space geometry of the A- and the B-model as well as their moduli spaces. We also examine period integrals on the Bmodel side, which give the flat coordinates as well as the closed genus zero and disk amplitudes. 2.1. Mirror symmetry and topological strings on toric Calabi-Yau threefolds. 2.1.1. A-model geometry. We consider the A-model topological string on a (non-compact) toric Calabi-Yau threefold, which can be described as a symplectic quotient M = Ck+3 //G, where G = U (1)k [19]. Alternatively, M may be viewed physically as the vacuum field configuration for the complex scalars X i , i = 1, . . . , k + 3 of chiral superfields in a 2d gauged linear, (2, 2) supersymmetric σ -model, transforming as α X i → ei Q i α X i , Q iα ∈ Z, α = 1, . . . , k under the gauge group U (1)k [47]. Without superpotential, M is determined by the D-term constraints Dα =

k+3

Q iα |X i |2 = r α , α = 1, . . . , k

(2.1)

i=1

modulo the action of G = U (1)k . The r α are the Kähler parameters and r α ∈ R+ defines a region in the Kähler cone. For this to be true Q iα have to fullfill additional constraints and for M to be smooth, field configurations for which the dimensionality of the gauge orbits drop have to be excluded. The Calabi-Yau condition c1 (T X ) = 0 holds if and only if the chiral U (1) anomaly is cancelled, that is [47] k+3

Q iα = 0,

α = 1, . . . , k.

(2.2)

i=1

Note from (2.1) that negative Q i lead to non-compact directions in M, so that all toric Calabi-Yau manifolds are necessarily non-compact. View the C’s with coordinates X k = |X k | exp(iθk ) as S 1 -fibrations over R+ . Then M can be naturally viewed as a T 3 -fibration over a non-compact convex and linearly bounded subspace B in R3 specified by (2.1), where the T 3 is parameterized by the three directions in the θ -space. The condition (2.2) allows an even simpler picture, capturing the geometry of M as a R+ × T 2 fibration over R3 . In this picture, the toric threefold M is constructed by gluing together C3 patches. In each patch, with coordinates (z 1 , z 2 , z 3 ), we can define, instead of rαi = |z i |2 — which would lead to the above picture — the three following hamiltonians: rα = |z 1 |2 − |z 2 |2 ,

rβ = |z 3 |2 − |z 1 |2 ,

rR = Im(z 1 z 2 z 3 ).

(2.3)

R3

and generate flows δrl xk = {rl , xk }ω , whose orbits The rl parameterize the base define the fiber. It is easy to see that rα , rβ generate S 1 ’s and rR , which is only well defined due to (2.2), generates R+ . The toric graph M describes the degeneration locus of the S 1 fibers. In B, |X i | ≥ 0, therefore B is bounded by |X i | = 0. The latter equations define two-planes in R3 whose normal vectors obey 3+k i=1

Q α ni = 0.

(2.4)

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Clearly, the S 1 parameterized by θi vanishes at |X i | = 0; and over the line segments L i j = {|X i | = 0} ∩ {|X j | = 0},

(2.5)

two S 1 ’s shrink to zero. If L i j is a closed line segment in ∂ B the open S 1 bundle over it make it a P1 ∈ M, while if L i j is half open in ∂ B it represents a non-compact line bundle direction C. So far we have defined the planes |X i | = 0 only up to parallel translation. Their relative location is determined by the Kähler parameters, simply by the condition that the length of the closed line segments L i j is the area of the corresponding P1 . Condition (2.2) and the T 2 fibration described above makes it possible to project all L i j into R2 without losing information about the geometry of M. This is how one constructs the two-dimensional toric graph M associated to M. 2.1.2. B-model mirror geometry. The mirror geometry W to the above non-compact toric Calabi-Yau threefold M was constructed by [30], extending [10,33]. Let w + , w − ∈ C. We further define homogeneous coordinates xi =: e yi ∈ C∗ , i = 1, . . . , k + 3 with the property |xi | = exp(−|X i |2 ); they are identified under the C∗ -scaling xi ∼ λxi , i = 1, . . . , k + 3, λ ∈ C∗ . The mirror geometry W is then given by w+ w− =

k+3

xi ,

(2.6)

i=1

subject to the exponentiated D-terms contraints, which become k+3

Q iα

xi

α

= e−t = q α , α = 1, . . . , k.

(2.7)

i=1

Note that these relations are compatible with the λ-scaling because of the Calabi-Yau condition. The parameters t α = r α + iθ α are the complexifications of the Kähler parameters r α , using the θ α -angles of the U (1)k group. After taking the λ-scaling and (2.7) into account the right-hand side of the defining equation (2.6) can be parameterized by two variables x = exp(u), y = exp(v) ∈ C∗ . In these coordinates the mirror geometry W becomes w + w − = H (x, y; t α ),

(2.8)

which is a conic bundle over C∗ ×C∗ , where the conic fiber degenerates to two lines over the (family of) Riemann surfaces : {H (x, y; t α ) = 0} ⊂ C∗ × C∗ .2 The holomorphic volume form on W is given by

=

dwd xd y . wx y

(2.9)

As an algebraic curve embedded in C∗ × C∗ , the Riemann surface has punctures, hence is non-compact. The fact that it is embedded in C∗ × C∗ rather than C2 like the usual specialization of a compact Riemann surface embedded in projective space to an affine coordinate patch will be crucial for us. Note that the Riemann surface is most 2 Note that for brevity in the following we will always talk about the Riemann surface ; it will always be understood that is in fact a family of Riemann surfaces parameterized by the Kähler parameters t α .


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easily visualized by fattening the toric diagram M associated to the mirror manifold M; the genus g of corresponds to the number of closed meshes in M , and the number of punctures n is given by the number of semi-infinite lines in M . It is standard to call the Riemann surface embedded in C∗ × C∗ the mirror curve. It is important to note that the reparameterization group G of the mirror curve is 01 , (2.10) G = S L(2, Z) × 10 which is the group of 2 × 2 integer matrices with determinant ±1. This is the group that preserves the symplectic form d x dy (2.11) ∧ x y on C∗ × C∗ . The action of G is given by

(x, y) → (x y , x y ), a b

c d

ab cd

∈ G .

(2.12)

2.1.3. Open string mirror symmetry. In this work we are interested in closed and open topological string amplitudes, hence we must consider branes, which are described in the A-model by special Lagrangian submanifolds. The Lagrangian submanifolds that we will be interested in were constructed by [6], as a generalization of Harvey-Lawson special Lagrangians [29] in C3 . Consider a toric Calabi-Yau threefold M constructed as a symplectic quotient as above, and denote by 1 d|X k |2 ∧ dθk 2 3

ω=

(2.13)

k=1

the canonical symplectic form. The idea is to determine a non-compact subspace L ⊂ M of three real dimensions by specifying a linear subspace V in the base k+3

qiα |X i |2 = cα , α = 1, . . . , r

(2.14)

i=1

and restricting the θk so that ω| L = 0. One shows that L becomes special Lagrangian with respect to = dz 1 ∧ dz 2 ∧ dz 3 in each patch if and only if k+3

qiα = 0,

α = 1, . . . , r

(2.15)

i=1

The relevant case for us is r = 2, i.e. V = R+ and L is an S 1 × S 1 -bundle over it. In a given patch, the restriction ω| L = 0 means that 3 i=1

θi (z i ) = 0 mod π.

(2.16)

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For one value of the θ -sum the Lagrangian is generically not smooth at the origin of R+ . It can be made smooth by “doubling,” which is done by allowing for instance 3 3 i=1 θi (z i ) = 0 and i=1 θi (z i ) = π [29]. If V passes through the locus in the base where one S 1 shrinks to zero, L splits into L ± , each of which have topology C × S 1 , where C is a fibration of the vanishing S 1 over R+ . L + (or L − ) is the relevant special Lagrangian. To make the notation simpler we denote L + by L henceforth. It has b1 (L) = 1 and its complex open modulus is given by the size of the S 1 and the Wilson line of the U (1) gauge field around it. Pictorially, it can be described as “ending on a leg of the toric diagram M of M,” since the half open line l defining L must end on a line L i j in M . We refer the reader to the figures in Sects. 4 and 5 for examples of toric diagrams with branes. Under mirror symmetry, the brane L introduced above maps to a one complex dimensional holomorphic submanifold of W , given by H (x, y) = 0 = w− .

(2.17)

That is, it is parameterized by w+ , and its moduli space corresponds to the mirror curve ⊂ C∗ × C∗ (w + = 0 corresponds to the equivalent brane L − ). 2.1.4. Topological open string amplitudes. Let us now spend a few words on topological open string theory to clarify the objects that we will consider in this paper. In the A-model, topological open string amplitudes can be defined by counting (in an appropriate way) the number of holomorphic maps from a Riemann surface g,h of genus g with h holes, to the Calabi–Yau target, satisfying the condition that the boundaries map to the brane L. Assuming for simplicity that b1 (L) = 1, the topological class of these maps is labeled by genus g, the bulk class β ∈ H2 (X, L) and the winding numbers wi , i = 1, · · · , h, specifying how many times the i th boundary wraps around the one-cycle in L. We can thus form the generating functionals Fg,w (Q) = N g,w,β e−β·t , (2.18) β∈H2 (X,L)

where N g,w,β are open Gromov–Witten invariants counting the maps in the topological class labeled by g, w = (w1 , · · · , wh ), and β. It is also convenient to group together the different boundary sectors with fixed g, h into a single generating func(g) tional Ah (z 1 , · · · , z h ) defined as (g)

Ah (z 1 , · · · , z h ) =

Fg,w (Q)z 1w1 · · · z hwh .

(2.19)

wi ∈Z

Here, the variables z i are not only formal variables. From the point of view of the underlying physical theory, they are open string parameters which parameterize the moduli space of the brane. In the B-model, as discussed earlier the moduli space of the brane is given by the mirror curve itself. The open string parameter z hence corresponds to a variable on the mirror curve (take for example the variable x). That is, fixing what we mean by open string parameter corresponds to fixing a parameterization of the embedding of the Riemann surface in C∗ × C∗ ; in other words, it corresponds to fixing a projection map → C∗ (the projection onto the x-axis in our case). Different parameterizations


125

will lead to different amplitudes. Once the open string parameter x is fixed, the disk amplitude is simply given by dx (0) A1 = log y , (2.20) x as will be explained in more details in the following sections. To fully understand open topological strings we need to include the notion of framing of the branes. The possibility of framing was first discovered in the context of A-model open string amplitudes in [5]. It is an integer choice f ∈ Z associated to a brane, which has to be made in order to define the open amplitudes. Framing has various interpretations. In the A-model, it corresponds to an integral choice of the circle action with respect to which the localization calculation is performed [34]. It can also be understood from the point of view of large N duality. A key idea in the large N approach is to relate open (and closed) string amplitudes to knot or link invariants in the Chern-Simons theory on a special Lagrangian cycle. As is well known the calculation of the Chern-Simons correlation functions requires a choice of the normal bundle of each knot. The framing freedom lies in a twist of this bundle, again specified by an integer f ∈ Z. We also want to understand framing from the B-model point of view. Recall that the moduli space of the brane is given by the mirror curve . As explained above, fixing the location on the brane on the A-model corresponds to fixing a parameterization of . It turns out that there is a one-parameter subgroup of the reparameterization group G of which leaves the location of the brane invariant; these transformations, which depend on an integer f ∈ Z, correspond precisely to the B-model description of framing [5]. More precisely, these transformations, which we will call framing transformations, are given by (x, y) → (x y f , y),

f ∈ Z.

(2.21)

As a result, fixing the location and the framing of the brane on the A-model side corresponds to fixing the parameterization of the mirror curve on the B-model side. 2.2. Moduli spaces, periods and flat coordinates. In this section we discuss the global picture of the open/closed moduli space of the A- and the B-model. We introduce the periods, which give us the open and closed flat coordinates, as well as the disk amplitude and the closed genus zero amplitude. 2.2.1. Moving in the moduli space. Mirror symmetry identifies the stringy Kähler moduli space of M with the complex structure moduli space of W , which are the A- and B-model closed string moduli spaces. Recall that generically, the stringy Kähler moduli space of M contains various phases corresponding to topologically distinct manifolds. Hence moving in the A-model closed string moduli space implies various topologically-changing phase transitions corresponding to flops and blowups of the target space. In fact, since we are interested in open topological strings, we want to consider the open/closed string moduli space, which also includes the moduli space associated to the brane. The B-model provides a natural setting for studying transitions in the open/closed string moduli space. Usually in mirror symmetry, we identify the A- and B-model moduli spaces locally by providing a mirror map, for example near large radius and for outer

126


branes. However, in the following we will propose a B-model formalism to compute open/closed amplitudes which can be applied anywhere in the open/closed string moduli space. Hence, to unleash the analytic power of this new B-model description one wishes to extend the identification between the moduli space to cover all regions of the open/closed string moduli space. In the B-model one simply wants to cover a suitable compactification of the open/ closed string moduli space with patches in which we can define convergent expansions of the topological string amplitudes in local flat coordinates. The latter are given by a choice of A-periods integrals, while the dual B-periods can be thought of as conjugated momenta. The closed string flat coordinates are given by integrals over closed cycles, while the open string flat coordinates are integrals over chains. Let us first discuss the closed string flat coordinates. If the genus of the B-model mirror curve is greater than one, one has non-trivial monodromy of the closed string periods. By the theory of the solution to differential equations with regular singular loci (normal crossing divisors), which applies in particular to periods integrals, the closed string moduli space can be covered by hyper-cylinders around the divisors with monodromy. The local holomorphic expansion of the amplitudes has to be invariant under the local monodromy around the corresponding divisor. In particular, this requires different choices of flat coordinates, or A-periods, in different regions in moduli space. These different choices of periods are related by symplectic Sp(2g, C) transformations, i.e. by changes of polarization. Invariance of the physical topological string amplitudes under the full monodromy group requires a non-holomorphic extension of the amplitudes and forces the closed string parameters to appear in terms of modular forms. In contrast there is no monodromy action on the open string flat coordinates. As a consequence, the amplitudes are in general rational functions of the open string parameters, and no non-holomorphic extension is needed to make the results modular. That is, there is no holomorphic anomaly equation involving the complex conjugate of an open string modulus. The situation for the open string moduli is hence similar to the closed string moduli for genus 0 mirror curves (for example the mirror of the resolved conifold M = O(−1) ⊕ O(−1) → P1 ), where there is no non-trivial monodromy. In such cases the holomorphic anomaly equations for the closed string moduli can be trivialized and the amplitudes are rational functions of the moduli. Let us now discuss the main features of the phase transitions3 between patches in the open/closed string moduli space in order of their complexity. In the easiest case adjacent patches are related by transitions merely in the open string moduli space. In the A-model these are referred to as open string phase transitions and correspond to moving the base of the special lagrangian submanifold over a vertex in the toric diagram, for example from an outer to an inner brane, see below. In the B-model they correspond simply to reparameterizations of the mirror curve by an element in G . More precisely, this type of phase transition is described by the reparameterization 1 y (x, y) → , (2.22) x x of the mirror curve — we will explain this in the next section. Since the amplitudes are rational functions in the open string moduli, there is no non-trival analytic continuation required and the amplitudes can be readily transformed. 3 Note that the term “phase transitions” is inspired from the classical A-model. In the B-model, the correlation functions are smoothly differentiable except at complex dimension one loci, so there are strictly speaking no phase boundaries.


127

The next type of transitions consists in closed string transitions between different large radius regions. In the A-model on non-compact toric Calabi-Yau threefolds those are all related to flops of P1 (in our examples they occcur only in the Hirzebruch surface F1 ). In these cases, the new flat closed string coordinates are given linearly in terms of the old ones and in particular the symplectic transformation in Sp(2g, Z) is trivial, in the sense that it does not exchange the A- and B-periods. The closed string parameters can be fixed in each large radius patch by the methods of [5], which are reviewed in the next section. The rather mild changes in the amplitudes can be described by wall crossing formulas. The more demanding transitions are the ones between patches which require a nontrival Sp(2g, C) transformation of the periods. The typical example is the expansion near a conifold point. Here a B-cycle — in the choice of periods at large radius — becomes small and will serve as a flat coordinate near the conifold point, while a cycle corresponding to a flat coordinate at infinity acquires a logarithmic term and will serve as dual momenta. In the A-model picture we enter a non-geometric phase, in which the α -expansion of the σ -model breaks down. In the B-model we are faced with the problem of analytic continuation and change of polarization when we transform the amplitudes, which involve modular transformations. Another interesting patch is the one of an orbifold divisor, i.e. one with a finite monodromy around it. This is likewise a region where the original geometric description breaks down due to a vanishing volume. However here we have a singular geometric description by a geometric orbifold. For example, for the O(−3) → P2 geometry, in the limit where the P2 shrinks to zero size we get simply the C3 /Z3 orbifold. Enumerative A-model techniques (orbifold Gromov-Witten invariants) have been developed to calculate closed string invariants on orbifolds, and in these phases we can still compare the closed B-model results with Gromov-Witten calculations on the A-model side. The behavior of the closed string amplitudes under this type of transition has been studied in [1]. In this paper, we will start investigating open amplitudes on orbifolds, which do not have, as far as we know, a Gromov-Witten interpretation. In particular, we will calculate the disk amplitude for C3 /Z3 in Sect. 6.4. Let us now describe in more detail the first type of phase transitions, involving only open string moduli. 2.2.2. Open string phase structure. Here we introduce classical open string coordinates and discuss the phases of the open string moduli, which arise when we “move” the Lagrangian submanifold over a vertex in the toric diagram. First, note that the open string variables generically get corrected by closed string instanton effects, when the latter are present and have finite volume; we will study this in the next section. However, the open string phase structure can already be understood directly in the large volume limit where the instanton corrections are suppressed. Hence, we will not bother for now with the instanton corrections; our analysis carries over readily to the instanton corrected variables. Recall from Sect. 2.1.1 that closed line segments in the toric diagram M correspond to compact curves, while half-open lines correspond to non-compact curves. Now, as explained in Sect. 2.1.3, the half open line l defining the Lagrangian submanifold L must end on a line L i j in M . Phase transitions in the open string moduli space then occur between Lagrangian submanifolds ending on half-open lines and Lagrangian submanifolds ending on closed line segments. One refers to the former as outer branes and to the latter as inner branes.

128


Fig. 1. Open string phase structure

Only maps which are equivariant with respect to the torus action contribute to the open string amplitudes. This means that disks must end on a vertex at one end of the line L i j intersecting l. Let this vertex be the locus where |X i | = |X j | = |X k | = 0. Branes ending on the three lines L i j , L ik and L jk meeting up at this vertex correspond to three different phases I , I I and I I I in the open string moduli space. The geometry of the open string phase structure is shown in Fig. 1. In phase I we can describe l by the equations |X j |2 − |X i |2 = 0, |X k |2 − |X i |2 = cr ,

r > cr > 0,

where r is the Kähler parameter of the P1 related to L i j and H, cr =

(2.23)

(2.24)

S1

where dH = ω parameterizes the size of the disk D, hence the radius of the S 1 = ∂ D. Recall that on the B-model side, the choice of location (or phase) of the brane corresponds to a choice of parameterization of the mirror curve defined by H (x, y) = 0. Generically, we can find the good parameterization of the curve as follows. We first use the fact from mirror symmetry (see Sect. 2.1.2) that by definition, |xi | = exp −|X i |2 , (2.25) to rewrite the Eqs. (2.23) fixing the location of the brane in terms of the C∗ -variables xi . We then use the C∗ -rescaling to fix one of them to 1, and we choose y to be the C∗ -variable which goes to 1 on the brane, and x to be the variable parameterizing the location of the brane on the edge (i.e |x| = ecr ). x becomes the open string parameter introduced earlier. Note that there is an ambiguity in this choice of parameterization; since y = 1 on the brane, we can reparameterize the variable x → x y f for any integer f ∈ Z without changing the discussion above. But since we change the meaning of the open string parameter x, we in fact change the physical setup and the open amplitudes.


129

This ambiguity precisely corresponds to the framing of the brane, and the transformation x → x y f is the framing transformation introduced in (2.21). For example, in phase I, the good choice of parameterization corresponds to first scaling xi = 1, and then identifying y := x j = x j /xi and x := xk = xk /xi . Indeed, the first equation in (2.23) says that y = 1 on the brane, while the second equation identifies x = exp cr + i A , (2.26) S1

where we complexified the disk size cr by the Wilson line. x hence agrees in the large Kähler parameter limit with the open string parameter, which appears in the superpotential. In fact, the superpotential — or disk amplitude — is given by the Abel-Jacobi map on H (x, y) = 0, as a curve embedded in C∗ × C∗ , with respect to the restriction of the holomorphic volume form to the mirror curve: x dx (0) (2.27) A1 (x) = log y , x x∗ (0)

i.e. x∂x A1 = log y(x), with y(x) a suitable branch of the solution of H (x, y) = 0. This gives the formula for the disk amplitudes presented earlier in (2.20). Note that we could also parametrize l in this phase by |X i |2 − |X j |2 = 0, |X k |2 − |X j |2 = cr ,

r > cr > 0,

(2.28)

which leads to parameters x = x y −1 and y = y −1 . In phase II the brane lˆ can be descibed by the equation |X j |2 − |X k |2 = 0, |X i |2 − |X k |2 = crˆ ,

crˆ > 0.

(2.29)

We fix the parameterization of the mirror curve by xk = 1, yˆ := x j = x j /xk and xˆ := xi = xi /xk , so that the open string parameter is xˆ and the superpotential is (2.27) with hatted variables. The relation to the previous parameters in phase I is xˆ = x −1 and yˆ = yx −1 ; this is the origin of the phase transformation proposed in (2.22). Again, we can also parametrize lˆ by |X k |2 − |X j |2 = 0, |X i |2 − |X j |2 = crˆ ,

crˆ > 0,

(2.30)

and get yˆ = x y −1 and xˆ = y −1 . Similarly, in phase III we can parameterize l˜ in two different ways, and introduce variables y˜ = x −1 and x˜ = x −1 y, or y˜ = x and x˜ = y. In this phase, r˜ is the Kähler parameter of the P1 related to L ik . Note however that different L nm can describe P1 ’s in the same Kähler class. Standard toric techniques allow to read the equivalences from the charge vectors Q α . 2.3. The open and closed mirror maps. We discussed in the previous section the phase structure of the open/closed moduli space. Here we discuss in detail how to find the flat coordinates (or open and closed mirror maps) in various phases in the moduli space. As an example we consider O(−3) → P2 , which is the simplest non-compact toric Calabi-Yau with non-trivial monodromy on the closed string moduli.

130


2.3.1. Closed flat coordinates. The closed string mirror map is given by finding flat coordinates T α , α = 1, . . . , k on the complex structure moduli space, which are mapped to the complexified Kähler parameters. The flat coordinates are generically defined by Tα = where the X α are the A-periods α

Xα , X0

(2.31)

X =

(2.32)

Aα α (A , Bα )

is a symplectic basis of three-cycles. of the holomorphic volume form , and Special geometry guarantees the existence of a holomorphic function F(X α ) of degree 2 — the so-called prepotential — such that the B-periods are ∂F Fα = =

. (2.33) ∂ Xα Bα

Fixing the flat coordinates involves a choice of basis (Aα , Bα ); it is well known that the choice of A-periods (and the B-periods, i.e. a polarization) is uniquely fixed at the point(s) of maximal unipotent monodromy q = 0, which are mirror dual to the large radius points in the stringy Kähler moduli space. This fixes the closed mirror map at these large radius points. α To be more precise, in the paramerization of the complex moduli q α = e−t determined by the Mori cone — spanned by the charge vectors Q α — these periods are singled out by their leading behaviour: X 0 = 1 + O(q),

X α (q) = log(q α ) + O(q).

(2.34)

In the non-compact cases there is a further simplification. First, = 1, and 1 T α = Xα = λ. (2.35) 2πi Aα The period F0 is absent and the dual periods are given by ∂F 1 Fα = = λ, (2.36) ∂T α 2πi B α where (Aα , Bα ) is now a canonical basis of one-cycles on the mirror curve , and λ is the meromorphic one form dx (2.37) λ = log y x on , which is the local limit of . In the A-model picture, the flat coordinate T α is the mass associated with a D2 brane wrapping the curve Cα ∈ H2 (M, Z). At a large radius point, it is given by the complexified volume tα = ω + i B. (2.38) X0

Cα

However, it is well known that it receives closed string worldsheet instanton corrections if the size of Cα is of the order of the string scale; the corrected volume α

T α = t α + O(e−t ), is the flat coordinate, which reduces in the local case to (2.35).

(2.39)


131

2.3.2. Open flat coordinates. The open string modulus is given by x = eu , which is a variable on the mirror curve defined by the equation H (x, y) = 0. In this section we will sometimes use the variables u and v instead of the C∗ -variable x and y, which are defined by the exponentiation x = eu , y = ev . Hopefully no confusion should occur. It was argued in [5] that in the A-model, the open string modulus u measures the tension W = W (x3 = −∞) − W (x3 = ∞)

(2.40)

of a domain-wall made from a D4-brane wrapping the disk of classical size u and extending at a point on the x3 -axis, say x3 = 0, over the subspace M2,1 of the four-dimensional Minkowski space M3,1 . In the large radius limit, this can be identified on the B-model side with the integral 1 2πi

1 v(u)du = 2πi αu

log y(x)

αu

dx , x

(2.41)

where αu is a not a closed cycle but rather a chain over which v jumps by 2πi. In analogy with (2.35), one expects that U=

1 2πi

αu

λ

(2.42)

is the exact formula for the flat open string parameter U , which includes all instanton corrections. Note that the above indeed depends on a choice of parameterization of the curve, which defines the location/phase and framing of the brane. In principle, the chain αu and the integral (2.42) can be obtained for branes in any phases. However, in practice, it turns out to be easier to start with outer branes, and use the open moduli phase transitions explained in the previous section, which relate the coordinates in various phases, to extract the flat open string parameters in other phases. Finally, note also that it is straightforward to show that both (2.35) and (2.42) receive only closed string worldsheet instanton corrections. The open string disk amplitude A(0) 1 can also be written as a chain integral. It is given by the Abel-Jacobi map (0) A1 (q, x)

=

βu

λ,

(2.43)

where βu is now the chain βu = [u ∗ , u]. Note that the disk amplitude has an integrality structure which may be exhibited by passing to the instanton-corrected coordinates (0) X = eU , Q = e−T . Then, it can be written in terms of the open BPS numbers Nn,m ∈ Z as follows: (0)

A1 (Q, X ) =

n∈N,m∈Z

(0) Nn,m Li2 (Q n X m ).

(2.44)

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2.3.3. Picard-Fuchs equations. On the Riemann surface it is possible to perform the period integrals (2.35), (2.42) and (2.43) directly. However, in practice it is simpler to derive Picard-Fuchs equations for general period integrals, construct a basis of solutions and find linear combinations of the solutions which reproduce the leading behaviour of the period integrals. When M is a toric threefold, the Picard-Fuchs operators annihilating the closed periods T α (2.35) can be defined in terms of the charge vectors defining M (see 2.1), as Lα = ∂xi − ∂xi . (2.45) Q iα >0

Q iα >0

The complex structure variables at the points of maximally unipotent monodromy α

q α = e−t are related to the xi by α

q α = (−1) Q 0

Qα

xi i .

(2.46)

i

Note that there are in general more x i than q α and C∗ -scaling symmetries are used to reduce to the q α variables. Solutions to (2.45) are easily constructed using the Fröbenius method. Defining w0 (q, ρ) =

1 Q α0 α n α α (n α + ρ α ) + 1] ((−1) q ) , [Q i i

nα

(2.47)

then X 0 = w0 (q, 0),

Tα =

∂ w0 (q, ρ)|ρ=0 ∂ρ α

(2.48)

are solutions. Higher derivatives X (αi1 ...αin ) =

∂ ∂ . . . αi w0 (q, ρ)|ρ=0 ∂ρ αi1 ∂ρ n

(2.49)

also obey the recursion imposed by (2.45), i.e. they fullfill (2.45) up to finitely many terms. However, only finitely many linear combinations of the X αi1 ...αin are actual solutions of the Picard-Fuchs system. Once the solutions T α to (2.45) are given, the period integrals (2.42) defining the flat open string parameters can be simply expressed in terms of them: U =u+

k

ruα (t α − T α ).

(2.50)

α=1

Here ruα ∈ Q, and most of them are zero. Note that only the combinations (t α − T α ) occur, which implies that the open string variables are invariant under the closed string B-field shift. Note that one can write down an extended Picard-Fuchs system, such that not only the closed periods but also the open periods (2.42) and (2.43) are annihilated by the differential operators [24,36]. The ruα are then related to entries in the charge vectors Q iα in (2.1). These relations are manifest in the extended Picard-Fuchs system and give an easy way to determine the ruα .


133

Finally, in the following we will always use the following notation. We always denote the flat, instanton corrected coordinates by uppercase letters, such as T , U and V , with their exponentiated counterparts Q = e−T , X = eU and Y = e V . The classical (or uncorrected) variables will always be denoted by lowercase lettes t, u and v, as well as q = e−t , x = eu and y = ev . 2.3.4. Open phase transitions. In the example above we have found the open mirror map in a particular parameterization corresponding to outer branes with zero framing. We could have done the same for branes in other phases, but in practice it is easier to simply follow the mirror map through the reparameterizations between different phases in order to obtain the mirror in other phases or framing. Here we simply write down an explicit example of such calculation. Let us start with a mirror curve H (x, ˜ y˜ ; q) in the parameterization corresponding to outer branes with zero framing. Following (2.50), we can write the open string mirror map, in terms of exponentiated coordinates, as X = xe ˜ u ,

(2.51)

where u =

k

ruα (t α − T α ).

(2.52)

α=1

Suppose that y˜ is not corrected, that is Y = y˜ , or in the notation above v = 0. Consider now the framing transformation (x, ˜ y˜ ) → (x, y) = (x˜ y˜ f , y˜ ) .

(2.53)

In this case, both the open and closed mirror maps are left unchanged by the framing reparameterization. Let us now consider a reparameterization corresponding to a phase transition to an inner brane phase: 1 y˜ , . (2.54) (x, ˜ y˜ ) → (x˜i , y˜i ) = x˜ x˜ In this case the open mirror maps becomes: X=

1 u e i = x˜i e−u , x˜

Y =

y˜ v e i = y˜i e−u . x˜

(2.55)

The fact that y˜i also gets renormalized in this phase implies that, under a framing reparameterization f

(x˜i , y˜i ) → (xi , yi ) = (x˜i y˜i , y˜i ),

(2.56)

the open flat coordinates acquire a non-trivial framing dependence: X = x˜i y˜i eui + f vi = xi e−( f +1)u , f

Y = y˜i e

vi

= yi e

−u

.

(2.57)

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2.3.5. Small radius regions. The more interesting case of phase transitions in the moduli space between patches which require a non-trivial symplectic transformation of the closed periods can be dealt with as follows. On the B-model side, these transitions simply correspond to moving in the complex structure moduli space beyond the radius of convergence of the large radius expansion, or more generally from one region of convergence into another. The flat open and closed coordinates in all regions are linear combinations of the closed periods (2.35) and chain integrals (2.42), (2.43). The right linear combinations that yield the flat open and closed coordinates in this new region can be found using the following requirements: • they should be small enough to be sensible expansion parameters around the singularity; • the amplitudes should be monodromy invariant when expanded in terms of the flat coordinates; • the linear combinations giving the flat closed coordinates should not involve the chain integrals. In simple cases this fixes the flat coordinates completely, up to scaling. This was the case, for instance, for the flat closed coordinates of the C3 /Z3 orbifold expansion of O(−3) → P2 , which was considered in [1]. A technical difficulty is that one has to find local expansions of the closed periods and chain integrals at various points in the moduli space. For the closed periods this can be done by solving the Picard-Fuchs system at the new points to obtain a basis of solutions everywhere. For the open periods, one uses the following observations. First, notice that (2.50) is a chain integral, while the T α are periods. Hence there is a linear combination uB = u +

k

ruα t α ,

(2.58)

α=1

which can be written as an elementary function of the global variables (x, q α ). Likewise the analytic continuation of A(0) 1 (q, x) is trivial since it is an elementary function in terms of the global variables. Hence, together with the closed string periods, u B (q, x) (0) and A1 (q, x) form a basis for the flat coordinates everywhere in the moduli space. 2.3.6. The O(−3) → P2 geometry. As an example, let us now discuss the open and closed mirror maps for the O(−3) → P2 geometry. Local P2 is defined by the charge Q = (−3, 1, 1, 1).

(2.59)

We start with the closed periods at large radius. Plugging this charge into (2.45) and changing variables to q = − x2 xx33 x4 , we get the Picard-Fuchs differential equation 1

D = [θt2 + 3q(3θt + 2)(3θt + 1)]θ, ∂ where θt = q ∂q = ∂t . This equation should annihilate the closed periods. 0 Clearly X = 1 and T := X (t) = λ = t − t (q), A

(2.60)

(2.61)


135

with t (q) =

∞ (−1)n (3n)! n q , n (n!)3

(2.62)

n=1

are solutions. It is easy to check that FT =

1 (t,t) 1 1 + T+ X 6 6 12

(2.63)

is a third solution, which corresponds to the integral FT = B λ over the B-cycle. Note that the particular combination of the Picard-Fuchs solutions giving the B-period is determined by classical topological data of the A-model geometry. The expression for the flat closed parameter (2.61) can be inverted to q = Q + 6 Q 2 + 9 Q 3 + 56 Q 4 − 300 Q 5 + 3942 Q 6 + · · · , e−t

(2.64)

e−T .

with q = and Q = We now consider an outer brane in this geometry. Applying (2.8) and the discussion in Sect. 2.2.2 we see that the parameterization of the mirror curve H (x, ˜ y˜ ; q) relevant for the outer brane with zero framing is H (x, ˜ y˜ ; q) = y˜ 2 + y˜ + y˜ x˜ + q x˜ 3 = 0.

(2.65)

(0)

The derivative of the superpotential is then given by x∂ ˜ x˜ A1 = log( y˜ ), with 1 + x˜ 1 y˜ = − (1 + x) ˜ 2 − q x˜ 3 . − 2 2

(2.66)

The special Lagrangian L in the A-model becomes a point on the Riemann surface; the exact domain-wall tension is then given by the period integral over the cycle (2.42) depicted in Fig. 2. The integral was performed in [5] and yields U = u˜ −

t−T , 3

(2.67)

Fig. 2. Toric base of O(−3) → P2 with an outer brane and the mirror curve with the open cycle defining the mirror map

136


or 1

X = xe ˜ − 3 t (q) ,

(2.68)

which defines the open flat coordinate at large radius. There is no mirror map for y˜ , that is, Y = y˜ . Consider now the framing transformation, (x, ˜ y˜ ) → (x, y) = (x˜ y˜ f , y˜ ).

(2.69)

Following this transformation, we get that the open mirror map for framed outer branes is still given by: 1

X = xe− 3 t (q) ,

Y = y,

(2.70)

and its inversion reads x = X 1 − 2 Q + 5 Q 2 − 32 Q 3 − 286 Q 4 + · · · .

(2.71)

We now move to inner branes. The phase transition from outer branes to inner branes consists in the transformation (x, ˜ y˜ ) → (x˜i , y˜i ) =

1 y˜ , x˜ x˜

,

(2.72)

which gives the curve H (x˜i , y˜i ; q) = y˜i2 x˜i + y˜i x˜i2 + y˜i x˜i + q,

(2.73)

parameterizing an inner brane with zero framing. Following the transformation (2.72), we get that the inner brane mirror map reads 1

1

X = x˜i e 3 t (q) , Y = y˜i e 3 t (q) .

(2.74)

In terms of the framed variables (xi , yi ), the mirror map becomes 1

1

X = xi e 3 (1+ f )t (q) , Y = yi e 3 t (q) ,

(2.75)

which can be inverted to

xi = X 1 + 2 (1 + f ) Q + −1 + f + 2 f

2

2 30 + 25 f −3 f 2 + 2 f 3 Q 3 Q + + ··· . 3 2

(2.76)


137

3. A New B-Model Formalism In this section we would like to propose a complete method for solving the open and closed B-model topological string on a Calabi-Yau threefold W which is the mirror of a toric Calabi-Yau threefold M. The method builds on and extends the proposal in [39], and it lies entirely in the B-model. It provides in this way a mirror formalism to the A-model topological vertex for toric Calabi-Yau threefolds [3]. However, our formalism differs from the topological vertex in one crucial aspect. The topological vertex is non-perturbative in gs , the string coupling constant, but it is a 2 perturbative expansion in Q = e−t/s around the large radius point Q = 0 of the moduli space. In the computation of open amplitudes, the vertex is also perturbative in the open moduli z i appearing for example in (2.19), and it provides an expansion around z i = 0. As mentioned earlier, the B-model is perfectly suited for studying the amplitudes at various points in the open/closed moduli space. In fact, our formalism provides a recursive method for generating all open and closed amplitudes at any given point in the moduli space. Basically, once one knows the disk and the annulus amplitude at this point, one can generate all the other open and closed amplitudes unambiguously. In particular, not only can we solve topological string theory at large radius points corresponding to smooth threefolds, but also at other points in the moduli space such as orbifold and conifold points. This is in contrast to the topological vertex, which is defined only for smooth toric Calabi-Yau threefolds. Our method is recursive in the genus and in the number of holes of the amplitudes, which is reminiscent of the holomorphic anomaly equations of [11]. However, a crucial point is that in contrast with the holomorphic anomaly equations, our equations are fully determined, that is, they do not suffer from the holomorphic ambiguities appearing genus by genus when one tries to solve the holomorphic anomaly equations. Our equations are also entirely different in nature from the holomorphic anomaly equations, although it was shown in [23] that the former imply the latter. More precisely, the resulting amplitudes admit a non-holomorphic extension fixed by modular invariance (as in [1]) which satisfies the holomorphic anomaly equations of [11] in the local case. The main ingredient that we will make use of is the fact that when W is mirror to a toric Calabi-Yau threefold, most of its geometry is captured by a Riemann surface, which is the mirror curve in the notation of the previous section. We will construct recursively an infinite set of meromorphic differentials and invariants living on the mirror curve, and show that the meromorphic differentials correspond to open topological string amplitudes, while the invariants give closed topological string amplitudes. The initial conditions of the recursion are fixed by simple geometric objects associated to the Riemann surface, which encode the information of the disk and the annulus amplitudes. Our method is in fact a generalization of the formalism proposed by Eynard and Orantin [22] for solving matrix models. Given a matrix model, one can extract its spectral curve, which is an affine curve in C2 . Eynard and Orantin used the loop equations of the matrix model to construct recursively an infinite set of meromorphic differentials and invariants on the spectral curve, which give, respectively, the correlation functions and free energies of the matrix model. However, the insight of Eynard and Orantin was that one can construct these objects on any affine curve, whether it is the spectral curve of a matrix model or not. The obvious question is then: what do these objects compute in general? As a first guess, one could try to apply directly Eynard and Orantin to the mirror curve and see what the objects correspond to in topological string theory. However, this would not be correct, since the mirror curve is embedded in C∗ × C∗ rather

138


than C2 ; this is a crucial difference which must be taken into account. But after suitably modifying the formalism such that it applies to curves in C∗ × C∗ , it turns out that the objects constructed recursively correspond precisely to the open and closed amplitudes of topological string theory. As argued in [39] and as we mentioned in the introduction, this is because the formalism of [22] gives the amplitudes of the chiral boson theory on a “quantum” Riemann surface constructed in [2], which should describe as well the B-model on mirrors of toric geometries (once the formalism is suitably modified). So let us first start by briefly reviewing the formalism of Eynard and Orantin.

3.1. The formalism of Eynard and Orantin for matrix models. Take an affine plane curve C : {E(x, y) = 0} ⊂ C2 ,

(3.1)

where E(x, y) is a polynomial in (x, y). Eynard and Orantin construct recursively an infinite set of invariants Fg of C, g ∈ Z+ , which they call genus g free energies, by analogy with matrix models. The formalism involves taking residues of meromorphic (g) differentials Wk ( p1 , . . . , pk ) on C, which are called genus g, k hole correlation functions. 3.1.1. Ingredients. The recursion process starts with the following ingredients: • the ramification points qi ∈ C of the projection map C → C onto the x-axis, i.e., the points qi ∈ C such that ∂∂Ey (qi ) = 0. Note that near a ramification point qi there are two points q, q¯ ∈ C with the same projection x(q) = x(q); ¯ • the meromorphic differential ( p) = y( p)dx( p)

(3.2)

on C, which descends from the symplectic form dx ∧ dy on C2 ; • the Bergmann kernel B( p, q) on C, which is the unique meromorphic differential with a double pole at p = q with no residue and no other pole, and normalized such that B( p, q) = 0, (3.3) AI

where (A I , B I ) is a canonical basis of cycles for C.4 The Bergmann kernel is related to the prime form E( p, q) by B( p, q) = ∂ p ∂q log E( p, q). We will also need the closely related one-form 1 q¯ B( p, ξ ), dE q ( p) = 2 q

(3.4)

(3.5)

which is defined locally near a ramification point qi . 4 Note that the definition of the Bergmann kernel involves a choice of canonical basis of cycles; hence the Bergmann kernel is not invariant under modular transformations — we will come back to that later.


139

For example, if C has genus 0, its Bergmann kernel is given, in local coordinate w, by B( p, q) =

dw( p)dw(q) . (w( p) − w(q))2

(3.6)

Note that the Bergmann kernel is defined directly on the Riemann surface, and does not depend on a choice of embedding in C2 , i.e. on the choice of parameterization of the curve. In contrast, by definition the ramification points qi and the differential ( p) depend on a choice of parameterization of the curve. Given these ingredients, we can split the recursion process into two steps. First, we (g) need to generate the meromorphic differentials Wk ( p1 , . . . , pk ), and then the invariants Fg . (g)

3.1.2. Recursion. Let Wh ( p1 , . . . , ph ), g, h ∈ Z+ , h ≥ 1, be an infinite sequence of meromorphic differentials on C. We first fix (0)

W1 ( p1 ) = 0,

(0)

W2 ( p1 , p2 ) = B( p1 , p2 ),

(3.7)

and then generate the remaining differentials recursively by taking residues at the ramification points as follows: dE q ( p) (g−1) (g) Wh+2 (q, q, Wh+1 ( p, p1 . . . , ph ) = Res ¯ p1 , . . . , p h ) q=qi (q) − (q) ¯ qi g (g−l) (l) + W|J |+1 (q, p J )W|H |−|J |+1 (q, ¯ p H \J ) . (3.8) l=0 J ⊂H

Here we denoted H = 1, · · · , h, and given any subset J = {i 1 , · · · , i j } ⊂ H we defined p J = { pi1 , · · · , pi j }. This recursion relation can be represented graphically as in Fig. 3. Now, from these correlation functions we can generate the invariants Fg . Let φ( p) be an arbitrary anti-derivative of ( p) = y( p)dx( p); that is, dφ( p) = ( p). We generate an infinite sequence of numbers Fg , g ∈ Z+ , g ≥ 1 by 1 (g) Res φ(q)W1 (q). (3.9) Fg = 2 − 2g q q=qi i

We refer the reader to [22] for the formula for the invariant F0 , which will not be needed in this paper.

Fig. 3. A graphic representation of the recursion relation (3.8)

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3.1.3. Symplectic transformations. As an affine curve in C2 , the reparameterization group G C of C is given by 01 G C = S L(2, C) × , (3.10) 10 that is the group of complex 2×2 matrices with determinant ±1, acting on the coordinates (x, y) by ab ∈ GC . (x, y) → (ax + by, cx + dy), (3.11) cd This is the group that preserves the symplectic form |dx ∧ dy| on C2 . It was shown in [22] that the free energies Fg constructed as above are invariants of the curve C, in the sense that they are invariant under the action of G C . However, (g) the correlation functions Wk ( p1 , . . . , pk ) are not invariant under reparameterizations, since they are differentials. 3.1.4. Interpretation. The definition of these objects was inspired by matrix models. When C is the spectral curve of a matrix model, the meromorphic differentials (g) Wk ( p1 , . . . , pk ) and the invariants Fg are respectively the correlation functions and free energies of the matrix model. To be precise, this is true for all free energies with g ≥ 1, and all correlation functions with (g, k) = (0, 1), (0, 2). We refer the reader to [22] for the definition of the genus 0 free energy F0 . In the case of matrix models, (0) the one-hole, genus 0 correlation function W˜ 1 ( p) is also known as the resolvent and depends on both the potential of the model and the spectral curve, 1 (0) W˜ 1 ( p) = (V ( p) − y( p))dx( p), 2

(3.12)

(0) while the two-hole, genus 0 correlation function W˜ 2 ( p1 , p2 ) is given by subtracting the double pole from the Bergmann kernel: (0) (0) W˜ 2 ( p1 , p2 ) = W2 ( p1 , p2 ) −

d p1 d p2 d p1 d p2 = B( p1 , p2 ) − . ( p 1 − p 2 )2 ( p 1 − p 2 )2

(3.13)

3.2. Our formalism. As noted earlier, when W is mirror to a toric Calabi-Yau threefold, there is a natural Riemann surface that pops out of the B-model geometry, which is the mirror curve. It is always given by an algebraic curve in C∗ × C∗ . Our strategy, extending the proposal in [39], will be to apply a recursive process analogous to the above to generate free energies and correlation functions living on the mirror curve. We will then check extensively that these objects correspond precisely to the open and closed topological string amplitudes. We start with an algebraic curve : {H (x, y) = 0} ∈ C∗ × C∗ ,

(3.14)

where H (x, y) is a polynomial in (x, y), which are now C∗ -variables. One can think of them as exponentiated variables (x, y) = (eu , ev ), and this is how they appeared for example in the derivation of mirror symmetry in [30]. The only difference with


141

Eynard-Orantin’s geometric setup is that our Riemann surfaces are embedded in C∗ ×C∗ rather than C2 . As such, their reparameterization group is the G of (2.12) (the group of integral 2 × 2 matrices with determinant ±1), which acts multiplicatively on the C∗ -coordinates of , rather than the G C of (3.11). Consequently, we want to modify the recursive formulae such that the free energies Fg constructed from our curve are invariant under the action of G given by (2.12). As such, they will be invariants of the Riemann surface embedded in C∗ × C∗ . 3.2.1. Ingredients. The recursion process now starts with the following ingredients: • The ramification points qi ∈ of the projection map → C∗ onto the x-axis, i.e., the points qi ∈ such that ∂∂Hy (qi ) = 0. Near a ramification point, there are again two points q, q¯ ∈ with the same projection x(q) = x(q). ¯ • The meromorphic differential ( p) = log y( p)

dx( p) x( p)

(3.15)

on , which descends from the symplectic form dx dy ∧ x y

(3.16)

on C∗ × C∗ . Note that the one-form ( p) controls complex structure deformations for the B-model. • The Bergmann kernel B( p, q) on , and the one-form dE q ( p) defined earlier. The main difference is in the meromorphic differential ( p), which differs from the previous differential ( p) because of the symplectic form on C∗ × C∗ . Again, both the ramification points qi and the differential ( p) depend on a choice of parameterization for the curve , while the Bergmann kernel is defined directly on the Riemann surface. 3.2.2. Recursion. As before, the recursion process is given in two steps by (3.8) and (3.9); however, we replace the differential ( p) by the new differential ( p), to make the formalism suitable for algebraic curves in C∗ × C∗ . Accordingly, in (3.9) φ( p) is replaced by an arbitrary anti-derivative θ ( p) of ( p) as defined in (3.15); that is, dθ ( p) = ( p). 3.2.3. Symplectic transformations. As a curve in C∗ ×C∗ , the reparameterization group of is given by the group G of integral 2 × 2 matrices with determinant ±1, acting on the coordinates (x, y) by ab ∈ G . (x, y) → (x a y b , x c y d ), (3.17) cd We claim that the Fg ’s constructed above are invariant under the action of this group, hence are invariants of the mirror curve . Computationally speaking, a direct consequence of this statement is that we can use the G reparameterizations above to write down the “simplest” embedding of the Riemann surface in C∗ × C∗ , and use this embedding to calculate the free energies. We will use this fact extensively in our computations. Note however again that the correlation functions are not invariant under G , which will turn out to be crucial.

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3.2.4. Interpretation. Suppose now that is the mirror curve of a toric Calabi-Yau threefold M. Our first claim is: 1. The free energies Fg constructed above are equal to the A-model closed topological string amplitudes on the mirror threefold M, after plugging in the closed mirror map. Our second claim is a little bit subtler. Recall that fixing the location and framing of a brane in the A-model corresponds to fixing the G parameterization of the mirror curve . Hence, the open amplitudes should depend on the parameterization of . We claim: (g) (g) 2. The integrated correlation functions Ak = Wk ( p1 , . . . , pk ) are equal to the A-model framed open topological string amplitudes on the mirror threefold M, after plugging in the closed and open mirror maps. This statement means that given a parameterization of , one can compute the correlation functions, integrate them, plug in the mirror maps, and one obtains precisely the A-model open amplitudes for a brane in the location and framing corresponding to this particular parameterization. Note that as for matrix models, these claims are true for closed amplitudes with g ≥ 1, and open amplitudes with (g, k) = (0, 1), (0, 2). The disk amplitude, that is (g, k) = (0, 1), is given by [5,6] dx (0) A1 = = log y , (3.18) x while the annulus amplitude, (g, k) = (0, 2), is given by removing the double pole from the Bergmann kernel: d p1 d p2 (0) B( p1 , p2 ) − A2 = . (3.19) ( p 1 − p 2 )2 The one-hole amplitude (3.18) can be interpreted as the one-point function of a chiral boson living on [2], and the Bergmann kernel (3.19) it just its two-point function [37], as expected from the identification of the recursive procedure with the theory of the “quantum” chiral boson on the mirror curve. We will not be concerned with the genus 0, closed amplitude in this paper. As a result, we get a complete set of equations, directly in the B-model, that generate unambiguously all genus (framed) open/closed topological string amplitudes for toric Calabi-Yau threefolds. These equations can be understood as some sort of gluing procedure in the B-model, with the building blocks corresponding basically to the disk and the annulus amplitudes. In other words, one only needs to know the disk and the annulus amplitudes, and every other amplitude can be computed exactly using the recursion solution. Let us finally point out that the approach of [39] is a particular case of our more general formalism in the case that the curve can be written as √ 2s a(x) + σ (x) y(x) = , σ (x) = (x − xi ). (3.20) c(x) i=1

The choice of x, x¯ is as usual a choice of sign in the square root, hence the differential (3.15) is given by √ 2 σ (x) −1 dx. (3.21) (x) − (x) ¯ = tanh x a(x)


143

Therefore, in this particular parameterization, our formalism could be regarded as identical to the formalism of [22], albeit for a nonpolynomial curve given by √ σ (x) 1 −1 . (3.22) yEO (x) = tanh x a(x) This was the point of view advocated in [37] (see for example Eq. (2.17) of that paper, where the extra factor of 2 comes from the contribution of x). ¯ Therefore, the results of [37] for outer branes with trivial framing are also a consequence of our formalism. As it will become clear in the following, curves of the form (3.20) describe only a very small class of D-branes, and the right point of view to work in general is precisely the one we are developing here. However, and as we will elaborate later on, the curve (3.20) is still a useful starting point to compute closed string amplitudes due to symplectic invariance.

3.3. Computations. Let us now spend some time describing how we will carry out calculations to provide various checks of our claims. We also present a more algorithmic version of this formalism that could be applied to compute higher genus/number of holes amplitudes. It could in principle be implemented in a computer code, which we hope to do in the near future. Most of our calculations will focus on open amplitudes; more precisely, on genus 0, one-hole (disk), two-hole (annulus) and three-hole amplitudes, and genus 1, one-hole amplitudes. Let us explain the general idea behind our computations. From mirror symmetry, we are given an algebraic curve : {H (x, y) = 0} in C∗ × C∗ , with a G group of reparameterizations acting as in (2.12). These reparameterizations correpond physically to changing the location and framing of the brane. 3.3.1. Disk amplitude. To compute the disk amplitude, which is given by dx (0) A1 = = log y , x

(3.23)

all we need to do is to write down y as a function of x; that is, we need to solve H (x, y) = 0 for y. This can be done, as a power series in x, in any parameterization of , and after plugging in the mirror map for the open string parameter x in a given parameterization we obtain the framed disk amplitudes for branes ending on any leg of the toric diagram of the mirror manifold. This case was studied in detail in [6,5]. 3.3.2. Annulus amplitude. To compute the annulus amplitude, we need to compute the Bergmann kernel of . This is trickier. Our strategy, which extends the analysis performed in [39], goes as follows. We first use the G reparameterizations to write down the curve in a simple form, such as hyperelliptic. This was the case considered in [39]. Generally, this will correspond physically to a brane ending on an outer leg of the toric diagram, with zero framing (but it does not have to be so). In such a parameterization, there exists explicit formulae to write down the Bergmann kernel of the curve, at least for curves of genus 0 and 1. For a curve of genus 0, the Bergmann kernel is simply given by B(x1 , x2 ) =

dy1 dy2 , (y1 − y2 )2

(3.24)

144


where the yi are defined implicitly in terms of the xi by yi := y(xi ), with the function y(x) determined by solving the curve H (x, y) = 0. When has genus 1, there is a formula, due to Akemann [7], which expresses the Bergmann kernel of an hyperelliptic curve of genus 1 in terms of the branch points of the projection map → C∗ onto the x-axis. Let λi ∈ C∗ , i = 1, . . . , 4 be the four branch points of the projection map. That is, if qi ∈ , i = 1, . . . , 4 are the ramification points, then λi := x(qi ). Then the Bergmann kernel is given by B(x1 , x2 ) =

(λ1 − λ3 )(λ4 − λ2 ) E(k)

K (k) 4 4 i=1 (x 1 λi − 1)(x 2 λi − 1)

1 (x1 λ1 − 1)(x1 λ2 − 1)(x2 λ3 − 1)(x2 λ4 − 1) + 2 4(x1 − x2 ) (x1 λ3 − 1)(x1 λ4 − 1)(x2 λ1 − 1)(x2 λ2 − 1) (x2 λ1 − 1)(x2 λ2 − 1)(x1 λ3 − 1)(x1 λ4 − 1) +2 , (3.25) + (x2 λ3 − 1)(x2 λ4 − 1)(x1 λ1 − 1)(x1 λ2 − 1)

where K (k) and E(k) are elliptic functions of the first and second kind with modulus k2 =

(λ1 − λ2 )(λ3 − λ4 ) . (λ1 − λ3 )(λ2 − λ4 )

(3.26)

Note that this expression involves an ordering of the branch points, which corresponds to choosing a canonical basis of cycles for the Riemann surface. Using these explicit formulae, we can integrate the two-point correlation function to (0) get the bare genus 0, two-hole amplitudes A2 (x1 , x2 ) in terms of the open string parameters x1 and x2 . We then plug in the open mirror map for that particular parameterization to obtain the open amplitude. However, this was done in a particular parameterization, or embedding, which exhibited in a simple form, such as hyperelliptic. To obtain the full framed annulus amplitude for branes in other locations, we need to be able to calculate the Bergmann kernel for other parameterizations. But we have seen that the Bergmann kernel is in fact defined directly on the Riemann surface, and does not depend on the particular embedding of the Riemann surface. Hence we can use our result above and simply reparameterize it to obtain the Bergmann kernel of the curve in another parameterization. For instance, suppose we are given the Bergmann kernel B(x˜1 , x˜2 ) for a curve H˜ (x, ˜ y˜ ) = 0, and that we reparameterize the curve with the framing transformations (x, y) = (x˜ y˜ f , y˜ ), f ∈ Z introduced earlier. We obtain a new embedding H (x, y) = 0 of the Riemann surface. To obtain its Bergmann kernel, we first compute x˜ = x(x) ˜ as a power series in x, and then reparameterize the Bergmann kernel to get B(x1 , x2 ) = B(x˜1 (x1 ), x˜2 (x2 )). In this way, we are able to compute the bare genus 0, two-hole amplitude for any framing and brane. To obtain the full result we must then plug in the open mirror map for the open string parameters, in the particular parameterization we are looking at. 3.3.3. Genus 0, three-hole amplitude. To compute the genus 0, three-hole amplitude, we use the recursion formula (3.8). We can also use the simpler formula for the threepoint correlation function proved by Eynard and Orantin in [22], which reads, for curves


145

embedded in C∗ × C∗ : (0)

W3 (x1 , x2 , x3 ) =

λi

Res B(x, x1 )B(x, x2 )B(x, x3 ) x=λi

x y(x) . dxdy(x)

(3.27)

Using our result for the Bergmann kernel in any parameterization, we can compute the three-point correlation function also in any parameterization. Note however that the branch points λi ∈ C∗ depend on the particular parameterization; hence, when we change parameterization, not only the Bergmann kernel gets reparameterized, but the branch points at which we take residues also change. Let us now spend a few lines on how to find the ramification points qi ∈ and the two points q and q¯ satisfying x(q) = x(q) ¯ in the neighborhood of a ramification point. First, standard geometry says that the ramification points qi are defined to be the points satisfying ∂H (qi ) = 0. ∂y

(3.28)

The x-projection of the ramification points qi defines the branch points λi := x(qi ) ∈ C∗ . The latter can also be found directly as solutions of dx = 0. We will also be interested in determining the branch points of the “framed” curve H (x, y), where (x, y) = (x˜ y˜ f , y˜ ); that is, the branch points of the projection on the x-axis of the framed curve. These are determined by dx = d(x˜ y˜ f (x)) ˜ = y˜ f −1 (x)( ˜ f x˜ y˜ (x) ˜ + y˜ (x))d ˜ x˜ = 0.

(3.29)

To find all the branch points λi , one has to solve (3.29) for all the different branches of y˜ (x). ˜ We can employ the above equation also to analyze the theory near the branch points: given a ramification point qi , and the associated branch point λi = x(qi ), of the projection on the x-axis, we can determine the two points q, q¯ ∈ with the same x-projection x(q) = x(q) ¯ near qi . Define x(q) ˜ = λi + ζ,

x( ˜ q) ¯ = λi + S(ζ ),

(3.30)

where S(ζ ) = −ζ +

ck ζ k .

(3.31)

k≥2

By definition, we have that x(q) = (λi + ζ ) y˜ (λi + ζ ) f = (λi + S(ζ )) y˜ (λi + S(ζ )) f = x(q), ¯

(3.32)

which can be used to determine S(ζ ). At the first orders, we get c2 = − c3 = −

2 −1+ f 2 y˜ (λi )+ f 2 λi 2 3 y˜ (λi )+ f λi y˜ (3) (λi ) , 2 2 3 f λi ((−1− f ) y˜ (λi )+ f λi y˜ (λi )) 2 2 2 2 (3) 2 −1+ f y˜ (λi )+ f λi 3 y˜ (λi )+ f λi y˜ (λi ) 9 f 2 λi 2 ((1+ f ) y˜ (λi )− f 2 λi 2 y˜ (λi ))

2

.

(3.33)

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3.3.4. Genus 1, one-hole amplitude. To compute the genus 1, one-hole amplitude, we also use the recursion formula (3.8), with the ramification points and the Bergmann kernel corresponding to the chosen parameterization. The general formula for W1(1) (q) is (1)

W1 ( p) =

Resq=qi

qi

dE q ( p) B(q, q). ¯ (q) − (q) ¯

(3.34)

3.3.5. Higher amplitudes. Computations at higher g, h can be readily made in this formalism, although they are more complicated. When the algebraic curve is of genus zero, the computations are straightforward, but they become more involved as soon as the curve has higher genus. Some simplifications arise however when the curve is of the form (3.20) and the differential (x) is of the form (3.21), since in this case one can adapt the detailed results of [21] to our context (see also [12] for examples of detailed computations). We will refer to this case as the hyperelliptic case, since the underlying geometry is that of a hyperelliptic curve. Let us briefly review this formalism, following [21] closely, in order to sketch how to compute systematically higher amplitudes. We first write (x) − (x) ¯ = 2M(x) σ (x)dx, (3.35) where σ (x) is defined in (3.20) and M(x) is called the moment function. In the formalism of [22] applied to conventional matrix models, M(x) is a polynomial. In our formalism for mirrors of toric geometries, in the parametrization of the curve given in (3.20), M(x) is given by √ 1 σ (x) M(x) = √ , (3.36) tanh−1 a(x) x σ (x) which is the moment function considered in [39] (again, up to a factor of 2 which comes from (3.35) and in [39] is reabsorbed in the definition of M(x)). When (x) is of the form (3.35) we are effectively working on the hyperelliptic curve of genus s − 1, y 2 (x) = σ (x),

(3.37)

with ramification points at x = xi , i = 1, · · · , 2s. We define the A j cycle of this curve as the cycle around the cut (x2 j−1 , x2 j ),

j = 1, · · · , s − 1.

(3.38)

There exists a unique set of s − 1 polynomials of degree s − 2, denoted by L j (x), such that the differentials ωj =

1 L j (x) dx √ 2π i σ (x)

(3.39)

satisfy ωi = δi j , Aj

i, j = 1, · · · , s − 1.

(3.40)


147

The ωi s are called normalized holomorphic differentials. The one-form (3.5) can then be written as [21] ⎞ ⎛ √ s−1 1 σ (x ) ⎝ 1 − C j (x )L j (x)⎠ dx, (3.41) dE x (x) = √ 2 σ (x) x − x j=1

where 1 C j (x ) := 2π i

Aj

1 dx . √ x − x σ (x)

(3.42)

In this formula, it is assumed that x lies outside the contours A j . One has to be careful when x approaches some branch point x j . When x lies inside the contour A j , then one has: δl j 1 dx 1 reg Cl (x ) + √ = , (3.43) √ 2π i A j σ (x) x − x σ (x ) which is analytic in x when x approaches x2 j−1 or x2 j . The Bergmann kernel is then given by: dx d B(x, x ) = dx + dE (x) , (3.44) x dx 2(x − x ) and it can be equivalently written as B( p, q) 1 σ ( p) = + √ √ 2 2 d pdq 2( p − q) 2( p − q) σ ( p) σ (q) A( p, q) σ ( p) + √ , − √ √ √ 4( p − q) σ ( p) σ (q) 4 σ ( p) σ (q)

(3.45)

where A( p, q) is a polynomial. In the elliptic case s = 2, there is one single integral C1 ( p) to compute, and one can find very explicit expressions in terms of elliptic integrals: 2 [(x2 −x3 )(n 4 , k) + ( p−x2 )K (k)] , √ π( p−x3 )( p−x2 ) (x1 −x3 )(x2 −x4 ) 2 reg C1 ( p) = [(x3 −x2 )(n 1 , k) + ( p−x3 )K (k)] , √ π( p−x3 )( p−x2 ) (x1 −x3 )(x2 −x4 ) (3.46) C1 ( p) =

where k2 =

(x1 − x2 )(x3 − x4 ) (x2 − x1 )( p − x3 ) (x4 − x3 )( p − x2 ) , n4 = , n1 = , (x1 − x3 )(x2 − x4 ) (x3 − x1 )( p − x2 ) (x4 − x2 )( p − x3 ) (3.47)

(n, k) is the elliptic integral of the third kind, 1 dt (n, k) = 2 0 (1 − nt ) (1 − t 2 )(1 − k 2 t 2 )

(3.48)

148


and K (k) is the standard elliptic integral of the second kind. With these ingredients one can compute the residues as required in (3.8). It is easy to see that dE q ( p)/y(q), as a function of q, has a pole at q = p but no pole at the branchpoints. It is then easy to see that all residues appearing in (3.8) will be linear combinations of the following kernel differentials: χi(n) ( p) = Resq=xi

dE q ( p) 1 y(q) (q − xi )n

(3.49)

which are explicitly given by ⎡ (n)

χi ( p) =

⎛

dn−1

1 1 ⎣ 1 ⎝ 1 − √ (n − 1)! σ ( p) dq n−1 2M(q) p − q

s−1

⎞⎤ L j ( p)C j (q)⎠⎦

j=1

.

q=xi

(3.50) Notice that in order to compute the kernel differentials, the only nontrivial objects to compute are dk C j /dq k . For a curve of genus one, they can be evaluated from the explicit expressions in (3.46). In order to compute the residues involved in (3.8), one has to take into account that the residues around branchpoints in terms of a local coordinate as in (3.8) are twice the residues around x = xi in the x plane [21]. One then finds, for example, 2s

(1)

(1)

(1)

M 2 (xi )σ (xi )χi ( p1 )χi ( p2 )χi ( p3 ), i=1

(3.51) 2s 2s 1 (1) A(xi ,xi ) 1 (2) 1 χi ( p) + 8 ( p), W1 ( p) = 16 2 σ (xi ) − χ i xi −x j

W 0 ( p1 , p2 , p3 ) =

1 2

i=1

i=1

j=i

where A( p, q) is the polynomial in (3.45). Therefore, in the hyperelliptic case, when (x) − (x) ¯ can be written as in (3.35), the computation of the amplitudes can be done by residue calculus and the only part of the calculation which is not straighforward is the evaluation of the integrals (3.42), (3.43). In the elliptic case, they reduce to elliptic functions, as we saw in (3.46). In the general case one can evaluate the integrals in terms of suitable generalizations of elliptic functions.

3.4. Moving in the moduli space. In Sect. 2.2 we discussed in some detail phase transitions in the open/closed string moduli space. We explained why the B-model was perfectly suited for studying such transitions. We now have a formalism, entirely in the B-model, that generates unambiguously all open/closed amplitudes for toric Calabi-Yau threefolds. An obvious application is then to use this formalism to study both open and closed phase transitions, which cannot be studied with A-model formalisms such as the topological vertex. Recall that the ingredients in our formalism consists in a choice of projection → C∗ (or equivalently a choice of parameterization of ), a differential ( p) corresponding to the disk amplitude, and the Bergmann kernel B( p, q) of the curve — which yields the


149

annulus amplitude. Note that once the parameterization is chosen, the one-form ( p) is canonically defined to be ( p) = log y( p)

dx( p) . x( p)

(3.52)

Hence ( p) really only depends on the choice of parameterization. We have seen that changing the parameterization of the curve corresponds to changing the location and framing of the branes, that is, moving in the open moduli space. This is the mildest type of transition that was considered in Sect. 2.2.1. Since the Bergmann kernel is really defined on the Riemann surface, it can simply be reparameterized, and open phase transitions are rather easy to deal with. As explained in Sect. 2.2.1, this is because the amplitudes are simply rational functions of the open string moduli, which we see explicitly in our formalism. The more interesting types of transitions are thus the transitions between different patches which require non-trivial Sp(2g, C) transformation of the periods. The only ingredient that is modified by these transitions in the closed string moduli space is the Bergmann kernel, since its definition involves a choice of canonical basis of cycles, which corresponds to a choice of periods. Modular properties of the Bergmann kernel have been studied in detail in [22,23]. Under modular transformations, the Bergmann kernel transforms with a shift as follows: B( p, q) → B( p, q) − 2πiω( p)(Cτ + D)−1 Cω(q), with

A B ∈ Sp(2g, Z), C D

(3.53)

(3.54)

and τ is the period matrix. Here, ω( p) is the holomorphic differentials put in vector form. In a sense, the Bergmann kernel is an open analog — since it is a differential in the open string moduli — of the second Eisenstein series E 2 (τ ), which also transforms with a shift under S L(2, Z) transformations and generates the ring of quasi-modular forms. The key point here is that we know how the Bergmann kernel transforms under phase transitions in the closed string moduli space. Hence not only can we use our formalism to generate the amplitudes anywhere in the open moduli space, but also in the full open/closed moduli space. This means that in principle, we can generate open and closed amplitudes for target spaces such as conifolds or orbifolds. We will explore this avenue further in Sect. 6. To end this section, let us be a little more precise. In this paper we will only consider S-duality transformations for curves of genus 1, which exchange the A- and the B-cycles. More precisely, the S-duality transformation acts on the basis of periods by 0 −1 ∈ S L(2, Z). (3.55) 1 0 When the curve has genus 1, we can use Akemann’s expression (3.25) to compute the Bergmann kernel. This expression depends on the branch points λi , i = 1, . . . , 4, and the choice of canonical basis (or periods) is encoded in the choice of ordering of the branch points. In terms of the elliptic modulus k 2 , the S-duality transformation is given by k 2 → 1 − k 2 .

(3.56)

150


Using the explicit expression for the modulus in terms of the branch points (3.26), we see that the S-transformation is given by exchanging the two branch points λ2 and λ4 . In other words, an S-duality transformation corresponds to the two cuts meeting at one point and then splitting again. Therefore, to determine the shifted Bergmann kernel after an S-duality transformation, we only need to use Akemann’s expression (3.25) again, but with λ2 and λ4 exchanged. Using this new Bergmann kernel we can generate all open and closed amplitudes after the phase transition corresponding to the S-duality transformation. We will exemplify this procedure in Sect. 6, where we use an S-duality phase transition to compute open and closed amplitudes at the point in the moduli space of local P1 ×P1 , where the two P1 ’s shrink to zero size. Using large N duality, we can compare the resulting amplitudes with the expectation values of the framed unknot in Chern-Simons theory on lens spaces, and we find perfect agreement.

4. Genus 0 Examples In this section we study two toric Calabi-Yau threefolds, C3 and the resolved conifold, for which the mirror curve has genus 0.

4.1. The vertex. Our first example is the simplest toric Calabi-Yau threefold, M = C3 . The mirror curve is P1 with three holes, and can be written algebraically as H˜ (x, ˜ y˜ ) = x˜ + y˜ + 1 = 0,

(4.1)

with x, ˜ y˜ ∈ C∗ .5 This parameterization corresponds to a brane ending on one of the three outer legs of the toric diagram, with zero framing (in standard conventions). The open mirror map, in this parameterization, is given simply by (X, Y ) = (−x, ˜ − y˜ ). 4.1.1. Framing. The framing transformation is given by (x, ˜ y˜ ) → (x, y) = (x˜ y˜ f , y˜ ),

(4.2)

where x is the framed bare open string parameter. From the transformation above, the open mirror map is now given by (X, Y ) = ((−1) f +1 x, −y). Under this reparameterization the mirror curve becomes H (x, y) = x + y f +1 + y f = 0,

(4.3)

which is a branched cover of C∗ . The framed vertex and its mirror curve are shown in Fig. 4. 5 In the following, tilde variables will always denote a curve in zero framing, while plain variables will denote a framed curve.


151

Fig. 4. The framed vertex and its mirror curve

4.1.2. Disk amplitude. The bare framed disk amplitude is given by dx (x) = log y(x) . A(0) 1 x

(4.4)

Thus, we need to find y = y(x). We can solve (4.3) for y as a power series of x, by using for example Lagrange inversion, and we get y(x) = −1 +

∞

(−1)n( f +1)

n=1

(n f + n − 2)! x n (n f − 1)! n!

= −1 − (−1) f x + f x 2 −

(−1) f ( f + 3 f 2 )x 3 + . . . . 2

Plugging in the map x = −(−1) f X , we thus get 1 1 (0) A1 (X ) = − X + (1 + 2 f )X 2 + (2 + 9 f + 9 f 2 )X 3 4 18 1 2 3 4 (3 + 22 f + 48 f + 32 f )X + . . . , + 48

(4.5)

(4.6)

up to an irrelevant constant of integration. This is precisely the result that is obtained on the A-model using the topological vertex. 4.1.3. Annulus amplitude. To compute the annulus amplitude we must compute the Bergmann kernel of the curve (4.3) in the bare open string parameters x1 and x2 . Let us first work in the zero framing parameterization. Since has genus 0, at zero framing the Bergmann kernel is simply given by B(x˜1 , x˜2 ) =

d y˜1 d y˜2 , ( y˜1 − y˜2 )2

(4.7)

˜ obtained where the y˜i are defined implicitly in terms of the x˜i by y˜i := y˜ (x˜i ), with y˜ (x) by solving H˜ (x, ˜ y˜ ) = 0, that is y˜ (x) ˜ = −1 − x. ˜

152


But the framing transformation sets y1 = y˜1 , y2 = y˜2 , hence we can reparameterize the Bergmann kernel and obtain immediately that B(x1 , x2 ) =

dy1 dy2 , (y1 − y2 )2

(4.8)

where now the yi are defined implicitly in terms of the xi by yi := y(xi ), with the function y(x) given by (4.5). The bare two-hole amplitude is given by removing the double pole and integrating:

dx1 dx2 (x1 − x2 )2 = log(−y1 (x1 ) + y2 (x2 )) − log(−x1 + x2 ).

(0)

A2 (x1 , x2 ) =

B(x1 , x2 ) −

(4.9)

Using the expansion (4.5) and the open mirror map X 1 = −(−1) f x1 , X 2 = −(−1) f x2 , we obtain (0)

A2 (X 1 , X 2 ) =

1 1 f ( f + 1)X 1 X 2 + f (1 + 3 f + 2 f 2 )(X 12 X 2 + X 1 X 22 ) 2 3 1 + f (1 + f )(1 + 2 f )2 X 12 X 22 4 1 + f (2 + 11 f + 18 f 2 + f 3 )(X 13 X 2 + X 1 X 23 ) + . . . , (4.10) 8

up to irrelevant constants of integration; this matches again the topological vertex result. (0)

4.1.4. Three-hole amplitude. To compute A3 , the additional ingredients needed are the ramification points of the projection map → C∗ onto the x-axis for the framed curve (4.3). Solving ∂H = 0, ∂y

(4.11)

f we find only one ramification point q1 at y(q1 ) = − f +1 . Denote by λ1 the associated branchpoint, which is given by the x-projection of q1 , that is λ1 = x(q1 ). The amplitude thus becomes

(0)

A3 (x1 , x2 , x3 ) =

Res B(x, x1 )B(x, x2 )B(x, x3 ) x=λ1

=

Res y=−

f f +1

x y(x) dxdx dy dx

x(y)ydydy1 (x1 )dy2 (x2 )dy3 (x3 ) (y−y1 (x1 ))2 (y−y2 (x2 ))2 (y−y3 (x3 ))2

dx dy

−1

. (4.12)

Since x = −y f (y + 1), we compute easily that

dx dy

−1

=−

y f −1 ( f

1 , + y( f + 1))

(4.13)


153

f which has a simple pole at y = − f +1 . Taking the residue and integrating, we get

A(0) 3 (x 1 , x 2 , x 3 ) = −

f 2 ( f + 1)2

3 i=1

=

f2 f +1

3 i=1

dyi (xi ) ( f + ( f + 1)yi (xi ))2

1 . f + ( f + 1)yi (xi )

(4.14)

Plugging in the expansion (4.5) and the open mirror map, we finally obtain (0) A3 (X 1 , X 2 , X 3 ) = − f 2 (1 + f )2 X 1 X 2 X 3 + f 2 (1 + f )2 (1 + 2 f )(X 12 X 2 X 3 + perms) 1 + f 2 (1 + f )2 (2 + 9 f + 9 f 2 )(X 13 X 2 X 3 + perms) 2 + f 2 (1 + 3 f + 2 f 2 )2 (X 12 X 22 X 3 + perms) + . . . ,

which is again in agreement with vertex computations. (1)

4.1.5. The genus one, one hole amplitude. In the computation of A1 (X ) we need some extra ingredients, besides the ones that we have already considered. For a curve of genus zero, 1 1 1 dE q ( p) = dy( p) − , (4.15) 2 y( p) − y(q) y( p) − y(q) ¯ where y is a local coordinate. To compute (3.34) in this example, we need q¯ near the ramification point q1 located at y(q1 ) = − 1+f f . Following the general discussion in Sect. 3.3, we write y(q) = −

f + ζ, 1+ f

y(q) ¯ =−

f + S(ζ ). 1+ f

(4.16)

By definition, x(q) = −y(q) f (y(q) + 1) = −y(q) ¯ f (y(q) ¯ + 1) = x(q), ¯

(4.17)

which we can use to solve for S(ζ ), which has the structure presented in (3.31). Its power series expansion can be easily determined, and the first few terms are 2 2 −1 + f 2 ζ 2 4 −1 + f 2 ζ 3 S(ζ ) = −ζ + − + O(ζ 4 ). (4.18) 3f 9 f2 We now compute (3.34) by using ζ as a local coordinate near the branchpoint. We need, B(q, q) ¯ =

(dζ )2 S (ζ ), (ζ − S(ζ ))2

as well as

(q) − (q) ¯ = log −

f +ζ 1+ f

− log −

f + S(ζ ) 1+ f

(4.19)

dx dζ. (4.20) dζ

154


The residue in (3.34) is easily evaluated, and we only need the expansion of S(ζ ) up to third order. One finds, (1)

W1 (y) =

(1 + f )4 y 2 + 2 f (1 + f )(2 + f + f 2 )y + f 4 24 ( f (1 + p) + p)4

dy.

(4.21)

After integration and expanding in X , we obtain X (1 + 2 f )( f 2 + f − 1)X 2 + 24 12 (1 + 3 f )(2 + 3 f )(−1 + 2 f + 2 f 2 ) X 3 + + O(X 4 ), (4.22) 16 which is in perfect agreement with the g = 1 piece of the exact formula in gs (but perturbative in X ) obtained from the topological vertex, (1)

A1 (y) = −

A1 (y, gs ) =

∞ g=0

(g)

2g−1

A1 (y)gs

=

∞ [m f + m − 1]! (−1)m f X m+1 , m[m]![m f ]!

(4.23)

m=0

where [n] denotes the q-number with parameter q = egs . To end this section, we mention that the framed vertex results can be written down in a nice way in terms of Hodge integrals, using the Mariño-Vafa formula [40]. The recursion relations proposed in this paper induce new recursion relations for the Hodge integrals. In turn, using the well known relation between the framed vertex geometry and Hurwitz numbers, one can obtain a full recursion solution for Hurwitz numbers. This is a nice mathematical consequence of the formalism proposed in this paper, which is studied in [15]. 4.1.6. Framed vertex in two legs. So far we assumed that all the branes ended on the same leg of the toric diagram of C3 (the vertex). However, when there is more than one hole, one can consider the case where there is one brane in one leg of the vertex and another brane in another leg; this is shown in Fig. 5. Let us now compute the annulus amplitude for two branes in two different legs. The strategy goes as usual: we start with the Bergmann kernel for two branes with zero framing in the same leg, and then reparameterize the Bergmann kernel to obtain two framed branes in different legs. To do so, we need to find the expansion y1 = y1 (x1 ) for a framed brane in one leg, which we found already in (4.5), but also y2 = y2 (x2 ), where x2 now corresponds to the open string parameter of a framed brane in a different leg. That is, we need to be able to relate the curves in the two different legs. As explained in Sect. 2.2, the phase transformation for moving from one leg of the toric diagram to another, at zero framing, reads: (x, ˜ y˜ ) → (x˜ , y˜ ) = (x˜ −1 , x˜ −1 y˜ ).

(4.24)

Now the framing transformation in this new leg reads (x˜ , y˜ ) → (x , y ) = (x˜ ( y˜ ) f , y˜ ),

(4.25)

where x and y now correspond to framed parameters in the new leg. Combining these two transformations we get (x, ˜ y˜ ) → (x , y ) = (x˜ −1− f y˜ f , x˜ −1 y˜ ).

(4.26)


155

Fig. 5. The framed vertex in two legs

Inversely, we have that (x, ˜ y˜ ) = ((x )−1 (y ) f , (x )−1 (y ) f +1 ).

(4.27)

Under this reparameterization the curve becomes x + (y ) f + (y ) f +1 = 0,

(4.28)

which is the same curve as before! Indeed, for the framed vertex, by symmetry changing the leg does not change the amplitudes. So we know y (x ) which is (4.5) as before. However, what we really want in order to reparameterize the Bergmann kernel is y˜ = y˜ (x ). Using the transformation above, we know that y˜ = y˜ (x ) = (x )−1 (y (x )) f +1 . As a power series, we get y˜ = −(−1) f

1 (−1) f f (1 + f )x + . . . . − (1 + f ) + x 2

(4.29)

Using these results, we can reparameterize the Bergmann kernel to get the framed annulus amplitude in two different legs. For the first open string parameter, we reparameterize using y˜1 = y1 = y1 (x1 ) given by (4.5), and for the second open string parameter we use (4.29) to get y˜2 = y˜2 (x2 ). The mirror map for the first parameter is X 1 = −(−1) f1 x1 , while for the second parameter from the transformations above we get the mirror map X 2 = −x2 . Removing the double pole and integrating as usual, we get6 (0)

(−1) f2 ( f 2 (1+3 f 2 ))X 1 X 23 2 1 (−1) f2 f 2 (2+ f 1 +3 f 2 f 1 )X 12 X 23 + . . . , (4.30) − (1+2 f 1 f 2 )X 12 X 22 − 2 2

A2 (X 1 , X 2 ) = −(−1) f2 X 1 X 2 − f 2 X 1 X 22 −

which again is in agreement with the vertex result.7 6 Note that here we have two framings f and f corresponding to the two different branes. 1 2 7 More precisely, to get the topological vertex result we need to redefine f → − f − 1, which is just a 1 1

redefinition of what we mean by zero framing.

156


Fig. 6. The resolved conifold and its mirror curve

4.2. The resolved conifold. Let us now turn to the resolved conifold, or local P1 . The mirror curve ⊂ C∗ × C∗ has genus 0, and reads H (x, ˜ y˜ ; q) = 1 + x˜ + y˜ + q x˜ y˜ , with x, ˜ y˜ ∈ C∗ and q size of the P1 . This is

(4.31)

e−t ,

= with t the complexified Kähler parameter controlling the shown in Fig. 6. There are two differences with the framed vertex. First, the mirror curve above has a one-dimensional complex structure moduli space, parameterized by q. Hence, we could consider phase transitions in the closed moduli space. However, as explained in Sect. 2.2.1, since the curve has genus 0, the amplitudes are rational functions of the closed moduli, that is there is no non-trivial monodromy for the periods. Hence, in this case these transitions are not very interesting. Another difference is that in contrast with the framed vertex, changing phase in the open moduli space, that is, moving the brane from one leg to another, yields different amplitudes. There are basically two types of amplitudes, corresponding to “outer” branes (ending on an outer leg of the toric diagram) and “inner” branes, as explained in Sect. 2.2. Since this type of transitions will be studied in detail for the local P2 example, for the sake of brevity we will not present here the calculations for the resolved conifold. Let us simply mention that we checked that both the framed outer and framed inner brane amplitudes at large radius (in the limit q → 0) reproduce precisely the results obtained through the topological vertex. The calculations are available upon request. 5. Genus 1 Examples We now turn to the more interesting cases where the mirror curve has genus 1. We will study two examples in detail: local P2 and local Fn , n = 0, 1, 2, where Fn is the n th Hirzebruch surface. Note that F0 = P1 × P1 . For the sake of brevity, we do not include here all the calculations; but we are happy to provide them with more detailed explanations to the interested reader. 5.1. Local P2 . The local P2 geometry is described by the charge vector (−3, 1, 1, 1). The mirror curve is an elliptic curve with three holes, and can be written algebraically as: H (x˜i , y˜i ; q) = x˜i y˜i + x˜i2 y˜i + x˜i y˜i2 + q,

(5.1)


157

Fig. 7. An outer brane in local P2 and its mirror

with x˜i , y˜i ∈ C∗ and q = e−t , with t the complexified Kähler parameter of local P2 . As for the resolved conifold, there are two distinct phases in the open moduli space, corresponding to outer and inner branes. The above parameterization of the curve corresponds to a brane ending on an inner leg of the toric diagram, with zero framing (in standard conventions), hence the i subscript. For an outer brane with zero framing, the curve reads (see Sect. 2.3.6) H (x, ˜ y˜ ; q) = y˜ 2 + y˜ + y˜ x˜ + q x˜ 3 = 0.

(5.2)

The outer brane geometry is shown in Fig. 7. Note that as for the resolved conifold, there are now more than one phases in the closed moduli space as well. Since the curve has genus 1, the periods now have non-trivial monodromy, and undergoing phase transitions in the closed moduli space becomes relevant. For instance, the closed moduli space contains a patch corresponding to the orbifold C3 /Z3 , in the limit where the P2 shrinks to zero size. However, in this section we will focus on the large radius limit q → 0 in order to compare with the topological vertex results on the A-model side. The mirror maps for this geometry at large radius were studied in Sect. 2.3.6, for both framed outer and framed inner branes. 5.1.1. Framed outer amplitudes. We start by computing the amplitudes for framed outer branes. To compute the disk amplitude we need y = y(x). We get f + 3 f 2 − 2 z x3 (1 + 4 f ) f + 2 f 2 − 3 z x 4 2 y =1+x − f x + − + .... 2 3 (5.3) By definition, the bare disk amplitude is given by dx (0) A1 (x) = log y(x) , x

(5.4)

and after expressing the result in flat open and closed coordinates using (2.71) we get precisely the topological vertex result for the disk amplitude of a framed brane in an outer leg.

158


We now turn to the annulus amplitude. The bare annulus amplitude is given by: (x , x ) = , x ) − log(−x1 + x2 ). (5.5) A(0) B(x 1 2 1 2 2 Hence, we need the Bergmann kernel B(x1 , x2 ) of the framed outer curve. As explained earlier, this is simply given by reparameterizing the Bergmann kernel of the unframed outer curve (5.2). It turns out that the unframed outer curve (5.2) is hyperelliptic. Consequently, we can use Akemann’s expression (3.25) for the Bergmann kernel in terms of the branch points of the x-projection ˜ — here we follow the calculation performed in [39]. To obtain these branch points, we first solve (5.2) for y˜ as: (x˜ + 1) ± (x˜ + 1)2 − 4q x˜ 3 y˜± = . (5.6) 2 It turns out to be easier to work with the inverted variable s = x˜ −1 . In this variable, the branch points of the curve are s1 = 0 and the roots of the cubic equation s(s + 1)2 − 4q = 0.

(5.7)

1 √ 3 ξ = 1 + 54 q + 6 3 q (1 + 27 q) ,

(5.8)

In terms of

they are given by 1 2 1 2 1 1 (ξ − 1)2 s2 = − + ωξ + , s3 = − + ω∗ ξ + ∗ , s4 = , 3 3 ωξ 3 3 ω ξ 3ξ

(5.9)

where ω = exp(2iπ/3). Plugging in these branch points in Akemann’s formula (3.25), we obtain the Bergmann kernel for the unframed outer curve, and the annulus amplitude in zero framing, as in [39]. We now want to implement the framing reparameterization. The reparameterization x˜ = x(x) ˜ can be computed using that x(x) ˜ = x y(x)− f with y(x) given in (5.3). We finally obtain, after reparameterizing the Bergmann kernel, plugging in the mirror maps (2.71), and integrating, that the framed annulus amplitude for outer branes reads: f2 f (0) A2 (X 1 , X 2 ) = + − 1 + 2 f + 2 f 2 Q + 4 + 7 f + 7 f 2 Q2 2 2 − 35 + 42 f + 42 f 2 Q 3 + . . . X 1 X 2 −f 2 f3 − f2 − + 1+4 f +6 f2 +4 f3 Q + 3 3 + −3 − 15 f − 27 f 2 − 18 f 3 Q 2 3 308 f 2 328 f 3 +164 f + Q + · · · (X 12 X 2 +X 1 X 22 ) + . . . . + 24+ 3 3 (5.10) This is again precisely the result obtained through the topological vertex.


159

The genus 0 three-hole amplitude for framed outer branes can be computed using the general formula (3.27), after reparameterizing the Bergmann kernel. However, to implement this formula we need to find the branch points of the framed curve — note that these are different from the branch points of the unframed curve found previously. As explained earlier, these branch points are given by the solutions of Eq. (3.29). In this case, (3.29) becomes a cubic equation in x, and the three branch points can be determined exactly by Cardano’s method. Note that it will be relevant which branch of (5.6) the branch points belong to; thus we will use the indices ± accordingly. The first orders of the q-expansion of the branch points read:8 2 2 (2 + 3 f )2 3 + 18 f + 37 f 2 + 30 f 3 + 9 f 4 q 2 + 6 f + 3 f 1 + 3 f + 2 f − λ+1 = + + ..., 3 1+3 f +2 f2 (2 + 3 f )2 q 1+3 f +2 f2 3 (2 + 3 f ) q (1 + 3 f )5 (2 + 3 f ) −1 + 2 f + 6 f 2 q 2 + 3 f 1 + 3 f (1 ) λ− − + ..., 2 = −1−2 f + f (1 + 2 f )3 f 3 (1 + 2 f )5 λ+3 = −

1 (−2−3 f ) q (2 + 3 f ) (1 + f (8 + 9 f )) q 2 + − + .... 1+ f f (1 + f )3 f 3 (1 + f )5

(5.11)

Taking into account the branches, plugging in the mirror map and integrating, we obtain the following result in flat coordinates: A(0) 3 (X 1 , X 2 , X 3 ) 2 = f 2 (1 + f )2 + 1 + 6 f + 12 f 2 + 12 f 3 + 6 f 4 Q − 3 1 + 3 f + 3 f 2 Q 2 +4 9 + 36 f + 77 f 2 + 82 f 3 + 41 f 4 Q 3 + · · · X 1 X 2 X 3 + · · · , (5.12) which reproduces again the topological vertex result. Note that we also computed the genus 1, one-hole amplitude, which also matches with topological vertex calculations. 5.1.2. Framed inner amplitudes. We can compute the amplitudes for framed inner branes in a way similar to the calculations above for outer branes. The main subtelty occurs in the reparameterization of the Bergmann kernel. Since we want to use Akemann’s formula for the Bergmann kernel, we start again with the curve in hyperelliptic form (5.2), which corresponds to the unframed outer brane. We then reparameterize that curve to obtain the Bergmann kernel corresponding to the curve associated to framed inner branes. Recall that the transformation which takes the unframed outer curve to the unframed inner curve is given by (2.72), 1 y˜i (x, ˜ y˜ ) = . (5.13) , x˜i x˜i The framing transformation for inner branes is −f

(x˜i , y˜i ) = (xi yi

, yi ).

(5.14)

8 Note that the branch points are not regular as f → 0, but the final expression of the three-hole amplitude will be.

160


Hence we obtain the combined transformation (x, ˜ y˜ ) = (xi−1 yi , xi−1 yi f

f +1

),

(5.15)

which we can use to reparameterize the Bergmann kernel. Note that this is similar to the calculation for the framed vertex in two legs. More explicitly, we obtain f f 1 2 f3 f2 2 x(x ˜ i) = f + − (5.16) + xi + + f + xi 2 + . . . . xi 2 2 3 3 Using this reparametrization and the mirror map (2.76) for framed inner branes we obtain the framed inner brane annulus amplitude: f f2 (0) + + −1 − 2 f − 2 f 2 − 2 f 3 − f 4 Q A2 (X 1 , X 2 ) = 2 2

35 f 81 f 2 157 f 3 93 f 4 27 f 5 5 f 6 + 11 + + + + + + Q 2 − 131 + 201 f 2 4 8 8 8 8 47 f 5 1429 f 6 1537 f 7 221 f 8 467 f 2 15023 f 3 781 f 4 + + − + − − + Q3 2 90 180 72 18 360 180 + . . . X1 X2 +

f2 3f + 1− 2 2

2 3 4 3 Q + −8 + 16 f −14 f + 6 f − f Q + ... 2

1 + .... X1 X2 (5.17)

This reproduces the topological vertex result, including both positive and negative winding numbers contributions. Note that we also computed the genus 0, three-hole and the genus 1, one-hole amplitudes for framed inner amplitudes and obtained a perfect match again. We also computed the annulus amplitude for one brane in an outer leg and one brane in an inner leg, paralleling the framed vertex in two legs calculation. We again obtained perfect agreement. 5.2. Local Fn , n = 0, 1, 2. We now study the local Fn , n = 0, 1, 2 geometries, where Fn is the n th Hirzebruch surface. Note that F0 = P1 × P1 . The local Fn geometries are described by the two charge vectors: Q 1 = (−2, 1, 1, 0, 0), Q 2 = (n − 2, 0, −n, 1, 1). C∗

(5.18)

C∗

× have genus 1 and four punctures. In the paramaThe mirror curves n ⊂ terization corresponding to a brane placed in an external leg (with zero framing), they read: Hn (x, ˜ y˜ ; qt , qs ) = y˜ x˜ + y˜ + y˜ 2 + qt x˜ 2 y˜ + qtn qs x˜ n+2 ,

(5.19)

with x, ˜ y˜ ∈ C∗ , qt = e−t and qs = e−s , with t and s the complexified Kähler parameters. The local F0 geometry is shown in Fig. 8.


161

Fig. 8. An outer brane in local F0 and its mirror

The closed moduli space is now two-dimensional, being spanned by qt and qs . However, these curves are still hyperelliptic, and we can apply our formalism exactly as we did for the local P2 case. Therefore, we will not do the full calculation here, but only highlight some interesting aspects. The large radius expansion for local F0 = P1 × P1 has been discussed in detail in [39], where several open amplitudes (for outer branes with canonical framing) were computed. Needless to say, we checked that our formalism can be used to complete the calculations by including framing and inner brane configurations. Besides the large radius point, our formalism allows to compute topological strings amplitudes at other points in the closed moduli space of local F0 , like the conifold point and the orbifold point. The latter corresponds to the point where the P1 × P1 shrinks to zero size. This special point will be discussed in great detail in the next section. For local F1 and F2 , the open and closed mirror maps together with the disk amplitudes for inner and outer branes were studied, for instance, in [36]. Again, we showed that our formalism allows to compute framed inner and outer higher amplitudes in the large radius limit, checking our results with the topological vertex ones. As an example, the outer annulus amplitude at zero framing for the local F1 geometry reads: 2 2 2 3 A(0) (X , X ) = −Q Q − 3Q Q + 4Q Q − 5Q Q + · · · X1 X2 1 2 s t s s t t s t 2 − −Q s Q t − 2Q 2t Q s + 3Q 2t Q 2s − 4Q s Q 3t + · · · X 1 X 22 + X 12 X 2 + Q s Q t − 2Q 2t Q s + 4Q 2t Q 2s − 3Q s Q 3t + · · · X 1 X 23 + X 13 X 2 7 + −Q s Q t −2Q 2t Q s + Q 2t Q 2s − 3Q s Q 3t + · · · X 12 X 22 + · · · , 2 (5.20) while for the local F2 geometry: (0) A2 (X 1 , X 2 ) = 2Q 2t Q s +4Q 3t Q s + · · · X 1 X 2 − Q 2t Q s +3Q 3t Q s + · · · X 1 X 22 +X 12 X 2 + Q 2t Q s + 2Q 3t Q s + · · · X 1 X 23 + X 13 X 2 + Q 2t Q s + 2Q s Q 3t + · · · X 12 X 22 + · · · . (5.21) Both of these coincide indeed with the topological vertex results.

162


There is also another interesting phase in the local F1 moduli space — see for instance [35]. By definition, F1 is a P1 bundle over P1 , where the P1 base is an exceptional curve. In fact, F1 is isomorphic to P2 blown up in one point, the base of the fibration corresponding to the blown up exceptional curve. Hence, we can blow down this exceptional P1 , and we should recover P2 . In other words, if we take the open amplitudes for local F1 and move to the phase in the moduli space where this exceptional P1 goes to zero size, we should recover the open amplitudes for local P2 . Going to this patch in fact corresponds to a mild transformation in the closed moduli space, since it does not involve a redefinition of the periods. The phase transition can then be directly implemented on the amplitudes as no modular transformation is needed. More specifically, it can be implemented in the local F1 annulus amplitude (5.20) by first defining Q˜ s = Q s Q t and then taking the limit Q t → 0. We get: (0) A2 (X 1 , X 2 ) = − Q˜ s + 4 Q˜ 2s + · · · X 1 X 2 + Q˜ s − 3 Q˜ 2s + · · · X 1 X 22 + X 12 X 2 + − Q˜ s + 4 Q˜ 2s + · · · X 1 X 23 + X 13 X 2 7 ˜2 ˜ (5.22) + − Q s + Q s + · · · X 12 X 22 + · · · , 2 which indeed coincides with the local P2 annulus amplitude at zero framing, see (5.10). 6. Orbifold Points As we already emphasized, one of the main features of our B-model formalism is that it can be used to study various phases in the open/closed moduli space, not just large radius points. In particular, there are two special points where we can use our formalism to generate open and closed amplitudes; the orbifold point of local P2 , which corresponds to the orbifold C3 /Z3 , and the point in the moduli space of local P1 × P1 , where the P1 × P1 shrinks to zero size (which we will call the local P1 × P1 orbifold point, although it is not really an orbifold). In the second example, we can use large N dualities to make a precise test of our formalism, and of its ability to produce results in all of the Kähler moduli space (and not only at the large radius limit). Indeed, it was argued in [4] that topological strings on A p−1 fibrations over P1 are dual to Chern–Simons theory on the lens space L( p, 1). In particular, the topological string expansion around the orbifold point of these geometries can be computed by doing perturbation theory in the Chern–Simons gauge theory. This was checked for closed string amplitudes in [4], for p = 2. We will extend this duality to the open string sector and make a detailed comparison of the amplitudes. In the first example, we would obtain open and closed orbifold amplitudes of C3 /Z3 . The closed amplitudes were already studied in [1]; by now some of the predictions of that paper for closed orbifold Gromov-Witten invariants have been proved mathematically. For the open amplitudes, to the best of our knowledge open orbifold Gromov-Witten invariants have not been defined mathematically, hence there is nothing to compare to. However, we proposed in Sect. 2.3.5 a method for determining the flat coordinates at all degeneration points in the moduli space which, as we will see, applies to the local P1 × P1 orbifold point. Therefore, we will assume that it should work at the C3 /Z3 orbifold point as well, and use it to make predictions for the disk amplitude for C3 /Z3 .


163

Let us start by studying the local P1 × P1 orbifold point. We first perform the Chern-Simons calculation, then explain the large N duality, and finally present our dual B-model calculation. 6.1. Chern–Simons theory and knots in lens spaces. In order to extend the duality of [4] to the open sector, we will need some detailed computations in Chern–Simons theory. In this subsection we review [4,37] and extend them slightly to include Wilson loops. Lens spaces of the form L( p, 1) can be obtained by gluing two solid 2-tori along their boundaries after performing the SL(2, Z) transformation, 10 Up = . (6.1) p1 This surgery description makes it possible to calculate the partition function of Chern– Simons theory on these spaces, as well as correlation functions of Wilson lines along trivial knots, in a simple way. To see this, we first recall some elementary facts about Chern–Simons theory. An SL(2, Z) transformation given by the matrix pi ri U ( pi ,qi ) = (6.2) qi si lifts to an operator acting on H(T2 ), the Hilbert space obtained by canonical quantization of Chern–Simons theory on the 2-torus. This space is the space of integrable representations of a WZW model with gauge group G at level k, where G and k are respectively the Chern–Simons gauge group and the quantized coupling constant. We will use the following notations: r denotes the rank of G, and d its dimension. y denotes the dual Coxeter number. The fundamental weights will be denoted by λi , and the simple roots by αi , with i = 1, · · · , r . The weight and root lattices of G are denoted by w and r , respectively. Finally, we put l = k + y. Recall that a representation given by a highest weight is integrable if the weight ρ + is in the fundamental chamber Fl (ρ denotes as usual the Weyl vector, given by the sum of the fundamental weights). The fundamental chamber is given by w /l r modded out by the action of the Weyl group. For example, in SU (N ) a weight p = ri=1 pi λi is in Fl if r

pi < l,

and pi > 0, i = 1, · · · , r.

(6.3)

i=1

In the following, the basis of integrable representations will be labeled by the weights in Fl . In the case of simply-laced gauge groups, the SL(2, Z) transformation given by U ( p,q) has the following matrix elements in the above basis [28,42]: 1 [i sign(q)]|+ | idπ Vol w 2 ( p,q) (U |β = exp − ) α|U (l|q|)r/2 12 Vol r ! iπ 2 2 pα − 2α(ln + w(β)) + s(ln + w(β)) . · (w) exp lq ( p,q)

n∈r /qr w∈W

(6.4)

164


In this equation, |+ | denotes the number of positive roots of G, and the second sum is over the Weyl group W of G. (U ( p,q) ) is the Rademacher function: p+s pr − 12s( p, q), (6.5) = q s q where s( p, q) is the Dedekind sum q−1 πn π np 1 cot cot . 4q q q

s( p, q) =

(6.6)

n=1

From the above description it follows that the partition function of the lens space L( p, 1) is given by Z (L( p, 1)) = ρ|U p |ρ,

(6.7)

where U p is the lift of (6.1) to an operator on H(T2 ). In order to make contact with the open sector, we need as well the normalized vacuum expectation value of a Wilson line along the unknot in L( p, 1), in the representation R, which is given by WR =

ρ|U p |ρ + , ρ|U p |ρ

(6.8)

where is the highest weight corresponding to R. The numerator can be written (up to an overall constant that will cancel with the denominator) ! iπ 2 ρ − 2ρ(ln + w(ρ + )) + (ln + w(ρ + ))2 . (6.9) (w) exp lp n∈r / pr w∈W

It is a simple exercise in Gaussian integration to check that this quantity can be written as ! r 1 2 (ww ) dλi exp − λ −n · λ + λ · (w(ρ)−w (ρ+)) , 2gˆ s n∈r / pr w,w ∈W

i=1

(6.10) where gˆ s =

2π i , pl

(6.11)

and dλ = ri=1 dλi and λi are the Dynkin coordinates of λ, understood as an element in w ⊗ R. This integral can be further written as ! λ·α 2 1 2 2 sinh λ − n · λ tr R e−λ , (6.12) dλ exp − 2gˆ s 2 α>0

where we have used Weyl’s formula for the character, −λ·w(ρ+) w∈W (w)e −λ tr R e = −λ·w(ρ) w∈W (w)e

(6.13)


165

as well as Weyl’s denominator formula. It follows that ! λ·α 2 1 1 2 2 sinh λ −n · λ tr R e−λ , (6.14) WR = dλ exp − Z (L( p, 1)) 2gˆ s 2 α>0

where Z (L( p, 1)) =

! 1 2 λ·α 2 λ − n · λ . dλ exp − 2 sinh 2gˆ s 2

(6.15)

α>0

This provides matrix integral representations for both the partition function (derived previously in [4,37]) and the normalized vacuum expectation value of a Wilson line around the unknot. Both expressions are computed in the background of an arbitrary flat connection labelled by the vector n. Notice that, when n = 0, one has that W R = egˆs /2(κ R +(R)N ) dimq R,

(6.16)

where dimq R is the U (N ) quantum dimension of R with q = egˆs . We can therefore regard (6.14) for arbitrary n as a generalization of quantum dimensions. As shown in [4,37], the partition function above can be written more conveniently in terms of a multi–matrix model for p Hermitian matrices. In the case of L(2, 1) (to which we will restrict ourselves), a generic flat connection can be specified by a breaking U (N ) → U (N1 ) × U (N2 ), or equivalently by a vector n with N1 +1 entries and N2 −1 entries. It is then easy to see [4] that the partition function (6.15) is given by the Hermitian two-matrix model, Z (N1 , N2 , gˆ s ) ! 1 1 TrM12 − TrM22 +V (M1 )+V (M2 )+W (M1 , M2 ) , = dM1 dM2 exp − 2gˆ s 2gˆ s (6.17) where V (M) = W (M1 , M2 ) =

1 2

∞

ak

k=1

∞

bk

k=1

2k

(−1)s

s=0

2k

(−1)s

s=0

2k s 2k−s , s TrM TrM

2k 2k−s s , s TrM1 TrM2

(6.18)

and ak =

B2k , k(2k)!

bk =

22k − 1 B2k . k(2k)!

(6.19)

The vacuum expectation value of the unknot in L(2, 1) is similarly given by W R (N1 , N2 , gˆ s ) =

1 tr R e M , Z (N1 , N2 , gˆ s )

(6.20)

where, in terms of the eigenvalues m i1 , m 2j of M1 , M2 , the matrix e M is given by 1

e M = diag(em 1 , · · · , e

m 1N

2

1

, −em 1 , · · · , −e

m 2N

2

).

(6.21)

166


The vev in (6.20) is defined by the weight given by the exponent in (6.17), and it can be easily computed in perturbation theory. In order to compare the results with the string theory results, we will need to compute the connected vevs W (c) in the k = (k1 , k2 , · · · ) basis, which are defined by k

log

"

# W R Tr R V

=

1 (c) W ϒ (V ), z k ! k k

(6.22)

k

R

where the notations are as in (A.3). Using the matrix model representation we can easily compute, for example, (c)

gˆ s 2 gˆ 2 (N1 − N22 )+ s (N1 − N2 ) 4N12 + 4N22 +10N1 N2 −1 + · · · , 2 24 gˆ s2 2 3N1 + 3N22 + 4N1 N2 = gˆ s (N1 + N2 ) + 2 gˆ s3 7(N13 + N23 ) + 15(N12 N2 + N1 N22 ) + 6 gˆ s4 15N14 + 47N13 N2 + {N1 ↔ N2 } + 63N12 N22 + 3N1 N2 + · · · , + 24 gˆ 2 = N1 + N2 + 2gˆ s (N12 + N22 ) + s (N1 + N2 ) 5N12 + 5N22 − 2N1 N2 + 1 3 gˆ s3 11N14 +18N13 N2 +6N12 N22 +5N12 +18N1 N2 +{N1 ↔ N2 } . + (6.23) 12

W(1,0,··· ) = N1 −N2 + (c) W(2,0,··· )

(c)

W(0,1,0,··· )

In order to compare with topological string amplitudes it is convenient to reorganize the connected vevs in terms of the ’t Hooft expansion. To do that, we introduce the ’t Hooft variables Si = gˆ s Ni , i = 1, · · · , p.

(6.24)

A diagrammatic argument based on fatgraphs says that the connected vevs have the structure W (c) (Si , gˆ s ) =

k

2g−2+|k|+ i hi

gˆ s

h

p h1 Fg,k,h i N1 · · · N p

g,h i

=

2g−2+|k|

gˆ s

h

p h1 Fg,k,h i S1 · · · S p .

(6.25)

g,h i (c)

The explanation for this is simple: in terms of fatgraphs, the connected vev W (Ni , gˆ s ) k but with varying is obtained by summing over fatgraphs with a fixed number of holes |k| genus g and number of “coloured” holes h i . We can sum over all coloured holes at fixed genus to obtain the amplitude (g) k

W (Si ) =

hi

h

p h1 Fg,k,h i S1 · · · S p .

(6.26)


167

Finally, in order to make contact with the open toplogical string amplitudes we notice that 1 (g) (g) W (Si )ϒk (V ) = Wh (z 1 , · · · , z h ), (6.27) z k k |k|=h k,

under the dictionary (A.6). From the above explicit computations we get the following results: 1 1 (0) A1 ( p) = p S1 − S2 + (S12 − S22 ) + (S1 − S2 )(4S12 + 10S1 S2 + 4S22 ) 2 24 ! 1 3 3 2 2 + (S1 − S2 )(S1 + S2 + 4(S1 S2 + S1 S2 )) + · · · 24 1 1 + p 2 S1 + S2 + 2(S12 + S22 ) + (S1 + S2 )(5S12 − 2S1 S2 + 5S22 ) 2 3 1 11(S14 + S24 ) + 18(S13 S2 + S23 S1 ) + 6S12 S22 + 12 ! 1 (S1 + S2 ) 69S14 + 126S13 S2 − 6S12 S22 + {S1 ↔ S2 } + · · · + 180 ! 9 1 1 3 + p S1 − S2 + (S12 − S22 ) + (S1 − S2 )(30(S12 + S22 ) + 39S1 S2 ) · · · , 3 2 4 (6.28) (1)

! 1 1 1 (S1 −S2 )− (S12 −S22 )− (S1 −S2 ) 4(S12 +S22 )+19S1 S2 + · · · 24 48 576 1 1 1 (S1 + S2 ) + 5(S12 + S22 ) + 18S1 S2 + p2 2 3 12 ! 3 ! 1 p 21 2 2 (S1 +S2 ) 55(S1 +S2 )+230S1 S2 + · · · + (S1 − S2 )+ · · · , + 180 3 8 (6.29)

A1 ( p) = p −

1 1 2 2 2 2 A(0) 2 ( p, q) = pq S1 + S2 + (3S1 + 3S2 + 4S1 S2 ) + (S1 + S2 )(7(S1 + S2 ) + 8S1 S2 ) 2 6 ! 1 4 4 3 3 2 2 15(S1 + S2 ) + 47(S1 S2 + S2 S1 ) + 63S1 S2 + · · · + 24 7 1 62(S13 −S23 ) + 51(S12 S2 −S1 S22 ) +( p 2 q+ pq 2 ) (S1 −S2 )+ (S12 −S22 )+ 2 12 ! 1 4 4 3 3 115(S1 − S2 ) + 201(S1 S2 − S1 S2 ) + · · · , (6.30) + 24 (0)

A3 ( p, q, r ) = pqr 3(S1 −S2 )+

! 17 2 2 1 (S1 −S2 )+ (46(S13 −S23 )+45(S12 S2 −S22 S1 ))+ · · · . 2 4 (6.31)

168


Finally, as explained in [45], Wilson loop operators in Chern–Simons theory need a choice of framing in order to be properly defined. The calculations above correspond to the framing coming naturally from the Gaussian integral in (6.20), and to change the framing by f units it is enough to multiply W R by exp{− f gˆ s κ R /2}.

(6.32)

The amplitudes computed above would change correspondingly. We would have, for example, ! 1 2 1 (0) 2 2 2 A1 ( p) = p S1 − S2 + (S1 − S2 ) + (S1 − S2 ) 4(S1 + S2 ) + 10S1 S2 + · · · 2 24 $ 2 p S1 + S2 + (2 − f )(S12 + S22 ) + 2 f S1 S2 + 2 ! 1 (6.33) + (S1 + S2 ) (5 − 3 f )(S12 + S22 ) + 2(3 f − 1)S1 S2 + · · · , 3 and (0) A2 ( p, q) = pq

! 1 2 2 2 2 (1− f )(S1 +S2 )+ (3−4 f + f )(S1 +S2 )+(4−2 f )S1 S2 + · · · + · · · . 2 (6.34)

6.2. The orbifold point and a large N duality. In [4] it was argued that topological string theory on X p , the symmetric A p−1 fibration over P1 , is dual to Chern–Simons theory in the lens space L( p, 1). This is a highly nontrivial example of a gauge theory/string theory duality which can be obtained by a Z p orbifold of the large N duality of Gopakumar and Vafa [25]. Equivalently, it can be understood as a geometric transition between T ∗ (S3 /Z p ) (which is equivalent to Chern–Simons theory on L( p, 1) [46]) and the X p geometry. Checking this duality is complicated because the perturbative regime of the gauge theory, where one can do computations easily, corresponds to string theory on X p near the point ti = 0, where the ti are the Kähler parameters. This is a highly stringy phase — a small radius region — where the α corrections are very important. It is conventional to refer to this point as an orbifold point (although the periods are still logarithmic) and we will do so in the following. This type of problem in testing the duality is well-known in the context of the AdS/CFT correspondence, where the perturbative regime of N = 4 Yang–Mills corresponds to a highly curved AdS5 × S5 target. In order to proceed, one has to either do computations in the strong ’t Hooft coupling regime of Chern–Simons theory, or to solve topological string theory near the orbifold point. Thanks to mirror symmetry and the B-model, the second option is easier, and this was the strategy used in [4] to test the duality in the closed string sector. How would we extend this story to the open sector? First we recall that, in the Gopakumar–Vafa duality, a knot K in S3 leading to a Wilson loop operator in Chern– Simons gauge theory corresponds to a Lagrangian submanifold LK in the resolved conifold [41]. Moreover, the connected vevs (6.27) become, under this duality, open string amplitudes with the boundary conditions set by LK . After orbifolding by Z p , the natural statement (generalizing the results of Ooguri and Vafa in [41]) is that a knot in L( p, 1) corresponds to a Lagrangian submanifold in X p . The simplest test of the


169

Fig. 9. The geometric transition between T ∗ (S3 /Z2 ) with a Lagrangian brane associated to the unknot, and local P1 × P1 with an outer brane

Ooguri–Vafa conjecture is the unknot, which corresponds to a toric D-brane in an outer edge of the resolved conifold (see for example [38] for details). It is then natural to conjecture that the unknot in L( p, 1) is dual to a toric D-brane in an outer edge of X p , and that the connected vevs for the corresponding Wilson line correspond to open string amplitudes for this brane. This should follow from the geometric transition for X p proposed in [4], and it is sketched in Fig. 9. Testing this conjecture is again difficult for the reasons explained above. In order to compare with the perturbative string amplitudes that we computed from the Chern–Simons matrix model, we need a way to compute open string amplitudes that makes it possible to go anywhere in the moduli space. But this is precisely one of the outcomes of the B-model formalism proposed in this paper! We will now explain how to compute open string amplitudes in the p = 2 case, i.e. local P1 × P1 , near the orbifold point, extending in this way the test of the duality performed in [4] to the open sector. This will verify not only our extension of the duality for knots in the lens space L(2, 1), but also the power of our B-model formalism.

6.3. Orbifold amplitudes. We now explain how to compute open string amplitudes at the orbifold point in the local P1 × P1 geometry, using the B-model formalism developed in this paper. We follow the general discussion in Sect. 3.4. Basically, to compute the open amplitudes at the orbifold point, one only needs to find the disk and the annulus amplitudes at this point, and then use our B-model formalism to generate the other amplitudes recursively. We also need to fix the open and closed mirror maps at the orbifold point in order to compare with the Chern-Simons results. Let us start by introducing the geometrical data, as in Sect. 5.2. The two charge vectors for local P1 × P1 are: Q 1 = (−2, 1, 1, 0, 0), Q 2 = (−2, 0, 0, 1, 1).

(6.35)

The mirror curve in the parameterization corresponding to an outer brane with zero framing is hyperelliptic and reads: H (x, ˜ y˜ ; qs , qt ) = y˜ 2 + y˜ (qs x˜ 2 + 1 + x) ˜ + qt x˜ 2 , with qs = e−ts and qt = e−tt .

(6.36)

170


This geometry was studied at large radius in [39]. Solving for y˜ we get: (1 + x˜ + x˜ 2 qs ) ± (1 + x˜ + x˜ 2 qs )2 − 4qt x˜ 2 y˜± = 2

(6.37)

from which we can construct the meromorphic differential (3.15). The Bergmann kernel can then be computed in terms of the branch points of the x-projection ˜ using Akemann’s formula (3.25). As in the case of local P2 , it is easier to work with s = x˜ −1 . In this variable, the branch points are given by: 1 √ 1 √ λ1,2 = − ∓ qt − (1 ± 2 qt )2 − 4qs , 2 2 1 √ 1 √ λ3,4 = − ∓ qt + (1 ± 2 qt )2 − 4qs . (6.38) 2 2 The large radius open flat coordinate for outer branes is given by the integral U = αu λ, where the cycle αu is analogous to the one in Fig. 2. This is evaluated to U = u˜ −

ts − Ts , 2

(6.39)

where Ts is the closed flat coordinate. We now have to implement the phase transition from large radius to the orbifold point. That is, we need to extract the disk and annulus amplitudes at the orbifold point from the large radius ones, as explained in Sect. 3.4. The disk transforms trivially, hence we just need to expand it in the appropriate variables at the orbifold point. However, the Bergmann kernel undergoes a non-trivial modular transformation. The phase transition from large radius to orbifold in the local P1 × P1 geometry is given by an S-duality transformation of the periods, corresponding to an exchange of the vanishing cycles.9 This is precisely the case that was studied in Sect. 3.4. This transformation can be implemented directly into the Bergmann kernel by permuting the branch points (λ1 , λ2 , λ3 , λ4 ) → (λ1 , λ4 , λ3 , λ2 )

(6.40)

in Akemann’s formula (3.25). All the other orbifold open amplitudes can then be generated by simply using the new Bergmann kernel (with the new ordering of the cuts) in the recursion. 6.3.1. Orbifold flat coordinates. We will now introduce the orbifold flat coordinates. Let us start with the closed ones. The appropriate variables to study the orbifold expansion were introduced in [4] and read: q1 = 1 −

qt , qs

1 q2 = √ qs 1 −

qt qs

.

(6.41)

9 In fact, this is not quite right. Going from large radius to the orbifold patch not only exchanges the cycles, but also changes the symplectic pairing by an overall factor of 2. Hence, the transformation is not quite symplectic; this is analogous to the transformation from large radius to the orbifold of local P2 considered in [1]. As was explained there, this change in the symplectic pairing can be taken into account by renormalizing the string coupling constant. In the present case, we get that gs = 2gˆ s , where gˆ s is the Chern-Simons coupling constant. This is also the origin of the 1/2 factors in (6.44).


171

In order to have q1 and q2 both small at the orbifold point, we have to take the following double—scaling limit: qt √ √ → ∞, (6.42) qs → ∞, qs 1 − qt → qs , qs corresponding to a blow up in the (qs , qt )-plane, which was described in detail in [4]. The flat coordinates s1 and s2 , are solutions of the Picard-Fuchs equations with a convergent local expansion in the variables q1 and q2 . The principal structure of the solutions of the orbifold Picard-Fuchs equations is ω0 = 1, s1 = − log(1 − q1 ), s2 = cm,n q1m q2n , m,n

Fs02

= s2 log(q1 ) +

dm,n q1m q2n ,

(6.43)

m,n

where the recursions of the cm,n and dm,n follow from the Picard-Fuchs operator. Note that the expansion coefficients cn,m have the property cm,n mod 2 = 0. The closed flat coordinates are related to the ’t Hooft parameters of Chern-Simons theory. The precise relation was found in [4] to be S1 =

1 (s1 + s2 ) T1 = , 2 4

S2 =

1 (s1 − s2 ) T2 = . 2 4

(6.44)

According to the location of the orbifold divisor at qs → ∞, described above, q2 picks up a phase under the orbifold monodromy MZ2 around it. Therefore, by definition (6.44) has the following behavior under orbifold monodromy: MZ2 : (S1 , S2 ) → (S2 , S1 ).

(6.45)

Notice that the closed string orbifold amplitudes, calculated in [4], are indeed invariant under the above MZ2 momodromy, as required for an orbifold expansion (see also [1]). Using the explicit form for the periods and the relation with the Chern-Simons variables we find the inverse mirror map for the closed parameters: 4 q1 = 2(S1 + S2 ) − 2(S1 + S2 )2 + (S1 + S2 )3 + · · · , 3 2 (S S2 − S1 S22 ) (S23 − S13 ) S1 − S2 1 q2 = + + ··· . + (S1 − S2 ) + 1 S1 + S2 2 12(S1 + S2 ) 24

(6.46)

We see from this expansion that, as already mentioned, q2 picks up a phase under orbifold monodromy. More precisely, we get the behavior: MZ2 : (q1 , q2 ) → (q1 , −q2 ).

(6.47)

Let us now consider the open flat coordinate. Recalling Sect. 2.3.5, the open flat coordinate should be a linear combination of u B = u˜ −

ts , 2

(6.48)

172


the disk amplitude A(0) ˜ qs , qt ) = 1 ( x;

βu

λ,

(6.49)

which according to (2.58) are globally defined integrals, and the closed string solutions (6.43). In this case, we fix the six coefficients in the definition of the open flat coordinates by matching the disk amplitude at the orbifold with the result from Chern-Simons theory. Defining as usual exponentiated coordinates X B = eu B and x˜ = eu˜ , we get for the open flat coordinate p := X or b = eu or b : √ p := X or b = X B = x˜ qs =

x˜ . q1 q2

(6.50)

Expanding the inverse relation x˜ = X or b q1 q2 , we get the open string inverse mirror map 1 2 x˜ = X or b 2(S1 − S2 ) − (S1 − S2 )(S1 + S2 ) + (S1 − S2 )(S1 + S2 ) + · · · . 3

(6.51)

(6.52)

Since x˜ is a globally defined variable on the curve, we see from (6.50) and (6.47) that under orbifold monodromy, MZ2 : X or b → −X or b .

(6.53)

This monodromy behavior of the open flat coordinate is crucial to ensure monodromy invariance of the topological string orbifold amplitudes. This mechanism is already visible in the first few terms of (6.52); under the orbifold MZ2 monodromy, the minus sign coming from S1 ↔ S2 cancels out with the minus sign coming from the action (6.53) on X or b , leaving the mirror map invariant. Furthermore, one can check that adding other periods si , Fs02 or the disk amplitude (0)

A1 to the definition of the open flat parameter would spoil this invariance property, so that we can fix the open flat parameter X or b uniquely, up to a scale. 6.3.2. Results. We have now all the ingredients required to compute open orbifold amplitudes. Let’s start with the disk amplitude: % S4 S 3 S2 S3 S 2 S2 (0) − 1 + S22 A1 ( p) = p 2 S1 − S12 + 1 − 1 − 2 S2 + 1 3 12 2 4 & S23 S1 S23 S24 S1 S22 − + + + ··· − 2 3 4 12 % 5 S13 3 S13 S2 11 S14 2 + p S1 − 2 S12 + − + S2 + S12 S2 − − 2 S22 3 12 2 & 3 S1 S23 11 S24 S12 S22 5 S23 2 + − − + ··· + S1 S2 − 2 3 2 12


173

%

59 S14 19 S13 S2 2 S1 2 S2 3 S12 S2 − 3 S12 + 5 S13 − − + − + 3 S22 3 12 3 2 4 & 3 S1 S22 19 S1 S23 59 S24 3 − 5 S2 + + ··· . (6.54) − 2 4 12 +p

3

Comparing (6.54) with the Chern-Simons result (6.28), we see that to match the two results we have to multiply (6.54) by − 21 and send S1 → −S1 , S2 → −S2 . With the above identifications we also checked that the higher amplitudes, such as the annulus, genus 0, three-hole and genus 1, one-hole, reproduce the Chern-Simons results. We notice that, as required, all the higher amplitudes are invariant under the MZ2 monodromy. Framing can also be taken into account; let us see how it goes for the disk amplitude. Higher amplitudes can be dealt with in a similar fashion. We start by computing the reparameterization x˜ = x(x) ˜ corresponding to the symplectic transformation (x, ˜ y˜ ) → (x, y) = (x˜ y˜ f , y˜ ) , which reads:

(6.55)

f + 3 f 2 − 2 f qs + 2 f qt x 3 x(x) ˜ =x− f x + + ... . 2 2

The bare framed disk amplitude is simply given by: dx A1(0) (x) = log y˜ (x(x)) . ˜ x

(6.56)

(6.57)

We could have computed y = y(x) by solving the framed mirror curve for y, rather than by reparameterizing y˜ (x); ˜ the reparameterization (6.56) is however required to compute the framed Bergmann kernel. We then have to expand the bare disk amplitude (6.57) in the orbifold variables (6.41), and express the result in flat coordinates using the inverse mirror maps (6.46) and (6.52). Doing so, we obtain a perfect matching with the Chern-Simons result (6.33) once the identification f cs = 2 f

(6.58)

between the Chern-Simons integer f cs and the integer f appearing in the symplectic transformation is taken into account. The matching holds for higher amplitudes with the above identification. 6.4. The C3 /Z3 orbifold. We studied in detail the open amplitudes at the local P1 × P1 orbifold point, and checked our results with Chern-Simons theory using large N duality. Here we will make a prediction for the disk amplitude at the local P2 orbifold point, which corresponds to the geometric orbifold C3 /Z3 . Basically, we use the same principles formulated in Sect. 2.3.5 to determine the flat parameters at the orbifold point, up to a scale factor. This is sufficient to predict the disk amplitude. To go to higher amplitudes, we would also need to understand the modular transformation of the annulus amplitude. We are presently working on that and hope to report on it in the near future.

174


Recall from Sect. 2.3.6 that the chain integral giving the open flat parameter at large radius of local P2 is given by t−T , 3

U = u˜ −

(6.59)

and the invariant combination of integrals is t u B = u˜ − . 3

(6.60)

Since this is globally defined, it provides a basis vector for the flat coordinates at the orbifold point. In terms of exponentiated coordinates X B = eu B , x˜ = eu˜ and q = e−t , we get 1

1

˜ 3 = (−) 3 X B = xq

x˜ , 3ψ

(6.61)

1 where we introduced the variable ψ on the moduli space defined by q = − (3ψ) 3 , so that the orbifold point is at q → ∞, or ψ → 0. To determine the open flat coordinate at the orbifold point, we can form combinations (0) of the closed periods, the chain integral u B and the disk amplitude A1 (x, ˜ q). But we find that 1

X or b = X B = (−) 3

x˜ 3ψ

(6.62)

is the only combination which leads to a monodromy invariant orbifold disk amplitude. Using this open flat parameter, we can write down explicitly the disk amplitude for C3 /Z3 . In [1], the closed flat parameter σ at the orbifold point was determined, using the Picard-Fuchs equations. We refer the reader to [1] for the explicit form of σ as an expansion in ψ around ψ = 0. Using this result and the open flat parameter (6.62), we get the following disk amplitude, up to a scale of X or b :

29 σ17 σ14 6607 (0) 10 − + σ + . . . X or b A1 = σ1 + 648 3674160 71425670400 1

σ15 197 σ18 σ12 5737 11 2 − + − σ + . . . X or + − b 4 1296 58786560 142851340800 1

σ6 σ19 1 σ3 3 4 . . . X or + − + 1 + 1 − (6.63) b + O(X or b ). 3 9 1944 544320 Notice that under the Z3 orbifold monodromy, given by 2πi 3

ψ,

MZ3 : (X or b , σ ) → (e−

2πi 3

ψ → e

(6.64)

we have that X or b , e

2πi 3

σ ),

which leaves the disk amplitude (6.63) invariant, as it should.

(6.65)


175

7. Conclusion and Future Directions The formalism proposed in this paper opens the way for various avenues of research. Let us mention a few specific ideas. • In this paper we proposed a complete B-model formalism to compute open and closed topological string amplitudes on local Calabi-Yau threefolds. An obvious question is whether we can extend this formalism to compact Calabi-Yau threefolds. At first sight this seems like a difficult task, since we relied heavily on the appearance of the mirror curve in the B-model geometry to implement the recursive formalism of Eynard and Orantin. However, there are various approaches that one could pursue. One could try to generalize the geometric formalism to higher-dimensional manifolds so that it applies directly to compact Calabi-Yau threefolds. Another idea, perhaps more promising, would be to formulate the recursion relations entirely in terms of physical objects in B-model topological string theory; in such a formalism it would not matter whether the target space is compact or non-compact. • We checked our formalism for all kinds of geometries, and have a rather clear understanding of the origin of the recursive solution based on the chiral boson interpretation of the B-model (see [39]). It was also proved in [23] that once the Bergmann kernel is promoted to a non-holomorphic, modular object, the amplitudes that we compute satisfy the usual holomorphic anomaly equations. But we do not have a proof that our formalism really is B-model topological string theory, not even a “physics proof”. It would be very interesting to produce such a proof, probably along the lines of [39]. • The recursion relations that we used were first found when the curve is the spectral curve of a matrix model. In the local geometries considered in this paper, there is no known matrix model corresponding to the mirror curves. Nevertheless, the recursion relations compute the topological string amplitudes. It would be fascinating to try to find a matrix model governing topological string theory on these local geometries. This could also provide a new approach towards a non-perturbative formulation of topological string theory. • Our formalism can be used to study phase transitions in the open/closed moduli spaces, and generate open and closed amplitudes at any point in the moduli space, including in non-geometric phases. We used this approach to study S-duality transformations and the orbifold point in the local P1 × P1 moduli space, and compared our results with Chern-Simons expectation values. We also proposed a prediction for the disk amplitude of C3 /Z3 , which corresponds to the orbifold point in the local P2 moduli space. However, while closed orbifold Gromov-Witten invariants are well understood mathematically, to our knowledge open orbifold Gromov-Witten invariants have not been defined mathematically. Hence, it would be fascinating to extend our analysis further and obtain a physics prediction for the higher open invariants of C3 /Z3 . In order to obtain these results, we would need to understand the Bergmann kernel at the orbifold point; this is more complicated than the P1 × P1 example studied in this paper since the tranformation from large radius to the orbifold is now in S L(2, C) — see [1]. We are presently working on that and should report on it in the near future. • Notice that the closed and open string amplitudes on X p provide the ’t Hooft resummation at strong coupling of the perturbative amplitudes of Chern–Simons gauge theory on L( p, 1), which is a nontrivial problem for p > 1. Already in the simple case of Chern–Simons theory on L(2, 1), the resummation problem involves considering a nontrivial moduli space, namely the moduli space of complex structures for

176


the mirror of X p , where the orbifold point corresponds to weak ’t Hooft coupling and the large radius point corresponds to strong ’t Hooft coupling. It would be interesting to see if the lessons extracted from this example have consequences for the problem of the ’t Hooft resummation of N = 4 SYM amplitudes, where a lot of progress has been made recently. At the very least, the topological example we have solved shows that the analytic structure of the ’t Hooft moduli space is very complicated, and that a clever parametrization of this space (by using an analogue of the mirror map) might simplify considerably the structure of the amplitudes. A. Useful Conventions In this appendix we recall some useful conventions necessary in order to compare the topological open string amplitudes (2.19) with the results of the topological vertex. In the formalism of the topological vertex [3], open string amplitudes are encoded in a generating functional depending on a U (∞) matrix V , F(V ) = log Z (V ), where Z (V ) =

(A.1)

Z R tr R V

(A.2)

R

is written as a sum over partitions R. It is often convenient to write the free energy F(V ) in terms of connected amplitudes in the basis labeled by vectors with nonnegative entries k = (k1 , k2 , · · · ). In this basis, 1 (c) W ϒ (V ), (A.3) F(V ) = z k ! k k k

where (see for example [38] for details) ϒk (V ) =

∞

(TrV j )k j ,

z k =

j=1

k j! jkj .

(A.4)

j

The functional (A.2) is related to the generating functions (2.19) as F(V ) =

∞ ∞

2g−2+h

gs

(g)

Ah (z 1 , · · · , z h ),

(A.5)

g=0 h=1

after identifying Tr V w1 · · · tr V wh ↔ m w (z) =

h σ ∈Sh i=1

i z σw(i) ,

(A.6)

where m w (z) is the monomial symmetric polynomial in the z i and Sh is the symmetric group of h elements. Under this dictionary we have that 1 (c) (g) W ϒ (V ), Ah (z 1 , · · · , z h ) = (A.7) z k ! k k k | |k|=h


177

where = |k|

kj.

(A.8)

j

Acknowledgements. It is a pleasure to thank Ron Donagi, Andy Neitzke, Tony Pantev, Cumrun Vafa and Johannes Walcher for helpful discussions. V.B., A.K. and S.P. would also like to thank the Theory Group at CERN for hospitality while part of this work was completed. The work of S.P. was partly supported by the Swiss National Science Foundation and by the European Commission under contracts MRTN-CT-2004-005104. The work of V.B. was partly supported by an NSERC postdoctoral fellowship.

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Communications in


Gauged Laplacians on Quantum Hopf Bundles Giovanni Landi1,2 , Cesare Reina3 , Alessandro Zampini4, 1 Dipartimento di Matematica e Informatica, Università di Trieste,

Via A. Valerio 12/1, I-34127 Trieste, Italy. E-mail: [email protected]

2 INFN, Sezione di Trieste, Trieste, Italy 3 Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014 Trieste, Italy.


4 Institut für Angewandte Mathematik, Universität Bonn, Wegelerstraße 6,

D-53115 Bonn, Germany. E-mail: [email protected] Received: 1 February 2008 / Accepted: 30 July 2008 Published online: 20 November 2008 – © The Author(s) 2008. This article is published with open access at Springerlink.com

Abstract We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect. 1. Introduction The motivation for the present paper is two-fold consisting in generalizing to noncommutative manifolds and bundles over them important mathematical and physical models. On the mathematical side we start from the following classical construction. Let (M, g) be a compact Riemannian manifold and P → M a principal bundle with compact structure Lie group G. With (ρ, V ) a representation of G, there is the well known identification of sections of the associated vector bundle E = P ×G V on M with equivariant maps from P to V , (M, E) C ∞ (P, V )G ⊂ C ∞ (P) ⊗ V . Given a connection on P one has a covariant derivative ∇ on (M, E), that is ∇ : (M, E) → (M, E) ⊗C ∞ (M) 1 (M). The Laplacian operator, E := −(∇∇ ∗ + ∇ ∗ ∇), where the dual ∇ ∗ is defined with the metric g, is a map from (M, E) to itself. This operator is related to the Laplacian operator on the total space P, P = −(dd∗ + d∗ d) acting on C ∞ (P), as E = P ⊗ 1 + 1 ⊗ C G C ∞ (P,V ) G (see e.g. [1, Prop. 5.6]). Here C G = a ρ(ξa )2 ∈ End(E) is the quadratic Casimir operator of G, with {ξa , a = 1, . . . , dim G} an orthonormal basis of the Lie algebra of G. Current address: Max Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany.

180

G. Landi, C. Reina, A. Zampini

The metric on P used for the dual d∗ is the canonical one obtained from the metrics on M and on the Lie algebra of G. We refer to E as the gauged Laplacian of M in the sense that the Laplacian M of M gets replaced by E when the exterior derivative d is replaced by the connection ∇. The above formula has several important applications, notably the study of the heat kernel expansion and index theorems on principal bundles. Our mathematical motivation is then to look for formulæ like the above relating Laplacian operators on noncommutative bundles and study their possible use. For the physical motivation the starting observation is the fact that the Laughlin wave functions [10] for the fractional quantum Hall effect (on the plane) are not translationally invariant. This problem was overcome in [7] with a model on a sphere with a magnetic monopole at the origin. The full Euclidean group of symmetries of the plane is recovered from the rotation group SO(3) of symmetries of the sphere. In fact, one is really dealing with the Hopf fibration of the sphere S3 = SU(2) over the sphere S2 with U(1) as gauge (or structure) group and needs to diagonalize the gauged Laplacian of S2 gauged with the monopole connection. For this bundle one has that P = SU(2) = CSU(2) and by the formula above the diagonalization is straightforward. It is then natural to seek models of Hall effect on noncommutative spaces. These will enjoy symmetry for quantum groups and lead to potentially interesting physical models. In the present paper we study a model of ‘excitations moving on a quantum 2-sphere’ and in the field of a magnetic monopole. The most striking fact is that the energies of the corresponding gauged Laplacian operator are not invariant under the exchange monopole/antimonopole, that is under inversion of the direction of the magnetic field, a manifestation of the phenomenon that ‘quantization removes degeneracy’. More specifically, the paper is organized as follows. In Sect. 2 we describe a quantum principal U(1)-bundle over a quantum sphere Sq2 having as total space the manifold of the quantum group SUq (2); there are natural associated line bundles classified by the winding number n ∈ Z. We describe in Sect. 5 the computation of the winding number of the bundles as well as a ‘twisted’ version of it. On the bundle one introduces differential calculi – recalled in Sect. 3 – that lead to the notion of gauge connection, given in Sect. 4. The gauged Laplacian operator on sections of associated bundles is introduced in Sect. 6. As mentioned, this operator describes ‘excitations moving on the quantum sphere’ and in the field of a magnetic monopole. There is symmetry under the action of the quantum universal enveloping algebra Uq (su(2)) – the Hopf algebra dual to the quantum group SUq (2) – that allows for a complete diagonalization of the gauged Laplacian. The energies of the Laplacian depend explicitly on the deformation parameter, an improvement with respect to a similar model of excitations in the field of instantons on a noncommutative 4-sphere [9] and, as mentioned, they are not invariant under the exchange monopole/antimonopole. The relation of the gauged Laplacian ∇ with the quadratic Casimir operator Cq of Uq (su(2)) is more involved than the classical one: 2 −1 K −2 q −1 K 2 − 2 + q K −2 2 1 1 qK − 2 + q , + q K ∇ = Cq + 4 − 2 (q − q −1 )2 (q − q −1 )2 with K the group-like element of Uq (su(2)), the ‘generator of the structure group’ U(1). In order to have a reasonable self-contained paper while making an effort to lighten it, we give in App. A some general constructions on differential calculi and quantum principal bundles with connections, and relegate to App. B some of the computations relevant to the U(1)-bundle over the quantum sphere Sq2 of Sect. 2. We leave to future work a more detailed study of potential applications to models of quantum Hall effect (with quantum group symmetries) as well as to heat kernel methods for noncommutative index theorems.

Gauged Laplacians on Quantum Hopf Bundles

181

2. The Quantum Hopf Bundle The quantum principal bundle we need is the well known U(1)-bundle over the standard Podle´s sphere Sq2 and whose total space is the manifold of the quantum group SUq (2). This bundle is an example of a quantum homogeneous space [2]. We recall in App. A the general framework of quantum principal bundles with nonuniversal calculi and endowed with a (gauge) connection. In this section we start with the algebras of the total and base space of the bundle and then we proceed to describe all bundles associated with the action of the group U(1). We also describe at length the related Peter-Weyl decomposition for the algebra A(SUq (2)) that we shall need later on in Sect. 6. 2.1. The algebras. The coordinate algebra A(SUq (2)) of the quantum group SUq (2) is the ∗-algebra generated by a and c, with relations, ac = qca ac∗ = qc∗ a cc∗ = c∗ c,

(2.1)

a ∗ a + c∗ c = aa ∗ + q 2 cc∗ = 1. The deformation parameter q ∈ R is taken in the interval 0 < q < 1, since for q > 1 one gets isomorphic algebras; at q = 1 one recovers the commutative coordinate algebra on the manifold SU(2). The Hopf algebra structure for A(SUq (2)) is given by the coproduct: a −qc∗ a −qc∗ a −qc∗ = ⊗ , c a∗ c a∗ c a∗ antipode:

S

and counit:

a −qc∗ c a∗

=

a ∗ c∗ , −qc a

a −qc∗ c a∗

=

10 . 01

The quantum universal enveloping algebra Uq (su(2)) is the Hopf ∗-algebra generated as an algebra by four elements K ,K −1 , E, F with K K −1 = 1 and subject to relations: K ± E = q ± E K ±,

K ± F = q ∓ F K ±,

[E, F] =

K 2 − K −2 . q − q −1

(2.2)

The ∗-structure is simply K∗ = K,

E ∗ = F,

F ∗ = E,

and the Hopf algebra structure is provided by coproduct , antipode S, counit : (K ± ) = K ± ⊗ K ± , (E) = E ⊗ K + K −1 ⊗ E, (F) = F ⊗ K + K −1 ⊗ F, S(K ) = K −1 , S(E) = −q E, S(F) = −q −1 F, (K ) = 1, (E) = (F) = 0. From the relations (2.2), the quadratic quantum Casimir element: Cq :=

q K 2 − 2 + q −1 K −2 + FE − (q − q −1 )2

1 4

(2.3)

182


generates the centre of Uq (su(2)). In the ‘classical limit q → 1’ the operator Cq goes to the Casimir element CSU(2) = H 2 + 21 (EF + FE), as can be seen by setting K = q H , expanding in the parameter =: log q and truncating at the 0th order in . In the limit the elements H, E, F are the generators of the Lie algebra su(2). There is a bilinear pairing between Uq (su(2)) and A(SUq (2)), given on generators by

K , a = q −1/2 , K −1 , a = q 1/2 , K , a ∗ = q 1/2 , K −1 , a ∗ = q −1/2 ,

E, c = 1, F, c∗ = −q −1 , and all other couples of generators pairing to 0. One regards Uq (su(2)) as a subspace of the linear dual of A(SUq (2)) via this pairing. There are [23] canonical left and right Uq (su(2))-module algebra structures on A(SUq (2)) such that

g, h x := gh, x , g, x h := hg, x ,

∀ g, h ∈ Uq (su(2)), x ∈ A(SUq (2)).

They are given by h x := (id ⊗h), x and x h := (h ⊗ id), x, or equivalently,

h x := x(1) h, x(2) , x h := h, x(1) x(2) , in the Sweedler notation. These right and left actions are mutually commuting:

(h a) g = a(1) h, a(2) g = g, a(1) a(2) h, a(3) = h g, a(1) a(2) = h (a g), and since the pairing satisfies

(Sh)∗ , x = h, x ∗ ,

∀ h ∈ Uq (su(2)), x ∈ A(SUq (2)),

the ∗-structure is compatible with both actions: h x ∗ = ((Sh)∗ x)∗ , x ∗ h = (x (Sh)∗ )∗ ,

∀ h ∈ Uq (su(2)), x ∈ A(SUq (2)).

We list both actions on powers of generators. Here and below we shall use the ‘q-number’, [x] = [x]q :=

q x − q −x , q − q −1

(2.4)

defined for q = 1 and any x ∈ R. First the left action; for s = 1, 2, . . . : s

s

s

s

K ± a s = q ∓ 2 a s , K ± a ∗s = q ± 2 a ∗s , K ± cs = q ∓ 2 cs , K ± c∗s = q ± 2 c∗s ; F a s = 0,

F a ∗s = q (1−s)/2 [s]ca ∗s−1 ,

F cs = 0,

F c∗s = −q −(1+s)/2 [s]ac∗s−1 ; E a = −q E c∗s = 0. s

(3−s)/2

[s]a

s−1 ∗

c ,

(2.5) E a

∗s

= 0,

E c =q s

(1−s)/2

[s]c

s−1 ∗

a ,

Then the right one: s

s

s

a s K ± = q ∓ 2 a s , a ∗s K ± = q ± 2 a ∗s , cs K ± = q ± 2 cs , s

c∗s K ± = q ∓ 2 c∗s ; a s F = q (s−1)/2 [s]ca s−1 , a ∗s F = 0, cs F = 0, c

∗s

F = −q

−(s−3)/2

a E = 0, a c∗s E = 0. s

∗s

∗ ∗s−1

[s]a c

E = −q

(2.6)

;

(3−s)/2

[s]c∗ a ∗s−1 , cs E = q (s−1)/2 [s]cs−1 a,


183

We also need to recall (see e.g. [11, Prop. 3.2.6]) that the irreducible finite dimensional ∗-representations σ J of Uq (su(2)) are labelled by nonnegative half-integers J ∈ 21 N (the spin); they are given by σ J (K ) |J, m = q m |J, m, σ J (E) |J, m = [J − m][J + m + 1] |J, m + 1, σ J (F) |J, m = [J − m + 1][J + m] |J, m − 1,

(2.7)

where the vectors |J, m, for m = J, J − 1, . . . , −J + 1, −J , form an orthonormal basis for the (2J + 1)-dimensional, irreducible Uq (su(2))-module V J , and the brackets denote the q-number as in (2.4). Moreover, σ J is a ∗-representation of Uq (su(2)), with respect to the hermitian scalar product on V J for which the vectors |J, m are orthonormal. In each representation V J , the Casimir (2.3) is a multiple of the identity with constant given by: Cq(J ) = [J + 21 ]2 − 41 .

(2.8)

Denote A(U(1)) := C[z, z ∗ ] < zz ∗ − 1 >; the map: π : A(SUq (2)) → A(U(1)),

π

a −qc∗ c a∗

z 0 = 0 z∗

(2.9)

is a surjective Hopf ∗-algebra homomorphism, so that A(U(1)) becomes a quantum subgroup of SUq (2) with a right coaction, R := (id ⊗π ) ◦ : A(SUq (2)) → A(SUq (2)) ⊗ A(U(1)).

(2.10)

The coinvariant elements for this coaction, that is elements b ∈ A(SUq (2)) for which R (b) = b ⊗ 1, form a subalgebra of A(SUq (2)) which is the coordinate algebra A(Sq2 ) of the standard Podle´s sphere Sq2 . It is straightforward to establish that R (a) = a ⊗ z, R (a ∗ ) = a ∗ ⊗ z ∗ , R (c) = c ⊗ z, R (c∗ ) = c∗ ⊗ z ∗ . As a set of generators (different from the original ones in [16]) for A(Sq2 ) we take B− := −ac∗ ,

B+ := qca ∗ ,

B0 :=

q2 − q 2 cc∗ , 1 + q2

for which one finds relations: B− B0 = q 2 B0 B− , B0 B+ = q 2 B+ B0 ,

B+ B− = q q −2 B0 − (1 + q 2 )−1 q −2 B0 + (1 + q −2 )−1 ,

B− B+ = q B0 + (1 + q 2 )−1 B0 − (1 + q −2 )−1 , and ∗-structure: (B0 )∗ = B0 ,

(B+ )∗ = −q B− .

(2.11)

184


The sphere Sq2 is a quantum homogeneous space of SUq (2) and the coproduct of A(SUq (2)) restricts to a left coaction of A(SUq (2)) on A(Sq2 ) which, on generators reads: (B− ) = a 2 ⊗ B− − (1 + q −2 )B− ⊗ B0 + c∗2 ⊗ B+ ,

(B0 ) = q ac ⊗ B− + (1 + q −2 )B0 ⊗ B0 − c∗ a ∗ ⊗ B+ ,

(B+ ) = q 2 c2 ⊗ B− + (1 + q −2 )B+ ⊗ B0 + a ∗2 ⊗ B+ .

The algebra inclusion A(Sq2 ) → A(SUq (2)) is a quantum principal bundle and can be endowed with compatible calculi [2], a construction that we shall illustrate below.

2.2. The associated line bundles. The left action of the group-like element K on A(SUq (2)) allows one [13, Eq. (1.10)] to give a vector basis decomposition A(SUq (2)) = ⊕n∈Z Ln , where, Ln := {x ∈ A(SUq (2)) : K x = q n/2 x}.

(2.12)

In particular A(Sq2 ) = L0 . Also, L∗n ⊂ L−n and Ln Lm ⊂ Ln+m . Each Ln is clearly a bimodule over A(Sq2 ). It was shown in [18, Prop. 6.4] that each Ln is isomorphic to a projective left A(Sq2 )-module of rank 1. These projective left A(Sq2 )-modules give modules of equivariant maps or of sections of line bundles over the quantum sphere Sq2 with winding numbers (monopole charge) −n (see Sect. 5 below). The corresponding projections (cf. [4,6]) can be written explicitly. They are elements p(n) and pˇ (n) in Mat|n|+1 (A(Sq2 )) (for n ≥ 0 and n ≤ 0 respectively) whose explicit form we give below. For n ≥ 0, (n) p(n) = (n) (n) , = βn,µ c∗µ a ∗n−µ , µ (2.13) µ−1 1 − q −2(n− j) , µ = 0, . . . , n . with βn,µ = q 2µ j=0 1 − q −2( j+1) Here and below we use the convention that p(n) µν =

j=0 (·)

= 1. The entries of p(n) are:

βn,µ βn,ν c∗µ a ∗n−µ a n−ν cν ∈ A(Sq2 ).

For n ≤ 0 one has instead, ˇ (n) ˇ (n) pˇ (n) = , with

−1

αn,µ =

√ ˇ (n) = αn,µ c|n|−µ a µ , µ

|n|−µ−1 1 − q 2(|n|− j) j=0

1 − q 2( j+1)

(2.14) , µ = 0, . . . , |n| .

The entries of pˇ (n) are: pˇ (n) µν =

√ αn,µ αn,ν c|n|−µ a µ a ∗ν c∗|n|−ν ∈ A(Sq2 ).


185

Both p(n) and pˇ (n) are self-adjoint by construction. Using the commutation relations (2.1) of A(SUq (2)) and the explicit form of the coefficients (2.13) and (2.14) a long but straightforward computation shows that: n (n) , (n) = βn,µ a n−µ cµ c∗µ a ∗n−µ = (aa ∗ + q 2 cc∗ )n = 1, (2.15) µ=0

and analogously |n| ˇ (n) , ˇ (n) = αn,µ a ∗µ c∗|n|−µ c|n|−µ a µ = (a ∗ a + c∗ c)|n| = 1, (2.16) µ=0

from which both p(n) and pˇ (n) are idempotents: (p(n) )2 = p(n) and (pˇ (n) )2 = pˇ (n) . Remark 2.1. The coefficients αn,µ and βn,µ above are q-binomial coefficients and their expression is so as to get the identities (2.15) and (2.16). When computing the q-winding number of the bundles in Sect. 5 below we shall need the relation q −2µ+2µ(n−µ) βn,µ = αn,n−µ ,

µ = 0, . . . , n,

(2.17)

which is obtained by a straightforward computation. The projections (2.13) and (2.14) play a central role throughout our paper. As a first application of their properties one shows that A(SUq (2)), A(Sq2 ), A(U(1)) is a quantum principal bundle: the relevant proof is in App. B.1. Next, we identify the spaces of equivariant maps Ln with the left A(Sq2 )-modules of sections (A(Sq2 ))n+1 p(n) or (A(Sq2 ))|n|+1 pˇ (n) , according to whether n is positive or negative. For this we write any element in the free module (A(Sq2 ))|n|+1 as f | = ( f 0 , f 1 , . . . , f n ) with f µ ∈ A(Sq2 ). This allows one to write equivariant maps as n φ f := f, (n) = f µ βn,µ c∗µ a ∗n−µ , for n ≥ 0, µ=0 |n| √ (n) ˇ φˇ f := f, = f µ αn,µ c|n|−µ a µ , for n ≤ 0. µ=0

Writing equivariant maps in the above form, the following proposition is easy to establish. Proposition 2.2. Let En := (A(Sq2 ))n+1 p(n) or Eˇn = (A(Sq2 ))|n|+1 pˇ (n) according to whether n ≥ 0 or n ≤ 0. There are left A(Sq2 )-modules isomorphisms: Ln −−→ En , φ f → σ f := φ f (n) = f | p(n) , with inverse

En −−→ Ln , σ f = f | p(n) → φ f := f, (n) ,

and similar maps for the case n ≤ 0. Remark 2.3. By exchanging the role of the previous projections p(n) and pˇ (n) one has also an identification of right modules: Ln pˇ (n) (A(Sq2 ))n+1 and Ln p(n) (A(Sq2 ))|n|+1 for n ≥ 0 and n ≤ 0 respectively.

186


In the sequel, to lighten the notation we shall simply write Ln En := (A(Sq2 ))|n|+1 p(n) with the positivity or negativity of the label n deciding which projection to be considered, the ones given in (2.13) and (2.14) for n ≥ 0 or n ≤ 0 respectively. The final result that we need in this section is the known decomposition of the modules Ln into representation spaces under the (right) action of Uq (su(2)). We have already mentioned the vector space decomposition A(SUq (2)) = ⊕n∈Z Ln . From the definition of the bundles Ln in (2.12) and the relations (2.2) of Uq (su(2)) one gets that, E Ln ⊂ Ln+2 ,

F Ln ⊂ Ln−2 .

(2.18)

On the other hand, commutativity of the left and right actions of Uq (su(2)) yields that Ln h ⊂ Ln ,

∀ h ∈ Uq (su(2)).

In fact, it was already shown in [18, Thm. 4.1] that there is also a decomposition, (n) Ln := VJ , (2.19) |n| |n| |n| J = 2 , 2 +1, 2 +2,... (n)

with V J the spin J -representation space (for the right action) of Uq (su(2)). Altogether we get a Peter-Weyl decomposition for A(SUq (2)) (already given in [23]). (n) More explicitly, the highest weight vector for each V J in (2.19) is c J −n/2 a ∗J +n/2 : K (c J −n/2 a ∗J +n/2 ) = q n/2 (c J −n/2 a ∗J +n/2 ), (c J −n/2 a ∗J +n/2 ) K = q J (c J −n/2 a ∗J +n/2 ), (n)

Analogously, the lowest weight vector for each V J

K (a J −n/2 c∗J +n/2 ) = q n/2 (a J −n/2 c∗J +n/2 ), (a J −n/2 c∗J +n/2 ) K = q −J (a J −n/2 c∗J +n/2 ),

(2.20) (c J −n/2 a ∗J +n/2 ) F = 0. in (2.19) is a J −n/2 c∗J +n/2 : (a J −n/2 c∗J +n/2 ) E = 0.

(n)

The elements of the vector spaces V J can be obtained by acting on the highest weight vectors with the lowering operator E, since clearly c J −n/2 a ∗J +n/2 E ∈ Ln , or explicitly,

K c J −n/2 a ∗J +n/2 E = q n/2 c J −n/2 a ∗J +n/2 E . To be definite, let us consider n ≥ 0. The first admissible J is J = n/2; the highest weight (n) element is a ∗n and the vector space Vn/2 is spanned by {a ∗n E l } with l = 0, . . . , n + 1: (n)

Vn/2 = span{a ∗n , c∗ a ∗n−1 , . . . , c∗n }. Keeping n fixed, the other admissible values of J (n)

are J = s + n/2 with s ∈ N. The vector spaces Vs+n/2 are spanned by {cs a ∗s+n E l } with l = 0, . . . , 2s + n + 1. Analogous considerations are valid when n ≤ 0. In these cases, the admissible values of J are J = s + |n| /2 = s − n/2, the highest weight vector in (n) Vs−n/2 is the element cs−n a ∗s , and a basis is given by the action of the lowering operator (n)

E, that is Vs−n/2 = span{ cs−n a ∗s E l , l = 0, . . . , 2s − n + 1}. We know from (2.18) that the left action F maps Ln to Ln−2 . If p ≥ 0, the element ( p) a ∗ p is the highest weight vector in V p/2 and one has that F a ∗ p ∝ ca ∗ p−1 . The element


187 ( p−2)

ca ∗ p−1 is the highest weight vector in V p/2 since one finds that (ca ∗ p−1 ) F = 0 and (ca ∗ p−1 ) K = q p/2 (ca ∗ p−1 ). In the same vein, the elements F t a ∗ p ∝ ct a ∗ p−t are ( p−2t) the highest weight elements in V p/2 ⊂ L p−2t , t = 0, . . . , p. Once again, a complete ( p−2t)

basis of each subspace V p/2 is obtained by the right action of the lowering operator E. With these considerations, the algebra A(SUq (2)) can be partitioned into finite dimensional blocks which are the analogues of the Wigner D-functions [21] for the group SU (2). To illustrate the meaning of this partition, let us start with the element a ∗ , the (1) highest weight vector of the space V1/2 . Representing the left action of F with a horizontal arrow and the right action of E with a vertical one, yields the box a∗ → c ↓ ↓ , −qc∗ → a (1) , while the second column is a basis where the first column is a basis of the subspace V1/2 (−1)

(2)

of the subspace V1/2 . Starting from a ∗2 –the highest weight vector of V1 –one gets: → q −1/2 [2] ca ∗ → [2] c2 a ∗2 ↓ ↓ ↓ −q 1/2 [2] c∗ a ∗ → [2] (aa ∗ − cc∗ ) → [2]2 q 1/2 ca . ↓ ↓ ↓ q 2 [2] c∗2 → −q 3/2 [2]2 ac∗ → q 3 [2]2 a 2 (2)

(0)

(−2)

The three columns of this box are bases for the subspaces V1 , V1 ,V1 , respectively. The recursive structure is clear. For a positive integer p, one has a box W p made up of ( p + 1) × ( p + 1) elements. Without explicitly computing the coefficients, one gets: a∗ p ↓ c∗ a ∗ p−1 ↓ ... ↓ c∗s a ∗ p−s ↓ ... ↓ c∗ p

→ ca ∗ p−1 ↓ → ... ↓ → ... ↓ → ... ↓ → ... ↓ → ac∗ p−1

→ ... ... → ... ... → ... ... → ... ... → ... ... → ...

→ ct a ∗ p−t ↓ → ... ↓ → ... ↓ → ... ↓ → ... ↓ → a t c∗ p−t

→ ... ... → ... ... → ... ... → ... ... → ... ... → ...

→

cp ↓ → ac p−1 ↓ → ... ↓ . → a s c p−s ↓ → ... ↓ → ap

The space W p is the direct sum of representation spaces for the right action of Uq (su(2)), p

( p−2t)

W p = ⊕t=0 V p/2

,

and on each W p the quantum Casimir Cq act in the same manner from both the right and the left, with eigenvalue (2.8), that is Cq w p = w p Cq = [ p + 21 ]2 − 41 w p , for all w p ∈ W p . The Peter-Weyl decomposition for the algebra A(SUq (2)) is given as p ( p−2t) . A(SUq (2)) = ⊕ p∈N W p = ⊕ p∈N ⊕t=0 V p/2

188


At the classical value of the deformation parameter, q = 1, one recovers the classical Peter-Weyl decomposition for the group algebra of SU (2). A straightforward computation shows that the elements in W p which are written (up to a multiplicative constant) as w p:t,r := F t a ∗ p E r , for t, r = 0, 1, . . . , p, become, in the classical limit, proporp/2 tional to the Wigner D-functions Dt− p/2;r − p/2 for SU (2)–the building blocks [21] of the Peter-Weyl decomposition of the algebra of functions on the manifold of the group SU (2). In order to get elements in the Podle´s sphere subalgebra A(Sq2 ) L0 out of a highest weight vector a ∗ p we need p = 2l to be even and left action of F l : F l a ∗2l ∝ cl a ∗l ∈ A(Sq2 ). Then, the right action of E yields a spherical harmonic decomposition, (0)

A(Sq2 ) = ⊕l∈N Vl ,

(2.21)

(0)

with a basis of Vl given by the vectors F l a ∗2l E r , for r = 0, 1, . . . , 2l. At the classical value of the deformation parameter these vectors become the standard spherical harmonics, {Ylr , l ∈ N, r = 0, 1, . . . , 2l}, which build up the spherical harmonics decomposition of the algebra of functions on the classical sphere S2 . 3. The Calculi on the Quantum Principal Bundle The principal bundle (A(SUq (2)), A(Sq2 ), A(U(1))) is endowed [2,3] with compatible nonuniversal calculi obtained from the 3-dimensional left-covariant calculus [23] on SUq (2) we present first. We then describe the unique left covariant 2-dimensional calculus [17] on the sphere Sq2 obtained by restriction and the projected calculus on the ‘structure Hopf algebra’ A(U(1)). All these calculi are compatible in a natural sense. 3.1. The left-covariant calculus on SUq (2). The first differential calculus we take on the quantum group SUq (2) is the left-covariant one already developed in [23]. It is three dimensional with corresponding ideal QSUq (2) generated by the 6 elements {a ∗ + q 2 a − (1 + q 2 ); c2 ; c∗ c; c∗2 ; (a − 1)c; (a − 1)c∗ }. The quantum tangent space X (SUq (2)) is generated by the three elements: Xz =

1 − K4 , 1 − q −2

X − = q −1/2 F K ,

X + = q 1/2 E K ,

whose coproducts are easily found: X z = 1 ⊗ X z + X z ⊗ K 4 ,

X ± = 1 ⊗ X ± + X ± ⊗ K 2 .

The dual space of 1-forms 1 (SUq (2)) has a basis ωz = a ∗ da + c∗ dc,

ω− = c∗ da ∗ − qa ∗ dc∗ ,

ω+ = adc − qcda,

(3.1)

(1) (1) the (left) coacof left-covariant forms, that is (1) L (ωs ) = 1 ⊗ ωs , with L = tion of A(SUq (2)) onto itself extended to forms. The differential d : A(SUq (2)) → 1 (SUq (2)) is

d f = (X + f ) ω+ + (X − f ) ω− + (X z f ) ωz ,

(3.2)


189

for all f ∈ A(SUq (2)). The above relations (3.1) can be inverted to da = −qc∗ ω+ + aωz , dc = a ∗ ω+ + cωz ,

da ∗ = −q 2 a ∗ ωz + cω− ,

dc∗ = −q 2 c∗ ωz − q −1 aω− ,

∗ = −ω and ω∗ = −ω . The bimodule structure is: from which one also gets that ω− + z z

ωz a = q −2 aωz , ωz a ∗ = q 2 a ∗ ωz , ω± a = q −1 aω± , ωz c = q

−2

∗

2 ∗

cωz , ωz c = q c ωz , ω± c = q

−1

ω± a ∗ = qa ∗ ω± , ∗

cω± ,

(3.3)

∗

ω± c = qc ω± ,

Higher dimensional forms can be defined in a natural way by requiring compatibility for commutation relations and that d2 = 0. One has: dωz = −ω− ∧ ω+ , dω+ = q 2 (1 + q 2 )ωz ∧ ω+ , dω− = −(1 + q −2 )ωz ∧ ω− , (3.4) together with commutation relations: ω+ ∧ ω+ = ω− ∧ ω− = ωz ∧ ωz = 0, ω− ∧ ω+ + q −2 ω+ ∧ ω− = 0, ωz ∧ ω− + q ω− ∧ ωz = 0, 4

(3.5) ωz ∧ ω+ + q

−4

ω+ ∧ ωz = 0.

Finally, there is a unique top form ω− ∧ ω+ ∧ ωz . 3.2. The standard calculus on Sq2 . The restriction of the above 3D calculus to the sphere Sq2 yields the unique left covariant 2-dimensional calculus on the latter [12]. An evolution of this approach has led [19] to a description of the unique 2D calculus of Sq2 in terms of a Dirac operator. The ‘cotangent bundle’ 1 (Sq2 ) is shown to be isomorphic to the direct sum L−2 ⊕ L2 , that is the line bundles with winding number ±2. Since the element K acts as the identity on A(Sq2 ), the differential (3.2) becomes, when restricted to the latter, d f = (X − f ) ω− + (X + f ) ω+

= (F f ) (q −1/2 ω− ) + (E f ) (q 1/2 ω+ ),

for f ∈ A(Sq2 ).

These leads to break the exterior differential into a holomorphic and an anti-holomorphic part, d = ∂¯ + ∂, with: ∂¯ f = (X − f ) ω− = (F f ) (q −1/2 ω− ), ∂ f = (X + f ) ω+ = (E f ) (q 1/2 ω+ ),

for

f ∈ A(Sq2 ).

An explicit computation on the generators (2.11) of Sq2 yields: ∂¯ B− = q −1 a 2 ω− , ∂¯ B0 = q ca ω− , ∂¯ B+ = q c2 ω− ,

∂ B+ = q 2 a ∗2 ω+ , ∂ B0 = −q 2 c∗ a ∗ ω+ , ∂ B− = q 2 c∗2 ω+ .

2 ¯ The above shows that: 1 (Sq2 ) = 1− (Sq2 )⊕1+ (Sq2 ), where 1− (Sq2 ) L−2 ∂(A(S q )) is the A(Sq2 )-bimodule generated by:

{∂¯ B− , ∂¯ B0 , ∂¯ B+ } = {a 2 , ca, c2 } ω− = q 2 ω− {a 2 , ca, c2 }

190


2 ¯ and 1+ (Sq2 ) L+2 ∂(A(S q )) is the one generated by:

{∂ B+ , ∂ B0 , ∂ B− } = {a ∗2 , c∗ a ∗ , c∗2 } ω+ = q −2 ω+ {a ∗2 , c∗ a ∗ , c∗2 }. That these two modules of forms are not free is also expressed by the existence of relations among the differential: ∂ B0 = q −1 B− ∂ B+ − q 3 B+ ∂ B− ,

∂¯ B0 = q B+ ∂¯ B− − q −3 B− ∂¯ B+ .

The 2D calculus on Sq2 has a unique top 2-form ω with ω f = f ω, for all f ∈ A(Sq2 ) and 2 (Sq2 ) is the free A(Sq2 )-module generated by ω, that is 2 (Sq2 ) = ωA(Sq2 ) = A(Sq2 )ω. Now, both ω± commutes with elements of A(Sq2 ) and so does ω− ∧ω+ , which is taken as the natural generator ω = ω− ∧ ω+ of 2 (Sq2 ). Writing any 1-form as α = xω− + yω+ ∈ L−2 ω− ⊕ L+2 ω+ , the product of 1-forms is: (xω− + yω+ ) ∧ (tω− + zω+ ) = (q −2 yt − x z)ω+ ∧ ω− . From (3.4) it is natural to ask that dω− = dω+ = 0 when restricted to Sq2 . Then, the exterior derivative of any 1-form α = xω− + yω+ ∈ L−2 ω− ⊕ L+2 ω+ is given by: dα = d(xω− + yω+ ) = ∂ x ∧ ω− + ∂¯ y ∧ ω+

= (X + x − q −2 X − y) ω+ ∧ ω− = q −1/2 (E x − q −1 F y) ω+ ∧ ω− , (3.6)

since K acts as q ∓ on L∓2 . Notice that in the above equality, both E x and F y belong to A(Sq2 ), as it should be. We summarize the above results in the following proposition: Proposition 3.1. The 2D differential calculus on the sphere Sq2 is given by: • (Sq2 ) = A(Sq2 ) ⊕ (L−2 ω− ⊕ L+2 ω+ ) ⊕ A(Sq2 )ω+ ∧ ω− , with multiplication rule

f 0 ; x, y; f 2 g0 ; t, z; g2 = f 0 g0 ; f 0 t + xg0 , f 0 z + yg0 ; f 0 g2 + f 2 g0 + q −2 yt − x z ,

and exterior differential d = ∂¯ + ∂: f → (q −1/2 F f, q 1/2 E f ), for f ∈ A(Sq2 ), (x, y) → q −1/2 (E x − q −1 F y), for (x, y) ∈ L−2 ⊕ L+2 . Remark 3.2. It is evident that the relevant operators for the calculus on A(Sq2 ) are {E, F} rather than {X + , X − }; indeed, it is proven in [19] that {E, F} span the quantum tangent space of the 2D calculus (2 (Sq2 ), d). We mantain {X + , X − } as well since these are the operators that will be lifted to the total space A(SUq (2)) via the connection and that will enter the gauged Laplacian later on.


191

3.3. The calculus on the structure group. From App. A.2 we know that a quantum principal bundle with nonuniversal calculi comes with compatible calculi on the ‘total space algebra’ P and on the ‘structure Hopf algebra’ H. The compatibility determines a calculus on the ‘base space algebra’ B. For the Hopf bundle over the sphere Sq2 we have already given the calculus on Sq2 starting from the left covariant one on SUq (2). Here we give the calculus on U(1) which makes them compatible from the quantum principal bundle point of view. The strategy [2] consists in defining the calculus on U(1) via the Hopf projection π in (2.9). In particular, out of the QSUq (2) which determines the left covariant calculus on SUq (2), one defines a right ideal QU(1) = π(QSUq (2) ) for the calculus on U(1). One finds that QU(1) is generated by the element {z ∗ + q 2 z − (1 + q 2 )} and the corresponding calculus is 1-dimensional, and bicovariant. Its quantum tangent space is generated by X = Xz =

1 − K4 , 1 − q −2

(3.7)

with dual 1-form given by ωz . Explicitly, one finds that ωz = z ∗ dz,

dz = zωz ,

dz ∗ = −q 2 z ∗ ωz ,

and noncommutative commutation relations, ωz z = q −2 zωz ,

ωz z ∗ = q 2 z ∗ ωz ,

zdz = q 2 (dz)z.

We know from Sect. 2.2 and App. B.1 that the datum (A(SUq (2)), A(Sq2 ), A(U(1))) is a ‘topological’ quantum principal bundle. We have now a specific differential calculus both on the total space A(SUq (2)) (the 3D left covariant calculus) and on A(U(1)) (obtained from it via the same projection π in (2.9) giving the bundle structure). Moreover, from the calculus on A(SUq (2)), we also obtained via restriction a calculus on the base space A(Sq2 ). It remains to show that these calculi are compatible so that the datum (A(SUq (2)), A(Sq2 ), A(U(1)); QSUq (2) , QA(U(1)) ) is a quantum principal bundle with nonuniversal calculi. As a result, the vector field X z is vertical. The details showing the compatibility are in App. B.2. In particular, from the analysis of the Appendix, the vector field (3.7) is a ‘vertical’ vector field for the fibration. 4. The Monopole Connection and its Curvature A connection on the quantum principal bundle with respect to the left covariant calculus (Sq2 ) will be the crucial ingredient for the gauged Laplacian operator. The connection will determine a covariant derivative on the module of equivariant maps Ln which will, in turn, be shown to correspond to the Grassmann connection on the modules (A(Sq2 ))|n|+1 p(n) . We shall also derive corresponding expressions for the curvature of the connection. 4.1. Enter the connection. The most efficient way to define a connection on a quantum principal bundle (with given calculi) is by splitting the 1-forms on the total space into horizontal and vertical ones [2,3]. Since horizontal 1-forms are given in the structure of the principal bundle, one needs a projection on forms whose range is the subspace of vertical ones. The projection is required to be covariant with respect to the right coaction of the structure Hopf algebra.

192


For the principal bundle over the quantum sphere Sq2 that we are considering, a principal connection is a covariant left module projection : 1 (SUq (2)) → 1ver (SUq (2)), that is 2 = and (xα) = x(α), for α ∈ 1 (SUq (2)) and x ∈ A(SUq (2)); equivalently it is a covariant splitting 1 (SUq (2)) = 1ver (SUq (2)) ⊕ 1hor (SUq (2)). The covariance of the connection is the requirement that (1)

(1)

R = ( ⊗ id) ◦ R , (1)

with R the extension to 1-forms of the coaction R (2.10) of the structure Hopf algebra. It is not difficult to realize that with the left covariant 3D calculus on A(SUq (2)), a basis for 1hor (SUq (2)) is given by ω− , ω+ (a proof is in App. B.2). Furthermore: (1)

R (ωz ) = ωz ⊗ 1,

(1)

R (ω− ) = ω− ⊗ z ∗2 ,

(1)

R (ω+ ) = ω+ ⊗ z 2 ,

and a natural choice of a connection is to define ωz to be vertical [2,12]: z (ωz ) := ωz ,

z (ω± ) := 0.

With a connection, one has a covariant derivative of equivariant maps, ∇ : E → (Sq2 ) ⊗A(Sq2 ) E,

∇ := (id −z ) ◦ d,

and one readily shows the Leibniz rule property: ∇( f φ) = f ∇(φ) + (d f ) ⊗ φ, for all φ ∈ E and f ∈ A(Sq2 ). We shall take for E the line bundles Ln of (2.12). Then, with the left covariant 2D calculus on A(Sq2 ) (coming from the left covariant 3D calculus on A(SUq (2)) as explained in Sect. 3.2) we have, ∇φ = (X + φ) ω+ + (X − φ) ω−

= q −n−2 ω+ (X + φ) + q −n+2 ω− (X − φ) ,

(4.1)

since X ± φ ∈ Ln±2 . The left action of Uq (su(2)) on (Sq2 ) is defined by requiring that it commutes with the exterior derivative d while on (Sq2 ) ⊗A(Sq2 ) E is defined in an obvious way as h (α ⊗ φ) = h (1) α ⊗ h (2) φ , for h ∈ Uq (su(2)), with notation (h) = h (1) ⊗ h (2) for the coproduct. Then, while X ± φ ∈ / Ln , one checks that K (∇φ) = q n ∇φ and ∇φ ∈ (Sq2 ) ⊗A(Sq2 ) E as it should be. It was shown in [6] that with the universal calculi on the principal bundle, the module E is projective and the covariant derivative ∇ corresponds to the Grassmann connection of the corresponding projection. We have a similar result for the left covariant calculi when taking as modules E the line bundles Ln of (2.12). Proposition 4.1. Let E be the line bundle Ln defined in (2.12). With the left A(Sq2 )-modules isomorphism Ln En := (A(Sq2 ))|n|+1 p(n) of Prop. 2.2 extended in a natural way to forms (Sq2 ) ⊗A(Sq2 ) Ln (Sq2 ) ⊗A(Sq2 ) En , the covariant derivative on Ln corresponds to the Grassmann connection on En , that is, (dσφ ) p(n) = σ∇φ , for φ ∈ Ln and the corresponding section σφ ∈ En .


193

Proof. We consider only the case n ≥ 0 since the proof for the case n ≤ 0 is the same. From Prop. 2.2, to any φ ∈ Ln there corresponds the section σφ = φ (n) ∈ En , having components (σφ )ν = (φ a n−ν cν ) βn,ν , ν = 0, 1, . . . , n. Similarly, to the covariant derivative ∇φ ∈ (Sq2 ) ⊗A(Sq2 ) Ln we associate the (Sq2 )-valued section σ∇φ = (∇φ) (n) ∈ (S2 ) ⊗A(S2 ) En . Explicitly: q

σ∇φ

q

= (X + φ) ω+ + (X − φ) ω− (n)

= q −n (X + φ) (n) ω+ + q −n (X − φ) (n) ω− ,

since (n) ∈ L−n and ω± L−n ⊂ q −n L−n ω± , from the commutation relations in the second column in (3.3). In components: σ∇φ µ = q −n (X + φ) (a n−µ cµ ) βn,µ ω+ + q −n (X − φ) (a n−µ cµ ) βn,µ ω− . On the other hand, for the Grassmann connection ∇σφ := d(σφ ) p(n) acting on the section σφ , using the commutativity of ω± with L0 = A(Sq2 ), one finds:

n n d σφ p(n) = d σφ ν p(n) = d φ a n−ν cν βn,ν p(n) νµ νµ ν=0 ν=0 µ n n−ν ν = X+ φ a c βn,ν p(n) νµ ω+ ν=0 n n−ν ν + X− φ a c βn,ν p(n) νµ ω− ν=0 n n−ν ν = c + q −n (X + φ) (a n−ν cν ) βn,ν p(n) φ X+ a νµ ω+ ν=0 n + q −n X − φ ) ( a n−ν cν βn,ν p(n) νµ ω− ν=0 n = q −n (X + φ) (a n−ν cν ) βn,ν p(n) νµ ω+ ν=0 n +q −n (X − φ) (a n−ν cν ) βn,ν p(n) νµ ω− ν=0 = q −n (X + φ) (a n−µ cµ ) βn,µ ω+ +q −n (X − φ) (a n−µ cµ ) βn,µ ω− = σ∇φ µ . Here the fourth equality follows from the vanishing n −n X + a n−ν cν βn,ν p(n) νµ = q [X + (1)] = 0. ν=0

As mentioned, in a similar fashion one proves the same result for the case n ≤ 0.

4.2. The curvature. Having the connection we can work out an explicit expression for its curvature, the A(Sq2 )-linear map ∇ 2 : E → 2 (Sq2 ) ⊗A(Sq2 ) E, by definition. Proposition 4.2. Let E be the line bundle Ln defined in (2.12) endowed with the connection ∇, for the canonical left covariant 2D calculus on A(Sq2 ), given in (4.1). Then, with φ ∈ Ln , its curvature is given by: ∇ 2 φ = −q −2n−2 ω+ ∧ ω− (X z φ) ,

194


with X z the vertical vector field in (3.7). As an element in HomA(Sq2 ) (Ln , (Sq2 ) ⊗A(Sq2 ) Ln ) one finds: ∇ 2 = q −n−1 [n] ω+ ∧ ω− . Proof. Using (4.1) and dω− = dω+ = 0 on Sq2 , we have ∇(∇φ) = −q −n−2 ω+ ∧ (X − X + φ) ω− − q −n+2 ω− ∧ (X + X − φ))ω+

= −q −2n−2 ω+ ∧ ω− (X − X + φ) − q −2n+2 ω− ∧ ω+ (X + X − φ) = −q −2n−2 ω+ ∧ ω− X − X + − q 2 X + X − φ,

and the first statement follows from the relation X − X + − q 2 X + X − = X z . The second statement comes from the computation of (X z φ) on φ ∈ Ln . Since X z (A(Sq2 )) = 0, it is evident that the curvature is A(Sq2 )-linear. On the other hand, from the action of the connection as ∇σ = (dσ ) p(n) on the module En := (A(Sq2 ))|n|+1 p(n) , a straightforward computation gives for its curvature F∇ = ∇ 2 as an element in (Sq2 ) ⊗A(Sq2 ) En the expression: F∇ = −dp(n) ∧ dp(n) p(n) .

(4.2)

To compare these two expressions for the curvature, we need an intermediate result. Lemma 4.3. Let p(n) denote the projection given in (2.13) and (2.14), for n ≥ 0 or n ≤ 0, respectively. With the standard 2D calculus on Sq2 of Sect. 3.2 one finds: dp(n) ∧ dp(n) p(n) = −q −n−1 [n] p(n) ω+ ∧ ω− , p(n) dp(n) ∧ dp(n) = −q −n−1 [n] p(n) ω+ ∧ ω− .

Proof. This is proved by explicit computation. We explicitly consider only the case n ≥ 0 since the proof for the case n ≤ 0 is the same. In components: p(n) dp(n)

σν

= =

n

p(n) dp(n) µ=0 σ µ µν

βn,σ βn,ν c∗σ a ∗n−σ

n µ=0

βn,µ a n−µ cµ d c∗µ a ∗n−µ a n−ν cν .

In the above the anti-holomorphic part vanishes: p(n) ∂¯ p(n) σν n = βn,σ βn,ν c∗σ a ∗n−σ βn,µ a n−µ cµ F c∗µ a ∗n−µ a n−ν cν q −1/2 ω− µ=0 n = βn,σ βn,ν c∗σ a ∗n−σ βn,µ a n−µ cµ F c∗µ a ∗n−µ q −n/2 a n−ν cν q −1/2 ω− µ=0

n = βn,σ βn,ν c∗σ a ∗n−σ βn,µ F a n−µ cµ c∗µ a ∗n−µ q −n a n−ν cν q −1/2 ω− µ=0 = βn,σ βn,ν c∗σ a ∗n−σ [F (1)] q −n a n−ν cν q −1/2 ω− = 0,


195

where the second and the third equality follow from the zero action of F on any power of a or c. As for the holomorphic part: p(n) ∂ p(n) σν n = βn,σ βn,ν c∗σ a ∗n−σ βn,µ a n−µ cµ E c∗µ a ∗n−µ a n−ν cν q 1/2 ω+ µ=0 n ∗σ ∗n−σ = βn,σ βn,ν c a βn,µ a n−µ cµ c∗µ a ∗n−µ E a n−ν cν q −n/2 q 1/2 ω+ µ=0 = βn,σ βn,ν c∗σ a ∗n−σ (1) E a n−ν cν q −n/2 q 1/2 ω+

= E βn,σ βn,ν c∗σ a ∗n−σ a n−ν cν q 1/2 ω+ = E pσ(n)ν q 1/2 ω+ , using now the zero action of E on any power of a ∗ or c∗ . Having established that: p(n) dp(n) = p(n) ∂p(n) = E p(n) q 1/2 ω+ , we have in turn: p(n) dp(n) ∧ d p(n) = p(n) ∂ p(n) ∧ ∂¯ p(n) = E p(n) ω+ ∧ F p(n) ω− = E p(n) F p(n) q −2 ω+ ∧ ω− . (n)

(n)

The last equality is easily found: Since pµσ ∈ L0 , one has F pµσ ∈ L−2 (see (2.18)), but ω+ L−2 ⊂ q −2 L−2 ω+ , from the second column in the commutation relations (3.3). We need to compute:

E p(n) F p(n) µν n n−ν ν −n ∗µ ∗n−µ F c∗σ a ∗n−σ a =q βn,µ βn,ν c a βn,σ E a n−σ cσ c . σ =0

For the term in the curly brackets we use the twisted derivation property of E in: 0 = E a n−σ cσ F c∗σ a ∗n−σ = q (n−2)/2 E a n−σ cσ F c∗σ a ∗n−σ + q n/2 a n−σ cσ E F c∗σ a ∗n−σ . Then, rearranging the terms and summing over σ one arrives at

E p(n) F p(n) µν n n−ν ν −n+1 a = −q βn,µ βn,ν c∗µ a ∗n−µ βn,σ a n−σ cσ E F c∗σ a ∗n−σ c . σ =0

With a direct computation one gets that: E F c∗σ a ∗n−σ = [n] c∗σ a ∗n−σ , giving in (n) turn, E p(n) F p(n) µν = −q −n+1 [n] pµν , and one finally obtains the claimed result: p(n) dp(n) ∧ d p(n) = −q −n−1 [n] p(n) µν ω+ ∧ ω− . µν

The case n ≤ 0 goes in a similar fashion. One has: ¯ (n) = F p(n) q −1/2 ω− , p(n) dp(n) = p(n) ∂p

196


leading to p(n) dp(n) ∧ dp(n) = F p(n) ω− ∧ E p(n) ω+ = −q |n|+1 [|n|] p(n) ω− ∧ ω+ = −q −n−1 [n] p(n) ω+ ∧ ω− . The second equality in the lemma is proved in a similar fashion, using (n) (n) (n) (n) ¯ p = F p(n) q −1/2 ω− , dp p = ∂p leading to (dp(n) ∧ d p(n) )p(n) = (∂p(n) ∧ ∂¯ p(n) ) p(n) = (E p(n) )(F p(n) ) q −2 ω+ ∧ ω− again, and similarly for the case n ≤ 0. Proposition 4.4. The curvature of the connection on En = (A(Sq2 ))|n|+1 p(n) for the canonical left covariant 2D calculus on A(Sq2 ), is given by F∇ = q −n−1 [n] p(n) ω+ ∧ ω− .

(4.3)

Moreover, with the left A(Sq2 )-modules isomorphism Ln En of Prop. 2.2 extended in a natural way to forms (Sq2 ) ⊗A(Sq2 ) Ln (Sq2 ) ⊗A(Sq2 ) En , the curvature on Ln corresponds to the curvature on En , that is, F∇ σφ = σ∇ 2 φ , for φ ∈ Ln and the corresponding section σφ ∈ En . Proof. The second statement is a direct consequence of the first one in (4.3) and the latter is evident once one substitutes the result of Lemma 4.3 in the expression (4.2). 5. The Winding Numbers The line bundles on the sphere Sq2 described in Sect. 2.2 are classified by their winding number n ∈ Z. In this section we first recall how to compute this number by means of a Fredholm module for the sphere. On the other hand, in order to integrate the gauge curvature on the quantum sphere Sq2 one needs a ‘twisted integral’; the result is not an integer any longer but rather its q-analogue. The projections p(n) given in Sect. 2.2, which describe the line bundles, are representatives of classes in the K -theory of Sq2 , i.e. [p(n) ] ∈ K 0 (Sq2 ). A way to compute the corresponding winding number is by pairing them with a nontrivial element in the dual K -homology, i.e. with (the class of) a nontrivial Fredholm module [µ] ∈ K 0 (Sq2 ). In fact, it is more convenient to first compute the corresponding Chern characters in the cyclic homology ch∗ ( p) ∈ HC∗ (Sq2 ) and cyclic cohomology ch∗ (µ) ∈ HC∗ (Sq2 ) respectively, and then use the pairing between cyclic homology and cohomology. The Chern character of the projections p(n) has a non-trivial component in degree zero ch0 (p(n) ) ∈ HC0 (Sq2 ) simply given by a (partial) matrix trace:

ch0 (p

(n)

) := tr(p

(n)

)=

⎧n n−µ−1 ∗ µ ⎪ (1 − q −2 j c∗ c), ⎨ µ=0 βn,µ (c c) j=0

n≥0

⎪ ⎩|n| α (c∗ c)|n|−µ µ−1 (1 − q 2 j c∗ c), µ=0 n,µ j=0

n≤0

,


197

and ch0 (p(n) ) ∈ A(Sq2 ). Dually, one needs a cyclic 0-cocycle, that is a trace on A(Sq2 ). This was obtained in [14] and it is a trace on A(Sq2 )/C, that is it vanishes on C ⊂ A(Sq2 ). On the other hand, its value on powers of the element (c∗ c) is: µ (c∗ c)k = (1 − q 2k )−1 , k > 0. The pairing was computed in [5] and results in [µ], [p(n) ] := µ ch0 (p(n) ) = −n. The integer above is a topological quantity that depends only on the bundle, both on the quantum sphere and on its classical limit, an ordinary 2-sphere. In this limit it could also be computed by integrating the curvature 2-form of a connection (indeed any connection) on the sphere. As mentioned, in order to integrate the gauge curvature on the quantum sphere Sq2 one needs a ‘twisted integral’; furthermore the result is not an integer any longer but rather a q-integer. To proceed we need some additional ingredients. Given a ∗-algebra A with a state ϕ, an automorphism ϑ of A is called a modular automorphism associated with ϕ if it happens that ϕ( f g) = ϕ(ϑ(g) f ), for f, g ∈ A. It is known [8, Prop. 4.15], that the modular automorphism associated with the Haar state h on the algebra A(SUq (2)) is: ϑ(g) = K −2 g K 2 .

(5.1)

The restriction of h to A(Sq2 ) yields a faithful, invariant—that is h( f X ) = h( f )ε(X ) for f ∈ A(Sq2 ) and X ∈ Uq (su(2))—state on A(Sq2 ) with modular automorphism ϑ(g) = g K 2 ,

for g ∈ A(Sq2 ),

(5.2)

the restriction of (5.1) to A(Sq2 ). It was proven in [19] that, with ω+ ∧ ω− the central generator of 2 (Sq2 ), h the Haar state on A(Sq2 ) and ϑ its modular automorphism in (5.2), the linear functional 2 2 : (Sq ) → C, f ω+ ∧ ω− := h( f ), (5.3) defines a non-trivial ϑ-twisted cyclic 2-cocycle τ on A(Sq2 ): τ ( f 0 , f 1 , f 2 ) := f0 d f1 ∧ d f2 .

(5.4)

That is bϑ τ = 0 and λϑ τ = τ, where bϑ is the ϑ-twisted coboundary operator: (bϑ τ )( f 0 , f 1 , f 2 , f 3 ) := τ ( f 0 f 1 , f 2 , f 3 ) − τ ( f 0 , f 1 f 2 , f 3 ) +τ ( f 0 , f 1 , f 2 f 3 ) − τ (ϑ( f 3 ) f 0 , f 1 , f 2 ), and λϑ is the ϑ-twisted cyclicity operator: (λϑ τ )( f 0 , f 1 , f 2 ) := λτ (ϑ( f 2 ), f 0 , f 1 ). The non-triviality means that there is no 1-cochain α on A(Sq2 ) such that bϑ α = τ and λϑ α = α. Here the operators bϑ and λϑ are defined by formulae like the above (and

198


directly generalizable in any degree). Thus τ is a class in HC2ϑ (Sq2 ), the degree 2 twisted cyclic cohomology of the sphere Sq2 . To couple the twisted cocycle τ with the bundles over Sq2 , one needs a twisted Chern character. In fact, for our specific case we do not need the full theory and it is enough to consider the lowest term, that of a twisted or ‘quantum trace’ [22]. If M ∈ Mat m+1 (A(Sq2 )), its (partial) quantum trace is the element trq (M) ∈ A(Sq2 ) given by M jl σm/2 (K 2 ) , trq (M) := tr Mσm/2 (K 2 ) := jl

lj

where σm/2 (K 2 ) is the matrix from (2.7) for the spin J = m/2 representation of Uq (su(2)). The q-trace is ‘twisted’ by the automorphism ϑ, that is trq (M1 M2 ) = trq (M2 K 2 )M1 = trq (ϑ(M2 )M1 ). For this one uses ‘right crossed product’ rules: xh = h (1) (x h (2) ), for x ∈ A(Sq2 ), h ∈ Uq (su(2)). Then,

trq (M1 M2 ) = tr M1 M2 σm/2 (K 2 ) = tr M1 σm/2 (K 2 )(M2 K 2 ) = tr (M2 K 2 )M1 σm/2 (K 2 ) = trq (M2 K 2 )M1 .

Lemma 5.1. Let p(n) be the projection given in (2.13) and (2.14) for n ≥ 0 or n ≤ 0, respectively. Then trq p(n) = q n . Proof. We prove this for the case n ≥ 0, the other case being similar. The matrix σn/2 (K 2 ) from the expression (2.7) for the spin J = n/2 representation is diagonal with entries (σn/2 (K 2 ))µµ = q n−2µ for µ = 0, . . . , n. One computes n trq p(n) = q n q −2µ βn,µ c∗µ a ∗n−µ a n−µ cµ µ=0 n = qn q −2µ+2µ(n−µ) βn,µ a ∗n−µ c∗µ cµ a n−µ µ=0 n = qn αn,n−µ a ∗n−µ c∗µ cµ a n−µ = q n (a ∗ a + c∗ c)n = q n , µ=0

having used the relation (2.17) for the coefficients α’s and β’s, and the identity (2.16). We are ready to integrate the gauge curvature. Proposition 5.2. Let F∇ be the curvature of the connection on En = (A(Sq2 ))|n|+1 p(n) for the canonical left covariant 2D calculus on A(Sq2 ). Then, for its integral one finds: − q trq (F∇ ) = −[n] . (5.5)


199

Proof. From Eq. (4.3) and Lemma 5.1 one has trq (F∇ ) = q −n−1 [n] trq (p(n) )ω+ ∧ ω− = q −1 [n]ω+ ∧ ω− , and the result follows from the definition of the integral in (5.3) and the fact that h(1) = 1 for the Haar state h on A(Sq2 ). Remark 5.3. From the definition (5.4) of the ϑ-twisted cyclic 2-cocycle τ and the expression (4.3) of the curvature F∇ , the (5.5) is also the coupling of the cocycle τ with the projection p(n) : (−q τ ) ◦ trq (p(n) , p(n) , p(n) ) = −[n]. In [15,22] this was obtained as the q-index of the Dirac operator on the sphere Sq2 . 6. The Gauged Laplacian Operator on the Sphere S2q Generalized Laplacian operators on the quantum sphere Sq2 were already studied in [17]. A Hodge -operator entered the game in [12]. We first recall the natural scalar Laplacian on Sq2 before going to its gauged version. 6.1. A Laplacian on the quantum Sq2 sphere. For the Hodge -operator on 1-forms one needs a left-covariant map : 1 (Sq2 ) → 1 (Sq2 ) whose square is the identity. In the description of the calculus as in Prop. 3.1, one defines: (∂ f ) = ∂ f,

(∂¯ f ) = −∂¯ f,

∀ f ∈ A(Sq2 ),

(6.1)

and shows its compatibility with the bimodule structure on forms. Thus is taken to have values ±1 on holomorphic or anti-holomorphic 1-forms respectively: 1± (Sq2 ) = ±1± (Sq2 ); in particular ω± = ±ω± . The calculus has one central top 2-form and the above operator is naturally extended by requiring that 1 = ω+ ∧ ω− ,

(ω+ ∧ ω− ) = 1.

(6.2)

We have all the ingredients to define a (scalar) Laplacian operator on Sq2 : ¯ f = ∂ ∂¯ f, f := − 21 d d f = −∂∂ Simple manipulations yield: f = 21 X + X − + q −2 X − X + f,

∀ f ∈ A(Sq2 ).

∀ f ∈ A(Sq2 ).

Indeed, with f ∈ A(Sq2 ), one has: d f = (X + f ) ω+ + (X − f ) ω− , and

(d f ) = (X + f ) ω+ − (X − f ) ω− ;

then, using the expression (3.6) for the exterior derivative of a 1-form: d(d f ) = − X + X − f + q −2 X − X + f ω+ ∧ ω− ,

200


and finally (d d f ) = − X + X − f + q −2 X − X + f . It comes as no surprise that the (scalar) Laplacian is the quadratic Casimir of Uq (su(2)) given in (2.3); more precisely, q = Cq +

1 4

− [ 21 ]2 .

(6.3)

Indeed, since K acts as the identity on elements on A(Sq2 ), on this algebra one has that X + X − + q −2 X − X + = q −1 (E F + F E) K 2 = q −1 (E F + F E) = 2q −1 (E F) , from the commutation relations (2.2). The claimed identity follows from direct comparison with the Casimir operator (2.3) restricted to A(Sq2 ). Relation (6.3) makes it also clear that the above Laplacian is related to the square of a Dirac operator [19,12]. The spectrum and the eigenspace decomposition are computed using the decompo(0) sition (2.21) for A(Sq2 ) = L0 = J ∈N V J , for the right action of Uq (su(2)). Since left and right action commute, this action clearly leaves invariant the eigenspaces of the Laplacian: ( f h) = ( f ) h, for all h ∈ Uq (su(2)). We know from (2.20) that for fixed J the highest weight vector is c J a ∗J on which a direct computation gives: (c J a ∗J ) = q −1 [J ][J + 1] (c J a ∗J ). The (2J + 1) corresponding eigenfunctions are obtained with the action of the raising operator F and are given by {(c J a ∗J ) F l , l = 0, 1, . . . , 2J }. This result is of course consistent with the equality (6.3).

6.2. The gauged Laplacian operator. Definition 6.1. Let ∇ : E → (Sq2 ) ⊗A(Sq2 ) E be a covariant derivative on the module E, with (Sq2 ) the left covariant calculus on the sphere Sq2 described in Sect. 3.2. And let be the Hodge operator on (Sq2 ) as given in (6.1) and (6.2). The gauged Laplacian operator ∇ : E → E is defined as: ∇ := − 21 ∇ ∇. We shall presently give it explicitly on the line bundle Ln and on the corresponding sections En := (A(Sq2 ))|n|+1 p(n) showing that they correspond to each other. Proposition 6.2. Let E be the line bundle Ln defined in (2.12), with connection ∇ given in (4.1). Then: ∇ φ = 21 q −2n X + X − + q −2 X − X + φ, for φ ∈ Ln .

(6.4)

Moreover, with the left A(Sq2 )-modules isomorphism Ln En := (A(Sq2 ))|n|+1 p(n) of Prop. 2.2, the Laplacian on Ln corresponds to the one on En for the Grassman connection, that is: ∇ (σφ ) = σ∇ φ , for φ ∈ Ln and corresponding section σφ ∈ En .


201

Proof. Using (4.1) and (6.1) we have, ∇φ = q −n−2 ω+ (X + φ) − q −n+2 ω− (X − φ) and (being dω− = dω+ = 0 on Sq2 ), ∇(∇φ) = −q −n−2 ω+ ∧ (X − X + φ) ω− + q −n+2 ω− ∧ (X + X − φ))ω+

= −q −2n−2 ω+ ∧ ω− (X − X + φ) + q −2n+2 ω− ∧ ω+ (X + X − φ)

= v − q −2n ω+ ∧ ω− q −2 (X − X + φ) + (X + X − φ) ,

and using (6.1) the first statement follows. For the second point, if σ ∈ En and ∇σ = d(σ )p(n) is the Grassmann connection, a computation similar to the previous one leads to:

∇ σ = 21 X + X − + q −2 X − X + σ p(n)

+ 21 q −2 (X + σ )X − p(n) − (X − σ )X + p(n) p(n) . (6.5) To continue, consider the case n ≥ 0, the case φ(n)∈

0 being similar. Prop. 2.2, to n≤ From dual Ln there corresponds the section σφ = φ (n) ∈ En , where (n) and its (n) (n) (n) (n) . are the vector valued functions in (2.13) with projection p = p =

(n) (n) (n) (n) −n/2 Using the X− = X + = 0 and that K = q vanishing

(n) n/2 (n) , we compute: and K =q

−2 −2 (n) 1 1 −2n X σ X φ X + q X X = q X + q X X + − − + φ + − − + 2 2 +q −n X − φ) X + (n) + 21 q −2 φ (X − X + (n) ,

and using X + (n) (n) = 0, we arrive at:

−2 1 X σ p(n) X + q X X + − − + φ 2

= 21 q −2n X + X − + q −2 X − X + φ (n) + 21 q −2 φ X − X + (n) p(n) . (6.6) On the know p(n) )p(n) = 0 and that (X − p(n) )p(n) = X − p(n) = that (X +(n) other hand,

we (n) −n (n) q X − . Using , X − (n) = 0, the second part in (6.5) becomes: − 21 q −n−2 φ

X + (n) X − (n) (n) ,

and this cancels the second term in (6.6) due to: X − X + (n) (n) + q −n X + (n) X − (n) = q −n X − X + (n) K 2 (n) + X + (n) X − (n)

= q −n X − X + (n) (n) = 0. Collecting all the above we arrive at:

∇ (σφ ) = 21 q −2n X + X − + q −2 X − X + φ (n) = σ∇ φ , and this ends the proof.

202


We are ready to diagonalize the gauged Laplacian ∇ . For this we use the decomposition (2.19) of the modules Ln for the right action of Uq (su(2)), as this action leaves invariant the eigenspaces of the gauged Laplacian: ∇ (φ h) = (∇ φ) h, for all h ∈ Uq (su(2)). From Sect. 2.1 we know that the integer n (the monopole charge) labels topological (n) sectors and given n the admissible values of J in the decomposition Ln = V J are (n) J = |n| 2 + s, with s ∈ N. Moreover, the highest weight elements in each V J is given in (n) (2.20) as φn,J = c J −n/2 a ∗J +n/2 , and the remaining 2J basis vectors in V J are obtained J −n/2 ∗J +n/2 l a ) E , with l = 0, . . . , 2J , via the right action E. The vectors φn,J,l = (c are eigenfunctions of ∇ with the same eigenvalue. Proposition 6.3. On the vectors φn,J,l the gauged Laplacian is diagonal, ∇ φn,J,l = λn,J φn,J,l , for l = 0, . . . , 2J , with the (2J + 1)-degenerate energies: λn,J = q −n−1 [J − n2 ] [J + n2 + 1] + 21 [n] 2 n−1 2 . = q −n−1 [J + 21 ]2 − 21 [ n+1 ] + [ ] 2 2

(6.7)

Proof. A short computation yields:

∇ (c J −n/2 a ∗J +n/2 ) = 21 q −2n X + X − + q −2 X − X + (c J −n/2 a ∗J +n/2 ) = λn,J (c J −n/2 a ∗J +n/2 ),

with λn,J given in the first line of (6.7). The second equality there is obtained with a direct algebraic manipulation. A remarkable fact is that, contrary to what happens in the classical limit, the energies are not symmetric under the exchange n ↔ −n, an additional example that ‘quantization removes degeneracy’. Writing J = |n| 2 + s, with s ∈ N, the energies become: −n−1 λn,s = q [s][n + s + 1] + 21 [n] , for n ≥ 0, with (n + 2s + 1) eigenfunctions φn,s,l = (cs a ∗n+s ) E l , and λn,s = q −n−1 [s − n][s + 1] + 21 [n] , for n ≤ 0, with (|n| + 2s + 1) eigenfunctions φn,s,l = (cs+|n| a ∗s ) E l . Having in mind a physics parallel with the quantum Hall effect, the integer s labels Landau levels and the φn,s,l are the (‘one excitation’) Laughlin wave functions with energies λn,s . The lowest Landau, s = 0, is |n|-degenerate with energy λn,0 = 21 q −n−1 [|n|]. It is worth spending a few words on the classical limit. At the value q = 1, the energies of the gauged Laplacian become λn,s (q → 1) = (J − n2 )(J +

n 2

+ 1) + 21 n = J (J + 1) − 41 n 2

= ( 21 |n| + s)( 21 |n| + s + 1) − 41 n 2 = |n| (s + 21 ) + s(s + 1), and coincide with the energies of the classical gauged Laplacian (see e.g. [7]). Clearly, they are symmetric under the exchange n ↔ −n which corresponds to inverting the direction of the magnetic field. The relation of the gauged Laplacian with the quadratic Casimir of Uq (su(2)) is more involved than its classical counterpart.


203

Proposition 6.4. When acting on the module Ln of equivariant maps the gauged Laplacian and the quadratic Casimir of Uq (su(2)) are related as: 2 n−1 2 q n+1 ∇ = Cq + 41 − 21 n+1 . (6.8) + 2 2 Moreover, the operators ∇ and Cq are related as: q K 2 − 2 + q −1 K −2 q −1 K 2 − 2 + q K −2 . + q K 2 ∇ = Cq + 41 − 21 (q − q −1 )2 (q − q −1 )2

(6.9)

Proof. We first note that the operator K commutes with all operators involved in the proposition and we can have it either on the left or on the right. Then, in (6.4) giving the action of the gauged Laplacian on an equivariant map, the factor q −2n can be traded for the action of the operator K −4 . In turn, ∇ = 21 K −4 (X + X − + q −2 X − X + ) = 21 q −1 K −4 (E F + F E)K 2 = 21 q −1 K −2 (E F + F E). Denote λ = (q − q −1 )−1 for simplicity. For the Casimir operator in (2.3) we get: Cq = 21 (F E + E F) + 21 [F, E] + λ2 (q K 2 − 2 + q −1 K −2 ) −

1 4

= 21 (F E + E F) + λ2 (q K 2 − 2 + q −1 K −2 ) − 21 λ(K 2 − K −2 ) − = 21 (F E + E F) + 21 λ2 (q K 2 − 2 + q −1 K −2 + q −1 K 2 − 2 + q K

1 4 −2

) − 41 .

A comparison between these two expressions establishes (6.9) which in turn, when computed on Ln gives (6.8). The above proposition has the expected classical limit. Again, by setting K = q H , expanding in the parameter =: log q and truncating at the 0th order in , from a direct computation the relation (6.9) becomes ∇ = CSU(2) − H 2 . This is an example of the general result recalled in the introduction, H being the generator of the structure group U(1) with Casimir element CU(1) = H 2 . Acknowledgements. This work was partially supported by the ‘Italian project Cofin06 - Noncommutative geometry, quantum groups and applications’. The research of AZ started at SISSA (Trieste, Italy) and went on at the IAM at Bonn University (Germany), thanks to a fellowship by the Alexander von Humboldt Stiftung; he thanks the Mathematical Physics Sector of SISSA and his host in Germany, Prof. Sergio Albeverio, for their warm hospitality. We thank Francesco D’Andrea for reading the comptu-script. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

A. An Appendix of Preliminaries We need to recall few facts about quantum principal bundles with (nonuniversal) differential calculi and connections on them. We briefly review their main properties here starting with covariant differential calculi.

204


A.1. Differential calculi. Given a C-algebra with unit A, any first order differential calculus (1 (A), d) on A can be obtained from the universal calculus (1 (A)un , δ). The space of universal 1-forms is the submodule of A ⊗ A given by 1 (A)un := ker(m : A ⊗ A → A), with m(a ⊗ b) = ab the multiplication map. The universal differential δ : A → 1un (A) is δa := 1 ⊗ a − a ⊗ 1. If N is any sub-bimodule of 1 (A)un with projection π : 1 (A)un → 1 (A) := 1 (A)un /N , then (1 (A), d), with d := π ◦ δ, is a first order differential calculus over A and any such calculus can be obtained in this way. If the algebra A is covariant for the coaction of a quantum group H = (H, , ε, S), one has a notion of covariant calculi on A as well. Then, let A be a (right, say) H-comodule algebra, with a right coaction R : A → A ⊗ H which is also an algebra map. In order to state the covariance of the calculus (1 (A), d) one needs to 1 1 extend the coaction of H. A map (1) R : (A) → (A) ⊗ H is defined by the requirement (1) R (d f ) = (d ⊗ id) R ( f ), and bimodule structure governed by (1)

(1)

(1)

R ( f d f ) = R ( f ) R (d f ),

(1)

R ((d f ) f ) = R (d f ) R ( f ).

The calculus is said to be right covariant if it happens that (1)

(1)

(1)

(id ⊗) R = ( R ⊗ id) R

and

(1)

(id ⊗ε) R = 1.

A calculus is right covariant if and only if for the corresponding bimodule N it is verified (1) (1) that R (N ) ⊂ N ⊗ H, where R is defined on N by formulæ as above with the universal derivation δ replacing the derivation d. Right covariance of the calculus implies that 1 (A) has a module basis {ηa } of right invariant 1-forms, that is 1-forms for which (1) R (ηa ) = ηa ⊗ 1. Similar consideration and formulæ holds for left covariance under a left coaction L . In particular, left covariance of a calculus similarly implies that 1 (A) has a module basis (1) {ωa } of left invariant 1-forms, that is 1-forms for which, L (ωa ) = 1 ⊗ ωa . Differential calculi on a quantum group H = (H, , ε, S) were already studied in [24]. Now : H → H ⊗ H is viewed as both a right and a left coaction of H on itself. Right and left covariant calculi on H will be defined as before with, in particular, a basis of invariant forms for the corresponding covariant calculus. In addition one has the notion of a bicovariant (that is both right and left covariant) calculus. Not surprisingly, on a quantum group there is more structure. Given the bijection r : H ⊗ H → H ⊗ H,

r (h ⊗ h ) := (h ⊗ 1)(h ),

one proves that r (1 (H)un ) = H ⊗ ker ε. Then, if Q ⊂ ker ε is a right ideal of ker ε, the inverse image, NQ = r −1 (H ⊗ Q), is a sub-bimodule contained in 1 (H)un . The differential calculus defined by such a bimodule, 1 (H) := 1 (H)un /NQ , is left-covariant, and any left-covariant differential calculus can be obtained in this way. Bicovariant calculi are in one to one correspondence with such right ideals Q which are in addition stable under the right adjoint coaction Ad of H onto itself, that is Ad(Q) ⊂ Q ⊗ H. Explicitly, one has Ad = (id ⊗m) (σ ⊗ id) (S ⊗ ) , with σ the flip operator,


205

or Ad(h) = h (2) ⊗ S(h (1) )h (3) using the Sweedler notation h =: h (1) ⊗ h (2) with summation understood, and higher numbers for iterated coproducts. The ideal Q also determines the quantum tangent space of the calculus. This is a collection {X a } of elements in Uq (H)—the Hopf algebra dual to H—which allows one to write the exterior differential as dh := (X a h) ωa , a

for h ∈ H and elements X a acting on the left on h. To be more specific, with the dual pairing , : Uq (H) × H → C, the quantum tangent space determined by the ideal Q is XQ := {X ∈ ker εUq (H) : X, Q = 0, ∀ Q ∈ Q}, where εUq (H) is the counit of Uq (H). The left action is given by X h := (id ⊗X ) h

or equivalently by X h := h (1) X, h (2) . The twisted derivation nature of elements in XQ is expressed by their coproduct, (X a ) = 1 ⊗ X a + b X b ⊗ f ba , with the elements f ab ∈ Uq (H) having specific properties [24]. These elements also control the commutation relation between the basis 1-forms and elements of H: ωa h = ( f ab h)ωb , hωa = ωb ( f ab ◦ S −1 ) h for h ∈ H. b

b

A.2. Quantum principal bundles and connections. The quantum principal bundles with nonuniversal calculi we are interested in were introduced in [2] (with refinements in [3]). As a total space we consider an algebra P (with multiplication m : P ⊗ P → P) and as structure group a Hopf algebra H. Thus P is a right H-comodule algebra with coaction R : P → P ⊗ H. The subalgebra of the right coinvariant elements, B = P H := { p ∈ P : R p = p ⊗ 1}, is the base space of the bundle. At the ‘topological level’ the principality of the bundle is the requirement of exactness of the sequence: χ 0 → P 1 (B)un P → 1 (P)un → P ⊗ ker εH → 0 (A.1) with 1 (P)un and 1 (B)un the universal calculi and the map χ is defined by, χ : P ⊗ P → P ⊗ H,

χ := (m ⊗ id) (id ⊗ R ) ,

(A.2)

or χ ( p ⊗ p) = p R ( p). The exactness of this sequence is equivalent to the requirement that the analogous ‘canonical map’ P ⊗B P → P ⊗ H (defined as the formula above) is an isomorphism. This is the definition that the inclusion B → P be a Hopf-Galois extension [20]. For quantum structure groups which are cosemisimple and have bijective antipodes, Th. I of [20] grants further nice properties. In particular, the surjectivity of the canonical map implies its bijectivity and faithfully flatness of the extension. The surjectivity of the map χ is the translation, for the deformed case, of the classical condition that the action of the structure group on the total space of the bundle is free. With differential calculi on both the total algebra P and the structure Hopf algebra H one needs compatibility conditions that eventually lead to an exact sequence like in (A.1) with the calculi at hand replacing the universal ones. Then, let 1 (P), d be a H-covariant differential calculus on P given via the subbimodule NP ∈ 1 (P)un , and 1 (H), d a bicovariant one on H given via the Ad-invariant right ideal QH ∈ ker εH .

206


The compatibility of the calculi are the requirements that χ (NP ) ⊆ P ⊗ QH and that the map ∼NP : 1 (P) → P ⊗ (ker εH /QH ), defined by the diagram 1 (P)un ↓χ

πN

−→ 1 (P) ↓∼NP

id ⊗πQ

(A.3)

P ⊗ ker εH −→ P ⊗ (ker εH /QH ) (with πP and πQ the natural projections) is surjective and has kernel ker ∼NP = P1 (B)P =: 1hor (P).

(A.4)

Here 1 (B) = BdB is the space of nonuniversal 1-forms on B associated to the bimodule NB := NP ∩ 1 (B)un . These conditions ensure the exactness of the sequence: ∼NP

0 → P1 (B)P → 1 (P) −→ P ⊗ (ker εH /QH ) → 0.

(A.5)

The condition χ (NP ) ⊆ P ⊗ QH is needed to have a map ∼NP well defined. In fact, with all conditions for a quantum principal bundle (P, B, H; NP , QH ) satisfied, this inclusion implies equality χ (NP ) = P ⊗ QH . Moreover, if (P, B, H) is a quantum principal bundle with the universal calculi, the equality χ (NP ) = P ⊗ QH ensures that (P, B, H; NP , QH ) is a quantum principal bundle with the corresponding nonuniversal calculi. Elements in the quantum tangent space XQH (H) giving the calculus on the structure quantum group H act on ker εH /QH via the pairing ·, · between Uq (H) and H. Then, with each ξ ∈ XQH (H) one defines a map ξ˜ : 1 (P) → P,

ξ˜ := (id ⊗ξ ) ◦ (∼NP ),

(A.6)

and declares a 1-form ω ∈ 1 (P) to be horizontal iff ξ˜ (ω) = 0, for all elements ξ ∈ XQH (H). The collection of horizontal 1-forms is easily seen to coincide with 1hor (P) in (A.4). Finally, we come to the notion of a connection. Recall that the covariance condition, R NP ⊂ NP ⊗ H allows one to extend the coaction R of H on P to a coaction of H (1) (1) on 1-forms, R : 1 (P) → 1 (P)⊗ H, by requiring that R ◦ d = (d ⊗ id)◦ R . A connection on the quantum principal bundle (P, B, H; NP , QH ) can be given as a map (1) ω : ker εH → 1 (P) such that ξ ◦ ω = 1 ⊗ id and R ◦ ω = (ω ⊗ id) ◦ Ad. Here the map Ad is the quotient of the right adjoint action on H to the space ker εH determined by Ad ◦πQ = (πQ ⊗id)◦Ad. It is shown in [2,3] that connections are in 1-1 correspondence with H-covariant complements to the horizontal forms 1hor (P) ⊂ 1 (P). B. Computations for the Quantum Hopf Bundle We collect here some of the computations toward showing the quantum principal bundle structure of the U(1)-bundle over the standard Podle´s sphere Sq2 with total space the quantum group SUq (2), given in Sect. 2.


207

B.1. The principal bundle condition. We prove here that the datum (A(SUq (2)), A(Sq2 ), A(U(1))) is a quantum principal bundle. This is done by showing exactness of the sequence 0 → A(SUq (2)) 1 (A(Sq2 ))un A(SUq (2)) → χ

→ 1 (A(SUq (2)))un −→ A(SUq (2)) ⊗ ker εA(U(1)) → 0 or equivalently that the map χ : 1 (A(SUq (2)))un → A(SUq (2))⊗ker εA(U(1)) defined as in (A.2) – and with the A(U(1))-coaction on A(SUq (2)) given in (2.10) – is surjective. Now, a generic element in A(SUq (2)) ⊗ ker εA(U(1)) is of the form f ⊗ (1 − z ∗n ) with n ∈ Z and f ∈ A(SUq (2)). To show surjectivity of χ it is enough to show that 1 ⊗ (1 − z ∗n ) is in its image since left A(SUq (2))-linearity of χ will give the general result: if γ ∈ 1 (A(SUq (2)))un is such that χ (γ ) = 1 ⊗ (1 − z ∗n ), then ∗n χ ( f γ ) = f (1 ⊗ (1 − z ∗n )) = f ⊗ (1

− z ). Firstly the case n ≥ 0. With (n) given in (2.13), if γ ∈ 1 (A(SUq (2)))un is n γ = (n) , δ (n) := βn,µ a n−µ cµ δ(c∗µ a ∗n−µ ) µ=0 n = 1⊗1− βn,µ a n−µ cµ ⊗ (c∗µ a ∗n−µ ), µ=0

n

⊗ z ∗n ) = 1 ⊗ (1 − z ∗n ). For ˇ (n) . The case n = 0 is trivial. n ≤ 0 the proof goes analogously with the vectors

we get χ (γ ) = 1 ⊗ 1 −

µ=0 βn,µ a

n−µ cµ (c∗µ a ∗n−µ

B.2. The compatibility of the calculi. In Sect. 3 we have given calculi on the principal bundle (A(SUq (2)), A(Sq2 ), A(U(1))). Out of the 3D left covariant calculus 1 (A(SUq (2))), with defining ideal QSUq (2) given in Sect. 3.1, a calculus on A(U(1)) was obtained by projection while a calculus on the subalgebra A(Sq2 ) was given by restriction. The ‘principal bundle compatibility’ of these calculi is established by showing that the sequence (A.5) is exact. For the case at hand, this sequence becomes, 0 → A(SUq (2)) 1 (Sq2 ) A(SUq (2)) → → 1 (A(SUq (2)))

∼NSU

q (2)

−→

A(SUq (2)) ⊗ ker εA(U(1)) /QA(U(1)) → 0,

where QA(U(1)) is the ideal given in Sect. 3.3 that defines the calculus on A(U(1)) and the map ∼NSUq (2) is defined as in the diagram (A.3) which now becomes, 1 (A(SUq (2))un ↓χ A(SUq (2)) ⊗ ker εA(U(1))

πQSU

q (2)

−→

id ⊗πQA(U(1))

−→

1 (A(SUq (2)) ↓∼NSUq (2) A(SUq (2)) ⊗ (ker εA(U(1)) /QA(U(1)) ) .

Having a quantum homogeneous bundle, that is a quantum bundle whose total space is a Hopf algebra and whose fiber is a Hopf subalgebra of it, with the differential calculus on the fiber obtained from the corresponding projection, for the above sequence to be exact it is enough [3] to check two conditions. The first one, (id ⊗π ) ◦ Ad R (QSUq (2) ) ⊂ QSUq (2) ⊗ A(U(1)),

208


with π : A(SUq (2)) → A(U(1)) the projection in (2.9), is easily established by a direct calculation and using the explicit form of the elements in QSUq (2) . The second condition amounts to the statement that the kernel of the projection π can be written as a right A(SUq (2))-module of the kernel of π , itself restricted to the base algebra A(Sq2 ). Then, one needs to show that ker π ⊂ (ker π |Sq2 )A(SUq (2)), the opposite implication being obvious. With π defined in (2.9), one has that ker π = {c f, c∗ g, with f, g ∈ A(SUq (2))}. Then c f = c(a ∗ a + c∗ c) f = ca ∗ (a f ) + c∗ c(c f ), with both ca ∗ and c∗ c in ker π |Sq2 . The same holds for elements of the form c∗ g, and the inclusion follows. We finish by showing that two of the generators in (3.1) of the 3D calculus on A(SUq (2)), that is ω± , are indeed the generators of the horizontal forms, ker ∼NSUq (2) , on the principal bundle as in (A.4). If f ∈ A with universal derivative δ f = 1⊗ f − f ⊗1, from the definition (A.2) we get χ (δ f ) = f (1) ⊗π( f (2) )− f ⊗1, with the usual Sweedler notation: f = f (1) ⊗ f (2) . In particular, for the generators of A(SUq (2)) we get: χ (δa) = a ⊗ (z − 1), χ (δa ∗ ) = a ∗ ⊗ (z ∗ − 1), χ (δc) = c ⊗ (z − 1), χ (δc∗ ) = c∗ ⊗ (z ∗ − 1). Given the two generators ω± and the specific QSUq (2) which determines the 3D calculus, corresponding universal 1-forms can be taken to be: aδc − qcδa ∈ [πQSUq (2) ]−1 (ω+ ),

c∗ δa ∗ − qa ∗ δc∗ ∈ [πQSUq (2) ]−1 (ω− ).

The action of the canonical map then gives: χ (aδc − qcδa) = (ac − qca) ⊗ (z − 1) = 0, χ (c∗ δa ∗ − qa ∗ δc∗ ) = (c∗ a ∗ − qa ∗ c∗ ) ⊗ (z ∗ − 1) = 0, which means that ∼NSUq (2) (ω+ ) = 0 =∼NSUq (2) (ω− ). For the third generator ωz , one shows in a similar fashion that ∼NSUq (2) (ωz ) = 1⊗(z −1). From these we may conclude that the elements ω± generate the module of horizontal forms. Finally, we know from (3.7) that the vector X = X z = (1−q −2 )−1 (1−K 4 ) is the dual generator to the calculus on the structure Hopf algebra A(U(1)). For the corresponding ‘vector field’ X˜ on A(SUq (2)) as in (A.6), one has that X˜ (ω± ) = X, ∼NSUq (2) (ω± ) = 0, while X˜ (ωz ) = X, ∼NSUq (2) (ωz ) = 1. These results identify X˜ as a vertical vector field. References 1. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Berlin-Heidelberg-New York: Springer, 1991 2. Brzezinski, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157 591–638 (1993) ; Erratum 167, 235 (1995) 3. Brzezinski, T., Majid, S.: Quantum differential and the q-monopole revisited. Acta Appl. Math. 54, 185–233 (1998) 4. Brzezinski, T., Majid, S.: Line bundles on quantum spheres. AIP Conf. Proc. 345, 3–8 (1998) 5. Hajac, P.M.: Bundles over quantum sphere and noncommutative index theorem. K-Theory 21, 141–150 (2000)


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6. Hajac, P.M., Majid, S.: Projective module description of the q-monopole. Commun. Math. Phys. 206, 247– 264 (1999) 7. Haldane, F.D.: Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states. Phys. Rev. Lett. 51, 605–608 (1983) 8. Klimyk, A., Schmudgen, K.: Quantum Groups and Their Representations. Berlin-Heidelberg-New York: Springer, 1997 9. Landi, G.: Spin-Hall effect with quantum group symmetries. Lett. Math. Phys. 75, 187–200 (2006) 10. Laughlin, R.B.: Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983) 11. Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge Univ. Press, 1995 12. Majid, S.: Noncommutative Riemannian and spin geometry of the standard q-sphere. Commun. Math. Phys. 256, 255–285 (2005) 13. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Ueno, K.: Representations of the Quantum Group SUq (2) and the Little q-Jacobi Polynomials. J. Funct. Anal. 99, 357–387 (1991) 14. Masuda, T., Nakagami, Y., Watanabe, J.: Noncommutative differential geometry on the quantum two sphere of P.Podle´s. I: An algebraic viewpoint. K-Theory 5, 151–175 (1991) 15. Neshveyev, S., Tuset, L.: A Local Index Formula for the Quantum Sphere. Commun. Math. Phys. 254, 323–341 (2005) 16. Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) 17. Podle´s, P.: Differential calculus on quantum spheres. Lett. Math. Phys. 18, 107–119 (1989) 18. Schmüdgen, K., Wagner, E.: Representations of cross product algebras of Podle´s quantum spheres. J. Lie Theory 17, 751–790 (2007) 19. Schmüdgen, K., Wagner, E.: Dirac operator and a twisted cyclic cocycle on the standard Podle´s quantm sphere. J. Reine Angew. Math. 574, 219–235 (2004) 20. Schneider, H.: Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72, 167–195 (1990) 21. Varshalovic, D.A., Moskalev, A.N., Khersonskii, V.K. : Quantum theory of angular momentum. Singapore: World Scientific, 1988 22. Wagner, E.: On the noncommutative spin geometry of the standard Podles sphere and index computations. http://arxiv.org/abs/0707.3403v2 [math. QA] 23. Woronowicz, S.L.: Twisted SUq (2) group An example of a noncommutative differential calculus. Publ. Rest. Inst. Math.Sci. Kyoto Univ. 23, 117–181 (1987) 24. Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys. 122, 125–170 (1989) Communicated by A. Connes


Communications in


Global Wellposed Problem for the 3-D Incompressible Anisotropic Navier-Stokes Equations in an Anisotropic Space Ting Zhang Department of Mathematics, Zhejiang University, Hangzhou 310027, China. E-mail: [email protected] Received: 5 February 2008 / Accepted: 8 June 2008 Published online: 16 September 2008 – © Springer-Verlag 2008

Abstract: In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (ANS). We prove the global wellposedness for ANS provided the initial horizontal data are sufficient small in the scaling invariant 1 Besov-Sobolev type space B 0, 2 . In particular, the result implies the global wellposedness of ANS with large initial vertical velocity. 1. Introduction 1.1. Introduction to the anisotropic Navier-Stokes equations. In this paper, we are going to study the 3-D incompressible anisotropic Navier-Stokes equations (ANS), namely, ⎧ ⎨ u t + u · ∇u − νh h u − ν3 ∂x23 u = −∇ P, (1.1) divu = 0, ⎩ u|t=0 = u 0 , where u(t, x) and P(t, x) denote the fluid velocity and the pressure, respectively, the viscosity coefficients νh and ν3 are two constants satisfying νh > 0, ν3 ≥ 0, x = (x h , x3 ) ∈ R3 and h = ∂x21 + ∂x22 . When νh = ν3 = ν, such a system is the classical (isotropic) Navier-Stokes system (NS). It appeared in geophysical fluids (see for instance [5]). In fact, instead of putting the classical viscosity −ν in NS, meteorologists often simulate the turbulent diffusion by putting a viscosity of the form −νh h − ν3 ∂x23 , where νh and ν3 are empiric constants, and ν3 usually is much smaller than νh . We refer to the book of J. Pedlosky [14], Chap. 4 for a more complete discussion. In particular, in studying of the Ekman boundary layers for rotating fluids [5,7,8], it makes sense to consider a anisotropic viscosity of the type −νh h − εβ∂x23 , where ε is a very small parameter. The system ( AN S) has been studied first by J.Y. Chemin, B. Desjardins,

212

T. Zhang

I. Gallagher and E. Grenier in [6] and D. Iftimie in [9], where the authors proved that such a system is locally wellposed for initial data in the anisotropic Sobolev space 0, 21 +ε 2 3 2 1+2ε 2 H = u ∈ L (R ); u 0, 1 +ε = |ξ3 | |u(ξ ˆ h , ξ3 )| dξ < +∞ , H˙

R3

2

for some ε > 0. Moreover, it has also been proved that if the initial data are small enough in the sense that u 0 εL 2 u 0 1−ε ≤ cνh 0, 1 +ε H˙

(1.2)

2

for some sufficiently small constant c, then the system (1.1) is global wellposed. Similar to the classical Navier-Stokes equations, the system (ANS) has a scaling invariance. Indeed, if u is a solution of ANS on a time interval [0, T ] with initial data u 0 , then the vector field u λ defined by u λ (t, x) = λu(λ2 t, λx) is also a solution of ANS on the time interval [0, λ−2 T ] with the initial data λu 0 (λx). But the norm · ˙ 0, 1 +ε is not scaling invariant. M. Paicu proved in [12] a similar result for H

2

1

the system (ANS) with ν3 = 0 in the case of the initial data u 0 ∈ B 0, 2 . This space has a scaling invariant norm. Then, J.Y. Chemin and P. Zhang [4] obtained a similar result − 12 , 21

in the scaling invariant space B4

. Recently, a similar result in the scaling invariant

−1+ 2 , 1 Bp p 2

space was obtained in [15]. Considering the periodic anisotropic Naiver-Stokes equations, M. Paicu obtained global wellposedness in [13]. These global results in [4,6,12,13,15] are obtained under the assumption that the initial data are sufficient small. The goal of our work is to prove that the system N S or AN S is globally wellposed when the initial horizontal data are sufficient small and the initial vertical velocity is large. Specially, we can prove that the system (N S) or (AN S) is globally wellposed for the initial data u ε0 defined by u ε0 (x) = (ε ln(− ln ε)φ h , ln(− ln ε)φ 3 )(x h , εx3 )

(1.3) 1

with small enough ε, where φ(x) = (φ h , φ 3 )(x) is a divergence free vector field in B 0, 2 . 1.2. Statement of the results.. As in [4], let us begin with the definition of the spaces which we will be going to work with. It requires an anisotropic version of dyadic decomposition of the Fourier space; let us first recall the following operators of localization in Fourier space, for (k, l) ∈ Z2 : ˆ lv a = F −1 (ϕ(2−l |ξ3 |)a), ˆ kh a = F −1 (ϕ(2−k |ξh |)a), h h v v Sk a = k a, Sl a = l a, k ≤k−1

l ≤l−1

where Fa or aˆ denotes the Fourier transform of the function a, and ϕ is a function in D(( 43 , 83 )) satisfying ϕ(2− j τ ) = 1, ∀ τ > 0. j∈Z 1

1

At first, we give the definitions of B 0, 2 and B 0, 2 (T ) as follows.

Anisotropic Navier-Stokes Equations

213 1

Definition 1.1 ( [4,10,12] ). We denote by B 0, 2 the space of distributions, which is the completion of S(R3 ) by the following norm: l 2 2 lv a L 2 (R3 ) . a 0, 1 = B

2

l∈Z

0, 21

We denote by B (T ) the space of distributions, which is the completion of C ∞ ([0, T ]; S(R3 )) by the following norm: j = 2 2 vj a L ∞ (L 2 (R3 )) a 0, 1 B

2 (T )

T

j∈Z

√ √ + νh ∇h vj a L 2 (L 2 (R3 )) + ν3 ∂3 vj a L 2 (L 2 (R3 )) . T

T

Now, we present the main result of this paper. Theorem 1.1. A positive constant C1 exists such that, if the divergence free vector field 1 u 0 ∈ B 0, 2 satisfies C1 νh−1 u 0h

1

B 0, 2

exp{C1 (νh−1 u 30

1

B 0, 2

+ 1)4 } ≤ 1,

(1.4) 1

then the system (1.1) with initial data u 0 has a unique global solution u ∈ B 0, 2 (∞) ∩ 1 C([0, ∞); B 0, 2 ), and u 0, 1 is independent of ν3 . B

2 (∞)

In what follows, we always use C to denote a generic positive constant independent of νh and ν3 . Remark 1.1. The positive constant C1 can be chosen in (3.9), and is independent of νh and ν3 . Remark 1.2. Paicu obtained the local wellposedness for large data in [12]. hε Remark 1.3. Let u ε0 = (u 0hε , u 3ε 0 ) be defined in (1.3). It is easy to show that u 0

u ε0

u 3ε 0 B 0, 21

1

B 0, 2

C ln(− ln ε). Thus, satisfies the condition (1.4) with Cε ln(− ln ε) and small enough ε. From Theorem 1.1, we obtain that the system N S or AN S is globally wellposed for the initial data u ε0 with a small enough ε. The reason may be that the initial data u ε0 almost only depend on the horizontal variable x h when ε is very small. From Proposition 1.1 in [3], one can obtain that u 3ε ≥ C ln(− ln ε), −1 0 B˙ ∞,∞ when φ 3 (x h , x3 ) = f (x h )g(x3 ), f ∈ S(R2 ) and g ∈ S(R). We should mention that the methods introduced by Chemin-Zhang in [4] and Paicu in [12] will play a crucial role in our proof here. Finally, we should also mention that there has been a large amount of investigation into the classical Navier-Stokes equations, see [1–3,11] and references therein. Specially, Chemin and Gallagher [3] proved that if ε is small enough, the initial data (v0h + εw0h , w03 )(x h , εx3 ) generates a unique global solution of NS.

214

T. Zhang

2. Anisotropic Littlewood-Paley Theory At first, we list anisotropic Berstein inequalities in the following. Please see the details in [4,12]. Lemma 2.1. Let Bh (resp. Bv ) be a ball of R2h (resp. Rv ), and Ch (resp. Cv ) be a ring of R2h (resp. Rv ). Then, for 1 ≤ p2 ≤ p1 ≤ ∞ and 1 ≤ q2 ≤ q1 ≤ ∞, there holds: 1. If the support of aˆ is included in 2k Bh , then β

∂h a L p1 (L qv1 ) 2

k(|β|+2( p1 − p1 )) 2

1

h

a L p2 (L qv1 ) , h

where ∂h := ∂xh . 2. If the support of aˆ is included in 2l Bv , then l(N + q1 − q1 )

∂3N a L p1 (L qv1 ) 2

2

1

h

a L p1 (L qv2 ) , h

where ∂3 := ∂x3 . 3. If the support of aˆ is included in 2k Ch , then β

a L p1 (L qv1 ) 2−k N sup ∂h a L p1 (L qv1 ) . |β|=N

h

h

4. If the support of aˆ is included in 2l Cv , then a L p1 (L qv1 ) 2−l N ∂3N a L p1 (L qv1 ) . h

h

Notations. In what follows, we make the convention that (dk )k∈Z denotes a generic element of the sphere of l 1 (Z) as in [4]. The following lemma is proved in [15] (Lemma 2.4). 1

Lemma 2.2. Let w be in B 0, 2 (T ). Then, we have vj w

2p 2p p−1 L T (L h (L 2v ))

− 21p − j 2

d j νh

2

w

1

B 0, 2 (T )

and w where w

2p 2p p−1 L T (L h (L ∞ v ))

2p 1 2p p−1

L T (

Lh (Bv2 ))

=

w

2p 1 2p p−1

L T (

Lh (Bv2 ))

− 21p

νh

j

2 2 vj w

j∈Z

2p p−1

2p

L T (L h

(L 2v ))

w

1

B 0, 2 (T )

,

and p ≥ 2.

Remark 2.1. We only use the following estimate in the proof of Theorem 1.1: −1

j

vj w L 4 (L 4 (L 2 )) d j νh 4 2− 2 w T

h

v

This estimate was proved in [4]. The following lemma is proved in [12] (Lemma 2.4). Lemma 2.3. Let w and ∇h w be in L 2 . Then, we have 1

1

w L 2 (L 4 ) w L2 2 ∇h w L2 2 . v

h

1

B 0, 2 (T )

.


215

3. The Proof of an Existence Theorem The purpose of this section is to prove the following existence theorem: 1

Theorem 3.1. A constant C1 exists such that, if the divergence free vector field u 0 ∈ B 0, 2 satisfies C1 νh−1 u 0h

1

B 0, 2


1

B 0, 2

+ 1)4 } ≤ 1, 1

then the system (1.1) with initial data u 0 has a global solution u ∈ B 0, 2 (∞) ∩ 1 C([0, ∞); B 0, 2 ), and u 0, 1 is independent of ν3 . B

2 (∞)

Proof. We shall use the classical Friedrichs’ regularization method to construct the approximate solutions to (1.1). For simplicity, we just outline it here (for the details, see [4,5,12]). In order to do so, let us define the sequence of operators (Pn )n∈N by Pn a := F −1 (1 B(0,n) a), ˆ and we define the following approximate system: ∂t u 1n + Pn (u n · ∇u 1n ) − νh h u 1n − ν3 ∂32 u 1n = −Pn ∂1 (−)−1 ∂ j ∂k (u n u kn ), (3.1) j

∂t u 2n + Pn (u n · ∇u 2n ) − νh h u 2n − ν3 ∂32 u 2n = −Pn ∂2 (−)−1 ∂ j ∂k (u n u kn ), (3.2) j

∂t u 3n + Pn (u n · ∇u 3n ) − νh h u 3n − ν3 ∂32 u 3n = −Pn ∂3 (−)−1 ∂ j ∂k (u n u kn ), (3.3) divu n = 0, (3.4) u n |t=0 = Pn u 0 , (3.5) j

where (−)−1 ∂ j ∂k is defined precisely by (−)−1 ∂ j ∂k a := −F −1 (|ξ |−2 ξ j ξk a). ˆ Then, the system (3.1)-(3.5) appears to be the ordinary differential equations in the space L 2n := {a ∈ L 2 (R3 )diva = 0, Suppaˆ ⊂ B(0, n)}. Such a system is globally wellposed because d u n 2L 2 ≤ 0. dt Now, the proof of Theorem 3.1 reduces to the following two propositions, which we shall admit for the time being. 1

1

Proposition 3.1. Let u be a divergence free vector field in B 0, 2 (T ) and w ∈ B 0, 2 (T ). Then, for any j ∈ Z, we have T v v F j (T ) := j (u · ∇w) j wd x dt d 2j νh−1 2− j w2 0, 1 u h 0, 1 . 0

R3

B

2 (T )

B

2 (T )

216

T. Zhang 1

1

Proposition 3.2. Let u be a divergence free vector field in B 0, 2 (T ) and w ∈ B 0, 2 (T ). Then, for any j ∈ Z, we have T v −1 l k v dt G j (T ) := (−) ∂ ∂ (u u ) ∂ wd x l k j j h 0 k,l R3 d 2j νh−1 2− j u h 2 0, 1 +d 2j 2− j +d 2j 2− j

2 (T )

B

T

T

1

B 0, 2 (T )

1

1

1

1

u h 2 0, 1 ∇h u h 2 0, 1 u 3 2 0, 1 ∇h u 3 2 0, 1 ∇h w B

0

w

B

2

∇h u h

0

B

2

1

B

0, 21

B

2

1

2

1

1

B 0, 2

dt

1

u 3 2 0, 1 ∇h u 3 2 0, 1 w 2 0, 1 ∇h w 2 0, 1 dt. B

B

2

B

2

B

2

2

Conclusion of the proof of Theorem 3.1. Applying the operator vj to (3.1) and taking the L 2 inner product of the resulting equation with vj u 1n , we have d vj u 1n 2L 2 + 2νh ∇h vj u 1n 2L 2 + 2ν3 ∂3 vj u 1n 2L 2 dt v 1 v 1 = −2 j (u n · ∇u n ) j u n d x + 2 vj (−)−1 ∂l ∂k (u ln u kn )vj ∂1 u 1n d x. R3

R3

k,l

From Propositions 3.1-3.2 and the Cauchy-Schwarz inequality, we get 2 j vj u 1n (t)2L 2 + 2νh ∇h vj u 1n 2L 2 (L 2 ) + 2ν3 ∂3 vj u 1n 2L 2 (L 2 ) t

t

≤ 2 j vj u 1n (0)2L 2 + Cd 2j νh−1 u nh 3 0, 1 B 2 (T ) t 1 1 1 3 +Cd 2j u 3n (s) 2 0, 1 ∇h u 3n (s) 2 0, 1 u nh (s) 2 0, 1 ∇h u nh (s) 2 0, 1 ds B

0

B

2

B

2

B

2

2

and √

√

ν3 ∂3 vj u 1n 2 0, 1 L t (B 2 ) √ 1 3 ν − h ∇h u nh 2 0, 1 ≤ Cu 1n (0) 0, 1 + Cνh 2 u nh 2 0, 1 + B 2 L t (B 2 ) 4 B 2 (T ) t

21 − 32 3 2 3 2 h 2 +Cνh u n (s) 0, 1 ∇h u n (s) 0, 1 u n (s) 0, 1 ds ,

u 1n (t)

B

0, 21

+

νh ∇h u 1n 2

L t (B

0

B

0, 21

)

+

B

2

where t ∈ (0, T ]. Similarly, we obtain √ u 2n (t) 0, 1 + νh ∇h u 2n 2

2

B

√ ν3 ∂3 vj u 2n 2 0, 1 L t (B L t (B 2 ) B 2 ) √ 1 3 ν − h ∇h u nh 2 0, 1 ≤ Cu 2n (0) 0, 1 + Cνh 2 u nh 2 0, 1 + B 2 L t (B 2 ) 4 B 2 (T ) t

21 − 32 3 2 3 2 h 2 +Cνh u n (s) 0, 1 ∇h u n (s) 0, 1 u n (s) 0, 1 ds 0

B

2

0, 21

(3.6)

2

+

B

2

B

2

(3.7)


217

and u 3n (t)

B

√ νh ∇h u 3n 2

+

0, 21

L t (B

≤ Cu 3n (0)

t

+C 0

B

0, 21

−1

)

+

√ ν3 ∂3 vj u 3n 2

1

+ Cνh 2 u nh 2 0, 1

0, 21

B

1

1

L t (B 0, 2 )

2 (T )

u 3n

1

B

0, 21

−1

(T )

3

1

u 3n 2 0, 1 ∇h u 3n 2 0, 1 ∇h u nh 2 0, 1 u nh 2 0, 1 ds B

B

2

B

2

where t ∈ (0, T ], since j Pn ∂3 (−)−1 ∂ j ∂k (u n u kn )u 3n d x = R3

B

2

3

+ Cνh 2 u nh 2 0, 1

21

2

B

2 (T )

,

(3.8)

Pn (−)−1 ∂ j ∂k (u n u kn )divh u nh d x. j

R3

Then, from (3.6)-(3.7) and Minkowski’s inequality, we have u nh 2 0, 1 B

2 (t)

≤ 2C0 u nh (0)2 0, 1 + Cνh−1 u nh 3 0, 1 B 2 B 2 (T ) t +Cνh−3 u 3n (s)2 0, 1 ∇h u 3n (s)2 0, 1 u nh 2 0, 1 B

0

B

2

B

2

2 (s)

ds, t ∈ (0, T ].

Gronwall’s inequality implies that u nh 2 0, 1 B 2 (T ) ≤ 2C0 u nh (0)2 0, 1 + Cνh−1 u nh 3 0, 1 B

2 (T )

B

2

exp{Cνh−4 u 3n 4 0, 1 B

2 (T )

}.

From (3.8) and the Cauchy-Schwarz inequality, we have u 3n

1

B 0, 2 (T )

≤ 2C0 u 3n (0)

B

− 12

+Cνh u nh

0, 21

1

−1

B

B 0, 2 (T )

≤ 2C0 u 3n (0)

B

0, 21

1

+ Cνh 2 u nh 2 0, 1 u 3n

2 (T )

u 3n

B

0, 21

B

0, 21

−1

3

(T )

+ Cνh 2 u nh 2 0, 1

(T )

+ Cνh 2 u nh 2 0, 1

B

1

B 0, 2 (T )

−1

1

+ Cνh 2 u nh 2 0, 1 B

2 (T )

u 3n

−1

3

B

Then, we obtain u nh 2 0, 1 B 2 (T )

≤e

2 (T )

1 2

Cνh−4 (4C0 u 30

B

0, 21

+νh )4

2C0 u 0h 2 0, 1 B

+Cνh−1 (4C0 u 0h 2 0, 1 ) e 3 2

B

2

2

−4 3 3 2 Cνh (4C 0 u 0 0, 1 B 2

+νh )4

2 (T )

.

218

T. Zhang

and u 3n

1

B 0, 2 (T )

≤ 2C0 u 30

B

0, 21

−1

3

+ Cνh 2 (4C0 u 0h 2 0, 1 ) 4 e B

− 12

1

+Cνh (4C0 u 0h 2 0, 1 ) 4 e B

−4 3 3 4 Cνh (4C 0 u 0 0, 1 B 2

2

−4 1 3 4 Cνh (4C 0 u 0 0, 1 B 2

+νh )4

2

+νh )4

4C0 u 30

B

0, 21

+ νh ,

for all T < Tn , where Tn := sup{t > 0; u nh 2 0, 1

2 (t)

B

≤ 4C0 u 0h 2 0, 1 e u 3n

B

1

Cνh−4 (4C0 u 30

B

0, 21

+νh )4

2

B 0, 2 (t)

≤ 4C0 u 30

1

B 0, 2

,

+ νh }.

Then, if u 0 satisfies C1 νh−1 u 0h

1

B 0, 2


1

B 0, 2

+ 1)4 } ≤ 1,

where C1 = 29 C 2 C04 ,

(3.9)

we get that for any n and for any T < Tn , u nh

B

0, 21

(T )

≤

Cνh−4 (4C0 u 30 0, 1 +νh )4 5 B 2 C0 u 0h 2 0, 1 e 2 B 2

and u 3n

B

0, 21

(t)

≤

5 1 C0 u 30 0, 1 + νh . 2 B 2 2

Thus, Tn = +∞. Then, the existence follows from the classical compactness method, the details of which are omitted (see [5,12]). Then, we prove the continuity of the solution u as follows. From (1.1), we have vj u t = νh vj h u + ν3 vj ∂32 u − vj (u · ∇u) − vj ∇ P. We can easily obtain that for any T > 0 and j ∈ Z, ν3 vj ∂32 u ∈ L ∞ ([0, T ]; L 2 ), νh vj h u ∈ L 2 (0, T ; L 2v ( H˙ h−1 )) and (νh vj h u + ν3 vj ∂32 u|vj u) L 2 ∈ L 1 ([0, T ]). From Proposition 3.1, we have (vj (u · ∇u)|vj u) L 2 ∈ L 1 ([0, T ]). Thus, we have that 1

d v 2 dt j u(t) L 2

∈ L 1 ([0, T ]), for any T > 0 and j ∈ Z. Combining 1

it with u ∈ B 0, 2 (∞), we can easily get that u ∈ C([0, ∞); B 0, 2 ). Then Theorem 3.1 is proved provided of course that we have proved Propositions 3.1–3.2.


219

To prove Propositions 3.1–3.2, we need the following lemma. 1

Lemma 3.1. Let w and u be in B 0, 2 (T ). We have vj (u∂h w)

4 4 L T3 (L h3 (L 2v ))

−3

j

d j νh 4 2− 2 w

1

B 0, 2 (T )

u

1

B 0, 2 (T )

.

Proof. Using Bony’s decomposition in the vertical variable, we obtain vj (u∂h w) = vj (S vj −1 u∂h vj w) + vj (vj u∂h S vj +2 w). | j− j |≤5

j ≥ j−N0

Using Hölder’s inequality and Lemma 2.2, we get vj (S vj −1 u∂h vj w)

4

4

L T3 (L h3 (L 2v ))

S vj −1 u L 4 (L 4 (L ∞ )) vj ∂h w L 2 (L 2 (R3 )) T

v

h

j 2

− 12

d j νh 2− w

T

− 14

ν 1 B 0, 2 (T ) h

u

1

B 0, 2 (T )

and vj (vj u∂h S vj +2 w)

4 4 L T3 (L h3 (L 2v ))

S vj +2 (∂h w) L 2 (L 2 (L ∞ )) vj u L 4 (L 4 (L 2 )) T

j 2

− 12

d j νh 2− w

v

h

T

− 14

ν 1 B 0, 2 (T ) h

u

v

h

1

B 0, 2 (T )

.

Then, we can immediately finish the proof.

Similarly, we can obtain the following lemma. 1

Lemma 3.2. Let ∇h w, u and ∇h u be in B 0, 2 . We have vj (u∂h w)

1

j

4 L h3 (L 2v )

d j 2− 2 ∇h w

B

0, 21

1

u 2 0, 1 ∇h u 2 0, 1 . B

B

2

2

1

Lemma 3.3. Let u, ∇h u, w and ∇h w be in B 0, 2 . We have j

1

1

1

1

vj (uw) L 2 (R3 ) d j 2− 2 u 2 0, 1 ∇h u 2 0, 1 w 2 0, 1 ∇h w 2 0, 1 . B

B

2

B

2

B

2

2

Proof. Using Bony’s decomposition in the vertical variable, we obtain vj (uw) = vj (S vj −1 uvj w) + vj (S vj +2 wvj u). | j− j |≤5

j ≥ j−N0

Using Hölder’s inequality and Lemma 2.3, we get vj (S vj −1 uvj w) L 2 (R3 ) S vj −1 u L ∞ (L 4 ) vj w L 2 (L 4 ) v

j 2

d j 2− u

v

h

1 2 1 B 0, 2

∇h u

1 2

h

1

1 B 0, 2

1

w 2 0, 1 ∇h w 2 0, 1 B

B

2

2

and j

1

1

1

1

vj (S vj +2 wvj u) L 2 (L 2 (R3 )) d j 2− 2 u 2 0, 1 ∇h u 2 0, 1 w 2 0, 1 ∇h w 2 0, 1 . T

Then, we can immediately finish the proof.

B

2

B

2

B

2

B

2

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T. Zhang

Proof of Proposition 3.1. We distinguish the terms with horizontal derivatives from the terms with vertical ones, writing F j (T ) ≤ F jh (T ) + F jv (T ), where F jh (T ) :=

T 0

vj (u h · ∇h w)vj wd x dt

R3

and F jv (T )

T

:= 0

R3

vj (u 3 ∂3 w)vj wd x dt.

Using Hölder’s inequality, Lemmas 2.2 and 3.1, we obtain F jh (T ) ≤ vj (u h · ∇h w)

4

4

L T3 (L h3 (L 2v ))

d 2j νh−1 2− j w2 0, 1

2 (T )

B

u h

vj w L 4 (L 4 (L 2 )) T

1

B 0, 2 (T )

h

v

.

Applying the trick from [4,12], using paradifferential decomposition in the vertical variable to vj (u 3 ∂3 w) first, then by a commutator process, one gets

vj (u 3 ∂3 w) = S vj−1 u 3 ∂3 vj w + +

v [vj ; Sl−1 u 3 ]∂3 lv w

| j−l|≤5 v (Sl−1 u3

− S vj−1 u 3 )∂3 vj lv w +

| j−l|≤5

v vj (lv u 3 ∂3 Sl+2 w).

l≥ j−N0

Correspondingly, we decompose F jv (T ) as F jv (T ) := F j1,v (T ) + F j2,v (T ) + F j3,v (T ) + F j4,v (T ). Using integration by parts and the fact that divu=0, we have F j1,v (T )

1 = 2 = 0

T 0

T

R3

R3

S vj−1 divh u h |vj w|2 d x dt

S vj−1 u h

· ∇h vj wvj wd x dt.

From Lemma 2.2 and Hölder’s inequality, we get F j1,v (T ) ≤ S vj−1 u h L 4 (L 4 (L ∞ )) vj w L 4 (L 4 (L 2 )) ∇h vj w L 2 (L 2 (R3 )) T

h

d 2j νh−1 2− j u h

v

1 B 0, 2 (T )

T

w2 0, 1 B

v

h

2 (T )

.

T


221

To deal with the commutator in F j2,v , we first use the Taylor formula to get F j2,v (T ) =

T

| j−l|≤5 0

R3

2j

1

h(2 j (x3 − y3 ))

R

0

v Sl−1 ∂3 u 3 (x h , τ y3 + (1 − τ )x3 )dτ

×(y3 − x3 )∂3 lv w(x h , y3 )dy3 vj w(x)d x dt, where h = F −1 ϕ. Using divu = 0 and integration by parts, we have F j2,v (T ) =

T

R3

| j−l|≤5 0

R

¯ j (x3 − y3 )) h(2

1 0

·∇h ∂3 lv w(x h , y3 )dy3 vj w(x)d x dt T ¯ j (x3 − y3 )) + h(2 R3

| j−l|≤5 0

R

v Sl−1 u h (x h , τ y3 + (1 − τ )x3 )dτ

1 0

v Sl−1 u h (x h , τ y3 + (1 − τ )x3 )dτ

×∂3 lv w(x h , y3 )dy3 · ∇h vj w(x)d x dt, ¯ 3 ) = x3 h(x3 ). Using Hölder’s inequality, Young’s inequality and Lemma 2.2, where h(x we obtain F j2,v (T ) v 2l− j Sl−1 u h L 4 (L 4 (L ∞ )) ∇h lv w L 2 (L 2 (R3 )) vj w L 4 (L 4 (L 2 )) T

| j−l|≤5

+

| j−l|≤5

v

h

T

T

v

h

v 2l− j Sl−1 u h L 4 (L 4 (L ∞ )) ∇h vj w L 2 (L 2 (R3 )) lv w L 4 (L 4 (L 2 )) T

d 2j νh−1 2− j u h

1

B 0, 2 (T )

h

v

T

w2 0, 1 B

2 (T )

T

h

v

.

It is easy to see that F j3,v (T )

≤

| j−l |≤5 | j−l|≤5

T 0

R3

lv u 3 ∂3 vj lv wvj wd x dt.

We can rewrite lv u 3 as follows: lv u 3

=

R

g v (2l (x3 − y3 ))∂3 lv u 3 (x h , y3 )dy3 ,

where g v ∈ S(R) satisfying F(g v )(ξ3 ) =

ϕ(|ξ ˜ 3 |) iξ3 ,

(3.10)

and ϕ˜ is a function in D(( 21 , 3))

satisfying ϕ(τ ˜ ) = 1 with τ ∈ ( 43 , 83 ). Using divu = 0, integration by parts, Young’s

222

T. Zhang

inequality and Lemma 2.2, we get F j3,v (T )

≤

| j−l |≤5 | j−l|≤5

T 0

R3 R

g v (2l (x3 − y3 ))lv u h (x h , y3 )dy3 · ∇h ∂3 vj lv wvj wd x

v l v h v v v + g (2 (x3 − y3 ))l u (x h , y3 )dy3 · ∇h j w∂3 j l wd x dt R3 R 2l−l lv u h L 4 (L 4 (L ∞ )) ∇h vj w L 2 (L 2 (R3 )) vj w L 4 (L 4 (L 2 )) T

| j−l |≤5 | j−l|≤5

d 2j νh−1 2− j u h

1

B 0, 2 (T )

v

h

T

w2 0, 1

2 (T )

B

T

v

h

.

Similarly, we have F j4,v (T )

T ≤ l≥ j−N0 0

R3

vj

v

R

g (2

l

(x3 − y3 ))lv u h (x h , y3 )dy3

v · ∇h ∂3 Sl+2 w

vj wd x

v v l v h v v + j g (2 (x3 − y3 ))l u (x h , y3 )dy3 ∂3 Sl+2 w · ∇h j wd x dt R3 R v h v l u L 4 (L 4 (L 2 )) ∇h Sl+2 w L 2 (L 2 (L ∞ )) vj w L 4 (L 4 (L 2 )) T

v

h

l≥ j−N0

+

T

v

h

T

v

h

v lv u h L 4 (L 4 (L 2 )) Sl+2 w L 4 (L 4 (L ∞ )) ∇h vj w L 2 (L 2 (L 2 )) T

v

h

l≥ j−N0

d 2j νh−1 2− j u h

1

B 0, 2 (T )

T

w2 0, 1 B

2 (T )

h

v

T

h

v

.

This completes the proof of Proposition 3.1.

Proof of Proposition 3.2. We distinguish the terms with horizontal derivatives from the terms with vertical ones, writing v2 G j (T ) ≤ G hj (T ) + 2G v1 j (T ) + G j (T ),

where G hj (T ) G v1 j (T )

:= :=

2 2

l=1 k=1 0 2 T k=1 0

and G v2 j (T )

T

:= 0

T

R3

R3

R3

vj (−)−1 ∂l ∂k (u l u k )vj ∂h wd x dt,

vj (−)−1 ∂3 ∂k (u 3 u k )vj ∂h wd x dt,

vj (−)−1 ∂3 (2u 3 ∂3 u 3 )vj ∂h wd x dt.


223

Using Hölder’s inequality and Lemma 3.3, we get

G hj (T )

2 2

T

vj (u l u k ) L 2 vj ∂h w L 2 dt 0 l=1 k=1 T j d j 2− 2 u h 0, 1 ∇h u h 0, 1 vj ∂h w L 2 dt B 2 B 2 0 2 −1 − j h 2 d j νh 2 u 0, 1 w 0, 1 . B 2 (T ) B 2 (T )

Similarly, we have G v1 j (T )

T 0

1

j

1

1

1

d j 2− 2 u h 2 0, 1 ∇h u h 2 0, 1 u 3 2 0, 1 ∇h u 3 2 0, 1 vj ∂h w L 2 dt

d 2j 2− j

B

T

u h

0

B

2

1 2

B

0, 21

∇h u h

B

2

1 2

B

0, 21

u 3

B

2

1 2

B

0, 21

∇h u 3

2

1 2 1

B 0, 2

∇h w

1

B 0, 2

dt.

Since divu = 0, we obtain

G v2 j (T )

=

T

0

=

0

T

v −1 3 h v 3 j (−) ∂3 (2u divh u ) j ∂h wd x dt R v 3 h v −1 dt. (2u div u ) (−) ∂ ∂ wd x h 3 h j 3 j R

Then, using Hölder’s inequality, Minkowski’s inequality, Lemmas 2.3 and 3.2, we get G v2 j (T )

T

0

vj (u 3 divh u h ) j

d j 2− 2

0

T

∇h u h

4

L h3 (L 2v )

vj (−)−1 ∂h ∂3 w L 4 (L 2 ) dt h

1

B

0, 21

v

1

u 3 2 0, 1 ∇h u 3 2 0, 1 B

2

B

2

1 2

1

×vj (−)−1 ∂h ∂3 w L 2 vj ∇h (−)−1 ∂h ∂3 w L2 2 dt T 1 1 1 1 d 2j 2− j ∇h u h 0, 1 u 3 2 0, 1 ∇h u 3 2 0, 1 w 2 0, 1 ∇h w 2 0, 1 dt. 0

B

2

B

This completes the proof of Proposition 3.2.

2

B

2

B

2

B

2

Using a similar argument as that in [12], one can easily obtain the uniqueness of the solution u. Thus, we finish the proof of Theorem 1.1. Acknowledgement. My research is supported by NSFC 10571158 and China Postdoctoral Science Foundation 20060400335.

224

T. Zhang

References 1. Cannone, M., Meyer, Y., Planchon, F.: Solutions autosimilaires des équations de Navier-Stokes. Séminaire “Équations aux Dérivées Partielles” de l’École Polytechnique, Exposé VIII, 1993-1994 2. Chemin, J.Y., Gallagher, I.: Wellposedness and stability results for the Navier-Stokes equations in R 3 . Annales de l’Institut Henri Poincaré - Analyse non linéaire (2008), doi:10.1016/j.anihpc.2007.05.008 3. Chemin, J.Y., Gallagher, I.: Large, global solutions to the Navier-Stokes equations, slowly varying in one direction. Trans. Amer. Math. Soc., accepted 4. Chemin, J.Y., Zhang, P.: On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations. Commun. Math. Phys. 272(2), 529–566 (2007) 5. Chemin, J.Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics. An introduction to rotating fluids and the Navier-Stokes equations. Oxford Lecture Series in Mathematics and its Applications, 32. Oxford: The Clarendon Press/Oxford University Press, 2006 6. Chemin, J.Y., Desjardins, B., Gallagher, I., Grenier, E.: Fluids with anisotropic viscosity. Math. Model. Numer. Anal. 34(2), 315–335 (2000) 7. Ekman, V.W.: On the influence of the earth’s rotation on ocean currents. Ark. Mat. Astr. Fys. 2(11), 1–52 (1905) 8. Grenier, E., Masmoudi, N.: Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Part. Differ. Eqs. 22(5-6), 953–975 (1997) 9. Iftimie, D.: A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity. SIAM J. Math. Anal. 33(6), 1483–1493 (2002) 10. Iftimie, D.: The resolution of the Navier-Stokes equations in anisotropic spaces. Rev. Mat. Iberoamericana 15(1), 1–36 (1999) 11. Koch, H. Tataru D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001) 12. Paicu, M.: Équation anisotrope de Navier-Stokes dans des espaces critiques. Rev. Mat. Iberoamericana 21(1), 179–235 (2005) 13. Paicu, M.: Équation periodique de Navier-Stokes sans viscosité dans une direction. Comm. Part. Differ. Eqs. 30(7-9), 1107–1140 (2005) 14. Pedlosky, J.: Geophysical Fluid Dynamics. 2nd Edition, Berlin Heidelberg - New York: Springer-Verlag, 1987 15. Zhang, T., Fang, D.Y.: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations. J. Math. Pures Appl. accepted, doi:10.1016/j.matpur.2008.06.008 Communicated by P. Constantin


Communications in


Continuity of Eigenfunctions of Uniquely Ergodic Dynamical Systems and Intensity of Bragg Peaks Daniel Lenz On leave from: Fakultät für Mathematik, D- 09107 Chemnitz, Germany. E-mail: [email protected] URL: http://www.tu-chemnitz.de/mathematik/analysis/dlenz Received: 8 February 2008 / Accepted: 18 March 2008 Published online: 2 September 2008 – © Springer-Verlag 2008

Abstract: We study uniquely ergodic dynamical systems over locally compact, sigma-compact Abelian groups. We characterize uniform convergence in Wiener/Wintner type ergodic theorems in terms of continuity of the limit. Our results generalize and unify earlier results of Robinson and Assani respectively. We then turn to diffraction of quasicrystals and show how the Bragg peaks can be calculated via a Wiener/Wintner type result. Combining these results we prove a version of what is sometimes known as the Bombieri/Taylor conjecture. Finally, we discuss various examples including deformed model sets, percolation models, random displacement models, and linearly repetitive systems. 1. Introduction This paper is devoted to two related questions. One question concerns (uniform) convergence in the Wiener/Winter ergodic theorem. The other question deals with calculating the intensities of Bragg peaks in the diffraction of quasicrystals and, in particular, with the so called Bombieri/Taylor conjecture. As shown below the calculation of Bragg peaks can be reduced to convergence questions in certain ergodic theorems. The ergodic theorems of the first part then allow us to prove a version of the Bombieri/Taylor conjecture. Let us give an outline of these topics in this section. More precise statements and definitions will be given in the subsequent sections. Consider a topological dynamical system (Ω, α) over a locally compact, σ -compact Abelian group G. Let ξ belong to the dual group of G and let (Bn ) be a van Hove sequence. We study convergence of averages of the form 1 (∗) (ξ, s) f (α−s ω)ds. |Bn | Bn Current Address: Department of Mathematics, Rice University, P. O. Box 1892, Houston, TX 77251, USA.

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Due to the von Neumann ergodic theorem and the Birkhoff ergodic theorem, it is known that these averages converge in L 2 and pointwise to the projection E T ({ξ }) f of f on the eigenspace of ξ . In fact (for G = R or G = Z) the set of ω ∈ Ω, where pointwise convergence fails, can be chosen uniformly in ξ . This is known as Wiener/Wintner ergodic theorem after [70]. Now, consider a uniquely ergodic (Ω, α). Unique ergodicity of (Ω, α) is equivalent to uniform (in ω ∈ Ω) convergence in (∗) for ξ ≡ 1 and continuous f . Thus, in this case one might expect uniform convergence for arbitrary ξ and continuous f on Ω. The first result of this paper, Theorem 1 in Sect. 3, characterizes validity of uniform convergence. It is shown to hold whenever possible, viz if and only if the limit E T ({ξ }) f is continuous. This generalizes earlier results of Robinson [57] for G = Rd and G = Zd . More precisely, Robinson’s results state uniform convergence in two situations, viz for continuous eigenvalues ξ and for ξ outside the set of eigenvalues. To us the main achievement of our result is not so much the generalization of Robinson’s results but rather our new proof. It does not require any case distinctions but only continuity of the limit. Our line of argument is related to work of Furman on uniform convergence in subadditive ergodic theorems [19]. We then study dependence of convergence on ξ . Here, again, our result, Theorem 2 in Sect. 4, gives a uniformity statement, provided the limit has strong enough continuity properties in ξ . This result generalizes a result of Assani [2], where the limit is identically zero (and thus has the desired continuity properties). In fact, Theorem 2 unifies the results of Assani and Robinson. While these results are of independent interest, here they serve as a tool in the study of aperiodic order. This is discussed next. Aperiodic order is a specific form of (dis)order intermediate between periodicity and randomness. It has attracted a lot of attention both in physics and in mathematics in recent years, see e.g. the monographs and conference proceedings [6,29,45,51,59]. This is due to its intriguing properties following from its characteristic intermediate form of (dis)order. In particular, the interest rose substantially after the actual discovery of physical substances, later called quasicrystals, which exhibit such a form of (dis)order [28,60]. These solids were discovered in diffraction experiments by their unusual diffraction patterns. These patterns have, on the one hand, many points, called Bragg peaks, indicating long range order. On the other hand these patterns have symmetries incompatible with a lattice structure. Hence, these systems are not periodic. Put together, these solids exhibit long range aperiodic order. Investigation of mathematical diffraction theory is a key point in the emerging theory of aperiodic order, see the survey articles [7,25,33,36,48] and references given there. The main object of diffraction theory is the diffraction measure γ associated to the structure under investigation (see e.g. the book [14]). This measure describes the outcome of a physical diffraction experiment. The sharp spots appearing in a diffraction experiment known as Bragg peaks are then given as the point part of γ and the intensity of a Bragg peak ξ is given by γ ({ξ }). According to these considerations central problems in mathematical diffraction theory are to • prove pure pointedness of γ or at least existence of a “large” point component of γ for a given structure, • explicitly determine ξ with γ ({ξ }) > 0 and calculate γ ({ξ }) for them. Starting with the work of Hof [23], these two problems have been studied intensely over the last two decades for various models (see references above). The two main classes

Continuity of Eigenfunctions

227

of models are primitive substitutions and cut and project models. Prominent examples such as the Fibonacci model or Penrose tilings belong to both classes [59]. From the very beginning the use of dynamical systems has been a most helpful tool in these investigations. The basic idea is to not consider one single structure but rather to assemble all structures with the “same” form of (dis)order (see e.g. [54]). This assembly will be invariant under translation and thus give rise to a dynamical system. There is then a result of Dworkin [17] showing that the diffraction spectrum is contained in the dynamical spectrum (see as well the results of van Enter/Mi¸ekisz [18] for closely related complementary results). This so-called Dworkin argument has been extended and applied in various contexts. In particular it has been the main tool in proving pure point diffraction by deducing it from pure point dynamical spectrum [25,58,62,64]. Recently it has even been shown that pure point dynamical spectrum is equivalent to pure point diffraction [3,22,40] and that the set of eigenvalues is just the group generated by the Bragg peaks [3]. These results can be understood as (at least) partially solving the first problem mentioned above by relating the set of Bragg peaks to the eigenvalues of the associated dynamical system. As for the second problem, the main line of reasoning goes as follows: Let the structure under investigation be given by a uniformly discrete relatively dense point set Λ ξ in Rd . For B ⊂ Rd bounded with non-empty interior and ξ ∈ Rd define c B (Λ) := 1 x∈Λ∩B exp(−2πiξ x), where | · | denotes Lebesgue measure. Then, the following |B| should hold: (∗∗)

ξ

γ ({ξ }) = lim |cCn (Λ)|2 , n→∞

where Cn denotes the cube around the origin with side length 2n. In fact, this is a crucial equality both in numerical simulations and in theoretical considerations. It is sometimes discussed under the heading of “Bombieri/Taylor conjecture”. In their work [10,11] Bombieri/Taylor state (for special one-dimensional systems) that the Bragg peaks are ξ given by those ξ for which limn→∞ cCn (Λ) = 0. They do not give a justification for their statement and it then became known as their conjecture [23,25]. ξ Since then various works have been devoted to proving existence of limn→∞ cCn (Λ) and rigorously justifying the validity of (∗∗). While the case of general uniquely ergodic systems is open, it has been shown by Hof in [23] that (∗∗) follows whenever a rather ξ uniform convergence of c Bn (Λ) for van Hove sequences (Bn ) is known (see the work [1] for a complementary result). This, in turn has been used to obtain validity of (∗∗) for model sets [25] and for primitive substitutions [20] (see Theorem 5.1 in [64] for related material as well). Also, it has been mentioned in various degrees of explicitness [23,25,33,63] that this ξ uniform convergence of c Bn (Λ) follows from or is related to continuity of eigenfunctions due to Robinson’s results [57] once one is in a dynamical system setting. So far no proofs for these statements seem to have appeared. These questions are addressed in the second part of the paper. Our results show that γ ({ξ }) is related to a specific eigenfunction. More precisely, we proceed as follows. In Sect. 5 we first discuss some background on diffraction and then introduce the measure dynamical setting from Baake/Lenz [3]. This setting has the virtue of embracing the two most common frameworks for the mathematical modeling of quasicrystals viz the framework of point sets used in mathematical diffraction theory starting with

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the work [23] and the framework of bounded functions brought forward in [8]. They are both just special cases of the measure approach. Using some material from [3] we derive Theorem 3 in Sect. 5. It shows that γ ({ξ }) is the norm square of a certain specific eigenfunction to ξ . This does not require any ergodicity assumptions and relies solely on the Stone/von Neumann spectral theorem for unitary representations and [3]. Depending on whether one has the von Neumann ergodic theorem, or the Birkhoff ergodic theorem or a uniform Wiener-Wintner type theorem at one’s disposal, one can then calculate γ ({ξ }) as an L 2 -limit, almost-sure limit or uniform limit of averages of the form (∗). These averages can then be related to the Fourier type averages in (∗∗) by Lemma 8 to give the corresponding L 2 , almost-sure pointwise and uniform convergence statement in (∗∗). This is summarized in Theorem 5 in Sect. 6. In particular, (b) of Theorem 5 shows that an almost sure justification of Bombieri/Taylor holds in arbitrary ergodic dynamical systems and does not require any continuity assumptions on the eigenfunction to ξ . As a consequence we obtain in Corollary 3 a variant of the Bombieri/Taylor conjecture valid for arbitrary uniquely ergodic systems. These abstract results can be used to reprove validity of (∗∗) for the two most common models of aperiodic order viz primitive substitution models and models arising from cut and project schemes (see the first remark in Sect. 6). More importantly, they can be used to prove validity of (∗∗) for a variety of new models. In particular, they allow one to obtain variants of the Bombieri/Taylor conjecture for several models arising from strictly aperiodically ordered ones by some randomization or smearing out process. In fact, based on the results of this paper a strong version of the Bombieri/Taylor conjecture is proven by Lenz/Strungaru [38] for the class of deformed model sets earlier studied in [4,9,22] and by Lenz/Richard [37] for dense Dirac combs introduced in [56]. These results are shortly sketched in Sect. 7. Moreover, as shown in Sect. 8 our abstract results yield almost sure validity of the Bombieri/Taylor conjecture for both percolation models and random displacement models based on aperiodic order. These models are more realistic in that they take into account defects and thermal motion in solids respectively. Percolation models based on aperiodic order were introduced by Hof [26]. There, equality of various critical probabilities is shown. An extension of Hof’s work to graphs together with an application to random operators is then given in recent work of Müller/Richard [50]. Random displacement in diffraction for a single object (rather than a dynamical system) is discussed by Hof in [24]. In fact, convergence of diffraction for both percolation and random displacement models (and quite some further models) has recently been studied by Külske [30,31]. His results give rather universal convergence of approximants. However, they deal with a smoothed version of diffraction. Thus, they do not seem to give validity of the Bombieri/Taylor conjecture. In this sense, our results complement the corresponding results of [30]. We refer to Sect. 8 for further details. In the final section, we study so called linearly repetitive Delone dynamical systems, introduced by Lagarias/Pleasants in [34], and their subshift counterparts, so-called linearly recurrent subshifts, studied e.g. by Durand in [16]. These examples have attracted particular attention in recent years and have been brought forward as models for perfectly ordered quasicrystals in [34]. Quite remarkably, continuity of eigenfunctions fails for these models in general as recently shown in [12]. Nevertheless, we are able to establish validity of (∗∗) for these models. More generally, we show that uniform convergence of the modules of the expressions in (∗) holds (while uniform convergence of the expressions themselves may fail) and this gives validity of (∗∗) as discussed above. This is based on the subadditive ergodic theorems from [15,35]. Let us emphasize that


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convergence in these cases does not hold uniformly in the van Hove sequences but only for so-called Fisher sequences. 2. Dynamical Systems Our general framework deals with actions of locally compact Abelian groups on compact spaces. Thus, we start with some basic notation and facts concerning these topics. These will be used throughout the paper. Whenever X is a σ -compact locally compact space (by which we include the Hausdorff property), the space of continuous functions on X is denoted by C(X ) and the subspace of continuous functions with compact support by Cc (X ). For a bounded function f on X , we define the supremum norm by f ∞ := sup{| f (x)| : x ∈ X }. Equipped with this norm, the space C K (X ) of complex continuous functions on X with support in the compact set K ⊂ G becomes a complete normed space. Then, the space Cc (G) is equipped with the locally convex limit topology induced by the canonical embedding C K (X ) → Cc (X ), K ⊂ G, compact. As X is a topological space, it carries a natural σ -algebra, namely the Borel σ -algebra generated by all closed subsets of X . The set M(X ) of all complex regular Borel measures on G can then be identified with the space Cc (X )∗ of complex valued, continuous linear functionals on Cc (G). This is justified by the Riesz-Markov representation theorem, compare [52, Ch. 6.5] for details. The space M(X) carries the vague topology, i.e., the weakest topology that makes all functionals µ → X f dµ, ϕ ∈ Cc (X ), continuous. The total variation of a measure µ ∈ M(X ) is denoted by |µ|. Now fix a σ -compact locally compact Abelian (LCA) group G. Denote the Haar and the pairing measure on G by θG or by ds. The dual group of G is denoted by G, between a character ξ ∈ G and t ∈ G is written as (ξ, t). Whenever G acts on the compact space Ω by a continuous action α : G × Ω −→ Ω, (t, ω) → αt (ω), where G ×Ω carries the product topology, the pair (Ω, α) is called a topological dynamical system over G. We will often write αt ω for αt (ω). An α-invariant probability measure is called ergodic if every measurable invariant subset of Ω has measure zero or measure one. The dynamical system (Ω, α) is called uniquely ergodic if there exists a unique α-invariant probability measure. We will need two further pieces of notation. A map between dynamical system (Ω, α) and (Ω , α ) over G is called a G - map if (αt (ω)) = αt ( (ω)) for all ω ∈ Ω and t ∈ G. A continuous surjective G - map is called a factor map. Given an α-invariant probability measure m, we can form the Hilbert space L 2 (Ω, m) of square integrable measurable functions on Ω. This space is equipped with the inner product f, g = f, g Ω := f (ω) g(ω) dm(ω). Ω

The action α gives rise to a unitary representation T = T (Ω,α,m) of G on L 2 (Ω, m) by Tt : L 2 (Ω, m) −→ L 2 (Ω, m) , (Tt f )(ω) := f (α−t ω) ,

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for every f ∈ L 2 (Ω, m) and arbitrary t ∈ G. An f ∈ L 2 (Ω, m) is called an eigenfunc if Tt f = (ξ, t) f for every t ∈ G. An eigenfunction tion of T with eigenvalue ξ ∈ G (to ξ , say) is called continuous if it has a continuous representative f with f (α−t ω) = (ξ, t) f (ω), for all ω ∈ Ω and t ∈ G. By Stone’s theorem, compare [42, Sect. 36D], there exists a projection valued measure −→ projections on L 2 (Ω, m) E T : Borel sets of G with

f, Tt f =

G

(ξ, t) d f, E T (ξ ) f :=

G

(ξ, t) dρ f (ξ ) ,

(1)

defined by ρ (B) := f, E T (B) f . where ρ f is the measure on G f We will be concerned with averaging procedures along certain sequences. To do so we define for Q, P ⊂ G the P boundary ∂ P Q of Q by ∂ P Q := ((P + Q) \ Q ◦ ) ∪ ((−P + G \ Q) ∩ Q)), where the bar denotes the closure of a set and the circle denotes the interior. As G is σ -compact, there exists a sequence {Bn : n ∈ N} of open, relatively compact sets Bn ⊂ G with Bn ⊂ Bn+1 , G = n≥1 Bn , and θG (∂ K Bn ) = 0, n→∞ θG (Bn ) lim

for every compact K ⊂ G see [62] for details. Such a sequence is called a van Hove sequence. The relevant averaging operator is defined next. B ⊂ G relatively Definition 1. Let (Ω, α) be a dynamical system over G. For ξ ∈ G, compact with non-empty interior and a bounded measurable f on Ω, the bounded ξ measurable function A B ( f ) on Ω is defined by 1 ξ (ξ, s) f (α−s ω)ds. A B ( f )(ω) := θG (B) B ξ

In particular, A B maps C(Ω) into itself. In this context the von Neumann ergodic theorem (see e.g. [32, Thm. 6.4.1]) gives the following. Lemma 1. Let (Ω, α) be a dynamical system over G with α-invariant probability measure m and associated spectral family E T . Then, ξ

A Bn ( f ) −→ E T ({ξ }) f, n → ∞, in L 2 (Ω, m) for any f ∈ C(Ω) and every van Hove sequence (Bn ). There is also a corresponding well-known pointwise statement.


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Lemma 2. Let (Ω, α) be a dynamical system and m an ergodic invariant probability measure on Ω. Let (Bn ) be a van Hove sequence in G along which the Birkhoff ergodic ξ theorem holds. Then, A Bn ( f ) converges almost surely (and in L 2 (Ω, m)) to E T ({ξ }) f for every f ∈ C(Ω). Remark. As shown by Lindenstrauss in [39], every amenable group admits a van Hove sequence along which the Birkhoff ergodic theorem holds. Proof of Lemma 2. Let T be the unit circle and consider the dynamical system Ω × T with action of G given by αs (ω, θ )) = (αs ω, (ξ, s)θ ). Then, the statement follows from the Birkhoff ergodic theorem applied to F(ω, θ ) = θ f (ω). In order to prove our abstract results, we need two more preparatory results. Lemma 3. Let (Ω, α) be a dynamical system over G. Let Q, P be open, relatively compact non-empty subsets of G. Then, ξ

ξ

ξ

A Q ( f ) − A P (A Q ( f )) ∞ ≤

θG (∂ P∪(−P) Q) f ∞ θG (Q)

for every f ∈ C(Ω). Proof. For t ∈ G a direct calculation shows ξ ξ |A Q ( f )(ω) − (ξ, t)A Q (α−t ω)|

1 = f (α−s ω)(ξ, s)ds θG (Q) Q f (α−s ω)(ξ, s)ds − t+Q

θG (Q \ (t + Q) ∪ (t + Q) \ Q) ≤ f ∞ . θG (Q) For t ∈ P, we have Q \ (t + Q) ∪ (t + Q) \ Q ⊂ ∂ P∪(−P) Q and the lemma follows. The following lemma is certainly well known. We include a proof for completeness. Lemma 4. Let (Ω, α) be uniquely ergodic with unique α-invariant probability measure m. Let K ⊂ Ω be compact and denote the characteristic function of K by χ K . Then, for every van Hove sequence (Bn ), 1 lim sup χ K (α−s ω)ds ≤ m(K ), n→∞ |Bn | Bn uniformly in ω ∈ Ω. Proof. As Ω is compact, the measure m is regular. In particular, for every ε > 0, we can find an open set V containing K with m(V ) ≤ m(K ) + ε. By Urysohns lemma, we can then find a continuous function h : Ω −→ [0, 1] with support contained in V and h ≡ 1 on K . By construction, Ω h(ω)dm(ω) ≤ m(K ) + ε and 1 1 χ K (α−s ω)ds ≤ h(α−s ω)ds 0≤ |Bn | Bn |Bn | Bn

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for every ω ∈ Ω. By unique ergodicity, |B1n | Bn h(α−s ω)ds converges uniformly on Ω to Ω h(ω)dm(ω). Putting this together, we obtain 1 χ K (α−s ω)ds ≤ h(ω)dm(ω) ≤ m(K ) + ε lim sup n→∞ |Bn | Bn Ω uniformly on Ω. As ε > 0 is arbitrary, the statement of the lemma follows. 3. Uniform Wiener-Wintner Type Results In this section we discuss the following theorem. Theorem 1. Let (Ω, α) be a uniquely ergodic dynamical system over G with and f ∈ C(Ω) be arbitrary. Then, α-invariant probability measure m. Let ξ ∈ G the following assertions are equivalent: (i) The function E T ({ξ }) f has a continuous representative g satisfying g(α−s ω) = (ξ, s) g(ω) for every s ∈ G and ω ∈ Ω. ξ (ii) For some (and then every) van Hove sequence (Bn ), the averages A Bn ( f ) converge uniformly, that is to say w.r.t. the supremum norm to a function g. Remark. (a) The hard part of the theorem is the implication (i) ⇒ (ii). Note that (i) comprises three situations: • ξ is an eigenvalue of T with a continuous eigenfunction. • ξ is not an eigenvalue at all (then g ≡ 0). • ξ is an eigenvalue and f is perpendicular to the corresponding eigenfunctions (then, again, g ≡ 0). (b) Our proof relies on the von Neumann ergodic theorem, Lemma 1, and unique ergodicity only. Thus, the proof carries immediately over to give a semigroup version e.g. for actions of N, as the von Neumann ergodic theorem is known then. (c) The statement (i) ⇒ (ii) is given for the first two situations of (a) separately by Robinson in [57] for actions of G = Zd and G = Rd . Actually, his proof also works for the third situation. Our proof is different in this case and works for all three situations at the same time. Robinson also has a version for actions of N. As mentioned in (b), this can be shown by our method as well. Proof of Theorem 1. (ii)⇒ (i): By assumption there exists a van Hove sequence (Bn ) ξ ξ such that the averages (A Bn ( f )) converge uniformly to a function g. As each A Bn ( f ) is continuous, so is g. Moreover, a direct calculation shows ξ

ξ

g(α−t ω) = lim A Bn ( f )(α−t ω) = lim (ξ, t)At+Bn ( f )(ω) = (ξ, t)g(ω) n→∞

n→∞ ξ

for all t ∈ G and ω ∈ Ω. By Lemma 1, A Bn ( f ) converges in L 2 (Ω, m) to E T ({ξ }) f . ξ

Thus, the uniform convergence of the A Bn ( f ) to g implies g = E T ({ξ }) f . This finishes the proof of this implication. (The fact that convergence holds for every van Hove sequence will be proven along the way of (i) ⇒ (ii).) (i)⇒ (ii): Let (Bn ) be an arbitrary van Hove sequence.


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Let ε > 0 be given. We show that ξ

() A Bn ( f ) − g ∞ ≤ 3(1 + f ∞ + g ∞ )ε for all sufficiently large n. ξ By assumption (i) and Lemma 1, A Bn ( f ) converges in L 2 (Ω, m) to the continuous function g. This implies limn→∞ µ(Ωn ) = 1, where ξ

Ωn := {ω ∈ Ω : |A Bn ( f )(ω) − g(ω)| < ε}. In particular, for sufficiently large N , we have m(Ω N ) ≥ 1 − ε. ξ

Fix such an N and set b := A B N ( f ). As both b and g are continuous, the set Ω N is open. Thus, its complement K := Ω \ Ω N is compact. Let χΩ N and χ K be the characteristic functions of Ω N and K respectively. Thus, in particular, • χΩ N (ω) > 0 implies ω ∈ Ω N , i.e. |b(ω) − g(ω)| < ε. • m(K ) ≤ ε (as m(Ω N ) ≥ 1 − ε). ξ

ξ

ξ

ξ

Note that A Bn (b) = A B N (A Bn ( f )) by Fubini’s theorem and A Bn (g) = g by the invariance assumption on g. Thus, we can estimate ξ

ξ

ξ

ξ

ξ

ξ

ξ

ξ

ξ

ξ

A Bn ( f ) − g ∞ ≤ A Bn ( f ) − A B N (A Bn ( f )) ∞ + A Bn (b) − A Bn (g) ∞ ≤ A Bn ( f ) − A B N (A Bn ( f )) ∞ + A Bn (χΩ N (b − g)) ∞ ξ

ξ

+ A Bn (χ K b) ∞ + A Bn (χ K g) ∞ . We estimate the last four terms: Term 1. By Lemma 3, this term can be estimated from above by θG (∂ B N ∪(−B N ) Bn ) f ∞ . θG (Bn ) As (Bn ) is a van Hove sequence, this term is smaller than ε for sufficiently large n. Term 2. As χΩ N (ω) > 0 implies |b(ω) − g(ω)| < ε, the estimate χΩ N (b − g) ∞ < ε ξ holds. This gives A Bn (χΩ N (b − g)) ∞ ≤ χΩ N (b − g) ∞ < ε and the second term is smaller than ε for every n ∈ N. Term 3. A short calculation shows 1 ξ A Bn (χ K b) ∞ ≤ χ K (α−s ω)ds b ∞ . θG (Bn ) Bn 1 As K is compact with m(K ) ≤ ε, we can then infer from Lemma 4 that θG (B n) χ (α ω)ds is uniformly bounded by 2ε for large enough n. As b ≤ f by K −s ∞ ∞ Bn definition of b, we conclude that the third term is smaller than 2ε f ∞ for sufficiently large n. Term 4. Using the same arguments as in the treatment of the third term, we see that the fourth term can be estimated above by 2ε g ∞ for sufficiently large n. Putting the estimates together we infer ().

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We can use the above method of proof to give a proof for the key technical lemma of [57] in our context. be arbitrary. Lemma 5. Let (Ω, α) be uniquely ergodic. Let f ∈ C(Ω) and ξ ∈ G √ ξ Then, limn→∞ A Bn ∞ = E T ({ξ }) f, E T ({ξ }) f . Proof. For each ε > 0 and N ∈ N, Lemma 3 gives ξ

ξ

ξ

ξ

A Bn ( f ) ∞ ≤ A B N (A Bn ( f )) ∞ + ε ≤ A B N ( f ) ∞ + ε ξ

for sufficiently large n ∈ N. This easily shows existence of the limit limn→∞ A Bn ( f ) ∞ . ξ

Now, by the von Neumann ergodic theorem, we have L2 -convergence of A Bn ( f ) to ξ

E T ({ξ }) f . This gives L2 -convergence of |A Bn ( f )| to |E T ({ξ }) f | and the latter is almost √ surely equal to c := E T ({ξ }) f, E T ({ξ }) f . As each L2 converging sequence contains ξ an almost surely converging subsequence, we infer that limn→∞ A Bn ( f ) ∞ ≥ c. ξ

It remains to show limn→∞ A Bn ( f ) ∞ ≤ c. Here, we mimic the previous proof: ξ

Choose ε > 0 arbitrary. By L2 -convergence of |A Bn ( f )| to the constant function c, we can find an N ∈ N such that m(Ω N ) ≥ 1 − ε, where ξ

Ω N := {ω ∈ Ω : ||A B N ( f )(ω)| − c| < ε}. Note that Ω N is open and hence Ω \ Ω N is compact. For n large enough we then have (see above) ξ

ξ

ξ

ξ

ξ

A Bn ( f ) ∞ ≤ A B N (A Bn ( f )) ∞ + ε ≤ A Bn (χΩ N A B N ( f )) ∞ ξ

ξ

+ A Bn (χΩ\Ω N A B N ( f )) ∞ + ε. ξ

For ω ∈ Ω N , we have |A B N ( f )(ω)| ≤ c + ε. Hence, the first term on the right-hand side can be estimated by c + ε. The second term on the right-hand side can be estimated by f ∞ |B1n | Bn χΩ\Ω N (αs ω)ds. By Lemma 4, |B1n | Bn χΩ\Ω N (αs ω)ds is smaller than 2ε = m(Ω \ Ω N ) + ε for sufficiently large n. This finishes the proof of the lemma. 4. Unifying Theorem 1 and a Result of Assani In this section we present the following consequence and in fact generalization of the hard part of Theorem 1, which generalizes a result of Assani as well. Theorem 2. Let (Ω, α) be a uniquely ergodic dynamical system over G with α-invariant be given such that, firstly, for every probability measure m. Let f ∈ C(Ω) and K ⊂ G ξ ∈ K the function E T ({ξ }) f is continuous with E T ({ξ }) f (α−s ω) = (ξ, s)E T ({ξ }) f for all ω ∈ Ω and s ∈ G and, secondly, K −→ C(Ω), ξ → E T ({ξ }) f , is continuous. Then, ξ

lim sup A Bn ( f ) − E T ({ξ }) f ∞ = 0

n→∞ ξ ∈K

for every van Hove sequence (Bn ).


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Remark. (a) Certainly the theorem contains the case K = {ξ } and we recover the implication (i) ⇒ (ii) of Theorem 1. (b) The proof of the theorem relies on the previous theorem and compactness. Thus, again, there is a semigroup version e.g. for actions of N. The theorem has the following immediate corollary. Corollary 1. Let G be discrete and let (Ω, α) be a uniquely ergodic dynamical system over G with α-invariant probability measure m. Let f ∈ C(Ω) be given such that Then, E T ({ξ }) f = 0 for every ξ ∈ G. ξ

lim sup A Bn ( f ) ∞ = 0

n→∞

ξ ∈G

for every van Hove sequence (Bn ). Remark. For G = Z the corollary was proven by Assani 1993 in an unpublished manuscript. A published proof can be found in his book [2]. In fact, the book gives the semigroup version for actions of N. ξ

Proof of Theorem 2. Define for each n ∈ N the function bn : K −→ R, ξ → A Bn ( f )− E T ({ξ }) f ∞ . Then, each bn is continuous by our assumptions and bn (ξ ) −→ 0, n → ∞,

(2)

for each ξ ∈ K by Theorem 1. Moreover, by the invariance assumption on E T ({ξ }) f , we ξ have A Bn (E T ({ξ }) f ) = E T ({ξ }) f for all n ∈ N. Thus, Lemma 3 and direct arguments give for all n, N ∈ N: ξ

ξ

ξ

bn (ξ ) = A Bn ( f ) − E T ({ξ }) f ∞ = A Bn ( f ) − A Bn (E T ({ξ }) f ) ∞ ξ

ξ

ξ

ξ

ξ

ξ

≤ A Bn ( f ) − A B N (A Bn ( f )) ∞ + A Bn (A B N ( f )) − A Bn (E T ({ξ }) f ) ∞ θG (∂ B N ∪(−B N ) Bn ) ξ f ∞ + A B N ( f ) − E T ({ξ }) f ∞ θG (Bn ) θG (∂ B N ∪(−B N ) Bn ) f ∞ + b N (ξ ). = θG (Bn ) ≤

As (Bn ) is a van Hove sequence this easily shows that the sequence (bn ) has the following monotonicity property: For each N ∈ N and ε > 0, there exists an n 0 (N , ε) ∈ N with bn (ξ ) ≤ b N (ξ ) + ε

(3)

for all n ≥ n 0 (N , ε) and all ξ ∈ K . Given (2) and (3), the theorem follows from compactness of K and continuity of the bn , n ∈ N, by standard reasoning.

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5. Diffraction Theory In this section we present a basic setup for diffraction (see e.g. [14]). For models with aperiodic order this framework has been advocated by Hof [23] and become a standard by now (see the introduction for references). The crucial quantity is a measure, called the diffraction measure and denoted by γ . It models the outcome of a diffraction experiment by representing the intensity (per unit volume). We begin the section with a short discussion of background material. We then discuss the measure based approach developed recently [3,4]. We also present a consequence of [3] viz Theorem 3 (and its corollary), which will be used in the next section. It may be of independent interest. We then finish this section by elaborating on how the usual approach via point sets fits into the measure approach. In a diffraction experiment a solid is put into an incoming beam of e.g. X rays. The atoms of the solid then interact with the beam and one obtains an outcoming wave. The intensity of this wave is then measured on a screen. When modeling diffraction, the two basic principles are the following: Firstly, each point x in the solid gives rise to a wave ξ → exp(−i xξ ). The overall wave w is the sum of the single waves. Secondly, the quantity measured in an experiment is the intensity given as the square of the modulus of the wave function. We start by implementing this for a finite set F ⊂ Rd . Each x ∈ F gives rise to a wave ξ → exp(−i xξ ) and the overall wavefunction w F induced by F is accordingly exp(−i xξ ). w F (ξ ) = x∈F

Thus, the intensity I F is I F (ξ ) =

⎛ exp(−i(x − y)ξ ) =

⎝

x,y∈F

⎞ δx−y ⎠ .

(4)

x,y∈F

Here, δz is the unit point mass at z and denotes the Fourier transform. When describing diffraction for a solid with many atoms it is common to model the solid by an infinite set in Rd . When trying to establish a formalism as above for an infinite set Λ, one faces the problem that exp(−i xξ ) wΛ = x∈Λ

diverges heavily and therefore does not make sense. This problem can not be overcome by interpreting the sum as a tempered distribution. The reason is that we are actually not interested in wΛ but rather in |wΛ |2 . Now, neither modulus nor products are defined for distributions. There is a physical reason behind the divergence: The intensity of the whole set Λ is really infinite. The correct quantity to consider is not the intensity but a normalized intensity viz the intensity per unit volume. It is given as I = lim

n→∞

1 IΛ∩Bn . |Bn |

Of course, existence of this limit is not clear at all. In fact, we will even have to specify in which sense existence of the limit is meant. It turns out that existence of the limit in the vague sense is equivalent to existence of the limit


237

1 n→∞ |Bn |

γ = lim

δx−y

x,y∈Λ∩Bn

in the vague sense. In this case, I is the Fourier transform γ of γ . Then, γ is known as an autocorrelation function and I = γ is known as a diffraction measure. We are particularly interested in the point part of γ . The points ξ ∈ Rd with γ ({ξ }) = 0 are called Bragg peaks. The value γ ({ξ }) is called the intensity of the Bragg peak. Particularly relevant questions in this context are the following: • When is γ a pure point measure? • Where are the Bragg peaks? • What are the intensities of the Bragg peaks? These questions have been discussed in a variety of settings by various people (see the introduction). Here, we will now present the framework and (part of) the results concerning the first two questions developed in [3,4]. The study of the last question is the main content of the remainder of the paper. We will be concerned with suitable subsets of the set M(G) of measures on G. There is a canonical map f : Cc (G) −→ C(M(G)), f ϕ (µ) := ϕ(−s) dµ(s). (5) G

The vague topology on M(G) is the smallest topology which makes all the f ϕ , ϕ ∈ Cc (G), continuous. Let C > 0 and a relatively compact open set V in G be given. A measure µ ∈ M(G) is called (C, V )-translation bounded if |µ|(t + V ) ≤ C for all t ∈ G. The set of all (C, V )-translation bounded measures is denoted by MC,V (G). The set MC,V (G) is a compact Hausdorff space in the vague topology. There is a canonical map f : Cc (G) −→ C(MC,V (G)), f ϕ (µ) := ϕ(−s) dµ(s). G

Moreover, G acts on MC,V (G) via a continuous action α given by α : G × MC,V (G) −→ MC,V (G) , (t, µ) → αt µ with (αt µ)(ϕ) := ϕ(s + t)dµ(s). G

Definition 2. (Ω, α) is called a dynamical system on the translation bounded measures on G (TMDS) if there exist a constant C > 0 and a relatively compact open set V ⊂ G such that Ω is a closed α-invariant subset of MC,V (G). Having introduced our models, we can now discuss some key issues of diffraction theory. Let (Ω, α) be a TMDS, equipped with an α-invariant probability measure m. Then, there exists a unique measure γ = γm on G, called the autocorrelation measure (often called Patterson function in crystallography [14], though it is a measure in our setting) with γ ∗ ϕ ∗ ψ (t) = f ϕ , Tt f ψ

(6)

for all ϕ, ψ ∈ Cc (G) and t ∈ G. The convolution ϕ∗ψ is defined by (ϕ∗ψ)(t) = ϕ(t − ∈ Cc (G) is defined by ψ (x) = ψ(−x). s)ψ(s) ds. For ψ ∈ Cc (G) the function ψ

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By (6) (applied with t = 0), the measure γ is positive definite. Therefore, its Fourier transform exists and is a positive measure γ . It is called the diffraction measure. As discussed in the beginning of this section, this measure describes the outcome of a diffraction experiment. Taking Fourier transforms in (6) and (1) with f = f ϕ , we obtain (see Proposition 7 in [3] for details) ρ fϕ = | ϕ |2 γ

(7)

for every ϕ ∈ Cc (G). This equation can be used to show that γ is a pure point measure if and only if T has pure point spectrum [3], see [40,62] as well. This equation also lies at the heart of the following theorem. Theorem 3. Let (Ω, α) be a TMDS with an α-invariant probability measure m and be arbitrary. Then, there exists a associated autocorrelation function γ . Let ξ ∈ G unique cξ ∈ L 2 (Ω, m) with (ξ )cξ E T ({ξ }) f ψ = ψ for every ψ ∈ Cc (G). The function cξ satisfies γ ({ξ }) = cξ , cξ . Proof. Uniqueness of such a cξ is clear. Existence and further properties can be shown as follows: From (7) we obtain by a direct polarization argument that d (#) f ϕ , E T (B) f ψ = χ B ϕˆ ψ γ measurable. Here, χ B denotes the characterisfor all ϕ, ψ ∈ Cc (G) and B ⊂ G tic function of B. Note that E T (B) is a projection and therefore f ϕ , E T (B) f ψ = measurable. E T (B) f ϕ , E T (B) f ψ for arbitraryϕ, ψ ∈ Cc (G) and B ⊂ G Now, choose σ ∈ Cc (G) with G σ (s)ds = 1 and define σ ∈ Cc (G) by σ := (ξ, ·)σ . Then, σ (ξ ) = 1. Define cξ := E T ({ξ }) f σ . Then, cξ , cξ = E T ({ξ }) f σ , E T ({ξ }) f σ = f σ , E T ({ξ }) f σ = γ ({ξ }), where we used (#) and σ (ξ ) = 1 in the last equality. Moreover, for arbitrary ψ ∈ Cc (G) a direct calculation using (#) and the definition of cξ shows cξ , E T ({ξ }) f ψ − ψ cξ = 0. E T ({ξ }) f ψ − ψ cξ for every ψ ∈ Cc (G). Thus, E T ({ξ }) f ψ = ψ Corollary 2. Let (Ω, α) be a TMDS with an α-invariant probability measure m and associated autocorrelation function γ . Let E be the set of eigenvalues of T . Then, the pure point part γ pp of γ is given as γ pp = cξ , cξ δξ . ξ ∈E

In particular, if T has pure point spectrum then γ =

ξ ∈E cξ , cξ δξ .


239

Proof. The characterizing property of cξ given in the previous theorem shows that a ξ with cξ = 0 is an eigenvalue with eigenfunction cξ . The formula for the norm of cξ in the previous theorem then gives the first statement. Now, the second statement follows by noting that (7) together with pure point spectrum of T implies pure point diffraction (see [3,40,62] as well) and, hence, γ = γ pp . Let us finish this section by discussing how the considerations from the beginning of this section dealing with point sets and diffraction as a limit fall into the measure framework. To do so we first note that it is possible to express γ (defined via the closed formula (6)) via a limiting procedure in the ergodic case. The following holds [3]. Theorem 4. Assume that the locally compact abelian G has a countable base of topology. Let (Ω, α) be a TMDS with ergodic measure m and (Bn ) a van Hove sequence along which the Birkhoff ergodic theorem holds. Then, |B1n | ω Bn ∗ ω Bn converges to γm vaguely for m-almost every ω ∈ Ω. Next, we show how to consider point sets as measures. The set of discrete point sets in G will be denoted by D. Then, δ : D −→ M(G), δΛ := δx , x∈Λ

is injective. In this way, D can and will be identified with a subset of M(G). In particular, it inherits the vague topology. A subset Λ of G is called relatively dense if there exists a compact C ⊂ G with

G= (x + C) x∈Λ

and it is called uniformly discrete if there exists an open neighbourhood U ⊂ G of the origin such that (x + U ) ∩ (y + U ) = ∅

(8)

for all x, y ∈ Λ with x = y. The set of uniformly discrete sets satisfying (8) is denoted by DU . An element Λ ∈ DU (considered as an element of M(G)) is in fact translation bounded. In particular, we can define the hull of Λ as the closure Ω(Λ) := {x + Λ : x ∈ G}. Then, Ω(Λ) is a compact TMDS. 6. The Bombieri/Taylor Conjecture for General Systems In this section we consider a TMDS (Ω, α) and ask for existence of certain Fourier type coefficients. These coefficients will be given as limits of certain averages. These averages will be defined next. and B ⊂ G relatively compact with Definition 3. Let (Ω, α) be a TMDS. For ξ ∈ G ξ non-empty interior the function c B : Ω −→ C is defined by 1 ξ (ξ, s)dω(s). c B (ω) := θG (B) B

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The conjecture of Bombieri/Taylor was originally phrased in the framework of point dynamical systems over Z, see [10,11,23,25]. In our context a specific version of it may be reformulated as saying that ξ

γ ({ξ }) = lim |c Bn (ω)|2 , n→∞

where the limit has to be taken in a suitable sense (see the introduction for further discussion). Our abstract result reads as follows. Theorem 5. Let (Ω, α) be a TMDS with an α-invariant probability measure m and be arbitrary and cξ ∈ L 2 (Ω, m) be associated autocorrelation function γ . Let ξ ∈ G given by Theorem 3. Then, the following assertions hold: ξ

(a) For every van Hove sequence (Bn ) the functions c Bn converge in L 2 (Ω, m) to cξ and ξ

ξ

γ ({ξ }) = cξ , cξ = lim c Bn , c Bn . n→∞

ξ

(b) If m is ergodic, the function |cξ |2 is almost everywhere equal to γ ({ξ }). Then, c Bn converge almost everywhere to cξ , whenever (Bn ) is a van Hove sequence along ξ which the Birkhoff ergodic theorem holds. In particular, the functions |c Bn |2 converge almost everywhere to γ ({ξ }). (c) If (Ω, α) is uniquely ergodic, then the following assertions are equivalent: ξ (i) c Bn converges uniformly for one (and then any) van Hove sequence. (ii) cξ is a continuous function satisfying cξ (α−s ω) = (ξ, s)cξ (ω) for every s ∈ G and ω ∈ Ω. (iii) cξ ≡ 0 or ξ is a continuous eigenvalue. ξ In these cases γ ({ξ }) = limn→∞ |c Bn |2 (ω) uniformly on Ω. Remark. Parts (a) and (b) of the theorem are new. As discussed in the introduction, validity of variants of (c) is hinted at in the literature, see e.g. [23,25,33,64]. However, so far no proof has been given. For so called model sets and primitive substitutions validity of (i) in (c) has been shown in [23] and [20] respectively. These proofs do not use continuity of eigenfunctions. On the other hand, continuity of eigenfunctions is known for so-called model sets [62] and primitive substitution systems [65,66] (see [27] for the one-dimensional situation). Thus, (c) combined with these results on continuity of eigenfunctions gives a new proof for the validity of the Bombieri/Taylor conjecture for these systems. The theorem gives immediately the following corollary, which proves one version of the Bombieri/Taylor conjecture. Corollary 3. Let (Ω, α) be a uniquely ergodic TMDS with an α-invariant probability measure m and associated autocorrelation function γ . Then, ξ

: : c does not converge uniformly to 0}. {ξ ∈ G γ ({ξ }) > 0} = {ξ ∈ G Bn


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Remark. The corollary gives an efficient method to prove γ ({ξ }) > 0, viz it suffices to ξ show that c Bn (ω) does not converge to zero for a single ω. The corollary is less useful in proving γ ({ξ }) = 0, as in this case one has to check uniform convergence to zero. This shortcoming will be addressed for special systems in the final section of the paper. Corollary 4. Let (Ω, α) be a uniquely ergodic TMDS with an α-invariant probability measure m and associated autocorrelation function γ . The following assertions are equivalent: ξ (i) γ is a pure point measure and c Bn converges uniformly for every ξ ∈ G. 2 (ii) L (Ω, m) has an orthonormal basis consisting of continuous eigenfunctions.

Proof. As shown in Theorem 7 of [3], γ is a pure point measure if and only if L 2 (Ω, m) has an orthonormal basis consisting of eigenfunctions. We are thus left with the statement on continuity of eigenfunctions. Here, (ii) ⇒ (i) follows from (c) of the previous theorem. The implication (i) ⇒ (ii) follows from Theorem 8 in [3] applied with V := Lin{ f ϕ : ϕ ∈ Cc (G)} ⊂ L 2 (Ω, m) after one notices that assumption (i) implies by the previous theorem that all eigenfunctions of the restriction of T to V are continuous. We will give the proof of the theorem at the end of this section. In order to do so, we need some preparatory results. Lemma 6. Let C > 0 and V ⊂ G be open, relatively compact and non-empty. Then, for every compact K ⊂ G and every van Hove sequence (Bn ), lim

1

n→∞ θG (Bn )

sup{|µ|(∂ K Bn ) : µ ∈ MC,V (G)} = 0.

Proof. For a fixed µ ∈ MC,V (G), the corresponding statement is shown by Schlottmann in Lemma 1.1 of [62]. Inspection of the proof shows that convergence to zero holds uniformly on MC,V (G) (see [37] for a different proof as well). Lemma 7. Let (Ω, α) be a TMDS with Ω ⊂ MC,V (G). Then, for every ϕ ∈ Cc (G), and B ⊂ G open relatively compact and non-empty the estimate Cϕ ξ ξ A B ( f ϕ ) − sup{|µ|(∂ S(ϕ) B) : µ ∈ MC,V (G)} + θG (∂ S(ϕ) B) ϕ (ξ )c B ∞ ≤ θG (B) holds, where S(ϕ) := supp(ϕ) ∪ (−supp(ϕ)) and Cϕ := |ϕ|dt + sup{ |ϕ| d|µ| : µ ∈ MC,V (G)}. ξ

ξ

Proof. Define D(ω) := θG (B)−1 |A B ( f ϕ )(ω) − ϕ (ξ )c B (ω)|. Then, 1 (ξ, t)ϕ(t − r )dtdω(r ) − (ξ, t)ϕ(t − r )dtdω(r ) D(ω) = θG (B) G B BG 1 ≤ (ξ, t)ϕ(t − r )dtdω(r ) + (ξ, t)ϕ(t − r )dtdω(r ) . θG (B) G\B B B G\B It is then straightforward to estimate the first term by term by

Cϕ S(ϕ) B). θG (B) θG (∂

Cϕ S(ϕ) B) θG (B) |ω|(∂

and the second

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The next lemma is the crucial link between Wiener/Wintner type averages and the ξ Fourier coefficient type averages. It shows that the behavior of A Bn ( f ϕ ) is “the same” ξ

as the behavior of ϕ (ξ )c Bn for large n ∈ N. Lemma 8. Let (Ω, α) be a TMDS with Ω and (Bn ) an arbitrary van Hove sequence. be arbitrary. Then, Let ξ ∈ G ξ

ξ

ϕ (ξ )c Bn ∞ −→ 0, n −→ ∞ A Bn ( f ϕ ) − for every ϕ ∈ Cc (G). Proof. This is a direct consequence of Lemma 7 and Lemma 6. Proof of Theorem 5. Choose σ ∈ Cc (G) with G σ (s)ds = 1 and define σ ∈ Cc (G) by σ := (ξ, ·)σ . Then, σ (ξ ) = 1 and according to Theorem 3 we have γ ({ξ }) = cξ , cξ with cξ = E T ({ξ }) f σ . This will be used repeatedly below. (a) From Lemma 1 and the definition of cξ , we infer ξ

ξ

γ ({ξ }) = cξ , cξ = lim A Bn ( f σ ), A Bn ( f σ ) . n→∞

Now, the statement follows from Lemma 8. (b) As cξ = E T ({ξ }) f σ , the function cξ is zero or an eigenfunction to ξ . Thus, for fixed s ∈ G, cξ (α−s ω) = (ξ, s)cξ (ω) for almost every ω ∈ Ω. In particular, the function |cξ |2 is invariant under α and thus, by ergodicity, almost surely equal to a constant. As m is a probability measure this constant is equal to cξ , cξ , which in turn equals γ ({ξ }). ξ The statement on almost sure convergence follows from Lemma 8 as A Bn ( f σ ) almost surely converges according to Lemma 2. (c) This follows from Lemma 8, Theorem 1 and the already shown part. As a by-product of our proof we obtain the following result. Corollary 5. Let γ be the autocorrelation of a TMDS (Ω, α). Let (Bn ) be an arbitrary be given. Then, van Hove sequence. Let ξ ∈ G 1 γ ({ξ }) = lim (ξ, s)dγ (s) n→∞ θG (Bn ) B n and, similarly, γ ({ξ }) = lim γ ∗ ϕn ∗ ϕ n (0) n→∞

for ϕn := G σ dt = 1.

1 θG (Bn ) (ξ, ·) χ Bn

∗ σ , where σ is an arbitrary element of Cc (G) satisfying

Remark. Results of this type play an important role in the study of diffraction on Rd [23,25,67]. They do not seem to be known in the generality of locally compact, σ -compact Abelian groups we are dealing with here. They can be inferred, however, from Theorem 11.4 of [21] whenever transformability of γ is known. This transformability in turn seems, however, not to be known in general.


243

Proof. We only consider the first equation. The second statement can be shown with a similar and in fact simpler proof. Choose σ ∈ Cc (G) and ϕn ∈ Cc (G) as in the statement, i.e. with G σ (s)ds = 1 1 (ξ, ·) χ Bn ∗ σ . Define σ ∈ Cc (G) by σ := (ξ, ·)σ . Then a direct and ϕn := θG (B n) calculation shows σ ∗ ϕn (t) = (ξ, t)

1 an (t) θG (Bn )

with an (t) = G

G

σ (s)σ (−r )χ Bn (t + s − r )dr ds.

Thus, with K := supp(σ ) − supp(σ ), we have an (t) = 0 for t ∈ G \ (Bn ∪ ∂ K Bn ), an (t) = 1 for t ∈ Bn \ ∂ K B and 0 ≤ |an (t)| ≤ 1 for t ∈ ∂ K B. As (Bn ) is a van Hove sequence and γ is translation bounded, we then easily infer from Lemma 6 that 1 γ ({ξ }) = lim (ξ, s)dγ (s) if and only if n→∞ θG (Bn ) B n γ ({ξ }) = lim γ ∗ σ ∗ ϕn (0). n→∞

ξ

The latter equality can be shown as follows: A direct calculation shows A Bn ( f σ ) = f ϕn . Thus, (6), Lemma 1 and Theorem 3 show ξ

γ ∗ σ ∗ ϕn (0) = f σ , f ϕn = f σ , A Bn ( f σ ) −→ f σ , E T ({ξ }) f σ = γ ({ξ }), and the proof is finished.

7. Cut and Project Models and Their Relatives In this section we apply the results of the preceding section to model sets and some variants thereof. Model sets were introduced by Meyer in [44] quite before the actual discovery of quasicrystals. A motivation of his work is the quest for sets with a very lattice-like Fourier expansion theory. In fact, model sets can be thought of to provide a very natural generalization of the concept of a lattice. Together with primitive substitutions they have become the most prominent examples of aperiodic order. Accordingly, they have received quite some attention. We refer the reader to [46,48,62] for background and further references. A cut and project scheme over G consists of a locally compact abelian group H , called the internal space, and a lattice L in G × H such that the canonical projection π : G × H −→ G is one-to-one between L˜ and L := π( L) and the image πint ( L) of the canonical projection πint : G × H −→ H is dense. Given these properties of the projec tions π and πint , one can define the -map (.) : L −→ H via x := πint ◦ (π | L )−1 (x), ˜ for all x ∈ L. where (π | L )−1 (x) = π −1 (x) ∩ L,

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We summarize the features of a cut- and project scheme in the following diagram: π

G ←−−− ∪ 1−1

L ←−−−

L

πint

G×H ∪ L˜

−−→ H ∪ dense

L

−−−−−−−−−−−→

−−→ L

We will assume that the Haar measures on G and on H are chosen in such a way that a fundamental domain of L˜ has measure 1. Given a cut and project scheme, we can associate to any W ⊂ H , called the window, the set

(W )

:= {x ∈ L : x ∈ W }.

A set of the form t + (W ) is called a model set if the window W is compact and the closure of its interior. Without loss of generality, we may assume that the stabilizer of the window, HW := {c ∈ H : c + W = W }, is the trivial subgroup of H , i.e., HW = {0}. A model set is called regular if ∂ W has Haar measure 0 in H . Any model set turns out to be uniformly discrete. A central result on model sets (compare [46,62] and references given there) states that regular model sets are pure point diffractive, i.e. γ is a pure point measure. In fact, the associated dynamical system (Ω(Λ), α) obtained by taking the closure Ω(Λ) of {t + Λ : t ∈ G} in M(G) is uniquely ergodic with pure point spectrum with continuous eigenfunctions and the diffraction measure can be calculated explicitly [23,25,62]. Given the material of the previous sections we can easily reproduce the corresponding results. This is discussed next. The underlying idea is that the dynamical system “almost agrees” (in the sense of being an almost one-to-one extension) with the so-called torus parametrization. A cut and project scheme gives rise to a dynamical system in the following way: Define T := (G × H )/ L. By assumption on L, T is a compact abelian group. Let G × H −→ T, (t, k) → [t, k], be the canonical quotient map. Then, there are canonical group homomorphisms κ : H −→ T, h → [0, h], and ι : G −→ T, t → [t, 0]. By the defining properties of a cut and project scheme the homomorphism ι has dense range and the homomorphism κ is injective. There is an action α of G on T via α : G × T −→ T, αt ([s, k]) := ι(−t) + [s, k] = [s − t, k]. The dynamical system (T, α ) is minimal and uniquely ergodic, as ι has dense range. Moreover, it has pure point spectrum. In fact, the dual group T gives a set of eigenfunctions, which form a complete orthonormal basis by the Peter-Weyl theorem. These eigenfunctions can be described in terms of characters on G and H via the the dual lattice L ⊥ of L given by × H : k(l)u(l ) = 1 for all (l, l ) ∈ L}. L ⊥ := {(k, u) ∈ G


245

More precisely, standard reasoning gives that T can naturally be identified with L ⊥ . In ⊥ this identification (k, u) ∈ L corresponds to ξ ∈ T with ξ([t, h]) = k(t)u(h). This ξ can then easily be seen to be an eigenfunction to the eigenvalue k. It turns out that k already determines ξ as will be shown next. Let L ◦ be the set of all for which there exists u ∈ H with (k, u) ∈ k∈G L ⊥ . As π2 ( L) is dense in H , we infer ⊥ that (k, u), (k, u ) ∈ L implies u = u . Thus, there exists a unique map : L ◦ −→ H such that L ⊥ , k → (k, k ), τ : L ◦ −→ is bijective. Having discussed the dynamical behavior of (T, α ) we now come to the connection between Ω(Λ) and T. This connection is known under the name of torus parametrization [49,62]. In Proposition 7 in [5] the following version is given. Proposition 1. There exists a continuous G-map β : Ω(Λ) −→ T such that β() = (t, h) + L if and only if t + (W ◦ − h) ⊂ ⊂ t + (W − h). For regular model sets, the Haar measure of the boundary of W is zero. Thus, the previous proposition shows that the set of points in T with more than one inverse image under β has measure zero. This gives easily (see e.g. [5,62]) that (Ω, α) inherits unique ergodicity and pure point spectrum with continuous eigenfunctions and eigenvalues k ∈ L ◦ from T. By Corollary 2 the diffraction measure can then be written as ck , ck δk . γ = k∈L ◦

It remains to determine the ck . By continuity of the eigenfunctions and Theorem 5, the ck arise as the uniform limit of the function ckBn , where (Bn ) is an arbitrary van Hove sequence. For k ∈ L ◦ the calculation of this limit can be performed using a convergence result for cut and project schemes known as uniform distribution. Using the uniform distribution result of [47] one obtains ck (Γ ) = τ (k)(β(Γ )) (k , y)dy W

for k ∈

L ◦.

Putting the previous two equations together we obtain 2 Ak δk with Ak = (k , y)dy . γ = k∈L ◦

W

We refrain from giving further details here but refer to the next subsection, where a more general situation is treated. 7.1. Deformed model sets. In this subsection we discuss a special form of perturbation of model sets leading to deformed model sets. These sets have attracted attention in recent years [4,9,22]. Based on the results of the previous sections and [4], it is possible to calculate diffraction measure and eigenfunctions. Details are worked out in [38]. Here, we only sketch the results. We keep the notation used so far. Let Λ := (W ) be a model set with a regular window. Let Ω be its hull. By the discussion above, Ω is uniquely ergodic with invariant

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probability measure m. Let now a continuous function ϑ : W −→ G be given. This map gives rise to the perturbed measure ωϑ := δx+ϑ(x ) . x∈Λ

Denote its hull (in M(G)) by Ωϑ . As shown in [4] there exists a unique G-invariant continuous map Φϑ : Ω −→ Ωϑ with Φϑ (Λ) = ωϑ and Ωϑ inherits unique ergodicity with pure point spectrum and continuous eigenfunctions ckϑ to the eigenvalues k ∈ L ◦ from Ω. Thus, again by Corollary 2, γ ϑ is a pure point measure which can be written as γ ϑ = k∈L ◦ ckϑ , ckϑ δk . By Theorem 5, each ckϑ is a limit of ckBn . Using uniform distribution [47], we infer that the limit exists and equals ck () = τ (k)(β()) (k , y)(k, ϑ(y))dy W

for k ∈ L ◦ . We also obtain that the limits are identically zero for k ∈ / L ◦ . Accordingly, we find 2 ϑ γ = Ak δk , with Ak := (k , y)(k, ϑ(y))dy . k∈L ◦

W

Note that with ϑ ≡ 0 we regain the case of regular model sets. 7.2. Cut and project models based on measures. In this subsection we shortly discuss the measure variant of model sets studied in [37] (see [56] as well). Based on the results of the previous sections, it is possible to calculate diffraction and eigenfunctions in this case. This is carried out in [37]. Here, we only sketch the main ideas. Definition 4. (a) A quadruple (G, H, L, ρ) is called a measure cut and project scheme if (G, H, L) is a cut and project scheme and ρ is an L-invariant Borel measure on G × H. (b) Let (G, H, L, ρ) be a measure cut and project scheme. A function f : H −→ C is called admissible if it is measurable, locally bounded and for arbitrary ε > 0 and ϕ ∈ Cc (G) there exists a compact Q ⊂ H with |ϕ(t + s) f (h + k)|(1 − 1 Q (h + k)) d|ρ|(t, h) ≤ ε G×H

for every (s, k) ∈ G × H , where 1 Q denotes the characteristic function of Q. An example of a measurecut and project scheme is given by a cut and project scheme (G, H, L) and ρ := δ L := x∈ L δx . It is not hard to see that then every Riemann integrable f : H −→ C is admissible. In this way regular model sets fall within this framework.


247

Given a measure cut and project scheme (G, H, L, ρ) with an admissible f the map ν f : Cc (G) −→ C, ϕ →

ϕ(t) f (h) dρ(t, h), G×H

is a translation bounded measure. Thus, we can consider its hull Ω(ν f ) := {αt (ν f ) : t ∈ G}. This hull is a TMDS. Assume for the remainder of this section that f is not only admissible but also continuous. Then, it turns out that (Ω(ν f ), α) is minimal, uniquely ergodic and has pure point spectrum with continuous eigenfunctions with a set of eigenvalues contained L ◦ . In fact, the map µ : T −→ Ω(ν f ), µ([s, k])(ϕ) =

f (h + k)ϕ(s + t) dρ(t, h)

is a continuous surjective G-map and (Ω(ν f ), α) inherits pure point spectrum with continuous eigenfunctions from (T, α). Note that in terms of factor maps the situation here is somehow opposite to the situation considered in the last subsection: The dynamical system in question (Ω(ν f ), α) is a factor of the torus and not the other way round! ◦ γ = By pure point spectrum with eigenvalues contained in L we can write c , c

δ by Corollary 2. Again, by Theorem 5, the c can be calculated as a ◦ k k k k k∈L uniform limit. The calculation of the limit requires some care. The outcome is ck (µ([s, h])) = τ (k)([s, h])

ρT (λ) (m G × m H )T (1)

f (u) (k , y) dy.

Here, ρT is the unique measure on T with

g(s, h) dρ(s, h) = G×H

for all g ∈ Cc (G × H ), where σξ (g) = Thus, we end up with γ =

k∈L ◦

Ak δk , with Ak = |

T

σξ (g) dρT (ξ )

(l,l )∈ L g(s

+ l, h + l ) for g ∈ Cc (G × H ).

ρT (τ (k)) (m G × m H )T (1)

f (u)(k , y) dy|2 . H

Note that (at least formally) we regain the formula for regular model sets by choosing ρ = δ L and f to be a characteristic function. Remark. As shown in [37], the set Ω(ν f ) carries a natural structure of a compact abelian group and µ is a continuous group homomorphism.

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8. Examples with Randomness In this section we study diffraction for randomizations of systems with aperiodic order. We will be particularly interested in models arising via percolation process and models arising via a random displacement (sometimes also known as Mott type disorder). Both percolation and random displacement models can be thought of to give a more realistic description of the solid in question: Percolation takes into account that defects arise. Random displacement takes into account the thermal movement of the atoms in the solid. Our results will show that in these cases validity of a (variant) of the Bombieri/Taylor conjecture is still true! Note that both models rely on perturbations via independent identically distributed random variables. As discussed in the introduction, Percolation and Random displacement models based on aperiodic order have been investigated earlier [24,26,30,31,50]. Here, we would like to emphasize the work of Külske [30,31]. This work gives strong convergence statements for approximants of the diffraction measure for rather general situations containing both percolation and random displacement models. In fact, [31] can even treat situations with non-i.i.d. random variables. Restricted to our setting this provides convergence for expressions of the form 1 IΛ∩Bn (k)ϕ(k)dk |Bn | for an arbitrary but fixed ϕ from the space of Schwartz functions. As this requires the smoothing with ϕ, it does not seem to give any Bombieri/Taylor type of convergence statement. In this sense, our results below provide a natural complement to his corresponding results of [30]. Our construction of the percolation model and the proof of its ergodicity seem to be new. In fact, we present a unified approach to construction and proof of ergodicity for percolation models and random displacement models. This may be of independent interest. We will be interested in point sets and measures in Euclidean space. Thus, our group is given as G = Rd . The σ -algebra generated by the vague topology is called Borel σ -algebra. It can be described as follows. A cylinder set is a finite union of sets of the form {µ ∈ M(G) : f ϕ j (µ) ∈ I j , j = 1, . . . , n} with n ∈ N, ϕ j ∈ Cc (G), and I j ⊂ C measurable. Here, f ϕ is defined in (5). The support of such a cylinder set is given as the union of the supports of all functions ϕ involved. In particular, the support of a cylinder set is always compact. Lemma 9. Let Ω ⊂ M(G) be compact. Then, the set of cylinder sets in Ω is an algebra (i.e. closed under taking complements and finite intersections) and generates the Borel-σ -algebra. Proof. The set of cylinder sets is obviously closed under taking finite intersections and complements. It generates the Borel-σ -algebra by its very definition. Lemma 10. Let Ω ⊂ M(G) be a compact α-invariant set and m an ergodic probability measure on (Ω, α). Let (ν ω ) be a family of probability measures on M(G) satisfying the following properties:


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(1) ω → ν ω ( f ) is measurable for any nonnegative measurable f on M(G). (2) ν αt ω ( f ) = ν ω ( f (αt ·)) for all t ∈ G and ω ∈ Ω. (3) There exists a constant D > 0 with ν ω (B ∩ C) = ν ω (B)ν ω (C) whenever B and C are cylinder sets with supports of distance bigger than D. Then, the measure m (ν) on M(G) with m (ν) ( f ) = f (µ)dν ω (µ) dm(ω) Ω

is ergodic. Proof. The proof is a variant of the well-known argument showing ergodicity (and, in fact, strong mixing of the Bernoulli shift). Let A be a measurable α-invariant set in M(G). Define f : Ω −→ [0, ∞) by f (ω) := ν ω (A). By the assumptions (1) and (2) on ν and the invariance of A the function f is invariant and measurable. Hence, by ergodicity of m, ν ω (A) = m (ν) (A) for m almost every ω ∈ Ω. By Lemma 9 the algebra of cylinder sets generates the Borel-σ -algebra. Thus, for any ε > 0, there exists a cylinder set B with m (ν) (A B) ≤ ε. Here, denotes the symmetric difference. Let t ∈ G be arbitrary. Then, the triangle inequality for symmetric differences and invariance of A give m (ν) (B αt B) ≤ m (ν) (B A) + m (ν) (A αt B) ≤ 2ε. As the cylinder set B has compact support we can choose t ∈ G so that the supports of B and αt B have distance at least D. Hence, (3) yields ν ω (B ∩ αt B) = ν ω (B)ν ω (αt B) for all ω ∈ Ω. Combining these formulas we obtain (ν) (ν) 2ε ≥ |m (B) − m (B ∩ αt B)| = ν ω (B)(1 − ν ω (αt B))dm(ω). Ω

As m (ν) (A B) ≤ ε, we infer ν ω (A)(1 − ν ω (αt B))dm(ω) ≤ 3ε. Ω

As

ν ω (A)

=

m (ν) (A)

almost surely and m is α-invariant, this gives m (ν) (A) − m (ν) (A)m (ν) (B) ≤ 3ε.

As this can be inferred for any ε > 0 we obtain m (ν) (A) − m (ν) (A)2 = 0. This shows m (ν) (A) = 1 or m (ν) (A) = 0. Remark. The proof shows that assumption (3) is stronger than needed. It suffices to find for each cylinder set with support B a t with ν ω (B ∩ αt B) = ν ω (B)ν ω (αt B). This type of condition could be required on arbitrary locally compact abelian groups, which are not compact. In fact, even an averaged version of this condition can be seen to be sufficient.

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Lemma 11. Let (cn ) be a bounded sequence of complex numbers and (X n ) a sequence of bounded identically distributed independent random variables with expectation value E ∈ C. Then, n 1 (c j X j − c j E) −→ 0, n → ∞, n j=1

almost surely. Proof. Without loss of generality we can assume that both (cn ) and (X n ) are real-valued. Now, the statement follows from the boundedness assumption on (cn ) and (X n ) by Kolmogorov criterion. We will now fix an open relatively compact neighborhood of the origin of Rd and consider the set DU of all uniformly discrete sets with “distance” at least U between different points. We will say that (Ω, α, m) with Ω ⊂ DU is a dynamical system if Ω is a compact α-invariant subset of DU and m is an α-invariant probability measure on Ω. 8.1. Percolation models. In this section we discuss diffraction for percolation models. Fix p ∈ (0, 1) and let ν p be the probability measure on {0, 1} with ν p ({1}) = p. To Λ ∈ DU we associate the product space SΛP := {0, 1} x∈Λ = with product measure νΛ

x∈Λ ν p .

The map

jΛP : SΛP −→ M(G), jΛ (s) :=

s(x)δx

x∈Λ to a measure ν Λ on M(G) viz we define allows one to push νΛ ν Λ ( f ) := νΛ ( f ◦ jΛP ).

The percolation associated to a dynamical system (Ω, α) with Ω ⊂ DU is then given by the measure m P = m (ν) with P Λ m (f) = f (µ)dν (µ) dm(Λ). Ω

The following theorem has been proven in [26] and extended in [50]. It also follows from Lemma 10 above. Theorem 6. The measure m P is ergodic with support contained in DU . Proof. It suffices to show that assumptions (1), (2) and (3) of Lemma 10 are satisfied. is a product measure. To show (1) it suffices Validity of (2) is clear. (3) follows as νΛ to show that Λ → ν Λ ( f ϕ1 . . . f ϕn )


251

is continuous (and hence measurable) for any n ∈ N and ϕ1 , . . . , ϕn ∈ Cc (G). Choose an open relatively compact set V with V = −V containing the supports of all ϕ j , j = 1, . . . , n. For s ∈ {0, 1}Λ∩V let 1 s and 0 s denote the number of 1’s and 0’s in s respectively. Then, a short calculation gives s(x)ϕ1 (−x) ν Λ ( f ϕ1 . . . f ϕn ) = ···

s∈{0,1}Λ∩V

x∈Λ∩V

s(x)ϕn (−x) p 1 s (1 − p)0 (s) .

x∈Λ∩V

This easily shows the desired continuity. We now turn to diffraction. The autocorrelation of γ P = γm P can easily be calculated and seen to be γ P = p 2 γm + p(1 − p)δ0 . In particular, γm + p(1 − p) 1 P = p 2 γ

(9)

contains an absolutely continuous component. be given. Let (Bn ) be a van Hove sequence in G. If Lemma 12. Let Λ ∈ DU and ξ ∈ G ξ ξ c Bn (Λ) converge to a complex number A, then c Bn (ω) converges to p A for ν Λ almost every ω ∈ DU . Proof. It suffices to consider ω of the form ω = jΛ (s) with s ∈ SΛP . The lemma then claims that 1 1 ξ c Bn (ω) = s(x)(ξ, x) −→ p lim (ξ, x), n → ∞. n→∞ |Bn | |Bn | x∈Λ∩Bn

x∈Λ∩Bn

is bounded. Thus, it suffices to show By uniform discreteness of Λ, the sequence B|Bn ∩Λ n| that 1 (s(x) − p)(ξ, x)) −→ 0, n → ∞. Bn ∩ Λ x∈Λ∩Bn

This in turn follows easily from Lemma 11. Putting these results together we obtain the following variant of the Bombieri/Taylor conjecture. Theorem 7. Let (Ω, α, m) with Ω ⊂ DU be a uniquely ergodic dynamical system with and Λ ∈ Ω the averages cξ (ω) continuous eigenfunctions. Then, for any ξ ∈ G, Bn converge for ν Λ almost every ω ∈ DU to a limit cξ (Λ). This limit depends only on Λ ξ (and not on ω) and satisfies |cξ (Λ)|2 = p 2 γm ({ξ }) = γ P ({ξ }). In particular, |c Bn (ω)|2 converge for m P almost every ω ∈ Ω to γ P ({ξ }). Proof. As (Ω, α) is uniquely ergodic with continuous eigenfunctions, Theorem 5 gives ξ convergence of c Bn (Λ) to continuous functions cξ (Λ) with γm ({ξ }) ≡ |cξ (Λ)|2 for The previous lemma then proves the ν Λ almost sure converall Λ ∈ Ω and ξ ∈ G. ξ gence of c Bn (ω) for each Λ. This lemma and the explicit formula (9) then show the last statement.

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8.2. Random displacement models. In this subsection we consider a random displacement in G = Rd . Fix a probability measure σ on Rd with compact support. To Λ ∈ DU we associate the space SΛR D = Rd with product measure

σΛ

=

x∈Λ x∈Λ σ .

The map

jΛR D : SΛR D −→ M(G), jΛR D (s) :=

δx+s(x)

x∈Λ

allows one to push σΛ to a measure σ Λ on M(G) viz we define σ Λ ( f ) := σΛ ( f ◦ jΛR D ). The random displacement model associated to a dynamical system (Ω, α, m) with Ω ⊂ DU , is then given by the measure m R D = m (σ ) with RD Λ m (f) = f (µ)dσ (µ) dm(Λ). Ω

As Ω is compact and σ has compact support, the support of m R D is compact. The following is a consequence of Lemma 10 above. It seems to be new. The proof is very similar to the proof of Theorem 6. We omit the details. Theorem 8. The measure m R D is ergodic. We now turn to diffraction. By ergodicity and Theorem 4, the autocorrelation γ R D = γm R D can be calculated as a limit almost surely. This limit has been calculated in [24] for a fixed Λ and shown to be γ R D = γm ∗ σ ∗ σ + n 0 (δ0 − σ ∗ σ ), where n 0 is the density of points. In particular, σ |2 γm + n 0 (1 − | σ |2 ) R D = | γ

(10)

contains an absolutely continuous component. The next lemma and the following theorem can now be proven along very similar lines as the corresponding results in the previous subsection. We omit the details. be given. Let (Bn ) be a van Hove sequence in G. Lemma 13. Let Λ ∈ DU and ξ ∈ G ξ

ξ

If c Bn (Λ) converge to a complex number A, then c Bn (ω) converges to σ (ξ )A for σ Λ almost every ω ∈ M(G).

Theorem 9. Let (Ω, α, m) with Ω ⊂ DU be a uniquely ergodic dynamical system with and Λ ∈ Ω the averages cξ (ω) continuous eigenfunctions. Then, for any ξ ∈ G, Bn converge for σ Λ almost every ω ∈ M(G) to a limit cξ (Λ). This limit depends only on Λ (and not on ω) and satisfies |cξ (Λ)|2 = γm ({ξ })| σ (ξ )|2 = γ R D ({ξ }). In particular, ξ 2 R D R D |c Bn (ω)| converge for m almost every ω ∈ Ω to γ ({ξ }). Remark. The above considerations rely essentially on the independent identical distribution of the randomness and the locality of the randomness. Therefore, various further models can be treated by the same line of reasoning. In particular, we could treat models, which combine random displacement with percolation.


253

9. Linearly Repetitive Systems In this section we discuss linearly repetitive Delone dynamical systems and their subshift counterparts known as linearly recurrent subshifts. We will refer to both classes as LR-systems. Such systems were introduced recently in [16,34] and have further been studied e.g. in [12,13,15]. In fact, LR-systems are brought as models for perfectly ordered quasicrystals [34]. Thus, validity of the Bombieri/Taylor conjecture for these systems is a rather relevant issue. LR-systems can be thought of as generalized primitive substitution systems [16]. As continuity of eigenfunctions is known for primitive substitutions [65,66], it is natural to assume that continuity holds for LR-systems as well. Somewhat surprisingly, this turns out to be wrong as discussed in [12]. Thus, validity of the Bombieri/Taylor conjecture can not be derived from the material presented so far for these models. It is nevertheless true as shown below. More generally, we show that for these sysξ tems the modules |A Bn ( f )| converge uniformly (while the averages themselves may not converge). The key to these results are the uniform subadditive ergodic theorems from [15,35]. Let us caution the reader that these results do not hold for arbitrary van Hove sequences (Bn ) but rather only for Fisher sequences. We focus on linearly repetitive Delone dynamical systems in this section and only shortly sketch the subshift case. A subset of Rd is called a Delone set if it is uniformly discrete and relatively dense (see the end of Sect. 5 for a definition of these notions). As usual we will identify a uniformly discrete subset of Rd with the associated translation bounded measure and this will allow us to speak about e.g. the hull Ω(Λ) of a Delone set. Definition 5. The open ball with radius R around the origin is denoted by U R (0). A Delone set Λ is called linearly repetitive if there exists a C > 0 such that for all R ≥ 1, x ∈ Rd and y ∈ Λ, there exists a z ∈ U RC (x) ∩ Λ with (−z + Λ) ∩ U R (0) = (−y + Λ) ∩ U R (0). Roughly speaking, linear repetitivity means that a local configuration of size R can be found in any ball of size C R. If Λ is linearly repetitive, then Ω(Λ) is minimal and uniquely ergodic. Linearly repetitive systems allow for a uniform subadditive ergodic theorem and this will be crucial to our considerations. The necessary details are given next. A subset of Rd of the form I1 × · · · × Id , with nonempty bounded intervals I j , j = 1, . . . , d, of R is called a box. The lengths of the intervals I j , j = 1, . . . , d, are called the side lengths of the box. The set of boxes with all side lengths between r and 2r is denoted by B(r ). The set of all boxes in Rd will be denoted by B. Lebesgue measure is denoted by | · |. Then Corollary 4.3 of Damanik/Lenz [15] can be phrased as follows (see [35] for related results as well). Lemma 14. Let Λ be linearly repetitive. Let F : B −→ R satisfy the following: (P0) There exists a C > 0 such that |F(B)| ≤ C|B| for all boxes with minimal side length at least 1.

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n) (P1) There exists a function b : B −→ [0, ∞) with limn→∞ b(Q |Q n | = 0 for any sequence of boxes (Q n ) with minimal side length going to infinity such that

F(∪nj=1 B j )

≤

n

(F(B j ) + b(B j )),

j=1

whenever ∪nj=1 B j is a box and the B j , j = 1, . . . , n, are boxes disjoint up to their boundary. (P2) There exists a function e : [1, ∞) −→ [0, ∞) with limr →∞ e(r ) = 0 such that |F(B)− F(x + B)| ≤ e(r )|B|, whenever x + B ∩Λ = (x + B)∩Λ and the minimal side length of B is at least r . Then, for any sequence of boxes (Q n ) with Q n ∈ B(rn ) and rn → ∞, the limit n) limn→∞ F(Q |Q n | exists and does not depend on this sequence. Remark. These conditions have simple interpretations. (P1) means that the function F is sub-additive up to a boundary term b and (P2) means that F has an asymptotic Λ-invariance property. Let now Λ be linearly repetitive. As Ω(Λ) is uniquely ergodic, the autocorrelation γ = γΓ exists for any Γ ∈ Ω(Λ). Define the set of local patches P(Γ ) of a Delone set Γ by P(Γ ) := {(−x + Γ ) ∩ U R (0) : R ≥ 0, x ∈ Γ }. Minimality implies Ω(Λ) = { : P(Γ ) = P(Λ)}.

(11)

d . For In the context of Rd , we can identify ξ ∈ Rd with the character exp(iξ ·) in R d d B ⊂ R relatively compact with non-empty interior, Λ Delone, and ξ ∈ R , we define accordingly ξ

ξ

C B (Λ) := c B (δΛ ) =

1 exp(−iξ x). |B| x∈B∩Λ

Now, our result reads as follows. Theorem 10. Let Λ be linearly repetitive and γ the associated autocorrelation. Then, ξ

γ ({ξ }) = lim |C Q n (Λ)|2 n→∞

for any sequence of boxes (Q n ) with Q n ∈ B(rn ) and rn → ∞. Proof. Define F : B −→ R by F(Q) = | x∈Q∩Λ exp(−iξ x)|. Then, F clearly satisfies the conditions (P0), (P1) and (P2) of Lemma 14. Therefore, by Lemma 14, n) the limit limn→∞ F(Q |Q n | exists for any sequence of cubes (Q n ) with Q n ∈ B(rn ) and rn → ∞ and the limit does not depend on this sequence. By (11), this means that the limit ξ

a(ξ ) := lim |C B (Γ )| n→∞

exists uniformly in Γ ∈ Ω(Λ) and does not depend on Γ . Now, Theorem 5 (a) gives the desired result.


255

We next come to a generalization for arbitrary eigenfunctions. Proposition 2. Let Λ be a Delone set and f : Ω(Λ) −→ C continuous. Then, there exists a function e : [1, ∞) −→ [0, ∞) with limr →∞ e(r ) = 0 and | f (α−s Γ ) − f (α−s Γ )|ds ≤ e(r )|B| B

whenever B is a box with minimal side length at least r and Γ, Γ ∈ Ω(Λ) with Γ ∩ B = Γ ∩ B. Proof. Define e(r ) to be the supremum of the set of terms 1 | f (α−s Γ ) − f α−s Γ )|ds, |B| B where B runs over all boxes with minimal side length at least r and Γ, Γ belong to Ω(Λ) and satisfy Γ ∩ B = Γ ∩ B. Choose ε > 0 arbitrary. As f is continuous, there exists R > 0 such that | f (Γ ) − f (Γ )| ≤ ε

(12)

whenever Γ ∩ C R = Γ ∩ C R , where C R denotes the cube centered at the origin with side length 2R. For a box B = [a1 , b1 ] × · · · [ad , bd ] with minimal side length bigger than 2R set B R := [a1 − R, b1 − R] × · · · [ad − R, bd − R]. Define B R accordingly if the intervals making up B are not closed. Choose r0 such that 2

|B \ B R | f ∞ ≤ ε |B|

(13)

for any box B with minimal side length at least r0 . Then, for such a box B and Γ , Γ ∈ Ω(Λ) with Γ ∩ B = Γ ∩ B we have | f (α−s Γ ) − f (α−s Γ )| ≤ ε

(14)

for all s ∈ B R by (12). By (13), we then easily infer e(r ) ≤ 2ε whenever r ≥ r0 . As ε > 0 is arbitrary, this proves the proposition. Theorem 11. Let Λ be linearly repetitive. Let f : Ω(Λ) −→ C be continuous. Then, ξ for any sequence of boxes (Q n ) with Q n ∈ B(rn ) and rn → ∞, the sequence |A Q n ( f )| √ converges uniformly to E T ({ξ }) f, E T ({ξ }) f . Proof. Choose ∈Ω(Λ) arbitrary. Define F : B −→ R by F(Q) = | Q exp(−iξ s) f (α−s (Γ )ds|. Clearly, F satisfies (P0) and (P1) of Lemma 14. Moreover, by the previous proposition it also satisfies (P2). Thus, the limit limn→∞ |Q n |−1 F(Q n ) exists. Another application of the previous proposition and (11) shows that the limit does not depend on the choice of Γ and is uniform in Γ . Now, the claim follows from Lemma 1. We finish this section with a short discussion of linearly repetitive subshifts. Let A be a finite set called the alphabet and equipped with the discrete topology. Let Ω be a subshift over A. Thus, Ω is a closed subset of AZ , where AZ is given the product topology and Ω is invariant under the shift operator α : AZ −→ AZ , (αa)(n) ≡ a(n+1).

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We consider sequences over A as words and use standard concepts from the theory of words ([16,43]). In particular, Sub(w) denotes the set of subwords of w and the length |w| of the word w = w(1) . . . w(n) is given by n. To we associate the set W = W() of finite words associated to given by W ≡ ∪ω∈ Sub(ω). A subshift is called linearly repetitive if there exists a D > 0 s.t. every v ∈ W is a factor of every w ∈ W with |w| ≥ D|v|. A function F : W −→ R is called subadditive if it satisfies F(ab) ≤ F(a) + F(b). If a subshift is linearly repetitive, the limit lim|x|→∞ F(x) |x| exists for every subadditive F (see [15,35]). This can be used to obtain the following analogue of the previous theorem. Theorem 12. Let (Ω, α) be a linearly repetitive subshift. Let f be a continuous function of Ω and z ∈ C with |z| = 1 be arbitrary. For n ∈ N define the function An ( f ) by −k f (α ω). Then, |A ( f )| converge uniformly to the constant An ( f )(ω) := n−1 −k n k=0 z √ function E T ({z}) f, E T ({z}) f . Proof. Recall that a function f on Ω is called locally constant (with constant L ∈ N) if f (ω) = f (ρ) whenever ω(−L) . . . ω(L) = ρ(−L) . . . ρ(L). It suffices to show the theorem for locally constant functions, as they are dense in the continuous functions. To a locally constant function f with constant L we associate the function F : W −→ R defined by F(w) ≡

2L f ∞ +|

|w| f ∞ : |w| ≤ 2L |w|−L −k z f (αk ω)| : for ω ∈ Ω with ω(1) . . . ω(|w|) = w if |w| > 2L. k=L

As f is locally constant this is well defined. It is not hard to see that F is subadditive. As discussed above, then the limit lim|x|→∞ F(x) |x| exists. This easily yields the statements. Let us finish this section by emphasizing the following subtle point: As continuity of eigenfunctions fails for general LR-systems, we can not appeal to the results of the previous section to obtain validity of (∗∗) for LR-systems. This does not exclude, however, the possibility that all eigenfunctions relevant to Bragg peaks are continuous. We consider this an interesting question. Acknowledgements. This work was initiated by discussions with Robert V. Moody at the “MASCOS Workshop on Algebraic Dynamics” in Sydney in 2005. I would like to thank Bob for generously sharing his insights. I would also like to take the opportunity to thank the organizers of this workshop for the stimulating atmosphere. This work was finished during a stay at Rice University. The hospitality of the department of mathematics is gratefully acknowledged. This work was partially supported by the German Research Council (DFG).

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Communications in


Spectral Properties of a q -Sturm–Liouville Operator B. Malcolm Brown1 , Jacob S. Christiansen2 , Karl Michael Schmidt3 1 Computer Science, Cardiff University, Queen’s Buildings, 5 The Parade, Roath,

Cardiff CF24 3AA, UK. E-mail: [email protected]

2 California Institute of Technology, Mathematics 253-37, Pasadena, CA 91125, USA.


3 School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG,

Wales, UK. E-mail: [email protected] Received: 8 February 2008 / Accepted: 29 April 2008 Published online: 20 September 2008 – © Springer-Verlag 2008

Abstract: We study the spectral properties of a class of Sturm-Liouville type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, we construct examples for isolated pure point, dense pure point, purely absolutely continuous and purely singular continuous spectrum. It is also shown that the last two spectral types are generic for analytic coefficients and for a class of positive, uniformly continuous coefficients, respectively. 1. Introduction In this paper we consider the q-Sturm–Liouville operator 1 T f = − Dq (ψDq f ) φ for suitable functions φ and ψ. More precisely, we study the spectral behaviour of T and seek to understand how σ (T ) depends on φ and ψ. The operator Dq will be defined in (1.1)–(1.3) below. It was introduced by Ismail [6] in his work on the q −1 -Hermite polynomials and is reminiscent of the Askey–Wilson operator [1]. One can think of Dq as playing the same role for the q −1 -Hermite polynomials as the Askey–Wilson operator plays for the q-Hermite polynomials. The main results of the paper are Theorems 11 and 13. We first prove that under mild conditions on φ and ψ, the spectrum of T is generically purely singular continuous. Next we show that more restrictive conditions on φ and ψ will lead to a spectrum that generically is purely absolutely continuous. Here, generic means in the sense of a dense G δ and, roughly speaking, mild and restrictive refer to uniformly continuous respectively analytic coefficient functions. The first result is obtained by applying the Wonderland

260

B. M. Brown, J. S. Christiansen, K. M. Schmidt

theorem of Simon [13] and relies in part on a theorem of Ponomarev [9]. The second result is proved more directly. To define the operator Dq , suppose that f is a function on R and write the variable as x = sinh y. Then one can think of f as a function of y and write, say f˜(y) = f (x). The divided difference operator Dq is now defined by (Dq f )(x) =

f˜(y − 21 log q) − f˜(y + 21 log q) , (q −1/2 − q 1/2 ) cosh y

(1.1)

and if we set e(x) = x, the denominator can also be written as e(y ˜ −

1 2

log q) − e(y ˜ +

1 2

log q).

(1.2)

It is convenient to set p = − 21 log q and simply view the operator as (Dq f )(x) =

f˜(y + p) − f˜(y − p) . e(y ˜ + p) − e(y ˜ − p)

(1.3)

Under the usual assumption of q ∈ (0, 1), we have p > 0. The operator Dq played a major role in [2]. We refer the reader to [7, Chap. 16] for more information about the Askey–Wilson operator. The paper is organized as follows. In Sect. 2 we set the stage by writing T as a direct integral of doubly-infinite Jacobi operators. This has the advantage that spectral properties of T now can be derived from the family of operators appearing in the integral. Next, in Sect. 3, we show how different coefficient functions can lead to different types of spectrum. In our first example, σ (T ) consists of a sequence of eigenvalues accumulating at zero. A slight modification leads to bands of purely absolutely continuous spectrum. Composing the coefficient functions with certain singular functions, we can obtain both dense pure point spectrum and purely singular continuous spectrum. In Sect. 4 we prove the two main results of the paper: 1) Among uniformly continuous coefficients, purely singular continuous spectrum is generic; and 2) Among analytic coefficients, purely absolutely continuous spectrum is generic. 2. A Direct Integral Decomposition of T Consider the q-Sturm–Liouville operator 1 T f = − Dq ψDq f , φ where Dq is the operator defined in (1.1) and φ, ψ are suitable functions. Our analysis of T relies on the fact that one can view this operator as a direct integral of Jacobi operators acting on 2 (Z). We explain the details below and also remind the reader of some basic facts about direct integrals, see, e.g. [10, Chap. XIII.16]. 2 A simple computation tells us that − (1−q) q (T f )(x) multiplied with the weight √ φ(x) 1 + x 2 can be written as ˜ − p) ˜ + p) ˜ − p) ˜ + p) ψ(y ψ(y ψ(y ψ(y + f˜(y + 2 p)− f˜(y) + f˜(y −2 p). cosh(y + p) cosh(y + p) cosh(y − p) cosh(y − p) Here, and throughout the paper, we always have x = sinh y ∈ R.

Spectral Properties of a q-Sturm–Liouville Operator

261

Now fix y ∈ [0, 2 p) and consider the sequence xn = (e y q −n − e−y q n )/2, n ∈ Z. Set f n := f (xn ) = f˜(y + 2np) and define an operator Ty by (Ty f )n =

(1 − q)2 (T f )(xn ), n ∈ Z. q

Here we slightly abuse notation since f on the left-hand side is the sequence { f n } whereas f on the right-hand side is a function on R. Note that ˜ + 2np) cosh(y + 2np)(Ty f )n = αn (y) f n+1 + βn (y) f n + αn−1 (y) f n−1 , n ∈ Z, φ(y where αn (y) = −

˜ + (2n + 1) p) ψ(y , βn (y) = −αn (y) − αn−1 (y). cosh(y + (2n + 1) p)

The operator Ty is symmetric on the weighted space 2 (Z, w(y)), with ˜ + 2np) cosh(y + 2np). wn (y) = φ(y ˜ If the function y → φ(y) cosh(y) is periodic with period 2 p, then Ty is even symmetric on 2 (Z). But in general, assuming that wn (y) > 0 for all n, Ty is only unitarily equivalent to a symmetric operator on 2 (Z). Defining a unitary operator U y : 2 (Z, w(y)) → 2 (Z) by (U y f )n = (−1)n f n wn (y), we see that Jy := U y Ty U y∗ is a doubly-infinite Jacobi operator on 2 (Z) given by Jy en = an (y)en+1 + bn (y)en + an−1 (y)en−1 , n ∈ Z with an (y) = − √

αn (y) βn (y) . , bn (y) = wn (y) wn (y)wn+1 (y)

As usual, {en } denotes the standard orthonormal basis for 2 (Z). We will only allow functions φ and ψ such that Jy is essentially self-adjoint for all y. As is well known, this is always the case if the sequence {an (y)} is bounded for each y. Later on we will put further restrictions on the functions φ and ψ or on the coefficient functions an and bn . We will always assume that φ and ψ are positive (> 0). This in turn implies that an > 0 for each n. Note that for all x ∈ 2 (Z) we then have an (y)xn xn+1 + bn (y)xn2 (Jy x, x)2 (Z) = 2 n∈Z

=

n∈Z

n∈Z

xn xn+1 an (y) √ +√ wn (y) wn+1 (y)

2

wn (y)wn+1 (y),

(2.1)

and Jy is therefore a positive operator. Note also that Jy+2 p and Jy are unitarily equivalent, since an (y + 2 p) = an+1 (y) and bn (y + 2 p) = bn+1 (y).

(2.2)

262


Hence it suffices to consider Jy for y ∈ [0, 2 p). Moreover, (2.2) shows that the coefficient functions a0 (y) and b0 (y) (y ∈ R) already determine all coefficients of all Jacobi operators involved. If each Jy is self-adjoint, the full operator J defined by J :=

⊕ [0,2 p)

Jy dy

(2.3)

⊕ is also self-adjoint, see, e.g. [10, Thm. XIII.85]. J is an operator on H := [0,2 p) 2 (Z), the Hilbert space consisting of all functions f : [0, 2 p) → 2 (Z) such that y → 2p ( f (y), g)2 (Z) is measurable for all g ∈ 2 (Z) and such that 0 f (y)22 (Z) dy < ∞. The inner product on H is given by

2p

( f, g) H =

f (y), g(y)

0

2 (Z)

dy,

and we require that y → (Jy f, g)2 (Z) is measurable for all f, g ∈ 2 (Z) and that ess sup y Jy < ∞. Then J is defined by (J f )(y) = Jy f (y) for f ∈ H and y ∈ [0, 2 p). We take J to be the self-adjoint realisation of the formal operator T . Its direct integral structure has the advantage that properties of the spectrum of J can be derived from the spectra of the Jacobi operators Jy , y ∈ [0, 2 p), as illustrated in the following lemma. Throughout the paper, m will denote the Lebesgue measure on R. Lemma 1. a) Consider the operator J defined in (2.3). Then λ ∈ σ (J ) if and only if m {y ∈ [0, 2 p) | σ (Jy ) ∩ (λ − ε, λ + ε) = ∅} > 0 for all ε > 0. Moreover, λ is an eigenvalue of J if and only if m {y ∈ [0, 2 p) | λ ∈ σp (Jy )} > 0. b) Every eigenvalue of J has infinite multiplicity. Proof. Part a) is taken from [10, Thm. XIII.85]. For part b), let λ be an eigenvalue of J and define Y := {y ∈ [0, 2 p) | λ ∈ σp (Jy )}; by a), m(Y ) > 0. For y ∈ Y , let f (y) be an ⊕ eigenvector of Jy for eigenvalue λ. Then, for any Y0 ⊂ Y with m(Y0 ) > 0, Y0 f (y) dy will be an eigenvector of J for eigenvalue λ. Clearly Y can be split into infinitely many disjoint subsets of positive measure, yielding infinitely many orthogonal eigenvectors.

One can also prove that if each Jy has purely absolutely continuous spectrum then so does J . However, we will always make sure to be in a situation where each Jy is a compact operator on 2 (Z). This is the case if an (y) → 0 and bn (y) → 0 as n → ±∞, for all values of y. Then the spectrum of Jy consists of the point 0 together with a sequence of positive eigenvalues that accumulate at 0. Moreover, each eigenvalue of Jy is simple since the singular spectrum of a Jacobi operator has spectral multiplicity one, see, e.g., [15, Lemma 3.6]. As we shall see, this does not rule out the possibility of J to have other types of spectrum (such as absolutely continuous, singular continuous or dense point spectrum).


263

Finding the spectrum of J is a matter of being able to control the eigenvalue branches of the family {Jy } y∈[0,2 p) of Jacobi operators. In what follows, we denote the eigenvalue branches by λ j , j = 1, 2, . . ., and assume that λ1 (y) > λ2 (y) > · · · > λn (y) > · · · for all y ∈ [0, 2 p). Because all eigenvalues of each of the Jy ’s are simple, it is impossible for the eigenvalue branches to cross each other. Furthermore, as a rule of thumb, the more regularity we require of the coefficient functions an and bn , the more regularity we get for the eigenvalue branches. In particular, we have the following two results. Lemma 2. Suppose that a0 and b0 are uniformly continuous on R and vanish at ±∞. Then the set σ (Jy ) depends continuously on y and is 2 p-periodic. In particular, the eigenvalue branches of the family {Jy } of compact operators are continuous and we have σ (J ) = σ (Jy ). y∈[0,2 p)

Moreover, the measure generated by the (non-decreasing) level function L(λ) =

∞ m {y ∈ [0, 2 p) | λ j (y) ≤ λ} j=1

is mutually absolutely continuous with the spectral measures of J . Proof. For every h > 0, we have Jy+h − Jy ≤ 2 sup |an (y + h) − an (y)| + sup |bn (y + h) − bn (y)|. n

n

Since an (y) = a0 (y + 2np) and bn (y) = b0 (y + 2np), we see that the right-hand side converges to 0 as h → 0 because a0 and b0 are assumed to be uniformly continuous on R. Hence, lim Jy+h − Jy = 0,

h→0

and this implies the desired continuity of σ (Jy ). For the last statement, cf. the proof of [10, Thm. XIII.85 (f)]. In the last but one step of the above proof we implicitly used the fact that for compact operators A, B > 0, one has |λn (A) − λn (B)| ≤ A − B,

(2.4)

where λn (A) and λn (B) denote the n th largest eigenvalue of A, respectively B. We mention that (2.4) follows directly from [14, Thm. 1.20] and also can be obtained from [14, Thm. 1.7]. Note that if a function f is continuous on R and vanishes at ±∞, it is automatically uniformly continuous. As uniform continuity will play a central role in our setting, we have decided to keep this extra assumption though it is redundant. The next result is taken from [8, Chap. 7, Sect. 3.5]. To keep the record straight, we mention that our use of the word ‘analytic’ refers to functions that are complex analytic in a neighbourhood of the real line and real-valued on R.

264


Lemma 3. Suppose that a0 and b0 are analytic functions on R vanishing at ±∞. If the null space of Jy is independent of y, then all eigenvalue branches of the family {Jy } of compact operators are analytic. The above lemma in particular applies if the point 0 is not an eigenvalue of Jy for any y. 3. Examples for Spectral Structures of σ (T ) In this section we consider a number of different choices of the functions φ and ψ or the coefficient functions an and bn . It turns out that the spectrum of J is very sensitive to even small changes of, say a0 and b0 . Our first example, explained in Sect. 3.1 below, is related to orthogonal polynomials, more precisely to the 1/q-Hermite polynomials. In fact, the operator Dq was introduced in connection with these polynomials. We encounter an isospectral family of operators, that is, the spectrum of Jy is independent of y. Hence, σ (J ) = σ (Jy ) for any value of y and besides the point 0, the spectrum of J thus only consists of isolated eigenvalues accumulating at 0. The next examples, presented in Sect. 3.2, are modifications of the first one which exhibit a totally different kind of spectra since the eigenvalue branches are no longer constant. We multiply a0 and b0 by certain 2 p-periodic functions and the resulting spectrum becomes purely absolutely continuous. To obtain more exotic types of spectra, such as dense pure point spectrum and singular continuous spectrum, we utilize the idea of composing a0 and b0 with certain singular continuous functions. Towards the end of the section, in Sect. 3.3, we will present examples of this type. 3.1. Isolated eigenvalues. Let us start out simply by setting φ = ψ = 1. The coefficient functions an and bn are then given by an (y) =

1 1 , √ cosh(y + (2n + 1) p) cosh(y + 2np) cosh(y + (2n + 2) p)

respectively bn (y) =

1 1 1 + . cosh(y + 2np) cosh(y + (2n + 1) p) cosh(y + (2n − 1) p)

In a somewhat different setup, Ismail [6] proved that the points 4q k−1/2 for k ≥ 1 are eigenvalues of T , see also [2, Sect. 9]. The corresponding eigenfunctions are closely related to the 1/q-Hermite polynomials. This result was later improved by Christiansen and Koelink in [3]. It follows as a special case of [3, Theorem 3.6] that the spectrum of Jy is independent of y and given by σ (Jy ) = 4q N−1/2 ∪ {0}, where the accumulation point 0 is not an eigenvalue. Hence we also have σ (J ) = 4q N−1/2 ∪ {0}, the only difference being that now all the isolated eigenvalues have infinite multiplicity.


265

3.2. Purely absolutely continuous spectrum. Let J be the operator from Sect. 3.1. If the coefficients a0 and b0 are multiplied by a positive function h which is periodic with period 2 p, then an (y) and bn (y) are multiplied by h(y) for each n. In other words, the operator Jy is multiplied by the number h(y). We let h J be short notation for the operator

⊕

[0,2 p)

h(y)Jy dy,

and note that if λ is an eigenvalue branch associated with J , then hλ is the corresponding eigenvalue branch associated with h J . Consider as an example the function h(y) = 1 + sin2 (yπ/2 p),

y ∈ R.

Not only is h positive and 2 p-periodic, it is also strictly increasing on the interval [0, p] and strictly decreasing on the interval [ p, 2 p]. So the eigenvalue branches associated with h J are no longer constant (as in the example of Sect. 3.1) but strictly increasing on [0, p] and strictly decreasing on [ p, 2 p]. Moreover, since h(2 p − y) = h(y), the spectrum of h(y)Jy is symmetric with respect to the point y = p. So it suffices to consider σ (h(y)Jy ) for y ∈ [0, p]. According to [10, Thm. XIII.86], the spectrum of h J is purely absolutely continuous and (besides the point 0) it consists of bands, [4q n−1/2 , 8q n−1/2 ] ∪ {0}. σ (h J ) = n∈N

Note that if we instead of h consider, for any ε > 0, the function h ε (y) = 1 + ε sin2 (yπ/2 p),

y ∈ R,

then the operator h ε J still has purely absolutely continuous spectrum and h ε J − J ≤ εJ . This technique of approximation will be useful in Sect. 4. We mention in passing two other examples that also lead to purely absolutely con˜ tinuous spectrum. The simplest way of making the function y → φ(y) cosh(y) periodic ˜ with period 2 p is to set φ(y) = 1/ cosh(y). Keeping ψ = 1, the coefficient functions an and bn take the form an (y) =

1 , cosh(y + (2n + 1) p)

respectively bn (y) =

1 1 + . cosh(y + (2n + 1) p) cosh(y + (2n − 1) p)

Since a0 and b0 are analytic, Lemma 3 tells us that each of the eigenvalue branches are analytic. There seems to be no straightforward way of finding an explicit expression for the eigenvalue branches. Numerical calculation (using Maple) indicates a simple behaviour: the n th largest eigenvalue branch is strictly increasing on [0, p] for n odd and strictly decreasing on [0, p] for n even (see Fig. 1 below). Since an−1 (2 p − y) = a−n−1 (y) and bn−1 (2 p − y) = b−n (y), we have σ (J2 p−y ) = σ (Jy ) for all y ∈ [0, p] and it suffices to consider the eigenvalue branches on the interval [0, p].

266


Fig. 1. Mapleplot of the eigenvalue branches

˜ Another and similar example can be obtained by setting φ = 1 and ψ(y) = cosh(y). Then 1 an (y) = √ cosh(y + 2np) cosh(y + (2n + 2) p) and bn (y) =

2 . cosh(y + 2np)

Again, the eigenvalue branches are analytic and Maple now shows that the n th largest eigenvalue branch is strictly decreasing on [0, p] if n is odd and strictly increasing if n is even. Numerically it looks like the eigenvalue branches coincide with the ones from the previous example, they are just shifted by p. We believe this identity relies on a hidden symmetry. 3.3. Exotic spectrum. Throughout this paragraph we will assume that a0 and b0 are uniformly continuous and vanish at ±∞. Moreover, it is convenient to assume that a0 (y − 2 p) = a0 (−y) and b0 (y − 2 p) = b0 (2 p − y) so that the spectrum of Jy is symmetric with respect to the point y = p. At some point we will also include the extra assumption that the continuous eigenvalue branches associated with J are strictly increasing on the interval [0, p]. The example we can base the following considerations on is the first one in Sect. 3.2, with sine modification of constant eigenvalue branches. Let c denote the Cantor function, just rescaled to the interval [0, p]. The Cantor function, see e.g. [5] for a precise definition, is a standard example of a singular function. In fact, c is continuous and non-decreasing on [0, p], with c(0) = 0 and c( p) = p, and yet c (x) = 0 for almost every x ∈ [0, p], since c is constant on each of the intervals [ p/3, 2 p/3], [ p/9, 2 p/9], [7 p/9, 8 p/9], (3.1) [ p/27, 2 p/27], [7 p/27, 8 p/27], [19 p/27, 20 p/27], [25 p/27, 26 p/27], . . . .


267

The corresponding values of c are, respectively, p/2, p/4, 3 p/4, p/8, 3 p/8, 5 p/8, 7 p/8, . . . . We extend c to a continuous non-decreasing function on R by setting c(2 p − y) = 2 p − c(y) for y ∈ [0, p] and by requiring that c(y + 2 p) = c(y) + 2 p for all y. Our first result on dense pure point spectrum now reads: Theorem 4. Set aˆ 0 := a0 ◦ c and bˆ0 := b0 ◦ c. Then σ ( Jˆ) = σ (J ) and σcont ( Jˆ) = ∅. In other words, Jˆ has dense pure point spectrum. Proof. The proof depends on basic properties of the Cantor function. It suffices to consider one of the bands in σ (J ), say , with associated eigenvalue branch λ. We only need to consider λ(y) for y ∈ [0, p]. The corresponding eigenvalue branch for Jˆ is given by λ ◦ c, and since c is constant on each of the intervals in (3.1), every x of the form λ( py), with y a dyadic rational in (0, 1), is an eigenvalue of Jˆ. In other words, Jˆ has dense point spectrum in . Consider now the function F(x) := m {y ∈ [0, p] | (λ ◦ c)(y) ≤ x} , x ∈ . Note that F is strictly increasing on but nowhere continuous since it has a jump at every x = λ( py), with y ∈ (0, 1) a dyadic rational. Therefore, by Lemma 2, σcont ( Jˆ)∩ = ∅.

For the above proof to work it is not essential that the eigenvalue branch λ is strictly monotonic on [0, p]. However, to obtain our next result we will assume that all eigenvalue branches are strictly increasing on the interval [0, p]. Let s denote the singular function of Salem [12], rescaled to [0, p] and then extended to R in the same way as the function c was extended above. While the Cantor function is constant on countably many intervals, the function s is continuous and strictly increasing on [0, p]. Nevertheless, s (x) = 0 for almost every x ∈ [0, p] (see [12] for details). Theorem 5. Set aˇ 0 := a0 ◦ s −1 and bˇ0 := b0 ◦ s −1 . Then σ ( Jˇ) = σ (J ) and σac ( Jˇ) = σpp ( Jˇ) = ∅. Thus, Jˇ has purely singular continuous spectrum. Proof. Since s is strictly increasing on [0, p], we immediately see that σ ( Jˇ) = σ (J ). As in the proof of the previous theorem, it is sufficient to consider one band ⊆ σ (J ) with eigenvalue branch λ. The corresponding eigenvalue branch for Jˇ is then λ ◦ s −1 . Consider the function F(x) := m {y ∈ [0, p] | (λ ◦ s −1 )(y) ≤ x} , x ∈ . Because λ is strictly increasing on [0, p], we have F(x) = (s ◦ λ−1 )(x) and since s is singular continuous on [0, p], so is F on . Therefore, Jˇ has purely singular continuous spectrum.

268


4. Results on Generic Spectrum of T As we have seen in the previous section, different choices of the coefficient functions an and bn can lead to very different types of spectra for J . It is therefore natural to ask which type of spectra is generic, in the sense of a dense G δ . This is closely related to the question of the stability of spectral type under small perturbations. In this section we consider two different classes of coefficient functions. First, we only assume that a0 and b0 are strictly positive and uniformly continuous on R, and that they vanish at ±∞. As explained in Sect. 2, this implies that each Jy is a compact, positive operator. Under these hypotheses on the coefficients, the full operator J generically has purely singular continuous spectrum. We prove this using Barry Simon’s Wonderland theorem, see [13, Sect. 2]. Next, we assume that a0 and b0 are strictly positive analytic functions on R that vanish at ±∞. Under this considerably stronger assumption we prove that the spectrum of J generically is purely absolutely continuous. 4.1. Singular continuous spectrum. Let Cu (R) denote the space of uniformly continuous functions on R vanishing at ±∞. Equipped with the supremum norm, denoted · ∞ , this is a Banach space (since a uniform limit of uniformly continuous functions is again uniformly continuous). We shall mainly be dealing with the subset

Cu+ (R) = f ∈ Cu (R) | f > 0 , consisting of all strictly positive functions in Cu (R). While Cu (R) is a complete metric space, Cu+ (R) is only an open subset of a complete metric space but still a Baire space (in the sense that every intersection of countably many dense open sets is dense). Let X be the metric space consisting of all operators J with coefficients a0 , b0 ∈ Cu+ (R). By definition, J (k) → J in X

⇐⇒

(k)

(k)

a0 → a0 and b0 → b0 in Cu (R).

To apply the Wonderland theorem, we need to check that convergence in X implies convergence in the strong resolvent sense. But if J (k) → J in X then {J (k) } is uniformly bounded in norm. So it suffices to prove that J (k) → J strongly and this is indeed the case since J (k) → J in operator norm: Given ε > 0 we can choose N so large that (k)

(k)

a0 − a0 ∞ , b0 − b0 ∞ < ε/3 for k ≥ N , and hence J − J (k) = ess sup Jy − Jy(k) < ε for k ≥ N . y∈[0,2 p)

We mention in passing that the Wonderland theorem applies to all metric spaces that are Baire spaces and not only to complete metric spaces. Our first goal is now to prove that the set A = {J | J has no continuous spectrum}

(4.1)

is dense in X . Let a0 , b0 ∈ Cu+ (R) be arbitrary coefficients. Following the lines of the proof of Theorem 4, we observe that composition of a0 and b0 with c, the Cantor function rescaled to the interval [0, 2 p] and then extended to R by setting c(y + 2 p) = c(y) + 2 p,


269

will lead to dense pure point spectrum. To show that A is dense in X we therefore just need to find a Cantor-like function which is arbitrarily close to the identity function on the interval [0, 2 p]. Divide the square [0, 2 p] × [0, 2 p] into N 2 smaller squares, all with the same side length, and place a rescaled Cantor function in each of the N squares on the diagonal. More explicitly, define a function c N by

2 pk 2 p(k + 1) 2 pk for y ∈ , and k = 0, 1, . . . , N − 1. c N (y) = c(N y − 2 pk) + N N N Then extend c N to R by requiring that c N (y + 2 p) = c N (y) + 2 p. Clearly, |c N (y) − y| ≤ 2 p/N for all y ∈ R. Moreover, composing a0 and b0 with c N instead of c will still lead to dense pure point spectrum. Proposition 6. Let a0 , b0 ∈ Cu+ (R) be given and let J (N ) denote the operator associated with the coefficients a0 ◦ c N and b0 ◦ c N . Then, as N → ∞, we have J (N ) → J in X . Proof. We restrict ourselves to proving that a0 ◦ c N → a0 in · ∞ -norm as N → ∞. The similar statement for b0 can be proven analogously. Let ε > 0 be given. Since a0 is uniformly continuous, there exists a δ > 0 such that |a0 (x) − a0 (y)| < ε for all x, y ∈ R with |x − y| < δ. Now choose N0 > 2 p/δ. Then |c N (y)− y| < δ for all y ∈ R whenever N ≥ N0 . Accordingly, a0 ◦ c N − a0 ∞ = sup |a0 ◦ c N (y) − a0 (y)| < ε for N ≥ N0 , y

and the result follows.

Corollary 7. The set A defined in (4.1) is dense in X . Our next goal is to prove that the set B = {J | J has purely absolutely continuous spectrum}

(4.2)

is also dense in X . The trick of composing a0 and b0 with suitable functions no longer works since eigenvalues will not necessarily disappear in this way. Instead we multiply a0 and b0 by certain 2 p-periodic functions. In what follows it will be convenient to deal only with operators J for which all the associated eigenvalue branches are differentiable a.e. on [0, 2 p]. Let Y denote the subset of operators having this property. We know from Lemma 2 that each eigenvalue branch is uniformly continuous when a0 and b0 are uniformly continuous on R. However, a uniformly continuous function need not be differentiable almost everywhere. Mimicking the proof of Lemma 2 one can show that the eigenvalue branches are Lipschitz continuous if a0 and b0 are uniformly Lipschitz continuous on R. Every Lipschitz continuous function is locally of bounded variation and hence differentiable almost everywhere by a theorem of Lebesgue, see, e.g. [11]. Since each f ∈ Cu+ (R) can be uniformly approximated by polynomials on compact subsets, the uniformly Lipschitz continuous functions are dense in Cu+ (R). Therefore, the set Y is dense in X . In order to show that B is dense in X , it thus suffices to show that B ∩ Y is dense in Y . Let J ∈ Y be given and let a0 , b0 ∈ Cu+ (R) be the associated coefficients. For δ > 0, let h δ denote the piecewise linear function defined by −βy + 1, 0 ≤ y < p, h δ (y) = βy − 2 pβ + 1, p ≤ y < 2 p,

270


√ √ where β = (1 − 1 + δ)/ p < 0. Note that h δ (0) = 1 and h δ ( p) = 1 + δ. We extend h δ to a function on R by requiring that h δ (y + 2 p) = h δ (y) for all y. Now multiply a0 and b0 by h 1+α for some α ∈ [0, 1]. The new eigenvalue branches have the form δ λh 1+α , where λ is an eigenvalue branch associated with J . By assumption, λh 1+α is δ δ differentiable a.e. on [0, 2 p] with derivative + λ(1 + α)h αδ h δ = h αδ λ h δ + λ(1 + α)h δ . λ h 1+α δ At this point, note that one can approximate a0 and b0 uniformly by functions of the 1+α form h 1+α δ a0 , resp. h δ b0 . Given ε > 0, simply choose δ > 0 so small that Mδ < ε, where

M := max a0 ∞ , b0 ∞ . Since 1 − h 1+α δ ∞ = (1 + δ)

1+α 2

− 1, the estimates

1+α a0 − h 1+α δ a0 ∞ , b0 − h δ b0 ∞ < ε

are valid for all α ∈ [0, 1]. We will show that to each δ > 0 there exists α˜ ∈ [0, 1] such α˜ that h 1+ δ J ∈ B, thus proving that B ∩ Y is dense in Y . The first step is to find a condition on the eigenvalues branches that leads to absolutely continuous spectrum. The lemma given below relies on a result of Ponomarev [9], of which, for the sake of completeness, we have included a proof in the Appendix. Lemma 8. Suppose that each of the eigenvalue branches is differentiable a.e. on [0, 2 p] with non-zero derivative a.e. Then the spectrum for J is purely absolutely continuous. Proof. Let {λ j } denote the eigenvalue branches. By Lemma 1, we know that a point x > 0 is an eigenvalue of J if and only if m λ−1 ({x}) > 0. j j

Since each λ j is continuous and differentiable a.e. with λj = 0 a.e., Theorem 14 tells us that m(A) = 0 ⇒ m λ−1 j (A) = 0 for all Borel sets A ⊆ R. In particular, m λ−1 j ({x}) = 0 for all x > 0 because m({x}) = 0. Hence, m λ−1 m λ−1 j ({x}) ≤ j ({x}) = 0 for all x > 0, j

j

and we conclude that J has no eigenvalues. A similar argument, using the second half of Lemma 2, shows that J has no singular continuous spectrum. 1+α has nonzero derivative a.e. on [0, 2 p] Lemma 8 tells us that h 1+α δ J ∈ B if λh δ for all eigenvalue branches λ associated with J . In this connection, note that λ h δ + λ(1 + α)h δ = 0 a.e. if and only if

λ h δ + (1 + α) = 0 a.e. λh δ


271

Lemma 9. Assume that f : [0, 2 p] → R is measurable. Then m( f −1 (α)) = 0 for a.e. α ∈ R. In fact, f = α a.e. on [0, 2 p] for all but at most countably many α ∈ R. Proof. Write the interval [0, 2 p] as the disjoint union ∪α∈R f −1 (α). For each n ∈ N, there are only finitely many α’s with m( f −1 (α)) > 1/n, since m([0, 2 p]) < ∞. Hence m( f −1 (α)) > 0 for at most countably many α ∈ R. Let λ1 , λ2 , . . . be the eigenvalue branches associated with the given J . Lemma 9 can be applied to each of the functions λn h δ /(λn h δ ). In fact, given δ > 0 we can find an α˜ ∈ [0, 1] such that, for all n, α˜ (h 1+ δ λn ) = 0 a.e. on [0, 2 p].

The statement holds for all n since each λn excludes at most countably many α’s. Hence, α˜ h 1+ δ J ∈ B and, as a consequence, B ∩ Y is dense in Y . We conclude that Corollary 10. The set B defined in (4.2) is dense in X . Combining Corollary 7 and Corollary 10, the Wonderland theorem leads to the following genericity result. Theorem 11. The set {J | J has purely singular continuous spectrum} is a dense G δ in X . In other words, among coefficients a0 , b0 ∈ Cu+ (R) the spectrum of J is generically purely singular continuous. 4.2. Absolutely continuous spectrum. Let S be a neighbourhood of the real line, contained in the horizontal strip | Im z| < 1. The space A(S) of all bounded analytic functions on S is a Banach space with respect to the supremum norm, f ∞ = sup{| f (z)| | z ∈ S}. As in Sect. 4.1, we are only interested in the subset

A+0 (S) = f ∈ A(S) | f (x) > 0 for x ∈ R and lim f (x) = 0 . x→±∞

The functions vanishing at ±∞ form a closed subspace of A(S) and as an open subset herein, A+0 (S) is a Baire space. Now, let X be the metric space of operators J with coefficients a0 , b0 ∈ A+0 (S). To be specific, we equip X with the metric d given by (1) (2) (1) (2) d J (1) , J (2) = 2a0 − a0 ∞ + b0 − b0 ∞ for J (1) , J (2) ∈ X . According to Lemma 3, the eigenvalue branches are analytic whenever a0 , b0 ∈ A+0 (S). So in order to prove that absolutely continuous spectrum is generic, we just need to argue that eigenvalues are rare. As a consequence of Lemma 8, singular continuous spectrum can never occur. Since the eigenvalue branches are non-intersecting, we can order them in such a way that, say, λn denotes the n th largest eigenvalue branch. Let n be the corresponding spectrum of J , that is, n = inf λn (y), sup λn (y) . y∈[0,2 p)

y∈[0,2 p)

272


Either n consists of only one point, which is then an eigenvalue of J , or n is a non-degenerate interval of purely absolutely continuous spectrum for J . Embedded eigenvalues cannot occur since λn is analytic and hence cannot take the same value on a set of positive measure unless it is constant. For each n, we introduce the set

G n = J | n ⊂ σac (J ) . (4.3) The intersection G :=

Gn

n

consists of all operators in X with purely absolutely continuous spectrum. Our goal is to prove the following proposition. Proposition 12. The set G n defined in (4.3) is open and dense in X . Proof. We first prove that G n is dense in X . Given J ∈ X , consider the operator h ε J , where h ε denotes the analytic function given by h ε (z) = 1 + ε 1 + sin(zπ/ p) , z ∈ S. Clearly, h ε is positive on R and periodic with period 2 p. Moreover, there is at most one value of ε > 0 such that h ε λn is constant. Since the sine function is bounded on the strip | Im z| < 1, there is a constant C > 0 such that the estimate h ε f − f ∞ ≤ εC f ∞ is valid for all f ∈ A+0 (S). Hence, h ε J → J in X as ε → 0 and we conclude that G n is dense. To prove that G n is open, choose any operator J ∈ G n and let a0 , b0 ∈ A+0 (S) be the associated coefficients. By assumption, λn is not constant, that is, δ := m(n ) > 0. We claim that J (1) ∈ G n as long as d J, J (1) < δ/4. Indeed, this implies that (1)

(1)

a0 (y) − a0 (y) < δ/8 and b0 (y) − b0 (y) < δ/4 for all y ∈ R. Hence, by (2.4) we have (1) |λn (y) − λ(1) n (y)| ≤ J y − J y < 2 · δ/8 + δ/4 = δ/2 for all y ∈ [0, 2 p). (1) ∈ G . This proves that G is So λ(1) n n n cannot possibly be constant and therefore J open.

Since X is a Baire space, the above proposition implies that G is a dense G δ . In other words, we have proved the following result. Theorem 13. The set {J | J has purely absolutely continuous spectrum} is a dense G δ in X .


273

Appendix The following theorem is a one-dimensional version of a result by Ponomarev [9]. For the reader’s convenience, we include a proof here; note that we use the term ‘null set’ to refer to what is also known as ‘set of measure zero’. Theorem 14. Let ⊂ R be open and bounded, f : → R continuous and almost everywhere differentiable, f = 0 almost everywhere. Then f −1 (A) is a null set whenever A ⊂ R is a null set. Proof. a) We first show that for compact sets K ⊂ of positive measure, f (K ) is of positive measure. As the set {x ∈ K | f is not differentiable in x} is a null set, we can cover it by countably many open intervals (Ii )i∈N of total length less than m(K )/2. Then K 0 := K \ i∈N Ii is compact, and m(K 0 ) ≥ m(K )(1 − 12 ) > 0. Clearly K 0 = j∈N K j , where

K j := x ∈ K 0 | f (x + h) − f (x)| ≤ j|h| (0 < |h| < 1/j) . As 0 < m(K 0 ) ≤ j∈N m(K j ), there is j0 ∈ N such that m(K j0 ) > 0. Moreover, there is x0 ∈ K j0 such that C := K j0 ∩ [x0 − 21j0 , x0 + 21j0 ] has positive measure, for otherwise K j0 would be a finite union of null sets. The restriction f |C is Lipschitz continuous with constant j0 . Hence there is a Lipschitz continuous extension g : R → R of f |C. To construct such an extension, we first note that K j0 (and hence C) is closed. Indeed, if (xn )n∈N is a sequence in K j0 and x = lim xn , then for 0 < |h| < j10 , n→∞

| f (x + h) − f (x)| ≤ | f (x + h) − f (xn + h)| + | f (xn + h) − f (xn )| + | f (xn ) − f (x)| ≤ j0 |h| + o(1) as n → ∞, by uniform continuity of f on K , so x ∈ K j0 . Now define g(x) := f (min C) if x < min C, g(x) := f (max C) if x > max C, g(x) := f (x) if x ∈ C and g(x) :=

f (x− )(x+ − x) + f (x+ )(x − x− ) x+ − x−

otherwise, where x+ := min{t ∈ C, t ≥ x} and x− := max{t ∈ C, t ≤ x}. It is not hard to show that g is Lipschitz continuous with constant j0 . Hence g is locally of bounded variation, so it is differentiable almost everywhere, and g = f a.e. on C. Now we use the identity (cf. [4, 3.2.3]) |g | = N (g|C, y) dy, C

g(C)

where N (g|C, y) is the cardinality of the preimage (g|C)−1 ({y}). As the left-hand integral is non-zero, the set g(C) has positive measure, so m( f (K )) ≥ m( f (C)) = m(g(C)) > 0. b) Now let E ⊂ R be a null set and assume, by way of contradiction, that f −1 (E) has positive measure. For each n ∈ N there is an open set E n with E ⊂ E n and m(E n ) < n1 ; let E 0 := n∈N E n ⊃ E. Then E 0 is a null set, and m( f −1 (E 0 )) ≥ m( f −1 (E)) > 0.

274


Clearly f −1 (E 0 ) = n∈N f −1 (E n ) and, since f is continuous, f −1 (E n ) is open. Also, m( f −1 (E n )) ≥ m( f −1 (E 0 )) > 0, and by regularity there is a compact set Mn ⊂ f −1 (E n ) such that m( f −1 E n ) \ Mn < 2−n m f −1 (E 0 ) . With M := n∈N Mn ⊂ f −1 (E 0 ), we have −1 m f −1 (E 0 )\M = m f −1 (E 0 )\Mn ≤ m f (E n )\Mn < m f −1 (E 0 ) . n∈N

n∈N

Consequently M ⊂ f −1 (E 0 ) has positive measure, so by part a) f (M) ⊂ E 0 has positive measure, a contradiction. Therefore f −1 (E) must be a null set. Acknowledgement. This work was supported by EPSRC grant EP/D03096X/1.

References 1. Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54, 319 (1985) 2. Christiansen, J.S., Ismail, M.E.H.: A moment problem and a family of integral evaluations. Trans. Amer. Math. Soc. 358(9), 4071–4097 (electronic) (2006) 3. Christiansen, J.S., Koelink, E.: Self-adjoint difference operators and symmetric Al-Salam–Chihara polynomials. Constr. Approx. 28(2), 199–218 (2008) 4. Federer, H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., 1969 5. Gelbaum, B.R., Olmsted, J.M.H.: Counterexamples in analysis, Mineola, NY: Dover Publications Inc., 2003, corrected reprint of the second (1965) edition 6. Ismail, M.E.H.: Ladder operators for q −1 -Hermite polynomials. C. R. Math. Rep. Acad. Sci. Canada 15(6), 261–266 (1993) 7. Ismail, M.E.H.: Classical and quantum orthogonal polynomials in one variable. In: Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge: Cambridge University Press, 2005, with two chapters by Walter Van Assche, With a foreword by Richard A. Askey 8. Kato, T.: Perturbation theory for linear operators, Classics in Mathematics, Berlin: Springer-Verlag, 1995, reprint of the 1980 edition 9. Ponomarëv, S.P.: Submersions and pre-images of sets of measure zero. Sibirsk. Mat. Zh. 28(1), 199–210 (1987) 10. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1978 11. Riesz, F., Sz.-Nagy, B.: Functional analysis, Dover Books on Advanced Mathematics, New York: Dover Publications Inc., 1990, translated from the second French edition by Leo F. Boron, Reprint of the 1955 original 12. Salem, R.: On some singular monotonic functions which are strictly increasing. Trans. Amer. Math. Soc. 53, 427–439 (1943) 13. Simon, B.: Operators with singular continuous spectrum. I. General operators. Ann. of Math. (2) 141(1), 131–145 (1995) 14. Simon, B.: Trace ideals and their applications, second ed., Mathematical Surveys and Monographs, Vol. 120,Providence, RI: Amer. Math. Soc. 2005 15. Teschl, G.: Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, Providence, RI: Amer. Math. Soc. 2000 Communicated by B. Simon


Communications in


A Translation-Invariant Renormalizable Non-Commutative Scalar Model R. Gurau1 , J. Magnen2 , V. Rivasseau1 , A. Tanasa3,4 1 Laboratoire de Physique Théorique, CNRS UMR 8627, bât. 210, Université Paris XI,

91405 Orsay Cedex, France. E-mail: [email protected]

2 Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique, 91128 Palaiseau, France 3 Institutul de Fizica si Inginerie Nucleara Horia Hulubei, P. O. Box MG-6,

077125 Bucuresti-Magurele, Romania

4 Max-Planck-Institut fur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Received: 8 February 2008 / Accepted: 9 June 2008 Published online: 22 October 2008 – © Springer-Verlag 2008

Abstract: In this paper we propose a translation-invariant scalar model on the Moyal space. We prove that this model does not suffer from the UV/IR mixing and we establish its renormalizability to all orders in perturbation theory.

1. Introduction and Motivation Space-time coordinates should no longer commute at the Planck scale where gravity should be quantized. This observation is a strong physical motivation for noncommutative geometry, a general mathematical framework developed by A. Connes and others [1]. Non-commutative field theory is the reformulation of ordinary quantum field theory on such a non-commutative background. It may represent a bridge between the current standard model of quantum fields on ordinary commutative R4 and a future formalism including quantum gravity which hopefully should be background independent. Initially there was hope that non-commutative field theory would behave better in the ultraviolet regime [2]. Later motivation came from string theory, because field theory on simple non-commutative spaces (such as flat space with Moyal-Weyl product) appear as special effective regimes of the string [3,4]. Finally another very important motivation comes from the study of ordinary physics in strong external field (such as the quantum Hall effect) [5–7]. Such situations which have not been solved analytically with the ordinary commutative techniques may probably be studied more fruitfully with non-commutative techniques. Renormalization is the soul of ordinary field theory and one would certainly want to extend it to the non-commutative setting. But the simplest non-commutative model, namely φ44 , whose action is given by (2.1) below, was found to be not renormalizable because of a surprising phenomenon called UV/IR mixing [8]. This mixing also occurs in non-commutative Yang-Mills theories. Roughly speaking the non-commutative theory still has infinitely many ultraviolet divergent graphs but fewer than the ordinary one.

276

R. Gurau, J. Magnen, V. Rivasseau, A. Tanasa

However some ultraviolet convergent two point graphs, such as the “non-planar tadpole” generate infrared divergences which are not of the renormalizable type 1 . The first path out of this riddle came when H. Grosse and R. Wulkenhaar introduced a modified φ44 model which is renormalizable [10,11]. They added to the usual propagator a marginal harmonic potential, which a posteriori appears required by Langmann-Szabo duality x˜µ = 2θµν x ν ↔ pµ [12]. The initial papers were improved and confirmed over the years through several independent methods [13,14]. The main property of the Grosse-Wulkenhaar model is that its β-function vanishes at all orders at the self-duality point = 1 [15–17]. The exciting conclusion is that this model is asymptotically safe, hence free of any Landau ghost, and should be a fully consistent theory at the constructive level. This is because wave function renormalization exactly compensates the renormalization of the four-point function, so that the flow between the bare and the renormalized coupling is bounded. Essentially most of the standard tools of field theory such as parametric [18,19] and Mellin representations, [20] dimensional regularization and renormalization [21] and the Connes-Kreimer Hopf algebra formulation of renormalization [22] have now been generalized to renormalizable non-commutative quantum field theory. Other renormalizable models have been also developed such as the orientable Gross-Neveu model [23]. For a general recent review on non-commutative physics including these new developments on non-commutative field theory, see [24]. However there are two shortcomings of the Grosse-Wulkenhaar (GW) model. Firstly it breaks translation invariance so that its relevance to physics beyond the standard model would be indirect at best; one should either use more complicated “covariant” models with harmonic potentials which are invariant under “magnetic translations”, such as the Langmann-Szabo-Zarembo model [25] or one should understand how many short distance localized GW models may glue into a translation-invariant effective model. Secondly it is not easy to generalize the GW method to gauge theories, which do present ultraviolet/infrared mixing. Trying to maintain both gauge invariance and LangmannSzabo duality one is lead to theories with non-trivial vacua [26–29], in which perturbation theory is difficult and renormalizability to all orders is therefore unclear up to now. Motivated by these considerations we explore in this paper another solution to the ultraviolet infrared mixing for the φ44 theory. It relies on the very natural idea to incorporate into the propagator the infrared mixing effects. This is possible because the sign of the mixing graphs is the right one. One can therefore modify the propagator to include from the start a 1/ p 2 term besides the ordinary p 2 term, and to define new renormalization scales accordingly. Adding the interaction and expanding into the coupling constant we prove in this paper that the model modified in this way is indeed renormalizable at all orders of perturbation theory. This is because the former infrared effects now just generate a flow (in fact a finite flow) for the corresponding 1/ p 2 term in the propagator. The “ordinary” φ44 is formally recovered in the case where the bare coefficient of the 1/ p 2 term is zero. The advantages of this “1/ p 2 − φ44 ” model are complementary to those of the GW model. The main advantage is that the model does not break translation invariance. The main inconvenience is that there is no analog of the Langmann-Szabo symmetry so that 1 This UV/IR mixing although quite generic may be avoided in some classes of “orientable models”. Remark also that in Minkowski space if one maintains a rigorous notion of causality, there are strong indications that ultraviolet/infrared mixing does not occur [9]. However the Minkowski theory has complications of its own which make it harder to study.

Translation-Invariant Renormalizable Non-Commutative Scalar Model

277

one should not expect this φ44 model to make sense non-perturbatively. However the real interest of this work is perhaps to offer an alternative road to the solution of ultraviolet/infrared mixing in the case of gauge theories. It may lead to gauge and translation invariant models with trivial vacua. Remark that since ordinary non-Abelian gauge theories are asymptotically free, there is no real need for the non-commutative version to behave better than the commutative case. This removes some of the motivation to implement Langmann-Szabo duality in that case. Therefore we hope the model studied here may be a step towards a better global proposal for a non-commutative generalization of the standard model. This proposal may perhaps have to combine different solutions of the ultraviolet/infrared mixing in the Higgs and gauge sectors of the theory. This model has already been used in [30]. Conjecturing that it would be renormalizable, the associated propagator (corresponding to√Coulomb’s law in a gauge theory) was proven to decay exponentially over a distance θ . Let us finally comment on the physics of this model. The propagator we use deviates significantly from the usual commutative one in the infrared2 . It is legitimate to ask whether commutative physics can be recovered as an effective limit of our model. If we couple the infrared and θ → 0 limits in a certain way, the singular part of the two point function becomes a mass counterterm. So letting the counterterms of the theory depend on θ we may reach a smooth commutative limit. The paper is organized as follows. Section 2 recalls useful facts about Feynman graphs and defines our model. The main result of the paper, Theorem 2.1 below is stated at the end of that section. The proof is through the usual renormalization group multiscale analysis. The definition of the renormalization group slices and the power counting is given in Sect. 3 and the proof of the theorem is completed in Sect. 4 using the momentum representation. Finally some low order renormalized amplitudes for this theory are computed in Appendix A. 2. Model and Main Result 2.1. The “naive” φ 4 model. It is obtained by replacing the ordinary commutative action by the Moyal-Weyl -product 1 1 λ (2.1) S[φ] = d 4 x( ∂µ φ ∂ µ φ + µ2 φ φ + φ φ φ φ), 2 2 4! with Euclidean metric. The commutator of two coordinates is [x µ , x ν ] = ı µν , where

⎛

0 ⎜−θ =⎝ 0 0

θ 0 0 0

0 0 0 −θ

⎞ 0 0⎟ . θ⎠ 0

In momentum space the action (2.1) becomes 1 1 λ S[φ] = d 4 p( pµ φp µ φ + µ2 φφ + φ φ φ φ). 2 2 4! 2 This is also the case of the Grosse-Wulkenhaar propagator.

(2.2)

(2.3)

(2.4)

278


The propagator is the same as in the commutative case 1 . p 2 + µ2

(2.5)

2.2. Feynman graphs: planarity and non-planarity, rosettes. In this subsection we give some useful conventions and definitions. Consider a φ 4 graph with n vertices, L internal lines and F faces. One has 2 − 2g = n − L + F,

(2.6)

where g ∈ N is the genus of the graph. If g = 0 the graph is planar, if g > 0 it is non-planar. Furthermore, we call a planar graph regular if it has a single face broken by external lines. We call B the number of such faces broken by external lines. The φ 4 graphs also obey the relation L=

1 (4n − N ), 2

(2.7)

where N is the number of external legs of the graph. In [31], T. Filk defined “contractions moves” on a Feynman graph. The first such move consists in reducing a tree line and gluing together the two vertices at its ends into a bigger one. Repeating this operation for the n − 1 lines of a tree, one obtains a single final vertex with all the loop lines hooked to it - a rosette (see Fig. 1). Note that the number of faces and the genus of the graph do not change under this operation. Furthermore, the external legs will break the same faces on the rosette. When one deals with a planar graph, there will be no crossing between the loop lines on the rosette. The example of Fig. 1 corresponds thus to a non-planar graph (one has crossings between e.g. the loop lines 3 and 5). Following [31] the rosette amplitude is ı

V˜ (external momenta) e 2

µ ν i j Ii j µν ki k j

Fig. 1. An example of a rosette

,

(2.8)


279

Fig. 2. The non-planar tadpole

where the intersection matrix Ii j is given by ⎧ ⎪ ⎨1, if line j crosses line i from right, Ii j = −1 if line j crosses line i from left, ⎪ ⎩0 if lines i and j do not cross,

(2.9)

where i and j correspond to an (arbitrary) numeration of the lines, independent of them being external or internal lines of the Feynman graph. An orientation is given by the sign convention chosen for the momenta in the conservation conditions. is the non-commutativity matrix (see Eq. (2.3)). Furthermore the overall phase factor corresponding to the external momenta is ı

V˜ (k1 , . . . , k N ) = δ(k1 + . . . + k N )e 2

N

µ ν i< j ki k j µν

,

(2.10)

which has exactly the form of a Moyal kernel. 2.3. UV/IR Mixing. The non-locality of the -product leads to a new type of divergence, the UV/IR mixing [8]. This can be seen already in the non-planar tadpole (see Fig. 2). Although this graph has zero genus, since it has two faces broken by external lines, it will lead to non-planarity when inserted into larger graphs. The amplitude of this non-planar tadpole with internal momentum higher than the external momentum is up to a constant

k −2

T =

dα

d 4 peık p e−α( p

2 +µ2 )

.

(2.11)

0

Integrating the Gaussian (and setting θ = 1) holds

k −2

T = 0

If k > 1 then

|T | < 0

dα − k 2 −αµ2 e αe . α2

∞

dα − 1 e α = 1. α2

(2.12)

(2.13)

Let k < 1. We have T = 0

k2

dα − k 2 −αµ2 e αe + α2

k −2 k2

dα − k 2 −αµ2 e αe . α2

(2.14)

280


Rescaling α = k 2 β we have for the first integral F(k) 1 1 dβ −β −1 −βk 2 m 2 e e = 2 , 2 2 k 0 β k

(2.15)

with F an analytic function of k. We separate again the second integral as µ−2 k −2 dα − k 2 −αµ2 dα − k 2 −αµ2 α e e + e αe . 2 α2 k2 µ−2 α

(2.16)

The second integral is bounded by a constant uniformly in k. In the first integral we 2 Taylor-expand e−αµ to get µ−2 µ−2 µ−2 dα − k 2 dα − k 2 dα − k 2 2 α α e − e αµ + e α O(α 2 ). (2.17) α2 α2 α2 k2 k2 k2 The first term integrates to k −2 F (k) with F analytic, the second computes to µ2 lnk 2 + F (k) with F analytic and the third is uniformly bounded. Thus the tadpole is c + c ln(k 2 ) + F(k) k2

T =

(2.18)

with c and c constants and F an analytic function at k = 0. We note that for a non-massive model the second term vanishes. One can include the contribution of the non-planar tadpole in the complete two-point function to obtain a dressed propagator. This motivates a modification of the kinetic part of the action (2.4) which leads to 2.4. The 1/ p 2 φ44 Model. This model is defined by the following action: 1 1 1 1 λ S[φ] = d 4 p( pµ φp µ φ + µ2 φφ + a 2 2 φφ + φ φ φ φ), (2.19) 2 2 2 θ p 4! with a some dimensionless parameter. The propagator is 1 p 2 + µ2 +

a θ 2 p2

,

(2.20)

and we choose a ≥ 0 so that this propagator is well-defined and positive. Using (2.8) and (2.20) the amplitude of a N -point graph is written A(G) = δ(

i=1...N

×

ki )e

ı 2

N

i< j ki k j

L

d 4 pi

i=1 ı

δ(q1v + q2v + q3v + q3v )e 2

1 pi2

+

v v i j Ii j qi q j

a θ 2 pi2

,

+ µ2 (2.21)

v=v¯

with k the external momenta, p the internal momenta, q v a generic notation for internal and external momenta at vertex v, and v¯ an external vertex of the graph chosen as root (to extract the global δ conservation on external momenta). The main result of this paper is


281

Theorem 2.1 (Main Result). The model defined by the action (2.19) is perturbatively renormalizable to all orders. The proof is given in the next two sections. We proceed now to the usual RG analysis by defining slices and establishing power counting.

3. Slices and Power counting In this section we establish the power counting of our model. For that purpose we shall use the very powerful tool of multiscale analysis. Power counting and the renormalization theory rely on some scale decomposition and the renormalization group is oriented: it integrates “fluctuating scales” (which we call here “high” scales) and computes an effective action for background scales (here called “low” scales). There are several technical different ways to define the RG scales, but in perturbation theory the best way is certainly to define the high scales as the locus where the denominator D of the propagator is big and the low scales as the locus where it is small, cutting the slices into a geometric progression. This certainly works well for the very different RGs of ordinary statistical mechanics (D = p 2 ), of condensed matter (D = i p0 + ( p)2 − 1) and of the Grosse-Wulkenhaar model (D = p 2 + 2 x 2 ). We use the same idea here again with D = p 2 + a/ p 2 . Power counting then evaluates contributions of connected subgraphs, also called “quasi local components” for which all internal scales are “higher” than all external scales in the sense above. We shall not rederive this basic principle here and shall use directly the particular version and notations of [32], in which these quasi-local compoj nents are labeled as G r . Before going into the detailed analysis of these contributions we first note a very important feature of our model: the term ap −2 changes the UV and IR regions. For the rest of this paper we set θ = 1. We employ the Schwinger trick and write: 1 = 2 p + ap −2 + µ2

∞

e−α( p

2 +ap −2 +µ2 )

dα .

(3.1)

0

Let M > 1. Slice the propagators as C( p) =

∞

C i ( p),

i=0

C i ( p) = C 0 ( p) =

M −2(i−1)

−2i M∞

dαe−α( p

dαe−α( p

2 +ap −2 +µ2 )

2 +ap −2 +µ2 )

≤ K e−cM

≤ K e−cp , 2

−2i ( p 2 +ap −2 +µ2 )

, i ≥ 1, (3.2)

1

with K and c some constants which, for simplicity, will be omitted from now on. To the i th slice corresponds either a momentum p ≈ M i or a momentum p ≈ M −i . Conversely, a momentum k ≈ M e for e ∈ Z has a scale α = M −2|e| . We have the following lemma:

282


Lemma 3.1. The superficial degree of convergence ω(G) of a Feynman graph G corresponding to the action (2.19) obeys N (G) − 4, if g(G) = 0 ω(G) ≥ , (3.3) N (G) + 4 if g(G) > 0 where N (G) is the number of external legs of G, and g(G) its genus. Proof. The first line is easy. It is enough to take absolute values in (2.21), and apply the momentum routing. We briefly recall this procedure. We fix a scale attribution for all propagators. As the sum over the scales is easy to perform (along the same lines as in [32]) we concentrate on the problem of summing the internal momenta at fixed scale attribution ν. At any scale i the graph G i made of lines with scales higher or equal to i splits into ρ connected components G ri , r = 1, . . . , ρ. We choose a spanning tree T compatible with the scale attribution, that is each Tri = T ∩ G ri is a tree in the connected component G ri . We define the branch b(l) associated to the tree line l as the set of all vertices such that the unique path of lines connecting them to the root contains l. We can then solve the delta functions for the tree momenta as

pl = − ql , (3.4) l ∈b(l)

where l ∈ b(l) denotes all loops or external momenta touching a vertex in the branch b(l). After integrating internal momenta we get the bound Aν ≤ M −2il M 4il , (3.5) l

l∈L

where L denotes the set of loop lines. The first factor comes from the prefactors of the propagators while the second comes from the integration of the loop momenta. We can reorganize the above product according to the scale attribution as i i i i Aν ≤ M −2L(G r ) M 4[L(G r )−n(G r )+1] = M −[N (G r )−4] , (3.6) i,k

i,k

where we have used (2.7). The second line of (3.3) is obtained using an argument similar to the one used in [14]. In fact if the graph is non-planar there will be two internal loop momenta p and q such that, after integrating all tree momenta with the delta functions, the amplitude contains a factor 2 −2 2 −2 I = d 4 p d 4 q e−α1 p −α1 ap −α2 q −α2 aq +ı p∧q . (3.7) A naive bound would be to bound the integral by M 4i1 M 4i2 . Instead we use ⎛ ⎞m

d2 1 ⎝1 + M 2i1 ⎠ eı p∧q = eı p∧q . 2 (1 + M 2i1 q 2 )m dp j j

(3.8)


283

Integrating by parts we get |I | ≤

d 4 pd 4 q −2i 2 2 −2i 2 −2 e−M q −M q (1 + M 2i1 q 2 )m ⎞m ⎛

d2 ⎠ e−M −2i1 p2 −M −2i1 p−2 . × ⎝1 + M 2i1 2 dp j j

(3.9)

The derivative acting on the exponential gives factors of order at most O(1). If we chose m = 3 we have a bound d 4 pd 4 q −2i 1 p 2 I ≤K e−M ≤ K . (3.10) (1 + M 2i1 q 2 )3 We have thus gained both factors M 4i1 and M 4i2 with respect to the naive bound.

4. Renormalization We have established that all possible divergences come from planar 2 or 4 point graphs. Note that they may have more that one broken face3 . We will prove that all divergences can be reabsorbed in a redefinition of the parameters in the action (2.19).

4.1. Two-point function. The single-broken-face 2-point graphs are ultraviolet divergent and as such give nontrivial mass and wave function renormalizations. By contrast the 2-point graphs with two broken faces are ultraviolet convergent. Nevertheless we will prove that they give a finite renormalization of the 1/ p 2 term. 4.1.1. 2-point function with a single broken face. From the standard multiscale analysis we know that power counting has to be computed only for connected components of the j G r type. Consider the case of such a planar, one particle irreducible, 2-point subgraph j S which is a component G r for j for a certain range of slices e < j ≤ i between e, its highest external scale and i, its lowest internal scale (and a particular value of r ), j A(G r )

= δ(k1 + k2 ) ×

L l=1

δ(q1v

4

d pl

M −2(il −1) M −2il

−2

dαl e−αl [ pl +apl

+ q2v + q3v + q3v ),

2

+µ2 ]

(4.1)

v=v¯

where we consider that all eventual subrenormalizations have been performed. We perj form the momentum routing for the subtree Tr . Let k1 enter into the root vertex of S. We define T 1 = {l ∈ T | k2 ∈ b(l)} , T 2 = T − T 1 .

(4.2)

3 This stands in contrast with the Grosse-Wulkenhaar theory in which only graphs with a single broken face diverge.

284


The amplitude is written, dropping the index on k2 and forgetting the overall δ function, A(G) =

|L|

d 4 pl

l=1

×

e

−αl (

×

M −2(il −1)

dαl

M −2il

l=1

l∈T 2

L

2

l∈L

2 l ∈b(l) pl ) +a( l ∈b(l)

−2

e−αl [ pl +apl

pl )−2 +µ2

e

+µ2 ]

−αl (k+ l ∈b(l) pl )2 +a(k+ l ∈b(l) pl )−2 +µ2

.

(4.3)

l∈T 1

We can now divide all integrals over pl ’s in regions for each l ∈ T1 , according to whether ( l ∈b(l) pl )2 < a 1/2 or ( l ∈b(l) pl )2 ≥ a 1/2 . The regions with at least one condition ( l ∈b(l) pl )2 < a 1/2 will count for O(1) instead of M 4i and using directly the power counting argument we bound such a contribution to (4.3) by M −2i per slice, for all k. In the following we will neglect all boundary terms on the sphere of radius a 1/2 as they are easy to bound uniformly in k. We conclude that only the case with all ( l ∈b(l) pl )2 ≥ a 1/2 have to be renormalized. In that case the factors

−2

e−αa(k+ l ∈b(l) pl ) give rise to an analytic behavior in k, and can be expanded around k = 0. We Taylor-expand the last line in (4.3) in that case. The odd terms in p are zero after integration, as the branch momenta are linearly independent. For each term we have a development of the form 1 −αl ( l ∈b(l) pl )2 +a( l ∈b(l) pl )−2 +µ2 2 e dt (1 − t)e−tαl k . (4.4) 1 − αl k 2 + αl2 k 4 0

Using the multiscale bound (3.2), we see that collecting the first terms we get a bound like (3.6), thus a quadratic mass divergence. If we have at least one factor in αl we gain at least M −2il ≤ M −2i and we pay a factor k 2 which is of order M 2e because the external momenta is of scale e. Thus for all scales j between i and e we have gained a factor M −2 and the power counting factor associated to the corresponding connected compoj j j nent G r , which was previously M −(N (G r )−4) = M 2 , has become M −(N (G r )−2) = 1. We get therefore a constant per slice as power counting for that connected component. As usual we recognize here the logarithmically divergent wave function renormalization associated to S. All other terms give convergent contributions, because a factor at least M −4 per slice between e and i is gained. 4.1.2. 2-point function with two broken faces. The amplitude of a one-particle irreducible 2-point graph with two broken faces is j A(G r )

= δ(k1 + k2 ) ×

v=v¯

L l=1

δ(q1v

4

d pl

M −2(il −1) M −2il

+ q2v + q3v + q3v )eık2 ∧(

−2

dαl e−αl [ pl +apl 2

l∈S

pl )

,

+µ2 ]

(4.5)


285

with S ∈ L the set of loop lines crossed by the second external line. Performing again the momentum routing, dropping the index in k2 and the global δ function yields j A(G r )

=

|L|

4

d pl

l=1

×

e

−αl (

l∈T 2

×

dαl

M −2il

M −2(il −1)

−2

e−αl [ pl +apl 2

l∈L

2 l ∈b(l) pl ) +a( l ∈b(l)

pl )−2 +µ2

+µ2 ]

e


eık∧(

l∈S

pl )

.

(4.6)

l∈T 1

We use the same decomposition as before according to whether ( l ∈b(l) pl )2 < a 1/2 or ( l ∈b(l) pl )2 ≥ a 1/2 . Again only the case with all ( l ∈b(l) pl )2 ≥ a 1/2 is potentially divergent and could give rise to a non-analytic behavior in k. We choose a line l ∈ S, use eık∧(

l∈S

pl )

=−

1 pl eık∧( l∈S pl ) 2 k

(4.7)

and integrate by parts in (4.6) to get j A(G r )

1 =− 2 k ×

|L|

×

d pl e

ık∧(

l∈S

pl )

e

−αl (

M −2(il −1) M −2il

l=1

l∈T 2

4

2 l ∈b(l) pl ) +a( l ∈b(l)

pl

dαl pl

)−2 +µ2


−2

e−αl [ pl +apl 2

+µ2 ]

l∈L

e

.

(4.8)

l∈T 1

The derivatives acting on the Gaussian will give rise to insertions scaling like α, α 2 p 2 , αp −2 , αp −4 , α 2 p −2 , α 2 p −4 . The first two terms scale as M −2i in a slice while the rest scale at least as M −4i . Using again the trick (4.7), and the power counting bound we get, when summing over all slices, a behavior like j

A(G r ) =

∞ 1 −2i 1 M −2 j M = 2 K, 4 k k M 2e

(4.9)

i= j

with K some constant, if k ≈ M e (and consequently of scale |e|). As the scale j is ultraviolet with respect to |e| we bound M −2 j−2e ≤ M −2(|e|+e) ≤ 1 .

(4.10)

We have thus proved that j

A(G r ) =

1 F(k) k2

(4.11)

286


with F(k) a function uniformly bounded by a constant for all k.4 We identify the terms F(0) as a finite renormalization for the coefficient a in the Lagrangian. Note that using this scale decomposition there are no logarithmic subleading divergences for this two point function with two broken faces. 4.2. Four-points function. The amplitude of a planar regular four-points graph is given by: ı

A(G r ) = δ(k1 + k2 + k3 + k4 )e 2 i< j ki ∧k j M −2(il −1) L −2 2 2 4 × d pl dαl e−αl [ pl +apl +µ ] δ(q1v + q2v + q3v + q3v ) . j

l=1

M −2il

v=v¯

(4.12) The first line reproduces exactly the Moyal four-points kernel. Power counting leads to bound the second line by a constant per slice, thus it corresponds to a logarithmic divergence, which in turn generates a logarithmic coupling constant renormalization. Some comments are in order for the planar four-points graphs with more than one broken face. Using (4.7) once, we get a bound like j

A(G r ) =

∞ 1 −2i M −2 j M = K, k2 M 2e

(4.13)

i= j

and by (4.10) we see that the amplitude of such a graph is a function of external momenta uniformly bounded by some constant. 5. Conclusions and Perspectives We have thus proved in this paper that the scalar model (2.19) is renormalizable at all orders in perturbation theory. The renormalization of the planar regular graphs goes along the same lines as the renormalization of the Euclidean φ 4 on a 4−dimensional commutative space. The non-planar graphs remain convergent and the main difference concerns the planar irregular graphs. The comparison with the action (2.1) (which is non-renormalizable, with UV/IR mixing) and with the Grosse-Wulkenhaar model is summarized in the following table:

planar regular planar irregular non-planar

model (2.1) 2-points 4-points ren ren UV/IR log UV/IR IR divergent IR divergent

GW model 2-points 4-points ren ren conv conv conv conv

1/ p 2 − φ44 model 2-points 4-points ren ren finite ren conv conv conv

where ren means renormalizable.and conv means convergent. Acknowledgement. We thank A. Abdesselam for indicating the proof of analyticity of F in (4.11), and F. Vignes-Tourneret for useful comments. Furthermore, Adrian Tanasa gratefully acknowledges the European Science Foundation Research Networking Program “Quantum Geometry and Quantum Gravity” for the Short Visit Grants 2219 and 2232. 4 In fact F(k) is analytic in k, as it is a sum of absolutely convergent integrals of analytic functions in α and k, in the region where all conditions ( l ∈b(l) pl )2 ≥ a 1/2 are fulfilled.


287

A. Examples of Graphs We illustrate the general results established in the previous section by some examples of two- and four-points graphs for which we analyze the Feynman amplitude.

A.1. A two-point graph example. Let us analyze the Feynman amplitude of the tadpole of Fig. 2. This graph has g = 0 but B = 2. Due to the new renormalization group slices the parameter α of an internal line obeys α < min{k 2 , k −2 }. Therefore the amplitude of the non planar tadpole is (up to a constant)

min{k 2 ,k −2 }

dα

d 4 peik∧ p e−α( p

2 +ap −2 +µ2 )

.

(A.1)

0

Applying (4.7) and integrating by parts holds: 1 − 2 k

min{k 2 ,k −2 }

dα

d 4 peik∧ p p e−α( p +ap 2

−2+µ2 )

0

2 −2 1 min{k ,k } = 2 dα d 4 peik∧ p k 0 2 −2 2 × 8α − α 2 (4 p 2 − 8ap −2 + 4a 2 p −6 ) e−α( p +ap +µ ) .

(A.2)

All but the first and second terms in (A.2) can be bounded by k 2 when taking absolute values such that the contribution to the amplitude of the tadpole is a constant. The coefficient of the k −2 divergences is therefore

min{k 2 ,k −2 }

c=

dα

2 −2 2 d 4 peik∧ p 8α + 4 p 2 α 2 e−α( p +ap +µ ) .

(A.3)

0

Applying again 4.7 and integrating again by parts holds only terms like 1 cn = 2 k

min{k 2 ,k −2 }

α dα 2

d 4 peik∧ p (αp 2 )n e−α( p

2 +ap −2 +µ2 )

with n = 0, 1, 2. Taking absolute values, using (αp 2 )n e−αp < e− irrelevant constants, 2

cn
0, and λ− an analytic function on I m( p) < 0. We also require the normalization 1 , p → ∞. (3.3) λ± = p + O p Let us define, on the real axis, the jump function f ( p, x, t) =

1 (λ− ( p, x, t) − λ+ ( p, x, t)) , 2πi

Hamiltonian Structures of Reductions of the Benney System

299

and suppose f is a function of real p which is Hölder continuous and satisfies the conditions +∞ p n f dp < ∞, n = 0, 1, . . . . −∞

Then, using Plemelj’s formula for boundary values of analytic functions, we may take +∞ f ( p ) λ± ( p) = p − π dp ∓ iπ f ( p). −∞ p − p What we obtained is that, with hypotheses above, the functions λ+ and λ− are Borel sums of the series (3.1) in the upper and lower half plane respectively. On the other hand, λ± will have, at p → ∞, the formal asymptotic series (3.1), where +∞ p n f ( p, x, t)dp. An (x, t) = −∞

Thus, to any solution of Benney’s equations we can associate a pair of functions λ± ( p; x, t). In particular, a real valued f leads to real valued moments. In this case, using the Schwarz reflection principle, we can restrict our attention to the function λ+ ; this is the case studied in [1–3,10,21]. On the other hand, it will be useful below to consider the analytic continuation of λ+ into the lower half plane, in the neighborhood of specified points in the real axis. Such a continuation may or may not coincide with λ− , the Schwarz reflection of λ+ . In particular important examples, such a continuation may be developed consistently, giving the structure of a Riemann surface. Remark 2. Other normalizations, more general than (3.3) are allowed, based on the fact that for any differentiable function ϕ of a single variable, the composed function ϕ(λ+ ) remains a solution of (3.2), the associated reduction being the same. In concrete examples, it is sometimes more convenient to use a different normalisation. Let us consider now the relations between solutions of (3.2) and the reduced Benney system. In this case, we have that λ+ is associated with an n component reduction if and only if it depends on the variables x, t via n independent functions. As any reduction is diagonalizable, it is not restrictive to take as these variables the Riemann invariants. Thus, we have λ+ ( p, x, t) = λ+ ( p, λ1 (x, t), . . . , λn (x, t)), (3.4) with λit = v i λix . (3.5) Remarkably, the characteristic velocities of the reduction turn out to be the critical points of the function λ+ associated with it. More precisely, we have Theorem 5. Let λ+ , solution of (3.2), satisfy conditions (3.4) with (3.5). Let us denote ϕ i (λ1 , . . . , λn ) = λ+ (v i , λ1 , . . . , λn ),

i = 1 . . . n,

and suppose that the ϕ i are not constant functions. Then, the velocities v i satisfy ∂λ+ i (v ) = 0, ∂p

i = 1 . . . n,

and the corresponding critical values ϕ i can be chosen as Riemann invariants for the system (3.5).

300

J. Gibbons, P. Lorenzoni, A. Raimondo

Proof. Considering Eq. (3.2) at p = v i , we obtain the system of n equations ϕti = v i ϕxi − A0x

∂λ+ i (v ). ∂p

As λ1 , . . . , λn can be chosen as coordinates, by the chain rule we get n n n ∂ϕ i j ∂ϕ i j ∂λ+ i ∂ A0 j i (v ) λ =v λx − λx , ∂λ j t ∂λ j ∂p ∂λ j j=0

j=0

j=0

and this, after substituting (3.5) into it, is equivalent to ∂λ 0 ∂ϕ i j + i ∂A i (v v + − v ) =0 i, j = 1 . . . , n, ∂λ j ∂p ∂λ j

(3.6)

j

due to the independence of the λx . Particularly, for i = j the system above reduces to ∂λ+ i ∂ A0 (v ) = 0. ∂p ∂λi

(3.7)

Further, if A0 does not depend on λi , the function λ+ is also independent of the same λi . In this case, the associated system (3.5) reduces to a n −1 reduction. On the other hand, if the system is a proper n−component reduction then ∂i A0 = 0 and the characteristic velocities are critical points for λ+ . Substituting back (3.7) into (3.6), we obtain ϕ i = ϕ i λi . Thus, if the critical values ϕ i are not constant functions, it is possible to choose them as Riemann invariants.

The converse of the theorem above is also true: if λ+ is a solution of (3.2) satisfying λ+ ( p, x, t) = λ+ ( p, λ1 (x, t), . . . , λn (x, t)), and with n distinct critical points obtain the diagonal system

v1, . . . , vm ,

(3.8)

then by evaluating Eq. (3.2) at p = v i we

ϕti = v i ϕxi , where ϕ i = λ+ (v i ). Thus, critical points are characteristic velocities. Moreover, the existence of a function λ associated with a reduction selects a natural set of Riemann invariants, the critical values of λ. Unless otherwise stated, these are the coordinates we will consider below. Remark 3. It might happen that the function λ+ possesses m critical points, with m > n. This is the case, for instance, in Remark 2, where critical points of the function ϕ have to be added. Then, substituting the critical points into (3.2) we obtain an m component diagonal system. However, in this case we have that m − n of the critical values have trivial dynamics for they are independent of x, t. Consequently, the m component system reduces to an n component one. Example 3.2. Consider u i = u i (x, t), i = 1, 2. The function λ+ = p +

u1 , p − u2

(3.9)

rational in p, satisfies Eq. (3.2) if and only if u 1 , u 2 satisfy the 2 component Zakharov reduction of Example 2.1.


301

Fig. 1. n−slit Loewner evolution on the upper half plane

3.2. Reductions and Loewner equations. It was shown in [13,21] that the solution of the initial value problem of an n reduction is given by a Inverse Scattering Transform procedure (which leads to a particular form of Tsarev’s generalized hodograph formula (1.7)), provided that ∂λ+ ( p) = 0, ∂p

I m( p) > 0.

It is thus necessary that λ+ ( p) be an univalent conformal map from the upper half plane to some image region. In [11,12] it was proved that these conformal maps have to be solutions of a system of so-called chordal Loewner equations. In fact, if a solution of Eq. (3.2) is associated with an n−component reduction of the Benney system, then conditions (3.4) hold. Substituting into Eq. (3.2), if v i are the characteristic velocities associated with the reduction, we obtain N ∂λ+ ∂ A0 ∂λ+ (v i − p) i + λix = 0. ∂λ ∂λi ∂ p

(3.10)

i=1

As the λix are independent, then it follows that ∂λ+ ∂i A0 ∂λ+ , = ∂λi p − vi ∂ p

i = 1, . . . , n.

(3.11)

This is a system of n chordal Loewner equations (see for example [8]). When the function λ+ is chosen with the normalization (3.3), this system describes the evolution of families of univalent conformal maps from the upper complex half plane to the upper half plane with n slits, when the end points of the slits are allowed to move along prescribed mutually non intersecting Jordan arcs. Using the implicit function theorem it is possible to show that the inverse function p satisfies an analogous system ∂p ∂i A0 = − , ∂λi p − vi

i = 1, . . . , n.

(3.12)

302


For n > 1, the consistency conditions of (3.11) (or (3.12) equivalently) turn out to be the Gibbons-Tsarev system. On the other hand, and more generally, we can consider a set of n Loewner equations, ∂λ+ ai ∂λ+ , = ∂λi p − bi ∂ p

i = 1, . . . , n,

(3.13)

for arbitrary functions ai , bi . The consistency conditions ∂ 2 λ+ ∂ 2 λ+ = ∂λi ∂λ j ∂λ j ∂λi are then equivalent to the set of equations ∂i a j = ∂ j ai , 2ai a j , ∂i a j = i (b − b j )2 ai ∂i b j = i , b − bj

(3.14) (3.15) (3.16)

where i = j. The first of these equations implies locally the existence of a function A0 (λ1 , . . . , λn ) such that ai = ∂i A0 . Consequently, Eqs. (3.15), (3.16) become the Gibbons-Tsarev system (2.6), with bi = v i . So, to any solution of a system of n chordal Loewner equations there corresponds an n-component reduction of the Benney chain. Example 3.3. The dispersionless Boussinesq reduction, which is a 2−component Gelfand-Dikii reduction, is given by A0t = A1x , A1t = −A0 A0x , can be described in the λ picture using the polynomial function λ+ = p 3 + 3A0 p + 3A1 . The characteristic velocities are

v 1 = − −A0 ,

v2 =

−A0 ,

and the Riemann invariants are given by 3

3

λ1 = λ+ (v 1 ) = 3A1 + 2(−A0 ) 2 , After the renormalization λ˜ (λ+ ) =

3

λ+ =

λ2 = λ+ (v 2 ) = 3A1 − 2(−A0 ) 2 . 3

p 3 + 3A0 p + 3A1

(3.17)

we obtain a family of Schwarz-Christoffel maps as in Fig. 2. It is easy to verify that the critical points of the function λ˜ (λ+√)( p) are the same as λ+ ( p), while the corresponding 3 new Riemann invariants are λ˜ i = λi . The higher moments can be found by identifying the Laurent series as p → ∞ of (3.17) with the formal series (3.1).


303

Fig. 2. The dispersionless Boussinesq reduction

As a consequence of the Loewner system (3.11) satisfied by a reduction, it follows immediately that the critical points of λ+ ( p) are simple. Indeed, taking the limit of the i th equation of the system, for p → v i gives 1=

∂ 2 λ+ i (v ) ∂i A0 , ∂ p2

(3.18)

where we used the identity ∂v i dλi ∂λ ∂λ | | = − = 1. i i ∂λi p=v dλi ∂ p p=v ∂λi Thus, ∂ 2 λ+ i (v ) = 0, ∂ p2 hence the v i are simple. Suppose now that λ+ admits an analytic continuation in some neighborhood of v i . Henceforth, to simplify the notations, the subscript + will be dropped from λ+ , and we will denote both the analytic function λ+ and its analytic continuation simply by λ. Moreover, we will write

λ ( p) =

∂λ ∂ 2λ ( p), λ ( p) = 2 ( p), . . . . ∂p ∂p

Then, the function p(λ) has the series development near λ = λi , p(λ) = v + i

√ 2

λ (v i )

λ − λi + O λ − λi ,

(3.19)

304


which becomes a Taylor expansion in the complex local parameter t = thermore, we have

√

λ − λi . Fur-

1 1 1 1 λ (v k ) = − 2 λ (v k )2 λ ( p) λ (v k ) p − v k +

1 λ (v k )2 1 λ (v k ) − 4 λ (v k )3 6 λ (v k )2

(3.20) ( p − v k ) + O ( p − v k )2 .

This expansion will be useful in Sect. 5. Finally, we introduce two sets of contours, in the p and λ plane respectively, that we will need later for describing the Hamiltonian structure of the reductions. We define i as a closed and sufficiently small contour in the p−plane around v i , and Ci as the image of i according to the analytical continuation of λ. Thus, i and Ci are well defined; in particular, it follows from expansion (3.19) that λi – the tip of the slit – is a square root branch point for p(λ), hence Ci encircles it twice.

3.3. Symmetries of reductions of the Benney chain. A well-known method [13] of obtaining a countable set of symmetries of a reduction of the Benney chain is based on the Lax representation of the dKP hierarchy, λtn = {λ , h n } = (h n ) p λx − (h n )x λ p ,

n = 1, 2, . . . ,

where h n = n1 (λn )≥0 . We assume, unless otherwise stated, that λ is normalized as in (3.3). If the solution λ possesses n critical values (v 1 , . . . , v n ) , the hierarchy can be reduced to λitn = wni λix ,

i = 1, . . . , n,

with

wni =

∂h n ∂p

n = 1, 2, . . . ,

| p=v i

.

(3.21)

These are, by construction, components of the symmetries of the reductions, the first few of them being w1i = 1, w2i = v i , w3i = (v i )2 + A0 . . .

i = 1, . . . , n.

Further, the above characteristic velocities can be obtained as the coefficients of the expansion at λ = ∞ of the generating functions W i (λ) =

1 . p(λ) − v i

(3.22)

Proposition 3.1. The functions W i (λ) are solutions of the linear system

W

∂ j W i (λ) j (λ) − W i (λ)

=

∂ j vi . v j − vi

(3.23)


305

Moreover, expanding W i (λ) at λ = ∞ we get W i (λ) =

∞ wni , λn

(3.24)

n=1

where the coefficients of the series are the symmetries wni given in (3.21). Remark 4. The system (3.23) holds in any normalization, but the expansion (3.24) is valid only with the normalization (3.3). Proof. In order to prove (3.23), knowing that ∂i A0 ∂p , = ∂λi vi − p we can write

−1 ∂p ∂v i ∂ j W (λ) = ∂ j = − ( p − v i )2 ∂λ j ∂λ j 0 ∂j A 1 1 − j =− i 2 j (p − v ) v − p v − vi i

=

(p

1 p − vi

∂ j A0 . − v j )(v j − v i )

− v i )( p

On the other hand W j (λ) − W i (λ) =

(v j − v i ) , ( p − v i )( p − v j )

and so ∂ j A0 ∂ j W i (λ) ∂ j vi = = . W j (λ) − W i (λ) (v j − v i )2 v j − vi In order to prove (3.24), having chosen the normalization (3.3), we have 1 (λn )+ d (λn )+ 1 = lim ( p − vi )2 res wni = dp . n p→vi dp ( p − vi )2 n p=vi ( p − vi )2 The function res

p=vi

(λn )+ ( p−vi )2

has poles only at p = vi and p = ∞ and therefore

(λn )+ (λn )+ (λn ) dp = − res dp = − res dp , p=∞ ( p − vi )2 p=∞ ( p − vi )2 ( p − vi )2

where the last identity is due to res

p=∞

(λn )− dp = 0. ( p − vi )2

306


Changing variables we obtain λn ddλp (λn ) 1 res dp = dλ p=∞ ( p − vi )2 2πi ∞ ( p(λ) − vi )2 n λn−1 = dλ, 2πi ∞ p(λ) − vi where ∞ is a sufficiently small contour around p = ∞. Thus, 1 wni = − W i (λ)λn−1 dλ; 2πi ∞ varying n we obtain the coefficients of the expansion (3.24).

Remark 5. It will be useful below to consider, as a generating function of the symmetries, ∞ n wni = − , w (λ) = ( p(λ) − v i )2 λn+1 i

∂p ∂λ

(3.25)

n=1

which is nothing but the λ− derivative of (3.22). We finally observe that, due to linearity of (3.23), the functions n i z = wi (λ)ϕk (λ)dλ k=1 Ck

still satisfy the system for the symmetries and therefore, applying the generalized hodograph method, we can write the general solution of the reduction in the implicit form z i = v i t + x, i = 1, . . . , n. The inverse scattering solutions found in [10,21] are of this form. 4. Hamiltonian Structure of the Reductions As it was shown in [11], any reduction of the Benney system is a diagonalizable and semi-Hamiltonian system of hydrodynamic type. However, very little is known about the Hamiltonian structure of these systems, whether local or nonlocal. A few examples are known explicitly. These include the Gelfand-Dikii and Zakharov reductions, which arise as dispersionless limits of known dispersive Hamiltonian systems. In this section, we use Ferapontov’s procedure sketched in Sect. 1.2 for semi-Hamiltonian diagonal systems in order to determine the metric associated with a generic reduction. Theorem 6 The general solution of the system (1.11) in the case of reductions of the Benney chain is ∂i A0 gii = , (4.1) ϕi (λi ) where the functions ϕi (λi ) are n arbitrary functions of one variable, the functions λ1 , . . . , λn being the Riemann invariants of the system.


307

Proof. From the system (1.11), and making use of both the Gibbons-Tsarev equations (2.6), which hold for any reduction, we obtain ∂ j ln

√

gii =

∂ j vi ∂ j A0 = = ∂ j ln ∂i A0 , j i 2 v −v (v j − vi )

from which we obtain the general solution (4.1). In the case ϕi = 1 the rotation coefficients √ ∂i g j j βi j := √ , gii are symmetric: βi j =

(4.2)

i = j

∂i A0 ∂ j A0 1 ∂i ∂ j A0 = . 2 ∂i A0 ∂ j A0 (vi − v j )2

(4.3)

(4.4)

We now focus our attention on this case, that is we consider the Egorov (potential) metric gii = ∂i A0 . (4.5) Remark 6. We notice that the choice of potential metric is not restrictive, as any other metric (4.1) can be written in potential form under a change of coordinates λi −→ ϕi (λi ),

(4.6)

which is exactly the freedom we have in the definition of the Riemann invariants. On the other hand, the choice of the Riemann invariants determines a unique metric which is potential with respect to those coordinates. In order to find the Christoffel symbols and the curvature tensor of the metric (4.5), one could compute these objects – as usual – starting from their definitions. However, for a reduction of the Benney chain, this procedure can be shortened. Indeed, using the Gibbons-Tsarev system (2.6), the connection and the curvature can be written as simple algebraic combinations of the quantities v i , ∂i A0 , δ(v i ), δ(log ∂i A0 ), i = 1, . . . , n, where we introduced the shift operator δ=

n ∂ . ∂λk k=1

We have Proposition 4.2. The symbols 1 ij j k = −g is sk = − g is g jl (∂s glk + ∂k gls − ∂l gsk ) , 2

308


where ikj are the Christoffel symbols associated to the diagonal metric gii = ∂i A0 , for a reduction of the Benney system are given by ij

k = 0,

i = j = k,

1 ij ji i = −i = , (vi − v j )2 1 ∂k A 0 kii = − , ∂i A0 (vk − vi )2 iii

=

i = j,

(4.7b)

i = k,

(4.7c)

1 δ(ln ∂i A0 ) − . ∂i A0 (v i − v k )2 ∂i A0

∂k A 0 k=i

(4.7a)

(4.7d)

Proof. For the metric gi j = δi j ∂i A0 , we get ij

k = −

1 1 0 0 0 δ , ∂ A + δ ∂ A − δ ∂ A jk ik i j k j ik jk 2 ∂i A0 ∂i A0

and Eqs. (4.7) are obtained from these by substituting, whenever allowed, the GibbonsTsarev equations (2.6b).

vi

In particular, the curvature can be expressed solely via the shift operator δ, acting on 0 and ln ∂i A .

Proposition 4.3. The non-vanishing components of the curvature tensor of the metric (4.5) for a reduction of the Benney system can be written in terms of the quantities δ(v i ), δ(ln ∂i A0 ). More precisely, we have the following identity: 0 ) + δ(ln ∂ A0 ) δ(ln ∂ A i j δ(v i ) − δ(v j ) ij Ri j = − + 2 . (4.8) (v i − v j )2 (v i − v j )3 Proof. The Riemann curvature tensor of the metric gi j and Levi-Civita connection ijk is given by i i R ijkh = ∂h ijk − ∂k ij h + αh αjk − αk αj h .

For a semi-Hamiltonian system (see [9]), the only non-vanishing components of the ij curvature are Ri j , i = j, moreover, since the rotation coefficients βi j of the metric (4.5) are symmetric, it is easy to check the classical result ij

Ri j =

n s=1

j

g is Rsi j = − √

1 δ βi j , gii g j j

i = j.

Thus, for the metric (4.5), we obtain ⎛ ⎛ ⎞ ⎞ 0 0 ∂ ∂ A i j ∂ A 1 1 1 ij ij ⎠=− ⎠, Ri j = − δ⎝ δ ⎝ 0 2 0 0 0 0 ∂ A i ∂i A ∂ j A ∂i A ∂ j A ∂i A0 ∂ j A0


309

using now the Gibbons-Tsarev system (2.6) we get ∂i A0 ∂ j A0 1 ij Ri j = − δ( i ) (v − v j )2 ∂ A0 ∂ A0 i

=−

δ(ln

j

∂i A0 ) + δ(ln ∂ j A0 ) (v i − v j )2

+2

δ(v i ) − δ(v j ) . (v i − v j )3

Formula (4.8) presents a compact way of describing the curvature tensor of the Poisson structure (1.13) associated with a reduction. However, we should notice here that the knowledge of the curvature is not sufficient to write the Poisson bracket. Indeed, what we need is a decomposition (1.12) of the curvature in terms of the symmetries of the system. From formula (4.8) this decomposition looks non-trivial; we will address this problem in the next section using a different approach. 5. Hamiltonian Structure in the λ Picture The purpose of this section is to derive an explicit formulation for the Hamiltonian structure of a reduction of the Benney chain in terms of the function λ( p), which defines the reduction itself. In particular, we will show how the differential geometric objects associated with Ferapontov’s Poisson operator of type (1.13) can be expressed, in the case of a reduction with associated λ, in terms of the set of data

v i , λ (v i ), λ (v i ), λ (v i ),

i = 1, . . . , n,

where v i , the characteristic velocities of the reduction, are the critical points of λ. Moreover, we will describe the quadratic expansion of the curvature associated with the metric. Let us consider a reduction with associated function λ( p). In this case, as already mentioned, a set of Riemann invariants is naturally selected, the critical values of λ( p). From (3.18) and (4.1), it is immediate to check that the components of the metric which is potential in those coordinates can be expressed in terms of λ as dp 1 gii = ∂i A0 = i = res . (5.1) λ (v ) p=v i λ ( p) This result was already known in the case of dispersionless Gelfand-Dikii [5] reductions. However, it holds for all reductions of the Benney chain. 5.1. Completing the Loewner system. We move now our attention from the metric to the Christoffel symbols and the associated curvature tensor, looking for a way to describe these objects in terms of λ and its critical points. However, this step is not immediate. Indeed, from Eqs. (4.7) and (4.8) we need to find an expression in the λ picture for the quantities δ(v i ), δ(log ∂i A0 );

310


we will see that the right object to look at is ∂p ∂p + , ∂λ ∂λi n

F( p) =

(5.2)

i=1

obtained from the inverse function of λ with respect to p, that is p = p(λ, λ1 , . . . , λn ).

(5.3)

The function F is thus determined once the function λ( p, λ1 , . . . , λn ) is known. Before discussing the Christoffel symbols and the curvature, we will consider the properties of this function in detail. First of all, using the Loewner equations (3.12) and the expression (3.18), we can write F( p) in the following form: ∂i A0 1 − λ ( p) i=1 p − v i n 1 1 1 − = res . λ ( p) i=1 p=vi λ ( p) p − v i n

F( p) =

From its expansion (3.20), the function

1 λ ( p)

(5.4)

in v i has a simple pole, therefore, F( p) is

analytic at p = v i . Using this fact, we can prove the following Theorem 7 Let λ( p, λ1 , . . . , λn ) be a solution of Eq. (3.2), and let v 1 , . . . , v n be its critical points. Defining the function F( p) as above (5.2), we have F(v i ) = δ(v i ),

(5.5)

∂F i (v ) = δ(ln ∂i A0 ). ∂p

(5.6)

Proof. We have already shown that F( p) is analytic at p = v i . In order to prove (5.5) and (5.6), we consider the system of n + 1 differential equations ∂p ∂i A0 , = ∂λi vi − p

i = 1, . . . , n, (5.7)

∂p = ∂λ

n k=1

∂k A 0 + F( p). p − vk

The conditions

∂2 p ∂2 p − =0 i = j (5.8) ∂λi ∂λ j ∂λ j ∂λi give nothing but the Gibbons-Tsarev system, hence are satisfied for any reduction. So, we concentrate our attention on the remaining n consistency conditions, ∂2 p ∂2 p − = 0, ∂λ∂λi ∂λi ∂λ

(5.9)


311

which – by construction – are satisfied, to obtain some information about F( p). Expanding both sides we obtain ∂2 p ∂i A0 ∂ p = i ∂λ∂λ ( p − vi )2 ∂λ n ∂k A 0 ∂i A0 = F( p) + , ( p − vi )2 p − vk k=1

n ∂ F ∂ p ∂ F ∂k ∂i A0 ∂k A0 ∂p ∂2 p ∂i vk − i = + + + ∂λi ∂λ ∂ p ∂λi ∂λi p − vk ( p − vk )2 ∂λ k=1 n ∂ F ∂i A0 ∂ F ∂k ∂i A0 ∂k A0 ∂i A0 = ∂ , + + + v + i k ∂ p v i − p ∂λi p − vk ( p − vk )2 p − vi k=1

substituting the Gibbons-Tsarev equations (2.6) in the above formulae and rearranging (5.9), we find that ∂F 0 F( p) − δ(v i ) ∂ p ( p) − 2δ(log ∂i A ) 1 ∂ F( p) + − = 0. (5.10) ( p − v i )2 p − vi ∂i A0 ∂λi Multiplying by ( p − v i )2 and taking the limit for p → v i , we get F(v i ) = δ(v i ). Then, taking the residue of the right-hand side of (5.10) at p = v i gives ∂F i (v ) = δ(log ∂i A0 ). ∂p

Thus, specifying the function F turns out to be the analogue, in the λ picture, of completing the system (2.6), from which we were able to express the Christoffel symbols and the curvature tensor of the metric. It follows from its definition, that F( p) is obtained from 1 by subtracting off its λ ( p)

singularities at p = v i . The importance of this function is that it describes the invariant properties of the reduction. For instance, if a reduction admits a function λ associated with it such that F( p) = 1, then the reduction is Galilean invariant. The case F( p) = p corresponds instead to the scaling invariance of the system. As seen in the example below, the function F, hence these invariances, are strongly related with the curvature of the Poisson bracket. Example 5.4. The 2−component Zakharov reduction is known to possess both Galilean and scaling invariance. Using the technique above, the former can be explained by saying that, for the function λ( p) given in Example 3.2, F( p) = 1 follows. Thus, we get δ(v i ) =

n ∂v i = 1, ∂λk k=1

312


where the Riemann invariants are the critical values of λ( p). For the scaling invariance, one can proceed as follows: define the function ϕ( p) = ln λ( p), where λ( p) is the same as before. This new function has the same critical points v i as λ, as well as the poles of λ( p), which play no role in the derivation of the reduction (see Remark 3). Hence, it is associated with the Zakharov reduction, and in this case F( p) = p. Thus δ(v i ) =

n ∂v i = vi , ∂ϕ k

(5.11)

k=1

where the ϕ i = ln λi are the natural Riemann invariants associated with ϕ( p), i.e. its critical values. It is elementary to show that (5.11) corresponds to the scaling invariance with respect to the λi . Remark 7. System (5.7) was considered for the first time in connection with a reduction of the Benney chain by Kokotov and Korotkin [14], in the particular case of the N −component Zakharov reduction, where F( p) = 1. In the next section, we will use the function F to describe the Christoffel symbols and the curvature in terms of λ. Nevertheless, the latter can be expressed directly in terms of F using the following residue formula Proposition 5.4. In terms of the function F, the nonvanishing components of the Riemann tensor are given by: F( p) F( p) 1 ij res dp = − dp, Ri j = − i 2 j 2 i p=vk ( p − v ) ( p − v ) 2πi i ∪ j ( p − v )2 ( p − v j )2 k=i, j

where i and j are two sufficiently small contours around p = v i and p = v j .

(5.12)

Proof. From (4.8) and Theorem 7 it follows immediately that ij Ri j

1 =− i (v − v j )2

∂F i ∂F j F(v i ) − F(v j ) . (v ) + (v ) − 2 ∂p ∂p vi − v j

(5.13)

It is easy to check the chain of identities: ∂F i ∂F j F(v i ) − F(v j ) (v ) + (v ) − 2 ∂p ∂p vi − v j F( p) F( p) d d + lim = lim p→v i dp ( p − v j )2 p→v j dp ( p − v i )2 d F( p) d F( p) i 2 j 2 + lim = lim ( p−v ) ( p−v ) ( p−v j )2 ( p−v i )2 p→v j dp ( p−v i )2 ( p−v j )2 p→v i dp F( p) F( p) = res dp + res dp . p=v i ( p − v i )2 ( p − v j )2 p=v j ( p − v i )2 ( p − v j )2 1 (v i − v j )2

In the last identity we used the fact that F( p) is regular at p = v i , for all i.


313

Remark 8. We notice that if λ ( p) is a rational function with respect to p, the quantity F( p) (p

− v i )2 ( p

− v j )2

dp

can be extended to a meromorphic differential on the Riemann sphere, with poles only at p = v i , p = v j . In this case, clearly, the curvature vanishes. The general case is much more involved, however. 5.2. Christoffel symbols and Curvature tensor. 5.2.1. Potential metric. We are now able to complete the description of the Poisson bracket associated with a reduction of the Benney chain, in the case where the metric is potential in the coordinates used. Proposition 5.5. The Christoffel symbols (4.7) of the potential metric (5.1) can be written, in terms of the function λ, as ij

k = 0, ij

i = j = k, 1 , (v i − v j )2

i = j,

(5.14b)

1 λ (v i ) , k λ (v ) (vk − v i )2

i = k,

(5.14c)

ji

i = −i =

kii = −

iii

(5.14a)

1 λ (v i ) 1 λ (v i )2 − = . 6 λ (v i ) 4 λ (v i )2

(5.14d)

The curvature tensor (4.8) of the potential metric (5.1) can then be written, in terms of the function λ( p) as λ (v i ) 1 1 1 3 λ (v j ) ij + + i Ri j = i − j 2 (v − v j )4 λ (v i ) λ (v j ) (v − v j )3 λ (v i )2 λ (v ) i 2 i j 2 1 λ (v ) 1 1 λ (v ) 1 λ (v ) 1 λ (v j ) (5.15) + i − + − (v − v j )2 4 λ (v i )3 6 λ (v i )2 4 λ (v j )3 6 λ (v j )2 1 1 + . λ (v k ) (v i − v k )2 (v j − v k )2 k=i, j

Proof. Starting from the definition (5.2) of F, and using (3.20), we can write

F(v i ) = −

1 1 λ (v i ) 1 − , 2 λ (v i )2 λ (v k ) v i − v k

(5.16)

k=i

∂F i 1 1 λ (v i )2 1 λ (v i ) 1 (v ) = − + . i i 3 i 2 k ∂p 4 λ (v ) 6 λ (v ) λ (v ) (v − v k )2 k=i

(5.17)

Then, by Theorem 7, and substituting the above expressions for F into (4.7) and (4.8), we obtain (5.14) and (5.15) respectively.

314


Remark 9. We note that (5.14d) is a constant multiple of the Schwarzian derivative of λ ( p), evaluated at p = v i . Recalling the expression (5.1) for the metric, from the proposition above we find that the whole Poisson operator associated with a reduction with symbol λ depends only on the critical points of λ and on the value of its second, third and fourth derivatives evaluated at these points. We have now all we need to write the nonlocal tail of the Hamiltonian structure associated with the metric gii = ∂i A0 . Proposition 5.6. The non-vanishing components of the Riemann tensor of the metric (5.1) admit the following quadratic expansion: 1 ij Ri j = − wi (λ)w j (λ)dλ, (5.18) 2πi C where C = C1 ∪ · · · ∪ Cn with Ci described as above, and the functions wi (λ) =

∂p ∂λ

( p(λ) − v i )2

,

are the generating functions of the symmetries (3.25). Consequently the nonlocal tail of the Hamiltonian structure associated with the metric gii = ∂i A0 is given by −1 d 1 j i i w (λ)λx w j (λ)λx dλ. 2πi C dx Proof. We prove the proposition showing that the integral in (5.18) is the same as the right hand side of (5.15). First, writing the integral 1 wi (λ)w j (λ)dλ 2πi C in terms of the variable p we obtain ij Ri j

1 =− 2πi

(p

1 λ ( p) − v i )2 ( p

− v j )2

dp = −

n k=1

⎛ res ⎝ (p

p=vk

1 λ ( p) − v i )2 ( p

⎞ − v j )2

Using (3.20), the integrand can be expanded, for k = 1. . . . , n, as

dp⎠ . (5.19)

1 ( p − v i )2 ( p − v j )2 1 1 λ (v k )2 1 1 λ (v k ) 1 λ (v k ) k × − + − ( p − v )+. . . . 2 λ (v k )2 4 λ (v k )3 6 λ (v k )2 λ (v k ) p − v k Thus, for k = i, j we get ⎛ res ⎝

p=vk

(p

1 λ ( p) − v i )2 ( p

⎞ − v j )2

dp ⎠ =

1 1 , λ (v k ) (v k − v i )2 (v k − v j )2


while

⎛

res ⎝

p=v i

(p

1 λ ( p) − v i )2 ( p

(p

1 λ ( p) − v i )2 ( p

res ⎝

p=v j

⎞

⎛

315

− v j )2

dp ⎠ =

1 1 λ (v i ) 3 + (v i − v j )4 λ (v i ) (v i − v j )3 λ (v i )2 1 λ (v i )2 1 1 λ (v i ) + i − , (v − v j )2 4 λ (v i )3 6 λ (v i )2

⎞

− v j )2

dp ⎠ =

1 1 λ (v j ) 3 + j j i 4 i 3 j (v − v ) λ (v ) (v − v ) λ (v j )2 1 λ (v j )2 1 1 λ (v j ) + j − . (v − v i )2 4 λ (v j )3 6 λ (v j )2

From these, formula (5.15) for the curvature tensor follows. The last statement of the proposition is a consequence of the general theory of Ferapontov.

Remark 10. Alternatively, one can prove the above result starting from the function F, namely deforming the integral 1 F( p) dp i 2πi i ∪ j ( p − v )2 ( p − v j )2 which is shown in Proposition 5.4 to be equal to the curvature, into (5.18). In order to do so, it is sufficient to verify the following identity: n ∂k A0 1 ∂λ k=1 p−v k 1 1 ∂p − dp = dp, 2πi i ∪ j ( p − v i )2 ( p − v j )2 2πi −(i ∪ j ) ( p − v i )2 ( p − v j )2 that can be proved by straightforward computation. Indeed, since the left-hand side is the sum of residues at p = v i and p = v j of a function having a pole of order 3 at p = v i , v j we obtain n 1 d2 ( p − v i )∂k A0 ∂k A 0 = l.h.s = − lim + lim ( p − v j )2 ( p − v k ) (v i − v j )2 p→v i p→v j 2 dp 2 k=1 k=i, j 1 1 2 2 + + × − i (v i − v j )(v i − v k ) (v i − v k )2 (v − v j )(v j − v k ) (v j − v k )2 ∂k A 0 = . (v i − v k )2 (v j − v k )2 k=i, j

On the other hand the right hand side is the sum of residues at p = v k , for k = i, j of a function having simple poles at p = v k : r.h.s. =

k=i, j

1 ( p − v k ) ∂λ

lim

p→v k

∂p

(p

− v i )2 ( p

− v j )2

=

k=i, j

(v i

Recalling the results above, we have the following

∂k A 0 k − v )2 (v j

− v k )2

= l.h.s.

316


Theorem 8 The reduction of the Benney chain associated with the function λ( p, λ1 , . . . , λn ) is Hamiltonian with the Hamiltonian structure −1 ∂ p j ∂p i d 1 ij k ∂λ λx ∂λ λx ij i ij d + k λ x − = λ (v )δ dλ, dx 2πi C ( p(λ) − v i )2 d x ( p(λ) − v j )2 (5.20) where j

ij

k λkx =

λix − λx (v i − v j )2

kii

λkx

=

i = j,

1 λ (v i ) 1 λ (v i )2 − 6 λ (v i ) 4 λ (v i )2

λix −

λ (v i ) λkx λ (v k ) (v i − v k )2 k=i

and C = C1 ∪ · · · ∪ Cn . Here, the v i are the critical points of λ( p), and the λi its critical ij values. In these coordinates, the metric gi j = δ i is a potential metric. λ (v )

5.2.2. The general case. As we pointed out previously, any metric (4.1) associated with a reduction can be put in potential form, after a suitable change of the Riemann invariants. However, it is often convenient to write the expression of the Poisson operators generated by these metrics in terms of the Riemann invariants selected by λ. Thus, we consider the metrics 1 ∂i A0 gii = = , (5.21) i i ϕi (λ ) ϕi (λ )λ (v i ) where ϕi = ϕi (λi ) are arbitrary functions, and we proceed as before. Proposition 5.7. The Christoffel symbols appearing in the Hamiltonian structure are given by ij

k = 0,

i = j = k,

ϕj ij , i = i (v − v j )2 ϕi ij , j = − i (v − v j )2

i = j, i = j,

ϕi λ (v i ) = − k , i = k, λ (v ) (v k − v i )2 1 λ (v i ) 1 λ (v i )2 1 ϕi iii = ϕi − . + 6 λ (v i ) 4 λ (v i )2 2 ∂i A0

kii

Here we denote dϕ i (λ ). dλi The nonlocal tail appearing in the Hamiltonian structure is then given by −1 ∂ p j n ∂p i d 1 ∂λ λx ∂λ λx ϕk (λ) dλ. i 2 2πi dx ( p(λ) − v j )2 Ck ( p(λ) − v )

ϕi =

k=1

(5.22)


317

ij

Proof. The proof of the formula for the k is a straightforward computation. Let us prove the second statement. Since nothing new is involved in such computations we will skip the details. For the metric (5.21), the non vanishing components of the curvature tensor are 2 3 ij 0 0 i j ϕ − ϕ Ri j = i ∂ A + ϕ ∂ A ∂ v − ϕ ∂ v i i j j i i j j (v − v j )4 (v i − v j )3 1 + i ϕi ∂i ln ∂i A0 + ϕ j ∂ j ln ∂ j A0 j 2 (v − v ) +

k=i, j

ϕk ∂k A0 1 ϕi + ϕ j + . i k 2 j k 2 (v − v ) (v − v ) 2 (v i − v j )2

(5.23)

Expression (5.23) can be written, in terms of λ, as ϕj ϕi λ (v i ) λ (v j ) 1 3 ij Ri j = i + + i ϕi i 2 − ϕi j 2 (v − v j )4 λ (v i ) λ (v j ) (v − v j )3 λ (v ) λ (v ) 1 λ (v i )2 1 λ (v j )2 1 λ (v i ) 1 λ (v j ) 1 − − + i ϕi + ϕj (v − v j )2 4 λ (v i )3 6 λ (v i )2 4 λ (v j )3 6 λ (v j )2 +

k=i, j

ϕk 1 1 ϕi + ϕ j + . i k 2 j k 2 k 2 (v i − v j )2 λ (v ) (v − v ) (v − v )

(5.24)

The equivalence between (5.22) and the right-hand side of (5.24) can be obtained by rewriting the integrals above in the p−plane, n ϕk (λ( p)) 1 1 λ ( p) dp, i 2 j 2 2πi k ( p − v ) ( p − v )

(5.25)

k=1

and using the same arguments of the main theorem, except that for every k, in the integral around k we have to consider also the contribution of the function ϕk (λ( p)) which expands, at p = v k , as

ϕ ϕk (λ( p)) = ϕk + k λ (v k ) ( p − v k )2 + . . . . 2

It follows from the above that we have Theorem 9. The reduction of the Benney chain associated with the function λ( p, λ1 , . . . , λn ) is Hamiltonian with the family of Hamiltonian structures

i j = ϕi λ (v i )δ i j

d ij + k λkx dx

−1 ∂ p j n ∂p i d 1 ∂λ λx ∂λ λx − ϕk (λ) dλ, i 2 2πi dx ( p(λ) − v j )2 Ck ( p(λ) − v ) k=1

(5.26)

318


with j

ij

k λkx =

ϕ j λix − ϕi λx (v i − v j )2

kii

λkx

= ϕi

i = j,

1 λ (v i ) 1 λ (v i )2 − 6 λ (v i ) 4 λ (v i )2

λ (v i ) ϕi λk 1 x , λix + ϕi λ (v i ) λix − 2 λ (v k ) (v i − v k )2 k=i

where ϕ1 , . . . , ϕn are arbitrary functions of a single variable. Here, the v i are the critical points of λ, the coordinates λi the corresponding critical values, and the Ck are the contours defined above. 6. Finite Nonlocal Tail: Some Examples In the expression (5.26) for the Poisson operator, the components of the curvature are expressed as integrals of functions around suitable contours in a complex domain. A natural question to ask is whether this integral can be reduced to a finite sum, and we will show now some examples where this is possible. For simplicity, we will consider the case when ϕ1 (λ) = · · · = ϕn (λ) = λk , for k ∈ Z. In this case the curvature can be expressed as 2 ∂p λk ∂λ 1 ij dλ, k ∈ Z. Ri j = − 2πi C ( p(λ) − v i )2 ( p(λ) − v j )2 Essentially, the finite expansion appears whenever it is possible to substitute the contour C with a contour around λ = ∞ and a finite number of other marked points. We illustrate this special situation in two simple examples. 6.1. 2-component Zakharov reduction. In this case (see Examples 2.1, 3.2), since λ is a single-valued rational function of p, it is convenient to work in the p-plane. In order to calculate the curvature, the non-vanishing components of the Riemann tensor are given by ⎞ ⎛ 2 λ( p)k 1 λ ( p) 12 R12 =− res ⎝ dp ⎠ , k ∈ Z. ( p − v1 )2 ( p − v2 )2 p=v i i=1

The abelian differential λ( p)k

1 λ ( p)

( p − v1 )2 ( p − v2 )2

dp

has poles at the points p = v1 , p = v2 , as well as: if k > 2, p = ∞,

p=

A1 A0

(poles of λ)


319

if k < 0, p = s1 =

1 A1 + (A21 − 4 A30 )1/2 1 A1 − (A21 − 4 A30 )1/2 , p = s2 = 2 A0 2 A0

(zeros of λ),

while for k = 0, 1, 2 there are no other poles. Since the sum of the residues of an abelian differential on a compact Riemann surface is zero, we can substitute the sum of residues at p = v1 , v2 with, respectively – zero if k = 0, 1, 2, – minus the sum of residues at p = ∞, p = AA01 – minus the sum of residues at p = s1 , p = s2 ,

if k > 2, if k < 0.

Summarizing, we have ij

Ri j = 0, ⎛

k = 0, 1, 2,

⎞

ij Ri j = ⎝ res + res ⎠ p=∞

1

p= A0

(p

λ( p)k 1 λ ( p) − v i )2 ( p − v j )2

k > 2,

dp,

A

ij Ri j

=

res + res

p=s1

p=s2

λ( p)k

1 λ ( p)

( p − v i )2 ( p − v j )2

k < 0.

dp,

Moreover, as a counterpart in the λ-plane of the above formulae we have ij

Ri j = 0,

k = 0, 1, 2,

ij Ri j = 2 res w 1 (λ)w 2 (λ)λk dλ , λ=∞ ij Ri j = 2 res w 1 (λ)w 2 (λ)λk dλ ,

k > 2, k < 0,

λ=0

and this shows that the residues have to be computed around marked points, which do not depend on the dynamics of the reduction. Expanding w1 (λ) and w 2 (λ) near λ = ∞, we get w (λ) = − 1

∞ k w1 k=1

k λk+1

3

3(2 A30 + A21 + 2 A1 A02 ) 1 1 2 v1 =− 2 − 3 − λ λ λ4 A20 9

3

4(A31 + 6A1 A30 + 3A02 + 3A21 A02 ) 1 − + ... λ5 A30 w (λ) = − 2

∞ k w2 k=1

k λk+1

3

3 (2 A30 + A21 − 2 A1 A02 ) 1 1 2 v2 =− 2 − 3 − λ λ λ4 A20 9

3

4 (A31 + 6A1 A30 − 3A02 − 3A21 A02 ) 1 − + ..., λ5 A30

320


and near λ = 0,

2 A20 (− A21 − 4 A30 + A1 ) 1 z −h λh = − w 1 (λ) = 3 2 h=0 A21 − 4 A30 A1 − A21 − 4 A30 + 2 A02 9 8A30 A21 − 4 A30 (A30 − A21 ) + A31 − 3A1 A30 + 2 A02 − λ + ... 3 ∞

(A21 − 4 A30 ) 2

3

A1 −

3

A1 −

3

A21 − 4 A30 + 2 A02 ∞ 2 A20 (− A21 − 4 A30 + A1 ) 2 w 2 (λ) = z −h λh = − 3 2 2 3 2 h=0 A1 − A1 − 4 A0 − 2 A0 A21 − 4 A30 9 3 2 3 3 2 3 3 2 8A0 A1 − 4 A0 (A0 − A1 ) + A1 − 3A1 A0 − 2 A0 − λ + ..., 3 (A21 − 4 A30 ) 2

3

A21 − 4 A30 − 2 A02

and taking into account that the coefficients of the expansion are characteristic velocities of symmetries, we easily obtain the quadratic expansion of the Riemann tensor. For k > 2 we have 12 R12 wi1 w 2j + w 1j wi2 , =− i+ j=k−1

while for k < 0, we obtain 12 R12 =−

z i1 z 2j + z 1j z i2 ,

i+ j=k+1

which can be put in the canonical form (1.12) after a linear change of basis of the symmetries. The expressions of these expansions in the Riemann invariants can be found by using formulae given in Example 2.1. We note that for this example, the cases k = 0, 1, 2 lead to three local Hamiltonian structures, in contrast to the general case. 6.2. Dispersionless Boussinesq reduction. The case of the dispersionless Boussinesq reduction can be treated in a similar way. From Example 3.3, we will consider a function λ which is polynomial in p, λ = p 3 + 3A0 p + 3A1 , thus meromorphic on the Riemann sphere. The choice of a different normalisation reflects in the expansions below, where we have to consider an expansion in the local 1 parameter t = λ− 3 . For simplicity let us consider only the case k ≥ 0. We observe that, apart from the poles at p = v1 and p = v2 , we have only an additional pole at


321

infinity (starting from k = 2). Following the same procedure used in the Zakharov case we obtain ij

Ri j = 0,

k = 0, 1,

Ri j = −3 res w 1 (t)w 2 (t) t −(3k+4) dt , ij

k > 2.

t=0

The expansions of w1 (t) and w 2 (t) near t = 0 are given by w 1 (t) =

∞

1 3 k wk1 t k+4 = t 4 + 2 (−A0 ) 2 t 5 + 4 A1 − (−A0 ) 2 t 7

k=0

w 2 (t) =

∞

1 + 5 2 A1 (−A0 ) 2 − A20 t 8 + . . . , 1 3 k wk2 t k+4 = t 4 − 2 (−A0 ) 2 t 5 + 4 A1 + (−A0 ) 2 t 7

k=0

1 + 5 −2 A1 (−A0 ) 2 − A20 t 8 + . . . .

From these formulas we immediately get the quadratic expansion of the Riemann tensor: 12 k = 0 : R12 = 0, 12 k = 1 : R12 = 0, 12 k = 2 : R12 = 3(v 1 + v 2 ) = 0.

More generally, we have 12 =− R12

3 2

1 2 2 (w3i w3 j + w31 j w3i )

k > 2.

i+ j=k−1

The expression in the Riemann invariants can be obtained from Example 3.3. Again we see that this approach finds both local and nonlocal Hamiltonian structures. Acknowledgement. We would like to thank the ESF grant MISGAM 1414, for its support of Paolo Lorenzoni’s visit to Imperial. We are also grateful to the FP6 programme of the European Commission for support of this work through the ENIGMA network, and particularly for their support of Andrea Raimondo, who also received an EPSRC DTA. We would like to thank Maxim Pavlov for valuable discussions, and the integrable system groups at Milano Bicocca and Loughborough universities for their hospitality to Andrea Raimondo.

References 1. Baldwin, S., Gibbons, J.: Hyperelliptic reduction of the Benney moment equations. J. Phys. A 36(31), 8393–8417 (2003) 2. Baldwin, S., Gibbons, J.: Higher genus hyperelliptic reductions of the Benney equations. J. Phys. A 37(20), 5341–5354 (2004) 3. Baldwin, S., Gibbons, J.: Genus 4 trigonal reduction of the Benney equations. J. Phys. A 39(14), 3607–3639 (2006)

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4. Benney, D.J.: Some properties of long nonlinear waves. Stud. Appl. Math. 52, 45–50 (1973) 5. Dubrovin, B., Liu, S.Q., Zhang, Y.: Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures. http://arXiv.org/abs/0710.3115v1[math.DG], 2007 6. Dubrovin, B.A., Novikov, S.P.: Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method. Dokl. Akad. Nauk SSSR 270(4), 781–785 (1983) 7. Dubrovin, B.A., Novikov, S.P.: Poisson brackets of hydrodynamic type. Dokl. Akad. Nauk SSSR 279(2), 294–297 (1984) 8. Duren, P.L.: Univalent functions. New York: Springer-Verlag, 1983 9. Ferapontov, E.V.: Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type. Funkts. Anal. i Pril., 25(3), 37–49, 95 (1991) 10. Gibbons, J., Kodama, Y.: Solving dispersionless Lax equations. In: Singular limits of dispersive waves (Lyon, 1991), vol. 320 of NATO Adv. Sci. Inst. Ser. B Phys., New York: Plenum 1994, pp. 61–66 11. Gibbons, J., Tsarev, S.P.: Reductions of the Benney equations. Phys. Lett. A 211(1), 19–24 (1996) 12. Gibbons, J., Tsarev, S.P.: Conformal maps and reductions of the Benney equations. Phys. Lett. A 258(4–6), 263–271 (1999) 13. Kodama, Y., Gibbons, J.: Integrability of the dispersionless KP hierarchy. In: Nonlinear world, vol. 1 (Kiev, 1989), River Edge, NJ: World Sci. Publ., 1990, pp. 166–180 14. Kokotov, A., Korotkin, D.: A new hierarchy of integrable systems associated to Hurwitz spaces. http:// arXiv.org/math-ph/0112051v3, 2001 15. Konopel chenko, B., Martines Alonso, L., Medina, E.: Quasiconformal mappings and solutions of the dispersionless KP hierarchy. Teoret. Mat. Fiz. 133(2), 247–258 (2002) 16. Kupershmidt, B.A., Manin, Yu.I.: Long wave equations with a free surface. I. Conservation laws and solutions. Funkt. Anal. i Pril. 11(3), 31–42 (1977) 17. Kupershmidt, B.A., Manin, Yu.I.: Long wave equations with a free surface. II. The Hamiltonian structure and the higher equations. Funkt. Anal. i Pril. 12(1), 25–37 (1978) 18. Lebedev, D.R., Manin, Yu.I.: Conservation laws and Lax representation of Benney’s long wave equations. Phys. Lett. A 74(3–4), 154–156 (1979) 19. Mokhov, O.I., Ferapontov, E.V.: Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature. Usp Mat. Nauk 45(3), 191–192 (1990) 20. Tsarëv, S.P.: Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Dokl. Akad. Nauk SSSR 282(3), 534–537 (1985) 21. Yu, L., Gibbons, J.: The initial value problem for reductions of the Benney equations. Inverse Problems 16(3), 605–618 (2000) 22. Zakharov, V.E.: Benney equations and quasiclassical approximation in the inverse problem. Funkt. Anal. i Pril. 14, 15–24 (1980) Communicated by L. Takhtajan


Communications in


Smooth Approximations and Exact Solutions of the 3D Steady Axisymmetric Euler Equations Quansen Jiu1, , Zhouping Xin2,3, 1 School of Mathematical Sciences, Capital Normal University,

Beijing 100048, PRC. E-mail: [email protected]

2 IMS and Department of Mathematics, The Chinese University of Hong Kong,

Shatin, N.T., Hong Kong. E-mail: [email protected]

3 Center for Nonlinear Studies, Northwest University, Xi’an 710069, PRC

Received: 18 February 2008 / Accepted: 22 August 2008 Published online: 20 November 2008 – © Springer-Verlag 2008

Abstract: In this paper, we prove that a class of C 1 -smooth approximate solutions {u ε , p ε } to the 3D steady axisymmetric Euler equations will converge strongly to 0 2 (R 3 ). The main assumptions are that the approximate solutions have uniformly in L loc finite energy and approach a constant state at far fields. We also show a Liouville type theorem that there are no non-trivial C 1 -smooth exact solutions with finite energy and uniform constant state at far fields. 1. Introduction The three-dimensional (3D) incompressible steady Euler equations in R 3 are (u · ∇)u + ∇ p = 0, x ∈ R 3 , div u = 0.

(1.1)

Here u = (u 1 (x), u 2 (x), u 3 (x)) represents the velocity field and p = p(x) is the pressure. By an axisymmetric solution of (1.1), we mean that, in the cylindrical coordinate system, the unknown functions u(x) and p(x) do not depend on θ -variable, that is, u(x) = u r (r, z)er + u θ (r, z)eθ + u z (r, z)ez , p(x) = p(r, z), where er = (cos θ, sin θ, 0), eθ = (− sin θ, cos θ, 0), ez = (0, 0, 1) The research is partially supported by National Natural Sciences Foundation of China (No. 10871133 & No. 10771177). The research is partially supported by Zheng Ge Ru Funds, Hong Kong RGC Emarked Research Grant CUHK4028/04P and CUHK4040/06P, RGC Central Allocation Grant CA 05/06. SC01, and a grant from Northwest University, Xi’an, PRC.

324

Q. Jiu, Z. Xin

form the standard orthogonal bases in the cylindrical coordinate system. Furthermore, when u θ ≡ 0, which means that the axisymmetric flow has no swirls, the corresponding 3-D steady axisymetric Euler equations can be written as u r ∂r u r + u z ∂z u r + ∂r p = 0, (1.2) u r ∂r u z + u z ∂z u z + ∂z p = 0. And the incompressibility condition becomes ∂r (r u r ) + ∂z (r u z ) = 0.

(1.3)

In this case, the vorticity of the velocity is given by ω = ∇ × u = ωθ eθ with ωθ = ∂z u r − ∂r u z . When the initial data is a vortex-sheets data, the 2D Euler equations have global (in time) weak solutions when the initial vorticity has a distinguished sign (see [2,7,16– 18,21]) or has a changing sign with reflection symmetry (see [14,15]). However, the global existence of weak solutions for both general 2D and 3D Euler equations for general vortex-sheets initial data is still an outstanding open problem. In particular, for three-dimensional unsteady axisymmetric flows without swirls, this problem remains to be solved even in the case that the initial vorticity is of one sign. It was shown in [3] that, for the 3D unsteady axisymmetric Euler equations without swirls, a sequence of approximate solutions generated by smoothing the initial data converges either strongly 2 (R 3 × (0, ∞)) or weakly in L 2 (R 3 × (0, ∞)) to a limit which is not a classical in L loc loc weak solution to the Euler equations under the additional assumption that the initial vorticity has a distinguished sign. In other words, there is no concentration-cancellation occurring for one-sign axisymmetric flows without swirls which is in sharp contrast to the 2-D theory (see [5]). The authors proved in [12] that the approximate solutions, 2 (R 3 )) progenerated by smoothing the initial data, converge strongly in L 2 ([0, T ]; L loc vided that they have strong convergence in the region away from the symmetry axis. This means that if there would appear singularity or energy lost in the process of limit for the approximate solutions, it then must happen in the region away from the symmetry axis. It is noted that there is no restriction on the signs of initial vorticity in [12]. The convergence properties of the viscous approximations were studied in [11]. When the initial vorticity has stronger assumptions (comparing with the vortex-sheets initial data), the global existence of weak solutions was proved in [1] and the references therein. For the two-dimensional steady Euler equations, DiPerna and Majda proved that, even though there exist approximate solutions with energy concentration, the weak limit of any approximate solutions is a weak solution, by using the shielding method (see [4]). That is, concentration-cancellation occurs in this case. The reader may refer to [6] for a more concise proof. However, for the three-dimensional steady equations, even for the axisymmetric case, it is not known whether or not there exist approximate solutions with energy concentration for the three-dimensional steady Euler equations. Recently, the authors studied some convergence properties of the approximate solutions of the 3D steady Euler equations (1.1) and the 3D steady axisymmetric Euler equations without swirls (1.2)–(1.3) (see [13]). In particular, in [13] the authors obtained a criterion for strong convergence for approximate solutions by establishing a relation between the energy distributions of the weak limit and the defect measure of the approximate solutions.

Approximations and Solutions of 3D Steady Axisymmetric Euler Equations

325

On the other hand, the existence of solutions of the 3D steady axisymmetric Euler equations without swirls (1.2)–(1.3) has been widely studied (see [8,9,19,20]). In particular, the vortex rings, which are steady, axisymmetric solutions without swirls of Eqs. (1.1), propagating with constant speed in the z-direction, has been extensively and systematically investigated, based mainly on the variational approaches (see [8,9,19] and references therein). In this paper, we are mainly concerned with the strong convergence of C 1 -smooth approximations and the existence of C 1 -smooth exact solutions with finite energy and uniform constant states at the far field of the 3D steady axisymmetric Euler equations. We will prove that any C 1 -approximations {u ε , p ε } to the 3D steady axisymmetric 2 (R 3 ) under appropriate assumptions Euler equations will converge strongly to 0 in L loc on approximate solutions and error terms (see Theorem 5.2). The main assumptions on approximate solutions are that the energy is finite and |u ε | → 0 and p ε → p0 as r 2 + z 2 → ∞, where p0 is a constant. These kinds of approximate solutions correspond to 3D steady vortex-sheets. At the end of the paper, we obtain a Liouville type theorem that there will be no non-trivial C 1 exact solutions with finite energy to the 3D steady axisymmetric Euler equations, which satisfy that |u| → 0 and p → p0 as r 2 + z 2 → ∞. The Liouville theorem can be seen as a direct result of one of our main results (Theorem 5.2) and can also be proved directly. Two proofs of the Liouville theorem are presented at the end of the paper. It should be noted that contrary to the 3D steady axisymmetric Euler equations, there exist non-trivial smooth exact solutions with finite energy and there exist smooth approximate solutions with finite energy and energy concentrations in the limit process to the 2D steady Euler equations (see [4]). Also, using the spherical vortex ring given in [10], an example of approximate solutions of the 3D steady axi2 (R 3 ) was constructed symmetric Euler equations which converge strongly to 0 in L loc in [13]. Our approach is mainly based on a deliberate construction of test functions and making full use of structures of the axisymmetric Euler equations. Let φr (r, z), φz (r, z) ∈ C0∞ ( H¯ ) be two usual test functions which have compact support in [0, ∞) × (−∞, ∞) and are divergence-free, that is, ∂r (r φr ) + ∂z (r φz ) = 0 or r ∂r φr + φr + r ∂z φz = 0. Here H = {(r, z)|(r, z) ∈ (0, ∞) × (−∞, ∞)} represents the (r, z)−plane. Then, it follows from (1.2)–(1.3) that (u r )2 φr r dr dz = [(u r )2 − (u z )2 ]∂z φz r dr dz r H H + u r u z (∂r φz + ∂z φr )r dr dz. (1.4) H

In particular, to study the convergence of the approximate solutions, (1.4) should be written as (u rε )2 φr r dr dz = [(u rε )2 − (u εz )2 ]∂z φz r dr dz r H H + u rε u εz (∂r φz + ∂z φr )r dr dz + h(ε), (1.5) H

where h(ε) is some error term satisfying h(ε) → 0 as ε → 0. In the limit ε → 0 (or its subsequence), there will appear more terms in the limit equation of (1.5), which corresponds to the defect measures of u ε . Denote by u the weak limit of u ε in L 2 (R 3 ). u2 A key point of this paper is to prove that {r ≥r0 >0} r 2r r dr dz = 0 for any r0 > 0, where

326

Q. Jiu, Z. Xin

{r ≥ r0 > 0} = {(r, z)|(r, z) ∈ (0, ∞) × (−∞, ∞), r ≥ r0 > 0} is a domain away from the symmetry axis in H . This can be obtained formally if we could choose the test functions as φr = 1 and φz = 1 in (1.4) or in the limit equation of (1.5). However, the test functions φr = 1 and φz = 1 do not satisfy the divergence-free condition and it is illegitimate to take the limit in (1.5) with φr = 1 and φz = 1. Thus, we should construct a new class of test functions which are divergence-free, decay at the far field and approximate the test functions φr = 1, φz = 1 in the appropriate sense such that the terms on the right-hand side of (1.4) or the limit of (1.5) will tend to zero in the approximation of the test functions. However, it is noted that the test functions denoted by ϕr , ϕz we construct in this paper do not belong to C0∞ ( H¯ ) which is required in the usual way. Especially, the test functions ϕr will have singularity o( r1 ) near the symmetry axis. Due to this singularity, new difficulties will arise in our subsequent and rigorous analysis. First, in integrations by parts, there will appear the boundary term of the pressure, which is H p(0, z)∂z ϕz r dr dz. Fortunately, by applying the special test functions we prove that the sign of this term is unchanged. Second, we need to investigate the properties of symmetry axis more carefully. Precisely, we will obtain the estimate u r 1nearur the 2 R 3 1+x 2 ( r ) d x ≤ C with C an absolute constant. In the unsteady case, this estimate is 3

naturally satisfied for the vortex-sheets initial data (see [1,11]). In steady case, however, it seems to be a nontrivial estimate. It is noted that other test functions such as those used in [12] and [13] (see also Sect. 2 of this paper) can provide us with some balance relations between the energy distributions of the velocity and the corresponding defect measures (see Theorems 2.1, 2.3 in Sect. 2) but can not yield the desired result of the vanishing of the right-hand side of (1.4). The Liouville theorem, which says that there are no non-trivial C 1 -smooth exact solutions with finite energy and uniform constant states at far fields of the 3D steady axisymmetric Euler equations, is proved at the end of the paper. It can be seen as a direct consequence of our results on the strong convergence of approximate solutions. And it can also be proved in a direct way, avoiding the technical construction of the test functions. It should be remarked that this direct method can not be applied to investigate the strong convergence of approximate solutions since one should take the limit first in the finite Radon space on both sides of (1.5) in order to study this problem. And in the process of the limit, we should use suitable test functions. The rest of this paper is organized as follows. In Sect. 2, we review a criterion for the strong convergence of approximate solutions for the 3D steady Euler equations, which has been obtained in [13]. In Sect. 3, we construct some special test functions which will be needed later. It should be noted that these test functions do not satisfy the conditions required in the usual definition of the weak solutions but they possess some special features which are crucial in the analysis of the strong convergence of the approximate solutions. In Sect. 4, we prove the strong convergence of u ε1 and u ε2 in the region away from the symmetry axis. In Sect. 5, we first prove the strong convergence of u ε1 and u ε2 2 (R 3 ), then applying the criterion established in [13] for the strong convergence in L loc of approximate solutions (see also Sect. 2), we obtain the strong convergence of u ε in 2 (R 3 ). Some appropriate conditions are imposed on the approximate solutions and L loc error terms. In the last, we prove the Liouville theorem which says that there are no non-trivial C 1 -smooth exact solutions with finite energy and uniform constant states at the far field to the 3D steady axisymmetric Euler equations. It can be seen as a direct result of the strong convergence of approximate solutions and it can also be proved in a direct way, avoiding the technical construction of the test functions.


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2. A Criterion on the Strong Convergence In this section, we give a brief review of the results in [13] on the strong convergence of approximate solutions to 3D steady Euler equations. Similar to the unsteady case, approximate solutions for the 3D steady Euler equations (1.1) can be defined in the usual way. Definition 2.1 (General Case). Smooth vector-valued functions {u ε } (ε ∈ J a parameter) are called approximate solutions of (1.1) if the following conditions are satisfied: (i) u ε (x) is uniformly bounded in L 2 (R 3 ) and divergence free (div u ε = 0); (ii) For any (x) = (1 , 2 , 3 ) ∈ C0∞ (R 3 ) satisfying div = 0, it holds that u ε · (u ε · ∇)d x = h(ε) (2.1) R3

with h(ε) → 0 as ε → 0. In particular, when the approximate solutions are axisymmetric, one can obtain approximate solutions for the 3D steady axisymmetric Euler equations (1.2)–(1.3). Definition 2.2 (Axisymmetric Case). Smooth vector-valued functions {u ε } (ε ∈ J a parameter) are called approximate solutions of the equations (1.2)–(1.3) if the following conditions are satisfied: (i) u ε (x) is uniformly bounded in L 2 (R 3 ) and divergence free (div u ε = 0); (ii) u ε = u rε er + u εz ez ; (iii) ωε = ∇ × u ε = ωθε eθ ; (iv) For φr (r, z), φz (r, z) ∈ C0∞ ( H¯ ), satisfying

one has

∂r (r φr ) + ∂z (r φz ) = 0,

(2.2)

[(u rε )2 ∂r φr + (u εz )2 ∂z φz ]r dr dz =− u rε u εz (∂r φz + ∂z φr )r dr dz + h(ε)

(2.3)

H

H

with h(ε) → 0 as ε → 0. Here H = {(r, z)|(r, z) ∈ (0, ∞) × (−∞, ∞)} represents the (r, z)−plane. Formally, multiplying r φr and r φz on both sides of (1.2)1 and (1.2)2 respectively, integrating the resulting equations on (0, ∞) × (−∞, ∞) with respect to r and z and summing over them, one obtains (2.3) with h(ε) = 0. It should be noted that the assumption that the approximate solutions u ε in Definitions 1.1–1.2 are smooth is only made for convenience and can be dispensed with. For a sequence of approximate solutions u ε = (u ε1 , u ε2 , u ε3 ) as in Definition 2.2, which is expressed by u ε = (u rε , 0, u εz ) in the cylindrical coordinates systems, there exists a subsequence of u ε , still denoted by itself, converging weakly in L 2 (R 3 ) and in L 2 (H ; r dr dz). Precisely, as ε → 0+ , one has u ε1 u 1 , u ε2 u 2 , u ε3 u 3

(2.4)

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weakly in L 2 (R 3 ), and, in the cylindrical coordinates, u rε u r , u εz u z

(2.5)

weakly in L 2 (H ; r dr dz). In what follows, a subsequence of approximate solutions will always be denoted by itself for convenience unless stated otherwise. Since (u ε (x))2 are uniformly bounded in L 1 (R 3 ), there exists a subsequence of ε (u (x))2 which converge weakly to a Radon measure. More precisely, as ε → 0+ , (u ε1 )2 u 21 + µ1 , (u ε2 )2 u 22 + µ2 , (u ε3 )2 u 23 + µ3

(2.6)

weakly in M(R 3 ) which is the space of finite Radon measures. Here µi ≥ 0(i = 1, 2, 3) is the defect measure of (u iε )2 (i = 1, 2, 3) respectively. The total variation of µi (i = 1, 2, 3), denoted by |µi |(i = 1, 2, 3), is finite. A criterion on strong convergence of approximate solutions to the 3D steady axisymmetric Euler equations is stated as (see [13]) Theorem 2.1. For any approximate solutions {u ε } defined as in Definition 2.2, there exists a subsequence of the approximate solutions satisfying (2.4)–(2.6). Moreover, it holds that 1 1 u 23 d x − (u 21 + u 22 )d x + |µ3 | − (|µ1 | + |µ2 |) = 0. (2.7) 2 R3 2 R3 2 (R 3 ), then Consequently, if u ε → u strongly in L loc 1 2 u3d x − (u 2 + u 22 )d x = 0. 2 R3 1 R3

(2.8)

Proof. We give a sketch of proof here and refer to [13] for more details. It suffices to prove (2.7). We choose the test functions in (2.3) as z − z0 z − z0 z − z0 1 r r χ+ ( )[χ ( )+ χ( )], 2 η η η η r r r z − z0 χ+ ( )](z − z 0 )χ ( ) φz = −[χ+ ( ) + η 2η η η φr =

(2.9)

for any η > 0 and any fixed z 0 ∈ R, where χ (s) and χ+ (s) are the same as (3.20) and (3.21) respectively. Then direct calculations lead to φr 1 z − z0 z − z0 z − z0 r = χ+ ( )[χ ( )+ χ( )], r 2 η η η η r z − z0 z − z0 z − z0 1 r r )+ χ( )], ∂r φr = (χ+ ( ) + χ+ ( ))[χ ( 2 η η η η η η r z − z0 z − z0 z − z0 r r χ+ ( )][χ ( )+ χ( )], ∂z φz = −[χ+ ( ) + η 2η η η η η z − z 0 z − z 0 1 r 2 z − z0 )+ )], ∂z φr = r χ+ ( )[ χ ( χ ( 2 η η η η2 η r 3 r r z − z0 ). ∂r φz = −[ χ+ ( ) + 2 χ+ ( )](z − z 0 )χ ( 2η η 2η η η

(2.10)


Letting ε → 0+ in (2.3), one can obtain 1 2 2 u 23 ∂z φz d x { (u + u 2 )∂r φr d x + 2π R 3 1 R3 + ∂r φr d(µ1 + µ2 ) + ∂z φz dµ3 } R3 R3 ≤ (u r2 + u 2z )(|∂z φr | + |∂r φz |)r dr dz H + (|∂z φr | + |∂r φz |)d(µ1 + µ2 + µ3 ).

329

(2.11)

H

Substituting (2.10) into (2.11), and then letting η → ∞ on both sides of (2.11), one has 1 1 u 23 d x − (u 21 + u 22 )d x + |µ3 | − (|µ1 | + |µ2 |) = 0. 2 R3 2 R3 Equation (2.7) thus follows. The proof of the theorem is completed.

If we choose the test functions in (2.1) as x3 x3 r x3 1 = α1 x1 χ+ ( )[χ ( ) + χ ( )], η η η η r x3 x3 x3 2 = α2 x2 χ+ ( )[χ ( ) + χ ( )], η η η η 2 α1 x1 + α2 x22 r x3 r χ+ ( )], 3 = x3 χ ( )[α3 χ+ ( ) − η η ηr η

(2.12)

3 where αi ∈ R(i = 1, 2, 3) satisfying i=1 αi = 0, and χ (s) and χ+ (s) are defined as in (3.20) and (3.21) respectively, then a similar approach gives Theorem 2.2. For any approximate solutions {u ε } defined as in Definition 2.1, there exists a subsequence of the approximate solutions satisfying (2.4) and (2.6). Moreover, we have 3

αi (E i + |µi |) = 0,

(2.13)

i=1

where, for i = 1, 2, or 3, E i = R 3 u i2 d x is the energy of the ith component of the limit, 3 µi is same as in (2.6), and αi is a real number satisfying i=1 αi = 0. Consequently, 2 (R 3 ), then if u ε → u strongly in L loc E1 = E2 = E3.

(2.14)

Theorem 2.3. Suppose that a vector function u = (u 1 , u 2 , u 3 ) is a weak solution of (1.1) in the sense that u · (u · ∇)d x = 0 (2.15) R3

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for any = (x) ∈ C0∞ (R 3 ) satisfying div = 0. Then E1 = E2 = E3,

(2.16)

where E i (i = 1, 2, 3) are the same as in Theorem 2.2. Therefore, suppose that u ε are exact solutions of (1.1) in the sense that (2.1) holds with h(ε) = 0. Then, where E iε =

E 1ε = E 2ε = E 3ε , ε 2 R 3 (u i ) d x(i

(2.17)

= 1, 2, 3).

The detail of the proofs of Theorem 2.2 and Theorem 2.3 is referred to [13] and omitted here. It should be remarked that Theorem 2.2 and Theorem 2.3 hold for any n-dimensional (n ≥ 2) steady Euler equations. 3. A Special Class of Test Functions and Estimates Suppose that the approximate solutions u ε , p ε ∈ C 1 (R 3 ) satisfy ε ε u r ∂r u r + u εz ∂z u rε + ∂r p ε = h rε (r, z),

(3.18)

u rε ∂r u εz + u εz ∂z u εz + ∂z p ε = h εz (r, z), and ∂r (r u rε ) + ∂z (r u εz ) = 0, h rε (r, z)

(3.19)

h εz (r, z)

where and are some error terms. To study the structures and properties of approximate solutions satisfying (3.18) and (3.19), we need to construct a special class of test functions. Let χ = χ (s) be a nonnegative smooth function satisfying χ (s) = 1, |s| ≤ 1, (3.20) χ (s) = 0, |s| > 2. Denote by χ+ (s) = χ (s)|s≥0 the restriction of χ (s) on {s ≥ 0}. Then χ+ (s) = 1, 0 ≤ s ≤ 1, χ (s) = 0,

s > 2.

(3.21)

For any η > 1, we define r ψ(r, z) = zχ+ ( ) f η (z), (r, z) ∈ H, η with

f η (z) =

|z| ≤ η,

1, a1

ηα1 |z|−α1

+ a2

ηα2 |z|−α2

+ a3

ηα3 |z|−α3 ,

|z| ≥ η.

(3.22)

Here 1 ≤ α1 < α2 < α3 and a1 , a2 , a3 are constants to be determined such that f η (z) is a C 2 −smooth function satisfying f η (z) + z f η (z) ≥ 0, z ∈ R,

(3.23)

|z|| f η (z)| + z 2 | f η (z)| ≤ C, z ∈ R

(3.24)

and


331

with C an absolute constant. To be more precise, we consider the case z ≥ 0 and the case z ≤ 0 can be treated similarly. Note that when z ≥ η > 1 we have f η (z) = a1 ηα1 z −α1 + a2 ηα2 z −α2 + a3 ηα3 z −α3 , f η (z) = −α1 a1 ηα1 z −α1 −1 − α2 a2 ηα2 z −α2 −1 − α3 a3 ηα3 z −α3 −1 , f η (z) = α1 (α1 + 1)a1 ηα1 z −α1 −2 + α2 (α2 + 1)a2 ηα2 z −α2 −2 +α3 (α3 + 1)a3 ηα3 z −α3 −2 . To guarantee that f η (z) ∈ C 2 (R), one requires that ⎧ ⎪ ⎨ a1 + a2 + a3 = 1, α1 a1 + α2 a2 + α3 a3 = 0, ⎪ ⎩ α1 (α1 + 1)a1 + α2 (α2 + 1)a2 + α3 (α3 + 1)a3 = 0. Solving (3.25), one has ⎧ a1 = ⎪ ⎪ ⎪ ⎨ a2 = ⎪ ⎪ ⎪ ⎩ a3 =

(3.25)

α2 α3 (α3 −α2 ) α2 α3 (α3 −α2 )+α1 α3 (α1 −α3 )+α1 α2 (α2 −α1 ) , α1 α3 (α1 −α3 ) α2 α3 (α3 −α2 )+α1 α3 (α1 −α3 )+α1 α2 (α2 −α1 ) ,

(3.26)

α1 α2 (α2 −α1 ) α2 α3 (α3 −α2 )+α1 α3 (α1 −α3 )+α1 α2 (α2 −α1 ) .

We note that (3.23) is clearly satisfied when z ≤ η. To guarantee that (3.23) is satisfied for all z ∈ R, we choose some particular 1 ≤ α1 < α2 < α3 , for example, α1 = 1, α2 = 2, α3 = 10. Then for any z = aη with a ≥ 1, direct calculations show that f η (z) + z f η (z) = a1 ηα1 z −α1 (1 − α1 ) + a2 ηα2 z −α2 (1 − α2 ) + a3 ηα3 z −α3 (1 − α3 ) =

α2 α3 (α3 − α2 )(1 − α1 )a −α1 + α1 α3 (α1 − α3 )(1 − α2 )a −α2 α2 α3 (α3 − α2 ) + α1 α3 (α1 − α3 ) + α1 α2 (α2 − α1 )

+

α1 α2 (α2 − α1 )(1 − α3 )a −α3 , α2 α3 (α3 − α2 ) + α1 α3 (α1 − α3 ) + α1 α2 (α2 − α1 )

and α2 α3 (α3 − α2 ) + α1 α3 (α1 − α3 ) + α1 α2 (α2 − α1 ) = 72, α2 α3 (α3 − α2 )(1 − α1 )a −α1 = 0, α1 α3 (α1 − α3 )(1 − α2 )a −α2 = 90a −2 , α1 α2 (α2 − α1 )(1 − α3 )a −α3 = −18a −10 . Therefore f η (z) + z f η (z) =

5a −2 − a −10 >0 4

for all z = aη with a ≥ 1 and (3.23) is satisfied for all z ∈ R. Moreover, (3.24) is clearly satisfied.

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Now we choose the test functions as follows: z r r ϕz = −∂r ψ = − χ+ ( ) f η (z), η η r r r ϕr = ∂z ψ = χ+ ( ) f η (z) + zχ+ ( ) f η (z). η η

(3.27) (3.28)

In view of (3.23), one has r ϕr ≥ 0. Note that the test functions defined in (3.27) and (3.28) do not satisfy the conditions required in Definition 2.2. Especially, the test functions ϕr has singularity o( r1 ) near the symmetry axis. But for these test functions, we have Theorem 3.1. Suppose that the approximate solutions u ε , p ε ∈ C 1 (R 3 ) satisfy (3.18)– (3.19) and the following conditions: u ε L 2 (R 3 ) ≤ C, 1 uε ( r )2 d x ≤ C, 2 R 3 1 + x3 r |u ε | → 0, p ε → p0 as r 2 + z 2 → ∞, where C(> 0) and p0 are some absolute constants. Suppose further that |h ε | |h ε | |h ε | (|h εz | + r )r dr dz ≤ C or ( z + r )r dr dz ≤ C, r r r H Hz h εz (0, z)dz ≤ 0 −∞

(3.29) (3.30) (3.31)

(3.32) (3.33)

for all z ∈ R. Then for the test functions defined as in (3.27)–(3.28), it holds that (u rε )2 ϕr dr dz H 1 r z r ≤ |[(u rε )2 − (u εz )2 ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]|dr dz η η η η H z r |u rε u εz [−ϕz − 2 χ+ ( ) f η (z)]|dr dz + η η H r r |u rε u εz [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|dr dz + h(ε), (3.34) + η η H where h(ε) = H |[h rε (r, z)ϕr + h εz (r, z)ϕz ]|r dr dz. Proof. Without loss of generality, we assume that p ε → 0 as r 2 + z 2 → ∞. Otherwise, one may replace p ε by p˜ ε = p ε − p0 in (3.18). Let p¯ ε = p ε − p ε (0, z). Then ε ε u r ∂r u r + u εz ∂z u rε + ∂r p¯ ε = h rε (r, z), u rε ∂r u εz + u εz ∂z u εz + ∂z p¯ ε + ∂z p ε (0, z) = h εz (r, z).

(3.35)

(3.36)


333

For the test functions r ϕr and r ϕz defined in (3.27) and (3.28), multiplying r ϕr and r ϕz on both sides of (3.36)1 and (3.36)2 respectively and integrating on H , we have [u rε ∂r u rε + u εz ∂z u rε + ∂r p¯ ε ]ϕr r dr dz = h rε (r, z)ϕr r dr dz, (3.37) H H [u rε ∂r u εz + u εz ∂z u εz + ∂z p¯ ε + ∂z p ε (0, z)]ϕz r dr dz = h εz (r, z)ϕz r dr dz. (3.38) H

H

uε

uε

C 1 (R 3 )

u rε er

u εz ez ,

u rε |r =0

∈ and = + so = 0. Formally, it follows from Since (3.37) and (3.38) through integrating by parts that [(u rε )2 ∂r ϕr + (u εz )2 ∂z ϕz ]r dr dz + p ε (0, z)∂z ϕz r dr dz H H ¯ =− u rε u εz (∂r ϕz + ∂z ϕr )r dr dz + h(ε), (3.39)

H

¯ where h(ε) = + h εz (r, z)ϕz ]r dr dz. It follows from (3.27) that ε H [h r (r, z)ϕr

r ∂r ϕz = −ϕz − with

z r χ ( ) f η (z), η2 + η

⎧ ⎪ ⎨ 0, − rzη χ+ ( ηr ) f η (z), ϕz = ⎪ ⎩ 0,

(3.40)

0 ≤ r ≤ η, η ≤ r ≤ 2η,

(3.41)

r ≥ 2η,

and 1 r z r r ∂z ϕz = − χ+ ( ) f η (z) − χ+ ( ) f η (z). η η η η

(3.42)

While (3.28) yields r ∂r ϕr = −ϕr +

1 r z r χ+ ( ) f η (z) + χ+ ( ) f η (z), η η η η

(3.43)

and r r r ∂z ϕr = 2χ+ ( ) f η (z) + zχ+ ( ) f η (z). η η Substitute (3.40)–(3.44) into (3.39) to obtain (u rε )2 ϕr dr dz = p ε (0, z)∂z ϕz r dr dz H H z r ε 2 ε 2 1 r + [(u r ) − (u z ) ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]dr dz η η η η H z r u rε u εz [−ϕz − 2 χ+ ( ) f η (z)]dr dz + η η H r r ¯ u rε u εz [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]dr dz + h(ε). + η η H

(3.44)

(3.45)

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In view of (3.36)2 , one has ∂z p ε (0, z) = −u εz (0, z)∂z u εz (0, z) + h εz (0, z). Thus 1 p ε (0, z) = − (u εz (0, z))2 + 2

z

−∞

h εz (0, z)dz ≤ 0,

(3.46)

(3.47)

where the assumptions (3.31) and (3.33) have been used. Thanks to (3.23), (3.42), we have 1 r z r r ∂z ϕz = − χ+ ( ) f η (z) − χ+ ( ) f η (z) ≥ 0, η η η η

(3.48)

since χ+ (s) ≤ 0 for s ≥ 0. Thus, combining (3.47), (3.48) with (3.45) shows (u rε )2 ϕr dr dz H 1 r z r ≤ |[(u rε )2 − (u εz )2 ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]|dr dz η η η η H z r ε ε |u r u z [−ϕz − 2 χ+ ( ) f η (z)]|dr dz + η η H r r |u rε u εz [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|dr dz + h(ε) + η η H ≡ I, (3.49) where h(ε) = H |[h rε (r, z)ϕr + h εz (r, z)ϕz ]|r dr dz. Each term on the right-hand side of (3.49) is well-defined. In fact, there exists a constant C = C(η) such that 1 r z r |[(u rε )2 − (u εz )2 ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]|dr dz ≤ C(η)u ε 2L 2 (R 3 ) ; η η η η H z r |u rε u εz [−ϕz − 2 χ+ ( ) f η (z)]|dr dz ≤ C(η)u ε 2L 2 (R 3 ) . η η H Moreover, by (3.24), one has 1 r r |(1 + z 2 ) 2 [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]| ≤ C, η η

and hence

(3.50)

|

r r u rε u εz [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]dr dz| η η H 1 1 1 u rε 2 2( ) ≤ C( ( r dr dz) (u εz )2 r dr dz) 2 . 2 r 1 + z H H

Due to (3.32), one has h(ε) ≤ C. Consequently, using (3.29), (3.30), one has |I | ≤ C, with C an absolute constant.

(3.51)


335

To obtain (3.49) rigorously, we should prove that the left-hand side of (3.49) is well-defined. To this end, we denote HM = (0, ∞) × [−M, M] for any M > 0. Multiplying r ϕr and r ϕz on both sides of (3.36)1 and (3.36)2 respectively and integrating on HM with respect to (r, z), we have [u rε ∂r u rε + u εz ∂z u rε + ∂r p¯ ε ]ϕr r dr dz = h rε (r, z)ϕr r dr dz, (3.52) HM HM [u rε ∂r u εz + u εz ∂z u εz + ∂z p¯ ε + ∂z p ε (0, z)]ϕz r dr dz HM = h εz (r, z)ϕz r dr dz. (3.53) HM

Integrating by parts in (3.52) and (3.53) and then adding the resulting equations show that ε 2 (u r ) ϕr dr dz = p ε (0, z)∂z ϕz r dr dz HM HM 1 r z r + [(u rε )2 − (u εz )2 ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]dr dz η η η η H M z r + u rε u εz [−ϕz − 2 χ+ ( ) f η (z)]dr dz η η H M r r + u rε u εz [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]dr dz η η HM + h M (ε) + SbM , where h M (ε) = HM [h rε (r, z)ϕr + h εz (r, z)ϕz ]r dr dz and ∞ M SbM = − [u εz u rε ∂z ϕr + (u εz )2 ∂z ϕz ]|z=−M r dr 0 ∞ M [( p¯ ε + p ε (0, z))∂z ϕz ]|z=−M r dr −

(3.54)

0

which is the boundary term. It follows from (3.47) and (3.48) that (u rε )2 ϕr dr dz HM 1 r z r ≤ [(u rε )2 − (u εz )2 ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]dr dz η η η η H M z r u rε u εz [−ϕz − 2 χ+ ( ) f η (z)]dr dz + η η H M r r + u rε u εz [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]dr dz η η HM + h M (ε) + SbM . Since |SbM | ≤ C max(|u ε |2 + | p ε |)|[

(3.55) 0

∞

M (∂z ϕr + ∂z ϕz )r dr ]|z=−M |,

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Q. Jiu, Z. Xin

it is clear to deduce that |SbM | → 0 for any fixed ε > 0 and η > 1 as M → ∞. Combining this with (3.51) and noting that |h M (ε)| ≤ C by (3.33), we obtain that the term on the left-hand side of (3.55) is uniformly bounded with respect to M. Therefore, taking the limit M → ∞ on both sides of (3.55), we obtain (3.49). The proof of the theorem is finished.

4. Strong Convergence in Region Away From the Symmetry Axis For any r0 > 0, we define r0 = {x|x ∈ R 3 , x12 + x22 > r02 }. Then we have Theorem 4.1. Suppose that the assumptions of Theorem 3.1 hold and h(ε) → 0 as ε → 0, where h(ε) is same as in (3.34). Then u ε1 → 0, u ε2 → 0

(4.1)

2 ( ) for any r > 0 as ε → 0. strongly in L loc r0 0

Proof. Due to (3.23), for any r > 0, we have ϕr =

1 r z r χ+ ( ) f η (z) + χ+ ( ) f η (z) ≥ 0. r η r η

For any r = rn = n1 > 0(n = 1, 2, · · · ), it follows from (4.2) and (3.34) that 1 r | (u rε )2 2 χ+ ( ) f η (z)r dr dz| r η {r ≥rn } r 1 |(u rε )2 zχ+ ( ) f η (z)|r dr dz ≤ 2 rn H η 1 r z r |[(u rε )2 − (u εz )2 ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]|r dr dz + r η η r η η H z ϕz r 1 |[(u rε )2 + (u εz )2 ][ + 2 χ+ ( ) f η (z)]|r dr dz + 2 H r rη η r r |u rε u εz [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|dr dz + h(ε) + η η H ≡ I1 + I2 + I3 + I4 + h(ε). Note that

(4.2)

(4.3)

1 1 u rε 2 r r ) (1 + z 2 ) 2 |[2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|r dr dz ( 2 r 1 + z η η H 1 r r 1 (u εz )2 (1 + z 2 ) 2 |[2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|r dr dz + 2 H η η 1 1 1 u rε 2 r r = ( ) (1 + z 2 ) 2 |[2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|r dr dz 2 2 {|z|≥η} 1 + z r η η 1 1 r r + (u ε )2 (1 + z 2 ) 2 |[2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|r dr dz. 2 {|z|≥η} z η η

|I4 | ≤

1 2


337

Equation (4.3) becomes |

1 r (u rε )2 2 χ+ ( ) f η (z)r dr dz| r η {r ≥rn } r 1 |(u rε )2 zχ+ ( ) f η (z)|r dr dz ≤ 2 rn H η 1 r z r + |[(u rε )2 − (u εz )2 ][ χ+ ( ) f η (z) + χ+ ( ) f η (z)]|r dr dz rη η rη η H ϕz r 1 z |[(u rε )2 + (u εz )2 ][ + 2 χ+ ( ) f η (z)]|r dr dz + 2 H r rη η 1 1 uε r r 1 ( r )2 (1 + z 2 ) 2 |[2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|r dr dz + 2 2 {|z|≥η} 1 + z r η η 1 1 r r + (u εz )2 (1 + z 2 ) 2 |[2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|r dr dz + h(ε) 2 {|z|≥η} η η ≡ I1 + I2 + I3 + I5 + I6 + h(ε).

(4.4)

Applying a diagonal procedure, taking the limit ε → 0, one can get {r ≥rn }

= ≡

1 2π 1 2π

+ for any rn = I0 →

1 2π

1 n

(u rε )2

1 2π

1 r χ+ ( ) f η (z)r dr dz 2 r η

R 3 \{r ≤rn }

R 3 \{r ≤rn }

[(u ε1 )2 + (u ε2 )2 ]

1 r χ+ ( ) f η (z)d x → I0 2 r η

[(u 1 )2 + (u 2 )2 ]

1 r χ+ ( ) f η (z)d x 2 r η

R 3 \{r ≤rn }

1 r χ+ ( ) f η (z)d(µ1 + µ2 ) 2 r η

(4.5)

> 0(n = 1, 2, · · · ) and η > 0. Then we obtain

R 3 \{r ≤rn }

[(u 1 )2 + (u 2 )2 ]

1 1 dx + 2 r 2π

R 3 \{r ≤rn }

1 d(µ1 + µ2 ) r2

(4.6)

as η → ∞. I1 , I2 and I3 can be treated in a similar way (see also the proof of Theorem 2.1). Taking the limit ε → 0 first for any η > 1 and then taking the limit η → ∞ in I1 , I2 and I3 , we can obtain I1 + I2 + I3 → 0. Now we consider the convergence of I5 and I6 . Due to (3.30), we have

(4.7)

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Q. Jiu, Z. Xin

1 u rε 2 ( ) r dr dz g + µw 1 + z2 r

(4.8)

weakly in M as ε → 0, where g ∈ L 1 (H ) and µw is a Radon measure. Note that for any fixed η > 1, 1 r r ηα1 |(1 + z 2 ) 2 [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]| = O( α ) η η |z| 1

as |z| → ∞. Then, taking the limit ε → 0 in I5 shows that 1 1 r r |g(1 + z 2 ) 2 [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|r dr dz I5 → I˜5 ≡ 2 {|z|≥η−1} η η 1 1 r r + |(1 + z 2 ) 2 [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]|dµw (4.9) 2 {|z|≥η−1} η η for any η > 1. Furthermore, thanks to (3.24), one has 1 r r |(1 + z 2 ) 2 [2χ+ ( ) f η (z) + zχ+ ( ) f η (z)]| ≤ C η η

with C an absolute constant, which yields I˜5 → 0

(4.10)

as η → ∞. Similarly, taking the limit ε → 0 first for any η > 1 and then taking the limit η → ∞ in I6 , we obtain I6 → 0.

(4.11)

Combining (4.5)–(4.7) and (4.9)–(4.11), taking the limit (up to a subsequence) ε → 0 first for any η > 1 and then taking the limit η → ∞ in (4.4) show 1 1 1 1 [(u 1 )2 + (u 2 )2 ] 2 d x + d(µ1 + µ2 ) = 0 (4.12) 2π R 3 \{r ≤rn } r 2π R 3 \{r ≤rn } r 2 for any rn = n1 (n = 1, 2, · · · ). Therefore, for any r0 > 0, in the region r0 = {x|x ∈ R 3 , x12 + x22 > r02 }, u 1 = u 2 = 0, x ∈ r0 , and µ1 (r0 ) = µ2 (r0 ) = 0. Consequently, u ε1 → 0, u ε2 → 0 2 ( ) as ε → 0. The proof of the theorem is finished.

strongly in L loc r0

(4.13)


339

5. Strong Convergence in R3 Theorem 5.1. Under the assumptions of Theorem 4.1, it holds that uε → 0

(5.1)

2 (R 3 ) as ε → 0. strongly in L loc

Proof. For any X 3 >> 1 large enough and r0 > 0, we have (u rε )2 r dr dz {|x3 |≤X 3 ,r ≥0} ε 2 (u r ) r dr dz + (u rε )2 r dr dz ≤ ≤ ≤ ≤

{|x3 |≤X 3 ,r >r0 }

{|x3 |≤X 3 ,r >r0 }

{|x3 |≤X 3 ,r >r0 } {|x3 |≤X 3 ,r >r0 }

{|x3 |≤X 3 ,0≤r ≤r0 }

(u rε )2 r dr dz + (1 + X 32 ) (u rε )2 r dr dz + r02 (1 +

(u rε )2 r dr dz 1 + x32

{|x3 |≤X 3 ,0≤r ≤r0 } 1 uε X 32 ) ( r )2 r dr dz 2 H 1 + x3 r

(u rε )2 r dr dz + r02 (1 + X 32 )C,

(5.2)

where (3.30) has been used. For any δ0 > 0 and X 3 >> 1, we choose r0 > 0 small enough such that r02 (1 + X 32 )C ≤ δ0 . Using (4.13) and taking the limit ε → 0 in (5.2) yield (u r )2 r dr dz + dµr ≤ δ0 . (5.3) {|x3 |≤X 3 ,r ≥0}

{|x3 |≤X 3 ,r >0}

Since δ0 is arbitrary, (5.3) shows that u r = 0 and µr = 0. Consequently, u ε1 → 0, u ε2 → 0

(5.4)

2 (R 3 ) as ε → 0. This, together with (2.7), shows that strongly in L loc

R3

u 23 d x + |µ3 | = 0,

which implies u 3 = µ3 = 0.

(5.5)

Consequently, combining (5.4) with (5.5) shows that uε → 0 2 (R 3 ) as ε → 0. The proof of the theorem is finished.

strongly in L loc

Now we investigate the validity of the condition (3.30).

(5.6)

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Q. Jiu, Z. Xin

Lemma 5.1. Suppose that the approximate solutions u ε , p ε ∈ C 2 (R 3 ) satisfy (3.18) and (3.19) with h rε , h εz some error terms satisfying ∂z h rε , ∂r h εz ∈ C(H ). Moreover, suppose that u ε L 2 (R 3 ) ≤ C,

(5.7)

|ωθε | ≤ C(ε), (r, z) ∈ H¯ = [0, ∞) × (0, ∞), ∂z h rε − ∂r h εz |r dr dz ≤ C, | r H

(5.8)

|u ε | → 0, as r 2 + z 2 → ∞,

(5.9) (5.10)

where C is an absolute constant and C(ε) is a constant which may depend on ε. Then (3.30) holds. Proof. It follows from (3.18) and (3.19) that u rε ∂r (

∂z h rε − ∂r h εz ωθε ωε ) + u εz ∂z ( θ ) = . r r r

(5.11)

x3 1 dτ . For any η > 0, we define ϕ(r, z) = χ+ ( ηr )ρ(z) with χ+ the Set ρ(x3 ) = −∞ 1+τ 2 same as in (3.21). In the following, we will multiply the test functions r ϕ(r, z) on both sides of (5.11) and make the integration on H with respect to r and z. Similar as in the proof of Theorem 3.1, especially as the rigorous derivation of (3.49), the proof can be completed rigorously by integrating on HM = (0, ∞) × [−M, M] instead of H and we will omit the details for conciseness. Multiplying r ϕ(r, z) on both sides of (5.11), integrating the resulting identity with respect to (r, z) over (0, ∞) × (−∞, ∞), and using (3.19) and (5.8), we obtain ∂z h rε − ∂r h εz ϕr dr dz. (5.12) u rε ωθε ∂r ϕdr dz + u εz ωθε ∂z ϕdr dz = − r H H H That is

Note that

r u εz ωθε χ+ ( )ρ (z)dr dz η H 1 r u rε ωθε χ+ ( )ρ(z)dr dz =− η η H ∂z h rε − ∂r h εz r − χ+ ( )ρ(z)r dr dz. r η H

r u εz ωθε χ+ (

r ρ u εz (∂z u rε − ∂r u εz )χ+ ( )dr dz η H H ∞ 1 1 r 1 ρ (u εz )2 (0, z)dz + ρ (u εz )2 χ+ ( )dr dz = 2 −∞ 2 H η η r − (ρ u εz u rε + ρ u rε ∂z u εz )χ+ ( )dr dz. η H η

(5.13)

)ρ (z)dr dz =

(5.14)


341

Therefore, one has

r u εz ωθε χ+ ( )ρ (z)dr dz η H 1 r 1 ρ (u εz )2 χ+ ( )dr dz ≥ 2 H η η uε r − (ρ u εz u rε + ρ u rε (− r − ∂r u rε ))χ+ ( )dr dz r η H

(u rε )2 r χ+ ( )dr dz − r η

r ρ u εz u rε χ+ ( )dr dz η H H 1 1 1 1 r r − ρ (u rε )2 χ+ ( )dr dz + ρ (u εz )2 χ+ ( )dr dz. 2 H η η 2 H η η ρ

=

(5.15)

It follows from (5.13) and (5.15) that

(u rε )2 r r χ+ ( )dr dz − ρ u εz u rε χ+ ( )dr dz r η η H H 1 1 1 1 r r ≤ ρ (u rε )2 χ+ ( )dr dz − ρ (u εz )2 χ+ ( )dr dz 2 H η η 2 H η η ∂z h rε − ∂r h εz 1 r r − χ+ ( )ρ(z)r dr dz. u rε ωθε χ+ ( )ρ(z)dr dz − η η r η H H ρ

(5.16)

For any N > 1, we choose η > N large enough such that |

r ρ u εz u rε χ+ ( )dr dz| η H

≤|

N −N

=|

N −N

N

0

N

0

+

N

N −N

N

0

N

H \(−N ,N )×(0,N )

ρ u εz u rε dr dz| + [

∞ ∞

+

≡|

ρ u εz u rε dr dz| +

N −N

∞ N

]|

−N

−∞

N

|ρ u εz u rε |dr dz

+ 0

∞ N

+ N

0

−N −∞

∞ N

2z u ε u ε |dr dz (1 + z 2 )2 z r

ρ u εz u rε dr dz| +

5 i=1

Ii .

(5.17)

342

Q. Jiu, Z. Xin

The following estimates are direct: I1 =

−N

−∞

0

∞ N

I2 =

N

N

|

0

−N

∞

|

1 2z u ε u ε |dr dz ≤ C max |u ε |2 ; (1 + z 2 )2 z r N

1 2z u εz u rε |dr dz ≤ C max |u ε |2 ; 2 2 (1 + z ) N

2z u ε u ε |dr dz (1 + z 2 )2 z r −∞ N −N ∞ 1 1 |u εz u rε |r dr dz ≤ C 4 u ε 2L 2 (R 3 ) ; ≤C 4 N −∞ N N ∞ ∞ 2z 1 I4 = | u ε u ε |dr dz ≤ C 4 u ε 2L 2 (R 3 ) ; 2 )2 z r (1 + z N N N N ∞ 2z 1 | u ε u ε |dr dz ≤ C 4 u ε 2L 2 (R 3 ) . I5 = 2 )2 z r (1 + z N −N N I3 =

|

Consequently, one has from (5.17) that |

r ρ u εz u rε χ+ ( )dr dz| η H N N 1 ≤| ρ u εz u rε dr dz| + C (max |u ε |2 + u ε 2L 2 (R 3 ) ) N −N 0

(5.18)

for any N > 1 and η > N . Combining (5.16) with (5.18), one has

N −N

N

0

≤|

(u rε )2 dr dz r

ρ

N

−N

0

|

+C

N

H

ρ u εz u rε dr dz| + C

1 (max |u ε |2 + u ε 2L 2 (R 3 ) ) N

∂z h rε − ∂r h εz |r dr dz + |J |, r

(5.19)

where 1 J ≡ 2 −

ρ

H

1 r (u rε )2 χ+ ( η

1 )dr dz − η 2

1 r u rε ωθε χ+ ( )ρ(z)dr dz. η η H

1 r ρ (u εz )2 χ+ ( )dr dz η η H (5.20)


343

The last term on the right-hand side of (5.20) can be rewritten as 1 r u rε ωθε χ+ ( )ρ(z)dr dz η η H 1 r u rε (∂z u rε − ∂r u εz ) χ+ ( )ρ(z)dr dz = η η H 1 1 1 r r =− (u rε )2 χ+ ( )ρ (z)dr dz + ∂r u rε u εz χ+ ( )ρ(z)dr dz 2 H η η η η H 1 r + u rε u εz 2 χ+ ( )ρ(z)dr dz η η H 1 1 uε r ε 21 r =− (u r ) χ+ ( )ρ (z)dr dz − ( r + ∂z u εz )u εz χ+ ( )ρ(z)dr dz 2 H η η r η η H 1 r u rε u εz 2 χ+ ( )ρ(z)dr dz + η η H 1 1 u rε ε 1 r r =− (u rε )2 χ+ ( )ρ (z)dr dz − u z χ+ ( )ρ(z)dr dz 2 H η η η η H r 1 1 r r 1 (u ε )2 χ ( )ρ (z)dr dz + u rε u εz 2 χ+ ( )ρ(z)dr dz. (5.21) + 2 H z η + η η η H It follows from (5.20) and (5.21) that |J | ≤ C

1 ε 2 u L 2 (R 3 ) → 0, η2

as η → ∞. Taking the limit η → ∞ on both sides of (5.19) yields N N N N (u ε )2 ρ r dr dz ≤ | ρ u εz u rε dr dz| r −N 0 −N 0 ∂z h rε − ∂r h εz 1 ε 2 ε 2 |r dr dz | +C (max |u | + u L 2 (R 3 ) ) + C N r H

(5.22)

(5.23)

for any N > 1. Since ρ (x3 ) > 0 for all x3 ∈ R, it follows from (5.23) and (5.9) that N N N N N N 1 1 (u ε )2 (u ε )2 (ρ )2 ρ r dr dz ≤ ( ρ r dr dz) 2 ( (u εz )2 r dr dz) 2 r r ρ −N 0 −N 0 −N 0 1 +C (max |u ε |2 + u ε 2L 2 (R 3 ) ) + C, N where C is an absolute constant independent of ε and N . By the Cauchy-Schwartz inequality, we obtain N N (u ε )2 1 ρ r dr dz ≤ C (max |u ε |2 + u ε 2L 2 (R 3 ) ) + C, (5.24) r N −N 0 where C is an absolute constant independent of ε and N . Letting N → ∞ on both sides of (5.24) yields (3.30) and the proof of the theorem is finished.

344

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Lemma 5.2. Suppose that the approximate solutions u ε , p ε ∈ C 1 (R 3 ) satisfy (3.18) and (3.19) with some error terms h rε and h εz satisfying h rε , h εz ∈ C 1 (H ) and h εz |r =0 = 0. Suppose further that (5.7),(5.9) and (5.10) are satisfied and p ε → p0 as r 2 + z 2 → ∞, where p0 is a constant. Then (3.30) holds. Proof. Without loss of generality, we assume that p ε → 0 as r 2 + z 2 → ∞. For any η > 0, we let ϕ(r, z) = χ+ ( ηr )ρ(z) be the same as in the proof of Lemma 5.1. Similar to the proof of Lemma 5.1, it is assumed that the following integrations make sense and the rigorous proof by integration on HM instead of H will be omitted for conciseness. Multiplying ∂z ϕ and ∂r ϕ on both sides of (3.36)1 and (3.36)2 respectively and integrating on H , one may get [u rε ∂r u rε + u εz ∂z u rε + ∂r p¯ ε ]∂z ϕdr dz = h rε (r, z)∂z ϕdr dz, (5.25) H H [u rε ∂r u εz + u εz ∂z u εz + ∂z p¯ ε + ∂z p ε (0, z)]∂r ϕdr dz H = h εz (r, z)∂r ϕdr dz, (5.26) H

where p¯ ε = p ε (r, z) − p ε (0, z). Since [u rε ∂r u rε + u εz ∂z u rε ]∂z ϕdr dz H ε ε = u r ∂z u r ∂r ϕdr dz + u εz ∂z u rε ∂z ϕdr dz, H

and

(5.27)

H

[u rε ∂r u εz + u εz ∂z u εz ]∂r ϕdr dz = u rε ∂r u εz ∂r ϕdr dz + u εz ∂r u εz ∂z ϕdr dz H H 1 ∞ ε 2 (u ) (0, z)∂z ϕ(0, z)dz, + 2 −∞ z

H

(5.28)

subtracting (5.26) from (5.25) and then integrating by parts, with help of (5.27) and (5.28), one has 1 ∞ ε 2 ε ε ε ε u r ωθ ∂r ϕdr dz + u z ωθ ∂z ϕdr dz − (u z ) (0, z)ρ (z)dz 2 −∞ H H ∂z h rε − ∂r h εz ε ϕr dr dz. (5.29) p (0, z)∂r ∂z ϕdr dz = − + r H H Moreover, since χ+ (s) ≤ 0 (s ∈ R), ρ > 0 and p ε (0, z) ≤ 0 due to (3.46), (3.47) and the assumption that h εz (0, z) = 0, it holds that 1 r p ε (0, z)∂r ∂z ϕ = p ε (0, z) χ+ ( )ρ dr dz ≥ 0. (5.30) η η H H


345

It follows from (5.29), (5.30) and (5.14) that

r (ρ u εz u rε + ρ u rε ∂z u εz )χ+ ( )dr dz η H ∂z h rε − ∂r h εz 1 1 r ≤− u rε ωθε ∂r ϕdr dz − ρ (u εz )2 χ+ ( )dr dz − ϕr dr dz. 2 H η η r H H (5.31)

−

Noting that the left-hand side of (5.31) is

u rε r − ∂r u rε ))χ+ ( )dr dz r η H 1 (u ε )2 1 r r r ρ r χ+ ( )dr dz − ρ u εz u rε χ+ ( )dr dz − ρ (u rε )2 χ+ ( )dr dz, = r η η 2 η η H H H (5.32)

−

(ρ u εz u rε + ρ u rε (−

one has

(u rε )2 r r ρ u εz u rε χ+ ( )dr dz χ+ ( )dr dz − r η η H H 1 1 1 r r 1 ρ (u rε )2 χ+ ( )dr dz − ρ (u εz )2 χ+ ( )dr dz ≤ 2 H η η 2 H η η ε ε ∂z h r − ∂r h z 1 r r − χ+ ( )ρ(z)r dr dz u rε ωθε χ+ ( )ρ(z)dr dz − η η r η H H ε ε ∂z h r − ∂r h z r ≡J− χ+ ( )ρ(z)r dr dz, (5.33) r η H ρ

where J is same as in (5.20). Using similar arguments as (5.17)–(5.22), we obtain (5.23) from (5.33) and hence (5.24) by the Cauchy-Schwartz inequality. Letting N → ∞ on both sides of (5.24) yields (3.30) and the proof of the theorem is finished.

Remark 5.1. For unsteady 3D axisymmetric Euler equations with vortex-sheets initial data, Chae and Imanuvilov proved in [1] that the smooth approximate solutions constructed through regularizing the initial data satisfy

T 0

R3

1 u rε 2 ) d x ≤ C, ( 1 + x32 r

where C is a constant depending on initial energy and total variation of initial vorticity. Corresponding viscous approximations can be found in [11]. Lemma 5.1 and Lemma 5.2 above concern the steady approximations with error terms and in particular in Lemma 5.2 we only need that approximate solutions are C 1 -smooth. Based on Theorem 5.1, Lemma 5.1 and Lemma 5.2, we have

346

Q. Jiu, Z. Xin

Theorem 5.2. i) Suppose that the approximate solutions u ε , p ε ∈ C 2 (R 3 ) satisfy (3.18) and (3.19) with error terms h rε and h εz satisfying ∂z h rε , ∂r h εz ∈ C(H ). Moreover, suppose that u ε L 2 (R 3 ) ≤ C,

(5.34)

|ωθε | ≤ C(ε), (r, z) ∈ H¯ = [0, ∞) × (0, ∞), |h ε | |h ε | |h rε | ε (|h z | + ( z + r )r dr dz ≤ C, )r dr dz ≤ C or r r r H H ε ε ∂z h r − ∂r h z |r dr dz ≤ C, | r H

(5.35)

|u ε | → 0, p ε → p0 , as r 2 + z 2 → ∞,

(5.38)

(5.36) (5.37)

where C, p0 are some constants and C(ε) is a constant which may depend on ε. 2 (R 3 ). Then u ε → 0 strongly in L loc ii) Suppose that the approximate solutions u ε , p ε ∈ C 1 (R 3 ) satisfy (3.18) and (3.19) with error terms h rε and h εz satisfying h rε , h εz ∈ C 1 (H ) and h εz |r =0 = 0. Assume fur2 (R 3 ). ther that (5.34) and (5.36)–(5.38) are satisfied. Then u ε → 0 strongly in L loc Remark 5.2. Contrary to the 3D steady axisymmetric Euler equations, there exist nontrivial smooth exact solutions with finite energy and there exist smooth approximate solutions with finite energy appearing energy concentrations in the limit process to the 2D steady Euler equations (see [4]). More precisely, in 2D steady case, choose a velocity field, r −2 −x 2 u(x) = r sω(s)ds, x1 0 1 satisfying suppω ⊂ {|x| ≤ 1} and 0 sω(s)ds = 0. Set u ε (x) = −1 u(x/). Then u are the exact solutions of the two-dimensional steady Euler equations. Moreover, 2 |u | d x + |∇u |d x ≤ C, R2

R2

and u 0 weakly in L 2 (R 2 ). However,

u ⊗ u C1

δ0

0

0 δ0

weakly in M(), the finite Radon space, where u ⊗ u = (u iε u εj ) is a 2 × 2 matrix, δ0 is a Dirac measure supported at the origin and C1 is a positive constant. Remark 5.3. Using the spherical vortex rings given in [10], an example of the approximate solutions of the 3D steady axisymmetric Euler equations which converge strongly 2 (R 3 ) was constructed in [13]. to 0 in L loc Based on Theorem 5.2 ii), we obtain a Liouville type theorem which reads:


347

Theorem 5.3. Suppose that u, p ∈ C 1 (R 3 ) are exact solutions of 3D steady axisymmetric Euler equations (1.2)-(1.3) satisfying u L 2 (R 3 ) ≤ C, |u| → 0,

p → p0 as r 2 + z 2 → ∞,

where C and p0 are some constants. Then u ≡ 0 and p ≡ p0 . Proof (I). Taking u ε = u, p ε = p, h rε , h εz = 0 in Theorem 5.2 ii), we obtain that u ≡ 0 directly. While (1.1) and the fact that p → p0 as r 2 + z 2 → ∞ shows that p ≡ p0 . The proof of the theorem is complete.

The following is a direct proof of Theorem 5.3. The merit of this proof is that we do not need the technical test functions above. Proof (II). Without loss of generality, we assume that p → 0 as r 2 + z 2 → ∞. Otherwise, one may replace p by p˜ = p − p0 in (1.2)–(1.3). Let p¯ = p − p(0, z). Then it follows from (1.2)–(1.3) that u r ∂r u r + u z ∂z u r + ∂r p¯ = 0, u r ∂r u z + u z ∂z u z + ∂z p¯ + ∂z p(0, z) = 0.

(5.39)

(5.40)

Note that (5.40)1 can be rewritten as (r u r )∂r

ur u r u r2 + (r u z )∂z + = −∂r p. ¯ r r r

(5.41)

Integrating (5.41) with respect to r over [0, R], and then with respect to z over [−Z , Z ], using (1.3) and the fact that u r (0, z) = 0, we have Z R Z R 2 ur Z dr dz u r2 (R, z)dz + u r (r, z)u z (r, z)|z=−Z dr + r −Z 0 −Z 0 Z =− p(R, ¯ z)dz. (5.42) −Z

Letting R → ∞ on both sides of (5.42), and using the fact that |u| → 0, p → 0 as r 2 + z 2 → ∞, one can obtain Z Z ∞ 2 ∞ ur Z dr dz = u r (r, z)u z (r, z)|z=−Z dr + p(0, z)dz. (5.43) r 0 −Z 0 −Z Taking r = 0 on both sides of (5.40)2 , one has ∂z p(0, z) = −u z (0, z)∂z u z (0, z), z ∈ (−∞, ∞). Thus 1 p(0, z) = − (u z (0, z))2 ≤ 0, z ∈ (−∞, ∞). 2

(5.44)

348

Q. Jiu, Z. Xin

Substitute (5.44) into (5.43) to obtain ∞ ∞ Z ∞ 2 ur dr dz ≤ |u r (r, Z )u z (r, Z )|dr + |u r (r, −Z )u z (r, −Z )|dr. r −Z 0 0 0 (5.45) Since u ∈ L 2 (R 3 ), we have ∞

∞

−∞ 0

Consequently, ∞ ∞ −∞ 1

|u r (r, z)u z (r, z)|r dr dz < ∞.

|u r (r, z)u z (r, z)|dr dz ≤

∞

∞

−∞ 1

|u r (r, z)u z (r, z)|r dr dz < ∞.

Thus there exists a sequence of number Z i > 0(i = 1, 2, · · · ) satisfying Z i → ∞ as i → ∞ such that ∞ ∞ |u r (r, Z i )u z (r, Z i )|dr + |u r (r, −Z i )u z (r, −Z i )|dr → 0 (5.46) 1

1

as i → ∞. Note that ∞ |u r (r, Z i )u z (r, Z i )|dr = ( 0

1

∞

+ 0

)|u r (r, Z i )u z (r, Z i )|dr.

(5.47)

1

Since u ∈ C 1 (R 3 ) and |u| → 0, we obtain that 1 |u r (r, Z i )u z (r, Z i )|dr → 0

(5.48)

0

as i → ∞. It follows from (5.46)–(5.48) that ∞ |u r (r, Z i )u z (r, Z i )|dr → 0

(5.49)

0

as i → ∞. Similarly, one has ∞

|u r (r, −Z i )u z (r, −Z i )|dr → 0

(5.50)

0

as i → ∞. Replacing Z by Z i in (5.45) and taking the limit i → ∞ on both sides of (5.45), we obtain ∞ ∞ 2 ur dr dz = 0 r −∞ 0 and u r = 0. This, combined with (1.3), implies that ∂z u z = 0, from which we have u r (r, z) = u z (r, z) = 0 for all (r, z) ∈ R+ × R. Equations (1.2) and the fact that p → p0 as r 2 + z 2 → ∞ show that p ≡ p0 . The proof of the theorem is complete.

Acknowledgements. This research was done when the first author was visiting the Institute of Mathematical Sciences (IMS) of The Chinese University of Hong Kong. He would like to thank the Zheng Ge Ru Funds, Hong Kong RGC Emarked Research Grant CUHK4028/04P and CUHK4040/06P, RGC Central Allocation Grant CA 05/06. SC01 for partial support, and the staff at the IMS for hospitality. The authors express gratitude to the anonymous referee for his/her valuable suggestions to improve the presentation and for pointing out the direct proof of the Liouville theorem (Theorem 5.3).


349

References 1. Chae, D., Imanuvilov, O.Y.: Existence of axisymmetric weak solutions of the 3D Euler equations for near-vortex-sheets initial data. Elect. J. Diff. Eq. 26, 1–17 (1998) 2. Delort, J.M.: Existence de nappes de tourbillon en dimension deux. J. Amer. Math. Soc. 4, 553–586 (1991) 3. Delort, J.M.: Une remarque sur le probleme des nappes de tourbillon axisymetriques sur R 3. J. Funct. Anal. 108, 274–295 (1992) 4. DiPerna, R.J., Majda, A.: Reduced Hausdorff dimension and concentration-cancellation for 2-D incompressible flow. J. Amer. Math. Soc. 1, 59–95 (1998) 5. DiPerna, R.J., Majda, A.: Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40, 301–345 (1987) 6. Evans, L.C.: Weak covergence methods for nonlinear partial differential equations. CBMS Regional Conf. Ser. in Math. no. 74, Providence, RI: Amer. Math. Soc., 1990 7. Evans, L.C., Müller, S.: Hardy space and the two-dimensional Euler equations with non-negative vorticity. J. Amer. Math. Soc. 7, 199–219 (1994) 8. Fraenkel, L.E., Burger, M.S.: A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 14–51 (1974) 9. Friedman, A., Turkington, B.: Vortex rings: existence and asymptotic Estimates. Trans. Amer. Math. Soc. 268(1), 1–37 (1981) 10. Hill, M.J.M.: On a spherical vortex. Philos. Trans. Roy. Soc. London Ser. A 185, 213–245 (1894) 11. Jiu, Q.S., Xin, Z.P.: Viscous approximation and decay rate of maximal vorticity function for 3D axisymmetric Euler equations. Acta Math. Sinica 20(3), 385–404 (2004) 12. Jiu, Q.S., Xin, Z.P.: On strong convergence to 3D axisymmetric vortex sheets. J. Diff. Eqs. 223, 33–50 (2006) 13. Jiu, Q.S., Xin, Z.P.: On strong convergence to 3D steady vortex sheets. J. Diff. Eqs. 239, 448–470 (2007) 14. Lopes Filho, M.C., Nussenzveig Lopes, H.J., Xin, Z.P.: Existence of vortex-sheets with reflection symmetry in two space dimensions. Arch. Rat. Mech. Anal. 158, 235–257 (2000) 15. Lopes Filho, M.C., Nussenzveig Lopes, H.J., Xin, Z.P.: Vortex sheets with reflection symmetry in exterior domains. J. Diff. Eqs. 229, 154–171 (2006) 16. Liu, J.G., Xin, Z.P.: Convergence of vortex methods for weak solutions to the 2-D Euler equations with vortex sheet data. Comm. Pure Appl. Math. 48, 611–628 (1995) 17. Majda, A.: Remarks on weak solutions for vortex sheets with a distinguished sigh. Indiana Univ. Math. J. 42, 921–939 (1993) 18. Majda, A., Bertozzi, A.: Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics 27, Cambridge: Cambridge University Press, 2002 19. Ni, W.M.: On the exitence of global vortex rings. J. Anal. Math. 17, 208–247 (1980) 20. Nishiyama, T.: Pseudo-advection method for the axisymmetric stationary Euler equations. Z. Angew. Math. Mech. 81(10), 711–715 (2001) 21. Schochet, S.: The weak vorticity formulation of the 2D Euler equations and concentration-cancellation. Comm. P. D. E. 20, 1077–1104 (1995) Communicated by P. Constantin

Commun. Math. Phys. 287, 351–382 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0633-z

Communications in


Harmonic Oscillators on Infinite Sierpinski Gaskets Edward Fan1, , Zuhair Khandker2, , Robert S. Strichartz3, 1 Mathematics Department, The Chinese University of Hong Kong, Shatin, Hong Kong.


2 Physics Department, Yale University, New Haven, CT, USA.


3 Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853, USA.

E-mail: [email protected] Received: 19 February 2008 / Accepted: 4 June 2008 Published online: 23 September 2008 – © Springer-Verlag 2008

Abstract: We study the analog of the quantum-mechanical harmonic oscillator on infinite blowups of the Sierpinski Gasket, using the standard Kigami Laplacian. Our main task is to find a class of potentials analogous to 21 (x − x0 )2 on the line. We describe a class of potentials u with the properties u = 1, u attains a minimum value zero, and u → ∞ at infinity. We show how to construct such potentials attaining the minimum value at any prescribed point, and we show how to parameterize the class of potentials by a certain surface in R3 . We obtain estimates for the growth rate of the eigenvalue counting function for − + u. We obtain numerical approximations to the eigenfunctions, and in particular observe that the ground-state eigenfunction resembles a Gaussian function.

1. Introduction The harmonic-oscillator Hamiltonian −

d2 + a(x − x0 )2 dx 2

(1.1)

on the line is one of the basic model operators in quantum mechanics that is completely understood, thanks to the fact that its eigenfunctions are given in terms of Hermite functions. Note that we can write (1.1) as − + u(x), Research supported by Chinese University Mathematics Alumni.

(1.2)

Research supported by the National Science Foundation through the Research Experiences for Under-

graduates program at Cornell. Research supported in part by the National Science Foundation, grant DMS 0652440.

352

E. Fan, Z. Khandker, R. S. Strichartz 2

d 1 2 interpreting dx 2 as the Laplacian on the line, and where u(x) = 2 (x −x 0 ) (for simplicity take a = 21 ) has the following properties:

u = 1, u(x) attains its minimum value zero at x = x0 , u(x) → ∞ as x → ∞.

(1.3) (1.4) (1.5)

Operators of the form (1.2) with potentials u(x) satisfying (1.3–5) may exist in any context where there is a Laplacian defined, and the underlying space is unbounded (so (1.5) makes sense). In particular, it is possible to define Laplacians on certain fractals. The simplest nontrivial example is the Sierpinski Gasket SG (also called the Sierpinski Triangle). Thanks to the work of Kigami [Ki], it is possible to give a completely explicit description of a Laplacian on SG that is self-similar and has all the dihedral-3 symmetries of an equilateral triangle (see [S3] for an elementary exposition of the construction). However, SG is compact and has boundary points, so it is analogous to the unit interval rather than the line. We will take an infinite blowup ([S1,T]) of SG as our underlying space, where we will construct harmonic-oscillator Hamiltonians. In a related paper [S4], a Schrödinger equation with Coulomb potential (Hydrogen atom type model) is studied on products of blowups of SG. We note previous numerical study of the analog of square-well potentials on SG in [CDS]. However, there is as yet no analog of classical mechanics on fractals. In Sect. 2 we begin the study of solutions of (1.3) on SG. Such functions, which we call Laplacian-1 functions, are uniquely determined by their boundary values (u(q0 ), u(q1 ), u(q2 )), where q0 , q1 , q2 are the vertices of an equilateral triangle that constitute the boundary points of SG. We denote by U0 the solutions of (1.3) that satisfy (1.4), and in place of (1.5) the assumption that u is positive at the boundary points. The main result of this section is the estimate 1 1 ≤ for u ∈ U0 . u(qi ) ≤ (1.6) 5 3 i

In Sect. 3 we construct potentials in U0 with minimum value at prescribed points. There are two interesting phenomena here that do not appear on the line. First, there are points where there is a one-parameter family of potentials. Second, there are potentials which vanish on a one-dimensional set of points (a cycle in SG). Both of these indicate that it is not reasonable to parametrize the space of potentials by the location of the zero. In Sect. 4 we begin the study of the infinite blowups, denoted SG w is ∞ , where w an infinite word that describes the sequence of directions in which we blow up. Each potential satisfying (1.3) on SG extends uniquely to a potential on SG w ∞ also satisfying (1.3). The main technical tool is an explicit formula describing how normal derivatives at boundary points evolve under blow up. In Sect. 5 we continue the study of the extension of functions in U0 to SG w ∞ . We w assume that the word w is not eventually constant, which means that SG ∞ has no boundary points. In that case we show that the analog of (1.5) holds automatically. Thus we have a characterization of all the harmonic-oscillator potentials on SG w ∞ with the minimum occurring inside SG. In Sect. 6 we study the parametrization of U0 by Dirichlet data (u(q0 ), u(q1 ), u(q2 )). We denote the set of all such data by S. By (1.6) we see that S lies between two planes that we call Pupper and Plower . We also consider the Neumann data

Harmonic Oscillators on Infinite Sierpinski Gaskets

353

(∂n u(q0 ), ∂n u(q1 ), ∂n u(q2 )), and show that the Dirichlet to Neumann map projects S onto a triangle in Pupper , and use this to show that S is a 2-dimensional surface. In Sect. 7 we introduce a dynamical-systems approach on Pupper and S to study blowup vectors w consisting of only two digits. We find that alternating the two digits successively in w yields, in some sense, minimal growth. w In Sect. 8 we study counting the eigenvalue function for the operator −+u on SG ∞ , namely, N (x) = # j : λ j ≤ x , where λ j are the eigenvalues (repeated if there is multiplicity). Using standard minimax arguments [RS], we are able to relate the growth of N (x) to the growth of the potential. In particular, for words w that do not have long strings omitting one of the indices {0, 1, 2}, we obtain upper and lower bounds on the order of x log9/log5 for N (x). In Sect. 9 we study numerical approximations to the solutions of the eigenvalue problem − ψ + uψ = λψ on SG w ∞.

(1.7)

We use a version of the Finite Element Method to approximate both the eigenfunctions ψ and the eigenvalues λ (data available in [FKS]). The ground-state eigenfunction, corresponding to the minimum eigenvalue, is the analog of a Gaussian function on the line, and indeed it appears to have the expected “bell-shaped curve.” Also, the distribution of eigenvalues seems much more regular than for the Dirichlet or Neumann eigenvalues of the Laplacian on SG. We believe that a more computationally intensive approach to the approximation problem should yield very interesting results, but we leave this to the future. We now present a brief outline of the properties of the Laplacian on SG and its blowups (see [Ki] or [S3] for more details). This will also set the notation for the rest of the paper. Let {q0 , q1 , q2 } denote the vertices of an equilateral triangle in the plane, and let Fi (x) =

1 1 x + qi 2 2

(1.8)

denote the similarity mapping (homothety) with fixed point qi and contraction ratio 21 . Then SG is the unique nonempty compact set in the plane satisfying the self-similar identity K = Fi K . (1.9) i

We call the sets Fi K 1-cells, and note that Fi K ∩ F j K = Fi q j = F j qi for i = j. We call the points Fi q j = F j qi junction points. The space K is connected, but becomes disconnected if we remove the junction points Fi q j . For this reason SG is called a finitely-ramified fractal. Note that if we perform the same construction on the line with two initial points q0 = 0 and q1 = 1, then we obtain the unit interval as the solution of (1.9). For this reason we may regard SG as a fractal analog of the interval. We may then iterate the above procedure. Let w = (w1 , ..., wm ) denote a finite word of length |w| = m, each w j = 0, 1 or 2. Let Fw = Fw1 ◦ Fw2 ◦ · · · ◦ Fwm . Then SG also satisfies K =

|w|=m

Fw K .

(1.10)

(1.11)

354

E. Fan, Z. Khandker, R. S. Strichartz

We call the sets Fw K m-cells and note that they are either disjoint or intersect at a single point. Such points have the form Fw qi and are also called junction points. Let Vm denote the set of all points of the form Fw qi for |w| = m. Then V0 ⊆ V1 ⊆ · · · ⊆ Vm . Denote V∗ = m Vm . Note that the points in Vm \V0 are the junction points for (1.11). Of course V0 = {q0 , q1 , q2 }, and we call these boundary points of SG. Note that these are not topological boundary points (SG has no interior), but rather serve as boundary points for the analytic theory of Laplacians. For a general m-cell Fw (SG) the boundary points are {Fw qi }. We use the symbol ∂ to denote “boundary,” e.g ∂ SG = V0 . We consider the points in Vm as vertices of a graph m , with edge relations x ∼m y if x and y belong to the same m-cell (equivalently Fw qi ∼m Fw q j for any i = j and |w| = m). We define the Laplacian on SG as a renormalized limit of graph Laplacians m on m . Note that each vertex x ∈ Vm\V0 has exactly four neighbors in m . Define m u(x) = (u(y) − u(x)) for x ∈ Vm \V0 (1.12) y∼m x

if u is defined on Vm . Then 3 (1.13) u(x) = limm→∞ 5m m u(x) for x ∈ V∗ \V0 . 2 More precisely, we say that u ∈ dom and u = f if both u and f are continuous on SG and the limit on the right side of (1.13) converges uniformly to f . (Note that V∗\V0 is dense in SG). The normalization factor 23 is relatively unimportant (chosen to make certain integration formulas come out nicely), but the factor 5m is crucial: if we tried bm for b > 5 the Laplacian would exist only for constant functions, while for b < 5 all functions would have Laplacian equal to zero. With the definition (1.13) we have the following theorem: the equation u = f with boundary conditions u(qi ) = ai has a unique solution for every continuous function f . Functions satisfying u = 0 are called harmonic. (These are the analog of linear functions on the interval.) The Laplacian is also self-similar, namely (u ◦ Fi ) =

1 (u) ◦ Fi , 5

(1.14)

and more generally 1 (u) ◦ Fw . (1.15) 5|w| We also need an integration theory on SG. This is given by the probability measure µ satisfying the self-similar identity 1 µ(Fi−1 A). µ(A) = (1.16) 3 (u ◦ Fw ) =

i

3−|w| .

In particular, µ(Fw K ) = There is also a weak formulation of the Laplacian in terms of an energy E(u), defined as a limit of graph energies 5 Em (u) = ( )m (u(x) − u(y))2 via (1.17) 3 x∼ y m

E(u) = limm→∞ Em (u) (increasing limit).

(1.18)


Then u = f if and only if

355

E(u, v) = −

f vdµ for all v with v|V0 = 0

(1.19)

(here E(u, v) = 41 (E(u + v) − E(u − v)) is the bilinear form associated to the quadratic form E(u)). More generally, we have the Gauss-Green Formula E(u, v) = − f vdµ + v(qi )∂n u(qi ), (1.20) i

where the normal derivatives ∂n u(qi ) are defined by (u(qi ) − u(y)) (two terms in the sum). ∂n u(qi ) = limm→∞ 5m

(1.21)

y∼m qi

We note that the harmonic functions minimize energy among all functions with given boundary values, and the limits in (1.18) and (1.21) are unnecessary, as both right hand sides are constant. More generally, functions that minimize energy subject to prescribed values on Vm are called harmonic splines of level m, and are the continuous functions that coincide with harmonic functions on each m-cell. They are the analog of piecewise linear functions on the interval. Harmonic functions are easily understood because there is a simple linear and local extension algorithm to determine the values on Vm from the values on Vm−1 . This is a consequence of the fact that harmonic functions on SG restrict to graph harmonic functions on Vm . There is an analogous theory for eigenfunctions of the Laplacian, called spectral decimation, due to Fukushima and Shima [FS] (inspired by earlier work [R,RT]). Basically, eigenfunctions of the Laplacian on SG restrict to eigenfunctions of m on Vm , but the eigenvalues change. This makes it possible to give a precise description of the spectrum of both the Dirichlet and Neumann Laplacian (−u = λu, u|V0 = 0 or ∂n u|V0 = 0). The eigenvalue counting function has growth rate x log3/log5 , but there is an additional oscillatory behavior. There exist eigenvalues with very high multiplicity, and large spectral gaps. In fact the spectrum asymptotically approaches a neighborhood of the Julia Set of the polynomial x(5 − x). Next we describe blowups. Given the infinite word w, we have the increasing sequence of sets SG ⊆ Fw−1 (SG) ⊆ Fw−1 Fw−1 (SG) ⊆ · · · ⊆ Fw−1 · · · Fw−1 (SG) ⊆ · · ·, m 1 1 2 1

(1.22)

and we define SG w ∞ to be the union. Each of the sets in the sequence (1.22) looks like an enlarged copy of SG. We can define m-cells in SG w ∞ for all m ∈ Z (there are infinitely many for each m). Different choices of words w lead to different blowups (they will not be homeomorphic unless the words are eventually equal). We will always assume that the blowup is nondegenerate in the sense that the word w is not eventually constant. w In that case SG ∞ has no boundary points (the analogous construction starting with the interval produces the line). It is easy to transfer all the analytic structures, Laplacian, energy, and measure from SG to SG w ∞ . The normal derivative may be localized to any m-cell. A complete description of the spectrum of on SG w ∞ (it has pure point spectrum even though the space is noncompact) is given in [T]. We conclude this introduction with some general remarks.

356


(i) The eigenfunctions of the harmonic oscillator are not “smooth” functions. In fact they do not even belong to the domain of 2 . The reason is that the eigenvalue equation (1.7) involves multiplication, and nontrivial products of functions in dom are never in dom [BST]. This is in contrast to the heat kernel, which is smooth. Both the groundstate eigenfunction and the heat kernel are analogs of Gaussian functions on the line, but clearly they are different. (ii) Because of the self-similar identity (1.15) for the Laplacian, the eigenfunction ψ in (1.7) satisfies −5−m (ψ ◦ Fw−1 ◦ · · · ◦ Fw−1 ) + (u ◦ Fw−1 ◦ · · · ◦ Fw−1 )ψ ◦ Fw−1 ◦ · · · ◦ Fw−1 m m m 1 1 1 = λψ ◦ Fw−1 ◦ · · · ◦ Fw−1 m 1

(1.23)

on SG. In other words, if we define ψ˜ = ψ ◦ Fw−1 ◦ · · · ◦ Fw−1 and u˜ = 5m u ◦ m 1 −1 −1 Fw1 ◦ · · · ◦ Fwm then − ψ˜ + u˜ ψ˜ = 5m λψ˜ on SG.

(1.24)

Note that u˜ no longer satisfies (1.3), but rather u˜ = 52m .

(1.25)

Also, since we expect eigenfunctions to decay rapidly at infinity, ψ˜ will be very close to zero on the boundary of SG. Thus we are very close to transforming the eigenvalue problem (1.7) on the blowup SG w ∞ to a Dirichlet eigenvalue problem on SG with a potential satisfying (1.25) and (1.4). In fact, this is essentially what we do in the numerical computations in Sect. 9. (iii) Harmonic-oscillator Hamiltonians on Euclidean space Rn can be written as sums of 1-dimensional harmonic-oscillator Hamiltonians in each of the coordinate directions, so the eigenfunctions are just tensor products of 1-dimensional eigenfunctions, and the eigenvalues add. The same is true in the setting of products of SG blowups. As shown in [S2], it makes sense to define a Laplacian on the product as the sum of Laplacians on each factor. If we take for our class of potentials the sum of U0 potentials in each variable, then we will have conditions (1.4) and (1.5) holding. This seems like a reasonable choice for defining harmonic-oscillator Hamiltonians in the product setting. (iv) It would be interesting to try to extend our results to other p.c.f. fractals, or even other Laplacians on SG, but it might be difficult to obtain such detailed information. There are also Laplacians defined on Sierpinski carpets, which are not finitely ramified, using probabilistic methods [Ba]. However, it is not at all clear that Eq. (1.3) suffices to characterize the class of potentials one would want to consider. 2. Harmonic-Oscillator Potentials on SG Definition 2-1. The set U0 consists of all u(x) defined on SG satisfying (1) u(x) = 1, for all x ∈ SG\∂ SG. (2) min x∈SG u(x) = 0. (3) The minimum is attained strictly inside SG. That is, u > 0 on ∂ SG.

Harmonic Oscillators on Infinite Sierpinski Gaskets x0

x´2

357 x0

y0

y0

x´1

x0 Cm

x1

x´0

x2

x1=y1

Cm

y2

x2

x1=y1

x2

y2

Cm

Fig. 1.

Remark. In Sect. 3, we show that a Laplacian-1 function that has min value zero at a non-junction point lying on a side of SG must vanish along the entire side. Since such a function vanishes on ∂ SG, it does not belong to U0 . Proposition 2-2. Suppose f (x) defined on SG satisfies f (x) = 1, for all x ∈ SG\∂ SG. Let Cm be an m-cell of SG ( possibly C0 = SG). Then f is uniquely determined by f |∂Cm , its values at the three boundary vertices of Cm . Proof. Recall from (1.12) the definition of the graph Laplacian m . We use the following lemma, whose proof can be found in [S3] (p.58). Lemma 2-3. Let f ∈ dom and let x ∈ Vm \V0 , that is, x is a level-m junction point not belonging to ∂ SG. Then, 3 3 (2.1) f (x) − 5m m f (x) = 3m ψx(m) (y)( f (x) − f (y))dµ(y), 2 2 (m)

where ψx is the harmonic spline of level m centered at x, µ is the standard measure on SG, and the integration is over all y ∈ SG. In our case, since f ≡ 1 everywhere inside SG, (2.1) simplifies to m f (x) =

2 , x ∈ Vm \V0 . 3 · 5m

(2.2)

Let x0 , x1 , x2 be the vertices of ∂Cm , and let x0 , x1 , x2 denote the three level-(m + 1) junction points inside Cm , with xi lying opposite xi as in the leftmost diagram of Fig. 1. Then (2.2) at level-(m + 1) yields three equations: 2 , 3 · 5m+1 2 m+1 f (x1 ) = f (x2 ) + f (x0 ) + f (x2 ) + f (x0 ) − 4 f (x1 ) = , 3 · 5m+1 2 m+1 f (x2 ) = f (x0 ) + f (x1 ) + f (x0 ) + f (x1 ) − 4 f (x2 ) = . 3 · 5m+1

m+1 f (x0 ) = f (x1 ) + f (x2 ) + f (x1 ) + f (x2 ) − 4 f (x0 ) =

(2.3)

If one knows the value of f at any three of the six vertices, then (2.3) can be solved to find the value of f at all six vertices. In particular, suppose we know f (x0 ), f (x1 ), f (x2 ). If Cm+1 = Fi (Cm ) for i ∈ {0, 1, 2} , and if y j = Fi (x j ), j = 0, 1, 2, are the vertices of ∂Cm+1 , (e.g. the middle diagram of Fig. 1), then 5 T , (2.4) ν ( f (y0 ), f (y1 ), f (y2 ))T = Ai · ( f (x0 ), f (x1 ), f (x2 ))T − 9 · 5m+1 i

358


where Ai and νi are the following matrices and vectors, respectively: ⎤ ⎡ ⎡2 1 2⎤ ⎡2 2 1⎤ 1 0 0 5 5 5 5 5 5 ⎥ ⎥ ⎢ ⎢ A0 = ⎣ 2 2 1 ⎦ , A1 = ⎣ 0 1 0 ⎦ , A2 = ⎣ 1 2 2 ⎦ , 5 5 5 2 1 2 5 5 5

1 2 2 5 5 5

5 5 5

0 0 1 ν0 = (0, 1, 1), ν1 = (1, 0, 1), ν2 = (1, 1, 0).

(2.5) (2.6)

Alternatively, if Cm−1 = Fi−1 (Cm ) for i ∈ {0, 1, 2} , and y j = Fi−1 (q j ), j = 0, 1, 2, now denote the vertices of ∂Cm−1 , (e.g the right diagram of Fig. 1), then ( f (y0 ), f (y1 ), f (y2 ))T = (Ai )−1 · ( f (x0 ), f (x1 ), f (x2 ))T +

5 νT . 9 · 5m i

(2.7)

Iterative application of (2.4) and (2.7) along with the continuity of f clearly imply that f |∂Cm determines f uniquely. If Cm = Fwm · · · Fw1 (SG), then (2.7) implies that −1 f |∂ SG = (A−1 w1 ) · · · (Awm ) f |∂Cm +

m k=1

5 (A−1 ) · · · (A−1 wk−1 )νwk . 9 · 5k w1

(2.8)

We use this formula in the next section. We conclude Sect. 2 by formulating Lemma 2-4 below about functions in U0 . The proof of this lemma uses properties of harmonic functions. Recall that a function h on SG is called harmonic if h(x) = 0 for all x ∈ SG\∂ SG and that if Cm is an m-cell of SG, then h is uniquely determined by h|∂Cm . Analogous to (2.3), if x0 , x1 , x2 are the boundary vertices of Cm and x0 , x1 , x2 are the three level-(m + 1) junction points inside Cm with xi lying opposite xi , then 1 (h(x0 ) + 2h(x1 ) + 2h(x2 )), 5 1 h(x1 ) = (2h(x0 ) + h(x1 ) + 2h(x2 )), 5 1 h(x2 ) = (2h(x0 ) + 2h(x1 ) + h(x2 )). 5

h(x0 ) =

(2.9)

Recall also that harmonic functions satisfy the following maximum principle: if h is non-constant, then h attains its maximum only on ∂ SG. Laplacian-1 functions also satisfy the maximum principle, because they are sub-harmonic. These properties of harmonic functions and their proofs can be found in [S3] (Chaps.1,2). Lemma 2-4. Suppose u ∈ U0 . Then 15 ≤ q∈∂ SG u(q) ≤ 13 . Proof. We have u min = 0, so u must be non-negative on SG. The lower bound follows from summing the three Eqs. (2.3) with m = 0. For the upper bound, let s = u(q0 ) + u(q1 ) + u(q2 ), where qi ∈ ∂ SG, and assume s = 13 + , with > 0. We will show that this leads to a contradiction. The idea of the proof is that if s > 13 , then u is too big in the sense that u min > 0. We write u = f + h. Here, f is the unique Laplacian-1 function with f |∂ SG = (0, 0, 0), while h is the unique harmonic function with h|∂ SG = (u(q0 ), u(q1 ), u(q2 )).


359

Claim. Let x0 , x1 , and x2 be the boundary vertices of an m-cell. Let the values of f at these points be f (x0 ), f (x1 ), and f (x2 ), respectively. Then at these vertices, h satisfies u(qi ) , 5m u(q j ) h(x1 ) ≥ −3s f (x1 ) + m , 5 u(qk ) h(x2 ) ≥ −3s f (x2 ) + m , 5

h(x0 ) ≥ −3s f (x0 ) +

(2.10)

where (u(qi ), u(q j ), u(qk )) is some permutation of (u(q0 ), u(q1 ), u(q2 )), the boundary values of u. Proof of Claim. We use induction. At level m = 0, recall that f (q0 ) = f (q1 ) = f (q2 ) = 0. We have h(q0 ) = u(q0 ), h(q1 ) = u(q1 ), while h(q2 ) = u(q2 ), so (2.10) are satisfied trivially at this level. We now assume that (2.10) hold for all m-cells and prove that they continue to hold for all (m + 1)-cells. Let x0 , x1 , and x2 be the vertices of an m-cell, Cm , where h takes on the values h(x0 ), h(x1 ), and h(x2 ), and f takes on the values f (x0 ), f (x1 ), and f (x2 ), respectively. Again, let x0 , x1 , and x2 be the level-(m + 1) junction points generated within Cm . Then (2.3) yields 1 1 , ( f (x0 ) + 2 f (x1 ) + 2 f (x2 )) − 5 3 · 5m+1 1 1 , f (x1 ) = (2 f (x0 ) + f (x1 ) + 2 f (x2 )) − 5 3 · 5m+1 1 1 f (x2 ) = (2 f (x0 ) + 2 f (x1 ) + f (x2 )) − . 5 3 · 5m+1

f (x0 ) =

(2.11)

On the vertices of C(m−1) , we have by assumption that u(qi ) , 5m u(q j ) h(x1 ) ≥ −3s f (x1 ) + m , 5 u(qk ) h(x2 ) ≥ −3s f (x2 ) + m , 5

h(x0 ) ≥ −3s f (x0 ) +

(2.12)

for (qi , q j , qk ) some permutation of (q0 , q1 , q2 ). From (2.12) it follows that u(q j ) 1 u(qi ) 2 2 u(qk ) (−3s f (x0 ) + m ) + (−3s f (x1 ) + m ) + (−3s f (x2 ) + m ) 5 5 5 5 5 5 u(q j ) + u(qk ) 2 2 1 1 = −3s( f (x0 ) + f (x1 ) + f (x2 )) + m+1 (u(qi ) + u(q j ) + u(qk )) + 5 5 5 5 5m+1 u(q j ) + u(qk ) 1 2 2 1 = −3s( f (x0 ) + f (x1 ) + f (x2 ) − )+ 5 5 5 3 · 5m+1 5m+1 u(q j ) + u(qk ) = −3s f (x0 ) + . 5m+1

h(x0 ) ≥

360


Similarly, u(qi ) + u(qk ) , 5m u(qi ) + u(q j ) . h(x2 ) ≥ −3s f (x2 ) + 5m

h(x1 ) ≥ −3s f (x1 ) +

Using these three inequalities, it is easy to verify that (2.15) continue to hold on the three (m + 1)-cells inside Cm . The claim says that if x is a junction point of SG, then h(x) ≥ −3s f (x). By assumption, s = 13 + , for > 0, which implies that u(x) = f (x) + h(x) ≥ −3 f (x),

whenever x is a junction point of SG. (2.13)

Since junction points are dense in SG, and since u and f are both continuous, we can further conclude that u(z) ≥ −3 f (z),

all z ∈ SG.

(2.14)

By the maximum principle, f (z) < 0 whenever z ∈ / {q0 , q1 , q2 }, so (2.14) implies u(z) > 0, whenever z ∈ / {q0 , q1 , q2 } .

(2.15)

Finally, since u ∈ U0 , we have u(q0 ), u(q1 ), u(q2 ) > 0. Therefore, u min > 0, contradicting our assumption that u ∈ U0 . This completes the proof of Lemma 2-4. 3. Construction of Laplacian-1 Functions With Min Value Zero In this section, we outline the construction of functions u ∈ U0 . There are two nondisjoint cases: 1) u vanishes at a junction point; 2) u vanishes at a non-junction point. Case 1. Choose q1 ∈ ∂ SG and let f be a Laplacian-1 function on SG such that f attains the minimum value zero at q1 and has vanishing normal derivative there. Note that f ∈ / U0 because it vanishes on ∂ SG. However, once an explicit formula for f is found, a scaled copy of it can be placed in any m-cell to create functions in U0 . This is why we require ∂n f (q1 ) = 0: we will ultimately be placing copies of f inside SG (or in later sections extending our fractal beyond SG), so there will be cells on both sides of q1 , and if ∂n f (q1 ) = 0, f will attain negative values on some side of q1 . We expand f in the basis of biharmonic polynomials Pki centered at q1 and write 3 cki Pki . Recall that Pki are defined for k = 0, 1; i = 1, 2, 3; and j ≤ k f = 1k=0 i=1 in the following way (see [S3] p.123): j Pki (q1 ) = δ jk δi1 , ∂n j Pki (q1 ) = δ jk δi2 ,

(3.1)

∂T Pki (q1 ) = δ jk δi3 , j

where ∂n and ∂T denote the normal and tangential derivatives, respectively. In this expansion, we must have c12 = c13 = 0, because only P12 has ∂n P12 (q1 ) = 0 and only P13 has ∂T P13 (q1 ) = 0, so including them would yield a non-constant Laplacian for f .


-1/2

1/6

1/30

-1/10

1/15

1/3

0

1/15

q1

q1 0

361

1/30

0

1/6

q1 1/10

1/2

P03

P11

1/15

0

0

0

P11 - (1/3)P03

Fig. 2.

Since only P02 has ∂n P02 (q1 ) = 0, we must also have c02 = 0. Finally, c11 = 1 because only P11 , with P11 (q1 ) = 1, has non-zero Laplacian. This leaves f = P11 + c03 P03 .

(3.2)

Proposition 3-1. A function f of the form (3.2) has a minimum at q1 if and only if − 13 ≤ c03 ≤ 13 . Proof. Clearly |c03 | cannot be greater than 13 , or else f attains negative values. If |c03 | = 1 3 we get the rightmost function in Fig. 2. This function is identically zero along the bottom line. By the skew-symmetry of P03 it now follows that if |c03 | < 13 then P11 + c03 P03 ≥ 0 on SG and equals zero only at q1 . If u ∈ U0 has min value zero at a general junction point x ∈ ∂Cm , then by the same reasoning, within Cm we have u = 51m (P11 + c03 P03 ), |c03 | ≤ 13 . The scale factor 51m is needed to maintain u = 1. Outside Cm , u is determined by (2.7) and (2.4). Case 2. Let z be a non-junction point. That is, there exists an infinite vector w = (w1 , w2 , ...) such that z = limm→∞ Fwm · · · Fw1 (SG), wi ∈ {0, 1, 2}, and at least two digits of w are repeated infinitely often. Let f m be the sequence of Laplacian-1 functions with f m |∂Cm = 3·51m+1 (1, 1, 1). Let || · ||1 denote the 1-norm of a vector or matrix. For a d-dimensional vector x, x||1 || x ||1 = |x1 | + |x2 | + · · · + |xd | and for a matrix M, ||M||1 = supx=0 ||M || x ||1 . By submultiplicativity of the 1-norm, we have that −1 limm→∞ ||(A−1 w1 ) · · · (Awm )

1 (1, 1, 1)T ||1 3 · 5m+1

−1 −1 −1 −1 −1 ≤ limm→∞ ||(A−1 w1 )(Aw2 )||1 ||(Aw3 )(Aw4 )||1 · · · ||(Awm−1 )(Awm )||1

(3.3) 1 5m+1

.

One can check that ||(Ai−1 )(Ai−1 )||1 = 25, whereas for i = j, ||(Ai−1 )(A−1 j )||1 ≈

18.03. Since w has two digits which repeat infinitely often, the product (Ai−1 )(A−1 j ), i = j, appears infinitely often in (3.3). It follows that the limit in (3.3) goes to zero exponentially fast. Hence the RHS of (2.8) converges as m → ∞ and we get that f m → f with f |∂ SG =

∞ k=1

5 (A−1 ) · · · (A−1 wk−1 )νwk . 9 · 5k w1

(3.4)

362


By construction, f has a local min at z with f (z) = 0 and f | Fwm ···Fw1 (SG) = O( 51m ). Theorem 3-2. The function f constructed above is the unique Laplacian-1 function with f (z) = 0 a local min. Proof. Any other Laplacian-1 function f˜ satisfying these properties can be written f˜ = f + h, where h is harmonic. We prove that h must be identically zero. Let Cm = (m) (m) (m) (m) (m) (m) Fwm · · · Fw1 (SG). Let f 0 , f 1 , f 2 and h 0 , h 1 , h 2 be the values of f and h, respectively, at the vertices of ∂Cm . We are assuming z is a local minimum of f˜ as well, so for all sufficiently large m, min x∈Cm f˜(x) = 0 and f˜ is positive on ∂Cm . Then a rescaling of Lemma 2-4 from SG to Cm implies (m) (m) 1 1 (m) ≤ ( fi + h i ) ≤ , with f˜i = 5 · 5m 3 · 5m

(3.5)

(m) (m) (m) f˜i = f i + h i > 0.

(3.6)

2

2

i=0

i=0

So we must have − fi(m) < h i(m) < 3·51 m − f i(m) . Recall that f i(m) = O( 51m ), so ||h(m) = (m) (m) (m) (h 0 , h 1 , h 2 )||1 = O( 51m ). If we assume h is non-zero, then limm→∞ ||(A−1 w1 ) · · · −1 (m) m (Awm )h ||1 must have at least 5 growth, but as shown above, this is not possible when w = (w1 , w2 , ...) is the address of a non-junction point. Remark. If z is a non-junction point located along a side of SG, then by the uniqueness stated in Theorem 3-2, we know f = P11 − 13 P03 ∈ / U0 (right diagram of Fig. 2). 4. Blowups of SG Definition 4-1. Given an infinite blowup vector w = (w1 , w2 , w3 , ..., wn , ...), where each wi ∈ {0, 1, 2}, define the corresponding fractal blowup to be the (non-compact) ∞ −1 −1 set SG w ∞ = m=1 Fw1 · · · Fwm (SG), as in (1.22). Corollary 4-2 (to Proposition 2-2). Suppose f is a Laplacian-1 function on SG and suppose we extend SG to SG w Then f , initially defined on ∞ for some blowup vector w. SG, has a unique extension to a Laplacian-1 function on SG w ∞. Proof. Although we initially assumed m > 0, Lemma 2-3 and (2.2) in fact hold for all m ∈ Z (with simply x ∈ Vm in (2.2)), so the corollary follows by the same arguments used in the proof of Proposition 2-2. For fixed w, we now consider functions defined on all of SG w ∞. Definition 4-3: Strictly Interior Min. Let f be a continuous function on SG w ∞ for some w w. Let C be a cell of SG ∞ . Then we say that f attains a minimum strictly inside C if min x∈C f (x) < min x∈∂C f (x). Theorem 4-4. Let f be a Laplacian-1 function on SG w Let C be a cell of ∞ for some w. w SG ∞ . Then f attains a minimum strictly inside C if and only if the normal derivatives ∂n f (x) at x ∈ ∂C are all positive. To prove Theorem 4-4, we need the following lemmas about normal derivatives. Lemma 4-5. Let f be a Laplacian-1 function on SG w Let Cm be an m-cell ∞ for some w. w of SG ∞ . Then, 1 ∂n f (x) = m . (4.1) 3 x∈∂Cm


363 a

y

b

z

x

c

Fig. 3.

Proof. immediately from the Gauss-Green Formula, which implies This lemma follows that x∈∂Cm ∂n f (x) = Cm f (x). The general Gauss-Green Formula and its proof can be found in [S3] (p. 41). Let Cm be an m-cell Lemma 4-6. Let f be a Laplacian-1 function on SG w ∞ for some w. w of SG ∞ . Let a, b, c be the values of f on ∂Cm and let Na , Nb , Nc be the corresponding normal derivatives. Then,

5 1 Na = ( )m (2a − b − c) + m+1 , 3 3 5 1 Nb = ( )m (2b − a − c) + m+1 , 3 3 5 1 Nc = ( )m (2c − b − a) + m+1 . 3 3

(4.2)

Proof. Let h be the unique harmonic function on SG w ∞ with values a, b, c on ∂C m . Then f = h + f 0 , where f 0 is a Laplacian-1 function that vanishes on ∂Cm . Since f 0 is symmetric, by Lemma 4-5 it has normal derivatives 13 · 31m at each point of ∂Cm . Recall that h has normal derivative ( 53 )m (2a − b − c) at the vertex with value a; ( 53 )m (2b − a − c) at the vertex with value b; and ( 53 )m (2c − b − a) at the vertex with value c. Lemma 4-7. Let f be a Laplacian-1 function on SG w Let C(m−1) be an ∞ for some w. w (m − 1)-cell of SG ∞ with values as shown in Fig. 3. Then,

Nb = Na + 3N y , and Nc = Na + 3Nz .

(4.3)

1 Proof. By (4.2), we know Na = ( 53 )m (2a − y − z) + 3m+1 and N y = ( 53 )m (2y − a − 1 4 z) + 3m+1 . So Na + 3N y = ( 53 )m (−a + 5y − 4z) + 3m+1 . On the other hand, Nb = 5 m−1 1 5 5 (2b − a − c) + 3m . By (2.4), though, we know b = − 23 a + 10 (3) 3 y − 3 z + 9·5m and 2 5 10 5 c = − 3 a − 3 y + 3 z + 9·5m . Substituting these values into the formula for Nb yields 4 Nb = ( 53 )m (−a + 5y − 4z) + 3m+1 = Na + 3N y . The formula for Nc is obtained similarly.

Proof of Theorem 4-4. The reverse direction is trivial. If ∂n f (x) > 0 for x ∈ ∂C then f (y) < f (x) for some point y ∈ C near x. For the forward direction, let z be a point in the interior of C where u attains its min.

364


+ 0

-

0

-

+

+

+

+ z C´

0 C´

Fig. 4.

Case 1. The point z is a non-junction point lying on a line inside C, that is, if α denotes the address of z within C, then exactly two digits of α are repeated infinitely often. The line on which z lies cannot be a side of C. If it were, then f would vanish along that entire side and its min would not be strictly inside C. Let C ⊆ C be the smallest cell whose interior has the line containing z. We know one function having local min at z is given by the left function in Fig. 4. This function is identically zero along the inverted triangle. By Theorem 3-2, there are no other possibilities, up to addition by a 1 constant. By symmetry and Lemma 4-5, then, we have ∂n f = 3m+1 at ∂C . By Lemma 4-7, positivity of ∂n f extends to C. Case 2. The point z is a non-junction point not lying on a line of C, that is, all three digits of α repeat infinitely often. Let C be a cell containing z that is disjoint from ∂C. This is always possible since z does not lie on a line and hence does not lie on ∂C. As shown in the previous section, f is a limit of functions from Case 1, so ∂n f |∂C ≥ 0, which in turn implies that ∂n f |∂C > 0. Case 3. The point z is a junction point. Let C be the smallest cell containing z in its interior, as shown in the right diagram of Fig. 4. In the two lower cells of the diagram, we have f = 51m (P11 + c03 P03 ), with |c03 | ≤ 13 , as derived in the previous section. If c03 = ± 13 , we are in Case 1. If |c03 | < 13 , then ∂n f > 0 at the remaining two vertices of the lower cells. By matching conditions, ∂n f < 0 in the lower two vertices of the top cell, and by Lemma 4-5 we must have ∂n f > 0 at the very top vertex. Thus ∂n f > 0 on ∂C and this remains true on any cell containing C . This completes the proof of Theorem 4-4. Suppose f attains Theorem 4-8. Let f be a Laplacian-1 function on SG w ∞ for some w. w a minimum strictly inside some m-cell Cm of SG ∞ . Let Cm−1 be the (m − 1)-cell that contains Cm . Then f |{Cm−1 \Cm } ≥ f |∂Cm . Lemma 4-9. Let f be a Laplacian-1 function on some m-cell C of SG w ∞ . Let a, b, c be the values of f at the boundary vertices of C, and let Na , Nb , Nc be the corresponding normal derivative values. Then Ni < N j implies i < j, for i, j ∈ {a, b, c}.

Proof of Lemma 4-9. The formulas for Na , Nb , Nc in terms of a, b, c are given in (4.2). If Na < Nb , then 2a − b − c < 2b − c − a, which gives a < b, etc. Proof of Theorem 4-8. Since f attains a minimum strictly inside Cm , the normal derivatives at ∂Cm are all positive by Theorem 4-4. By Lemma 4-7, the normal derivatives at ∂Cm−1 are also all positive. Matching conditions tell us that the other two m-cells inside Cm−1 have negative normal derivatives at the juntion points x1 and x2 in common with


365

x4 + -

x1

x3

+ +

Cm

+

x2

+ x5

Fig. 5.

Cm (Fig. 5). By Theorem 4-4, the minimum value of f over these two m-cells occurs at their boundary. By Lemma 4-9, the minimum cannot occur at x4 or x5 . Moreover, the normal derivative at x3 must be non-negative along some side, so by Lemma 4-9, either f (x1 ) < f (x3 ) or f (x2 ) < f (x3 ). We conclude that the minimum value of f over these other two m-cells is attained at either x1 , x2 , or both x1 and x2 . This proves the theorem. Suppose f attains Corollary 4-10. Let f be a Laplacian-1 function on SG w ∞ for some w. w a min value, s, strictly inside some cell C of SG ∞ . Then (i) s is the global minimum of f over SG w ∞ and f > s outside of C. (ii) min f |C = min f |∂C for any cell C lying outside of C. (iii) f has no local minima outside C.

Proof. C is an m-cell, so we can call it Cm ; then (i) follows immediately from iterative applications of Theorem 4-8. To show (ii), suppose min f |C < min f |∂C for some cell C lying outside of C, i.e. f attains a minimum value s strictly inside C . Then (i) implies that both s and s are the global minimum of f and hence that s = s . But this also contradicts (i), because f > s outside C. To show (iii), suppose f has a local minimum at the point z ∈ / C. If z is a non-junction point not lying on a line, then there is a cell C lying outside C such that z is a strict interior min of C , which contradicts (ii) (it follows from the explicit construction in Sect. 3 that z is a strict interior min for any cell that contains it, and it suffices to choose a sufficiently small cell to avoid containing C). If z is a non-junction point lying on a line or a junction point, then recall from Cases 1 and 3 in the proof of Theorem 4-4 that we can find a cell C where f , up to an additive constant, looks either like the left or right diagram of Fig. 4. Note that z is in fact a strict interior min in C . C = C because z ∈ / C, and C does not contain C because the three cells inside C have a non-positive normal derivative at some boundary point, and any cell containing C should have all positive normal derivatives (Theorem 4-4). Therefore, C lies outside C and has a strict interior min, which contradicts (ii). Corollary 4-11. Suppose u ∈ U0 is extended to SG w ∞ . Then u vanishes at either a single non-junction point not lying on a line, along a single inverted triangle, or at a single junction point and has no other local minima.

Proof. Let z be a point where u vanishes. If z is a non-junction point not lying on a line, then we can find arbitrarily small cells C such that z is a strict interior min in C and apply Corollary 4-10. It follows that z is the unique zero of u and that u has no other local minima. Therefore, we may assume that z is a non-junction point lying on a line or a junction point. Then again, we can find a cell C such that u looks either like

366


a´ a

b´

C (-n)

b

c

c´

Fig. 6.

the left or right diagram of Fig. 4. Outside of C, we apply Corollary 4-10 to see that u doesn’t vanish or have any local minima outside C. Inside C, suppose there were another distinct local minimum, possibly another zero, at y. By the preceding discussion, we can rule out the possibility that y is a non-junction point not lying on a line. Moreoever, y cannot be a boundary vertex of C since normal derivatives are non-zero there, and the same reasoning rules out the two remaining junction points shown in the right diagram. Therefore, y must be contained in one of the three subcells of C as a junction point or as part of an inverted triangle. This yields a contradiction: as with z, we should be able to find a cell C like the ones in Fig. 4 containing y as a strict interior min, but then positivity of the normal derivatives at ∂C would extend to the appropriate subcell of C, which is false. 5. Harmonic-Oscillator Potentials on SG Blowups In the previous section, we have shown that if we start with a function u ∈ U0 , initially defined on SG, and extend it (uniquely) to a Laplacian-1 function on SG w ∞ , for any w, then the resulting function has global minimum zero, vanishes (within SG) either at a single junction point, a single non-junction point not lying on a line, or a single inverted triangle and has no other local minimum. Such functions on SG w ∞ , then, are very close to satisfying the characteristics of a harmonic-oscillator potential. Qualitatively, though, a harmonic-oscillator potential on SG w ∞ should also grow to infinity as we move far away from the original SG, the location of the minimum. This is what we investigate next. Definition 5-1: Growth. Let u ∈ U0 . Choose a blowup vector w and ∞Unbounded −1 −1 let SG w ∞ = n=1 Fw1 · · · Fwn (SG) as usual. In addition, define the partial blowups −1 −1 w SG w n = Fw1 · · · Fwn (SG). Extend u to SG ∞ . Then we say that u has unbounded growth on SG w ∞ if min u|∂ SG wn → ∞ as n → ∞.

(5.1)

Definition 5-2: Uw the set Uw ∞ . Given a blowup vector w, ∞ consists of all u(x) defined on SG satisfying

(1) u ∈ U0 . (2) u extended to SG w ∞ has unbounded growth. w We consider functions in Uw ∞ to be harmonic-oscillator potentials on SG ∞ . For any w we would like to know which u in U0 are also in Uw fixed w, U∞ ⊆ U0 . For given w, ∞. that has at least two of the three digits {0, 1, 2} Theorem 5-3. U0 = Uw ∞ for any w appearing infinitely often.


367

w Proof. Suppose u ∈ U0 . Fix w and extend u to SG w ∞ . Consider the partial blowup SG n , which is a (−n)-cell with boundary vertices a, b, c, and consider also the next step of the blowup SG w n+1 , which is a (−n − 1)-cell with boundary vertices a , b, c and which w contains SG n (Fig. 6). We want to show that u|∂ SG wn → ∞ as n → ∞. n

n

5 5 We first note that Lemma 2-4 scaled to SG w n implies 5 ≤ u(a) + u(b) + u(c) ≤ 3 . The values of u at a , b, c are obtained from the values of u at a, b, c using (2.7): u(b) does not change, but the formulas for u(a ) and u(c ) give

5 10 5 · 5n 2 u(a ) = − u(b) − u(c) + u(a) + 3 3 3 9 5 · 5n 5 ≥ − (u(a) + u(b) + u(c)) + u(b) + 5u(a) + 3 9 ≥ 5u(a) + u(b), where we have used the upper bound implied by Lemma 2-4. Similarly u(c ) ≥ 5u(c) + u(b). The fact that u(a ) ≥ 5u(a) and u(c ) ≥ 5u(c) tells us that when we apply an extra w blowup step to get from SG w n to SG n+1 , the value of u at one boundary vertex, b, does not change, but the value of u at the other two vertices of ∂ SG w n+1 are larger than their w counterparts in ∂ SG n by a factor of at least 5 (by Corollary 4-10 u is always positive outside of the interior of SG). Clearly, then, as long as w has two digits appearing infinitely often to avoid a stagnant boundary vertex, u will have unbounded growth. 6. Dirichlet/Neumann Maps Recall that a Laplacian-1 function f defined on SG is completely specified by its three boundary values f |∂ SG = { f (q0 ), f (q1 ), f (q2 )}. So to every Laplacian-1 function f we can associate the boundary-value vector ( f (q0 ), f (q1 ), f (q2 )) ∈ R3 . Conversely, every vector (a, b, c) in R3 is associated with a Laplacian-1 function, namely, the function f which has boundary values f |∂ SG = { f (q0 ) = a, f (q1 ) = b, f (q2 ) = c} and which is extended throughout SG using (2.4). In this way there exists a one-to-one correspondence between Laplacian-1 functions and vectors in R3 . Since we are interested in Laplacian-1 functions that also have min value zero, we define the following set S ⊂ R3 : S = points in R3 that represent Laplacian-1 functions with min value zero . (6.1) In Sect. 3, we outlined the construction of Laplacian-1 functions with min value zero. Using those constructions, it is straightforward to numerically generate points in R3 which belong to S in order to get an idea of what the set S looks like. Figure 7 shows 500 vectors which belong to S. By definition, the points in S cannot have negative coordinates and must lie in the first octant of R3 . Moreover, S lies between the planes x + y + z = 15 and x + y + z = 13 (Lemma 2-4). We call these planes Plower and Pupper , respectively. S touches Plower at 1 1 1 the point ( 15 , 15 , 15 ) (bold dot in Fig. 7). S touches Pupper along the planes x = 0, y = 0, and z = 0. These latter lines of intersection (shown in bold in Fig. 7) correspond to the functions f = P11 + c03 P03 , |c03 | ≤ 13 , constructed in Sect. 3.

368


Vectors in S 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0

0.4

0.1

0.2

0.3

0.4

Fig. 7.

We show below that S is a continuous surface without “holes”, but first we interpret Fig. 7 from the perspective of Dirichlet and Neumann maps. Let f be a Laplacian-1 function defined on SG. Define the Dirichlet map D [ f ] : f → f |∂ SG ,

(6.2)

which maps f to its boundary-value vector in R3 . Also, define the Neumann map N [f] : f →

1 ∂n f |∂ SG , 3

(6.3)

which maps f to its 3-vector of normal derivatives at ∂ SG. Proposition 6-1. N [ f ] is the projection of D [ f ] onto Pupper . Proof. Suppose D [ f ] = (a, b, c). If proj{Pupper } (a, b, c) denotes the projection of (a, b, c) onto Pupper , then proj{Pupper } (a, b, c) =

1 1 1 1 (2a − b − c + , 2b − a − c + , 2c − b − a + ). (6.4) 3 3 3 3

We note from Lemma 4-6 that the RHS of (6.4) is precisely N [ f ] = 13 (∂n f (q0 ), ∂n f (q1 ), ∂n f (q2 )), where ∂n f (qi ) is the normal derivative of f at the vertex qi ∈ ∂ SG. functions f satisfy the property that Recall from Lemma 4-5 that Laplacian-1 1 q∈∂ SG ∂n f (q) = 1, so every vector 3 (∂n f (q0 ), ∂n f (q1 ), ∂n f (q2 )) corresponding to the boundary normal derivatives of a Laplacian-1 function must lie on Pupper . Now consider an arbitrary point p = ( p0 , p1 , p2 ) ∈ Pupper and ask which vectors (a, b, c) ∈ R3 represent Laplacian-1 functions with normal derivatives specified by 3 p ? In other words,


we want to find all (a, b, c) which satisfy the three equations 1 1 2a − b − c + = p0 , 3 3 1 1 2b − a − c + = p1 , 3 3 1 1 2c − b − a + = p2 . 3 3

369

(6.5) (6.6) (6.7)

The solution is the line l p , which is perpendicular to Pupper and runs through p. That is, the points on l p represent all Laplacian-1 functions f with normal derivatives 1 1 1 3 ∂n f (q0 ) = p0 , 3 ∂n f (q1 ) = p1 , and 3 ∂n f (q2 ) = p2 . To summarize, in Fig. 7, if f is a Laplacian-1 function with min value zero, then D maps f to a point in S while N maps f to a point p on Pupper , which is the projection of D [ f ] onto Pupper . The line l p running through D [ f ] and N [ f ] represents all Laplacian-1 functions with normal derivatives 3 p at ∂ SG. We now show that S is indeed a surface. From here on, we restrict our attention to the first octant of R3 and consider Pupper only in this octant. Choose any point p = ( p0 , p1 , p2 ) on Pupper and let l p again be the line orthogonal to Pupper running through p. Starting from p, begin moving below Pupper along l p . We ask, is it true that one will eventually reach a point (a, b, c) ∈ S, and that this is the only point of S on l p ? Proposition 6-2. ∀ p ∈ Pupper , ∃ unique (a, b, c) ∈ l p , lying on or below Pupper , such that (a, b, c) ∈ S, that is, the Laplacian-1 function f represented by (a, b, c) satisfies f min = 0. Proof. We know one point on l p , namely, p itself. Consider the Laplacian-1 function g represented by this vector. That is, g has g|∂ SG = ( p0 , p1 , p2 ). Let gmin denote the minimum value of g. Any other vector on l p is of the form (a, b, c) = p + t (1, 1, 1), t ∈ R. Thus, l p represents all Laplacian-1 functions that differ from g by a constant function, t. Setting t = −gmin , we see that there is a unique vector, (a, b, c) = p − gmin (1, 1, 1), corresponding to the Laplacian-1 function f = g − gmin , which satisfies f min = 0. To confirm that (a, b, c) ∈ l p lies on or below Pupper , we simply note that according to Lemma 2-4, this must be the case if the corresponding Laplacian-1 function f satisfies f min = 0. Let F be the map that takes p to the corresponding (a, b, c) ∈ S on l p . In light of Proposition 6-2, we see that F works in the following way. A vector p ∈ Pupper represents a Laplacian-1 function g. F takes p → (a, b, c) = p + t p (1, 1, 1), where t p = −gmin . In this view, we have F( p) = t p : Pupper → R. Proposition 6-3. The function F( p) = t p : Pupper → R is continuous. That is, ∀ > 0, ∃δ > 0 such that | pα − pβ | < δ ⇒ |t pα − t pβ | < . Proof. Recall that pα represents a Laplacian-1 function gα with gα |∂ SG = (( pα )0 , ( pα )1 , ( pα )2 ). Similarly, pβ represents a Laplacian-1 function gβ with gβ |∂ SG = (( pβ )0 , ( pβ )1 , ( pβ )2 ). We know that t pα = −(gα )min and that t pβ = −(gβ )min . The idea is that if pα and pβ are close together, then (gα )min and (gβ )min differ only slightly, so that t pα and t pβ are also close together. This is because gα and gβ are both Laplacian-1 functions, so they differ by a harmonic function h with h|∂ SG = (( pα )0 − ( pβ )0 , ( pα )1 − ( pβ )1 , ( pα )2 − ( pβ )2 ). By the

370


Maximum Principle for Harmonic Functions, then, we know that |h(z)| ≤ max |( pα )0 − ( pβ )0 |, |( pα )1 − ( pβ )1 |, |( pα )2 − ( pβ )2 )| ,

for all z ∈ SG. (6.8)

By assumption, | pα − pβ | < δ, so |h(z)| < δ throughout SG. This implies that |(u α )min − (u β )min | < δ, hence |t pα − t pβ | < δ. So, ∀ > 0, letting δ = , one has | pα − pβ | < δ ⇒ |t pα − t pβ | < . Thus S is indeed a continuous surface that has a bijective correspondence with Pupper (in the first octant). 7. Dynamical Systems and Minimality of the 0,1,0,1,... Blowup Consider u ∈ U0 defined on SG with boundary values (x (0) , y (0) , z (0) ) and correspond(0) (0) (0) ing normal derivatives (N x , N y , Nz ). Extend u to SG w ∞ for a given blowup vector (m) , y (m) , z (m) ) are obtained using w. On SG w m (Definition 5-1) the boundary values (x (2.7). Meanwhile, the corresponding normal derivatives (N x(m) , N y(m) , Nz(m) ) are obtained using the recursion relation (Lemma 4-7): (N x(m) , N y(m) , Nz(m) )T = 3 · Mi · (N x(m−1) , N y(m−1) , Nz(m−1) )T , where i = wm ∈ {0, 1, 2} and the matrices Mi ⎤ ⎡1 ⎡ 1 1 3 3 0 0 ⎥ ⎢ ⎢ M0 = ⎣ 13 1 0⎦ , M1 = ⎣0 13 1 3

01

0

1 3

are as follows: ⎤ ⎤ ⎡ 0 1 0 13 ⎥ ⎥ ⎢ 0⎦ , M2 = ⎣0 1 13 ⎦ . 1 0 0 13

(7.1)

(7.2)

By Lemma 4-5, N x(m) + N y(m) + Nz(m) = 3m , so in R3 we have that ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (m) (0) (1) N x N N 1⎜ x ⎟ 1 ⎜ x ⎟ 1 ⎜ (m) ⎟ ⎟ p0 ≡ ⎝ N y(0) ⎠ , p1 ≡ 2 ⎝ N y(1) ⎠ , ..., pm ≡ m+1 ⎜ ⎝ N y ⎠ , ... ∈ Pupper . (7.3) 3 3 3 (0) (1) (m) Nz Nz Nz We have that (x (0) , y (0) , z (0) ) ∈ S, where S ⊂ R3 is the surface described in Sect. 6. For m > 0, (x (m) , y (m) , z (m) ) ∈ / S, but sm ≡ 51m (x (m) , y (m) , z (m) ) ∈ S. This follows from the scaling of the Laplacian, (1.15). (When we take u defined on SG w m and interpret it as a new function u˜ on SG, u˜ still has mininum value zero, but u˜ = 5m , so dividing by 5m yields a function in U0 , which is represented by a point on the surface S.) Consequently, ⎛ (0) ⎞ ⎛ (m) ⎞ ⎛ (1) ⎞ x x x 1 1 s0 ≡ ⎝ y (0) ⎠ , s1 ≡ ⎝ y (1) ⎠ , ..., sm ≡ m ⎝ y (m) ⎠ , ... ∈ S, (7.4) 5 5 z (0) z (1) z (m) and pm is the projection of sm onto Pupper , as discussed in Sect. 6. w w As we extend u ∈ U0 on SG to SG w 1 , SG 2 , SG 3 , etc., the sequences { pm } and {sm } represent dynamical systems which depend on the particular digits of w. We now use


371

(0,0,1/3)

1

0 2

(1/3,0,0)

(0,1/3,0) Fig. 8.

this observation to examine blowup vectors w consisting of only two digits, say 0 and 1, where each digit appears infinitely often. The matrices Mi map Pupper (restricted to the first octant) to the subtriangles i shown in Fig. 8. For instance, M0 fixes the vertices (0, 0, 13 ) and (0, 13 , 0) of Pupper and maps is made up only of infinitely many 0s and 1s, the third vertex ( 13 , 0, 0) → ( 19 , 19 , 19 ). If w then clearly pm → (0, 0, 13 ) as m → ∞, and hence sm → (0, 0, 13 ) as well since Pupper and S coincide there. In other words, the vertex (0, 0, 13 ) is an attractor for our dynamical (m) system when w has only 0s and 1s. It follows that z (m) → 13 5m and Nz → 3m as m → ∞, regardless of the particular sequence of 0s and 1s appearing in w. Now, however, we ask, which sequence of 0s and 1s minimizes the growth of the remaining boundary values x (m) and y (m) ? Lemma 4-6 (with m replaced by −m since SG w m are really (−m)-cells) implies 1 5 x (m) = z (m) + ( )m (N x(m) − Nz(m) ), 3 3 1 5 y (m) = z (m) + ( )m (N y(m) − Nz(m) ). 3 3 (m)

As m → ∞, the z (m) and Nz

(7.6)

terms cancel, and we have 1 5 m (m) ( ) Nx 3 3 1 5 → ( )m N y(m) , 3 3

x (m) →

(7.7)

y (m)

(7.8)

(m)

(m)

so we need to understand the growth of N x and N y . In general, ⎛ ⎞ ⎛ ⎞ (m) (0) Nx Nx ⎜ (m) ⎟ ⎜ (0) ⎟ m ⎝ N y ⎠ = 3 Mwm · · · Mw1 ⎝ N y ⎠ . (m) (0) Nz Nz (0)

(7.5)

(0)

(0)

(7.9)

The matrix multiplying (N x , N y , Nz ) always has one eigenvector e1 = (0, 0, 1)T (m) (m) with eigenvalue λ1 = 3m . The growth of N x and N y , then, depends on the larger of the two remaining eigenvalues λ2 and λ3 . To find λ2 and λ3 , it suffices to consider the upper right 2 × 2 submatrices of M0 and M1 , namely, 1 1 13 0 3 M¯ 0 = 1 . (7.10) , M¯ 1 = 0 13 3 1

372


(0,1) T M 0T M 1T

(1,0)

(0,0) Fig. 9.

Both M¯ 0 and M¯ 1 map T → T , where T is the triangle with vertices (0, 0), (1, 0), and (0, 1) (Figure 9). In particular, M¯ 0 fixes (0, 0) and (0, 1) and takes (1, 0) → ( 13 , 13 ) while M¯ 1 fixes (0, 0) and (1, 0) and takes (0, 1) → ( 1 , 1 ). 3 3

Claim. If x ∈ M¯ 1 T , then | M¯ 0 x| ≤ | M¯ 1 x|. Proof. It suffices to show | M¯ 0 x|2 ≤ | M¯ 1 x|2 . But M¯ 0 (x1 , x2 )T = ( 13 x1 , 13 x1 + x2 )T , so | M¯ 0 x|2 = ( 13 x1 )2 + ( 13 x1 + x2 )2 = 29 x12 + 23 x1 x2 + x22 . And similarly, | M¯ 1 x|2 = x 2 + 2 x1 x2 + 2 x 2 . Thus, | M¯ 1 x|2 − | M¯ 0 x|2 = 7 (x 2 − x 2 ) and x2 ≤ x1 in T . 1

3

9 2

9

1

2

Corollary. For any x ∈ T , | M¯ w2k · · · M¯ w2 M¯ w1 x| ≥ |( M¯ 0 M¯ 1 )k x| if w1 = 1, and ≥ |( M¯ 1 M¯ 0 )k x| if w1 = 0. Of course, the same is true for x ∈ Q = positive quadrant (x1 ≥ 0 and x2 ≥ 0). Given the matrix M, let its spectral radius be defined as spect rad(M) = max {|λi |}, where {λi } are the eigenvalues of M. Theorem 7-1. spect rad( M¯ w2k · · · M¯ w2 M¯ w1 ) ≥ spect rad( M¯ 0 M¯ 1 )k = spect rad ( M¯ 1 M¯ 0 )k . Proof. Because all these matrices have non-negative entries, the Perron-Frobenius x| Theorem says spect rad(M) = max x∈Q |M |x| . The result follows from this and the corollary above. Corollary. If w contains only infinitely many 0s and 1s, then N x(m) and N y(m) , and hence x (m) and y (m) , grow slowest when 0 and 1 alternate successively in w. (m)

(m)

Proof. Recall that the growth of N x and N y depends on max {λ2 , λ3 }, which is minimized for the alternating blowup by Theorem 7-1. 8. Estimates for the Eigenvalue Counting Function Let u be any potential on SG w ∞ with unbounded growth. This implies that (− + u) has a compact resolvant, hence a discrete spectrum. The eigenvalues of (− + u) are the critical values of the Rayleigh quotient

E( f ) + u f 2 dµ Ru ( f ) = f 2 dµ

(8.1)


373

and are given by the min-max formula λ j = mindimL= j max f ∈L Ru ( f ), where L varies over the subspaces of L2 ∩ domE. The eigenvalue counting function is defined by N (x) = # j : λ j ≤ x . Consider any partition of SG w ∞ into cells Ci of level −n i , SG w Ci , ∞ =

(8.2)

(8.3)

(8.4)

i

where Ci ∩ C j is at most a point for i = j. Let Mi = maxCi u,

m i = minCi u.

Theorem 8-1. There exist constants c1 , c2 , such that 3n i (x − Mi )α ≤ N (x) ≤ c2 3n i (x − m i )α c1

(8.5)

(8.6)

m i ≤x

Mi ≤x

for α = log3/log5. Proof. Define potentials u + and u − by u+ = Mi χCi , u − = m i χCi . i

(8.7)

i

Note that u − ≤ u ≤ u + and also Ru − ( f ) ≤ Ru ( f ) ≤ Ru + ( f ),

(8.8) (8.9)

where Ru ± are defined by (8.1) with u replaced by u ± . Let D denote the subspace of L2 ∩ domE consisting of functions vanishing on the boundary of all the cells Ci . Let N denote the space of functions in L2 with possible discontinuities on the boundary points of the cells Ci such that f |Ci ∈ domCi E and i ECi ( f ) < ∞. Define E( f ) = ECi ( f ) for f ∈ N . (8.10) i

Then Ru ( f ) and Ru ± ( f ) are defined for f ∈ N . Note that D ⊆ L2 ∩ domE ⊆ N , hence if we define λ−j = mindimL= j,L⊆N max f ∈L Ru − ( f ) and

(8.11)

λ+j = mindimL= j,L⊆D λ−j ≤ λ ≤ λ+j .

(8.12)

max f ∈L Ru + ( f ), we have

(8.13)

374


f jD be an orthonormal basis of Dirichlet eigenfunctions on SG with eigen values λ Dj , and f jN and λ Nj be the same for the Neumann eigenfunctions. We can construct a basis of eigenfunctions in D by transporting f jD to each cell Ci . If we call these functions f iDj , then Ru + ( f iDj ) = 5−n i λ Dj + Mi . In other words, the values λ+j coincide with the values 5−n i λ Dj + Mi arranged in increasing order. Sim ilarly, a basis of eigenfunctions in N is f iNj , where each f jN is transported to each Ci , and Ru − ( f iNj ) = 5−n i λ Nj + m i . Again, these values λ−j coincide with the values 5−n i λ Nj + m i arranged in increasing order. It follows that

Let

N D (5n i (x − Mi )) ≤ N (x) ≤

i

N N (5n i (x − m i )),

(8.14)

i

where N D (x) and N N (x) denote the eigenvalue counting function for Dirichlet and Neumann eigenvalues on SG. Note that the sums in (8.14) are finite, extending only over Mi ≤ x on the left and m i ≤ x on the right, and m i → 0, Mi → 0 since u has unbounded growth. We then use the known estimates c1 x α ≤ N D (x) ≤ c2 x α , c1 x α ≤ N N (x) ≤ c2 x α , and 5α = 3 to transform (8.14) into (8.6).

(8.15)

Now suppose the potential u belongs to Uw ∞ . We consider a partition (8.4) consisting of three cells of order 0 and two cells of orders −1, −2, −3, ... defined as follows: take the cells of order 0 in Fw−1 (SG), and the two cells of order −n in Fw−1 F −1 · · · 1 n+1 wn Fw−1 (SG)\Fw−1 · · · Fw−1 (SG). It is clear that the fastest growth rate for Mn and m n is n 1 1 n O(5 ). We will see that this is actually achieved for many choices of the word w. To be specific, suppose we can show

Mn ≤ a5n , m n ≥ b5n

(8.16)

for positive constants a and b, for both cells of order −n (when n = 0 we will have m 0 = 0 for one or two of the cells). Suppose 5 N ≤ x ≤ 5 N +1 . Substituting (8.16) in (8.6) yields essentially 3n (5 N − a5n )α ≤ N (x) ≤ 2c2 3n (5 N +1 − b5n )α (8.17) 2c1 n bρ , 3 w Let u be a harmonic-oscillator potential on SG w ∞ and consider the (−m)-cell SG m ⊂ w w SG ∞ . In light of (8.27), on SG m we can write u = u 1 + u 2 , where u 1 and u 2 are the Laplacian-1 and harmonic functions, respectively, shown in the left two diagrams of Fig. 10 ( p = ρ in the figure). logx Suppose x ≈ ρ m , i.e. m ≈ logρ , and define k so that ρ m = 5m−k , i.e. k ≈ log

1 1 x( logρ − log5 ). Successively subdivide SG w m as in the rightmost diagram of Fig. 10. The dotted line is drawn at step k of the subdivision, at which there are 2k cells of size −(m − k). The function u 1 is just a multiple of P11 − 13 P03 (Fig. 2), which decreases by a factor of 5 at each subdivision level shown in Fig. 10. Since u is already at size x or greater on ∂ SG w m , we know from the proof of Theorem w 5-3 that u ≥ x outside SG m as well. Hence, by (8.6), there is no contribution to N (x) w outside SG w m . The same is true above the dotted line in SG m , where u 1 dominates u 2


377

so that u ≥ 5m−k ≈ x. The contribution to N (x), then, comes from the 2k cells of size −(m −k) below the dotted line, where (x − Mi ) and (x −m i ) are on the order x ≈ 5m−k . The steps between (8.16) and (8.18) now give c1 x 2α+β ≤ N (x) ≤ c2 x 2α+β ,

(8.28)

log2 − log2 where 2α = log9/log5 ≈ 1.365, and β = logρ log5 ≈ 0.085. Since w = (0, 1, 0, 1, ...) represents slowest growth among words consisting of only two digits (each repeated infinitely often), and since words that use all three digits with regularity, i.e. (8.21), satisfy (8.18), we make the following conjecture: Conjecture 8-4. ∀w, ∃a, b ∈ (2α, 2α + β) ≈ (1.365, 1.450) and constants c1 , c2 such that

c1 x a ≤ N (x) ≤ c2 x b .

(8.29)

We have seen that a = b for many w, but this may not always be the case. 9. Eigenvalues and Eigenfunctions For an infinite blowup vector w having at least two digits repeating infinitely often and a potential function u ∈ Uw odinger Equation ∞ , we wish to study the Schr¨ − ψ + uψ = λψ.

(9.1)

To study this equation numerically, instead of considering the infinite blowup vector w, we consider a finite truncation (w1 , ..., wn ) of length n. Instead of SG w ∞ , then, we w −1 −1 take the underlying space for (9.1) to be the compact set SG n = Fw1 · · · Fwn (SG). For the remainder of this section, we assume that (9.1) is formulated on SG w n , for some fixed n, and use the Finite Element Method (FEM) outlined in [B] to approximate eigenvalues and eigenfunctions. As in [S3] (p.20), we let H0 denote the space of harmonic functions and Vm the set of level-m junction points in SG w n , level zero being the level of the original SG. Definition 9-1. S(H0 , Vm ) is defined to be the space of continuous functions f on SG w n w such that f vanishes on ∂ SG n and such that f ◦ Fw is harmonic for all |w| = n + m. S(H0 , Vm ) is called the space of level-m piecewise harmonic splines. Functions f ∈ S(H0 , Vm ) are obtained by specifying arbitrary values for f on Vm \∂ SG w n and then extending f harmonically within each m-cell. It is clear that S(H0 , Vm ) has dimension #(Vm \∂ SG w n ) and that a basis can be constructed as fol lows. Label the junction points of Vm \∂ SG w n as x 1 , ..., xr (m) , where the final index r depends on m. Let φi ∈ S(H0 , Vm ) be the function with φi (x j ) = δi j . Then {φi }ri=1 is a basis for S(H0 , Vm ). We fix a spline level, m, and expand ψ approximately using the basis {φi }ri=1 for S(H0 , Vm ), ψ≈

r i=1

ci φi .

(9.2)

378


Remark. In (9.2) all the φi vanish on ∂ SG w n by definition, and hence so does our approx imation for ψ. Thus we are requiring that eigenfunctions of (9.1) vanish on ∂ SG w n . This w assumption is reasonable for large n, because u ∈ Uw ∞ grows without bound on ∂ SG n , and analogous to the real line, one expects eigenfunctions of (9.1) to vanish where u gets infinitely large. Multiplying (9.1) by an arbitrary φ j and integrating over SG w n gives − (ψ)φ j + uψφ j = λψφ j .

(9.3)

Inserting (9.2) now gives −

n

ci

(φi )φ j +

i=1

n

ci

uφi φ j = λ

i=1

n

φi φ j .

ci

(9.4)

i=1

This can be simplified using the weak formulation for the Laplacian [S] (p.31). Recall that the energy E(u) of a function u is defined as 3 E(u) = limm→∞ ( )−m (u(x) − u(y))2 , 5 x∼y

(9.5)

where the sum is over all adjacent vertices x, y ∈ Vm . If u ∈ dom, then E(u, v) = − (u)v for all v ∈ dom0 E.

(9.6)

The integral above is over SG w n and the subscript 0 means that v must vanish on w ∂ SG n . Using (9.6) in (9.4) yields n i=1

ci E(φi , φ j ) +

n

ci

uφi φ j = λ

i=1

n

φi φ j .

ci

(9.7)

i=1

Equation (9.7) holds for all j = 1, ..., n, and can be expressed compactly as: c · (E + U − λP) = 0,

(9.8)

where we have defined the matrices E i j = E(φi , φ j ), Ui j = uφi φ j , Pi j = φi φ j , and c = (c1 , ..., cn ). For a specified blowup SG w n and spline space S(H0 , Vm ), the matrix elements of E and P can be derived analytically, but the elements of U need to be approximated using numerical integration techniques, many of which can be found in [B]. We rewrite (9.8) as c · (E + U )P −1 = λ c · I.

(9.9)

Thus, studying (9.1) numerically using FEM amounts to finding the eigenvalues and eigenvectors of (9.9). An eigenvector of (9.9) provides the coefficients in (9.2) for the approximate expansion of an eigenfunction ψ of (9.1). We have used FEM on the underlying space SG w 3 with a level-4 spline basis to study four different potentials u. We include partial data for one of these potentials below (the remaining data is available online at [FKS]). Fig. 11 shows one of the potentials


379

3 Fig. 11. Harmonic-Oscillator Potential u: u|∂ SG w 3 = 5 · (0.130, 0.087, 0.029)

Fig. 12. Groundstate Eigenfunction for u

Table 1. First 75 Eigenvalues for u 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.0654 0.1944 0.2528 0.2757 0.3699 0.3840 0.4578 0.5218 0.5397 0.6071 0.6166 0.6516 0.7049 0.8236 0.8351

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.8663 0.9601 0.9877 1.0954 1.1597 1.1642 1.1900 1.2213 1.2426 1.2444 1.2963 1.3331 1.4067 1.4509 1.4783

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1.4929 1.5423 1.5518 1.5710 1.6195 1.6640 1.7280 1.7459 1.7816 1.7907 1.8078 1.8953 1.9003 1.9349 1.9584

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.9856 1.9910 2.0846 2.1019 2.1260 2.1550 2.1559 2.2285 2.2350 2.2850 2.3037 2.3112 2.3176 2.3342 2.3492

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

2.3526 2.3928 2.4355 2.4533 2.4751 2.4885 2.5161 2.5288 2.5676 2.5796 2.5916 2.6152 2.6176 2.7021 2.7047

380


Fig. 13. 1st Excited Eigenfunction for u

80 70 60

N(x)

50 40 30 20 10 0 0

0.5

1

1.5

2

2.5

3

x

Fig. 14. N (x) for Potential u

4.4 y = 1.448*x + 2.831

4.2

data 1 linear

4

log N(x)

3.8 3.6 3.4 3.2 3 2.8 0

0.2

0.4

0.6

0.8

1

log x

Fig. 15. N (x) on Log-Log Scale

studied, while Figures 12 and 13 show the groundstate and first excited eigenfunctions, respectively, corresponding to this potential. We note that qualitatively the groundstate eigenfunction, denoted ψ0 , resembles a gaussian, which is the groundstate eigenfunction for (9.1) on R1 . Analogous to R1 , the maximum value of ψ0 occurs where the corresponding potential u attains its minimum value of zero. As one moves away from the


381

2.95

log W(x)

2.9

2.85

2.8

2.75

2.7

0

0.2

0.4

0.6

0.8

1

log x

Fig. 16. W (x) for Potential u, using 2α = 1.448 (Log-Log Scale)

location where u vanishes, ψ0 decays quickly to zero. This qualitatively supports the initial assumption that eigenfunctions ψ vanish on ∂ SG w 3 . The quantitative similarities 1 between ψ0 and a gaussian on R remain to be explored. Table 1 lists the first seventy-five eigenvalues for the potential shown in Fig. 11. Figure 14 shows the corresponding eigenvalue counting function, N (x). Figure 15 shows N (x) on a log-log scale for logx ≥ 0, along with the best-fit line in that domain. The slope for the best-fit line (≈ 1.448) approximates the exponential growth of N (x). Substituting the value of the best-fit slope for the exponent 2α in (8.23) then yields an approximation for the Weyl Ratio, W (x), which we plot on a log-log scale in Fig. 16. In Table 1, we note that the spacing between eigenvalues does not appear to be fixed as in the case of R1 . The additional data in [FKS] suggests that the spacing between eigenvalues tends to decrease for larger eigenvalues. Acknowledgement. We are grateful to Luke Rogers for useful discussions. The first author would like to thank Professor Ka Sing Lau for providing such a good opportunity and arranging the financial aid.

References [B]

Bockelman, B.: Dynamic Equations on the Sierpinski Gasket. Available at http://www.math.unl.edu/ ~s-bbockel1/dsweb/index.php [Ba] Barlow, M.: Diffusion on Fractals, Lecture Notes Math., Vol. 1690. Berlin-Heidelberg-New York: Springer, 1998 [BST] Ben-Bassat, O., Strichartz, R.S., Teplyaev, A.: What is not in the domain of the laplacian on a sierpinski gasket type fractal. J. Funct. Anal. 166, 197–217 (1999) [CDS] Coletta, K., Dias, K., Strichartz, R.: Numerical analysis on the sierpinski gasket, with applications to schrödinger equations, wave equation, and gibbs phenomenon. Fractals 12, 413–449 (2004) [FKS] Fan, E., Khandker, Z., Strichartz, R.: Harmonic Oscillators on Infinite Sierpinski Gaskets: Finite Element Method, http://www.math.cornell.edu/~khandker [FS] Fukushima, M., Shima, T.: On a spectral analysis for the sierpinski gasket. Potential Anal. 1, 1–35 (1992) [Ki] Kigami, J.: Analysis on fractals, Cambridge Tracts in Math. 143, Cambridge: Cambridge University Press, 2001 [R] Rammal, R.: Spectrum of harmonic excitations on fractals. J. Physique 45, 191–206 [RS] Reed, M., Simon, B.: Methods of mathematical physics, Vols. I-IV, Academic Press, 1978 [RT] Rammal, R., Toulouse, G.: Random walks on fractal structure and percolation cluster. J. Phys. Lett. 44, L13–L22 (1983) [S1] Strichartz, R.: Fractals in the large. Canad. J. Math. 50, 638–657 (1998)

382

[S2] [S3] [S4] [T]


Strichartz, R.: Analysis on products of fractals. Trans. Amer. Math. Soc. 357, 571–615 (2005) Strichartz, R.: Differential Equations on Fractals: A Tutorial. Princeton, NJ: Princeton University Press, 2006 Strichartz, R.: A fractal quantum mechanical model with Coulomb potential. Comm. Pure Appl. Anal. to appear Teplyaev, A.: Spectral analysis on infinite sierpinski gaskets. J. Funct. Anal. 159, 537–567 (1998)

Communicated by B. Simon


Communications in


Energy and Volume: A Proof of the Positivity of ADM Energy Using the Yamabe Invariant of Three-Manifolds Martin Reiris Mathematics Department, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail: [email protected] Received: 28 February 2008 / Accepted: 7 October 2008 Published online: 8 January 2009 – © Springer-Verlag 2008

Abstract: We give a new proof of the positivity (non-negativity) of ADM energy1 using the Yamabe invariant of three-manifolds. From a physical point of view, the new proof is motivated by a formula (explicitly non-negative) for the total ADM energy of emerging (asymptotically flat) stationary solutions on maximally expanding compact cosmologies. Mathematically, the proof is an application of the Thurston Geometrization of three-manifolds. Introduction Remark 1. The Yamabe invariant of a compact three-manifold as it is used in the present article, means the supremum over all conformal classes of three-metrics, of the infimum of the Yamabe functional (see below) at each conformal class. In the literature, the Yamabe invariant is also known under the name of sigma constant [1]. Also, it is common to call the Yamabe invariant to the infimum of the Yamabe functional on a given conformal class [6]. Our terminology shouldn’t be confused with it. The proof of the positivity of the ADM mass on asymptotically flat Riemannian three-manifolds2 of non-negative scalar curvature has a rich and long history. The first proof, given by Schoen and Yau [11] in 1979 (see also [12]), was partially motivated to conclude the proof of the Yamabe problem. Later, in 1981, Witten [13] gave another proof using four-spinors which explicitly displayed the non-negativity of mass through a Bochner-type formula. In 1997, Lohkamp [7] gave a different geometric proof, studying deformations of the scalar curvature on localized regions. More recently, the positivity of mass has been proved through the sharper lower estimates for the mass provided by 1 Properly speaking, we give a new proof of the Riemannian positive energy Theorem. Namely, we prove that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature cannot have negative mass. 2 We will always restrict to dimension three.

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the Penrose inequality [5]. Here, we will provide a proof of the non-negativity of mass which in a vague sense closes a circle of concepts opened since the original proof by Schoen and Yau motivated by the Yamabe problem. Being still a little imprecise, the present proof shows that the non-negativity of mass is implied by the solvability of the Yamabe problem on manifolds with non-positive Yamabe invariant and the computation [2] (after the proof of the Thurston Geometrization Conjecture) of the Yamabe invariant of three-manifolds, for manifolds whose Yamabe invariant is also non-positive.3 We will give thus a new proof of the following well known statement. Theorem 1. Say (M, g) is an asymptotically flat Riemannian three-manifold of nonnegative scalar curvature. Then, if m = 0 it is m > 0. Remark 2. The theorem doesn’t tell (and we won’t prove) that the manifold is flat and topologically R3 when m = 0. By asymptotically flat we mean |∂ (i) (g−g S )| = O(r −2−i ) m 4 with i = 0, 1, 2 and g S the Schwarzschild metric g S = (1 + 2r ) (dr 2 + r 2 d2 ). Let us give below a brief introduction of the main elements involved in the proof. We will end up explaining the ideas behind the main argument. Say (M, g) is a compact Riemannian three-manifold, define the Yamabe functional on the conformal class [g] of g as ˜ g˜ Rdv Y (g) ˜ = M 1 , Vg˜3 where g˜ is a metric in the conformal class of g, i.e. g˜ = e2 f g, Vg˜ is the volume of M under the volume form of g˜ and R˜ the scalar curvature of g. ˜ Denote by λ([g]) the infimum of Y in [g] and define the Yamabe invariant Y (M) of M as the supremum of λ([g]) over all conformal classes [g]. A landmark in geometric analysis is the resolution of the Yamabe problem (see [6] for a survey). Theorem 2. (Yamabe, Aubin, Trudinger, Schoen). Say g is not conformal to the standard sphere. Then, λ([g]) < λ(S 3 ) and there is a metric in [g] of constant scalar curvature reaching λ([g]). Observe that the Yamabe functional is scale invariant therefore if g˜ realizes λ([g]) so does any scaling of it. Observe too that if Y (M) < 0 the Yamabe invariant is equal to minus the infimum of the two-third power of the volumes of Yamabe metrics of scalar curvature negative one. Therefore maximizing the scalar curvature among unit volume Yamabe metrics is equivalent to minimizing the volume among Yamabe metrics of constant scalar curvature minus one. What is the relation between the signature of Y (M) and the topology of M, and how much is its value? This question was partially answered after the proof of the Thurston Geometrization Conjecture [3,9] via the Ricci flow. Let us briefly review how to obtain Thurston’s geometrization on three-manifolds (see [1] for a summary) as it is relevant to the article and to give a partial answer to the question before. Given a three-manifold M, to obtain the geometric decomposition one first performs the prime decomposition, i.e. factors M into a unique (up to reordering) connected sum of prime three-manifolds Pi . A prime three-manifold is one which is not the three 3 It is worth to remark that (in dimension three) the Positivity Energy Theorem is needed to settle the Yamabe problem in the case that the infimum of the Yamabe functional on the given conformal class is positive. Thus it is not needed to solve the Yamabe problem on manifolds with non-positive Yamabe invariant.

Energy and Volume: A Proof of the Positivity of ADM Energy

385

sphere and it is either S 2 × S 1 or it is irreducible (i.e. every two sphere bounds a disc). On each one of the resulting (prime) three-manifolds one performs the torus decomposition (JSJ) by excising incompressible tori (those whose fundamental group injects). In this way we obtain a set of manifolds with toric boundaries. Thurston’s geometrization asserts that each one of the resulting pieces admits a geometric structure among eight possible [1]. In particular, after this decomposition is carried out, there is a possibly empty set of manifolds with or without boundary admitting a complete hyperbolic metric of finite volume. We will denote such summands as Hi . An important property [8] of the prime decomposition is that if M = P1 . . . Pm and N = P˜1 . . . Pñ , then MN = P1 . . . Pm P˜1 . . . Pñ , where each denotes a connected sum. This property will be used in Step 3 inside the proof of Theorem 1. The following partial answer to the question above was proved in [2]. Theorem 3. Say M is a three-manifold whose Thurston decomposition has at least one component with hyperbolic geometry. Then, Y (M) < 0 and 2 3 Y (M) = −6 V (Hi ) . In this case the Yamabe invariant is only sensitive to the hyperbolic sector of the Thurston decomposition. We will exploit this fact to give a proof of the positivity of mass proceeding by contradiction and showing that if not the volume of a certain Yamabe 3 metric of scalar curvature negative one is below (−Y (M)) 2 with M a three-manifold of negative Yamabe invariant. Let us explain the main steps to carry out this program. Roughly speaking a metric is asymptotically flat at spatial infinity if asymptotically it looks as the Schwarzschild metric 1 g= (1) dr 2 + r 2 d, 1 − 2m r where a priori the mass m can have arbitrary signature.4 Let S(r ) be the sphere of constant coordinate radius r , and let n be the outgoing normal. Being spherically symmetric the Schwarzschild metric is well known to be conformally flat. In fact defining 1 − 2m 1 + m r v= , 2 1 − 1 − 2m r

we get explicitly m 4 g = 1+ (dv 2 + v 2 d2 ), 2v which after the change of variables u = ln v transforms into (u − ln m/2) (du 2 + d2 ), 2

(2)

(u − ln −m/2) (du 2 + d2 ), 2

(3)

g = 4m 2 cosh4 if m > 0 and g = 4m 2 sinh4

4 We will assume throughout that the mass m is different from zero.

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if m < 0. The conformal presentation above (Eqs. (2) and (3)) clearly display the dependence of the normal component of the Ricci curvature with respect to the signature of mass. In fact recalling that Ric(n, n) = −2e−2 f ∂u2 f with f half the logarithm of the conformal factor in Eqs. (2) and (3) respectively, we get Ric(n, n) =

−1 8m 2 cosh6 u−ln2m/2

,

if m > 0 and Ric(n, n) =

1 8m 2 sinh6 u−ln 2−m/2

,

if m < 0. Thus if the mass is positive the normal component of the curvature is negative while it is positive if the mass is negative.5 We will use this fact to make a volume comparison. Observe that the mean curvature of a sphere S(r ) (as a two-spheres embedded 2 1− 2m

r in Schwarzschild space) is θ = . As a consequence, the surrounding geometry r of the spheres S(r ) inside the Schwarzschild space gets closer and closer to the surrounding geometry of the spheres S E (r ) of radius r inside Euclidean space. Thus, large spheres provide a starting point from which to compare volumes between the Schwarzschild and the Euclidean spaces. Let us describe this in more detail. Fix a center o in Euclidean three-space and denote by S E (r ) the two-spheres with center o and radius r . In Schwarzschild space let d(r1 , r0 ) (r1 < r0 ) be the Riemannian distance between the spheres S(r1 ) and S(r0 ). Finally denote by VE (r, r0 ) the Euclidean volume lying between the spheres S E (r ) and S E (r0 ) and denote by VS (r1 , r0 ) the volume lying between S(r1 ) and S(r0 ) inside the Schwarzschild space. In the sense of the classical volume comparison, one would like to compare the volumes VS (r1 , r0 ) and VE (r0 − d(r1 , r0 ), r0 ) as r1 decreases, starting from r1 = r0 , with r0 a large radius. According to the BishopGromov volume comparison, as r1 decreases the volume VS (r1 , r0 ) increases faster than the volume VE (r0 − d(r1 , r0 ), r0 ) when m > 0 (because Ric(n, n) < 0) and slower if m < 0 (because Ric(n, n) > 0). Let us quantify this volume comparison. From the 1 expansion of √1−x we get

1 1−

2m r

=1+

m 3 m 2 5 m 3 35 m 4 + + + + O(r −5 ). r 2 r2 2 r3 8 r4

(4)

The distance d(r1 , r0 ) is estimated as d(r1 , r0 ) = r0 − r1 + m ln

r0 + O(r1 , r0 ), r1

(5)

where O(r1 , r0 ) → 0 as r1 , r0 → ∞ and the volume VS (r1 , r0 ) as 1 3 m 3m 2 5m 3 r0 (r0 − r13 ) + (r02 − r12 ) + (r0 − r1 ) + ln + O(r0 , r1 ) , VS (r1 , r0 ) = 1 3 2 2 2 r1 (6) 5 One may interpret that by saying that if m < 0 the Riemannian slice focuses into a naked singularity (at u = ln − m/2) while if m > 0 the slice gets thickened (forming a horizon at u = lnm/2) preventing any singularity.


387

with O(r0 , r1 ) as above and 1 the volume of the unit two-dimensional sphere. Formulas (5) and (6) show that given r1 the difference in volume between the Schwarzschild and Euclidean space increases to infinity if m > 0 as r0 goes to infinity (and comparing them from r0 ) and to negative infinity if m < 0. As said before, we will use this fact, proceeding by contradiction and assuming there exists an asymptotically flat metric with negative mass, to find a Yamabe metric of constant scalar curvature negative one whose 3 volume is below (−Y (M)) 2 for M certain compact three-manifold with negative Yamabe invariant. In other words we increase the Yamabe invariant of M. The idea is the following. Pick a (any) compact hyperbolic manifold H . Say g H is the (unique) hyperbolic metric of sectional curvature −1. Scale g H to get a metric g K of sectional curvature −K (and scalar curvature −6K ) in such a way that the local geometry is almost flat. Pick the (hypothetical) asymptotically flat metric of negative mass and, loosely speaking, place it inside H in replacement of a big ball (in the metric g K ) whose geometry is not far from flat. In this way, as has been argued above, one obtains a new metric g (and a new manifold M) whose volume is substantially below the volume of (H, g K ). The crucial point is to show that the gluing can be done with enough care to guarantee that the Yamabe metric of scalar curvature −6K (in the conformal class of g in M) has a volume still below the volume of (H, g K ). It is then shown that −Y (M) > −Y (H ) > 0 which gives the mentioned contradiction. A cosmological motivation. The reasoning above aroused in the study of the long time evolution of constant mean curvature cosmological solutions having maximal rate of expansion, i.e. expanding as the K = −1 Robertson-Walker cosmological model does. We mention here (sketchily) the main lines of the motivation6 . For a detailed analysis of the presentation below see [10]. Say H = M is a hyperbolic manifold. For any CMC (constant mean curvature) state (g, K ), where g is a three-metric and K the second fundamental form, define the reduced volume7 V(g, K ) = H3 Vg , where H = −k 3 is the Hubble parameter. It is known that the infimum of V in the set of all CMC 3 states is given by (− 16 Y (M)) 2 , and therefore Vin f = Vg H . Define the CMC energy as 1 (V − Vin f ). E= (7) 4π H Say N is the lapse when we take the mean curvature k as time. Define the Newtonian 2 potential φ = N3k − 1. Then φ satisfies the Poisson like equation φ + |K |2 φ = | Kˆ |2 , where Kˆ is the traceless part of K . Consider for simplicity an empty-matter universe which is expanding and suppose that some gravitational energy collapsed into a set of black holes that, asymptotically in time, become asymptotically flat and away from each other. Using the Einstein equations it is seen that the CMC energy evolves as

dE 1 =− N˜ | Kˆ |2 dvg + E, dσ 4π H 6 Although the mathematics can be made rigorous assuming suitable hypotheses, the hypotheses themselves are, mathematically speaking speculative [10]. 7 Under the name of Reduced Hamiltonian the quantity (up to a constant) V was first introduced and studied in the context of long time evolution by Fischer and Moncrief [4].

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where σ = − ln −k is the logarithmic time. Making the assumption that the volumes enclosed by the black holes grow slower than H12 and that at big balls surrounding them i (centers of mass) the potential is approximately φ ∼ −m r we get after integration by dE parts that after a long time dσ ∼ 0 and

E∼ mi + | Kˆ |2 (1 + φ)dvg . (8) (∪Bi )c

The formula above can be interpreted as E ∼ M + R, where M is the total mass of the black holes and R the total energy of radiation. Observe that the expression R coincides with the kinetic term in the linearized ADM energy (see [10] for a more complete description). A remarkable fact about the formula (8) is that the energy E on the left-hand side is explicitly positive (because of its definition in (7)). This is the fact that inspired the present proof of the positivity of mass. Proof of Theorem 1. As it was explained in the introduction, we will proceed by contradiction and assume we have an asymptotically flat Riemannian manifold (M, g) with non-negative scalar curvature and negative mass. We will use fresh notation in this section. Fix any compact hyperbolic three-manifold H , with hyperbolic metric g H of sectional curvature minus one having volume V (H ). The proof of Theorem 1 is made in four steps. In the first three steps we will assume that outside a compact set in M the metric g is exactly Schwarzschild. We will explain in the fourth step the necessary modifications to account for the general case. The first step consists in gluing the metric g to a hyperbolic metric g K of sectional curvature −K in the hyperbolic disc model, carefully controlling the quotient −R/(6K ) with R the scalar curvature of the resulting Riemannian metric. Once this is done we place it inside the hyperbolic manifold H with metric K1 g H of sectional curvature −K . Call the resulting Riemannian manifold (MH, g ). The second step proceeds to construct a barrier for the solution φ of the Yamabe problem (with scalar curvature −6K ) with base metric g . In the third step we show, using the barrier found in the second step, that the volume of (MH, φ 4 g ) is below the minimum possible provided by the Yamabe invariant of MH . Step 1. As said we assume that outside a compact set in M the metric g is of the form m 4 g = 1+ (dr 2 + r 2 d2 ). (9) 2r Recall that the disk representation of a hyperbolic space of sectional curvature −K (and therefore scalar curvature −6K ) is gK = 1−

1 Kr2 4

2 2 2 2 (dr + r d ).

(10)

We represent a conformally flat metric by e2 f g F with g F = dr 2 + r 2 d2 a flat metric. We start the gluing of the metrics (9) and (10) by linearly deforming the exponent f from some radius r0 to some radius r1 . A schematic representation can be seen in Fig. 1. Denote by f S and f K half the logarithm of the conformal factors in the metrics (9) and (10) respectively, namely Kr2 f K = − ln 1 − , 4


389

Fig. 1. A schematic representation of the gluing

and

m . f S = 2 ln 1 + 2r

Denote by f L the linear function tangent to f S at r0 . There are unique K and r1 such that f L is tangent to f K at r1 . We get them solving the system of equations −m K r1 = , m Kr2 r02 1 + 2r0 2 1 − 41 K r12 m m (r1 − r0 ) + 2 ln 1 + . − ln 1 − =− 4 2r0 r2 1 + m 0

(11)

(12)

2r0

We display now the dependence of r1 and K with respect to r0 and m. Make δ = r1 /r0 . From Eq. (11) we get 1 1 m = −δ + 1. 2 2r0 1 + m Kr 1 − 41 2r0 Putting this into Eq. (12) and making U = 1 + 2rm0 we get 1−U 1−U + 1 − ln U 2 = 2 (δ − 1), ln δ U U and rearranging terms 1 ln δ 1−U U + 1 U2 2 1−U U

= δ − 1.

When r0 → ∞, U → 1 and δ → 4 (use the equivalent ln x ∼ x − 1 as U tends to one). 2m Note that K ∼ − δr 3 as r 0 → ∞. 0

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M. Reiris

Let R be the scalar curvature of e2 f L g F considered in the domain where r lies between r0 and r1 . As will be explained in Step 2 we need to estimate the maximum value of −R/(6K ) in the interval [r0 , r1 ], we do that below. Being spherically symmetric, the scalar curvature R is expressed as 2f f s2 R = −4e−2 f f + + , (13) r 2 d . Thus, as f = f L is linear and increasing, −R is a decreasing function of where = dr r in the interval [r0 , r1 ]. Therefore −R/6K is maximum at r0 . We have ⎛ ⎞

m2 ⎟ 2 ⎠ m 3 m 4 r0 1 + 2r0 2r0 1 + 2r0 (1 − U ) (1 − U ) 1+ . = 16e2 f (r0 ) 2 2U r0 U

⎜ −R(r0 ) = 4e2 f (r0 ) ⎝−

2m

+

From Eq. (11) we get K = 4δ

(1 − U ) 1 U r12 1 +

1 δ U (1 − U )

.

Thus −R/(6K ) → (2/3)δ → 8/3 as r0 → ∞. So far we constructed a C 1 exponent, to ¯ get a smooth exponent (that we will denote f¯) in the conformal factor (e2 f ) we need to deform (slightly) the functions f L and f S at r0 and the functions f L and f K at r1 . As will be explained later we want to do so without changing much the maximum of 8/3 for the quotient −R/(6K ). This is a delicate operation as the scalar curvature Equation involves the second derivative of f . Let us explain how the deformation is performed at r0 . The deformation at r1 proceeds along similar lines. Pick a function ξ of one variable, positive and symmetric around the origin with support in [−1, 1] and total integral one. Define the kernel ξ = 1 ξ( x ). Say f is the function equal to f S before r0 and the function f L after r0 . We smooth it out by convolving it with the kernel ξ , i.e. we consider the function

f¯(r ) = f (t)ξ (r − t)dt. r Integrating by parts we get f¯ (r ) = f (t)ξ (r −t)dt and f¯ (r ) = 0 f (t)ξ (r −t)dt (note that f (r ) = 0 if r > r0 ). This shows that given β there is γ such that we can modify f on [r0 − γ , r0 + γ ] to get | f¯ − f (r0 )| ≤ β, | f¯ − f (r0 )| ≤ β on [r0 − γ , r0 + γ ] while making the second derivative f¯ increasing (observe that f S is increasing). In particular we can see from Eq. (13) that, up to β, −R/(6K ) passes from zero to 8/3 in the interval [r0 − γ , r0 + γ ]. This finishes the construction. Summarizing, we have constructed a metric on M equal to g until r0 − γ , hyperbolic of sectional curvature −K after r1 + γ and conformally flat with linear exponent in the conformal factor in the interval [r0 + γ , r1 − γ ] in such a way that the quotient −R/(6K ) has a maximum in M approaching 8/3 as r0 tends to infinity if we choose γ (r0 ) → 0 as r0 → ∞ conveniently. We place now the metric constructed above inside the scaled hyperbolic manifold (H, K1 g H ). Observe that the annulus [r1 + γ , r1 + 1] is isometric to an annulus


391

B( p, s1 )/B( p, s0 ) in (H, K1 g H ), where B( p, s) denotes a ball centered at p and of radius s in (H, K1 g H ). After excising the ball B( p, s0 ) we identify both annulus by the given isometry, thus constructing a new manifold denoted as MH and a Riemannian metric on it denoted as g . Step 2. We study now the constant scalar curvature equation 6K φ 5 = 8φ − R φ,

(14)

for a metric gY = φ 4 g . We will construct an upper barrier φT for the solution φ allowing us to estimate the total volume of (MH, gY ). Let φ K be the conformal factor such that 4 g = g on [r − 2γ , r + 1]. Define the upper barrier φ to the solution φ of Eq. (14) φK K 0 1 T as (we will prove soon this is actually a barrier) ⎧ i) 1 on the H − side of the two − sphere {r = r1 }, ⎪ ⎪ ⎪ ii) φ K (r ) for r in [r0 − 2γ , r1 + 1], ⎪ ⎪ ⎨ iii) in such a way that F = 6K φT5 + R φT − 8φT ≥ 0, φT = (15) for r in [r0 − 3γ , r0 − 2γ ], ⎪ ⎪ ⎪ iv) φ (r − 2γ ) ≤ φ (constant) ≤ φ (r − 2γ ) + γ , ⎪ ⎪ T 0 T T 0 ⎩ on the M − side of {r = r0 − 3γ }. The construction in the third step in the definition (15) above can be done following the next argument. For a function φT depending only on the radius, write8 φ = 1 ds m 2 A(s) (A(s)φT (s)) , where s is the radial distance, i.e. dr = (1 + 2r ) , A(s) is the area

of the two sphere with s constant and = φT = φ K therefore φT = 1−

d ds .

For r > r0 − 2γ we have defined φT as

1 1 Kr2 2 1+ 4

m 2r

,

(16)

for r greater but close to r0 − 2γ . A straightforward computation gives φT r0 − 2γ ) ∼ 1 1 ) r 2 < 0 as r0 → ∞. Pick a function of one variable ξ , being zero for x < −1 m( 21 − 2δ 0

and one for x > 0 with a graph as is represented in Fig. 2. Define φT on [r0 −3γ , r0 −2γ ] by running the ODE 5 (s)A(s) r − (r0 − 2γ ) 6K φ K (A(s)φT (s)) = ξ γ 8 backwards and starting from r0 − 2γ . Say that at r = r0 − 2γ and at r = r0 − 3γ it is s = s0 and s = s1 respectively. We have

s0 r − (r0 − 2γ ) 1 5 A(s)ξ 6K φ K A(s)φT (s) = A(s0 )φT (s0 ) − (s)ds. (17) 8 γ s Fix (see Fig. 2) in such a way that A(s1 )φT (s1 ) = 0. Then it is φT = 0 and φT constant for s < s1 . Observe that from Eq. (17) it is φT ≤ 0 and consequently φT is increasing in the decreasing direction of s. Therefore F = 6K φT5 + R φT − 8φT = 6K φT5 −

8 5 (AφT ) ≥ 6K φ K (s0 )(1 − ξ ) ≥ 0. A

8 This formula can be seen easily by integration by parts in a region between two spheres (say S(s ) and 1 S(s)) and then differentiating with respect to s.

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M. Reiris

Fig. 2. A schematic representation of ξ

This finishes the construction. Note that φT ≥ 1 everywhere. To show that φT is an upper barrier for φ, we proceed by contradiction and assume that the infimum of φT − φ is less than zero. Then at the point q where it takes place it is φ(q) > φT (q) ≥ 1. From the equation 8(φT − φ) = 6K (φT5 − φ 5 ) + R (φT − φ) − F, we get 6K (φT5 (q) − φ 5 (q) +

R (φT (q) − φ(q))) ≥ F(q) ≥ 0. 6K

(18)

Express φT5 − φ 5 as (φT − φ)(φT4 + φT3 φ + φT2 φ 2 + φT φ 3 + φ 4 ). Plugging it into Eq. (18) gives R 4 3 2 2 3 4 6K (φT (q) − φ(q)) (φT + φT φ + φT φ + φT φ + φ )(q) + ≥ 0. 6K R The factor (φT4 + φT3 φ + φT2 φ 2 + φT φ 3 + φ 4 )(q) + 6K is greater or equal to 5 − 8/3 as r0 goes to infinity, thus if r0 is chosen sufficiently large φT (q) ≥ φ(q) which gives a contradiction. Step 3. We compare now the volumes of V (MH, gY ) and V (H, K1 g H ). Fix a coordinate radius r = r2 with s = s2 . We observe that the ball B( p, s2 ) in (H, K1 g H ) and the compact side of the two sphere r = r2 in (M, g) have bounded volume as r0 → ∞. Also observe that because φT → 1 as r0 → ∞, the volume of the annulus [r2 , r1 ] in (MH, gY ) minus the volume of annulus B( p, s1 )/B( p, s2 ) in (H, K1 g H ) tends to −∞ as r0 tends to ∞. Thus for r0 sufficiently large it is V (MH, gY ) < V (H, K1 g H ). Now the manifold MH has (independently of the nature of M) the hyperbolic piece H in its Thurston decomposition. Therefore by Theorem 3 it must be V (MH, gY ) ≥ V (H, K1 g H ), which is absurd. This finishes the proof of Theorem 1 in the case the metric g is exactly Schwarzschild outside a compact set in M. Step 4. We treat now the case when the metric g is not exactly Schwarzschild outside a compact set. We start by introducing a notation. We say that a function h is an O(r −α )


393

as r0 → ∞ if sup{h/r −α } is bounded as r0 → ∞. Now we observe that the argument carried out in steps one, two and three above can be repeated if we can (slightly) deform 1

+

the metric g to a metric g˜ inside the annulus [r0 −r02 , r0 ] to get an exact Schwarzschild metric of mass m after r0 in such a way that Rg˜ is an O(r −3−2 ) as r0 → ∞ and in particular with sup{−Rg˜ /r 3 } tending to zero as r0 tends to ∞. Let ϕ be a smooth nonnegative function of one variable x with range in [0, 1], being one for x < 0 and zero 1 +

for x > 1. Consider the metric g˜ = g S + ϕ A computation gives

r −r0 +r02 1 +

r02

(g − g S ). Write ∇˜ = ∇ S + .

1 ρ ν Rg˜ = Ric(S)αβ g˜αβ − g˜ αβ ∇αS ∇βS ln |g| ˜ + µρ ν ρ¯ g˜ µρ¯ , 2

(19)

where |g| ˜ is the quotient of the volume forms of g˜ and g S , i.e. dvg˜ = |g|dv ˜ g S . The Christoffel symbols are computed as µ

αβ =

1 S mµ (∇ (g˜βm − g(S)βm ) + ∇βS (g˜αm − g(S)αm ) − ∇mS (g˜αβ − g(S)αβ ))g S . 2 α (20) − 25 −

Substituting g˜ − g S for (ϕ)(g − g S ) in Eq. (20) it is seen that and are O(r ) and O(r −3−2 ) as r0 → ∞ respectively. This implies that the last term of Eq. (19) is an O(r −5−2 ) as r0 → ∞. To analyze the second term in Eq. (19) we recall that ν = ∂ ln |g|, νρ ˜ which makes it an O(r −3−2 ) as r0 → ∞. The first term in Eq. (19) ρ is seen to be an O(r −5 ) as r0 → ∞ by noting that it can be written as Ric(S)αβ (g˜αβ − g(S)αβ ). ∇S

References 1. Anderson, M.T.: Scalar curvature and geometrization conjectures for 3-manifolds. In: Comparison geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ., 30, Cambridge: Cambridge Univ. Press, 1997, pp 49–82 2. Anderson, M.T.: Canonical metrics on 3-manifolds and 4-manifolds. Asian J. Math. 10(1), 127–163 (2006) 3. Cao, H.-D., Zhu, X.-P.: A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10(2), 165–492 (2006) 4. Fischer, A.E., Moncrief, V.: The reduced hamiltonian of general relativity and the σ -constant of conformal geometry. In: Mathematical and quantum aspects of relativity and cosmology (Pythagoreon, 1998), Lecture Notes in Phys. 537, Berlin: Springer, 2000, pp 70–101 5. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001) 6. Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) 17(1), 37–91 (1987) 7. Lohkamp, J.: Scalar curvature and hammocks. Math. Ann. 313(3), 385–407 (1999) 8. Milnor, J.: A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84, 1–7 (1962) 9. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http://arXiv.org/ list/math/0211159, 2002 10. Reiris, M.: General K = −1 Friedman-Lemaître models and the averaging problem in cosmology. Class. Quant. Grav. 25 (8), 085001, 2008, 26 pp 11. Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979) 12. Schoen, R., Yau, S.T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981) 13. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981) Communicated by G. W. Gibbons


Communications in


Scalar Curvature and Asymptotic Symmetric Spaces Mario Listing Abteilung für Reine Mathematik, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg, Germany. E-mail: [email protected] Received: 22 September 2006 / Accepted: 15 October 2008 Published online: 5 February 2009 – © Springer-Verlag 2009

Abstract: A well known fact is that a complete Riemannian (spin) manifold which is strongly asymptotically flat and has nonnegative scalar curvature must be isometric to the Euclidean space. Using Witten’s positive mass argument this paper proves the analogous rigidity result for certain asymptotic symmetric spaces. Contents 1. 2. 3. 4. 5.

Noncompact Bochner–Weitzenböck Technique Spin Symmetric Lie Algebras . . . . . . . . . Symmetric Spaces with Spin Killing Structures Proof of the Main Theorem . . . . . . . . . . Irreducible Examples . . . . . . . . . . . . .

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397 405 417 422 425

Introduction In [20] R. Schoen and S.T. Yau proved the positive mass conjecture in general relativity by showing an interesting scalar curvature rigidity result of the 3–dimensional Euclidean space. In [21] E. Witten used spin geometry and a non–compact Bochner technique to give an easier proof of Schoen and Yau’s result. R. Bartnik generalized in [3] Witten’s argument to n– dimensions. In particular, he showed that a strongly asymptotically flat spin manifold of non–negative scalar curvature must be isometric to the Euclidean space. This lead to studies on the scalar curvature rigidity of asymptotic hyperbolic manifolds which was started by M. Min–Oo in [19] and continued by M. Herzlich in [11]. Moreover, we presented in [16] the analogous result for hyperbolic product manifolds. The present paper now proves scalar curvature rigidity of certain asymptotic symmetric spaces. Supported by the German Science Foundation.

396

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Let (G/H, g0 ) be a simply connected symmetric space of non– positive sectional curvature and n = dim G/H ≥ 3. For the Euclidean space the asymptotic assumptions to obtain rigidity differ slightly from the asymptotic assumptions needed in this paper and since the case G/H = Rn has been extensively studied, G/H is always supposed to be not flat. Furthermore, in order to make H “as large as possible”, G will be the full (orientation preserving) isometry group of the symmetric space, i.e. Rm will be represented by SO(m) Rm /SO(m). Thus, G/H is given by k m (G/H, g0 ) = SO(m) R /SO(m) × G j /H j , j=1

spaces of noncompact type and where m ≥ 0 and G j /H j are irreducible symmetric minimal sectional curvature −κ j < 0. We define κ := κ j which of course depends on the choice of g0 . A Riemannian (G, H )–structure on a Riemannian manifold (M n , g) is a reduction of the SO–frame bundle of (M, g) to an H –principal bundle PH M where H → SO(n) is the isotropy representation of the symmetric space G/H . The Levi Civita connection reduces to PH M if and only if the intrinsic torsion tensor T(PH M) of PH M vanishes (cf. [12]). Suppose that PH M is a Riemannian (G, H ) structure on (M, g), then (M, g, PH M) is said to be strongly asymptotic to (G/H, g0 ) if there is a compact set C ⊂ M, a diffeomorphism f : M −C → G/H − B R (0) (for some R > 0), and a torsion free Riemannian (G, H )–structure PH0 (M −C) ⊆ PSO (M −C, f ∗ g0 ) of (M −C, f ∗ g0 ) together with an isomorphism A : PH0 (M − C) → PH M|M−C in such a way that 1 cg

| A˜

≤ f ∗ g0 ≤ c · g

− Id| + |∇ 0 g|

∈

L1

for some c > 0 , ∩ L 2 (M − C; eκr vol0 ).

In this case r is the f ∗ g0 –distance to a fixed point in M − C, ∇ 0 means the Levi Civita connection of f ∗ g0 and A˜ is the section in End(T M|M−C ) which is uniquely given by A (note that PH0 (M − C) and PH M|M−C are reductions of the linear frame bundle of M − C and that the torsion free H – structure PH0 (M − C) exists and is unique up to equivalence). Unfortunately, the method presented in this paper is not going to provide scalar curvature rigidity of all symmetric spaces with sectional curvature K ≤ 0. The proof needs an additional property of the model space G/H which will be called spin Killing. In fact, 0 := ∇ 0 +T0 on a subG/H is spin Killing if G/H is spin and there is a flat connection ∇ bundle V0 ⊆ /S G/H preserved by the Levi– Civita connection ∇ 0 such that ∇ 0 T0 = 0. 0 are called T0 –Killing spinors since they are closely related Spinors parallel w.r.t. ∇ to Killing vector fields on G/H . It is not hard to see that a symmetric space G/H is spin Killing if and only if all of its irreducible components are spin Killing. However, deciding spin Killing for irreducible symmetric spaces is a nontrivial representation theoretical problem. So far the following simply connected examples are known: S n , CP2n−1 , RHn , CH2n−1 , symmetric spaces of type II and IV are spin Killing, whereas CP2n , HPn (n > 1), CaP2 , G 2 /SO(4), CH2n , HHn (n > 1), CaH2 , G2C /SO(4) are not spin Killing.

Scalar Curvature and Asymptotic Symmetric Spaces

397

Main Theorem. Suppose (G/H, g0 ) is a simply connected symmetric space which is spin Killing and of sectional curvature 0 ≡ K ≤ 0. Let (M, g, PH M) be a complete Riemannian spin manifold of dimension n ≥ 3 with a Riemannian (G, H )–structure PH M ⊆ PS O (M, g). If (M, g, PH ) is strongly asymptotic to (G/H, g0 ) and the scalar curvature satisfies

n + 1 κ · |T(PH M)|, scal(g) ≥ scal(g0 ) + 3n 2 (1) 2 then PH M is torsion free and (M, g) is isometric to G/H . In particular, this theorem gives scalar curvature rigidity of symmetric space G/H if each of its irreducible components G j /H j is one of the following spaces: RHm , CH2m−1 or a type IV symmetric space. In what way other symmetric spaces with K ≤ 0 are scalar curvature rigid is unclear at the moment. Although CH2n is not spin Killing, a scalar curvature rigidity result has been proved for the complex hyperbolic space in even complex dimension by H. Boualem and M. Herzlich in [5], respectively by us in [17]. However, in these cases the assumptions are stronger (cf. [5]) or there are more terms involved on the right hand side of the above scalar curvature inequality (cf. [17]). In order to show the main result we are going to use the noncompact Bochner– Weitzenböck technique which is presented in detail in the first section. In contrast to the compact case a second order elliptic operator will not be sufficient to use this method, we really need first order elliptic operators which are known as Dirac operators. In particular, under a certain curvature inequality this method yields Killing spinors from asymptotic Killing spinors which are obtained by the asymptotic assumptions and the Killing spinors on the model space G/H . These Killing spinors will provide the Riemannian curvature. Here we have to take much more care than previous works on the subject (like in [11,19]) since we do not start with a torsion free H –structure; it is part of the proof to show T(PH M) = 0. In the second section we develop a theory on Lie algebra level for the existence of Killing structures on symmetric spaces. In the third section we apply these results and provide the known examples of irreducible spin Killing symmetric spaces. In the last section we discuss the main theorem for the three known irreducible spin symmetric spaces, in fact, the main theorem reduces to Min-Oo’s rigidity result for the real hyperbolic space (cf. [19]) and to Herzlich’s, respectively our, rigidity results for the complex hyperbolic space in odd complex dimensions (Herzlich proved scalar curvature rigidity for CH2n−1 under the additional assumption T(PH M) = 0 ⇔ ∇ J = 0). A new example is presented by the scalar curvature rigidity of type IV symmetric spaces. 1. Noncompact Bochner–Weitzenböck Technique 1.1. Integrated Bochner–Weitzenböck formula. Suppose S → M is a Dirac bundle over M (cf. [9,15]), then the corresponding Dirac operator D : Γ (S) → Γ (S) satisfies the general Bochner identity (cf. [15, Ch. II Thm. 8.2]) D2 = ∇ ∗ ∇ + R, where R ∈ Γ (End(S)) is given by R(ϕ) :=

1 e j · ek · Re j ,ek (ϕ). 2 j,k

(2)

398

M. Listing

= ∇ + T on S, In order to prove the main theorem we have to consider connections ∇ where ∇ is the Dirac connection and T is a section in T ∗ M ⊗ End(S). The following proposition is the main tool to prove the general positive mass theorem as well as the main result of this paper. Proposition 1.1. Suppose S is a complex Dirac bundle over (M n , g). Let T be a section of T ∗ M ⊗ End(S) such that T X is a selfadjoint endomorphism for all X ∈ T M. Define

:= D + z · Id on S with z ∈ iR. := ∇ + T and the elliptic operator D the connection ∇ Then the integrated Bochner–Weitzenböck formula

ψ =

Dψ

+ Rϕ, ψ ν ϕ + ν · Dϕ, ∇ψ − Dϕ, ∇ ∇ϕ, ∂N

N

holds for all ϕ, ψ ∈ Γ (S) and compact N ⊂ M, where ∂ N is the boundary of N , ν is is given by the outward normal vector field on ∂ N and R = R + |z|2 − R

n

Te j ◦ Te j + δT

j=1

(e1 , . . . , en is an orthonormal basis of T M and δT := j (∇e j T)e j ). Furthermore, the

is selfadjoint: ν + ν · D boundary operator ∇

ψ =

. ν ϕ + ν · Dϕ, ν ψ + ν · Dψ ∇ ϕ, ∇ ∂N

∂N

follows from ν + ν · D Proof. The selfadjointness of the boundary operator ∇ ∇ν ϕ + ν · Dϕ, ψ = ∇ϕ, ∇ψ − Dϕ, Dψ + Rϕ, ψ ∂N

N

ϕ, ∇ν ψ + ν · Dψ ,

= ∂N

and the fact that T X and zγ (X ) are selfadjoint endomorphisms of S. Since ∇ is a Dirac connection and (T X )∗ = T X , we obtain ∇ψ = ∇ϕ, ∇ψ + Tϕ, ∇ψ + ∇ϕ, Tψ + Tϕ, Tψ ∇ϕ, N

N

∇ν ϕ + Tν ϕ, ψ +

= ∂N

∇ ∗ ∇ϕ, ψ − δTϕ, ψ + Tϕ, Tψ .

N

Moreover, we conclude from z ∈ iR and [15, Chp.II Eq. (5.7)]:

Dψ

=−

ψ + D2 ϕ, ψ + |z|2 ϕ, ψ. Dϕ, ν · Dϕ, N

∂N

N


399

Hence, the general Bochner identity (2) provides the integrated Bochner–Weitzenböck formula with = R + |z|2 − R

n

T∗e j ◦ Te j + δT

j=1

and (T X )∗ = T X proves the proposition.

1.2. Killing structures. In general a Killing structure is determined by a Lie algebra representation and together with a Riemannian connection it yields a flat connection on a certain vector bundle. In order to motivate a definition of Killing structures on Dirac on a subbundle V of S. We set bundles S, we start with a flat connection ∇ − ∇ ∈ Γ (T ∗ M ⊗ End(S)) , T := ∇ where the trivial extension of T X to End(S) is considered and ∇ is the Dirac connection on S. Define Im(T) := span{T X ϕ| X ∈ T M, ϕ ∈ S} ⊆ S , then Im(T) is a subbundle of S if ∇T = 0. Moreover, if T is parallel w.r.t. the Dirac and ∇ satisfy on V: connection ∇, the curvatures of ∇ X,Y = R X,Y + [T X , TY ]. 0=R Definition 1.2. Suppose S is a complex Dirac bundle over (M, g) and T is a section in T ∗ M ⊗ End(S). A spinor ϕ is called T–Killing spinor if ϕ is parallel w.r.t. the con := ∇ + T. If V is a subbundle of S, we denote by K(V, T) the vector space nection ∇ of T–Killing spinors ψ ∈ Γ (V). The pair (V, T) is called a Killing structure if V is trivialized by T–Killing spinors. In particular, if (V, T) is a Killing structure, we have rk(V) = dim K(V, T). A Killing structure is said to be selfadjoint if T X is a selfadjoint endomorphism for all X ∈ T M and it is said to be Riemannian if (T X )∗ = −T X for all X . Furthermore, T is a parallel Killing structure if additionally ∇T = 0. = ∇ + T which satisfy an integrated Selfadjoint Killing structures yield connections ∇ Bochner–Weitzenböck formula like in the previous section while Riemannian Killing = ∇ + T. The notation T–Killing spinor is structures yield Riemannian connections ∇ originated from the following observations: Let S be a complex Dirac bundle and T be a section in T ∗ M ⊗ End(S) such that ∇T = 0 and (T X )∗ = −T X for all X ∈ T M. Suppose ϕ, ψ ∈ Γ (S) are T–Killing spinors. Then the vector fields V1 and V2 defined by V1 , X := Re T X ϕ, ψ ,

V2 , X := Im T X ϕ, ψ

are Killing vector fields. Note that V1 = 0 if ϕ = ψ. Killing spinors of selfadjoint Killing structures supply in a similar way Killing vector fields on the base manifold. Let T ∈ Γ (T ∗ M ⊗ End(S)), then dT ∈ Γ (Λ2 M ⊗ End(S)) and δT ∈ Γ (End(S)) are defined as follows: dT(X, Y ) := (∇ X T)Y − (∇Y T) X ,

δT :=

n j=1

(∇e j T)e j .

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Proposition 1.3. Suppose V is a subbundle of S and T is a section in T ∗ M ⊗ End(S) with Im(T) ⊆ V. If the rank of V satisfies rk(V) = dim K(V, T), then the Dirac connection ∇ preserves V, in particular ∇π = 0, where π denotes the = ∇ + T defines a flat connection on orthogonal projection S → V. Furthermore, ∇ ∗ V. If T is additionally selfadjoint (i.e. (T X ) = T X for all X ∈ T M), then dT vanishes and the curvature of the Dirac connection ∇ satisfies on V R X,Y + T X TY − TY T X = 0.

(3)

Proof. Set m := dim(K(V, T)). By definition of K(V, T), there are m–linear indepen = ∇ + T. dent spinors ϕ1 , . . . , ϕm ∈ Γ (V) which are parallel w.r.t. the connection ∇ j = 0 and The spinors ϕ1 , . . . , ϕm are linear independent in each fiber of S, since ∇ϕ ϕ1 , . . . , ϕm are linear independent. Hence, ϕ1 , . . . , ϕm ∈ Γ (V) and m = rk(V) yield that ϕ1 , . . . , ϕm provide in each point a basis of the fiber of V, i.e. V is trivialized by ϕ1 , . . . , ϕm . In particular, each spinor ψ ∈ Γ (V) can be (uniquely) written as ψ = f j ϕ j , where f 1 , . . . , f m are smooth functions on M. Thus, ∇ X ϕ j = −T X ϕ j ∈ Γ (V) shows ∇X ψ =

m

(X f j )ϕ j − f j T X ϕ j ∈ Γ (V).

j=1

In particular, (∇ X π )(ψ) = 0 for all ψ ∈ V where π is the orthogonal projection S → V. Furthermore, since π is selfadjoint, ∇ X π is a selfadjoint endomorphism for all X . Let π ⊥ be the orthogonal projection S → V ⊥ to the orthogonal complement of V, then we conclude from 0 = (∇ X π )π ⊥ + π(∇ X π ⊥ ) the fact π ⊥ (∇ X π )π ⊥ = 0. Thus, we obtain for any φ ∈ V ⊥ and ψ ∈ S, (∇ X π )φ, ψ = (∇ X π )φ, π ψ = φ, (∇ X π )π ψ = 0 , is flat, the curvature of ∇ vanishes: which proves ∇ X π = 0 for all X ∈ T M. Since ∇ X,Y = R X,Y + T X TY − TY T X + dT X,Y . 0=R Since T is selfadjoint, i.e. (T X )∗ = T X , (∇ X T)Y is a selfadjoint (Hermitian) endomorphism of S: (∇ X T)Y ϕ, ψ = (∇ X TY − T∇ X Y − TY ∇ X )ϕ, ψ = − TY ϕ, ∇ X ψ − ϕ, T∇ X Y ψ + ϕ, ∇ X TY ψ = ϕ, (∇ X T)Y ψ , and therefore, dT X,Y is a selfadjoint (Hermitian) endomorphism of S. Contrary, R X,Y and [T X , TY ] are skew Hermitian endomorphisms on S which yields Eq. (3) and (dT) X,Y ψ = 0 for all ψ ∈ V. Since T is selfadjoint, T X φ vanishes for any X ∈ T M and φ ∈ V ⊥ (we assumed Im(T) ⊆ V). Hence, if ψ is a section in V ⊥ , we obtain (∇ X T)Y ψ = ∇ X TY ψ − T∇ X Y ψ − TY ∇ X ψ = 0 which leads to dT = 0.


401

1.3. Some analysis. Proposition 1.4. Suppose (M, g) is a complete Riemannian manifold (without boundary). Let S be a Dirac bundle over M, D be the corresponding Dirac operator and z ∈ iR. If R [from Eq. (2)] satisfies 1 Id ≤ |z|2 + R ≤ c · Id c

(4)

for some constant c > 0, then the elliptic operator

= D + z : W 1,2 (M, S) → L 2 (M, S) D is an isomorphism of Hilbert spaces.

are bounded on W 1,2 Proof. This follows by a straightforward argument since D, D

is coercive by inequality (4) (compare also [9]). and D Let M = K ∪ E be a disjoint decomposition of M into a connected compact manifold K with boundary ∂ K and a manifold E without boundary such that for every connected compact manifold K with K ⊂ K the manifolds M − K and E are diffeomorphic. The connected components of E are called ends of M. An exhaustion of M is a family of compact manifolds Mr with boundary ∂ Mr such that Mr ⊆ Mr for r < r and M = Mr . Let g be a (complete) Riemannian metric on M, then (M, g) is said to be L 1 –regular if for any (smooth) L 1 (M) function b, there is an exhaustion {Mr } of M with lim b = 0. r →∞ ∂ Mr

In fact, if (M n , g) is a complete simply connected Riemannian manifold (without boundary) of sectional curvature K ≤ 0, then (M, g) is L 1 –regular. 1.4. General positive mass theorem. The previous considerations lead to the following setup. General Setup. Suppose S is a complex Dirac bundle over (M n , g) and T is an uniformly bounded section of T ∗ M ⊗ End(S) such that (T X )∗ = T X for all X ∈ T M. Let z ∈ iR and define ⎛ ⎞ n V := ker ⎝z · Id − γ (e j )Te j ⎠ . (5) j=1

We assume that V is a subbundle of S with rk(V) > 0. Note that the fibers of V are vector spaces and that V is a subbundle of S if the dimensions of the fibers are constant. We denote by πV : S → V the orthogonal projection to V. Suppose further that T preserves V and V ⊥ , i.e. T X ϕ ∈ V for all X ∈ T M and ϕ ∈ V. Denote by TV the restriction := ∇ + T and the operator of T to V, i.e. TVX := T X πV . We define the connection ∇

D := D + z on S. Let D be the Dirac operator of ∇, then condition (5) means that the

and D coincide on Γ (V). Let R denote the curvature operator appearing in operators D the integrated Bochner–Weitzenböck formula in Proposition 1.1.

402

M. Listing

the constant z ∈ iR will later be chosen to be a In order to get a non–negativeR, maximal absolute eigenvalue of γ (ei )Tei . We have to make the difference between

:= D + z in order to use the integrated Bochner– := ei and D the operators D ei · ∇ Weitzenböck formula from Proposition 1.1. A section ψ ∈ Γ (S) is called an asymptotic T–Killing spinor if ∈ L 2 (M, T ∗ M ⊗ S) . ∇ψ An asymptotic T–Killing spinor ψ is called non– trivial if ψ ∈ / W 1,2 (M, S). Denote by K∞ (V, T) the vector space which is generated by all non–trivial asymptotic T–Killing spinors ψ ∈ Γ (V), in particular K∞ (V, T) =

{asymptotic T–Killing spinor ψ ∈ Γ (V)} . W 1,2 (M, S) ∩ Γ (V)

= Dψ yields Hence, if ψ is an asymptotic T–Killing spinor with ψ ∈ Γ (V), then Dψ

∈ L 2 (M, S). Dψ

∈ L 2 (M, S) and R ≥ 0 Suppose ψ T–Killing spinor with Dψ is an asymptotic 1 respectively Rψ, ψ ∈ L (M), then

ψ ∈ R ∪ {+∞} ν ψ + ν · Dψ, M(ψ) = lim inf (6) ∇ r →∞

∂ Mr

is well defined for any exhaustion {Mr } of M. The integrated Bochner–Weitzenböck formula (Prop. 1.1) proves that M(ψ) exists and is independent of the chosen exhaustion. is nonnegative, M(ψ) is bounded from below by If R 2 2 Dψ ∇ψ − M

ψ ∈ L 1 (M). The selfadjointness of the and moreover, M is finite if and only if Rψ,

(cf. Prop. 1.1) leads to the following lemma which is one ν + ν · D boundary operator ∇ of the crucial facts using the non–compact Bochner technique. Lemma 1.5. Let (M, g) be complete and L 1 –regular. Suppose ϕ is an asymptotic T–

∈ L 2 (M, S). If R is non–negative definite, we obtain Killing spinor with Dϕ M(ϕ) = M(ϕ + ξ ) for any ξ ∈ W 1,2 (M, S) ∩ Γ (S) [note that ϕ + ξ

+ ξ ) ∈ L 2 (M, S)]. D(ϕ

is an asymptotic T–Killing spinor with

Proof. ξ ∈ W 1,2 (M, S) yields ∇ξ, Dξ ∈ L 2 (M, S). Moreover, z ∈ iR and T are uniformly bounded which leads to

∈ L 2 (M, S). ∈ L 2 (M, T ∗ M ⊗ S) and Dξ ∇ξ

we obtain for any compact ν + ν · D Using the selfadjointness of the boundary operator ∇ manifold Mr with boundary ∂ Mr :

(ϕ + ξ ), ϕ + ξ =

ϕ + b(ϕ, ξ ) , ν + ν · D ν ϕ + ν · Dϕ, ∇ ∇ ∂ Mr

∂ Mr


403

where

ξ + ξ, ∇

+ ∇

ξ . ν ϕ + ν · Dϕ ν ξ + ν · Dξ, ν ϕ + ν · Dϕ, b(ϕ, ξ ) = ∇

∈ L 2 (M, S) and the above arguments show b(ϕ, ξ ) ∈ L 1 (M). ν ϕ, Dϕ In particular, ∇ Hence, there is an exhaustion {Mr } of M with lim

r →∞ ∂ Mr

(ϕ + ξ ), ϕ + ξ = lim ν + ν · D ∇

r →∞ ∂ Mr

ϕ ν ϕ + ν · Dϕ, ∇

(here we use that (M, g) is L 1 –regular). Because M(ϕ + ξ ) and M(ϕ) are independent ≥ 0, we conclude the claim. of the chosen exhaustion if R Thus, supposing one of the conditions is non–negative definite, 1. R ψ ∈ L 1 (M) for all asymptotic T–Killing spinors ψ ∈ Γ (V), 2. Rψ, the mass of an asymptotic T–Killing spinor ψ is the well defined value in (6). Moreover, the functional

ψ ν ψ + ν · Dψ, M : K∞ (V, T) → R ∪ {+∞} , [ψ] → lim inf ∇ r →∞

∂ Mr

is well defined, where {Mr } is an arbitrary exhaustion of M. If M consists of a finite set of ends E 1 , . . . , E k , we define the mass of ψ with respect to the end E j by M E j (ψ) := M( f · ψ), where f : M → [0, 1] is some cut off function for E j , i.e. f = 0 in M − E j , f = 1 at infinity of E j as well as supp(d f ) compact. Since the ends are disconnected, we obtain: M(ψ) = M E j (ψ). In most cases the mass functional is an invariant for subspaces of K∞ (V, T). In the asymptotic flat case, for instance, the mass is just a real number and does not depend on the choice of the spinor if we normalize |ψ| = 1 at infinity (cf. [3]). Moreover, on asymptotically hyperbolic manifolds (M, g), the mass functional M can be reduced to a (n + 1)–dimensional real vector space instead of the 2[n/2] – dimensional complex vector space K∞ (S / M, 2i γ (.)) (cf. [7]). Proposition 1.6. Consider the above setup where (M, g) is a complete Riemannian

has no negative eigenvalues, every D–harmonic manifold. If R spinor ψ ∈ Γ (V) of = 0. zero (total) mass is a T–Killing spinor: ∇ψ Proof. This follows immediately from the integrated Bochner–Weitzenböck formula for We have ∇.

Dψ

ψ = ψ − Dψ, ν ψ + ν · Dψ, ∇ψ + Rψ, ∇ ∇ψ, ∂ Mr

Mr

and M(ψ) vanish, an exhaustion of M yields for any compact Mr ⊂ M. Thus, if Dψ the claim.

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Since we assume (T X )∗ = T X , Te j ◦ Te j is non–negative definite. Thus, let t : M → [0, ∞) be the maximal eigenvalue function of Te j ◦ Te j , i.e. t is point wise given by the operator norm: n max Tφ, Tφ . Te j ◦ Te j t := = |φ|=1 j=1 Then t is non–negative and vanishes at a point p ∈ M if and only if T = 0 at p. The expression . means throughout the paper the point wise operator norm. Theorem 1.7. Let (M, g) be a complete L 1 –regular Riemannian manifold. Consider the above setup and assume t ≥ c for some constant c > 0. Suppose that the curvature operator R is uniformly bounded with R ≥ −|z|2 + t + δT.

(7)

If ϕ ∈ Γ (V) is an asymptotic T– Killing spinor, then the mass of ϕ is non–negative and

= 0, ϕ − ψ ∈ W 1,2 (M, S) there is an asymptotic T–Killing spinor ψ ∈ Γ (S) with Dψ and 0 ≤ M(ϕ) = M(ψ). = 0. Moreover, we If M(ϕ) vanishes, ψ is a section in V and a T–Killing spinor: ∇ψ have ψ = 0 if and only if 0 = [ϕ] ∈ K∞ (V, T). Proof. Inequality (7) yields = R + |z|2 − R

n

Te j ◦ Te j + δT ≥ R + |z|2 − t − δT ≥ 0

j=1

and R + |z|2 ≥ t ≥ c.

= Dϕ ∈ L 2 (M, S). ∈ L 2 (M, T ∗ M ⊗ S) and ϕ ∈ Γ (V), we obtain Dϕ Since ∇ϕ 1,2

= Dϕ.

Set Hence, Proposition 1.4 supplies a smooth spinor ξ ∈ W (M, S) with Dξ

ψ := ϕ − ξ , then ψ is D–harmonic. Moreover, ψ is an asymptotic T–Killing spinor (use ξ ∈ W 1,2 (M, S) and the fact that T is uniformly bounded). Thus, the above lemma shows M(ϕ) = M(ξ + ψ) = M(ψ). is non–negative definite, the integrated Bochner–Weitzenböck formula yields Since R = 0, i.e. ψ M(ψ) ≥ 0. If the mass of ϕ vanishes, the previous proposition shows ∇ψ is a T–Killing spinor. We obtain Dψ = 0 from ∇ψ = 0. In particular, ⎛ ⎞ n

− Dψ = ⎝z · Id − 0 = Dψ γ (e j )Te j ⎠ ψ j=1

proves ψ ∈ Γ (V) (use the definition of V). If ϕ is a non–trivial asymptotic T–Killing / W 1,2 (M, S). Conspinor, ψ = ϕ − ξ is non–trivial, since ξ ∈ W 1,2 (M, S) and ϕ ∈

is injective on versely, if ϕ ∈ W 1,2 (M, S), then ψ is contained in W 1,2 (M, S). Since D

= 0. W 1,2 (M, S), we conclude the claim from Dψ


405

Corollary 1.8. Let (M, g) be a complete L 1 –regular Riemannian manifold and consider the assumptions of the previous theorem including inequality (7). If the mass functional = ∇ + T trivializes V, the M : K∞ (V, T) → R ∪ {+∞} vanishes identically, then ∇ Dirac connection ∇ preserves sections of V (i.e. ∇πV = 0) and the restriction of T to V satisfies dTV = 0. Proof. We assumed Im(TV ) ⊆ V in the general setup, where TVX := T X ◦ πV . Hence, if the mass functional vanishes, the above theorem shows that rk(V) = dim K(V, TV ). In fact, Proposition 1.3 yields ∇πV = 0 and dTV = 0. 2. Spin Symmetric Lie Algebras 2.1. Notations. Suppose G/H is a symmetric space and let g be the Lie algebra of G as well as h be the Lie algebra of H . Then there is an involution τ : g → g with τ = Id on h. We denote by p the eigenspace of τ to the eigenvalue −1, then g = h ⊕ p and we obtain the well known relation [h, h] ⊆ h, [h, p] ⊆ p, [p, p] ⊆ h.

(8)

Thus, the decomposition g = h ⊕ p is Z2 –graded, where h is the even part and p is the odd part. The pair (g, τ ) is called an (effective) orthogonal symmetric Lie algebra in [10]. Moreover, (g, τ ) is said to be of Euclidean type if [p, p] = 0, it is said to be of compact type if g is semi simple and compact, and it is said to be of non–compact type if g is semi simple, non–compact and g = h ⊕ p is a Cartan decomposition of g. In fact, each (g, τ ) decomposes orthogonal into g = g0 ⊕ g+ ⊕ g− , τ = τ0 ⊕ τ+ ⊕ τ− , where (g0 , τ0 ) is of Euclidean type, (g+ , τ+ ) is of compact type and (g− , τ− ) is of non– compact type (cf. [10, Ch. V,Thm. 1.1]). Let gC be the complexification of the orthogonal symmetric Lie algebra (g = h⊕p, τ ). Denote by g ⊂ gC the subspace h⊕ip and define τ (h + ix) := h − ix for h ∈ h as well as x ∈ p. Then (g , τ ) is a (real) orthogonal symmetric Lie algebra called the dual of (g, τ ). As shown in [10, Ch. V Prop. 2.1] taking the dual of an orthogonal symmetric Lie algebra interchanges compact and non–compact type. For notational convenience /Sp will in this section always denote a Z2 –graded complex spinor space of p which means that /Sp = /S0 ⊗ Ck for some k ∈ N, where /S0 means the fundamental (irreducible) complex spinor space of p. Because of analytic difficulties using the noncompact Bochner technique, in some cases we have to enlarge the spinor space like in [16]. We will denote by γ : Cl(p) → gl(S / p) the Clifford action on /Sp and by Π ∈ gl(S / p) the involution which determines the Z2 grading of /Sp. In particular, Π acts as Id on /S0 p (even spinors) and as −Id on /S1 p (odd spinors). Note that if /Sp is a Z2 –graded complex spinor space of p, /Sp ⊗ Ck (k ∈ N) is a Z2 –graded complex spinor space of p with Clifford multiplication γ (.) ⊗ 1. We consider only Z2 graded spinor spaces since Killing structures on g1 ⊕ g2 are obtained by tensor products of Killing structures on g1 and g2 where of course /Sp1 ⊗ /Sp2 does not have to be a spinor space if /Sp1 , /Sp2 are not Z2 graded. Remark 2.1. If p is of even dimension, the irreducible complex spinor space admits a Z2 –grading which is induced by the volume form of Cl(p). If p is odd–dimensional,

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there are (up to equivalence) two irreducible complex representations of Cl(p) which are not Z2 –graded. If S denotes a complex spinor space, we equip /S p := S ⊗ C2 with the following Clifford multiplication: γ (σ0 + σ1 ) := γ (σ0 ) ⊗ 1 + γ (σ1 ) ⊗ w , where σ0 ∈ Cl 0 (p), σ1 ∈ Cl 1 (p) and z, w : C2 → C2 are given by z=

1 0 0 −1

w=

0 −i . i 0

(9)

The Z2 –grading of /S p is determined by the involution Π = 1 S ⊗z and since zw+wz = 0, we obtain γ (σ j )Π = (−1) j Π γ (σ j ) for any σ j ∈ Cl j (p). The definition of the graded tensor products and the proof of the following fact are presented in [15, p. 11 and 40]. Suppose p1 and p2 are (real) inner product spaces. Then Cl(p2 ). Moreover, if /Sp1 and the Clifford algebra of p1 ⊕ p2 is isomorphic to Cl(p1 )⊗ /Sp2 are Z2 –graded complex spinor spaces of p1 and p2 , /Sp1 ⊗ /Sp2 together with γ (σ1 ⊗ σ2 )ϕ ⊗ ψ := (−1)|ϕ|·|σ2 | γ1 (σ1 )ϕ ⊗ γ2 (σ2 )ψ is a Z2 –graded complex spinor space of p1 ⊕ p2 (where σ j ∈ Cl(p j ) and ϕ ∈ /Sp1 , ψ ∈ /Sp2 are homogeneous and |.| means the degree of the corresponding element). /Sp2 . The This spinor space of p1 ⊕ p2 is denoted by the graded tensor product: /Sp1 ⊗ /Sp2 is given by the involution Π = Π1 ⊗ Π2 , where Π j are the Z2 –grading of /Sp1 ⊗ involutions which determine the Z2 –grading of /Sp j . 2.2. Spin Killing structures. The adjoint representation adg : g → gl(g) restricted to h is reducible to adh : h → gl(h) and the isotropy representation ρ : h → so(p), where on p the usual inner product is considered. Suppose /Sp is the irreducible complex spinor space of p, then 21 γ : Cl(p) → gl(S / p) restricted to so(p) ∼ = Λ2 p ⊂ Cl(p) is the / p) complex spin representation spin(p) = so(p) → gl(S / p). Hence, 21 γ ◦ ρ : h → gl(S is a representation of the lie algebra h on the irreducible complex spinor space of p. Let 1 γ ◦ ρ = 1 ⊕ · · · ⊕ m 2

(10)

be an irreducible decomposition, i.e. j : h → gl((S / p) j ) is irreducible. Since h is the Lie algebra of a compact Lie group, this decomposition is unique up to equivalence (use the orthogonality relations of the character). The irreducible representations j are called spin isotropy representations of h. In particular, if we have a decomposition like in (10), there are at most m non equivalent spin isotropy representations of h. In fact if rank(g) = rank(h), i is not equivalent to j for all i = j, i.e. every spin isotropy representation appears exactly once in (10). Moreover, if g is semi simple, j is nontrivial for any j. In the case that /Sp is a complex spinor space of p which is not irreducible, then 1 2 γ ◦ ρ restricted to h is also a direct sum of spin isotropy representations of h, however each j appears more than once.


407

Definition 2.2. A Lie algebra representation λ : g → gl(W ) is called a spin Killing structure of the orthogonal symmetric Lie algebra (g, τ ) if the restriction of λ to h is injective and equivalent to a direct sum of spin isotropy representations of h: λ|h ∼ = i1 ⊕ · · · ⊕ il (i s ∈ {1, . . . , m} and j can appear more than once). By definition W is a subspace of a complex spinor space /Sp of p, where /Sp does not have to be irreducible, i.e. /Sp = /S0 ⊗Ck for some k and /S0 being the fundamental (irreducible) spinor space of p. Hence, λ yields a representation λ˜ : g → gl(S / p) such that the restriction of λ˜ to W coincides with λ and the action of λ˜ on W ⊥ ⊂ /Sp is trivial. We need the injectivity of λ|h in order to describe the complete geometry of the symmetric space. However, the following observations show that it is sufficient to assume the injectivity of λ|h0 , where h0 ⊂ g0 and g0 means the Euclidean part of g. In fact, the remarks below prove that the assumption on the injectivity of λ|h is superfluous as long as W = /Sp0 ⊗ W with W ⊂ /Sp± and p0 ⊂ g0 , p± ⊂ g± . Remark 2.3. Let χ : g → gl(V ) be a representation of a semi– simple Lie algebra g. If g is simple, χ is injective or χ = 0. In particular, if g = g1 ⊕ · · · ⊕ gm is a decomposition of g into a direct sum of simple Lie algebras g j , then the restriction of χ to g j is injective or trivial. Remark 2.4. Let (g, τ ) be of Euclidean type, i.e. [p, p] = 0. Then a spin Killing structure is given by λ : g = h ⊕ p → gl(S / p), h + x →

1 γ (ρ(h)) 2

[h ∈ h, x ∈ p and ρ as usual the isotropy representation h → so(p)]. Of course, if λ : g → gl(W ) is a spin Killing structure of (g, τ ), its trivial extension λ˜ : g → gl(S / p) is not compatible with Clifford multiplication. Note that we always consider the inner product on /Sp which makes γ (x) a skew– Hermitian endomorphism for all x ∈ p. In fact reducing this inner product to W yields by definition an H –invariant inner product. Definition 2.5. Let λ : g = h ⊕ p → gl(W ) be a spin Killing structure of (g, τ ). Then λ is said to be Riemannian if λ(x) = −λ(x)∗ for all x ∈ p and it is said to be selfadjoint if λ(x) = λ(x)∗ for all x ∈ p. A standard argument and the following lemma show that we can always ”gauge transform” our spin Killing structure to be Riemannian, respectively selfadjoint. In fact, if (g, τ ) is an orthogonal symmetric Lie algebra of compact type [non–compact type] and λ : g → gl(W ) is a spin Killing structure of (g, τ ), then there is A ∈ Gl(W ) in such a way that λr : g → gl(W ), x → Aλ(x)A−1

(11)

is a Riemannian [selfadjoint] spin Killing structure of (g, τ ) with λr (h) = λ(h) for all h ∈ h.

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Lemma 2.6. Suppose λ : g → gl(W ) is a spin Killing structure of (g, τ ). Then λ : g = h ⊕ p → gl(W ), h + ix → λ(h) + iλ(x) defines a spin Killing structure of the dual orthogonal symmetric Lie algebra (g , τ ). Moreover, if λ is Riemannian [selfadjoint], λ is selfadjoint [Riemannian]. Lemma 2.7. Suppose (g1 , τ1 ) and (g2 , τ2 ) are orthogonal symmetric Lie algebras. Then the direct sum (g, τ ) := (g1 ⊕ g2 , τ1 ⊕ τ2 ) is an orthogonal symmetric Lie algebra and (g, τ ) admits a spin Killing structure λ if and only if there exist spin Killing structures λ1 and λ2 of (g1 , τ1 ) and (g2 , τ2 ). Proof. Suppose λ : g → gl(W ) is a spin Killing structure of (g, τ ), then the restriction of λ to λ j : g j → gl(W ) is a Lie algebra representation. The restrictions of λ j to h j are injective since λ is injective on h = h1 ⊕ h2 . Moreover, each spin isotropy representation of h1 ⊕ h2 is a tensor product of a spin isotropy representation of h1 with a spin isotropy representation of h2 , i.e. λ j restricted to h j is a direct sum of spin isotropy representations of h j . Hence, λ j is a spin Killing structure of (g j , τ j ). Conversely, assume λ j : g j → gl(W j ) ( j = 1, 2) are spin Killing structures of (g j , τ j ). We consider the tensor product representation λ = λ1 λ2 : g → gl(W1 ⊗ W2 ), a1 + a2 → λ1 (a1 ) ⊗ 1 + 1 ⊗ λ2 (a2 ) ,

(12)

where a1 ∈ g1 and a2 ∈ g2 . λ is a Lie algebra representation and λ restricted to h is a direct sum of spin representations of h = h1 ⊕ h2 . Moreover, λ|h is injective which follows from the facts that λ j is injective on h j and λ j (h) = κ1 for all h ∈ h j − {0} and κ ∈ C. Hence, λ is a spin Killing structure of (g, τ ). 2.3. Spin symmetric Lie algebras. Unfortunately in order to prove the main theorem the existence of a spin Killing structure is not quite sufficient, we have to make an additional assumption on the spin Killing structure. Let λ : g → gl(W ) be a spin Killing structure of the orthogonal symmetric Lie algebra (g, τ ) and let λ˜ : g → gl(S / p) be its trivial extension to the corresponding complex spinor space (remember that /Sp does not have to be the fundamental spinor space of p). Choose an H –invariant inner product on p and define Υλ :=

n

γ (ei )λ˜ (ei ) ∈ gl(S / p) ,

i=1

where e1 , . . . , en is an orthonormal basis of p and γ : Cl(p) → End(S / p) means the Clifford action. In general Υλ does not commute with Clifford multiplication, hence, the boundary operator = ∇ν + λ˜ (ν) + ν · D + ν · Υλ ν + ν · D ∇ which would appear in the integrated Bochner–Weitzenböck formula (cf. Prop. 1.1) can not be selfadjoint. In order to avoid this problem we will assume that λ˜ (a)Υλ = Υλ λ˜ (a)

(13)


409

for all a ∈ g. In fact, if V is an eigenspace of Υλ to an eigenvalue z = 0, V is a g– invariant subspace, i.e. λ restricts to a spin Killing structure µ : g → gl(V ) with the property that Υµ = z · pr V ∈ gl(S / p). Still Υµ does not commute with Clifford multiplication in

= D + z instead of D = D + Υµ . general, hence, we consider the Dirac operator D

and D coincide This yields a selfadjoint boundary operator and the Dirac operators D in Proposition 1.1, we will choose on sections of V . Because of the curvature operator R V to be the eigenspace to the maximal absolute eigenvalue of Υλ . These observations lead to the following definition. Definition 2.8. Let (g, τ ) be an orthogonal symmetric Lie algebra and g0 ⊕ g+ ⊕ g− its decomposition into the Euclidean part and semi-simple part of compact and noncompact type. (g, τ ) is called a spin symmetric Lie algebra if there is a spin Killing structure λ : g → gl(W ) of (g, τ ) in such a way that 1. Υλ is g–invariant: ˜ ˜ λ(a)Υ λ = Υλ λ(a) for all a ∈ g where λ˜ is the trivial extension of λ to /Sp ⊇ W . 2. λ+ := λ|g+ is a Riemannian spin Killing structure on (g+ , τ+ ) and λ− := λ|g− is a selfadjoint spin Killing structure on (g− , τ− ). The next problem is that condition (13) does not have to hold when considering tensor products of spin Killing structures which satisfy (13): If λ j : g j → gl(V j ), j = 1, 2, are spin Killing structures of the orthogonal symmetric lie algebras (g1 , τ1 ) and (g2 , τ2 ), then the tensor product representation λ = λ1 λ2 is a spin Killing structure of (g1 ⊕ g2 , τ1 ⊕ τ2 ) (cf. the above lemma) with /Sp2 ) , / p1 ⊗ Υλ = Υλ1 ⊗ 1 + Π1 ⊗ Υλ2 ∈ gl(S where Π1 means the involution which determines the Z2 – grading of /Sp1 . If (13) holds for the spin Killing structures λ1 and λ2 , then λ satisfies (13) if and only if λ˜ 1 (a1 )Π1 = Π1 λ˜ 1 (a1 ) for all a1 ∈ g1 . But the standard Killing structure of a rank one symmetric space satisfies λ(x)Π = −Π λ(x) for all x ∈ p, hence, in order to get useful spin Killing structures on Riemannian products, we have to enlarge the spinor space like in [16]. The following remark proves that we can always enlarge the representation space to obtain a spin Killing structure that preserves the Z2 – grading of /Sp. Remark 2.9. Suppose λ : g → gl(W ) is a spin Killing structure of (g, τ ) and W ⊆ /Sp. Consider the spinor space /S p = /Sp⊗C2 with Clifford multiplication γ from Remark 2.1 and define λ : g → gl(W ⊗ C2 ), a → λ(a) ⊗ 1. Since the Z2 –grading of /S p is determined by Π = 1 ⊗ z, λ is a spin Killing structure of (g, τ ) with λ (a)Π = Π λ (a) for all a ∈ g. Proposition 2.10. 1. An orthogonal symmetric Lie algebra (g, τ ) is spin symmetric if and only if its dual (g , τ ) is a spin symmetric Lie algebra. 2. Let (g = g1 ⊕ g2 , τ ) be the direct sum of (g1 , τ1 ) and (g2 , τ2 ). Then (g, τ ) is a spin symmetric Lie algebra if and only if (g1 , τ1 ) and (g2 , τ2 ) are spin symmetric Lie algebras.

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Proof. (1) follows from Lemma 2.6. (2) Suppose (g1 , τ1 ) and (g2 , τ2 ) are spin symmetric Lie algebras as well as λ1 : g1 → gl(W1 ) and λ2 : g2 → gl(W2 ) are the corresponding spin Killing structures. According to Remark 2.9 we can assume that λ1 (x) preserves the Z2 – grading of /Sp1 , i.e. λ1 (x)Π1 = Π1 λ1 (x). Let λ := λ1 λ2 be the tensor product representation on /Sp2 . Then λ is a spin Killing structure which restricted to g+ is Riemannian and /Sp1 ⊗ restricted to g− is selfadjoint. Moreover, we obtain Υλ = Υλ1 ⊗ 1 + Π1 ⊗ Υλ2 . Hence, λ˜ 1 (a)Π1 = Π1 λ˜ 1 (a) (λ1 is even) and λ˜ j (a)Υλ j = Υλ j λ˜ j (a) show λ˜ (a)Υλ = Υλ λ˜ (a) for all a ∈ g. In particular, g = g1 ⊕ g2 is a spin symmetric Lie algebra. Conversely, suppose g = g1 ⊕ g2 is a spin symmetric Lie algebra and λ : g → gl(W ) is the corresponding spin Killing structure. Denote by λ j the restriction of λ to g j . Then λ j is a spin Killing structure of (g j , τ j ) which is Riemannian on g j,+ and selfadjoint on g j,− . λ is a direct sum of irreducible representations of the Lie algebra g, and since each irreducible representation of g = g1 ⊕ g2 is the tensor product of irreducible representations of g1 and g2 (cf. [6, Prop. 4.14]), we conclude ˜ 2 ) = λ(x ˜ 2 )γ (x1 ), γ (x1 )λ(x

˜ 1 ) = λ(x ˜ 1 )γ (x2 ) γ (x2 )λ(x

on /Sp for all x1 ∈ p1 and x2 ∈ p2 . This shows λ˜ 1 (a1 )Υλ1 = Υλ1 λ˜ 1 (a1 ) , λ˜ 2 (a2 )Υλ2 = Υλ2 λ˜ 2 (a2 ) for all a j ∈ g j , i.e. (g1 , τ1 ) and (g2 , τ2 ) are spin symmetric Lie algebras.

This proposition reduces the classification of spin symmetric Lie algebras to the classification of irreducible compact spin symmetric Lie algebras. However, since this turns out to be a non–trivial representation theoretical problem, we can only give a few examples of irreducible spin symmetric Lie algebras below. 2.4. Application to the problem. Lemma 2.11. Suppose (G/H, g0 ) is a symmetric space such that the associated orthogonal symmetric Lie algebra (g, τ ) is spin symmetric. Consider the H –invariant product g0 on p and let λ : g → gl(W ) be the spin Killing structure which satisfies the assumptions of the above definition. Then (Υλ )2 ∈ gl(W ) with scal(g0 ) = 2

γ (ei )γ (e j )λ([e j , ei ]) = 4(Υλ )2 + 4

i, j

n

λ(ei )λ(ei )

(14)

i=1

on W , where e1 , . . . , en is a g0 orthonormal basis of p. eigenbasis of the Ricci endomorProof. We assume that e1 , . . . , en is an orthonormal phism ric : p → p, then the Jacobi identity ( j e j ∧ ρ([e j , x]) = 0) gives 2

n

γ (ei )γ (e j )λ˜ ([e j , ei ]) =

i, j=1

=

γ (ei )γ (e j )γ (ρ[e j , ei ]) ◦ pr W

i, j

γ (ei )γ ([[e j , ei ], e j ]) ◦ pr W = −

i, j

= tr(ric)pr W = scal(g0 )pr W .

i

γ (ei )γ (ric(ei )) ◦ pr W


411

˜ ˜ Moreover, using the fact that λ˜ is a representation, that λ(x)Υ and that λ = Υλ λ(x) γ (ei )γ (e j ) = −γ (e j )γ (ei ) − 2δi j leads to

γ (ei )γ (e j )λ˜ ([e j , ei ]) = 2(Υλ )2 + 2

i, j

λ˜ (e j )λ˜ (e j ).

j

Hence, λ = λ˜ |W and λ˜ |W ⊥ = 0 show (Υλ )2 ∈ gl(W ) and Eq. (14).

This identity yields the basic assumptions for the general setup (1.4). Suppose (G/H, g0 ) is a symmetric space with Ric(g0 ) ≤ 0 and suppose its associated orthogonal symmetric Lie algebra (g = g0 ⊕ g− , τ ) is spin symmetric. If λ : g → gl(W ) is the spin Killing structure of (g, τ ) which satisfies the assumptions for a spin symmetric Lie algebra, then all eigenvalues of Υλ are purely imaginary. Hence, if z ∈ iR is the maximal absolute eigenvalue of Υλ and V := ker (z − Υλ ) = ker z − γ (ei )λ˜ (ei ) is the corresponding eigenspace, then V ⊆ W ⊆ /Sp is g–invariant and λ˜ (ei )λ˜ (ei ) ∈ gl(S scal(g0 ) ≥ 4z 2 + 4 / p) , where equality is attained on V . Note that V = W if and only if Υλ = z · pr W . Lemma 2.12. Suppose that λ : g → gl(W ) is a spin Killing structure of an arbitrary orthogonal symmetric Lie algebra (g, τ ). Define Σ ∈ gl(W ) by Σ(φ) := pr W (Υλ φ) for all φ ∈ W (pr W : /Sp → W is the orthogonal projection, note that Υλ (φ) = 0 for all φ ∈ W ⊥ but the image of Υλ does not have to be in W ). Then Σ is h–invariant: λ(h)Σ = Σλ(h) for all h ∈ h. Proof. Suppose η ∈ Λ2 p and x ∈ p, then [γ (x), γ (η)] = −2γ (xη). Consider h embedded in Λ2 p, then by definition of a spin Killing structure, W is invariant under Clifford multiplication with γ (h). Thus, we obtain 1 γ (h)pr W γ (ei )λ(ei ) 2 1 = pr W γ (ei h)λ(ei ) + γ (ei )γ (h)λ(ei ) 2 γ ( [h, ei ], e j e j )λ(ei ) + γ (ei )λ(ei )λ(h) + γ (ei )λ([h, ei ]) = pr W

λ(h)Σ =

= Σλ(h) , where in the last line we used [h, x], y = − x, [h, y].

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2.5. Irreducible examples. 2.5.1. Standard sphere S n [cf. real Killing spinors in [4]]. The orthogonal symmetric Lie algebra of the standard sphere S n is given by g = so(n +1) and h = so(n) ⊂ g. Since R2 and each spin representation of so(n + 1) is so(k) ⊂ Cl(Rk ), Cl(Rn+1 ) = Cl(Rn )⊗ induced by 21 γ , where γ is a representation of the Clifford algebra, the restriction of a spin representation of so(n + 1) to so(n) is a direct sum of spin representations of so(n). To be more precise, let /Sp be an irreducible complex Z2 –graded spinor space of p and γ be the Clifford multiplication on it. If we identify so(n) = h with Λ2 p, then / p), x + y ∧ z → λ : p ⊕ Λ2 p → gl(S

1 1 1 γ (x) + γ (y)γ (z) − γ (z)γ (y) 2 4 4

is a representation of g which restricted to h is a direct sum of 2 spin representations of so(n) (the representations are equivalent if n is odd, and not equivalent if n is even). Note that λ is equivalent to a spin representation of so(n + 1). Since λ is injective and λ(x)∗ = −λ(x) for all x ∈ p, λ is a Riemannian spin Killing structure. Moreover, Υλ = − n2 Id proves that so(n + 1) = so(n) ⊕ Rn is a spin symmetric Lie algebra. Note that this spin Killing structure does not preserve the Z2 –grading of /Sp since Π λ(x) = −λ(x)Π for all x ∈ p. 2.5.2. Complex projective space CP2n−1 [cf. Kähler Killing spinors in [13]]. Let g = su(m + 1), h = u(m) be the orthogonal symmetric Lie algebra of the complex projective space CPm in odd complex dimensions m = 2n − 1. The vector space p admits a complex structure J and the complexification of p denoted by pC decomposes orthogonal into p1,0 and p0,1 . As usual if x ∈ p, we denote by x 1,0 the projection to p1,0 and by x 0,1 the projection to p0,1 , i.e. x 1,0 = 21 (x − i J x) and x 0,1 = 21 (x + i J x). The irreducible complex spinor space of p is isomorphic to /Sp = Λ∗,0 pC ⊗

Λ0,m pC ,

where Λ∗,0 pC = Λ∗ (p1,0 ) = Λ∗C p is the exterior form bundle of the complex vec ∗ tor space p and Λ0,m pC = Λm C p means the following complex one dimensional representation space: i i u(m) → gl( Λ0,m pC ), A → − trace(A) = − g(ω, .) . 2 2 In particular, /Sp decomposes orthogonal into /S0 ⊕ /S1 ⊕ · · · ⊕ /Sm with /S j = Λ j,0 pC ⊗ Λ0,m pC . Denote by π j : /Sp → /S j the orthogonal projection. If e1 , J e1 , . . . , em , J em is an J – orthonormal basis of p, the Kähler form m e j ∧ J e j ∈ u(m) satisfies γ (ω) = i (2 j − m)π j . Clifford multiplication with ω= j=1

x ∈ p on /S decomposes into γ (x) = γ (x 1,0 ) + γ (x 0,1 ), where γ (x 1,0 ) : /S j → /S j+1


413

and γ (x 0,1 ) : /S j → /S j−1 with /S j = {0} if j < 0 or j > m. Using the identification u(m) = Λ1,1 p ⊂ Λ2 p, / p), λ : p ⊕ Λ1,1 p = su(2n) → gl(S 1 1 1 x + h → γ (x 1,0 )πn−1 + γ (x 0,1 )πn + γ (h)(πn−1 + πn ) 2 2 2 is a representation of su(2n). Consider the inclusion su(2n) ⊂ so(p ⊕ R2 ), then a simple computation shows that λ is equivalent to the following fundamental representation of su(2n): Λnsu(2n) : su(2n) → gl(Λn C2n ). Since λ = 0 and su(2n) is simple, λ has to be injective (Remark 2.3). Moreover, (γ (x 1,0 )πn−1 )∗ = −γ (x 0,1 )πn and (γ (x 0,1 )πn )∗ = −γ (x 1,0 )πn−1 yield λ(x)∗ = −λ(x) and thus, λ is a Riemannian spin Killing structure. Using the fact that ω acts as −i on /Sn−1 and as +i on /Sn , we obtain Υλ = −n · (πn−1 + πn ). Hence, su(m + 1) = u(m) ⊕ R2m is a spin symmetric Lie algebra if m is odd. 2.5.3. Symmetric spaces of type II. Let h be a compact simple real Lie algebra. Then g = h ⊕ h together with τ : g = h ⊕ h → g, τ (a, b) = (b, a) is an irreducible orthogonal symmetric Lie algebra with h = {(a, a)| a ∈ h } and p = {(a, −a)| a ∈ h }. (g, τ ) is the orthogonal symmetric Lie algebra of a type II symmetric space. We obtain as usual the symmetric relations (8) which means that p is not a Lie subalgebra of g. However, if we equip p with the Lie bracket [(a, −a), (b, −b)]p := ([a, b]h , −[a, b]h ), then (p, [., .]p) is a Lie algebra. Moreover, we obtain the property [[x, y]g, z]g = [[x, y]p, z]p for x, y, z ∈ p. Define the following linear transformation ξ : h → p with ξ(a, a) = (a, −a). Then a straightforward calculation shows that ξ is a Lie algebra isomorphism: ξ([(a, a), (b, b)]h) = ξ ([a, b]h , [a, b]h ) = [ξ(a, a), ξ(b, b)]p with [x, h]g = [x, ξ(h)]p ∈ p for x ∈ p and h ∈ h. Note that the inner product on p is induced by the Killing form of h and a straightforward calculation shows adp(x) ∈ so(p) for all x ∈ p. Hence, let adp : p → so(p) be the adjoint representation of p and ρ : h → so(p) as usual, then the above relations yield ρ = adp ◦ ξ and ρ([x, y]g) = adp([x, y]p).

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Proposition 2.13. Let (g, τ ) be the orthogonal symmetric Lie algebra of a type II symmetric space. Then λ : g = p ⊕ h → gl(S / p), x + h →

1 1 γ (adp(x)) + γ (ρ(h)) 2 2

is a Riemannian spin Killing structure of (g, τ ) [as usual we imbed so(p) into Cl(p) by the identification Λ2 p = so(p)]. Moreover, n 1 Υλ = γ ei ∧ adp(ei ) 2 i=1

commutes with λ(x) for all x ∈ g, i.e. (g, τ ) is a spin symmetric Lie algebra. Proof. λ is certainly injective on h since ρ is injective and γ is injective on so(p). 21 γ is a Lie algebra representation of so(p), i.e. using the above facts, we conclude for x1 , x2 ∈ p and h 1 , h 2 ∈ h: 1 1 γ ◦ adp([x1 , x2 ]p) + γ ([adp(x1 ), ρ(h 2 )]so(p) ) 2 2 1 1 + γ ([ρ(h 1 ), adp(x2 )]so(p) ) + γ ◦ ρ([h 1 , h 2 ]h) 2 2 1 = γ ◦ ρ [x1 , x2 ]g + [h 1 , h 2 ]g 2 1 + γ ◦ adp [x1 , ξ(h 2 )]p + [ξ(h 1 ), x2 ]p 2 = λ([x1 + h 1 , x2 + h 2 ]g).

[λ(x1 + h 1 ), λ(x2 + h 2 )] =

Thus, λ is a spin Killing structure. λ is Riemannian since Clifford multiplication with a real 2–form is a skew Hermitian action on /Sp and adp(x) ∈ so(p), in particular λ : g → so(S / p). Hence, it remains to show that λ(x) commutes with Υλ = 21 γ (η) for all x ∈ g where η := i ei ∧ adp(ei ) ∈ Λ3 p. Lemma 2.12 yields this claim for x ∈ h and if x ∈ p, then: 1 λ(x)Υλ = γ (adp(x))γ (ei )λ(ei ) 2 i 1 γ (ei adp(x))λ(ei ) + γ (ei )γ (adp(x))λ(ei ) = 2 i γ ([x, ei ]p)λ(ei ) + γ (ei )λ([x, ei ]p) + γ (ei )λ(ei )λ(x) = Υλ λ(x) = i

completes the proof (in the last equation use [x, ei ]p = −

j

[x, e j ]p, ei e j ).

Theorem 2.14. There are no spin Killing structures for the following orthogonal symmetric Lie algebras: 1. 2. 3. 4.

su(2n + 1) = u(2n) ⊕ R4n . sp(n + 1) = sp(1) ⊕ sp(n) ⊕ R4n in case n > 1. f4 = so(9) ⊕ R16 . g2 = so(4) ⊕ R8


415

Proof. If λ : g → gl(W ) is a spin Killing structure and V ⊂ W is an irreducible λ– module, then λ : g → gl(V ) restricted to h is a direct sum of spin isotropy representations of h. In particular, there is a spin Killing structure λ if and only if there is an irreducible representation of g which restricted to h is a direct sum of spin isotropy representations of h. Furthermore, since the representation space is complex, a spin Killing structure of g yields a complex representation of gC = g⊗C. In particular, if g = h⊕p admits a spin Killing structure, there is an irreducible representation gC → gl(V ) which restricted to h ⊗ C is a direct sum of (complex) extensions of spin isotropy representations. We first consider the orthogonal symmetric Lie algebra of the Cayley plane. The isotropy representation ρ : so(9) → so(R16 ) is nothing but the (real) spin representation of V = R9 . Using the fact that in dimension 16 ≡ 0 (mod 8) the complex spin representation is the complexification of the real spin representation, the (irreducible) complex spinor space decomposes under 21 γ ◦ ρ into (cf. [8]) /Sp = Sym20 V ⊗ C ⊕ Λ3 V ⊗ C ⊕ /S− . In particular, Sym20 V ⊗ C, Λ3 V ⊗ C and /S− are all the spin isotropy representations of so(9) ⊂ f4 . In order to obtain a spin Killing structure, we need an irreducible (complex) representation of f4 which decomposes under the restriction to so(9) in less than 4 non-equivalent representation. Thus, we are looking for a complex Lie algebra representation of F4 = f4 ⊗ C which restricted to B4 ∼ = so(9, C) ⊂ F4 decomposes in less than 4 non-equivalent factors. The only candidates are the representations with maximal weight (0001) and (1000) of F4 (the dimension of the representation space is too big in any other case with less than 4 non-equivalent representations of B4 ). However, the representations (0001) and (1000) decompose under B4 ⊂ F4 as follows (cf. [18, p. 306]) (0001) = (0001) ⊕ (1000) ⊕ (0000) = /S ⊕ C9 ⊕ C , (1000) = (0100) ⊕ (0001) = Λ2 C9 ⊕ /S , where C9 is the standard complex representation of so(9) and /S ∼ = C16 means the (complex) spin representation of so(9). Hence, there is no representation of f4 which restricted to so(9) is a direct sum of spin isotropy representations of so(9) ⊂ f4 . In particular, the orthogonal symmetric Lie algebra f4 = so(9) ⊕ R16 is not spin symmetric. Consider the case g2 = so(4) ⊕ p with p = R8 . Using the Lie algebra isomorphism so(4) = sp(1) ⊕ sp(1) with sp(1) ∼ = sp(1), the complexification of the isotropy representation ρC : so(4) → so(C8 ) is determined by the tensor product representation p⊗C = H ⊗ E, where H ∼ = C2 is the standard representation of sp(1) and E ∼ = C4 is the induced representation of the imbedding sp(1) ⊂ sp(2) (note that sp(1)⊕sp(2) ⊂ so(8)). , where H ∼ Since E is an irreducible sp(1) module, we can conclude E = Sym3 H = C2 is the standard representation of sp(1) (the difference in the notation of sp(1) and sp(1) is made to make sure which sp(1) ⊂ so(4) factor acts, i.e. which as well as of H and H sp(1) factor induces the (unique) Quaternionic Kähler structure on the symmetric space). The complex spinor space /Sp = /S+ ⊕ /S− decomposes as follows: ⊕ Sym2 H , /S+ = Λ20 E ⊕ Sym2 H = Sym4 H . /S− = H ⊗ Λ10 E = H ⊗ Sym3 H Thus, if λ is an irreducible (complex) representation of g2 which yields a spin Killing structure, then the restriction of λ to sp(1) ⊕ sp(1) decomposes in less than 4 nonequivalent representations. We conclude from dimension reasons that the only candidates for a spin Killing structure are the fundamental representation of G2. However,

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under the imbedding A1 ⊕ A1 ⊂ G2 the fundamental representations decompose as follows (cf. [18, p. 308]): ⊕ Sym2 H , (01) = (1, 1) ⊕ (0, 2) = H ⊗ H (10) = (1, 3) ⊕ (0, 2) ⊕ (2, 0) ⊕ Sym2 H ⊕ Sym2 H. = H ⊗ Sym3 H Hence, (01) does not yield spin isotropy representations, but (10) decompose into a . Although Sym2 H is a sp(1) representa2 spin isotropy representation and Sym2 H 2 2 tion the same as Sym H , Sym H and Sym2 H are non equivalent representations of is not a spin isotropy representation of so(4) = sp(1) ⊕ sp(1). In particular, Sym2 H so(4) ⊂ g2 . Hence, g2 = so(4) ⊕ R8 does not admit a spin Killing structure. Let g be a simple Lie algebra of type Ak , Bk , Ck or Dk and α1 , . . . , αk be irreducible fundamental representations of g, i.e. the representation ring of g is isomorphic to Z[α1 , . . . , αk ], then we denote as usual the maximal weight of α j by (0, . . . 0, 1, 0, . . . , 0) = e j ∈ Zk . If µ =(µ1 , . . . , µk ) is the maximal weight of an irreducible representation, define |µ| = µ j . Suppose λ : g → gl(W ) is an irreducible representation of maximal weight µ, and λ|k is the restriction of λ to a simple Lie subalgebra k ⊂ g of the same type as g. Then λ|k splits into irreducible k– representations such that at least one representation has maximal weight ν with |ν| ≥ |µ|. This is certainly true for the fundamental representations and the general case follows from the multiplication property of the representation rings. Consider the orthogonal symmetric Lie algebra su(m + 1) = u(m) ⊕ p with p = R2m and u(m) = su(m) ⊕ R · ω. The spin isotropy representations of u(m) are determined by /S j := Λ j,0 pC ⊗

Λ0,m pC , j = 0 . . . m

and restricted to su(m) they are equivalent to 2 trivial representations Λ0m , Λm m and j the fundamental representations Λm , j = 1, . . . , m − 1 of su(m). Since these representations have maximal weight e j = (0, . . . , 0, 1, 0, . . . , 0) with |e j | = 1 for all j = 1, . . . , m − 1, the only candidates for an irreducible Lie algebra representation λ : su(m + 1) → gl(V ) (which is a spin Killing structure) must have maximal weight µ with |µ| ≤ 1. In particular, λ can only be a fundamental representation of su(m +1) (note that spin Killing structures are assumed to be injective on h, i.e. λ can not be trivial). Let Λkm+1 be the fundamental representation of su(m + 1), then the restriction of Λkm+1 k to su(m) is isomorphic to Λk−1 m ⊕ Λm , i.e. the candidate for extending spin representations of u(m) is /Sk−1 ⊕ /Sk . The Kähler form ω acts under the Clifford representation as i( m2 + 1 − k) on /Sk−1 and as i( m2 − k) on /Sk [we have to consider 21 γ (ω)]. The action of ω ∈ u(m) on Λkm+1 depends on the imbedding u(m) ⊂ su(m + 1), i.e. it depends on a choice of orientation and on a scaling factor. In particular, a straightforward calculation shows that ω ∈ su(m + 1) has eigenvalues ic · k and ic(k − m − 1) for some c ∈ R. Thus, in order to have equivalent representations /Sk−1 ⊕ /Sk and Λkm+1 |u(m) of u(m), we need all possible solutions of the following two linear systems: c · k = j + 1, c · k = j, c(k − m − 1) = j, c(k − m − 1) = j + 1,


417

1 where j = m2 −k. The first system has only the solution c = m+1 which means that k must m+1 1 be 2 . The second system is only solved by c = − m+1 and k = m+1 2 . However, k is supposed to be an integer, i.e. the orthogonal symmetric Lie algebra su(m +1) = u(m)⊕R2m does admit a spin Killing structure λ : su(m + 1) → gl(S / R2m ) if and only if m is odd. Consider the orthogonal symmetric Lie algebra sp(n + 1) = sp(1) ⊕ sp(n) ⊕ R4n . In case n = 1 this orthogonal symmetric Lie algebra is isomorphic to so(5) = so(4) ⊕ R4 and hence spin symmetric. Let pC be the complexification of p = R4n , then pC = H ⊗ E, where E ∼ = C2n means the standard representation of sp(n) and H ∼ = C2 is the standard representation of sp(1) = su(2). The spin isotropy representations of sp(1) ⊕ sp(n) are determined by (cf. [2])

Symk H ⊗ Λn−k 0 E j

for k = 0, . . . , n. Λ0 E is an (irreducible) fundamental representation of sp(n) for j = 1, . . . , n. In particular, if λ is an irreducible representation of sp(n +1) with maximal weight µ, then the restriction of λ to sp(n) admits at least one irreducible representation of maximal weight ν with |ν| ≥ |µ|. Hence, |µ| ≤ 1 and λ can only be one of the fundamental representations of sp(n + 1). However, in case n > 1 the restriction of Λk0 E is the standard representation space of sp(n + 1)] always contains to sp(1) ⊕ sp(n) [ E a representation which is not a spin isotropy representation. For instance, if n = 2, we obtain the following decompositions (cf. [18, p. 230]): = E = E ⊕ H, Λ10 E

= E ⊕ H ⊗ Λ2 E, Λ20 E 0

= C ⊕ H ⊗ E ⊕ Λ2 E Λ30 E 0

(note that the trivial representation is not a spin isotropy representation of sp(1)⊕sp(2)). As a matter of fact it turns out that Λn+1 0 E is the only fundamental representation of sp(n + 1) that contains spin isotropy representations of sp(1) ⊕ sp(n). However, Λn+1 0 E decomposes as follows (cf. [14]): n−1 n−2 n Λn+1 0 E = Λ0 E ⊕ H ⊗ Λ0 E ⊕ Λ0 E , 4n and Λn−2 0 E is not a spin isotropy representation. Hence, sp(n +1) = sp(1)⊕sp(n)⊕R is not spin symmetric if n > 1.

3. Symmetric Spaces with Spin Killing Structures Since it does not play any part in the proof we can drop the assumption on the Z2 – grading of the spinor spaces. In the previous section we used Z2 –graded spinor spaces for notational simplicity and to show the second claim of Proposition 2.10. 3.1. A glimpse at the main ideas. The key points for the definitions in the previous section are the following. If G/H is a symmetric space such that its associated orthogonal symmetric Lie algebra (g, τ ) is spin symmetric, then the Lie algebra representation λ : g → End(V ) is H –equivariant (V ⊆ /Sp), and hence, λ|p : p → End(S / p) extends to a smooth bundle map T : T M → End(S / M) on any spin manifold (M, g) with H –structure PH M ⊆ PSO (M, g), where H → SO(n) is the isotropy representation of G/H . In particular, assuming PH M to be torsion free, then T is parallel and the connection ∇ + T is flat on the subbundle defined by V if and only if (M, g) is locally

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symmetric. For instance, if G is simply connected and Λ : G → Gl(V ) the Lie group version of λ, then for each v ∈ V , the map G → V ⊂ /Sp, g → Λ(g −1 )v is H –equivariant, i.e. each v ∈ V yields a section in /S G/H which is parallel w.r.t. ∇ +T. In this way the subbundle corresponding to V is trivialized by these spinor fields. Moreover, the Dirac operator D / acts on these parallel sections as the endomophism −Υλ . The condition [λ(a), Υλ ] = 0 for all a ∈ g (cf. Eq. (13)) ensures that the eigenspaces of Υλ are preserved by λ. In fact, if v is an eigenvector of Υλ , the H –equivariant map g → Λ(g −1 )v yields a Dirac eigenspinor parallel w.r.t. ∇ + T. Thus, the formula (cf. Eq. (14)) (Υλ )2 = −

1 λ(ei )λ(ei ) + scal(g0 ) 4

follows simply by applying these parallel Dirac eigenspinors to the Lichnerowicz formula D / 2 = ∇ ∗ ∇ + 41 scal(g0 ). 3.2. Riemannian (G, H )–structures. Let (g = h⊕p, τ ) be the orthogonal symmetric Lie algebra of a symmetric space G/H with dim G/H = n. We denote by ρ : H → GL(p) the isotropy representation of G/H , in particular Ad G (h) = Ad H (h) ⊕ ρ(h) for all h ∈ H . Since ρ is a faithful representation, Hρ := ρ(H ) is a closed Lie subgroup of GL(n) and isomorphic to H . A (G, H )–structure on a manifold M n is a Hρ –structure on M, i.e. a reduction of the linear frame bundle PGl M to a principal bundle PHρ M = PH M. Since g admits an H invariant product, ρ restricts to a representation ρ : H → SO(p). Hence, a Riemannian (G, H )–structure on a Riemannian manifold (M, g) is a reduction of the SO– frame bundle of (M, g) to a principal bundle PH M ⊆ PSO (M, g), where ρ

H → SO(n). Since β : H → G, a (Riemannian) (G, H )–structure leads to a G– principal bundle PG M = PH M ×β G. Let PH M be a (Riemannian) (G, H )–structure and consider the following associated Lie algebra bundles: HM := ad(PH M) = PH M ×Ad H h,

G M := PH M ×ξ g = ad(PG M)

on M, where Ad H means the adjoint representation of H on its Lie algebra and ξ is the restriction of Ad G to H . The smooth Lie algebra structure on HM and G M is induced by the Lie algebra structure on h and g. Since ξ is reducible to Ad H ⊕ ρ, we obtain G M = HM ⊕ T M. The Lie algebra version of ρ extends to a smooth bundle map HM → End(T M) which is fiber-wise a faithful Lie algebra representation. In particular, HM can be considered as a Lie subbundle of End(T M) which acts in the obvious way on T M. Moreover, if PH M is a Riemannian (G, H )–structure, HM can be cong sidered as a Lie subbundle of Λ2 M ∼ = so(T M) ⊂ End(T M). Since g = h ⊕ p is an orthogonal symmetric Lie algebra, we obtain the well known relations: [HM, HM]G ⊆ HM, [HM, T M]G ⊆ T M, [T M, T M]G ⊆ HM. If the Levi Civita connection reduces to a connection on a Riemannian (G, H )–structure PH M (i.e. PH M is torsion free), then the Riemannian curvature operator R : Λ2 M → Λ2 M has its image in HM ⊆ Λ2 M. If G/H is a simply connected symmetric space, G/H admits a (G, H )–structure PH ⊆ PS O (G/H ) which is determined by the Lie


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group G. Moreover, since the holonomy group of G/H is contained in H , the Levi– Civita connection is reducible to PH . If PSpin (M, g) is a spin structure of (M, g) and /S M = PSpin (M) ×µ S is a complex spinor module of M (remember that /S M = /S0 ⊗ Ck for the irreducible complex spinor bundle /S0 of M and some k ≥ 1), the bundle End(S / M) = /S M ∗ ⊗ /S M is determined by: End(S / M) = PSO (M, g) ×κ gl(S) , where κ : SO(n) → GL(gl(S)) is induced from µ∗ ⊗ µ (cf. [15, Ch. I, Prop. 5.18]). In particular, if PH M ⊆ PSO (M, g) is a Riemannian H –structure, we conclude End(S / M) = PH ×χ gl(S) with χ := κ ◦ ρ : H → GL(gl(S)). This leads to the following proposition. Proposition 3.1. Let G/H (H connected) be a symmetric space such that the corresponding orthogonal symmetric Lie algebra (g, τ ) admits a spin Killing structure λ. Suppose (M, g) is a spin manifold and PH M ⊆ PSO (M, g) is a Riemannian (G, H )– structure on M. Then there is a complex spinor module /S M = /S0 ⊗ Ck (S / 0 irreducible complex spinor bundle and k ≥ 1) and a smooth bundle map T : G M = T M ⊕ HM → End(S / M) which is in each fiber equivalent to the Lie algebra representation λ : g → gl(S / p), where /Sp is a complex spinor space of p = g/h. In particular, T satisfies [T(A), T(B)] = T([A, B]),

T(W )ψ =

1 γ (W )ψ 2

for all A, B ∈ G M, W ∈ HM ⊆ Λ2 T M and ψ ∈ VT := span{T(A)(ϕ)| A ∈ G M, ϕ ∈ /S M} ⊆ /S M.

(15)

Suppose additionally that M is simply connected, g is complete and PH M is torsion free. Then T ∈ Γ (T ∗ M ⊗ End(S / M)) defines a parallel spin Killing structure if and only if (M, g) is a symmetric space with orthogonal symmetric Lie algebra (g, τ ). By definition T is a parallel spin Killing structure if T is parallel w.r.t. the Levi–Civita connection ∇ and the subbundle VT is trivialized by T–Killing spinors (i.e. spinors which are parallel := ∇ + T). w.r.t. the connection ∇ Proof. The subspace V := span{λ(a)ϕ|ϕ ∈ /Sp, a ∈ g} ⊆ /Sp is λ invariant and V ∗ ⊗ V ⊆ gl(S / g) is χ invariant, i.e. the bundle VT ⊆ /S M is well defined. The Lie algebra version χ∗ of χ is given by ( 21 γ ◦ ρ∗ )∗ 21 γ ◦ ρ∗ , where ρ∗ : h → so(p) is the isotropy representation. 21 γ ◦ ρ∗ restricted to V equals λ|h, i.e. χ∗ restricted to V ∗ ⊗ V is determined by χ∗ : h → gl(V ∗ ⊗ V ), b → (λ∗ λ)(b) = λ(b)∗ ⊗ 1 + 1 ⊗ λ(b) ,

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and since the Lie algebra version of ξ : H → GL(g) is the restriction of adg to h, we conclude the H –invariance of λ: χ (h)λ(a) = λ(ξ(h)a) for all h ∈ H and a ∈ g. Thus, the following diagram is commutative: PH × g PH × g

λ

/ PH × gl(V )

(x, a)

λ

/ PH × gl(V )

(x · h −1 , ξ(h)a)

λ

/ (x, λ(a))

λ

/ (x · h −1 , χ (h)λ(a))

and there exists a smooth bundle map T : G M → End(S / M) with the claimed properties. Suppose that (M, g) is a simply connected symmetric space and the Levi–Civita connection is reducible to PH M, then T is parallel by construction. In fact, VT is preserved := ∇ + T satisfies on VT, by the Levi Civita connection and the curvature of ∇ X,Y = R sX,Y + [T X , TY ] = 1 γ (R(X ∧ Y )) + T([X, Y ]G ) R 2 1 = γ (−[X, Y ]G ) + T([X, Y ]G ) = 0 , 2 where R : Λ2 T M → Λ2 T M means the Riemannian curvature operator (note that is a flat connection T(h) = 21 γ (h) on VT for all h ∈ HM by definition of λ). Thus, ∇ on VT and since M is simply connected, VT is trivialized by T–Killing spinors. Conversely, suppose that T defines a parallel Killing structure, i.e. ∇T vanishes and = ∇ + T trivializes the bundle VT by spinors ψ with ∇ψ = 0. In this the connection ∇ case ∇ preserves VT and the curvature of ∇ satisfies for all ϕ ∈ VT: R sX,Y ϕ = −[T X , TY ]ϕ = −T([X, Y ]G )(ϕ).

(16)

Since PH M is torsion free, the curvature operator R : Λ2 M → Λ2 M takes only values in HM ⊆ Λ2 M. In particular, 21 γ (R(X ∧ Y )) = R sX,Y and 21 γ (h) = T(h) for all h ∈ HM show: T(R(X ∧ Y ))ϕ = R sX,Y ϕ = −T([X, Y ]G )ϕ for all ϕ ∈ VT. Moreover, by assumption T : HM → End(VT) is injective which proves R(X ∧ Y ) = −[X, Y ]G , i.e. (M, g) is a symmetric space with orthogonal symmetric Lie algebra (g, τ ).

3.3. Estimating ∇T. Let PH M ⊆ PSO (M, g) be a Riemannian (G, H )– structure, then any connection on PH M induces a Riemannian connection ∇ on the tangent bundle T M. If T(∇ ) ∈ Λ2 M ⊗ T M denotes the torsion of ∇ , then the orthogonal projection of T(∇ ) to the subspace (HM)⊥ ⊗ T M ⊆ Λ2 M ⊗ T M does not depend on the choice of ∇ , it depends only on PH M. Thus, we denote the orthogonal projection of T(∇ ) to (HM)⊥ ⊗ T M by T(PH M) and call T(PH M) the intrinsic torsion of PH M (cf. [12]). Suppose ∇ is the projection of the Levi– Civita connection to PH M, then T(∇ ) = T(PH M). Moreover, we conclude for α(X ) := ∇ X − ∇ X , α(X )Y − α(Y )X = T(∇ )(X, Y ).


421

Conversely, using the fact α(X ) ∈ so(T M) ∼ = Λ2 M, we obtain α(X ), Y ∧ Z = T(∇ )(X, Y ), Z − T(∇ )(Y, Z ), X + T(∇ )(Z , X ), Y which means that α is completely determined by T(PH M) and |α| ≤ 3|T(PH M)| ≤ 6|α|. The lift of to the spinor bundle is given by ∇ X = ∇ X + 21 γ (α(X )). By construction of T we have ∇ T = 0 which supplies the following: ∇

0 = (∇ X T)Y ϕ = ∇ X TY ϕ − T∇ X Y ϕ − TY ∇ X ϕ 1 γ (α(X ))TY ϕ − Tα(X )Y ϕ − TY γ (α(X ))ϕ . = (∇ X T)Y ϕ + 2 Hence, we obtain the estimate 1 (∇ X T)Y ≤ TY · γ (α(X )) + Tα(X )Y 2

1 n + |X | · |Y | · |α| · sup TV ≤ 2 2 |V |=1 √ (. is the operator norm and for all 2–forms η: γ (η) ≤ [n/2] · |η| [cf. Lem. 5.5]), in particular, δT = (∇e j T)e j satisfies:

1 n δT ≤ 3n + · |T(PH M)| · sup TV . (17) 2 2 |V |=1 3.4. Examples of symmetric spaces which are spin Killing. A symmetric space G/H is spin Killing if G/H is spin and its associated orthogonal symmetric Lie algebra (g, τ ) is a spin symmetric Lie algebra. A simply connected symmetric space is spin Killing if and only if all of its irreducible components are spin Killing. Hence, it is sufficient to consider irreducible symmetric spaces. Since symmetric spaces of non–compact type are diffeomorphic to Rm , these spaces are spin Killing if and only if the corresponding orthogonal symmetric Lie algebra is spin symmetric. In the compact case however, there are examples of symmetric spaces which are not spin and the corresponding orthogonal symmetric Lie algebra is spin symmetric. For instance, RPn is spin if and only if n ≡ 3(mod 4), but the corresponding orthogonal symmetric Lie algebra so(n + 1) = so(n) ⊕ Rn is spin symmetric. In particular, RPn is spin Killing if and only if n ≡ 3(mod 4). Using the results in Sect. 2.5 the following examples are known: – Simply connected compact irreducible spin Killing symmetric spaces: S n , CP2n+1 , symmetric spaces of type II. – Non–compact irreducible spin Killing symmetric spaces: RHn , CH2n+1 , symmetric spaces of type IV. – Simply connected symmetric spaces which are not spin Killing: compact CP2n , HPm , CaP2 , G 2 /SO(4), non–compact CH2n , HHm , CaH2 , G 22 /SO(4).

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M. Listing

4. Proof of the Main Theorem Although not necessary for the proof we first remark that inequality (1) is well behaved under rescaling of g0 and g. Consider g 0 := cg0 and g := cg for some constant c > 0. Since connections are invariant under rescaling, the intrinsic torsion of the H –structure is invariant, but T(PH M) is a (2, 1) tensor which leads to 1 |T(PH M)|g = √ |T(PH M)|g c

and

1 κ = √ κ. c

Hence, inequality (1) is equivalent to

n + 1 κ · |T(PH M)|g . scal(g) ≥ scal(g 0 ) + 3n 2 2

Since G/H is spin Killing, there is a spin Killing structure T0 : G 0 → End(S / G/H ) which is induced from a Lie algebra representation λ : g → gl(S / p) (cf. Prop. 3.1) ∗ that satisfies λ(x) = λ(x) for all x ∈ p (note that g = g0 ⊕ g− ). Define as before Υλ = γ (ei )λ(ei ) ∈ gl(S / p), where e1 , . . . , en is an orthonormal base of p, then (cf. 14) 0 ≥ scal(g0 ) = 2 γ (ei )γ (e j )λ([e j , ei ]) = 4(Υλ )2 + 4 λ(e j )λ(e j ) i, j

j

on the subspace Im(λ) = span{λ(a)φ|φ ∈ /Sp, a ∈ p}. Together with λ(x)∗ = λ(x) we conclude that all eigenvalues of (Υλ )2 are real and ≤ 0. In fact, each eigenvalue of Υλ is purely imaginary. Let z ∈ iR be an eigenvalue of Υλ with |z| = Υλ and let V ⊂ /Sp be the corresponding eigenspace. Using Υλ λ(a) = λ(a)Υλ for all a ∈ g, we obtain that V is λ–invariant. Hence V defines a nontrivial subbundle V 0 ⊆ /S G/H which is preserved by the Levi–Civita connection and T0 . Moreover, the bundle V 0 is trivialized by T0 – Killing spinors. Without further noting we consider now the pair (V 0 , T0 ) on M − C instead on G/H (simply use the pull back by f ). In fact, ∇ 0 + T0 is a flat connection on the vector bundle V 0 → M − C. The Riemannian (G, H )–structure PH M ⊆ PSO (M, g) yields a map T : G M → End(S / M) and a subbundle V ⊆ /S M (induced by the H – invariant subspace V ⊆ /Sp). Although V is preserved by T it is not yet preserved by the Levi– Civita connection of g. In the following we will show that V is trivialized by T–Killing spinors. We note that the inequality in the main theorem yields the necessary inequality (7) to apply the general positive mass theorem. Since T X ≤ 21 κ|X | (see below), inequality (17) supplies

3 n + 1 · κ · |T(PH M)|. δT ≤ n 2 4 2 Thus, if t is the maximal eigenvalue of λ(ei )λ(ei ) and |z|2 is the maximal eigenvalue of −(Υλ )2 ≥ 0, we obtain

1 n 3 1 + 1 · κ · |T(PH M)| R = scal(g) ≥ scal(g0 ) + n 2 4 4 4 2 ≥ −|z|2 + t + δT .

(18)


423

Let us consider the isomorphism A between the H –principal bundles from the definition in the introduction. A extends uniquely to an isomorphism on the corresponding G and SO–frame principal bundles. Moreover, since M −C is simply connected and n ≥ 3, A also extends uniquely to an isomorphism on the corresponding Spin(n)–principal bundles. In fact, if W is a representation space of K ∈ {H, SO(n), G, Spin(n)}, then A : PK0 (M − C) → PK M|M−C extends in the usual way to an isomorphism A˜ on the associated vector bundle. In particular, A yields an isometric Lie algebra isomorphism: ˜ Ay) ˜ = f ∗ g0 (x, y), g( Ax,

A˜ : G 0 M|M−C → G M|M−C

and a bundle isometry A˜ : /S0 M|M−C → /S M|M−C . Here is a small difference from previous works on the subject, since [1,19] used the (unique) positive definite g–symmetric ˜ but in our case A˜ does not have to be symmetric w.r.t. g. Howgauge transformation A, ever, since we already start with a bundle isomorphism A, we do not need the symmetry of A˜ in order to get a unique (smooth) bundle map. Thus, A˜ maps V 0 to V ⊂ /S M and by construction of T0 , T: ˜ A˜ ◦ T0X = T AX ˜ ◦ A. ˜ 0 A˜ −1 yields a connection If ∇ 0 denotes the Levi Civita connection of f ∗ g0 , ∇ X := A∇ X on T M|M−C respectively on /S M|M−C which is Riemannian w.r.t. g. In fact, considering the torsion of ∇, we conclude the estimate: ˜ · |Y | , |∇ X Y − ∇ X Y | ≤ const · | A˜ −1 | · |∇ 0 A| ˜ · |ψ| |∇ X ψ − ∇ X ψ| ≤ const · | A˜ −1 | · |∇ 0 A| for all vector fields X, Y and spinors ψ. Moreover, since g and f ∗ g0 are uniformly bounded against each other, A˜ and A˜ −1 are uniformly bounded from below and above. In particular, ˜ AZ ˜ ) ˜ ˜ ) + g( AY, ˜ (∇ X0 A)Z ˜ ) = −(∇ X0 g)( AY, AZ g((∇ X0 A)Y, ˜ ≤ const · |∇ 0 g| as well as leads to |∇ 0 A| |∇ X ψ − ∇ X ψ| ≤ const · |∇ 0 g| · |ψ| . Lemma 4.1. The mass functional M : K∞ (V, T) → R ∪ {+∞} is non–negative and vanishes on the image of the well defined injective map ˜ , Ξ : K(V 0 , T0 ) → K∞ (V, T), ψ → [h · Aψ] where h : M → [0, ∞) is a smooth cut off function for the asymptotic symmetric end, i.e. h = 0 in C, h = 1 at infinity and dh has compact support.

424

M. Listing

Proof. We start by showing that each nontrivial T0 –Killing spinor yields a nontrivial 0 = ∇ 0 + T0 on /S M|M−C asymptotic T–Killing spinor. Consider the connections ∇ 0 = ∇ + T on /S M, where ∇ , ∇ mean the Levi–Civita connections of f ∗ g0 and ∇ 0 , then ψ0 is not contained in and g respectively. Let ψ0 ∈ Γ (V 0 ) be parallel w.r.t. ∇ 2 2 κr L and |ψ0 |0 grows asymptotically like e (see the estimate below). Moreover, set ˜ 0 ∈ Γ (V), then ψ := h Aψ ˜ 0 − ∇ X Aψ ˜ 0 | + |∇ X Aψ ˜ 0 + T X Aψ ˜ 0 |, X ψ| ≤ (X h)|ψ0 |0 + |∇ X Aψ |∇ ˜ 0X ψ0 | ˜ 0 − AT X0 ψ0 | + |T X Aψ ≤ const (X h) + |∇ 0 g| · |X | · |ψ0 |0 + | A˜ ∇ ˜ = const (X h) + |∇ 0 g| · |X | · |ψ0 |0 + |T X − AX ˜ Aψ0 | ≤ const |dh| + |∇ 0 g| + | A˜ − Id| · sup TV · |X | · |ψ0 |0 . |V |=1

Thus, 2TV ≤ κ|V | and our asymptotic assumptions supply ∈ L 2 (M, T ∗ M ⊗ /S M) ∇ψ as well as

X ψ, ψ ∈ L 1 (M; volg ) ∇

X ·D / ψ, ψ ∈ L 1 (M; volg )

for all X with |X | = 1. In this case D / is defined on Γ (S / M) by D / := D / + z · Id (note

ei on Γ (V) by definition, since z ∈ iR was defined to be the that D / =D / = ei · ∇ maximal absolute eigenvalue of γ (ei )λ(ei )). Thus, ψ is an asymptotic T–Killing spinor with mass M(ψ) = 0. This proves that Ξ is well defined and injective (since ψ∈ / L 2 , it is nontrivial). Using the general positive mass theorem and inequality (18) we conclude that M vanishes on the image of the linear map Ξ . Hence, it remains to show that |ψ0 |20 ≤ c · eκr for some constant c > 0. Suppose G/H is an irreducible symmetric space of noncompact type and minimal sectional curvature −κ0 < 0. Then the maximal √ κ0 0 (absolute) eigenvalue of T X is given by 2 |X | for all X ∈ T M (just consider the Lie algebra representation λ). Suppose G/H is the symmetric space from the theorem, and −κ j are the minimal sectional curvatures of the irreducible components of G/H . Let X ∈ T G/H and denote by X j the orthogonal projection of X to the j th irreducible component G j /H j of G/H . Since T0 comes from the tensor product representation of the spin Killing structures on G j /H j , we obtain (note |X |0 = |X j |20 ) ! " m " m 1 1 1 √ 0 κ j |X j |0 ≤ |X |0 # κ j = κ|X |0 . T X ≤ 2 2 2 j=1

j=1

In fact, T0X admits the eigenvalue ± 21 κ|X | for some vectors X (note that the (parallel) spinors which come from the Euclidean part of G/H are constant, i.e. these spinors do not increase the growth rate of the Killing spinors on G/H ). Thus, if φ is a T 0 –Killing spinor on G/H , we conclude |Xg0 (φ, φ)| ≤ 2|∇ X0 φ, φ| = 2|T X0 φ, φ| ≤ κ · |X |0 · |φ|20 , and integrating along a radial geodesic supplies |φ|20 ∈ O(eκr ), where r is the g0 –distance to a fixed point.


425

Because rk(V 0 ) = dim K(V 0 , T0 ) = rk(V), we conclude from Corollary 1.8 that V is trivialized by T–Killing spinors and that the curvature on V is given by: R sX,Y = −[T X , TY ] = −T([X, Y ]G ). Hence, if ϕ is a nontrivial T–Killing spinor, the scalar curvature of g satisfies (cf. 14): γ (ei )γ (e j )Resi ,e j ϕ = 2 γ (ei )γ (e j )T([e j , ei ]G )ϕ = scal(g0 ) · ϕ. scal(g) · ϕ = 2 i, j

i, j

Since ϕ has no zeros, we obtain scal(g) = scal(g0 ) and therefore, inequality (1) shows that PH M is torsion free. Thus, T and TV = T ◦ πV are parallel and (V, TV ) is a parallel Killing structure. In particular, Proposition 3.1 proves that (M, g) is isometric to (G/H, g0 ) (the restriction of λ to V is also a spin Killing structure of (g, τ )). 5. Irreducible Examples The following examples are applications of the main theorem. Without further noting all the irreducible symmetric spaces are scaled in such a way that the minimal sectional curvature is K min = −1. 5.1. The real hyperbolic space. Scalar curvature rigidity of the real hyperbolic space was shown by M. Min–Oo in [19]. Since the real hyperbolic space is given by RHn = SO(1, n)/SO(n), the H –structure on M n is already determined by the choice of the Riemannian metric. Thus, a Riemannian manifold (M n , g) is said to be strongly asymptotically hyperbolic if there is a compact set C ⊂ M and a diffeomorphism f : M −C → RHn − B R (0) in such a way that 1 c

· g ≤ f ∗ g0 ≤ c · g,

|g − f ∗ g0 | + |∇ 0 g| ∈ L 1 ∩ L 2 (M − C; er volg ) for some constant c > 0 (g0 means the hyperbolic metric and r is the g0 –distance to a fixed point). In particular, (M, g) is strongly asymptotically hyperbolic if and only if (M, g, PSO (M)) is strongly asymptotic to the real hyperbolic space SO(1, n)/SO(n) in the sense of the definition in the introduction. Thus, the main theorem reduces to Min–Oo’s rigidity result. Theorem 5.1 (Min–Oo [19]). Let (M, g) be a complete spin manifold of dimension n ≥ 3. If (M, g) is strongly asymptotically hyperbolic of scalar curvature scal(g) ≥ −n(n − 1), then (M, g) is isometric to RHn . The bundle G(M) is given by T M ⊕ Λ2 T M and the bundle map T by 1 1 i γ (X ) + γ (Y )γ (Z ) − γ (Z )γ (Y ) , 2 4 4 where γ denotes as usual the Clifford multiplication on the spinor bundle. Since ∇γ = 0, T is parallel. Suppose ϕ ∈ Γ (S / M) is a non–trivial spinor which is parallel w.r.t. the = ∇ + T, then (M, g) is an Einstein space and ϕ is called an imaginary connection ∇ Killing spinor (cf. [4]). Moreover, the complex spinor bundle is trivialized by imaginary Killing spinors if and only if (M, g) ∼ = RHn (in this case we assumed (M, g) to be complete). T(X + Y ∧ Z ) =

426

M. Listing

5.2. The complex hyperbolic space in odd complex dimensions. The first scalar curvature rigidity result of the complex hyperbolic space is due to M. Herzlich. He proved in [11] the main theorem for CH2n−1 in the presence of a torsion free H – structure. Herzlich’s result was generalized in [17] by assuming an almost Hermitian structure instead of a Kähler structure. The complex hyperbolic space is given by CHm = SU(1, m)/U(m) = G/H. An U(m) ⊂ SO(2m) structure on a manifold is uniquely determined by an almost Hermitian structure (g, J ) on the tangent bundle (g is a Riemannian metric and J is a g– compatible almost complex structure). Thus, an almost Hermitian manifold (M 2m , g, J ) is strongly asymptotic to CHm (called strongly asymptotically complex hyperbolic) if there is a compact set C ⊂ M and a diffeomorphism f : M − C → CHn − B R (0) in such a way that |g − f ∗ g0 | + |J −

1 ∗ c · g ≤ f g0 ≤ c · g, f ∗ J0 | + |∇ 0 g| ∈ L 1 ∩ L 2 (M

− C; er · volg )

(g0 means the complex hyperbolic metric, J0 is the standard complex structure on CHm , and r is the g0 –geodesic distance to a fixed point). We showed in one of the previous sections that CHm is spin Killing if and only if m is odd. Hence, we obtain the following theorem which was already proved in [17] (including the estimate 4δT ≤ c(m)|∇ J |). As already mentioned, the case ∇ J = 0 is due to Herzlich (cf. [11]). Theorem 5.2 ([11,17]). Let (M 2m , g, J ), m odd, be a complete almost Hermitian spin manifold which is strongly asymptotically complex hyperbolic. If the scalar curvature satisfies √ √ scal(g) ≥ −m(m + 1) + 8m + 4m 4m − 2 |∇ J |, then (M,g,J) is Kähler and isometric to CHm . Using spinc methods a similar result can be shown in even complex dimension although there are more terms involved on the right-hand side of the scalar curvature inequality (cf. [17]). The bundle G(M) is determined by T M ⊕ Λ1,1 M. The irreducible complex spinor bundle of an almost Hermitian manifold decomposes orthogonal into /S M = /S0 ⊕ · · · ⊕ /Sm , where /S j is an eigenspace of Ω = g(., J.) to the eigenvalue i(m − 2 j) and /S j is com√ plex isomorphic to Λ0, j ⊗ Λm,0 . If X 1,0 = 21 (X − i J X ) and X 0,1 = 21 (X + i J X ) denote the usual decompositions of X in TC M, we obtain γ (X 1,0 ) : /S j → /S j+1 and γ (X 0,1 ) : /S j → /S j−1 . Let π j : /S M → /S j be the orthogonal projection to /S j and define 1 i γ (X 1,0 )πn−1 + γ (X 0,1 )πn + γ (θ )(πn−1 + πn ) T(X + θ ) = 2 2 for θ ∈ Λ1,1 M ⊂ Λ2 M and X ∈ T M, where n := m+1 2 . Then T is parallel if (g, J ) is Kähler. Moreover, T–Killing spinors are called imaginary Kähler Killing spinors (cf. [13]). The bundle VT = span{T(A)ϕ| A ∈ G(M), ϕ ∈ /S M} = /Sn−1 ⊕ /Sn is trivialized by imaginary Kähler Killing spinors on the complex hyperbolic space.


427

5.3. Symmetric spaces of type IV. Symmetric spaces of type IV are dual to symmetric spaces of type II. Let (g, τ ) be the orthogonal symmetric Lie algebra of a type IV symmetric space. Then g = h ⊕ ih for a simple real Lie algebra h and τ is the complex conjugation. In particular, p = ih has the following Lie algebra structure: [ih 1 , ih 2 ]p := i[h 1 , h 2 ]h = −i[ih 1 , ih 2 ]g for ih 1 , ih 2 ∈ p. Moreover, ξ : h → p, h → ih is a Lie algebra isomorphism with [x, h]g = [x, ξ(h)]p for x ∈ p and h ∈ h. Hence, adp ◦ ξ = ρ, where ρ : h → so(p) (induced by adg) and a straightforward calculation shows: adp([x, y]p) = −ρ([x, y]g). If g := −B(., τ.) denotes the positive definite inner product induced by the Killing form B on the orthogonal symmetric Lie algebra (g, τ ) of noncompact type, then adg(a) is symmetric w.r.t. g for a ∈ p and skew symmetric for a ∈ h (cf. [10, Ch. VI, Lem. 1.2]). Moreover, adp(x) is skew symmetric w.r.t. g for all x ∈ p (note that ix ∈ h ⊂ g): g([x, y]p, z) = g(−[ix, y]g, z) = g(y, [ix, z]g) = g(y, −[x, z]p), and thus, adp : p → so(p) yields a selfadjoint spin Killing structure of (g, τ ): λ : p ⊕ h → gl(S / p), x + h →

i 1 γ (adp(x)) + γ (ρ(h)) 2 2

(note that for an orthogonal symmetric Lie algebra of type II the adjoint map also satisfies adp : p → so(p) when using the standard product induced by the Killing form of g). Let G/H be a type IV or a type II symmetric space and (M, g) be a Riemannian manifold. Then a Riemannian (G, H )–structure PH ⊂ PSO (M, g) on M is determined by a smooth skew symmetric Lie algebra structure on T M which is fiberwise equivalent to the Lie algebra of H . Definition 5.3. Let (M, g) be a Riemannian manifold, then a smooth Lie algebra structure [., .]T on T M is called skew symmetric if the corresponding adjoint map ad T : T M → End(T M) has its image in so(T M, g), i.e. ad T (X ) is skew symmetric w.r.t. g for all X ∈ T M. We call the triple (M, g, [., .]T ) an H -simple Riemannian manifold if [., .]T defines a smooth skew symmetric Lie algebra structure on T M which is fiberwise equivalent to the Lie algebra of a simple real Lie group H . If the adjoint map ad T is parallel w.r.t. the Levi– Civita connection, then (M, g) is locally isometric to a type II or a type IV symmetric space. Note that ad T can not determine the type, it only yields local symmetry and the holonomy group if ∇ad T = 0. A natural definition of an asymptotic type IV symmetric space would be |∇ad T | ∈ O(e−αr ). However, it seems difficult to show rigidity if we control just ∇ad T instead of the two expressions ∇ and ad T independently. Thus, an H - simple Riemannian manifold (M, g, [., .]T ) is strongly asymptotic to a type IV symmetric space (G/H, g0 ) if there

428

M. Listing

is a compact set C ⊂ M and a diffeomorphism f : M − C → G/H − B R (0) in such a way that 1 ∗ c · g ≤ f g0 ≤ c · g, f ∗ ad0 | + |∇ 0 g| ∈ L 1 ∩

|g − f ∗ g0 | + |ad T −

L 2 (M − C; er · volg )

(r is the f ∗ g0 –geodesic distance to a fixed point on M and ad0 is the adjoint map induced by the canonical Lie algebra structure on T G/H ). Since H is simple, there is an unique A˜ ∈ Γ (End(T M|M−C )) with ˜ ˜ Ay) ˜ = f ∗ g0 (x, y) and A˜ ◦ f ∗ ad0 (x) ◦ A˜ −1 = ad T ( Ax), g( Ax, in fact A˜ is equivalent to an isomorphism A : PH0 (M − C) → PH M|M−C . Theorem 5.4. Let (G/H, g0 ) be a type IV symmetric space of dimension n ≥ 3 and (M, g, [., .]T ) be a complete H -simple Riemannian spin manifold which is strongly asymptotic to (G/H, g0 ). If the scalar curvature satisfies

n · |∇ad T |, scal(g) ≥ scal(g0 ) + 2 n · 2 then (M, g) is isometric to G/H and ad T is parallel. In this case the bundle G M is given by T M ⊕ T M, where one of the tangent bundles is considered as a subbundle of Λ2 M using the adjoint transformation ad T . The map T is defined by T X := 2i γ (ad T (X )) for X ∈ T M, where ad T (X ) is considered as a ∼ 2 2–form using the isomorphism so(T M) =TΛ T M. Hence, ∇γ = 0 yields the following i T δT = 2 γ (δad T ). Since δad = (∇ei ad )(ei ) is considered as 2– form, we conclude the estimate 4δT ≤ 2 [n/2] · |δad T | ≤ 2 n · [n/2] · |∇ad T | from the lemma below. The proof of the above theorem follows now from the proof of the main theorem replacing the intrinsic torsion tensor T(PH M) by ∇ad T . 2 Lemma 5.5. Suppose p is an n–dimensional vector space and ϑ ∈ Λ p, then γ (ϑ) ≤ |ϑ| [ n2 ] with equality for some nontrivial ϑ ∈ Λ2 p.

Proof. Consider ϑ ∈ Λ2 p, then 2γ (ϑ) =

n

γ (e j ∧ (e j ϑ)) =

j=1

n

γ (e j )γ (e j ϑ)

j=1

for an orthonormal basis e1 , . . . , en of p. Since γ (x) = |x| for any x ∈ p, we obtain ! " n √ " n 2γ (ϑ) ≤ |e j ϑ| ≤ #n |e j ϑ|2 = 2n|ϑ|. j=1

j=1

This inequality is an equality if n is even and ϑ = e1 ∧ e2 + e3 ∧ e4 + e5 ∧ e6 + · · · + en−1 ∧ en .


429

If n is odd and ϑ ∈ Λ2 p, then ϑ is degenerated and we can choose en in such a way that en ϑ = 0. Using the same estimates as above but with the difference that we only have n − 1 nonzero summands yields: 2γ (ϑ) ≤ |ϑ| 2(n − 1) for all ϑ ∈ Λ2 p and n = dim p odd. As before this inequality is an equality for certain 2–forms of maximal rank.

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Communications in


Analytic Continuation of Eigenvalues of a Quartic Oscillator Alexandre Eremenko , Andrei Gabrielov Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. E-mail: [email protected]; [email protected] Received: 21 February 2008 / Accepted: 30 June 2008 Published online: 22 October 2008 – © Springer-Verlag 2008

Abstract: We consider the Schrödinger operator on the real line with even quartic potential x 4 +αx 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane. 1. Introduction Consider the boundary value problem on the real line: − y + (βx 4 + x 2 )y = λy,

y(−∞) = y(∞) = 0.

(1)

If β > 0, then this problem is self-adjoint, it has a discrete spectrum of the form λ0 < λ1 < . . . → +∞, and every eigenspace is one-dimensional. The eigenvalues λn are real analytic functions of β defined on the positive ray. In 1969, Bender and Wu [5] studied analytic continuation of λn to the complex β-plane. Their main discoveries are the following: (i) For every non-negative integer m and n of the same parity, the function λm can be obtained by an analytic continuation of the function λn along some path in the complex β-plane. (ii) The only singularities encountered in the analytic continuation of λn in the punctured β-plane C\{0} are algebraic ramification points.

Supported by NSF grant DMS-0555279. Supported by NSF grant DMS-0801050.

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(iii) These ramification points accumulate to β = 0, in such a way that no analytic continuation of any λn to 0 is possible. The last statement gives a good reason why the formal perturbation series of λn in powers of β is divergent. Bender and Wu also studied the global structure of the Riemann surfaces of the functions λn spread over the β-plane, that is the position of their ramification points, and how the sheets of these Riemann surfaces are connected at these points. Bender and Wu used a combination of mathematical and heuristic arguments with numerical computation. Since the publication of their paper, several results about analytic continuation of λn were proved rigorously. We state some of these results. A broader context in which this problem arises is described in the survey [34]. First of all, we recall a change of the variable which substantially simplifies the problem [32]. Consider the family of differential equations H (α, β)y = λy, where H (α, β) = −d 2 /d x 2 + (βx 4 + αx 2 ). The change of the independent variable w(x) = y(t x)

(2)

gives the differential equation H (t 4 α, t 6 β)w = t 2 λw. Thus, if α and t are real, and β > 0, we have λn (t 4 α, t 6 β) = t 2 λn (α, β). If α = 1 we can take t = β −1/6 > 0 and obtain λn (1, β) = β 1/3 λn (β −2/3 , 1).

(3)

This reduces our problem to the study of the analytic continuation of eigenvalues of the one-parametric family − y + (x 4 + αx 2 )y = λy,

y(−∞) = y(∞) = 0,

(4)

depending on the complex parameter α. The study of this family of quartic oscillators is equivalent to the study of the family (1). Indeed, if we know an analytic continuation of λn (α, 1) along some curve in the α-plane, then Eq. (3) gives an analytic continuation of λn (1, β) in the β-plane and vice versa. The main advantage of restating the problem in the form (4) is that any analytic continuation of an eigenvalue (eigenfunction) of (4) remains an eigenvalue (eigenfunction) of (4). This is not so for the operator in (1): when we perform an analytic continuation of an eigenfunction the result may no longer be an eigenfunction, because it may fail to satisfy the boundary condition, see [32,4]. From now on we consider only the family (4), and slightly change our notation: the λn will be real analytic functions of α > 0 representing the eigenvalues of (4). Notice that the λn have immediate analytic continuations1 from the positive ray to the whole real line. Loeffel and Martin [24] proved that the functions λn have immediate analytic continuations to the sector | arg α| < 2π/3. They conjectured that the radius of convergence of the power series of λn at α = 0 tends to infinity as n → ∞. 1 We say that a function analytic on a set X ⊂ C has an immediate analytic continuation to a set Y ⊂ C if X ∩ Y has limit points in X and there exists an analytic function g on Y such that f (z) = g(z) for z ∈ X ∩ Y .

Analytic Continuation of Eigenvalues of a Quartic Oscillator

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Simon [32,33] proved that the singularities of the λn accumulate to ∞ in the asymptotic direction of the negative ray. More precisely, for every n and for every η ∈ (0, π ), there exists B = Bn (η) such that λn has an immediate analytic continuation from the positive ray to the region {α : | arg α| < η, |α| > B}. On the other hand, he also proved that the λn do not have immediate analytic continuations to full punctured neighborhoods of ∞. This proves statement (iii) of Bender and Wu and implies divergence of the perturbation series for λn at β = 0. Delabaere, Dillinger and Pham in their interesting papers [7–9] used a version of the WKB method to study operator (4) for large α. They claim to confirm all conclusions of Bender and Wu, however it is not clear to us what is proved rigorously in [7–9], which statements are heuristic and which are verified numerically. In particular, we could not determine whether these papers contain a complete proof of the statement (i). Another study of λn for large α is [20]. It is not clear whether statements of Theorem 1 can be derived from the results of [20]. In this paper we give complete proofs of statements (i) and (ii) of Bender and Wu and of the conjecture of Loeffel and Martin. Our methods are different from those of all papers mentioned above. Theorem 1. a) All λn are branches of two multi-valued analytic functions i , i = 0, 1, of α, one for even n, another for odd n. b) The only singularities of i over the α-plane are algebraic ramification points. c) For every bounded set X in the α-plane, there are only finitely many ramification points of i over X . Statements a) and b) prove (i) and (ii) of Bender and Wu. Statement c) implies the conjecture of Loeffel and Martin stated above. Statements b) and c) actually hold in greater generality. Let P(a, z) = z d +ad−1 z d−1 + · · · + a1 z be any monic polynomial of even degree d, with complex coefficients a = (a1 , . . . , ad−1 ) ∈ Cd−1 . Consider the boundary value problem −y + P(a, z)y = λy, y(+∞) = y(−∞) = 0,

(5) (6)

where the boundary condition is imposed on the real axis. For non-real a, this problem is not self-adjoint, however it is known [31] that the spectrum of this problem is infinite and discrete, eigenspaces are one-dimensional, and the eigenvalues tend to infinity in the asymptotic direction of the positive ray. Let be the multi-valued function of a ∈ Cd−1 which to every a puts into correspondence the set of eigenvalues of (5), (6). We write the set (a) as (a) = {µ0 (a), µ1 (a), . . .}, where |µ0 (a)| ≤ |µ1 (a)| ≤ · · · → +∞. So for real a we have µn (a) = λn (a) but this does not have to be the case for complex a. (The functions µn are not expected to be analytic; they are only piecewise analytic). It is known [31] that there exists an entire function F of d variables with the property that λ is an eigenvalue of the problem (5), (6) if and only if F(a, λ) = 0.

(7)

Equation (7) may be called the characteristic equation of the problem (5), (6). It is the equation implicitly defining our multi-valued function .

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Theorem 2. The only singularities of are algebraic ramification points. For every R > 0 there exists a positive integer N such that for n > N the µn are single-valued analytic functions on the set {a : |a| < R} with disjoint graphs. Here are some general properties of implicit functions λ(a) defined by equations of the form (7) with arbitrary entire function F. Let g0 be an analytic germ of such function at some point a0 , and γ0 : [0, 1] → Cd−1 a curve in the a-space beginning at a0 . Then for every > 0 there is a curve γ beginning at a0 , satisfying |γ (t) − γ0 (t)| < , t ∈ [0, 1], and such that an analytic continuation of g0 along γ is possible. In other words, “the set of singularities” of is totally discontinuous. This is called the Iversen property, and it was proved by Julia [21], see also [35]. However, the “set of singularities” of in general can have non-isolated points, as can be shown by examples, [15]. Theorem 2 is in fact an easy consequence of the following known result. Theorem A. For every R > 0 there exists a positive integer N such that for all a in the ball |a| ≤ R we have the strict inequalities |µn+1 (a)| > |µn (a)| for all n ≥ N . For a complete proof of this result we refer to Shin [28, Thm 1.7] who used his earlier paper [29] and the results of Sibuya [31]. Theorem A is derived from the asymptotic expansion for the eigenvalues µn in powers of n which is uniform with respect to a for |a| ≤ R [28, Theorem 1.2]. A similar asymptotic formula for the eigenvalue problem (5), (6) is also given in [17, Ch. III, Sect. 6] where it is derived with a different method. To deduce Theorem 2 from Theorem A, we also need the Weierstrass Preparation Theorem [6,19]: Theorem B. Let Z ⊂ Cm be the set of solutions of Eq. (7), and (a0 , λ0 ) ∈ Z . Suppose that F(a0 , λ) ≡ 0. Then there is a neighborhood V of (a0 , λ0 ) such that in V we have F(a, λ) = (λ − λ0 )k + Fk−1 (a)(λ − λ0 )k−1 + · · · + F0 (a) G(a, λ), where F j and G are analytic functions in V , and G(a0 , λ0 ) = 0, and F j (a0 ) = 0 for 0 ≤ j ≤ k − 1. So for each a close to a0 the equation F(a, λ) = 0 with respect to λ has k roots close to λ0 , and these roots tend to λ0 as a → a0 . Proof of Theorem 2. Application of Theorem A shows that for every R > 0 there exists N such that for n > N the functions µn have disjoint graphs over {a : |a| < R}. Application of Theorem B to the solutions µn (a) of the equation F(a, µn (a)) = 0 with n > N shows that k = 1 for all points (a, µn (a0 )), and then the implicit function theorem implies that the µn are analytic for n > N . This proves the second part of Theorem 2. To prove the first part, consider a curve γ : [0, 1] → {a : |a| ≤ R} ⊂ Cd−1 such that has an analytic continuation gt , 0 ≤ t < 1. Here gt is an analytic germ of at the point γ (t). Suppose that g0 (γ (0)) = µ j (γ (0)). By Theorem A there exists N > j such that |gt (γ (t))| < |µ N (γ (t))| for all t ∈ [0, 1). As µ N (a) is bounded for |a| ≤ R, we conclude that gt (γ (t)) is a bounded function on [0, 1), and there exists a sequence tk → 1 such that gtk (γ (tk )) has a finite limit λ1 . Application of the Weierstrass Preparation theorem to the point (γ (1), λ1 ) shows that in fact gt (γ (t)) → λ1 as t → 1, and gt either has an analytic continuation to the point γ (1) along γ or γ (1) is a ramification point of some order k. This completes the proof.


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An alternative proof can be given by using perturbation theory of linear operators instead of the Weierstrass Preparation theorem as is done in [32]. However we notice that an analog of Theorem 2 does not hold for general linear operators analytically dependent of parameters, as examples in [22, pp. 371–372] show. The crucial property of our operators (5), (6) is expressed by Theorem A. Theorem 2 implies statements b) and c) of Theorem 1. In the rest of the paper we prove statement a). We briefly describe the idea of the proof. Equation (7) which we now write as F(α, λ) = 0,

(8)

defines an analytic set Z ⊂ C2 which consists of all pairs (α, λ) for which the problem (4) has a solution. We are going to show that this set Z consists of exactly two irreducible components, which are also its connected components. To do this we introduce a special parametrization of the set Z by a (not connected) Riemann surface G. As this parametrization : G → Z comes from the work of Nevanlinna [26], we call it the Nevanlinna parametrization. To study the Riemann surface G we introduce a function W : G → C, which has the property that it is unramified over C\{0, 1, −1, ∞}. More precisely, this means that W : G\W −1 ({0, 1, −1, ∞}) → C\{0, 1, −1, ∞} is a covering map. Then we study the monodromy action on the generic fiber of this map W . Using a description of G and W which goes back to Nevanlinna, we label the elements of the fiber by certain combinatorial objects (cell decompositions of the plane) and explicitly describe the monodromy action on this set of cell decompositions. Our explicit description shows that there are exactly two equivalence classes of this action, thus the Riemann surface G consists of exactly two components. In fact we not only prove that G consists of two components but give in some sense a global topological description of the surface G, and thus of the set Z . The plan of the paper is the following. In Sects. 2 we collect all necessary preliminaries and construct G, and W . In Sects. 3 and 4 we discuss the cell decompositions of the plane needed in the study of the monodromy of the map W : G → C. In Sect. 5 we compute this monodromy and complete the proof of statement a) in Theorem 1. In Sect. 6 we briefly mention several other one-parametric families of linear differential operators with polynomial potentials which can be treated with the same method. We thank the referee for many helpful suggestions. 2. Preliminaries Some parts of our construction apply to the general problem (5), (6) so we explain them for this general case. We include more detail than it is strictly necessary for our purposes because the papers [26] and [25] are less known nowadays than they deserve. First we recall some properties of solutions of the differential equation (5). The proofs of all these properties can be found in Sibuya’s book [31]. Every solution of this differential equation is an entire function of order (d + 2)/2, where d is the degree of P. To avoid trivial exceptional cases, we always assume that d > 0. We set q = d + 2 and divide the plane into q disjoint open sectors S j = {z : | arg z − 2π j/q| < π/q},

j = 0, . . . , q − 1.

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In what follows we will always understand the subscript j as a residue modulo q, so that, for example, Sq = S0 , etc. We call S j the Stokes sectors of Eq. (5). 1. For each solution y = 0 of Eq. (5) and each sector S j we have either y(z) → 0 or y(z) → ∞ as z → ∞ along each ray from the origin in S j . We say that y is subdominant in S j in the first case and dominant in S j the second case. 2. Of any two linearly independent solutions of (5), at most one can be subdominant in a given Stokes sector. Let y1 and y2 be two linearly independent solutions, and consider their ratio f = y2 /y1 . Then f is a meromorphic function of order q/2. (The order of a meromorphic function f can be defined as the minimal number ρ such that f is a ratio of two entire functions of order at most ρ.) 3. For each S j , we have f (z) → w j ∈ C as z → ∞ along any ray in S j starting at the origin. 4. w j = w j+1 for all j mod q. 5. w j ∈ {0, ∞} if and only if one of the solutions y1 , y2 is dominant and another is subdominant in S j . A curve γ : [0, 1) → C is called an asymptotic curve of a meromorphic function f if γ (t) → ∞ as t → 1, and f (γ (t)) has a limit, finite or infinite, as t → 1. This limit is called an asymptotic value of f . A classical theorem of Hurwitz says that the singularities of the inverse function f −1 are exactly the critical values and the asymptotic values of f . Returning to the function f = y2 /y1 , where y1 and y2 are linearly independent solutions of Eq. (5), we notice that f does not have critical points. Indeed, all poles of f are simple because y1 can have only simple zeros, and f (z) = 0 because the Wronskian determinant of y1 , y2 is constant. Thus f has no critical values, and the only singularities of f −1 are the asymptotic values of f . Next we describe these asymptotic values and associated asymptotic curves. By property 3 above, for each sector S j , every ray from the origin in S j is an asymptotic curve. Thus all w j are asymptotic values. Function f has no other asymptotic values except the w j . By Hurwitz theorem we conclude that the only singularities of f −1 lie over the points w j . More precisely, f : C\ f −1 ({w0 , . . . , wq−1 }) → C\{w0 , . . . , wq−1 }

(9)

is an unramified covering. Let D j be discs centered at w j and having disjoint closures. Each component B of the preimage f −1 (D j ) is either a topological disc in the plane which is mapped by f onto D j homeomorphically, or an unbounded domain such that f : B → D j \{w j } is a universal covering. Such unbounded domains are called tracts over w j . 6. The tracts are in bijective correspondence with the sectors S j . More precisely, for each j there exists a unique tract B j which contains each ray from the origin in S j , except a bounded subset of this ray, and the total number of tracts is q. The Schwarzian derivative of a function f is f 3 f 2 Sf = − . f 2 f The main fact about S f that we need is its relation with Eq. (5):


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7. A ratio f = y2 /y1 of two linearly independent solutions of (5) satisfies the differential equation S f = −2(P − λ),

(10)

and conversely, every non-zero solution of the differential equation (10) is a ratio of two linearly independent solutions of (5). Let Z d ⊂ Cd be the set of all pairs (a, λ) such that λ is an eigenvalue of the problem (5), (6). Consider the class G d of meromorphic functions with the following two properties: −

1 S f is a monic polynomial of degree d 2

(11)

and f (z) → 0, z ∈ R, z → ±∞.

(12)

The set G d is equipped with the usual topology of uniform convergence on compact subsets of C with respect to the spherical metric in the target. Now we define a map : G d → Cd by ( f ) = (a1 , . . . , ad−1 , λ), where −λ, a1 , . . . , ad−1 are the coefficients of the polynomial −(1/2)S f . This map is evidently continuous. Proposition 1. The map sends G d to Z d surjectively. Proof. First we prove that the image of is contained in Z d . Let f be an element of G d . By property 7 above, f = y/y1 , a ratio of two linearly independent solutions of (5). By (12) and property 5 above, y should be subdominant in S0 and Sq/2 . So y satisfies the boundary condition (6) and thus ( f ) = (a, λ) is an element of Z d . Now we prove that maps G d to Z d surjectively. Let (a, λ) ∈ Z d and let y be the corresponding eigenfunction. Let y1 be any solution of Eq. (5) which is linearly independent of y. Then f = y/y1 satisfies the differential equation (10), thus (11) holds. Now in view of the boundary condition (6), y is subdominant in S0 and Sq/2 , so by properties 1 and 2 above y1 must be dominant in S0 and Sq/2 . So f = y/y1 satisfies (12). Thus f ∈ G d , and (10) gives (a, λ) = ( f ). Proposition 2. A meromorphic function g satisfies g(z) = f (cz) for some f ∈ G d and c ∈ C∗ if and only if it has the following three properties: (i) g has no critical points, (ii) g has q = d + 2 tracts, (iii) There is a tract B0 such that if the tracts are ordered counterclockwise as B0 , B1 , . . . , Bq−1 , and if w j are the asymptotic values of g in B j , then w0 = wq/2 = 0. The sufficiency of these conditions is a deep result of R. Nevanlinna [26]. The proof becomes much simpler if one adds the condition (iv) g is a meromorphic function of finite order.

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This additional condition will be easy to verify in our setting and we sketch a simpler proof of Proposition 2, using condition (iv). This proof is based on F. Nevanlinna’s work [25]. Proof of Proposition 2 with condition (iv). The necessity of conditions (i)–(iv) has already been established. Now we prove sufficiency. Condition (i) implies that Sg is an entire function (in general, Schwarzian derivative of a meromorphic function has poles exactly at its critical points). Condition (iv) combined with the lemma on the logarithmic derivative [18,27] gives a growth estimate of Sg which implies that Sg is a polynomial. Now by property 7 above, g = y/y1 , where y, y1 are two linearly independent solutions of the differential equation y +

1 Sg y = 0. 2

Property 6 above shows that g has deg Sg + 2 tracts, so we conclude from (ii) that deg Sg = d. Now we can find c ∈ C∗ so that for f (z) = g(z/c) the polynomial (−1/2)S f is monic. Such c is defined up to multiplication by a q th root of unity. Using (iii), we choose this root of unity in such a way that (12) is satisfied. q Now we define a map W : G d → C , W ( f ) = (w0 , . . . , wq−1 ), whose image is evidently contained in the subspace H of codimension 2 given by the equations w0 = wq/2 = 0. This map is known to be a local homeomorphism into H [2], and its image can be described precisely using a result of R. Nevanlinna [26]. We do not use these results in our paper. We only remark that the local homeomorphism W permits to define a structure of a complex analytic manifold of dimension d on G d , so that W becomes holomorphic. The map we introduced earlier is also holomorphic with respect to this analytic structure. Proposition 3. Let f 0 be an element of G d , and γ : [0, 1] → H , γ (t) = (w0 (t), . . . , wq−1 (t)), w0 ≡ wq/2 ≡ 0, be a path with the properties γ (0) = W ( f 0 ), and w j (0) = wk (0)

⇐⇒ w j (t) = wk (t)

for all j = k and all t ∈ [0, 1]. Then there is a lift of the path γ to G d , that is a continuous family f t ∈ G d , t ∈ [0, 1] such that W ( f t ) = γ (t). Proof. There is a continuous family of diffeomorphisms ψt : C → C, ψ0 = id, such that ψt (w j (0)) = w j (t), 0 ≤ j ≤ q − 1. These diffeomorphisms are quasiconformal [1]. Then the Fundamental existence theorem for quasiconformal maps [1, Chap. V] implies the existence of a continuous family of quasiconformal maps φt , φ0 = id, such that gt = ψt ◦ f 0 ◦ φt are meromorphic functions. These meromorphic functions evidently have no critical points, because g0 does not. They have the same number of tracts as g0 and their tracts satisfy the condition (iii) of Proposition 2. Thus all conditions of Proposition 2 are satisfied. We can also check the additional condition (iv): it follows from the general property of quasiconformal mappings |φt (z)| ≤ |z|C , |z| > r0 , where C is a constant. Thus by Proposition 2, gt = f t (ct z), f t ∈ G d , where the constants ct are determined from the condition that the polynomials −(1/2)S ft are monic. Evidently the correspondence t → ct is continuous, and we have W ( f t ) = γ (t) by construction.


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Centrally symmetric case. Suppose now that the polynomial P in (5) is even. We write it as P(a, z) = z d +ad−2 z d−2 +· · ·+a2 z 2 and consider the set Z de of all pairs (a, λ) ∈ Cd/2 such that λ is an eigenvalue of the problem (5), (6). Then each eigenfunction y of the problem (5), (6) is either even or odd. Indeed, y(−z) is also an eigenfunction with the same eigenvalue, so y(z) = cy(−z) because the eigenspace is one-dimensional. Putting z = 0 we obtain that either c = 1 (so the eigenfunction is even) or y(0) = 0. In the latter case, differentiate to obtain y (z) = −cy (−z) and put z = 0 to conclude that c = −1, so the eigenfunction is odd. Equation (5) with even P always has even and odd solutions: to obtain an even solution we solve the Cauchy problem with the initial conditions y1 (0) = 1, y1 (0) = 0; to obtain an odd solution we use the initial conditions y1 (0) = 0, y1 (0) = 1. Let y be an eigenfunction, and y1 a solution of (5) of the opposite parity to y. Then y and y1 are linearly independent, and the ratio f = y/y1 is odd. Let G od be the set of all odd functions in G d . Then maps G od to Z de because the Schwarzian derivative of an odd function is even. Thus we have a centrally symmetric version of Proposition 1: the map

: G od → Z de is well defined and surjective. Similarly, Proposition 2 has a centrally symmetric analog: for an odd meromorphic function g to be of the form f (cz), where f ∈ G od , it is necessary and sufficient that conditions (i)–(iii) (or (i)–(iv)) be satisfied. Finally, Proposition 3 has a centrally symmetric analog: Proposition 3 . Let f 0 be an element of G od , and γ : [0, 1] → H , γ (t) = (w0 (t), . . . , wq (t)), w0 ≡ wq/2 ≡ 0, be a path with the properties γ (0) = W ( f 0 ), w j (t) = −w j+q/2 (t) and w j (0) = wk (0)

⇐⇒ w j (t) = wk (t)

for all j = k mod q and all t ∈ [0, 1]. Then there is a lift of the path γ to G od , that is a continuous family f t ∈ G od , t ∈ [0, 1] such that W ( f t ) = γ (t). The proof is the same as that of the original Proposition 3: one can choose all homeomorphisms ψt and φt to be odd, then gt = ψt ◦ f 0 ◦ φt will be odd. Case a) of Theorem 1 which we are proving corresponds to the even potential with d = 4. To prove a) we only need to show that G o4 consists of two components: one containing the functions with f (0) = 0 and another containing the functions with f (0) = ∞. Notice that C∗ acts on G d and on G od by the rule f → c f , and that the map is invariant with respect to this action. Introducing the equivalence relation f ∼ cg, c ∈ C∗ on ˜ which maps the equivalence classes to Z d . G od we obtain a factor-map

˜ : G o / ∼→ Z e is a homeomorphism. Proposition 4. The map

d d Proof. We only have to show that it is injective, that is that any two non-zero odd solutions f 1 and f 2 of the Schwarz differential equation S f = −2P, which tend to zero as z → ±∞ on the real line, are proportional. All non-zero solutions of a Schwarz differential equation are related by fractional-linear transformations. So we have f 1 = T ◦ f 2 , where T is a fractional-linear transformation. Changing z to −z we conclude that T is

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odd. Every odd fractional-linear transformation has the form cz or c/z. The latter case is excluded by the condition that f 1 (z) and f 2 (z) both tend to zero as z → ∞ on the real line. Thus f 1 = c f 2 . q

In the next section we study the map W : G d → C , and in particular, the monodromy of this map. For this we need a description of the general fiber of this map by certain cell decompositions of the plane. Notice that our map W commutes with multiplication by constants c ∈ C∗ . 3. Some Cell Decompositions of the Plane By a cell decomposition of a surface X we understand its representation as a locally finite union of disjoint subsets called cells. The cells can be of dimension 0 (points or vertices), 1 (edges) or 2 (faces). The edges and faces are homeomorphic images of an open interval or of an open disc, respectively, and they satisfy the following condition: the boundary (in X ) of each cell is a locally finite union of cells of smaller dimension of this decomposition. We do not assume that the homeomorphisms of the open discs defining faces have extensions to the closed discs. Let w = f (z) be a meromorphic function without critical points and with finitely many asymptotic values. Consider a fixed cell decomposition 0 of the sphere Cw such that all asymptotic values are contained in the faces and each face contains one asymptotic value. Then the preimage f = f −1 (0 ) is a cell decomposition of the plane Cz with connected 1-skeleton. (The 1-skeleton of a cell decomposition is the union of edges and vertices.) That the 1-skeleton is connected is seen from (9), which is a covering, and from the fact that every path in C\{w0 , . . . , wq−1 } can be deformed to a path in the 1-skeleton of 0 . The closures of the edges of f are mapped by f onto the closures of the edges of 0 homeomorphically. Each face B of f is mapped by f onto a face D of 0 either homeomorphically or as a universal covering over D\{w}, where w is the asymptotic value in D. In the former case the face B of f is bounded, in the latter case it is unbounded. We label the faces of f by the names of their image faces under f . Labeled cell decompositions of the plane Cz are considered up to equivalences, orientation-preserving homeomorphisms of the plane preserving the labels. If f is an odd function, it is reasonable to choose 0 to be invariant under the map w → −w. Then f will be also invariant under z → −z. For such cell decompositions of Cz , the natural equivalence relation is that they are mapped one onto another by an odd orientation-preserving homeomorphism of Cz respecting the face labels. We call two such cell decompositions symmetrically equivalent. For a given set of asymptotic values and a given 0 , the labeled cell decomposition f almost completely determines f . Namely, we have the following Proposition 5. Let f 1 and f 2 be two meromorphic functions without critical points and with the same finite set of asymptotic values. Fix a cell decomposition 0 of Cw such that the asymptotic values are contained in faces of 0 and each face contains one asymptotic value. If i = f i−1 (0 ) are equivalent cell decompositions of Cz then f 1 (z) = f 2 (cz + b), c = 0. If f i are odd, 0 is centrally symmetric, and the i are symmetrically equivalent then b = 0. Proof. Let ψ be the orientation-preserving homeomorphism of the plane Cz that maps 1 onto 2 preserving the face labels. We are going to define another homeomorphism


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ψ with the same properties, and in addition, f1 = f2 ◦ ψ .

(13)

Let B1 and B2 = ψ(B1 ) be two faces such that f i map Bi onto a face B0 of 0 . If one of the B1 , B2 is bounded then another is also bounded and the maps f i : Bi → B0 are homeomorphisms. So there exists a unique homeomorphism ψ : B1 → B2 such that (13) holds. If both B1 and B2 are unbounded, then f i : Bi → B0 \{w0 } are universal coverings, and there are infinitely many homeomorphisms B1 → B2 that satisfy (13). To choose one, we first notices that every homeomorphism B1 → B2 with property (13) has a continuous extension to the boundary ∂ B1 and sends boundary edges of B1 to boundary edges of B2 . We choose ψ in B1 so that it maps the boundary edges in the same way as ψ. This is possible to do as ψ preserves orientation. Now ψ is defined on all faces of 1 , and it is easy to check that the boundary extensions from different faces to edges match. Thus ψ is a homeomorphism of the plane satisfying (13), and (13) implies that it is conformal. So ψ (z) = az + b. It is easy to check that in the centrally symmetric case the above construction gives an odd homeomorphism ψ . Now we consider a special class of cell decompositions 0 which is convenient for our purposes.2 Let f : Cz → Cw be a meromorphic function of order q/2 without critical points and with the asymptotic values w0 , . . . , wq−1 , ordered according to the cyclic order of their Stokes sectors. We assume that the set J = { j : w j = 0} is a fixed subset of {0, . . . , q − 1}, and that all nonzero asymptotic values of f are finite and distinct. Let J = { j1 , . . . , jk } with j1 < · · · < jk , and let cν = w jν . Then the cyclic order of the Stokes sectors with nonzero asymptotic values c1 , . . . , ck agrees with that of {1, . . . , k} mod k. We assume 3 ≤ k < q, so f has at least three distinct non-zero asymptotic values. We define a cell decomposition 0 of Cw with a single vertex at ∞ as follows (see Fig. 1). In the complex plane C•w = Cw \{0}, fix a system of directed loops γc1 , . . . , γck , starting and ending at ∞, intersecting only at their endpoints, and such that each loop γcν is an oriented boundary of an open domain Dcν containing cν and not containing other asymptotic values of f . Let D0 be the connected component of 0 in Cw \(γc1 ∪· · ·∪γck ). The domains D0 , Dc1 , . . . , Dck are the faces of 0 , and the open loops γ˙ν = γν \{∞} are its edges. There is a natural cyclic order ν1 ≺ · · · ≺ νk ≺ ν1 of the loops γcν , and of the corresponding domains Dcν . It is defined by the order in which the domains Dcν cross the oriented boundary of a small disk in C•w centered at ∞. Alternatively, it is opposite to the order in which the loops γcν appear in the boundary of D0 . So we introduced two cyclic orders on c1 , . . . , ck , the first one coming from the cyclic order of the Stokes sectors, and the second one from the cyclic order of the loops in the cell decomposition 0 . These two cyclic orders are in general different. Let f = f −1 (0 ) be the corresponding cell decomposition of Cz . An example of such cell decomposition is shown at the top of Fig. 2. The vertices of f are the poles of f . The faces of f are labeled with 0, c1 , . . . , ck and edges with c1 , . . . , ck (only labels of unbounded faces corresponding to the Stokes 2 The usual choice of as in [13,18,26,27] leads to the cell decompositions of the plane which are called 0 line complexes. We prefer a different choice of 0 , as in [16], which is better compatible with the symmetries of our problem.

442


-i -i

-1

1

-1 1

i

i Fig. 1. Cell decomposition 0 for asymptotic values 0, i, 1, 0, −i, −1

sectors are shown in Fig. 2). The edges of f are directed, being preimages of directed loops γcν . Since f has no critical points, the restriction of f to any face of f is either a homeomorphism or a universal covering over the image face of 0 minus the asymptotic value in this image face. Accordingly, f has the following properties: (1) Unbounded faces of f are in one-to-one correspondence with the Stokes sectors of f . This follows from property 6 in Sect. 2. For each ν = 1, . . . , k there is exactly one unbounded face Bcν labeled with cν . The faces Bc1 , . . . , Bck have the cyclic order {1, . . . , k} mod k at infinity. (2) Edges of f may be either links (having two distinct vertices) or loops (having both ends at the same vertex). Each edge labeled by cν separates two faces, labeled with cν and 0, respectively. Its orientation agrees with that of the boundary of its adjacent face labeled with cν . If it is a link, it is adjacent to the unbounded face Bcν . (3) Each bounded face of f labeled with cν has as its boundary a loop labeled by cν . The unbounded face Bcν has as its boundary an infinite chain of links labeled with cν , and no loops. A bounded face labeled with 0 has as its boundary k edges labeled with cν , (oriented opposite to their natural orientation) in the cyclic order opposite to the order ν1 , . . . , νk of the loops γcν . An unbounded face labeled with 0 has as its boundary an infinitely repeated sequence of k edges labeled with cν , in the cyclic order opposite to ν1 , . . . , νk . (4) The cyclic order of the values cν labeling the links in the boundary of a bounded face labeled by 0 is the same as the cyclic order of the unbounded faces Bcν adjacent to these links, which agrees with the cyclic order {1, . . . , k} mod k of the Stokes sectors. (5) Each vertex v of f has degree 2k, with the directed edges labeled with cν1 , . . . , cνk consecutively exiting and entering v, where ν1 , . . . , νk is the cyclic order of the loops γcν . This means that, as an edge labeled with cν1 exits v, the next in the cyclic


f

443

i

1

0

0 -i

-1

T 1

1

i

0

0 -i

-1

i 0

0 -i

-1

Fig. 2. Cell decomposition f for asymptotic values 0, i, 1, 0, −i, −1, and the corresponding graphs and T of type A1

order is (the same or another) edge labeled with cν1 entering v, then an edge labeled with cν2 exiting v, and so on, till an edge labeled with cνk entering v, followed by the initial edge labeled with cν1 exiting v. The one-skeleton of f is an infinite directed graph properly embedded in Cz . It is connected. Removing from the 1-skeleton of f all loops, we obtain a directed graph (see Fig. 2). From property (3) of f , all bounded components of the complement of are labeled by 0, and the unbounded components are in one-to-one correspondence with the Stokes sectors of f . Moreover, the unbounded components corresponding to the Stokes sectors with nonzero asymptotic values are exactly the faces Bcν of f . Replacing two edges in the boundary of each two-gon of by one undirected edge inside the twogon connecting its two vertices, and forgetting orientation of all remaining edges of , we obtain a properly embedded graph T without loops or multiple edges (see Fig. 2), with the components of its complement labeled with 0 and cν . That T has no multiple edges follows from property (2) of f : each link in f belongs to the boundary of some unbounded face, thus there are at most two links in f between any pair of vertices. Each edge of T separates two components with different labels. All bounded components are labeled with 0.

444


Proposition 6. Suppose that the cyclic order ν1 , . . . , νk of the loops γcν in C•w agrees with the cyclic order {1, . . . , k} mod k of the Stokes sectors with the asymptotic values c1 , . . . , ck in Cz . Then each bounded face of f labeled with 0 has exactly two vertices, and its boundary contains exactly two links. Each bounded component of the complement of is a two-gon, and T is an embedded planar tree. Proof. First, a bounded face C of f labeled with 0 must have at least two vertices. Otherwise, its boundary would consist of loops labeled with cν = 0, which should be also boundaries of bounded faces labeled with cν . This is impossible since the union of this face with the loops and the vertices would be a sphere, and could not be embedded in Cz . Hence there are at least two links in the boundary of C. From property (3) of f , the cyclic order of the links labeled with cν in the boundary of C should be opposite to the cyclic order of the loops γcν . From property (4), it should agree with the cyclic order {1, . . . , k} modk of the faces Bcν . If the loops γcν and the faces Bcν have the same cyclic order, this is impossible when the boundary of C contains more than two links. Since is obtained by removing loops from the one-skeleton of f , its complement has no bounded components other than two-gons, hence the complement of T has no bounded components, so T is a forest (a union of disjoint trees). To prove that it is connected, we notice that the 1-skeleton of f is connected, and the removal of loops and multiple edges from f does not affect this connectedness. Proposition 7. The embedded planar directed graph and the cell decomposition f are determined by the embedded planar graph T uniquely up to an orientationpreserving homeomorphism of Cz preserving their common vertices. Proof. The components of the complement of T labeled with 0 coincide with the components of the complement of labeled with 0. A unique unbounded component Cν of the complement of T labeled by cν contains the unbounded face Bcν of f . Each of its boundary edges separates Cν either from a component of the complement of T labeled with 0 or from Cµ with µ = ν. In the first case, the edge belongs to . We make it directed as part of the boundary of Cν and label it with cν . Otherwise, we connect the two vertices of the edge by a new edge labeled with cν inside Cν , directed according to the orientation of the boundary of Cν , so that all these new edges are disjoint (this can be done one edge at a time, in any order). As a result, each edge of T separating two components Cν and Cµ is included inside a two-gon. Let be the embedded planar directed graph obtained by removing all such edges of T . We want to show that can be obtained from by an orientation-preserving homeomorphism of Cz preserving all vertices and labels. First, two-gons of have the same vertices and the same labeling of edges as the two-gons of . Hence there exists an orientation-preserving homeomorphism between each two-gon of and the two-gon of with the same vertices, preserving their common vertices. These homeomorphisms define a homeomorphism between the union of all two-gons of and the union of all two-gons of , preserving their common vertices and the labeling of their edges. It can be extended to unbounded components to obtain a homeomorphism of Cz . From property (5) of f , the cyclic order of the components Bcν adjacent to a vertex v of agrees with the cyclic order ν1 , . . . , νk of the loops γcν . The cyclic order of the edges of exiting and entering v is determined by the cyclic order of the unbounded components Bcν of its complement adjacent to v. Since this order agrees with the cyclic order of the faces Dcν of 0 (same as the order of the loops γcν ), one can add to non-intersecting loops, having both ends at v, labeled with the missing values cν inside


445

the connected components of the complement of labeled by 0 adjacent to v, so that the cyclic order of the edges consecutively exiting and entering v becomes ν1 , . . . , νk . Labeling the interiors of these loops by the corresponding values cν , we obtain a cell decomposition of Cz that can be obtained from f by an orientation-preserving homeomorphism of Cz preserving vertices and labels (this can be done first for the loops, in any order, then extended to components labeled with 0). Centrally symmetric case. Suppose now that the set of asymptotic values w j (and the corresponding set of non-zero values cν ) is centrally symmetric. In this case, k is even and cν+k/2 = −cν . We can choose the loops γcν centrally symmetric (e.g., by selecting k/2 loops about cν2 and taking square roots of them). If f is odd, the cell decomposition f is centrally symmetric, and so are the graphs and T (assuming the edges of T inside centrally symmetric two-gons of the complement of are chosen centrally symmetric). The origin 0 ∈ Cz is either a vertex of f when f (0) = ∞, or the center of a bounded face labeled with 0 when f (0) = 0. This bounded face may correspond either to a bounded face of the complement of T (if its boundary has more than two links) or to the middle of an edge of T . 4. Case q = 6, k = 4: Classification of Trees Consider the centrally symmetric case q = 6, k = 4, with w0 = w3 = 0. We assume that the Stokes sector S0 contains a ray of the positive real axis. We have c1 = w1 = −c3 = −w4 and c2 = w2 = −c4 = −w5 satisfying {c1 , c2 } ∩ {0, ∞} = ∅ and c1 = ±c2 . Since multiplication of all asymptotic values by a nonzero constant corresponds to multiplication of f by the same constant, we can assume c2 = 1 and c1 = c = 0, ∞, ±1. Suppose that the cell decomposition 0 is centrally symmetric and satisfies conditions of Proposition 6, so that T is a centrally symmetric tree. The two components of its complement labeled by 0 and opposite via central symmetry, cannot have a common boundary edge. Proposition 8. Consider centrally symmetric embedded trees in C with six ends, two opposite components of the complement labeled by zero, and such that these two components do not have a common edge. Any such tree is equivalent to either one of the trees shown in Fig. 3, or its complex conjugate. The integer parameters satisfy the following restrictions: for Ak : k ≥ 0, for Dk,l : k ≥ 0, l ≥ 1, and for E k,l : k ≥ 1, l ≥ 0. Proof. Let X be a graph in Cz with the vertices corresponding to components of the complement of T and the edges connecting two of its vertices if the corresponding components have a common edge in T . Then X is combinatorially a hexagon with some of the chords connecting its vertices, so that the chords do not intersect and the vertices 0 and 3 not connected. We can also assume that X is centrally symmetric and that its vertex labeled with 0 is on the positive real axis. There are ten possible cases. Six of them are listed in Fig. 4. They correspond to embedded planar trees Ak , Dk,l , E k,l shown in Fig. 3. Here indices k and l denote the number of edges in a chain of edges and simple vertices of T corresponding to a chord of X (with l corresponding to the chord passing through the origin). In particular, a tree with even l has a vertex at the origin, while a tree with odd l has the origin as the middle of its edge. The remaining four cases are obtained by complex conjugation. The trees Ak are symmetric with respect to complex conjugation. The complex conjugates of the other trees are denoted by D¯ k,l and E¯ k,l .

446


Ak

Ek,l

k

0

0

0 k

l

D k,l 0

0 l

k

0

Fig. 3. Classification of trees

If the condition of Proposition 6 is not satisfied (i.e., the cyclic order of the loops in 0 is different from the cyclic order of the Stokes sectors) the graph T may still be a tree when no three components of its complement labeled by nonzero values are adjacent to any of its vertices. This is the case for the trees E k,l and their complex conjugates E¯ k,l . To distinguish them from the same trees with the correct cyclic order of face labels we and E¯ , respectively. Examples of non-tree graphs T are shown denote them by E k,l k,l in Fig. 5. These graphs are Ok and Q k,l , with k, l ≥ 0, and the complex conjugates Q¯ k,l of Q k,l . We will later show that these graphs T can really occur. Moreover, these are all possible cases of non-tree graphs T corresponding to cell decompositions f with the reverse cyclic order of non-zero labels, but we do not use this fact in the proof; it will rather come as a consequence of our arguments. 5. Monodromy. Proof of Theorem 1 a Let be the space of all four-point sequences ξ = {c1 , . . . , c4 } in Cw such that c3 = −c1 , c4 = −c2 , cµ = 0, ∞, ±cν for µ = ν. Let 0 be the quotient space of with respect to multiplication by a nonzero constant. A point of 0 can be represented by a sequence {c, 1, −c, −1} such that c = 0, ∞, ±1. Let b = c2 = 0, ∞, 1. The fundamental group of C\{0, ∞, 1} with a base point −1 is a free group with two generators s0 and s∞ corresponding to the loops starting at −1, going along the negative real axis towards 0 (resp. ∞), going counterclockwise about 0 (resp. ∞) and returning to −1 along the negative real axis. Each of these generators defines two paths in the space 0 (denoted also by s0 and s∞ ) starting at c = i (resp. c = −i) and ending at c = −i (resp. c = i) (see Fig. 6).


A0

2

1

3

D 0,l

2

0

4 2

2

1

Dk,l 2

2

0

4

5

1

Ek,0 0

4

1

3

5

3

5

Ak 0

4

0

1

3

5

3

4

447

5 1

3

Ek,l 0

4

5

Fig. 4. To the proof of Proposition 8

The fundamental group of 0 with the base point i is generated by q0 = s02 , 2 , q −1 −1 q ∞ = s∞ −1 = (s0 s∞ ) , and q1 = (s∞ s0 ) , with the relation q0 q1 q∞ q−1 = 1. Suppose that f has nonzero asymptotic values c1 = i, c2 = 1, c3 = −i, c4 = −1. Define four loops γi , γ1 , γ−i , γ−1 in C•w by following either real or imaginary axis from ∞ to one of the points i, 1, −i, −1, moving about that point counterclockwise, and returning to ∞ along the same axis (Fig. 1). These four loops generate the free group π1 (C•w \{±i, ±1}). We can assume that the cell decomposition 0 of C•w defined by these four loops is both centrally symmetric and invariant with respect to complex conjugation. The cyclic order of the loops γi , γ1 , γ−i , γ−1 at the point ∞ agrees with the cyclic order of the corresponding Stokes sectors in Cz . According to Propositions 6 and 7, the cell decomposition f of Cz can be defined by a tree T . Each path {c(t), 1, −c(t), −1} in 0 starting at c(0) = i defines a continuous deformation γi (t), γ1 (t), γ−i (t), γ−1 (t) of the original four loops, each of the deformed loops starting and ending at ∞ and avoiding 0, ∞, ±1, ±c(t). This deformation is unique up to isotopy. We can choose the deformation so that the corresponding cell decomposition 0 (t) of C•w remains centrally symmetric. For the paths corresponding to s0 and s∞ , we have c(1) = −i. Hence the loops γˆ−i = γi (1), γˆ1 = γ1 (1), γî = γ−i (1), γˆ−1 = γ−1 (1) belong to the same space C•w \{±i, ±1} as the original loops γi , γ1 , γ−i , γ−1 . See Figs. 7, 8.

448


Ok

0

0 k

Q k,l

0 k l

0

Fig. 5. Non-tree examples of graphs T

-i s0 s

-1

1

s s0 i Fig. 6. Paths s0 and s∞

The paths in 0 corresponding to s0 and s∞ can be considered as elements of the braid group B4 on four strands in C•w (leaving ±1 fixed). From the classical formulas for the action of Bk on the fundamental group of the plane without k points [23], we have (see Fig. 7 for s0 and Fig. 8 for s∞ ),


449

-i ^

-i

i

^ -1

-1

-1

1 ^ 1

1

^

i

-i

i Fig. 7. Action of s0

^ 1

-i

^ -i

-i

-1

1

-1 1

^ i

i

i ^ -1

Fig. 8. Action of s∞

450


γî = (γ1 )−1 γi γ1 , γˆ1 = γ1 , γˆ−i = (γ−1 )−1 γ−i γ−1 , γˆ−1 = γ−1 for s0 ;

(14)

γî = γi , γˆ1 = (γ−i )−1 γ1 γ−i , γˆ−i = γ−i , γˆ−1 = (γi )−1 γ−1 γi , for s∞ .

(15)

Conversely, the original loops can be expressed as products of the new loops: γi = γˆ1 γî (γˆ1 )−1 , γ1 = γˆ1 , γ−i = γˆ−1 γˆ−i (γˆ−1 )−1 , γ−1 = γˆ−1 for s0 ;

(16)

γi = γî , γ1 = γˆ−i γˆ1 (γˆ−i )−1 , γ−i = γˆ−i , γ−1 = γî γˆ−1 (γî )−1 , for s∞ .

(17)

Let f t be the family of functions constructed in Proposition 3 , for the path {c(t), 1, −c(t), −1} in 0 with c(0) = i and c(1) = −i, and let ft = f t−1 (0 (t)) be the corresponding cell decompositions. Note that the cell decomposition 0 (1) is defined by the loops γˆ−i = γi (1), γˆ1 = γ1 (1), γî = γ−i (1), γˆ−1 = γ−1 (1). Then f 1 has the nonzero asymptotic values −i, 1, i, −1. The cell decomposition f1 is obtained by a continuous deformation exchanging i and −i from the cell decomposition f0 . Accordingly, the directed graph without loops ˆ corresponding to f1 is the same as the graph corresponding to f0 , with the labels i and −i of its edges exchanged. Let f1 = f 1−1 (0 ), and let be the corresponding directed graph without loops. Embedding of the graph in the plane Cz is determined, up to an orientation-preserving homeomorphism of Cz , by its combinatorial structure and the cyclic order of its labeled directed edges at each of its vertices of degree greater than 2 [23]. Since this cyclic order agrees with the cyclic order of directed edges of the cell decomposition 0 at its vertex ∞, to define it is enough to specify its combinatorial structure, i.e., for a vertex v of ˆ since both graphs have f −1 (∞) as their (which can be identified with a vertex of , 1 vertices) to determine whether an edge of labeled with one of i, 1, −i, −1 exits v, and if yes, which vertex of does it enter. Two vertices v and v of ˆ are connected by a directed edge labeled by i (resp., 1, −i, −1) if and only if the monodromy of f 1−1 along the loop γî (resp., γˆ1 , γˆ−i , γˆ−1 ) maps v to v = v. For , the same holds for the monodromy of f 1−1 along the loop γi (resp., γ1 , γ−i , γ−1 ). Let us denote the corresponding monodromy transformations by σi , σ1 , σ−i , σ−1 , and by σˆ i , σˆ 1 , σˆ −i , σˆ −1 , respectively. Then the transformations σˆ can be determined from the transformations σ from the relations (16) and (17) between the two sets of loops. Note that the monodromy is an anti-representation of the fundamental group, i.e., the transformation corresponding to the product γ γ of two elements of the fundamental group is σ ◦ σ , where σ and σ are the monodromy transformations corresponding to γ and γ . For s0 , the edges of labeled with 1 and −1 are the same as the edges of ˆ with the same labels. For each vertex v, the edge of labeled with i that starts at v ends at the vertex obtained from v by moving along an edge of ˆ labeled with 1 (or staying at v if there is no such edge), then along an edge of ˆ labeled with i, then backward along an edge of ˆ labeled with 1. (There is no edge when the last vertex coincides with v.) The edge of labeled with −i that starts at v ends at the vertex obtained from v by moving along an edge of ˆ labeled with −1, then along an edge of ˆ labeled with −i, then backward along an edge of ˆ labeled with −1. For s∞ , the edges of labeled with i and −i are the same as the edges of ˆ with the same labels. For each vertex v, the edge of labeled with 1 that starts at v ends at the


451

T 0

0

0

-1

-i

i

1

i

1

0 -i

-1

^

-i

1 0

0 i

-i

1 0

-1

0 -1

i

T

-i 1

0 i

0 -1

Fig. 9. Action of s0 transforms A1 to Q 1,0

vertex obtained from v by moving along an edge of ˆ labeled with −i, then along an edge of ˆ labeled with 1, then backward along an edge of ˆ labeled with −i. The edge of labeled with −1 that starts at v ends at the vertex obtained from v by moving along an edge of ˆ labeled with i, then along an edge of ˆ labeled with −1, then backward along an edge of ˆ labeled with i. Figure 9 shows the action of s0 for the tree T and graph of type A1 from Fig. 2. The graph corresponds to an undirected graph T of type Q 1,0 . Similarly, for the paths (s0 )−1 and (s∞ )−1 , the transformations σˆ can be determined from the transformations σ from the relations (14) and (15). However, we do not need this, since they can also be obtained from the symmetry with respect to complex conjugation, sending a function f (z) to f¯(¯z ) and the asymptotic values 0, c, 1, 0, −c, −1 (after normalization) to 0, 1/c, ¯ 1, 0, −1/c, ¯ −1. Since the cell decomposition 0 in Fig. 1 is symmetric with respect to complex conjugation (acting as w → i w¯ after normalization), the cell decomposition f and the corresponding graph is exchanged with its complex −1 . conjugate, c with 1/c, ¯ α with α, ¯ and the action of s0 with the action of s∞ The action of s0 and s∞ and their inverses on all trees of Proposition 8 is summarized in the following tables (see also Figs. 10, 11, where this action is represented graphically): Completion of the proof of Theorem 1 a). To prove that the Riemann surface G has exactly two connected components, corresponding to functions with a pole at the origin and the functions with a zero at the origin, respectively, it is enough to show that, for each point c0 ∈ C over which the mapping W : G → C is not ramified (i.e., c0 = 0, ∞, ±1)

_ Q2,4 _ Q2,2 _ D2,4 _ Q1,4

_ Q1,2 _ D1,4 _ Q0,4

_ Q0,2 _ D0,4

A1

1,4

_ E´

_ E 2,2

_ _ E 1,2 E ´2,2

s

s0

_ E 2,0

E 2,0

_ E 1,4

_ E´1,2

E´2,0

_ _ E 1,0 E´2,0

E 1,0

_ ´ E 1,0

E´1,0

A0

_ Q0,0 _ D0,2

Q0,0

D0,2 E ´1,2 E 1,2 E ´2,2

E 2,2

_ Q3,2 _ D3,4

A4 A3 A2 _ Q1,0 _ D1,2

Q1,0

D1,2 Q0,2

D0,4

_ Q2,0 _ D2,2

Q2,0

D2,2 Q1,2

D1,4 Q0,4 E´1,4 E 1,4

_ Q3,0 _ D3,2

Q3,0

Q2,2

Q1,4

D2,4

D3,2

Q2,4

Q3,2


D3,4

452

Fig. 10. Monodromy action on the even part of G

the monodromy of W −1 acts transitively on the fiber W −1 (c0 ) of W . Let us choose a point c0 = i. With the loops γi , γ1 , γ−i , γ−1 defined above, the fiber of W −1 (i) consists of the functions f such that the cell decomposition f of Cz corresponds to one of the trees Ak , Dk,l , E k,l , D¯ k,l , E¯ k,l . From the tables for the action of s0 and s∞ , for the trees with a vertex at the origin we have: (i) (ii) (iii) (iv) (v)

Ak can be obtained from A0 applying (s0 )2k ; Dk,2l can be obtained from Ak applying (s0 s∞ )l ; E k,2l can be obtained from D0,2l applying (s0 )−2k ; E k,0 can be obtained from A0 applying (s0 )−2k ; The points corresponding to D¯ k,2l and E¯ k,2l can be obtained from A0 combining the paths in (i–iv) with the complex conjugation.

For the trees with no vertex at the origin we have:

O3 _ D3,1 O2

_ D2,1

O1

_ D1,1

_ Q2,3

D3,1 D2,1

_ Q3,1 _ D3,3

Q3,1

453

O0 _ D0,1 s

s0

_ E 2,1

E 2,1

´ E 2,1

_ _ E 1,1 E ´2,1

_ E 1,3

_ ´ E 1,1

_ E´1,3

_ Q0,1 _ D0,3

_ Q0,3

D1,1 D0,1

_ Q1,1 _ D1,3

_ Q1,3

_ Q2,1 _ D2,3

Q2,1 Q1,1 Q0,1 E ´1,1

E 1,1

E 1,3

E ´1,3

D0,3

Q0,3

D1,3

Q1,3

D2,3

Q2,3

D3,3


Fig. 11. Monodromy action on the odd part of G

(i) (ii) (iii) (iv) (v)

Dk,1 can be obtained from D0,1 applying (s0 )2k ; Dk,2l+1 can be obtained from Dk,1 applying (s0 s∞ )l ; E k,2l+1 can be obtained from D0,2l+1 applying (s0 )−2k ; D¯ k,1 can be obtained from Dk,1 applying s∞ s0 ; The points corresponding to D¯ k,2l+1 and E¯ k,2l+1 can be obtained from D0,1 combining the paths in (i-iv) with the complex conjugation.

Eigenfunctions of self-adjoint operators. Suppose that α in (4) is real, i.e., the problem is self-adjoint and the eigenvalues are real. Let λ0 < λ1 < · · · be the eigenvalues of (4) and y1 (z), y2 (z), . . . the corresponding eigenfunctions. Then yn (z) has n real zeros (see [31]). Let λ = λn be one of these eigenvalues. Since α and λ are real, the function f (z)

454


-i -a-

a

-1

1

a-

-a i Fig. 12. Cell decomposition for a real function

defined in Sect. 2 is a real odd meromorphic function. Hence its nonzero asymptotic values satisfy c2 = −c¯1 , c3 = −c1 , c4 = c¯1 . These asymptotic values can be neither real nor pure imaginary (see [16]). Let a = c1 , and let a be a cell decomposition of Cw defined similarly to the cell decomposition 0 in Fig. 1, except the four loops of a contain the asymptotic values ±a, ±a¯ of f (see Fig. 12). We assume that a is centrally symmetric and invariant under complex conjugation. Then the cell decomposition f = f −1 (a ) of Cz is also centrally symmetric and invariant under complex conjugation. Proposition 9. The cell decomposition f is of the type Ak for n = 2k, and of the type Ok for n = 2k + 1. Proof. Since f is invariant under complex conjugation in Cz , the corresponding graphs and T are symmetric under complex conjugation. If a is in the second or fourth quadrant (as in Fig. 12) the cyclic order of the nonzero asymptotic values of f in C•w is consistent with the cyclic order of the corresponding Stokes sectors S1 , S2 , S4 , S5 . Thus conditions of Proposition 6 are satisfied, and the graph T should be one of the trees listed in Proposition 8. Since the only such trees symmetric under complex conjugation are Ak , we have T = Ak for some k. A real function f with the cell decomposition f of the type Ak has 2k real zeros (one zero in each two-gon of f corresponding to an edge of T on the real line). Hence n = 2k in this case. If a is either in the first or third quadrant, the cyclic order of the asymptotic values of f is opposite to the cyclic order of the corresponding Stokes sectors. In this case, similar to s and s (Figs. 7 and 8) can be applied to reverse the operations s0 and s∞ 0 ∞ cyclic order. More precisely, for a = ±eiπ/4 , the function g(z) = eiπ/4 f (z) has the asymptotic values ±1, ±i and the cell decomposition g = g −1 (0 ), where 0 is on is the cell decomposition in Fig. 1, coincides with f . The action of s0 and s∞ f defined as the action of s0 and s∞ on g . For any a in the first or third quadrant, one can deform ±a to ±eiπ/4 without crossing the real and imaginary axes, apply the two on operations, and deform ±a back to initial values. Thus the action of s0 and s∞ f is the same as the action of s0 and s∞ described in Tables 1 and 2. In particular, the


455

Table 1. The action of s0 . s0 (Ak ) = Q k,0 , k ≥ 0; s0 (Dk,l ) = Q k,l , k, l ≥ 0; , k ≥ 1, l ≥ 0; s0 (E k,l ) = E k,l s0 ( D¯ k,l ) = Q¯ k,l−2 , k ≥ 0, l ≥ 2; , k ≥ 1, l ≥ 0; s0 ( E¯ k,l ) = E¯ k,l ¯ s0 ( Dk,1 ) = Ok , k ≥ 0 ) = D , l ≥ 0; s0 (E 1,l 0,l

s0 (Q k,0 ) = Ak+1 , k ≥ 0; s0 (Q k,l ) = Dk+1,l , k, l ≥ 0; )= E s0 (E k,l k−1,l , k ≥ 1, l ≥ 0; s0 ( Q¯ k,l ) = D¯ k,l+2 , k, l ≥ 0; ) = E¯ , k ≥ 1, l ≥ 0; s0 ( E¯ k,l k,l s0 (Ok ) = D¯ k,1 , k ≥ 0; )= A . s0 (E 1,0 0

Table 2. The action of s0 . s∞ (Ak ) = Q¯ k−1,0 , k ≥ 1; s∞ (Dk,l ) = Q k,l−2 , k ≥ 0, l ≥ 2; , k ≥ 1, l ≥ 0; s∞ (E k,l ) = E k,l s∞ ( D¯ k,l ) = Q¯ k−1,l , k ≥ 1, l ≥ 1; s∞ ( E¯ k,l ) = E¯ k+1,l , k ≥ 1, l ≥ 0; s∞ (Dk,1 ) = Ok , k ≥ 0; , l ≥ 0; s∞ ( D¯ 0,l ) = E¯ 1,l

s∞ ( Q¯ k,0 ) = Ak , k ≥ 0; s∞ (Q k,l ) = Dk,l+2 , k, l ≥ 0; ) = E , k ≥ 1, l ≥ 0; s∞ (E k,l k,l s∞ ( Q¯ k,l ) = D¯ k,l , k ≥ 1, l ≥ 1; ) = E¯ , k ≥ 1, l ≥ 0; s∞ ( E¯ k,l k,l s∞ (Ok ) = Dk,1 , k ≥ 0; . s∞ (A0 ) = E¯ 1,0

result of this action is a cell decomposition satisfying the condition of Proposition 6, hence having the type of one of the trees listed in Proposition 8. Accordingly, the cell decomposition f itself should correspond to a graph T obtained from one of these −1 . From Tables 1 and 2, all such graphs are of the trees by the operations s0−1 and s∞ ¯ ¯ types Ok , Q k,l , E k,l , Q k,l , E k,l . The only graphs in this list symmetric under complex conjugation are Ok . A real function f with the cell decomposition f of the type Ok has 2k + 1 real zeros (one zero at the origin and one inside each two-gon of f corresponding to an edge of T on the real line). Hence n = 2k + 1 in this case. 6. Other Potentials The following three one-parametric families of potentials can be treated with the same method. The details will appear elsewhere. 1. PT-symmetric cubic [11]. A differential equation −y + P(z)y = λy, with a general cubic polynomial P, by an affine change of the independent variable and a shift of λ can be brought to the form − y + (i z 3 + iαz)y = λy.

(18)

We impose the boundary condition y(z) → 0, as y ∈ R,

y → ±∞.

(19)

This problem is not self-adjoint but for real α it has the so-called PT-symmetry property. It is known [12,30] that for α ≥ 0 the spectrum is real. Theorem 3. Let Z 3 ∈ C2 be the set of all pairs (α, λ) such that the problem (18), (19) has a non-trivial solution. Then Z 3 is an irreducible non-singular curve.

456


2. Quasi-exactly solvable sextic family [36,37]. Consider the equation − y + (z 6 + 2αz 4 + {α 2 − (4m + 2 p + 3)}z 2 )y = λy,

(20)

with the same boundary condition (19). This problem is self-adjoint for real α. It was shown by Turbiner and Ushveridze that for real α this problem has exactly m + 1 linearly independent “elementary” eigenfunctions of the form Qe T with polynomials Q and T . The degree of Q is 2m + p, so the eigenfunction has 2m + p zeros in the complex plane. If p = 0 then these elementary eigenfunctions correspond to the first m +1 even-numbered eigenvalues, and if p = 1 to the first m + 1 odd-numbered eigenvalues. Theorem 4. Let m be a non-negative integer and p ∈ {0, 1}. Let Z 6,m, p ∈ C2 be the set of all pairs (α, λ) such that λ is an eigenvalue of the problem (20), (19) corresponding to an elementary eigenfunction. Then Z 6,m, p is a non-singular irreducible curve. 3. Quasi-exactly solvable PT-symmetric quartic family [3]. Consider the equation − y + (−z 4 − 2αz 2 − 2imz)y = λy.

(21)

Here the boundary condition is y(r eiθ ) → 0, as r → ∞, θ ∈ {−π/6, −π + π/6}.

(22)

Similarly to the previous example, this problem has m elementary eigenfunctions. Theorem 5. Let m be a non-negative integer, and Z 4,m ⊂ C2 be the set of pairs (α, λ) such that λ is an eigenvalue of the problem (21), (22), with an elementary eigenfunction. Then Z 4,m is a smooth irreducible curve.

References 1. Ahlfors, L.: Lectures on quasiconformal mappings. Second edition, Providence, RI: Amer. Math. Soc., 2007 2. Bakken, I.: A multiparameter eigenvalue problem in the complex plane. Amer. J. Math. 99(5), 1015–1044 (1977) 3. Bender, C., Boettcher, S.: Quasi-exactly solvable quartic potential. J. Phys. A: Math. Gen. 31, L273–L277 (1998) 4. Bender, C., Turbiner, A.: Analytic continuation of eigenvalue problems. Phys. Lett. A 173(6), 442–446 (1993) 5. Bender, C., Wu, T.: Anharmonic oscillator. Phys. Rev. (2) 184, 1231–1260 (1969) 6. Bochner, S., Martin, W.: Several Complex Variables. Princeton, NJ: Princeton University Press, 1948 7. Delabaere, E., Pham, F.: Unfolding the quartic oscillator. Ann. Physics 261(2), 180–218 (1997) 8. Delabaere, E., Dillinger, H., Pham, F.: Exact semiclassical expansions for one-dimensional quantum oscillators. J. Math. Phys. 38(12), 6126–6184 (1997) 9. Delabaere, E., Pham, F.: Resurgence de Voros et periodes de curbes hyperelliptiques. Ann. Inst Fourier 43(1), 163–199 (1993) 10. Delabaere, E., Pham, F.: Resurgent methods in semi-classical asymptotics, Annales de l’Inst. Poincaré, Sect. A 71, 1–94 (1999) 11. Delabaere, E., Trinh, D.T.: Spectral analysis of the complex cubic oscillator. J. Phys. A 33, 8771–8796 (2000) 12. Dorey, P., Dunning, C., Tateo, R.: Spectral equivalences, Bethe ansatz equations and reality properties in PT-symmetric quantum mechanics. J. Phys. A 34, 5679–5704 (2001) 13. Drape, E.: Über die Darstellung Riemannscher Flächen durch Streckenkomplexe. Deutsche Math. 1, 805– 824 (1936)


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14. Eremenko, A.: Geometric theory of meromorphic functions. In: In the tradition of Ahlfors–Bers III, Contemp. Math. 355, Providence, RI, Amer. Math. Soc., 2004, pp. 221–230. (Expanded version available at http://www.math.purdue.edu/~eremenko/dvi/mich.pdf) 15. Eremenko, A.: Exceptional values in holomorphic families of entire functions. Michigan Math. J. 54(3), 687–696 (2006) 16. Eremenko, A., Gabrielov, A., Shapiro, B.: Zeros of eigenfunctions of some anharmonic oscillators. Ann. Inst. Fourier 58(2), 603–624 (2008) 17. Fedoryuk, M.: Asymptotic Analysis. New York: Springer, 1993 18. Goldberg, A., Ostrovskii, I.: Distribution of values of meromorphic functions, Moscow: Nauka, 1970, (in Russian. English translation: Value Distribution of Meromorphic Functions, Providence, RI: Amer. Math. Soc., 2008) 19. Gunning, R., Rossi, H.: Analytic functions of several complex variables. Englewood Cliffs, HJ: Prentice-Hall, 1965 20. Gurarii, V., Matsaev, V., Ruzmatova, N.: Asymptotic behavior of solutions of second-order ordinary differential equation in the complex domain, and the spectrum of an anharmonic oscillator. In: Analytic methods in probability theory and operator theory, Kiev: Naukova Dumka, 1990, pp. 145–154 21. Julia, G.: Sur le domain d’existence d’une fonction implicite définie par une relation entière G(x, y) = 0. Bull. Soc. Math. France 54, 26–37 (1926) 22. Kato, T., Perturbation theory for linear operators. Berlin-New York: Springer-Verlag, 1976 23. Lando, S., Zvonkin, A.: Graphs on surfaces and their applications. Berlin-Heidelberg-New York: Springer, 2004 24. Loeffel, J., Martin, A.: Propriétés analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Pade. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 415–429 25. Nevanlinna, F.: Über eine Klasse meromorpher Funktionen. 7 Congr. Math. Scand., Oslo, 1929 26. Nevanlinna, R.: Über Riemannsche Flächen mit endlich vielen Windungspunkten. Acta Math. 58, 295– 373 (1932) 27. Nevanlinna, R.: Eindeutige analytische Funktionen, Berlin: Springer, 1953 28. Shin, Kwang C.: Schrödinger type eigenvalue problems with polynomial potentials: asymptotics of eigenvalues. http://arXiv.org/list/math.SP/0411143v1, 2004 29. Shin, Kwang C.: Eigenvalues of PT-symmetric oscillators with polynomial potentials. J. Phys. A 38, 6147– 6166 (2005) 30. Shin, Kwang C.: On the reality of the eigenvalues for a class of PT-symmetric operators. Commun. Math. Phys. 229, 543–564 (2002) 31. Sibuya, Y.: Global theory of a second order linear ordinary differential equation with a polynomial coefficient. Amsterdam: North Holland, 1975 32. Simon, B.: Coupling constant analyticity for the anharmonic oscillator. Ann. Phys. 58, 76–136 (1970) 33. Simon, B.: The anharmonic oscillator: a singular perturbation theory. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 383–414 34. Simon, B.: Large order and summability of eigenvalue perturbation theory: a mathematical overview. Intl. J. Quantum Chem. 21, 3–25 (1982) 35. Stoïlov, S.: Leçons sur les principes topologiques de la théorie des fonctions analytiques. Paris: Gauthier-Villars, 1956 36. Turbiner, A., Ushveridze, A.: Spectral singularities and the quasi-exactly solvable problem. Phys. Lett. 126 A, 181–183 (1987) 37. Ushveridze, A.: Quasi-exactly solvable models in quantum mechanics. Bristol and Philadelphia: Inst. of Phys. Publ., 1994 Communicated by B. Simon


Communications in


Local Geometry of the G 2 Moduli Space Sergey Grigorian1 , Shing-Tung Yau2 1 DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA,

United Kingdom. E-mail: [email protected]

2 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

Received: 21 February 2008 / Accepted: 6 March 2008 Published online: 7 August 2008 – © Springer-Verlag 2008

Abstract: We consider deformations of torsion-free G 2 structures, defined by the G 2 -invariant 3-form ϕ and compute the expansion of ∗ϕ to fourth order in the deformations of ϕ. By considering M-theory compactified on a G 2 manifold, the G 2 moduli space is naturally complexified, and we get a Kähler metric on it. Using the expansion of ∗ϕ, we work out the full curvature of this metric and relate it to the Yukawa coupling.

1. Introduction One of the possible approaches to M-theory is to consider compactifications of the 11-dimensional spacetimes of the form M4 × X , where M4 is the 4-dimensional Minkowski space and X is a 7-dimensional manifold. If X is a compact manifold with G 2 holonomy, then this gives a vacuum solution of the low-energy effective theory, and moreover, since X has one covariantly constant spinor, the resulting theory in 4 dimensions has N = 1 supersymmetry. The physical content of the 4-dimensional theory is given by the moduli of G 2 holonomy manifolds. Such a compactification of M-theory is in many ways analogous to Calabi-Yau compactifications in String Theory, where much progress has been made through the study of the Calabi-Yau moduli spaces. In particular, as it was shown in [1] and [2], the moduli space of complex structures and the complexified moduli space of Kähler structures are both in fact, Kä hler manifolds. Moreover, both have a special geometry - that is, both have a line bundle whose first Chern class coincides with the Kä hler class. However until recently, the structure of the moduli space of G 2 holonomy manifolds has not been studied in that much detail. Generally, it turned out that the study of G 2 manifolds is quite difficult. Firstly, unlike in the Calabi-Yau case [3], there is no general theorem for existence of G 2 manifolds. Although there are constructions of compact G 2 manifolds such as those that can be found in [4] and [5], they are not explicit (a non-compact construction was also given in [6]). Another difficulty is that the G 2 -invariant 3-form which defines the G 2 -structure

460

S. Grigorian, S.-T. Yau

and the metric corresponding to it are related in a non-linear fashion. This makes the study of G 2 manifolds more difficult from a computational point of view. We first start with an overview of G 2 structures in Sect. 2, where we state the basic facts about G 2 manifolds and set up the notation. A G 2 -structure is defined by a G 2 -invariant 3-form ϕ, and in Sect. 3 we review some of the computational properties of ϕ and its Hodge dual ∗ϕ, which we will need later on. Since one of our main motivations to study G 2 manifolds comes from physics, in Sect. 4, we review the role of G 2 manifolds in M-theory, and in particular we consider the Kaluza-Klein compactification of the effective M-theory low-energy action on a G 2 manifold. It turns out that in the reduced action, the moduli of the M-theory 3-form Cmnp and the G 2 moduli naturally combine, to effectively give a complexification of the G 2 moduli space. Moreover, the metric on this complexified space turns out to be Kähler, and the Kähler potential is essentially the logarithm of the volume of the G 2 manifold. The aim of this paper is to gain more information about the geometry of the moduli space, and so the aim is to compute the curvature of this Kähler metric. This involves calculation of the fourth derivative of the Kähler potential. The method which we use for this requires us to know the expansion of ∗ϕ to third order in the deformations of ϕ . So in section 5, we in fact explicitly give the expansion of ∗ϕ to fourth order in the deformations of ϕ. Previously, only the full expansion to first order was known [4], and only partially to second order [7]. However, there are approaches to calculating higher derivatives of the Kähler potential without explicitly computing an expansion of ∗ϕ - for example the third derivative has been computed by de Boer et al in [8] and by Karigiannis and Leung in [9]. Finally, in section 6, we use our expansion of ∗ϕ from section 5 to calculate the full curvature of the G 2 moduli space, and then the Ricci curvature as well. As it has already been noted in [8] and [9], the third derivative of the Kähler can be interpreted as a Yukawa coupling, and it bears a great resemblance to the Yukawa coupling encountered in the study of Calabi-Yau moduli spaces. At the end of section 6 we consider look at some properties of covariant derivatives on the moduli space. 2. Overview of G 2 Structures We will first review the basics of G 2 structures on smooth manifolds. The main references for this section are [4,7] and [10]. The 14-dimensional Lie group G 2 can be defined as a subgroup of G L (7, R) in the following way. Suppose x 1 , . . . , x 7 are coordinates on R7 and let ei jk = d x i ∧d x j ∧d x k . Then define ϕ0 to be the 3-form on R7 given by ϕ0 = e123 + e145 + e167 + e246 − e257 − e347 − e356 .

(2.1)

Then G 2 is defined as the subgroup of G L (7, R) which preserves ϕ0 . Moreover, it also fixes the standard Euclidean metric 2 2 g0 = d x 1 + . . . + d x 7 (2.2) on R7 and the 4-form ∗ϕ0 which is the corresponding Hodge dual of ϕ0 : ∗ ϕ0 = e4567 + e2367 + e2345 + e1357 − e1346 − e1256 − e1247 .

(2.3)

Local Geometry of the G 2 Moduli Space

461

Now suppose X is a smooth, oriented 7-dimensional manifold. A G 2 structure Q on X is a principal subbundle of the frame bundle F, with fibre G 2 . However we can also uniquely define Q via 3-forms on X. Define a 3-form ϕ to be positiveif we locally can choose coordinates such that ϕ is written in the form (2.1) - that is for every p ∈ X there is an isomorphism between T p X and R7 such that ϕ| p = ϕ0 . Using this isomorphism, to each positive ϕ we can associate a metric g and a Hodge dual ∗ϕ which are identified with g0 and ∗ϕ0 under this isomorphism, and the associated metric is written (2.2). It is shown in [4] that there is a 1 − 1 correspondence between positive 3-forms ϕ and G 2 structures Q on X . So given a positive 3-form ϕ on X , it is possible to define a metric g associated to ϕ and this metric then defines the Hodge star, which in turn gives the 4-form ∗ϕ. Thus although ∗ϕ looks linear in ϕ, it actually is not, so sometimes we will write ψ = ∗ϕ to emphasize that the relation between ϕ and ∗ϕ is very non-trivial. In general, any G-structure on a manifold X induces a splitting of bundles of p-forms into subbundles corresponding to irreducible representations of G. The same is of course true for G 2 -structure. From [4] we have the following decomposition of the spaces of p-forms p : 1 = 17 ,

(2.4a)

=

(2.4b)

2

= 3

= 4

= 5

= 6

27 ⊕ 214 , 31 ⊕ 37 ⊕ 327 , 41 ⊕ 47 ⊕ 427 , 57 ⊕ 514 , 67 .

(2.4c) (2.4d) (2.4e) (2.4f)

p

Here each k corresponds to the k -dimensional irreducible representation of G 2 . Morep 7− p over, for each k and p, k and k are isomorphic to each other via Hodge duality, p and also 7 are isomorphic to each other for n = 1, 2, . . . , 6. Note that ϕ and ∗ϕ are G 2 -invariant, so they generate the 1-dimensional sectors 31 and 41 , respectively. Define the standard inner product on p , so that for p -forms α and β, α, β =

1 αa ...a β a1 ...a p . p! 1 p

(2.5)

This is related to the Hodge star, since α ∧ ∗β = α, β vol, where vol is the invariant volume form given locally by vol = det gd x 1 ∧ . . . ∧ d x 7 .

(2.6)

(2.7)

Then it turns out that the decompositions (2.4) are orthogonal with respect to (2.5). This will be seen easily when we consider these decompositions in more detail in the next section. As we already know, the metric g on a manifold with G 2 structure is determined by the invariant 3-form ϕ. It is in fact possible to write down an explicit relationship between ϕ and g. Let u and v be vector fields on X . Then u, v vol =

1 (uϕ) ∧ (vϕ) ∧ ϕ. 6

(2.8)

462


Here denotes interior multiplication, so that (uϕ)bc = u a ϕabc .

(2.9)

The definition (2.8) is rather indirect because vol depends on g via (2.7). To make more sense of it, rewrite in components 1 ϕamn ϕbpq ϕr st εˆ mnpqr st , gab det g = 144

(2.10)

where εˆ mnpqr st is the alternating symbol with ε12...7 = +1. Define Bab =

1 ϕamn ϕbpq ϕr st εˆ mnpqr st 144

(2.11)

so that then, after taking the determinant of (2.10) we get 1

gab = (det B)− 9 Bab .

(2.12)

This gives a direct definition, but because det s may be awkward to compute, (2.12) is not always the most practical definition. For us, it will be more useful to take the trace of (2.10) with respect to g, which gives 1 det g = Tr B, 7

(2.13)

and hence gab =

7Bab . Tr B

(2.14)

Although this is also an indirect definition, it is sometimes easier to handle this expression. There are in fact a total of 16 torsion classes of G 2 structures, each of which places certain restrictions on dϕ or d ∗ ϕ [11]. One of the most important classes of manifolds with G 2 structure are manifolds with G 2 holonomy. The group G 2 appears as one of two exceptional holonomy groups - the other one is Spin (7) for 8-dimensional manifolds. The list of possible holonomy groups is limited and they were fully classified by Berger [12] . Specifically, if (X, g) is a simply-connected Riemannian manifold which is neither locally a product nor is symmetric, the only possibilities are shown in the table below. Dimension 2k 2k 4k 4k 7 8

Holonomy U (k) SU (k) Sp (k) Sp (k) Sp (1) G2 Spin (7)

Type of Manifold Kähler Calabi-Yau HyperKähler Quaternionic Exceptional Exceptional

It turns out that the holonomy group H ol (X, g) ⊆ G 2 if and only if X has a torsionfree G 2 structure [4]. In this case, the invariant 3-form ϕ satisfies dϕ = d ∗ ϕ = 0

(2.15)


463

and equivalently, ∇ϕ = 0, where ∇ is the Levi-Civita connection of g. So in fact, in this case ϕ is harmonic. Moreover, if H ol (X, g) ⊆ G 2 , then X is Ricci-flat. For a torsion-free G 2 structure, the decompositions (2.4) carry over to de Rham cohomology [4], so that we have 2 H 2 (X, R) = H72 ⊕ H14 ,

H (X, R) = 3

H (X, R) = 4

H (X, R) = 5

H13 H14 H75

⊕ ⊕ ⊕

H73 ⊕ H74 ⊕ 5 H14 .

(2.16a) 3 H27 , 4 H27 ,

(2.16b) (2.16c) (2.16d)

p p Define the refined Betti numbers bk = dim Hk . Clearly, b13 = b14 = 1 and we also have b1 = b7k for k = 1, . . . , 6. Moreover, it turns out that b1 = 0 if and only if H ol (X, g) = G 2 . Therefore, in this case the H7k component vanishes in (2.16). An example of a construction of a manifold with a torsion-free G 2 structure is to consider X = Y × S 1 , where is a Calabi-Yau 3-fold. Define the metric and a 3-form on X as g X = dθ 2 × gY , ϕ = dθ ∧ ω + Re ,

(2.17) (2.18)

where θ is the coordinate on S 1 . This then defines a torsion-free G 2 structure, with ∗ϕ =

1 ω ∧ ω − dθ ∧ Im . 2

(2.19)

However, the holonomy of X in this case is SU (3) ⊂ G 2 . From the Künneth formula we get the following relations between the refined Betti numbers of X and the Hodge numbers of Y : b7k = 1 fork = 1, . . . , 6, k b14 = h 1,1 − 1 fork = 2, 5, k b27 = h 1,1 + 2h 2,1 fork = 3, 4.

3. Properties of ϕ The invariant 3-form ϕ which defines a G 2 structure on the manifold X has a number of useful and interesting properties. In particular, contractions of ϕ and ψ = ∗ϕ are very useful in computations. From [7,13] and [14], we have ϕabc ϕmnc = gam gbn − gan gbm + ψabmn , ϕabc ψmnp c = 3 ga[m ϕnp]b − gb[m ϕnp]a .

(3.20) (3.21)

Essentially, these identities can be derived straight from the definitions of ϕ and ψ = ∗ϕ in flat space - (2.1) and (2.3) respectively. For more details, please refer to [7] and [13]. Note that we are using a different convention to [13], and hence some of the signs are different.

464


Consider the product ψabcd ψ mnpq . Expanding ψ as the Hodge star of ϕ and then using the usual identity for a product of Levi-Civita tensors and then applying (3.20) gives ψabcd ψ mnpq = 24δa[m δbn δc δd + 72ψ[ab[mn δc δd] − 16ϕ[abc ϕ [mnp δd] . p q]

p q]

q]

(3.22)

Contracting over d and q gives ψabcd ψ mnpd = 6δa[m δbn δc + 9ψ[ab [mn δc] − ϕabc ϕ mnp , p]

p]

(3.23)

which agrees with the expression given in [14]. Of course the above relations can be further contracted to obtain ϕabc ϕm bc = 6gam ,

(3.24)

ϕabc ψmn

bc

= 4ϕamn ,

(3.25)

ψabcd ψmn

cd

= 4gam gbn − 4gan gbm + 2ψabmn .

(3.26)

Contracting even further, we are left with ϕabc ϕ abc = 42, ϕabc ψm abc = 0, ψabcd ψm bcd = 24gam , ψabcd ψ

abcd

= 168.

(3.27) (3.28) (3.29) (3.30)

|ϕ|2

= 7 in the inner product (2.5). So in fact The relations (3.27) and (3.30) both yield we have 1 V = ϕ ∧ ∗ϕ, (3.31) 7 where V is the volume of the manifold X . Now look in more detail at the decompositions (2.4). We are in particular interested in decompositions of 2-forms and 3-forms since the decompositions for 4-forms and 5-forms are derived from these via Hodge duality. From [7] and [10], we have 27 = {ωϕ : ω a vector field} , 1 214 = α = αab d x a ∧ d x b : (αab ) ∈ g2 , 2 31 = { f ϕ : f a smooth function} , 37 327

= {ω ∗ ϕ : ω a vector field} ,

= χ ∈ 3 : χ ∧ ϕ = 0 andχ ∧ ∗ϕ = 0 .

(3.32) (3.33) (3.34) (3.35) (3.36)

Following [7], it is enough to consider what happens in R7 in order to understand these decompositions. Consider first the Lie algebra so (7), which is the space of antisymmetric 7 × 7 matrices. For a vector ω ∈ R7 , define the map ρϕ : R7 −→ so (7) by ρϕ (ω) = ωϕ, and this map is clearly injective. Conversely, define the map τϕ : so (7) −→ R7 such that τϕ (αab )c = 16 ϕ c ab α ab . From (3.24), we get that τϕ ρϕ (ω) = ω,


465

so that τϕ is a partial inverse of ρϕ . Now the Lie algebra g2 can be defined as the kernel of τϕ [13], that is

(3.37) g2 = ker τϕ = α ∈ so (7) : ϕabc α bc = 0 . This further implies that we get the following decomposition of so (7): so (7) = g2 ⊕ ρϕ R7 .

(3.38)

The group G 2 acts via the adjoint representation on the 14 -dimensional vector space g2 and via the natural, vector representation on the 7-dimensional space ρϕ R7 . This is a G 2 -invariant irreducible decomposition of so (7) into the representations 7 and 14. Hence follows the decomposition of 2 (2.4a and also the characterizations (3.32) and (3.33)). ∗ Following [7] again, let us look at 327 in more detail. Consider Sym 2 R7 - the ∗ ∗ space of symmetric 2-tensors and define a map iϕ : Sym 2 R7 −→ 3 R7 by iϕ (h)abc = h d[a ϕbc]d .

(3.39)

Clearly, iϕ (g)abc = ϕabc . ∗ ∗ = Rg ⊕ Sym 20 R7 , where Rg is the set of Now, we can decompose Sym 2 R7 ∗ symmetric tensors proportional to the metric g and Sym 20 R7 is the set of traceless ∗ symmetric tensors. This is a G 2 -invariant irreducible decomposition of Sym 2 R7 into 1-dimensional and 27 -dimensional components. The map iϕ is also G 2 -invariant and is injective on each summand of this decomposition. Looking at the first summand, ∗ 3 7 . Now look at the second summand and consider we get that iϕ (Rg) = 1 R ∗ 2 7 iϕ Sym 0 R . This is 27-dimensional and irreducible, so by dimension count ∗ ∗ = 327 R7 . All of this carries over to it follows easily that iϕ Sym 20 R7 3-forms on our G 2 manifold X , and so we get

(3.40) 327 = χ ∈ 3 : χabc = h d[a ϕbc]d forh ab traceless and symmetric . From the identities for contraction of ϕ and ∗ϕ, it is possible to see that this is equivalent to the description (3.36) of 327 . Thus we see that 1-dimensional components correspond to scalars, 7-dimensional components correspond to vectors and 27 -dimensional components correspond to traceless symmetric matrices. Now suppose we have χ ∈ 3 , then it is always useful to be able to compute the different projections of χ into 31 , 37 and 327 . Denote these projections by π1 , π7 and π27 , respectively. As shown in Appendix 1, we have the following relations: 1 1 χabc ϕ abc = χ , ϕ and |π1 (χ )|2 = 7a 2 , (3.41) π1 (χ ) = aϕ where a = 42 7 1 π7 (χ ) = ω ∗ ϕ where ωa = − χmnp ψ mnpa and |π7 (χ )|2 = 4 |ω|2 , (3.42) 24 3 2 π27 (χ ) = iϕ (h) where h ab = χmn{a ϕb}mn and |π27 (χ )|2 = |h|2 . (3.43) 4 9

466


Here {a b} denotes the traceless symmetric part. 4. G 2 manifolds in M-Theory Special holonomy manifolds play a very important role in string and M-theory because of their relation to supersymmetry. In general, if we compactify string or M-theory on a manifold of special holonomy X the preservation of supersymmetry is related to existence of covariantly constant spinors (also known as parallel spinors). In fact, if all bosonic fields except the metric are set to zero, and a supersymmetric vacuum solution is sought, then in both string and M-theory, this gives precisely the equation ∇ξ = 0

(4.44)

for a spinor ξ . As lucidly explained in [15], condition (4.44) on a spinor immediately implies special holonomy. Here ξ is invariant under parallel transport, and is hence invariant under the action of the holonomy group H ol (X, g). This shows that the spinor representation of H ol (X, g) must contain the trivial representation. For H ol (X, g) = S O (n), this is not possible since the spinor representation is reducible, so H ol (X, g) ⊂ S O (n). In particular, Calabi-Yau 3-folds with SU (3) holonomy admit two covariantly constant spinors and G 2 holonomy manifolds admit only one covariantly constant spinor. Consider the bosonic action of eleven-dimensional supergravity [16], which is supposed to describe low-energy M-theory: 1 S= 2

1 1 d x −gˆ 2 R (11) − 4 11

1 G ∧ ∗G − 12

C ∧ G ∧ G,

(4.45)

where gˆ is the metric on the 11-dimensional space M and C is a 3-form potential with field strength G = dC. From (4.45), the equation of motion for C is found to be 1 G ∧ G. (4.46) 2 Suppose we fix M = M4 × X , where M4 is the 4-dimensional Minkowski space and X is a space with holonomy equal to G 2 . Then M is Ricci flat, so from Einstein’s equation, G has to vanish. However, it turns out that the assumption that G X = G| X = 0 is not an obvious one to make. In fact, as explained in [17], Dirac quantization on X gives a shifted quantization condition and gives the statement GX λ − ∈ H 4 (X, Z) , (4.47) 2π 2

where G2πX is the cohomology class of G2πX and λ = 21 p1 (X ), where p1 (X ) is the first d∗G =

Pontryagin class on X . So if λ were not even in H 4 (X, Z), then the ansatz G X = 0 would not be consistent. Nonetheless, it was shown in [18] that if X is a seven dimensional spin manifold (or in particular G 2 holonomy manifold), then in fact λ is even, and setting G X = 0 is consistent. So overall the simplest, Ricci-flat vacuum solutions are given by (4.48) gˆ = η × g7 , C = 0, (4.49) G = 0, (4.50)


467

where · denotes the vacuum expectation value and g7 is some metric with G 2 holonomy while η is the standard metric on the four dimensional Minkowski space. However, we know that a G 2 structure and hence the metric g7 is defined by a G 2 -invariant 3-form ϕ0 , so we have ϕ = ϕ0 .

(4.51)

Now consider small fluctuations about the vacuum, gˆ = gˆ + δ g, ˆ C = C + δC = δC, ϕ = ϕ + δϕ = ϕ0 + δϕ.

(4.52) (4.53) (4.54)

So a Kaluza-Klein ansatz for C can be written as C=

b3

c (x) φ N + N

N =1

b2

A I (x) ∧ α I ,

(4.55)

I =1

where {φ N } are a basis for harmonic 3-forms on X , {α I } are a basis for harmonic 2-forms on X , c N (x) are scalars on M4 and A I (x) are 1-forms on M4 which describe the fluctuations of C. Also b2 and b3 are the Betti numbers of X . Since we assume that X has holonomy equal to G 2 , b1 = 0, so in (4.55) we do not have a contribution from harmonic 1-forms on X . Now, deformations of the metric on X are encoded in the deformations of ϕ and since ϕ is harmonic on X , we parameterize ϕ as ϕ=

b3

s N (x) φ N .

(4.56)

N =1

Overall, in 4 dimensions we get b3 real scalars c N and b3 real scalars s N . Together these combine into b3 massless complex scalars z N : zN =

1 N s + ic N . 2

(4.57)

In the 4-dimensional supergravity theory this gives b3 massless chiral superfields. The 1-forms A I in (4.55) give rise to b2 massless Abelian gauge fields, and together with superpartners arising from the gravitino fields, these form b2 massless vector superfields [15]. Thus overall, in four dimensions the effective low energy theory is N = 1 supergravity coupled to b2 abelian vector supermultiplets and b3 massless chiral supermultiplets. The physical theory is not very interesting from a phenomenological point of view, since the gauge group is abelian and there are no charged particles. However the combination (4.57) proves to be very useful for studying the moduli space of G 2 manifolds, since it provides a natural, physically motivated complexification of the pure G 2 moduli space—something very similar to the complexified Kähler cone used in the study of Calabi-Yau moduli spaces. Let us now use our Kaluza-Klein ansatz to reduce the 11-dimensional action (4.45) to 4 dimensions. Here we follow [19,20] and [14]. The term which interests us is the kinetic term for the z N . The kinetic term for the c N , L kin (c) comes from the reduction

468


of the G ∧ ∗G term in (4.45). After switching to the Einstein frame by gµν −→ V −1 gµν we immediately see this gives us 1 M µ N L kin (c) = − φ M ∧ ∗φ N . (4.58) ∂µ c ∂ c 4V X The kinetic term for the s M appears from the reduction of the R (11) term in (4.45). This is less straightforward than the derivation of L kin (c), but the calculation was shown explicitly in [14]. From the general properties of the Ricci scalar we can decompose the eleven-dimensional Einstein-Hilbert action as 1 1 1 ∂µ gmn ∂ µ g mn d 11 x −gˆ 2 R (11) = d 11 x −gˆ 2 V R (4) + R (7) + 4V − Tr ∂µ g Tr ∂ µ g . (4.59) Then, using deformation properties of the G 2 metric gmn from Sect. 5, and switching to the Einstein frame gµν −→ V −1 gµν , we eventually get 1 ∂µ s M ∂ µ s N φ M ∧ ∗φ N . (4.60) L kin (s) = − 4V X The kinetic term of the dimensionally reduced action is in general given in the Einstein frame by L kin = −G M N¯ ∂µ z M ∂ µ z¯ N .

(4.61)

Comparing (4.61) with (4.58) and (4.60), we can read off the moduli space metric G M N¯ as 1 G M N¯ = φ M ∧ ∗φ N¯ . (4.62) V X Note that the Hodge star implicitly depends on the coordinates z M , so this metric is quite non-trivial. The bosonic part of fully reduced 4-dimensional Lagrangian is given in this case by [21,20] L = −G M N¯ ∂µ z M ∂ µ z¯ N −

1 1 I I Re h I J Fmn F J mn + Im h I J Fmn ∗ F J mn , (4.63) 4 4

where G M N¯ is as in (4.62), and I Fmn = ∂m AnI − ∂n AmI .

The couplings Re h I J and Im h I J are given by 1 M 1 Re h I J (s) = αI ∧ α J ∧ φM , α I ∧ ∗α J = − s 2 2 1 Im h I J (c) = − c M α I ∧ α J ∧ φ M . 2

(4.64) (4.65)

2 for manifolds with G To get the second equality in (4.64) we have used that H 2 = H14 2 holonomy and that for a 2 -form α, 2 ∗ π7 (α) − ∗π14 (α) = α ∧ ϕ. Proof of this fact can be found in [10].


469

5. Deformations of G 2 Structures As we already know, the G 2 structure on X and the corresponding metric g are all determined by the invariant 3 -form ϕ. Hence, deformations of ϕ will induce deformations of the metric. These deformations of metric will then also affect the deformation of ∗ϕ. Since the relationship (2.8) between g and ϕ is non-linear, the resulting deformations of the metric are highly non-trivial, and in general it is not possible to write them down in closed form. However, as shown by Karigiannis in [10], metric deformations can be made explicit when the 3 -form deformations are either in 31 or 37 . We now briefly review some of these results. First suppose ϕ˜ = f ϕ.

(5.1)

1 g˜ ab det g˜ = ϕãmn ϕ˜ bpq ϕ˜r st εˆ mnpqr st 144 = f 3 gab det g.

(5.2)

Then from (2.10) we get

After taking the determinant on both sides, we obtain det g˜ = f

14 3

det g.

(5.3)

Substituting (5.3) into (5.2), we finally get 2

g˜ ab = f 3 gab ,

(5.4)

and hence ∗˜ ϕ˜ = f

4 3

∗ ϕ.

(5.5)

Therefore, a scaling of ϕ gives a conformal transformation of the metric. Hence deformations of ϕ in the direction 31 also give infinitesimal conformal transformation. Suppose f = 1 + εa , then to fourth order in ε, we can write 4 4 5 4 4 2 ∗˜ ϕ˜ = 1 + aε + a 2 ε2 − a 3 ε3 + a ε + O ε5 ∗ ϕ. (5.6) 3 9 81 243 Now, suppose in general that ϕ˜ = ϕ + εχ for some χ ∈ 3 . Then using (2.8) for the definition of the metric associated with ϕ, ˜ 1 = (uϕ) u, vvol ˜ ∧ (vϕ) ˜ ∧ ϕ˜ 6 1 = (uϕ) ∧ (vϕ) ∧ ϕ (5.7) 6 1 + ε [(uχ ) ∧ (vϕ) ∧ ϕ + (uϕ) ∧ (vχ ) ∧ ϕ + (uϕ) ∧ (vϕ) ∧ χ ] 6 1 + ε2 [(uχ ) ∧ (vχ ) ∧ ϕ + (uϕ) ∧ (vχ ) ∧ χ + (uχ ) ∧ (vϕ) ∧ χ ] 6 1 + ε3 (uχ ) ∧ (vχ ) ∧ χ . 6

470


After some manipulations, we can rewrite this as: 1 u, vv ol = (uϕ) ∧ (vϕ) ∧ ϕ 6 1 + ε [(uχ ) ∧ ∗ (vϕ) + (vχ ) ∧ ∗ (uϕ)] 2 1 + ε2 (uχ ) ∧ (vχ ) ∧ ϕ 2 1 + ε3 (uχ ) ∧ (vχ ) ∧ χ . 6 Rewriting (5.8) in local coordinates, we get det g˜ 1 1 = gab + εχmn(a ϕb)mn + ε2 χamn χbpq ψ mnpq g˜ ab √ 2 8 det g 1 + ε3 χamn χbpq (∗χ )mnpq . 24

(5.8)

(5.9)

Now suppose the deformation is in the 37 direction. This implies that χ = ω ∗ ϕ

(5.10)

for some vector field ω. Look at the first order term. From (3.41) and (3.43) we see that this is essentially a projection onto 31 ⊕ 327 —the traceless part gives the 327 component and the trace gives the 31 component. Hence this term vanishes for χ ∈ 37 . For the third order term, it is more convenient to study it in (5.8). By looking at ω ((uω ∗ ϕ) ∧ (vω ∗ ϕ) ∧ ∗ϕ) = 0, we immediately see that the third order term vanishes. So now we are left with 1 2 c d mnpq g˜ ab det g˜ = gab + ε ω ω ψcamn ψdbpq ψ det g 8 = gab 1 + ε2 |ω|2 − ε2 ωa ωb det g, (5.11) where we have used the contraction identity for ψ (3.26) twice. Taking the determinant of (5.11) gives 2 3 det g˜ = 1 + ε2 |ω|2 det g, (5.12) − 2 3 gab 1 + ε2 |ω|2 − ε2 ωa ωb , g˜ ab = 1 + ε2 |ω|2

(5.13)

and eventually, − 1 3 ∗˜ ϕ˜ = 1 + ε2 |ω|2 ∗ϕ + ∗ε (ω ∗ ϕ) + ε2 ω ∗ (ωϕ) .

(5.14)

The details of these last steps can be found in [10]. Notice that to first order in ε, both √ det g and gab remain unchanged under this deformation. Now let us examine the last term in (5.14) in more detail. Firstly, we have ω ∗ (ωϕ) = ∗ ω ∧ (ωϕ)


and

471

ω ∧ (ωϕ) mnp = 3ω[m ωa ϕ|a|np] = 3iϕ (ω ◦ ω) ,

(5.15)

where (ω ◦ ω)ab = ωa ωb . Therefore, in (5.14), this term gives 41 and 427 components. So, we can write (5.14) as − 1 3 3 ∗˜ ϕ˜ = 1 + ε2 |ω|2 1 + ε2 |ω|2 ∗ ϕ 7 2 + ∗ ε (ω ∗ ϕ) + ε ∗ iϕ ((ω ◦ ω)0 ) . (5.16) Here (ω ◦ ω)0 denotes the traceless part of ω ◦ ω, so that iϕ ((ω ◦ ω)0 ) ∈ 327 and thus, in (5.16), the components in different representations are now explicitly shown. As we have seen above, in the cases when the deformations were in 31 or 37 directions, there were some simplifications, which make it possible to write down all results in a closed form. Now however we will look at deformations in the 327 directions, and we will work to fourth order in ε. So suppose we have a deformation ϕ˜ = ϕ + εχ , where χ ∈ 327 . Now let us set up some notation. Define 1 1 ϕãmn ϕ˜bpq ϕ˜r st εˆ mnpqr st √ 144 det g det g˜ = g˜ ab √ . det g

sãb =

(5.17) (5.18)

From (2.10), the untilded sab is then just equal to gab . We can rewrite (5.18) as det g˜ = gab + δsab , (5.19) (gab + δgab ) √ det g where δgab is the deformation of the metric and δsab is the deformation of sab , which from (5.9) is given by δsab =

1 1 1 εχmn(a ϕb)mn + ε2 χamn χbpq ψ mnpq + ε3 χamn χbpq (∗χ )mnpq . 2 8 24

Also we introduce the following short-hand notation: sk = Tr (δs)k , tk = Tr (δg)k ,

(5.20)

(5.21) (5.22)

where the trace is taken using the original metric g. From (5.20), note that since χ ∈ 327 , when taking the trace the first order term vanishes, and hence s1 is second-order in ε. Further, after taking the trace of (5.19) using g ab and rearranging, we have −1 1 1 det g˜ 1 + t1 = 1 + s1 , (5.23) det g 7 7

472


and hence g˜ ab

−1 1 1 1 + s1 = sãb 1 + t1 . 7 7

(5.24)

As shown in Appendix B, we can also expand det g˜ as 1 12 det g˜ = 1 + t1 + t1 − t2 + t13 − 3t1 t2 + 2t3 (5.25) det g 2 6 1 4 t1 − 6t12 t2 + 3t22 + 8t1 t3 − 6t4 + O |δg|5 , + 24 and hence 1 1 3 1 det g˜ 1 2 1 1 = 1 + t1 + t − t2 + t − t1 t2 + t3 (5.26) det g 2 81 4 48 1 8 6 1 4 1 1 1 1 t1 − t12 t2 + t22 + t1 t3 − t4 + O |δg|5 . + 384 32 32 12 8 Thus we can equate (5.23) and (5.26). Suppose t1 is first order in ε. Then the only first order term in (5.26) is 21 t1 , but since s1 is second-order, the only first order term in (5.23) is − 17 t1 . It therefore follows that first order terms vanish, and so in fact t1 is also second-order in ε. This has profound consequences in that we can ignore some of the terms in (5.26), as they give terms higher than fourth order: 1 1 det g˜ 1 1 2 1 1 1 =1+ t1 − t2 + t3 + t1 − t1 t2 + t22 − t4 + O ε5 . det g 2 4 6 8 8 32 8 (5.27) From (5.24) we can write down δgab to fourth order in ε in terms of t1 and quantities related to δsab and from this get t2 , t3 and t4 in terms of t1 and δsab . So we have 1 1 2 1 1 t1 − s 1 + s − s 1 t1 δgab = gab 7 7 49 1 49 1 1 + O ε5 t1 − s 1 +δsab 1 + (5.28) 7 7 and then from this,

1 2 2 −s1 + t1 + 2t1 s2 − 2s1 s2 + O ε5 , 7 3 t3 = s3 + (t1 s2 − s1 s2 ) + O ε5 , 7

t2 = s 2 +

t4 = s 4 + O ε 5 .

Substituting, (5.29)-(5.31) into (5.27), we obtain 1 1 det g˜ 1 1 1 1 = 1 + − s2 + t1 + s3 + − s4 − s2 t1 + s12 det g 4 2 6 8 8 28 1 5 + s22 + t12 + O ε5 . 32 56

(5.29) (5.30) (5.31)

(5.32)


473

After expanding (5.23) to fourth order in ε and equating with (5.32), we are left with a quadratic equation for t1 : 27 2 9 1 1 1 1 1 t1 + t 1 + s1 − s2 + − s1 − s2 + s3 392 14 49 8 7 4 6 1 1 2 1 2 (5.33) − s4 + s1 + s2 + O ε 5 . 8 28 32 Obviously there are two solutions, but it turns out that one of them has a term which is zero order in ε, so this does not fit our assumptions, and hence we are only left with one solution, which to fourth order in ε is given by 2 7 7 7 1 11 2 7 2 t1 = s 1 + s 2 − s 3 + s4 + s1 s2 − s1 + s2 + O ε5 . (5.34) 9 18 27 36 81 162 648 Now that we have t1 = Tr (δg), from (5.23) we have 1 1 det g˜ 1 1 2 1 = 1+ s1 − s2 + s3 + s − s1 s2 det g 9 18 27 162 1 162 1 1 2 − s4 + s2 + O ε 5 . 36 648

(5.35)

Using this and (5.19) we can immediately get the deformed metric. The precise expression however is not very useful for us at this stage. What we want is to be able to calculate the Hodge star with respect to the deformed metric. So let α be a 3-form, and consider the Hodge dual of α with respect to the deformed metric: 1 1 ∗˜ α mnpq = εˆ abcdr st g˜ ma g˜ nb g˜ pc g˜ qd αr st 3! det g˜ √ det g = (∗α)abcd g˜ ma g˜ nb g˜ pc g˜ qd det g˜ 5 det g 2 = (∗α)abcd s˜ma sñb s˜ pc s˜qd det g˜ 5 det g 2 = (∗α)mnpq + 4 (∗α)[mnpd δsq]d + 6 (∗α)[mncd δs p|c| δsq]d det g˜ +4 (∗α)[m bcd δsn|b| δs p|c| δsq]d + (∗α)abcd δsam δsbn δscp δsdq . From (5.35), the prefactor

det g det g˜

5 2

det g det g˜

5 2

is given to fourth order by

5 5 5 5 s4 = 1 + − s1 + s2 − s3 + 9 18 27 36 25 25 2 25 2 s1 s2 + s1 + s2 + O ε 5 . − 162 162 648

(5.36)

474


Finally, consider how ∗ϕ deforms:

∗˜ ϕ˜

mnpq

= ∗˜ ϕmnpq + ε∗˜ χmnpq 5 det g 2 = (∗ϕ)mnpq +4 (∗ϕ)[mnpd δsq]d +6 (∗ϕ)[mncd δs p|c| δsq]d det g˜

(5.37)

+4 (∗ϕ)[m bcd δsn|b| δs p|c| δsq]d + (∗ϕ)abcd δsam δsbn δscp δsdq +ε (∗χ )mnpq + 4ε (∗χ )[mnpd δsq]d + 6ε (∗χ )[mncd δs p|c| δsq]d +4ε (∗χ )[m bcd δsn|b| δs p|c| δsq]d + O ε5 We ignored the last term, because overall it is at least fifth order. So far, the only property of 327 that we have used is that it is orthogonal to ϕ, thus in fact, up to this point everything applies to 37 as well. Now however, let χ be of the form χabc = h d[a ϕbc]d ,

(5.38)

where h ab is traceless and symmetric, so that χ ∈ 327 . Let us first introduce some further notation. Let h 1 , h 2 , h 3 , h 4 be traceless, symmetric matrices, and introduce the following shorthand notation: be (ϕh 1 h 2 ϕ)mn = ϕabm h ad 1 h 2 ϕden ,

ϕh 1 h 2 h 3 ϕ = (ψh 1 h 2 h 3 ψ)mn = ψh 1 h 2 h 3 h 4 ψ =

be c f ϕabc h ad 1 h 2 h 3 ϕde f , be c f ψabcm ψde f n h ad 1 h2 h3 , be c f mn ψabcm ψde f n h ad 1 h2 h3 h4 .

(5.39) (5.40) (5.41) (5.42)

It is clear that all of these quantities are symmetric in the h i and moreover (ϕh 1 h 2 ϕ)mn and (ψh 1 h 2 h 3 ψ)mn are both symmetric in indices m and n . Then, it can be shown that 4 h ab , 3 4 16 2 4 h = − |χ |2 gab + − (ϕhhϕ){ab} , {ab} 7 9 9 32 8 Tr h 3 gab − ϕhh 2 ϕ = , {ab} 189 9

χ(a|mn| ϕb)mn = χamn χbpq ∗ ϕ mnpq χamn χbpq ∗ χ mnpq

where as before {a b} denotes the traceless symmetric part. Using this and (5.20), we can now express δsab in terms of h: δsab

1 2 2 2 4 3 3 ε Tr h = εh ab + gab − ε |χ | + 3 14 567 2 2 ε3 1 h ϕhh 2 ϕ +ε2 − , (ϕhhϕ){ab} − {ab} {ab} 9 18 27

(5.43)


475

and hence 1 4 (5.44) s1 = Tr (δs) = − ε2 |χ |2 + ε3 Tr h 3 , 2 81 8 2 s2 = Tr δs 2 = 2ε2 |χ |2 + ε3 Tr h 3 − (5.45) (ϕhhhϕ) 27 27 1 2 7 1 Tr h 4 − ϕhhh 2 ϕ + +ε4 − |χ |4 + (ψhhhhψ) , 16 162 81 324 3 8 8 ε3 Tr h 3 + ε4 − |χ |4 + Tr h 4 s3 = Tr δs 3 = 27 2 27 2 ϕhhh 2 ϕ , (5.46) − 27 16 s4 = Tr δs 4 = ε4 Tr h 4 . (5.47) 81 To get the full expression for ∗˜ ϕ, ˜ (5.44)-(5.47) have to be substituted into the expression 5 det g 2 (5.36), and then both (5.36) and (5.43) have to be substituted for the prefactor det g˜ into the expression for ∗˜ ϕ˜ (5.37). Obviously, the expressions involved quickly become absolutely gargantuan. Thankfully, we were able to use Maple and the freely available package "Riegeom" [22] to help with these calculations. After all the substitutions, the resulting expression still has dozens of terms which are not of much use. In order for the expression for ∗˜ ϕ˜ to be useful, the terms in it have to be separated according to which representation of G 2 they belong to. Thus the final step is to apply projections onto 41 , 47 and 427 (3.41)-(3.43). When applying these projections, many of the terms have ϕ and ψ contracted in some way, so the contraction identities (3.20)-(3.23) have to be used to simplify the expressions. The package "Riegeom" lacks the ability to make such substitutions, so a few simple custom Maple programs based on "Riegeom"had to be written in order to facilitate these calculations. Overall, the expansion of ∗˜ ϕ˜ to third order is 1 1 |χ |2 ∗ ϕ ∗˜ ϕ˜ = ∗ϕ − ε ∗ χ + ε2 ∗ iϕ (φhhφ)0 − (5.48) 6 42 2 1 5 |χ |2 ∗ χ − ∗ iϕ h 30 −ε3 (ϕhhhϕ) ∗ ϕ + 1701 24 18 1 1 u ∗ ϕ + O ε4 , + ∗ iϕ (ψhhhψ)0 + 36 324 where (φhhφ)0 , h 30 and (ψhhhψ)0 denote the traceless parts of (φhhφ)ab , h 3 ab and (ψhhhψ)ab , respectively, and u a = ψ amnp ϕr st h mr h ns h pt .

(5.49)

Although above we did all calculations to fourth order, we will really only need the expansion of ∗˜ ϕ˜ to third order. However for possible future reference here is the G 2 singlet piece of the fourth order 5 5 25 |χ |4 − Tr h 4 . (5.50) π1 ∗˜ ϕ˜ ε4 = (ψhhhhψ) + 13608 2016 6804

476


In fact, using the homogeneity property of ϕ ∧ ∗ϕ , it is possible to relate 427 terms with a higher order 41 term, so calculating higher order terms is also a way to make sure that all the coefficients are consistent. Now that we have expansions of ∗˜ ϕ˜ for 1- and 27-dimensional deformations, it is not difficult to combine them together. Suppose we want to combine conformal transformation and 27-dimensional deformations. As in the case with 7-dimensional deformations consider ϕ˜ = ϕˆ + εχ , where ϕˆ = f ϕ and χ ∈

327 .

Consider only up to second order in (5.48), 1 2 1 | ∗ˆ ϕˆ + ∗ˆ iϕˆ ϕˆ hˆ hˆ ϕˆ ∗˜ ϕ˜ = ∗ˆ ϕˆ − ε∗ˆ χ + ε2 − |χ + O ε3 . 0 42 6

Note that since h ab = 43 χmn{a ϕb}mn , 3 mn 3 mr ms hˆ ab = χmn{a ϕˆ b } = 4 gˆ gˆ χmn{a ϕˆ b }r s 4 1

= f − 3 h ab , and hence

ϕˆ hˆ hˆ ϕˆ

ab

= ϕâbm ϕˆden hˆ ad hˆ be 4

= f − 3 (ϕhhϕ)ab . Moreover, iϕˆ

ϕˆ hˆ hˆ ϕˆ = f −1 iϕ ((ϕhhϕ)0 ) . 0

Therefore, overall,

2 1 ∗˜ ϕ˜ = f ∗ ϕ − ε f ∗ χ + ε − f − 3 |χ |2 ∗ ϕ 42 1 −2 + f 3 ∗ iϕ ((ϕhhϕ)0 ) + O ε3 . 6 4 3

1 3

2

Let f = 1 + εa, and expand in powers of ε to third order to get 4 2 2 1 2 2 |χ | ∗ ϕ a ∗ ϕ − ∗χ + ε a − ∗˜ ϕ˜ = ∗ϕ + ε 3 9 42 1 1 − a ∗ χ + ∗ iϕ ((ϕhhϕ)0 ) 3 6 1 1 4 3 3 2 +ε a |χ | − a ∗ ϕ − a ∗ iϕ ((ϕhhϕ)0 ) 63 81 9 1 2 5 |χ |2 ∗ χ + a − 9 24 1 1 3 +ε ∗ iϕ h 30 − ∗ iϕ (ψhhhψ)0 18 36 2 1 u ∗ ϕ + O ε4 . − (ϕhhhϕ) ∗ ϕ − 1701 324

(5.51)

(5.52)


477

6. Moduli Space In Sect. 4 we described how M-theory can be used to give a natural complexification of the G 2 moduli space—denote this space by MC . The metric (4.62) on MC arises naturally from the Kaluza-Klein reduction of the M-theory action. As shown in [19], it turns out that this metric is in fact Kähler, with the Kähler potential K given by K = −3 log V, where as before, V is the volume of X , V =

1 7

(6.1)

ϕ ∧ ∗ϕ.

Note that sometimes K is given with a different normalization factor. Here we follow [19], but in [20] and [9], in particular, a different convention is used. Let us show that K is indeed the Kähler potential for G M N¯ . Clearly, V, K and G M N¯ only depend on the parameters s M for the G 2 3-form—that is, only the real part s M of the complex coordinates z M on MC . So let us for now just look at the s M derivatives. Note that under a scaling s M −→ λs M , ϕ scales as ϕ −→ λϕ and from (5.5), ∗ϕ scales 4 as ∗ϕ −→ λ 3 ∗ ϕ , and so V scales as 7

V −→ λ 3 V. 7 3

So V is homogeneous of order sM

in the s M , and hence

∂V 7 = V ∂s M 3 1 = s M φ M ∧ ∗ϕ, 3

and thus, ∂V 1 = M ∂s 3

Hence, ∂K 1 =− ∂s M V

φ M ∧ ∗ϕ.

(6.2)

φ M ∧ ∗ϕ.

(6.3)

Here the dependence on the s M is encoded in V and in ∗ϕ, which depends non-linearly on the s M . Thus we have, ∂2 K 3 ∂V ∂V 3 ∂2V = 2 M N − M N ∂z ∂ z¯ V ∂s ∂s V ∂s M ∂s N 1 1 1 = ∧ ∗ϕ ∧ ∗ϕ − φ φ(M ∧ ∂ N ) (∗ϕ) . φ M N 3 V2 V As we know from Sect. 5 , the first derivative of ∗ϕ is given by ∂ N (∗ϕ) =

4 ∗ π1 (φ N ) + ∗π7 (φ N ) − ∗π27 (φ N ) , 3

(6.4)

478


so therefore, 4 φ(M ∧ ∂ N ) (∗ϕ) = (π1 (ϕ M ) ∧ ∗π1 (ϕ N )) + (π7 (ϕ M ) ∧ ∗π7 (ϕ N )) 3 − (π27 (ϕ M ) ∧ ∗π27 (ϕ N )) . Also using (3.41), we get 1 1 71 ∧ ∗ϕ ∧ ∗ϕ = φ π1 (ϕ M ) ∧ ∗π1 (ϕ N ) . φ M N 3 V2 3V

(6.5)

Thus overall, 1 ∂2 K = M N ∂z ∂ z¯ V +

(π1 (ϕ M ) ∧ ∗π1 (ϕ N )) − (π27 (ϕ M ) ∧ ∗π27 (ϕ N )) .

(π7 (ϕ M ) ∧ ∗π7 (ϕ N )) (6.6)

Note that if H ol (X ) = G 2 then all the seven-dimensional components vanish, and hence we get ∂2 K 1 = φ M ∧ ∗φ N¯ = G M N¯ , (6.7) ∂z M ∂ z¯ N V X as claimed. Since the negative definite part of (6.6) vanishes, the resulting metric is positive definite. In general, there is at least one other good candidate for the metric on the G 2 moduli space. The Hessian of V , ratherthan of log V , can be used as a Kähler potential and gives 3 . This metric is in particular used in [23] and [9]. There a metric with signature 1, b27 are some advantages to using V as the Kähler potential, because some computations give more elegant results. However if we use the supergravity action as a starting point for the study of the moduli space, our choice of the Kähler potential is very natural. Now we have a complex manifold MC , equipped with the Kähler metric G M N¯ , so it is now interesting to study the properties of this metric, and the geometry which it gives. We will use the metric G M N¯ to calculate the associated curvature tensor R M N¯ P Q¯ of the manifold MC . Note that calculation of the curvature of the moduli space but for a different choice of metric is done in [24]. Let us introduce local special coordinates on MC . Let φ0 = aϕ and φµ ∈ 327 for 3 , so s 0 defines directions parallel to ϕ and s µ define directions in 3 . µ = 1, . . . , b27 27 Since our metric is K ähler, the expression for R M N¯ P Q¯ is given by ¯ R K¯ L M¯ N = ∂ M¯ ∂ N ∂ L ∂ K¯ K − G R S ∂ M¯ ∂ R ∂ K¯ K ∂ N ∂ L ∂ S¯ K .

(6.8)

Also define AM N R =

∂3 K ∂z M ∂z N ∂z R

(6.9)

so that we can rewrite (6.8) as ¯

R K¯ L M¯ N = ∂ M¯ ∂ N ∂ L ∂ K¯ K − G R S A M¯ R K¯ A N L S¯ .

(6.10)


479

Now it only remains to work out the third and fourth derivatives of K . Starting from (6.3) we find that 1 ∂2 ∂ 1 φ(M ∧ ∗ϕ φ M ∧ N R (∗ϕ) + 2 φ N ∧ R) (∗ϕ) V ∂s ∂s V ∂s 2 − 3 (6.11) φ M ∧ ∗ϕ φ N ∧ ∗ϕ φ R ∧ ∗ϕ , 9V

AM N R = −

and from the power series expansion of ∗ϕ (5.52), we can extract the higher derivatives of ∗ϕ: ∂0 ∂0 (∗ϕ) = ∂0 ∂µ (∗ϕ) = ∂µ ∂ν (∗ϕ) = ∂0 ∂µ ∂ν (∗ϕ) = ∂µ ∂ν ∂κ (∗ϕ) =

4 2 8 a ∗ ϕ, ∂0 ∂0 ∂0 (∗ϕ) = − a 3 ∗ ϕ, 9 27 1 2 − a ∗ φµ , ∂0 ∂0 ∂µ (∗ϕ) = a 2 ∗ φµ , 3 9 1 1 φµ , φν ∗ ϕ + ∗ iϕ ϕh µ h ν ϕ 0 , − 21 3 2 2 a φµ , φν ∗ ϕ − a ∗ iϕ ϕh µ h ν ϕ 0 , 63 9 1 5 − φµ , φν ∗ φκ + ∗ iϕ (h µ h ν h κ )0 4 3 1 4 − ∗ iϕ ψh µ h ν h κ ψ 0 − ϕh µ h ν h κ ϕ ∗ ϕ, 6 567

(6.12a) (6.12b) (6.12c) (6.12d)

(6.12e)

where h µ ,h ν and h κ are traceless symmetric matrices corresponding to the 3-forms φµ ,ϕν and φκ , respectively. Using these expressions, we can now write down all the components of A M N R : 3 A00 ¯ 0¯ = −14a , A00 ¯ µ¯ = 0, 2a A0µ¯ = − φµ ∧ ∗φν¯ = −2aG µ¯ν , ¯ ν V 2 Aµν ϕh µ¯ h ν h ρ¯ ϕ d V. ¯ ρ¯ = − 27V

(6.13a) (6.13b) (6.13c) (6.13d)

Now also look at the fourth derivative of K . From (6.12), we get ∂4 K ∂z 0 ∂ z¯ 0 ∂z 0 ∂ z¯ 0 ∂4 K 0 ∂z ∂ z¯ 0 ∂z 0 ∂ z¯ µ ∂4 K 0 ∂z ∂ z¯ 0 ∂z µ ∂ z¯ ν ∂4 K 0 ∂z ∂ z¯ µ ∂z ν ∂ z¯ ρ

= 42a 4 ,

(6.14a)

= 0,

(6.14b)

4 a2 4 φµ ∧ ∗φν¯ = a 2 G µ¯ν , 3V 3 2a = ϕh µ h ν h ρ ϕ vol = −3a Aµν ¯ ρ¯ , 9V =

(6.14c) (6.14d)

480


1 1 ∂4 K 1 G µν = φκ ∧ ∗φν φµ¯ ∧ ∗φρ¯ ¯ G κ ρ¯ + G µκ ¯ G ν ρ¯ + ∂z κ ∂ z¯ µ ∂z ν ∂ z¯ ρ 3 3 V2 1 + ψh κ h µ¯ h ν h ρ¯ ψ − 2 Tr h κ h µ¯ h ν h ρ¯ 27V 5 + Tr h (κ h µ¯ Tr h ν h ρ) vol, (6.14e) ¯ 3 Note that it can be shown using the identity (3.22) that 2 ψhhhhψ = 12 ϕh 2 hhϕ + 3 Tr h 2 − 6 Tr h 4 . Now define CM N =

∂2 K . ∂z M ∂z N

(6.15)

This is the second derivative of K but with pure indices, rather than the derivative with mixed indices which gives the metric G M N¯ . Note that since K = K (Im z), we have ∂2 K ∂2 K = ∂z M ∂z N ∂z M ∂ z¯ N so numerically, C M N and G M N¯ are in fact equal, and in particular, 1 Cµν = φµ ∧ ∗φν . V

(6.16)

(6.17)

So while C M N is not technically part of the metric, it inherits some similar properties. This happens due to the fact that while the complexification of the moduli space comes naturally, the holomorphic structure is artificial to some extent, because the G 2 and C-field moduli do not really mix. Using this definition, we can rewrite (6.14e) as 1 ∂4 K 1 G µν = ¯ G κ ρ¯ + G µκ ¯ G ν ρ¯ + C µ¯ ρ¯ C κν ∂z κ ∂ z¯ µ ∂z ν ∂ z¯ ρ 3 3 1 − 2 Tr h κ h µ¯ h ν h ρ¯ − ψh κ h µ¯ h ν h ρ¯ ψ 27V 5 vol. − Tr h (κ h µ¯ Tr h ν h ρ) ¯ 3 ¯

Taking into account that G 00 = 7a1 2 and G 0µ¯ = 0, we have enough information to be able to write down the full expressions for the components of the curvature tensor: 4 R000 ¯ 0¯ = 14a , R000 ¯ µ¯ = 0,

(6.18) (6.19)

2 R00µ¯ ¯ ν = 2a G µ¯ν , R0µν ¯ ρ¯ = −Aµν ¯ ρ¯ a,

(6.20) (6.21)


Rκ µν ¯ ρ¯ =

481

1 5 τ σ¯ G µν Cµ¯ ρ¯ Cκν ¯ G κ ρ¯ + G µκ ¯ G ν ρ¯ − G Aµτ ¯ ρ¯ Aκν σ¯ − 3 21 1 + ψh κ h µ¯ h ν h ρ¯ ψ − 2 Tr h κ h µ¯ h ν h ρ¯ 27V 5 + Tr h (κ h µ¯ Tr h ν h ρ) vol. ¯ 3

(6.22)

Let us look at more detail at the expression for Aµ¯ν ρ¯ : 2 Aµν = − ϕh µ¯ h ν h ρ¯ ϕvol ¯ ρ¯ 27V 2 bn cp =− ϕabc ϕmnp h am µ¯ h ν h ρ¯ vol. 27V Define h aµ = h µa m d x m . Then cp

bn a b c ϕabc ϕmnp h am µ¯ h ν h ρ¯ vol = 6ϕabc h µ¯ ∧ h ν ∧ h ρ¯ ∧ ∗ϕ

and so, Aµν ¯ ρ¯ = −

4 9V

ϕabc h aµ¯ ∧ h bν ∧ h ρc¯ ∧ ∗ϕ.

(6.23)

This is the precise analogue of the Yukawa coupling which is defined on the Calabi-Yau moduli space. Similar expressions have appeared previously in [25,8] and [9]. Similarly, we can write bn cp dq ψh κ h µ¯ h ν h ρ¯ ψ vol = ψabcd ψmnpq h am κ h µ¯ h ν h ρ¯ vol = 24 ψabcd h aκ ∧ h bµ¯ ∧ h cν ∧ h ρd¯ , ∗ψ vol = 24ψabcd h aκ ∧ h bµ¯ ∧ h cν ∧ h dρ¯ ∧ ϕ.

(6.24)

Hence, we can rewrite (6.22) as Rκ µν ¯ ρ¯ =

1 5 τ σ¯ G µν Cµ¯ ρ¯ Cκν (6.25) ¯ G κ ρ¯ + G µκ ¯ G ν ρ¯ − G Aµτ ¯ ρ¯ Aκν σ¯ − 3 21 81 + ψabcd h aκ ∧ h bµ¯ ∧ h cν ∧ h ρd¯ ∧ ϕ 9V 1 1 + 5 Tr h (κ h µ¯ Tr h ν h ρ) ¯ − 6 Tr h κ h µ¯ h ν h ρ¯ vol. 81 V

Note that because in the 327 directions the first derivative of V vanishes, some of these terms which appear in the curvature expression can also be expressed as derivatives of V : ∂3V 1 = − Aµν ¯ ρ¯ µ ν ρ ∂ z¯ ∂z ∂ z¯ 3 ∂4V 8 =− ψabcd h aκ ∧ h bµ¯ ∧ h cν ∧ h ρd¯ ∧ ϕ κ µ ν ρ ∂z ∂ z¯ ∂z ∂ z¯ 27 1 vol. + 6 Tr h κ h µ¯ h ν h ρ¯ − 5 Tr h (κ h µ¯ Tr h ν h ρ) ¯ 243

482


So alternatively, we can write Rκ µν ¯ ρ¯ =

1 5 τ σ¯ G µν Cµ¯ ρ¯ Cκν ¯ G κ ρ¯ + G µκ ¯ G ν ρ¯ − G Aµτ ¯ ρ¯ Aκν σ¯ − 3 21 ∂4V 3 − . V ∂z κ ∂ z¯ µ ∂z ν ∂ z¯ ρ

Define U M¯ = Then ∂ K U M¯ =

3 V

∂3V 3 ¯ G N R. V ∂ z¯ M ∂z N ∂ z¯ R

∂4V ∂3V N R¯ N R¯ G − A . ∂z K ∂ z¯ M ∂z N ∂ z¯ R ∂ z¯ M ∂z N ∂ z¯ R K

We can use this to express the Ricci curvature 1 3 1 b (X ) − G κ µ¯ − ∂κ Uµ¯ , Rκ µ¯ = 3 63

(6.26)

(6.27)

(6.28)

3 + 1 is the third Betti number of X . Also, where b3 (X ) = b27 ν ρ¯ R0µ¯ = −a Aµν = −∂0 Uµ¯ , ¯ ρ¯ G

(6.29)

R00¯ = 2a b (X ) .

(6.30)

2 3

Although here we have certain similarities with the structure of the Calabi-Yau moduli space, we are lacking a key feature of Calabi-Yau moduli space—a particular line bundle over the moduli space. For example, the holomorphic 3-form on a Calabi-Yau 3-fold defines a complex line bundle over the complex structure moduli space. In the G 2 case, we could try and see what happens if we look at the real line bundle L defined by ϕ over the complexified G 2 moduli space MC . So consider the gauge transformations ϕ −→ f (Re z) ϕ,

(6.31)

where each f (z) is a real number. Then, as in [8], define a covariant derivative D on L by DM ϕ = ∂M ϕ +

1 (∂ M K ) ϕ. 7

(6.32)

Under the transformation (6.31), 7

V −→ f 3 V, K −→ K − 7 log f, and so ∂ M K −→ ∂ M K −

7 ∂ M f. f

Hence D M ϕ −→ f D M ϕ.

(6.33)


483

Moreover, from the expression for ∂ M K (6.3), we find that D0 ϕ = 0

Dµ ϕ = ∂µ ϕ.

So as noted in [8], this covariant derivative projects out the G 2 singlet contribution. It also gives a covariant way in which to extract the 27 contributions so we can use D M ϕ when we just need to extract ∂µ ϕ. Also consider 1 1 D M ϕ, ∗D N¯ ϕ = (6.34) D M ϕ ∧ ∗D N¯ ϕ V V 1 = G M N¯ − ∂ M K ∂ N¯ K . 7 When one of the indices is equal to zero, the whole expression vanishes. However if both refer to the 27-dimensional components, then we just get G µ¯ν . A similar expression holds for C M N . More generally, we can extend the covariant to any quantity which transforms under (6.31). Suppose Q (z) is a function on MC , which under (6.31) transforms as Q (z) −→ f (z)a Q (z) . Then define the covariant derivative on it by DM Q = ∂M Q +

a (∂ M K ) Q. 7

(6.35)

From this we get D M V = 0, D M (∗ϕ) = ∂ M (∗ϕ) +

14 (∂ M K ) (∗ϕ) , 73

and in particular, D0 (∗ϕ) = 0

Dµ (∗ϕ) = − ∗ ∂µ ϕ

so, in fact D M (∗ϕ) = − ∗ D M ϕ. Further we can extend D M to objects with moduli space indices by replacing ∂ by ∇—the metric-compatible covariant derivative with respect to the moduli space metric G M N¯ , for which the Christoffel symbols are given by ¯

MNQ = G N P ∂ M G P¯ Q = A NM Q .

(6.36)

With these Christoffel symbols the covariant derivative of C M N is hence ∇ Q C M N = −A Q M N .

(6.37)

484


Then we also find that 1 1 D M D N ϕ = ∂ M ∂ N ϕ + (∂ N K ) ϕ − A PN M D P ϕ + ∂ M K D N ϕ 7 7 1 2 1 C M N − ∂ M K ∂ N K ϕ − A PN M D P ϕ + ∂(M K D N ) ϕ (6.38) = 7 7 7 11 2 P D M ϕ, ∗D N ϕ ϕ − A N M D P ϕ + ∂(M K D N ) ϕ, = (6.39) 7V 7 and for mixed type derivatives, we have 1 2 D M¯ D N ϕ = ∂ M¯ ∂ N ϕ + (∂ N K ) ϕ + ∂( M¯ K D N ) ϕ 7 7 1 1 2 D M¯ ϕ, ∗D N ϕ ϕ + ∂( M¯ K D N )ϕ = 7V 7 1 2 G M¯ N ϕ + ∂ M¯ K ∂ N K ϕ + ∂( M¯ K ∂ N ) ϕ . = 7 7 Note that here the covariant derivatives commute, so this connection is in fact flat. Now look at the third covariant derivative of ϕ: D R D M D N ϕ, ∗ϕ = D R D M D N ϕ, ∗ϕ − D M D N ϕ, D R ∗ ϕ = D R D M ϕ, ∗D N ϕ + D M D N ϕ, ∗D R ϕ .

(6.40)

First look at the second term in (6.40). Since D R ϕ ∈ 327 , we basically get the projection π27 (D M D N ϕ): 2 D M D N ϕ, ∗D R ϕ = −A PN M D P ϕ, ∗D R ϕ + ∂(M K D N ) ϕ, ∗D R ϕ 7 1 P 2 2 = −A M N R + A M N ∂ R K ∂ P K + C R(N ∂ M) K − ∂ R K ∂ M K ∂ N K . 7 7 49 In the first term of (6.40), we have

D R D M ϕ, ∗D N ϕ = = = =

1 D M ϕ, ∗D N ϕ V DR V V ∇ R D M ϕ, ∗D N ϕ 1 V ∇ R C M N − ∇ R (∂ M K ∂ N K ) 7 2 2 P V −A R M N − C R(M ∂ N ) K + A R(M ∂ N ) K ∂ P K . 7 7

Combining, we overall obtain 2 3 1 D R D M D N ϕ, ∗ϕ = −2 A R M N − ∂ R K ∂ M K ∂ N K + A(M NP ∂ R) K ∂ P K . V 49 7 (6.41)


485

Decomposing this into components, we have 1 Dρ Dµ Dν ϕ, ∗ϕ = −2 Aρµν , V 1 D0 Dµ Dν ϕ, ∗ϕ = 2Cµν , V 1 D0 D0 Dν ϕ, ∗ϕ = 0, V 1 D0 D0 D0 ϕ, ∗ϕ = 0. V Therefore, the quantity V1 Dρ Dµ Dν ϕ, ∗ϕ essentially gives the Yukawa coupling, again giving a result analogous to the case of Calabi-Yau moduli spaces. 7. Concluding Remarks In this paper, we have computed the curvature of the complexified G 2 moduli space and found that while it has terms which are similar to the curvature of the Calabi-Yau moduli, there are a number of new terms. In future work it would be interesting to interpret these new terms geometrically. If we consider a 7-manifold of the form CY3 × S 1 , where CY3 is a Calabi-Yau 3-fold, then we can define a torsion-free G 2 structure on it. The relationship between the Calabi-Yau moduli space and the G 2 moduli space is however very non-trivial, because the complex structure moduli and the Kähler structure moduli become intertwined with each other. So it could turn out to be illuminating to try and relate the curvature of the G 2 moduli space to the curvatures of complex and Kä hler moduli spaces. In that case, however, b73 = 1, so in fact the second derivative of our Kähler potential would give a pseudo-Kähler metric with signature (− + . . . +) (6.6). Moreover, the ansatz for the C-field (4.55) would also have to be different. Understanding how the Calabi-Yau moduli space is related to the G 2 moduli space could also enable us to find a manifestation of mirror symmetry from the G 2 perspective. Moreover, it would be interesting to see how existing approaches to mirror symmetry on G 2 manifolds (such as [26]) affect the geometric structures on the moduli space. Another possible direction for further research is to look at G 2 manifolds in a slightly different way. Suppose we have type I I A superstrings on a non-compact Calabi-Yau 3-fold with a special Lagrangian submanifold which is wrapped by a D6 brane which also fills M4 . Then, as explained in [27], from the M-theory perspective this looks like a S 1 bundle over the Calabi-Yau which is degenerate over the special Lagrangian submanifold, but this 7-manifold is still a G 2 manifold. The moduli space of this manifold will then be determined by the Calabi-Yau moduli and the special Lagrangian moduli. This possibly could provide more information about mirror symmetry on Calabi-Yau manifolds [28]. A. Appendix A: Projections of 3-Forms Here will prove the formulae (3.41) to (3.43) which give the projections of 3-forms into 1-dimensional, 7 -dimensional and 27-dimensional components. Let χ ∈ 3 . Since 31 , 37 and 327 are all orthogonal to each other, we immediately get 1 1 π1 (χ ) = aϕ where a = χabc ϕ abc = χ , ϕ and |π1 (χ )|2 = 7a 2 . 42 7

486


To work out π7 (χ ), suppose π7 (χ ) = u ∗ ϕ, then consider (u ∗ ϕ) ∧ ∗ (v ∗ ϕ) = (u ∗ ϕ) ∧ ϕ ∧ v = 4 ∗ u ∧ v = 4 u, v vol.

(A.1)

|π7 (χ )|2 = 4 |ω|2 .

(A.2)

So this gives

However (A.1) can also be expressed as 1 π7 (χ )mnp va ψ amnp vol 6 1 = − π7 (χ )mnp ψ mnpa va vol. 6

(u ∗ ϕ) ∧ ∗ (v ∗ ϕ) =

(A.3)

Equating (A.1) and (A.3), we get ua = −

1 π7 (χ )mnp ψ mnpa = ωa . 24

Finally we look at π27 (χ ). Consider χabc = π1 (χ )abc + π7 (χ )abc + h d[a ϕbc]d . Then, π1 (χ )mn{a ϕb}mn = aϕmn{a ϕb}mn = 6g{ab} = 0,

(A.4)

π7 (χ )mn{a ϕb}mn

(A.5)

=ω

p

ψ pmn{a ϕb}mn

= 4v ϕ p{ab} = 0. p

Therefore, 3 3 χmn{a ϕb}mn = h d[m ϕn{a]d ϕb}mn 4 4 1 1 = h dm ϕn{a|d| ϕb}mn + ϕmnd h d{a ϕb}mn 2 4 3 1 d m m = h m g{ab} δd − δ{a gb}d − ψ{ab}d m + h ab 2 2 = h ab

(A.6)

as required. Moreover, 1 d h ϕbc]d h ea ϕ bce 6 [a 1 d 1 h a ϕbcd h ea ϕ bce + h dc ϕabd h ea ϕ bce = 18 9 1 2 1 d ea c δa gde − gae δqc + ∗ϕ cade = |h| − h c h 3 9 2 2 = |h| . 9

|π27 (χ )|2 =

(A.7)


487

B. Appendix B: Determinants In this section, we will review deformations of determinants. Let I be the n × n identity matrix, and let h be a symmetric n × n matrix. Suppose λ1 , . . . , λn are eigenvalues of h. Then det (I + εh) =

n

(1 + ελi )

i=1

= 1+ε

λi + ε2

i

λi λ j + ε3

i< j

λi λ j λk

i< j 0 for all j = 1, . . . , k −1 while ker Pk = Span( f δ ). The manifold has a special conformal geometry at infinity that makes the resonance δ disappear and transform into a 0-eigenvalue for the conformal Laplacian Pk . The proof uses methods of Tang-Zworski [38] together with information on the closest resonance to the critical line, that is δ when δ ∈ / n/2 − N (the physical sheet for the resolvent R(λ) := ( − λ(n − λ))−1 is {Re(λ) > n/2}) this last fact has been proved by Patterson [29] using Poincaré series and Patterson-Sullivan measure. The powerful dynamical theory of Dolgopyat [7] has been used by the second author [26] (for surfaces) and Stoyanov [36] (in higher dimension) to prove the existence of a strip with no zero on the left of the first zero λ = δ for the Selberg zeta function. Using results of Patterson-Perry [30], this implies a strip {δ − < Re(λ) < δ)} with no resonance. Then we can view u(t) as a contour integral of the resolvent R(λ) and move the contour up to δ and apply the residue theorem. This involves obtaining rather sharp estimates on the truncated (on compact sets) resolvent near the line {Re(λ) = δ}. This is achieved by combining the non-vanishing result with an a priori bound that results from a precise parametrix of the truncated resolvent. A second result of this article is the proof of the existence of an explicit strip with infinitely many resonances. Theorem 1.2. Let X = \Hn+1 be a convex co-compact hyperbolic manifold and let δ ∈ (0, n) be the Hausdorff dimension of its limit set. Then for all ε > 0, there exist infinitely many resonances in the strip {−nδ − ε < Re(s) < δ}. If moreover is a Schottky group, then there exist infinitely many resonances in the strip {−δ 2 − ε < Re(s) < δ}. Note that the existence of infinitely many resonances in some strips was proved by Guillopé-Zworski [21] in dimension 2 and Perry [33] in higher dimension, but in both cases, they did not provide any geometric information on the width of these strips. Our proof is based on a Selberg-like trace formula and uses all previously known counting estimates for resonances. An interesting consequence is an explicit Omega lower bound for the remainder in (1.4) for generic compactly supported initial data. Corollary 1.3. Let K ⊂ X be a relatively compact open set, then there exists a generic set ⊂ L 2 (K ) such that for all f 1 ∈ , f 0 = 0 and all > 0, the remainder in (1.4) is n n not a O L 2 (e−( 2 +nδ+)t ) as t → ∞. If X is Schottky, O L 2 (e−( 2 +nδ+)t ) can be improved n 2 to O L 2 (e−( 2 +δ +)t ). The meaning of “generic” above is in the Baire category sense, i.e. it is a G δ -dense subset. We point out that when n = 1, all convex-cocompact surfaces are Schottky, i.e. are obtained as \H2 , where is a Schottky group. For a definition of Schottky groups in our setting we refer for example to the introduction of [17]. In higher dimensions, not all convex co-compact manifolds are obtained via Schottky groups. For more details and references around these questions we refer to [15].

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

493

The rest of the paper is organized as follows. In § 2, we review and prove some necessary bounds on the resolvent in the continuation domain. In § 3 we prove the estimate on the strip with finitely many resonances. In § 4, we derive the asymptotics by using contour deformation and the key bounds of § 2. We also show how to relate § 3 to an Omega lower bound of the remainder. Section § 5 is devoted to the analysis of the special cases δ ∈ n2 − N in terms of the conformal theory of the infinity. 2. Resolvent We start in this section by analyzing the resolvent of the Laplacian for the convex co-compact quotient of Hn+1 and we give some estimates of its norms. 2.1. Geometric setting. We let be a convex co-compact group of isometries of Hn+1 with Hausdorff dimension of its limit set satisfying 0 < δ < n/2, we set X = \Hn+1 , its quotient equipped with the induced hyperbolic metric, and we denote the natural projection by π : Hn+1 → X = \Hn+1 , π¯ : → ∂ X¯ = \ .

(2.1)

By assumption on the group , for any element h ∈ there exists α ∈ Isom(Hn+1 ) such that for all (x, y) ∈ Hn+1 = Rn × R+ , α −1 ◦ h ◦ α(x, y) = el(γ ) (Oγ (x), y), where Oγ ∈ S On (R), l(γ ) > 0. We will denote by α1 (γ ), . . . , αn (γ ) the eigenvalues of Oγ , and we set n

1 − e−kl(γ ) αi (γ )k . G γ (k) = det I − e−kl(γ ) Oγk =

(2.2)

i=1

The Selberg zeta function of the group is defined by ⎛ ⎞ ∞ −λml(γ ) 1e ⎠; Z (λ) = exp ⎝− m G γ (m) γ m=1

the sum converges for Re(λ) > δ and admits a meromorphic extension to λ ∈ C by results of Fried [9] and Patterson-Perry [30]. 2.2. Extension of resolvent, resonances and zeros of Zeta. The spectrum of the Laplacian X on X is a half line of absolutely continuous spectrum [n 2 /4, ∞), and if we take for the resolvent of the Laplacian the spectral parameter λ(n − λ), R(λ) := ( X − λ(n − λ))−1 , this is a bounded operator on L 2 (X ) if Re(λ) > n/2. It is shown by Mazzeo-Melrose [25] and Guillopé-Zworski [19] that R(λ) extends meromorphically in C as continuous operators R(λ) : L 2comp (X ) → L 2loc (X ), with poles of finite multiplicity, i.e. the rank of the polar part in the Laurent expansion of R(λ) at a pole is finite. The poles are called

494

C. Guillarmou, F. Naud

resonances of X , they form the discrete set R included in Re(λ) < n/2, where each resonance s ∈ R is repeated with the mutiplicity m s := rank(Resλ=s R(λ)). A corollary of the analysis of divisors of Z (λ) by Patterson-Perry [30] and Bunke-Olbrich [4] is Proposition 2.1 (Patterson-Perry, Bunke-Olbrich). Let s ∈ C \ (−N0 ∪ (n/2 − N)), then Z (λ) is holomorphic at s, and s is a zero of Z (λ) if and only if s is a resonance of X . Moreover its order as zero of Z (λ) coincide with the multiplicity m s of s as a resonance. We insist on the fact that the correspondence between zeros of Z (λ) and poles of the resolvent R(λ) will be a crucial argument in our estimates for the solutions of the wave equation. This correspondence can be understood as a Selberg trace formula and comes from the fact that the logarithmic derivative of Selberg zeta function is given by Z (λ) = (2λ−n)FP→0 (R(λ; m, m )− RHn+1 (λ; m, m ))|m=m dvol(m) , Z (λ) F ∩{ρ(m)>} where RHn+1 (λ; m, m ) is the resolvent kernel of the Laplacian HN+1 on Hn+1 , F ⊂ Hn+1 is a fundamental domain of the group , ρ is a boundary defining function of X = \Hn+1 and FP means finite part (i.e. the 0 coefficient of the asymptotic expansion as → 0). The core of the proof of Patterson-Perry is to use the meromorphic extension of R(λ) to λ ∈ C to prove meromorphic extension of s(λ) := Z (λ)/Z (λ) to λ ∈ C, and then to show that the poles of s(λ) are first order, located at the resonances (except for the negative integer points) and with integer residues given by the multiplicity of the resonance. This analysis strongly uses the scattering operator S(λ) defined in Sect. 5. 2.3. Estimates on the resolvent R(λ) in the non-physical sheet. The series Pλ (m, m ) defined in (1.3) converges absolutely in Re(λ) > δ, is a holomorphic function of λ there, with local uniform bounds in m, m , which clearly gives ∀ > 0, ∃C (m, m ) > 0, ∀λ with Re(λ) ∈ [δ + , n], |Pλ (m, m )| ≤ C,m,m , and C,m,m is locally uniform in m, m . We show Proposition 2.2. With previous assumptions, there exists > 0 and a holomorphic family in {Re(λ) > δ − } of continuous operators K (λ) : L 2comp (X ) → L 2loc (X ) such that the resolvent satisfies in Re(λ) > δ, n

(2π )− 2 (λ) R(λ) = P(λ) + K (λ), (λ − n2 ) where P(λ) is the operator with Schwartz kernel Pλ (m, m ). Moreover there exists M > 0 such that for any χ1 , χ2 ∈ C0∞ (X ), there is a C > 0 such that ||χ1 K (λ)χ2 ||L(L 2 (X )) ≤ C(|λ| + 1) M , Re(λ) > δ − .


495

Proof. We choose a fundamental domain F for with a finite number of sides paired by elements of . By standard arguments of automorphic functions, the resolvent kernel R(λ; m, m ) for m, m ∈ F and Re(λ) > δ is the average G(λ; m, γ m ) = σ (dh (m, γ m ))λ kλ (σ (dh (m, γ m ))), R(λ; m, m ) = γ ∈

γ ∈ −1

σ (d) := (cosh d)

= 2e

−d

(1 + e−2d )−1 ,

where G(λ; m, m ) is the Green kernel of the Laplacian on Hn+1 and kλ ∈ C ∞ ([0, 1)) is the hypergeometric function defined for Re(λ) > n−1 2 , kλ (σ ) :=

2

3−n 2

π

(λ −

n+1 2

(λ)

n+1 2

+ 1)

1

(2t (1 − t))λ−

n+1 2

(1 + σ (1 − 2t))−λ dt,

0

which extends meromorphically to C and whose Taylor expansion at order 2N can be written kλ (σ ) = 2−λ−1

N

α j (λ)

σ 2 j

j=0

2

n

+ kλN (σ ), α j (λ) :=

π − 2 (λ + 2 j) (λ − n2 + 1)( j + 1)

with kλN ∈ C ∞ ([0, 1)) and the estimate for any 0 > 0 |kλN (σ )| ≤ σ 2N +2 C N (|λ| + 1)C N , σ ∈ [0, 1 − 0 ), Re(λ) >

n −N 2

(2.3)

for some C > 0 depending only on 0 , see for instance [13, Lem. B.1]. Extracting the first term with α0 in kλ , we can then decompose ⎞ ⎛ n 2 (λ) π ⎝ e−λdh + e−(λ+1)dh f λ (e−dh )⎠ R(λ; m, m ) = 2(λ − n2 + 1) γ ∈ γ ∈ λ 0 + σ (dh ) kλ (σ (dh )) γ ∈

f λ (x) :=

(1 + x 2 )−λ − 1 , x

and where dh means dh (m, γ m ) here. Thus to prove the proposition, we have to analyze the term K (λ) := 2−1 α0 (λ)K 1 (λ) + K 2 (λ) with K 1 (λ) := e−(λ+1)dh f λ (e−dh ), K 2 (λ) := σ (dh )λ kλ0 (σ (dh )). γ ∈

γ ∈

The first term K 1 is easy to deal with since | f λ (x)| ≤ C(|λ| + 1) for x ∈ [0, 1], thus we can use the fact that Pλ+1 (m, m ) converges absolutely in Re(λ) > δ − 1, is holomorphic there, and is locally uniformly bounded in (m, m ), thus n

|α0 (λ)χ1 (m)χ2 (m )K 1 (λ)| ≤ C(|λ| + 1) 2 +1 ; the same bound holds for the operator in L(L 2 (X )) with Schwartz kernel χ1 (m)χ2 (m) F1 (λ). Note that α0 (λ) has no pole in Re(λ) > 0, thus no pole in Re(λ) > δ/2 > 0.

496


For K 2 (λ) we can decompose it as follows: for m ∈ Supp(χ1 ), m ∈ Supp(χ2 ) (which are compact in F), for 0 > 0 fixed there is only a finite number of elements 0 = {γ0 , . . . , γ L ∈ } such that dh (m, γ m ) > 0 for any γ ∈ / 0 and any m, m in any fixed compact set K of F; this is because the group acts properly discontinuously on Hn+1 . Thus we split the sum in K 2 (λ) into σ (dh )λ kλ0 (σ (dh )) + σ (dh )λ kλ0 (σ (dh )). (2.4) K 2 (λ) = γ ∈0

γ ∈ / 0

We first observe that the second term is a convergent series, holomorphic in λ, for Re(λ) > δ − 1 and uniformly bounded in m, m ∈ K. Indeed it is easily seen to be bounded by C N (|λ| + 1)

N

|α j (λ)|PRe(λ)+2 j (m, m ) + C N (|λ| + 1)C N PRe(λ)+2N +1 (m, m )

(2.5)

j=1

by assumption on 0 and using (2.3), C depending on 0 only. Moreover since α j (λ) is polynomially bounded by C(|λ| + 1)2 j we have a polynomial bound for (2.5) of degree depending on N . The first term in (2.4) has a finite sum thus it suffices to estimate each term, but because of the usual conormal singularity of the resolvent at the diagonal, it explodes as dh (m, m ) → 0. We want to use Schur’s lemma for instance, so we have to bound

sup |χ1 (m)χ2 (m )K 2 (λ; m, m )|dm Hn+1 , m∈F F

|χ1 (m)χ2 (m )K 2 (λ; m, m )|dm Hn+1 . sup m ∈F

First we recall that

Hn+1

F

= (0, ∞)x × Rny has a Lie group structure with product

y 1 (x, y).(x , y ) = (x x , y + x y ), (x, y)−1 = ( , − ) x x and neutral element e := (1, 0). Then if (u, v) := (x , y )−1 .(x, y) = (x/x , (y−y )/x ) we get (cosh(dh (x, y; x , y )))−1 =

2x x x2

+

x 2

+ |y −

y |2

=

2u 1 + u + |v|2

= (cosh(dh (u, v; 1, 0)))−1 .

(2.6)

Moreover the diffeomorphism (u, v) → m = m.(u, v)−1 on Hn+1 pulls the hyperbolic measure dm Hn+1 = x −n−1 d x dy back into the right invariant measure u −1 dudv for the group action. This is to say that we have to bound

dudv , (2.7) |χ1 (m)χ2 (m.(u, v)−1 )K 2 (λ; m, m.(u, v)−1 )| sup −1 u m∈F F .m where F −1 .m := {m −1 .m; m ∈ F} and similarly

dudv . |χ1 (m .(u, v)−1 )χ2 (m )K 2 (λ; m .(u, v)−1 , m )| sup −1 u m ∈F F .m

(2.8)


497

Because m, m are in compact sets, the estimate (2.5) with N = n gives a polynomial bounds in λ in {Re(λ) > δ − } for the terms coming from γ ∈ / 0 . To deal with the term of (2.4) containing elements γ ∈ 0 , we use Lemma B.1 of [13] which proves that for any compact K of Hn+1 , there exists a constant C K such that

C N (|λ| + 1)n−1 n dudv ≤ K , Re(λ) > − N . (2.9) |G(λ; (u, v), e)| u dist(λ, −N ) 2 0 K Now to bound (2.7) with K 2 (λ, •, •) replaced by σ (dh (•, γ •))λ kλ (σ (dh (•, γ •))) we note that before we did our change of variable in (2.7), we can make the change of variable m → γ −1 m which amounts to bound

sup χ1 (m)χ2 (γ −1 m.(u, v)−1 ) G(λ; (u, v), e) m∈F

(γ F )−1 .m

dudv , −2−λ−1 α0 (λ)σ λ (dh ((u, v), e)) u

where we used (2.6). But again, since χ1 , χ2 have compact support, we get a polynomial bound in λ using (2.9) and a trivial polynomial bound for kλ (0). The term (2.8) can be dealt with similarly and we finally deduce that for some M, ||χ1 K 2 (λ)χ2 ||L(L 2 (X )) ≤ C(|λ| + 1) M , Re(λ) > δ − , and the proposition is proved.

This clearly shows that the resolvent extends to {Re(λ) > δ} analytically. Actually, Patterson [29] (see also [31, Prop 1.1]) showed the following. Proposition 2.3 (Patterson). The family of operators (λ − n/2 + 1)R(λ) is holomorphic in {Re(λ) > δ}, has no pole on {Re(λ) = δ, λ = δ} and has a pole of order 1 at λ = δ with rank 1 residue given by Resλ=δ (λ − n/2 + 1)R(λ) = A X u δ ⊗ u δ , where A X = 0 is some constant depending on and u δ is the Patterson generalized eigenfunction defined by

π∗ u δ (m) = (2.10) (P(m, y))δ dµ (y), ∂∞ Hn+1

P being the Poisson kernel of Hn+1 and dµ the Patterson-Sullivan measure associated to on the sphere ∂∞ Hn+1 = Rn ∪ {∞} = S n . We can but notice that δ ∈ n/2 − N is a special case since the resolvent becomes holomorphic at λ = δ. We postpone the analysis of this phenomenon to § 5. A rough exponential estimate in the non-physical sheet also holds using the determinant method (used for instance in [20]). Lemma 2.4. For χ1 , χ2 ∈ C0∞ (X ), j ∈ N0 , and η > 0 there is C > 0 such that for |λ| ≤ N /16 and dist(λ, R} > η, j

||∂λ χ1 R(λ)χ2 ||L(L 2 (X )) ≤ eC(N +1)

n+3

,

498


Proof. We apply the idea of [20, Lem. 3.6] with the parametrix construction of R(λ) written in [19]. Let x be a boundary defining function of ∂ X¯ in X¯ , which can be considered as a weight to define Hilbert spaces x α L 2 (X ), for any α ∈ R. For any large N > 0 that we suppose in 2N for convenience, Guillopé and Zworski [19] construct operators PN (λ, λ0 ) : x N L 2 (X ) → x −N L 2 (X ),

K N (λ, λ0 ) : x N L 2 (X ) → X N L 2 (X )

meromorphic with finite multiplicity in O N := {Re(λ) > (n − N )/2}, whose poles are situated at −N0 , and such that ( X − λ(n − λ))PN (λ, λ0 ) = 1 + K N (λ, λ0 ) with λ0 large depending on N , take for instance λ0 = n/2 + N /8. Moreover K N (λ, λ0 ) is compact with characteristic values satisfying in O N ,η := O N ∩ {dist(λ, −N0 ) > η}, CN 1 if j ≤ C N n+1 e (2.11) µ j (K N (λ, λ0 )) ≤ C(1 + |λ − λ0 |) j − n + −N /C 2 e j if j ≥ C N n+1 for some 0 < η < 1/4 and C > 0 independent of λ, N . They also have ||K N (λ0 , λ0 )|| ≤ 1/2 in L(x N L 2 (X )), thus by the Fredholm theorem R(λ) = PN (λ, λ0 )(1 + K N (λ, λ0 ))−1 : x N L 2 (X ) → x −N L 2 (X ) is meromorphic with poles of finite multiplicity in O N . By a standard method as in [20, Lem. 3.6] we define d N (λ) := det(1 + K N (λ, λ0 )n+2 ) which exists in view of (2.11), and we have the rough bound ||(1 + K N (λ, λ0 ))−1 ||L(x N L 2 (X )) ≤

det(1 + |K N (λ, λ0 )|n+2 ) |d N (λ)|

(2.12)

1

in O N ,η , and where |A| := (A∗ A) 2 for A compact. The term in the numerator is easily shown to be bounded by exp(C(N + 1)n+2 ) in O N ,η from (2.11); actually this is written in [19, Lem. 5.2]. It remains to have a lower bound of |d N (λ)|. In Lemma 3.6 of [20], they use the minimum modulus theorem to obtain the lower bound of a function using its upper bound, but this means that the function has to be analytic in C. Here there is a substitute which is Cartan’s estimate [23, Th. I.11]. We first need to multiply d N (λ) by a holomorphic function J N (λ) with zeros of sufficient multiplicity at {−k; k = 0, . . . , N /2} to make J N (λ)d N (λ) holomorphic in O N , for instance the polynomial J N (λ) :=

N /2

(λ − k)C N

n+2

k=0

for some large integer C > 0 suffices in view of the order (≤ C N n+2 ) of each −k as a pole of d N (λ) proved in [19, Lem. A.1]. Then clearly f N (λ) := J N (λ + λ0 )d N (λ + λ0 )/(J N (λ0 )d N (λ0 )) is holomorphic in {|λ| ≤ N /4} and satisfies in this disk | f N (λ)| ≤ eC(N +1)

n+3

,

f N (0) = 1,


499

where we used the maximum principle in disks around each −k to estimate the norm there. Thus we may apply Cartan’s estimate for this function in |λ| < N /4: for all α > 0 small enough there exists Cα > 0 such that, outside a family of disks the sum of whose radii is bounded by α N , log | f N (λ)| > −Cα log

sup | f N (λ)|

|λ|≤N /4

and |λ| ≤ N /4. Fixing α sufficiently small, there exists β N ∈ (3/4, 1) so that |d N (λ)| > e−C(N +1)

n+3

N . 4

for |λ − λ0 | = β N

Note that we can also choose β N so that dist(β N N /4, N) > η for some small η uniform with respect to N . Thus the same bound holds for ||(1 + K N (λ, λ0 ))−1 ||L(x N L 2 (X )) using (2.12). Now we need a bound for PN (λ, λ0 ) and it suffices to get back to its definition in the proof of Proposition 3.2 of [19]: it involves operators of the form ι∗ ϕ RHn+1 (λ)ψι∗ for some cut-off functions ψ, ϕ ∈ C ∞ (Hn+1 ) and isometry ι : U ⊂ X → {(x, y) ∈ (0, ∞) × Rn ; x 2 + |y|2 < 1} ⊂ Hn+1 , and operators whose norm is explicitly bounded in [19, Sect. 4] by eC(N +1) in O N ,η . Appendix B of [13] gives an estimate of the same form for ||ϕ RHn+1 (λ)ψ|| as an operator in L(x N L 2 (X ), x −N L 2 (X )) for λ ∈ O N ,η (this is actually a direct consequence of (2.9) and (2.3)), thus we have the bound ||R(λ)||L(x N L 2 (X ),x −N L 2 (X )) ≤ eC(N +1)

3

in {|λ − λ0 | = β N N /4}. Let R N be the set of poles of R(λ) in O N , each pole being repeated according to its order; R N has at most C N n+2 elements so we may multiply R(λ) by E(λ/s, n + 2), FN (λ) := s∈R N

where E(z, p) := (1 − z) exp(z + · · · + p −1 z p ) is the Weierstrass elementary function. It is rather easy to check that for all > 0 small, we have the bounds eC (N +1)

n+3

≥ |FN (λ)| ≥ e−C (N +1)

(n+3)

(2.13)

for some C and for all λ ∈ O N such that dist(λ, R) > . Thus R(λ)FN (λ) is holomorphic in {|λ−λ0 | ≤ β N N /4} and we can use the maximum principle which gives an upper bound ||FN (λ)R(λ)||L(x N L 2 ,x −N L 2 ) ≤ exp(C (N + 1)n+3 ) in {|λ − λ0 | ≤ β N N /4}. We get our conclusion using (2.13), the fact that χi is bounded by eC N as an operator from L 2 to x N L 2 , and the Cauchy formula for the case j > 0 (estimates of the derivatives with respect to λ). Remark. Notice that similar estimates are obtained independently by Borthwick [3]. In the case of surfaces the second author [26] used the powerful estimates developed by Dolgopyat [7] to prove that the Selberg zeta function Z (λ) is analytic and non-vanishing in {Re(λ) > δ − , λ = δ} for some > 0. In higher dimension, the same result holds, as was shown recently by Stoyanov [36].

500


Theorem 2.5 (Naud, Stoyanov). There exists > 0 such that the Selberg zeta function Z (λ) is holomorphic and non-vanishing in {λ ∈ C; Re(λ) > δ − , λ = δ}. Using Proposition 2.1, this result about the zeta function implies that the resolvent R(λ) is holomorphic in a similar set (possibly by taking > 0 smaller). Then an easy consequence of the maximum principle as in [38,2] together with a rough exponential bound for the resolvent allows to get a polynomial bound for ||χ1 R(λ)χ2 || on the {Re(λ) = δ; λ = δ}. Corollary 2.6. There is > 0 such that the resolvent R(λ) is meromorphic in Re(λ) > δ − with the only possible pole the simple pole λ = δ, the residue of which is given by Resλ=δ R(λ) =

( n2

AX uδ ⊗ uδ , − δ + 1)

where u δ is the Patterson generalized eigenfunction of (2.10), A X = 0 a constant. Moreover for all χ1 , χ2 ∈ C0∞ (X ), there exists L ∈ N, C > 0 such that for |λ − δ| > 1 and all j ∈ N0 j

||∂λ χ1 R(λ)χ2 ||L(L 2 (X )) ≤ C(|λ| + 1) L+(n+3) j in {Re(λ) ≥ δ}. Proof. This is a consequence of Proposition 2.2, Proposition 2.3, Theorem 2.5 and the maximum principle as in [2, Prop. 1]. First we remark from Proposition 2.2 and Proposition 2.3 that Pλ has a first order pole with rank one residue at λ = δ and, since |Pλ (m, m )| ≤ |PRe(λ) (m, m )|, we have the estimate ||χ1 R(λ)χ2 ||L(L 2 (X )) ≤ |Re(λ) − δ|−1 C(|λ| + 1) M for Re(λ) ∈ (δ, n/2). This implies by the Cauchy formula that ||∂λ χ1 R(λ)χ2 ||L(L 2 (X )) ≤ |Re(λ) − δ|−1− j C(|λ| + 1) M . j

Let A > 0, and ϕ, ψ ∈ L 2 (X ), we can apply the maximum principle to the function f (λ) = ei A(−i(λ−δ))

n+4

j

∂λ χ1 R(λ)χ2 ϕ, ψ

which is holomorphic in the domain bounded by the curves + : = {δ + u −n−3 + iu; u > 1}, − := {δ − + iu; u > 1}, 0 : = {i + u; δ − 0 large enough | f (λ)| < C(|λ| + 1) L+(n+3) j ||ϕ|| L 2 ||ψ|| L 2 in for some L depending only on M. In particular, applying the same method in the ¯ := {λ; ¯ λ ∈ }, we obtain the polynomial bound ||∂ j χ1 R(λ)χ2 || ≤ symmetric domain λ C(|λ| + 1) L+(n+3) j on {Re(λ) = δ, |Im(λ)| > 1}.


501

3. Width of the Strip with Finitely Many Resonances As stated in Theorem 2.5, we know that there exists a strip {δ − < Re(λ) < δ} with no resonance for g , or equivalently no zero for Selberg zeta function. However the proof of this result does not provide any effective estimate on the width of this strip (i.e. on above). More generally it is of interest to know the following: ρ := inf {s ∈ R; Z (λ) has at most finitely many zeros in {Re(λ) > s}} , or equivalently ρ = inf {s ∈ R; R(λ) has at most finitely many poles in {Re(λ) > s}} . In this work, we give a lower bound for ρ : Theorem 3.1. Let X = \Hn+1 be a convex co-compact hyperbolic manifold and let δ ∈ (0, n) be the Hausdorff dimension of its limit set. Then for all ε > 0, there exist infinitely many resonances in the strip {−nδ − ε < Re(s) < δ}. If moreover is a Schottky group, then there exist infinitely many resonances in the strip {−δ 2 − ε < Re(s) < δ}. Remark. In particular, we have ρ ≥ −δn in general and ρ ≥ −δ 2 for Schottky manifolds. The limit case δ → 0 may be viewed as a cyclic elementary group 0 , and resonances of the Laplace operator on 0 \H2 are given explicitly [18, App.], they form a lattice {−k + iα; k ∈ N0 , ∈ Z} for some α ∈ R, in particular there are infinitely many resonances on the vertical line {Re(s) = 0}. This heuristic consideration suggests that for small values of δ, our result is rather sharp. Proof. The proof is based on the trace formula of [15] and estimates on the distribution of resonances due to Patterson-Perry [30], Guillopé-Lin-Zworski [17] (see also Zworski [40] for dimension 2). To make some computations clearer (Fourier transforms), we will use the spectral parameter z with λ = n2 +i z and Imz > 0 in the non-physical half-plane. We set β := δ if X is Schottky, while β := n if X is not Schottky. We proceed by contradiction and assume that there is ρ = n/2 + βδ + ε for some ε > 0 such that there are at most finitely many resonances in Im(z) < ρ. Let us first recall the trace formula of [15]: as distributions of t ∈ R \ {0}, we have the identity ⎞ ⎛ n ∞ 1 ⎝ i z|t| (γ )e− 2 m(γ ) δ(|t| − m(γ )) e + dk e−k|t| ⎠ = 2 n 2G γ (m) 2 +i z∈R

k∈N

γ ∈P m=1

+

χ ( X¯ ) cosh (2 sinh

t 2 , |t| n+1 2)

(3.1)

where P denotes the set of primitive closed geodesics on X = \Hn+1 , (γ ) stands for the length of γ ∈ P, G γ (m) is defined in (2.2), dk := dim ker Pk if Pk is the kth GJMS conformal Laplacian on the conformal boundary ∂ X¯ , R is the set of resonances of X counted with multiplicity and χ ( X¯ ) denotes the Euler characteristic of X¯ . Next we choose ϕ0 ∈ C0∞ (R) a positive weight supported on [−1, +1] with ϕ0 (0) = 1 and 0 ≤ ϕ0 ≤ 1. We set t −d , ϕα,d (t) = ϕ0 α

502


where d will be a large positive number and α > 0 will be small when compared to d (typically α = e−µd ). Plugging it into the trace formula (3.1) and assuming that d coincides with a large length of a closed geodesic, we get that for d large enough, (γ )e− n2 m(γ ) 2G γ (m)

γ ,m

n

ϕα,d (ml(γ )) ≥ Ce− 2 d ,

with a constant C > 0, whereas the other term can be estimated by αχ ( X¯ )

1 −1

ϕ(t)

n cosh((d + tα)/2) dt = O(α)e− 2 d . (2 sinh(|d + tα|/2))n+1

The key part of the proof is to estimate carefully the spectral side of the formula, i.e. we must examinate ϕα,d (−z) + dk ϕα,d (−z), n 2 +i z∈R

n 2 +i z=−k k∈N0

where ϕ is the usual Fourier transform. Standard formulas for Fourier transform on the Schwartz space show that for all integer M > 0, there exists a constant C M > 0 such that e−dIm(z)+α|Im(z)| ϕα,d (−z) ≤ αC M . (1 + α|z|) M

(3.2)

the set {z ∈ C; n + i z ∈ R ∪ iN}, where each element z is To simplify, we denote by R 2 repeated with the multiplicity

m n/2+i z if z ∈ / iN . m n/2−k + dk if z = ik with k ∈ N

Our assumption now is that {0 ≤ Im(z) ≤ ρ} ∩ R is finite for ρ = n2 + βδ + ε. We set ρ > ρ ≥ 0. The idea is to split the sum over resonances as ϕα,d (−z) = ϕα,d (−z) + ϕα,d (−z) + ϕα,d (−z), z∈R X

n 2 −δ≤Im(z)≤ρ

ρ≤Im(z)≤ρ

ρ≤Im(z)

and estimate their contributions using dimensional and fractal upper bounds. Using (3.2) we can bound the last term (for d large) by

+∞ dN (r ) −ρ(d−α) ϕα,d (−z) ≤ C M αe , (1 + αr ) M ρ ρ≤Im(z)


503

|z| ≤ r }. By [30, Th. 1.10] (see also [15, Lemma 2.3] for where N (r ) = #{z ∈ R; a discussion about the dk terms), we know that N (r ) = O(r n+1 ), thus we can choose M = n + 2 and obtain, after a Stieltjes integration by parts, the following upper bound. ϕα,d (−z) = O(α −n e−ρd ). ρ≤Im(z) Similarly, we have the estimate (for d large and α small)

+∞ (r ) dN −ρ(d−α) ϕα,d (−z) ≤ C M αe , (1 + αr ) M ρ ρ≤Im(z)≤ρ (r ) = #{z ∈ R : ρ ≤ Im(z) ≤ ρ, |z| ≤ r }. This counting function is known where N (r ) = O(r 1+δ ) when X is Schottky [17] (see also to enjoy the “fractal” upper bound N (r ) = O(r 1+β ) where β is defined above. In [40] when n = 1), thus we can write N other words, one obtains by choosing M = n + 2, = O(α −β e−ρd ). ϕ (−z) α,d ρ≤Im(z)≤ρ is finite, and using the fact that Since we have assumed that {0 ≤ Im(z) ≤ ρ} ∩ R resonances (in the z plane) have all imaginary part greater than n2 − δ, we also get n ϕα,d (−z) = O(αe(δ− 2 )d ). n −δ≤Im(z)≤ρ 2

Gathering all estimates, we have obtained as d → +∞, n

n

e− 2 d (C + O(α)) = O(αe(δ− 2 )d ) + O(α −β e−ρd ) + O(α −n e−ρd ), where all the implied constants do not depend on d and α. If we now set α = e−µd , we get a contradiction as d → +∞, provided that ⎧ ⎨ nµ − ρ < − n2 δ < µ ⎩ ρ − βµ > n . 2 The last inequality is satisfied if we set µ := δ + ε/(2β) since ρ = n/2 + βδ + ε by assumption, then we can then choose ρ := nµ + n/2 + ε which is larger than ρ and we have our contradiction for all ε > 0. The proof reveals that any precise knowledge in the asymptotic distribution of resonances in strips has a direct impact on resonances with small imaginary part.

504


4. Wave Asymptotic 4.1. The leading term. Let f, χ ∈ C0∞ (X ), it is sufficient to describe the large time asymptotic of the function 2 sin(t X − n4 ) u(t) := χ f 2 X − n4 and ∂t u(t). We proceed using the same ideas as in [6]. We first recall that from the Stone formula the spectral measure is i n n R( + iv) − R( − iv) dv d(v 2 ) = 2π 2 2 in the sense that for h ∈ C ∞ ([0, ∞)) we have ⎛ ⎞

∞ 2 n h ⎝ X − ⎠ = h(v)d(v 2 )2vdv. 4 0 Since sin is odd, then it is clear that u(t) can be expressed by the integral

∞

1 n n u(t) = eitv χ R( + iv) f − χ R( − iv) f dv 2π −∞ 2 2

(4.1)

which is actually convergent since f ∈ C0∞ (X ) (this is shown below). We want to move the contour of integration into the non-physical sheet {Im(v) > 0} (which correponds with λ = n/2 + iv to {Re(λ) < n/2}) for the part with eitv and into the physical sheet {Im(v) < 0} for the part with e−itv . Let us define the operator L(v) : L 2 (X ) → L 2 (X ) by

n n L(v)ϕ := χ R( + iv)ϕ − χ R( − iv)ϕ 2 2 and let η > 0 be small. We study the following integral for β := n/2 − δ:

I1 (R, η, t) := eitv L(v) f dv, I2 (R, t) := eitv L(v) f dv, |Re(v)|=R 0 0 such that ||∂vj L(v) f || L 2 ≤ C j,N (1 + |v|2 )−N || f || H 2N

n n × max ||∂v χ R( + iv)χ ||L(L 2 ) + ||∂v χ R( − iv)χ ||L(L 2 ) . ≤ j 2 2 To prove the statement about the last limit, it suffices to take j = 0 and N M large enough. Now we get estimates in t for I1 (R, η, t). j

Lemma 4.2. If for all j ∈ N0 , there exists C > 0, M ∈ N such that ||∂v L(v)||L(L 2 ) ≤ C(|v| + 1) M in |Im(v)| ≤ β, then in the L 2 sense, I1 (R, η, t) and ∂t I1 (R, η, t) have a limit as R → ∞, η → 0 and lim lim I1 (R, η, t) = πie−βt Resv=iβ (L(v) f ) + O L 2 (e−βt t −∞ ), t → ∞,

η→0 R→∞

lim lim ∂t I1 (R, η, t) = −πβie−βt Resv=iβ (L(v) f ) + O L 2 (e−βt t −∞ ), t → ∞,

η→0 R→∞

Proof. Let us first consider I1 (R, η, t), it can clearly be written as

−tβ e eitu L(u + iβ) f du. η δ} and has a simple pole at δ with residue Resλ=δ S(λ) = A X

2−2k+1 fδ ⊗ fδ , (k − 1)!

f δ := (x −δ u δ )|x=0 .


509

Note that Perry [32] proved that f δ is well defined and in C ∞ (∂ X¯ ). The functional equation S(λ)S(n − λ) = Id (see for instance Sect. 3 of [12]) and the fact that S(λ) is analytic in {Re(λ) > δ} clearly imply that ker S(λ) = 0 for Re(λ) ∈ (δ, n − δ), thus in particular ker P j = 0 for any j ∈ N with j < n/2 − δ. Moreover, using [30, Lemma 4.16] and the fact that m n/2 = 0 since R(λ) is holomorphic in {Re(λ) > δ}, one obtains S(n/2) = Id thus S(λ) > 0 for all λ ∈ (δ, n − δ) by continuity of S(λ) with respect to λ. We also deduce from the functional equation and the holomorphy of S(s) at n − δ that S(n − δ) f δ = 0. We thus see from this discussion and Proposition 2.3 that, in (5.1), the relation m(δ) = ν(δ) = 1 holds when δ ∈ / n/2 − N while ν(δ) = dim ker Pk when δ = n/2 − k with k ∈ N, since m(δ) = 0 in that case by holomorphy of R(λ) at δ = n/2 − k. To compute dim ker Pk when δ = n/2 − k, one can use for instance Selberg’s zeta function. Indeed by Proposition 2.1 of [32], Z (λ) has a simple zero at δ but it follows from Theorems 1.5-1.6 of Patterson-Perry [30] that Z (λ) has a zero at λ = n/2 − k of order ν(n/2 − k) if k ∈ N, k < n/2, therefore ν(n/2 − k) = 1 and thus dim ker Pk = 1. One can now describe a bit more precisely the function f δ . The Poisson kernel of Proposition 2.3 in the half-space model Rny × R+yn+1 of Hn+1 is P(λ; y, yn+1 , y ) =

2 yn+1

yn+1 , + |y − y |2

thus if x is the boundary defining function used to define S(λ) and if (π∗ x/yn+1 )| yn+1 =0 = k(y) (recall π , π¯ are the projections of (2.1)) for some k(y) ∈ C ∞ (Rn ), so we can describe rather explicitly f δ , we have

π¯ ∗ f δ (y) = k(y)−δ |y − y |−2δ dµ (y ), y ∈ . (5.3) Rn

To summarize the discussion, if δ < n/2, the Patterson function u δ is an eigenfunction for X with eigenvalue δ(n − δ), it is not an L 2 eigenfunction though and it has leading asymptotic behaviour u δ ∼ x δ f δ as x → 0, where f δ ∈ C ∞ (∂ X¯ ) is in the kernel of the boundary operator S(n − λ). When δ ∈ / n/2 − N, this is a resonant state for X with associated resonance δ while when δ ∈ n/2 − N it is still a generalized eigenfunction of X but not a resonant state anymore, and δ is not a resonance yet in that case: the resonance disappears when δ reaches n/2 − k and instead the k th GJMS at ∂ X¯ gains an element in its kernel given by the leading coefficient of u n/2−k in the asymptotic at the boundary. Remark. Notice that the positivity of P j for j < n/2−δ has been proved by Qing-Raske [35] and assuming a positivity of Yamabe invariant of the boundary. Our proof allows to remove the assumption on the Yamabe invariant, which, as we showed, is automatically satisfied if δ < n/2. Acknowledgement. Both authors are supported by ANR grant JC05-52556. C.G ackowledges support of NSF grant DMS0500788, ANR grant JC0546063 and thanks the Math department of ANU (Canberra) where part of this work was done.

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References 1. Bony, J.F., Häfner, D.: Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric. Commun. Math. Phys. 282(3), 697–719 (2008) 2. Bony, J.F., Petkov, V.: Resolvent estimates and local energy decay for hyperbolic equations. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52(2), 233–246 (2006) 3. Borthwick, D.: Upper and lower bounds on resonances for manifolds hyperbolic near infinity. Commun. Part. Diff. Eqs. 33(8), 1507–1539 (2008) 4. Bunke, U., Olbrich, M.: Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. Ann. Math 149, 627–689 (1999) 5. Burq, N., Zworski, M.: Resonances expansion in semi-classical propagation. Commun. Math. Phys. 223(1), 1–12 (2003) 6. Christiansen, T., Zworski, M.: Resonance wave expansions: two hyperbolic examples. Commun. Math. Phys. 212(2), 323–336 (2000) 7. Dolgopyat, D.: On decay of correlations in Anosov flows. Ann. of Math. (2) 147(2), 357–390 (1998) 8. Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.J.: Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc. (2) 46, 557–565 (1992) 9. Fried, D.: The zeta functions of Ruelle and Selberg. Ann. Ecole Norm. 19(Supp. 4), 491–517 (1986) 10. Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. Rend. Circ. Mat. Palermo, Ser.II, Suppl. 63, 31–42 (2000) 11. Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87(2), 186–225 (1991) 12. Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003) 13. Guillarmou, C.: Résonances sur les variétés asymptotiquement hyperboliques. PhD Thesis, 2004 Available online at http://tel.ccsd.cnrs.fr/tel-00006860 14. Guillarmou, C.: Resonances and scattering poles on asymptotically hyperbolic manifolds. Math. Research Letters 12, 103–119 (2005) 15. Guillarmou, C., Naud, F.: Wave 0-Trace and length spectrum on convex co-compact hyperbolic manifolds. Commun.in Analy. and Geom. 14(5), 945–967 (2006) 16. Guillopé, L.: Fonctions Zêta de Selberg et surfaces de géométrie finie. Adv. Stud. Pure Math. 21, 33–70 (1992) 17. Guillopé, L., Lin, K., Zworski, M.: The Selberg zeta function for convex co-compact Schottky groups. Commun. Math. Phys. 245(1), 149–176 (2004) 18. Guillopé, L., Zworski, M.: Upper bounds on the number of resonances for non-compact complete Riemann surfaces. J. Funct. Anal. 129, 364–389 (1995) 19. Guillopé, L., Zworski, M.: Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity. Asymp. Anal. 11, 1–22 (1995) 20. Guillopé, L., Zworski, M.: Scattering asymptotics for Riemann surfaces. Ann. Math. 145, 597–660 (1997) 21. Guillopé, L., Zworski, M.: The wave trace for Riemann surfaces. G.A.F.A. 9, 1156–1168 (1999) 22. Lax, P.D., Phillips, R.S.: Scattering theory. 2nd edition. London: New york: Academic Press, 1989 23. Levin B.Ja.: Distribution of Zeros of Entire Functions. Transl. Math. Monogr. vol 5. Providence, RI: Am. Math. Soc. 1964 24. Joshi, M., Sá Barreto, A.: Inverse scattering on asymptotically hyperbolic manifolds. Acta Math. 184, 41–86 (2000) 25. Mazzeo, R., Melrose, R.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J.Funct.Anal. 75, 260–310 (1987) 26. Naud, F.: Expanding maps on Cantor sets and analytic continuation of zeta functions. Ann. Sci. Ecole Norm. 1(Supp. 38), 116–153 (2005) 27. Naud, F.: Classical and Quantum lifetimes on some non-compact Riemann surfaces, Special issue on “Quantum chaotic scattering”. Journal of Physics A. 49, 10721–10729 (2005) 28. Patterson, S.J.: The limit set of a Fuchsian group. Acta Math 136, 241–273 (1976) 29. Patterson, S.J.: On a lattice-point problem for hyperbolic space and related questions in spectral theory. Arxiv för Math. 26, 167–172 (1988) 30. Patterson, S., Perry, P.: The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein. Duke Math. J. 106, 321–391 (2001) 31. Perry, P.: The Laplace operator on a hyperbolic manifold II, Eisenstein series and the scattering matrix. J. Reine. Angew. Math. 398, 67–91 (1989) 32. Perry, P.: Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. GAFA 11, 132–141 (2001) 33. Perry, P.: A Poisson formula and lower bounds for resonances on hyperbolic manifolds. Int. Math. Res. Notices 34, 1837–1851 (2003)


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Communications in


The Time Slice Axiom in Perturbative Quantum Field Theory on Globally Hyperbolic Spacetimes Bruno Chilian, Klaus Fredenhagen II. Institut für Theoretische Physik, Universität Hamburg, D-22761 Hamburg, Germany. E-mail: [email protected]; [email protected]; [email protected] Received: 29 February 2008 / Accepted: 8 July 2008 Published online: 6 November 2008 – © Springer-Verlag 2008

Abstract: The time slice axiom states that the observables which can be measured within an arbitrarily small time interval suffice to predict all other observables. While well known for free field theories where the validity of the time slice axiom is an immediate consequence of the field equation it was not known whether it also holds in generic interacting theories, the only exception being certain superrenormalizable models in 2 dimensions. In this paper we prove that the time slice axiom holds at least for scalar field theories within formal renormalized perturbation theory. 1. Introduction It is an important feature of hyperbolic differential equations that they typically admit a well posed initial value problem. From the point of view of physics, this allows to predict the future from the present and has an enormous impact, both philosophically and in view of technical applications. It is therefore an important question whether this property remains valid in quantum field theory. In non-interacting theories the field equations are linear and the quantum fields obey the same equations as the classical fields which allow the same conclusions. Note however that a formal proof was not given earlier as in [4] (cf. also [9]. ). Note also that the proof of the well-posedness of the Cauchy problem for classical linear hyperbolic field equations on generic hyperbolic spacetimes was available up to recently only via mimeographed notes of Leray [16] (see [1] for a recent monograph). In the case of quantum field theory the field equation has a somewhat unclear status due to the singular nature of quantum fields. Roughly speaking, the quantum fields are operator valued distributions and their nonlinear functions are notoriously ill defined. In renormalized perturbation theory quantum fields including composite fields can be defined as formal power series. The renormalized field equation, however, does not seem to be sufficient to answer the question of predictability. Since the time zero fields are not always well defined even the formulation of the initial value problem seems to pose

514

B. Chilian, K. Fredenhagen

problems. The situation is better in P(ϕ)2 -models [11] and in the Yukawa2 -model [18] where the time slice axiom follows from the finite speed of propagation. In axiomatic quantum field theory one therefore has weakened the requirement of a well posed initial value problem. Instead one requires that the algebra generated by all fields which can be measured within an arbitrarily small time slice is already the algebra of all observables. It is obvious that a precise formulation of the axiom needs a definition of an “algebra generated by a set of fields”. In algebraic field theory one may associate von Neumann algebras to spacetime regions with compact closure. The time slice axiom then corresponds to the axiom of primitive causality [13] which was first introduced by Haag and Schroer in [14]. In [10], it was shown that the diamond property is equivalent to the time slice axiom for the generalized free field. In perturbative quantum field theory the difficulty of equipping the algebra with a suitable topology was a big obstacle for progress. But while studying the generic features of quantum field theories on curved spacetimes an algebra of functionals of classical fields was discovered [6,15] which carries a natural topology induced by the so-called microlocal spectrum condition [17,3]. This algebra, on the one hand, is an extension of the algebra of canonical commutation relations for the free field, hence the first step in our proof is to show that the time slice axiom holds also on the larger algebra. On the other hand, the algebra contains the time ordered products of all Wick polynomials of the free field and therefore the coefficients of the formal power series defining the interacting fields. We show in the second step of our proof that the property of causal factorization of time ordered products which is the crucial starting point of causal perturbation theory à la Epstein-Glaser [8] implies that the time slice property remains valid for interacting field theories provided it holds true for free field theories. Note that we prove the validity of the axiom within the framework of algebras of observables. The axiom then holds automatically in every Hilbert space representation of the theory. Since on a generic spacetime there is no distinguished representation, and since even in Minkowski space the choice of the vacuum representation of the interacting theory would require a control of the adiabatic limit, this is an enormous advantage compared to the formulation as irreducibility in the vacuum representation (i.e. triviality of the weak commutant) which was chosen in [19]. Notations and conventions: Throughout this text, M denotes a globally hyperbolic spacetime. We denote by V± the union x∈M Vx± ⊂ T ∗ M of all forward or backward lightcones Vx± , respectively. By J± (N ) we denote the causal future / past of any N ⊂ M, i.e. for any point y ∈ J± (N ), there exists a point z ∈ N and a smooth, causal (future / past directed) curve from z to y. By I± , we denote the chronological future resp. past (same definition, except that “causal” is replaced by “timelike”). We call a subset P ⊂ M past compact if J− (x) ∩ P is contained in a compact set for all x ∈ M. 2. The Time Slice Axiom for the Algebra of Wick Polynomials In this section we prove the validity of the time slice axiom for the net W of algebras W(M) of Wick polynomials in the globally hyperbolic spacetimes M. This will set the stage for the treatment of the time slice axiom in perturbatively constructed interacting theories in Sect. 3, since W(M) also contains the time-ordered products which are used in the perturbation series. The technical preliminaries which are neccessary for this construction were given by Brunetti, Fredenhagen and Köhler in [3], and the construction of W(M) was carried out by Dütsch and Fredenhagen in [6] and Hollands and Wald in [15].

Time Slice Axiom in Perturbative Quantum Field Theory on Globally Hyperbolic Spacetimes

515

Let A(M) be the algebra of the free field which is generated by smeared fields ϕ( f ), f ∈ D(M), which satisfy the relations f → ϕ( f ) is linear , ϕ( f )∗ = ϕ( f ) , ϕ(K f ) = 0 ,

(2.1)

[ϕ( f ), ϕ(g)] = i∆( f, g) . Here K = − κR + m 2 denotes the Klein-Gordon operator with the covariant wave operator = ∇µ ∇ µ for the Levi-Civita connection ∇ on M, the coupling κ to the scalar curvature R and the mass m. In the following, K i f , i = 1, . . . , n will denote the application of K to an f ∈ D(M n ) with respect to the i th argument. By ∆ we denote the difference of the uniquely determined advanced and retarded fundamental solutions of K . Now, for the construction of the algebra of Wick polynomials W(M), let ω be a quasifree Hadamard state on A(M) and let ω2 be its two-point distribution. Normally ordered products are defined as the operator-valued distributions δn 1 de f (2.2) : ϕ(x1 ) · · · ϕ(xn ) : = n exp ω2 ( f ⊗ f ) + iϕ( f ) i δ f (x1 ) · · · δ f (xn ) 2 f =0 on the GNS Hilbert space corresponding to ω. The algebra of Wick polynomials W(M) is now defined to consist of smeared normally ordered products, φ ⊗n ( f ) = : ϕ(x1 ) · · · ϕ(xn ) : f (x1 , . . . , xn )d x1 . . . d xn , (2.3) where the space of admissible test distributions f is T n (M) = f ∈ D (M n ) symm. , supp f compact, W F( f ) ∩ V−n ∪ V+n = ∅ . (2.4) By Wick’s theorem, the product of smeared normally ordered products is φ ⊗n ( f )φ ⊗m (g) =

min(n,m)

φ ⊗(n+m−2k) ( f ⊗k g) ,

(2.5)

k=0

with the symmetrized, k-times contracted tensor product ( f ⊗k g)(x1 , . . . , xn+m−2k ) n!m! de f = S dy1 · · · dy2k (n − k!)(m − k)!k! M 2k ω2 (y1 , y2 ) · · · ω2 (y2k−1 , y2k ) f (x1 , . . . , xn−k , y1 , y3 , . . . , y2k−1 )× g(xn−k+1 , . . . , xn+m−2k , y2 , y4 , . . . , y2k ) ,

(2.6)

where S denotes symmetrization in x1 , . . . , xn+m−2k . The condition on the wavefront set of the smearing distributions f guarantees that the elements (2.3) and the product are well defined. Moreover, the expressions (2.6) are again in T n+m−2k (M) [3].

516


At this point, we would like to emphasize the observation made in [15], that the algebraic structure of this construction is independent of the particular Hadamard state ω, and therefore, of the associated GNS-Hilbert space representation. In fact, (2.5) may be de f ∞ n taken as the definition of an associative product on the space T • (M) = n=0 T (M), i.e. f m ⊗k gl . (2.7) ( f g)n = m+l−2k=n

Now Eq. (2.3) may be interpreted as the definition of a representation map π : T • (M) → W(M), f → φ ⊗• ( f ) .

(2.8)

We denote by T K• (M) the ideal in T • (M) which is generated by elements SK 1 f for f ∈ T • (M). Due to the validity of the field equation, we see that T K• (M) is just the kernel of π . Therefore, we have a faithful representation πω of the quotient algebra T • (M)/T K• (M) by elements in W(M): πω : f + T K• (M) → φ ⊗• ( f ) .

(2.9)

It has been found that it is sometimes more convenient to work in the abstract algebra T • (M) than in its realization W(M), especially when treating problems of renormalization. This is referred to as the off-shell formalism, since the validity of the field equation is not enforced in T • (M). However, since the validity of the time slice axiom obviously depends on the field equation, in this paper we work in the on-shell formalism, i.e. in the isomorphic algebras W(M) and T • (M)/T K• (M). The algebra T • (M) carries a natural topology inherited from the Hörmander topologies on the spaces of distributions T n (M) with compact support and the given restrictions on the wave front set [5]. In this topology the space of test functions D(M n ) is sequentially dense in T n (M), the product (2.7) is sequentially continuous and the ideal T K• (M) is sequentially closed. As a consequence, convergence of sequences in W(M) is well defined and A(M) is sequentially dense in W(M). The key ingredient that we need for the first part of our proof is the following proposition which follows from an extension of the argument of [4,9] to the case of Wick polynomials and relies on a slight generalization of the proof of Lemma 4.1 in [15]. Proposition 1. Let f ∈ T n (M). Choose a compact C ⊂ M such that supp f ⊂ C n and let N be a neighborhood of some Cauchy surface such that C is in the future of N . Then there exists a g ∈ T n (M) with the following properties: (i) g = f + i where i ∈ T K• (M) ∩ T n (M), (ii) supp g ⊂ N n . Proof. Choose two Cauchy surfaces Σ0 , Σ1 ⊂ N such that Σ0 is in the past of Σ1 . Let χ ∈ C ∞ (M) be such that

χ (x) =

1

for x ∈ J+ (Σ1 ) ,

0

for x ∈ J− (Σ0 ) .

(2.10)


517

Using the advanced fundamental solution ∆av of the Klein-Gordon equation, we define an operator αi : T n (M) → T n (M) by (αi f )(x1 , . . . , xn ) = f (x1 , . . . , xn ) − K i χ (xi ) (2.11) dy ∆av (xi , y) f (x1 , . . . , xi−1 , y, xi+1 , . . . , xn ) . We now claim that for g = α1 · · · αn f , we have g ∈ T n (M), and that g has the properties (i) and (ii). To prove that g ∈ T n (M), we have to check the symmetry, support and wavefront set of g. The symmetry is obvious from the definition of g. That g has compact support, follows from Proposition 3, together with the compactness of supp f , the support properties of ∆av , and the fact that supp χ is past compact. For the wave front property, it is sufficient to show that W F ∆iav f ∩ V−n ∪ V+n = ∅ for av ∆i f (x1 , . . . , xn ) = dy∆av (xi , y) f (x1 , . . . , xi−1 , y, xi+1 , . . . , xn ), (2.12) since multiplication by a smooth function and application of a differential operator do not enlarge the wavefront set of a distribution. K i is a properly supported differential operator on M n with real principal part ki (X, Ξ ) = gxi (ξi , ξi )

(2.13)

for (X, Ξ ) ∈ T ∗ (M n ), X = (x1 , . . . , xn ), Ξ = (ξ1 , . . . , ξn ) and the metric g on M. By definition, there holds K i ∆iav f = f , so we may apply Hörmander’s theorem on the propagation of singularities [5, Theorem 6.1.1]. Suppose that (X 0 , Ξ0 ) ∈ W F(∆iav f )∩ V−n ∪ V+n . By Hörmander’s theorem, it follows that (X 0 , Ξ0 ) ∈ Char K i , i.e. ki (X 0 , Ξ0 ) = 0. Let γ be the uniquely determined curve in T ∗ (M n ) such that γ (0) = (X 0 , Ξ0 ) and γ (t) = Hk (γ (t)) with the Hamiltonian vector field Hk associated to the Hamiltonian function ki . γ has the form (2.14) γ (t) = ((q1 (t), . . . , qn (t)) , ( p1 (t), . . . , pn (t))) with q j (t), p j (t) ∈ T ∗ (M) ∀ j ∈ {1, . . . , n}, t ∈ R. Following [17, Prop. 4.2] (qi , pi ) is a null geodesic strip in T ∗ (M). Since k(X, Ξ ) depends only on xi and ξi , only the i th component of Hk (X, Ξ ) is nonzero, i.e. q j (t), p j (t) = q j (0), p j (0) ∀t ∈ R for i = j. So we have γ (t) ∈ V−n ∪ V+n ∀t ∈ R. Due to the assumption on W F( f ) we therefore have γ (R) ∩ W F( f ) = ∅. So from Hörmander’s theorem it follows that γ (R) ⊂ W F(∆iav f ) .

(2.15)

n There exists a Cauchy surface Σ with supp ∆iav f ⊂ J− (Σ ) . Since qi is a causal curve in M, J+ (Σ ) ∩ qi (R) = ∅. So (q1 (R), . . . , qn (R)) supp ∆iav f . But this contradicts (2.15), so there must hold W F(∆iav f ) ∩ V−n ∪ V+n = ∅.

(2.16)

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Property (i) is now obvious from the definition of the αi . To prove (ii), we define a family of open subsets Jis (i,s)∈{1,...,n}×{+,−} by

Ji+ = (x1 , . . . , xn ) ∈ M n xi ∈ J+ (Σ1 ) \ Σ1 , (2.17)

J − = (x1 , . . . , xn ) ∈ M n xi ∈ J− (Σ0 ) \ Σ0 . i

n n This s is an open cover of M \ (J− (Σ1 ) ∩ J+ (Σ0 )) so there exists a partition of unity εi (i,s)∈{1,...,n}×{+,−} , such that εis ∈ C ∞ M n \ (J− (Σ1 ) ∩ J+ (Σ0 ))n , εis (x) = 1 ∀x ∈ M n \ (J− (Σ1 ) ∩ J+ (Σ0 ))n , (2.18) i,s

supp εis ⊂ Jis . For Φ ∈ C ∞ (M n ) with supp Φ ∩ (J− (Σ1 ) ∩ J+ (Σ0 ))n = ∅, we have (α1 . . . αn f n )(Φ) = (α1 . . . αn f n )(εi+ Φ) + (α1 . . . αn f n )(εi− Φ) . i

(2.19)

i

Let i ∈ {1, . . . , n}, then (α1 . . . αn f n )(εi+ Φ)

=(α1 . . . αi . . . αn f n )(εi+ Φ) − K i χi (∆iav α1 . . . αi . . . αn f n ) (εi+ Φ) =(α1 . . . αi . . . αn f n )(εi+ Φ) − K i (∆iav α1 . . . αi . . . αn f n ) (εi+ Φ)

(2.20)

=0 , where we used the notation χi (x1 , . . . , xn ) = χ (xi ) and the fact that χi εi+ = εi+ . Now we consider (α1 . . . αn f n )(εi− Φ)

=(α1 . . . αi . . . αn f n )(εi− Φ) − K i ∆iav α1 . . . αi . . . αn f n ) (χi εi− Φ) .

(2.21)

The terms on the right-hand side vanish individually. The first one vanishes, because supp εi− Φ ⊂ M i−1 × J− (Σ0 )× M n−i , but supp α1 . . . αi . . . αn f n ⊂ M i−1 × X × M n−i and X ∩ J− (Σ0 ) = ∅. The second term vanishes, since χi εi− = 0. So all terms in (2.19) vanish. By using the retarded fundamental solution of the Klein-Gordon operator one proves the analogous result for the case when C is in the past of N . Combining both results one obtains the conclusion for every compact set C and every neighbourhood N of a Cauchy surface. As a consequence, we get the validity of the time slice axiom for the algebra W(M) of Wick polynomials: Theorem 2. Let M be a globally hyperbolic spacetime and let N be a neighborhood of some Cauchy surface of M. Then, W(N ) = W(M) . (2.22) k Proof. Let W(M) F(ϕ) = n=0 : ϕ(x1 ) · · · ϕ(xn ) : f n (x1 , . . . , xn )d x1 . . . d xn . Using Proposition 1 for each f n , we can find gn and i n with supp gn ⊂ N n and i n ⊂ TK such that f n = gn + i n . The gn define an element G(ϕ) = kn=0 : ϕ(x1 ) · · · ϕ(xn ) : gn (x1 , . . . , xn )d x1 . . . d xn of W(N ). But because of the validity of the wave equation in W(M), we have F(ϕ) = G(ϕ).


519

3. The Time Slice Axiom for the Interacting Case In causal perturbation theory, the interacting fields are constructed in terms of time ordered products of polynomials of fields. Most easily one describes the time ordered products in terms of their generating functions, the so-called local S-matrices. They are unitary formal power series with coefficients in the algebra of Wick polynomials, and they are functionals S(g) of test functions g ∈ D(M, V ), where V denotes the finite dimensional space of possible interaction terms, including the free field itself. The crucial property of the S-matrix is the causal factorization property S( f + g + h) = S( f + g)S(g)−1 S(g + h)

(3.23)

if the supports of f and h are causally separated in the sense that there is a Cauchy surface such that supp f is in the future and supp h is in the past of the surface. Together with the normalization condition S(0) = 1 this is the only property of perturbation theory which we need in our proof. The proof is therefore also valid beyond perturbation theory once a solution to the causal factorization property has been found. The algebra A0 (O) of the free field associated to the region O is the sequentially closed unital ∗-algebra generated by the elements S(g), where supp g ⊂ O. We assume that this net of algebras satisfies the time slice axiom. (In perturbation theory this follows from the fact that this net of algebras is identical to the net of algebras discussed in the previous section.) Interacting fields for localized interactions parametrized by a spacetime dependent coupling constant g ∈ D(M, V ) can be defined in terms of relative S-matrices Sg ( f ) := S(g)−1 S(g + f ). It is a nice feature of the causal factorization property that the relative S-matrices Sg satisfy also the functional equation (3.23). Therefore another interaction may be added. In particular the original interaction may be compensated so that the free field can be expressed in terms of the interacting fields. This is the starting idea for the extension of the time slice property to the interacting case. But there is an obstacle, namely we want to describe interactions which do not vanish outside of a compact region. They can no longer be described by test functions g with compact support. Instead we have to admit smooth functions g ∈ E(M, V ). Fortunately, due to the functional equation (3.23), it turns out that the structure of the algebras Ag (O), which are taken to be the sequentially closed unital ∗-algebras generated by the relative S-matrices Sg ( f ) with supp f ⊂ O, is independent of the behaviour of g outside of O. This allows to perform the limit to interactions with noncompact support in a purely algebraic way (“algebraic adiabatic limit” [2]). Actually, again due to (3.23) the relative S-matrices Sg can be defined if the support of g is past compact. In this case, we may use Proposition 3 (see the Appendix) to conclude that the past of every compact set K ⊂ M intersects supp g only within a relatively compact region. Consequently, we can find a test function b ∈ D(M, V ) which coincides with g in J− (K ). Then for every f with support in the interior of K we set Sg ( f ) = Sb ( f ) .

(3.24)

The right-hand side does not depend on the choice of b, for let b˜ also satisfy the condition on b, then for c = b˜ − b, supp c does not intersect the past of K . So we have Sb˜ ( f ) = S(b + c)−1 S(b + c)S(b)−1 S(b + f ) = Sb ( f ) .

(3.25)

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Fig. 1. Sketch of the geometrical configuration

We now want to prove the time slice axiom. Let O be a relatively compact region and let Σ be a Cauchy surface. We choose a second Cauchy surface Σ1 such that Σ and O lie in the interior of the future of Σ1 . Now let g be a smooth function with support in the interior of the future of Σ1 . We want to prove that Ag (O) ⊂ Ag (N ) for every open neighbourhood N of Σ . We already know by construction that Ag (O) ⊂ A0 (M) and that by the time slice property for the free field A0 (M) = A0 (N ) for every open neighbourhood of Σ. Let N ⊂ N be an open neighbourhood of Σ whose past boundary is again a Cauchy surface (see Fig. 1). We choose a function g with supp g ⊂ N which coincides with g on N . We then construct the relative S-matrices Sg,g ( f ) = Sg (−b )−1 Sg (−b + f ) with a test function b with support in N and which coincides with g on J− (K ), where supp f is contained in the interior of K and K ⊂ N . Again, the right hand side is independent of the choice of b . Inserting the definition of Sg we obtain Sg,g ( f ) = S(b − b )−1 S(b − b + f ), where b coincides with g on J− (L) and where L is a compact region containing supp b in its interior. Now we may split b − b = b+ + b− such that supp b+ does not intersect the past and supp b− not the future of supp f , hence the second factor factorizes, and we obtain Sg,g ( f ) = S(b− )−1 S( f )S(b− ) . We draw two conclusions: First we see that for every A ∈ A0 (O1 ) with O1 relatively compact and with closure K ⊂ N we have S(b− )−1 AS(b− ) ∈ Ag (N ),

(3.26)

provided b and b satisfy the conditions above. Moreover, due to the validity of the time slice axiom for the free theory, we have ˜ S(b− )−1 AS(b− ) ∈ A0 (O),

(3.27)

with O˜ = N ∩ J+ (J− (K ) ∩ N ), where N is an open neighbourhood of Σ whose closure is contained in N . We see that the map S( f ) → Sg,g ( f ) , supp f ⊂ N

(3.28)


521

extends to an endomorphism α of A0 (M) whose image is contained in Ag (N ). Moreover, for each relatively compact region O2 there is an invertible element U ∈ A0 (M) such that α(A) = U AU −1 for all A ∈ A0 (O2 ). In particular, α is injective. It remains to show that α is surjective. Since g − g vanishes within N , we may decompose it into g − g = g+ + g− , where g+ has support in the future and g− in the past of N . For f with supp f ⊂ N we have α(S( f )) = Sg− ( f ). It amounts to the statement that an interaction in the past causes an endomorphism of the algebra of observables which can be approximated by inner automorphisms αb implemented by invertible elements S(b− ), where b− coincides with g− in the past of a sufficiently large compact subregion of Σ. We may now choose b− such that it coincides with g− also in the future of J− (K ) ∩ S, where S = J− (Σ1 ) ∩ J+ (Σ2 ), and Σ2 is a Cauchy surface in the past of Σ1 . By the time slice axiom for the algebra of the free field, we have A0 (K ) ⊂ A0 (J− (K ) ∩ S). But A0 (J− (K ) ∩ S) is generated by elements S(h), where h ∈ D(M, V ) with supp h ⊂ J− (K ) ∩ S. For each such h, supp h is in the past of Σ1 and supp b− in the future, so αb−1 (S(h)) = S(b− )S(h)S(b− )−1 = S(b− + h)S(b− )−1 − does not depend on the choice of b− by an argument analogous to the one used in (3.25). So the inverse of α exists on A0 (J− (K ) ∩ S) ⊃ A0 (K ) for all compact regions K ⊂ N and hence everywhere. 4. Appendix The following simple proposition is used in two separate arguments of our proof, in Sects. 2 and 3, respectively. Proposition 3. Let M be a globally hyperbolic spacetime, P ⊂ M past compact and K ⊂ M compact. Then J− (K ) ∩ P is contained in a compact set. Proof. Choose a Cauchy surface Σ in the future of K . The family (I− (y)) y∈Σ is an open cover of K , since I± (z) is open for any z ∈ M. Since K is compact, there exists a finite subset Y = {y1 , . . . , yn } ⊂ Σ such that (I− (y)) y∈Y is an open cover of K . Let x be any point in I− (K ). Then, x ∈ I− (yi ) for some i ∈ {1, . . . , n}. (This is true because there exists a timelike, future directed curve γ , from x to some k ∈ K . Since k ∈ I− (yi ) for some i, γ can be extended timelike to yi .) So we have that I− (y) ⊂ J− (y) . (4.29) I− (K ) ⊂ y∈Y

y∈Y

Since J± (S) ⊂ I± (S) for any S ⊂ M, and since J− (z) is closed for any z ∈ M, J− (y) = J− (y) = J− (y) . (4.30) J− (K ) ⊂ I− (K ) ⊂ y∈Y

Intersecting with P, we get

⎛

J− (K ) ∩ P ⊂ ⎝

y∈Y

y∈Y

⎞ J− (y)⎠ ∩ P ⊂

y∈Y

(J− (y) ∩ P) ,

(4.31)

y∈Y

where the right-hand side is by assumption contained in a finite union of compact sets.

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References 1. Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. Providence, RI: Amer. Math. Soc., 2007 2. Brunetti, R., Fredenhagen, K.: Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds. Commun. Math. Phys. 208, 623 (2000) 3. Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996) 4. Dimock, J.: Algebras of Local Observables on a Manifold. Commun. Math. Phys. 77, 219–228 (1980) 5. Duistermaat, J.J., Hörmander, L.: Fourier Integral Operators II. Acta Math. 128, 183–269 (1972) 6. Dütsch, M., Fredenhagen, K.: Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion. Commun. Math. Phys. 219, 5 (2001) 7. Dütsch, M., Fredenhagen, K.: Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity. Rev. Math. Phys. 16, 1291–1348 (2004) 8. Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann Inst. H. Poincaré A 19, 211 (1973) 9. Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved space-time. II. Ann. Phys. (NY) 136, 243–272 (1981) 10. Garber, W.-D.: The Connexion of Duality and Causal Properties for Generalized Free Fields. Commun. Math. Phys. 42, 195–208 (1975) 11. Glimm, J., Jaffe, A.: A λϕ 4 Quantum Field Theory without Cutoffs I. Phys. Rev. 176, 1945–1951 (1968) 12. Haag, R.: Local Quantum Physics: Fields, particles and algebras. Berlin: Springer-Verlag, 2nd ed., 1996 13. Haag, R., Kastler, D.: An algebraic approach to field theory. J. Math. Phys. 5, 848 (1964) 14. Haag, R., Schroer, B.: Postulates of quantum field theory. J. Math. Phys. 3, 248–256 (1962) 15. Hollands, S., Wald, R.M.: Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys. 223, 289–326 (2001) 16. Leray, J.: Hyperbolic Differential Equations. Lecture Notes, Princeton, N.J.: Institute for Advanced Study, 1963 17. Radzikowski, M.J.: Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time. Commun. Math. Phys. 179, 529–553 (1996) 18. Schrader, R.: Yukawa Quantum Field Theory in Two Space-Time Dimensions Without Cutoffs. Ann. Phys. 70, 412–457 (1972) 19. Streater, R.F., Wightman, A.S.: PCT Spin and Statistics, and All That Reading, MA: Addison Wesley Longman Publishing Co, 1989 Communicated by M. Salmhofer


Communications in


Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors Romeo Brunetti1 , Giuseppe Ruzzi2 1 Dipartimento di Matematica, Università di Trento, Via Sommarive 14,

I-38050 Povo (TN), Italy. E-mail: [email protected]

2 Dipartimento di Matematica, Università di Roma “Tor Vergata,”

Via della Ricerca Scientifica, I-00133 Roma, Italy. E-mail: [email protected] Received: 7 March 2008 / Accepted: 27 July 2008 Published online: 6 November 2008 – © Springer-Verlag 2008

Dedicated to Klaus Fredenhagen on the occasion of his sixtieth birthday Abstract: A new form of superselection sectors of topological origin is developed. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C∗ –algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts’ cohomological analysis to the case where 1–cocycles bear non-trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in the case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1–cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I . All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1–cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part much resembles what in literature is known as geometric phases. Indeed, by the very geometrical origin of the 1–cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations. Contents 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Category of Net Representations . . . . . . . . . . . . . . . Charged Sectors Induced by Topology . . . . . . . . . . . . . . Net Cohomology and the Localization of the Fundamental Group Charge Structure . . . . . . . . . . . . . . . . . . . . . . . . . . The Topological Content . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction A large class of effects in physics can be explained using the features and language of topology. Starting from those due to non-trivial topology of configuration spaces in classical and quantum physics, one also finds intriguing topological effects in general relativity and quantum field theory. To name just a few of the most interesting ones we recall the Ehrenberg-Siday-Aharonov-Bohm effect [1,21] in quantum mechanics and its generalization named Pancharatnam-Berry phase [7,43], or its classical counterpart called Hannay’s angle [29], the Jahn-Teller effect [33], Wheeler’s geons [58] and the Casimir effects [15]. The aim of the present paper is to trace the route for a rigorous attack to the dependence on topology of certain structures in quantum field theory. In particular, we aim at a complete model independent description, at least in a preliminary case where the topological effects of interest are those related to specific topological features of spacetimes. The language is that of local quantum theory [9,28] (otherwise called algebraic quantum field theory) where, as it is well known, one finds the best understanding about the nature and properties of structural properties of quantum field theories. The proper setting is that related to the prominent case of superselection sectors where charged quantum numbers find their definition as attributes associated to unitary equivalence classes of representations of the net of local observable algebras satisfying certain selection conditions. The traditional analysis of such selection criteria – and associated equivalence classes of representations thereof – is associated mainly with the names of Doplicher, Haag, and Roberts [18]. They worked out the structure of charges localizable into bounded regions, whilst the study of charges that can be localized in unbounded regions, i.e. spacelike cones is due to Buchholz and Fredenhagen [13]. All that was done for quantum field theories on Minkowski spacetime in dimensions d ≥ 3. Other groups of researchers have been able to follow the same route in various directions, especially in the direction of conformal quantum field theory in two dimensions, and besides the crucial results of Fredenhagen, Rehren and Schroer [24], the main success was obtained by Kawahigashi and Longo in [34], where they have been able to completely classify theories with central charge less than one. The authors of the present paper have recently put forward an analysis of the structure of superselection sectors [12] that provides a new perspective both by the adopted techniques and in the fact that superselection theory is now applicable to the larger setting of locally covariant quantum field theories [11]. The obtained results confirm that sectors of the kind that Doplicher, Haag and Roberts studied long ago find their most natural position in the locally covariant framework. In fact, we can associate with any 4-dimensional globally hyperbolic spacetime a unique, symmetric, tensorial C∗ –category (that possesses conjugates in case of finite statistics) and that to any isometric embedding between such spacetimes the previous categories can be contravariantly related so as to guarantee that charges are preserved under the embedding.

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The local covariance of sectors comes from the analysis of 1–cocycles associated with the Roberts’ cohomology of posets [45–48,53] that carry a trivial unitary representation of the fundamental group of the spacetime. It is natural to try to understand the kind of 1–cocycles that carry a non-trivial unitary representation and to see whether one can associate with them a different kind of superselection sector and charge, now attributable to the possible non-trivial topology of the spacetime. The main results of the following analysis show that this is indeed possible and fruitful. We have now a clear relation between topological properties of a spacetime and structural properties of 1–cocycles carrying non-trivial representations of the fundamental group of the spacetime. The analysis that follows, however, is not yet cast into a locally covariant form, although our initial aim was in that direction. We will be working on a fixed, but otherwise arbitrary, 4-dimensional globally hyperbolic spacetime. We hope to return elsewhere to the locally covariant analysis, and consider this paper as the third one of the announced series in [12]. The main ingredients are the following: first, a generalization of the usual notion of representations of a net of local algebras, something that we termed “unitary net representations;” secondly, the association of this new notion with a 1–cocycle, in the sense of Roberts’ cohomology of posets; thirdly, a new selection criterion that generalizes that of Doplicher, Haag and Roberts, for the sake of attributing a non-trivial dependence on the spacetime topology to the superselection sectors so defined; fourthly, one defines a von Neumann algebra which is the group algebra generated by a 1–cocycle evaluated on all loops over a fixed bounded region, as a starting and ending point, and proves that this algebra is localized, i.e. it is a subalgebra of the von Neumann algebra of the net that is localized in the chosen region. This last ingredient is the key structural element of the analysis that follows. It allows us to attribute to each 1–cocycle generating its own group von Neumann algebra a new invariant, called “topological dimension,” that resembles much, and has similar properties of a charge quantum number, and carries non-trivial information about the topology of the spacetime. Furthermore, we prove that any 1–cocycle can be split into a part that carries only information on the charge content of the sector and a part that carries the topological information of spacetime. This last resembles much of what we cited at the beginning as geometric phases. For more on that see also Sect. 7, where we also prove the existence of net representations. Abstract as they are, one would like to have concrete examples of constructions of this non-Abelian kind of geometric phases. A recent work, done by one of the authors in collaboration with Franceschini and Moretti [8], provides a first explicit example of a 1–cocycle induced by the non-trivial topology of spacetime in the simple case of a massive bosonic quantum field theory on the 2-dimensional Einstein cylinder. A further glance at models may indicate other situations where our analysis may apply. For instance, in cosmology one looks for visible effects of the non-trivial topology of spacetimes by searching for additional images in the sky of the same galaxy, due to the possible presence of a cosmic string. Besides that, we mention also that there is a large class of physically meaningful multiply connected spacetimes. These spacetimes are a class of Friedmann-Lamaître models, solutions of the Einstein equations which are used as cosmological models (see [36,55]). We finally point the reader to a recent interesting paper by Morchio and Strocchi [40] where, in the case of quantum mechanics on manifolds, they describe a classification of topological effects in close analogy to our results. Also, papers by Döbner et al. [17] and Landsman [37], have a similar flavour.

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The paper has been structured in such a way to mantain a decent ratio between size and completeness, hence many results are presented in the simplest form that we could think of. We refer the reader to [12,19,38,53] for a deeper introduction to some of the mathematical notions that we use.

2. The Category of Net Representations We introduce the notion of net representations for nets of C∗ –algebras. We analyze in particular the class of unitary net representations pointing out their topological content. The importance of this new notion resides in the fact that a new class of superselection sectors induced by the topology of spacetimes is described in terms of net representations (see the next sections). We shall use the tool of cohomology of posets to make explicit the topological information carried by net representations. Within this section we shall also discuss, very briefly, preliminary information on the simplicial set associated with a poset and the first degree of its cohomology. Details can be found in [12,48,50,53]. Let K be a poset with order relation ≤. We consider the simplicial set Σ∗ (K ) of singular simplices associated with K . We use the standard symbols ∂i and σi to denote the face and degeneracy maps, and denote the compositions ∂i ∂ j , σi σ j respectively by ∂i j and σi j . We pass now to a brief definition of the set Σn (K ) of n–simplices. A 0–simplex is just an element of K . Inductively, for n ≥ 1, an n–simplex x is formed by n + 1, (n − 1)–simplices ∂0 x, . . . , ∂n x and by an element of the poset |x|, called the support of x, such that |∂i x| ≤ |x| for i = 0, . . . , n. We shall denote 0–simplices either by a or by o, 1–simplices by b, and 2–simplices by c. Given a 1–simplex b the reverse b is the 1–simplex having the same support as b and such that ∂0 b = ∂1 b, ∂1 b = ∂0 b. Composing 1–simplices one gets paths. A path p is a finite ordered set of 1–simplices bn ∗ · · · ∗ b1 satisfying the relations ∂0 bi−1 = ∂1 bi for i = 2, . . . , n. We define the . . 0–simplices ∂1 p = ∂1 b1 and ∂0 p = ∂0 bn and call them, respectively, the starting and the ending point of p. By p : a → a˜ we mean a path starting from a and ending at a. ˜ . The reverse of p is the path p : a˜ → a defined by p = b1 ∗ · · · ∗ bn . If q is a path from a˜ to a, ˆ then we can define, in an obvious way, the composition q ∗ p : a → a. ˆ The poset K is said to be pathwise connected whenever for any pair a,a˜ of 0–simplices there is a path from a to a. ˜ Let p = bn ∗ · · · ∗ b1 be a path. An elementary deformation of p consists in replacing a 1–simplex ∂1 c of the path by the pair ∂0 c ∗ ∂2 c, where c ∈ Σ2 (K ), or conversely in replacing a consecutive pair ∂0 c∗∂2 c by a single 1–simplex ∂1 c. Two paths with the same endpoints are homotopic if they can be obtained from one another by a finite sequence of elementary deformations. Homotopy defines an equivalence relation ∼ on the set of paths with the same endpoints which is compatible with reverse and composition. The first homotopy group of the poset π1 (K , a), with base point a, is the quotient of the set Loops K (a) of all paths p : a → a in K with respect to the homotopy equivalence relation. If K is pathwise connected the first homotopy group does not depend, up to isomorphism, on the base point. The isomorphism class is the fundamental group π1 (K ) of the poset and we will say that K is simply connected whenever the fundamental group is trivial. We conclude this introductory part with two remarks. First, we recall that if K is upward directed, namely if for any a1 , a2 ∈ K there is a3 ∈ K with a1 , a2 ≤ a3 , then it is pathwise and simply connected. Secondly, let M be a topological space and consider a basis of its topology formed by open arcwise and simply-connected open sets of M.


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If K is a poset formed by the elements of this basis with the inclusion order relation ⊆, then π1 (M) = π1 (K ). We now turn to the definition of net representations. From now on we fix a poset K and assume that it is pathwise connected. A net of C∗ –algebras A K on a poset K is given by the following data: there is mapping a → A(a) from K to unital C∗ –algebras; for any pair a, ˜ a ∈ K with a˜ ≤ a, there is an injective ∗–morphism ja a˜ : A(a) ˜ → A(a). The morphisms ja a˜ are called inclusion morphisms. These morphisms are required to satisfy the following coherence property: ja a˜ ja˜ aˆ = ja aˆ ,

aˆ ≤ a˜ ≤ a.

(2.1)

A net representation of A K is a pair {π, ψ}, where π denotes a function that associates a representation πa of A(a) on a Hilbert space Haπ with any a ∈ K ; ψ denotes a function that associates an injective linear operator ψa a˜ : Haπ˜ → Haπ with any pair a, a˜ ∈ K , with a˜ ≤ a. The functions π and ψ are required to satisfy the following relations: ψa a˜ πa˜ = πa ja a˜ ψa a˜ , a˜ ≤ a, and ψa a˜ ψa˜ aˆ = ψa aˆ , aˆ ≤ a˜ ≤ a.

(2.2)

An intertwiner from {π, ψ} to {ρ, φ} is a function T associating a bounded linear operator ρ Ta : Haπ → Ha with any a ∈ K , and satisfying the relations Ta πa = ρa Ta , and Ta ψa a˜ = φa a˜ Ta˜ , a˜ ≤ a.

(2.3)

We denote the set of intertwiners from {π, ψ} to {ρ, φ} by ({π, ψ} , {ρ, φ}), and say that the net representations are unitarily equivalent if they have a unitary intertwiner T , that is, Ta is a unitary operator for any a. The definition of net representation is suggested by the theory of bundles over posets [50]. There is, in fact, an underlying structure of a net bundle and, as we shall point out, some results on net representations are analogous to those of net bundles. The contact point resides in the defining properties of the function ψ, which are the same as the net structure of a net bundle. However, net representations already appeared in the literature of algebraic quantum field theory, although not in this general form. They have been considered by Buchholz, Haag and Roberts in an unpublished paper.1 Fredenhagen and Haag encountered this class of representations in the reconstruction of a theory from its germs [23]. An argument used in that paper allows us to show how net representations arise. We call a net state a function ω associating a state ωa of A(a) with any a ∈ K , and which is compatible with the inclusion morphisms, i.e., ωa ja a˜ = ωa˜ ,

a˜ ≤ a.

(2.4)

Given a net state ω, denote the GNS–construction of ωa by {πa , Ha , Ωa }, and define . ψa a˜ πa˜ (A) Ωa˜ = πa (ja a˜ (A)) Ωa , a˜ ≤ a. (2.5) By using (2.4), we have that ψa a˜ : Ha˜ → Ha is an isometry. Furthermore, it is a routine calculation to check that ψa a˜ πa˜ = πa ja a˜ ψa a˜ for any a˜ ≤ a. Finally observe that, by the defining equation of ψ and by (2.1) we have ψa a˜ ψa˜ aˆ πaˆ (A) Ωaˆ = ψa a˜ πa˜ (ja a˜ (A)) Ωa˜ = πa (ja a˜ ja˜ aˆ (A)) Ωa = πa (ja aˆ (A)) Ωa = ψa aˆ πaˆ (A) Ωaˆ 1 Private communication by J. E. Roberts.

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for any aˆ ≤ a˜ ≤ a and A ∈ A(a), ˆ and this implies that ψa a˜ ψa˜ aˆ = ψa aˆ . So the pair {π, ψ} is a net representation. In the present paper we are interested in unitary net representations. A net representation {π, ψ} is said to be unitary whenever ψa a˜ is a unitary operator for any a˜ ≤ a. An interesting feature is that, since K is pathwise connected, any unitary net representation is equivalent to a unitary net representation on a fixed Hilbert space. The argument is the same as that used to prove that any net bundle has a standard fibre (see [50, Prop. 4.5]). Since the operators ψa a˜ are unitary, by a standard argument, one defines a unitary operator Va from a fixed Hilbert space H to Haπ for any a ∈ K . Afterwards one defines . . φa a˜ = Va∗ ψa a˜ Va˜ , a˜ ≤ a. ρa = Va∗ πa Va , a ∈ K , The pair {ρ, ψ} is a unitary net representation on the Hilbert space H; the function V : K a → Va defines a unitary intertwiner from {ρ, φ} to {π, ψ}. From now on we will consider only unitary net representations in a fixed Hilbert space. We denote by Repnet (A) the set of unitary net representations of A and by the same symbol the category having unitary net representations of A as objects and the corresponding intertwiners as arrows. We call this one the category of unitary net representations of A. If the target of an arrow T is equal to the source of S, the composition S · T is defined by (S · T )a = Sa Ta for any a ∈ K . The identity arrow 1{π,ψ} is (1{π,ψ} )a = 1Hπ for any a. Furthermore, Repnet (A) is a C∗ –category. The adjoint ∗ is defined as the identity on objects, while on arrows T it is defined as (T ∗ )a = Ta∗ for any a. Finally, given an . arrow T , then T = supa∈K Ta is a norm which makes Repnet (A) a C∗ –category. Note that by (2.3), since K is pathwise connected, Ta is constant. We now use the cohomology of posets to make explicit the topological content of unitary net representations. We give only a brief introduction of this topic and refer the reader to the papers quoted at the beginning for details. Consider a Hilbert space H. A 1–cocycle z of the poset K , with values in B(H), is a function z : Σ1 (K ) b → z(b) ∈ B(H) of unitary operators of H satisfying the equation z(∂0 c) z(∂2 c) = z(∂1 c),

c ∈ Σ2 (K ).

(2.6)

The trivial 1–cocycle ı is defined by ı(b) = 1H for any 1–simplex b. A 1–cocycle is a 1–coboundary if there is a function v : Σ0 (K ) a → va ∈ B(H) of unitary operators such that z(b) = v∂∗0 b v∂1 b for any 1–simplex b. We denote the set of 1–cocycles by Z1 (K , B(H)). Given a pair z, z˜ of 1–cocycles an intertwiner t from z to z˜ is a function t : Σ0 (K ) a → ta ∈ B(H) satisfying t∂0 b z(b) = z˜ (b) t∂1 b ,

b ∈ Σ1 (K ).

(2.7)

We denote the set of intertwiners from z to z˜ by (z, z˜ ). The category of 1–cocycles is the category whose objects are 1–cocycles and whose arrows are the corresponding set of intertwiners. We denote this category by the same symbol Z1 (K , B(H)) as that used to denote the corresponding set of objects. This is a C∗ −category: composition of arrows and the adjoint are defined in the same way as in Repnet (A K ) (see [47,53] for details). Two 1–cocycles z, z˜ are unitarily equivalent if there exists a unitary arrow t ∈ (z, z˜ ). Observe that any 1–coboundary is unitarily equivalent to the trivial 1–cocycle ı. We extend a 1–cocycle z from 1–simplices to paths by setting . z( p) = z(bn ) · · · z(b2 ) z(b1 ), p = bn ∗ · · · ∗ b1 .


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It is easily seen that z( p) = z( p)∗ for any path p, and if p and q are homotopic then z( p) = z(q) (homotopic invariance). These properties imply that any 1–cocycle defines a unitary representation, denoted by z, in H of the fundamental group of the poset. Using this result the topological content of a unitary net representation is easily analyzed. Indeed, given a unitary net representation {π, ψ}, define . ∗ ζ π (b) = ψ|b|,∂ ψ|b|,∂1 b , b ∈ Σ1 (K ). (2.8) 0b One can check that ζ π is a 1–cocycle of K with values in the group of unitary operators of Hπ ([50]). This 1–cocycle defines a representation of the fundamental group of the poset. Thus, we say that a net representation {π, φ} is topologically trivial whenever ζ π is a 1–coboundary. Thus, if K is simply connected then any unitary net representation is topologically trivial. Lemma 1. Assume that {π, ψ} and {ρ, φ} are unitarily equivalent. Then the corresponding 1–cocycles ζ π and ζ ρ are equivalent. Proof. Let W ∈ ({π, ψ}, {ρ, φ}) be unitary. Then ∗ ψ|b|,∂1 b = (ψ|b|,∂0 b W∂∗0 b )∗ ψ|b|,∂1 b W∂0 b ζ π (b) = W∂0 b ψ|b|,∂ 0b ∗ ∗ = (W|b| φ|b|,∂0 b )∗ ψ|b|,∂1 b = φ|b|,∂ W|b| ψ|b|,∂1 b 0b

= ζ ρ (b) W∂1 b .

Hence W ∈ (ζ π , ζ ρ ) and this proves the assertion. Thus, equivalent unitary net representations have the same topological content (the converse, in general, does not hold as we will see at the end of this section). Lemma 2. Let {π, ψ} be a topologically trivial unitary net representation. Then {π, ψ} is equivalent to a unitary net representation of the form {ρ, 1}. Proof. Since ζ π is a 1–coboundary there exists a family of unitary operators Wa : H → H . such that ζ π (b) = W∂0 b ∗ W∂1 b . For any 0–simplex a, define ρa (A) = Wa πa (A) Wa ∗ , with A ∈ A(a). It is clear that Wa πa = ρa Wa . Moreover, Wa ψa,a˜ = Wa ζ π (a; a, a) ˜ = Wa˜ , where (a; a, a) ˜ is the 1–simplex whose support is a, and whose 0– and 1–face are respectively a and a. ˜ This completes the proof. A unitary net representation can be easily defined starting from a representation χ of the fundamental group of K with values in the complex numbers C. It is shown in [53] that there is a 1–cocycle, associated with χ . We maintain the symbol χ to denote this 1–cocycle and define (see the previous proof for notation) χ . ψaa ˜ a, ˜ a), a ≤ a. ˜ (2.9) ˜ = χ (a; Consider now a topologically trivial net representation {π, 1}. Since ψ χ takes values in the complex numbers, the pair {π, ψ χ } is a unitary net representation (note that {π, ψ χ } and {π, 1} are not equivalent because of Lemma 1). So, if the fundamental group of the poset is Abelian, then there are topologically non-trivial unitary net representations (clearly if the net is not trivial). For the non-Abelian case, topologically non-trivial examples, which are of interest for the theory of superselection sectors, will be given in Sect. 7. Finally, note that the above example shows that there are non equivalent unitary net representations whose 1–cocycles are equivalent. We conclude this section with some observations.

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(1) A net is nothing but a precosheaf. By reversing the arrows, the results of this section apply also to the duals, i.e. to presheaves, either of C∗ –algebras or of groups. Briefly, given a presheaf over a poset, if we have a presheaf representation on a Hilbert space whose restriction morphisms are implemented by unitary operators satisfying relations corresponding to (2.2), then the presheaf representation carries a representation of the fundamental group of the poset. Proceeding as is done for net representations, one gets a 1–cocycle of the dual poset K ◦ (the poset having the same elements as K with opposite order relation). However, as shown in [50], the fundamental group of K is isomorphic to that of K ◦ . (2) The notion of representation of a net of C∗ –algebras, usually considered in the applications to quantum fields theory (see for instance [12,13,18,26,30,53]), corresponds, in our framework, to a topologically trivial net representation. To see this note that, in the cited papers, by a representation of a net A K it is meant a function π associating to any a ∈ K a representation of A(a) in a fixed Hilbert space Hπ , and such that πa ja a˜ = πa˜ for any a˜ ≤ a. An intertwiner S from a representation π to a representation ρ, is a bounded linear operator S : Hπ → Hρ such that Sπa = ρa S for any a ∈ K . Now, it is clear that a representation is a unitary net representation of the form {π, 1}. Moreover, if T is an intertwiner from {π, 1} to {ρ, 1}, then, according to (2.3), we have Ta = Ta˜ for any a˜ ≤ a. Since K is pathwise connected, we have Ta = Taˆ for any pair a, a. ˆ So T is constant. This shows that our definition is indeed a generalization, and in particular that the category of representations is equivalent to the full subcategory of Repnet (A K ) whose net representations are topologically trivial. We will denote this category by Repnet t (A K ). (3) Carpi, Longo and Kawahigashi have considered representations of net over the covering spaces of S 1 [14]. We think that unitary net representations are of the same nature as those considered by them. We prefer however not to explore this topic in the present paper. 3. Charged Sectors Induced by Topology In [18], Doplicher, Haag and Roberts were able to select a class of superselection sectors of the observable net which manifest a covariant charge structure. These sectors, known as DHR–sectors, are representations of the observable net, in Minkowski spacetime, which are sharp excitations of the vacuum representation. This feature has been used to extend the notion of a DHR–sector to curved spacetimes [26]. In 4–dimensional globally hyperbolic spacetimes, DHR–sectors have a charge structure [26,48,53], which is generally covariant [12]. It is now clear that these sectors are not induced by the topology of spacetimes since they are associated with representations of the observable net which are, in the terminology introduced in the present paper, topologically trivial. In this section, by taking into account unitary net representations, we try to check whether they lead to genuine superselection structures. We start by discussing aspects of the causal structure of globally hyperbolic spacetimes, introduce the observable net, and the reference unitary net representation of the theory. The definition of sharply localized unitary net representations concludes the section.

3.1. The observable net. We start by discussing some aspects of the causal structure of globally hyperbolic spacetimes. The focus is on the set of diamonds, the class of regions that we will use as indices of the observable net. Standard properties of globally


531

hyperbolic spacetimes can be found [4,22,42,54,57]. Some advanced aspects can be found in [5,6,39]. Consider a 4–dimensional connected globally hyperbolic spacetime M. The causal disjointness relation is a symmetric binary relation ⊥ defined on subsets of M as follows: o ⊥ o˜

⇐⇒

o˜ ⊆ M \ J (o),

(3.1)

where J denotes the causal set of o. The causal complement of a set o is the open set . o⊥ = M \ cl(J (o)). An open set o is causally complete whenever o = o⊥⊥ . Now, the observable net over the spacetime M is a correspondence from open sets of the spacetime to the observables localized within these regions. In general not all the open sets are suited for this scope, since one needs a family of sets which fits very well both the topological and the causal properties of M. Moreover, additional conditions are imposed by the study of the observable nets derived from models of quantum fields [52,56]. A family of sets that satisfies all these requirements is the set K (M) of diamonds of M [53]. A diamond of M is a subset o of M such that there is a spacelike Cauchy surface C, a chart (U, φ) of C, and an open ball B of R3 such that o = D(φ −1 (B)), cl(B) ⊂ φ(U ) ⊂ R3 ,

(3.2)

where D(φ −1 (B)) is the domain of dependence of φ −1 (B), and such that cl(o) is compact. We will say that o is based on C and call φ −1 (B) the base of o. It turns out that a diamond is an open, relatively compact, connected and simply connected subset of M. Any diamond o is causally complete, and the causal complement o⊥ is connected. The set of diamonds K (M) of M is a base for the topology of M. Some technical properties of diamonds are shown in Appendix B. Notice that our present definition of diamonds differs, by the requirement of compactness of the closure, from the original one in [53]. The results provided there and in [12] do not change after restriction to this smaller class. We call a subspacetime of M any globally hyperbolic open connected subset of M. Diamonds and their causal complements are examples of subspacetimes of M. Another example is the causal puncture x ⊥ in a point x ∈ M. This is nothing but the causal complement of the point x. Now, it is an easy consequence [12] of a powerful result on the deformation of Cauchy surfaces [6], that K (N ) = {o ∈ K (M) : cl(o) ⊂ N },

(3.3)

for any subspacetime N of M. We now move toward the definition of the observable net, and consider the poset formed by the set of diamonds of M ordered under inclusion ⊆. Some topological information of the spacetime can be deduced from the poset K (M). First of all, the poset K (M) is pathwise connected since M is connected. Secondly, the first homotopy group of K (M) is isomorphic to the first homotopy group of M. Furthermore, recall that if a poset is upward directed, then it is simply connected (see Sect. 2). Thus K (M) is not upward directed, when M is not simply connected. The same happens when M has compact Cauchy surfaces. The observable net in the Minkowski spacetime is defined according to the HaagKastler axioms [30] (see also [28]). A generalization to a 4–dimensional globally hyperbolic spacetime M has been provided in [26]. The observable net A K (M) is defined as a correspondence o → A(o),

(3.4)

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associating with any diamond o of M a unital C∗ -algebra A(o) representing the algebra generated by all the observables localized within o, and satisfying the isotony relation, o ⊆ o˜ ⇒ A(o) ⊆ A(o). ˜

(3.5)

Isotony implies that the observable net is a net of C∗ –algebras over the poset K (M). Now, the Haag-Kastler axioms include Einstein’s causality principle, saying that observables localized in causally disjoint (spacelike separated) regions must commute. However, when the indices of the observable net is a non-upward directed poset this principle cannot be fully implemented. Recall that K (M) fails to be upward directed when the spacetime is not simply connected or when it has compact Cauchy surfaces. Following [26], we restore this principle to the level of net representations of the observable net. A net representation {π, ψ} of A K (M) is said to be causal whenever ˜ , o ⊥ o˜ ⇒ πo (A(o)) ⊆ πo˜ (A(o))

(3.6)

where the prime stands for the commutant of the algebra. 3.2. Sharply localized net representations: a selection criterion. We start by introducing the reference net representation. This net representation, that turns out to be a DHR– like representation because of the requirement of topological triviality, shall play for the theory the same rôle as the vacuum one in Minkowski spacetime. We conclude by giving the definition of net representations which are a sharp excitation of the reference one. As a reference net representation of the observable net we consider a faithful, causal and topologically trivial net representation in an infinite separable complex Hilbert space H0 . Thus, according to Lemma 2, we take a net representation of A K (M) of the form {ι, 1}. Moreover, let R K (M) be the net of von Neumann algebras o → R(o), . where R(o) = ιo (A(o)) , that is, the observable net in the reference representation. Note that because of causality if o ⊥ o˜ then R(o) ⊆ R(o) ˜ . Then, we require that R K (M) satisfies the following properties: Irreducibility : C 1 = ∩{R(o) | o ∈ K (M)}; Outer regularity : R(o) = ∩{R(o) ˜ | cl(o) ⊂ o}; ˜ Borchers property : Given o ∈ K (M) and a non-zero orthogonal projection E ∈ R(o), for any o˜ with cl(o) ⊂ o˜ there exists an isometry V ∈ R(o) ˜ such that V V ∗ = E; Punctured Haag duality : Given a point x ∈ M there holds ˜ ⊥ {x} , R(o) = ∩ R(o) ˜ : o˜ ∈ K (M), o˜ ⊥ o, cl(o) for any o ∈ K (M) with cl(o) ⊥ {x}. Apart from the explicit requirement of topological triviality and outer regularity, the reference representation is defined in the same way as in [53]. Outer regularity, in particular, enters the theory only at one point, namely at the equivalence between sharply localized representations and net cohomology (see Appendix A). We stress that physically meaningful examples of representations satisfying the defining properties of the reference representation are the representations of a free scalar field which satisfy the


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microlocal spectrum condition [52,56], a generalization to globally hyperbolic spacetimes of the spectrum condition [10,44]. As a consequence ofthe above assumptions (see [53]) the net R K (M) satisfies Haag duality, i.e., R(a) = ∩ R(a) ˜ : a˜ ⊥ a , for any a ∈ K (M); and it is locally definite, i.e., C 1 = ∩{R(o) : x ∈ o} for any x ∈ M. Punctured Haag duality can be better understood by looking at the restriction of the theory to the causal punctures of the spacetime (see Sect. 3.1). Let R K (x ⊥ ) be the net obtained by restricting R K (M) to the set of diamonds K (x ⊥ ) with x ∈ M. Then R K (x ⊥ ) is an irreducible net satisfying Haag duality. We recall that the restriction to the causal punctures was the key idea for the understanding of the charge structure of DHR-sectors on globally hyperbolic spacetimes, mainly because the point x plays for the set K (x ⊥ ) the same rôle as the spacelike infinity in Minkowski spacetime. The purpose now is to generalize the criterion, used in [26,53] to select DHR-sectors, to net representations. DHR-sectors are topologically trivial net representations {π, 1} of A K (M) which are a sharp excitation of the reference representation, in symbols π o⊥ ∼ = ι o⊥ ,

(3.7)

for any o ∈ K (M). This means that for any o there is a unitary operator U o : Hπ → H such that U o πa = ιa U o for any a ⊥ o. This definition does not work if one takes into account net representations which are not topologically trivial. In fact assume that {π, ψ} is topologically non-trivial. If {π, ψ} were equivalent to {ι, 1} on o⊥ , by Lemma 1 the 1–cocycle ζ π would be trivial on o⊥ and this leads to a contradiction. Indeed, let be a loop of K (M) over a 0–simplex whose closure is contained in o⊥ . By Corollary 6 is homotopic to a loop whose support has closure contained in o⊥ . Then by homotopic invariance of 1–cocycles ζ π () = ζ π ( ) = 1Hπ . Hence ζ π should be trivial on K (M). This observation suggests how to modify (3.7): we shall require that the above criterion is satisfied only in restriction to simply connected subspacetimes of M (see Sect. 3.1). To be precise, we say that a causal net representation {π, ψ} is a sharp excitation of the reference one, if for any o ∈ K (M) and for any simply connected subspacetime N of M, such that cl(o) ⊂ N , there holds {π, ψ} o⊥ ∩ N ∼ (3.8) = {ι, 1} o⊥ ∩ N . . This amounts to saying that there is a family W N o = {WaN o : cl(a) ⊂ N , a ⊥ o} of π unitary operators from H to H0 such that 1. WaN o πa = ιa WaN o ; 2. WaN o ψa a˜ = WaÑ o for any a˜ ⊆ a; 3. W N o = W N o for any simply connected subspacetime N with N ⊆ N . These three equations represent the selection criterion. Observe that while equations 1 and 2 derive from (3.8) and from the definition of equivalent net representations, equation 3 does not. The latter equation is a compatibility requirement. Our next aim is to prove that this criterion is indeed a generalization of (3.7). Consider a causal representation of the form {π, 1} satisfying the above selection criterion. Given o and N as above, since N is pathwise connected then WaN o is constant, i.e. independent of a (see the first observation at the end of Sect. 2). So we can rewrite it as W N o . By the third equation of the selection criterion we have WN

o

˜

= W N o = W N o,

N ⊆ N ∩ N˜ .

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This observation and, once again, pathwise connectedness of M implies that W N o is independent of the region N . So we have obtained the DHR notion of sharp excitation for topologically trivial representations. Denote the set of representations satisfying the selection criterion by SC(A K (M) ), and consider the C∗ –subcategory of Repnet (A K (M) ) whose set of objects is SC(A K (M) ). We denote this category by the same symbol SC(A K (M) ) used to denote the corresponding set of objects. We denote the full C∗ –subcategory of SC(A K (M) ) whose objects are topologically trivial net representations by SCt (A K (M) ). Because of the Borchers property, a routine calculation shows that these two categories are closed under direct sums and subobjects. Unitary equivalence classes of irreducible objects of SC(A K (M) ) are the superselection sectors of the theory, and the analysis of their charge structure and topological content will be our scope from now on. Note that the superselection sectors of the subcategory SCt (A K (M) ) are the DHR-sectors analyzed in [53]. 4. Net Cohomology and the Localization of the Fundamental Group The category of the net representations satisfying the selection criterion admits an equivalent description in terms of the net cohomology of the poset K (M) with values in the observable net. In the present section we prove a property of net cohomology which is at the base of this equivalence, the localization of the fundamental group, i.e., the representation of the first homotopy group defined by a 1–cocycle is localized. As we shall see in the following, this property is the key for understanding both charge structure and topological content of superselection sectors. Consider the observable net in the reference representation R K (M) , and denote the category of 1–cocycles of the poset K (M) with values in the net R K (M) by Z1 (R K (M) ). This is the C∗ –subcategory of Z1 (K (M), B(H0 )) whose objects z and whose arrows t ∈ (z, zˆ ) satisfy the locality condition, i.e., z(b) ∈ R(|b|),

b ∈ Σ1 (K (M)),

(4.1)

and ta ∈ R(a),

a ∈ Σ0 (K (M)).

(4.2)

Now, as any 1–cocycle defines a representation of the fundamental group of the poset K (M), we denote the full C∗ –subcategory of Z1 (R K (M) ) whose objects are trivial representations of the fundamental group by Zt1 (R K (M) ). Sometimes we shall refer to the elements of Zt1 (R K (M) ) as topologically trivial 1–cocycles. Note that topologically trivial 1–cocycles are nothing but the 1–coboundaries in Z1 (K (M), B(H0 )). Before diving further into the deep sea of net cohomology, some words of explanation are in order. As pointed out in Sect. 3.1, the set of indices of the observable net in a globally hyperbolic spacetime is non-directed under inclusion when the spacetime either is multiply connected or has compact Cauchy surfaces. In such situations it is not possible to define the C∗ –algebra of all local observables, i.e., the C∗ -inductive limit. This does not happen in Minkowski spacetime where there is a canonical choice of the set of indices, the set of double cones, which is directed under inclusion. This fact reflects in the way DHR-sectors have been analyzed in these two situations. A key step of DHR analysis was the understanding that all the information of sectors is encoded in a unique Hilbert space: one related to the vacuum. The category of DHRsectors in Minkowski spacetime turns out to be equivalent to the category of localized


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and transportable endomorphisms of the algebra of all local observables defined in the vacuum representation [18]. In globally hyperbolic spacetimes DHR-sectors are still encoded in the vacuum Hilbert space (the reference) but in a different form. There is a notion of localized and transportable endomorphisms of the observable net defined in the vacuum representation, but the corresponding category is not equivalent to the category of DHR-sectors anymore when the set of indices is non-directed under inclusion [26]. The functor from the latter to the former category is not full, and it is not clear how to define the functor in the opposite direction. However, as pointed out by Roberts, in Minkowski spacetime DHR-sectors can be equivalently described in terms of the operators that in DHR analysis play the rôle of charge transporters: 1–cocycles of double cones taking values in the observable net defined in the vacuum representation [45,47]. As shown in [26], this equivalence generalizes in arbitrary globally hyperbolic spacetimes: the category SCt (A K (M) ) is equivalent to the category Zt1 (R K (M) ) (see also [53]). Analyzing the category Zt1 (R K (M) ), the charge structure and the general covariance of DHR-sectors have been understood [12,26,48,53]. Our first step for understanding sectors introduced in the present paper according to the selection criterion (3.8) consists in proving that the category SC(A K (M) ) is equivalent to Z1 (R K (M) ). However, in this case, we have to pay attention, since non trivial topological objects are involved. To begin with, we show the property of net cohomology which underlies this equivalence. Let π1 (K (M), a) be the first homotopy group of the poset K (M) based on the 0–simplex a. Given a 1–cocycle z of Z1 (R K (M) ), we call the von Neumann algebra defined by . Rz (M, a) = {z() : ∈ Loops K (M) (a)} , (4.3) the group algebra associated with z, where the double prime stands for the bicommutant. The next theorem, to which we shall refer as the localization of the fundamental group, asserts that this von Neumann algebra is localized. Theorem 1. Given z ∈ Z1 (R K (M) ), the following assertions hold: (i) z() ∈ R(a), ∀ ∈ Loops K (M) (a); (ii) Rz (M, a) ⊆ R(a). Proof. Note that if we prove that z() ∈ R(o) for any diamond o such that o ⊥ a, then Haag duality implies that z() ∈ R(a). To this end, observe that if o ⊥ a then, by Corollary 6, the loop is homotopic to a loop 1 whose support2 is contained in the causal complement of o. By homotopic invariance of 1–cocycles we have z() = z(1 ) ∈ R(o) completing the proof. In [26] it was proved that any 1–cocycle z of Zt1 (R K (M) ) satisfies the following localization properties: First, given a path p, then z( p) ∈ R(o) ,

o ⊥ ∂ p,

(4.4)

where ∂ p denotes the boundary of the path, i.e., ∂ p = {∂0 p, ∂1 p}; secondly, let p, q be paths with ∂0 p = ∂0 q, and let o be a diamond such that ∂1 p, ∂1 q ⊥ o, then z( p) A z( p)∗ = z(q) A z(q)∗ ,

A ∈ R(o).

(4.5)

Thanks to the localization of the fundamental group we now are able to prove that these properties hold in full generality. 2 The support of a path is the union of the supports of the 1–simplices that form the path.

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Corollary 1. Any 1–cocycle of Z1 (R K (M) ) satisfies (4.4) and (4.5). Proof. Let us prove (4.4). As the causal complement of o is pathwise connected, there is a path q with ∂q = ∂ p and |q| ⊥ o. Observe that q ∗ p is a loop whose endpoint, say a, is causally disjoint from o. Since z(q ∗ p) ∈ R(a) because of Theorem 1, we have z( p) A = z(q) z(q ∗ p) A = z(q) A z(q ∗ p) = A z(q) z(q ∗ p) = A z( p), for any A ∈ R(o) and this proves relation (4.4). Let p, q and o be as in (4.5). As q ∗ p satisfies, with respect to o, the hypotheses of (4.4), we have z( p) A z( p)∗ = z(q) z(q ∗ p) A z(q ∗ p)∗ z(q)∗ = z(q) A z(q)∗ , for any A ∈ R(o). Now, the first application of this corollary is the crucial equivalence between sharply localized net representations and net cohomology. Theorem 2. SC(A K (M) ) and Z1 (R K (M) ) are equivalent categories. We prefer to postpone the, rather technical, proof of this equivalence in Appendix. We only point out that the functors that define the equivalence are an extension of the functors that define the equivalence between SCt (A K (M) ) and Zt1 (R K (M) ). On these grounds the superselection sectors are described by the unitary equivalence classes of the irreducible objects of the category Z1 (R K (M) ). So from now on, the analysis of superselection sectors will be carried out on Z1 (R K (M) ). Corollary 1 applies also to the analysis of the charge structure of superselection sectors which is the subject of the next section. 5. Charge Structure The purpose of the present section is to show that the superselection sectors previously introduced manifest a charge structure. As observed in the previous section, this analysis will be performed on the category Z1 (R K (M) ). At this point it is worth recalling that the charge structure of topologically trivial cocycles has been completely understood: the C∗ –category Zt1 (R K (M) ) has a tensor product, a permutation symmetry and a conjugation. This amounts to saying that the quantum numbers, i.e., the labels of sectors, have a composition law, a particle-antiparticle symmetry and an additional number saying that a sector has either the para-Bose or the para-Fermi statistics. The study of the category Zt1 (R K (M) ) resembles a standard argument of differential geometry. One first restricts the attention to the causal punctures of the spacetimes, namely to the categories Zt1 (R K (x ⊥ ) ). The advantage is that in these regions the point x has properties similar to the spacelike infinite in Minkowski space. So one can prove the existence of a tensor product, permutation symmetry, left inverses and conjugated object in Zt1 (R K (x ⊥ ) ) for any point x. Then, one observes that for 1–cocycle z of Zt1 (R K (M) ) the local definitions, i.e. on the causal punctures, of tensor product, permutation symmetry and conjugated objects can be glued together to form the corresponding global notions. It is now important to note that most of the constructions made for Zt1 (R K (M) ), the tensor product, the permutation symmetry and the conjugation do not involve the topological triviality of 1–cocycles directly but rather relations (4.4) and (4.5). Thus, by Corollary 1, these constructions can be straightforwardly applied to Z1 (R K (M) ). Only one point of that analysis cannot be extended to the general case: the proof of the existence of left inverses (and consequently the definition of statistics) because it relies on the fullness of the restriction functor from Zt1 (R K (M) ) to Zt1 (R K (x ⊥ ) ), a property that


537

does not hold for general 1–cocycles. However, although we will not prove the existence of left inverses for all the objects of Z1 (R K (M) ) we will be able to define objects with finite statistics via a detour. We assume that the reader is familiar with symmetric tensor C∗ –categories and related notions. Two references for this topic are [19,38]. Other references whose focus is on the theory of superselection sectors are [3,32,47,48].

5.1. DHR-like endomorphisms. Any 1–cocycle defines a class of endomorphisms that are localized and transportable in the same sense of those used in DHR analysis, but live on a presheaf associated with the observable net. Although these endomorphisms do not contain all the information about superselection sectors (see Remark 1), they enter the definitions of tensor product, permutation symmetry and conjugation. Given a diamond o, the algebra of its causal complement is the C∗ -algebra R⊥ (o) generated by all the algebras R(a) with a ⊥ o. The presheaf R K⊥(M) associated with the observable net is the correspondence o → R⊥ (o). Consider a 1–cocycle z of Z1 (R K (M) ). Fix a 0–simplex o, and let a be a 0–simplex such that a ⊥ o. Define . yaz (o)(A) = z( p) A z( p)∗ ,

A ∈ R⊥ (a),

(5.1)

where p is path with ∂1 p ⊆ a and ∂0 p = o. By (4.4) and (4.5) this definition does not depend on the path chosen p and on the choice of the starting point ∂1 p. Therefore yaz˜ (o) R⊥ (a) = yaz (o),

a˜ ⊆ a.

(5.2)

Fix a point x of the spacetime M. Since K is a base for the topology of M, the . collection of 0–simplices K (x) = {a˜ : x ∈ a} ˜ is downward directed. The stalk in a point x can be seen either as the C∗ -inductive limit of the system R⊥ (o) with o ∈ K (x) or as the C∗ −algebra generated by the algebras R(o) for any o in K (x ⊥ ). Then, by property (5.2), the collection . yxz (o) = {yaz (o) | a ∈ K (x)},

o ∈ K (x ⊥ ),

(5.3)

is extendible to a morphism of the stalk R⊥ (x). Lemma 3. On the premises outlined before, we have that yxz (o) is an endomorphism of R⊥ (x) satisfying the following properties: (i) (ii) (iii) (iv)

⊥ ˜ = idR(o) yxz (o) R(o) ˜ for any o˜ ∈ K (x ) with o˜ ⊥ o ; z z z( p) yx (∂1 p) = yx (∂0 p) z( p) for any path p in K (x ⊥ ); to y z (o) = y z 1 (o) to , with t ∈ (z, z 1 ); y z (o)(R(o)) ˜ ⊆ R(o) ˜ for any o˜ ∈ K (x ⊥ ) with o ⊆ o. ˜

The proof of these properties is the same as the proof of [53, Lemma 4.5]. We only observe that the first three properties are a consequence of the localization of the fundamental group (Properties (4.4) and (4.5)). Property (iv) derives from property (i) and from punctured Haag duality, because the restriction of R K (M) to K (x ⊥ ) satisfies Haag duality (see Sect. 3.2).

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Remark 1. The family {yxz (o) | o ∈ K (x ⊥ )} of endomorphisms of R⊥ (x) are localized (i) and transportable (ii) in the same sense of DHR analysis. So, using the interpretation given in DHR analysis we may think of yxz (o) as a charge localized within the diamond o and the 1–cocycle z as the transporter of these charges. As said at the beginning of this section, these endomorphisms do not contain all the information of superselection sectors. Among the various difficulties, the simplest one is to observe that if z is an irreducible object but carries a non-trivial representation of the fundamental group then, by property (ii) of Lemma 3, the corresponding endomorphism is not irreducible. We finally point out two useful relations. The first one, an obvious consequence of (5.2), says that yxz (o) R⊥ (a) = yxz˜ (o) R⊥ (a),

(5.4)

where x, x˜ ∈ a and o ∈ K (x ⊥ ) ∩ K (x˜ ⊥ ). This, in turn, implies that yxz (o)(z( p)) = yxz˜ (o)(z( p)),

(5.5)

for any pair x, x˜ of points and any path p such that | p| ⊆ K (x ⊥ ) ∩ K (x˜ ⊥ ). The proof of these two properties is given in [53] where they are called gluing conditions, because they allow to extend cocycles and arrows defined on causal punctures over all K (M). 5.2. Tensor structure. Thanks to the localization of the fundamental group, we shall define the tensor product and the permutation symmetry in Z1 (R K (M) ) by the same formulas as those used to define the corresponding notions in Zt1 (R K (M) ) [53]. In that paper these formulas are first defined on the causal punctures and then extended globally by the gluing conditions. For brevity we shall give directly the global definitions. Clearly, most of the proofs are omitted, with some exceptions because they need modifications from the original ones. We start by introducing a preliminary definition. Given z, z 1 ∈ Z1 (R K (M) ) and t ∈ (z, z 1 ), s ∈ (z 2 , z 3 ), define . z( p) ×x z 1 (q) = z( p) yxz (∂1 p)(z 1 (q)), p, q paths in K (x ⊥ ), (5.6) . a ∈ Σ0 (K (x ⊥ )). ta ×x sa˜ = ta yxz (a)(sa˜ ), As a consequence of properties (ii) and (iii) of localized transportable endomorphisms (see Lemma 3) we have z( p ∗ p)× ˆ x z 1 (q ∗ q) ˆ = z( p)×x z 1 (q) z( p)× ˆ x z 1 (q), ˆ

(5.7)

t∂0 p ×x s∂0 q z( p)×x z 1 (q) = z 2 ( p)×x z 3 (q) t∂1 p ×x s∂1 q ,

(5.8)

and

(cf. [53, Lemma 4.6]). Furthermore we have the following Lemma 4. Given a pair of paths p, q of K (x ⊥ ). Then z( p) ×x z 1 (q) = z 1 (q) ×x z( p), whenever ∂i p ⊥ ∂i q for i = 0, 1.


539

Proof. There are in K (x ⊥ ) two paths p1 = b jn ∗ · · · ∗ b j1 and q1 = bkn ∗ · · · ∗ bk1 such that |b ji | ⊥ |bki | for i = 1, . . . , n and ∂ p1 = ∂ p, ∂q1 = ∂q, see [53, Sect. 3.2.1]. For these paths we have z( p1 ) ×x z 1 (q1 ) = z 1 (q1 ) ×x z 1 ( p1 ) (cf. [53, Lemma 4.8]). Using this and (5.7) we have z( p) ×x z 1 (q) = z( p1 ) ×x z 1 (q1 ) z( p1 ∗ p) ×x z 1 (q1 ∗ q) = z 1 (q1 ) ×x z( p1 ) z 1 (q1 ∗ q) ×x z( p1 ∗ p) = z 1 (q) × z( p), where we have used the fact that z( p1 ∗ p) ×x z 1 (q1 ∗ q) = z( p1 ∗ p) z 1 (q1 ∗ q) = z 1 (q1 ∗ q) z( p1 ∗ p) = z 1 (q1 ∗ q) ×x z( p1 ∗ p), because z( p1 ∗ p) ∈ R(∂1 p), z 1 (q1 ∗ q) ∈ R(∂1 q), ∂1 p ⊥ ∂1 q, and property (i) of localized and transportable endomorphisms of stalks. The tensor product is a particular case of the expressions (5.6). Given z, z 1 ∈ Z1 (R K (M) ) and t, s arrows of Z1 (R K (M) ) define . (z ⊗ z 1 )(b) = z(b) ×x z 1 (b), b ∈ Σ1 (K (M)), (5.9) . (t ⊗ s)a = ta ×x˜ sa , a ∈ Σ0 (K (M)), where x and x˜ are points of M such that x ⊥ cl(|b|) and x˜ ⊥ cl(a). One first observes that these definitions behave as a tensor product when restricted to the causal puncture K (x ⊥ ) of M in x. Afterwards one observes that by (5.5) the definitions are independent of the choice of the point ([53, Prop. 4.7 and Lemma 4.17]). The following lemma characterizes the morphisms of stalks associated with the tensor product of two 1–cocycles. Lemma 5. Let z, z 1 ∈ Z1 (R K (M) ). Then yxz⊗z 1 (o) = yxz (o)yxz 1 (o), for any o ∈ K (x ⊥ ). Proof. Let a be a 0–simplex in K (x) (see Sect. 5.2) such that cl(a) ⊥ o. Let p be a path in K (x ⊥ ) such that cl(∂1 p) ⊂ a and ∂0 p = o. Take A ∈ R⊥ (a). According to the definition (5.3) we have yxz⊗z 1 (o)(A) = = = =

yaz⊗z 1 (o)(A) (z ⊗ z 1 )( p) A (z ⊗ z 1 )( p)∗ (z( p) ×x z 1 ( p)) A (z( p) ×x z 1 ( p))∗ ∗ z( p) yxz (∂1 p)(z 1 ( p)) A z( p) yxz (∂1 p)(z 1 ( p)) ∗ = z( p) z(q) z 1 ( p) z(q)∗ A z( p) z(q) z 1 ( p) z(q)∗ ,

where q is a path in K (x ⊥ ) such that ∂0 q = ∂1 p and cl(∂1 q) ⊂ a and ∂1 q ⊥ ∂1 p.3 Applying (4.4) we have z(q)∗ A z(q) = A; thus yxz⊗z 1 (o)(A) = z( p) z(q) z 1 ( p) A(z( p) z(q) z 1 ( p))∗ = z( p ∗ q) z 1 ( p) A (z( p ∗ q) z 1 ( p))∗ = z( p ∗ q) yxz 1 (o)(A) z( p ∗ q)∗ = yxz (o) yxz 1 (o)(A) , 3 Note that such a path exists since cl(∂ p) ⊂ a, by Lemma 14, there are two diamonds o and o such 1 1 2 that cl(o2 ) ⊂ a and cl(o1 ) ⊥ cl(∂1 p). So we can take q to be the 1–simplex whose support is o2 , the 1–face is o2 , and the 0–face is ∂1 p.

540


where we have used the relation yxz 1 (o)(A) ∈ R⊥ (a) which is a consequence of property (iv) of localized and transportable endomorphisms of stalks. The permutation symmetry ε is defined, for any pair z, z 1 in Z1 (R K (M) ), by . a ∈ Σ0 (K (M)), (5.10) ε(z, z 1 )a = z 1 (q)∗ ×x z( p)∗ z( p) ×x z 1 (q), where x is any point of M with x ⊥ cl(a), and p, q are two paths of K (x ⊥ ) with ∂0 p ⊥ ∂0 q and ∂1 p = ∂1 q = a. Once again one restricts the attention to the causal punctures. First one observes that the above definition does not depend on the choice of the paths p and q, and shows ε is indeed a symmetry in restriction to the causal punctures. Afterwards, one checks that the above definition does not depend on the choice of the point x (see [53]). A useful relation for analyzing the topological content of 1–cocycles is provided in the following lemma. Lemma 6. Given o and x ∈ M with cl(o) ⊥ x, then ε(z, z 1 )o z() ×x z 1 ( ) = z 1 ( ) ×x z() ε(z, z 1 )o , where , ∈ Loops K (x ⊥ ) (o). Proof. Consider a point x and two paths p and q as in the definition of ε. By (5.7) we have ε(z, z 1 )o z() ×x z 1 ( ) = = = = =

z 1 (q)∗ ×x z( p)∗ z( p)×x z 1 (q) z()×x z 1 ( ) z 1 (q)∗ ×x z( p)∗ z( p ∗ )×x z 1 (q ∗ ) z 1 ( )z 1 (q ∗ )∗ ×x z()z( p ∗ )∗ z( p ∗ ) ×x z 1 (q ∗ ) z 1 ( )×x z() z 1 (q ∗ )∗ ×x z( p ∗ )∗ z( p ∗ )×x z 1 (q ∗ ) z 1 ( ) ×x z() ε(z, z 1 )o

because p ∗ and q ∗ are paths satisfying the definition of ε. Note that if we take in this lemma as the trivial loop, i.e., = σ0 o, then ε(z, z 1 )o z() = yxz 1 (o)(z()) ε(z, z 1 )o . Since the unitaries z() generate the algebra Rz (M, o), and since yxz 1 (o) is normal on this algebra we have A = ε(z 1 , z)o yxz 1 (o)(A) ε(z, z 1 )o ,

A ∈ Rz (M, o),

(5.11)

with o ∈ K (x ⊥ ). 5.3. Statistics and Conjugation. Our purpose now is to identify the objects of Z1 (R K (M) ) having conjugates. The first step will be to understand what the objects with finite statistics are. To reach this goal we shall not follow the traditional way, rather we shall identify a C∗ –subcategory Z1 (R K (M) ) closed under tensor product, direct sums, subobjects and having left inverses, and containing all the simple objects of Z1 (R K (M) ). Within this category we shall define the objects with finite statistics in the same way as in DHR analysis. Afterwards, we prove that any object with finite statistics has conjugates. We recall that a left inverse φ of an object z of a tensor C∗ –category is a family of linear mappings φz 1 ,z 2 : (z ⊗ z 1, z ⊗ z 2 ) → (z 1, z 2 ), for pair any z 1 , z 2 of objects, satisfying the following relations: given X ∈ (z ⊗ z 1 , z ⊗ z 2 ), then φz 1 ⊗˜z ,z 2 ⊗˜z (X ⊗ 1z˜ ) = φz 1 ,z 2 (X ) ⊗ 1z˜ , φz ,z (1z ⊗ S · X · 1z ⊗ R) = S · φz 1 ,z 2 (X ) · R,

S ∈ (z 2 , z ), R ∈ (z 1 , z ).


541

A left inverse of z is said to be positive whenever, for any object z˜ , φz˜ ,˜z sends positive elements of (z⊗ z˜ , z⊗ z˜ ) into positive elements of (˜z , z˜ ); normalized whenever φι,ι (1z ) = 1ι . A positive left inverse φ of z is said to be faithful whenever, for any object z˜ , φz˜ ,˜z (X ) = 0 for any positive and non-zero element X of (z ⊗ z˜ , z ⊗ z˜ ). From now on, by a left inverse we will always mean a positive and normalized left inverse. An object u ∈ Z1 (R K (M) ) is said to be simple whenever ε(u, u) = χ (u) · 1u⊗u , where χ (u) ∈ {1, −1}. yxu (o)

(5.12)

: R (x) → R (x) is an automorphism for any If u is simple, it turns out that o ∈ Σ0 (K (x ⊥ )) (cf. [53, Prop. 4.12, Theorem 4.22]). If we denote the inverse of yxu (o) by yxu (o)−1 it is easily seen that . φz 1 ,z 2 (t)o = yxu (o)−1 (to ), t ∈ (u ⊗ z 1 , u ⊗ z 2 ), (5.13) ⊥

⊥

where x is a point of M causally disjoint from the closure of o, is a faithful left inverse of u. So any simple object of Z1 (R K (M) ) has faithful left inverses. Denote by Z1 (R K (M) ) the full C∗ –subcategory of Z1 (R K (M) ) whose objects have faithful left inverses. By applying formulas (A.1) (A.2) and (A.3) in [51], it easily follows that this category is closed under tensor product, direct sum, subobjects and equivalence. Furthermore, this category is not trivial. In fact, as observed above, any simple object of Z1 (R K (M) ) belongs to this category. Since the category Z1 (R K (M) ) has left inverses and since it is closed under tensor product, direct sums and subobjects, it is possible to apply the mathematical machinery of DHR analysis to define and classify the objects with finite statistics (see references quoted at the beginning of Sect. 5). An object z of Z1 (R K (M) ) has finite statistics if it admits a standard left inverse φ, that is φz,z (ε(z, z))2 = c · 1z ,

c > 0.

Z1 (R K (M) ) whose objects with finite Let Z1 (R K (M) )f be the full C∗ –subcategory of 1 statistics. Then, Z (R K (M) )f is closed under tensor product, direct sum and subobjects. Any object of this category is a finite direct sum of irreducible objects with finite statistics. Given an irreducible object z of Z1 (R K (M) )f and a left inverse φ, then φz,z (ε(z, z)) = λ(z) · 1z , where λ(z) is an invariant of the equivalence class of z, called the statistics parameter, and it is the product of two invariants: λ(z) = κ(z) · d(z)−1 where κ(z) ∈ {1, −1}, d(z) ∈ N. The possible statistics of z are classified by the statistical phase κ(z) distinguishing paraBose (1) and para-Fermi (−1) statistics and by the statistical dimension d(z) giving the order of the para-statistics. Ordinary Bose and Fermi statistics correspond to d(z) = 1. In a symmetric tensor C∗ -category an object z has conjugates if there exists an object z and a pair of arrows r ∈ (ι, z ⊗ z) and r ∈ (ι, z ⊗ z) satisfying the conjugate equations r ∗ ⊗ 1z · 1z ⊗ r = 1z ,

r ∗ ⊗ 1z · 1z ⊗ r = 1z .

(5.14)

It is a well known fact that if an object has conjugates, then it has a faithful left inverse and finite statistics. So any object of z having conjugates belongs to Z1 (R K (M) )f . We

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now show that any object of this category has conjugates. To this end it is enough to prove that simple objects have conjugates. So, consider a simple object u. Define . u(b) = yxu (∂0 b)−1 (u(b)∗ ),

b ∈ Σ1 (K (M)),

(5.15)

for some x with x ⊥ cl(|b|). Again, one first observes that the definition is independent of the choice of the point x. Finally, one checks that (u ⊗u)(b) = 1 and that (u ⊗u)(b) = 1. So, if we take r = r = 1, then r and r satisfy the conjugate equations for u and u, cf. [53]. Then, the above observation leads to the following conclusion. Theorem 3. Any object of Z1 (R K (M) )f has conjugates. 6. The Topological Content The twofold information contained in 1–cocycles, the charge and the topological content, can be split. We shall see that any 1–cocycle z can be written as a suitable composition (the joining) of two 1–cocycles: the charge component z, a topologically trivial 1–cocycle having the same charge quantum numbers as z; the topological component χz , a 1–cocycle that carries the same representation of the fundamental group of M as z but it does not take values in the observable net. This decomposition holds for any 1–cocycle of Z1 (R K (M) ). When we specialize to the finite statistics case, we shall find a relation between the statistics and the topological content of 1–cocycles. This relation shall lead us to discover a new invariant: the topological dimension. In order to decompose 1–cocycles into charge and topological components, we introduce the notion of path-frame which assigns to any 0–simplex a path-coordinate with respect to a fixed 0–simplex, the pole. To be precise we fix a 0–simplex o, the pole. For any 0–simplex a, we pick a path p(a,o) from o to a such that p(o,o) is homotopic to the trivial loop over o, i.e., the degenerate 0–simplex σ0 o. We call the collection . Po = { p(a,o) | a ∈ Σ0 (K (M))} a path-frame with pole o. The translation of a pathframe Po is the path-frame Po ∗ o1 whose elements, denoted by p(a,o1 ) , are of the form p(a,o) ∗ p(o1 ,o) . Note that the translation Po ∗ o1 ∗ o can be identified with Po since they have homotopic elements. Once a path-frame Po is given, two paths are uniquely associated with any 1–simplex b: the first path defined as p(∂0 b,o) ∗ b ∗ p(∂1 b,o) is a loop over o; the second path defined as p(∂0 b,o) ∗ p(∂1 b,o) is a path from ∂1 b to ∂0 b. These two different ways of “representing” 1–simplices shall allow us to split the topological content of a 1–cocycle from its charge content. 6.1. Splitting. We now show the splitting of a 1–cocycle into charge and topological components. Fix a path-frame Po with pole o. Given a 1–cocycle z of Z1 (R K (M) ) define . z(b) = z( p(∂0 b,o) ∗ p(∂1 b,o) ), b ∈ Σ1 (K (M)). (6.1) We call z the charge component of z. It is very easy to see that z is a topologically trivial 1–cocycle, i.e., z ∈ Zt1 (R K (M) ). In fact, it follows straightforwardly from the definition that z is a 1–coboundary of Z1 (K (M), B(H0 )). Moreover, given a 1–simplex b for any 0–simplex a with |b| ⊥ a by (4.4) we have z(b) ∈ R(a) . Thus z(b) ∈ R(|b|) by Haag duality.


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We now show that definition (6.1) is independent, up to equivalence, of the choice of the path-frame and of the choice of the pole. Given another path-frame Q o , define . sa = z(q(a,o) ∗ p(a,o) ) for any 0–simplex a. Then s∂0 b z( p(∂0 b,o) ∗ p(∂1 b,o) ) = z(q(∂0 b,o) ∗ p(∂1 b,o) ) = z(q(∂0 b,o) ∗ q(∂1 b,o) ) s∂1 b , for any 1–simplex b. Furthermore, since q(a,o) ∗ p(a,o) is a loop over a, then sa ∈ R(a) because of the localization of the fundamental group. Thus, different choices of pathframes with the same pole lead to equivalent charge components of z. Now, consider another pole o1 and the translation Po ∗ o1 of Po . Since 1–cocycles are homotopic invariant we have z( p(∂0 b,o) ∗ p(∂1 b,o) ) = z( p(∂0 b,o1 ) ∗ p(∂1 b,o1 ) ),

(6.2)

because the paths p(∂0 b,o) ∗ p(∂1 b,o) and p(∂0 b,o) ∗ p(o,o1 ) ∗ p(o,o1 ) ∗ p(∂1 b,o) = p(∂0 b,o1 ) ∗ p(∂1 b,o1 ) are homotopic. This completes the proof of our claim. We now go deep inside the relation between z and its charge component. We fix a path-frame Po . The first important observation is that the morphisms of stalks associated with z and z are equal. According to (5.3) and (5.1), it is enough to see that given a path q and a 0–simplex a, with |q| ⊥ a, and A ∈ R(a), then z(q) A z(q)∗ = z( p(∂0 q,o) ∗ p(∂1 q,o) ) Az( p(∂0 q,o) ∗ p(∂1 q,o) )∗ = z(q) A z(q)∗ , where we have applied (4.5) since the paths q and p(∂0 q,o) ∗ p(∂1 q,o) satisfy the hypotheses of that relation. Lemma 7. The mapping Pc : Z1 (R K (M) ) → Zt1 (R K (M) ), which sends an object z to its charged component z, with respect to a fixed path-frame Po , and acts as the identity on arrows t → t, defines a faithful and symmetric, covariant ∗ -functor. Proof. It is easily seen that Pc is a faithful and covariant ∗ –functor. We only observe that if t ∈ (z, z 1 ), then t∂0 b z(b) = t∂0 b z( p(∂0 b,o) ∗ p(∂1 b,o) ) = z 1 ( p(∂0 b,o) ∗ p(∂1 b,o) ) t∂1 b = z 1 (b) t∂1 b , for any 1–simplex b. We now prove that Pc preserves the tensor product. Given a 1–simplex b, pick x ∈ M such that |b| ∈ K (x ⊥ ). Moreover, pick pole o1 in K (x ⊥ ). Given z, z 1 , and using (6.2) we have Pc (z ⊗ z 1 )(b) = z ⊗ z 1 (b) = z ⊗ z 1 ( p(∂0 b,o) ∗ p(∂1 b,o) ) = z ⊗ z 1 ( p(∂0 b,o1 ) ∗ p(∂1 b,o1 ) ), where p(a,o1 ) is the path associated with the translation Po ∗ o1 . Since |b|, o1 are in K (x ⊥ ), there are paths q0 and q2 in K (x ⊥ ) which are homotopic respectively to p(∂0 b,o1 ) and p(∂1 b,o1 ) (Corollary 6). The previous equation and the homotopic invariance of 1–cocycles lead to Pc (z ⊗ z 1 )(b) = z ⊗ z 1 (q0 ∗ q1 ). Finally, by definition of the tensor product, by (5.7) and by applying again homotopic invariance of 1–cocycles we have Pc (z ⊗ z 1 )(b) = = = =

z ⊗ z 1 (q0 ∗ q1 ) = z(q0 ∗ q1 ) ×x z 1 (q0 ∗ q1 ) z( p(∂0 b,o1 ) ∗ p(∂1 b,o1 ) ) ×x z 1 ( p(∂0 b,o1 ) ∗ p(∂1 b,o1 ) ) z(b) ×x z 1 (b) z ⊗ z 1 (b),

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for any 1–simplex b. Note that we have used the fact that z and z define the same morphisms of stalks, as observed just before this lemma. Finally we prove that ε(z, z 1 ) = ε(z, z 1 ). To this end, using the same notation in (5.10), recall that ε(z, z 1 )a does not depend on the choice of the paths p and q in (5.10). Now the paths p(∂0 p,o) ∗ p(a,o) and p(∂0 q,o) ∗ p(a,o) have the same endpoints as p and q respectively. However we cannot replace in the definition of ε(z, z 1 )a the paths p and q by p(∂0 p,o) ∗ p(a,o) and p(∂0 q,o) ∗ p(a,o) respectively, since the latter do not belong to K (x ⊥ ) in general. However, we note that by Corollary 6 (see the observation below the corollary) there are two paths p1 and q1 lying in K (x ⊥ ) which are homotopic to p(∂0 p,o) ∗ p(a,o) and p(∂0 q,o) ∗ p(a,o) respectively. So we have ε(z, z 1 )a = z 1 (q)∗ ×x z( p)∗ z( p) ×x z 1 (q) = z 1 (q1 )∗ ×x z( p1 )∗ z( p1 ) ×x z 1 (q1 ). Observing (5.6) and applying homotopic invariance of 1–cocycles z( p1 ) ×x z 1 (q1 ) = z( p1 ) yxz (∂0 p1 )(z 1 (q1 )) = z( p(∂0 p,o) ∗ p(a,o) ) yxz (∂0 p1 )(z 1 ( p(∂0 q,o) ∗ p(a,o) )) = z( p) yxz (∂0 p)(z 1 (q)) = z( p) yxz (∂0 p)(z(q)) = z( p) ×x z 1 (q), where we have used the identities z( p(∂0 p,o) ∗ p(a,o) ) = z( p) and z 1 ( p(∂0 q,o) ∗ p(a,o) ) = z 1 (q), which derive from (6.1), and that z and z define the same endomorphisms of stalks. Remark 2. Two observations on Pc are in order. First, it easily follows from the definition that Pc is a projection, i.e. Pc Pc = Pc . Secondly, the functor Pc is not full in general. In fact, assume that z carries a non-trivial representation of the fundamental group. . Take a loop over o, define ta = z( p(a,o) ∗ ∗ p(a,o) ) for any a ∈ Σ0 (K (M)). By the localization of the fundamental group ta ∈ R(a) for any 0–simplex. Moreover, t∂0 b z(b) = z( p(∂0 b,o) ∗ ∗ p(∂0 b,o) ) z( p(∂0 b,o) ∗ p(∂1 b,o) ) = z( p(∂0 b,o) ) z()z( p(∂1 b,o) ) = z(b) t∂1 b . So, t ∈ (z, z), while t ∈ (z, z) in general. We now introduce a second 1–cocycle encoding the topological content of z. Fix a path-frame Po . Define . χz (b) = z( p(∂0 b, o) ∗ b ∗ p(∂1 b, o) ), b ∈ Σ1 (K (M)). (6.3) We call χz the topological component of z. The topological component χz is a 1–cocycle of K (M) which takes values, because of the localization of the fundamental group, in R(o). Moreover χz (∂0 c) χz (∂2 c) = = = =

z( p(∂00 c,o) ∗ ∂0 c ∗ p(∂10 c,o) ) z( p(∂02 c,o) ∗ ∂2 c ∗ p(∂12 c,o) ) z( p(∂01 c,o) ) z(∂0 c) z( p(∂02 c,o) ) z( p(∂02 c,o) )z(∂2 c)z( p(∂11 c,o) ) z( p(∂01 c,o) ) z(∂1 c)z( p(∂11 c,o) ) χz (∂1 c),

for any 2–simplex c. Hence χz is a 1–cocycle of the category Z1 (K (M), R(o)). We now observe that z and χz contain the same topological information, namely χz () = z(),

∈ Loops K (M) (o).

(6.4)


545

In words z and χz define the same representation of the first homotopy group. In fact assume that is of the form = bn ∗ · · · ∗ b1 . Then χz () = z( p(∂0 bn ,o) ∗ bn ∗ p(∂1 bn ,o) ) z( p(∂0 bn−1 ,o) ∗ bn ∗ p(∂1 bn−1 ,o) ) · · · ··· z( p(∂0 b2 ,o) ∗ b2 ∗ p(∂1 b2 ,o) ) z( p(∂0 b1 ,o) ∗ b1 ∗ p(∂1 b1 ,o) ) = z( p(∂0 bn ,o) ) z(bn ) z(bn−1 )z( p(∂1 bn−1 ,o) ) · · · ··· z( p(∂0 b2 ,o) ) z(b2 )z(b1 ) z( p(∂1 b1 ,o) ) = z(σ0 o) z() z(σ0 o) = z(). Note that Eq. (6.4) says that the representation of the fundamental group carried by the topological component of a 1–cocycle depends neither on the choice of the path-frame nor on the choice of the pole. Remark 3. We point out the geometrical meaning of the topological component of a 1– cocycle z of Z1 (R(K (M) ). We recall that 1–cocycles of a poset taking values in a group are flat connections of the poset (see [49,50]). So the topological component χz is a holonomy of the flat connection z. We also note that the definition of χz is the same as the definition of the reduced connection in the Ambrose-Singer theorem for posets [49, Theorem 4.28]. Further information about the relation between z and its topological and charge components will be obtained by means of the following embedding theorem. Theorem 4. Given z ∈ Z1 (R K (M) ), fix a path-frame Po and, for any X ∈ Rz (M, o), define . a (X ) = z( p(a,o) ) X z( p(a,o) )∗ ,

a ∈ Σ0 (K ).

(6.5)

Denote the family of mappings X → a (X ), a ∈ Σ0 (K (M)), by . Then (i) : Rz (M, o) → (z, z) is a injective ∗ –morphism; (ii) : Z(Rz (M, o)) → (z, z), . where Z(Rz (M, o)) = Rz (M, o) ∩ Rz (M, o) , the centre of Rz (M, o). Proof. Let us start by observing that a (X ) ∈ R(a). To this end, let us consider a loop over o. Observe that a (z()) = z( p(a,o) ∗ ∗ p(a,o) ), and that p(a,o) ∗ ∗ p(a,o) is a loop over a. Then a (z()) belongs to the von Neumann algebra Rz (M, a). Since the unitaries z() generate Rz (M, o) and since a is, clearly, normal we have a (X ) ∈ Rz (M, a). By Theorem 1 we have a (X ) ∈ R(a). (i) Given a 1–simplex b we have ∂0 b (X ) z(b) = z( p(∂0 b,o) ) X z( p(∂0 b,o) )∗ z( p(∂0 b,o) ) z( p(∂1 b,o) )∗ = z( p(∂0 b,o) ) X z( p(∂1 b,o) )∗ = z(b) ∂1 b (X ). Thus (X ) ∈ (z, z).

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(ii) Assume that X belongs to the centre of Rz (M, o). Given a 1–simplex b we have ∂0 b (X ) z(b) = = = = =

z( p(∂0 b,o) ) X z( p(∂0 b,o) )∗ z(b) z( p(∂0 b,o) ) X z( p(∂0 b,o) ∗ b ∗ p(∂1 b,o) ) z( p(∂1 b,o) )∗ z( p(∂0 b,o) ) z( p(∂0 b,o) ∗ b ∗ p(∂1 b,o) ) X z( p(∂1 b,o) )∗ z(b) z( p(∂1 b,o) ) X z( p(∂1 b,o) )∗ z(b) ∂1 b (X ),

because p(∂0 b,o) ∗ b ∗ p(∂1 b,o) is a loop over o. The first application of Theorem 4 derives from the following observation. Given a . 1–cocycle z, define (z, z)a = {ta | t ∈ (z, z)} as the component a of the set of intertwiners of the 1–cocycle. It is easily seen that any component of the algebra of intertwiners is a von Neumann algebra and that, since the poset is pathwise connected, this algebra is isomorphic to the full algebra of intertwiners. Now, since, by definition, p(o,o) is homotopic to σ0 o we have (X )o = X for any X ∈ Rz (M, o). Then by Theorem 4 we have Rz (M, o) ⊆ (z, z)o , Z(Rz (M, o)) ⊆ (z, z)o .

(6.6) (6.7)

The next result is a direct consequence of (6.7). Corollary 2. Let z be a 1–cocycle of Z1 (R K (M) ). If z is irreducible, then the representation of π1 (M, o) associated with z is a factor representation, i.e., Z(Rz (M, o)) = C1. 6.2. Joining. We learned that a 1–cocycle z splits in a pair (χz , z) of 1–cocycles: z is topologically trivial but contains the charge structure of z (this will be clearly shown in Subsect. 6.3); χz encodes the topological content of z. We now show that two such 1–cocycles can be joined together to form a 1–cocycle of Z1 (R K (M) ). Consider a topologically trivial 1–cocycle z ∈ Zt1 (R K (M) ), and a 1–cocycle ϕ of the category Z1 (K (M), R(o)). We say that ϕ and z are joinable whenever ϕ(b) ∈ (z, z)o for any 1–simplex b, and define the join of ϕ and z, with respect to a path-frame Po , as . (ϕ z)(b) = z(b) z( p(∂1 b,o) ) ϕ(b) z( p(∂1 b,o) )∗,

b ∈ Σ1 (K (M)).

(6.8)

Our aim is to show that the join ϕ z is a 1–cocycle of Z1 (R K (M) ), whose topological and charge component are equivalent to ϕ and z, respectively, and that any 1–cocycle of Z1 (R K (M) ) arises as the join of its topological component and its charge component (note that, because of (6.6), these are joinable). We start by showing a property of the join. Lemma 8. Fix a path-frame Po . Let z ∈ Zt1 (R K (M) ) and ϕ ∈ Z1 (K (M), R(o)) be joinable. Then (ϕ z)(q) = z( p(∂0 q,o) ) ϕ(q) z( p(∂1 q,o) )∗ , for any path q.


547

Proof. We start by observing that a 1–cocycle z is topologically trivial if, and only if, it is path-independent, that is, z( p) = z( p) ˜ for any pair p, p˜ of paths with the same endpoints. With this in mind, note that the above formula holds for paths which are formed by a single 1–simplex. By induction, assume that the formula holds for paths formed by n 1–simplices. Let q be a path formed by n 1–simplices and consider the path q1 = q ∗ b. Then z)(q) (ϕ z)(b) (ϕ z)(q1 ) = (ϕ = z(q) z( p(∂1 q,o) ) ϕ(q) z( p(∂1 q,o) )∗ z(b) z( p(∂1 b,o) ) ϕ(b) z( p(∂1 b,o) )∗ = z(q) z( p(∂1 q,o) ) ϕ(q) z( p(∂1 q,o) ∗ b ∗ p(∂1 b,o) ) ϕ(b) z( p(∂1 b,o) )∗ = z(q) z( p(∂1 q,o) ) ϕ(q) ϕ(b) z( p(∂1 b,o) )∗ = z( p(∂0 q,o) ) ϕ(q ∗ b) z( p(∂1 b,o) )∗ = z( p(∂0 q1 ,o) ) ϕ(q1 ) z( p(∂1 q1 ,o) )∗ , where we have used topological triviality of z (z gives the same value on paths having the same endpoints). Note that as a consequence of the above lemma, the join does not depend on the choice of the path-frame. This follows directly from the formula in the statement of Lemma 8 and from the topological triviality of the 1–cocycle z. We now are in a position to prove the main result of this section. Theorem 5. Fix a path-frame Po . Let z ∈ Zt1 (R K (M) ) and ϕ ∈ Z1 (K (M), R(o)) be joinable. Then the following assertions hold. (i) The join ϕ z is a 1–cocycle of Z1 (R K (M) ) whose topological and charge components are equivalent to ϕ and z, respectively. (ii) Any 1–cocycle z of Z1 (R K (M) ) is the join χz z, with respect to Po , of its topological component χz with its charged component z. Proof. (i) Clearly by Lemma 8 the join satisfies the 1–cocycle identity. Moreover ϕ z is localized. In fact given a 1–simplex b, take a point x of M such that b lies K (x ⊥ ). Let a be a 0–simplex in K (x ⊥ ) such that a ⊥ |b|. Then (ϕ z)(b) A = = = = = =

z(b) z( p(∂1 b,o) ) ϕ(b) z( p(∂1 b,o) )∗ A z(b) z( p(∂1 b,o) ) ϕ(b) yxz (o)(A) z( p(∂1 b,o) )∗ z(b) z( p(∂1 b,o) ) yxz (o)(A) ϕ(b) z( p(∂1 b,o) )∗ z(b) yxz (∂1 b)(A) z( p(∂1 b,o) ) ϕ(b) z( p(∂1 b,o) )∗ yxz (∂0 b)(A) z(b) z( p(∂1 b,o) ) ϕ(b) z( p(∂1 b,o) )∗ A (ϕ z)(b),

where we have used the properties of localized transportable endomorphisms of stalks z)(b) ∈ R(|b|); hence ϕ z ∈ and the fact that ϕ ∈ (z, z)o . By Haag duality (ϕ Z1 (R K (M) ). We now prove that the topological component χϕ z is equivalent to ϕ. First of all, given a loop = bn ∗ · · · ∗ b1 over o, observe that

548


χϕ z)() z () = (ϕ = z(bn ) z( p(∂1 bn ,o) ) ϕ(bn ) z( p(∂1 bn ,o) ∗ bn−1 ∗ p(∂1 bn−1 ,o) ) ϕ(bn−1 ) · · · · · · ϕ(b2 ) z( p(∂1 b2 ,o) ∗ b1 ∗ p(∂1 b1 ,o) ) ϕ(b1 ) z( p(∂1 b1 ,o) ) = z(bn ∗ p(∂1 bn ,o) ) ϕ(bn ) ϕ(bn−1 ) · · · ϕ(b2 ) ϕ(b1 )z( p(∂1 b1 ,o) ) = z( p(o,o) ) ϕ()z( p(o,o) ) = ϕ(), because z( p(∂1 bk+1 ,o) ∗ bk ∗ p(∂1 bk ,o) ) = 1, with k = 1, . . . , n − 1, since z is topologically . trivial. So define sa = ϕ( p(a,o) ) for any 0–simplex a. Then sa ∈ R(a) and by the above identity we have z)( p(∂0 b,o) ∗ b ∗ p(∂1 b,o) ) s∂0 b χϕ z (b) = ϕ( p(∂0 b,o) ) (ϕ = ϕ( p(∂0 b,o) ) ϕ( p(∂0 b,o) ∗ b ∗ p(∂1 b,o) ) = ϕ(b) ϕ( p(∂1 b,o) ) = ϕ(b) s∂1 b , 1 for any 1–simplex b; thus χϕ z is equivalent to ϕ in Z (K (M), R(o)). We prove that . 1 ϕ z is equivalent to z in Zt (R K (M) ). Define ta = z( p(a,o) ) ϕ( p(a,o) )∗ z( p(a,o) )∗ , for any 0–simplex a. We first observe that ta ∈ R(a). In fact, since the topological z (o) = Rϕ (o). Now, by Lemma 2 and component of the join is ϕ we have that Rϕ using topological triviality of z we have

z (ϕ( p(a,o) )∗ ) aϕ

= (ϕ z)( p(a,o) ) ϕ( p(a,o) )∗ (ϕ z)( p(a,o) ) = z( p(a,o) )ϕ( p(a,o) ) z( p(o,o) )∗ ϕ( p(a,o) )∗ z( p(o,o) ) ϕ( p(a,o) )∗ z( p(a,o) )∗ = z( p(a,o) ) ϕ( p(a,o) )∗ z( p(a,o) )∗ = ta , z is the embedding (6.5) associated with the join. The for any 0–simplex a, where ϕ preceding observation and Theorem 4 imply that ta ∈ R(a). According to (6.1) and by Lemma 8 we have

z(b) = t∂0 b (ϕ z)( p(∂0 b,o) ∗ p(∂1 b,o) ) t∂0 b ϕ = t∂0 b z( p(∂0 b,o) )ϕ( p(∂0 b,o) ∗ p(∂1 b,o) ) z( p(∂1 b,o) )∗ = z( p(∂0 b,o) ) ϕ( p(∂0 b,o) )∗ ϕ( p(∂0 b,o) ∗ p(∂1 b,o) ) z( p(∂1 b,o) )∗ = z( p(∂0 b,o) ) ϕ( p(∂1 b,o) )∗ z( p(∂1 b,o) )∗ = z( p(∂0 b,o) ) z( p(∂1 b,o) )∗ z( p(∂1 b,o) ) ϕ( p(∂1 b,o) )∗ z( p(∂1 b,o) )∗ = z(b) t∂1 b , for any 1–simplex b, and this proves the equivalence. (ii) As already observed if z ∈ Z1 (R K (M) ), then χz and z are joinable. Then z)(b) = z(b) z( p(∂1 b,o) ) χz (b) z( p(∂1 b,o) ) (χz = z( p(∂0 b,o) ∗ p(∂1 b,o) ) z( p(∂1 b,o) ) z( p(∂0 b,o) ∗ b ∗ p(∂1 b,o) ) z( p(∂1 b,o) )∗ = z(b), for any 1–simplex b, and this completes the proof.


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As an easy consequence of this theorem, we have the following Corollary 3. Assume that the fundamental group of M is Abelian. Fix a path-frame Po . Then for any irreducible 1–cocycle z we have that χz ∈ Z1 (K (M), C) and z(b) = χz (b) z(b),

b ∈ Σ1 (K (M)).

Proof. Since π1 (M, o) is Abelian the algebra Rz (M, o) is Abelian. Hence Rz (M, o) = C1 because of Corollary 2(ii). The proof now follows from Theorem 5. 6.3. The topological dimension. We now focus on irreducible 1–cocycles having finite statistics. We shall see that the charge component completely encodes the charge structure of 1–cocycles and find a relation between the statistical dimension and the topological content of 1–cocycles: the representation of the fundamental group carried by the 1–cocycle is, up to infinite multiplicity, irreducible and finite dimensional. The dimension of this representation is bounded from above by the statistical dimension, and is a new invariant of sectors: the topological dimension. Proposition 1. Let z be an irreducible object of Z1 (R K (M) )f whose statistical parameter λ(z) is equal to κ(z) d(z)−1 . Fix a path-frame Po . Then the following assertions hold. (i) If r and r solve the conjugate equations for z and z, then the same arrows solve the conjugate equations for z and z. (ii) z is an object with finite statistics which is, in general, a finite direct sum z = z 1 ⊕ · · · ⊕ z m , with m ≤ d(z), of irreducible objects of Zt1 (R K (M) )f having the same statistical phase as z and whose statistical dimension d(z i ) satisfies d(z) = d(z 1 ) + · · · + d(z m ). Proof. (i) The mapping Pc : Z1 (R K (M) ) → Zt1 (R K (M) ), sending an object z into its charge component z, with respect to Po , and acting as the identity on arrows is a faithful and symmetric tensor ∗ –functor (Lemma 7). It follows straightforwardly from these properties that if r and r solve the conjugate equations for z and z, then Pc (r ) = r and Pc (r ) = r solve the conjugate equations for z and z. This, in particular, implies that z has finite statistics. (ii) Given z , z ∈ Zt1 (R K (M) )f , define . ψz ,z (X ) =

1 Pc (r )∗ ⊗ 1z · 1Pc (z) ⊗ X · Pc (r ) ⊗ 1z , (6.9) d(z) . with X ∈ (Pc (z)⊗z , Pc (z)⊗z ).4 By [38, Prop.4.5], the collection ψ = {ψz ,z | z , z ∈ 1 Zt (R K (M) )f }, defines a standard left inverse of z, within the category Zt1 (R K (M) )f . Moreover z has the same statistical dimension as z. In addition, since Pc is symmetric we have ψz,z (ε(z, z)) = d(z)−1 Pc (r )∗ ⊗ 1z · 1Pc (z) ⊗ ε(z, z) · Pc (r ) ⊗ 1z = d(z)−1 Pc (r ∗ ⊗ 1z ) · Pc (1z ⊗ ε(z, z) · Pc (r ⊗ 1z ) = d(z)−1 Pc r ∗ ⊗ 1z · 1z ⊗ ε(z, z) · r ∗ ⊗ 1z = Pc (λ(z) 1z ) = λ(z) 1z . 4 Note that ψ (X ) = d(z)−1 r ∗ ⊗ 1 · 1 z ⊗ X · r ⊗ 1z with X ∈ (z ⊗ z , z ⊗ z ), according to z ,z z the definition of Pc .

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It is a well known fact, see for instance [18], that this relation implies that z is a finite direct sum of irreducible objects of Zt1 (R K (M) )f which have the same statistical phase as z and the sum of their statistical dimensions is equal to d(z). Proposition 2. The following assertions hold for any irreducible 1–cocycle z with finite statistics. (i) Let r, r be arrows solving the conjugate equations for z and z. Then the functional . ωoz (X ) =

1 ∗ z r y (o)(X ) ro , d(z) o x

X ∈ Rz (M, o),

(6.10)

is a normal and faithful tracial state of Rz (M, o). (ii) The algebra Rz (M, o) is a type Im factor with m ≤ d(z). Proof. Fix a path-frame Po and consider the charge component z, with respect to Po , of z. (i) We have seen in the proof of Proposition 6.9 that z has a standard left inverse ψ in Zt1 (R K (M) ), defined by Eq. (2). This implies that ψι,ι is a faithful tracial state of the algebra (z, z). Observe in particular that ψι,ι (t)o =

1 1 ∗ z (r ∗ ⊗ 1ι )o (1z ⊗ t)o (r ⊗ 1ι )o = r y (o)(to ) ro = ωoz (to ). d(z) d(z) o x

Hence ωoz is a faithful tracial state of (z, z)o because so is ψι,ι (z, z)o . Now the proof follows by (6.6) and by observing that the morphisms of stalks of 1–cocycles are locally normal. (ii) Note that (z, z)o is a finite dimensional algebra having at most d(z) minimal mutually orthogonal projections. According to the definition of a finite type I factor (see [35]), the proof follows by Corollary 2 and by (6.6). On the grounds of this result we can introduce the following notion. Definition 1. Given a 1–cocycle z with finite statistics, the topological dimension τ (z) of z is the dimension of the factor Rz (M, o). Let us see which are the main properties of this new notion and its meaning. First, the topological dimension is a quantum number of superselection sectors, i.e., it is an invariant of the equivalence class of a 1–cocycle, since Rz (M, o) is spatially equivalent to Rz 1 (M, o) whenever z is equivalent to z 1 . Secondly, the topological dimension is bounded from above by the statistical dimension ((ii) of Proposition 2). Thirdly, the topological dimension and the tracial state defined in (6.10) characterize the topological content of a 1–cocycle. To explain this point we need a preliminary result which is an easy consequence of Proposition 2. Corollary 4. Let z be an irreducible object with finite statistics. Then: (i) z, as a representation of π1 (M, o), is equivalent to a representation of the form 1H0 ⊗ σz where σz is a τ (z)-dimensional irreducible representation of π1 (M, o); (ii) the normalized character cσz of σz satisfies the equation cσz ([]) = ωoz (z()), where ωoz is the functional (6.10)

∈ Loops K (M) (o),


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Proof. (i) Since Rz (M, o) is a type Iτ (z) factor, there is a unitary operator U : H0 → H0 ⊗ Cτ (z) such that Rz (M, o) ∼ = 1H0 ⊗ Mτ (z) . Then the representation σz of π1 (M, o) defined by . 1H0 ⊗ σz ([]) = U z() U ∗

∈ Loops K (M) (o),

is irreducible, because 1Ho ⊗ (σz ) = (1Ho ⊗ σz ) = U Rz (M, o) U ∗ = 1 ⊗ Mτ (z) . (ii) follows from (i) and from uniqueness of the trace for the algebra Mτ (z) . Now recall that the category of finite dimensional representations of a topological group is equivalent to the category of finite dimensional representations of its Bohrcompactification [16, Prop.16.1.3]. Accordingly, finite dimensional representations of a topological group are classified by their characters, and this shows our claim. Fourthly, the topological dimension is stable under conjugation, this is the content of the next result. Lemma 9. Let z be an irreducible object with finite statistics. Then z and the conjugate z have the same topological dimension. Proof. Since z is irreducible, z is irreducible and d(z) = d(z). Let σz and σz be the τ (z)and τ (z)-dimensional representations of π1 (M, o) associated, respectively, with z and z by Corollary 4, and let cσz and cσz be the corresponding normalized characters. Given a loop over o, using Eq. (5.11) we have d(z) ωoz (z()) = ro∗ y(o)(z()) ro = ro∗ y(o)(z()) z() z()∗ ro = ro∗ (z × z)() z()∗ ro = ro∗ z()∗ ro = ro∗ ε(z, z)o y(o)(z()∗ ) ε(z, z)o ro = r ∗o y(o)(z()∗ ) r o = d(z) ωoz (z()∗ ), where we have used the relation that r = κ(z) · ε(z, z) · r (see [38]). Hence ωoz (z()) = ωoz (z())∗ ,

∈ Loops K (M) (o).

(6.11)

This relation and (ii) of Corollary 4 imply that cσz is equal to the adjoint of cσz . Hence σz is equivalent to the conjugated representation of σz ; the latter is irreducible and has dimension τ (z). Remark 4. We do not define the topological dimension for reducible objects. This can be understood by the following observation. If z is an irreducible 1–cocycle with finite statistics, then the representation of the fundamental group associated with z ⊕ z and z are equivalent. In fact by Corollary 4 we have (1H0 ⊗ σz ) ⊕ (1H0 ⊗ σz ) ∼ = 1H0 ⊗ σz , because of the infinite multiplicity. Thus, it is not possible to extend the topological dimension to reducible objects by additivity. Finally, we show the structure of those 1–cocycles whose topological dimension equals the statistical dimension. Lemma 10. Let z be an irreducible object with finite statistics. Assume that τ (z) = d(z). (i) Then z = u ⊕τ (z) , the τ (z)–fold direct sum of a simple object u of Zt1 (R(K (M)).

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(ii) If d(z) = 1 then τ (z) = 1, so z is a simple object, while χz takes values in C; hence z(b) = χz (b) z(b),

b ∈ Σ1 (K (M)).

Proof. (i) If τ (z) = d(z), then (z, z) has d(z) mutually orthogonal projections. Since z has statistical dimension equal to d(z) and since the statistical dimension is additive, z is a finite direct sum of d(z) simple subobjects u i . Moreover, all the subobjects of z are equivalent. Note that Rz (M, o) is a type Iτ (z) factor. So as a linear vector space it has dimension τ (z)2 . Observe that if there were a subobject u 1 , say, which is not equivalent to the other subobjects of z, then the algebra (z, z)o would have, as linear vector space, dimension less than (τ (z) − 1)2 + 1. This leads to a contradiction because Rz (M, o) ⊆ (z, z)o . (ii) follows from (i). 7. Existence and Physical Interpretation The first aim of this section is to show that for any irreducible finite dimensional representation of the fundamental group of the spacetime M, there is an irreducible object with finite statistics carrying, up to infinite multiplicity, this representation. This says that the category Z1 (R K (M) )f describes the Bohr-compactification of the fundamental group of the spacetime. The second aim is to show how the topology of spacetime affects the charges associated with Z1 (R K (M) )f and to point out the analogy with the Ehrenberg-Siday-Aharonov-Bohm effect. Recall that any representation of the fundamental group of the spacetime M defines, up to equivalence, a unique representation of the fundamental group of K (M). Thus, let σ be an irreducible n–dimensional representation of π1 (K (M), o). Now, pick a simple . object u, possibly u = ι, of Zt1 (R K (M) ) with, say, Bose-statistics. Define z = u ⊕n and note that the algebra (z, z)o is spatially equivalent to 1H0 ⊗Mn . Let U : H0 → H0 ⊗Cn be the unitary operator yielding this equivalence. Define . σ ([]) = U ∗ 1H0 ⊗ σ ([]) U,

[] ∈ π1 (K (M), o).

Observe that σ is a factor representation taking values in (z, z)o . Following [53], by this representation we can define a 1–cocycle ϕσ of K (M) taking values in (z, z)o . The representation of π1 (K (M), o) associated with ϕσ is equal to σ . Now, define . z σ = ϕσ z. By Theorem 5 z σ is a 1–cocycle of Z1 (R K (M) ) such that the representation of the fundamental group associated with z σ is σ (see within the proof the cited theorem). Hence, this fact and Theorem 2 prove the existence of topologically non-trivial net representations for the observable net A K (M) . We now show that z σ is an irreducible object with Bose-statistics and statistical dimension equal to n. We need a preliminary observation. Since χz σ is equivalent to ϕσ and z σ ∼ = z (Theorem 5), without loss of generality, from now on we assume χz σ = ϕσ and z σ = z. We now prove that z σ is irreducible. Assume that there is a non-zero projection t ∈ (z σ , z σ ) such that t = 1. Clearly t ∈ (z, z) and to commutes with ϕσ . Moreover, U to U ∗ is a non-zero projection of 1H0 ⊗ Mn which is different from the identity of this algebra and commutes with σ . This leads to a contradiction because σ is irreducible. Hence z σ is irreducible.


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Concerning the statistics, one can deduce by the particular form of z, that there is an isometry t ∈ (v, z ⊗n ) such that t · t ∗ = azn , where azn is the totally antisymmetric projection of (z ⊗n , z ⊗n ) defined by the representation εzn of the permutation group P(n) associated with z, and v is a Bosonic simple subobject z ⊗n (see [18]). Now, we observe that as a consequence of Lemma 7 the representation εzn equals the representation εznσ for any n (note that the functor Pc in Lemma 7 acts as the identity on arrows). This, in particular, implies that aznσ = azn and ε(z σ⊗n , z σ⊗n ) = ε(z ⊗n , z ⊗n ). On these premises, using the defining properties of a permutation symmetry, since v is a Bosonic simple object, we have 1 = ε(v, v) = t ∗ ⊗ t ∗ · ε(z ⊗n , z ⊗n ) · t ⊗ t, which is equivalent to saying that azn ⊗ azn = azn ⊗ azn · ε(z ⊗n , z ⊗n ). Hence aznσ ⊗ aznσ = aznσ ⊗aznσ ·ε(z σ⊗n , z σ⊗n ). Accordingly, the subobject w of z σ⊗n associated with the projection aznσ is a simple object. This is enough to prove that z σ has finite statistics [51]. By Proposition 1 z σ is Bosonic and has statistical dimension equal to n. This leads to the following existence theorem. Theorem 6. Let σ be an irreducible and n-dimensional representation of the fundamental group of the spacetime M. Then, there exists a 1–cocycle z of Z1 (R K (M) ) having the following properties: (i) z is an irreducible object with finite statistics whose statistical and topological dimension are equal to n; (ii) the representation of the fundamenal group of the spacetime M associated with z is equivalent to 1H0 ⊗ σ . We draw on some consequences of this result. First, we point out that Theorem 6 implies the existence of topologically non-trivial superselection sectors even when the only DHR-sector of the net R K (M) is the vacuum, i.e., Zt1 (R K (M) ) is formed by finite direct sums of the trivial 1–cocycle ι. Secondly, since any finite dimensional representation of π1 (M, o) appears, up to infinite multiplicity, as the representation of π1 (M, o) associated with a 1–cocycle with finite statistics, what we have shown is that the topological content of the category Z1 (R K (M) )f is the Bohr-compactification of the fundamental group of M. We now turn to explain how the topology of the spacetime M affects, if not trivial, the charges Z1 (R K (M) )f . Consider an irreducible 1–cocycle z with finite statistics. Fix a point x of the spacetime, and consider the family yxz (o), with o ∈ K (x ⊥ ), of localized transportable endomorphisms of the stalk R(x ⊥ ) associated with z. According to the interpretation given in DHR analysis, yxz (o) is a representation describing a charge within the region o; the 1–cocycle z is the transporter of this charge (see Remark 1). Then, transport this charge from o to another region o˜ along two different paths p and q such that the loop p ∗ q over o is not homotopic to the trivial loop. Then z( p ∗ q) yxz (o) = z( p) yxz (o)z(q) ˜ = yxz (o)z( p ∗ q) = yxz (o). This means that, analogously to the Ehrenberg-Siday-Aharonov-Bohm effect [21,1] the final state differs from the initial one of the unitary z( p ∗ q). The analogy is actually tighter if one thinks of 1–cocycles as flat connections of a principal bundle over the poset K (M) [49,50]). Then the difference from the initial to the final state is the parallel transport of the flat connection z along the loop p ∗ q.

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8. Comments and Outlook The present paper is, in our opinion, an interesting contribution to the topic of the existence of quantum effects induced by the topology of spacetimes. We have shown the existence of a new class of superselection sectors having well defined charge and topological contents, in the case in which the spacetime is multiply connected. These sectors are sharply localized in the same sense of those discovered by Doplicher, Haag and Roberts [18], but they are affected by the spacetime topology in a way similar to the quantum geometric phases. In a certain sense, the results of the present paper bring some support to the idea, suggested by Ashtekar and Sen [2], that non-trivial topologies may induce the existence of a new kind of particles (see also [31]). We think that the main and the new contribution given by the present paper to that idea resides in the fact that the expounded results are model-independent and based on a few, physically reasonable, assumptions. We have shown the charge and the topological content of sectors of Z1 (R K (M) ) in a fixed spacetime background M. As said at the beginning, it would be interesting to understand the locally covariant [11] behavior of these sectors. This may clarify some issues in the analysis of the locally covariant structure of DHR-sectors [12]. We now point out a central question arising from our results. Is there an underlying gauge theory giving rise to the charged sectors of Z1 (R K (M) )f ? We are asking whether it is possible either to provide models of gauge fields giving rise to sectors Z1 (R K (M) )f and, in general, whether one is able to reconstruct the fields and the gauge group underlying the charges Z1 (R K (M) )f , as Doplicher and Roberts have shown to happen for DHR-sectors and for BF-sectors (the Buchholz-Fredenhagen charges [13]) in Minkowski spacetime [20]. As far as models are concerned, a first positive, but very preliminary step has been provided in [8]: there, it is proven that a massive bosonic quantum field in a 2-dimensional spacetime (the Einstein cylinder) has a non-trivial topological cocycle that gives rise to non-trivial unitary representations of the fundamental group of the circle. Note that, on 2dimensional Minkowski spacetime, the model does not have any DHR sector of the usual kind besides the vacuum [41]. It is an intriguing question whether our selection criterion gives something really different from DHR, or new perspectives, in low dimensions. We hope to return elsewhere to this question too. As far as the abstract construction is concerned, the question about the DoplicherRoberts Reconstruction Theorem has, in our opinion, two different answers according to whether the fundamental group of the spacetime is Abelian or not. In the Abelian case, we think that there are no important differences from the scenery suggested by the Doplicher-Roberts reconstruction: there should exist a field net acted upon globally by the gauge group (the example in [8] goes in this direction). Conversely, in the nonAbelian case, we expect the scenery of the Doplicher-Roberts reconstruction to break down. It is reasonable to think that this happens because of the joining operation (6.8) that couples the charge and the topological component of sectors. When the fundamental group is not Abelian the coupling between the topological and the charge component is, in general, local: the function describing that coupling within Definition (6.8) depends on 1–simplices. Conversely, this coupling is global in the Abelian case because this function reduces to the identity for any 1–simplex (Corollary 3). It is not clear what could be the mathematical structure of the field theory underlying the charges Z1 (R K (M) )f in the non-Abelian case. One wonders whether it may resemble, at least in a certain sense, what people describe as topological field theories. Indeed,


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in that framework, theories of flat connections (for instance, Chern-Simons [25]) are considered, much as we do at the quantum level, by considering 1-cocycles with values in operator algebras. Acknowledgement. We gratefully acknowledge discussions with Klaus Fredenhagen and John E. Roberts. We wish to warmly thank Miguel Sánchez for sharing with us his insights in Lorentzian geometry.

A. Proof of Theorem 2 We prove the equivalence between SC(A K (M) ) and Z1 (R K (M) ). Let {π, ψ} and element of SC(A K (M) ). Given a 1–simplex b, take a simply connected subspacetime N such that cl(|b|) ⊂ N , and define ∗ . z π (b) = WaN ∂0 b WaN ∂1 b ,

cl(a) ⊂ N , a ⊥ |b|,

(A.1)

where W N o is the unitary satisfying the selection criterion (3.8). First of all we prove this definition is well posed. By property 2 of (3.8), if a˜ ⊆ a, then ∗

∗

∗

WaÑ ∂0 b WaÑ ∂1 b = WaN ∂0 b ψa a˜ ψa∗a˜ WaN ∂1 b = WaN ∂0 b WaN ∂1 b . Since cl(|b|) ⊂ N we have that |b| is a diamond of N . So the causal complement of |b|, relative to N , is pathwise connected. This and the above identity leads to the independence of the choice of a. Concerning the independence of the choice of N , consider a second simply connected subspacetime N such that cl(|b|) ⊆ N ∩ N . By Lemma 14 there is a diamond O and a diamond a such that O ⊆ N ∩ N , cl(|b|), cl(a) ⊂ O and |b| ⊥ a. Note that O is a simply connected subspacetime of M. Property 3 of (3.8) implies that WaN ∂i b = WaN ∂i b = WaO∂i b for i = 0, 1. This and the independence of the choice of a, leads to the independence of the choice of N . Hence z π is well defined. Now, the cocycle identity follows from the definition of z π and from the independence of the choice of N and a. In fact given a 2–simplex c take N such that cl(|c|). By Lemma 14 there is a such that cl(a) ⊂ N and c ⊥ a. Then ∗

∗

∗

∗

z π (∂0 c)z π (∂2 c) = WaN ∂00 c WaN ∂10 c WaN ∂02 c WaN ∂12 c = WaN ∂01 c WaN ∂10 c WaN ∂10 c WaN ∂11 c = WaN ∂01 c WaN ∂11 c = z π (∂1 c).

∗

Concerning the locality condition, note that by outer regularity and Haag duality it is easily seen that R(o) = ∩{R(o) ˜ | cl(o) ⊥ cl(o)} ˜ (see for instance [52]). So given a 1– simplex b pick a 0–simplex o such that cl(|b|) ⊥ cl(o). By the smoothability argument [6] there is a spacelike Cauchy surface C which contains the closure of the bases of the diamonds |b| and o. Moreover, since the closure of the bases of |b| and o are disjoint there is a simply connected open subset G of C which contains the closure of both the bases. Then, the domain of dependence of G is a simply connected subspacetime N which contains cl(|b|) and cl(o). Since (A.1) is independent of the choice of a, by property 1 of (3.8) we have ∗

∗

z π (b) ιo (A) = WoN ∂0 b WoN ∂1 b ιo (A) = ιo (A) WoN ∂0 b WoN ∂1 b = ιo (A) z π (b),

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for any A ∈ A(o). Since this holds for any cl(o) ⊥ cl(|b|), the above observation implies that z π (b) ∈ A(|b|). This proves that z π ∈ Z1 (R K (M) ). Consider now {σ, φ} ∈ SC(A K (M) and an arrow T ∈ ({π, ψ}, {σ, φ}). For any 0–simplex a take o ⊥ a and N such that cl(a), cl(o) ⊂ N . Define ∗ . ta = VoN a To WoN a , (A.2) where W and V are the unitary associated with {π, ψ} and {σ, φ}, respectively, by the selection criterion. As above t is well defined and independent of the choices of o and N . Given a 1–simplex b, pick N with cl(|b|) ⊂ N and let o be such that |b| ⊥ o and cl(o) ⊂ N . Then ∗

∗

∗

t∂0 b z π (b) = VoN ∂0 b To WoN ∂0 b WoN ∂0 b WoN ∂1 b = VoN ∂0 b To WoN ∂1 b = z σ (b) t∂1 b . One can also easily see that ta ∈ R(a), hence t ∈ (z π , z σ ). We now construct the map from net cohomology to net representations satisfying the selection criterion. Given a 1–cocycle z ∈ Z1 (R K (M) ), define . πaz (A) = z(qa ) ιa (A) z(qa )∗ , a ∈ K (M), A ∈ A(a), (A.3) z . ˜ a˜ ⊆ a, ψa a˜ = z(a, a), where qa is a path with ∂0 qa = a and ∂1 qa ⊥ a, and (a, a) ˜ denotes the 1–simplex such that ∂1 (a, a) ˜ = a, ˜ ∂0 (a, a) ˜ = |(a, a)| ˜ = a. The independence of the chosen path qa is a consequence of (4.5). This also implies that πaz = z(a, a) ˜ πa˜z z(a, a) ˜ ∗ = ψaza˜ πa˜z ψaza˜ ∗ z z for any a˜ ⊆ a. Hence the pair {π , ψ } is a net representation. Note that the 1–cocycle z ζ π , associated with this net representation by (2.8), is equivalent to z. In fact, we have ∗ z z ζ π (b) = ψ|b|∂ ψ|b|∂1 b = z(|b|, ∂0 b)∗ z(|b|, ∂1 b) = z(b) 0b z

for any 1–simplex b. We now prove that this net representation satisfies the selection criterion (3.8). Given o ∈ K (M), let N be simply connected subspacetime such that cl(o) ⊂ N . Given a ⊥ o, with cl(a) ⊂ N , define . WaN o = z( po,a ), (A.4) where po,a is the path from a to o whose support has closure contained in N . Clearly this definition does not depend on p since N is simply connected. Furthermore, given a path qa as in definition (A.3), by relation (4.4) we have WaN o πaz (A) = z( po,a ) z(qa ) ιa (A) z(qa )∗ = z( po,a ∗ qa ) ιa (A) z(qa )∗ = ιa (A) z( po,a ) = ιa (A) WaN o, for any A ∈ A(a), because the endpoints of po,a ∗ qa are causally disjoint from a. Hence {π z , ψ z } is a sharp excitation of the reference representation. Finally, given t ∈ (z, zˆ ) define . Ta = ta , a ∈ Σ0 (K (M)). (A.5) Given A ∈ A(a) we have Ta πaz (A) = ta z(qa ) ιa (A) z(qa )∗ = zˆ (qa ) t∂1 qa ιa (A) z(q)∗ = zˆ (qa ) ιa (A) t∂1 qa z(qa )∗ = zˆ (qa ) ιa (A) zˆ (qa )∗ ta = πazˆ (A) Ta .


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Moreover, it is easily seen that Ta ψaza˜ = ψazâ˜ Ta˜ , if a˜ ⊆ a. Thus, T is an intertwiner from {π z , ψ z } to {π zˆ , ψ zˆ }. Now, the functor F : SC(A K (M) ) → Z1 (R K (M) ) defined by means of Eqs. (A.1) and (A.2): given {π, ψ}, {σ, φ} and T ∈ ({π, ψ}, {σ, φ}) define . F({π, ψ})(b) = z π (b), b ∈ Σ1 (K (M)), (A.6) ∗ . Na F(T )a = Vo To WoN a , a ∈ Σ0 (K (M)), where W and V are the operators associated with {π, ψ} and {σ, φ}, respectively, by the selection criterion. The functor G : Z1 (R K (M) ) → SC(A K (M) ) is defined by Eqs. (A.3) and (A.4): given z, zˆ ∈ Z1 (R K (M) ) and t ∈ (z, zˆ ), define . G(z) = {π z , ψ z }, . (A.7) G(t) = t. We first study the composition F G. Consider a 1–simplex b. Pick a 0–simplex a with cl(a) ⊂ N and a ⊥ |b|. By using (A.4) and (A.1) we have F G(z)(b) = z( p∂0 b,a ) z( p∂1 b,a )∗ = z( p∂0 b,a ∗ p∂1 b,a ) = z(b), because p∂0 b,a ∗ p∂1 b,a is a path from ∂1 b to ∂0 b and its closure is in N . Hence it is homotopic to b because N is simply connected. Moreover by (A.5), (A.4), (A.2), we have F G(t)a = zˆ ( pa,o ) to z( p˜ a,o )∗ = ta for any 0–simplex a, because pa,o and p˜ a,o are paths from o to a. Hence F G = idZ1 (R K (M) ) . We now show that there is a natural isomorphism from G F and idSC(A K (M) ) . First of all, given a, take a simply connected region N which contains cl(a) and pick o, a˜ such that o ⊥ a, a˜ and a˜ ⊥ o. Define ∗ ∗ . (A.8) Sa ({π, ψ}) = WaN a˜ WoN a˜ WoN a , where W are the unitaries associated with {π, ψ} by the selection criterion. By the definitions of G and F we have G F({π, ψ})a (A) = F({π, ψ})(qa ) ιa (A) F({π, ψ})(qa )∗ ∗

∗

= WoN a WoN a˜ ιa (A) WoN a˜ WoN a , where a˜ ⊥ a. In fact, the path qa is from a to a ⊥ . Since, the definition of F({π, ψ}) does depend neither on the choice of the path nor on the choice of the 0–simplex in a ⊥ , we have considered a path which lies on N from a to a. ˜ Finally since N is simply connected, ∗ F({π, ψ})(qa ) depends only on the endpoints of qa . Thus F({π, ψ})(qa ) = WoN a WoN a˜ . Using this and (A.8) we have ∗

Sa ({π, ψ}) G F({π, ψ})a (A) = Sa ({π, ψ}) WoN a WoN a˜ ιa (A) WoN a˜ WoN a ∗

= WaN a˜ ιa (A) WoN a˜ WoN a ∗

∗

= πa (A) WaN a˜ WoN a˜ WoN a = πa (A) Sa ({π, ψ}).

∗

Furthermore, given a1 ⊆ a we have ∗

∗

∗

ψaa1 Sa1 ({π, ψ}) = ψaa1 WaN1 a˜ WoN a˜ WoN a1 = WaN a˜ WoN a˜ WoN a1 ∗

∗

= WaN a˜ WoN a˜ WoN a WoN a WoN a1 π

z = Sa (π ) ψaa . 1

∗

∗

∗

558


This proves that S({π, ψ}) is a unitary intertwiner from G F({π, ψ}) to {π, ψ}. Finally, given T ∈ ({σ, φ}, {π, ψ}) then Sa ({π, ψ}) G F(T )a = Sa ({π, ψ}) F(T )a ∗

∗

∗

∗

∗

= WaN a˜ WoN a˜ WoN a WoN a To VoN a = WaN a˜ WoN a˜ To VoN a˜ VoN a˜ VoN a ∗

∗

∗

∗

= WaN a˜ WaN a˜ Ta VaN a˜ VoN a˜ VoN a = Ta VaN a˜ VoN a˜ VoN a = Ta Sa ({σ, φ}), ∗

∗

∗

∗

where the property WoN a˜ To VoN a˜ = WaN a˜ Ta VaN a˜ has been used; W and V denote the unitaries associated, respectively, with {π, ψ} and {σ, φ} by the selection criterion. This completes the proof of Theorem 2. B. Miscellanea on the Causal Structure In this section we provide some results about the causal structure of globally hyperbolic spacetimes. In what follows M denotes a connected globally hyperbolic spacetime with dimension d = 4. References for this Appendix are the same as those quoted in Sect. 3.1. To begin with, we recall that the future (past) domain of dependence D ± (A) of a set A is the union of those points such that any inextendible past (forward) directed causal curve starting from these points meets A. It is worth observing that, if A is a closed achronal set then the closure of D ± (A) is the union of those points having the same property (relative to A) for timelike curves. The first result is due to the courtesy of Miguel Sánchez. We are indebted with him for the nice short proof. Lemma 11 (M. Sánchez). Let C be a spacelike Cauchy surface of M, and let K be a nonempty closed subset of C. Then D ± (K ) is closed. Proof. Let q be a point in the boundary of D + (K ) not included in D + (K ). Then there is a past-directed inextendible causal curve through q which does not cross K . Nevertheless, it will cross C at some point p. As K is closed in C, a neighborhood U of p in C does not intersect K . Easily, there are points in U chronologically related to q ( p lies in J − (q) which is the closure I − (q)). Let p1 be one such point. The past directed timelike curve from q to p1 cannot intersect K (as it cannot intersect C again). This contradicts the fact that cl(D + (K )) is the set of points x of M such that every inextendible timelike curve through x crosses K . If A ⊆ C let us denote by intC (A) the internal part of A in the relative topology of C. Note that as C is a closed subset of M, the closure of A in the relative topology of C coincides with its closure in the topology of M. Lemma 12. Let C be a spacelike Cauchy surface of M, and let K be a nonempty closed subset of C. Assume that K ⊆ cl(intC (K )). Then cl(D ± (intC (K ))) = D ± (K ). Proof. As far as the first inclusion is concerned, as D + (intC (K )) ⊂ D + (K ), we have cl(D + (intC (K ))) ⊆ D + (K ) by Lemma 11. For the opposite inclusion, take x ∈ D + (K ). Note that if x ∈ K , then x ∈ cl(D + (intC (K ))). In fact as K ⊆ cl(intC (K )) we have that K ⊂ cl(D + (intC (K ))). So assume


559

that x ∈ D + (K )\K . Since, by Lemma 11, D + (K ) is closed, we have that I − (x)∩C ⊂ K . However, I − (x) is open in M, hence I − (x) ∩ C is open in the relative topology of C; thus I − (x) ∩ C ⊆ intC (K ). Take a sequence of points {yn } converging to x such that yn ∈ I − (x) ∩ I + (K ) for any n ∈ N. Note that J − (yn ) ⊂ I − (x) because yn ∈ I − (x). Hence J − (yn ) ∩ C ⊂ I − (x) ∩ C ⊆ intC (K ). Thus yn ∈ D + (intC (K )) for any n. Since yn → x, then x ∈ cl(D + (intC (K ))). The same argument applies to the past development and the proof is over. Corollary 5. Let C be a spacelike Cauchy surface of M, and let O be a nonempty open subset of C in the relative topology of C. Assume that intC (cl(O)) ⊆ O. Then cl(D ± (O)) = D ± (cl(O)). . Proof. Note that the closed set K = cl(O) satisfies the hypothesis of Lemma 12. In fact intC (cl(O))= O, because O is open; hence cl(intC (K ))= cl(intC (cl(O)))= cl(O) = K , and the proof follows by Lemma 12. This result applies to diamonds. Let o be a diamond whose base G lies on a spacelike Cauchy surface C. G is an open and relatively compact subset in the relative topology of C. Moreover, intC (cl(G)) = G. Then as a consequence of the previous result we have cl(o) = cl(D(G)) = D(cl(G)).

(B.1)

We now are in a position to prove the following property of diamonds. Lemma 13. Let o be a diamond of M. Then there is a sequence {on , an } of pairs of diamonds based on the same Cauchy surface as C and satisfying the following properties: (i) cl(on+1 ) ⊂ on for any n ∈ N and ∩n on = cl(o); (ii) cl(an ) ⊂ on and cl(an ) ⊥ cl(on+1 ) for any n ∈ N. ⊥ (iii) on+1 ∩ on and an⊥ ∩ on are connected for any n ∈ N. Proof. The proof is divided into three parts. In the first step we construct, by induction, a sequence of pairs of “formal” diamonds satisfying the properties (i) and (ii) in the statement. By a formal diamond we mean a set satisfying all the properties to be a diamond with the exception that it might be non-relatively compact. In the second step we prove that from the above sequence it is possible to extract a subsequence of diamonds. In the third step we prove that these sequences of diamonds satisfy property (iii). First step. According to the definition of diamonds there is a spacelike Cauchy surface C, a chart (U, φ) of C, and an open ball B of R3 such that cl(B) ⊂ φ(U ), o = D(G), . . where G = φ −1 (B). Define W1 = U . Since cl(B) is a proper compact subset of the open φ(W1 ) the distance d of B from R3 \φ(W1 ) is strictly positive. Accordingly, denoting the radius of B by r , define B1 as the open ball having the same centre as B and radius . r1 = r + d/2, and let L 1 be an open ball such that cl(L 1 ) ∩ cl(B) = ∅ and cl(L 1 ) ⊂ B1 . . . Observe that if we define G 1 = φ −1 (B1 ) and H1 = φ −1 (L 1 ), then by construction and by Lemma 12, we have cl(o), cl(D(H1 )) ⊂ D(G 1 ), cl(o) ⊥ cl(D(H1 )), cl(G 1 ) ⊂ W1 .

(*)

By induction, for n ≥ 1, assume that we have a 3-tuple (G n , Hn , Wn ) satisfying (∗). . Define Wn+1 = (Wn\cl(Hn )) ∩ G n . Applying the above construction with respect to Wn+1 we get a pair G n+1 and Hn+1 . This procedure leads to a sequence {G n , Hn , Wn }

560


satisfying (∗) for any n. Furthermore, by construction we have that cl(Hn ) ⊥ cl(G n+1 ) and cl(G n+1 ) ⊂ G n for any n; moreover ∩n G n = cl(G). (**) . . Define on = D(G n ) and an = D(Hn ). Hence we have cl(on+1 ) ⊂ o; cl(an ) ⊂ on and cl(an ) ⊥ cl(on+1 ) for any n. We now show that ∩n on = cl(o). Clearly cl(o) ⊆ ∩n on . Let x ∈ ∩n on . Since the sets G n belong to the same Cauchy surface C, there are two possibilities: either x ∈ D + (G n ) for all n, or x ∈ D − (G n ) for all n. Assume x ∈ D + (G n ) for all n. Then, J − (x) ∩ C ⊆ G n for all n; hence by (∗∗) J − (x) ∩ C ⊆ cl(G), thus x ∈ D + (cl(G)) = cl(D + (G)) ⊂ cl(o). The same argument applies if x ∈ D − (G n ) for all n. This proves our claim. Second step. We now prove that the sequence {on , an } is definitively formed by diamonds of M. To start with, note that there is a smooth foliation F : Σ × R → M of the spacetime, by spacelike Cauchy surfaces, such that F(Σ, 0) = C [6]. To be precise, F : Σ × R → M is a diffeomorphism such that F(Σ, t) is a spacelike Cauchy surface . for any t ∈ R and the curve γ y (t) = F(y, t) as t varies in R, for a fixed y ∈ Σ, is an inextendible forward directed timelike curve. The inverse function F −1 is equal to (h(x), τ (x)), where h and τ are smooth functions from M to Σ and R, respectively. On these grounds, since cl(o) is compact there are t1 , t2 ∈ R such that t1 < t2 and t1 < {τ (x) ∈ R | x ∈ cl(o)} < t2 ; . in other words cl(o) is a proper subset of F(Σ, (t1 , t2 )). Let K Σ = {h(x) ∈ Σ | x ∈ Σ cl(G 1 )}, where G 1 is the first set constructed above. K is a compact subset of Σ. Note that, by the properties of the foliation, we have cl(on ) = cl(D(G n )) is contained in F(K Σ , R) for any n. Consider now the cylinder F(K Σ , [t1 , t2 ]). We claim that there exists a k such that cl(on ) ⊆ F(K Σ , [t1 , t2 ]) for n ≥ k. If it were not so, there should be a sequence of points xn ∈ cl(on ) ∩ (F(K Σ , t1 ) ∪ F(K Σ , t2 )). So, extract a subsequence {xsn } which belongs say to cl(osn ) ∩ F(K Σ , t1 ). This is a compact set because so is F(K Σ , t1 ). So there is a subsequence {xrsn } converging to a point x. Clearly x ∈ F(K Σ , t1 ). Moreover x ∈ cl(o) because x ∈ cl(orsn ) for any n. This leads to a contradiction because by construction cl(o) ⊂ F(K Σ , (t1 , t2 )). Finally, for n ≥ k the sets on , an are contained in the compact set F(K Σ , [t1 , t2 ]), so they are relatively compact, and the proof follows. Third step. Since the sequence {on , an }n≥k is formed by diamonds based on the same ⊥ Cauchy surface, by (B.1), it is easily seen that on+1 ∩ on = D(G n\cl(G n+1 )) and ⊥ an ∩ on = D(G n \cl(Hn )). Then the proof of property (iii) follows since the sets G n\cl(G n+1 ) and G n\cl(Hn ) are, by construction, connected. Lemma 14. Let o be a diamond of M and let W be an open subset of M such that cl(o) ⊂ W . Then there are two diamonds o and a based on the same Cauchy surface as o such that: (i) cl( o) ⊂ W , cl(o), cl(a) ⊂ o and cl(o) ⊥ cl(a); (ii) the sets o⊥ ∩ o and a ⊥ ∩ o are connected. Proof. Let {on , an } be a sequence of diamonds satisfying the properties of Lemma 13. The idea of the proof is the same as the second step of the proof of Lemma 13. Briefly, assume that cl(on ) ∩ (M\W ) = ∅ for any n. Take a sequence of points {xn } with xn ∈ cl(on ) ∩ (M\W ). Since cl(on ) ∩ (M\W ) ⊂ cl(o1 ), for any n, and since the latter


561

is compact, there is a subsequence {xsn } converging to a point x. The limit point x is in cl(o) ∩ (M \ W ) and this leads to a contradiction. So there is k such that for n ≥ k cl(on ) ⊂ W , and the proof follows by the properties of the sequence {on , an }. We now show a homotopy deformation result whose proof is very similar to the proof of the Van Kampen theorem about the fundamental groups of topological spaces (see [27]). Lemma 15. Let o be a diamond of M. If γ : [0, 1] → M is a curve with γ (0), γ (1) ∈ o⊥ , then γ is homotopic to a curve γ˜ lying in o⊥ . Proof. Let x = γ (0) and y = γ (1). Since o⊥ is connected, there is a curve γ˜ : [0, 1] → o⊥ such that γ˜ (0) = y, γ˜ (1) = x. So, the composition γ1 ∗ γ is a loop over x. By the definition of causal complement and since diamonds are relatively compact, we have that o⊥ = (cl(o))⊥ . So the point x is causally disjoint from cl(o). Let G be the base of the diamond o. By Lemma 14 there is a diamond o1 whose base G 1 is contained in the same Cauchy surface as G and such that cl(G) ⊂ G 1 , G 1\cl(G) is connected and cl(o1 ) ⊥ x. By [6], there is a Cauchy surface C which contains G, G 1 and x and cl(G 1 ) is disjoint from x. Since any Cauchy surface is a deformation retract of M the loop γ1 ∗ γ is homotopic to a loop γ2 lying in C. The loop γ2 meets cl(G) (otherwise the proof would be complete), hence it meets G 1\cl(G). Let z be a point where γ1 meets G 1 \cl(G). Since C\cl(G) is connected there is a curve τ , in C\cl(G), from x to z. Then τ ∗ γ2 ∗ τ is a loop over z (τ is the reverse of τ ). Since G 1 and C\cl(G) form an open cover of C, we have that τ ∗ γ1 ∗ τ ∼ βn ∗ βn−1 ∗ · · · ∗ β1 , where βi is a curve in C, contained either in G 1 or C \cl(G) (the Lebesgue’s covering lemma for [0, 1]). However we can assume that any βi is a loop over z, either in G 1 or C\cl(G), because G 1\cl(G) is connected. Hence observing that G 1 is contractible, we have that τ ∗ γ1 ∗ τ ∼ β, where β is a loop over z in C\cl(G). Thus γ2 ∼ τ ∗ β ∗ τ . . Clearly τ ∗ β ∗ τ lies in o⊥ . Since γ2 ∼ γ1 ∗ γ , then γ ∼ γ˜ , where γ˜ = γ1 ∗ τ ∗ β ∗ τ ⊥ is a curve from x to y lying in o . It is obvious but worth observing that the same result holds if we replace in the statement of this lemma o⊥ with the causal complement x ⊥ of a point x of M. We now provide a version of this lemma in terms of the poset K (M). Corollary 6. Let o be a diamond of M. (i) If p is a path with ∂1 p, ∂0 p in o⊥ , then p is homotopic to a path q whose support is contained in o⊥ . (ii) If p is a path with cl(∂1 p), cl(∂0 p) in o⊥ , then p is homotopic to a path q whose support has closure contained in o⊥ . Proof. The proof is an easy consequence of Lemma 15 and [53, Lemma 2.17]. As before we observe that the results of this corollary hold if we replace in the statement o⊥ with the causal complement x ⊥ of a point x of M. Remark 5. The results of this section hold for any globally hyperbolic spacetime with dimension d ≥ 3. They continue to hold in dimension 2 apart from the following exceptions: item (iii) of Lemma 13, item (ii) of Lemma 14 and, consequently, Lemma 15 and Corollary 6.

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Commun. Math. Phys. 287, 565–588 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0688-x

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Hölder Continuity of the Rotation Number for Quasi-Periodic Co-Cycles in SL(2, R) Sana Hadj Amor Ecole Nationale d’Ingénieurs de Monastir, Avenue Ibn Eljazzar, 5019 Monastir, Tunisia. E-mail: [email protected] Received: 10 March 2008 / Accepted: 27 August 2008 Published online: 20 November 2008 – © Springer-Verlag 2008

Abstract: We prove two results on the rotation number of the skew-product system (ω, A) : (θ, y) ∈ Td × R2 → (θ + ω, A(θ )y) ∈ Td × R2 , where ω is Diophantine and A(θ ) ∈ S L(2, R) is homotopic to the identity. On the one hand, we prove that this function has the behavior of a 21 − Hölder function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. We give also an estimate of the complement of the spectrum. These results are obtained by studying the reducibility of the quasi-periodic co-cycle (ω, A).

1. Introduction and Main Results Consider a co-cycle of the form X n+1 = (A + F(θ + nω))X n , n ∈ Z,

(∗)

where A ∈ S L(2, R), A + F(θ + nω) ∈ S L(2, R), X n ∈ R2 and θ ∈ Td (T = R/2π Z). The transformation F : Td → gl(2, R) is assumed to be analytic in a complex neighbourhood |x| < r ≤ 1 of the d−torus Td , and we shall use the norm |F|r = sup|x| 0 and K > 0 such that the bound < n, ω > − π j| ≥ K |n|−τ 2 j∈Z

inf |

DC(K , τ )

holds for any integer n ∈ Zd \{0}.( is the usual Euclidean product in Rd .) Such ω exists for τ > d − 1.

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1.1. Diophantine number and rational number. A real number ρ is said to be Diophantine (with respect to ω2 ) if there exist L > 0 and s > 0 such that inf |ρ − j∈Z

< n, ω > − π j| ≥ L|n|−s , n ∈ Zd \{0}, 2

and it is said to be rational (with respect to

ω 2)

if inf j∈Z |ρ −

DCω (L , s), 2

− π j| = 0.

a(θ ) b(θ ) ∈ S L(2, R) and we c(θ ) d(θ ) define the map T(ω,A+F) : (θ, ϕ) ∈ Td × 21 T → (θ + ω, φ(ω,A+F) (θ, ϕ)) ∈ Td × c(θ)+d(θ) tan ϕ 1 1 2 T, ( 2 T = R/π Z), where φ(ω,A+F) (θ, ϕ) = arctan( a(θ)+b(θ) tan ϕ ). Assume that a, b, c and d are continuous on Td and A + F(θ ) is homotopic to the identity, then the same is true for the map T(ω,A+F) and therefore it admits a continuous lift T˜(ω,A+F) : (θ, ϕ) ∈ Td × R → (θ + ω, φ˜ (ω,A+F) (θ, ϕ)) ∈ Td × R such that φ˜ (ω,A+F) (θ, ϕ) mod π = φ(ω,A+F) (θ, ϕ mod π ). The function (θ, ϕ) → φ˜ (ω,A+F) (θ, ϕ)−ϕ is (2π )d -periodic in θ and π -periodic in ϕ. We define now ρ(φ˜(ω,A+F) ) by ρ(φ˜ (ω,A+F) ) = lim supn→+∞ n1 ( p2 ◦ n T˜(ω,A+F) (θ, ϕ) − ϕ) ∈ R, where p2 (θ, ϕ) = ϕ. This limit exists for all θ ∈ Td and for all ϕ ∈ R and the convergence is uniform in (θ, ϕ). (For the existence of this limit and its properties we refer to [9].) The class of the number ρ(φ˜ (ω,A+F) ) in 21 T, which is independent of the chosen lift, is called the rotation number of the skew-product system (ω, A + F) : (θ, y) ∈ Td × R2 → (θ + ω, (A + F(θ ))y) ∈ Td × R2 and we denote it by ρ(ω,A+F) . However, in the analysis of results we must consider rotation numbers of different systems, so we describe this concept and some elementary properties in an Appendix. 1.2. The rotation number. Let A + F : θ ∈ Td →

1.3. Description of results. We shall formulate our result for the co-cycle (∗) : Theorem 1. There exists a constant ε = ε(r, K , τ ) such that if |F|r < ε, then the following hold: If ρ(ω,A+F) is Diophantine or rational with respect to ω2 , then there exists a matrix B in S L(2, R) and an analytic matrix valued function Z : (2T)d → S L(2, R),1 such that X n = Z (θ + nω)B n Z −1 (θ )X 0 is a solution of (∗) for any vector X 0 ∈ R2 . Theorem 1 is a statement about reducibility of co-cycle (∗). The associated first order system at the discrete, one-dimensional Schrödinger operator with real analytic small potential and one Diophantine frequency is a particular instance of a quasi-periodic co-cycle (∗). Indeed, if we consider the discrete, one-dimensional Schrödinger operator Hθ , acting on the Hilbert space 2 (Z), (Hθ u)n = −(u n+1 + u n−1 ) + V (θ + nω)u n , n ∈ Z, where V : Td → R is an analytic function (the potential) in a complex neighbourhood |x| < r ≤ 1 of Td , θ ∈ Td and ω is Diophantine, then the second order differential 1 Z is 4π periodic in each variable.

Hölder Continuity of Rotation Number for Quasi-Periodic Co-Cycles

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equation, (Hθ − E)u n = 0 with n ∈ Z and E ∈ R (the energy), can be written as a first order system u n+1 = (A + F(θ + nω))X n , (∗∗) X n+1 = un −E −1 V (θ + nω) 0 where A = and F(θ + nω) = . In this case, the matrix B, 1 0 0 0 the matrix valued function Z and the rotation number ρ(ω,A+F) will depend on E. In the KAM (Kolmogorov-Arnold-Moser) theory it is known that quasi-periodic Schrödinger co-cycles are reducible for a set of energies in a part of the spectrum of large measure. In [4], E. I. Dinaburg and Ja. G. Sina˘ı showed that the continuous quasi-periodic Schrödinger co-cycle with small potential and Diophantine frequency is reducible to constant coefficients for a set of energies E under the assumption that ρ(ω,A+F) (E) is Diophantine but this set is not of full measure in the spectrum. J. Moser and J. Pöschel, in [11], gave an extension of a result by Dinaburg and Sina˘ı. They constructed a set of energies E for which the Schrödinger co-cycle is reducible and ρ(ω,A+F) (E) is rational but this set was also not as large as one could hope for. Both sets of energies are defined by certain arithmetic conditions on the rotation number, it is not of full measure. In [6], L.H. Eliasson, generalizing the KAM approach, showed that the smallness condition on V is completely freed from any dependence of the arithmetic properties of ρ(ω,A+F) (E) other than being Diophantine or rational. Reducibility of the quasi-periodic co-cycle (∗∗) is a discrete version of the work of L.H. Eliasson. This subject is of great interest in “perturbative” theory and a similar result on S O(3, R) and on more general compact groups is given in [10]. (To show the present knowledge in this statement, we refer the reader to [5].) Consider now two families of matrices {A(E)} E∈R ⊂ S L(2, R) and {F(E, θ )} E∈R ⊂ gl(2, R) such that {A(E) + F(E, θ )} E∈R ⊂ S L(2, R) for each θ ∈ Td and consider the co-cycle (∗), X n+1 = (A(E) + F(E, θ + nω))X n , n ∈ Z, also depending on E. We shall assume that for every E ∈ R, F(E, .) : Td → gl(2, R) is real analytic in a complex neighbourhood |x| < r ≤ 1 of the d−torus Td . The rotation number of the skew-product system (ω, A(E) + F(E, .)) : (θ, y) ∈ Td × R2 → (θ + ω, (A(E) + F(E, θ ))y) ∈ Td × R2 will depend on E. Let ω ∈ DC(K , τ ), K ≤ 1 and let A and F be C 2 on I (I is an interval of R). We assume that for every E ∈ I , we have 1 | ∂∂E trA(E)| ≥ ε 600 where C is a constant that doesn’t depend on E, and ν | ∂∂ E ν trA(E)| ≤ C ν ν = 1, 2, that ρ(ω,A+F) is monotone on I. Then, the following holds: Theorem 2. There exists a constant c that only depends on τ, d, K and ν ν max0≤ν≤2 sup E∈I | ∂∂E ν A(E)| such that if | ∂∂E ν F(E)|r = ε ≤ c r 600(21τ +12+d) ; ν = 0, 1, 2, then κ

−1 ( <m,ω> + π Z)) ≤ c e−γ |m| 2 , where 1. For every m ∈ Zd \{0}, we have Leb(ρ(ω,A+F) 2 c and κ are two numerical constants and γ is a constant that only depends on K , τ and r. √ 2. For every E 0 and E in I, we obtain |ρ(ω,A+F) (E) − ρ(ω,A+F) (E 0 )| ≤ c |E − E 0 |, where c is a numerical constant. mod π, n ∈ Zd − half the frequency module of V. As a conLet now M = 2 sequence of the above theorem and with the same assumptions, we have the following result:

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Theorem 3. There exists a constant c that only depends on τ, d, K and ν ν max0≤ν≤2 sup E∈I | ∂∂E ν A(E)| such that if | ∂∂E ν F(E)|r = ε ≤ c r 600(21τ +12+d) ; ν = 0, 1, 2, then κ 1 4τ

−1 Leb([E 1 , E 1 + δ] ∩ ρ(ω,A+F) (M)) ≤ β1 e−β2 ( δ ) , for all δ > 0, −1 where E 1 is the upper bound of the connected component [E 0 , E 1 ] of ρ(ω,A+F) (M), κ is a numerical constant, β1 is a constant that only depends on K , d, τ, r, C and M and β2 is a constant that only depends on K , d, τ, r and C.

For the associated first order system at the discrete, one-dimensional Schrödinger operator and in order to give a unique representation for the rotation number in the case of system (∗∗), we introduce the integrated density of states k : R → [0, 1] defined by ⎧ ⎨ k(E) = 0 if E ≤ E 0 ρ (E) k(E) = (ω,A+F) mod 1 if E0 < E < E1 π ⎩ k(E) = 1 if E ≥ E 1 , where E 0 and E 1 are respectively the lower and the upper bound of the spectrum. The I.D.S. is a monotone and continuous function, and σ (H ) = R\k −1 ( M π ), (see [3]). It is well known that the spectrum of the bounded self-adjoint operator Hθ , which we shall denote by σ (Hθ ) or just σ (H ), is a closed non-empty subset of the interval [inf V − 2, sup V + 2], which is independent of θ. The connected components of int(k −1 ( M π )) are the gaps. So the resolvent set of Hθ is the union of all gaps. A col−1 lapsed gap is a point {E} for which k(E) ∈ M π and k (k(E)) = {E}. By means of Theorem 2 and Theorem 3, we give a sub-exponential estimate of the complement of the spectrum and we prove that the rotation number has the behavior of a 21 −Hölder function of the energy parameter E. In Theorem 2, the modulus of continuity cannot be improved following the behavior of the rotation number at the boundaries of gaps. Also, for generic ω, the rotation number is not Hölder because the Lyapunov exponent is discontinuous at rational ω. Finally, we note that, for d = 1, the smallness condition on the perturbation can be significantly improved and made independent of the Diophantine parameters (see [1]). Previously, in [2], J. Bourgain proved (in the “perturbative” regime) that for large λ the I.D.S. for the Almost Mathieu operator Hλ = λ cos 2π(kω + θ ) + ∆ is Hölder continuous of exponent κ < 21 . Assuming positivity of the Lyapunov exponent, M. Goldstein and W. Schlag proved also in [8] that the I.D.S. for the discrete quasi-periodic Schrödinger equation with analytic potential is Hölder continuous in the energy. An estimate of the complement of the spectrum (as in Theorem 3) was proven by Ja. G. Sina˘ı, in [13], in the case where E 1 is the lower bound of the spectrum. 1.4. Notations. In this paper we shall use the following notations: Let Br be the space of all analytic functions F : Td → gl(2, R), for which |F|r = sup|I mθ|0 Br . Let Br (X ) be the space of matrix F ∈ Br such that F(θ ) ∈ X for each θ ∈ Td . We denote < n, ω >=< n > and x = inf j∈Z |x − π j|, x ∈ R. We also denote DCωN (L , s) the restriction at the ball 0 < |n| ≤ N of the Diophantine condition on ρ with respect to ω2 . So we write ρ −

≥ L|n|−s , 0 < |n| ≤ N , 2

DCωN (L , s).


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In all this paper, the sign ∂ stands for the derivative with respect to the parameter E and for any smooth function A : E ∈ I → A(E) ∈ S L(2, R) we use the norm ν |A|C 2 (I ) = max0≤ν≤2 sup E∈I | ∂∂E A(E)|. Outline of the paper. The paper is organized in the following way. The reducibility will be studied by the K.A.M. method of faster convergence: In Sect. 2, we prove a small divisor proposition and in Sect. 3 an inductive proposition - these are based in K.A.M. techniques. In Sect. 4, we describe the iteration process and in Sect. 5, we finish the proof of Theorem 1. In Sect. 6, we prove Theorem 2 and Theorem 3. For the first, we show that the rotation number of our skew-product system has the behavior of a 21 −Hölder function of the energy parameter E and that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. From these results, we deduce a sub-exponential estimate of the complement of the spectrum and this proves Theorem 3. Finally, in Sect. 7, we describe, in an Appendix, the concept of rotation number and some of its elementary properties. 2. The Small Divisor Proposition The aim of this section is to find a solution Y : Td → sl(2, R) of the equation ˆ Y (θ + ω)A − AY (θ ) = A(G N (θ ) − G(0)).

i , where |.| is the Euclidˆ Here, G N (θ ) is the truncated Fourier series |n|≤N G(n)e dθ ˆ . In this equation, ω ∈ Rd is Diophanean norm and G(n) = Td G(θ )e−i (2π )d

tine, the matrix A ∈ S L(2, R) has two eigenvalues e±iα , α ∈ (R ∪ iR) with α ∈ DCωN (L , s) and G ∈ Br (sl(2, R)), which is equivalent to find a solution of the equaˆ tion Y (θ + ω)A − AY (θ ) = A(G(θ ) − G(0)) with the error A(G N (θ ) − G(θ )). In the following lemma we establish an estimate of the upper bound of this error to determine the number N : Lemma 1. Let 0 < r < r, f ∈ Br and let f N (θ ) be the truncated Fourier series of f 1 d at order N . We get | f N − f |r ≤ c| f |r e−N (r −r ) (N + r −r ) , where c is a constant that doesn’t depend on r, r and N .

Proof. First, an easy calculation yields | f N − f |r ≤ c1 | f |r

e−k(r −r ) ≤ c2 | f |r

+∞

k d−1 e−k(r −r ) dk.

N

k>N k=|n|

By multiple integrating by parts, we obtain

+∞ N

k d−1 e−k(r −r ) dk = e−N (r −r )

d−1 (d − 1)! j=0

j!

N j(

1 )d− j . (r − r )

(d−1)! j 1 d− j 1 d Clearly now, inequality d−1 ≤ c(N + r −r ) finishes the proof. j=0 j! N ( r −r ) Here c1 , c2 and c are constants that don’t depend on r, r and N . Before we introduce the small divisor proposition, we need the following result which is a well known small divisor lemma introduced, in its sharpest form, by Rüssmann [12]:

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S. Hadj Amor

Lemma 2. Let ω ∈ DC(K , τ ), f ∈ Br and let f N (θ ) be the truncated Fourier series of f at the order N . Let also α ∈ (R ∪ iR) such that α ∈ DCωN (L , τ ), (N ≥ 1, L ≤ K ). There exists a unique g : Td → C analytic such that

g(θ + ω) − e2iα g(θ ) = f N (θ ) − fˆ(0), g(0) ˆ = 0 and g(n) ˆ = 0 for |n| > N .

1 )| f |r , 0 < r < r, Moreover, the function g satisfies the estimate |g|r ≤ c( L(r −r )τ +2 where c is a constant that only depends on τ and d.

Proof. This equation can be solved in Fourier series, and this proves the existence of g and its uniqueness under conditions g(0) ˆ = 0 and g(n) ˆ = 0 for |n| > N . For the estimate see [7]. We will establish now the first step of the proof of Theorem 1: Proposition 1. Let ω ∈ DC(K , τ ), A ∈ S L(2, R) with two eigenvalues e±iα , α ∈ (R ∪ iR) and G ∈ Br (sl(2, R)). We assume that α ∈ DCωN (L , τ ), (N ≥ 1, L ≤ K ). Then, there exists a unique Y : Td → sl(2, R) such that

ˆ Y (θ + ω)A − AY (θ ) = A(G N (θ ) − G(0)) ˆ ˆ Y (0) = 0 and Y (n) = 0 for |n| > N .

2 2 r Y verifies |Y |r ≤ cτ,d |G| M , 0 < r < r. Moreover, if A and F are C or p.w. C on I ⊂ R, then Y ∈ Br is also C 2 or p.w. C 2 on I, and

⎧ ⎨ |∂Y |r ≤ cτ,d ( |∂G|r + |A|C2 (I ) |G|r,I |∂ A| ) M M2 ⎩ |∂ 2 Y | ≤ c ( |∂ 2 G|r + |A|C2 (I ) (|∂ 2 A||G| r

τ,d

M

M2

r,I

+ |∂ A||∂G|r ) +

|A|2 2

C (I )

|∂ A|2 |G|r,I M3

)

for all 0 < r < r, where we denote |G|r,I = sup E∈I sup|Imθ| 2 < L|m| ). Let now C A be the matrix of normalized eigenvectors associated to the eigenvalues of A. We <m,θ> i 2 0 e −1 define Hm,A (θ ) = C A C −1 A , A<m> = Hm,A (θ + ω) AHm,A (θ ) = −i <m,θ> 2 0 e +i(α− <m> ) 2 0 e −1 ˜ C −1 CA <m> A and F(θ ) = Hm,A (θ + ω) F(θ )Hm,A (θ ). Then 0 e−i(α− 2 ) Br Hm,A : (2T)d → S L(2, R), r˜ = N1 < r, F˜ and A<m> will be C 2 or p.w. C 2 on I and we have ⎧ τ ⎪ |Hm,A |r˜ ≤ c3 er˜ N |A|C 2 (I ) NL , ⎪ ⎪ ⎪ N τ 1+3ν 1+ν ν ⎪ , ν = 1, 2, ⎪ ⎨ |∂ Hm,A |r˜ ≤ c3 |A|C 2 (I ) a( L ) L |A<m> − I | ≤ c3 |A|C 2 (I ) K , ⎪ N τ 2+5ν 2+3ν νA ⎪ |∂ , ν = 1, 2, ⎪ <m> |r˜ < c3 |A|C 2 (I ) a( L ) ⎪ ⎪ ⎪ N τ 2+5ν ⎩ |∂ ν F| ˜ r˜ < c |A|2+3ν ( ) a|F|r , ν = 0, 1, 2, 2 3

C (I )

L

where c3 is a constant that only depends on τ and d and a = sup{E∈I } (1 + |∂ A| + |∂ A|2 + K K ˜ − |∂ 2 A|). Moreover, if L ≤ 2(2N )τ , then the vector m is unique and α 2 ≥ 2|n|τ , for every n such that 0 < |n| ≤ 2N , where α˜ = α − <m> 2 . The matrix A<m> and the matrix valued function Hm,A depend only on < m > and A. Remark 2. From the fact that ω ∈ DC(K , τ ), and α ∈ / DCωN (L , τ ) we get α ≥ K 2|m|τ . Remark 3. Let A ∈ S L(2, R) and let C be the matrix of normalized eigenvectors associated to the eigenvalues of A. By an easy calculation, we find that |C −1 | ≤

2|A|C 2 (I ) α .

We start now the proof of Proposition 3. Proof. First, we note that there exists a matrix D ∈ sl(2, R) such that A = e D . Let {±iα} = σ (D) and C A be the matrix of normalized eigenvectors associated to iα 0 the eigenvalues of the matrix D. It follows that A = e D = C e 0 −iα C −1 = A

CA e

eiα 0 C −1 A . Since α = 0, we write Hm,A (θ ) = e 0 e−iα

<m,θ> 2α D

|C −1 A |

, where D ∈ sl(2, R) and

≤

2|A|C 2 (I ) α ,

<m,θ> 2α

<m,θ> 2α C A

iα 0 0 −iα

A

C −1 A

:=

∈ R. Hence Hm,A (θ ) ∈ S L(2, R). Since ˜ r˜ (by Remark 3), we clearly obtain the estimates for |Hm,A |r˜ , | F|


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and |A<m> − I |. If T (E) is the normalized eigenvectors associated to the eigenvalues 2 c e−iα(E) of the matrix A(E), then |∂ T | ≤ α (|∂ A| + |∂α|) and |∂ 2 T | ≤ c( (|∂ A|+|∂α|) + α2

|∂ 2 A|+|∂ 2 α| A| c ), where |∂α| ≤ c |∂α and |∂ 2 α| ≤ α (|∂ 2 A| + 2|∂α|2 ). Now, differentiating α the transformation Hm,A gives the estimate of |∂ ν Hm,A |r˜ for ν = 1, 2. These estimates ˜ r˜ and |∂ ν A<m> |r˜ for ν = 1, 2 using the enable us to give the upper bound of |∂ ν F| K ˜ ˜ definition of A and F. Assume now that L ≤ 2(2N )τ and suppose that there exists a vector m = 0 and a vector m = 0 such that m = m and |m|, |m | ≤ N , and verifying

−τ and α − <m > < L|m |−τ . By an easy calculation and α − <m> 2 < L|m| 2 using the Diophantine condition on ω we obtain that

K (2N )−τ ≤ K |m − m |−τ ≤

< m − m > ≤ L|m|−τ + L|m |−τ ≤ 2L , 2

<m> which is impossible. Finally, we have α˜ − 2 ≥ 2 − α − 2 . Since L K K α − <m> 2 < |m|τ < L < 2(2N )τ and 2 ≥ |n|τ , we obtain the result for |n| ≤ 2N .

Combining Proposition 2 and 3, we obtain: K Proposition 4. Assume that N ≥ r2 , 0 < L ≤ max( 2(2N )τ , 1) and δ ∈]0, 1[ fixed. There exists a constant c4 that only depends on τ and d such that if, for ν = 0, 1, 2,

|∂ ν F|r ≤ c4

1 |A|3

C 2 (I )

L2 N 2τ +d

(

1 L2 Lδ τ +2 3 ) := c4 3 M, 2 |A|C 2 (I ) |A|C 2 (I ) N 2τ +d

then there exist Z : (2T)d → S L(2, R), A ∈ S L(2, R) and F ∈ B, all C 2 or p.w. C 2 on I, such that Z (θ + ω)−1 (A + F(θ ))Z (θ ) = A + F (θ ), verifying ⎧ |A|4+ν N d+3τ (1+ν) ⎪ ⎪ |∂ ν (Z − Hm,A )|r < c C2 (I ) ⎪ a|F|r ⎪ 4 M 1+ν L 3(ν+1) ⎨ d+τ (11+10ν) |A|5+10ν N 2 |∂ ν (A − A<m> )| < c4 C (IL) 11+10ν a|F|r ⎪ ⎪ 10(1+ν) 4(1+ν) 2(1+2ν)τ 10(1+ν)τ +2d ⎪ |A| N |A| N ⎪ ν 2 ) ⎩ C 2 (I ) |∂ F |r < c4 ( CM(I2+ν |F| + (N + 1δ )d e−N (r −r ) )a|F|r , r 10(1+ν) 2ν L M L 2(1+2ν) for ν = 0, 1, 2, where Hm,A and A<m> are as defined in Proposition 3, c4 is a constant that only depends on τ and d and a = sup{E∈I } (1 + |∂ A| + |∂ A|2 + |∂ 2 A|). With Case I. If α ∈ DCωN (L , τ ), then m = 0 and 0 < r < r − δ. / DCωN (L , τ ), then m is the unique vector |m| ≤ N such that Case II. If α ∈ <m> α − 2 < L|m|−τ and r = N1 − δ. Here {e±iα } = σ (A). Proof. Case I. By Proposition 2, this case is obvious provided c4 ≤ c4

≥

2 c2 NL2τ

.

c2 |A|2 2

C (I ) L2

N 2τ +d

and

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S. Hadj Amor

Case II. By Proposition 3, there exists a transformation Hm,A verifying |Hm,A |r˜ ≤ τ c3 er˜ N |A|C 2 (I ) NL for all r˜ . There exists also A<m> and F˜ N (θ ) = H −1 (θ +ω)FN (θ )H (θ ) defined on the Td and verifying |A<m> − I | ≤ c3 |A|C 2 (I )

L , K

(5)

and | F˜ N |r˜ ≤ c3 er˜ N |A|2C 2 (I ) NL 2 |FN |r˜ for all r˜ < r. But, by Lemma 1, it follows that 2τ

|FN |r˜ ≤ |F|r˜ + |F − FN |r˜ ≤ |F|r + c |F|r e−N (r −˜r ) (N +

1 d ) , r − r˜

where c is a constant that only depends on d. We clearly have | F˜ N |r˜ ≤ c3 er˜ N |A|2C 2 (I )

N 2τ (1 + ∆)|F|r , L2

(6)

1 d 1 2 d where ∆ = e−N (r −˜r ) (N + r −˜ r ) . Since r˜ = N and r ≥ N it follows that 1 + ∆ ≤ cN , where c is a constant that only depends on d. Using now the hypothesis on |F|r , com˜ bined with (5), the estimate (6) becomes | F˜ N |r˜ ≤ cc4 |A|M2 ≤ c c4 |A<m>M| 2 , where c C (I )

is a constant that only depends on d and τ and where M˜ = ( |A

Lδ τ +2 2 <m> | 2

C (I )

)3 . Moreover, by

C (I )

2N Proposition 3, we deduce that α˜ = α− <m> 2 ∈ DC ω (L , τ ). We can now apply Propc2 d osition 2 provided c4 ≤ c . Hence, there exist Z˜ : T → S L(2, R), A ∈ S L(2, R) and FN ∈ B, all C 2 or p.w. C 2 on I, such that Z˜ (θ +ω)−1 (A<m> + F˜ N (θ )) Z˜ (θ ) = A + FN (θ ) with the estimates involved in the proposition. We define Z (θ ) = Hm,A (θ ) Z˜ (θ ) and F (θ ) = FN + Z (θ + ω)−1 (F(θ ) − FN (θ ))Z (θ ). Now, as in the first case, all the requirements are fulfilled and the estimates follow after some estimates (see [6]).

4. Iteration Process Let ε1 be fixed. We define a decreasing sequence {ε j } j≥1 by ε j+1 = ε1+σ j , where 1 σ ≤ 150 . Let 0 < r1 ≤ 1 be fixed and let {r j } j≥1 be a decreasing sequence of positive numbers such that ⎧ rj ⎨ r j − r j+1 ≥ 4 j and (7) ⎩r 1 j+1 ≥ 2N j , 1 where N j = 4 j 4σ r j log( ε j ) and this for all j ≥ 1. Here, constraints (7) are compatible provided ε1 is sufficiently small. We have the following result:

Lemma 3. 1. e−N j (r j −r j+1 ) ≤ ε4σ j . 2. For all c > 0, σ > 2 and α > 0, there exists γ = γ (c, σ, α) > 0 such that if 4 ε1 ≤ γ b σ r12α for some b > 0, then ε j ≤ cb j r αj for all j ≥ 1.


575

Proof. The proof of (a) is evident. For (b), by the definition of the sequence {r j } j≥1 , it follows that r αj ≥ (

1 8(4σ ) ln( ε11 )

)( j−1)α (

1 2 )α( j−1) r1α , 4(1 + σ ) (1+σ ) j−1

and using the definition of sequence {ε j } j≥1 , we get ε j = ε1

. So, to obtain the

4

result, it is sufficient to have ε1 ≤ c2 r12α , ε1 ≤ b σ , ε1 ≤ γ (α, σ ), and ε1 ≤ γ (α, σ ), where γ (α, σ ) = 0 is a positive constant that only depends on α and σ. The first and the second estimate are deduced using, respectively, that 21 (1+σ ) j−1 ≥ 21 and 41 (1+σ ) j−1 ≥ jσ 1 j−1 ≥ ( j−1)σ and σ ln( 1 ) ≤ ( 1 )σ . 4 . The third estimate is proven using 8 (1 + σ ) 8 ε1 ε1 σ Finally, we prove the last estimate by observing that (1 + σ ) j−1 ≥ ( j − 1)2 2! . Hence, 4

it is sufficient to have ε1 ≤ γ (c, σ, α)b σ r12α . We are now ready to present the iterative result. Proposition 5. Let ⎧ ω ∈ DC(K , τ ), K ≤ 1, ⎪ ⎪ ⎪ ⎨ A1 ∈ S L(2, R), F1 ∈ Br1 , ⎪ d ⎪ ⎪ A1 + F1 (θ ) ∈ S L(2, R) for all θ ∈ T , ⎩ 2 A1 and F1 be C on I ⊂ R.

(8)

There exists a constant c5 that only depends on τ, d, σ, K and |A1 |C 2 (I ) such that, if 2

(42τ +24+2d)

|∂ ν F1 |r = ε1 ≤ c5r1σ , ν = 0, 1, 2, then, for all j ≥ 1, there exist Z j+1 : (2T)d → S L(2, R), A j+1 ∈ S L(2, R) and F j+1 ∈ B, all C 2 or p.w. C 2 on I such that Z j+1 (θ + ω)−1 (A j + F j (θ ))Z j+1 (θ ) = A j+1 + F j+1 (θ ), and verifying 1

|∂ ν (Z j+1 − Hm j ,A j )|r j+1 < ε j2 , ν = 0, 1, 2; ⎧ ⎪ ⎨ |A j+1 | ≤ 2|A1 |C 2 (I ) , 2 |∂ ν (A j+1 − A<m j > )| < ε j3 , ν = 0, 1, 2, ⎪ ⎩ |∂ ν A | < ε−νσ , ν = 1, 2, j+1 j |∂ ν F j+1 |r j+1 < ε j+1 , ν = 0, 1, 2,

Γ j+1 , Θ j+1 , Λ j+1 .

With N Case I. If α j ∈ DCω j (εσj , τ ), then m j = 0 and r j − r j+1 = N

rj . 4j

/ DCω j (εσj , τ ), then m j is the unique vector that verifies |m j | ≤ N j Case II. If α j ∈ and such that α j −

<m j > 2

< εσj |m j |−τ , r j+1 =

1 2N j

εσ

and |A<m j > − I | ≤ 4|A j | Kj .

576

S. Hadj Amor

Proof. Under the hypothesis, we need to prove the existence of Z j+1 , F j+1 and A j+1 with the required properties. For the case j = 0, if we let A0 = A<m 0 > = A1 , F0 = F1 and Z 1 = Hm 0 ,A0 = I, then Γ1 , Θ1 and Λ1 are verified by hypothesis. This is the first step of the iteration. Assume now that we have the result for all n ≤ j − 1, ( j ≥ 1). We denote that 2

(42τ +24+2d)

the sequence {r j } j≥1 verifies the condition (7) for all j, if ε1 ≤ c5 r1σ apply Proposition 4, we let L j = εσj . So, we need that L j = εσj ≤ Nj ≥

2 rj

2

i.e. that ε j ≤ ( 25τK+1 ) σ (

ε1 ≤ γ (K , σ, τ )(

1 4

2τ σ

4τ σ

4

1 2τ 4σ

2τ

) j r jσ and N j ≥

2 rj

K 2(2N j )τ

. To and

. By Lemma 3, this is true if

) σ r1 .

Case I. Let δ j = r j − r j+1 . Using Lemma 3 we conclude that to apply Proposition 4 and to show that all the estimates are verified, it is sufficient to have 4 2 (22τ +24) 1 ε1 ≤ γ (|A1 |C 2 (I ) , σ, τ, d)( 1 ) σ r1σ . 4 σ (22τ +24) Case II. Under the hypothesis, Remark 2 and Remark 3, it follows that |A<m j > − I | ≤ εσ

4|A j | Kj . We let δ j = 2N1 j . To apply Proposition 4 and to show that the estimates Γ j+1 , Λ j+1 and the second estimate of Θ j+1 are verified , it is sufficient to have ε1 ≤ γ (|A1 |C 2 (I ) , σ, τ, d)(

1 4 2

1 σ (42τ +24+2d)

4

2

) σ r1σ

(42τ +24+2d)

.

2τ

The vector m j is unique since ε j < K σ r jσ . Finally, by Proposition 3, we have that 2

εσ

|A j+1 | ≤ ε j3 + 1 + c3 Kj |A j | ≤ 2|A1 |C 2 (I ) because |A1 |C 2 (I ) ≥ 1, which prove the first estimate of Θ j+1 . Assume now that for all 1 ≤ k ≤ j, |∂ Ak | < εk−σ . Let j1 < j2 < · · · < Nj

/ DCω k (L jk , τ ). jn be the finite sequence of indices jk , 1 ≤ k ≤ n for which α jk ∈ So using Proposition 3 and the induction hypothesis we obtain |∂ A j+1 | ≤ |∂(A j+1 − A<m jn > )| + |∂ A<m jn > | < ε−σ j+1 , and this because j + 1 = jn+1 + 1 jn , (see the

next remark). We also suppose that for 1 ≤ k ≤ j, |∂ 2 Ak | < εk−2σ . By the hypothesis and Proposition 3, we obtain |∂ 2 A j+1 | < ε−2σ j+1 , since j + 1 = jn+1 + 1 jn , (see the next remark). Hence we get the third estimate of Θ j+1 . This finishes the proof of Proposition 5. Now we proceed with the remark used for the proof of Proposition 5. Nj

Remark 4. If { jk }k≥1 is the sequence of indices for which α jk ∈ DCω k (L jk , τ ), then jk+1 jk for all k ≥ 1. Indeed, we denote, for all k ≥ 1, 1 r jk +1 But r jk ≥ 1 )r .) 4l jk−1 +1

3 4 r jk−1 +1

= N jk =

4σ (4 + 4σ ) jk 1 log( ). 1+σ r jk ε1

for all k ≥ 2. (This is true by the fact that r jk ≥ 3(1+σ ) r jk−1 +1 ε 42 σ (4+4σ ) jk 1

jk

l= jk−1 +1 (1

−

So, it follows that r jk +1 ≥ for all k ≥ 2. By an immediate induction, we deduce that for all k ≥ 2, 3(1 + σ )ε1 k 1 r jk +1 ≥ ( ) ( ) jk + jk−1 +···+ j1 r1 . (9) 42 σ 4 + 4σ


577

On the other hand, we have by the second estimate of Θ jk , that for all k ≥ 2, |A jk − σ

I | ≤ ε j2k−1 . Then, by Lemma 9 of the Appendix and since ρ(0,A jk ) = α jk mod π and σ

ρ(0,I ) = 0 mod π, it follows that α jk ≤ 2ε j4k−1 . This together with Remark 2 give, for all k ≥ 2, N jk ≥ (

K 1 1 σ )τ ( ) 4τ . 4 ε jk−1

(10)

Combining (9) and (10), we obtain (

1 K 1 1 σ 42 σ )τ ( ) 4τ ≤ ( )k (4 + 4σ ) jk + jk−1 +···+ j1 , 4 ε jk−1 3(1 + σ )ε1 r1

which gives, for all k ≥ 2, (4 + 4σ ) jk + jk−1 +···+ j1 ≥ r1 (

K 1 3(1 + σ )ε1 k )τ ( ) ( 4 42 σ

1 σ 4τ (1+σ )

ε1

)(1+σ )

jk−1

.

This inequality implies that the sequence { jk }k≥1 grows quickly, in particular for all D > 0, if ε1 is sufficiently small (depending on D), then jk+1 ≥ D jk for all k ≥ 1. 5. Floquet Solutions-Proof of Theorem 1 We now come to the conclusions of Proposition 5. Under its hypothesis, it is clear that r j → r∞ ≥ 0, |F j |r j → 0 and A j → A∞ as j → ∞. We establish that if m j = 0 for all j sufficiently large, then Z j converges (see [6]). Consequently, if X n is a solution of (∗) then, by Proposition 5, we have a representation X n = Z (nω)An∞ Z (0)−1 X 0 , n ∈ Z, where X 0 is an arbitrary vector. Let ρ(ω,A+F) be the rotation number of the skew-product

j system (ω, A + F) and let ρ j+1 = α j+1 + 21 k=1 < m k >, where ρ(0,A j ) = α j mod π for all j ≥ 1. That the sequence {ρ j mod π }j≥2 converges uniformly in E is the content of the following lemma: 1

Lemma 4. 1. ρ j+1 − ρ j < cε j3 for all j, where c is a numerical constant. In particular, the sequence {ρ j mod π }j≥2 converges uniformly to the limit ρ(ω,A+F) . 2. If ρ(ω,A+F) is Diophantine or rational, then m j = 0 for all j sufficiently large. <m >

1

Proof. For a) it suffices to show that α j+1 −(α j − 2j ) ≤ cε j3 , where c is a numerical constant, which follows from Lemma 9 of the Appendix and Θ j+1 by an explicit compu

<m > tation. Then the sequence {ρ j mod π }j≥2 converges to the limit α + j≥1 2j , where α mod π = limj→∞ αj mod π = ρ(0,A∞ ) . Finding a Floquet representation implies solving, in every step of the iteration, the linear differential equation of Proposition 1 and transforming the system X n+1 = (A j + F j (θ ))X n by Z j+1 (θ ) as explained in Proposition 5. By Lemma 6, Lemma 7 and Lemma 8 (of the Appendix) and if we let ρ˜(ω,A j +F j ) be the rotation number of the skew product (ω, A j + F j ) for all j ≥ 1, we obtain that

k=j <m > ρ˜(ω,A j+1 +F j+1 ) = ρ˜(ω,A j +F j ) − 2j mod π = ρ(ω,A+F) − k=1 <m2k > mod π. So, we can conclude that the limit of the sequence {ρ j mod π }j≥2 is ρ(ω,A+F) . For b) see [6]. Now, Theorem 1 follows from Proposition 5 and Lemma 4.

578

S. Hadj Amor

6. Proof of Theorem 2 To begin the proof of Theorem 2 we need a lemma: Lemma 5. Let I =]E 0 − δ, E 0 + δ[⊂ R, f : E ∈ I → 2 cos α(E) ∈ R, be C 2 on I such that f (E) > −2 for all E ∈ I and α be continuous on I. Assume that ∂ 2 f (E) ≤ −γ , γ > 0, for all E ∈ I. We have: 1. The function f has a unique maximum or it doesn’t have an extremum on I. 2 2. Let I ⊆ I ⊂ R. We have Var I f = Max I f − Min I f ≥ γ |I8 | , where |I | is the length of interval I . 3. If f doesn’t have an extremum in ]E 0 − 2δ , E 0 + 2δ [, then |∂ f (E 0 )| ≥ δγ 2 . δ δ 4. If f has a unique maximum E 1 ∈]E 0 − 2 , E 0 + 2 [ and α is a function that verifies α(E) − α(E ) ≥ −ε for all E < E , (E, E ) ∈ I 2 , 2 (11) ε ≤ δ64γ , then f (E 1 ) = a ≥ 2, and if |∂ f (E)| ≤ M on I and f (E 0 ) = 2 − η, (η > 0), then ∂ f (E 0 ) ≥ γMη . Proof. For 1, 2 and 3 the proof is obvious. For 4, we denote I1 =]E 0 − δ, E 1 [ and I2 =]E 1 , E 0 + δ[. If f (E 1 ) < 2, then α(I ) ⊆] − π, 0[ mod 2π or α(I ) ⊆]0, π [mod 2π. Suppose that α(I ) ⊆] − π, 0[mod 2π. By 2, we have δ2 γ . 32 On the other hand | cos α(E 1 ) − cos α(E 0 + δ)| ≤ |α(E 1 ) − α(E 0 + δ)|. So, it follows that 2 2 α(E 1 ) − α(E 0 + δ) ≥ δ64γ or α(E 0 + δ) − α(E 1 ) ≥ δ64γ . The first inequality is impossible by (11) and the second is also impossible under the fact that f is a decreasing function on I2 . If we suppose that α(I ) ⊆]0, π [mod 2π, then by the same reasoning we also obtain a contradiction. Hence, f (E 1 ) ≥ 2. For the second part of 4, we assume that E 0 ≤ E 1 . Using the hypothesis, we get ∂ f (E) ≥ γ (E 1 − E) for all E ∈ I. In particular, Var I2 f = 2(cos α(E 1 ) − cos α(E 0 + δ)) ≥

∂ f (E 0 ) ≥ γ (E 1 − E 0 ).

(12)

Under the hypothesis, we also have a − 2 + η ≤ f (E 1 ) − f (E 0 ) ≤ sup (∂ f (E))(E 1 − E 0 ) ≤ M(E 1 − E 0 ).

(13)

E∈I

Combining (12) and (13), the result follows.

Let us remember that we consider co-cycle (∗), where A = A1 ∈ S L(2, R) and A + F = A1 + F1 (θ + nω) ∈ S L(2, R). The transformation F = F1 : Td → gl(2, R) is assumed to be analytic in a complex neighbourhood |x| < r and ω ∈ Rd is Diophantine. We let ρ(ω,A+F) = ρ(ω,A1 +F1 ) be the rotation number of the skew-product system (ω, A1 + F1 ). It is determined modulo π and depends continuously on A1 + F1 . In particular ρ(ω,A1 ) = α1 mod π , where {e±iα1 , 0 ≤ α1 < π } are the eigenvalues of A1 . In the previous section, we obtain reducibility near constant coefficients by means of K.A.M. theory and in Proposition 5 we describe every step of the iteration process. Suppose now that co-cycle (∗) depends smoothly on a parameter E ∈ I0 ⊂ R, then any almost reducing transformations will be only piece-wise smooth on I0 . The control of the size of intervals of continuity and some other basic estimates will contain the following proposition which is the key part for the proof of Theorem 2:


579 σ

Proposition 6. Under hypothesis (8), assume that for all E ∈ I0 , |∂trA1 (E)| ≥ ε14 and |∂ ν trA1 (E)| ≤ C ν , ν = 1, 2, where C is a constant that doesn’t depend on E and also assume that ρ(ω,A1 +F1 ) is monotone on I0 . There exists a constant c6 that 2

(42τ +24+2d)

, only depends on τ, d, σ, K and |A1 |C 2 (I0 ) such that, if |∂ ν F1 |r = ε1 ≤ c6r1σ ν = 0, 1, 2, then for all j ≥ 1 and for all E 0 ∈ I0 , transformations Z j+1 (E 0 ) : (2T)d → S L(2, R), F j+1 (E 0 ) ∈ B and A j+1 (E 0 ) ∈ S L(2, R) of Proposition 5 have respectively (E) ∈ B and A (E) ∈ S L(2, R), prolongations Z j+1 (E) : (2T)d → S L(2, R), F j+1 j+1 all C 2 on I j an open interval which is symmetric of center E 0 , and verifying for all E ∈ Ij, 1

|∂ ν (Z j+1 − Hm j ,A j )(E)|r j+1 < ε j2 , ν = 0, 1, 2, ⎧ |A | ≤ 2|A1 |C 2 (I0 ) , ⎪ ⎨ j+1 2 |∂ ν (Aj+1 − A<m j > )(E)| < ε j3 , ν = 0, 1, 2, ⎪ ⎩ ν , ν = 1, 2, |∂ A j+1 (E)| < ε−νσ j |∂ ν F j+1 (E)|r j+1 < ε j+1 , ν = 0, 1, 2,

Γ j+1

Θ j+1 Λj+1

with N Case I. If α j (E 0 ) ∈ DCω j (εσj , τ ), then m j (E 0 ) = 0 and r j − r j+1 = N

rj ; 4j

Case II. If α j (E 0 ) ∈ / DCω j (εσj , τ ), then m j (E 0 ) is a unique vector verifying <m (E )>

0 j |m j (E 0 )| ≤ N j and such that α j (E 0 ) − < εσj |m j (E 0 )|−τ and 2 r j+1 = 2N1 j . Let { jk }k≥1 be the sequence of indices for which m jk (E 0 ) = m jk = 0. Then I j is the most long symmetric interval included in I j−1 and such that for all E ∈ I j , we have ⎧ εσj <m> ⎪ ⎪ ⎨ α j (E) − 2 ≥ 2|m|τ for all 0 < |m| ≤ N j if j ∈ { jk }k≥1 and (14) ⎪ 3εσj ⎪ ⎩ α (E) − <m j (E 0 )> < if j ∈ { j } . τ k k≥1

j

2

2|m j (E 0 )|

The following properties are also fulfilled: 1. |∂trAj (E)| < C + 1 := C1 for all E ∈ I j−1 ; 2. The function (α j )2 is C 2 on I j−1 and |∂(α j )2 (E)| ≤ πC1 := C2 for all E ∈ I j−1 ; εσ

−1 3. |I j | ≥ 2( 8√Cj N τ )2 and |I jk −1 | ≥ N −10τ ; jk 2 j 1 + C 2 if j ≤ j1 for all E ∈ I j−1 ; 4. |∂ 2 trAj (E)| ≤ 12C12 N 4τ j 1 + C 2 + K 2 k if jk < j ≤ jk+1 , k ≥ 1 σ

11σ

5. |∂trAjk (E)| ≥ ( 21 )2k−1 ε j4k −1 for all E ∈ I jk −1 such that |E − E 0 | < ε jk4−1 , k ≥ 1; √ √ 6. α j (E) − α j (E 0 ) ≤ 2 C2 |E − E 0 | for all E ∈ I j−1 . Proof. The existence of the constant c6 that warrants the prolonged verification of , Θ Γ j+1 j+1 and Λ j+1 is proved in the same way as in Proposition 5. To see this, it is sufficient to change the parameter of the Diophantine condition in order to remove discontinuities, which is done in (14). In particular, each E ∈ I j verifies (14).

580

S. Hadj Amor

For 1, first we note that using Θ j and for jk < j ≤ jk+1 , k ≥ 1, |∂trAj | ≤ 2 | sin α˜ j |

j−1 2 |∂trA<m j > | + 2 l= jk εl3 ≤ |∂trA<m j > | + 4ε j3k . Secondly, since | sin α k | ≤ 1, we k

k

i α˜ j

jk

−i α˜ j

k )| ≤ |∂trA |. This gives the result. = |∂(e k + e have jk For 2, suppose that there exists E ∈ I j−1 such that α j (E) = 0 mod π then

|∂trA<m j > | k

|∂(α j )2 (E)| = 2|α j (E)||∂α j (E)| = |

α j (E) sin α j (E)

||∂trAj (E)|.

α (E)

Since α j (E) = 0 mod π, then | sin αj (E) | ≤ π, and the function (α j )2 is differentiated j

in E. If now α j (E) = 0 mod π, then we can prove that (α j )2 is also differentiated and that we have the same estimate. εσ εσ For 3, suppose that I˜j =]E 0 − ( 8√Cj N τ )2 , E 0 + ( 8√Cj N τ )2 [⊆ I j−1 . We denote εσj

E 1 = E 0 + ( 8√C

τ 2Nj

2

)2 .

2

j

j

Let E be such that E 0 < E < E 1 . Two cases occur:

– α j (E 0 ) and α j (E) have the same sign. Hence, √ α j (E) − α j (E 0 ) ≤ 2 sup I˜j |∂(α j (E))2 | |E − E 0 |. Recall that 2 and the definition of I˜j give α j (E) − α j (E 0 ) ≤ –

α j (E 0 )

and

α j (E)

εσj 4N τj

.

have two different signs. Then there exists E 2 such that E 0
2 ≥

εσj 2|m|τ

.

<m j (E 0 )> 2

(15)
< n(E 0 ) > − ≤ 2 C2 E 2 − E 0 + εl2 . 2 2

<m jk (E 0 )> − ≥ 2 σ 2 suppose that K 2−τ −1 N −τ jk ≥ 2εl ,

On the other hand, we have

K 2−τ N −τ jk , using that

then E 2 − E 0 ≥ N −3τ ω ∈ DC(K , τ ). Hence, jk , provided ε1 is sufficiently small. If not, then by the first estimate of 3 we have E 2 − E 0 = εσ |Il | −7τ 2 √ l 2 ≥ ( 8 C N τ ) , which gives that E 2 − E 0 ≥ N jk , if ε1 is sufficiently small. 2

l

<m jk−1 (E 0 )> 3εσ = 2|m j l(E 0 )|τ . Using the first estimate 2 k−1 −1 of 3 and (10) we obtain E 2 − E 0 ≥ N −10τ , if ε1 is sufficiently small. jk For 4, Let j be such that jk < j ≤ jk+1 , k ≥ 1. By Θ j and for all E ∈ I j−1 , 2 sin α˜ j we have |∂ 2 trAj | ≤ |∂ 2 trA<m j > | + 4ε j3k . But, using the fact that | sin α k | ≤ 1 and k j

Case 2. l = jk−1 : so αl (E 2 )−

k

| cos α jk −cos α˜ jk | ≤ 23 , we obtain |∂ 2 trA<m j > | ≤ |∂ 2 trAjk |+3(∂α jk )2 . So, we deduce, k by induction, that |∂ 2 trAj | ≤ |∂ 2 trAj1 | + 3

k

(∂α jl )2 + 4

l=1 2

≤ C 2 + 4ε13 + 3 By Remark 2, we have | sin α jl | ≥

α j l 2

≥

k l=1

|∂trAjl | 2 ) ( 2| sin α jl | l=1

k

K 4N τj

l

by a simple calculation, we obtain the result.

2

ε j3l 2

+ 8ε j31 . 1

, if ε jl ≤ ( K3 ) σ which is the case. So, 11σ

For 5, let E ∈ I jk , k ≥ 1 such that |E − E 0 | < ε jk4 . We know that α jk is C 2 on this interval because Ajk is C 2 on I jk and α jk (E) = 0 mod π for all E ∈ I jk . By the fact that | cos α˜ jk | ≥ 21 , we obtain |∂ 2 trA<m j

k

sin α˜ j

2 > | ≥ |∂α jk | − |

It is clear that | sin α k | ≤ jk

2α˜ j k α j k

sin α˜ jk

sin α jk

||∂ 2 trAjk + 2 cos α jk (∂α jk )2 |.

. Moreover, Remark 2 and (14) imply that 11σ

α˜ j k α j k

≤

3εσj k K

.

Hence, for all E ∈ I jk such that |E − E 0 | < ε jk4 , we obtain |∂ 2 trA<m j

k

if

|∂trAjk |

≥ 4

6εσj k K

(1 + C 2 +

>|

12C12 N 4τ j

k−1

K2

≥

1 |∂trAjk |2 , 16

(16)

), for k ≥ 1, which is true by the induction

hypothesis. Then by Θ jk+1 and using (16), we obtain |∂ 2 trAjk+1 (E)| ≥ 11σ

1 2 32 |∂trA jk (E)|

for all E ∈ I˜ = I jk+1 −1 ∩ {E; |E − E 0 | < ε jk4 }. We apply now Lemma 5 with

582

S. Hadj Amor

−1 f = trAjk+1 defined on I = I˜jk+1 −1 =]E 0 − δ, E 0 + δ[, where δ = N −10τ (I ⊂ I˜ jk+1 1 2 if ε1 is sufficiently small). We denote γ = 32 inf E∈I |∂trA jk (E)| and M = C1 . We suppose for this proof that ρ(ω,A1 +F1 ) is an increasing function on I0 . (The opposite case is deduced from the first case if we replace E by −E.) Then, using Lemma 4, we obtain 2

α jk+1 (E ) − α jk+1 (E) ≥ −cε j3k+1 , where c is a numerical constant and this for all E ∈ I 2

2

and E ∈ I such that E > E. We denote ε = c ε j3k+1 and we see that c ε j3k+1 ≤ δ64γ for all k ≥ 1, provided ε1 is sufficiently small. We denote I =]E 0 − 2δ , E 0 + 2δ [. We shall assume that ∂ 2 trAjk+1 (E) ≤ −γ for all E ∈ I . (If not we replace Ajk+1 by −Ajk+1 .) By Lemma 5, the function trAjk+1 doesn’t have an extremum or it has a unique maximum on I . For the first case and using Lemma 5-3 and Remark 4, it follows that 2

σ

|∂trAjk+1 (E 0 )| ≥ ( 21 )2k ε j4k+1 −1 . For the second case and if we denote this maximum E 1 then, by Lemma 5-4, it follows that |trAjk+1 (E 1 )| = a ≥ 2. We also have |trAjk+1 (E 0 )| = 2| cos(α jk+1 (E 0 ))| = 2

∞ (α j (E 0 ))2l (−1)l k+1 . (2l)! l=0

But

∞

l l=0 (−1)

(α j (E 0 ))2l k+1 (2l)!

≤ 1−

|trAjk+1 (E 0 )| = 2 − η ≤ 2 −

(α j (E 0 ))2 k+1

4 (α j (E 0 ))2 k+1 , 2

, if α jk+1 (E 0 ) ≤

with η ≥

(α j

(E 0 ))2 2

k+1

√ 6. So, we obtain ≥

K2 8N 2τ j

. We con-

k+1

σ

clude now, using the second part of Lemma 5-4, that |∂trAjk+1 (E 0 )| ≥ ( 21 )2k ε j4k+1 −1 for all k ≥ 1, which is true if ε1 is sufficiently small. Finally, the two cases give σ

|∂trAjk+1 (E 0 )| ≥ ( 21 )2k ε j4k+1 −1 . Hence, we deduce for all E ∈ I jk+1 −1 with |E − E 0 | < 11σ

σ

1 2k+1 4 4 ε jk+1 −1 , for all k ≥ 1, provided ε1 is sufficiently ε jk+1 −1 , that |∂trA jk+1 (E)| ≥ ( 2 ) small, which proves 5. For 6, we have for all E ∈ I j−1 , α j (E) − α j (E 0 ) ≤ α j (E) − α j (E 0 ). We √ now note that we have α j (E) − α j (E 0 ) ≤ 2 sup I j−1 |∂(α j (E))2 | |E − E 0 | and that 2 gives the result.

As a first consequence of this proposition we get Corollary 1. With the same assumptions as in Proposition 6, the following holds: for κ

−1 −γ |m| 2 , where c and κ are ( <m> all m ∈ Zd \{0} we have Leb(ρ(ω,A+F) 2 mod π )) ≤ c e two numerical constants and γ is a constant that only depends on K , σ, τ and r1 . −1 Proof. Let m ∈ Zd \{0} and denote IG = ρ(ω,A+F) ( <m> mod π ). Let E 0 be a cen2 ter of IG and j L ∈ Z the smallest indices for which |m| ≤ N jL −1 . Let also I˜G ⊆ 11σ

11σ

IG ∩ ( I˜jL −1 ) ⊂ I jL −1 , where I˜jL −1 =]E 0 − ε jL4−1 , E 0 + ε jL4−1 [ and let E ∈ I˜G .

jL −1 <mj (E0 )> We note that ρ(ω,A+F) (E) = <m> mod π +ρ˜ jL (E), where j=1 2 mod π = 2 ρ˜ jL denote the rotation number of the skew-product system (ω, AjL + F jL ). Recall 1

that, by Lemma 9 of the Appendix, we have ρ˜ jL (E) − α jL (E) ≤ 2ε j2L . Conse jL −1 <m j (E 0 )> √ quently α jL (E) − ρ(ω,A+F) (E) + j=1 ≤ 2 ε jL . From this we conclude 2


583

√ that Var I˜G α jL − Var I˜G ρ(ω,A+F) ≤ 4 ε jL . Since Var I˜G α jL ≥ inf I˜G |∂α jL (E)| ≥ σ

inf I˜G |∂trAjL (E)| ≥ | I˜G | ≥ ( 21 )2L ε j4L −1 | I˜G |, which is clear by Lemma 5-5, it hence σ √ √ follows that 0 = |Var I˜G ρ(ω,A+F) | ≥ Var I˜G α jL − 4 ε jL ≥ ( 21 )2L ε j4L −1 | I˜G | − 4 ε jL . This implies that −σ 11σ −σ √ √ | I˜G | ≤ 22L+2 ε 4 ε jL . It is clear that | I˜jL −1 | = 2ε 4 > 22L+2 ε 4 ε jL ≥ 1 2

j L −1

j L −1

11σ

j L −1

| I˜G |. So, we get IG = I˜G and |IG | < 2ε jL4−1 . Hence, if m ∈ Zd \{0}, then there exists 11σ

−1 ( <m> mod π )) ≤ 2εjL4−1 . jL ∈ Z, L ≥ 2, such that |m| ≤ N jL −1 and Leb(ρ(ω,A+F) 2 Now, we have

ε jL −1 = e where κ =

log(1+σ ) log(8+8σ ) .

−(1+σ ) j L −2 log( ε1 ) 1

≤e

−( (8+8σr )

j L −2

jL

log( ε1 ))κ ( 34 r j L−1 +1 )κ 1

,

Using (9), we obtain

ε jL −1 ≤ e

j L − j L−1 −···− j1 r j (8+8σ )2 L

−( (4+4σ )

2 j L log( ε1 )( 34 )( 1

3(1+σ )ε1 (L−1) ) r 1 )κ 42 σ

.

By Remark 4, we have that jn+1 ≥ 4 jn for all n provided ε1 is sufficiently small. Hence κ

ε jL −1 ≤ e

κ

3(1+σ )ε 3 )κ (2 j L ( 2 1 )(L−1) r1 )κ 4 σ L 4(8+8σ )2

2 2 −( 4+4σ 4σ ) N j (

1 τ

σ 4τ

. j2

Summing up the definition of N j2 and (10), gives ( K4 ) ( ε1j ) ≤ 4σ 4r j log( ε1j ). Fur1 2 2 thermore, from the definition of the sequence {r j } j≥1 and (9), we now get j2 ≥ c(K , τ, σ, r1 ) log ε1 .

(17)

Moreover, by Remark 4, we have for all n ≥ 2, jn ≥ 4n−2 j2 . L−1 j

(18) L−1

Combining (17) and (18), we obtain ε1 L ≥ (c(K , τ, σ, r1 )) 4 L−2 . Since L−1

lim L→+∞ (c(K , τ, σ, r1 )) 4 L−2 = 1 this gives, for L sufficiently large, ε1L−1 ≥ ( 21 ) jL . Us) ≥ 1, we obtain ε jL −1 ≤ e ing this last inequality and since 3(1+σ 42 σ which gives the result.

κ

κ

3 )κ r1κ L 4(8+8σ )2

2 2 −( 4+4σ 4σ ) N j (

,

As a second consequence of Proposition 6 we get Corollary 2. With the same assumptions as in Proposition 6, √ the following holds: let √ √ E 0 ∈ I0 , then |ρ(ω,A+F) (E) − ρ(ω,A+F) (E 0 )| ≤ 4 π C + 1 |E − E 0 | for all E ∈ I0 . Proof. Let E ∈ I0 . It is clear that |ρ(ω,A+F) (E) − ρ(ω,A+F) (E 0 )| = |ρ˜ j (E) − ρ˜ j (E 0 )|, where ρ˜ j = ρ(ω,Aj +F j ) . By Lemma 9 of the Appendix, we get that ρ˜ j (E)−α j (E) ≤ √ √ 2 ε j and ρ˜ j (E 0 )−α j (E 0 ) ≤ 2 ε j . Consequently, |ρ(ω,A+F) (E)−ρ(ω,A+F) (E 0 )| ≤ √ 4 ε j + α j (E) − α j (E 0 ). This, together with Proposition 6-6, now yields √ |ρ(ω,A+F) (E) − ρ(ω,A+F) (E 0 )| ≤ 4 ε j + 2 C2 |E − E 0 | ≤ 4 C2 |E − E 0 |,

584

S. Hadj Amor

provided |E − E 0 | ≥ εj 2C2

εσj−1

≤ ( 8√C

τ 2 N j−1

εj 2C2 .

Since, for ε1 sufficiently small, we have

)2 , the result follows.

By Corollary 1 and 2 we complete the proof of Theorem 2. Now, the question of the sub-exponential estimate of the complement of the spectrum follows as an immediate consequence of Theorem 2: Corollary 3. With the same assumptions as in Theorem 2, the following holds: let −1 −1 [E 0 , E 1 ] be a connected component of ρ(ω,A+F) (M), then Leb(ρ(ω,A+F) (M)∩[E 1 , E 1 + 1

κ

δ]) ≤ β1 e−β2 ( δ ) 4τ , for all δ > 0, where κ is a numerical constant and β1 and β2 are two constants that only depend on K , d, σ, τ, r1 and C. Moreover, the constant β1 also depends on M. −1 Proof. Let m ∈ Zd \{0} be such that [E 0 , E 1 ] = ρ(ω,A+F) ( <m> 2 mod π ), δ > 0 and let −1 n ∈ Zd \{0} be such that n = m and ρ(ω,A+F) ( 2 mod π ) ∩ [E1 , E1 + δ] = ∅. Let

−1 E 2 ∈ ρ(ω,A+F) ( π ) ∩ [E1 , E1 + δ]. Using Corollary 2, we get |ρ(ω,A+F) (E 1 ) − 2 mod √ √ √ ρ(ω,A+F) (E 2 )| ≤ 4 π C + 1 δ. On the other hand |ρ(ω,A+F) (E 1 ) − ρ(ω,A+F) (E 2 )| ≥ K K τ √1 √ 2|m−n|τ . Then |m − n| ≥ 8√π C+1 δ . So, for δ sufficiently small (that depends on m), we obtain |n| |m|. Using Corollary 1, it follows that

+∞ κ −1 d−1 −γ t 2 mod π )) ≤ c Leb(ρ(ω,A+F) ( e dt, 1 t 2 α1 ( 1δ ) 2τ

N

−1 ( where α1 = 21 ( √ K√ ) τ , N = {n; ρ(ω,A+F) 2 mod π ) ∩ [E1 , E1 + δ] = ∅} and 8 π C+1 c is a constant that only depends on K , τ, d and C. For δ sufficiently small and that depends on m, (i.e. there exists δ(m) sufficiently small such that δ ≤ δ(m)), we obtain 1

N

−1 Leb(ρ(ω,A+F) (

κ

c − 1 γ α 2 ( 1 ) 4τκ mod π )) ≤ e 2 1 δ , 2 κγ

which gives −1 Leb(ρ(ω,A+F) (

κ 1 4τ <m> mod π ) ∩ [E1 , E1 + δ]) ≤ β1 e−β2 ( δ ) , 2

where κ is a numerical constant and β1 and β2 are two constants that only depend on K , τ, σ, d, r1 and C. If δ isn’t sufficiently small (i.e. δ ≥ δ(m)), then β1 also depends on m. 7. Appendix. On the Rotation Number a(θ ) b(θ ) d Let A : θ ∈ T → ∈ S L(2, R) be continuous. We define the conc(θ ) d(θ ) tinuous function T(ω,A) : (θ, ϕ) ∈ Td × 21 T → (θ + ω, φ(ω,A) (θ, ϕ)) ∈ Td × 21 T, where φ(ω,A) (θ, ϕ) = arctan( c(θ)+d(θ) tan ϕ ). Let T˜(ω,A) : (x, y) ∈ Rd × R → (x + a(θ)+b(θ) tan ϕ

ω, φ˜ (ω,A) (x, y)) ∈ Rd × R, be a lift of T(ω,A) to Rd × R, where φ˜ (ω,A) (x, y) mod π =


585

φ(ω,A) (x mod π, y mod π ). We denote φ˜ (ω,A),x the function y ∈ R → φ˜ (ω,A) (x, y) ∈ R. Then, we have the following properties: (P1 ): The function y ∈ R → φ˜ (ω,A),x (y) − y ∈ R is π −periodic for all x ∈ Rd . (P2 ): There exists a unique m ∈ Zd such that the function x ∈ Rd → φ˜ (ω,A) (x, y) − <m,x> ∈ Rd is (2π )d −periodic for all y ∈ R. 2 : (x, y) ∈ Rd × R → (x + ω, φ˜ (ω,A) (x, y) + nπ ) ∈ Rd × R, n ∈ Z is (P3 ): T˜(ω,A) also a lift of T(ω,A) to Rd × R. (P4 ): If T˜(ω,A) : (x, y) ∈ Rd × R → (x + ω, φ˜ (ω,A) (x, y)) ∈ Rd × R is also a lift of (x, y) + nπ T(ω,A) to Rd × R, then there exists n ∈ Z such that φ˜ (ω,A) (x, y) = φ˜ (ω,A) d for all x ∈ R and y ∈ R. (P5 ): If A : Td → S L(2, R) is continuous and T˜(ω ,A ) : (x, y) ∈ Rd × R → (x + ω , φ˜ (ω ,A ) (x, y)) ∈ Rd × R is a lift of T(ω ,A ) to Rd × R, then T˜(ω ,A )◦(ω,A) is a lift of T(ω ,A )◦(ω,A) and φ˜ (ω ,A )◦(ω,A),x (y) = φ˜ (ω ,A ),x+ω ◦ φ˜ (ω,A),x (y). Furthermore, if A : Td → S L(2, R) is homotopic to the identity, then the same is true for the function T(ω,A) and therefore it admits a lift T˜(ω,A) : (θ, y) ∈ Td × R → (θ + ω, φ˜ (ω,A) (θ, y)) ∈ Td × R, where φ˜ (ω,A) (θ, y) mod π = φ(ω,A) (θ, y mod π ). For a Diophantine ω and by [9] the sequence φ˜ (ω,A),(θ+( j−1)ω) ◦ · · · ◦ φ˜ (ω,A),θ (y) − y j converges to the limit ρ(φ˜ (ω,A) ). This limit exists for all θ ∈ Td and for all y ∈ R and the convergence is uniform in (θ, y). We define now the rotation number of the skew-product system (ω, A) : (θ, y) ∈ Td × R2 → (θ + ω, A(θ )y) ∈ Td × R2 by ρ(ω,A) = ρ(φ˜ (ω,A) ) mod π which is independent of the chosen lift. For A : Td → S L(2, R) which is homotopic to the identity, properties Pi , 1 ≤ i ≤ 5 are available in the case of a lift to Td × R and m = 0 in P2 . We need now to discuss some properties of this concept: Lemma 6. Let ω ∈ Rd be Diophantine. If A, Z : Td → S L(2, R) are continuous and homotopic to the identity, then ρ(ω,Z −1 (.+ω)AZ ) = ρ(ω,A) . Proof. It is obvious that Z −1 (. + ω)AZ : Td → S L(2, R) is also homotopic to the identity. Let now T˜(0,Z ) , T˜(0,Z −1 ) and T˜(ω,A) be lifts of T(0,Z ) , T(0,Z −1 ) and T(ω,A) to Td × R, respectively. By property P5 , T˜(ω,Z −1 (.+ω)AZ ) is a lift of T(ω,Z −1 (.+ω)AZ ) to Td × R and we have φ˜ (ω,Z −1 (.+ω)AZ ) (θ, y) = φ˜ (0,Z −1 ),θ+ω ◦ φ˜ (ω,A),θ ◦ φ˜ (0,Z ),θ (y) for all θ ∈ Td and y ∈ R. By property P5 , T˜(0,Z )◦(0,Z −1 ) is also a lift of T(0,I ) to Td × R and then there exists n ∈ Z such that φ˜ (0,Z )◦(0,Z −1 ),θ (y) = y + nπ. But since φ˜ (0,Z −1 ) − nπ is also a lift, we can suppose that n = 0. Consequently, φ˜ (ω,Z −1 (.+ω)AZ ),θ+( j−1)ω ◦ · · · ◦ φ˜ (ω,Z −1 (.+ω)AZ ),θ (y) = φ˜ (0,Z −1 ),θ+ jω ◦ (φ˜ (ω,A),θ+( j−1)ω ◦ · · · ◦ φ˜ (ω,A),θ ) ◦ φ˜ (0,Z ),θ (y) = f (θ + jω, φ˜ (ω,A),θ+( j−1)ω ◦ · · · ◦ φ˜ (ω,A),θ (y )) + φ˜ (ω,A),θ+( j−1)ω ◦ · · · ◦ φ˜ (ω,A),θ (y ), where y = φ˜ (0,Z ),θ (y) and f is bounded. So, we deduce that ρ(φ˜ (ω,Z −1 (.+ω)AZ ) ) = ρ(φ˜ (ω,A) ) and this finishes the proof.

586

S. Hadj Amor

Lemma 7. Let ω ∈ Rd be Diophantine. If A : Td → S L(2, R) is continuous and homotopic to the identity, then B(θ ) = Z −1 (θ + ω)A(θ )Z (θ ), θ ∈ Td is homotopic to the identity and < m, ω > mod π. ρ(ω,B) = ρ(ω,A) − 2 Proof. It is clear that B(θ ) is homotopic to the identity. Let now T˜(0,Z ) , T˜(0,Z −1 ) and T˜(ω,A) be lifts of T(0,Z ) , T(0,Z −1 ) and T(ω,A) to Rd × R, respectively. By property P5 , T˜(ω,B) is a lift of T(ω,B) to Rd × R and we have φ˜ (ω,B) (x, y) = φ˜ (0,Z −1 ),x+ω ◦ φ˜ (ω,A),x ◦ φ˜ (0,Z ),x (y)

(19)

for all x ∈ Rd and y ∈ R. By property P5 , T˜(0,Z )◦(0,Z −1 ) is also a lift of T(0,I ) to Rd × R and then there exists n ∈ Z such that φ˜ (0,Z )◦(0,Z −1 ),x (y) = y + nπ. But since φ˜ (0,Z −1 ) − nπ is also a lift, we can suppose that n = 0. By property P1 and P2 , there exists a unique m ∈ Zd such that functions (i) (x, y) → φ˜ (ω,A) (x, y) − y, − y, (ii) (x, y) → φ˜ (0,Z ) (x, y) − <m,x> 2 (iii) (x, y) → φ˜ (0,Z −1 ) (x, y) + <m,x> − y, 2

are all × π −periodic. Combining (i), (ii) and (iii) with (19) yields φ˜ (ω,B) (x + ˜ 2kπ, y) = φ(ω,B) (x, y), where k ∈ Zd and this for all x ∈ Rd and y ∈ R, which proves that the function x → φ˜ (ω,B) (x, y) is (2π )d −periodic. So T˜(ω,B) is a lift of T(ω,B) to Td × R. Now, (2π )d

φ˜ (ω,B),x mod π +(j−1)ω ◦ · · · ◦ φ˜ (ω,B),x mod π (y) = φ˜ (0,Z −1 ),x+ jω ◦ (φ˜ (ω,A),x mod π +(j−1)ω ◦ · · · ◦ φ˜ (ω,A),x = f (x + jω, φ˜ (ω,A),x +φ˜ (ω,A),x

mod π +(j−1)ω

mod π +(j−1)ω

◦ · · · ◦ φ˜ (ω,A),x

◦ · · · ◦ φ˜ (ω,A),x

mod π ) ◦ φ˜ (0,Z ),x (y)

mod π (y

)) < m, x + jω > , mod π (y ) − 2

where y = φ˜ (0,Z ),x (y). Since f is bounded, the result follows.

Lemma 8. Let ω be Diophantine. If A : Td → S L(2, R) is homotopic to the identity and <m,x> Z (x) = e 2 J , where m ∈ Zd and x ∈ Rd , then B(x) = Z −1 (x +ω)A(x mod π )Z(x) is defined on Td , homotopic to the identity and < m, ω > mod π. ρ(ω,B) = ρ(ω,A) − 2 Proof. Since B(x + 2kπ ) = B(x) for all x ∈ Rd and k ∈ Zd , then B is defined on Td . Now, we verify that T˜(0,Z ) and T˜(0,Z −1 ) are lifts of T(0,Z ) and T(0,Z −1 ) to Rd × R with φ˜ (0,Z ) (x, y) = y + <m,x> and φ˜ (0,Z −1 ) (x, y) = y − <m,x> . So, by applying the same 2 2 argument as in lemma 8 we obtain the result. Lemma 9. Let ω be Diophantine and let a skew-product system (ω, A + F) : (θ, y) ∈ Td × R2 → (θ + ω, (A + F(θ ))y) ∈ Td × R2 , where A ∈ S L(2, R) is constant and A√+ F(θ ) ∈ S L(2, R) is homotopic to the identity. If |F| < ε, then |ρ(ω,A+F) −ρ(ω,A) | < c ε, where c is a numerical constant.


587

Proof. Let T˜(0,A) and T˜(ω,A+F) be lifts of T(0,A) and T(ω,A+F) to Td × R, respectively. If A = I, we denote φ˜ (0,I ) (θ, ϕ) = f (0) and φ˜ (ω,I +F) (θ, ϕ) = f (ε). By the finite φ˜

(θ,ϕ)−φ˜

(θ,ϕ)

increase theorem, we obtain | f (ε)−ε f (0) | = | (ω,I +F) ε (0,I ) | = | f (t)| ≤ 1 for d t ∈]0, ε[ and for all θ ∈ T and all ϕ ∈ R. If A = I and A − I is nilpotent, then the same calculation gives |

φ˜ (ω,A+F) (θ, ϕ) − φ(0,A) (θ, ϕ) |≤2 ε

for all θ ∈ Td and all ϕ ∈ R. If A ∈ S L(2, R) with two eigenvalues e±iρ(0,A) , ρ(0,A) ∈ R\{0}, then there exists a matrix C of eigenvectors such that cos(ρ(0,A) ) sin(ρ(0,A) ) C −1 := C Rρ(0,A) C −1 = C Rθ+ρ(0,A) Rθ−1 C −1 . A=C sin(ρ(0,A) ) cos(ρ(0,A) ) F

−1 So, we can write A + F(θ ) = C Rθ+ρ(0,A) (I + Rθ+ρ F Rθ )Rθ−1 C −1 , here (0,A)

= C −1 FC. By Lemma 6 and Lemma 8, we obtain that ρ(ω,A+F) = ρ(ω,I +R −1

θ+ρ(0,A) F

1 2

+ρ(0,A) , since |F | ≤ |F| . It follows by Case 2, that |ρ(ω,I +R −1

θ+ρ(0,A)

F Rθ )

R

θ)

−1 − ρ(0,I ) | ≤ 2|Rθ+ρ F Rθ | ≤ 2|F | ≤ 2 |F|. (0,A)

√ Then |ρ(ω,A+F) − ρ(0,A) | ≤ 2 ε, which gives the result.

Acknowledgement. I am indebted to Håkan ELIASSON for his interesting help.

References 1. Avila A., Jitomirskaya S.: Almost localization and almost reducibility. arXiv: 0805. 1761v1 [math.DS], 2008, to appear in J. Eur. Math. Soc. 2. Bourgain, J.: Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime. Lett. Math. Phys. 51(2), 83–118 (2000) 3. Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commu. Math. Phys. 89(3), 415–426 (1983) 4. Dinaburg, E.I., Sina˘ı, Ja.G.: The one-dimensional Schrödinger equation with quasiperiodic potential. Funk. Anal. Prilož. 9-4, 8–21 (1975) 5. Eliasson, L.H.: Almost reducibility of linear quasi-periodic systems. Proc. Sympos. Pure Math. 69, 679–705 (2001) 6. Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commu. Math. Phys. 146-3, 447–482 (1992) 7. Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Annali Della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 15-1, 115–147 (1988) 8. Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. (2) 154-1, 155–203 (2001) 9. Herman, M.-R.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol d et de Moser sur le tore de dimension 2. Comm. Math. Helv. 58-3, 453–502 (1983) 10. Krikorian, R.: Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts. Astérisque 259 (1999) 11. Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sina˘ı on quasiperiodic potentials. Comm. Math. Hel. 59-1, 39–85 (1984)

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12. Rüssmann, H.: On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash. (1974) Lecture Notes in Phys. 38, Berlin-Heidelberg-New York: Springer, 1975, pp. 598–624 13. Sina˘ı, Ja.G.: Structure of the spectrum of a Schrödinger difference operator with almost periodic potential near the left boundary. Funk. Anal. Priloz. 19-1, 34–39 (1985) Communicated by B. Simon


Communications in


On the Partial Regularity of a 3D Model of the Navier-Stokes Equations Thomas Y. Hou1 , Zhen Lei2 1 Applied and Comput. Math, Caltech, Pasadena, CA 91125, USA. E-mail: [email protected] 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China.

E-mail: [email protected] Received: 10 March 2008 / Accepted: 5 September 2008 Published online: 20 November 2008 – © Springer-Verlag 2008

Abstract: We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In [11], we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations. 1. Introduction The question of whether the solution of the 3D Navier-Stokes equations can develop singularities in a finite time from a large smooth initial data with finite energy is one of the most outstanding open mathematical problems [9]. The main difficulty in obtaining the global regularity of the 3D Navier-Stokes equations is due to its super-criticality and the presence of the vortex stretching. Many researchers have contributed to the understanding of the 3D Navier-Stokes equations. The pioneering work was done by Leray in the classical paper [21] in which the author established weak solutions of the 3D Navier-Stokes equations (which is in general called the Leray-Hopf weak solutions due to the important contributions made by Hopf [13] in the case of bounded domains). To understand the regularity properties of the Leray-Hopf weak solutions of the 3D Navier-Stokes equations, Scheffer [27–30] began the study of localizing Leray’s results

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T. Y. Hou, Z. Lei

in space. Scheffer’s program was developed further by Caffarelli-Kohn-Nirenberg [2], who proved that the one-dimensional Hausdorff measure of the singularity set is zero. Later, F. Lin [22] gave a simplified proof of the Caffarelli-Kohn-Nirenberg theory (see also [19]). The Caffarelli-Kohn-Nirenberg theory provides an important characterization of the nature of possible singularities of the 3D Navier-Stokes equations. Another important development on the global regularity of the 3D Navier-Stokes equations is to establish some non-blowup criteria for the solution of the 3D Navier-Stokes equations. Many researchers have made important contributions in this area, see, e.g. [1,8,17,26,31,32]. In [11], we studied the stabilizing effect of the convection term for the 3D incompressible Euler and Navier-Stokes equations. We demonstrated the stabilizing effect of convection by constructing a new 3D model for the axisymmetric Navier-Stokes equations with swirl. This model is formulated in terms of a set of new variables related to the angular velocity, the angular vorticity, and the angular stream function. The only difference between our 3D model and the reformulated Navier-Stokes equations in terms of these new variables is that we neglect the convection term in the model. If we add the convection term back to the model, we will recover the full Navier-Stokes equations. This new 3D model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations. In particular, the strong solution of the model satisfies an energy identity similar to that of the full 3D Navier-Stokes equations. We also proved a non-blowup criterion of Beale-Kato-Majda type [1] as well as a non-blowup criterion of Prodi-Serrin type [26,32] for the model. Despite the striking similarity at the theoretical level between the 3D model and the Navier-Stokes equations, our model has a completely different behavior from the full Navier-Stokes equations. In [11], we provided numerical evidence which seems to support that the model develops finite time singularities from smooth initial data with finite energy. The mechanism for developing these finite time singularities is due to the strong alignment between the variables that contribute to the vortex stretching term. But when we add the convection term back to our model, the mechanism for generating the finite time singularities in the model is destroyed. In this paper, we study the local behavior of the solutions to the 3D model equations and establish an analogue of the Caffarelli-Kohn-Nirenberg partial regularity theory for our model. We prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The proof of this partial regularity result is similar in spirit to that of Lin in [22], but there are some new technical difficulties associated with our model. One of the difficulties is to handle the singularity induced by the cylindrical coordinates. This makes it difficult to analyze the partial regularity of our model in R × R3 . To overcome this difficulty, we perform our partial regularity analysis in R × R5 . By working in R5 , we avoid the problem associated with the coordinate singularity. Another difficulty in obtaining our partial regularity result is that we do not have an evolution equation for the entire velocity field. We need to reformulate our model in terms of a new vector variable. This new variable can be considered as a “generalized velocity field” in R5 . We remark that the partial regularity theory for Navier-Stokes equations in R5 is still open due to the lack of certain compactness. For our model reformulated in R × R5 , we find a 3D structure which has the same scaling as that of the 3D Navier-Stokes equations. This is why the partial regularity analysis can be carried out for the model in R × R5 using a strategy similar to that of Lin in [22]. The results presented in this paper and our previous work [11] may have some important implication to the global regularity of the 3D Navier-Stokes equations. We believe

On Partial Regularity of a 3D Model of Navier-Stokes Equations

591

that a successful strategy in analyzing the global regularity of the 3D Navier-Stokes equations should take advantage of the stabilizing effect of the convection term in an essential way. So far most of the regularity analyses for the 3D Navier-Stokes equations do not use the stabilizing effect of the convection term. In many cases, the same results can be also obtained for our model. In [11], we have presented numerical evidence which shows that the 3D model is much more singular than the corresponding 3D Navier-Stokes equations. New analytical tools that exploit the local geometric structure of the solution and the stabilizing effect of convection may be needed to prove the global regularity of the 3D Navier-Stokes equations. We remark that the stabilizing effect of convection has been studied by Hou and Li in a recent paper [10]. They showed that the convection term has a surprising stabilizing effect which cancels the destabilizing term from vortex stretching in a new 1D model problem. This observation enabled them to obtain a crucial a priori pointwise estimate for a high order norm for the 1D model. Using this a priori estimate, they proved the global regularity of the 3D Navier-Stokes equations for a family of large initial data, whose solutions can lead to large dynamic growth. The stabilizing effect of convection has also been used by Deng-Hou-Yu in [6,7] in deriving localized non-blowup criteria for the 3D Euler equations using a Lagrangian formulation. Recently, Okamoto and Ohkitani [25] investigated the role of the convection term in preventing singularity formation by studying several one-dimensional models and a 2D model derived from the 2D Euler equations. There has been some interesting development in the study of the 3D incompressible Navier-Stokes equations and related models. In particular, by exploiting the special structure of the governing equations, Cao and Titi [3] proved the global well-posedness of the 3D viscous primitive equations which model large scale ocean and atmosphere dynamics. For the axisymmetric Navier-Stokes equations, Chen-Strain-Tsai-Yau [4,5] and KochNadirashvili-Seregin-Sverak [15] recently proved that if |u(x, t)| ≤ C∗ |t|−1/2 , where C∗ is allowed to be large, then the velocity field u is regular at time zero. The rest of this paper is organized as follows. In Sect. 2 we derive our 3D model and describe some of its important properties. In Sect. 3, we introduce the notations to be used in this paper. We then give the definition of weak and suitable weak solutions of the 3D model. In Sect. 4, we establish an important property which asserts that if the generalized velocity field is “sufficiently small” on the unit cylinder Q 1 , then the solution of the 3D model is regular on the smaller cylinder Q 1 . In Sect. 5, we establish 2 a “decay estimate” to study the local behavior of the gradient of the generalized velocity field and prove our partial regularity result. 2. The Derivation of the 3D Model and its Properties In this section, we will give a derivation of the 3D model that we introduced in [11]. Consider the 3D axisymmetric incompressible Navier-Stokes equations with swirl ⎧ ⎨ ut + (u · ∇)u = −∇ p + νu, ∇ · u = 0, (2.1) ⎩ u| t=0 = u0 (x), x = (x 1 , x 2 , z). Let er =

x

1

r

,

x x x2 2 1 , 0 , eθ = − , , 0 , ez = (0, 0, 1) , r r r

(2.2)

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T. Y. Hou, Z. Lei

be the three orthogonal unit vectors along the radial, the angular, and the axial directions respectively, r =

x12 + x22 . We will decompose the velocity field as follows:

u = u r (r, z, t)er + u θ (r, z, t)eθ + u z (r, z, t)ez ,

(2.3)

where u r , u θ , u z are called the radial, angular and axial velocity respectively. The component u θ is also referred to as the swirl component of the velocity. One can derive the following axisymmetric form of the Navier-Stokes equations in the cylindrical coordinates [24]: ⎧ r θ ⎪ ∂t u θ + u r ∂r u θ + u z ∂z u θ = ν x − r12 u θ − u ru , ⎪ ⎪ ⎨ θ 2 r θ ∂t ωθ + u r ∂r ωθ + u z ∂z ωθ = ν x − r12 ωθ + ∂z (ur ) + u rω , (2.4) ⎪ ⎪ ⎪ ⎩− x − 12 ψ θ = ωθ , r where u r = −∂z ψ θ , u z =

1 ∂r (r ψ θ ). r

(2.5)

The incompressible constraint in cylindrical coordinates is given by ∂r u r + ∂z u z +

ur = 0 or ∂r (r u r ) + ∂z (r u z ) = 0. r

(2.6)

Introduce the following new variables: u1 =

uθ ωθ ψθ , ω1 = , ψ1 = . r r r

(2.7)

In [10], Hou and Li derived an equivalent reformulation of the Navier-Stokes equations in terms of these new variables as follows: ⎧ 2 3 r z 2 ⎪ u , ⎨∂t u 1 + u ∂r u 1 + u ∂z u 1 = ν ∂r + r ∂r + ∂z u 1 + 2∂z ψ 1 1 (2.8) ∂t ω1 + u r ∂r ω1 + u z ∂z ω1 = ν ∂r2 + r3 ∂r + ∂z2 ω1 + ∂z (u 1 )2 , ⎪ ⎩ − ∂ 2 + 3 ∂ + ∂ 2 ψ = ω , 1 1 r z r r where u r = −∂z (r ψ1 ), u z =

1 ∂r (r 2 ψ1 ). r

(2.9)

As observed by Liu and Wang [23], any smooth solution of the Navier-Stokes equations must satisfy the following compatibility condition: u θ |r =0 = ωθ |r =0 = ψ θ |r =0 = 0. Thus the variables u 1 , ω1 and ψ1 are well defined. Our 3D model is derived by simply dropping the convection term from (2.8): ⎧ 2 3 2 ⎪ ⎨∂t u 1 = ν ∂r + r ∂r + ∂z u 1 + 2∂z ψ1 u 1 , (2.10) ∂ ω = ν ∂ 2 + 3 ∂ + ∂z2 ω1 + ∂z (u 21 ), ⎪ t 1 2 3 r 2r r ⎩ − ∂r + r ∂r + ∂z ψ1 = ω1 . Note that (2.10) is already a closed system. The main difference between our 3D model and the original Navier-Stokes equations is that we neglect the convection term in


593

our model. If we add the convection term back to our 3D model, we will recover the Navier-Stokes equations. This 3D model shares many important properties with the axisymmetric NavierStokes equations. First of all, there is an intrinsic incompressible structure in the model equations (2.10). To see this, we define a generalized velocity field as u(t, x) = u r (t, r, z)er + u θ (t, r, z)eθ + u z (t, r, z)ez , (r 2 ψ1 )r u θ = r u 1 , u r = −(r ψ1 )z , u z = , r

(2.11) (2.12)

where x = (x1 , x2 , z), r = x12 + x22 , er , eθ and ez are the three orthogonal unit vectors defined by (2.2). It is easy to check that ∇ · u = ∂r u r + ∂z u z +

ur = 0, r

(2.13)

which is the same incompressibility condition for the original incompressible Euler or Navier-Stokes equations. Our model also enjoys the following properties (see [11] for more details and their proofs): i) Energy identity. The strong solution of (2.10) satisfies the following energy identity: 1 d 2 dt

∞

∞ 2 2 3 |u 1 | + 2|Dψ1 | r dr + dz |Du 1 |2 +2|D 2 ψ1 |2 r 3 dr = 0, dz 0

0

(2.14) which has been proved to be equivalent to that of the Navier-Stokes equations. Here D is the first order derivative operator defined in R5 , see Sect. 3 for definition. ii) A non-blowup criterion of Beale-Kato-Majda type. A smooth solution (u 1 , ω1 , ψ1 ) of the model (2.10) for 0 ≤ t < T blows up at time t = T if and only if

T ∇ × uBMO(R3 ) dt = ∞,

(2.15)

0

where u is defined in (2.11)–(2.12). iii) A non-blowup criterion of Serrin-Prodi type. A weak solution (u 1 , ω1 , ψ1 ) of the model (2.10) is smooth on [0, T ] × R3 provided that u θ L qt L xp ([0,T ]×R3 ) < ∞ for some p, q satisfying

3 p

+

2 q

≤ 1 with 3 < p ≤ ∞ and 2 ≤ q < ∞.

(2.16)

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3. The Main Result and Suitable Weak Solutions We first state the main result of this paper which is summarized in the following theorem: Theorem 3.1. For any suitable weak solution of the 3D model equations (2.10) on an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. Before we define the suitable weak solutions of our 3D model, we introduce some notations to be used in this paper. Due to the special nature of the 3D model equations (2.10), we give a slightly different definition of weak and suitable weak solutions from the conventional one. The existence of such weak and suitable weak solutions are sketched at the end of this section. 3.1. Notations. Throughout the paper, we often switch our physical space between R3 and R5 . We denote by x = (x , z) = (x1 , x2 , z) a point in R3 and y = (y , z) = (y1 , y2 , y3 , y4 , z) a point in R5 . A space-time point in R × R3 and R × R5 will be denoted by ξ = (t, x) and ζ = (t, y), respectively. We use ∂t to denote the derivative of a function with respect to time, and ∇ = (∂x , ∂z )T = (∂x1 , ∂x2 , ∂z )T and D = (∂ y , ∂z )T = (∂ y1 , ∂ y2 , ∂ y3 , ∂ y4 , ∂z )T to denote the derivatives of a function with respect to space variables in R3 and R5 , respectively. Similarly, we will use Laplacians x = ∂x21 + ∂x22 + ∂z2 ,

y = ∂ y21 + ∂ y22 + ∂ y23 + ∂ y24 + ∂z2 .

Throughout this paper, if the function is axi-symmetric, we will denote its space var 2 iable by (r, z), where the z-axis is the symmetry axis and r = x1 + x22 in R3 and r = y12 + y22 + y32 + y42 in R5 , respectively. In particular, for an axi-symmetric function f = f (r, z), we have 3 y f = (∂r2 + ∂r + ∂z2 ) f. r

(3.1)

Further, we denote the domain consisting of a ball or a parabolic cylinder as follows: ⎧ ⎪ B(x0 , θ ) = x ∈ R3 | |x − x0 | < θ } , ⎪ ⎪ ⎨ Q(ξ , θ ) = ξ ∈ R × R5 |x − x | < θ, t − θ 2 < t < t , 0 0 0 0 5 | |y − y | < θ } , ⎪ B(y , θ ) = y ∈ R 0 ⎪ 0

⎪ ⎩ Q(ζ0 , θ ) = ζ ∈ R × R5 |y − y0 | < θ, t0 − θ 2 < t < t0 . In the case when the ball and cylinder are centered at the origin, we will use the following abbreviation: Bθ = B(0, θ ) and Q θ = Q(0, θ ). Their dimension will be indicated in the context without creating confusion. Further, we will denote the time interval (0, T ) by I . q p q p We will use L t L y (I × ) (resp. L t L x ) to denote the space-time norm

q/ p 1/q | f | p dy dt , f L qt L yp (I × ) = I


595

with the usual modifications when p or q is equal to infinity, or the domain is replaced by Q(ζ0 , θ ) or Q(ξ0 , θ ). The mean value of a function will be denoted by [ f ] y0 ,θ = |B(y10 ,θ)| B(y0 ,θ) f (t, y)dy, ( f )ζ0 ,θ = |Q(y10 ,θ)| Q(y0 ,θ) f (t, y)dy. When the domain is Bθ or Q θ , we use the abbreviation [ f ]θ or ( f )θ , respectively. 3.2. Weak and suitable weak solutions. Let R > 0 be a constant. We use x to denote one of the following domains in R3 : • the whole space R3 ;

• x ∈ R3 x12 + x22 < R, z ∈ R ;

• x ∈ R3 x12 + x22 < R, z ∈ T1 , T1 is the 1-dimensional torus. Similarly, we use y to denote one of the following domains in R5 : • the whole space R5 ;

• y ∈ R5 y12 + y22 + y32 + y42 < R, z ∈ R ;

• y ∈ R5 y12 + y22 + y32 + y42 < R, z ∈ T1 . The initial condition for our 3D model is of the form: u 1 (0, x) = u 10 (r, z), ω1 (0, x) = ω10 (r, z), ψ1 (0, x) = ψ10 (r, z), in x . (3.2) When we consider the 3D model in R5 , the initial condition becomes u 1 (0, y) = u 10 (r, z), ω1 (0, y) = ω10 (r, z), ψ1 (0, y) = ψ10 (r, z), in y . (3.3) In both cases, we require that ω10 and ψ10 satisfy the following compatibility condition: 3 ω10 (r, z) = − ∂r2 + ∂r + ∂z2 ψ10 (r, z). r

(3.4)

Note that for the initial-boundary value problem in a cylindrical domain with r < R 1 = ψ1 − 1 ψ1 (t, R, z)dz, then (u 1 , ψ 1 , ω1 ) also satisfies and z ∈ T1 , if we let ψ 0 1 1 (t, R, z)dz = 0. Thus, without loss of generality, we assume that (2.10) with 0 ψ 1 0 ψ1 (t, R, z)dz = 0 in our 3D model equations (2.10) for the initial-boundary value problem in a cylindrical domain with r < R and z ∈ T1 (if not, we turn to consider the 1 , ω1 )). We will impose an analogue of the no-slip, no-flow boundary system for (u 1 , ψ conditions for the generalized velocity field: u |r =R = 0,

(3.5)

which is equivalent to u r = u θ = u z = 0, i.e.

− r ∂z ψ1 = r u 1 = 2ψ1 + r ∂r ψ1 = 0, on r = R. (3.6)

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1 Since 0 ψ1 (t, R, z)dz = 0, it is easy to see that the boundary conditions (3.6) and (3.5) are equivalent to u 1 = ψ1 = ∂r ψ1 = 0 on r = R.

(3.7)

When x = x ∈ R3 x12 + x22 < R, z ∈ R or y = y ∈ R5 y12 + y22 + y32 + y42
0 and Br be a ball with radius r in R5 . Then there exists a positive constant C(q) independent of r such that for all f ∈ H01 (Br ) or f ∈ H 1 (Br ) with Br f dy = 0, the multiplicative inequality 5

−3

5

−5

f L q (Br ) ≤ C(q) f Lq 2 (B2 ) D f L2 2 (Bq

r)

r

1 is valid with q ∈ [2, 10 3 ]. If f ∈ H (Br ) with 5

−3

Br

(3.14)

f dy = 0, then multiplicative inequality 5

−5

f L q (Br ) ≤ C(q) f Lq 2 (B2 ) D f L2 2 (Bq ) + r

r

C(q) 5

r2

− q5

f L 2 (Br )

(3.15)

is valid with q ∈ [2, 10 3 ]. We will also use the following Sobolev-Poincarè imbedding inequality in our analysis f L 5 (Br ) ≤ CD f

5

L 2 (Br )

,

(3.16)

5 for all f ∈ W 1, 2 (R5 ) satisfying Br f dy = 0. First of all, we prove the following lemma which is similar in spirit to Theorem 2.2 in [22]:

On Partial Regularity of a 3D Model of Navier-Stokes Equations (n)

599

(n)

Lemma 3.4. Let (u 1 , ψ1 ) be a sequence of suitable weak solutions to the 3D model (n) (n) (n) (n) equations (2.10) on Q 1 . Denote v(n) = (u 1 , Dψ1 ). Assume that (u 1 , ψ1 ) and v(n) satisfy the following inequality: v(n) L ∞ + Dv(n) L 2 L 2 (Q 1 ) ≤ C0 , 2 t L y (Q 1 ) t

(3.17)

y

for some constant C0 > 0. Further we assume that (u 1 , Dψ1 ) is the weak limit of (n) (n) (u 1 , Dψ1 ) in L 2 [−1, 0], H 1 (B1 ) . Then (u 1 , ψ1 ) is also a suitable weak solution of the 3D model equations (2.10) on Q 1 . (n)

(n)

Proof. For each n, since (u 1 , ψ1 ) is a suitable weak solution to the 3D model equations (2.10) on Q 1 , we have

1 (n) 2 (n) 2 2 (n) 2 |u | |Du 1 | + 2|D ψ1 | hdζ ≤ 2 1 Q1

Q1

(n) (n) (n) + Dψ1 |2 ∂t h + y h − 2ψ1 (u 1 )2 ∂z h (n) + D(− y )−1 ∂z [(u 1 )2 ]Dh dζ, for all 0 ≤ h ∈ C0∞ (Q 1 ).

Using Fatou’s lemma, we can further deduce from the above inequality that

(n) (n) |Du 1 |2 + 2|D 2 ψ1 |2 hdζ ≤ lim inf n→∞ |Du 1 |2 + 2|D 2 ψ1 |2 hdζ Q1

Q1

≤ lim inf n→∞

1 (n) 2 (n) |u 1 | + |Dψ1 |2 ∂t h + y h 2

Q1

(n) −2ψ1

(n) (u 1 )2 ∂z h (n)

+ D(− y )−1 ∂z [(u 1 )2 ]Dh

dζ,

(3.18)

for all 0 ≤ h ∈ C0∞ (Q 1 ). To estimate the last term on the right-hand side of (3.18), we (n) may assume that B1 ψ1 dy = 0 since

(n) (n) (u 1 )2 ∂z h + D(− y )−1 ∂z [(u 1 )2 ]Dh dζ = 0,

(3.19)

Q1

which can be proved by performing integration by parts. Using the Hölder inequality, the Calderón-Zygmund theorem and the Sobolev-Poincarè imbedding inequality (3.16), we obtain

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(n) 2 2 −1 ψ1(n) (u (n) ) ∂ h + D(− ) ∂ [(u ) ]Dh dζ z y z 1 1 Q1

0 ≤C

2 ψ1(n) L 5 (B1 ) u (n) 1 L 5/2 (B ) dt 1

−1

0 ≤C

v(n) 3L 5/2 (B ) dt = Cv(n) 3 3

5/2

L t L y (Q 1 )

1

−1

. 5

If we can prove that v(n) strongly converges to v = (u 1 , Dψ1 ) in L 3t L y2 (Q 1 ), then we can pass the limit inside the integral on the right-hand side of (3.18) and the lemma follows. 5 We now prove the strong convergence of v(n) in L 3t L y2 (Q 1 ). Using the assumption (3.17) and the interpolation inequality (3.15) with q = 25 , we can show that v(n)

5

L 4t L y2 (Q 1 )

≤ C.

(3.20)

Next, we will show that ∂t v(n) is uniformly bounded in L 2t Hy−2 (Q 1 ). To see this, let φ be any smooth, compactly supported test function in H 2 (B1 ). Using (3.8), the Hölder inequality, and performing integration by parts, we obtain (n)

(n)

(n)

(n)

|(∂t u 1 , φ)| = |(2u 1 ∂z ψ1 , φ) + (νu 1 , y φ)| (n)

(n)

(n)

≤ 2u 1 L 5/2 (B ) ∂z ψ1 L 5/2 (B ) φ L 5y (B1 ) + νu 1 L 2y (B1 ) φ H 2 (B1 ) 1 1 y y ≤ 2v(n) 2 5/2 + νu (n) (3.21) 2 L y (B1 ) φ H 2 (B1 ) , 1 L y (B1 )

where we have used the Sobolev imbedding inequality φ L 5 (B1 ) ≤ Cφ H 2 (B1 ) in five space dimensions. Similarly, using (3.8) and the Calderón-Zygmund theorem, we can prove that (n) (n) |(∂t Dψ1 , φ)| ≤ C v(n) 2 5/2 + νDψ1 L 2y (B1 ) φ H 2 (B1 ) . (3.22) L y (B1 )

Combining (3.21) with (3.22), we obtain ∂t v(n) H −2 (B1 ) ≤ C v(n) 2 5/2

L y (B1 )

+ νv(n) L 2y (B1 ) .

(3.23)

It follows from (3.20) and (3.17) that ∂t v(n) is uniformly bounded in L 2t Hy−2 (Q 1 ). Thus each v(n) ∈ C([−1, 0], H −2 (B1 )). Now we can apply the well-known compactness theorem (see Theorem 2.1 on p. 184 in [34] with X 1 = H −2 (B1 ), X 0 = H 1 (B1 ), X = L 2 (B1 ), and α0 = α1 = 2) to conclude that v(n) lies in a compact subset of L 2t L 2y (Q 1 ). Using (3.17) and the following Sobolev interpolation inequality: 1

2

v(n) − v L 6 L 2 (Q 1 ) ≤ Cv(n) − v L3 2 L 2 (Q ) v(n) − v L3 ∞ L 2 (Q ) , t

y

t

y

1

t

y

1


601

we conclude that v(n) (up to a subsequence) converges strongly to v in L 6t L 2y (Q 1 ). On the other hand, the bound (3.17) and the Sobolev interpolation inequality (3.15) with 10

(n) in L 2 L 3 (Q ). Finally, the following interpolation q = 10 1 t y 3 give a uniform bound for v inequality

v(n) − v

1

5 L 3t L y2 (Q 1 )

1

≤ Cv(n) − v L2 6 L 2 (Q ) v(n) − v 2 t

y

1

10

L 2t L y3 (Q 1 )

5

implies that v(n) strongly converges to v = (u 1 , Dψ1 ) in L 3t L y2 (Q 1 ). It is easy to show 5

using the strong convergence of v(n) in L 3t L y2 (Q 1 ) that v is a weak solution of the 3D model equations (2.10). This completes the proof of the lemma.

Now we consider the following problem which is an approximation to the 3D model. Let n > 0 be any given integer. We set δ = Tn and solve the following problem: (n) (n) (n) ∂t u (n) 1 = 2∂z ψ1 δ (u 1 ) + y u 1 , (3.24) (n) (n) (n) (n) ∂t Dψ1 = D(− y )−1 ∂z [u 1 δ (u 1 )] + y Dψ1 , (n)

(n)

where δ (u 1 ) is a “retarded mollification” of u 1 , whose value at time t depends only (n) on the values of u 1 at times prior to t − δ (see [2]). For fixed n, solving (3.24) amounts to solving a linear equation on each strip mδ ≤ t ≤ (m + 1)δ, 0 ≤ m ≤ n − 1. One can construct suitable weak solutions of (3.24) for each n and prove its weak convergence using a proof similar to that of [2]. By using an argument similar to the proof of Lemma 3.4, we can prove the existence of suitable weak solutions of the 3D model equations (2.10). 4. Partial Regularity Theory: Part I This and the next section are devoted to proving the partial regularity result of our 3D model. The partial regularity analysis we present here uses a strategy similar to that of [22]. In this section, we prove Theorem 4.1 concerning the minimum rate at which a singularity can develop. This is an analogue of Theorem 3.1 in [22]. In Sect. 5, we will prove Theorem 5.1 which is similar to the gradient estimate of Theorem 4.1 in [22]. By using the classical covering lemma and Theorem 5.1, we can prove Theorem 3.1 in exactly the same way as that of the partial regularity theory of the Navier-Stokes equations. We will omit the detail of this part of the proof of Theorem 3.1 and refer the reader to [2] for more details. In the remaining part of this paper, we will focus on proving Theorem 4.1 and Theorem 5.1, which are two key estimates whose analysis is different from that of the Navier-Stokes equations. Theorem 4.1. There exist two positive constants 0 and κ0 such that if (u 1 , ψ1 ) is a suitable weak solution of the 3D model equations (2.10) which satisfies 1 θ2

0 v3L 5/2 (B ) dt ≤ 0 θ

−θ 2

for some θ > 0, then v = (u 0 , Dψ1 ) is regular at (0, 0).

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T. Y. Hou, Z. Lei

We note that the 3D model (2.10) has an important scaling property. If (u 1 , ω1 , ψ1 ) is a solution of (2.10), so is λ u 1 (t, x), ω1λ (t, x), ψ1λ (t, x) = λ2 u 1 (λ2 t, λx), λ3 ω1 (λ2 t, λx), λψ1 (λ2 t, λx) , for any λ > 0. Therefore, to prove Theorem 4.1, it suffices to prove Lemma 4.2. There exist two positive constants 0 and κ0 such that if (u 1 , ψ1 ) is a suitable weak solution of the 3D model that satisfies

0 v3L 5/2 (B ) dt ≤ 0 ,

(4.1)

vC α (Q δ ) ≤ κ0 ,

(4.2)

1

−1

then we have

for some 0 < α < 1 and 0 < δ < 1, where v = (u 1 , Dψ1 ). To prove Lemma 4.2, we first prove the following lemma: Lemma 4.3. For given 0 < γ < δ ≤ 21 , there exists a positive constant 0 depending only on γ such that if (u 1 , ψ1 ) is a suitable weak solution of the 3D model that satisfies

0 v3L 5/2 (B ) dt ≤ 0 ,

(4.3)

1

−1

then we have 1 γ8

0 v

− (v)γ 3L 5/2 (B ) dt γ

≤γ

0

α0

v3L 5/2 (B ) dt,

(4.4)

1

−1

−γ 2

for some positive constant α0 ∈ (0, 15 ), where v = (u 1 , Dψ1 ) and (v)γ =

1 |Q γ |

Qγ

vdζ .

Proof. We will prove the lemma by contradiction. Suppose that the statement of Lemma 4.3 is false. This means that one can find a sequence of suitable weak solutions (u k1 , ψ1k ) of the 3D model equations (2.10) on Q 1 such that

0 vk 3L 5/2 (B ) dt = k3 → 0 as k → ∞,

(4.5)

1

−1

and 1 γ8

v − (v k

−γ 2

k

)γ 3L 5/2 (B ) dt γ

>γ

α0

0 vk 3L 5/2 (B ) dt, 1

−1

(4.6)


603

for any α0 ∈ (0, 15 ), where vk = (u k1 , Dψ1k ). Below we will show that (4.5) and (4.6) would lead to a contradiction. Let u k1 =

u k1 ωk ψk vk 1k = 1 , , ω1k = 1 , ψ vk = k . k k k

(4.7)

k ) satisfies Then ( u k1 , ψ 1 ⎧ k k ⎪ u k1 = y u k1 + 2k ∂z ψ ⎨∂t 1 u1, k 2 k k + k D(− y )−1 ∂z ( = D y ψ u1) , ∂t D ψ 1 1 ⎪ ⎩− ψ k = k ω , y 1

(4.8)

1

in the sense of distribution. By (4.5), we have vk L 3 L 5/2 (Q t

y

1)

= 1.

(4.9)

Consequently, there exists a subsequence of { vk } (still denoted by { vk }) such that 1k ) 1 ) in L 3t L 5/2 u k1 , D ψ v = ( u1, Dψ vk = ( y (Q 1 )

(4.10)

1 ) ∈ L 3t L 5/2 for some v = ( u1, Dψ y (Q 1 ). Using (4.8)–(4.9) and the Calderón-Zygmund theorem, we can show that the weak limit v satisfies the following equations: ⎧ ⎪ u 1 = y u1, ⎨∂t 1 , 1 = D y ψ (4.11) ∂t D ψ ⎪ ⎩− ψ = ω y 1 1 1 ) is in the sense of distribution. The classical parabolic estimates imply that ( u1, ψ smooth and 1 γ8 for 0 < γ < δ ≤ has

1 2

0 v − ( v)γ 3L 5/2 (B ) dt ≤ Cγ 2α0 γ

−γ 2

and some constant C > 0. By choosing a smaller δ if necessary, one

1 γ8

0 v − ( v)γ 3L 5/2 (B ) dt ≤ γ

1 α0 γ . 2

(4.12)

−γ 2

k ) is a suitable weak solution of system On the other hand, it is easy to see that ( u k1 , ψ 1 k dζ = 0 without loss of (4.8) on Q 1 . In view of (3.19), we may assume that B1 ψ 1

604

T. Y. Hou, Z. Lei

generality. Consequently, by the Sobolev-Poincaré imbedding inequality (3.16) and the Calderón-Zygmund theorem, we obtain 0

(k) (k) (k) ( ψ u 1 )2 ∂z h + D(− y )−1 ∂z [( u 1 )2 ]Dh dyds 1 −1 B1

0 ≤C

k 2 1k L 5 (B ) u ψ | | L 5/4 (B ) ds 1 1 y 1

y

−1

≤ C vk 3 3

5/2

L t L y (Q 1 )

≤ C,

(4.13)

where we have used (3.16) in the last inequality. Combining (4.13) with the generalized energy inequality (3.13), we get vk L ∞ + D vk L 2 L 2 (Q 1/2 ) ≤ C, 2 t L y (Q 1/2 ) t

y

(4.14)

for some absolute positive constant C. Using (4.14) and the dynamic equations (4.8), we can argue exactly as we did in the last part of the proof of Lemma 3.4 to show that 5/2 vk converges strongly to v in L 3t L y (Q 1/2 ), i.e. 5/2

vk → v in L 3t L y (Q 1/2 ).

(4.15)

Passing to the limit k → ∞ in (4.6) gives 1 γ8

v − ( v)γ 3L 5/2 (B ) dt ≥ γ α0 , γ

(4.16)

Qγ

which contradicts (4.12). This completes the proof of Lemma 4.3.

Now we are ready to prove Lemma 4.2. Let (u 1 , ω1 , ψ1 ) be a suitable weak solution of the 3D model equations (2.10). We assume that (4.1) is satisfied. For 0 < γ < δ ≤ 21 , we define ⎧ 2 −α /3 ⎪ ⎨u 11 (t, y) = γ 0 u 1 − (u 1 )γ (γ t, γ y), ω11 (t, y) = γ 1−α0 /3 ω1 (γ 2 t, γ y), ⎪ ⎩ψ (t, y) = γ −1−α0 /3 ψ − [ψ ] (γ 2 t, γ y). 11 1 1 γ Obviously, (u 11 , ψ11 ) also forms a suitable weak solution of ⎧ α /3 2 3 2 2 ⎪ ⎨∂t u 11 = ∂r + r ∂r + ∂z u 11 + 2γ ∂zψ11 γ 0 u 11 + (u 1 )γ , ∂t ω11 = ∂r2 + r3 ∂r + ∂z2 ω11 + γ 2 ∂z γ α0 /3 u 211 + 2u 11 (u 1 )γ , ⎪ ⎩− ∂ 2 + 3 ∂ + ∂ 2 ψ = ω 11 11 r z r r


605

on Q 1 . Moreover, using Lemma 4.3 and the assumption of Lemma 4.2, we obtain

0 (u 11 , Dψ11 )3L 5/2 (B ) dt 1

−1

0

−8−α0

=γ

(u 1 , Dψ1 ) − (u 1 , Dψ1 )γ 3 5/2 L

(Bγ )

(u 11 , Dψ11 ) − (u 11 , Dψ11 )γ 3 5/2

dt

dt

−γ 2

0 ≤

v3L 5/4 (B ) dt ≤ 0 . 1

−1

Applying Lemma 4.3 one more time to (u 11 , ψ11 ), we get 1 γ8

0

L

(Bγ )

−γ 2

≤γ

α0

0

(u 11 , Dψ11 )3L 5/2 (B ) dt ≤ γ α0 0 , 1

−1

which is equivalent to

0

1 (γ 2 )8

(u 1 , Dψ1 ) − (u 1 , Dψ1 )γ 2 3 5/2 L

−γ 4

(Bγ 2 )

dt ≤ 0 (γ 2 )α0 .

(4.17)

dt ≤ 0 (γ k )α0 ,

(4.18)

A simple iteration yields that

0

1 (γ k )8

(u 1 , Dψ1 ) − (u 1 , Dψ1 ) k 3 5/2 γ

−γ 2k

L

(Bγ k )

which gives 1 |Q γ k | ≤C

(u 1 , Dψ1 ) − (u 1 , Dψ1 ) k 5/2 dζ γ

(4.19)

Qγ k

1 (γ k )8

0

(u 1 , Dψ1 ) − (u 1 , Dψ1 ) k 3 5/2 γ

−γ 2k

L

(Bγ k

dt )

5/6

≤ C0 (γ k )5α0 /6 5/6

for all integer k ≥ 0. By Campanato’s condition, v = (u 1 , Dψ1 ) is Hölder continuous on Q δ . This proves Lemma 4.2.

606

T. Y. Hou, Z. Lei

5. Partial Regularity Theory: Part II We begin with the following theorem: Theorem 5.1. Let (u 1 , ψ1 ) be any suitable weak solution of the 3D model equations (2.10) on Q(ζ0 , R0 ) and v = (u 1 , Dψ1 ). There exists a positive constant such that if

1 lim supρ→0 |Dv|2 dζ ≤ , (5.1) ρ Q(ζ0 ,ρ)

then ζ0 = (t0 , y0 ) is a regular point of (u 1 , ψ1 ). As we explained at the beginning of Sect. 3, the proof of Theorem 3.1 follows by the classical covering lemma and the above Theorem 5.1. We refer the reader to [2] for details. Below we will present the proof of Theorem 5.1, which is in spirit similar to that of [22]. To prove Theorem 5.1, let us first define several functionals: ⎧ 1 2 ⎪ ⎨ A1 (θ ) = supt0 −θ 2 ≤t≤t0 θ B(y0 ,θ) |v(t, y)| dy, t A2 (θ ) = θ12 t 0−θ 2 v(t, ·)3L 5/2 (B(y ,θ)) dt, (5.2) 0 ⎪ 0 ⎩ 1 2 E(θ ) = θ Q(ζ0 ,θ) |Dv(t, y)| dζ. We need the following lemma: Lemma 5.2. There holds

3/2 3/4 A2 (θ ) ≤ C (θ/γ )6 A1 (γ ) + A1 (γ )(γ /θ )3 E 3/4 (γ )

(5.3)

for 0 < θ ≤ γ < R0 , where C is an absolute positive constant. Proof. First of all, we recall the Poincaré inequality:

2 2 |v| − [|v| ] y0 ,γ dy ≤ Cγ B(y0 ,γ )

|v||Dv|dy.

(5.4)

B(y0 ,γ )

By using (5.4) and the Hölder inequality, we can show by a straightforward calculation that

2 2 [|v| dy + (5.5) |v|2 dy ≤ ] |v| − [|v|2 ] y0 ,γ dy y0 ,γ B(y0 ,θ)

B(y0 ,θ)

≤ (θ/γ )5

B(y0 ,θ)

|v|2 dy + Cγ B(y0 ,γ )

|v||Dv|dy B(y0 ,γ )

⎛

1/2 ⎜ ≤ (θ/γ )5 γ A1 (γ ) + Cγ γ A1 (γ ) ⎝

⎞1/2

⎟ |Dv|2 dy ⎠

B(y0 ,γ )

Thus, using (5.5) and the standard multiplicative inequality (see Lemma 3.3) v

1

5 L 2 (Bθ )

1

1

≤ Cθ − 2 v L 2 (Bθ ) + Cv L2 2 (B ) Dv L2 2 (B ) , θ

θ

.


607

we get

C

|v|5/2 dy ≤ B(y0 ,θ)

θ 5/4

|v|2 dy

+C

|v|2 dy

5/4

B(y0 ,θ)

5/8

B(y0 ,θ)

|Dv|2 dy

5/8

B(y0 ,θ)

1/2 C 5 ≤ 5/4 (θ/γ ) γ A1 (γ ) + Cγ γ A1 (γ ) θ 5/8

+ Cγ 5/8 A1 (γ )

|Dv|2 dy

1/2 5/4

B(y0 ,γ )

|Dv|2 dy

5/8

B(y0 ,θ)

5/4 5/8 ≤ C(θ/γ )5 A1 (γ )+C A1 (γ ) γ 5/8 +γ 15/8 /θ 5/4

5/8 |Dv|2 dy . (5.6)

B(y0 ,γ )

We finally arrive at

t0

3/2

v3L 5/2 (B(y

0 ,θ))

dt ≤ Cθ 2 (θ/γ )6 A1 (γ )

t0 −θ 2

+

3/4 C A1 (γ )

t0

3/4 9/4 3/2 γ + γ /θ t0

≤

3/2 Cθ 2 (θ/γ )6 A1 (γ )

+

3/4 C A1 (γ )

γ

3/4

+γ

9/4

/θ

3/2

−θ 2

θ

1/2

|Dv|2 dy

3/4

dt

B(y0 ,γ )

Q(ζ0 ,γ )

|Dv|2 dζ

3/4

3/2 3/4 ≤ Cθ 2 (θ/γ )6 A1 (γ ) + C A1 (γ ) γ 3/4 + γ 9/4 /θ 3/2 θ 1/2 γ 3/4 E 3/4 (γ ) 3/2 3/4 ≤ Cθ 2 (θ/γ )6 A1 (γ ) + A1 (γ )(γ /θ )3 E 3/4 (γ ) . (5.7)

This proves the lemma.

We also need the following lemma:

608

T. Y. Hou, Z. Lei

Lemma 5.3. For suitable weak solution (u 1 , ψ1 ) of the 3D model equations (2.10) on Q 1 , we have 2/3 1/2 1/3 A1 (θ/2)+ E(θ/2) ≤ C A2 (θ ) + A1 (θ )E 1/2 (θ )+ A1 (θ )A2 (θ )E 1/2 (θ ) , (5.8)

where C is an absolute positive constant. Proof. Let h ∈ C0∞ (Q θ ) be any smooth cut-off function. Integration by parts gives

[ψ1 ] y0 ,θ u 21 ∂z h + D(− y )−1 ∂z (u 21 )Dh dζ = 0.

Q1

Now we choose a smooth cut-off function h with the following properties:

h = 0 on Q(ζ0 , θ )c , h = 1 in Q(ζ0 , θ/2), 0 ≤ h ≤ 1, θ |∇h| + θ 2 |∂t h| + |D 2 h| ≤ C.

Using the generalized energy inequality (3.13) and the Calderón-Zygmund theorem, we get A1 (θ/2) + 2E(θ/2)

t0 C supt0 −θ 2 1 x→∞

(1.2)

and it is no accumulation point if lim sup −4x 2 q(x) < 1.

(1.3)

x→∞

The proof consists of the observation that the Euler equation τ0 = −

d2 µ + 2 2 dx x

(1.4)

is explicitly solvable plus an application of Sturm’s comparison theorem between τ0 and τ. The corresponding result within spectral gaps is Rofe-Beketov’s theorem for periodic operators ([22], see also the recent monograph [23]). We state it in the version of [24],

Relative Oscillation Theory

615

where a detailed proof plus extensions are given. Suppose lim x→∞ x 2 (q(x) − q0 (x)) exists. Then a boundary point E n of the essential spectrum is an accumulation point of eigenvalues if lim κn x 2 (q(x) − q0 (x)) > 1

x→∞

(1.5)

and no accumulation point if lim κn x 2 (q(x) − q0 (x)) < 1,

x→∞

(1.6)

where the critical constant κn will be defined in Sect. 4 below. Comparing this result with Kneser’s result, there is an obvious mismatch: While Kneser’s result covers all potentials which are above respectively below the critical case −1 q(x) = 4x 2 , the above result only covers the case where q(x) has a precise asymptotic q(x) = q0 (x) + xc2 + o(x −2 ). As a short application we will show how a comparison argument analogous to the one for Kneser’s theorem, can be used to fill this gap. While the ingredients for such a comparison argument can essentially be found in the results by Weidmann [41] (see the paragraph after Theorem 3.8 for a more detailed discussion of earlier results), we still feel there is a need to advocate their use. In addition, we hope that our novel interpretation as weighted zeros of Wronskians will lead to new relative oscillation criteria and stimulate further research in this direction. Some possible extensions are listed at the end of Sect. 3. Furthermore, the arguments used in [41] typically lead to spectral estimates (compare Satz 4.1 in [41] respectively Lemma 3.11 below). On the other hand, the results in [8] and [5] give not only estimates, but precise equalities between spectral data and zeros of solutions respectively Wronskians of solutions. Hence, as our main results we will establish precise equalities between spectral shifts and the number of weighted zeros of Wronskians of certain solutions in Theorem 3.13 and Theorem 3.16. 2. Weighted Zeros of Wronskians, Prüfer Angles, and Regular Operators The key ingredient will be roughly speaking weighted zeros of Wronskians of solutions of different Sturm–Liouville operators. However, this naive definition has a few subtle problems which we need to clarify first. This will be done by giving an alternate definition in terms of Prüfer angles and establishing equivalence for those cases where both definitions are well-defined. To set the stage, we will consider Sturm–Liouville operators on L 2 ((a, b), r d x) with −∞ ≤ a 0.

(2.2)

We will use τ to describe the formal differentiation expression and H the operator given by τ with separated boundary conditions at a and/or b. If a (resp. b) is finite and p −1 , q, r are in addition integrable near a (resp. b), we will say a (resp. b) is a regular endpoint. We will say τ respectively H is regular if both a and b are regular.

616

H. Krüger, G. Teschl

For every z ∈ C\σess (H ) there is a unique (up to a constant) solution ψ− (z, x) of τ u = zu which is in L 2 near a and satisfies the boundary condition at a (if any). Similarly there is such a solution ψ+ (z, x) near b. One of our main objects will be the (modified) Wronskian Wx (u 0 , u 1 ) = u 0 (x) p(x)u 1 (x) − p(x)u 0 (x) u 1 (x)

(2.3)

of two functions u 0 , u 1 and its zeros. Here we think of u 0 and u 1 as two solutions of two different Sturm–Liouville equations d d 1 τj = j = 0, 1. (2.4) − p + qj , r dx dx Note that we have chosen p0 = p1 ≡ p here. The case p0 = p1 will be given in [14]. Classical oscillation theory counts the zeros of solutions u j , which are always simple since u and pu cannot both vanish except if u ≡ 0. This is no longer true for the zeros of the Wronskian Wx (u 0 , u 1 ). In fact, it could even happen that the Wronskian vanishes on an entire interval if q0 = q1 on this interval (cf. (2.8)). Such a situation will be counted as just one zero, in other words, we will only count sign flips. Furthermore, classical oscillation theory involves the spectrum of only one operator, but we want to measure the difference between the spectra of two operators. That is, we need a signed quantity. Hence we will weight the sign flips according to the sign 1 of q1 − q0 . Of course this definition is not good enough for the case q1 − q0 ∈ L loc considered here. Hence we will give a precise definition in terms of Prüfer variables next. We begin by recalling the definition of Prüfer variables ρu , θu of an absolutely continuous function u: u(x) = ρu (x) sin(θu (x)),

p(x)u (x) = ρu (x) cos(θu (x)).

(2.5)

If (u(x), p(x)u (x)) is never (0, 0) and u, pu are absolutely continuous, then ρu is positive and θu is uniquely determined once a value of θu (x0 ) is chosen by requiring continuity of θu . Notice that Wx (u, v) = −ρu (x)ρv (x) sin( v,u (x)),

v,u (x) = θv (x) − θu (x).

(2.6)

Hence the Wronskian vanishes if and only if the two Prüfer angles differ by a multiple of π . We will call the total difference #(c,d) (u 0 , u 1 ) = 1,0 (d)/π − 1,0 (c)/π − 1

(2.7)

the number of weighted sign flips in (c, d), where we have written 1,0 (x) = u 1 ,u 0 for brevity. Next, let us show that this agrees with our considerations above. We take two solutions u j , j = 1, 2, of τ j u j = λ j u j and associated Prüfer variables ρ j , θ j . Since we can replace q → q − λr , it is no restriction to assume λ0 = λ1 = 0. Under these assumptions Wx (u 0 , u 1 ) is absolutely continuous and satisfies Wx (u 0 , u 1 ) = −(q0 (x) − q1 (x))u 0 (x)u 1 (x).

(2.8)


617

Lemma 2.1. Abbreviate 1,0 (x) = θ1 (x) − θ0 (x) and suppose 1,0 (x0 ) ≡ 0 mod π . If q0 (x) − q1 (x) is (i) negative, (ii) zero, or (iii) positive for a.e. x ∈ (x0 , x0 + ε) respectively for a.e. x ∈ (x0 −ε, x0 ) for some ε > 0, then the same is true for ( 1,0 (x)− 1,0 (x0 ))/(x − x0 ). Proof. By (2.8) we have Wx (u 0 , u 1 ) = −ρ0 (x)ρ1 (x) sin( 1,0 (x)) x =− (q0 (t) − q1 (t))u 0 (t)u 1 (t)dt,

(2.9)

x0

and there are two cases to distinguish: Either u 0 (x0 ), u 1 (x0 ) are both different from zero or both equal to zero. If both are different from zero, u 0 (t)u 1 (t) does not change sign at t = x0 . If both are equal to zero, they must change sign at x0 and hence again u 0 (t)u 1 (t) does not change sign at t = x0 . Now the claim is evident. Hence #(c,d) (u 0 , u 1 ) counts the weighted sign flips of the Wronskian Wx (u 0 , u 1 ), where a sign flip is counted as +1 if q0 − q1 is positive in a neighborhood of the sign flip, it is counted as −1 if q0 − q1 is negative in a neighborhood of the sign flip. If q0 − q1 changes sign (i.e., it is positive on one side and negative on the other) the Wronskian will not change its sign. In particular, we obtain Lemma 2.2. Let u 0 , u 1 solve τ j u j = 0, j = 0, 1, where q0 −q1 ≥ 0. Then #(a,b) (u 0 , u 1 ) equals the number sign flips of W (u 0 , u 1 ) inside the interval (a, b). In the case q0 − q1 ≤ 0 we get of course the corresponding negative number except for the fact that zeros at the boundary points are counted as well since −x = − x . That is, if q0 − q1 > 0, then #(c,d) (u 0 , u 1 ) equals the number of zeros of the Wronskian in (c, d) while if q0 − q1 < 0, it equals minus the number of zeros in [c, d]. In the next theorem we will see that this is quite natural. In addition, note that #(u, u) = −1. Finally, we establish the connection with the spectrum of regular operators. For this let H0 , H1 be self-adjoint extensions of τ0 , τ1 with the same separated boundary conditions. Theorem 2.3. Let H0 , H1 be regular Sturm–Liouville operators associated with (2.4) and the same boundary conditions at a and b. Then dim Ran P(−∞,λ1 ) (H1 )− dim Ran P(−∞,λ0 ] (H0 ) = #(a,b) (ψ0,± (λ0 ), ψ1,∓ (λ1 )). (2.10) The proof will be given in Sect. 5 employing interpolation between H0 and H1 , using Hε = (1 − ε)H0 + ε H1 together with a careful analysis of Prüfer angles. It is important to observe that in the special case H1 = H0 , the left-hand side equals dim Ran P(λ1 ,λ0 ) (H0 ) if λ1 > λ0 and − dim Ran P[λ0 ,λ1 ] (H0 ) if λ1 < λ0 . This is of course in accordance with our previous observation that #(ψ0,± (λ0 ), ψ1,∓ (λ1 )) equals the number of zeros in (a, b) if λ1 > λ0 while it equals minus the numbers of zeros in [a, b] if λ1 < λ0 . 3. Relative Oscillation Theory After these preparations we are now ready to develop relative oscillation theory. For the connections with earlier work we refer to the discussion after Theorem 3.8.

618


Definition 3.1. For τ0 , τ1 possibly singular Sturm–Liouville operators as in (2.4) on (a, b), we define #(u 0 , u 1 ) = lim inf #(c,d) (u 0 , u 1 ) and #(u 0 , u 1 ) = lim sup #(c,d) (u 0 , u 1 ), (3.1) d↑b, c↓a

d↑b, c↓a

where τ j u j = λ j u j , j = 0, 1. We say that #(u 0 , u 1 ) exists, if #(u 0 , u 1 ) = #(u 0 , u 1 ), and write #(u 0 , u 1 ) = #(u 0 , u 1 ) = #(u 0 , u 1 )

(3.2)

in this case. By Lemma 2.1 one infers that #(u 0 , u 1 ) exists if q0 −λ0 r −q1 +λ1r has the same definite sign near the endpoints a and b. On the other hand, note that #(u 0 , u 1 ) might not exist even if both a and b are regular, since the difference of Prüfer angles might oscillate around a multiple of π near an endpoint. Furthermore, even if it exists, one has #(u 0 , u 1 ) = #(a,b) (u 0 , u 1 ) only if there are no zeros at the endpoints (or if q0 − λ0 r − q1 + λ1r ≥ 0 at least near the endpoints). Remark 3.2. Note that cases like #(u 0 , u 1 ) = −∞ and #(u 0 , u 1 ) = +∞ can occur. To 2 construct such a situation let τ j = − ddx 2 + q j on (0, ∞) and λ0 = λ1 = 0. Let u j (x) be the solutions satisfying a Neumann boundary condition at u j (0) = 0.

π Choose q0 (x) = 0, q1 (x) = 1 on (0, π ) such that θ1 (π )−θ0 (π ) = 3π 2 − 2 = π . Next, 5π choose q0 (x) = 1, q1 (x) = 0 on (π, 3π ) such that θ1 (3π ) − θ0 (3π ) = 3π 2 − 2 = −π . 9π Next, choose q0 (x) = 0, q1 (x) = 1 on (3π, 6π ) such that θ1 (6π )−θ0 (6π ) = 2 − 5π 2 = 2π . etc.

We begin with our analog of Sturm’s comparison theorem for zeros of Wronskians. We will also establish a triangle-type inequality which will help us to provide streamlined proofs below. As with Sturm’s comparison theorem, the proofs are elementary. Theorem 3.3. (Comparison theorem for Wronskians) Suppose u j satisfies τ j u j = λ j u j , j = 0, 1, 2, where λ0 r − q0 ≤ λ1r − q1 ≤ λ2 r − q2 . If c < d are two zeros of Wx (u 0 , u 1 ) such that Wx (u 0 , u 1 ) does not vanish identically, then there is at least one sign flip of Wx (u 0 , u 2 ) in (c, d). Similarly, if c < d are two zeros of Wx (u 1 , u 2 ) such that Wx (u 1 , u 2 ) does not vanish identically, then there is at least one sign flip of Wx (u 0 , u 2 ) in (c, d). Proof. Let c, d be two consecutive zeros of Wx (u 0 , u 1 ). We first assume that Wc (u 0 , u 2 ) = 0 and consider τε = (2 − ε)τ1 + (ε − 1)τ2 , ε ∈ [1, 2], restricted to (c, d) with boundary condition generated by the Prüfer angle of u 0 at c. Set u ε = ψε,− , then we have u 0 ,u ε (c) = 0 for all ε. Moreover, we will show in (5.8) that u 0 ,u ε (d) is increasing, implying that Wx (u 0 , u ε ) has at least one sign flip in (c, d) for ε > 1. To finish our proof, let u˜ 2 be a second linearly independent solution. Then, since W (u 2 , u˜ 2 ) is constant, we can assume 0 < u˜ 2 ,u 2 (x) < π . This implies u˜ 2 ,u 0 (c) = u˜ 2 ,u 2 (c) < π and u˜ 2 ,u 0 (d) = u˜ 2 ,u 2 (d) + u 2 ,u 0 (d) > π . Consequently Wx (u 0 , u˜ 2 ) also has at least one sign flip in (c, d). The second claim is proven analogously.


619

Theorem 3.4. (Triangle inequality for Wronskians) Suppose u j , j = 0, 1, 2 are given functions with u j , pu j absolutely continuous and (u j (x), p(x)u j (x)) = (0, 0) for all x. Then #(u 0 , u 1 ) + #(u 1 , u 2 ) − 1 ≤ #(u 0 , u 2 ) ≤ #(u 0 , u 1 ) + #(u 1 , u 2 ) + 1,

(3.3)

and similarly for # replaced by #. Proof. Take a < c < d < b. By definition #(c,d) (u 0 , u 2 ) = 2,0 (d)/π − 2,0 (c)/π − 1, and using x + y ≤ x + y ≤ x + y + 1 respectively x + y − 1 ≤ x + y ≤ x + y and 2,0 = 2,1 + 1,0 , we obtain #(c,d) (u 0 , u 2 ) ≤ #(c,d) (u 0 , u 1 ) + #(c,d) (u 1 , u 2 ) + 1. Thus the result follows by taking the limits c ↓ a and d ↑ b. We recall that in classical oscillation theory τ is called oscillatory if a solution of τ u = 0 has infinitely many zeros. Definition 3.5. We call τ1 relatively nonoscillatory with respect to τ0 , if the quantities #(u 0 , u 1 ) and #(u 0 , u 1 ) are finite for all solutions τ j u j = 0, j = 0, 1. We call τ1 relatively oscillatory with respect to τ0 , if one of the quantities #(u 0 , u 1 ) or #(u 0 , u 1 ) is infinite for some solutions τ j u j = 0, j = 0, 1. Note that this definition is in fact independent of the solutions chosen as a straightforward application of our triangle inequality (cf. Theorem 3.4) shows. Corollary 3.6. Let τ j u j = τ j v j = 0, j = 0, 1. Then |#(u 0 , u 1 ) − #(v0 , v1 )| ≤ 4, |#(u 0 , u 1 ) − #(v0 , v1 )| ≤ 4.

(3.4)

Proof. By our comparison theorem we have |#(u j , v j )| ≤ 1, j = 0, 1. Now use the triangle inequality, twice. The bounds can be improved using our comparison theorem for Wronskians to be ≤ 2 in the case of perturbations of definite sign. If τ0 is nonoscillatory our definition reduces to the classical one. Lemma 3.7. Suppose τ0 is a nonoscillatory operator, then τ1 is relatively nonoscillatory (resp. oscillatory) with respect to τ0 , if and only if τ1 is nonoscillatory (resp. oscillatory). Proof. This follows by taking limits in |#(c,d) (u, v) − #(c,d) (u) + #(c,d) (v)| ≤ 2, where #(c,d) (u) = θu (d)/π − θu (c)/π − 1 is the number of zeros of u inside (c, d). To demonstrate the usefulness of Definition 3.5, we now establish its connection with the spectra of self-adjoint operators associated with τ j , j = 0, 1. Theorem 3.8. Let H j be self-adjoint operators associated with τ j , j = 0, 1. Then

620


(i) τ0 − λ0 is relatively nonoscillatory with respect to τ0 − λ1 if and only if dim Ran P(λ0 ,λ1 ) (H0 ) < ∞. (ii) Suppose dim Ran P(λ0 ,λ1 ) (H0 ) < ∞ and τ1 − λ is relatively nonoscillatory with respect to τ0 − λ for one λ ∈ [λ0 , λ1 ]. Then it is relatively nonoscillatory for all λ ∈ [λ0 , λ1 ] if and only if dim Ran P(λ0 ,λ1 ) (H1 ) < ∞. Proof. (i) This is item (i) of [5, Thm.7.5]. (ii) Let λ, λ˜ ∈ [λ0 , λ1 ], τ j u j (λ) = λu j (λ), j = 0, 1 and suppose we are relatively nonoscillatory at λ. Then applying our triangle inequality twice implies #(u 0 (λ˜ ), u 1 (λ˜ )) ≤ #(u 0 (λ˜ ), u 1 (λ)) + #(u 1 (λ), u 1 (λ˜ )) + 1 ≤ #(u 0 (λ˜ ), u 0 (λ)) + #(u 0 (λ), u 1 (λ)) + #(u 1 (λ), u 1 (λ˜ )) + 2, and similar estimates with the roles of λ and λ˜ interchanged and # replaced by #. Hence if dim Ran P(λ0 ,λ1 ) (H1 ) < ∞ we are relatively nonoscillatory by (i). The converse direction is proven analogously. We remark that item (i), which corresponds to the case of equal operators H0 = H1 but different spectral parameters λ0 = λ1 , is what was called renormalized oscillation theory by Gesztesy, Simon, and Teschl in [5]. It also follows from earlier results by Hartman [8], Weidmann [41, Satz 4.1] (see also [42]), which was pointed out in the appendix of [24] and was called a relative oscillation theorem there. An argument in the spirit of item (ii) was also one of the key ingredients in Rofe-Beketov’s original work (cf. [22]). For a practical application of this theorem one needs of course criteria when τ1 − λ is relatively nonoscillatory with respect to τ0 − λ for λ inside an essential spectral gap. Lemma 3.9. Let lim x→a r (x)−1 (q0 (x) − q1 (x)) = 0 if a is singular, and similarly, lim x→b r (x)−1 (q0 (x) − q1 (x)) = 0 if b is singular. Then σess (H0 ) = σess (H1 ) and τ1 − λ is relatively nonoscillatory with respect to τ0 − λ for λ ∈ R\σess (H0 ). Proof. Since τ1 can be written as τ1 = τ0 + q˜0 + q˜1 , where q˜0 has compact support near singular endpoints and |q˜1 | < ε, for arbitrarily small ε > 0, we infer that R H1 (z)−R H0 (z) is the norm limit of compact operators. Thus R H1 (z) − R H0 (z) is compact and hence σess (H0 ) = σess (H1 ). Let δ > 0 be the distance of λ to the essential spectrum and choose a < c < d < b, such that |r −1 (q1 (x) − q0 (x))| ≤ δ/2,

x ∈ (c, d).

Clearly #(c,d) (u 0 , u 1 ) < ∞, since both operators are regular on (c, d). Moreover, observe that q0 − λ+r ≤ q1 − λr ≤ q0 − λ−r,

λ± = λ ± δ/2,

on I = (a, c) or I = (d, b). Then Theorem 3.8 (i) implies # I (u 0 (λ− ), u 0 (λ+ )) < ∞ and invoking Theorem 3.3 shows # I (u 0 (λ± ), u 1 (λ)) < ∞. From Theorem 3.4 and 3.8 (i) we infer # I (u 0 (λ), u 1 (λ)) < #(u 0 (λ), u 0 (λ+ )) + # I (u 0 (λ+ ), u 1 (λ)) + 1 < ∞, and similarly for # I (u 0 (λ), u 1 (λ)). This shows that τ1 − λ is relatively nonoscillatory with respect to τ0 .


621

Our next task is to reveal the precise relation between the number of weighted sign flips and the spectra of H1 and H0 . The special case H0 = H1 is covered by [5]: Theorem 3.10 ([5]). Let H0 be a self-adjoint operator associated with τ0 and suppose [λ0 , λ1 ] ∩ σess (H0 ) = ∅. Then dim Ran P(λ0 ,λ1 ) (H0 ) = #(ψ0,∓ (λ0 ), ψ0,± (λ1 )).

(3.5)

We will provide an alternate proof in Sect. 6. Combining this result with our triangle inequality already gives some rough estimates in the spirit of Weidmann [41] who treats the case H0 = H1 . Lemma 3.11. For j = 0, 1 let H j be a self-adjoint operator associated with τ j and separated boundary conditions. Suppose that (λ0 , λ1 ) ⊆ R\(σess (H0 ) ∪ σess (H1 )), then dim Ran P(λ0 ,λ1 ) (H1 ) − dim Ran P(λ0 ,λ1 ) (H0 ) ≤ #(ψ1,∓ (λ1 ), ψ0,± (λ1 ))−#(ψ1,∓ (λ0 ), ψ0,± (λ0 ))+2, (3.6) respectively, dim Ran P(λ0 ,λ1 ) (H1 ) − dim Ran P(λ0 ,λ1 ) (H0 ) ≥ #(ψ1,∓ (λ1 ), ψ0,± (λ1 ))−#(ψ1,∓ (λ0 ), ψ0,± (λ0 ))−2.

(3.7)

Proof. By the triangle inequality (cf. Theorem 3.4) we have #(c,d) (ψ1,− (λ1 ), ψ1,+ (λ0 )) − #(c,d) (ψ0,− (λ1 ), ψ0,+ (λ0 )) ≤ #(c,d) (ψ1,− (λ1 ), ψ0,+ (λ1 )) + #(c,d) (ψ1,− (λ1 ), ψ0,+ (λ1 )) + 2. The result now follows by taking limits using that lim #(c,d) (ψ1,− (λ1 ), ψ1,+ (λ0 )) = dim Ran P(λ0 ,λ1 ) (H1 )

c↓a,d↑b

and lim #(c,d) (ψ0,− (λ0 ), ψ0,+ (λ1 )) = − dim Ran P(λ0 ,λ1 ) (H0 )

c↓a,d↑b

by the previous theorem. The second claim follows similarly. To turn the inequalities into equalities in Lemma 3.11 will be one of our remaining goals. Observe that for semibounded operators we can choose λ0 below the spectra of H0 and H1 , causing #(ψ1,∓ (λ0 ), ψ0,± (λ0 )) to vanish: Lemma 3.12. Let r −1 (q0 − q1 ) ≤ δ for some δ ∈ R. Furthermore, suppose that the operator H0 associated with τ0 is bounded from below, H0 ≥ E 0 , and the form domains of H0 and H1 are equal. Then #(ψ1,∓ (λ), ψ0,± (λ)) = 0, λ < E 0 − δ.

(3.8)

622


Proof. Suppose there is c ∈ (a, b) such that Wc (ψ0,+ (λ), ψ1,− (λ)) = 0. Then, there is a γ such that ψ1,− (λ, x), x ≤ c, ϕ(x) = γ ψ0,+ (λ, x), x ≥ c, is continuous and hence in the form domain of H0 (see the remark after Theorem A.3 in [5]). But then ϕ, H0 ϕ ≤ (λ + δ)ϕ2 < E 0 ϕ2 , contradicting H0 ≥ E 0 .

Our first approach will use approximation by regular problems. However, the standard approximation technique only implies strong convergence, which is not sufficient for our purpose. Hence our argument is based on a refinement of a method by Stolz and Weidmann [31] which will provide convergence of spectral projections in the trace norm for suitably chosen regular operators (see [43] for a nice overview). Theorem 3.13. Let H0 , H1 be self-adjoint operators associated with τ0 , τ1 , respectively, and separated boundary conditions. Suppose (i) q1 ≤ q0 , near singular endpoints, (ii) lim x→a r (x)−1 (q0 (x) − q1 (x)) = 0 if a is singular and lim x→b r (x)−1 (q0 (x) − q1 (x)) = 0 if b is singular, (iii) H0 and H1 are associated with the same boundary conditions near a and b, that is, ψ0,− (λ) satisfies the boundary condition of H1 at a (if any) and ψ1,+ (λ) satisfies the boundary condition of H0 at b (if any). Suppose λ0 < inf σess (H0 ). Then dim Ran P(−∞,λ0 ) (H1 ) − dim Ran P(−∞,λ0 ] (H0 ) = #(ψ1,∓ (λ0 ), ψ0,± (λ0 )). (3.9) Suppose σess (H0 ) ∩ [λ0 , λ1 ] = ∅. Then τ1 − λ0 is nonoscillatory with respect to τ0 − λ0 and dim Ran P[λ0 ,λ1 ) (H1 ) − dim Ran P(λ0 ,λ1 ] (H0 ) = #(ψ1,∓ (λ1 ), ψ0,± (λ1 )) − #(ψ1,∓ (λ0 ), ψ0,± (λ0 )).

(3.10)

The proof will be given in Sect. 6. Remark 3.14. Note that condition (ii) implies σess (H0 ) = σess (H1 ) as pointed out in Lemma 3.9. In addition, (ii) implies that any function which is in D(τ0 ) near a (or b) is also in D(τ1 ) near a (or b), and vice versa. Hence condition (iii) is well-posed. Our second approach will connect our theory with Krein’s spectral shift function. Given the regular case in Theorem 2.3, we can extend this result to operators whose resolvent difference is trace class by replacing the left-hand side in (2.10) by the spectral shift function. In order to fix the unknown constant in the spectral shift function, we will require that H0 and H1 are connected via a path within the set of operators whose resolvent difference with H0 are trace class. Hence we will require


623

Hypothesis H. 3.15. Suppose H0 and H1 are self-adjoint operators associated with τ0 and τ1 and separated boundary conditions. Abbreviate q˜ = r −1 |q0 − q1 |, and assume that: (i) q˜ is relatively bounded with respect to H0 with H0 -bound less than one or (i’) H0 is bounded from below and q˜ is relatively form bounded with respect to H0 with relative form bound less than one and (ii) q˜ R H0 (z) is Hilbert–Schmidt for one (and hence for all) z ∈ ρ(H0 ). It will be shown in Sect. 8 that these conditions ensure that we can interpolate between H0 and H1 using operators Hε , ε ∈ [0, 1], such that the resolvent difference of H0 and Hε is continuous in ε with respect to the trace norm. Hence we can fix ξ(λ, H1 , H0 ) by requiring ε → ξ(λ, Hε , H0 ) to be continuous in L 1 (R, (λ2 + 1)−1 dλ), where we of course set ξ(λ, H0 , H0 ) = 0 (see Lemma 8.5). While ξ is only defined a.e., it is constant on the intersection of the resolvent sets R ∩ ρ(H0 ) ∩ ρ(H1 ), and we will require it to be continuous there. In particular, note that by Weyl’s theorem the essential spectra of H0 and H1 are equal, σess (H0 ) = σess (H1 ). Then we have the following result: Theorem 3.16. Let H0 , H1 satisfy Hypothesis 3.15. Then for every λ ∈ R ∩ ρ(H0 ) ∩ ρ(H1 ) we have ξ(λ, H1 , H0 ) = #(ψ0,± (λ), ψ1,∓ (λ)).

(3.11)

The proof will be given in Sect. 7. In particular, this result implies that under these assumptions τ1 − λ is relatively nonoscillatory with respect to τ0 − λ for every λ in an essential spectral gap. The main idea is to interpolate between H0 and H1 . Under proper assumptions it is then possible to control ξ(λ, Hε , H0 ). However, it seems extremely hard to control the zeros of the Wronskian. To do this we will have to assume that q1 − q0 has compact support. We will then remove this restriction by extending the support first to one and then to the other side. The details will be given in Sect. 7. Finally, we remark that since the results from [5] extend to one-dimensional Dirac operators [34] (see also [25]) and Jacobi operators [33] (compare also [35, Chap. 4]), similar results are expected to hold for these operators and will be given in [37] respectively [38]. Furthermore, it will of course be interesting to develop relative oscillation criteria for Sturm–Liouville operators! What are the analogs of some classical oscillation criteria? Is there an analog of [4]? Some results concerning these questions will be given in [15]. 4. Applications In this section we want to look at the classical problem of the number of eigenvalues in essential spectral gaps of perturbed periodic operators [1,18] (see also [3]), [46]. The precise critical case was first determined by Rofe-Beketov in a series of papers [19,22] with later additions by Khryashchev [12] and Schmidt [24]. The purpose of this section is to show how the results in [24] can be extended using our methods. For convenience of the reader, we recall some basic facts of the theory of periodic differential operators first (see for example [2] or [42]). Let p, q0 be α-periodic, that is, p(x + α) = p(x), q0 (x + α) = q0 (x), and consider the corresponding Sturm–Liouville expressions τ0 = −

d d p + q0 . dx dx

(4.1)

624


Denote by c(λ, x), s(λ, x) a fundamental system of solutions corresponding to the initial conditions c(λ, 0) = p(0)s (λ, 0) = 1, s(λ, 0) = p(0)c (λ, 0) = 0. In particular, their Wronskian reads W (c, s) = 1. We then call c(λ, α) s(λ, α) M(λ) = (4.2) p(0)c (λ, α) p(0)s (λ, α) the monodromy matrix. The discriminant D(λ) is given by D(λ) = tr(M(λ)). Since we are only interested in the question whether the number of eigenvalues are finite or not, it suffices to look at the half-line case (1, ∞) with a, for example, Dirichlet boundary condition at 1. Denote the corresponding self-adjoint operator by H0 . The essential spectrum of H0 is given by σess (H0 ) = σac (H0 ) = {λ | |D(λ)| ≤ 2} =

∞

[E 2n , E 2n+1 ].

(4.3)

n=0

The critical coupling constant at the endpoint E n of an essential spectral gap introduced in Sect. 1 is given by ([24]) κn =

α2 . 4|D| (E n )

(4.4)

It is related to the effective mass m(E n ) used in solid state physics via κn = (8m(E n ))−1 (see [22]). We note that κ2n+1 > 0 for a lower endpoint of a spectral gap, and κ2n < 0 for an upper endpoint. Before coming to our applications, we ensure that our hypotheses from the previous section are satisfied. We begin by computing the form domain of H0 . Lemma 4.1. Abbreviate Q = { f ∈ L 2 (1, ∞) | f ∈ AC[1, ∞),

√

p f ∈ L 2 (1, ∞)}.

(4.5)

The form domain of H0 is given by Q(H0 ) = { f ∈ Q | f (1) = 0}.

(4.6)

Moreover, for every ε > 0 there is some C > 0 such that sup

x0 ≤x≤x0 +α

| f (x)| ≤ ε 2

x0 +α

p(x)| f (x)| d x + C 2

x0

x0 +α

| f (x)|2 d x,

f ∈ Q. (4.7)

x0

In particular, q1 − q0 is infinitesimally form bounded with respect to H0 if

x0 +α

|q1 (x) − q0 (x)|d x < C0 ,

x0

where C0 is independent of x0 .

x0 ∈ (1, ∞),

(4.8)


625

Proof. Equation (4.7) is a standard Sobolev estimate. For the case p(x) = 1 required here compare for example [5, Lem A.2]. Next, set A=

√

p

d , dx

D(A) = { f ∈ Q | f (1) = 0}

(4.9)

and note that A is then a closed operator with adjoint given by d √ p, dx √ √ D(A∗ ) = { f ∈ L 2 (1, ∞) | p f ∈ AC[1, ∞), ( p f ) ∈ L 2 (1, ∞)}. (4.10) A∗ = −

Hence, d d p , dx dx D(A∗ A) = { f ∈ L 2 (1, ∞) | f, p f ∈ AC[1, ∞), ( p f ) ∈ L 2 (1, ∞), f (1) = 0} (4.11) A∗ A = −

is self-adjoint with Q(A∗ A) = D(A) and by (4.7) q0 is infinitesimally form bounded with respect to A∗ A. Since the same is true for q1 by assumption, the lemma is proven. √ Lemma 4.2. Let H0 be an arbitrary Sturm–Liouville operator on (a, b). Then q R H0 (z) is Hilbert–Schmidt if and only if √ q R H0 (z)2J2 =

1 Im(z)

b

|q(x)|Im(G 0 (z, x, x))r (x)d x

(4.12)

a

is finite. Here G 0 (z, x, y) = (H0 − z)−1 (x, y) denotes the Green’s function of H0 . Proof. From the first resolvent identity we have

G 0 (z, x, y) − G 0 (z , x, y) = (z − z )

b

G 0 (z, x, t)G 0 (z , t, y)r (t)dt.

a

Setting x = y and z = z ∗ we obtain Im(G 0 (z, x, x)) = Im(z)

b

|G 0 (z, x, t)|2 r (t)dt.

a

Using this last formula to compute the Hilbert–Schmidt norm proves the lemma. We recall that in the case of periodic operators G(z, x, x) is a bounded function of x. In fact, we have |G(z, x, y)| ≤ const (z) exp(−γ (z)|x − y|), where γ (z) > 0 denotes the Floquet exponent. Now we are ready to apply our theory:

(4.13)

626


Theorem 4.3. Let p, q0 α-periodic and q1 a perturbed potential which is regular at 1, be such that either q0 − q1 ∈ L 1 (1, ∞) or lim x→∞ (q0 (x) − q1 (x)) = 0. Define the differential expressions on (1, ∞) by τ0 = −

d d d d p(x) + q0 (x), τ1 = − p(x) + q1 (x), dx dx dx dx

(4.14)

and let H0 , H1 be the corresponding self-adjoint operators L 2 (1, ∞). Let E n be an endpoint of a gap in the essential spectrum of H0 with corresponding κn given by (4.4). Then E n is an accumulation point of eigenvalues of H1 if lim inf κn x 2 (q1 (x) − q0 (x)) > 1,

(4.15)

x→∞

and E n is no accumulation point of eigenvalues of H1 if lim sup κn x 2 (q1 (x) − q0 (x)) < 1,

(4.16)

x→∞

and κn (q1 − q0 ) ≥ 0 near infinity. Proof. Lemma 4.2 together with (4.13) shows that Hypothesis 3.15 is satisfied if q0 − q1 ∈ L 1 (1, ∞). Thus we can either apply Theorem 3.13 or Theorem 3.16 to conclude that τ1 − λ is nonoscillatory with respect to τ0 − λ for any λ ∈ R\σess (H0 ). Hence, Theorem 3.8 (ii) is applicable and it suffices to show that τ1 is relatively oscillatory (resp. nonoscillatory) with respect to τ0 . Without restriction, we assume κn > 0. For the first statement, note that we can find c, ε > 0 such that q0 (x) < q0 (x) +

1 < q1 (x), (κn − ε)x 2

x > c.

Now since perturbations with compact support only add finitely many eigenvalues, it is no restriction to assume c = 1. Next, [24, Thm. 1] shows that τ0 + (κn − ε)−1 x −2 − E n is relatively oscillatory with respect to τ0 − E n . Hence τ1 − E n being relatively oscillatory with respect to τ0 − E n now follows using our comparison theorem for Wronskians (cf. Theorem 3.3). For the second statement, we first note that our conditions imply q0 (x) ≤ q1 (x) < q0 (x) +

1 (κn + ε)x 2

near infinity, and then one proceeds as before. We note that even the second order term was also computed in [24]. So we obtain: Theorem 4.4. Assume lim κn x 2 (q1 (x) − q0 (x)) = 1

x→∞

(4.17)

in addition to the assumptions in Theorem 4.3. Then E n is an accumulation point of eigenvalues if lim inf log2 (x)(κn x 2 (q1 (x) − q0 (x)) − 1) > 1, x→∞

(4.18)

and E n is not an accumulation point of eigenvalues if lim sup log2 (x)(κn x 2 (q1 (x) − q0 (x)) − 1) < 1. x→∞

(4.19)


627

Proof. Similarly to that of Theorem 4.3, except, one now uses [24, Thm. 2]. The main argument in [24] is a perturbation argument for the difference of Prüfer angles for the solution of the unperturbed and the perturbed equation. This corresponds exactly to calculating the asymptotics of the Prüfer angle of the Wronskian.

5. More on Prüfer Angles and the Case of Regular Operators Now let us suppose that τ0,1 are both regular at a and b with boundary conditions cos(α) f (a) − sin(α) p(a) f (a) = 0, cos(β) f (b) − sin(β) p(b) f (b) = 0. (5.1) (λ, a) = cos(α) Hence we can choose ψ± (λ, x) such that ψ− (λ, a) = sin(α), p(a)ψ− respectively ψ+ (λ, b) = sin(β), p(b)ψ+ (λ, b) = cos(β). In particular, we may choose

θ− (λ, a) = α ∈ [0, π ), −θ+ (λ, b) = π − β ∈ [0, π ).

(5.2)

Next we introduce ε τε = τ0 + (q1 − q0 ) r

(5.3)

and investigate the dependence with respect to ε ∈ [0, 1]. If u ε solves τε u ε = 0, then the corresponding Prüfer angles satisfy θ˙ε (x) = −

Wx (u ε , u˙ ε ) , ρε2 (x)

(5.4)

where the dot denotes a derivative with respect to ε. Lemma 5.1. We have Wx (ψε,± , ψ˙ ε,± ) =

b 2 x (qx0 (t) − q1 (t))ψε,+ (t) dt , − a (q0 (t) − q1 (t))ψε,− (t)2 dt

(5.5)

where the dot denotes a derivative with respect to ε and ψε,± (x) = ψε,± (0, x). Proof. Integrating (2.8) we obtain b (q (t) − q1 (t))ψε,+ (t)ψε˜ ,+ (t)dt, Wx (ψε,± , ψε˜ ,± ) = (˜ε − ε) x x0 − a (q0 (t) − q1 (t))ψε,− (t)ψε˜ ,− (, t)dt.

(5.6)

Now use this to evaluate the limit ψ±,ε − ψε˜ ,± . lim Wx ψε,± , ε − ε˜ ε˜ →ε

(5.7)

628


Denoting the Prüfer angles of ψε,± (x) = ψε,± (0, x) by θε,+ (x), this result implies for q0 − q1 ≥ 0, b (q (t) − q1 (t))ψε,+ (t)2 dt ˙θε,+ (x) = − x 0 ≤ 0, ρε,+ (x)2 x (q0 (t) − q1 (t))ψε,− (t)2 dt θ˙ε,− (x) = a ≥ 0, (5.8) ρε,− (x)2 with strict inequalities if q1 ≡ q0 . Now we are ready to investigate the associated operators H0 and H1 . In addition, we will choose the same boundary conditions for Hε as for H0 and H1 . Lemma 5.2. Suppose q0 − q1 ≥ 0 (resp. q0 − q1 ≤ 0). Then the eigenvalues of Hε are analytic functions with respect to ε and they are decreasing (resp. increasing). Proof. First of all the Prüfer angles θε,± (x) are analytic with respect to ε since τε is by a well-known result from ordinary differential equations (see e.g., [39, Thm. 13.III]). Moreover, λ ∈ σ (Hε ) is equivalent to θε,+ (a) ≡ α mod π (respectively θε,− (b) ≡ β mod π ), where α (respectively β) generates the boundary condition (cf. (5.1)). In particular, this implies that dim Ran P(−∞,λ) (Hε ) is continuous from below (resp. above) in ε if q0 − q1 ≥ 0 (resp q0 − q1 ≤ 0). Now we are ready for the Proof of Theorem 2.3. It suffices to prove the result for #(ψ0,+ , ψε,− ). Again we can assume λ0 = λ1 = 0 without restriction. We split q0 − q1 according to q0 − q1 = q+ − q− ,

q+ , q− ≥ 0,

and introduce the operator τ− = τ0 − q− /r . Then τ− is a negative perturbation of τ0 and τ1 is a positive perturbation of τ− . Furthermore define τε by τ0 + 2ε(τ− − τ0 ), ε ∈ [0, 1/2], τε = τ− + 2(ε − 1/2)(τ1 − τ− ), ε ∈ [1/2, 1]. Let us look at N (ε) = #(ψ0,+ , ψε,− ) = ε (b)/π − ε (a)/π − 1, ε (x) = ψ0,+ ,ψε,− (x) and consider ε ∈ [0, 1/2]. At the left boundary ε (a) remains constant whereas at the right boundary ε (b) is increasing by Lemma 5.1. Moreover, it hits a multiple of π whenever 0 ∈ σ (Hε ). So N (ε) is a piecewise constant function which is continuous from below and jumps by one whenever 0 ∈ σ (Hε ). By Lemma 5.2 the same is true for P(ε) = dim Ran P(−∞,0) (Hε ) − dim Ran P(−∞,0] (H0 ), and since we have N (0) = P(0), we conclude N (ε) = P(ε) for all ε ∈ [0, 1/2]. To see the remaining case ε = [1/2, 1], simply replace increasing by decreasing and continuous from below by continuous from above.


629

6. Approximation by Regular Operators Now we want to extend our results to singular operators. We will do so by approximating a singular operator by a sequence of regular ones following [42, Chap. 14]. Abbreviate in the following L 2 ((c, d), r d x) as L 2 (c, d). Fix functions u, v ∈ D(τ ) and pick cn ↓ a, dn ↑ b. Define H˜ n ,

where

H˜ n : D( H˜ n ) → L 2 (cn , dn ) , f → τ f.

(6.1)

D( H˜ n ) = f ∈ L 2 (cn , dn )| f, p f ∈ AC(cn , dn ), τ f ∈ L 2 (cn , dn ), Wcn (u, f ) = Wdn (v, f ) = 0 .

(6.2)

Take Hn = α1l ⊕ H˜ n ⊕ α1l on L 2 (a, b) = L 2 (a, cn ) ⊕ L 2 (cn , dn ) ⊕ L 2 (dn , b), where α is a fixed real constant. Then we have the following result: Lemma 6.1. Suppose that either H is limit point at a or that u = ψ− (λ0 ) for some λ0 ∈ R and similarly, that either H is limit point at b or v = ψ+ (λ1 ) for some λ1 ∈ R. Then Hn converges to H in the strong resolvent sense as n → ∞. Furthermore, if H is limit circle at a (resp. b), we can replace u (resp. v) with any function in D(τ ), which generates the boundary condition. However, strong resolvent convergence is not sufficient for our purpose here. In addition we will need the following result from [31] (see also [42]). We give a slightly refined analysis which allows eigenvalues at the boundary of the spectral intervals and the possibility of infinite-dimensional projections. We remark that for a self-adjoint projector P we have dim Ran(P) = tr(P) = PJ 1 ,

(6.3)

where .J 1 denotes the trace class norm. If P is not finite-rank, all three numbers equal ∞. Then we have the following result ([36, Lem. 2], see also [31]): Lemma 6.2. Let An → A in strong resolvent sense and suppose tr(P(λ0 ,λ1 ) (An )) ≤ tr(P(λ0 ,λ1 ) (A)). Then, lim tr(P(λ0 ,λ1 ) (An )) = tr(P(λ0 ,λ1 ) (A)),

n→∞

(6.4)

and if tr(P(λ0 ,λ1 ) (A)) < ∞, we have lim P(λ0 ,λ1 ) (An ) − P(λ0 ,λ1 ) (A)J 1 = 0.

n→∞

(6.5)

Proof. This follows from (see e.g. [5, Lem. 5.2]) tr(P(λ0 ,λ1 ) (A)) ≤ lim inf tr(P(λ0 ,λ1 ) (An )), n→∞

(6.6)

together with Grümm’s theorem ([28, Thm. 2.19]). The key result of Stolz and Weidmann is that this lemma is applicable if certain Weyl solutions ψ± (λ) are used to generate the boundary conditions of H˜ n . As already pointed out, the version below is slightly refined since it allows λ0 , λ1 to be eigenvalues of H .

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Lemma 6.3 ([31]). Suppose [λ0 , λ1 ] ∩ σess (H ) = ∅ and let Hn be defined as in (6.1) with u = ψ− (λ− ), v = ψ+ (λ+ ) and λ± ∈ [λ0 , λ1 ]. Then, tr(P(λ0 ,λ1 ) ( H˜ n )) ≤ tr(P(λ0 ,λ1 ) (H )).

(6.7)

Proof. Abbreviate P = P(λ0 ,λ1 ) (H ), Pn = P(λ0 ,λ1 ) ( H˜ n ). For ψ˜ 1 , . . . , ψ˜ k ∈ Ran Pn being eigenfunctions of H˜ n , construct ⎧ ⎨ γ j,u u(x), x < cn , ψ j (x) = ψ˜ (x), cn ≤ x ≤ dn , ⎩ j γ j,v v(x), x > dn , where γ j,u , γ j,v are chosen such that ψ j and pψ j are continuous. A computation shows that (H −

λ1 + λ 0 λ1 − λ0 )ψ < ψ 2 2

for any ψ in the linear span of the ψ j ’s, which yields the first result. This version is sufficient to give an alternative proof for the main theorem in [5]. Proof of Theorem 3.10. Approximate H0 by regular operators Hn defined as in (6.1) with n (λ) the solutions of the approximating u = ψ0,− (λ0 ), v = ψ0,+ (λ1 ). Denote by ψ0,± n n (λ , x) = problems. Then, by construction ψ0,− (λ0 , x) = ψ0,− (λ0 , x) respectively ψ0,+ 1 ψ0,+ (λ1 , x) for x ∈ (cn , dn ) and Theorem 2.3 in the special case H1 = H0 implies tr(P(λ0 ,λ1 ) ( H˜ n )) = #(cn ,dn ) (ψ0,− (λ0 ), ψ0,+ (λ1 )). Letting n → ∞ the left-hand side converges to tr(P(λ0 ,λ1 ) (H0 )) by the first part of Lemma 6.3. Hence the right-hand side converges as well and, according to Definition 3.1, is given by #(ψ0,∓ (λ0 ), ψ0,± (λ1 )). However, the proof of our Theorem 3.13 requires some further extensions. In fact, in [31] Stolz and Weidmann point out that the Weyl functions of a different operator H˜ will also do, as long as H˜ is not too far away from H . Again the version below is slightly improved to allow for some border line cases. ˜ where q˜ is bounded, Lemma 6.4 ([31]). Suppose [λ0 , λ1 ] ∩ σess (H ) = ∅. Let τ˜ = τ + q, and pick the same boundary conditions for H˜ as for H (if any). Abbreviate Q a = [lim inf q(x), ˜ lim sup q(x)], ˜ Q b = [lim inf q(x), ˜ lim sup q(x)], ˜ x→a

x→a

x→b

and choose λ− such that one of following conditions holds: (i) λ− − Q a ⊆ (λ0 , λ1 ), or (ii) λ− − Q a ⊆ [λ0 , λ1 ) and q˜ ≤ 0 near a, or (iii) λ− − Q a ⊆ (λ0 , λ1 ] and q˜ ≥ 0 near a.

x→b

(6.8)


631

Similarly, choose λ+ to satisfy one of these conditions with a replaced by b. Then, Hn defined as in (6.1) with u = ψ˜ − (λ− ), v = ψ˜ + (λ+ ), satisfies lim sup tr(P(λ0 ,λ1 ) ( H˜ n )) ≤ tr(P(λ0 ,λ1 ) (H )).

(6.9)

n→∞

Furthermore, if we just require λ− − Q a ⊆ [λ0 , λ1 ],

λ+ − Q b ⊆ [λ0 , λ1 ],

(6.10)

we at least have lim sup tr(P[λ0 ,λ1 ] ( H˜ n )) ≤ tr(P[λ0 ,λ1 ] (H )).

(6.11)

n→∞

0 0 − q(x)| ˜ ≤ λ1 −λ for x Proof. Since any of our three conditions implies |λ− − λ1 +λ 2 2 λ1 +λ0 λ1 −λ0 sufficiently close to a and similarly |λ+ − 2 − q(x)| ˜ ≤ 2 for x sufficiently close to b, we can proceed as in the previous lemma to prove the first claim. −1 (λ +λ )| ≤ For the second claim choose n sufficiently large such that |λ± −q(x)−2 ˜ 1 0 (2−1 (λ1 − λ0 ) + ε) for x < cn , respectively, x > dn . Then, with the same argument as in the previous lemma, we have H − λ1 + λ0 ψ ≤ λ1 − λ0 + ε ψ , 2 2

and hence the second claim follows. Since our results involve projections to half-open intervals, we need one further step. Lemma 6.5. Suppose [λ0 , λ1 ] ∩ σess (H ) = ∅. Let τ˜ = τ + q, ˜ where lim x→a q(x) ˜ = 0 if a is singular and lim x→b q(x) ˜ = 0 if b is singular. Furthermore, pick the same boundary conditions for H˜ as for H (if any). Define Hn as in (6.1) with u = ψ˜ − (λ), v = ψ˜ + (λ), λ ∈ {λ1 , λ2 }. If λ = λ1 and q˜ ≤ 0 (near a and b), then lim tr(P(λ0 ,λ1 ] ( H˜ n )) = tr(P(λ0 ,λ1 ] (H )),

(6.12)

n→∞

and if λ = λ0 and q˜ ≥ 0, then lim tr(P[λ0 ,λ1 ) ( H˜ n )) = tr(P[λ0 ,λ1 ) (H )).

(6.13)

n→∞

Proof. Without restriction, we just prove the first claim. For a sufficiently small ε > 0 we still have [λ0 , λ1 + ε] ∩ σess (H ) = ∅ and thus by the previous lemma lim tr(P(λ0 ,λ1 +ε) ( H˜ n )) = tr(P(λ0 ,λ1 +ε) (H ))

n→∞

and lim tr(P(λ1 ,λ1 +ε) ( H˜ n )) = tr(P(λ1 ,λ1 +ε) (H )).

n→∞

Hence the result follows from P(λ0 ,λ1 ] = P(λ0 ,λ1 +ε) − P(λ1 ,λ1 +ε) . Finally, we note:

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Lemma 6.6. Suppose τ j u j = λ j u j , j = 0, 1, with q1 ≤ q0 near singular endpoints and λ0 < λ1 . If τ j u j,n = λ j,n u j,n , where λ j,n → λ j and u j,n → u j , uniformly on compact sets [c, d] ⊆ (a, b), then lim inf #(u 0,n , u 1,n ) ≥ #(u 0 , u 1 ). n→∞

(6.14)

Proof. Let N ∈ N0 be any finite number with N ≤ #(u 0 , u 1 ). Choose a compact set [c, d] containing N sign flips of W (u 0 , u 1 ). Then, for n sufficiently large, W (u 0,n , u 1,n ) has N sign flips in [c, d]. Hence #(u 0,n , u 1,n ) ≥ #(c,d) (u 0,n , u 1,n ) = N and the claim follows. Now, we are ready for the Proof of Theorem 3.13. It suffices to show the #(ψ1,− (λ j ), ψ0,+ (λ j )) case. Define H˜ j,n , j = 0, 1, as in (6.1) with u = ψ1,− (λ0 ) and v = ψ0,+ (λ0 ). Denote by ψ nj,± (λ), j = 0, 1, the solutions of the approximating problems. Then, by Theorem 2.3, n n tr(P(−∞,λ0 ) ( H˜ 1,n )) − tr(P(−∞,λ0 ] ( H˜ 0,n )) = #(cn ,dn ) (ψ1,− (λ0 ), ψ0,+ (λ0 )),

and we need to investigate the limits as n → ∞. n (λ , x) = ψ n First of all ψ1,− 0 1,− (λ0 , x), ψ0,+ (λ0 , x) = ψ0,+ (λ0 , x) for x ∈ (cn , dn ) implies n n (λ0 ), ψ0,+ (λ0 )) = lim #(cn ,dn ) (ψ1,− (λ0 ), ψ0,+ (λ0 )) lim #(cn ,dn ) (ψ1,−

n→∞

n→∞

= #(ψ1,− (λ0 ), ψ0,+ (λ0 )). This takes care of the number of sign flips and it remains to look at the spectral projections. Let λ0 < σess (H0 ), that is H0 and hence also H1 are bounded from below. Replacing P(−∞,λ0 ) (H j ) by P(λ,λ0 ) (H j ) with some λ below the spectrum of both H0 and H1 we infer from Lemma 6.5, lim tr(P(−∞,λ0 ) ( H˜ 1,n )) = tr(P(−∞,λ0 ) (H1 ))

n→∞

and lim tr(P(−∞,λ0 ] ( H˜ 0,n )) = tr(P(−∞,λ0 ] (H0 )).

n→∞

This settles the first claim (3.9), where λ0 < σess (H0 ). For the second claim (3.10), we first note that τ1 − λ0 is relatively nonoscillatory with n (λ , .) → ψ (λ , .) pointwise, respect to τ0 − λ0 by Lemma 3.9. Next note that ψ0,+ 1 0,+ 1 since n (λ1 , x) = c0 (λ1 , x) + m n0,+ (λ1 )s0 (λ1 , x), ψ0,+

(6.15)

where c0 (λ, x), s0 (λ, x) is a fundamental system of solutions for τ0 − λ, and m n0,+ (λ) are the corresponding Weyl–Titchmarsh m-functions. Next, strong resolvent convergence implies convergence of the Weyl m-function and hence uniform convergence of n (λ , x) → ψ (λ , x) on compact sets. Clearly the same applies to ψ n (λ , x) → ψ0,+ 1 0,+ 1 1,− 1 ψ1,− (λ1 , x). Thus, by Lemma 6.2, Lemma 6.5, and Lemma 6.6, tr(P[λ0 ,λ1 ) (H1 )) − tr(P(λ0 ,λ1 ] (H0 )) ≥ #(ψ1,− (λ1 ), ψ0,+ (λ1 )) − #(ψ1,− (λ0 ), ψ0,+ (λ0 )).

(6.16)


633

Repeating the argument with u = ψ1,− (λ1 ) and v = ψ0,+ (λ1 ) shows that tr(P[λ0 ,λ1 ) (H1 )) − tr(P(λ0 ,λ1 ] (H0 )) ≤ #(ψ1,− (λ1 ), ψ0,+ (λ1 )) − #(ψ1,− (λ0 ), ψ0,+ (λ0 )).

(6.17)

This proves the second claim. 7. Approximation in Trace Norm Now we begin with an alternative approach toward singular differential operators by proving the case where q1 − q0 has compact support. The next lemma would follow from Theorem 3.13, but to demonstrate that this approach is independent of the last, we will provide an alternative proof. Lemma 7.1. Let H j , j = 0, 1, be Sturm–Liouville operators on (a, b) associated with τ j , and suppose that r −1 (q1 − q0 ) has support in a bounded interval (c, d) ⊆ (a, b), where a < c if a is singular and d < b if b is singular. Moreover, suppose H0 and H1 have the same boundary conditions (if any). Suppose λ0 < inf σess (H0 ). Then, dim Ran P(−∞,λ0 ) (H1 ) − dim Ran P(−∞,λ0 ] (H0 ) = #(ψ1,∓ (λ0 ), ψ0,± (λ0 )). (7.1) Suppose σess (H0 ) ∩ [λ0 , λ1 ] = ∅. Then, dim Ran P[λ0 ,λ1 ) (H1 ) − dim Ran P(λ0 ,λ1 ] (H0 ) = #(ψ1,∓ (λ1 ), ψ0,± (λ1 )) − #(ψ1,∓ (λ0 ), ψ0,± (λ0 )).

(7.2)

Proof. By splitting r −1 (q1 − q0 ) into a positive and negative part as in the proof of the regular case (Theorem 2.3), we can reduce it to the case where r −1 (q1 − q0 ) is of one sign, say r −1 (q1 − q0 ) ≥ 0. Define Hε = ε H1 + (1 − ε)H0 and observe that ψε,− (z, x) = ψ0,− (z, x) for x ≤ c, respectively, ψε,+ (z, x) = ψ0,+ (z, x) for x ≥ d. Furthermore, ψε,± (z, x) is analytic with respect to ε and λ ∈ σ p (Hε ) if and only if Wd (ψ0,+ (λ), ψε,− (λ)) = 0. Now the proof can be done as in the regular case. Lemma 7.2. Suppose H0 , H1 satisfy the same assumptions as in the previous lemma and set Hε = ε H1 + (1 − ε)H0 . Then, ε ∈ [0, 1]. (7.3) r −1 |q0 − q1 |R Hε (z)J2 ≤ C(z), In particular, the resolvent difference of H0 and H1 is trace class and ξ(λ, H1 , H0 ) = #(ψ1,∓ (λ), ψ0,± (λ))

(7.4)

for every λ ∈ R ∩ ρ(H0 ) ∩ ρ(H1 ). Here ξ(H1 , H0 ) is assumed to be constructed such that ε → ξ(Hε , H0 ) is a continuous mapping from [0, 1] → L 1 (R, (λ2 + 1)−1 dλ). Proof. Denote by G ε (z, x, y) = (Hε − z)−1 (x, y) =

ψε,− (z, x< ), ψε,+ (z, y> ) , W (ψε,− (z), ψε,+ (z))

634


where x< = min(x, y), y> = max(x, y), the Green’s function of Hε . As pointed out in the proof of the previous lemma, ψε,± (z, x) is analytic with respect to ε and hence a simple estimate shows b b |G ε (z, x, y)|2 |r (y)−1 (q1 (y) − q0 (y))|r (x)d x r (y)dy ≤ C(z)2 a

a

for ε ∈ [0, 1], which establishes the first claim. Furthermore, a straightforward calculation (using (2.8)) shows b G ε (z, x, y) = G ε (z, x, y)+(ε−ε ) G ε (z, x, t)r −1 (t)(q1 (t)−q0 (t))G ε (z, t, y)r (t)dt. a

(Note that this does not follow from the second resolvent identity unless r −1 (q1 − q0 ) is relatively bounded with respect to H0 .) Hence, R Hε (z) − R Hε (z) can be written as the product of two Hilbert–Schmidt operators whose norm can be estimated by the first claim, R Hε (z) − R Hε (z)J1 ≤ |ε − ε|C(z)2 .

(7.5)

Thus ε → ξ(Hε , H0 ) is continuous by Lemma 8.3. The rest follows from (8.4). Now we come to the Proof of Theorem 3.16. We first assume that we have compact support near one endpoint, say a. Furthermore, abbreviate V = r −1 (q0 − q1 ) which satisfies Hypothesis 3.15 (and thus also Hypothesis 8.4). Define by K ε the operator of multiplication by χ(a,bε ] with bε ↑ b as ε ↑ 1. Then K ε satisfies the assumptions of Lemma 8.5. Introduce Hε = H0 − K ε V , and denote by ψε,− (λ, x) the corresponding solutions satisfying the boundary condition at a. By Lemma 8.5 we have ξ(., Hε , H0 ) → ξ(., H1 , H0 ) as ε → 1 in L 1 (R, (λ2 + −1 1) dλ). Moreover, Hε → H1 in (trace) norm resolvent sense and hence λ ∈ ρ(H1 ) implies λ ∈ ρ(Hε ) for ε sufficiently close to 1. Since ξ(λ, Hε , H0 ) ∈ Z is constant near every λ ∈ R ∩ ρ(H0 ) ∩ ρ(Hε ), we must have ξ(λ, Hε , H0 ) = ξ(λ, H1 , H0 ) for ε ≥ ε0 with some ε0 sufficiently close to 1. Now let us turn to the Wronskians. We first prove the #(ψ1,− (λ), ψ0,+ (λ)) case. By Lemma 7.2 we know ξ(λ, Hε , H0 ) = #(ψε,− (λ), ψ0,+ (λ) for every ε < 1. Concerning the right-hand side observe that Wx (ψε,− (λ), ψ0,+ (λ)) = Wx (ψ1,− (λ), ψ0,+ (λ)) for x ≤ bε and that Wx (ψε,− (λ), ψ0,+ (λ)) is constant for x ≥ bε . This implies that for ε ≥ ε0 we have ξ(λ, H1 , H0 ) = ξ(λ, Hε , H0 ) = #(ψε,− (λ), ψ0,+ (λ)) = #(a,bε ) (ψε,− (λ), ψ0,+ (λ)) = #(a,bε ) (ψ1,− (λ), ψ0,+ (λ)). In particular, the last item #(a,bε ) (ψ1,− (λ), ψ0,+ (λ)) is eventually constant and thus has a limit which, by Definition 3.1, is #(ψ1,− (λ), ψ0,+ (λ)).


635

For the corresponding #(ψ1,+ (λ), ψ0,− (λ)) case one simply exchanges the roles of H0 and H1 . Hence the result holds if the perturbation has compact support near one endpoint. Now one repeats the argument to remove the compact support assumption near the other endpoint as well. 8. Appendix: Some Facts on the Spectral Shift Function In this appendix we collect some facts on Krein’s spectral shift function which are of relevance to us. Most results are taken from [44] (see also [29] for an easy introduction). Two operators H0 and H1 are called resolvent comparable, if R H1 (z) − R H0 (z)

(8.1)

is trace class for one z ∈ ρ(H1 ) ∩ ρ(H0 ). By the first resolvent identity (8.1) then holds for all z ∈ ρ(H1 ) ∩ ρ(H0 ). Theorem 8.1 (Krein [13]). Let H1 and H0 be two resolvent comparable self-adjoint operators, then there exists a function ξ(λ, H1 , H0 ) ∈ L 1 (R, (λ2 + 1)−1 dλ)

(8.2)

such that tr( f (H1 ) − f (H0 )) =

∞ −∞

ξ(λ, H1 , H0 ) f (λ)dλ

(8.3)

for every smooth function f with compact support. Note. Equation (8.3) holds in fact for a much larger class of functions f . See [44, Thm. 8.7.1] for this and a proof of the last theorem. The function ξ(λ) = ξ(λ, H1 , H0 ) is called Krein’s spectral shift function and is unique up to an additive constant. Moreover, ξ(λ) is constant on every interval (λ0 , λ1 ) ⊂ ρ(H0 ) ∩ ρ(H1 ). Hence, if dim Ran P(λ0 ,λ1 ) (H j ) < ∞, j = 0, 1, then ξ(λ) is a step function and dim Ran P(λ0 ,λ1 ) (H1 ) − dim Ran P(λ0 ,λ1 ) (H0 ) = lim (ξ(λ1 − ε) − ξ(λ0 + ε)) . ε↓0

(8.4)

This formula explains the name spectral shift function. Before investigating further the properties of the SSF, we will recall a few things about trace ideals (see for example [28]). First, for 1 ≤ p < ∞ denote by J p the Schatten p-class, and by .J p its norm. We will use . for the usual operator norm. Using AJ p = ∞ if A ∈ / J p , we have the following inequalities for all operators: ABJ p ≤ ABJ p , ABJ 1 ≤ AJ 2 BJ 2 .

(8.5)

Furthermore, we will use the notation of J p -converges to denote convergence in the respective .J p -norm. The following result from [6, Thm IV.11.3] will be needed.

636

H. Krüger, G. Teschl s

s

Lemma 8.2. Let p > 0, A ∈ J p , Tn − → T , Sn − → S be sequences of strongly convergent bounded linear operators in some separable Hilbert space, then Tn ASn∗ − T AS ∗ J p → 0.

(8.6)

We will also need the following continuity result for ξ . It will also allow us to fix the unknown constant. Lemma 8.3. Suppose Hε , ε ∈ [0, 1], is a family of self-adjoint operators, which is continuous in the metric ρ(A, B) = R A (z 0 ) − R B (z 0 )J 1 ,

(8.7)

for some fixed z 0 ∈ C\R and abbreviate ξε = ξ(Hε , H0 ). Then there exists a unique choice of ξε such that ε → ξε is a continuous map [0, 1] → L 1 (R, (λ2 + 1)−1 dλ) with ξ0 = 0. If Hε ≥ λ0 is bounded from below, we can also allow z = λ ∈ (−∞, λ0 ). Proof. The first statement can be found in [44, Lem. 8.7.5]. To see the second statement, let λ < λ0 and |λ − z| < λ0 − λ for some z ∈ C\R. Abbreviate Rε (z) = R Hε (z). Now using the first resolvent identity gives Rε (z) − Rε (z)J 1 ≤ Rε (λ) − Rε (λ)J 1 + |z − λ|Rε (z)Rε (λ) − Rε (λ)J 1 + |z − λ|Rε (λ)Rε (z) − Rε (z)J 1 and our conditions imply |z − λ|Rε (λ) ≤

|z − λ|

b a

we have (see [10, Sec. VI.3] or [27, Thm. II.12]) Rε (−λ) = R0 (−λ)1/2 (1 + Cε )−1 R0 (−λ)1/2 , Cε = (U R0 (−λ)1/2 )∗ (K ε W R0 (−λ)1/2 ).

Hence, a straightforward calculation shows Rε (−λ) = R0 (−λ) − (U R0 (−λ))∗ (1 + C˜ ε )−1 (K ε W R0 (−λ)), C˜ ε = K ε W R0 (−λ)1/2 (U R0 (−λ)1/2 )∗ . By C˜ ε ≤ a < 1, one concludes that (1 + C˜ ε )−1 exists as a bounded operator. Then Dε,ε ψ = (−C˜ ε (1 + C˜ ε )−1 − C˜ ε (1 + C˜ ε )−1 )ψ = (Cε − Cε )(1 + Cε )−1 ψ − Cε Dε,ε ψ, where Dε,ε = (1 + C˜ ε )−1 − (1 + C˜ ε )−1 . Taking norms we obtain Dε,ε ψ =

1 (Cε − Cε )(1 + Cε )−1 ψ, 1−a

where the last term converges to 0 as ε → ε. This implies that (1 + C˜ ε )−1 is strongly continuous. Now, we obtain for the difference of resolvents Rε (−λ) − Rε (−λ) = (U R0 (−λ))∗ ((1 + C˜ ε )−1 K ε − (1 + C˜ ε )−1 K ε ))(W R0 (−λ)) which J 1 -converges to 0 as ε → ε by Lemma 8.2 and by U R0 (−λ) and W R0 (−λ) being Hilbert–Schmidt. Acknowledgements. We thank F. Gesztesy for several helpful discussions and hints with respect to the literature. In addition, we are indebted to the referee for constructive remarks.

References 1. Birman, M.Sh.: On the spectrum of singular boundary value problems. AMS Translations (2) 53, 23–80 (1966) 2. Eastham, M.S.P.: The spectral theory of periodic differential equations. Edinburgh: Scottish Academic Press, 1973 3. Gesztesy, F., Simon, B.: A short proof of Zheludev’s theorem. Trans. Am. Math. Soc. 335, 329–340 (1993) 4. Gesztesy, F., Ünal, M.: Perturbative oscillation criteria and Hardy-type inequalities. Math. Nachr. 189, 121–144 (1998) 5. Gesztesy, F., Simon, B., Teschl, G.: Zeros of the Wronskian and renormalized oscillation Theory. Am. J. Math. 118, 571–594 (1996) 6. Gohberg, I., Goldberg, S., and Krupnik, N., Traces and Determinants of Linear Operators. Basel: Birkhäuser, 2000


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Communications in


Local Semicircle Law and Complete Delocalization for Wigner Random Matrices László Erd˝os1 , Benjamin Schlein1, , Horng-Tzer Yau2, 1 Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany.

E-mail: [email protected]; [email protected]

2 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.

E-mail: [email protected] Received: 18 March 2008 / Accepted: 19 June 2008 Published online: 24 September 2008 – © Springer-Verlag 2008

Abstract: We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N . Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales η N −1 (log N )8 . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ∞ -norm of the corresponding eigenvectors is of order O(N −1/2 ), modulo logarithmic corrections. The upper bound O(N −1/2 ) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements. 1. Introduction The Wigner semicircle law states that the empirical density of the eigenvalues of a random matrix is given by the universal semicircle distribution. This statement has been proved for many different ensembles, in particular for the case when the distributions of the entries of the matrix are independent, identically distributed (i.i.d.). To fix the scaling, we normalize the matrix so that the bulk of the spectrum lies in the energy interval [−2, 2], i.e., the average spacing between consecutive eigenvalues is of order 1/N . We now consider a window of size η in the bulk so that the typical number of eigenvalues is of order N η. In the usual statement of the semicircle law, η is a fixed number Supported by Sofja-Kovalevskaya Award of the Humboldt Foundation. On leave from Cambridge University, UK. Partially supported by NSF grant DMS-0602038.

642

L. Erd˝os, B. Schlein, H.-T. Yau

independent of N and it is taken to zero only after the limit N → ∞. This can be viewed as the largest scale on which the semicircle law is valid. On the other extreme, for the smallest scale, one may take η = k/N and take the limit N → ∞ followed by k → ∞. If the semicircle law is valid in this sense, we shall say that the local semicircle law holds. Below this smallest scale, the eigenvalue distribution is expected to be governed by the Dyson statistics related to sine kernels. The Dyson statistics was proved for many ensembles (see [1,4] for a review), including Wigner matrices with Gaussian convoluted distributions [7]. In this paper, we establish the local semicircle law up to logarithmic factors in the energy scale, i.e., for η ∼ N −1 (log N )8 . The result holds for any energy window in the bulk spectrum away from the spectral edges. In [5] we have proved the same statement for η N −2/3 (modulo logarithmic corrections). Prior to our work the best result was obtained in [2] for η N −1/2 . See also [6] and [8] for related and earlier results. As a corollary, our result also proves that no gap between consecutive bulk eigenvalues can be bigger than C(log N )8 /N , to be compared with the expected average 1/N behavior given by Dyson’s law. It is widely believed that the eigenvalue distribution of the Wigner random matrix and the random Schrödinger operator in the extended (or delocalized) state regime are the same up to normalizations. Although this conjecture is far from the reach of the current method, a natural question arises as to whether the eigenvectors of random matrices are extended. More precisely, if v = (v1 , . . . , v N ) is an 2 -normalized eigenvector, v = 1, we say that v is completely delocalized if v∞ = max j |v j | is bounded from above by C N −1/2 , the average size of |v j |. In this paper, we shall prove that all eigenvectors with eigenvalues away from the spectral edges are completely delocalized (modulo logarithmic corrections) in probability. Similar results, but with C N −1/2 replaced by C N −1/3 were proved in [5]. Notice that our new result, in particular, answers (up to logarithmic factors) the question posed by T. Spencer whether v4 is of order N −1/4 . Denote the (i, j)th entry of an N × N matrix H by h i, j = h i j . When there is no confusion, we omit the comma between the two subscripts. We shall assume that the matrix is Hermitian, i.e., h i j = h ji . These matrices form a Hermitian Wigner ensemble if √ h i j = N −1/2 [xi j + −1 yi j ], (i < j), and h ii = N −1/2 xii , (1.1) where xi j , yi j (i < j) and xii are independent real random variables with mean zero. We assume that xi j , yi j (i < j) all have a common distribution ν with variance 1/2 and with a strictly positive density function: dν(x) = (const.)e−g(x) dx. The diagonal elements, g (x) dx, that may be different xii , also have a common distribution, d ν(x) = (const.)e− from dν. Let P and E denote the probability and the expectation value, respectively, w.r.t the joint distribution of all matrix elements. We need to assume further conditions on the distributions of the matrix elements in addition to (1.1). C1) The function g is twice differentiable and it satisfies sup g (x) < ∞.

x∈R

C2) There exists a δ > 0 such that

ν(x) < ∞. eδx d 2

(1.2)

(1.3)

Local Semicircle Law

643

C4) The measure ν satisfies the logarithmic Sobolev inequality, i.e., there exists a constant Csob such that for any density function u > 0 with u dν = 1, √ u log u dν ≤ Csob |∇ u|2 dν. (1.4) Here we have followed the convention in [5] to use the label C4) for the logarithmic Sobolev bound and reserved C3) for a spectral gap condition in [5]. We will also need the decay condition (1.3) for the measure dν, i.e., for some small δ > 0, 2 eδx dν(x) < ∞. (1.5) This condition was assumed in the earlier version of the manuscript, but J.-D. Deuschel and M. Ledoux kindly pointed out to us that (1.5) follows from C4), see [9]. Condition C1) is needed only because we will use Lemma 2.3 of [5] in the proof of the following Theorem 1.1. J. Bourgain has informed us that this lemma can also be proved without this condition. We include the precise statement and his proof in the Appendix. Notation. We will use the notation |A| both for the Lebesgue measure of a set A ⊂ R and for the cardinality of a discrete set A ⊂ Z. The usual Hermitian scalar product for vectors x, y ∈ C N will be denoted by x · y or by (x, y). We will use the convention that C denotes generic large constants and c denotes generic small positive constants whose values may change from line to line. Since we are interested in large matrices, we always assume that N is sufficiently large. Let H be the N × N Wigner matrix with eigenvalues µ1 ≤ µ2 ≤ . . . ≤ µ N . For any spectral parameter z = E + iη ∈ C, η > 0, we denote the Green function by G z = (H − z)−1 . Let F(x) = FN (x) be the empirical distribution function of the eigenvalues F(x) =

1 | {α : µα ≤ x}| . N

(1.6)

We define the Stieltjes transform of F as 1 dF(x) Tr G z = , N R x−z

(1.7)

N 1 η Im m(z) 1 = Im Tr G z = π Nπ Nπ (µα − E)2 + η2

(1.8)

m = m(z) = and we let ρ = ρη (E) =

α=1

be the normalized density of states of H around energy E and regularized on scale η. The random variables m and also depend on N ; when necessary, we will indicate this fact by writing m N and N . For any z = E + iη we let sc (x)dx m sc = m sc (z) = R x−z

644


be the Stieltjes transform of the Wigner semicircle distribution function whose density is given by sc (x) =

1 4 − x 2 1(|x| ≤ 2). 2π

For κ, η > 0 we define the set η ≤ η ≤ 1} , S N ,κ, η := {z = E + iη ∈ C : |E| ≤ 2 − κ, and for η = N −1 (log N )8 we write (log N )8 S N ,κ := z = E + iη ∈ C : |E| ≤ 2 − κ, ≤η≤1 . N The following two theorems are the main results of this paper. Theorem 1.1. Let H be an N × N Wigner matrix as described in (1.1) and assume the conditions (1.2), (1.3) and (1.4). Then for any κ > 0 and ε > 0, the Stieltjes transform m N (z) (see (1.7)) of the empirical eigenvalue distribution of the N × N Wigner matrix satisfies P

sup |m N (z) − m sc (z)| ≥ ε ≤ e−c(log N )

2

(1.9)

z∈S N ,κ

where c > 0 depends on κ, ε. In particular, the density of states η (E) converges to the Wigner semicircle law in probability uniformly for all energies away from the spectral edges and for all energy windows at least N −1 (log N )8 . Furthermore, let η∗ = η∗ (N ) such that (log N )8 /N η∗ 1 as N → ∞, then we have the convergence of the counting function as well:

Nη∗ (E)

2

P sup

− sc (E) ≥ ε ≤ e−c(log N ) (1.10) ∗ |E|≤2−κ 2N η for any ε > 0, where Nη∗ (E) = |{α : |µα − E| ≤ η∗ }| denotes the number of eigenvalues in the interval [E − η∗ , E + η∗ ]. This result identifies the density of states away from the spectral edges in a window where the typical number of eigenvalues is of order bigger than (log N )8 . Our scale is not sufficiently small to identify individual eigenvalues, in particular we do not know whether the local eigenvalue spacing follows the expected Dyson statistics characterized by the sine-kernel or some other local statistics, e.g., that of a Poisson point process. Theorem 1.2. Let H be an N × N Wigner matrix as described in (1.1) and satisfying the conditions (1.2), (1.3) and (1.4). Fix κ > 0, and assume that C is large enough. Then there exists c > 0 such that C(log N )9/2 P ∃ v with H v = µv, v = 1, µ ∈ [−2 + κ, 2 − κ] and v∞ ≥ N 1/2 ≤ e−c(log N ) . 2


645

We now sketch the key idea to prove Theorem 1.1; Theorem 1.2 can be proved following similar ideas used in [5]. Let B (k) denote the (N − 1) × (N − 1) minor of H after removing the k th row and k th column and let m k (z) denote the Stieltjes transform of the eigenvalue distribution function associated with B (k) . It is known that m(z), defined in (1.7), satisfies a recurrence relation m(z) =

N 1 1 , N h − z − 1 − N1 m (k) (z) − X k k=1 kk

(1.11)

where X k (defined precisely in (2.4)) is an “error” term depending on B (k) and the k th column and row elements of the random matrix H . If we neglect X k (and h kk which is of order N −1/2 by definition) and replace m (k) by m, we obtain an equation for m and this leads to the Stieltjes transform of the semi-circle law. So our main task is to prove that X k is negligible. Unfortunately, X k depends crucially on the eigenvalues and eigenfunctions of B (k) . In an earlier work [2], the estimate on X k was done via an involved bootstrap argument (and valid up to order N −1/2 ). The bootstrapping is needed in [2] since X k depends critically on properties of B (k) for which there was only limited a priori information. In our preceding paper [5], we split m and m (k) into their means and variances; the variances were then shown to be negligible up to the scale N −2/3 . (The variance control of m up to the scale N −1/2 was already in [6].) On the other hand, the means of m and m (k) are very close due to the fact that the eigenvalues of H and B (k) are interlaced. Finally, X k was controlled via an estimate on its fourth moment. We have thus arrived at a fixed point equation for the mean of m whose unique solution is the Stieltjes transform of the semi-circle law. In the current paper, we avoid the variance control by viewing m and m (k) directly as random variables in the recurrence relation (1.11). Furthermore, the moment control on X k is now improved to an exponential moment estimate. Since our estimate on the fourth moment of X k was done via a spectral gap argument, it is a folklore that moment estimates usually can be lifted to an exponential moment estimate provided the spectral gap estimate is replaced by a logarithmic Sobolev inequality. In this paper, we use a concentration inequality to avoid all bootstrap arguments appearing both in [2] and [8]. In the previous version of this paper we obtained the concentration inequality by using the logarithmic Sobolev inequality together with the (Gibbs) relative entropy inequality. We would like to thank the referee who pointed out to us that the concentration inequality of Bobkov and Götze [3] can be used directly to shortcut our original proof. The applicability of the Bobkov-Götze inequality as well as some of the heuristic arguments presented here, however, depends crucially on an a priori upper bound on |m(z)|; this was obtained via a large deviation estimate on the eigenvalue concentration [5]. 2. Proof of Theorem 1.1 The proof of (1.10) follows from (1.9) exactly as in Corollary 4.2 of [5], so we focus on proving (1.9). We first remove the supremum in (1.9). For any two points z, z ∈ S N ,κ,η we have |m N (z) − m N (z )| ≤ N 2 |z − z |, since the gradient of m N (z) is bounded by |Im z|−2 ≤ N 2 on S N ,κ . We can choose a set of at most Q = Cε−2 N 4 points, z 1 , z 2 , . . . , z Q , in S N ,κ,η such that for any z ∈ S N ,κ,η

646


there exists a point z j with |z − z j | ≤ 14 εN −2 . In particular, |m N (z) − m N (z j )| ≤ ε/4 if N is large enough and |m sc (z) − m sc (z j )| ≤ ε/4. Since Im z j ≥ η, under the condition that η ≥ N −1 (log N )8 we have

Q

ε . P |m N (z j ) − m sc (z j )| ≥ P sup |m N (z) − m sc (z)| ≥ ε ≤ 2 z∈S N ,κ j=1

Therefore, in order to conclude (1.9), it suffices to prove that P {|m N (z) − m sc (z)| ≥ ε} ≤ e−c(log N )

2

(2.1)

for each fixed z ∈ S N ,κ . Let B (k) denote the (N − 1) × (N − 1) minor of H after removing the k th row and k th column. Note that B (k) is an (N − 1) × (N − 1) Hermitian Wigner matrix with (k) (k) (k) a normalization factor off by (1 − N1 )1/2 . Let λ1 ≤ λ2 ≤ . . . ≤ λ N −1 denote its (k)

(k)

eigenvalues and u1 , . . . , u N −1 the corresponding normalized eigenvectors. Let a(k) = (h k,1 , h k,2 , . . . h k,k−1 , h k,k+1 , . . . h k,N )∗ ∈ C N −1 , i.e. the k th column after removing the diagonal element h k,k = h kk . Computing the (k, k) diagonal element of the resolvent G z , we have G z (k, k) =

h kk

−1 N −1 (k) 1 1 ξα = h kk −z − , (k) N − z − a(k) · (B (k) − z)−1 a(k) α=1 λα − z

(2.2)

where we defined

2

√

ξα(k) := N a(k) · uα(k) . Similarly to the definition of m(z) in (1.7), we also define the Stieltjes transform of the density of states of B (k) , m

(k)

=m

(k)

1 dF (k) (x) 1 Tr (k) (z) = = N −1 B −z R x−z

with the empirical counting function F (k) (x) =

1

≤ x

.

α : λ(k) α N −1

The spectral parameter z is fixed throughout the proof and we will often omit it from the argument of the Stieltjes transforms. It follows from (2.2) that m = m(z) =

N N 1 1 1 G z (k, k) = . (2.3) (k) N N h kk − z − a · (B (k) − z)−1 a(k) k=1

k=1


647

Let Ek denote the expectation value w.r.t the random vector a(k) . Define the random variable X k (z) = X k := a(k) ·

1 B (k) − z (k)

a(k) − Ek a(k) ·

1 B (k) − z

a(k) =

N −1 (k) 1 ξα − 1 , (k) N α=1 λα − z

(2.4)

(k)

where we used that Ek ξα = uα 2 = 1. We note that 1 1 1 1 m (k) . = 1 − a(k) = Ek a(k) · (k) N α λ(k) N B −z α −z With this notation it follows from (2.2) that m=

N 1 1 . N h − z − 1 − N1 m (k) − X k k=1 kk

(2.5)

We use that

(k) (x)

dF(x) 1 1 dF

(k)

m − 1 − m

=

− 1−

x−z N N x−z

1

N F(x) − (N − 1)F (k) (x)

= dx .

N

(x − z)2 We recall that the eigenvalues of H and B (k) are interlaced, (k)

(k)

(k)

µ1 ≤ λ1 ≤ µ2 ≤ λ2 ≤ . . . ≤ λ N −1 ≤ µ N ,

(2.6)

(see e.g. Lemma 2.5 of [5]), therefore we have max x |N F(x) − (N − 1)F (k) (x)| ≤ 1. Thus

C dx

m − 1 − 1 m (k) ≤ 1 . (2.7) ≤

N

N |x − z|2 Nη We postpone the proof of the following lemma: Lemma 2.1. Suppose that vα and λα are eigenvectors and eigenvalues of an N × N random matrix with a law satisfying the assumption of Theorem 1.1. Let X=

1 ξα − 1 N α λα − z

with z = E + iη, ξα = |b · vα |2 , where the components of b are i.i.d. random variables satisfying (1.4). Then there exist sufficiently small positive constants ε0 and c such that in the joint product probability space of b and the law of the random matrices we have P[|X | ≥ ε] ≤ e−cε(log N ) for any ε ≤ ε0 and η ≥ (log N )8 /N .

2

648


For a given ε > 0 and z = E + iη ∈ S N ,κ , we set z n = E + i2n η and we define the event =

N [log N 2 (1/η)] {|X k (z n )| ≥ ε/3} ∪ {|h kk | ≥ ε/3} , k=1

n=0

k=1

where [ · ] denotes the integer part. Since h kk = N −1/2 bkk with bkk satisfying (1.3), we have P{|h kk | ≥ ε/3} ≤ Ce−δε

2 N /9

.

We now apply Lemma 2.1 for each X k (z n ) and conclude that P() ≤ e−cε(log N )

2

with a sufficiently small c > 0. On the complement c we have from (2.5), m(z n ) =

N 1 1 , N −m(z n ) − z n + δk k=1

where

1 m k (z n ) − X k (z n ) δk = δk (z n ) = h kk + m(z n ) − 1 − N

are random variables satisfying |δk | ≤ ε by (2.7). After expansion, the last equation implies that

ε 1

m(z n ) +

≤ , (2.8)

m(z n ) + z n (Im (m(z n ) + z n )) (Im (m(z n ) + z n ) − ε) if Im (m(z n ) + z n ) > ε, using that | − m(z n ) − z n + δk | ≥ Im (m(z n ) + z n ) − ε. We note that for any z ∈ S N ,κ the equation M+

1 =0 M+z

(2.9)

has a unique solution M with Im M > 0, namely M = m sc (z), the Stieltjes transform of the semicircle law. Note that there exists c(κ) > 0 such that Im m sc (E + iη) ≥ c(κ) for any |E| ≤ 2 − κ, uniformly in η. Equation (2.9) is stable in the following sense. For any small δ, let M = M(z, δ) be a solution to M+

1 =δ M+z

with Im M > 0. Explicitly, we have M=

−z +

√

z 2 − 4 + 2zδ + δ 2 δ + , 2 2

(2.10)


649

where we have chosen the square root so that ImM > 0 when δ = 0 and Imz > 0. On the compact set z ∈ S N ,κ , |z 2 − 4| is bounded away from zero and thus |M − m sc | ≤ Cκ δ

(2.11)

for some constant Cκ depending only on κ. Now we perform a bootstrap argument in the imaginary part of z to prove that | m(z) − m sc (z)| ≤ C ∗ ε

(2.12)

uniformly in z ∈ S N ,κ with a sufficiently large constant C ∗ . Fix z = E + iη with |E| ≤ 2 − κ and let z n = E + i2n η. For n = [log2 (1/η)], we have Im z n ∈ [ 21 , 1], (2.12) follows from (2.8) with some small ε, since the right hand side of (2.8) is bounded by Cε. Suppose now that (2.12) has been proven for z = z n , for some n ≥ 1 with ηn = Im z n ∈ [2N −1 (log N )8 , 1]. We want to prove it for z = z n−1 , with Im z n−1 = ηn /2. By integrating the inequality ηn ηn /2 1 ≥ (x − E)2 + (ηn /2)2 2 (x − E)2 + ηn2 with respect to dF(x) we obtain that Im m(z n−1 ) ≥

1 1 c(κ) Im m(z n ) ≥ c(κ) − C ∗ ε > 2 2 4

for sufficiently small ε, where (2.12) and Im m sc (z n ) ≥ c(κ) were used. Thus the right hand side of (2.8) with z n replaced with z n−1 is bounded by Cε, the constant depending only on κ. Applying the stability bound (2.11), we get (2.12) for z = z n−1 . Continuing the induction argument, finally we obtain (2.12) for z = z 0 = E + iη. 3. Proof of Lemma 2.1 Let In = [nη, (n + 1)η] and K 0 be a sufficiently large number. We have [−K 0 , K 0 ] ⊂ ∪m n=−m In with m ≤ C K 0 /η. Denote by the event

:= max N In ≥ N η(log N )2 ∪ {max |λα | ≥ K 0 }, α

n

where N In = |{α : λα ∈ In }| is the number of eigenvalues in the interval In . From Theorem 2.1 and Lemma 7.4 of [5], the probability of is bounded by P() ≤ e−c(log N )

2

for some sufficiently small c > 0. Therefore, if Pb denotes the probability w.r.t. the variable b, we find P[|X | ≥ ε] ≤ e−c(log N ) + E [1c Pb [|X | ≥ ε]] 2

≤ e−c(log N ) + E [1c · Pb [ReX ≤ −ε/2]] + E[1c · Pb [ReX ≥ ε/2]] (3.1) +E [1c · Pb [ImX ≤ −ε/2]] + E [1c · Pb [ImX ≥ ε/2]] . 2

650


The last four terms on the r.h.s. of the last equation can all be handled with similar arguments; we show, for example, how to bound the last term. For any T > 0, we have (3.2) E [1c · Pb [ImX ≥ ε/2]] ≤ e−T ε/2 E 1c Eb e T ImX , where Eb denotes the expectation w.r.t. the variable b. Using the fact that the distribution of the components of b satisfies the log-Sobolev inequality (1.4), it follows from the concentration inequality (Theorem 2.1 from [3]) that Eb e T ImX ≤ Eb exp

Csob T 2 |∇(ImX )|2 , 2

(3.3)

where

∂ (ImX )

2

∂ (ImX )

2

|∇(ImX )| =

∂ (Re b ) + ∂ (Im b )

k k k ⎛

2

η

1 ⎝

= (b · vα ) vα (k) + b · vα vα (k)

N α |λα − z|2

2

k

=4 ≤

2 ⎞

η

1

(b · vα )vα (k) − b · vα vα (k) ⎠ +

2

N

|λ − z| α α

η2 ξα N 2 α |λα − z|4

4 Y. Nη

(3.4)

Here we defined the random variable Y =

1 ξα . N α |λα − z|

From (3.2), choosing T = 2(log N )2 and using that N η ≥ (log N )8 , we obtain 8Csob 2 E [1c · Pb [ImX ≥ ε/2]] ≤ e−ε(log N ) E 1c Eb exp Y . (log N )4

(3.5)

Let ν = 8Csob /(log N )4 . By Hölder inequality, we can estimate Eb e

νY

= Eb

α

where

1 α cα

exp

ν N |λα − z|

ξα

≤ Eb exp α

= 1. We shall choose cα =

N |λα − z| , ν

νcα ξα N |λα − z|

1/cα

, (3.6)


651

where is given by =

ν log N 1 8Csob ν ≤ max N In ≤ ν(log N )3 = . n N α |λα − z| Nη log N

Here we have used maxn N In ≤ N η(log N )2 because we are in the set c . Notice that with this choice, νcα 8Csob ≤ N |λα − z| log N is a small number. In the proof of Lemma 7.4 of [5] (see Eq. (7.13) of [5]) we showed that Eb eτ ξα < K with a universal constant K if τ is sufficiently small depending on δ in (1.3). From (3.5), it follows that E [1c · Pb [ImX ≥ ε/2]] ≤ K e−ε(log N ) . 2

(3.7)

Since similar bounds hold for the other terms on the r.h.s. of (3.1) as well, this concludes the proof of the lemma. 4. Delocalization of Eigenvectors Here we prove Theorem 1.2, the argument follows the same line as in [5] (Prop. 5.3). Let η∗ = N −1 (log N )9 and partition the interval [−2+κ, 2−κ] into n 0 = O(1/η∗ ) ≤ O(N ) intervals I1 , I2 , . . . In 0 of length η∗ . As before, let N I = |{β : µβ ∈ I }| denote the eigenvalues in I . By using (1.10) in Theorem 1.1, we have

2 P max N In ≤ εN η∗ ≤ e−c(log N ) , n

if ε is sufficiently small (depending on κ). Suppose that µ ∈ In , and that H v = µv. Consider the decomposition ∗ ha , (4.1) H= a B where a = (h 1,2 , . . . h 1,N )∗ and B is the (N −1)×(N −1) matrix obtained by removing the first row and first column from H . Let λα and uα (for α = 1, 2, . . . , N − 1) denote the eigenvalues and the normalized eigenvectors of B. From the eigenvalue equation H v = µv, and from (4.1) we find that hv1 + a · w = µv1 , and

av1 + Bw = µw

(4.2)

with w = (v2 , . . . , v N )t . From these equations we obtain w = (µ − B)−1 av1 and thus w2 = w · w = |v1 |2 a · (µ − B)−2 a.

652


Since w2 = 1 − |v1 |2 , we obtain |v1 |2 =

1 = 1 + a · (µ − B)−2 a 1+

1

ξα α (µ−λα )2

1 N

4N [η∗ ]2 ≤ , λα ∈In ξα

(4.3)

√ where in the second equality we set ξα = | N a · uα |2 and used the spectral representation of B. By the interlacing property of the eigenvalues of H and B, there exist at least N In − 1 eigenvalues λα in In . Therefore, using that the components of any eigenvector are identically distributed, we have C(log N )9/2 P ∃ v with H v = µv, v = 1, µ ∈ [−2 + κ, 2 − κ] and v∞ ≥ N 1/2 C(log N )9 2 ≤ N n 0 sup P ∃ v with H v = µv, v = 1, µ ∈ In and |v1 | ≥ N n ⎛ ⎞ 4N η∗ ⎠ ≤ const N 2 sup P ⎝ ξα ≤ (4.4) C n λα ∈In ⎛ ⎞ ∗ 4N η and N In ≥ εN η∗ ⎠ ξα ≤ ≤ const N 2 sup P ⎝ C n λα ∈In +const N 2 sup P N In ≤ εN η∗

n

2 −c(log N )9

≤ const N e

+ const N 2 e−c(log N ) ≤ e−c (log N ) , 2

2

by choosing C sufficiently large, depending on κ via ε. Here we used Corollary 2.4 of [5] that states that under condition C1) in (1.2) there exists a positive c such that for any δ small enough, P

ξα ≤ δm

≤ e−cm

(4.5)

α∈A

for all A ⊂ {1, · · · , N − 1} with cardinality |A| = m. We remark that by applying Lemma 4.1 from the Appendix instead of Lemma 2.3 in [5], the bound (4.5) also holds without condition C1) if the matrix elements are bounded random variables. It is clear that the boundedness assumption in Lemma 4.1 can be relaxed by performing an appropriate cutoff argument; we will not pursue this direction in this article. Appendix: Removal of the Assumption C1) Jean Bourgain School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA.

The following lemma shows that the assumption C1) in Lemma 2.3 and its corollary in [5] can be removed.


653

Lemma 4.1. Suppose that z 1 , . . . , z N are bounded, complex valued i.i.d. random variables with E z i = 0 and E |z i |2 = a > 0. Let P : C N → C N be a rank-m projection, and z = (z 1 , . . . , z N ). Then, if δ is small enough, there exists c > 0 such that ! P |Pz|2 ≤ δm ≤ e−cm . Lemma 2.3 in [5] stated that the same conclusion holds under Condition C1), but it required no assumption on the boundedness of the random variables. Proof. It is enough to prove that

! 2

P |Pz|2 − am > τ m ≤ e−cτ m

(4.6)

#1/q " . Since for all τ sufficiently small. Introduce the notation X q = E|X |q $ $

! $ |Pz|2 − am $q q

2 , P |Pz| − am > τ m ≤ (τ m)q

(4.7)

the bound (4.6) follows by showing that √ √ |Pz|2 − amq ≤ C q m

for all q < m

(4.8)

(and then choosing q = ατ 2 m with a small enough α). To prove (4.8), observe that (with the notation ei = (0, . . . , 0, 1, 0 . . . , 0) for the standard basis of C N ) |Pz|2 =

N

|z i |2 |Pei |2 +

N

z i z j Pei · Pe j = am

i= j

i=1

+

N

! |z i |2 − E|z i |2 |Pei |2 + z i z j Pei · Pe j ,

(4.9)

i= j

i=1

and thus

$ $ $ $ N $ N $ $ $ $ $ ! $ $ $ $ $ 2 2 2 2$ $ $ |z |Pe − am ≤ | − E|z | | + z z Pe · Pe $|Pz| $ $ i i i $ i j i j $ . (4.10) $ $ $ $ q $ i=1 i= j q q

To bound the first term, we use that for arbitrary i.i.d. random variables x1 , . . . , x N with 2 E x j = 0 and E eδ|x j | < ∞ for some δ > 0, we have the bound √ X q ≤ C q X 2 (4.11) N for X = j=1 a j x j , for arbitrary a j ∈ C. The bound (4.11) is an extension of Khintchine’s inequality and it can be proven as follows using the representation ∞ q X q = q dy y q−1 P(|X | ≥ y) . (4.12) 0

Writing a j = |a j |eiθ j , θ j ∈ R, and decomposing eiθ j x j into real and imaginary parts, it is clearly sufficient to prove (4.11) for the case when a j , x j ∈ R are real and x j ’s are

654


independent with E x j = 0 and E eδ|x j | < ∞. To bound the probability P(|X | ≥ y) we observe that 2

P(X ≥ y) ≤ e−t y E et X = e−t y

N

E eta j x j ≤ e−t y eCt

2

N

2 j=1 a j

,

j=1

because E eτ x ≤ eCτ from the moment assumptions on x j with a sufficiently large C depending on δ. Repeating this argument for −X , we find 2

P(|X | ≥ y) ≤ 2 e−t y eCt

2

N

2 j=1 a j

≤ e−y

2 /(2C

N

2 j=1 a j )

after optimizing in t. The estimate (4.11) follows then by plugging the last bound into (4.12) and computing the integral. 2 Applying (4.11) with xi = |z i |2 − E |z i |2 (E eδxi < ∞ follows from the assumption z i ∞ < ∞), the first term on the r.h.s. of (4.10) can be controlled by $ N $ N N 1/2 1/2 $ $ ! √ √ $ 2 2 2$ 4 2 |z i | − E|z i | |Pei | $ ≤ C q |Pei | ≤C q |Pei | $ $ $ i=1 i=1 i=1 q √ √ = C q m. (4.13) As for the second term on the r.h.s. of (4.10), we define the functions ξ j (s), s ∈ [0, 1], j = 1, . . . , N by %2 j−1 −1 2k 2k+1 ! , 2j 1 if s ∈ k=0 2j . ξ j (s) = 0 otherwise Since

1

ds ξi (s)(1 − ξ j (s)) =

0

1 4

for all i = j, the second term on the r.h.s. of (4.10) can be estimated by $ $ $ $ $ N $ N $ $ 1 $ $ $ $ $ $ $ $ z z Pe · Pe ≤ 4 ds ξ (s) (1 − ξ (s)) z z Pe · Pe i j i j$ i j i j i j $ . (4.14) $ $ 0 $ i= j $ i= j $ $ q

q

For fixed s ∈ [0, 1], set I (s) = {1 ≤ i ≤ N : ξi (s) = 1}

and

J (s) = {1, . . . , N }\I (s) .

Then $ $ $ $ $ $ $ $ $ $ $ $ N $ $ $ ξi (s) (1 − ξ j (s)) z i z j Pei · Pe j $ = $ z i z j Pei · Pe j $ $ $ $ i∈I (s), j∈J (s) $ $ $ i= j q q $ $ ⎛ ⎞ $ $ $ $ $ ⎝ ⎠ · e =$ z z Pe j i i j$ . $ $ j∈J (s) $ i∈I (s) q


655

Since by definition I ∩ J = ∅, the variable {z i }i∈I and the variable {z j } j∈J are independent. Therefore, we can apply Khintchine’s inequality (4.11) in the variables {z j } j∈J (separating the real and imaginary parts) to conclude that $⎛ $

⎛ $ ⎞ 2 ⎞1/2 $ $ $ $

$ $ $

⎟ $ $ N √ ⎜ $ $

⎝ $ ξi (s) (1 − ξ j (s)) z i z j Pei · Pe j $ z i Pei⎠ · e j

⎠ $ ⎝ $ ≤C q$

$ $ $ $

$ i= j $ j∈J (s) i∈I (s) $ q q $ $ $ $ $ √ $ √ ≤C q$ z i Pei$ $ $ ≤ C qPzq (4.15) $i∈I (s) $ q

for every s ∈ [0, 1]. It follows from (4.14) that $ $ $ $ $ N $ √ $ z i z j Pei · Pe j $ $ $ ≤ C q Pzq . $ i= j $ q

Inserting the last equation and (4.13) into the r.h.s. of (4.10), it follows that $ $ √ √ $ $ m + Pzq . $|Pz|2 − am $ ≤ C q q

Since clearly

$ $1/2 √ $ $ Pzq ≤ $|Pz|2 − am $ + am, q

the bound (4.8) follows immediately. Acknowledgement. We thank the referee for very useful comments on earlier versions of this paper. We also thank J. Bourgain for the kind permission to include his result in the Appendix.

References 1. Anderson, G.W., Guionnet, A., Zeitouni, O.: Lecture notes on random matrices. Book in preparation 2. Bai, Z.D., Miao, B., Tsay, J.: Convergence rates of the spectral distributions of large Wigner matrices. Int. Math. J. 1(1), 65–90 (2002) 3. Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163(1), 1–28 (1999) 4. Deift, P.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics 3, Providence, RI: American Mathematical Society, 1999 5. Erd˝os, L., Schlein, B., Yau, H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. http://arxiv.org/abs/0711.1730, 2007 6. Guionnet, A., Zeitouni, O. Concentration of the spectral measure for large matrices. Electronic Comm. in Probability 5, Paper 14 (2000) 7. Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001) 8. Khorunzhy, A.: On smoothed density of states for Wigner random matrices. Random Oper. Stoch. Eq. 5(2), 147–162 (1997) 9. Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89, Providence, RI: American Mathematical Society, 2001 10. Quastel, J., Yau, H.-T.: Lattice gases, large deviations, and the incompressible Navier-Stokes equations. Ann. Math 148, 51–108 (1998) Communicated by M. Aizenman


Communications in


On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics Paolo Antonelli, Pierangelo Marcati Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Via Vetoio, 67010 Coppito (AQ), Italy. E-mail: [email protected]; [email protected] Received: 19 March 2008 / Accepted: 28 April 2008 Published online: 23 September 2008 – © Springer-Verlag 2008

Abstract: In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung [44], have been extensively used in Physics to investigate Superfluidity and Superconductivity phenomena [19,38] and more recently in the modeling of semiconductor devices [20] . Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method which iterates a Schrödinger Madelung picture with a suitable wave function updating mechanism. Therefore several a priori bounds of energy, dispersive and local smoothing type, allow us to prove the compactness of the approximating sequences. No uniqueness result is provided. 1. Introduction In this paper we study the Cauchy problem for the Quantum Hydrodynamics (QHD) system: ⎧ ∂t ρ + div J = 0 ⎪ ⎪ ⎪ ⎪ ⎨ ∂t J + div J ⊗J + ∇ P(ρ) + ρ∇V ρ√ (1) ⎪ = 2 ρ∇ ∆√ ρ + f (√ρ, J, ∇ √ρ, ∇ 2 √ρ, ∇(J/√ρ)) ⎪ ⎪ 2 ρ ⎪ ⎩ −∆V = ρ − C(x), with initial data ρ(0) = ρ0 ,

J (0) = J0 .

(2)

We will discuss in particular the case f = −J , however we are able to treat a more general collision term, as it will be clarified in the following Remark 5. Indeed, we show that it is possible to consider a general collision term of the form f = −α J +ρ∇g, where

658

P. Antonelli, P. Marcati

√ √ α ≥ 0 and g is a nonlinear function of ρ, J, ∇ ρ, satisfying certain Carathéodory-type conditions (see Remark 5 for more precise conditions). We are interested to study the global existence in the class of finite energy initial data without higher regularity hypotheses or smallness assumptions. The analysis of uniqueness of weak solutions in some restricted classes will be done in a forthcoming paper. There is a formal analogy between (1) and the classical fluid mechanics system, in particular when = 0, the system (1) formally coincides with the (nonhomogeneous) Euler-Poisson incompressible fluid system. The theoretical description of microphysical systems is generally based on the wave mechanics of Schrödinger, the matrix mechanics of Heisenberg, or the path-integral mechanics of Feynman (see [19]). Another approach to Quantum mechanics was taken by Madelung and de Broglie (see [44]), in particular the hydrodynamic theory of quantum mechanics has been later extended by de Broglie (the idea of “double solution”) and used as a scheme for quasicausal interpretation of microphysical systems. There is an extensive literature (see for example [12,32–34,36,38], and references therein) where superfluidity phenomena are described by means of quantum hydrodynamic systems. Another example is provided by the so called Schrödinger-Langevin equation in chemical physics [37]. Furthermore, the Quantum Hydrodynamics system is well known in literature since it has been used in modeling semiconductor devices at nanometric scales (see [20]). The hydrodynamical formulation for quantum mechanics is quite useful with respect to other descriptions for semiconductor devices, such as those based on Wigner-Poisson or Schrödinger-Poisson, since kinetic or Schrödinger equations are computationally very expensive. For a derivation of the QHD system we refer to [2,13– 15,21,29,31]. The unknowns ρ, J represent the charge and the current densities respectively, P(ρ) p+1 2 (here and throughout the classical pressure which we assume to satisfy P(ρ) = p−1 p+1 ρ the paper we assume 1 ≤ p ≤ 5). The function V is the self-consistent electric potential, given by the Poisson equation, the function C(x) represents the density of the background positively charged ions. In the paper we will treat extensively the case C(x) = 0, but all the results can be extended to more general cases. For instance we can assume 3 ). C ∈ W 1,1 (R3 ) ∩ W1,3 (R √ 2 ∆ ρ The term 2 ρ∇ √ρ can be interpreted as the quantum Bohm potential, or as a quantum correction to the pressure, indeed with some regularity assumptions we can write the dispersive term in different ways: √ ∆ ρ 2 2 2 √ √ ρ∇ √ div(ρ∇ 2 log ρ) = ∆∇ρ − 2 div(∇ ρ ⊗ ∇ ρ). (3) = 2 4 4 ρ There is a formal equivalence between the system (1) and the following nonlinear Schrödinger-Poisson system: 2 i∂t ψ + 2 ∆ψ = |ψ| p−1 ψ + V ψ + V˜ ψ, (4) −∆V = |ψ|2 where V˜ = 2i1 log ψ , in particular the hydrodynamic system (1) can be obtained by ψ

2 defining ρ = |ψ| , J = Im ψ∇ψ and by computing the related balance laws. This problem has to face a serious mathematical difficulty connected with the need to solve (4) with the ill-posed potential V˜ . Presently there are no mathematical results

Finite Energy Weak Solutions to QHD

659

concerning the solutions to (4), except small perturbations around constant plane waves or local existence results under various severe restrictions (see [26,28,40]). In the paper by Li and the second author [41] there is a global existence result for the system (1), regarding small perturbations in higher Sobolev norms of subsonic stationary solutions, with periodic boundary conditions. A possible way to circumvent this type of difficulty could be to develop a theory regarding wave functions taking values on Riemann manifolds, but we will not pursue this direction in this paper (see also [10,51]). Another nontrivial problem concerning the derivation of solutions to (1) starting from the solutions to (4), regards the reconstruction of the initial datum ψ(0) in terms of the observables ρ(0), J (0). Actually this is a case of a more general important problem in physics, pointed out by Weigert in [58]. He named it the Pauli problem (since this question originated from a footnote in Pauli’s article in Handbuch der Physik, see [48]), and it regards the possibility of reconstructing a pure quantum state, just by knowing a finite set of measurements of the state (in our case, the mass and current densities). Here the possible existence of nodal regions, or vacuum in fluid terms, namely where ρ = 0, forbids in general this reconstruction in a classical way, and in any case some additional requirements (quantization rules like the Bohr-Sommerfeld rule) would be necessary. In any case, various authors (see [58] and references therein) showed that knowledge of only position and momentum distribution does not specify any single state. The opposite direction, namely the derivation of solutions of (4) starting from solutions of (1) also can face severe mathematical difficulties in various points. In particular if we prescribe ψ(0), we can define ρ(0) and J (0), however from the evolution of the quantities ρ(t) and J (t), we cannot reconstruct the wave function ψ(t). Furthermore, from the moment equation in (1) we cannot derive the quantum eikonal equation √ 1 2 ∆ ρ 2 ∂t S + |∇ S| + h(ρ) + V + S = (5) √ 2 2 ρ which is the key element to reconstruct a solution of (4) via the WKB ansatz for the √ wave function, ψ = ρei S/. Similar difficulties arise when approaching with Wigner functions, which has been recently quite popular to deduce quantum fluid systems in a kinetic way (see [59]). Even in the case when we know the initial data ρ(0), J (0) originated from a wave function ψ(0), it is very difficult to show that the solutions ρ(t), J (t) coincide for all times with the first and second momenta of the Wigner function obtained by solving the Wigner quantum transport equation. Moreover in our case there is also the difficulty due to the non-classical potential V˜ . A related question has been investigated by Bourgain, Brezis, Mironescu in several papers (see [6] and references therein) regarding the lifting problem for harmonic maps. Another formal approach is provided by the following transport equation: 2 √ √ ρ∆ ρ (6) 2 which is obtained by multiplying Eq. (5) by ρ. Unfortunately the lack of regularity of the solutions does not allow to apply even the more recent advances of the theory of transport equations [1,16] or the somehow related approach developed by Teufel and Tumulka [56] in the study of trajectories of Bohmian mechanics. A related problem arises in the study of Nelson stochastic mechanics (see for instance Nelson [47], Guerra and Morato [24]), where the mathematical theory is based on the ρ∂t S + J ∇ S + ρh(ρ) + ρV + ρ S =

660


analysis of the velocity fields (see Carlen [7]). The application of this approach to our problem presents the same level of difficulty of the previously mentioned methods from transport theory. A natural framework to study the existence of the weak solutions to (1) is given by the space of finite energy states. Here the energy associated to the system (1) is given by 2 1 1 E(t) := |∇ ρ(t)|2 + |Λ(t)|2 + f (ρ(t)) + |∇V (t)|2 dx, (7) 2 2 R3 2 p+1 √ 2 where Λ := J/ ρ, and f (ρ) = p+1 ρ 2 . The function f (ρ) denotes the internal energy, which is related to the pressure through the identity P(ρ) = ρ f (ρ) − f (ρ). Therefore our initial data are required to satisfy 2 √ 2 1 1 E 0 := (8) |∇ ρ0 | + |Λ0 |2 + f (ρ0 ) + |∇V0 |2 dx < ∞, 3 2 2 2 R

or equivalently (if we have 1 ≤ p ≤ 5), √ √ ρ0 ∈ H 1 (R3 ) and Λ0 := J0 / ρ0 ∈ L 2 (R3 ).

(9)

Definition 1. We say the pair (ρ, J ) is a weak solution of the Cauchy problem (1), (2) 3 2 3 locally integrable in [0, T ) × √ R with Cauchy√data (ρ2 0 , J0 ) ∈ L 1(R ),3 if there exist 2 ([0, T ); L 2 (R3 )), functions ρ, Λ, such that ρ ∈ L loc ([0, T ); Hloc (R )), Λ ∈ L loc loc √ √ and by defining ρ := ( ρ)2 , J := ρΛ, one has − for any test function η ∈ C0∞ ([0, T ) × R3 ) we have T ρ∂t η + J · ∇ηdxdt + ρ0 η(0)dx = 0; 0

R3

R3

(10)

− for any test function ζ ∈ C0∞ ([0, T ) × R3 ; R3 ), T J · ∂t ζ + Λ ⊗ Λ : ∇ζ + P(ρ) div ζ − ρ∇V · ζ − J · ζ 0

R3

2 √ √ +2 ∇ ρ ⊗ ∇ ρ : ∇ζ − ρ∆ div ζ dxdt + J0 · ζ (0)dx = 0; (11) 4 R3

− (Generalized irrotationality condition) for almost every t ∈ (0, T ), √ ∇ ∧ J = 2∇ ρ ∧ Λ

(12)

holds in the sense of distributions. Remark 2. Suppose we are in the smooth case, so that we can factorize J = ρu, for some current velocity field u, then the last condition (12) simply means ρ∇ ∧ u = 0, the current velocity u is irrotational in ρdx. This is why we will call it generalized irrotationality condition. Definition 3. We say that the weak solution (ρ, J ) to the Cauchy problem (1), (2) is a finite energy weak solution (FEWS) in [0, T ) × R3 , if in addition for almost every t ∈ [0, T ), the energy (7) is finite.


661

In the sequel we restrict our attention only to FEWS with ρ0 ∈ L 1 (R3 ). The hydrodynamic structure of the system (1), (2) should not lead to conclude that the solutions behave like classical fluids. Indeed the connection with Schrödinger equations suggests that ρ0 , J0 should in any case be seen as momenta related to some wave function ψ0 . The main result of this paper is the existence of FEWS by assuming the initial data ρ0 , J0 are momenta of some wave function ψ0 ∈ H 1 (R3 ). Theorem 4 (Main Theorem). Let ψ0 ∈ H 1 (R3 ) and let us define ρ0 := |ψ0 |2 ,

J0 := Im(ψ0 ∇ψ0 ).

Then, for each 0 < T < ∞, there exists a finite energy weak solution to the QHD system (1) in [0, T ) × R3 , with initial data (ρ0 , J0 ) defined as above. Remark 5. As we said above, we can extend the result of the previous main Theorem 4 to a more general possible to√consider nonlinear terms √ type of√collision term. Indeed it is √ of the type f ( ρ, J, ∇ ρ) = −α J + ρ∇g(t, x, ρ, Λ, ∇ ρ), where α ≥ 0 and g satisfies the following Carathéodory type conditions: √ √ √ − for all ( ρ, Λ) ∈ [0, ∞) × R3 , the function (t, x) → g(t, x, ρ, Λ, ∇ ρ) is Lebesgue measurable; − for almost all (t, x) ∈ [0, ∞) × R3 , the function (u, v, w) → g(t, x, u, v, w) is continuous for (u, v, w) ∈ [0, ∞) × R3 × R3 ; − there exists C ∈ L ∞ ([0, ∞) × R3 ) such that |g(t, x, u, v, w)| ≤ C(t, x)(1 + |u|4 + |v|4/3 + |w|4/3 ).

(13)

The case α = 0 requires a slight modification of the method presented in Sect. 4, as we are going to explain below. The case α > 0 can be treated with √ √ minor modifications of the finite difference scheme in Sect. 5 defined for f ( ρ, J, ∇ ρ) = J . See Remark 17 below in Sect. 5. We remark that the condition needed in order to ensure g ∈ L ∞ ([0, ∞); √ (13) is p 3 ∞ 3 ∞ L (R ) + L (R )) as long as ρ ∈ L ([0, T ); H 1 (R3 )), Λ ∈ L ∞ ([0, T ); L 2 (R3 )). Note that with this definition of f we need to change the definition given before for the weak solutions. Indeed the balance law for the current density becomes ∞ J · ∂t ζ + Λ ⊗ Λ : ∇ζ + P(ρ) div ζ − ρ∇V · ζ − J · ζ R3 0 √ + g(t, x, ρ, Λ)(∇ρ · ζ + ρ div ζ ) 2 √ √ J0 · ζ (0)dx = 0. + 2 ∇ ρ ⊗ ∇ ρ : ∇ζ − ρ∆ div ζ dxdt + 4 R3 In Sect. 5 and 6 we will explain how to modify our methods to include this case (see Remarks 17 and 23). At the end of this section we would like to remark that the system (1) is closely related to the Quantum Drift-Diffusion (QDD) equation

2 ∆x ρ˜ − V − h(ρ) ˜ = 0, (14) ∂t ρ˜ + divx ρ∇ ˜ x 2 ρ˜

662


where V is again the electrostatic potential and the enthalpy h(ρ) is such that ρh (ρ) = P (ρ). Indeed if we scale the collision term J in (1) with a relaxation time ε and we write ⎧ ε Jε = 0 ⎪ ⎨ ∂t ρ + div √ ε ε ε 2 ε + ∇ P(ρ ε ) + ρ ε ∇V ε + 1ε J ε = 2 ρ ε ∇ ∆√ρρε , (15) ∂t J + div J ρ⊗J ε ⎪ ⎩ −∆V ε = ρ ε − C(x), then the rescaled quantities (ρ˜ ε (t , x ), J˜ε (t , x )) = (ρ ε (t /ε, x ), 1ε J ε (t /ε, x )) (see [39,45]) in the formal limit ε → 0 satisfy Eq. (14). A partial result on this limit was given by Jüngel, Li and Matsumura in [27], while in [22] the global existence of non-negative variational solutions to (14) is shown without the electrostatic potential and the pressure term. The global existence of (14) without pressure and potential was also proven for periodic solutions in [30]. In a forthcoming paper we deal with the relaxation limit from solutions to (15) to solutions of (14). 2. Preliminaries and Notations 2.1. Notations. For convenience of the reader we will set some notations which will be used in the sequel: If X , Y are two quantities (typically non-negative), we use X Y to denote X ≤ CY , for some absolute constant C > 0. We will use the standard Lebesgue norms for complex-valued measurable functions f : Rd → C,

f L p (Rd ) :=

1 p | f (x)| dx . p

Rd

If we replace C by a Banach space X , we will adopt the notation

f L p (Rd ;X ) :=

1/ p Rd

f (x) X dx

to denote the norm of f : Rd → X . In particular, if X is a Lebesgue space L r (Rn ), and d = 1, we will shorten the notation by writing

f L q L r (I ×Rn ) := t

x

q

I

f (t) L r (Rn ) dt

1/q =

( I

1/q Rn

| f (t, x)|r dx)q/r dt

to denote the mixed Lebesgue norm of f : I → L r (Rn ); moreover, we will write q L t L rx (I × Rn ) := L q (I ; L r (Rn )). For s ∈ R we will define the Sobolev space H s (Rn ) := (1 − ∆)−s/2 L 2 (Rn ). Definition 6 (see [8]). We say that (q, r ) is an admissible pair of exponents if 2 ≤ q ≤ ∞, 2 ≤ r ≤ 6, and 3 1 1 1 = − . (16) q 2 2 r


663

Now we will introduce the Strichartz norms. For more details, we refer the reader to [11,55]. Let I × R3 be a space-time slab, we define the Strichartz norm S˙ 0 (I × R3 ),

u S˙ 0 (I ×R3 ) := sup

N

1/2

PN u 2L q L r (I ×R3 ) t x

,

where the sup is taken over all the admissible pairs (q, r ). Here PN denotes the Paley-Littlewood projection operator, with the sum taken over diadic numbers of the form N = 2 j , j ∈ Z. For any k ≥ 1, we can define

u S˙ k (I ×R3 ) := ∇ k u S˙ 0 (I ×R3 ) . Note that, from the Paley-Littlewood inequality we have

u L qt L r (I ×R3 ) (

x

|PN u| )

2 1/2

L qt L r (I ×R3 )

x

N

N

1/2

PN u 2L q L r (I ×R3 ) t x

,

and hence for each admissible pair of exponents, one has

u L qt L r (I ×Rn ) u S˙ 0 (I ×R3 ) . x

Lemma 1 ([11]).

u L 4 L ∞ (I ×R3 ) u S˙ 1 (I ×R3 ) . t

x

2.2. Schrödinger equations. In this paragraph we recall some important results concerning the nonlinear Schrödinger equations that will be used throughout the paper. First of all, we recall a global well-posedness theorem for the nonlinear Schrödinger-Poisson system (see [8] and the references therein): Theorem 7. Let us consider the following nonlinear Schrödinger-Poisson system: 2 i∂t ψ + 2 ∆ψ = |ψ| p−1 ψ + V ψ (17) −∆V = |ψ|2 , with the initial datum ψ(0) = ψ0 ∈ H 1 (R3 ).

(18)

There exists a unique globally defined strong solution ψ ∈ C(R; H 1 (R3 )), which depends continuously on the initial data. Furthermore, the total energy E(t) :=

2 2 |∇ψ(t, x)|2 + |ψ(t, x)| p+1 + V (t, x)|ψ(t, x)|2 dx 3 p+1 R 2

(19)

is conserved, namely E(t) = E 0 ,

for all t ∈ R.

(20)

664


Moreover, the dispersive nature of the Schrödinger equation provides also some further integrability and regularity properties of the solutions. The first result in this direction regards some space-time integrability properties, the most important of them are the well known Strichartz estimates for the Schrödinger equation in R1+3 . We refer to the classical paper of Ginibre and Velo [23], the paper of Keel and Tao [35], the books of Cazenave [8] and Tao [55] and the references therein. Let ψ be the unique (strong) solution of the Cauchy problem for the free Schrödinger equation i∂t ψ + ∆ψ = 0 ψ(0) = ψ0 , In the following we shall denote by U (·) the free Schrödinger group, defined by the indentity U (t)ψ0 = ψ(t). The next result is taken from [35] and the estimates are obtained in a very general setting. Theorem 8 (Keel, Tao [35]). Let (q, r ), (q, ˜ r˜ ) be two arbitrary admissible pairs of exponents, and let U (·) be the free Schrödinger group. Then we have

U (t) f L qt L r f L 2 (R3 ) ,

(21)

x

s 0, 1/2

u ∈ L 2 ([0, T ]; Hloc (R3 )). We have a similar result also for the nonhomogeneous case: Theorem 10 (Constantin, Saut [9]). Let u be the solution of i∂t u + ∆u = F u(0) = u 0 ∈ L 2 (R3 ),

(25)

where F ∈ L 1 ([0, T ]; L 2 (R3 )). Then it follows 1/2

u ∈ L 2 ([0, T ]; Hloc (R3 )). Moreover, let χ ∈ C0∞ (R1+3 ) be of the form χ (t, x) = χ0 (t)χ1 (x1 )χ2 (x2 )χ3 (x3 ) with χ j ∈ C0∞ (R), supp χ0 ⊂ [0, T ]. Then the following local smoothing estimate holds

1 2

χ (t, x)|(I − ∆) u(t, x)| dxdt R1+3 ≤ C u 0 L 2 (R3 ) + F L 1 L 2 ([0,T ]×R3 ) . 2

1/4

2

t

x

(26)

These results imply that the free Schrödinger group U (·) fulfills the following inequalities

U (·)u 0 L 2 ([0,T ];H 1/2 ) u 0 L 2 (R3 ) ,

(27)

loc

t

0

U (t − s)F(s)ds L 2 ([0,T ];H 1/2 (R3 )) F L 1 ([0,T ];L 2 (R3 )) .

(28)

loc

2.3. Compactness tools. Here we recall some compactness theorems in function spaces, which will be relevant in Sect. 6 to prove the convergence of the approximate solutions. Let us recall in particular a compactness result due to Rakotoson, Temam [49] in the spirit of classical results of Aubin, Lions and Simon, see [3,43,52]. Theorem 11 (Rakotoson, Temam [49]). Let (V, · V ), (H ; · H ) be two separable Hilbert spaces. Assume that V ⊂ H with a compact and dense embedding. Consider a sequence {u ε }, converging weakly to a function u in L 2 ([0, T ]; V ), T < ∞. Then u ε converges strongly to u in L 2 ([0, T ]; H ), if and only if in H for a.e. t; 1. u ε (t) converges to u(t) weakly 2. lim|E|→0,E⊂[0,T ] supε>0 E u ε (t) 2H dt = 0.

666


3. Polar Decomposition In this section √ we will explain how to decompose an arbitrary wave function ψ into its amplitude ρ = |ψ| and its unitary factor φ, namely √ a function taking its values in the unitary circle of the complex plane, such that ψ = ρφ. The idea is similar in spirit to that one used by Y. Brenier in [5], to find the measure preserving maps needed to write vector-valued functions as compositions of gradients of convex functions and measure preserving maps. Our case is much simpler than [5] and it can be studied directly in a simpler setting. Brenier’s idea looks for projections of L 2 functions u onto the set of measure preserving maps S, contained in a given sphere of L 2 , i.e. one has to find s ∈ S, which minimizes the distance u − s L 2 , or equivalently, which maximizes (u, s) L 2 . In our case we should maximize Re(u, s) L 2 , within complex-valued functions with the constraint to take value in the unit ball of L ∞ . Actually in this case the maximizer turns out to be trivially determined and the previous variational argument has only to be considered a motivation to the subsequent considerations. However, in the case we assume u ∈ H 1 (R3 ) (that is the relevant situation in physics), then it would be useful to know the regularity of the maximizers in the light of possible use of the considerations explained in Remark 21. Let us consider a wave function ψ ∈ L 2 (R3 ) and define the set: √ P(ψ) := φ measurable | φ L ∞ (R3 ) ≤ 1, ρφ = ψ a. e. in R3 , (29) √ where ρ = |ψ|. Of course if we consider φ ∈ P(ψ), then by the definition of the √ set P(ψ) it is immediate that |φ| = 1 a.e. ρ − dx in R3 and φ ∈ P(ψ) is uniquely √ determined a.e. ρ − dx in R3 . Remark 12. Let B R be the ball in R3 with radius R, centered at the origin and let us define the set √ PR (ψ) := φ measurable | φ L ∞ (B R ) ≤ 1, ρφ = ψ a. e. in B R . Then it is easy to see that φ ∈ PR (ψ) if and only if φ is a maximizer for the functional Φ R [φ] := Re ψφdx BR

over the set

S R := φ ∈ L 2 (B R ) | φ L ∞ (B R ) ≤ 1 .

The connects the structure of the bilinear term 2 Re(∇ψ ⊗ ∇ψ) with √ next lemma √ 1 ∇ ρ ⊗ ∇ ρ and J ⊗J ρ . Moreover it shows this structure is H -stable. √ Lemma 3. Let ψ ∈ H 1 (R3 ), ρ := |ψ|, then there exists φ ∈ L ∞ (R3 ) such that √ √ √ ψ = ρφ a.e. in R3 , ρ ∈ H 1 (R3 ), ∇ ρ = Re(φ∇ψ). If we set Λ := Im(φ∇ψ), one has Λ ∈ L 2 (R3 ) and moreover the following identity holds √ √ (30) 2 Re(∂ j ψ∂k ψ) = 2 ∂ j ρ∂k ρ + Λ( j) Λ(k) . Furthermore, let ψn → ψ strongly in H 1 (R3 ), then it follows √ √ ∇ ρn → ∇ ρ, Λn → Λ in L 2 (R3 ), where Λn := Im(φn ∇ψn ).

(31)


667

Proof. Let us consider a sequence {ψn } ⊂ C ∞ (R3 ), ψn → ψ in H 1 (R3 ), defined as before: ψn (x) if ψn (x) = 0 (32) φn (x) = |ψn (x)| 0 if ψ(x) = 0. ∗

Then, there exists φ ∈ L ∞ (R3 ) such that φn φ in L ∞ (R3 ) and ∇ψn → ∇ψ in L 2 (R3 ), hence

in L 2 (R3 ). Re φn ∇ψn Re φ∇ψ Since by (32), one has

Re φn ∇ψn = ∇|ψn |

a.e. in R3 , it follows

√ ∇ ρn Re φ∇ψ in L 2 (R3 ). √ √ Moreover, one has ∇ ρn ∇ ρ in L 2 (R3 ), therefore

√ ∇ ρ = Re φ∇ψ ,

where φ is a unitary factor of ψ. The identity (30) follows immediately from the following: 2 Re(∂ j ψ∂k ψ) = 2 Re((φ∂ j ψ)(φ∂k ψ)) = 2 Re(φ∂ j φ)Re(φ∂k ψ) − 2 Im(φ∂ j ψ)Im(φ∂k ψ) √ √ = 2 ∂ j ρ∂k ρ + Λ( j) Λ(k). Now we prove (31). Let us take a sequence ψn → ψ strongly in H 1 (R3 ), and consider

∗ √ ∇ ρn = Re φn ∇ψn , Λn := Im φn ∇ψn . As before, φn φ in L ∞ (R3 ), with φ

√ √ √ a polar factor for ψ; then ∇ ρn ∇ ρ, Re φn ∇ψn Re φ∇ψ , and ∇ ρ =

Re φ∇ψ . Moreover, Λn := Im φn ∇ψn Im φ∇ψ =: Λ. To upgrade the weak convergence into the strong one, simply notice that by (30) one has √ √ 2 ∇ψ 2L 2 = 2 ∇ ρ 2L 2 + Λ 2L 2 ≤ lim inf 2 ∇ ρn 2L 2 + Λn 2L 2 n→∞

= lim inf 2 ∇ψn 2L 2 = 2 ∇ψ 2L 2 . n→∞

Corollary 13. Let ψ ∈ H 1 (R3 ), then

√ ∇ψ ∧ ∇ψ = 2i∇ ρ ∧ Λ.

(33)

Proof. It suffices to note that we can write ∇ψ ∧ ∇ψ = (φ∇ψ) ∧ (φ∇ψ),

√ where φ ∈ L ∞ (R3 ) is the polar factor of ψ such that ∇ ρ = Re(φ∇ψ), Λ := Im(φ∇ψ), as in Lemma 3. By splitting φ∇ψ into its real and imaginary part, we get the identity (33).

668


Now we state a technical lemma which will be used in the next sections. It summarizes the results of this section we use later on and will be handy for the application to the fractional step method. Lemma 4. Let ψ ∈ H 1 (R3 ), and let τ, ε > 0 be two arbitrary (small) real numbers. Then there exists ψ˜ ∈ H 1 (R3 ) such that ρ˜ = ρ, Λ˜ = (1 − τ )Λ + rε ,

√ ˜ ψ), ˜ φ, φ˜ are polar ˜ Λ := Im(φ∇ψ), Λ˜ := Im(φ∇ where ρ := |ψ|, ρ˜ := |ψ|, ˜ respectively, and factors for ψ, ψ,

rε L 2 (R3 ) ≤ ε. Furthermore we have τ ∇ ψ˜ = ∇ψ − i φ Λ + rε,τ ,

(34)

rε,τ L 2 (R3 ) ≤ C(τ ∇ψ L 2 (R3 ) + ε).

(35)

where φ L ∞ (R3 ) ≤ 1 and Proof. Consider a sequence {ψn } ⊂ C ∞ (R3 ) converging to ψ in H 1 (R3 ), and define ψn (x) if ψn (x) = 0 φn (x) := |ψn (x)| 0 if ψn (x) = 0 as polar factors for the wave functions ψn . Since ψn ∈ C ∞ , then φn is piecewise smooth, and Ωn := {x ∈ R3 : |ψn (x)| > 0} is an open set, with smooth boundary. Therefore we can say there exists a function θn : Ωn → [0, 2π ), piecewise smooth in Ωn and φn (x) = eiθn (x) ,

for any x ∈ Ωn . ∗

Moreover, by the previous lemma, we have φn φ in L ∞ (R3 ), where φ is a polar factor of ψ, and Λn := Im(φn ∇ψn ) → Λ := Im(φ∇ψ) in L 2 (R3 ). Thus there exists n ∈ N such that

ψ − ψn H 1 (R3 ) + Λ − Λn L 2 (R3 ) ≤ ε. Now we can define

√ ψ˜ := ei(1−τ )θn ρn .

(36)

˜ Let us remark that √ (36) is a good definition for ψ, since outside Ωn , where θn is not defined, we have ρn = 0, and hence also ψn = 0. Furthermore, τ ∇ ψ˜ = e−iτ θn ∇ψn − i ei(1−τ )θn Λn ⎛ ⎞ ∞ j τ j−1 (−iθ ) τ i(1−τ )θn n ⎠ ∇ψn = ∇ψ − i e Λ+τ⎝ j! j=1

τ +(∇ψn − ∇ψ) − i ei(1−τ )θn (Λn − Λ),


669

and thus we obtain (34) and (35), with rε,τ given by ⎛ ⎞ ∞ j j−1 (−iθn ) τ ⎠ ∇ψn + (∇ψn − ∇ψ) − i τ ei(1−τ )θn (Λn − Λ). rε,τ = τ ⎝ j! j=1

Moreover ˜ = (1 − τ )Λn = (1 − τ )Λ + (1 − τ )(Λn − Λ), Λ˜ = Im(e−i(1−τ )θn ∇ ψ) and clearly rε := (1 − τ )(Λn − Λ) has L 2 -norm less than ε by assumption.

(37)

4. QHD Without Collisions Now let us summarize some key points regarding the existence of weak solutions to the Quantum Hydrodynamic system, in the collisionless case. The balance equations can be written in the following way: ⎧ J = 0 ⎪ ⎨ ∂t ρ + div √ ∆ ρ 2 √ + ∇ P(ρ) + ρ∇V = (38) ρ∇ ∂t J + div J ⊗J ρ 2 ρ ⎪ ⎩ −∆V = ρ, where P(ρ) =

p−1 ( p+1)/2 , p+1 ρ

1 ≤ p < 5.

Definition 14. We say the pair (ρ, J ) is a weak solution in [0, T ) × R3 to the system 1 (R3 ) if and only if there exist locally integrable (38) with Cauchy data (ρ0 , J0 ) ∈ L loc √ √ 2 1 (R3 )), Λ ∈ L 2 ([0, T ); L 2 (R3 )) functions ρ, Λ such that ρ ∈ L loc ([0, T ); Hloc loc loc √ √ 2 and if we define ρ := ( ρ) , J := ρΛ, then − for any test function ϕ ∈ C0∞ ([0, T ) × R3 ) we have

T 0

R3

ρ∂t ϕ + J · ∇ϕdxdt +

R3

ρ0 ϕ(0)dx = 0

(39)

and for any test function η ∈ C0∞ ([0, T ) × R3 ; R3 ) we have

T 0

R3

J · ∂t η + Λ ⊗ Λ : ∇η + P(ρ) div η − ρ∇V · η

2 √ √ +2 ∇ ρ ⊗ ∇ ρ : ∇η − ρ∆ div ηdxdt + J0 · η(0)dx = 0; (40) 4 R3

− (Generalized irrotationality condition) for almost every t ∈ (0, T ) √ ∇ ∧ J = 2∇ ρ ∧ Λ holds in the sense of distributions. We say that (ρ, J ) is a finite energy weak solution if and only if it is a weak solution and the energy E(t) is finite a.e. in [0, T ).

670


The next existence result roughly speaking shows how to get a weak solution to the system (38) out of a strong solution to the Schrödinger-Poisson system 2 i∂t ψ + 2 ∆ψ = |ψ| p−1 ψ + V ψ . (41) −∆V = |ψ|2

The quadratic nonlinearities in (38) are originated by a term of the form Re ∇ψ ⊗ ∇ψ since formally

J⊗J √ √ 2 Re ∇ψ ⊗ ∇ψ = 2 ∇ ρ ⊗ ∇ ρ + . ρ However this identity can be justified in the nodal region {ρ = 0} only by means of the polar factorization discussed in the previous section. Indeed, we stress that in the previous identity the right hand side is written in terms of ρ and J which do exist in the whole space R3 , via the Madelung transform. However, interpreted as Λ⊗Λ, where Λ is the Radon-Nikodym derivative the term J ⊗J ρ should be√ of J dx with respect to ρdx. Unfortunately the Madelung transformations are unable to define Λ on the whole R3 , hence one needs to use the polar factorization discussed in the previous section to define Λ in the whole R3 . Furthermore, the study of the existence of weak solutions of (38) is done with Cauchy data of the form (ρ0 , J0 ) = (|ψ0 |2 , Im(ψ0 ∇ψ0 )), for some ψ0 ∈ H 1 (R3 ; C). These special initial data yield to consider the Cauchy problem for the QHD system only for solutions compatible with a wave mechanics point of view. Proposition 15. Let 0 < T < ∞, let ψ0 ∈ H 1 (R3 ) and define the initial data for (38), (ρ0 , J0 ) := (|ψ0 |2 , Im(ψ0 ∇ψ0 )). Then there exists a finite energy weak solution (ρ, J ) to the Cauchy problem (38) in the space-time slab [0, T ) × R3 . Furthermore the energy E(t) defined in (7) is conserved for all times t ∈ [0, T ). The idea behind the proof of this proposition is the following. Let us consider the Schrödinger-Poisson system (41) with initial datum ψ(0) = ψ0 . It is well known that it is globally well-posed for initial data in H 1 (R3 ) (see [8]), and the solution satisfies ψ ∈ C 0 (R; H 1 (R3 )). Then it makes sense to define for each time t ∈ [0, T ) the quantities ρ(t) := |ψ(t)|2 , J (t) := Im(ψ(t)∇ψ(t)) and we can see that (ρ, J ) is a finite energy weak solution of (38): indeed for the solution ψ of (41) we have i i ∂t ∇ψ = ∆∇ψ − ∇ (|ψ| p−1 + V )ψ 2 in the sense of distributions. Thus formally we get the following identities for (ρ, J ):

∂t ρ = −Im(ψ∆ψ) = − div Im(ψ∇ψ) , i i p−1 ∂t J = Im ∇ψ − ∆ψ + (|ψ| + V )ψ 2 i i i p−1 p−1 ∆∇ψ − ∇(|ψ| + V )ψ − (|ψ| + V )∇ψ + Im ψ 2

p−1 2 = ∇∆|ψ|2 − 2 div Re(∇ψ ⊗ ∇ψ) − ∇(|ψ| p+1 ) − |ψ|2 ∇V. 4 p+1


671

Thanks to Lemma 3 we can write

√ √ 2 Re ∇ψ ⊗ ∇ψ = 2 ∇ ρ ⊗ ∇ ρ + Λ ⊗ Λ √ with ρ, Λ as in statement of Lemma 3, and clearly they also satisfy √ J = ρΛ. Hence formally the following identity holds: ∂t J + div(Λ ⊗ Λ) + ∇ P(ρ) + ρ∇V =

2 √ √ ∆∇ρ − 2 div(∇ ρ ⊗ ∇ ρ). 4

(42)

Of course these calculations are just formal, since ψ doesn’t have the necessary regularity to implement them. Furthermore, it is well known that the energy 2 1 2 1 2 p+1 E(t) := |∇ψ| + |ψ| dx + |ψ(t, x)|2 dxdy |ψ(t, y)|2 3 3 3 2 p + 1 2 |x − y| R R x ×R y (43) is conserved for the Schrödinger-Poisson system (41). Thus, it only remains to note that by Lemma 3 the energy in (43) is equal to that in (7). Proof. Let us consider the Schrödinger-Poisson system (41). From standard theory about nonlinear Schrödinger equations it is well known that (41) is globally well-posed for initial data in the space of energy: ψ(0) = ψ0 ∈ H 1 (R3 ). Now, let us take a sequence of mollifiers {χε } converging to the Dirac mass and define ψ ε := χε ψ, where ψ ∈ C(R; H 1 (R3 )) is the solution to (41). Then ψ ε ∈ C ∞ (R1+3 ) and moreover i∂t ψ ε +

2 ∆ψ ε = χε |ψ| p−1 ψ + V ψ , 2

(44)

where V is the classical Hartree potential. Therefore differentiating |ψ ε |2 with respect to time we get i i ∂t |ψ ε |2 = 2Re(ψ ε ∂t ψ ε ) = 2Re ψ ε ( ∆ψ ε − χε (|ψ| p−1 ψ + V ψ)) 2

2 = − div Im(ψ ε ∇ψ ε ) + Im ψ ε χε (|ψ| p−1 ψ + V ψ) . If we differentiate with respect to time Im(ψ ε ∇ψ ε ) we get

i i i ∂t Im(ψ ε ∇ψ ε ) = Im − ∆ψ ε + χε (|ψ| p−1 ψ) + χε (V ψ) 2 i i i ε p−1 ε ∆∇ψ − χε ∇(|ψ| ψ) − χε ∇(V ψ) + Im ψ 2 2

= Re −∇ψ ε ∆ψ ε +ψ ε ∆∇ψ ε + Re χε (|ψ| p−1 ψ)∇ψ ε − ψ ε χε ∇(|ψ| p−1 ψ) 2

+ Re χ (V ψ)∇ψ ε − χε ∇(V ψ)ψ ε =: A + B + C.

672


Let us discuss these three terms separately. For the first term A it is immediate that

2 2

Re −∇ψ ε ∆ψ ε + ψ ε ∆∇ψ ε = 2 div Re(∇ψ ε ⊗ ∇ψ ε ) + ∆∇|ψ ε |2 . 2 4 Let us discuss the second and third terms, B and C. We can recast B in the following way: χε (|ψ| p−1 ψ)∇ψ ε − ψ ε χε ∇(|ψ| p−1 ψ) = |ψ ε | p−1 ψ ε ∇ψ ε − ψ ε χε ψ∇|ψ| p−1 + |ψ| p−1 ∇ψ + R1ε = −ψ ε χε ψ∇|ψ| p−1 + R1ε + R2ε = −|ψ ε |2 ∇|ψ ε | p−1 + R1ε + R2ε + R3ε , where

R1ε : = χε (|ψ| p−1 ψ) − |ψ ε | p−1 ψ ε ∇ψ ε , R2ε : = ψ ε |ψ ε | p−1 ∇ψ ε − χε (|ψ| p−1 ∇ψ) , R3ε : = ψ ε ψ ε ∇|ψ ε | p−1 − χε (ψ∇|ψ| p−1 ) .

Now it only remains to analyze how the remainder terms R εj go to zero. We will prove it in the next lemma. Furthermore, the following identity holds: |ψ ε |2 ∇|ψ ε | p−1 =

p−1 ∇|ψ ε | p+1 . p+1

For the third term C after similar computations we get C = |ψ ε |∇(χε V ) + R4ε , where the remainder term R4ε will be analyzed in the next lemma. Hence we can conclude that Im(ψ ε ∇ψ ε ) satisfies

2 ∂t Im(ψ ε ∇ψ ε ) = −2 div Re(∇ψ ε ⊗ ∇ψ ε ) + ∆∇|ψ ε |2 4 p−1 ∇|ψ ε | p+1 + |ψ ε |2 ∇ (χε V ) + R ε , + p+1 where R ε := R1ε + R2ε + R3ε + R4ε . Now, as ε → 0, we get

2 ∂t Im(ψ∇ψ) = −2 div Re(∇ψ ⊗ ∇ψ) + ∆∇|ψ|2 4 p−1 p+1 2 ∇|ψ| + + |ψ| ∇V. p+1 This identity is equivalent to (42) since by Lemma 3 we have √ √ 2 Re (∇ψ ⊗ ∇ψ) = 2 ∇ ρ ⊗ ∇ ρ + Λ ⊗ Λ, √ where Λ is defined as in Lemma 3 and ρΛ = J , where J := Im(ψ∇ψ). Finally, let us note that, by the definition of J , we have ∇ ∧ J = Im(∇ψ ∧ ∇ψ), √ then by Corollary 13, we get ∇ ∧ J = 2∇ ρ ∧ Λ.


673

Lemma 5. Let 0 < T < ∞, then

R εj L 1

t,x ([0,T ]×R

3)

→ 0,

as ε → 0,

j = 1, 2, 3, 4. Proof. It is a direct consequence of Strichartz estimates for the solution of the Schrödinger-Poisson system (41),

ψ L qt W 1,r ([0,T ]×R3 ) ≤ C(E 0 , ψ0 L 2 (R3 ) , T ),

(45)

where (q, r ) is a pair of admissible exponents and E 0 is the initial energy for (41). The first error can be controlled in the following way:

R1ε L 1

t,x ([0,T ]×R

3)

≤ χε (|ψ| p−1 ψ) − |ψ ε | p−1 ψ ε

4( p+1) p+7

Lt

p+1 p

Lx

([0,T ]×R3 )

p+7 ≤ T 4( p+1) χε (|ψ| p−1 ψ) − |ψ| p−1 ψ L ∞ L p+1 x t p−1 ε p−1 ε ε + |ψ| ψ − |ψ | ψ L ∞ L p+1 ∇ψ 4( p+1) x

t

3( p−1)

Lt

∇ψ ε

p+1

4( p+1) 3( p−1)

Lt

p+1

Lx

([0,T ]×R3 )

.

Lx

The remaining error terms can be computed in the same way. We remark that by using 4( p+1)

p+1

the Strichartz estimates it follows ψ∇|ψ| p−1 lies in L t3( p−1) L x ([0, T ] × R3 ). 5. The Fractional Step: Definitions and Consistency In this section we make use of the results of the previous sections to construct a sequence of approximate solutions of the QHD system ⎧ ⎪ ⎨ ∂t ρ + divJ = 0 √ ∆ ρ 2 √ + ∇ P(ρ) + ρ∇V + J = (46) ρ∇ ∂ J + div J ⊗J ρ 2 ρ ⎪ ⎩ −∆V = ρ with Cauchy data (ρ(0), J (0)) = (ρ0 , J0 ).

(47)

Definition 16. Let 0 < T < ∞. We say {(ρ τ , J τ )} is a sequence of approximate solutions 1 (R3 ), if there exist to the system (1) in [0, T ) × R3 , with initial data (ρ0 , J0 ) ∈ L loc √ τ √ τ τ 2 1 (R3 )), Λτ ∈L 2 locally integrable functions ρ , Λ , such that ρ ∈ L loc ([0, T ); Hloc loc √ τ 2 τ √ τ τ 2 3 τ ([0, T ); L loc (R )), and if we define ρ := ( ρ ) , J := ρ Λ , then − for any test function η ∈ C0∞ ([0, T ) × R3 ) one has

T 0

as τ → 0,

R3

τ

τ

ρ ∂t η + J · ∇ηdxdt +

R3

ρ0 η(0)dx = o(1)

(48)

674


− for any test function ζ ∈ C0∞ ([0, T ) × R3 ; R3 ) we have T J τ · ∂t ζ + Λτ ⊗ Λτ : ∇ζ + P(ρ τ ) div ζ − ρ τ ∇V τ · ζ − J τ · ζ 0

R3

2 +2 ∇ ρ τ ⊗ ∇ ρ τ : ∇ζ − ρ τ ∆ div ζ dxdt + J0 · ζ (0)dx = o(1) (49) 4 R3 as τ → 0; − (Generalized irrotationality condition) for almost every t ∈ (0, T ) we have

∇ ∧ J τ = 2∇ ρ τ ∧ Λτ . Our fractional step method is based on the following simple idea. We split the evolution of our problem into two separate steps. Let us fix a (small) parameter τ > 0, then in the former step we solve a non-collisional QHD problem, while in the latter one we solve the collisional problem without QHD, and at this point we can start again with the non-collisional QHD problem, the main difficulty being the updating of the initial data at each time step. Indeed, as remarked in the previous section we are able to solve the non-collisional QHD only in the case of Cauchy data compatible with the Schrödinger picture. This restriction imposed to reconstruct a wave function at each time step. Now we explain how to set up the fractional step procedure which generates the approximate solutions. We first remark that, as in the previous section, this method requires a special type of initial data (ρ0 , J0 ), namely we assume there exists ψ0 ∈H 1 (R3 ), such that the hydrodynamic initial data are given via the Madelung transforms ρ0 = |ψ0 |2 ,

J0 = Im(ψ0 ∇ψ0 ).

This assumption is physically relevant since it implies the compatibility of our solutions to the QHD problem with the wave mechanics approach. The iteration procedure can be defined in the following way. First of all, we take τ > 0, which will be the time mesh unit; therefore we define the approximate solutions in each time interval [kτ, (k + 1)τ ), for any integer k ≥ 0. At the first step, k = 0, we solve the Cauchy problem for the Schrödinger-Poisson system ⎧ 2 ⎨ i∂t ψ τ + 2 ∆ψ τ = |ψ τ | p−1 ψ τ + V τ ψ τ τ = |ψ τ |2 (50) ⎩ −∆V τ ψ (0) = ψ0 by looking for the restriction of the unique strong solution ψ ∈ C(R; H 1 (R3 )) (see [8]) in [0, τ ). Let us define ρ τ := |ψ τ |2 , J τ := Im(ψ τ ∇ψ τ ). Then, as shown in the previous section, (ρ τ , J τ ) is a weak solution to the non-collisional QHD system. Let us assume that we know ψ τ in the space-time slab [(k − 1)τ, kτ ) × R3 , we want to set up a recursive method, hence we have to show how to define ψ τ , ρ τ , J τ in the strip [kτ, (k + 1)τ ). In order to take into account the presence of the collisional term f = −J we update ψ in t = kτ , namely we define ψ τ (kτ +). The construction of ψ τ (kτ +) will be done by means of the polar decomposition described in Sect. 3. Let us apply Lemma 4, with ψ = ψ τ (kτ −), ε = τ 2−k ψ0 H 1 (R3 ) , then we can define ˜ ψ τ (kτ +) = ψ,

(51)


675

by using the wave function ψ˜ defined in Lemma 4. Therefore we have ρ τ (kτ +) = ρ τ (kτ −), Λτ (kτ +) = (1 − τ )Λτ (kτ −) + Rk ,

(52) (53)

where Rk L 2 (R3 ) ≤ τ 2−k ψ0 H 1 (R3 ) and τ ∇ψ τ (kτ +) = ∇ψ τ (kτ −) − i φ ∗ Λτ (kτ −) + rk,τ ,

(54)

with φ ∗ L ∞ ≤ 1 and 1

rk,τ L 2 ≤ C(τ ∇ψ τ (kτ −) + τ 2−k ψ0 H 1 (R3 ) ) τ E 02 . We then solve the Schrödinger-Poisson system with initial data ψ(0) = ψ τ (kτ +). We define ψ τ in the time strip [kτ, (k + 1)τ ) as the restriction of the Schrödinger-Poisson solution just found in [0, τ ), furthermore, we define ρ τ := |ψ τ |2 , J τ := Im(ψ τ ∇ψ τ ) as the solution of the non-collisional QHD (38). With this procedure we can go on every time strip and then construct an approximate solution (ρ τ , J τ , V τ ) of the QHD system. √ √ Remark 17. To √ cover the √ general case where f ( ρ, J, ∇ ρ) = −α J + ρ∇g(t, x, ρ, Λ, ∇ ρ), α ≥ 0, g satisfies the conditions in Remark 5 and one has a non-zero doping profile, we solve the following Cauchy problem: ⎧ 2 ⎨ i∂t ψ + 2 ∆ψ = |ψ| p−1 ψ + V ψ + gψ 2 . ⎩ −∆V = |ψ| − C(x) ψ(0) = ψ0

(55)

The global well-posedness follows from the result in [8]. Theorem 18 (Consistency of the approximate solutions). Let us consider a sequence of approximate solutions {(ρ τk , J τk√ )}k≥0 constructed via the fractional step method, and 2 ([0, T ); H 1 (R3 )) and Λ ∈ L 2 ([0, T ); assume there exists 0 < T < ∞, ρ ∈ L loc loc loc 2 (R3 )), such that L loc

ρ τk → τk

Λ

√

→Λ

ρ

1 in L 2 ([0, T ); Hloc (R3 )),

in L

2

2 ([0, T ); L loc (R3 )).

Then the limit function (ρ, J ), where as before J = QHD system, with Cauchy data (ρ0 , J0 ).

√

(56) (57)

ρΛ, is a weak solution of the

Proof. Along this proof we omit the index k. Let us plug the approximate solutions (ρ τ , J τ ) in the weak formulation and let ζ ∈ C0∞ ([0, T ) × R3 ), then we have

676


∞

R3

0

J τ · ∂t ζ + Λτ ⊗ Λτ : ∇ζ + P(ρ τ ) div ζ − ρ τ ∇V τ · ζ − J τ · ζ

+ ∇ 2

=

∞

ρτ

⊗∇

(k+1)τ

R3

k=0 kτ

+ ∇ 2

=

∞

ρτ

(k+1)τ

ρτ

2 τ : ∇ζ − ρ ∆ div ζ dxdt + J0 · ζ (0)dx 4 R3

J τ · ∂t ζ + Λτ ⊗ Λτ : ∇ζ + P(ρ τ ) div ζ − ρ τ ∇V τ · ζ − J τ · ζ

⊗∇

2 τ : ∇ζ − ρ ∆ div ζ dxdt + J0 · ζ (0)dx 4 R3 τ −J · ζ dxdt + J τ ((k + 1)τ −) · ζ ((k + 1)τ )

ρτ

R3

kτ

k=0

− J τ (kτ ) · ζ (kτ )dx + =

∞

(k+1)τ

R3

k=0 kτ

=

∞

(k+1)τ

R3

k=0 kτ

+

=

∞

3 k=1 R

∞

+ (1 − τ )

R3

J0 · ζ (0)dx

−J τ · ζ dxdt + −J τ · ζ dxdt +

∞

3 k=1 R

∞ τ k=1

R3

J τ (kτ −) − J τ (kτ ) · ζ (kτ )dx J τ (kτ −) · ζ (kτ )dx

(1 − τ ) ρ τ (kτ )Rk · ζ (kτ )dx

(k+1)τ

k=0 kτ

R3

R3

∞ k=0

R3

J τ ((k + 1)τ −) · ζ ((k + 1)τ ) − J τ (t) · ζ (t)dxdt

ρ τ (kτ )Rk · ζ (kτ )dx

= o(1) + O(τ ) as τ → 0. 6. A Priori Estimates and Convergence In this section we obtain various a priori estimates necessary to show the compactness of the sequence of approximate solutions (ρ τ , J τ ) in some appropriate function√spaces. As we stated in Theorem 18, we wish to prove the strong convergence of { ρ τ } in 2 ([0, T ); H 1 (R3 )) and that one of {Λτ } in L 2 ([0, T ); L 2 (R3 )). To achieve this L loc loc loc loc goal we use a compactness result in the class of the Aubin-Lions’s type lemma, due to Rakotoson-Temam [49] (see Sect. 2). The plan of this section is the following, first of all we get a discrete version of the (dissipative) energy inequality for the system (1), later we use the Strichartz estimates for ∇ψ τ by means of the formula (61) below. Consequently via the Strichartz estimates and by using the local smoothing results of Theorems 9, 10, we deduce some further regularity properties of the sequence {∇ψ τ }.


677

This fact will be stated in Proposition 25. In this way it is possible to get the regularity τ } which are needed to apply Theorem 11 and hence to properties of the sequence √{∇ψ τ get the convergence for { ρ and {Λτ }}. Let us begin with the energy inequality. First of all note that if we have a sufficiently regular solution of the QHD system, one has t E(t) = − |Λ|2 dxdt + E 0 , (58) 0

R3

where the energy is defined as in (7). Now we would like to find a discrete version of the energy dissipation for the approximate solutions. Proposition 19 (Discrete Energy Inequality). Let (ρ τ , J τ ) be an approximate solution of the QHD system, with 0 < τ < 1. Then, for t ∈ [N τ, (N + 1)τ ) we have E τ (t) ≤ −

N τ

Λ(kτ −) L 2 (R3 ) + (1 + τ )E 0 . 2

(59)

k=1

Proof. For all k ≥ 1, we have 1 τ 1 |Λ (kτ +)|2 − |Λτ (kτ −)|2 dx E τ (kτ +) − E τ (kτ −) = 2 2 1 = (−2τ + τ 2 )|Λτ (kτ −)|2 2 + 2(1 − τ )Λτ (kτ −) · Rk + |Rk |2 dx 1 ≤ (−2τ + τ 2 )|Λτ (kτ −)|2 dx + (1 − τ )α|Λτ (kτ −)|2 2 1−τ |Rk |2 + |Rk |2 dx + α 1 = (−2τ + τ 2 + α − ατ )|Λτ (kτ −)|2 2 1−τ +α |Rk |2 dx. + α Here Rk denotes the error term as in expression (53). If we choose α = τ , it follows τ 1

Rk 2L 2 E τ (kτ +) − E τ (kτ −) ≤ − Λτ (kτ −) 2L 2 + 2 2τ τ ≤ − Λτ (kτ −) 2L 2 + τ 2−k−1 ψ0 H 1 . 2

(60)

The inequality (59) follows by summing up all the terms in (60) and by the energy conservation in each time strip [kτ, (k + 1)τ ). Unfortunately the energy estimates are not sufficient to get enough compactness to show the convergence of the sequence of the approximate solutions. Indeed from the discrete energy inequality, we get only the weak convergence of ∇ψ τ in L ∞ ([0, ∞); L 2 (R3 )), and therefore the quadratic terms in (49) could exhibit some concentration phenomena in the limit.

678


More precisely, from the energy inequality we get the sequence {ψ τ } is uniformly bounded in L ∞ ([0, ∞); H 1 (R3 )), hence there exists ψ ∈ L ∞ ([0, ∞); H 1 (R3 )), such that, up to subsequences ψτ ψ

in L ∞ ([0, ∞); H 1 (R3 )).

Therefore we get

ρτ

√ ρ

in L ∞ ([0, ∞); H 1 (R3 )),

in L ∞ ([0, ∞); L 2 (R3 )), Λτ Λ √ 1 J τ ρΛ in L ∞ ([0, ∞); L loc (R3 )). The need to pass into the limit the quadratic expressions leads us to look for a priori estimates in stronger norms. Relationships with the Schrödinger equation brings naturally into this search the Strichartz-type estimates. The following results are concerned with these estimates. However they are not an immediate consequence of the Strichartz estimates for the Schrödinger equation since we have to take into account the effects of the updating procedure which we implement at each time step. Remark 21 will clarify in detail why we stated Lemma 4 in place of using the exact factorization property provided by Lemma 3. Lemma 6. Let ψ τ be the wave function defined by the fractional step method (see Sect. 5), and let t ∈ [N τ, (N + 1)τ ). Then we have ∇ψ τ (t) = U (t)∇ψ0 − i −i

τ U (t − kτ ) φkτ Λτ (kτ −) N

k=1

t

U (t − s)F(s)ds +

0

N

U (t − kτ )rkτ ,

(61)

k=1

where as before U (t) is the free Schrödinger group,

φkτ L ∞ (R3 ) ≤ 1, rkτ L 2 (R3 ) ≤ τ ψ0 H 1 (R3 ) ,

(62)

and F = ∇(|ψ τ | p−1 ψ τ + V τ ψ τ ). √ √ Remark 20. Clearly, with a general collision term f ( ρ, J, ∇ ρ) (see Remark 5) and a non-zero doping profile C(x) one has F defined by F = ∇(|ψ τ | p−1 ψ τ +V τ ψ τ +g τ ψ τ ), where now V τ solves −∆V τ = |ψ τ |2 − C(x) √ √ and g τ = g(t, x, ρ τ , Λτ , ∇ ρ τ ). Remark 21. Now we can see why the updating step in (51) has been defined through Lemma 4. Indeed, one could √ possibly try to √ use the exact unitary factor as in Lemma 3, namely ψ τ (kτ +) := φ (1−τ ) ρ, where φ, ρ are respectively the√unitary factor and the√amplitude of ψ τ (kτ −) (note that φ is uniquely determined ρdx, and |φ| = 1, ρdx-a.e.). The meaning of this definition is clear if we approximate ψ τ (kτ −) √ (as in the proof of Lemma 4) with smooth ψn = eiθn ρn , and then we update ψn with


679

√ ψ˜ n := ei(1−τ )θn ρn . Hence ψ(kτ +) := limn→∞ ψ˜ n in H 1 (R3 ). Moreover in this case we get the exact updating formulas |ψ τ (kτ +)|2 = |ψ τ (kτ )|2 , Im(ψ τ ∇ψ τ )(kτ +) = (1 − τ )Im(ψ τ ∇ψ τ )(kτ −), without the small error rε in the expression for the density current; however instead of formula (61) we would find ∇ψ τ (t) = U (t − N τ )σ Nτ τ U (τ ) . . . σττ U (τ )∇ψ0 τ τ − i U (t − N τ )σ Nτ τ Λτ (N τ −) + . . . − i U (t − N τ ) . . . U (τ )φττ Λτ (τ −) t Nτ −i U (t − s)F(s)ds − iU (t − N τ )σ Nτ τ U (N τ − s)F(s)ds Nτ (N −1)τ τ U (τ − s)F(s)ds, + . . . − iU (t − N τ )σ Nτ τ U (τ ) . . . σττ 0

τ where σkτ

τ )−t , φ τ (φkτ kτ

= being the polar factor of ψ τ (kτ −). Now, to recover an expression similar to (61), we have to calculate the commutators between the free Schrödinger τ . The estimates of these comevolution operator and the multiplication operators by σkτ τ mutators are not known for non-smooth σkτ . Proof. Since ψ τ is solution of the Schrödinger-Poisson system in the space-time slab [N τ, (N + 1)τ ) × R3 , then we can write t U (t − s)F(s)ds, (63) ∇ψ τ (t) = U (t − N τ )∇ψ τ (N τ +) − i Nτ

where F is defined in the statement of Lemma 6. Now there exists a piecewise smooth function θ N , as specified in the proof of Lemma 4, such that √ ψ(N τ +) = ei(1−τ )θ N ρn , and furthermore all the estimates of Lemma 4 hold, with ψ = ψ τ (N τ −), ψ˜ = ψ τ (N τ +) and ε = 2−N τ ψ0 H 1 (R3 ) . Therefore, we have τ ∇ψ τ (N τ +) = ∇ψ τ (N τ −) − i ei(1−τ )θ N Λτ (N τ −) + r Nτ ,

(64)

where r Nτ L 2 ≤ τ ψ0 H 1 . By plugging (64) into (63) we deduce τ ∇ψ τ (t) = U (t − N τ )∇ψ τ (N τ −) − i U (t − N τ )(ei(1−τ )θ N Λτ (N τ −)) t + U (t − N τ )r Nτ − i U (t − s)F(s)ds. Nτ

Let us iterate this formula, repeating the same procedure for ∇ψ τ (N τ −), then (61) holds. At this point we can use formula (61) to obtain Strichartz estimates for ∇ψ τ , simply by applying Theorem 8 to each of the terms in (61). After some computations, the following result holds.

680


Proposition 22 (Strichartz estimates for ∇ψ τ ). Let, 0 < T < ∞, and let ψ τ be as in the previous section, then one has 1

∇ψ τ L qt L r ([0,T ]×R3 ) ≤ C(E 02 , ρ0 L 1 (R3 ) , T ) x

(65)

for each admissible pair of exponents (q, r ). Remark 23. As we showed in Remark 20, in the general case the non-homogeneous term F is slightly different. Anyway by the Strichartz estimates the same result of Proposition 22 holds even in the case of the modified non-homogeneous term F. Proof. First of all, let us prove Proposition 22, for a small time 0 < T1 ≤ T and let (q, r ) be an admissible pair of exponents. We choose T1 > 0 later. Let N be the smallest positive integer such that T1 ≤ N τ . By applying Theorem 8 to formula (61), we get ∇ψ τ q r ≤ U (t)∇ψ0 L qt L r ([0,T1 ]×R3 ) L t L x ([0,T1 ]×R3 ) x N τ + U (t − kτ ) ei(1−τ )θk Λτ (kτ −) q r L t L x ([0,T1 ]×R3 ) k=1 +

N U (t − kτ )r τ q r k L L ([0,T t

k=1

t + U (t − s)F(s)ds 0

=: A + B + C + D.

1 ]×R

x

3

)

L t L rx ([0,T1 ]×R3 ) q

Now we estimate term by term the above expression. The estimate of A is straightforward, since

U (t)∇ψ0 L q L r ([0,T1 ]×R3 ) ∇ψ0 L 2 (R3 ) . t

x

The estimate of B follows from N τ U (t − kτ ) ei(1−τ )θk Λτ (kτ −) q r L t L x ([0,T1 ]×R3 ) k=1

τ

N

1

Λτ (kτ −) L 2 (R3 ) T1 E 02 .

(66)

k=1

The term C can be estimated in a similar way, namely N

U (t − kτ )rkτ L qt L r

N

x

k=1

rkτ L 2 (R3 ) T1 ψ0 H 1 (R3 ) .

(67)

k=1

The last term is a little bit trickier to estimate. First of all we decompose F into three terms, F = F1 + F2 + F3 , where F1 = ∇(|ψ τ | p−1 ψ τ ), F2 = ∇V τ ψ τ and F3 = V τ ∇ψ τ . By the Strichartz estimates (Theorem 8), we have t U (t − s)F(s)ds q r 0 L t L x ([0,T1 ]×R3 ) + F2 q2 r2 + F3 q3 r3 , (68) F1 q1 r1 L t L x ([0,T1 ]×R3 ) L t L x ([0,T1 ]×R3 ) L t L x ([0,T1 ]×R3 )


681

where (qi , ri ) are pairs of admissible exponents. Let us start with the first term. Lemma 7. There exists α > 0, depending on p, such that

|ψ τ | p−1 ∇ψ τ

q˜

L t L rx˜ ([0,T1 ]×R3 )

T1α ψ τ S˙ 1 ([0,T1 ]×R3 ) .

(69)

Proof. First of all let us apply the Hölder inequality on the left-hand side of (69). Then we have

|ψ τ | p−1 ∇ψ τ = Now we want

q˜

L t L rx˜ ([0,T1 ]×R3 ) α τ p−1 q r T1 |ψ |

L 1 L x1 ([0,T1 ]×R3 ) ∇ψ L q2 L rx2 ([0,T1 ]×R3 ) t t α τ p−1 T1 ψ q1 ( p−1) r1 ( p−1)

∇ψ τ L q2 L rx2 ([0,T1 ]×R3 ) . t Lt Lx ([0,T1 ]×R3 )

1 q1 ( p−1)

=

3 2

1 6

−

1 r1 ( p−1)

and

=

1 q2

3 2

1 2

−

1 r2

(70)

, in such a way that

f q1 ( p−1) r1 ( p−1) , ∇ f L q2 L rx2 ≤ f S˙ 1 = ∇ f S˙ 0 . We already know q˜1 = 1 + t

Lt 1 Lx 3 1 ˜ r˜ ), (q j , r j ), it follows 2 2 − r˜ , then putting together the conditions on (q, 1 1 1 1 1 3 1 1 = + = + + ( p − 1) − q˜ α q1 q2 α 2 6 r1 ( p − 1) 3 1 1 1 3 1 =1+ − − , + 2 2 r2 2 2 r˜ and when 1 ≤ p < 5, α=

5− p > 0. 4

This means that we can always choose pairs (q˜ , r˜ ), (q1 , r1 ), (q2 , r2 ) satisfying the previous conditions so that inequality (70) holds, with α > 0. 1 1 For instance, if 1 ≤ p ≤ 3, we can choose r11 = p−1 6 , r2 = 2 , therefore q1 = q2 = ∞, hence we have 1 r1

=

p−1 1 6 , r2

=

1 6,

=

1 r˜

2+ p 1 6 , q˜

=

5− p 4 .

In the case 3 ≤ p < 5, we take

then q1 = ∞, q2 = 2, hence we have

p 1 6 , q˜

=

1 r˜

7− p 4 .

=

Now let us consider the second term: V τ ∇ψ τ , here we choose (q2 , r2 ) = (1, 2), so by the Hölder and the Hardy-Littlewood-Sobolev (see [54]) inequalities one has 1

τ

V τ ∇ψ τ L 1 L 2 ([0,T1 ]×R3 ) ≤ T12 V τ L ∞ 2 ∇ψ L 2 L 6 t Lx t

t

x

x

1 2

T1 ψ τ 2L ∞ L 2 ∇ψ τ L 2 L 6 . t x t x

2 2 For the third term, we choose (q3 , r3 ) = ( 2−3ε , 1+2ε ) and again by using the HardyLittlewood-Sobolev and the Hölder inequalities, we have

∇V τ ψ τ

1

2 L t2−3ε

2 L x1+2ε

([0,T1 ]×R3 )

1

T12 ∇|ψ τ |2

≤ T12 ∇V τ

2

1

L t1−3ε L xε

ψ τ L ∞ 2 t Lx

1

2 L t1−3ε

3 L x2+3ε

τ 2 τ 2

ψ˜ τ L ∞ 2 T1 ψ ∞ 2 ∇ψ L L t Lx t

x

2

6

L t1−3ε L x1+6ε

.

682


Now, we summarize the previous estimates by using (61) in the following way 1

∇ψ τ S˙ 0 ([0,T1 ]×R3 ) ∇ψ0 L 2 (R3 ) + T1 E 02 + T1α ∇ψ τ S˙ 0 ([0,T p

1 ]×R

3)

1 2

+T1 ψ0 2L 2 (R3 ) ∇ψ τ S˙ 0 ([0,T1 ]×R3 ) 1

(1 + T )E 02 + T1α ∇ψ τ S˙ 0 ([0,T

1

p

1

]×R3 )

+ T12 ψ0 2L 2 (R3 ) ∇ψ τ S˙ 0 ([0,T1 ]×R3 ) . (71)

Lemma 8. There exist T1 (E 0 , ψ0 L 2 (R3 ) , T ) > 0 and C1 (E 0 , ψ0 L 2 (R3 ) , T ) > 0, independent on τ , such that

∇ψ τ S˙ 0 ([0,T˜ ]×R3 ) ≤ C1 (E 0 , ψ0 L 2 (R3 ) , T )

(72)

for all 0 < T˜ ≤ T1 (E 0 , ψ0 L 2 (R3 ) ). Let us recall that E τ (t) = E 0 and ψ τ (t) L 2 (R3 ) = ψ0 L 2 (R3 ) , hence we are in the situation in which we can repeat our argument to every time interval of length T1 , depending always on the same parameters E 0 , ψ0 L 2 . The consequence of this fact is the following inequality on [0, T ]:

∇ψ τ S˙ 0 ([0,T ]×R3 )

≤ C1 (E 0 , ψ0 L 2 , T )

T + 1 = C( ψ0 L 2 , E 0 , T ). T1

(73)

Proof of Lemma 8. Let us consider the non-trivial case ψ0 L 2 > 0. Assume that X ∈ (0, ∞) satisfies X ≤ A + µX + λX p = φ(X ),

(74)

with p > 1, A > 0 and for all 0 < µ < 1, λ > 0. Let X ∗ be such that φ (X ∗ ) = 1, 1 p−1 , hence one has φ(X ∗ ) < X ∗ each time the following inequality namely X ∗ = 1−µ pλ is satisfied: p 1 (1 − µ) p−1 1 − > A. (75) p 1 1 p p−1 p p−1 λ p−1 Therefore the convexity of φ implies that, if condition (75) holds, there exist two roots X ± , X + (µ, λ, A) > X ∗ > X − (µ, λ, A), to the equation φ(X ) = X . It then fol1/2 lows either 0 ≤ X ≤ X − , or X ≥ X + . In our case µ = T1 ψ0 2L 2 , λ = T1α , 1/2

A = (1 + T )E 0 , hence we assume 1/2

1 , 2 − 1 p−1 − p p p−1 − p p−1 < . p 1/2 2 p−1 (1 + T )E 0

µ = T1 ψ0 2L 2 < 5− p 4

λ = T1α = T1


Therefore we choose

683

⎡

⎢ T1 := min ⎣(2 ψ0 L 2 )−2 ,

p 2

1 − p−1

p p−1

−p

(1 +

p − p−1

p−1) 4(5− p

1/2 T )E 0

⎤ ⎥ ⎦.

(76)

Clearly we cannot have

1 − T1 ψ0 L 2 x∗ = pT1α

1 p−1

≤ x+ ≤ ∇ψ τ S˙ 0 ([0,T1 ]×R3 ) ,

since we get a contradiction as T1 → 0, hence

∇ψ τ S˙ 0 ([0,T1 ]×R3 ) ≤ X − .

(77)

√ Corollary 24. Let 0 < T < ∞ and let ρ τ , Λτ be as in the previous section, then 1

∇ ρ τ L qt L r ([0,T ]×R3 ) + Λτ L qt L r ([0,T ]×R3 ) ≤ C(E 02 , ρ0 L 1 (R3 ) , T ), x

x

(78)

for each admissible pair of exponents (q, r ). Unfortunately this is not enough to achieve the convergence of the quadratic terms. We need some additional compactness estimates on the sequence {∇ψ τ } in order to apply Theorem 11. In particular we need some tightness and regularity properties on the sequence {∇ψ τ }, therefore we apply some results concerning local smoothing due to Vega [57] and Constantin, Saut [9]. Proposition 25 (Local smoothing for ∇ψ τ ). Let 0 < T < ∞ and let ψ τ be defined as in the previous section. Then one has

∇ψ τ L 2 ([0,T ];H 1/2 (R3 )) ≤ C(E 0 , T, ρ0 L 1 ).

(79)

loc

Proof. Using the Strichartz estimates obtained above, we can apply Theorems 9, 10 about local smoothing. Indeed, by using again formula (61) it follows

∇ψ τ L 2 ([0,T ];H 1/2 (R3 )) ∇ψ0 L 2 (R3 ) loc

τ τ

Λ (kτ −) L 2 (R3 ) N

+

k=1

+τ

N

∇ψn k L 2 (R3 )

k=1 N τ + ∇ψn k − ∇ψ τ (kτ −) + (Λn k − Λτ (kτ −)) 2 3 L (R ) k=1

+ F L 1 ([0,T ];L 2 (R3 )) . The first three terms are clearly estimated by a constant C(E 0 , T ) depending only on the initial energy and on time. The fourth term is O(τ ). The last term will be estimated using

684


the previous Strichartz estimates. As before, we split F into three parts F = F1 + F2 + F3 , then one has

|ψ τ | p−1 ∇ψ τ L 1 L 2 ([0,T ]×R3 ) t

≤T

4 5− p

≤T

4 5− p

x

τ p−1

|ψ |

Lt

4 p−1

3 L∞ x ([0,T ]×R )

∇ψ τ L ∞ 2 3 t L x ([0,T ]×R )

( p−1)

ψ τ L 4 L ∞ ([0,T ]×R3 ) ∇ψ τ L ∞ 2 3 , t L x ([0,T ]×R ) t x

while 1

∇V τ ψ τ L 1 L 2 ([0,T ]×R3 ) ≤ T 2 ∇V τ t

x

2 L t1−2ε

1 L xε ([0,T ]×R3 )

ψ τ

2

2

L t3ε L x1−2ε ([0,T ]×R3 )

,

and now the remaining calculations are similar to those already done for the Strichartz estimates. Regarding the term V τ ∇ψ τ we already estimated its L 1t L 2x norm. 1/2

2 , we can apply Theorem 11 due to Rakotoson, Since Hloc is compactly embedded in L loc Temam [49].

Proposition 26. The sequence {∇ψ τ } is, up to subsequences, relatively compact in 2 (R3 )), namely L 2 ([0, T ]; L loc ∇ψ := s − lim ∇ψ τk k→∞

2 in L 2 ([0, T ]; L loc (R3 )).

(80)

√ √ 2 (R3 )). In particular, one has ∇ ρ τ → ∇ ρ and Λτ → Λ in L 2 ([0, T ]; L loc Proof. The previous Proposition 25 implies that the sequence {∇ψ τ }τ >0 , is uniformly 1/2 bounded in L 2 ([0, T ]; Hloc (R3 )) and then, up to subsequences, ∇ψ τ ∇ψ in that 1/2 2 , ∇ψ τ (t) ∇ψ(t) for almost every space. Now Hloc is compactly embedded in L loc t ≥ 0 and lim

sup

|E|→0,E⊂[0,T ] τ >0 E

∇ψ τ (t) 2L 2 dt = 0, loc

since ∇ψ τ ∈ L ∞ ([0, T ]; L 2 (R3 )), then we can apply Theorem 11 by Rakotoson and Temam [49] and we get (80). Proposition 27. The limit functions (ρ, J ) are weak solutions to the Cauchy problem (1), (2). Proof. It follows directly by combining Theorem 18 in Sect. 5 and Proposition 26. As for the collisionless QHD system we should note that the generalized irrotationality condition holds by the definition of the current density and Corollary 13. Acknowledgement. The authors wish to thank Prof. Luigi Ambrosio for some useful comments.


685

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37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

Communicated by P. Constantin


Communications in


Finite-Time Singularities of an Aggregation Equation in Rn with Fractional Dissipation Dong Li1 , Jose Rodrigo2 1 School of Mathematics, Institute For Advanced Studies, Einstein Drive,

Princeton, NJ 08540, USA. E-mail: [email protected]

2 Warwick University, Coventry CV4 7AL, UK. E-mail: [email protected]

Received: 20 March 2008 / Accepted: 29 July 2008 Published online: 29 October 2008 – © Springer-Verlag 2008

Abstract: We consider an aggregation equation in Rn , n ≥ 2 with fractional dissipation, namely, u t + ∇ · (u∇ K ∗ u) = −ν(−)γ /2 u , where 0 ≤ γ < 1 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K (x) = e−|x| . We prove that for a class of smooth initial data, the solutions develop blow-up in finite time. 1. Introduction and Main Results We consider the following aggregation equation in Rn with fractional dissipation: u t + ∇ · (u∇ K ∗ u) = −ν(−)γ /2 u,

(1)

where K is a nonnegative radial decreasing kernel with a Lipschitz point at the origin, e.g. K (x) = e−|x| . As usual, ∗ denotes spatial convolution. Here ν ≥ 0 and 0 ≤ γ < 1 are parameters controlling the strength of the dissipative term. For any (reasonable) function f on Rn , the fractional Laplacian (−)γ /2 is defined via the Fourier transform: γ /2 u(ξ ) = |ξ |γ u(ξ (−) ˆ ). Aggregation equations of the form (1), with more general kernels (and other modifications) arise in many problems in biology, chemistry and population dynamics. In particular, these types of equations have applications in modeling the swarming phenomenon in biology. We use the term swarm here to describe the collective behavior of an aggregation of similar biological individuals cruising in the same direction. An overview of the modeling aspects of swarming can be found in [15,32 and 36]. Some Lagrangian type models in which each individual is regarded as a discrete point are studied in [1,11,13,14,26,30,41,44 and 45]. In the Eulerian setting, in which the individuals are approximated by a continuum population density field, several earlier models are constructed in [16,17,26,31,32,44 and 35]. As it has already been pointed out by several authors (see [43 and 39] ) the challenge with these continuum models has been

688

D. Li, J. Rodrigo

obtaining biologically realistic swarm solutions with sharp boundaries (often referred to as clumping, see [40 and 39]), relatively constant internal population densities and long life times. In one space dimension, some analytic studies have been conducted by Mogilner and Edelstein-Keshet [31], where they considered an integro-differential population model of the form (based on traditional population models, see [32,35 and 18]): ∂f ∂ ∂f ∂ = D( f ) − (2) (V ( f ) f ) + B( f ), ∂t ∂x ∂x ∂x where D( f ) is the density-dependent diffusion coefficient, B( f ) is the growth-rate of the population and V ( f ) is the advection velocity which takes the form V ( f ) = ae f + Aa (K a ∗ f ) − Ar f (K r ∗ f ), with constants ae , Aa and Ar representing density-dependent motion, attraction and repulsion respectively. Here the kernels K a and K r are called attraction and repulsion kernels (they belong to the so-called social interaction kernels). Based on perturbation analysis and numerical studies, they identified the conditions when aggregation occurs and also the stability of travelling swarm profiles. As noted in [31], the clumping behavior does not seem to be supported in the one-dimensional model (2) under realistic assumptions on the social interaction kernels. We refer the reader to [16,19–23,31,33,38,46 and 34] and the references therein for more extensive background and reviews on these one-dimensional models. As a multi-dimensional generalization of the model (2), Topaz and Bertozzi [43] constructed a kinematic two-dimensional swarming model which takes the form u t + ∇ · (u (G ∗ u)) = 0,

(3)

where the (vector-valued) kernel G is called the social interactional kernel which is spatially decaying. By applying the Hodge decomposition theorem [29], one can write G = G (I ) + G (P) := ∇ ⊥ N + ∇ P, where N and P are scalar functions. In the language of [43], the kernel G (I ) introduces incompressible motion which leads to pattern formation (e.g. vortex patterns), while the potential kernel G (P) models repulsion or attraction between biological organisms which in turn leads to either dispersion or aggregation. In a related paper, Topaz, Bertozzi and Lewis [42] modified the classical model of Kawasaki [23] and derived a model similar to [31], which takes the form u t + ∇ · (u K ∗ ∇u − r u 2 ∇u) = 0,

(4)

where the kernel K has fast decay in space. We remark that the clumping can be observed in these two-dimensional models (3) and (4) which were also found numerically in Levine, Rappel and Cohen [26]. We refer the reader to [24 and 3] and references therein for more details about aggregation models in this context. Aggregation equations have also been applied to image processing (see for example [2 and 37] for more details). From the mathematical point of view the aggregation equations have been studied extensively (see e.g. [3,5–8,24 and 43]). In one space dimension with C 1 initial data,

Blow-Up for the Generalized Surface Quasi-Geostrophic Equation

689

Bodnar and Velázquez [6] proved global well-posedness for some classes of interaction potentials and finite-time blow-up for others. Burger and Di Francesco [7] and also Burger, Capasso and Morale [8] studied the well-posedness of the model with an additional smoothing term. In connection with the problem we study here, Laurent [24] has developed the existence theory for a general class of equations containing the nondissipative version of (1) (i.e. ν = 0) and studied the connections between the regularity of the potential K and the global existence of the solution. More recently, Bertozzi and Laurent [3] have obtained finite-time blow-up of solutions for (1) without dissipation (ν = 0). The goal of this paper is to extend this result to the dissipative equation for the range 0 ≤ γ < 1. Additionally, we show that if the dissipation is sufficiently strong, i. e., 1 < γ ≤ 2, the solutions don’t develop any singularities. Aggregation equations with a dissipation term have been considered by several authors (see [24] and references therein for more details). For example, Topaz, Bertozzi and Lewis [42] have considered the equation u t = −∇ · [u(u ∗ ∇G)] + ∇ · (u 2 ∇u)

(5)

in cell-based models for the case in which we have a long range social attraction and short range dispersal. We remark that (5) contains the same type of aggregation term considered here and a local, nonlinear, diffusion term. We have chosen a diffusion term that contains different features, namely it is linear (which will translate into a milder diffusion process) and nonlocal. We believe the nonlocality should be an interesting feature for many applications. It is the interest in these features, linearity and nonlocality that leads directly into the use of the Laplacian for the dissipative term. We introduce fractional powers of the Laplacian to have a scale of strength for the dissipative terms against which we can study well-posedness. Given the natural scales of Eq. (1) we have 3 different ranges to the parameter γ . Namely 0 ≤ γ < 1, γ = 1 and 1 < γ ≤ 2, known as the supercritical, critical and subcritical regimes. We motivate the choice of the three regimes as follows. Since the kernel ∇ K x scales as |x| near the origin, heuristically our Eq. (1) which is not scale invariant can be approximated by the homogeneous version γ x ut + ∇ · u ∗ u = −ν(−) 2 u. (6) |x| Equation 6 has a scaling symmetry in the sense that if u is a solution, then for any λ > 0, u λ (t, x) = λn+γ −1 u(λγ t, λx) is also a solution with initial data u λ (0, x) = λn+γ −1 u 0 (λx). Here n is the space dimension where we are considering the problem. For positive initial data, it follows from Lemma 1 that the L 1x norm of the solutions of Eq. (1) is preserved for all time. The critical threshold of γ is then determined by the relation u λ L ∞ 1 = u L ∞ L 1 . t Lx t x Solving this equations yields γ = 1, which is then referred to as the critical case. For γ > 1, the a priori control of the L 1x norm then allows us to prove the global wellposedness of the solution (with L 1x initial data, see Theorem 3 below) and hence the name subcritical. In the supercritical case γ < 1, we prove the blow up of solutions in finite time (see Theorem 2 below). We refer the reader to [9,10 and 27] where this type

690

D. Li, J. Rodrigo

of dissipation has been used in the context of the surface quasi-geostrophic equation and other one dimensional models, for a more detailed explanation of the 3 regimes. A detailed study of the well-posedness issues, regularity of solutions will be contained in a forthcoming paper [28]. We state our results starting with an extension of the local existence theorem and continuation result proved by Bertozzi and Laurent [3] in the case ν = 0. It is an analogy of the Beale-Kato-Majda result for the 3D Euler case [4]. In this case we have the following Theorem 1 (Local existence and continuation [3]). Let ν ≥ 0 and 0 ≤ γ ≤ 2. Given initial data u 0 ∈ H s (Rn ), n ≥ 2, for positive integer s ≥ 2, there exists a unique solution u of (1) with life span [0, T ∗ ) such that either T ∗ = +∞ or limt→T ∗ sup0≤τ ≤t u(τ, ·) L qx = +∞. The result holds for all q ≥ 2 for n > 2 and q > 2 for n = 2. Proof. We refer the reader to [3] for the proof of the inviscid case ν = 0. We sketch here the main modification needed to prove the general result. Notice that the changes needed are very similar to the ones used to prove local existence and continuation for Euler and Navier-Stokes. We refer the reader to [28] for a detailed explanation of the necessary modifications introduced by the presence of viscosity. As in the case of Euler and Navier-Stokes, the main difference appears at the level of energy estimates. The presence of the viscosity term produces a regularizing effect and consequently a gain of derivatives. More precisely we have the following energy estimates for the approximate solutions u d 1 2 u H 2 + νu 2 s+ γ ≤ cs u H s−1 u 2H s , dt 2 H 2

(7)

which provides control of a higher norm, u s+ γ2 , than in the inviscid case (see PropoH sition 1 in [3] for the inviscid energy estimate). From these estimates, Theorem 1 follows easily. In the inviscid case ν = 0, Bertozzi and Laurent [3] proved the existence of finitetime blow up for a class of compactly supported smooth initial data. It is conceivable that when there is some amount of weak diffusion term, the blow-up phenomenon should still persist. Indeed we show that, in the case of supercritical dissipation 0 ≤ γ < 1, there exist finite-time singularities of Eq. (1) for a suitable class of initial conditions (subset of H s , s ≥ 2). Postponing the definition of this class of initial data (denoted below by Aδ,C,w , see (26), (27)) and the technical definition of admissible weight (see 1) we state our result in the supercritical case Theorem 2 (Blow-up for the supercritical case). Let w be an admissible weight function and let ν ≥ 0 and 0 ≤ γ < 1. There exist constants δ = δ(n) > 0, C = C(n, w, ν, γ ) > 0 such that if u 0 ∈ H s ∩ Aδ,C,w , s ≥ 2, then there exists a finite time T ∗ and a unique local solution u ∈ C([0, T ∗ ); H s ) ∩ C 1 ([0, T ∗ ); H s−1 ) for (1) that blows up at time T . Furthermore, we have, for every q ≥ 2 (q > 2 for n = 2), sup0≤τ ≤t u(·, τ ) L q → ∞, as t ↑ T ∗ . In contrast with the above theorem, when the dissipation power is bigger, that is, in the subcritical regime, the solutions don’t develop a singularity. More precisely, we have the following result.


691

Theorem 3 (Global wellposedness for positive initial data in the subcritical case). Let ν > 0 and 1 < γ ≤ 2. Assume the initial data u 0 ∈ L 1x (Rn ) and u 0 ≥ 0 for a.e. x. Then there exists a unique global solution u ∈ C([0, ∞), L 1x ) ∩ C((0, ∞), Wx1,1 ) of Eq. (1). 2. Proof of Theorem 2 We will argue by contradiction. Under the assumption that there is global existence for all initial data in H s , s ≥ 2 we will prove contradicting estimates for the energy of the system. As in the context of gradient flows and following Bertozzi and Laurent [3] (see also Topaz, Bertozzi and Lewis [42]), it is convenient to define the (free) energy as E(t) = u(x, t)(K ∗ u)(x, t)d x. (8) We will restrict our attention to positive initial data, and since the kernel K is positive, E is also positive. We recall the following lemma Lemma 1 (Persistence of positivity and L 1 norm [24]). Let ν ≥ 0 and 0 ≤ γ ≤ 2. Assume u 0 ≥ 0 for a.e. x. Let u be the solutions as described in Theorem 1. Then for each t ∈ [0, T ∗ ), the solution u is nonnegative and u(t) L 1x = u 0 L 1x . By using Hölder’s inequality, together with Young’s inequality and Lemma 1, it is easy to see that the energy has an a priori bound E(t) ≤ u2L 1 . The main estimate that we will obtain is a growth estimate for the energy, more precisely we will prove E (t) > c(u 0 L 1 ) > 0, for t up to some time T .

(9)

We will arrive at a contradiction by showing that at time T (from (9)) the energy E(T ) exceeds the a priori bound. In order to obtain (9) we notice that an elementary calculation yields (using the fact that K is radial) 2 E (t) = 2 u|∇ K ∗ u| d x − 2ν (−)γ /2 u(K ∗ u)d x. (10) Rn

Rn

We will explicitly describe a set of initial conditions for which the first term dominates the second, that is the nonlinear term controls the difussion. The bulk of estimate (9) is obtaining a lower bound for the first integral coming from the nonlinear term. Dealing with the second integral, involving the diffusion term is elementary. We have γ /2 γ /2 2ν (−) u(K ∗ u)d x ≤ 2ν u(−) K L ∞ u L 1 d x Rn

≤ 2ν(−)γ /2 K L ∞ u o L 1 ≤ C K u o L 1 , (11)

where 2ν(−)γ /2 K L ∞ ≤ 2ν |ξ |γ K (ξ ) L 1 =: C K .

(12)

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Remark 1. We notice that C K , given by |ξ |γ K (ξ ) L 1 , is only finite for 0 ≤ γ < 1. This is precisely where the argument for the existence of singularities breaks down for γ = 1. Notice that if we take K to be exactly e−|x| , its Fourier transform is given by the Poisson Kernel, which up to a constant multiple equals ((2π )−2 + (ξ )2 )−

n+1 2

,

making the function (γ = 1) |ξ |1 K (ξ )((2π )−2 + (ξ )2 )−

n+1 2

not integrable in Rn . We return now to the estimate for the first term in (10). Since we are only considering potentials K that are nonnegative, decreasing, radial and with a Lipschitz point at the origin, we can rewrite the gradient of K as ∇ K (x) = a

x + S(x), |x|

(13)

where a = 0 is a constant, S ∈ L ∞ (Rn ) is continuous at x = 0 with S(0) = 0. In order for the nonlinearity to generate a singularity it is clear we need ∇ K ∗ u sufficiently large. Since for positive functions the L 1 norm is preserved, the main problem x is the cancellation arising in |x| ∗ u if u is essentially constant over a large ball centered at the origin. It is clear from this observation, and the work of Bertozzi and Laurent [3] on the inviscid equation that we need to consider solutions that are highly concentrated near the origin. x We will now estimate several integrals arising in the evolution of E involving |x| ∗u and ∇ K ∗ u, for functions highly concentrated around the origin. The right definition of highly concentrated is made precise in Lemma 3. x Define N (x) = |x| . We have the following lemma which gives a lower bound of the contribution due to the homogeneous kernel N (a multiple of the homogeneous part of ∇ K (see (13)). Lemma 2 (Lower bound for the homogenous kernel). There exists a constant 1 (Rn ) we C1 = C1 (n) > 0 such that for any nonnegative radial function g ∈ L rad have g(x)|(N ∗ g)(x)|d x ≥ C1 g2L 1 . Proof. It is clear that we can assume that g L 1 = 1. By the Cauchy-Schwartz inequality we have g(x)|(N ∗ g)(x)|d x x d x ≥ g(x)(N ∗ g)(x), |x| (x − y) · x = g(x)g(y) d xd y. (14) |x − y| · |x| By symmetrizing in the integral in x and y and using the fact that g is nonnegative, we obtain


693

x y x−y · − d xd y RHS of (14) = 2 g(x)g(y) |x − y| |x| |y| x−y x y d xd y = g(x)g(y) · − |x − y| |x| |y| |y|≤|x| x·y ) (|x| + |y|) · (1 − |x||y| = d xd y g(x)g(y) |x − y| |y|≤|x| ≥ C2 g(x)g(y)d xd y

|y|≤|x| x·y≤0

≥

C2 2

g(x)g(y)d xd y,

(15)

x·y≤0

where C2 is a constant depending only on n. In the last inequality we symmetrized again in the variables x, y. To bound this last integral, we now use the fact that g is a radial function. Denoting by dσ as the surface measure on S n−1 , with a simple scaling argument we obtain C2 ∞ ∞ RHS of (15) = g(ρ1 )g(ρ2 ) dσ (x)dσ (y)dρ1 dρ2 |x|=ρ1 , |y|=ρ2 2 0 0 x·y≤0 2 ∞ C2 n−1 ≥ g(ρ)ρ dρ dσ (x)dσ (y) |x|=1, |y|=1 2 0 x·y≤0

≥

C1 g2L 1 , x

(16)

where C1 is a positive constant depending only on n. Remark 2. The proof of Lemma 2 is the only place in our blow-up argument where we need the radial assumption of the solution u. It is possible to remove the radial assumption although we shall not do it here. In the next lemma we establish a similar conclusion for the whole kernel ∇ K . Because of the presence of the inhomogeneous part, we need to consider functions having mass localized near the origin so that the contribution due to S(x) (see (13)) is small and the whole integral is still bounded below by a large constant. Lemma 3 (Lower bound for the kernel ∇ K for mass localized functions). There exists a constant δ = δ(n, K ) > 0 such that the following holds true: For any nonnegative radial function f on Rn with the property f (x)d x ≤ δ f L 1 , (17) |x|≥δ

we have

Rn

f |∇ K ∗ f |2 d x ≥

(aC1 )2 f 3L 1 , 2

where C1 is the same constant as in Lemma 2 and a is defined in the decomposition (13).

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Proof. Without loss of generality we will assume that f L 1 = 1. Recall the decomposition (13), since S(x) is continuous at x = 0 with S(0) = 0, we know that for any 1 > 0, there exists δ1 = δ1 (K , 1 ), such that |S(x)| ≤ 1 , ∀ |x| ≤ δ1 . On the other hand since S is assumed to be bounded, we have |S(x)| ≤ D1 , ∀ |x| ≥ 0,

(18)

where D1 is another constant depending only on K . Take 1 = sufficiently small such that aC1 δ1 (1 , K ) . δ < min , 100D1 4

aC1 100

and let δ > 0 be (19)

Fix this δ and assume that f satisfies the localization property (17). For |x| ≤ δ, by splitting the integral and using the fact that f L 1 = 1, we have | f (x − y)||S(y)|dy + | f (x − y)||S(y)|dy |(S ∗ f )(x)| ≤ |x|≤2δ |y|>2δ ≤ 1 + D 1 | f (y)|dy |y|>δ

≤ 1 + δ D 1 ,

(20)

where the last inequality follows from the localization assumption (17). For any |x| ≥ 0, we have by Young’s inequality and (18), |(S ∗ f )(x)| ≤ D1 .

(21)

In view of our choice of 1 , δ (see (19)) and the pointwise bounds on (S ∗ f )(x) (20) (21), we have f |(S ∗ f )(x)|d x ≤ | f (x)|d x(1 + δ D1 ) + | f (x)|d x D1 Rn

|x|≤δ

|x|≥δ

≤ 1 + 2δ D1 aC1 . ≤ 10 Now by the Cauchy-Schwartz inequality and Lemma 2, we have 1 2 2 f |∇ K ∗ f | d x Rn

= ≥

f |∇ K ∗ f | d x 2

Rn

1 2 Rn

1 f dx

2

f |∇ K ∗ f |d x ≥ aC1 − f |S ∗ f |d x Rn

Rn

aC1 ≥ √ , 2 where the last inequality follows from the bound (22). The lemma is proved.

(22)


695

We remark that both Lemma 2 and Lemma 3 deal with time independent estimates but require high concentration of mass near the origin. It is crucial for our proof that we show that if u 0 is concentrated near the origin, then the solution u(·, t) remains concentrated near the origin for at least some short time t. In the inviscid case ν = 0, Bertozzi and Laurent [3] showed that if one starts with compactly supported data then it remains compactly supported during the time of existence. The situation changes dramatically in the dissipative case ν > 0. In the case we considered here, even if the initial data is compactly supported, the solution at any t > 0 will have nonzero support on the whole space due to the infinite speed of propagation of the fractional heat semigroup γ /2 e−t (−) . It is for this reason that we need to prove the non-evacuation of mass for a short time. As we shall see later, the mass localization will follow from a weighted estimate for u. To this end, we need the following definition Definition 1 (Admissible weight functions). A function w ∈ C ∞ (Rn ) is said to be an admissible weight function if w is a nonnegative radial function such that w(0) = 0 and w(x) = 1 for all |x| ≥ 1. An admissible weight function can be regarded as a smoothed out version of the spatial cut-off function χ{|x|≥1} . Let w be an admissible weight function and let δ > 0 be the same constant as in Lemma 3. We define x I (t) = d x. u(t, x)w δ Rn Intuitively speaking, the integral I (t) quantifies the mass of u outside of a small ball of size δ near the origin. The growth of I (t) provides an upper bound of the mass of u away from the origin. Let w1 (x) = w(x) − 1. Clearly by definition w1 ∈ Cc∞ (Rn ). By integration by parts, Young’s inequality and Lemma 1, we compute d x x I (t) = − ∇ · (u∇ K ∗ u)w( )d x − ν (−)γ /2 u(x)w( )d x n n dt δ δ R R x γ 1 x 1 = u∇ K ∗ u · (∇w1 )( )d x − ν u(x) γ (−) 2 w1 ( )d x δ δ δ δ Rn Rn γ 1 1 ≤ ∇w1 L ∞ |u∇ K ∗ u|d x − νu L 1x γ (−) 2 w1 L ∞ x x n δ δ R 1 ν ≤ ∇w1 L ∞ ∇ K L ∞ u 0 2L 1 − u 0 L 1x · γ |ξ |γ wˆ 1 (ξ ) L 1 x x ξ x δ δ 2 ≤ C3 · (u 0 L 1 + 1), (23) where C3 = C3 (n, ν, γ , w, δ) is a constant. Now if we choose T =

δu 0 L 1 , 2C3 · (u 0 2L 1 + 1)

then we have δ sup I (t) ≤ I (0) + u 0 L 1 , 2 0≤t≤T

(24)

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where

I (0) =

x u 0 (x)w( )d x. δ Rn

Since w(x/δ) = 1 for |x| ≥ δ, (24) implies the bound, δ x sup u(t, x)d x ≤ u 0 (x)w( )d x + u 0 L 1 . δ 2 Rn 0≤t≤T |x|≥δ Now if we choose u 0 such that Rn

δ x u 0 (x)w( )d x ≤ u 0 L 1 , δ 2

then clearly sup 0≤t≤T

|x|≥δ

u(t, x)d x ≤ δu 0 L 1 .

(25)

This is the mass localization property we need. Based on the results above we will specify the set of initial conditions for which one can easily obtain blow-up. Let δ > 0, C > 0 be two constants. We define A = 1 (Rn ) to be the class of nonnegative radial functions u satisfying the Aδ,C,w ⊂ L rad following properties: 1. The mass of u is comparable to its energy: |K (0)|u2L 1 < u(K ∗ u)d x + 1. Rn

(26)

2. u is localized near the origin:

δ x u(x)w( )d x C. For any δ > 0, C > 0 and any admissible weight w, it is not too difficult to see that the 1 (Rn ) such that f class Aδ,C,w is nonempty. Indeed one can take any f ∈ L rad L 1 > C, −n −1 then define f λ (·) = λ f (λ ·). For all sufficiently small λ > 0, one can check directly that u = f λ satisfies (26) and (27) due to the assumption that K (0) = K L ∞ and w(0) = 0. We are now ready to complete the proof of the main theorem. Proof. (Proof of Theorem 2) Take δ to be the same constant as in Lemma 3 and choose a constant C sufficiently large such that C > max{

4C3 + C K , 1}, (a C1 )2

(28)

where C3 was defined in (23) and C K is given in (12) in the estimate for the difussion term.


697

Take u 0 ∈ H s ∩ Aδ,C,w and recall that E(t) = u(t, x)(K ∗ u)(t, x)d x. Rn

Then obviously E(t) ≤ u 0 2L 1 K L ∞ = u 0 2L 1 K (0). On the other hand we have d 2 u|∇ K ∗ u| d x − 2ν (−)γ /2 u(K ∗ u)d x. E(t) = 2 dt Rn Rn Let T =

δu 0 L 1 , 2C3 · (u 0 2L 1 + 1)

then by the mass localization property (25) and Lemma 3, together with the estimate (11) for the diffusion term we have d E(t) ≥ (a C1 )2 u 0 3L 1 − C K u 0 2L 1 . dt By our choice of u 0 and the choice of the constant C (see (28)), it is not difficult to check that (a C1 )2 u 0 3L 1 − C K u 0 2L 1 >

2C3 · (u 0 2L 1 + 1) 1 = . T δu 0 L 1

This gives us E(T ) ≥ E(u 0 ) + 1. But this is impossible since we have E(T ) ≤ u 0 2L 1 K L ∞ = u 0 2L 1 K (0) < E(u 0 ) + 1, where the last inequality is due to the fact that u 0 ∈ Aδ,C,w . The theorem is proved. 3. Global Well-Posedness and Smoothing for the Subcritical Case 1 < γ ≤ 2 In this section we consider the aggregation equation in the subcritical regime 1 < γ ≤ 2. We first prove local well-posedness in L 1x (Rn ). We shall do this by constructing mild solutions. This is Theorem 4 (Local well-posedness in L 1x for the subcritical case). Let ν > 0 and 1 < γ ≤ 2. Assume the initial data u 0 ∈ L 1x (Rn ). Then there exists a time T = T (u 0 L 1x , ν, γ , ∇ K L ∞ ) > 0 and a unique mild solution of (1) in the space C([0, T ), x 1 n L x (R )). In fact the uniqueness of mild solutions holds in a slightly stronger sense: for any T > 0, there exists at most one solution in the space C([0, T ), L 1x (Rn )) with initial data u 0 ∈ L 1x .

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Remark 3. As we shall see in the proof of Theorem 4, the time of existence of the constructed mild solution has an upper bound of the form T
0 such that 4·

γ − 1 1− 1 · ν γ T γ ∇ K L ∞ u 0 L 1x < 1. x γ −1

Then by the inequality S(t) ∗ u 0 X T ≤ u 0 L 1x , the strong continuity of the semigroup S(t) in L 1x , the boundedness of the bilinear operator Lemma 5 and the fixed point Lemma 4, we conclude that there exists a solution of Eq. (29) in the space X T . It only remains for us to prove the uniqueness part of Theorem 4. Let T > 0 be arbitrary and u 1 , u 2 be two solutions of (29) with the same initial data u 0 . Denote

M = max u 1 X T , u 2 X T . Let T be sufficiently small such that γ 1 −1 1− 1 · ν γ (T ) γ ∇ K L ∞ . M< x γ −1 10 Then since u 1 and u 2 has the same initial data u 0 , we have by Lemma 5, u 1 − u 2 X T ≤ B(u 1 , u 1 − u 2 ) X T + B(u 1 − u 2 , u 2 ) X T 1 ≤ u 1 − u 2 X T . 2 This implies that u 1 ≡ u 2 on [0, T ). A finite iteration of the argument then gives u 1 ≡ u 2 on the whole time interval [0, T ). The theorem is proved. We now show that our constructed mild solution has additional regularity. This is achieved by another contraction argument in the subspace of X T . We first formulate a two-normed version of the fixed point Lemma 4.

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Lemma 6 (Two-normed fixed point lemma). Assume that Z is a Banach space endowed with the norms · Z , · X and seminorm · Y such that · Z = max{ · X , · Y }. Let B : Z × Z → Z be a bilinear map such that for any x1 , x2 ∈ Z , we have B(x1 , x2 ) Z ≤ C(x1 Z x2 X + x1 X x2 Z ), and B(x1 , x2 ) X ≤ Cx1 X x2 X . Then for any y ∈ Z such that 8Cy X < 1, the equation x = y + B(x, x) has a solution in Z with x Z ≤ 2y Z . Moreover by Lemma 4 the solution is unique in the ball {z : z X ≤ C2 }. Proof. Again we construct the solution x by iteration. Define x0 xn = y + B(xn−1 , xn−1 ) for n ≥ 1. Then since

=

y and

xn Z ≤ y Z + 2xn−1 Z xn−1 X ≤ y Z + 4Cy X xn−1 Z , it is easy to prove by induction that xn Z ≤ 2y Z . To show (xn ) is Cauchy in Z we calculate xn+1 − xn Z ≤ B(xn , xn − xn−1 ) Z + B(xn − xn−1 , xn−1 ) Z ≤ 4Cy Z xn − xn−1 X + 4Cy X xn − xn−1 Z . From the proof of Lemma 4 we know that xn − xn−1 X ≤ θ n for some constant 0 < θ < 1. This together with the fact that 4Cy X < 1 and a few elementary manipulations implies that xn+1 − xn Z ≤ (θ )n for another constant 0 < θ < 1. This immediately shows that xn is Cauchy in Z and hence converges to a fixed point x. In what follows, it is useful to consider the · YT norm of u defined by 1

uYT := t γ ∇u L ∞ 1 n . t L x ([0,T )×R ) We first prove that the · YT norm of the bilinear operator (30) is bounded. Lemma 7 ( · YT norm boundedness of the bilinear operator). The bilinear operator (30) is bounded in the following sense: γ −1

−1 B( f, g)YT ≤ f YT g X T + f X T gYT · ∇ K L ∞ · C1 ν γ · T γ , x

where C1 = C1 (γ ) is a positive constant depending only on γ .


701

Proof. We have 1

B( f, g)YT = t γ ∇ B( f, g) L ∞ L 1 ([0,T )×Rn ) t t x 1 1 − γ1 − γ ≤ν (t − τ ) γ (∇ f · ∇ K ∗ g)(τ ) L 1x dτ t 0

L ∞ ([0,T ))

t t 1 1 1 − − γ +ν γ (t − τ ) γ ( f ∇ K ∗ ∇g)(τ ) L 1x dτ t ∞ 0 L t ([0,T ))

≤ f YT g X T + f X T gYT · ∇ K L ∞ x t 1 1 1 − γ1 − − t γ ·ν (t − τ ) γ τ γ dτ ∞ L t ([0,T ))

0

− 1 γ −1 ≤ f YT g X T + f X T gYT · ∇ K L ∞ · C1 ν γ T γ , x where C1 is an constant depending only on γ . The lemma is proved. We can now upgrade the regularity of our constructed mild solution. We define Z T ⊂ C([0, T ), L 1x ) as a Banach space with the norm u Z T = max{u X T , uYT } 1

= max{u L ∞ 1 n , t γ ∇u L ∞ L 1 ([0,T )×Rn ) }. t L x ([0,T )×R ) t x Theorem 5 (Local well-posedness in Z T for the subcritical case). Let ν > 0 and 1 < γ ≤ 2. Assume the initial data u 0 ∈ L 1x (Rn ). Then there exists a time T = T (u 0 L 1x , ν, γ , ∇ K L ∞ ) > 0 and a unique mild solution of (1) in the space Z T . x By Theorem 4 the uniqueness of the mild solutions holds in a larger space: for any T > 0, there exists at most one solution in the space C([0, T ), L 1x (Rn )) with initial data u 0 ∈ L 1x . Remark 4. As we will see in the proof below, the time of existence of the constructed mild solution has an upper bound of the form − γ 1 γ −1 T < C2 · ν γ −1 · ∇ K L ∞ u , 0 L 1x x where C2 = C2 (γ ) is a positive constant depending only on γ . Proof. (Proof of Theorem 5) We only need to prove the existence. The uniqueness part is already in Theorem 4. Choose T > 0 such that 8C1 · ν

− γ1

T

1− γ1

∇ K L ∞ u 0 L 1x < 1, x

where C1 is the same constant as in Lemma 7. By the inequality ∇ S(t) ∗ u 0 L 1x ≤ −1

t γ u 0 L 1x , the boundedness of the bilinear operator Lemma 7 and the two-normed fixed point Lemma 6, we conclude that there exists a solution of Eq. (29) in the space ZT . By a standard bootstrap argument, we can obtain the following corollary.

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Corollary 1 (Maximal time of existence of solutions). Let ν > 0 and 1 < γ ≤ 2. Assume the initial data u 0 ∈ L 1x (Rn ). Then there exists a maximal time of existence T ∗ ∈ (0, ∞] and a unique solution u ∈ C([0, T ∗ ), L 1x ) ∩ C((0, T ∗ ), Wx1,1 ). Moreover if T ∗ < ∞, then necessarily limt→T ∗ u(·, t) L 1x = ∞. Proof. This is a standard argument which follows from Theorem 5. By Corollary 1, to obtain a global solution, it suffices for us to control the L 1x (Rn ). Concerning positive initial data, the following result was originally proved by Laurent [24] for the inviscid case ν = 0 and with different assumptions on the initial data. By using the time splitting approximation, it is straightforward to obtain the same result for the dissipative case ν > 0. By another approximation argument, we obtain the following Lemma 8 (Persistence of positivity and L 1 norm [24]). Let ν ≥ 0 and 1 < γ ≤ 2. Assume u 0 ∈ L 1x and u 0 ≥ 0 for a.e. x. Then for each t ∈ [0, T ∗ ), the solution u is nonnegative and u(t) L 1x = u 0 L 1x . We are now ready to complete Proof. (Proof of Theorem 3) It follows directly from Corollary 1 and Lemma 8.

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Communications in


Stellar Collapse in the Time Dependent Hartree-Fock Approximation Christian Hainzl1 , Benjamin Schlein2 1 Departments of Mathematics and Physics, UAB, Birmingham, AL 35294, USA.


2 Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany.

E-mail: [email protected] Received: 25 March 2008 / Accepted: 16 July 2008 Published online: 11 November 2008 – © Springer-Verlag 2008

Abstract: We prove blow-up in finite time for radially symmetric solutions to the pseudo-relativistic Hartree-Fock equation with negative energy. The non-linear HartreeFock equation is commonly used in the physics literature to describe the dynamics of white dwarfs. We extend thereby recent results by Fröhlich and Lenzmann, who established in [3,4] blow-up for solutions to the pseudo-relativistic Hartree equation. As key ingredient for handling the exchange term we use the conservation of the expectation of the square of the angular momentum operator. 1. Introduction According to the theory of Chandrasekhar [1], white dwarfs can be described by a model of electrically neutral atoms interacting through classical Newtonian gravitation. Atoms consist of nuclei, which are responsible for the main part of the potential energy, and electrons, which, on the other hand, give the leading contribution to the kinetic energy of the star. Because of local charge neutrality, we can assume that the space and momentum distributions of the nuclei coincide with the ones of the electrons; in this approximation, we only keep track of the electronic degrees of freedom. Considering a relativistic dispersion E( p) = p 2 + m 2 for electrons with mass m, and assuming a single species of nuclei with mass m Z m and charge Z e (where −e denotes the charge of the electron), this simplified model is described, on the microscopic level, by the quantum mechanical Hamiltonian HN =

N j=1

−x j + m 2 − κ

N i< j

1 , |xi − x j |

(1.1)

where N is the number of electrons and κ = Gm 2Z /Z 2 (G denotes here the gravitational constant). We use units with = c = 1.

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C. Hainzl, B. Schlein

Since the electron spin does not play an important role, we neglect it, and describe electrons by wave functions in L 2 (R3 ). In accordance with Pauli’s principle, the N Hamiltonian (1.1) acts then on the antisymmetric tensor product space H N = L 2 (R3 , dx). In [5,6] Lieb and Yau derived the Chandrasekhar equation for the ground state of the Hamiltonian (1.1) in the limit of large N with κ N 2/3 kept fixed. Additionally they reproduced Chandrasekhar critical number of particles Nc ∼ O(κ −3/2 ) proving the instability of (1.1) for all N > Nc (the white dwarf is supposed to undergo gravitational collapse for N > Nc ). Up to a factor 4 the correct value of Nc had already been established by Lieb and Thirring in [7]. See [4–7] for a more thorough discussion on white dwarfs. In principle (1.1) can also be used to describe neutron stars; however, in this case, a correct understanding of the collapse requires the inclusion of general relativity effects. In the physical relevant regime of very small κ and very large N , one expects the ground state of (1.1) to be approximated by a Slater determinant 1 σπ ψ1 (xπ 1 )ψ2 (xπ 2 ) . . . ψ N (xπ N ) (ψ1 ∧ ψ2 ∧ · · · ∧ ψ N ) (x) = √ N ! π ∈S N

of N orthonormal one-particle wave functions ψ j ∈ L 2 (R3 ) (here the sum runs over all permutations of the N particles; moreover, σπ = 1 if the permutation π is even while σπ = −1 if it is odd). It is simple to verify that the energy of the Slater determinant ∧ Nj=1 ψ j is given by the so called Hartree-Fock functional EHF ({ψ j } Nj=1 ) =

N

dx |(−+m 2 )1/2 ψ j (x)|2

j=1

−

N |ψi (x)|2 |ψ j (y)|2 −ψi (x)ψ i (y)ψ j (y)ψ j (x) κ . dx dy 2 |x − y| i, j=1

(1.2) Within the range of its applicability, one also expects the Hartree-Fock theory to describe the time-evolution of Slater determinants. In other words, one expects that, in an appropriate sense, and in a suitable limit of large N and small κ, e−i HN t (ψ1 ∧ · · · ∧ ψ N ) ψ1 (t) ∧ ψ2 (t) ∧ · · · ∧ ψ N (t), where the wave functions evolve according to the time-dependent Hartree-Fock equations ( j = 1, . . . , N ) N N 1 1 2 2 i∂t ψ j = − + m ψ j − κ ∗ |ψi | ψ j + κ ∗ ψ j ψ i ψi . (1.3) |.| |.| i=1

i=1

Note that this system of non-linear equations, which can be formally obtained computing the variation of (1.2), preserves the orthonormality relations ψi (t), ψ j (t) = δi j and the energy EHF . The Hartree-Fock theory can be formulated in a more compact form in terms of the N orthogonal projection Q = j=1 |ψ j ψ j | onto the subspace spanned by the wave functions {ψ j } Nj=1 . The Hartree-Fock energy (1.2) is given, in terms of Q and its kernel Q(x, y), by

Finite Time Blowup for the Hartree-Fock Equation

EHF (Q) = tr

− + m 2

κ Q− 2

707

dxdy

Q(x, x)Q(y, y) − |Q(x, y)|2 . |x − y|

(1.4)

Also the time-dependent Hartree-Fock system (1.3) can be translated into an evolution equation for the time dependent density Q t = Nj=1 |ψ j (t) ψ j (t)|. It is easy to obtain the nonlinear Hartree-Fock equation

1 (1.5) ∗ ρ Qt + κ R Qt , Q t , i∂t Q t = (− + m 2 )1/2 − κ |.| where ρ Q t (x) = Q t (x, x) (we denote by Q t (x, y) the kernel of the projection Q), and where the operator R Q t is defined by its kernel R Q t (x, y) = Q t (x, y)/|x − y|. By construction, it is clear that (1.5) preserves the trace N = tr Q t , and the energy (1.4). In the present paper we are interested in solutions to the nonlinear Hartree-Fock equation (1.5); in particular we prove the existence of solutions to (1.5) which exhibit blow up in finite time. Within the framework of the Hartree-Fock approximation, the blow up of solutions to (1.5) is interpreted as evidence for the dynamical collapse of white dwarfs. The last contribution in the commutator on the r.h.s. of (1.5) (the term containing the operator R Q ) is known as the exchange term (while the second contribution, containing the density ρ Q , is known as the direct term). The presence of the exchange term is a consequence of the Pauli principle. Since, in the relevant limit of large N and small κ, the exchange term is expected to be of smaller order compared with the direct term, it is often neglected in the physics literature. In this approximation one obtains the Hartree equation,

1 (1.6) ∗ ρ Qt , Q t . i∂t Q t = (− + m 2 )1/2 − κ |.| For bosonic systems (boson stars) this equation has been in fact rigorously derived from many body quantum dynamics in [2]. Recently, blow-up in finite time has been proven to occur for solutions to the Hartree equation (1.6) by Fröhlich and Lenzmann in [3,4]. To obtain this result, they consider the non-negative observable 3 M = x − + m 2 x = x j − + m 2 x j , j=1

and they estimate the expectation value tr(M Q t ), where Q t is a solution to (1.6) with spherical symmetry (in the sense that Q t (Rx, Ry) = Q t (x, y) for all R ∈ S O(3)). Under this assumption, they show that tr M Q t ≤ 2t 2 EHartree (Q) + O(t),

(1.7)

where O(t) denotes error terms growing at most linearly in t. For initial data with negative energy EHartree (Q) < 0, Eq. (1.7) leads to a contradiction to the non-negativity of the observable M (choosing t sufficiently large). This implies that the solution Q t cannot exist globally in time. The spherical symmetry of the density Q plays a very important role in the analysis developed by Fröhlich and Lenzmann; it allows them to control error terms arising from the commutator of M with the interaction (|.|−1 ∗ ρ Q t ) (the time derivative of tr M Q t contains the term tr [M, (|.|−1 ∗ ρ Q t )]Q t ). The same approach can be applied

708


to solutions of the Hartree-Fock equation (1.5); it turns out, however, that the error terms arising from the commutator of M with the exchange term R Q cannot be handled like the errors arising from the direct term. This is the reason why the approach of Fröhlich and Lenzmann does not extend in a simple way to the Hartree-Fock equation (1.5) (the method does extend to the Hartree-Fock equation if one assumes not only that Q is spher ically symmetric, but also that each orbital in its decomposition Q = Nj=1 |ψ j ψ j | is spherically symmetric; however, as pointed out in [4], this is a physically unnatural condition). In our analysis, we use √ the strategy of Fröhlich and Lenzmann; we study the evolution of the observable M = x − + m 2 x on spherically symmetric solutions Q t of (1.5) and we obtain a bound like (1.7). The novelty of our approach lies in the estimate of the error terms arising from the commutator of M with the exchange term R Q ; to control this error terms, we make use of the expectation of the square of the angular momentum operator, which is a conserved quantity due to the radial symmetry. 2. The Main Result and its Proof The local well-posedness of the Hartree-Fock system (1.3) has been established by Fröhlich and Lenzmann in [4, Theorem 1]. The well-posedness of (1.5) follows along the same line. For s ≥ 0 we define the space Hs = {Q ∈ L1 (L 2 (R3 )) : Q Hs < ∞} with the norm

Q Hs = tr (1 − )s/2 Q(1 − )s/2 .

It turns out that (1.5) is locally well-posed in Hs for all s ≥ 1/2. Theorem 1 (Local Well-Posedness, [4]). Fix s ≥ 1/2. For every orthogonal projection Q ∈ Hs (L 2 (R3 )) with N (Q) = tr Q < ∞ there exists a maximal existence time T > 0 and a unique solution Q t ∈ C([0, T ), Hs (L 2 (R3 ))) of (1.5) with Q t ≥ 0, and tr Q t = N for all t ∈ [0, T ). Moreover, if T < ∞, then Q t s = tr(1 − )s Q t → ∞ as t → T − (blow-up alternative). For sufficiently small values of N = tr Q, Fröhlich and Lenzmann also proved global well-posedness of the Hartree-Fock equation in [4, Theorem 2]. Our goal here is to prove that, for sufficiently large values of N = tr Q, blow up in finite time can occur. In particular, we show that the time evolution of an arbitrary spherically symmetric density with negative energy and with finite expectation for the square of the angular momentum operator exhibits blow-up in finite time. To prove this result, we need a few simple preliminary lemmas. First of all, we need to prove that the spherical symmetry is preserved by the time-evolution; this follows easily from the local uniqueness of the solution to (1.5). Lemma 2.1. Let Q ∈ H1/2 (L 2 (R3 )), Q ≥ 0, and assume that Q is spherically symmetric in the sense that Q(Rx, Ry) = Q(x, y)

for all R ∈ S O(3).

For t ∈ [0, T ) denote by Q t the local in time solution to (1.5) with Q t=0 = Q. Then, for every t ∈ [0, T ), we have Q t (Rx, Ry) = Q t (x, y)

for all R ∈ S O(3).

(2.1)


709

t (x, y) = Q t (Rx, Ry). It is then simple Proof. For arbitrary R ∈ S O(3), we define Q t is also a solution to (1.5), characterized by the same initial data. By the to verify that Q local uniqueness of the solution, we immediately obtain (2.1). The main reason why spherical symmetry is so important to prove the blow up of solutions of (1.5) is Newton’s Law, as stated in the following lemma. 1 3 Lemma 2.2. (Newton’s Law)

Suppose that ρ ∈ L (R , dx) is non-negative and spherically symmetric with N = dy ρ(y). Then N ρ(y) ρ(y) N ∇ x ≤ dy (2.2) dy ≤ and |x − y| |x| |x − y| |x|2

for a.e. x ∈ R3 . Proof. The proof relies on the explicit formula for radial functions ρ(y) ρ(y) 1 dy ρ(y) + dy dy = . |x − y| |x| |y|≤|x| |y| |y|>|x| For ρ ∈ L 1 (R3 ) ∩ C 0 (R3 ), explicit differentiation leads to 1 N ρ(y) ∇ x = dy ρ(y) ≤ . dy 2 |x − y| |x| |y|≤|x| |x|2 For general spherically symmetric ρ ∈ L 1 (R3 ) the statement follows using a simple density argument. An important tool in the proof of the finite time blow-up of solutions of (1.5) is the fact that the expectation of the square of the angular momentum operator L = x ∧ p is preserved by the time evolution for spherically symmetric solutions. Lemma 2.3. Let L = x ∧ p denote the angular momentum operator, with p = −i∇x . Let Q ∈ H1/2 (L 2 (R3 )), Q ≥ 0 be a spherically symmetric density with tr L 2 Q < ∞. For t ∈ [0, T ), denote by Q t the local in time solution of (1.5) with Q t=0 = Q. Then tr L 2 Q t = tr L 2 Q

for all t ∈ [0, T ).

Proof. Observe that the angular momentum operator L generates rotations in the sense that ei L·α ψ (x) = ψ(Rα x) for all ψ ∈ L 2 (R3 ). Here Rα ∈ S O(3) denotes the rotation around the axis α, ˆ with angle |α|. This implies that, for an arbitrary spherically symmetric density Q, we have ei L·α Qe−i L·α (x, y) = Q(Rα x, Rα y) = Q(x, y) for all α. Differentiating with respect to α, we obtain that [L , Q] = 0 and thus that [L 2 , Q] = 0 for any spherically symmetric density Q. Now, since the time evolution Q t

710


of the spherically symmetric initial data Q is spherically symmetric (by Lemma 2.1), it follows that

d 1 tr L 2 Q t = tr L 2 ∗ ρ Qt + κ R Qt , Q t p2 + m 2 − κ dt |.| 1 2 2 Q t , L 2 = 0. ∗ ρ Qt + κ R Qt = tr p +m −κ |.| We are now ready to state and prove our main theorem. Theorem 2 (Blow-up for Hartree-Fock). Let Q ∈ H1/2 (L 2 (R3 )) be a spherically symmetric orthogonal projection with N (Q) = tr Q < ∞, with negative energy EHF (Q) < 0, with finite expectation of the square of the angular momentum operator L2 (Q) = tr L 2 Q < ∞, such that tr x 4 Q < ∞,

and

tr (−) Q < ∞.

(2.3)

For t ∈ [0, T ), let Q t denote the maximal local in time solution to (1.5) with Q t=0 = Q. Then T < ∞ and Q t 2H 1/2 = tr (1 − )1/2 Q t → ∞

as t → T −.

Remarks. • According to [4, Theorem 4] our main Theorem 2 implies that when approaching the time of blow-up any blow-up solution exhibits a concentration of particles at the origin, with lim inf dx ρ Q t (x) > 0 for any R > 0. t→T − |x|≤R

• The condition EHF (Q) < 0 requires N = tr Q to be sufficiently large. Proof. Throughoutthe proof, we will usethe notation p = −i∇x . Consider the observable M = x p 2 + m 2 x = 3j=1 x j p 2 + m 2 x j . We are interested in the time evolution of the expectation tr(M Q t ). Step 1. There exists a constant C, only depending on N (Q) and L2 (Q), such that d tr M Q t ≤ tr ( p · x + x · p) Q t + C. dt To prove (2.4) we start by computing

d 1 tr M Q t = − i tr M ∗ ρ Qt + κ R Qt , Q t p2 + m 2 − κ dt |.|

= − i tr x p 2 + m 2 x, p 2 + m 2 Q t

1 Qt ∗ ρ Qt + iκ tr x p 2 + m 2 x, |.|

− iκ tr x p 2 + m 2 x, R Q t Q t .

(2.4)

(2.5)


711

Using that x = i∇ p , the first term on the r.h.s. of the last equation is given by

2 2 2 2 −i tr x p + m x, p + m Q t = tr ( p · x + x · p) Q t . To control the second term on the r.h.s. of (2.5), let VQ t (x) = (|.|−1 ∗ ρ Q t )(x). Then

2 2 iκ tr x p + m x, VQ t Q t 2 2 2 2 = iκ tr x p + m x VQ t (x) − VQ t (x)x p + m x Q t

2 2 2 = iκ tr p + m , x VQ t (x) Q t p p −κ tr · x VQ t (x) + VQ t (x)x · Qt . p2 + m 2 p2 + m 2 Therefore, using Lemma 2.4, we find

iκ tr x p 2 + m 2 x, VQ Q t ≤ κ N p 2 + m 2 , x 2 VQ (x) + κ N x VQ (x) t t t ≤ κ N x VQ t (x) + x 2 ∇VQ t . (2.6) By Newton’s law (Lemma 2.2), we have x VQ t = sup |x|

dy

x∈R3

and

ρ Q t (y) ≤ CN |x − y|

ρ Q t (y) ≤ C N. x ∇VQ t = sup |x| ∇x dy |x − y| x∈R3 2

2

(2.7)

(2.8)

From (2.6), it follows that the second term on the r.h.s. of (2.5) is bounded, in absolute value, by

iκ tr x p 2 + m 2 x, VQ Q t ≤ C κ N 2 . (2.9) t Finally, we consider the third term on the r.h.s. of (2.5). To this end, we decompose the density Q t in a sum over orthogonal projections Qt =

N

|ψ j,t ψ j,t |

with

ψ j,t , ψi,t = δi j .

(2.10)

j=1

It is easy to see that the wave functions {ψ j,t } Nj=1 are actually the solution of the Hartree-Fock system (1.3) with initial data {ψ j } Nj=1 chosen so that, at time t = 0, ψi , ψ j = δi j and Q = Nj=1 |ψ j ψ j | (it is then easy to show that the r.h.s. of (2.10)

712


is a solution to (1.5); from the local uniqueness of the solution to (1.5), we obtain (2.10)). Using this decomposition of the density Q t , we find

2 2 tr x p + m x, R Q t Q t N 2 2 2 2 2 2 = ψ j,t , p + m x R Q t − R Q t x p + m ψ j,t j=1

N +i ψ j,t ,

p

· x R Qt + R Qt x ·

p

ψ j,t p2 + m 2

N 1 2 2 2 = ∗ ψ j,t ψ i,t p +m ,x ψ j,t , ψi,t |.| i, j=1 N 1 p +i Re ·x ψ j,t , ∗ ψ j,t ψ i,t ψi,t . |.| p2 + m 2 j=1

p2 + m 2

(2.11)

i, j=1

To bound the last term on the r.h.s. of the last equation, we observe that N p 1 ∗ ψ j,t ψ i,t ψi,t ·x ψ j,t , |.| p2 + m 2 i, j=1 N |ψ j,t (y)| |ψi,t (y)| p ≤ ψ j,t (x) |ψi,t (x)| |x| dxdy p2 + m 2 |x − y| i, j=1 2 N p ρ Q t (y) ρ Q t (y) + ψ j,t (x) |x| ≤ dxdy ρ Q t (x) |x| dxdy 2 2 |x − y| |x − y| p +m j

≤ CN , 2

where we used Newton’s law (Lemma 2.2) to perform the y-integration, and then, in the second term, we estimated tr( p/ p 2 + m 2 )Q t ( p/ p 2 + m 2 ) ≤ tr Q t = N . As for the first term on the r.h.s. of (2.11), we can bound its absolute value using Lemma 2.4. We find N

1 ∗ ψ j,t ψ i,t ψ j,t , ψi,t p2 + m 2 , x 2 |.| i, j=1 N p 2 + m 2 , x 2 1 ∗ ψ j,t ψ i,t ≤ |.| i, j=1

N 2 1 1 ≤ x |.| ∗ ψ j,t ψ i,t + x ∇ |.| ∗ ψ j,t ψ i,t . i, j=1

(2.12)


To bound the first contribution, we observe that, by Lemma 2.2, 1 x 1 ∗ ψ j,t ψ i,t = sup |x| dy ψ (y)ψ (y) j,t i,t |.| |x − y| x |ψ j,t (y)|2 + |ψi,t (y)|2 1 ≤ sup |x| dy |x − y| 2 x ρ Q t (y) ≤ sup |x| dy ≤ C N. |x − y| x

713

(2.13)

Next, we consider the second term on the r.h.s. of (2.12). For x ∈ R3 , we can find a rotation R ∈ S O(3) such that x = R(r e3 ), where r = |x| and e3 = (0, 0, 1). Therefore 2 (x − y) (R(r e3 ) − y) 2 ψ j,t (y)ψ i,t (y) = r ψ j,t (y) ψ i,t (y) dy |x| dy |x − y|3 |R(r e3 ) − y|3 2 2 yi ≤ ψ j,t (Ry) ψ i,t (Ry) dy r |r e3 − y|3 i=1 2 r − y3 +r ψ j,t (Ry) ψ i,t (Ry) . dy |r e3 − y|3 (2.14) The last term on the r.h.s. of (2.14) can be estimated by 2 r − y3 |r − y3 | 2 r ≤ r ψ (Ry) ψ (Ry) |ψ j,t (Ry)| |ψi,t (Ry)| dy dy j,t i,t |r e3 − y|3 |r e3 − y|3 |r − y3 | ρt (Ry) ≤ r 2 dy |r e3 − y|3 |r − y3 | ≤ r 2 dy ρt (y). (2.15) |r − y|3 Introducing spherical coordinates for y = s yˆ for yˆ ∈ S 2 , we find ∞ 2 |1 − s ryˆ3 | r − y3 2 r ≤ ≤ C N. ψ (Ry) ψ (Ry) dss ρ (s) d y ˆ dy j,t t i,t |r e3 − y|3 S2 0 |e3 − sryˆ | (2.16) Here we used the fact that, as we prove in Lemma 2.5 below, |1 − λ yˆ3 | < ∞. d yˆ sup 2 |e 3 − λ yˆ | λ>0 S

(2.17)

As for the first term on the r.h.s. of (2.14), we remark that, for example, the summand with i = 1 can be controlled as follows: 2 (r − y3 )y1 y1 r r dy ≤ ψ (Ry)ψ (Ry) ψ (Ry)ψ (Ry) dy j,t j,t i,t i,t 3 3 |r e3 − y| |r e3 − y| y1 . + r dy y3 ψ (Ry)ψ (Ry) j,t i,t 3 |r e3 − y| (2.18)

714


Observing that y1 r dy y3 ψ j,t (Ry) ψ i,t (Ry) 3 |r e3 − y| 1 = r dy y3 ∂ y1 ψ j,t (Ry) ψ i,t (Ry) |r e3 − y| 1 ≤ r dy y3 ∂ y1 − y1 ∂ y3 ψ j,t (Ry)ψ i,t (Ry) |r e3 − y| |r − y3 | + r dy |y1 | |ψ j,t (Ry)||ψi,t (Ry)|, |r e3 − y|3 it follows from (2.18) that (using the notation ϕ j,t (y) = ψ j,t (Ry)) 2 y1 r ψ j,t (Ry)ψ i,t (Ry) dy 3 |r e3 − y| 1 1 ≤ r dy L 2 ϕ j,t (y)ϕ i,t (y) + r dy L 2 ϕ i,t (y)ϕ j,t (y) |r e3 − y| |r e3 − y| 1 + r dy |ψ j,t (Ry)||ψi,t (Ry)| |r e3 − y|3 ρt (y) 1 |L 2 ϕ j,t (y)||ϕi,t (y)|+|L 2 ϕi,t (y)||ϕ j,t (y)| +r dy . ≤ r dy |r e3 − y| |r e3 − y|3 (2.19) The last term is bounded by C N by Lemma 2.2. The first term on the r.h.s. of the last equation, can be controlled by 1 |L 2 ϕ j,t (y)||ϕi,t (y)| + |L 2 ϕi,t (y)||ϕ j,t (y)| r dy |r e3 − y| N 1 |L 2 ϕ j,t (y)|2 + |ϕ j,t (y)|2 . ≤r (2.20) dy |r e3 − y| j=1

Note that

N

N

j=1 |ϕ j,t (y)|

2

= ρ Q t (Ry) = ρ Q t (y). Moreover, we have

|L 2 ϕ j,t (y)|2 =

j=1

N

t L 2 )(y, y), L 2 |ϕ j,t ϕ j,t |L 2 (y, y) = (L 2 Q

j=1

t = j |ϕ j,t ϕ j,t |. Since where we defined Q t (x, y) = Q ϕ j,t (x)ϕ j,t (y) = ψ j,t (Rx)ψ j,t (Ry) = Q t (Rx, Ry) = Q t (x, y) j

j

t = Q t . Therefore, from (2.19) and (2.20) we conclude for all x, y ∈ Q that 2 1 y1 r ≤ C N + r dy (L 2 Q t L 2 )(y, y) ψ (Ry)ψ (Ry) dy j,t i,t 3 |r e3 − y| |r e3 − y| 3 1 (L Q t L )(y, y). r dy ≤ CN + |r e3 − y| R3 , it follows that

=1


715

Since 3 =1 (L Q t L )(y, y) is invariant w.r.t. rotations of y, we can apply Lemma 2.2 to obtain that 3 2 y1 r ≤ C N + C ψ (Ry)ψ (Ry) dy dy (L Q t L )(y, y) j,t i,t |r e3 − y|3 =1

= C N + C tr L 2 Q t . Using Lemma 2.3 and inserting the last bound and (2.16) back in (2.14) we find that (x − y) 2 ≤ C N + C tr L 2 Q, |x| dy ψ (y)ψ (y) j,t i,t |x − y|3 and thus, from (2.12), that

N 1 ψ j,t , ψi,t ≤ C N 3 (Q) + C N 2 (Q)L2 (Q). ∗ ψ j,t ψ i,t p2 + m 2 , x 2 |.| i, j=1 From (2.5), we obtain (2.4). Step 2. If Q t is a solution of the Hartree-Fock equation (1.5), we have d tr ( p · x + x · p) Q t ≤ 2 EHF (Q). dt

(2.21)

To show (2.21), we compute d tr( p · x + x · p) Q t dt

1 ∗ ρ Qt + κ R Qt , Q t = −i tr ( p · x + x · p) p2 + m 2 − κ |.|

1 ∗ ρ Qt + κ R Qt Q t . = −i tr ( p · x + x · p), p 2 + m 2 − κ |.|

(2.22)

Now we observe that

p2 − i tr ( p · x + x · p), p 2 + m 2 Q t = 2 tr Q t ≤ 2 tr p 2 + m 2 Q t p2 + m 2 (2.23) and

1 1 ∗ ρ Q t Q t = 2κ tr x · ∇ ∗ ρ Qt Q t iκ tr ( p · x +x · p), |.| |.| (x − y) = −2κ dx dy x · ρ Q t (y) ρ Q t (x) |x − y|3 (x − y) ρ Q t (y) ρ Q t (x) = 2κ dx dy y · |x − y|3 1 = −κ dx dy ρ Q (y) ρ Q t (x). (2.24) |x − y| t

716


As for the contribution to (2.22) from the term with R Q t , we have −iκ tr (x · p + p · x), R Q t Q t = −2iκ tr x · p , R Q t Q t = −2iκ tr x · p R Q t Q t − R Q t x · p Q t Q t (x, y) Q t (y, x) − Q t (y, x)x · ∇x Q t (x, y) = −2κ dx dy x · ∇x |x − y| x−y |Q t (x, y)|2 = 2κ dx dy x · |x − y|3 |Q t (x, y)|2 = κ dx dy . (2.25) |x − y| From (2.22), (2.23), (2.24), and (2.25), we obtain (2.21). Step 3. If Q t is a spherically symmetric solution to (1.5), there exists a constant C, only depending from N (Q) and L2 (Q) such that tr M Q t ≤ t 2 EHF (Q) + t (tr (x · p + p · x)Q + C) + tr M Q.

(2.26)

Equation (2.26) follows directly from the statements proven in Step 1 and Step 2, integrating twice over time. Step 4. Conclusion of the proof. From the assumption (2.3) on the initial density Q, it follows immediately that tr M Q = tr x p 2 + m 2 x Q ≤ tr (1 + x 4 + p 2 )Q < ∞ and that |tr (x · p + p · x)Q| ≤ tr (x 2 + p 2 )Q < ∞. Thus, if EHF (Q) < 0, (2.26) contradicts, for t large enough, the non-negativity of the expectation tr M Q t . This implies immediately that the maximal existence time T for the local solution Q t is finite. From the blow-up alternative (see Theorem 1), it follows that there exists T < ∞ with Q t H1/2 = tr (1 − )1/2 Q t → ∞ as t → T − . In the proof of Theorem 2, we had to control commutators of the pseudo-differential operator p 2 + m 2 with certain multiplication operators (see for example (2.12)). In this respect, it turns out that the Calderon-Zygmund theory of singular integrals is very useful. Lemma 2.4. Suppose m > 0, p = −i∇. Then, for every f ∈ W 1,∞ (R3 ), we have

p 2 + m 2 , f (x) ≤ C ∇ f ∞ . (2.27) A proof of this lemma can be found in [8]; see in particular the corollary on p. 309. The statement of this corollary does not give an effective bound on the norm of the commutator. However, the corollary is based on Theorem 3, on p. 294 of [8], whose proof provides the effective control we need (a remark in this sense can be found in paragraph 3.3.5, on p. 305 of [8]). Finally, in the next lemma we give a proof of the bound (2.17).


Lemma 2.5. We have

sup λ≥0

Proof. First, we observe that

S2

d yˆ

|1 − λ yˆ3 | < ∞. |e3 − λ yˆ |3

sup

717

λ 2 S 2

dy

|1 − λy3 | ≤C |e3 − λy|3

(2.28)

(2.29)

because, in this regime of λ, there is no singularity from the denominator. On the other hand, for arbitrary λ ∈ [1/2, 2], we have π |1 − λ yˆ3 | |1 − λ cos θ | d yˆ = 2π dθ sin θ 3/2 3 |e3 − λ yˆ | S2 0 λ2 sin2 θ + (1 − λ cos θ )2 π |1 − λ cos θ | = 2π dθ sin θ 3/2 λ2 −1 0 + 1 − λ cos θ 2 1+λ 2π |z| 2π 2λ |x + (1 − λ)| = dz dx 3/2 = 3/2 . 2 λ 1−λ λ 0 (λ−1)2 λ −1 + z + x 2 2 Therefore 2λ 2λ |1 − λ yˆ3 | |x| (1 − λ) d yˆ ≤C dx dx 3/2 + C 3/2 3 2 2 |e − λ y ˆ | (λ−1) (λ−1)2 3 S 0 0 + x + x 2 2 4 ∞ dx 1 ≤C +C dx ≤ C. 1/2 (1 + x)3/2 0 |x| 0 Acknowledgments. The authors would like to thank Lei Zhang and Enno Lenzmann for very helpful remarks. B. S. is on leave from the University of Cambridge, UK. His research is supported by a Sofja Kovalevskaja Award of the Alexander von Humboldt Foundation.

References 1. Chandrasekhar, S.: Phil. Mag. 11, 592 (1931); Astrophys. J. 74, 81 (1931); Rev. Mod. Physics 56, 137 (1984) 2. Elgart, A., Schlein, B.: Mean field dynamics of boson stars. Comm. Pure Appl. Math. 60(4), 500–545 (2007) 3. Fröhlich, J., Lenzmann, E.: Blowup for nonlinear wave equations describing boson stars. Commun. Pure Appl. Math. 60(11), 1691–1705 (2007) 4. Fröhlich, J., Lenzmann, E.: Dynamical collapse of white dwarfs in Hartree- and Hartree-Fock theory. Commun. Math. Phys. 274(3), 737–750 (2007) 5. Lieb, E.H., Yau, H.T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112(1), 147–174 (1987) 6. Lieb, E.H., Yau, H.T.: A rigorous examination of the Chandrasekhar theory of stellar collapse. Astrophys. J. 323, 140–144 (1987) 7. Lieb, E.H., Thirring, W.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984) 8. Stein, E.: Harmonic Analyis. Princeton, NJ: Princeton University Press, 1993 Communicated by I. M. Sigal


Communications in


Contact Spheres and Hyperkähler Geometry Hansjörg Geiges1, , Jesús Gonzalo Pérez2, 1 Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany.


2 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain.

E-mail: [email protected] Received: 27 March 2008 / Accepted: 9 May 2008 Published online: 25 September 2008 – © Springer-Verlag 2008

Abstract: A taut contact sphere on a 3-manifold is a linear 2-sphere of contact forms, all defining the same volume form. In the present paper we completely determine the moduli of taut contact spheres on compact left-quotients of SU(2) (the only closed manifolds admitting such structures). We also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles. This moduli problem leads to a new viewpoint on the Gibbons-Hawking ansatz in hyperkähler geometry. The classification of taut contact spheres on closed 3-manifolds includes the known classification of 3-Sasakian 3-manifolds, but the local Riemannian geometry of contact spheres is much richer. We construct two examples of taut contact spheres on open subsets of R3 with nontrivial local geometry; one from the Helmholtz equation on the 2-sphere, and one from the Gibbons-Hawking ansatz. We address the Bernstein problem whether such examples can give rise to complete metrics. 1. Introduction We begin with the definition of our basic objects of interest. Recall that a contact form on a 3-manifold is a differential 1-form α such that α ∧ dα = 0. Definition 1. A contact sphere is a triple of 1-forms (α1 , α2 , α3 ) on a 3-manifold such that any non-trivial linear combination of these forms (with constant coefficients) is a contact form. In other words, we require that the 3-form (λ1 α1 + λ2 α2 + λ3 α3 ) ∧ (λ1 dα1 + λ2 dα2 + λ3 dα3 ) Partially supported by DFG grant GE 1245/1-2 within the framework of the Schwerpunktprogramm “Globale Differentialgeometrie”. Partially supported by grants MTM2004-04794 and MTM2007-61982 from MEC Spain.

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H. Geiges, J. Gonzalo Pérez

be nowhere zero, i.e. a volume form, for any λ1 , λ2 , λ3 ∈ R with λ21 + λ22 + λ23 = 0. The name ‘contact sphere’ derives from the fact that it suffices to check this condition for points (λ1 , λ2 , λ3 ) on the unit sphere S 2 ⊂ R3 . Definition 2. A contact sphere (α1 , α2 , α3 ) is called taut if the contact form λ1 α1 + λ2 α2 + λ3 α3 defines the same volume form for all (λ1 , λ2 , λ3 ) ∈ S 2 . The requirement for a contact sphere to be taut is equivalent to the system of equations (for i = j) αi ∧ dαi = α j ∧ dα j = 0, αi ∧ dα j = −α j ∧ dαi . A straightforward calculation shows that one can then find a 1-form β and a nowhere zero function Λ such that dαi = β ∧ αi + Λ α j ∧ αk , (1) where (i, j, k) runs over the cyclic permutations of (1, 2, 3). Notice that Λ is defined by αi ∧ dαi = Λ α1 ∧ α2 ∧ α3 . The analogous structure of a (taut) contact circle, defined in terms of two contact forms (α1 , α2 ), was studied in our previous papers [15–17]. In [15] we gave a complete classification of the closed, orientable 3-manifolds1 that admit a taut contact circle or a taut contact sphere: Theorem 3. Let M be a closed 3-manifold. (a) M admits a taut contact circle if and only if M is diffeomorphic to a quotient of the Lie group G under a discrete subgroup Γ acting by left multiplication, where G is one of the following: (i) S 3 = SU(2), the universal cover of SO(3). 2 , the universal cover of PSL2 R. (ii) SL (iii) E2 , the universal cover of the Euclidean group (that is, orientation preserving isometries of R2 ). (b) M admits a taut contact sphere if and only if it is diffeomorphic to a left-quotient of SU(2). In the course of this paper we shall present a new proof, more self-contained than the one given in [15], of the fact that the universal cover of a closed 3-manifold admitting a taut contact sphere is diffeomorphic to S 3 . In [16] we showed that every closed, orientable 3-manifold admits a (non-taut) contact circle, and we gave examples of contact spheres. For instance, S 1 × S 2 ⊂ S 1 × R3 , described in terms of coordinates (θ, x, y, z), does not admit any taut contact circles by Theorem 3, but it admits the contact sphere α1 = x dθ + y dz − z dy, α2 = y dθ + z d x − x dz, α3 = z dθ + x dy − y d x. 1 Our manifolds are always understood to be connected.

Contact Spheres and Hyperkähler Geometry

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In [17] we described deformation spaces for taut contact circles and gave a complete classification of taut contact circles. The present paper achieves the corresponding classification for taut contact spheres. The investigation of taut contact spheres amounts to a systematic study of hyperkähler metrics with a homothety, in a sense made precise below. Constructions of complete hyperkähler metrics with translational invariance have played a prominent role in general relativity and supersymmetric field theories, beginning with the Gibbons-Hawking ansatz [18]. This ansatz will be discussed in Sect. 5.2 in the context of explicit constructions of taut contact spheres with nontrivial local geometry. See [2] for a fairly recent discussion of several constructions related to the Gibbons-Hawking ansatz (Eguchi-Hanson metric, Taub-NUT metric, Atiyah-Hitchin metric). Hyperkähler metrics with a homothety are equally important in physical applications, see [11] and [19]. The latter pays particular attention to homotheties that are hypersurface orthogonal (such homotheties are called dilatations in [19]). This is equivalent to saying that the hyperkähler metric on U × R, where U is some 3-manifold, is the cone metric over a 3-Sasakian metric on U , cf. [3]. For more general information on Sasakian and in particular 3-Sasakian geometry, see the definitive survey [6] or the monograph [7]; some of the definitions will be recalled in Sect. 7. There we use our methods to recover the classification of the closed 3-manifolds admitting 3-Sasakian structures. In contrast with 3-Sasakian structures, taut contact spheres do not, in general, give rise to a cone metric on U × R (in other words, the relevant homothety is not a dilatation; such general homotheties also appear in [11]). This implies that taut contact spheres are definitely more general than 3-Sasakian structures. We elaborate on this point in Sect. 7. By comparison, Kähler metrics on U × R admitting a dilatation correspond to a Sasakian structure on U . (For a classification of the closed 3-dimensional manifolds admitting Sasakian structures see [14].) In [20] it is shown that if the Sasakian analogue of the Kähler potential satisfies a Monge-Ampère equation, then the metric on U is Sasakian-Einstein. This happens in particular if U × R is Ricci-flat, cf. [19]. A taut contact sphere on a closed manifold M always gives rise to a flat hyperkähler metric on M × R (Theorem 10). This must be read as a global rigidity phenomenon, because the theorem fails for open 3-manifolds. In Sect. 5.1 we use a Monge-Ampère equation to construct a taut contact sphere on an open subset U of R3 giving rise to a non-flat hyperkähler metric on U × R. In the Appendix we use a contact transformation to relate this construction to the Helmholtz equation on the 2-sphere. In Sect. 5.2 we use the Gibbons-Hawking ansatz to give an even simpler construction of a non-flat example. In Sect. 6 we discuss the question, known as a Bernstein problem, whether such non-flat examples can give rise to complete metrics. The answer to this question depends on the choice of one of the two natural metrics associated with a taut contact sphere (see Definition 15, Theorem 16, and the comments following it). This paper supersedes our preprint “Contact spheres and quaternionic structures”. 2. Statement of Results We now describe in outline some of the main results of the present paper. Our notational convention throughout will be that M denotes a closed, orientable 3-manifold; U will denote a 3-manifold (without boundary) that need not be compact. The relation (α1 , α2 , α3 ) ∼ (vα1 , vα2 , vα3 ) for some smooth function v : U → R+ is easily seen to be an equivalence relation within the set of (taut) contact spheres.

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Definition 4. Two (taut) contact spheres are conformally equivalent if one is obtained from the other by multiplying each contact form by the same positive function. Definition 5. We call a contact sphere naturally ordered if αi ∧ dαi is a positive multiple of α1 ∧ α2 ∧ α3 . Throughout this paper we shall assume our contact spheres to satisfy this condition. Since αi ∧ dαi and α1 ∧ α2 ∧ α3 scale with the second and third power of v, respectively, it is obvious that every conformal equivalence class of naturally ordered taut contact spheres contains, for any c ∈ R+ , a unique representative satisfying αi ∧ dαi = c α1 ∧ α2 ∧ α3 , i = 1, 2, 3.

(2)

Definition 6. We call a taut contact sphere (α1 , α2 , α3 ) c-normalised if it satisfies Eq. (2). Remark 7. This condition is equivalent to Λ ≡ c in Eq. (1); beware that β in that equation is not an invariant of the conformal equivalence class. It is implicit in [13] and follows by a simple extension of the ideas from [15] that a taut contact sphere on U gives rise to a hyperkähler structure on U × R. In Sect. 3 we analyse this situation a little more carefully. One of the results proved there is the following. Proposition 8. A contact sphere on U determines an oriented conformal structure on U × R. A naturally ordered taut contact sphere on U determines a hyperkähler structure on U × R. Conformally equivalent (taut) contact spheres determine isomorphic conformal (resp. hyperkähler) structures. As we shall see in Sect. 3, the hyperkähler structure (g, J1 , J2 , J3 ) on U × R induced by a taut contact sphere (α1 , α2 , α3 ) on U is given by the equations −g(·, Ji ·) = d(et αi ) =: Ωi , i = 1, 2, 3, where t denotes the R-coordinate, and the complex structures Ji are ∂t -invariant. Often we write the hyperkähler structure as the triple of symplectic forms (Ω1 , Ω2 , Ω3 ). These symplectic forms are homogeneous of degree 1 with respect to the vector field ∂t , that is, L ∂t Ωi = Ωi . The hyperkähler metric g has the same homogeneity. For reversing this construction, it is useful to make the following definition. Definition 9. A vector field Y on a hyperkähler manifold is called tri-Liouville if it is a Liouville vector field for every parallel self-dual 2-form Ω, that is, L X Ω = Ω. Notice that a tri-Liouville vector field is automatically homothetic for the hyperkähler metric, but the converse is not true. The construction of taut contact spheres in this paper uses two ingredients: a hyperkähler metric g and a nowhere zero tri-Liouville vector field Y . For any conformal basis (Ω1 , Ω2 , Ω3 ) of parallel self-dual 2-forms, the corresponding taut contact sphere is defined on any transversal to the flow of Y by restricting the triple of 1-forms (i Y Ω1 , i Y Ω2 , i Y Ω3 ) to that transversal. For the formal statement see Proposition 21. We also show that for a naturally ordered taut contact sphere (α1 , α2 , α3 ) on U there is the following pointwise model on U × R for the triple of symplectic forms


723

(Ω1 , Ω2 , Ω3 ), expressed in quaternionic notation: At any point x ∈ U × R, there is a quaternionic coordinate dqx for the tangent space Tx (U × R) such that d(et (iα1 + jα2 + kα3 ))x = −dqx ∧ dq x . The key to the classification of taut contact spheres is then the following statement. Theorem 10 (Global rigidity). Let M be a closed 3-manifold. Then any hyperkähler structure (g, J1 , J2 , J3 ) on M × R with ∂t as a homothetic vector field must be flat. In particular, this applies to hyperkähler metrics arising from a taut contact sphere on M. Proof. Hyperkähler metrics are always Ricci flat [4, 14.13], and any Ricci flat Kähler manifold of complex dimension 2 is anti-self-dual [1], that is, the self-dual part W + of the Weyl tensor of the metric g vanishes. Since the Weyl tensor is an invariant of the conformal class of a metric [4, 1.159], W + also vanishes for the metric g/g(∂t , ∂t ), which descends to the quotient M × S 1 of M × R under the map ( p, t) → ( p, t + 1), say. (In fact, the hyperhermitian structure (g/g(∂t , ∂t ), J1 , J2 , J3 ) descends to that quotient, and one may also appeal to a result of Boyer [5] saying that a hyperhermitian metric on a 4-manifold is anti-self-dual.) Then the signature formula for (M × S 1 , g/g(∂t , ∂t )) yields 1 τ (M × S 1 ) = W + 2 − W − 2 12π 2 M×S 1 1 =− W − 2 . 12π 2 M×S 1 But τ (M × S 1 ) = 0 for purely topological reasons. So the Weyl tensor W = W + + W − vanishes for g/g(∂t , ∂t ). Again appealing to the conformal invariance of the Weyl tensor, we deduce that it also vanishes for g. For Ricci flat metrics (thus, in particular, for the hyperkähler metric g) the full curvature tensor equals its Weyl part [4, 1.116]. This proves the theorem.

This theorem implies that on a closed manifold the pointwise model above for the triple of symplectic forms d(et αi ), i = 1, 2, 3, coming from a taut contact sphere, is actually a local model, since the three symplectic forms and the hypercomplex structure (J1 , J2 , J3 ) are all parallel with respect to the flat hyperkähler metric g. It is then not very difficult, using properties of the ∂t -flow, to derive the following classification statement. The proof will be given in Sect. 4. Theorem 11. If M is diffeomorphic to a lens space L(m, m − 1), including the 3-sphere L(1, 0), then the (naturally ordered) taut contact spheres on M, up to diffeomorphism and conformal equivalence, are given by the following family of Zm -invariant quaternionic 1-forms on S 3 ⊂ H: 1 (dq · q − q · dq) − ν d(qiq), ν ∈ R, 2 where ν and −ν determine equivalent structures. If M is diffeomorphic to Γ \SU(2) with Γ ⊂ SU(2) a non-abelian group, there is a unique equivalence class of taut contact spheres on M, described by the formula above with ν = 0. All these taut contact spheres are homogeneous under a natural SO(3)-action. In particular, all great circles in a given taut contact sphere are isomorphic taut contact circles. iα1 + jα2 + kα3 =

724


Note that the manifolds listed in this theorem exhaust all the possible left-quotients of SU(2), cf. [15]. The Zm -action on S 3 that produces the quotient L(m, m − 1) is generated by right multiplication with cos(2π/m) + i sin(2π/m). Remark 12. As we shall see in the construction of the moduli space of taut contact spheres, all taut contact spheres on a given closed 3-dimensional manifold yield the same 4-dimensional metric g up to global isometry. So it is not this induced metric which determines the modulus, but in fact the different possible homothetic vector fields ∂t . In the cases where non-trivial moduli exist, neither the vector field ∂t nor the function et (which turns out to equal g(∂t , ∂t )) are unique for that g. With our coordinate conventions in Sect. 3 below, which seem natural in that context, left multiplication on C2 by a −b ∈ SU(2), b a where a = a1 + ia2 , b = b1 + ib2 , |a|2 + |b|2 = 1, corresponds to right multiplication on H by the unit quaternion u = a1 + ia2 + jb1 + kb2 . The quaternionic 1-form dq · q − q · dq is invariant under this right multiplication q → qu by unit quaternions u, and therefore descends to all left-quotients of SU(2). The 1-form d(qiq) is invariant under right multiplication by unit quaternions of the special form a1 + ia2 , hence it descends to all abelian quotients. Here are some details about what we mean by ‘homogeneity under a natural SO(3)action’ in Theorem 11. The action of SO(3) on iα1 + jα2 + kα3 rotates the contact forms and is given by conjugating the quaternionic 1-form by elements u ∈ S 3 ⊂ H. This action is induced from the S 3 -action q → uq on H, and that latter action cannot be replaced by an SO(3)-action. This amounts to a spinor phenomenon, see also the proof of Proposition 20. Since this left multiplication by unit quaternions commutes with all quaternionic right multiplications, it descends to all left-quotients of SU(2). Remark 13. With q = x0 + ix1 + jx2 + kx3 we obtain the following real expression for the taut contact sphere in Theorem 11 corresponding to ν = 0, where (i, j, k) runs over the cyclic permutations of (1, 2, 3): αi = x0 d xi − xi d x0 + x j d xk − xk d x j . Notice that it satisfies (1) with β ≡ 0 and Λ ≡ 2. In particular, it is a 2-normalised taut contact sphere. Restricting this to the hyperplane {x0 = 1}, we obtain a simple expression for a taut contact sphere on R3 : αi = d xi + x j d xk − xk d x j . See Proposition 21 for the principle behind this observation. One might suspect that taut contact spheres constitute such a rigid structure that Theorem 10 would also hold locally and for open manifolds. However, this turns out to be false, even conformally. Theorem 14. There are examples of taut contact spheres on open domains U in R3 inducing a metric on U × R that is not conformally flat.


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This theorem will be proved in Sect. 5, where we present two methods for constructing such examples. Our first construction, in Sect. 5.1, starts from the observation that (locally) the conditions for a taut contact sphere lead to a complex MongeAmpère equation. After imposing an additional homogeneity amounting to the existence of a tri-Hamiltonian vector field, we are able to find concrete solutions of that equation whose associated 4-dimensional metric is non-flat. In fact, subject to this extra homogeneity, the Monge-Ampère equation can be linearised to yield the Helmholtz equation ∆u + 2u = 0 for the Laplacian ∆ on the 2-sphere. Solutions of that Helmholtz equation then give rise to taut contact spheres with a tri-Hamiltonian symmetry. Our second construction, in Sect. 5.2, starts from the Gibbons-Hawking ansatz, which is essentially a construction of an R-invariant hyperkähler metric on U0 × R, starting from a harmonic function on an open subset U0 of R3 . For an appropriate choice of such a harmonic function, one obtains a non-flat hyperkähler metric giving rise to a taut contact sphere on a suitable hypersurface U ⊂ U0 × R. Beware that the R-factor in this splitting U0 × R is not the one corresponding to the tri-Liouville vector field. Taut contact spheres come associated with two natural metrics on the 3-manifold. In order to describe these, we observe that the 2-forms Ωi = d(et αi ) can be written with the help of the structure equation (1) as follows: Ωi = et Λ−1/2 (dt + β) ∧ Λ1/2 αi + Λ1/2 α j ∧ Λ1/2 αk . So the hyperkähler metric g is given by g = et Λ−1 (dt + β)2 + Λ (α12 + α22 + α32 ) .

(3)

In particular, if the contact sphere is 1-normalised, we have g = et (dt + β)2 + α12 + α22 + α32 .

(4)

This motivates the following definition. Definition 15. The short metric associated with a 1-normalised taut contact sphere (α1 , α2 , α3 ) on U is the metric gs = α12 + α22 + α32 . The long metric associated with this contact sphere is gl = β 2 + α12 + α22 + α32 . Observe that gl is simply the restriction of g to U ≡ U × {0}, so from the viewpoint of hyperkähler geometry this is the more natural metric to consider. In Sect. 4 we shall must be S 3 prove that in the case of a compact 3-manifold M, whose universal cover M by Theorem 3, the long metric is spherical while the short metric is a Berger metric. In either of the above-mentioned constructions of taut contact spheres on U giving rise to a non-flat hyperkähler metric on U × R, the induced long metric gl on U is incomplete, and so is, a fortiori, the short metric gs . In Sect. 6 we raise the question whether one can find such examples where gl , at least, is complete. This type of question is known as a Bernstein problem [9]. Concerning gs , we provide a partial answer to this problem for contact spheres with additional symmetries. For gl there is a positive answer to the Bernstein problem, even subject to the additional symmetry requirement:

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Theorem 16. Let (α1 , α2 , α3 ) be a 1-normalised taut contact sphere on U giving rise to the short metric gs and the long metric gl on U . (a) If gs is complete and admits a non-trivial Killing vector field that preserves (α1 , α2 , α3 ), then U is necessarily compact, and hence (α1 , α2 , α3 ) belongs to the family described in Theorem 11 (and in particular gives rise to a flat hyperkähler metric). (b) There are examples of S 1 -invariant taut contact spheres on U = D × S 1 , where D is the open unit disc, for which gl is complete. The induced hyperkähler metrics on U × R are not flat. Part (a) will be proved in Sect. 6. Theorem 24 in that section is a more explicit reformulation of this part. We reserve the proof of part (b) for a forthcoming paper. It turns out that in the R-or S 1 -invariant context one can use the Gibbons-Hawking ansatz in order to develop a complete theory of such contact circles. An infinite-dimensional family of examples giving rise to complete long metrics is then found with the help of Blaschke products on D ⊂ C. 3. Hyperkähler Linear Algebra In this section we discuss the linear algebraic aspects of contact spheres, leading to a proof of Proposition 8. This prepares the ground for the proof of Theorem 11. Let (α1 , α2 , α3 ) be a contact sphere on a 3-manifold U . This gives rise to the symplectic forms Ωi = d(et αi ), i = 1, 2, 3, on U × R. At any point x of U × R, these symplectic forms span a definite 3-plane in the space 2 Tx∗ (U × R) of skew-symmetric bilinear forms on the tangent space Tx (U × R). If the contact sphere is taut, we have in addition the identities (for i = j) Ωi ∧ Ωi = Ω j ∧ Ω j = 0, Ωi ∧ Ω j = 0. First we are going to study the linear algebra of this situation. Thus, let V4 be a 4-dimensional real vector space and write V6 = 2 V4∗ . Consider the quadratic form Q : V6 −→ R, Q(A) = A ∧ A, of signature (3, 3). We call a triple (A1 , A2 , A3 ) of elements of V6 a symplectic triple on V4 if it spans a definite 3-plane for Q in V6 , and a conformal symplectic triple if the stronger condition Ai ∧ Ai = A j ∧ A j = 0, Ai ∧ A j = 0, is satisfied for i = j, cf. [13]. The same terminology will be used for triples of symplectic forms (Ω1 , Ω2 , Ω3 ) on a 4-manifold as described above. Remark on Notation. In the sequel, any equation (or other statement) involving the indices i, j, k is meant to be read as three equations, with √ (i, j, k) ranging over the cyclic permutations of (1, 2, 3). We write boldface i for −1 ∈ C, and boldface i, j, k for the standard quaternionic units with ij = k. The relation between real, complex, and quaternionic coordinates will be given by z 1 = x0 + ix1 , z 2 = x2 + ix3 ; q = x0 + ix1 + jx2 + kx3 = z 1 + z 2 j.


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( p,q) For J a complex structure on V4 , we denote by J the space of exterior forms of type ( p, q) on V4 . As shown in [15], a conformal symplectic couple (Ai , A j ) determines (2,0) a unique complex structure Jk on V4 for which Ai + i A j ∈ Jk . Thus, a conformal symplectic triple (A1 , A2 , A3 ) induces three complex structures J1 , J2 , J3 . In [13] it was shown that these complex structures satisfy the quaternionic identities as well as the relations g(v, w) = Ai (v, Ji w), i = 1, 2, 3, for all v, w ∈ V4 , for a unique definite symmetric bilinear form g. We call a conformal symplectic triple naturally ordered if this g is positive definite. Notice that the sign of g is well defined, since the (−J1 , −J2 , −J3 ) do not satisfy the quaternionic identities. (In particular, hyperkähler structures are always naturally ordered.) An alternative way to define this definite bilinear form g is via the identity (v

A1 ) ∧ (v

A2 ) ∧ (v

A3 ) =

1 g(v, v) v 2

(Ai2 ) for all v ∈ V4 .

(5)

This is an obvious consequence of the SO(3)-homogeneous normal form for conformal symplectic triples discussed in the next two propositions. Proposition 17. Let (A1 , A2 , A3 ) be a naturally ordered conformal symplectic triple on V4 . Then there are real linear coordinates d x0 , d x1 , d x2 , d x3 on V4 such that Ai = d x 0 ∧ d x i + d x j ∧ d x k . In terms of the corresponding complex and quaternionic coordinates we have i (dz 1 ∧ dz 1 + dz 2 ∧ dz 2 ), 2 A2 + i A3 = dz 1 ∧ dz 2 , A1 =

and 1 i A1 + jA2 + k A3 = − dq ∧ dq. 2 Remark 18. Because of the non-commutativity of H some care is necessary in interpreting the wedge product of H-valued 1-forms α, β on a vector space V . Our convention is to read it as (α ∧ β)(v, w) = α(v)β(w) − α(w)β(v) for all v, w ∈ V. This ensures α ∧ qβ = αq ∧ β, and that α ∧ α is always purely imaginary. and A1 ∧ (A2 + i A3 ) = 0 we Proof of Proposition 17. From 0 = A2 + i A3 ∈ (2,0) J1 conclude that the (0, 2)-part of A1 with respect to J1 is zero. The form A1 being real, its (2, 0)-part must also vanish, hence A1 ∈ (1,1) J1 . 1 (1,0) Let {1 , 2 } be a basis for J1 and set = . Then we can write 2

A1 = i( )T ∧ A1 ,

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with A1 a hermitian (2 × 2)-matrix. By our assumption on (A1 , A2 , A3 ) being naturally ordered, the matrix A1 is positive definite. Hence there is a matrix C ∈ GL2 (C) such that 1/2 0 T , C A1 C = 0 1/2 and so in terms of the basis {1 , 2 } for A1 =

(1,0) J1

defined by = C we have

i (1 ∧ 1 + 2 ∧ 2 ) 2

and A2 + i A3 = c 1 ∧ 2 for some c ∈ C. We then find |c|2 1 ∧ 2 ∧ 1 ∧ 2 = (A2 + i A3 ) ∧ (A2 − i A3 ) = A22 + A23 = 2 A21 = −1 ∧ 1 ∧ 2 ∧ 2 , from which we conclude |c| = 1. The linear complex coordinates z 1 , z 2 corresponding to {c 1 , 2 } then give the desired complex normal form. The real and quaternionic normal forms can be derived easily from the complex one.

Remark 19. (1) Notice that in terms of these pointwise coordinates we have g = d x12 + d x22 + d x32 + d x42 and (J1 , J2 , J3 ) = (i, j, k). Moreover, we recognise the 3-plane in V6 spanned by A1 , A2 , A3 as the space of self-dual 2-forms for the metric g and √ the orientation of V4 defined by Ai ∧ Ai . The length of the Ai equals 2. (2) Here is another characterisation of taut contact spheres that can be read off from the preceding proposition: The purely imaginary 1-form α = iα1 + jα2 + kα3 defines a taut contact sphere if and only if α1 is a contact form and at each point x of the manifold there is an H-valued linear form βx on the tangent space at x such that d(et α)x = ±βx ∧ β x . The case d(et α)x = −βx ∧ β x corresponds to (α1 , α2 , α3 ) being naturally ordered. The following proposition will be an important ingredient in the proof of Proposition 8, while the last part of its proof is used in the proof of Theorem 11. Proposition 20 (The spinor equivariance). There is a one-to-one correspondence between oriented conformal structures on V4 and definite 3-planes V3 in V6 (with respect to Q). Proof. Given an oriented conformal structure on V4 , define V3 as the corresponding space of self-dual 2-forms on V4 . For the converse, we recall some quaternionic linear algebra. Under the identification of the purely imaginary quaternions with R3 , any element φ of SO(3) can be written as quaternionic conjugation φ = φu , R3 x −→ φu (x) = uxu,


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with some unit quaternion u ∈ S 3 ⊂ H. The map u → φu is the standard double covering S 3 → SO(3). Given a definite 3-plane in V6 , choose a naturally ordered conformal symplectic triple (A1 , A2 , A3 ) spanning it. An orientation on V4 is then given by Ai ∧ Ai . Let q be a quaternionic coordinate for V4 as in Proposition 17, and g = d x12 + d x22 + d x32 + d x42 the inner product on V4 determined by (A1 , A2 , A3 ). Suppose that A1 , A2 , A3 is another naturally ordered conformal symplectic triple spanning the same 3-plane V3 and satisfying Ai ∧ Ai = Ai ∧ Ai . Notice that Q defines an inner product on V3 for which (A1 , A2 , A3 ) and (A1 , A2 , A3 ) are orthogonal bases consisting of vectors of equal length, and defining the same orientation. Hence there is an element φu ∈ SO(3) such that ⎛ ⎞ ⎛ ⎞ A1 A1 ⎝ A2 ⎠ = φu ⎝ A2 ⎠ . A3 A3 By the preceding discussion this can be written as i A1 + jA2 + k A3 = u(i A1 + jA2 + k A3 )u 1 = − u dq ∧ dq u 2 1 = − d(uq) ∧ d(uq). 2 So a quaternionic coordinate corresponding to (A1 , A2 , A3 ) is given by uq, which gives rise to the same inner product g on V4 since left multiplication on H by a unit quaternion is an isometry for d x12 + d x22 + d x32 + d x42 . If the conformal symplectic triple (A1 , A2 , A3 ) is replaced by (v A1 , v A2 , v A3 ), v ∈ R+ , then the induced inner product changes to vg. This proves the proposition.

Proof of Proposition 8. A (taut) contact sphere (α1 , α2 , α3 ) on U gives rise to a (conformal) symplectic triple (Ω1 , Ω2 , Ω3 ) on U × R as described at the beginning of this section. The complex structures J1 , J2 , J3 defined on each tangent space Tx (U × R) depend smoothly on the point x, and thus define almost complex structures on U × R, which are integrable in the taut case, see [15]. So the statement concerning contact spheres and conformal structures is immediate from the preceding proposition. Notice that the orientation and conformal class of the metric on U × R are completely characterised as the unique ones for which the Ωi are self-dual. The statement about taut contact spheres and hyperkähler structures follows similarly; see [13] for an explicit argument. A non-taut contact sphere determines at each point a linear 2-sphere worth of complex structures, since pointwise the symplectic triple (Ω1 , Ω2 , Ω3 ) can be replaced by a conformal symplectic triple spanning the same 3-plane of skew-symmetric forms; this does not change the space of corresponding almost complex structures. Although this replacement can be done globally, leading again to triples (J1 , J2 , J3 ) satisfying the quaternionic identities, there does not seem to be a canonical choice for doing it. If two contact spheres are related by multiplication by the function v : M → R+ , the induced structures on U × R are related by the diffeomorphism ( p, t) → ( p, t + log v( p)).

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The following proposition explains how to go back from a hyperkähler structure and a tri-Liouville vector field to a taut contact sphere. Proposition 21. Let (Ω1 , Ω2 , Ω3 ) be a hyperkähler structure on a 4-manifold W with a nowhere zero tri-Liouville vector field Y . Then the equations αi = Y Ωi define a naturally ordered taut contact sphere (α1 , α2 , α3 ) on any transversal of Y . Shifting the points of the transversal along the orbits of Y will change the taut contact sphere within its conformal class.

The proof is a straightforward computation, using Ωi ∧ Ωi = Ω j ∧ Ω j = 0 and Ωi ∧ Ω j = 0 for i = j, cf. [13], as well as the identities Y Ωi2 = 2αi ∧ dαi , cf. identity (5). If the hyperkähler structure comes from a taut contact sphere, then the above construction with Y = ∂t recovers that contact sphere. 4. Classification of Taut Contact Spheres This section is largely devoted to the proof of Theorem 11. As promised, this includes a new proof that the universal cover of a closed 3-manifold carrying a taut contact sphere is diffeomorphic to S 3 . The strategy for the proof is as follows. From Theorem 10 we know that a taut contact sphere on a closed 3-manifold M induces a flat metric on M × R. With the help of the developing map of this metric we are led to study hyperkähler structures on Euclidean 4-space E4 , with the hyperkähler metric being the flat Euclidean metric. Parallel 2-forms for this metric have constant coefficients in Euclidean coordinates. The classification of taut contact spheres is thus reduced to a straightforward problem of determining the possible tri-Liouville vector fields. Proof of Theorem 11. Let M be a closed 3-manifold with a taut contact sphere, and let Write g for the (α1 , α2 , α3 ) be the lifted taut contact sphere on its universal cover M. induced metric on M × R. The fact that g is flat and M × R simply-connected implies × R → E4 for this metric which is a local that we have a developing map Φ : M isometry. × R is a domain on which Φ restricts to a diffeomorphism, then (Φ|W )∗ ∂t If W ⊂ M is a vector field YW on the domain Φ(W ) ⊂ E4 generating a 1-parameter group of homotheties of the Euclidean metric (because L ∂t g = g by the construction of g). Since homothetic transformations of a Riemannian manifold are affine transformations (this is easy to see for the Euclidean metric), YW is the restriction Y |Φ(W ) of a homothetic vector field Y defined on all of E4 . Then ∂t and Φ ∗ Y are homothetic vector fields for g × R. that coincide on the open set W , which forces ∂t = Φ ∗ Y on all of M A homothetic vector field on E4 vanishes at a single point, and without loss of generality we may assume that Y vanishes at 0. Let π : E4 \ {0} → SE3 be the projection onto the orbit space of Y (which we can identify with the unit sphere SE3 ⊂ E4 , for Y is a genuinely expanding homothetic vector field and hence transverse to any sphere centred at 0). Then the composition Φ π × {0} −→ E4 \ {0} −→ SE3 M

being simply-connected, a diffeomorphism. is a local diffeomorphism and therefore, M 3 . Moreover, the property ∂ = Φ ∗ Y implies that Φ is a ∼ So we have proved M S = t diffeomorphism from S 3 × R to E4 \ {0}, hence a global isometry.


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Now Φ sends S 3 × {0} to some transversal From now on we write S 3 instead of M. 4 of Y in E , and by Proposition 21 the original contact sphere is equivalent to the one induced on SE3 . To simplify notation, we continue to write et αi , Ωi , Ji for the push-forwards of these objects to E4 \ {0}, and we identify ∂t with Y . Thus (J1 , J2 , J3 ) defines a hyperkähler structure with respect to the Euclidean metric gE , and (Ω1 , Ω2 , Ω3 ) are the corresponding Kähler forms. In particular, the Ji and Ωi are parallel with respect to gE , and thus have constant coefficients in any linear coordinate system for E4 . As a consequence, there are linear coordinates x0 , x1 , x2 , x3 on E4 with respect to which the formulae of Proposition 17 hold (with Ai replaced by Ωi ). Observe that this forces x0 , x1 , x2 , x3 to be an orthonormal coordinate system with respect to gE . Write ψt for the flow of Y . This flow commutes with the Ji and satisfies ψt∗ gE = et gE and ψt∗ Ωi = et Ωi , in particular ψt∗ (dz 1 ∧ dz 2 ) = et dz 1 ∧ dz 2 . Since the flow of Y preserves J1 , we have a holomorphic vector field YC on E4 = C2 with Y = 2 Re(YC ). As a homothetic vector field vanishing at zero, Y can be represented as a linear map, and this implies that YCcorresponds to a complex linear map z → YC z z1 . with respect to the coordinate z = z2 A straightforward calculation shows that the condition ψt∗ gE = et gE translates into the matrix YC being of the form 1/2 0 + ZC YC = 0 1/2 with ZC a skew-Hermitian matrix; the condition ψt∗ (dz 1 ∧ dz 2 ) = et dz 1 ∧ dz 2 forces ZC to have zero trace. After a special unitary change of coordinates (which does not change the expressions for the Ωi ) we may assume that ZC is in diagonal form, i.e. iν 0 ZC = 0 −iν with ν ∈ R. In the usual notation for vector fields this means

1

1 + iν z 1 ∂z 1 + − iν z 2 ∂z 2 . YC = 2 2 We conclude et α1 = ∂t =Y

Ω1 i (dz 1 ∧ dz 1 + dz 2 ∧ dz 2 ) 2

i ν (z 1 dz 1 − z 1 dz 1 + z 2 dz 2 − z 2 dz 2 ) − d |z 1 |2 − |z 2 |2 , 4 2 et (α2 + iα3 ) = ∂t (Ω2 + iΩ3 ) = Y (dz 1 ∧ dz 2 )

1

1 + iν z 1 dz 2 − − iν z 2 dz 1 . = 2 2 =

Remark 22. This analysis can be carried out locally. As a result, the quaternionic formula in Theorem 11 is a universal local model for (naturally ordered) taut contact spheres inducing a flat hyperkähler metric.

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To translate the preceding equations into quaternionic notation, we observe q · dq = z 1 dz 1 + z 2 dz 2 − z 1 dz 2 j + z 2 dz 1 j and dq · q = z 1 dz 1 + z 2 dz 2 + z 1 dz 2 j − z 2 dz 1 j (using jz = zj for z ∈ C and z 1 + z 2 j = z 1 − z 2 j), hence dq · q − q · dq = z 1 dz 1 − z 1 dz 1 + z 2 dz 2 − z 2 dz 2 + 2(z 1 dz 2 − z 2 dz 1 )j. We also observe

qiq = |z 1 |2 − |z 2 |2 i − 2iz 1 z 2 j.

Putting all this together, we find et (iα1 + jα2 + kα3 ) = et (iα1 + (α2 + iα3 )j) 1 ν = (dq · q − q · dq) − d(qiq). 4 2 This shows that, up to conformal equivalence and diffeomorphism, the taut contact spheres on S 3 are as described in Theorem 11. The fact that different non-negative values of ν give non-isomorphic contact spheres follows from the corresponding classification of taut contact circles in [15,17]. We shall be a bit more explicit about this point below, where we give a pictorial description of the moduli spaces in question. This will include a synthetic method for determining the modulus, independent of our previous papers. Next we want to show that ν and −ν correspond to equivalent structures. The diffeomorphism of S 3 given by q → qj is isotopic to the identity and pulls the quaternionic 1-form with parameter value ν to that with value −ν, because j anticommutes with i. ∼ After having determined the possible lifted taut contact spheres on M = S 3 , we now consider the taut contact sphere on M itself. From Theorem 3 we know that M has to be a left-quotient of SU(2). The induced flat hyperkähler structure on M × R lifts to just such a structure on S 3 × R, invariant under the deck transformation group Γ . As was already argued in [15] for taut contact circles, this implies that in the complex coordinates (z 1 , z 2 ) which give a normal form as described above, one has Γ ⊂ SU(2). Moreover, again as in [15], the parameter ν is forced to be zero for non-abelian Γ , and it can take any value for the group Γ = Zm ⊂ SU(2) generated by ε 0 , 0 ε−1 where ε is some m th root of unity. Now we address the homogeneity issue in the statement of Theorem 11. For any unit quaternion u we have

1 1 u (dq · q − q · dq) − ν d(qiq) u = (d(uq) · uq − uq · d(uq)) − ν d(uqiuq). 2 2 Therefore, if two taut contact spheres are related by an element φu of SO(3) as in the proof of Proposition 20, then one is the pull-back of the other under a diffeomorphism of Γ \SU(2) induced by the map q → uq. In particular, this SO(3)-action on taut contact spheres shows that any taut contact sphere can be swept out by great circles, all defining isomorphic taut contact circles. This concludes the proof of Theorem 11.


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This theorem allows us to define a map from the space of taut contact spheres to that of taut contact circles: simply pick any great circle. Theorem 23. Let M be any left-quotient of SU(2). The map sending a taut contact sphere on M to any of its great circles induces an embedding of the moduli space of taut contact spheres on M into the moduli space of taut contact circles. Proof. Recall the classification (up to diffeomorphism and homothety, i.e. conformal equivalence and rotation) of taut contact circles (α2 , α3 ) on left-quotients of S 3 , see [15,17]. On the lens spaces L(m, m − 1) we have the continuous family of taut contact circles induced by the Zm -invariant complex 1-form

1

1 α2 + iα3 = + δ z 1 dz 2 − − δ z 2 dz 1 2 2 (restricted to S 3 ⊂ C2 ) with δ ∈ C, −1/2 < Re(δ) < 1/2, modulo replacing δ by −δ, which corresponds to the diffeomorphism defined by (z 1 , z 2 ) → (z 2 , z 1 ) and changing from (α2 , α3 ) to (−α2 , −α3 ). So from Theorem 11 we see that – only the taut contact circles with δ purely imaginary extend to taut contact spheres, – this extension is unique up to automorphisms of α2 + iα3 , and – two taut contact circles giving rise to isomorphic taut contact spheres must be isomorphic. This proves the theorem in the given case. For the abelian left-quotients the map on moduli spaces amounts to the inclusion R+0 −→ {δ ∈ C : − 1/2 < Re(δ) < 1/2}/δ∼−δ , ν −→ iν. The moduli space of taut contact circles on L(m, m − 1) also includes a discrete family described by α2 + iα3 = nz 1 dz 2 − z 2 dz 1 + z 2n dz 2 , where n ranges over the natural numbers congruent −1 mod m. These contact circles, however, do not extend to any taut contact sphere, so they are not being considered here.

The moduli spaces described in the foregoing proof (without the discrete family) are illustrated on the left-hand side of Fig. 1. The moduli space of taut contact circles is the orbifold {δ ∈ C : − 1/2 < Re(δ) < 1/2}/δ∼−δ . The half-line {δ ∈ iR}/δ∼−δ constituting the moduli space of taut contact spheres is shown as a dashed line. The real part {δ ∈ R : − 1/2 < δ < 1/2}/δ∼−δ of this moduli space, shown in bold, corresponds to so-called Cartan structures. The origin represents the unique 3-Sasakian structure. These structures will be discussed in Sect. 7.

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Fig. 1. The moduli spaces of taut contact circles and spheres

Under the mapping δ → δ 2 , the moduli space of taut contact circles is mapped bijectively to the interior of the parabola

x + iy ∈ C : x =

1 − y2 4

shown on the right-hand side of Fig. 1. So this singular mapping flattens the cone point and yields a representation of the moduli space as an open subset of C. Here is the promised synthetic characterisation of the modulus ν of a taut contact sphere on S 3 . One can define a canonical slice inside the hyperkähler manifold S 3 × R as the subset { ∂t = 1}. A straightforward calculation shows that for the taut contact sphere of modulus ±ν we have ∂t 2 =

1 4

+ ν 2 · |z 1 |2 + |z 2 |2 .

√ So the canonical slice is isometric with the 3-sphere of radius 2/ 1 + 4ν 2 . The canonical slice of maximal radius corresponds to the unique 3-Sasakian structure. The canonical slice has the property that the 1-forms et αi induce on it a 1-normalised taut contact sphere. We conclude that the family in Theorem 11 induces on any Euclidean sphere a c-normalised taut contact sphere, for some constant c. Recall now Definition 15. Once the contact sphere is 1-normalised, the canonical slice is given by {t = 0} and we see that the hyperkähler metric induces the long metric gl on it. Thus, on a closed 3-manifold the long metric is always spherical. The short metric can be written as gs = gl − β 2 . We claim that β is invariant under the maps q → uq with u ∈ S 3 , which shows that gs is a Berger metric. To see this invariance property, notice that the system of structure equations (1) is invariant under rotations of the triple (α1 , α2 , α3 ), and we have seen that left multiplication by unit quaternions u induces such rotations. The contact sphere with ν = 0 in Theorem 11 is invariant under right multiplication by unit quaternions, which makes the corresponding β bi-invariant, but recall that in this case β ≡ 0 and so there is no contradiction here.


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We briefly expand on the point in the proof of Theorem 23 concerning the automorphisms of α2 + iα3 . From the arguments in [15, Sect. 5.2] it follows that automorphisms φ of α2 + iα3 are C-linear maps of C2 , in fact, elements of SL2 C. Given an α1 extending (α2 , α3 ) to a taut contact sphere, the other extensions are given by φ ∗ α1 . For S 3 and ν = 0, any element of SL2 C defines an automorphism of α2 + iα3 , so the possible extensions are parametrised by SL2 C/SU(2). For S 3 and ν = 0, the condition that φ preserve α2 + iα3 forces it to be a diagonal map φc (z 1 , z 2 ) = (cz 1 , c−1 z 2 ). The same is true for the lens spaces Zm \SU(2) other than S 3 , even in the case ν = 0; here the condition for φ to be diagonal follows from the fact that it has to lie in the normaliser of Zm . In these two cases, the possible extensions are parametrised by R+ , since φc∗ α1 =

ν i 2 |c| (z 1 dz 1 −z 1 dz 1 ) + |c|−2 (z 2 dz 2 − z 2 dz 2 ) − d |c|2 |z 1 |2 −|c|−2 |z 2 |2 . 4 2

Finally, for the quotients of Γ \SU(2) with Γ non-abelian, the fact that φ lies in the normaliser of Γ forces it to be plus or minus the identity map. Thus the extension α1 is unique. 5. Non-Flat Metrics In this section we describe two constructions of taut contact spheres giving rise to non-flat hyperkähler metrics. These examples constitute a proof of Theorem 14, since a non-flat hyperkähler metric is not even conformally flat (cf. the proof of Theorem 10). 5.1. The Helmholtz equation on the 2-sphere. We have seen in Proposition 21 how to construct a taut contact sphere corresponding to a suitable hyperkähler structure (Ω1 , Ω2 , Ω3 ). Notice that in terms of the holomorphic structure given by J1 , an equivalent description of this hyperkähler structure is given by a holomorphic symplectic form Ω = Ω2 + iΩ3 and a closed real (1, 1)-form Ω1 with 2Ω12 = Ω ∧ Ω. We now make the following ansatz: Let z 1 , z 2 be complex coordinates on C2 and identify C2 with R3 × R by equating the R-direction with the real part of 2z 1 , so that ∂t = Re(∂z 1 ) and ψt (z 1 , z 2 ) = (z 1 + t/2, z 2 ). Set Ω = Ω2 + iΩ3 = λ(z 1 , z 2 ) dz 1 ∧ dz 2 with λ(z 1 , z 2 ) a nowhere zero holomorphic function, and Ω1 =

i ∂∂ H (z 1 , z 2 ) 2

with H (z 1 , z 2 ) a real-valued function. To satisfy the conditions ψt∗ Ωi = et Ωi it is sufficient to have ψt∗ λ = et λ, ψt∗ H = et H.

(6)

Further, the identity 2Ω12 = Ω ∧ Ω is equivalent to the complex Monge-Ampère equation Hz 1 z 1 Hz 1 z 2 = λλ. det (7) Hz 2 z 1 Hz 2 z 2

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If all these conditions are met, Proposition 21 tells us how to recover the corresponding taut contact sphere. We satisfy condition (6) by taking λ = e2z 1 and simplifying the ansatz further to H (z 1 , z 2 ) = e2Re(z 1 ) h(2 Im(z 1 ), 2 Re(z 2 )) = e z 1 +z 1 h(iz 1 − iz 1 , z 2 + z 2 ), with h(s1 , s2 ) a function of two real variables. Then, writing h i for the partial derivatives h si = ∂h/∂si , etc., we have ∂∂ H = e z 1 +z 1 (h + h 11 ) dz 1 ∧ dz 1 + (h 2 − ih 12 ) dz 1 ∧ dz 2 + (h 2 + ih 12 ) dz 2 ∧ dz 1 + h 22 dz 2 ∧ dz 2 , and Eq. (7) becomes

h + h 11 h 2 − ih 12 h 11 h 12 h h 2 = + . 1 = h 2 + ih 12 h 22 h 12 h 22 h 2 h 22

(8)

The reader may check directly that the function s2 / cos s1 h(s1 , s2 ) = cos s1 ξ 2 − 1 dξ 1

satisfies this equation. In the Appendix we shall explain how to derive this sample solution, and verify that it leads to a non-flat hyperkähler metric. We arrive at this example after proving a general result that relates solutions of (8) to solutions of the Helmholtz equation on the 2-sphere. 5.2. The Gibbons-Hawking ansatz. The Gibbons-Hawking ansatz [18] starts with a positive function V in three variables x1 , x2 , x3 (locally on R3 ) and a triple of functions b1 , b2 , b3 in the same variables, satisfying the condition ∇V = −curl(b1 , b2 , b3 ),

(9)

so that in particular ∆V = div(∇V ) = −div(curl(b1 , b2 , b3 )) = 0, i.e. V is harmonic. Set β = b1 d x1 + b2 d x2 + b3 d x3 and consider the triple of 2-forms on R3 × R (with θ denoting the R-coordinate) defined by Ωi = (dθ + β) ∧ d xi + V d x j ∧ d xk . The relation between V and β implies that these 2-forms are closed. Writing θ0 := V −1/2 (dθ + β) and θi := V 1/2 d xi , i = 1, 2, 3, we have Ωi = θ0 ∧ θi + θ j ∧ θk , which shows that (Ω1 , Ω2 , Ω3 ) is a conformal symplectic triple. The corresponding hyperkähler metric is


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θ02 + θ12 + θ22 + θ32 = V −1 (dθ + β)2 + V (d x12 + d x22 + d x32 ). Observe the formal similarity with the metric in (3). The translational invariance of this metric in the θ -direction is obvious.2 A homothety can be built into this ansatz by choosing β appropriately. Here is an example. As domain U0 ⊂ R3 we take the half-space given by the condition x1 > 0, and V (x1 , x2 , x3 ) := x1 is our positive function on that domain. Set (b1 , b2 , b3 ) = (0, x3 , 0), that is, β = x3 d x2 . Then ∇V = (1, 0, 0) = −curl(b1 , b2 , b3 ), i.e. condition (9) is satisfied. Then the Gibbons-Hawking ansatz yields the hyperkähler structure3 Ω1 = (d x0 + x3 d x2 ) ∧ d x1 + x1 d x2 ∧ d x3 , Ω2 = d x0 ∧ d x2 + x1 d x3 ∧ d x1 , Ω3 = (d x0 + x3 d x2 ) ∧ d x3 + x1 d x1 ∧ d x2 , with corresponding hyperkähler metric g=

1 (d x0 + x3 d x2 )2 + x1 (d x12 + d x22 + d x32 ). x1

A tri-Liouville vector field for this hyperkähler structure is Y :=

2 1 x0 ∂x0 + (x1 ∂x1 + x2 ∂x2 + x3 ∂x3 ). 3 3

By Proposition 21, the equations αi = Y Ωi , i = 1, 2, 3, define a taut contact sphere on any transversal U to Y . By construction, g is in turn the hyperkähler metric on U × R induced by this taut contact sphere, with the R-factor now corresponding to the flow lines of Y . The surface Σ := {x2 = x3 = 0} ⊂ U0 × R is totally geodesic for the metric g, since it is the fixed point set of the isometric involution (x0 , x1 , x2 , x3 ) → (x0 , x1 , −x2 , −x3 ). The metric on Σ induced by g is 1 d x 2 + x1 d x12 , x1 0 and, from the well-known formula for computing the Gauß curvature of a metric in diagonal form, one obtains K Σ = −1/x13 . This proves that g is non-flat. 2 In particular, the metric descends to R3 × S 1 ; this is the usual form of the Gibbons-Hawking ansatz. 3 The change in notation from θ to x is meant to emphasise that we want this to be an R-, not an 0 S 1 -coordinate.

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5.3. Comparison of the two examples. We observe that our ansatz from Sect. 5.1 can also be related to a harmonic function on R3 : 2 3 S Write ∆ for the Laplacian on the unit 2-sphere in R3 , and ∆R for the Laplacian on R3 . Moreover, let ρ be the radial coordinate on R3 . Given a differentiable function u˜ : R3 → R, we have the relation ∆ S (u| ˜ S 2 ) = (∆R u˜ − u˜ ρρ − 2u˜ ρ )| S 2 ; 2

3

the last summand reflects the fact that both principal curvatures of S 2 are equal to 1. If u˜ is of the form u(x ˜ 1 , x2 , x3 ) = ρu(x1 /ρ, x2 /ρ, x3 /ρ), then this last equation simplifies to ˜ S2 . ∆ S u + 2u = (∆R u)| 2

3

In that particular case, u˜ is homogeneous of degree 1 in ρ, hence ∆R u˜ is homogeneous 3 ˜ S 2 is sufficient for the vanishing of degree −1. This implies that the vanishing of (∆R u)| 3 of ∆R u˜ on all R3 . In conclusion, solutions to our Monge-Ampère equation (8) are — by the preceding discussion and Proposition 36 in the Appendix — in direct correspondence with harmonic functions on R3 that are homogeneous of degree 1 in ρ. Finally, notice that both examples admit a tri-Hamiltonian vector field: in the example of Sect. 5.1, this is the vector field ∂x3 , with x3 := Im(z 2 ); in the example of Sect. 5.2, it is ∂x2 . As we plan to show in a forthcoming paper, all taut contact spheres with such a symmetry can be related to a Gibbons-Hawking ansatz, although this relation, in the Helmholtz case, is far from straightforward. 3

6. A Bernstein Problem Are there any 1-normalised taut contact spheres—on a suitable open domain U — inducing a non-flat metric g on U × R such that the long metric gl (see Definition 15) induced on U ≡ U × {0} is complete? This kind of completeness question is known as a Bernstein problem, see [9]. In order to appreciate the difficulty of this question, it is helpful to consider how the ansatz in Sect. 5.1 has to be extended if it is to be of any use in providing an answer. First of all, we ensured the homogeneity of ∂∂ H by taking H to be homogeneous. However, the example H = z 1 ez1 + z 1 ez1 , where ∂ H = z 1 e z 1 + e z 1 dz 1 and ∂∂ H = e z 1 + e z 1 dz 1 ∧ dz 1 , shows that it is perfectly possible for ∂∂ H to be homogeneous in eRe(z 1 ) without either H or ∂ H having this property. Secondly, we took H to be independent of Im(z 2 ), but to discuss the general case one needs to allow the auxiliary function h to depend on all three variables Im(z 1 ), Re(z 2 ), and Im(z 2 ). Thirdly, while a potential H always exists locally, it need not exist globally.


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The question we just raised has a venerable history. In [10] Calabi describes a construction of Kähler-Einstein metrics, and in particular Ricci-flat Kähler metrics, on complex tubular domains D × iRn ⊂ Cn , where D is some connected, open subset of Rn . The construction is based on a real Monge-Ampère equation (with a constant on the right-hand side), or equivalently, a complex Monge-Ampère equation for a function depending only on the real parts of n complex variables. The resulting metrics are invariant under the group of translations along iRn . Earlier results of Calabi [8] allowed him to show that, in the Ricci-flat case, metrics obtained via this construction can never be complete, except for the trivial case with D = Rn and a flat metric. Here is a partial answer to this Bernstein problem, dealing with contact spheres that possess additional symmetries. In place of the long metric gl we consider the short metric gs . Completeness of gs is a stronger condition than completeness of gl . The following is part (a) of Theorem 16. Theorem 24. Let (ω1 , ω2 , ω3 ) be a 1-normalised taut contact sphere on a 3-manifold U with the property that the short metric gs := ω12 + ω22 + ω32 is complete and admits a Killing field X ≡ 0 that preserves each form, i.e. L X ω1 = L X ω2 = L X ω3 = 0. Then U is compact and hence a left-quotient of SU(2), and (ω1 , ω2 , ω3 ) is isomorphic to one of the taut contact spheres described in Theorem 11. The proof of this theorem will take up the remainder of this section. We begin by observing that X does not have any zeros, which can be seen as follows. Arguing by contradiction, assume that p ∈ U is a point with X ( p) = 0. Then the flow of X preserves the distance spheres from p, and is then necessarily a rotation about an axis through p (in geodesic normal coordinates). This is incompatible with the fact that this flow preserves the coframe (α1 , α2 , α3 ). The fact that X does not have any zeros means that 1/2 Λ := ω1 (X )2 + ω2 (X )2 + ω3 (X )2 defines a function Λ : U → R+ . Set αi := ωi /Λ, i = 1, 2, 3, so that α1 (X )2 + α2 (X )2 + α3 (X )2 ≡ 1. Notice that the αi satisfy the Eqs. (1) with that very Λ, and they are likewise invariant under the flow of X . Consider the map Ψ := (x1 , x2 , x3 ) := α1 (X ), α2 (X ), α3 (X ) : U −→ S 2 . It is clear that the differential of Ψ satisfies T Ψ (X ) = 0, i.e. each flow line of X is mapped to a single point in S 2 . Write . s for the length of tangent vectors to U with respect to the short metric gs , and . S 2 for the length of tangent vectors to S 2 ⊂ R3 with respect to the standard metric d x12 + d x22 + d x32 .

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Lemma 25. If Z ∈ T p U is a tangent vector gs -orthogonal to X , then T Ψ (Z ) S 2 ≥ Z s . Proof. The X -invariance of the αi gives, with the Cartan formula for the Lie derivative, d xi = d(αi (X )) = L X αi − X

dαi = −X

dαi .

All the following computations are made at the single point p. Rotate the contact sphere so that at that point p we have α1 (X ) = 1 and α2 (X ) = α3 (X ) = 0, i.e. Ψ ( p) = (1, 0, 0). Then, with (1), d x1 = −β(X )α1 + β, d x2 = −β(X )α2 + Λ α3 , d x3 = −β(X )α3 − Λ α2 . The condition that Z be gs -orthogonal to X means that α1 (Z ) = 0. Hence d x1 (Z ) = β(Z ), d x2 (Z ) = −β(X )α2 (Z ) + Λ α3 (Z ), d x3 (Z ) = −β(X )α3 (Z ) − Λ α2 (Z ). From Ψ ( p) = (1, 0, 0) we have d x1 (Z ) = 0. Then T Ψ (Z ) 2S 2 = d x1 (Z )2 + d x2 (Z )2 + d x3 (Z )2 = d x2 (Z )2 + d x3 (Z )2 = β(X )2 + Λ2 · α2 (Z )2 + α3 (Z )2 = β(X )2 + Λ2 · α1 (Z )2 + α2 (Z )2 + α3 (Z )2 ≥ Λ2 α1 (Z )2 + α2 (Z )2 + α3 (Z )2 = Z 2s .

This lemma implies in particular that T Ψ has full rank at every point, so Ψ (U ) is an open subset of S 2 . Lemma 26. The map Ψ : U → S 2 is surjective. Proof. Since Ψ (U ) ⊂ S 2 is open, it suffices to show that Ψ (U ) is complete, i.e. that every path γ : [0, 1) → Ψ (U ) of finite length has a limit point inside Ψ (U ). Let γ˜ : [0, t0 ) → U be a maximal lift of such a path, gs -orthogonal to X . By the previous lemma, this lift is non-empty and of finite length in the short metric. Since U is complete, we deduce that t0 = 1 and that the lift γ˜ has a limit point in U . Therefore γ has a limit point in Ψ (U ).

Lemma 27. For each q ∈ S 2 , the preimage Ψ −1 (q) ⊂ U is a single orbit of X . Proof. Arguing by contradiction, we assume that q ∈ S 2 is a point in the image of two distinct orbits O0 , O1 of X . Let γ˜ : [0, 1] → U be a path joining these two orbits; γ := Ψ ◦ γ˜ : [0, 1] → S 2 is then a loop based at q. By the argument in the proof of


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the preceding lemma we may assume that γ˜ is orthogonal to X . Let γs , s ∈ [0, 1], be a homotopy of γ = γ0 rel {0, 1} to the constant path γ1 at q, and γ˜s the corresponding homotopy of lifts orthogonal to X with initial point γ˜s (0) = γ˜ (0) for all s ∈ [0, 1]. The endpoints γ˜s (1) form a smooth path in Ψ −1 (q). The completeness of (U, gs ) entails that the flow of X is complete, and this in turn ensures that γ˜s (1) ∈ O1 for all s. But the constant path γ1 lifts to the constant path γ˜1 at γ˜ (0) ∈ O0 , hence γ˜1 (1) ∈ O0 ∩ O1 , which is impossible.

As preimages of single points, the orbits of X are closed subsets of U . This means that every orbit is either periodic or a proper embedding of R in U . If there is a periodic orbit O0 , all other orbits remain at a bounded distance from O0 , since the flow of X is by isometries. This precludes proper embeddings of R, i.e. in this case all orbits must be periodic, and U is compact, as asserted in Theorem 24. It remains to consider the complementary case, when all orbits are proper embeddings of R. In this case, the time-1 map of the flow of X will disjoin any sufficiently small compact set K from itself, since each single orbit through K is proper and the flow is by isometries. It follows that this time-1 map defines a free and properly discontinuous Z-action on U . The quotient manifold under this action is, by the first case, a compact manifold with a taut contact sphere, and hence a left-quotient of SU(2). Such manifolds do not admit infinite covers. In other words, this second case cannot occur. This completes the proof of Theorem 24. 7. 3-Sasakian Structures In this section we consider taut contact spheres (α1 , α2 , α3 ) on a 3-dimensional domain U that satisfy the stronger condition αi ∧ dα j = 0 for i = j. The conditions for a naturally ordered taut contact sphere then imply that dαi = Λ α j ∧αk for some Λ : U → R+ . Differentiation of this equation yields 0 = d(dαi ) = dΛ ∧ α j ∧ αk + Λ dα j ∧ αk − Λ α j ∧ dαk = dΛ ∧ α j ∧ αk . It follows that dΛ ≡ 0, so Λ ≡ c for some constant c and the contact sphere is c-normalised. Recall that the Reeb vector field R of a contact form α is defined by α(R) ≡ 1 and R dα ≡ 0. We now have the following simple lemma, where Ri denotes the Reeb vector field of αi , and β is the 1-form from Eq. (1). The proof is left to the reader. Lemma 28. For a taut contact sphere (α1 , α2 , α3 ), the following conditions are equivalent: (i) αi ∧ dα j = 0 for i = j. (ii) The Reeb vector fields (R1 , R2 , R3 ) constitute a frame dual to the coframe (α1 , α2 , α3 ). (iii) β = 0. (iv) The short metric gs equals the long metric gl . If any of these conditions holds and c is the normalisation constant, then [Ri , R j ] = −c Rk . This implies that the metric α12 + α22 + α32 has constant curvature equal to c2 /4. In particular, the metric gs = gl has curvature identically equal to 1/4.

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In analogy with our terminology in [15] we call a taut contact sphere satisfying any of the conditions in this lemma a Cartan structure. Remark 29. Of the taut contact spheres in the statement of Theorem 11 exactly those with ν = 0 are Cartan structures. Those with ν = 0 are not even conformally equivalent to a Cartan structure; this follows for instance from [15, Prop. 6.1]. We now want to relate such Cartan structures to 3-Sasakian structures. Recall the definition of these structures. Definition 30. A metric g on a 3-manifold U is called 3-Sasakian if the cone metric C = e2s (g + ds 2 ) on U × R is a hyperkähler metric. ∂s

Given such a cone metric C, we observe that ∂s is a homothetic vector field and C = e2s ds is a closed 1-form.

Remark 31. Given a Riemannian metric g and a vector field Z , the following are equivalent: (i) ∇ Z is the identity on every tangent space. (ii) L Z g = g and Z g is closed. (iii) L Z g = g, and where Z is non-vanishing it is orthogonal to a codimension 1 foliation. (iv) On the open set {Z = 0} we have local descriptions g = e2s (g + ds 2 ) with ∂s = Z and g being the metric induced by g in a transversal orthogonal to Z . Following [19], we call a vector field satisfying either of these conditions a dilatation. Thus, any particular description of g as a cone metric corresponds to a non-vanishing dilatation. Proposition 32. A 1-normalised taut contact sphere on a 3-dimensional manifold U is a Cartan structure if and only if the induced hyperkähler metric g on U × R is a cone metric with the induced tri-Liouville vector field ∂t as dilatation. Proof. By formula (4) for the metric g, the 1-form ∂t closed if and only if β ≡ 0.

g equals et (dt + β), which is

Lemma 33. Given any metric g on U , any endomorphism field on U × R parallel with respect to the corresponding cone metric is invariant under the flow of ∂s . Proof. Let (x1 , x2 , x3 ) be local coordinates on U . Computing the Levi-Civita connection of the cone metric in the coordinates (x1 , x2 , x3 , s), one finds that e−s ∂x1 , e−s ∂x2 , e−s ∂x3 , e−s ∂s are parallel along the radii. The coefficients of a parallel endomorphism field in this frame are constant along the radii. Those coefficients are the same in the frame ∂x1 , ∂x2 , ∂x3 , ∂s .

It is a fact in Riemannian geometry that a family of metrics of the form g λ = λg, λ ∈ R+ , in general gives rise to a family of non-isometric cone metrics. We use the methods of this paper to give a simple proof of the following result, well-known in Sasakian geometry, which can be read as saying that for g having constant positive curvature, only the g λ of curvature equal to 1 gives rise to a hyperkähler cone. Proposition 34. A 3-Sasakian metric in dimension 3 has constant curvature equal to 1. Therefore, if a 4-dimensional hyperkähler metric admits a dilatation, it must be flat.


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Proof. Let g be a Riemannian metric on a 3-manifold U and assume that the cone metric C = e2s (ds 2 + g) is hyperkähler. By the preceding lemma the flow of the dilatation (1/2)∂s is tri-holomorphic and also tri-Liouville. Introduce the new coordinate t = 2s, so that ∂t = (1/2)∂s . By the theory we have developed, there is a 1-normalised taut contact sphere (α1 , α2 , α3 ) on U such that C = et (dt 2 + α12 + α22 + α32 ). Notice that g is recovered from the cone metric and the dilatation ∂s via the formula g = C|{s=0} = C|{ ∂s =1} . But ∂s = 2 ∂t , so g = et (dt 2 + α12 + α22 + α32 )|{et =1/4} = (α1 /2)2 + (α2 /2)2 + (α3 /2)2 . We see that g has a 2-normalised Cartan structure as an orthonormal frame, making it a metric of constant curvature equal to 1. Then e2s (g + ds 2 ) describes the 4-dimensional Euclidean metric in spherical coordinates.

This result must be read as local rigidity of 3-Sasakian structures. As our constructions of taut contact spheres giving rise to non-flat hyperkähler metrics show, taut contact spheres have much richer local geometry. In particular, their associated hyperkähler metrics admit homotheties, but no dilatations. So they are not cone metrics in any way. As we have mentioned in Sect. 4, the c-normalised taut contact spheres on closed 3-manifolds are orthonormal for Berger metrics, which are spherical only in the Cartan case. The long metrics, on the other hand, are always spherical. By the discussion in this section, our Theorems 3 and 11 can be read as a classification of the closed 3-Sasakian 3-manifolds: Corollary 35. The closed 3-Sasakian 3-manifolds are precisely the left-quotients of SU(2).

For the parametric family of taut contact spheres in Theorem 11, the induced hyperkähler metric is always the standard Euclidean metric on E4 \ {0}, so it is always a cone metric. But only for the parameter value ν = 0 (corresponding to the unique class containing Cartan structures) does the tri-Liouville vector field Y point in the radial direction (i.e. the direction of the only non-vanishing dilatation of E4 \ {0}). In fact, only the tri-Liouville vector field changes within this parametric family. The dαi obviously do not depend on ν, and a little computation shows that neither do the Reeb vector fields Ri of the αi . A proof of Corollary 35 previous to the one given above was indeed based on the observation that a 3-Sasakian 3-manifold is a space of constant curvature 1; the classification of 3-Sasakian manifolds among the 3-dimensional space forms was achieved by Sasaki [22] (who still spoke of normal contact metric 3-structures). Appendix: A Contact Transformation Leading to Helmholtz’ Equation The following intriguing proposition relates solutions of Eq. (8) from Sect. 5.1 to solutions of the Helmholtz equation (cf. [12] for this terminology) ∆u + 2u = 0 on the

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2-sphere S 2 with its metric of constant curvature 1. In geodesic polar coordinates this metric is given by ds 2 = dr 2 + sin2 r dθ 2 . Hence the gradient of a differentiable function f : S 2 → R is computed via ∇ f = fr ∂r +

1 f θ ∂θ . sin2 r

The area element is A = sin r dr ∧ dθ , so from d(X

A) = L X A = div(X ) · A and ∆f = div(∇ f ) 2

we see that the spherical Laplacian ∆ = ∆ S takes the form ∆f =

1 cos r fr + frr + f θθ . sin r sin2 r

Proposition 36. Let h(s1 , s2 ) be a solution of Eq. (8). Set ⎧ ⎨ θ = s1 , (T ) r = arccot(h s2 ) ∈ (0, π ), ⎩ u = −s cos r + h sin r. 2 If h s2 s2 = 0, then u = u(r, θ ) is a function of the independent variables r and θ that solves the spherical Helmholtz equation ∆u + 2u = 0.

(10)

Conversely, a solution u(r, θ ) of (10) satisfying u + u rr = 0 gives rise to independent variables s1 , s2 and a solution h(s1 , s2 ) of (8) via the inverse transformation ⎧ ⎨ s1 = θ, (T −1 ) s2 = u r sin r − u cos r, ⎩ h = u cos r + u sin r. r Proof. Given h(s1 , s2 ), let (r, θ, u) be defined by (T ). The condition h s2 s2 = 0 is obviously necessary and sufficient for (s1 , s2 ) → (r, θ ) to be an invertible coordinate transformation. From du = − cos r ds2 + s2 sin r dr + (h s1 ds1 + h s2 ds2 ) sin r + h cos r dr = (s2 sin r + h cos r ) dr + h s1 sin r dθ we see that u is a function of r and θ with u r = s2 sin r + h cos r, (T ) u θ = h s1 sin r. Conversely, given u(r, θ ), let (s1 , s2 , h) be defined by (T −1 ). Since ds2 = (u + u rr ) sin r dr + (u r θ sin r − u θ cos r ) dθ,


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the condition u + u rr = 0 is necessary and sufficient for (r, θ ) → (s1 , s2 ) to be a coordinate transformation, because r ∈ (0, π ). From dh = (u + u rr ) cos r dr + (u r θ cos r + u θ sin r ) dθ uθ ds1 + cot r ds2 = sin r we infer that h is a function of s1 and s2 with h s1 = u θ / sin r, −1 (T ) h s2 = cot r. It is then a straightforward check that (T ) and (T −1 ) are indeed inverse transformations of each other. In particular, one finds that h s2 s2 = −

1 , (u + u rr ) sin3 r

which shows that (T ) is defined on h if and only if (T −1 ) is defined on u. In order to show that solutions of (8) correspond to solutions of (10) under the transformation (T ), it is convenient to rewrite (8) as an exterior differential system dh = p1 ds1 + p2 ds2 , ds1 ∧ ds2 = dp1 ∧ dp2 + h ds1 ∧ dp2 − p22 ds1 ∧ ds2 .

(11)

We then compute, using (T −1 ) and (T −1 ) , dp1 ∧ dp2 + h ds1 ∧ dp2 − (1 + p22 ) ds1 ∧ ds2 uθ ∧ d cot r + (u r cos r + u sin r ) dθ ∧ d cot r =d sin r − (1 + cot 2 r ) dθ ∧ d(u r sin r − u cos r ) 1 1 cos r 1 u θθ · + + ur +u (u + u rr ) sin r dr ∧ dθ = sin r sin2 r sin r sin2 r sin2 r 1 cos r 1 u r + u rr + 2u dr ∧ dθ = u θθ + sin r sin2 r sin r 1 = (∆u + 2u) dr ∧ dθ. sin r This completes the proof of Proposition 36.

Since the reader is bound to wonder how we arrived at the transformation (T ), we present, in nuce, our chain of discovery: First, one can get rid of mixed derivatives in the exterior differential system (11) by introducing p2 as an independent variable. The identity dh − p1 ds1 − p2 ds2 = d(h − p2 s2 ) − p1 ds1 + s2 dp2 suggests the contact transformation h(s1 , s2 ) k(t1 , t2 ) given by t1 = s1 , t2 = p2 = h s2 , k = h − s2 h s2 ,

746


with inverse transformation s1 = t1 , s2 = −kt2 , h = k − t2 kt2 . (Here one needs h s2 s2 = 0 or kt2 t2 = 0, respectively, for these to be honest, i.e. invertible, transformations.) We compute dp1 ∧ dp2 = dkt1 ∧ dt2 = kt1 t1 dt1 ∧ dt2 , h ds1 ∧ dp2 = (k − t2 kt2 ) dt1 ∧ dt2 , −(1 + p22 ) ds1 ∧ ds2 = (1 + t22 ) dt1 ∧ dkt2 = (1 + t22 )kt2 t2 dt1 ∧ dt2 . So the contact transformation takes Eq. (8) to the following equation, where we now write ki for kti etc.: k11 + (1 + t22 )k22 − t2 k2 + k = 0. (12) The first order term in this equation can be made to disappear by setting x = t1 , sinh y = t2 and w(x, y) = k(x, sinh y)/ cosh y. This turns Eq. (12) into wx x + w yy +

2 w = 0, cosh2 y

which is the Helmholtz equation for the metric 1 0 cosh2 y

1 cosh2 y

0

(13)

.

The Gauß curvature of this metric turns out to be identically equal to 1. Indeed, it is the spherical metric in Mercator coordinates. Therefore, we pass from (13) to (10) by making the substitution cosh y = 1/ sin r, sinh y = cot r, θ = x, and u(r, θ ) = w(x, y). We now want to find an explicit solution of (10). The ansatz u(r, θ ) = cos θ sin r g(r ) leads to 3gr cos r + grr sin r = 0, which has the solution gr = 1/ sin3 r ; that in turn integrates to 0 g(r ) = 1 + t 2 dt. cot r

The resulting u corresponds under (T −1 ) to the solution s2 / cos s1 h(s1 , s2 ) = cos s1 ξ 2 − 1 dξ 1

of Eq. (8). As domain of definition we may take


747

U = {(s1 , s2 ) ∈ R2 : |s1 | < π/2, s2 > cos s1 }. The Kähler potential H corresponding to this solution gives rise to a hyperkähler metric g on U × R inducing a taut contact sphere on U , with U = {(s1 , s2 , s3 ) ∈ R3 : (s1 , s2 ) ∈ U }. The relation with the complex coordinates is given by t + is1 = 2z 1 and s2 + is3 = 2z 2 , say. We claim that g is non-flat. The coefficients of this metric are 1 Hz z . 2 α β Write G for the (2 × 2)-matrix (gαβ ). Then the curvature tensor K αβγ δ , read for fixed γ , δ as a (2 × 2)-matrix indexed by α and β, is computed by gαβ = g(∂z α , ∂z β ) = Ω1 (∂z α , J1 ∂z β ) =

(K αβγ δ ) = Gz γ z δ − Gz γ G−1 Gz δ , cf. [21, p. 159]. We now want to show that K 2222 is non-zero. In the following computations we write ∗ for any matrix entry that is irrelevant for the final result: 1 z 1 +z 1 h + h 11 h 2 − ih 12 G= e , h 2 + ih 12 h 22 2 1 ∗ ∗ , Gz 2 = e z 1 +z 1 + ih h h 22 122 222 2 1 ∗ h 22 − ih 122 , Gz 2 = e z 1 +z 1 ∗ h 222 2 1 ∗ ∗ . Gz 2 z 2 = e z 1 +z 1 ∗ h 2222 2 On the hyperplane {2 Im(z 1 ) = z 1 − z 1 = 0}, corresponding to the line {s1 = 0}, we have s2 h 1 = 0 and h = ξ 2 − 1 dξ. 1

Writing s22 − 1 = σ , for short, we have along that same line h 2 = σ, h 22 = s2 /σ, h 222 = −1/σ 3 , h 2222 = 3s2 /σ 5 . √ At (s1 , s2 ) = (0, 2) we thus find √ √ h 2 = 1, h 22 = 2, h 222 = −1, h 2222 = 3 2, h 12 = h 122 = 0, and h + h 11 =

√ 2.

√ That last equality can be computed from (8). Hence, at (z 1 , z 2 ) = (0, 2/2) we obtain √ √ 1 ∗ √ 1 √∗ ∗ ∗ ∗ ∗ 2 √ −1 ∗ 2 = , − ∗ K 2222 ∗ −1 2 −1 −1 2 2 ∗ 3 2 2 √ which yields K 2222 = − 2 = 0. Thus g is indeed non-flat.

748


Acknowledgements. We are immensely grateful to Nigel Hitchin for drawing our attention to the similarities between our original treatment of taut contact spheres and the Gibbons-Hawking ansatz. Hitchin’s observation inspired a large part of the present paper. We thank Martin Lübke for useful hints concerning the curvature calculations in the Appendix.

References 1. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser. A 362, 425–461 (1978) 2. Bakas, I., Sfetsos, K.: Toda fields of SO(3) hyper-Kähler metrics and free field realizations. Int. J. Mod. Phys. A 12, 2585–2611 (1997) 3. Bär, C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993) 4. Besse, A.: Einstein Manifolds, Ergeb. Math. Grenzgeb. (3). Berlin: Springer-Verlag, 1987 5. Boyer, C.P.: A note on hyper-Hermitian four-manifolds. Proc. Amer. Math. Soc. 102, 157–164 (1988) 6. Boyer, C.P., Galicki, K.: 3-Sasakian manifolds. In: Surveys in Differential Geometry: Essays on Einstein Manifolds, Surv. Differ. Geom. VI. Boston: Int. Press, 1999, pp. 123–184 7. Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford Math. Monogr. Oxford: Oxford University Press, 2008 8. Calabi, E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K Jörgens. Mich. Math. J. 5, 105–126 (1958) 9. Calabi, E.: Examples of Bernstein problems for some nonlinear equations. In: Global Analysis (Berkeley, 1968), Proc. Sympos. Pure Math. 15. Providence RI: Amer. Math. Soc. 1970, pp. 223–230 10. Calabi, E.: A construction of nonhomogeneous Einstein metrics. In: Differential Geometry (Stanford, 1973), Proc. Sympos. Pure Math. 27, Part 2. Providence RI: Amer. Math. Soc. 1975, pp. 17–24 11. Chave, T., Tod, K.P., Valent, G.: (4, 0) and (4, 4) sigma models with a tri-holomorphic Killing vector. Phys. Lett. B 383, 262–270 (1996) 12. Evans, L.C.: Partial Differential Equations, Grad. Stud. Math. 19. Providence RI: Amer. Math. Soc. 1998 13. Geiges, H.: Symplectic couples on 4-manifolds. Duke Math. J. 85, 701–711 (1996) 14. Geiges, H.: Normal contact structures on 3-manifolds. Tôhoku Math. J. 49, 415–422 (1997) 15. Geiges, H., Gonzalo, J.: Contact geometry and complex surfaces. Invent. Math. 121, 147–209 (1995) 16. Geiges, H., Gonzalo, J.: Contact circles on 3-manifolds. J. Differ. Geom. 46, 236–286 (1997) 17. Geiges, H., Gonzalo, J.: Moduli of contact circles. J. Reine Angew. Math. 551, 41–85 (2002) 18. Gibbons, G.W., Hawking, S.W.: Gravitational multi-instantons. Phys. Lett. B 78, 430–432 (1978) 19. Gibbons, G.W., Rychenkova, P.: Cones, tri-Sasakian structures and superconformal invariance. Phys. Lett. B 443, 138–142 (1998) 20. Godliński, M., Kopczyński, W., Nurowski, P.: Locally Sasakian manifolds. Class. Quant. Grav. 17, L105–L115 (2000) 21. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry II. New York: Interscience, 1969 22. Sasaki, S.: Spherical space forms with normal contact metric 3-structure. J. Differ. Geom. 6, 307–315 (1972) Communicated by G. W. Gibbons

Commun. Math. Phys. 287, 749–767 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0635-x

Communications in


Superstring Scattering Amplitudes in Higher Genus Samuel Grushevsky Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA. E-mail: [email protected] Received: 8 April 2008 / Accepted: 20 May 2008 Published online: 16 September 2008 – © Springer-Verlag 2008

Abstract: In this paper we continue the program pioneered by D’Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots. 1. Introduction The problem of finding an explicit expression for the string measure to an arbitrary loop (genus) order is one of the major open problems in string perturbation theory. For the bosonic string, chiral expressions in any genus in terms of theta functions and additional points on the worldsheet have been proposed by Manin in [18], Beilinson and Manin in [1], and Verlinde and Verlinde in [26]. Non-chiral expressions in terms of the WeilPetersson measure and Selberg zeta functions have also been proposed by D’Hoker and Phong in [7]. The problem is much more difficult for the superstring and the heterotic string. Although the one-loop scattering amplitudes had been derived by Green and Schwarz in [5] for the superstring and by Gross, Harvey, Martinec, and Rohm in [6] for the heterotic string, the case of genus g ≥ 2 remained inaccessible until relatively recently. The difficulty in obtaining a formula for the chiral superstring measure in higher genus is the occurrence of odd supermoduli in any amplitude starting from genus g ≥ 2. A solution to this problem was proposed by D’Hoker and Phong, who introduced a gauge-fixing procedure respecting the local supersymmetry of the worldsheet. Research is supported in part by National Science Foundation under the grant DMS-05-55867.

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Using this procedure, they managed to compute the genus 2 superstring measure from first principles in a series of papers [8–11] and to verify that the corresponding result θ []4 6 [] times the bosonic measure produced vanishing cosmological constant, 2- and 3-point scattering amplitudes, and other expected physical properties [14,15]. With the new insight from 6 [](), they started a modern program of identifying the higher genera superstring measure from factorization constraints and syzygy/asyzygy conditions [12,13]. In [13] they also proposed an ansatz for the superstring measure in genus 3, including a θ []4 factor, subject to the condition of certain linear combinations of modular forms having a square root. However, to date such a linear combination has not been found, and may not exist (see below). In [2] Cacciatori and Dalla Piazza used the combinatorics of the action of the symplectic group on the set of theta characteristics in genus 2 to identify D’Hoker and Phong’s modular form 6 []() from certain invariance properties. Recently Cacciatori, Dalla Piazza, and van Geemen [3] proposed an ansatz for the chiral superstring measure in genus 3, by constructing an appropriate modular form 8 [], which has a θ []2 rather than the θ []4 factor, satisfying the factorization constraints on the locus of products of abelian varieties of lower genera. They also promise to determine in a forthcoming paper the dimension of the appropriate space of modular forms, and to show that their form is the unique one satisfying the factorization constraints (and thus there would be no solution in the form suggested in [13]). They also say that their constraints appear to have a solution in genus 4 as well. In this paper we rewrite the ansatz of D’Hoker and Phong in genus 2, and of Cacciatori, Dalla Piazza, and van Geemen in genus 3 in terms of modular forms associated to isotropic spaces of theta characteristics, which have been studied since the times of Krazer [17] and in particular used by Salvati Manni [23]. This allows us to propose a straightforward generalization of the chiral superstring measure to higher genera, which for genus 4 is an appropriate holomorphic modular form satisfying the necessary factorization constraints and producing vanishing cosmological constant. For higher genera we conjecture a possible ansatz, satisfying the factorization constraints, contingent on certain monomials in theta constants admitting holomorphic roots. An expression for higher genus superstring amplitudes was also proposed by Matone and Volpato [19]. Their formulas depend on the choice of points on the worldsheet and do not seem to give an explicit modular form. It would be interesting to understand the relation of our work to theirs. The structure of this work is as follows: in Sect. 2 we fix notations and introduce basic notions of modular forms. In Sect. 3 we review the orbits of the action of the symplectic group on sets of theta characteristics. In Sect. 4 we reinterpret the modular form G defined in [3] in terms of syzygies. Though strictly speaking this computation is not needed to define our modular forms and construct an ansatz, this is our motivation for considering, in Sect. 5, modular forms corresponding to vectors subspaces of the space of theta characteristics and reviewing what is known about them. In Sect. 6 we prove the crucial Theorem 15 describing the restrictions of these modular forms to loci of products. In Sect. 7 we obtain a new expression for the ansätze in genera 2 and 3 in terms of our modular forms, and also verify that in genus 4 there is a unique modular form that is a linear combination of ours that has correct factorization properties, thus giving an ansatz in genus 4. In Sect. 8 we describe a possible generalization to arbitrary genus, proving in Theorem 22 that it satisfies the factorization constraints (this also gives another proof of this for the ansatz in genus 4), and describe further tests and questions that could be used to study the validity and uniqueness of the ansatz.

Superstring Scattering Amplitudes in Higher Genus

751

2. Notations and Definitions Definition 1. We denote by Ag the moduli space of complex principally polarized abelian varieties of dimension g. We denote by Hg the Siegel upper half-space of symmetric complex matrices with positive-definite imaginary part, called period matrices. The right action of the symplectic group Sp(2g, Z) on Hg is given by A B ◦ τ := (Cτ + D)−1 (Aτ + B), C D

where we think of elements of Sp(2g, Z) as of consisting of four × g blocks, and g 0 1 they preserve the symplectic form given in the block form as . We then have −1 0 Ag = Sp(2g, Z)\Hg . Definition 2. Given a period matrix τ ∈ Hg we denote the abelian variety corresponding to [τ ] ∈ Ag by Aτ := Cg /(Zg + τ Zg ). The theta function is a function of τ ∈ Hg and z ∈ Cg given by θ (τ, z) := exp(πi(n t τ n + 2n t z)). n∈Zg

We denote by τ the line bundle on Aτ of which the theta function is a section. Definition 3. Given a point of order two on Aτ , which can be uniquely represented as g τ ε+δ 2 for ε, δ ∈ Z2 (where Z2 denotes the abelian group Z/2Z = {0, 1} and we use the additive notations throughout the text), the associated theta function with characteristic is ε (τ, z) := θ exp(πi((n + ε)t τ (n + ε) + 2(n + ε)t (z + δ)). δ n∈Zg

ε As a function of z, θ is odd or even depending on whether the scalar product ε·δ ∈ Z2 δ is equal to 1 or 0, respectively. Theta constants are restrictions of theta functions to z = 0, and all odd theta constants vanish identically in τ . Definition 4. A modular form of weight k with respect to a subgroup ⊂ Sp(2g, Z) is a function f : Hg → Cg such that f (γ ◦ τ ) = det(Cτ + D)k f (τ ) ∀γ ∈ , ∀τ ∈ Hg . We define the level subgroups of Sp(2g, Z) as follows: A B 10 ∈ g | M ≡ mod n , g (n) := M = C D 01 g (n, 2n) := M ∈ g (n) | diag(At B) ≡ diag(C t D) ≡ 0 mod 2n . These are normal subgroups of Sp(2g, Z) for n > 1; however, g (1, 2) is not normal.

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Remark 5. Theta constants with characteristics are not algebraically independent, and satisfy a host of algebraic identities. Their squares can be expressed algebraically in terms of a smaller set of modular forms, called theta constants of the second order, by using Riemann’s bilinear addition theorem—see [16] for details. Theta constants of the second order are algebraically independent for g ≤ 2. The only identity among them for g = 3 was known classically at least since the time of Schottky, and is discussed in [4], while the ideal of relations among them for g > 3 is not known, and presumably very complicated. 3. The Action of Sp(2g, Z) on Theta Characteristics Proposition 6 (see [16]). Theta constants with characteristics are modular forms of weight one half with respect to g (4, 8). Moreover, the full symplectic group acts on theta constants with characteristics as follows: 1 ε ε (M · τ ) = φ(ε, δ, M, τ, z) det(Cτ + D) 2 θ (τ ), θ M δ δ where φ is some explicit eighth root of unity, and the action on the characteristic is diag(C t D) ε D −C ε + := M , δ −B A δ diag(At B)

(1)

where the addition in the right-hand-side is taken in Z2 . Notice that by this formula the action of the subgroup g (2) on the set of characteristics is trivial, and thus the action of the entire group Sp(2g, Z) on the set of characteristics factors through the action of Sp(2g, Z)/ g (2) = Sp(2g, Z2 ). One can thus study the orbits of characteristics or sets of characteristics under the symplectic group action. This was done by Salvati Manni in [24], where all of the following results can be found. One first observes that the action of g (4, 8)\g (2) on the set of theta constants differs from the modular one by extra signs, while g (2) in addition permutes the characteristics. It is also clear that the action of Sp(2g, Z) on the set of characteristics (which factors through the action of Sp(2g, Z2 )) is transitive. To study the action on tuples of characteristics (i.e. the orbits of the Sp(2g, Z2 ) acting on

2g n

Z2

diagonally), we need more definitions. 2g

Definition 7. The Weil symplectic form on the space Z2 of characteristics is defined to be ε α := α · δ + β · ε. , δ β Notice that this symplectic form is not preserved by the action of Sp(2g, Z) on the set of characteristics: the pairing of the zero characteristic with any characteristic is zero, and it is the only such characteristic, while the action Sp(2g, Z) given by (1) is affine and does not preserve zero.


Definition 8. A triple of characteristics

753

ε ε ε1 , 2 , 3 is called syzygetic or azygetic δ1 δ2 δ3

depending on whether the sum ε1 · δ1 + ε2 · δ2 + ε3 · δ3 + (ε1 + ε2 + ε3 ) · (δ1 + δ2 + δ3 ) ε ε2 ε ε3 ε ε1 , 2 + , 3 + , 1 ∈ Z2 = δ1 δ2 δ2 δ3 δ3 δ1 is 0 or 1, respectively. Notice in particular that a triple of even characteristics is syzygetic or azygetic if their sum is even or odd, respectively. This notion is in fact invariant under the symplectic group action. The orbits of the action (1) of the symplectic group on sets of characteristics are completely described by the following Theorem 9 ([16], p. 212; [24]). There exists an element of the symplectic group mapping a set of n characteristics to another set of n characteristics if and only if there exists a way to number the characteristics in the first set a1 . . . an , and the characteristics in the second set b1 . . . bn in such a way that • for any i the parity of ai and bi is the same, • for any linear relation among ai with an even number of terms, i.e. if ai1 +. . .+ai2k = 0, there is a corresponding linear relation bi1 + . . . + bi2k = 0 and vice versa, • any triple ai , a j , ak is a/syzygetic if and only if the corresponding triple bi , b j , bk is a/syzygetic. 4. Results of Cacciatori, Dalla Piazza, van Geemen in Terms of Syzygy Conditions The main new ingredient of the superstring measure in genus 3 proposed by Cacciatori, Dalla Piazza, and van Geemen in [3] is the modular form G, of weight 8 with respect to the group (1, 2) ⊂ Sp(6, Z)—it is described there in terms of certain quadrics on Z62 . However, since G is a polynomial in theta constants with characteristics, from Theorem 9 it follows that the monomials appearing in it should be characterized by the syzygy properties and linear dependencies of the characteristics involved. We now obtain such a description of the modular form G, by unraveling the definition of G given in [3] in terms of syzygies. We would like to thank Eric D’Hoker and Duong Phong for encouraging us to do this translation—which then gave a formula amenable to generalizing to higher genus. abc α Given an even characteristic = —or in our notations—one can define de f β a corresponding quadratic form on the set of characteristics, i.e. ([3], p. 12) for v ∈ Z62 one defines q (v) := v1 v4 + v2 v5 + v3 v6 + av1 + bv2 + cv3 + dv4 + ev5 + f v6 . ε , this is simply If we write v = δ q (v) = ε · δ + α · ε + β · δ.

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α is an even characteristic, we have α · β = 0, and thus β ε = (ε + β) · (δ + α). q α δ β

This looks strange as the α and β are strangely swapped, and I believe that there is a small typo in [3] of interchanging the top and the bottom vector of the characteristic. 000 that is used for explicit calculations in [3] there is of course For the characteristic 000 no difference, but otherwise modularity would not hold. The correct definition should thus be ε q α = (ε + α) · (δ + β). (2) δ β In [3] a quadric is now introduced Q := {v | q (v) = 0}.

ε ε+α such that is even, δ δ+β α i.e. this is just the set of even characteristics, to which is added. The symplectic β 6 form v, w on Z2 is denoted E(v, w) in [3]. Notice that if both characteristics are even, this is the same as the quadratic form q. Considering the Lagrangian (also called the maximal isotropic classically, see [16] and [23]) subspaces of Z62 with respect to ·, ·

ε such that the Weil means choosing three linearly independent characteristics i δi i=1..3 pairing is zero on any pair, i.e. such that the sum of any pair of characteristics is again even. This is equivalent to saying that the triple of characteristics consisting of this pair and zero is syzygetic. The set of all even quadrics containing a Langrangian subspace α is now considered in [3]. This means considering the set of all characteristics = β such that q | L = 0. Thus ε+α ε is even ∀ ∈ L. Q ⊃ L ⇐⇒ δ+β δ From (2) it follows that this is the set of characteristics

We now notice (following 8.4 in [3], essentially) that if and are two characteristics such that Q ∩ Q ⊃ L, then we must have − ∈ L. Thus the definition of G at the top of p. 13 in [3] becomes G[] = θ [v + ]2 . L⊂Q v∈L

The condition L ⊂ Q means that the sum in definition of G istaken over linear the εi α + εi are even spaces generated by triples of characteristics such that all δi i=1..3 β + δi


755

ε εi , j = ε j · δi + εi · δ j = 0. Adding these conditions together shows that δi δj α + εi α + εi + ε j α α + εj + all characteristics = + are even. Thus we get the β β + δi + δ j β + δi β + δj following alternative formula

and all

Proposition 10. The modular form G[] defined in [3] (of weight 8 with respect to 3 (1, 2)) is equal to the sum over all sets of 8 even characteristics {u i }i=1..8 such that any pair of characteristics together with form a syzygetic triple, of the products θ [u i ]2 . Denote now vi := u i + and observe vi , v j = u i , u j + u i , + , u j = 0, since the right-hand-side is exactly the condition that the triple , u i , u j is syzygetic. We thus get yet another formula Corollary 11. The modular form G can be written as θ [v + ]2 . G[] = V ⊂Z62

(3)

dim V =3 v∈V

We remark that a given summand on the right-hand-side is not identically zero if and only if the set V + contains only even characteristics (is a purely even coset in the language of [23]), in which case as described there it follows that V is totally isotropic. This expression for G yields itself to a straightforward generalization, and in these terms the restriction of G to the locus of decomposable abelian varieties is easy to understand. We undertake this study in the next two sections. 2g

5. Modular Forms Corresponding to Subspaces of Z2

Motivated by the study of quartic relations among theta constants undertaken by Salvati Manni in [23] and by our reinterpretation of the form G above, in this section we investigate the properties of products of theta constants with characteristics forming a translate of an isotropic subspace. We thank Riccardo Salvati Manni for telling us about [23] and encouraging us to explore the behavior of these modular forms. Following [23], we denote 2g PM (τ ) := θ [v](τ ) for any M ⊂ Z2 . v∈M

Notice that if M contains any odd characteristics, then PM is identically equal to zero, 2g as all odd theta constants vanish identically. Let V ⊂ Z2 be a vector subspace with basis v1 . . . vn . V is called isotropic if the symplectic form restricts to zero on it, i.e. if v, w = 0 for any v, w ∈ V . Since for even characteristics v and w the value of the symplectic form v, w is equal to the parity of v + w, the space with basis {vi } is isotropic if and only if it only contains even characteristics, or equivalently, if PV (τ ) is not identically zero.

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Definition 12. We define a function Pi,s on Hg as the sum

(g)

Pi,s (τ ) :=

PV (τ )s .

2g

V ⊂Z2 ; dim V =i

Here the sum is taken over all i-dimensional (and thus of cardinality 2i ) vector subspaces, but note that if V is not isotropic, it contains odd characteristics, and the corresponding summand is zero. We will only consider these functions subject to the condition 2i s = 2k for some integer k ≥ 4. For the case of the superstring measure k = 4. (g)

Proposition 13. The function Pi,s is a modular form (assuming 2i s = 2k for k ≥ 4) of weight 2i−1 s for the subgroup g (1, 2). Proof. This can be seen from the discussion in [23,16,17]. To see this, one notes that the action ofSp(2g, Z) on the set of characteristics is affine; however, for an element A B g γ = ∈ g (1, 2) by definition diag(C t D) = diag(At B) = 0 ∈ Z2 , and C D thus the action of g (1, 2) on the set of characteristics fixes zero and is linear. Thus the summands in the definition of Pi,s get permuted—vector subspaces are mapped to vector subspaces by a linear action. The multiplicative factors are det(Cτ + D)1/2 for i−1 each theta constant, giving the overall factor of det(Cτ + D)2 for each PV . The other factor in the transformation formula (1) is the 8th root of unity φ. Since 2i s = 2k is divisible by 16, it can be shown that the product of the 8th roots will turn out to be equal to 1. We refer to [25] for a complete discussion and rigorous proof. The action of the full group Sp(2g, Z) on the set of characteristics is affine—it shifts (g) the zero to some characteristic . Acting by Sp(2g, Z) on the modular form Pi,s we then get Corollary 14. Assuming 2i s = 2k for k ≥ 4, for any even characteristic the function

(g)

Pi,s [](τ ) :=

PV + (τ )s

2g

V ⊂Z2 ; dim V =i

is a modular form of weight 2i−1 s with respect to the subgroup [] ⊂ Sp(2g, Z) that stabilizes under the action (1). This subgroup is conjugate to g (1, 2) (which, recall, is not a normal subgroup of Sp(2g, Z)). The conjugation is provided by any element of Sp(2g, Z) that maps the zero characteristic to under the action (1). Note that if is odd, the corresponding expression would be zero, as all summands would contain θ []. (g)

(g)

Proof. To prove modularity, one can observe that Pi,s = Pi,s [0], which we know to (g)

be a modular form, is conjugated to Pi,s [] by the Sp(2g, Z) action. For a direct proof, note that v + is an even characteristic for all v ∈ V if and only if for any triple v1 , v2 , for v1 , v2 ∈ V is syzygetic—this is the argument used to obtain the expression (3) in genus 3 at the end of the previous section—and the syzygy is preserved by the Sp(2g, Z) action. See [25] for more discussion, especially on the possible 8th roots of unity.


757

6. Restrictions of Modular Forms Corresponding (g)

In this section we determine the restrictions of modular forms Pi,s to the loci of decomposable abelian varieties (products of lower-dimensional ones). Note that it is (g) enough to determine the restriction of the modular form Pi,s with zero characteristic to Hk × Hg−k —its restrictions to the conjugates of this locus under g (1, 2) can be obtained by acting by g (1, 2), which preserves the form. Furthermore, the restrictions of (g) Pi,s [] with non-zero characteristic can then be obtained by conjugating by an element (g)

(g)

of Sp(2g, Z) that maps Pi,s to Pi,s []. (g)

Theorem 15. The modular form Pi,s restricts to the locus of decomposable abelian varieties (reducible period matrices in [3]) as follows: (g) (g−k) (k) Nn,m; i Pn,2i−n s · Pm,2i−m s , (4) Pi,s |Hk ×Hg−k = 0≤n,m≤i≤n+m

where Nn,m; i =

n+m−i−1 j=0

(2n − 2 j )(2m − 2 j ) . (2n+m−i − 2 j )

(5)

(c) Remark 16. Notice that many of the summands can be zero, as Pa,b ≡ 0 for a > c. In (g)

particular for Pg,s (the case of maximal isotropic subspaces; the form G constructed in (g−k) (3) (k) [3] is P3,2 in our notations) the only non-zero term on the right is Pk,2g−k s Pg−k,2k s . Proof. Indeed, let V ∼ = Zi2 be a vector subspace of Z2 (notice that we never need to worry about parity or isotropy—the summands for non-isotropic subspaces will vanish automatically). If a period matrix is a product of two lower-dimensional ones, τg = τk × τg−k , the group of points of order two on Aτ is the direct sum of the groups of 2g 2(g−k) points of order two on the factors. Let Z2 = Z2k be this decomposition, and 2 ⊕ Z2 let π1 and π2 denote the projections onto the two summands. Since V ⊆ π1 (V )⊕π2 (V ), we must have #V ≤ #π1 (V ) · #π2 (V ). Since these are all vector spaces over Z2 , denoting by n, m the dimensions of π1 (V ) and π2 (V ) respectively, this implies the inequality 2i ≤ 2n · 2m or, equivalently, i ≤ n + m. The projections maps π1 : V → π1 (V ) and π2 : V → π2 (V ), being group homomorphisms, are then 2i−n -to-1 and 2i−m -to-1, respectively. Since any theta constant with characteristic restricts to the product 2g

θ [v](τg ) = θ [π1 (v)](τk ) · θ [π2 (v)](τg−k ), it follows that for n and m fixed we have PV (τg )s = θ [v](τg )s = θ [π1 (v)](τk )s · θ [π2 (v)](τg−k )s v∈V

=

v1 ∈π1 (V ) (g)

v∈V i−n s

θ [v1 ](τk )2

v2 ∈π2 (V )

i−m s

θ [v2 ](τg−k )2

i−n

i−m

= Pπ21 (Vs) · Pπ22 (V )s .

To compute Pi,s , we need to sum over all V . Let us first sum over all the spaces V for which the spaces π1 (V ) and π2 (V ) are fixed. Notice that the product on the right is the same for all V with fixed projections. The number of V with fixed projections only depends on the dimensions, and we compute it in the following combinatorial lemma.

758

S. Grushevsky

Lemma 17. The number of i-dimensional vector subspaces V of Zn2 ⊕ Zm 2 surjecting onto both summands n+m−i−1 j=0

(2n − 2 j )(2m − 2 j ) , (2n+m−i − 2 j )

(where this number is understood to be zero if n + m < i, and to be one if n + m = i. Note also that the product has a zero factor if n > i or m > i). Proof. Fix a scalar product on Zn+m such that the chosen decomposition Zn+m = 2 2 n m Z2 ⊕Z2 is orthogonal. The projections from V to Zn2 and Zm are then the orthogonal pro2 jections; the image of such a projection misses a vector v if and only if it is orthogonal to V . Thus what we need to count is (for a fixed scalar product) the number of V ∼ = Zi2 such n+m−i n m ⊥ does not intersect the coordinate subspaces Z2 and Z2 away from that V ∼ = Z2 zero. Let us construct such a V ⊥ by choosing a basis v1 , . . . , vn+m−i for it. To choose such a basis, we will choose independently the projections π1 (v1 ), . . . , π1 (vn+m−i )— note that in order to have V ⊥ ∩Zn2 = {0}, these vectors must be linearly independent—and similarly choosing linearly independent π2 (v1 ), . . . , π2 (vn+m−i ) ∈ Zm 2 . Thus π1 (v1 ) can be chosen in 2n − 1 ways, after which π1 (v2 ) can be chosen in 2n − 2 ways, and so on, and similarly we choose π2 (v1 ) in 2m − 1 ways and so on. Thus the total number of spaces V ⊥ ⊂ Zn+m not intersecting Zn2 and Zm 2 2 , together with a choice of an ordered basis of it, is equal to n+m−i−1

(2n − 2 j )(2m − 2 j ),

j=0

while the number of ordered basis in a fixed space V ⊥ ∼ is = Zn+m−i 2 n+m−i−1

(2n+m−i − 2 j ),

j=0

and dividing one by the other gives the lemma. Now observe that if we take the sum over all V for fixed dimensions n and m, the projections π1 (V ) and π2 (V ) range over all n-dimensional subspaces of Z2k 2 , and 2(g−k) , respectively. We thus get m-dimensional subspaces of Z2 PV (τg )s 2g

V ⊂Z2 ; dim V =i; dim π1 (V )=n; dim π2 (V )=m

⎛

⎞ ⎛

⎜ = Nn,m; i ⎝

V1 ⊂Z2k 2 dim V1 =n

=

(k) Nn,m; i Pn,2i−n s

·

i−n s

PV21

⎞

⎟ ⎜ ⎠·⎝

2(g−k)

V2 ⊂Z2

i−m s

PV22

⎟ ⎠

dim V2 =m

(g−k) Pm,2i−m s .

This is of course zero if n > k or m > g − k. Taking the sum over all possible n and m (recall that we have n + m ≥ i and n, m ≤ i) gives the theorem.


759

7. Ansätze for Genera ≤ 4 in Terms of Vector Subspaces We now rewrite the low genus superstring measure proposed by D’Hoker and Phong [8] for genus 2 and by Cacciatori, Dalla Piazza, and van Geemen [3] for genus 3 in terms (g) of the modular forms Pi,s constructed and studied above. Following the earlier works, we try to find the superstring measure as the product of the bosonic measure (some formulas for which are known, but explicit point-independent expressions for which are apparently not known for high genus—see the discussion in [8]) and a function (g) [] depending on the characteristic . As argued in [3], p.5 the factorization property can then be rewritten as a condition on (g) []. The arguments in [3], 2.7 show (essentially arguing that if (g) [] is equal to γ (g) for some γ ∈ Sp(2g, Z), then its degenerations can be computed by acting by γ on the degenerations of (g) ) that for g ≤ 3 to satisfy these constraints it is enough to construct a holomorphic modular form (g) of weight 8 with respect to g (1, 2) satisfying the factorization constraint (g) |Hk ×Hg−k = (k) · (g−k) for any k < g. The reason this statement is only proven for g ≤ 3 is that the superstring measure a priori may only be defined on the moduli space of curves Mg and not on the entire space Ag . For genus high enough the locus of decomposable abelian varieties Ak × Ag−k does not lie in (the closure of) the locus of Jacobians, and the superstring measure on Mg may not give rise to a modular form on Ag . However, for genus 4 any decomposable abelian variety is still a product of Jacobians (possibly of nodal curves), and the statement still holds. We will now rewrite the known low genus superstring measures in terms of our forms (g) Pi,s . Notice that we are looking for a form (g) of weight 8, and thus we will look for it as a linear combination of (g)

Gi

(g)

:= Pi,24−i ,

(6)

which are the only G s of appropriate weight. Since the only 0-dimensional vector space is zero, and all the 1-dimensional spaces over Z2 consist of zero and another vector, we have (g) (g) G 0 = θ [0]16 ; G 1 = θ [0]8 θ [v]8 . 2g

v∈Z2 \{0}

In genus 1 the superstring measure is known to be given by 4 8 4 4 0 0 0 1 (1) = θ η12 = θ θ θ . 0 0 1 0 The modular forms we have are 16 0 (1) G0 = θ and 0

G (1) 1

8 8 8 0 1 0 =θ +θ θ . 0 0 1

(7)

Using the Jacobi relation—the only algebraic identity among the three even theta constants with characteristics in genus 1—we can express (1) in terms of these, obtaining 1 (1) (1) G0 − G1 . (1) = 2

760

S. Grushevsky

In fact looking in [3], p.9 we see that (1) G0

4 1 0 12 =θ f 21 + η 0 3

and

(1) G1

4 1 0 12 =θ . f 21 + η 0 3

The genus 2 superstring measure was computed in [8], but let us try to look for it in the form (2)

(2)

(2)

(2) = a0 G 0 + a1 G 1 + a2 G 2 . The decomposable locus in this case is H1 ×H1 (together with its Sp(4, Z2 ) conjugates), the restriction of (2) by Theorem 15 is

(1) (1) (1) (1) (1) (1) (1) (1) G + a2 G (1) · G + a · G + G · G + G · G a0 G (1) 1 0 0 0 1 1 0 1 1 1 · G1 . Notice that here all the combinatorial coefficients Nn,m; i from Theorem 15 are equal to one, which is easy to see geometrically by remembering that this is the count of the number of i-dimensional subspaces of Zn2 ⊕ Zm 2 projecting onto both factors. Some of the coefficients for the restrictions we compute for higher genus are not as obvious, and we make substantial use of the theorem. For this restriction to be equal to (1) · (1) =

1 (1) 1 (1) (1) 1 (1) 1 (1) (1) (1) (1) G · G0 − G0 G1 − G1 · G0 + G1 · G1 4 0 4 4 4

we must choose a0 = 41 , a1 = − 41 , a2 = 21 , thus verifying Proposition 18. The following ansatz: (2) =

1 (2) (2) G 0 − G (2) , 1 + 2G 2 4

while being a holomorphic modular form of weight 8 with respect to 2 (1, 2), satisfies the factorization constraint. Remark 19. Note that unlike the formula for the genus 2 amplitude obtained in [8], and the various expressions for it studied in [12], our formula for (2) involves syzygetic rather than azygetic sets—and they all include the zero characteristic, so that the θ [0]4 factors appears naturally. To show that our ansatz is equal to the formulas given in [8] and [3] one expresses all theta functions with characteristics in terms of theta functions of the second order using the bilinear addition theorem and verifies that an identity is obtained (using Maple, not by hand). We now search for a genus 3 ansatz in the form (3)

(3)

(3)

(3)

(3) = a0 G 0 + a1 G 1 + a2 G 2 + a3 G 3 . Notice that this form is equivalent to the one used in [3]: our G (3) 3 is their G, and (3) (3) (3) their Fi ’s can be expressed as linear combinations of our G 0 , G 1 , G 2 , as can be verified by implementing the bilinear addition theorem in Maple. In our form we can


761

use Theorem 15 to easily compute the restriction of (3) to the locus of decomposable abelian varieties, which again has only one component, H1 × H2 , to be

(1) (2) (1) (2) (1) (2) (1) (2) (3) |H1 ×H2 = a0 G 0 · G 0 + a1 G 0 · G 1 + G 1 · G 0 + G 1 · G 1

(2) (1) (2) (1) (2) (2) + a2 G (1) + a3 G (1) 0 · G 2 + G 1 · G 1 + 3G 1 · G 2 1 · G2 . Requiring this to be equal to

1 (1) (2) (2) G 0 − G (1) G (2) 1 0 − G 1 + 2G 2 8 1 (1) (2) (1) (2) (1) (2) G0 · G0 − G0 · G1 − G1 · G0 = 8

(1) · (2) =

(1)

(2)

(1)

(2)

(1)

(2)

+2G 0 · G 2 + G 1 · G 1 − 2G 1 · G 2

(we arranged the terms to be in the same order as in the formula for the restriction, (2) (1) (2) where we note that G (1) 1 · G 1 and G 1 · G 2 appear twice) allows us to compute the coefficients ai starting from i = 0 uniquely to get Proposition 20. The following ansatz: (3) =

1 (3) (3) (3) (3) G 0 − G 1 + 2G 2 − 8G 3 , 8

while being a holomorphic modular form of weight 8 with respect to 3 (1, 2), satisfies the factorization constraints. We now look for a genus 4 ansatz in the form (4) (4) (4) (4) (4) = a0 G (4) 0 + a1 G 1 + a2 G 2 + a3 G 3 + a4 G 4 .

This is the first case when we have two different reducible loci: H1 × H3 and H2 × H2 . We again use Theorem 15 to compute the restrictions. To unclutter the formulas we drop the upper indices on the P’s and compute (4) |H1 ×H3 = a0 G 0 · G 0 + a1 (G 0 · G 1 + G 1 · G 0 + G 1 · G 1 ) + a2 (G 0 · G 2 + G 1 · G 1 + 3G 1 · G 2 ) + a3 (G 0 · G 3 +G 1 · G 2 +7G 1 · G 3 )+a4 G 1 · G 3 . Requiring this to be equal (1) · (3) =

1 (G 0 − G 1 ) · (G 0 − G 1 + 2G 2 − 8G 3 ), 16

we can solve for ai term by term (and the solution is unique!). On the other hand, we must also have a correct factorization on H2 × H2 : we must have (4) |H2 ×H2 = a0 G 0 · G 0 + a1 (G 0 · G 1 + G 1 · G 0 + G 1 · G 1 ) + a2 (G 0 · G 2 + G 1 · G 1 + 3G 1 · G 2 + G 2 · G 0 + 3G 2 · G 1 + 6G 2 · G 2 ) + a3 (G 1 · G 2 + G 2 · G 1 + 9G 2 · G 2 ) + a4 G 2 · G 2

762

S. Grushevsky

equal to 1 (G 0 − G 1 + 2G 2 ) · (G 0 − G 1 + 2G 2 ). 16 This can again be solved term by term to give a unique solution for all ai . Miraculously these solutions are the same (this is best checked by Maple, or, very tediously, by hand)) and we thus get (2) · (2) =

Theorem 21. The expression 1 (4) (4) (4) (4) (4) (4) := G 0 − G 1 + 2G 2 − 8G 3 + 64G 4 16 is a modular form of weight 8 with respect to 4 (1, 2), and satisfies the factorization constraints. This is the unique such linear combination of G i(4) , and is thus a natural candidate for the genus 4 superstring measure. 8. Further Directions There seem to be three natural further questions to ask, which we now discuss one by one. Question 1. Propose an ansatz for the superstring measure in any genus The computations above seem to work miraculously, but this is of course not a coincidence. By working carefully with the combinatorics of the coefficients in the restriction formula (g) in Theorem 15 one can always get a unique linear combination of G i that restricts correctly. Theorem 22. For any genus g the (possibly multivalued) function

(g)

g i(i−1) 1 (g) := g (−1)i 2 2 G i 2 i=0

is a modular form of weight 8 (up to a possible inconsistency in the choice of roots of unity for the different summands) with respect to g (1, 2), such that its restriction to Hk × Hg−k is equal to (k) · (g−k) . (g) Moreover, (g) is the unique linear combination of G i that restricts to the decomposable locus in this way. (g)

Proof. Uniqueness is easy to see: indeed, the coefficients of G i can be computed inductively; notice that the coefficient of G i has to be the same for all genera, as there is no dependence on g in the formula for restrictions—see (8) below. Thus in genus g (g) there is in fact only one new coefficient to compute, that in front of G g , and this can (g−1) (1) be computed from the coefficient of G 1 · G g−1 when restricting to H1 × Hg−1 . The hard part is verifying that this ansatz works—a priori it could happen that imposing the restriction constraint for some Hk × Hg−k would be incompatible with the constraint for some other k . Thus we need to verify that for the ansatz above we have for all k, (g) |Hk ×Hg−k = (k) · (g−k) .


763 (k)

(g−k)

Notice that the product on the right-hand-side is a sum of products of the type G n ·G m with coefficients given by the lower-genus ansätze. Theorem 15 shows that the left-handside is also a sum of terms of the same kind, and thus what we need to prove is that the (g−k) on both sides agree, i.e. that (notice that the powers of 21 all coefficients of G (k) n · Gm cancel) n+m

(−1)i 2

i(i−1) 2

Nn,m; i = (−1)n 2

n(n−1) 2

· (−1)m 2

m(m−1) 2

,

(8)

i=0

where Nn,m; i is the number of i-dimensional subspaces of Zn2 ⊕ Zm 2 surjecting onto both summands, given explicitly by (5). Note that the summands for i < max(n, m) are automatically zero, but it is convenient to include them formally. Notice that this (g) identity does not depend on g and k, which is why the coefficient of G i in (g) does not depend on i. While the quantity N has a geometric interpretation and thus (8) seems amenable to a geometric inclusion-exclusion proof, we give an easy proof by induction, still using some geometry for the inductive step. Note that n and m enter the formula symmetrically, so we can induct in either, and note that identity (8) is obviously true for n = 0 or m = 0, when there is only one summand on the left, and there is no product to take. To perform induction, we use the following combinatorial Lemma 23. The counting functions Nn,m: i given explicitly by (5) satisfy the following recursion: Nn,m+1; i+1 = Nn,m; i + (2i+1 − 2m )Nn,m; i+1 .

(9)

Proof. Recall that Nn,m+1; i+1 is the number of (i + 1)-dimensional subspaces V of surjecting onto both summands under the projection maps π1 and π2 . Let Zn2 ⊕ Zm+1 2 → Zn2 ⊕ Zm p : Zn2 ⊕ Zm+1 2 2 be the projection forgetting the last basis vector (denote this vector by em+1 ). Since p is a homomorphism, the map p : V → p(V ) is either 2-to-1, in which case dim p(V ) = i and V = p −1 (V ), or it is 1-to-1 and dim p(V ) = i + 1. We now count how many different V can give rise to a given p(V ) ⊂ Zn2 ⊕ Zm 2 (which still surjects onto both summands). If dim p(V ) = i, then V = p −1 (V ) is unique—thus we get the first summand in the lemma, with no coefficient. For the second case, choose ( p1 , . . . , pm ) ∈ p(V ) such that p◦π2 ( pk ) = ek is the k th m basis vector of Zm 2 (this is possible since p ◦ π2 : V Z2 ). The vectors { p1 , . . . , pm } ∈ p(V ) are linearly independent since their projections are. We can complete them to a basis { p1 , . . . , pi+1 } of p(V ) such that p ◦ π2 ( pk ) = 0 for m < k ≤ i + 1: to accomplish this, take any basis of p(V ) and subtract the appropriate sums of p1 . . . pm from the rest to make p ◦ π2 zero. To determine V given this p(V ) it suffices to choose v1 , . . . , vi+1 such that p(vk ) = pk , which amounts to choosing the em+1 -coordinate of each vk —thus there are 2i+1 choices. For any such choice V will surject onto Zn2 and onto Zm 2 , but for V to surject it is necessary and sufficient for there to exist m + 1 vectors in V with linearly onto Zm+1 2 independent π2 projections. We can choose v1 , . . . , vm as m of these vectors, and thus the condition for π2 : V → Zm+1 to be surjective is for there to exist some vector in the 2 span of vm+1 , . . . , vi+1 with non-zero em+1 -coordinate. Thus unless the em+1 coordinate of all vk is zero, i.e. unless vk = pk for all k = m +1, . . . , i (and then we have two choices for each of v1 , . . . , vm —so there are 2m such cases), V surjects onto both summands

764

S. Grushevsky

of Zn2 ⊕ Zm+1 and is counted in Nn,m+1; i+1 . Thus for each p(V ) of dimension i + 1 2 there are exactly 2i+1 − 2m different subspaces V projecting to it that are counted in Nn,m+1; i+1 . Remark 24. Note that this proof works also in the case when some of the N ’s appearing in the formula are zero. Another proof of the lemma can be obtained (but not so easily guessed!) by writing out the formulas (5) for N ’s in terms of products and manipulating them using 2k+1 − 2 j+1 = 2(2k − 2 j ), etc. Then one also has to check the cases when some N is zero separately, while the geometric argument works in all cases. We now complete the proof of the theorem by inducting from m to m +1. We substitute the recursive expression (9) from the lemma into the left-hand-side of (8) to get (we use I = i + 1 for the index of summation) n+m

(−1) I 2

I (I −1) 2

Nn,m+1; I

I =0

=

n+m+1

(−1) I 2

I (I −1) 2

Nn,m; I −1 + (2 I − 2m )Nn,m; I

I =0

= −2m

n+m

(−1) I 2

I (I −1) 2

Nn,m; I

I =0

+

n+m

(−1)i+1 2

(i+1)i 2

Nn,m; i +

n+m

(−1) I 2

I (I −1) 2

2 I Nn,m; I ,

I =0

i=0

where we used the fact that the i = −1 and I = n + m + 1 summands are zero in the last two sums, and the fact that formula (9) works for N ’s, some of which are zero as well. Now we note that the two expressions in the last line are the same up to sign and renaming the variable from i to I , and thus they cancel, so that we finally use the inductive assumption that (8) holds for m to obtain −2m

n+m

(−1) I 2

I (I −1) 2

Nn,m; I = −2m (−1)n+m 2

n(n−1) m(m−1) + 2 2

,

I =0

which is equal to the expression (−1)n+m+1 2

n(n−1) m(m+1) + 2 2

,

the right-hand-side of (8) for n and m + 1. The step of the induction is thus proven. Remark 25. This ansatz is a direct generalization of the formulas we obtained above for g ≤ 4 (and of course agrees with those). The potential problem with the multivaluedness here stems from the fact that for example G 5 = P5, 1 is the sum of square roots of products 2 of thetas. It could well happen, and seems perhaps not quite unlikely in view of Riemann’s quartic relations and Schottky-Jung identities (for an example of the Riemann quartic relation and the identities for theta constants on the Schottky locus, see the discussion of the genus 3 situation in [4]), that the product of 32 theta constants with characteristics in a vector subspace may indeed admit a holomorphic root over Mg . In this case the


765

expression above would be a natural candidate for the superstring measure. Note that the product of all theta constants has a holomorphic square root in genus 3 by results of Igusa; this kind of condition for the square root to be holomorphic was also encountered in the first attempts to compute the genus 3 superstring measure in [12,13]. Question 2. Verify that the proposed ansatz satisfies further physical constraints, for example that it yields a vanishing cosmological constant and vanishing 2- and 3-point functions Showing that the cosmological constant vanishes is equivalent to showing that the sum (g) [] is identically zero. This has been verified for genus 2 in [8] and for genus 3 in [3]. In general notice that this sum, if non-zero, is a modular form with respect to the entire group Sp(2g, Z) of weight 8. From the factorization constraint being satisfied we know that it restricts to the locus of decomposable abelian varieties as the product of the corresponding lower-dimensional sums, which we can inductively assume to vanish. In particular in genus 4 this sum vanishes on the locus A3 × A1 and thus on the boundary of Ag , which implies that this sum, if non-zero, is a modular form of slope at most 8. 12 However, it is known that the slope of the effective cone of M4 is equal to 6+ 4+1 > 8, and it is in fact known that the Schottky locus M4 ⊂ A4 is the zero locus of the unique modular form of slope 8 on A4 , the Schottky equation. Thus the form (4) [] must be a (possibly zero) multiple of this Schottky equation, and thus vanishes identically on M4 , so our ansatz does produce a vanishing cosmological constant in genus 4. It seems very hard to extend a similar kind of argument to higher genus, where the slopes of effective divisors on Mg and Ag are not known. Another constraint on the measure is to verify that all the 2- and 3-point functions vanish. This was verified for the genus 2 measure in [14,15], and checking this for the proposed ansätze in genera 3 and 4 would be a good indication of their potential validity. Question 3. Investigate whether the above restrictions are sufficient to guarantee the uniqueness of the solution for the superstring measure If we restrict ourselves to looking for the superstring measure as a product of the bosonic measure and a modular form of weight 8, suppose g is the lowest genus for which the ansatz is not unique, i.e. when there exist two distinct modular forms of weight 8 for g (1, 2) with the identical restriction to the decomposable locus. Then their difference F would be a modular form F of weight 8 with respect to g (1, 2) vanishing on all the components Ak × Ag−k of the locus of decomposable abelian varieties in Ag . If it could be shown from the theory of modular forms that such an F is then identically zero, then uniqueness of the ansatz in genus g would follow. This seems to be a really hard question, as the ring of modular forms for g (1, 2) for genus g ≥ 4 is not generated by theta constants—see the recent results in [21,20]. In general describing the ring of modular forms for g (1, 2) and obtaining conditions guaranteeing the vanishing of such a form seems very hard—see [22]. The authors of [3] indicate that they will give a proof of uniqueness in genus 3 in a forthcoming paper. Also note that while in [8–11] the formula for the genus 2 superstring measure was derived from the first principles and as such has to be unique, the derivations in higher genus are based on the assumption of the superstring measure being a product of a modular form and the bosonic measure, which then needs to be justified in some physical way.

766

S. Grushevsky

Acknowledgements. I learned about the problem from Duong Phong, to whom I am very grateful for his constant encouragement, for explaining the basic questions and computations for the string measure, and for detailed comments on the draft of this text. I am very thankful to Eric D’Hoker and Duong Phong for many conversations on the subject, for sharing their Maple code, for ideas on how the higher genus superstring measure could be constructed, and especially for suggesting that I express the genus 3 ansatz of Cacciatori, Dalla Piazza, and van Geemen in terms of syzygy conditions, which allowed me to then further generalize it. I am also very grateful to Riccardo Salvati Manni for bringing to my attention his work [23] and the literature on polynomials P(N m) corresponding to even cosets, and for the encouragement in exploring whether these can be used to obtained an ansatz. I am very thankful to Riccardo Salvati Manni for pointing out (g) and giving a rigorous proof in [25] that Pi,s are only known to be modular forms for 2i s = 2k for k ≥ 4 (Proposition 13), and that Pi,s [] are modular with respect to conjugates of g (1, 2) rather than g (1, 2) itself, and for comments on the manuscript.

References 1. Beilinson, A., Manin, F.: The Mumford form and the Polyakov measure in string theory. Commun. Math. Phys. 107, 359–376 (1986) 2. Cacciatori, S.L., Dalla Piazza, F.: Two loop superstring amplitudes and S6 representations. Lett. Math. Phys. 83(2), 127–138 (2008) 3. Cacciatori, S.L., Dalla Piazza, F., van Geemen, B.: Modular Forms and Three Loop Superstring Amplitudes. http://arXiv.org/abs/0801.2543v2[hepth], 2008 4. van Geemen, B., van der Geer, G.: Kummer varieties and the moduli spaces of abelian varieties. Amer. J. of Math. 108, 615–642 (1986) 5. Green, M.B., Schwarz, J.H.: Supersymmetrical string theories. Phys. Lett. B 109, 444–448 (1982) 6. Gross, D.J., Harvey, J.A., Martinec, E.J., Rohm, R.: Heterotic String Theory (II). The interacting heterotic string. Nucl. Phys. B 267, 75 (1986) 7. D’Hoker, E., Phong, D.H.: Multiloop amplitudes for the bosonic Polyakov string. Nucl. Phys. B 269, 205– 234 (1986) 8. D’Hoker, E., Phong, D.H.: Two-Loop Superstrings I, Main Formulas. Phys. Lett. B 529, 241–255 (2002) 9. D’Hoker, E., Phong, D.H.: Two-Loop Superstrings II, The chiral Measure on Moduli Space. Nucl. Phys. B 636, 3–60 (2002) 10. D’Hoker, E., Phong, D.H.: Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities. Nucl. Phys. B 636, 61–79 (2002) 11. D’Hoker, E., Phong, D.H.: Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms. Nucl. Phys. B 639, 129–181 (2002) 12. D’Hoker, E., Phong, D.H.: Asyzygies, modular forms, and the superstring measure I. Nucl. Phys. B 710, 58 (2005) 13. D’Hoker, E., Phong, D.H.: Asyzygies, modular forms, and the superstring measure. II. Nucl. Phys. B 710, 83 (2005) 14. D’Hoker, E., Phong, D.H.: Two-Loop Superstrings V, Gauge Slice Independence of the N-Point Function. Nucl. Phys. B 715, 91–119 (2005) 15. D’Hoker, E., Phong, D.H.: Two-Loop Superstrings VI, Non-Renormalization Theorems and the 4-Point Function. Nucl. Phys. B 715, 3–90 (2005) 16. Igusa, J.-I.: Theta functions. Die Grundlehren der mathematischen Wissenschaften, Band 194. New YorkHeidelberg: Springer-Verlag, 1972 17. Krazer, A.: Lehrbuch der Thetafunktionen. Leipzig: B. G. Teubner, 1903 18. Manin, Y.: The partition function of the Polyakov string can be expressed in terms of theta functions. Phys. Lett. B 172, 184–185 (1986) 19. Matone, M., Volpato, R.: Higher genus superstring amplitudes from the geometry of moduli space. Nucl. Phys. B 732, 321–340 (2006) 20. Oura, M., Poor, C., Yuen, D.S.: Toward the Siegel ring in genus four. Int. J. Number Th. 4(4), 563– 586 (2008) 21. Oura, M., Salvati Manni, R.: On the image of code polynomials under theta map. http://arXiv.org/abs/ 0803.4389v1[math-NT], 2008 22. Poor, C., Yuen, D.S.: Linear dependence among Siegel modular forms. Math. Ann. 318, 205–234 (2000) 23. Salvati Manni, R.: On the dimension of the vector space C[θm ]4 . Nagoya Math. J. 98, 99–107 (1985) 24. Salvati Manni, R.: Modular varieties with level 2 theta structure. Amer. J. Math. 116, 1489–1511 (1994)


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25. Salvati Manni, R.: Remarks on Superstring amplitudes in higher genus. Nucl. Phys. B, to appear, http:// arXiv.org/abs/0804.0512v2[hep-th], 2008 26. Verlinde, E., Verlinde, H.: Chiral Bosonization, determinants and the string partition function. Nucl. Phys. B 288, 357–396 (1987) Communicated by A. Kapustin


Communications in


Quantum Enveloping Algebras with von Neumann Regular Cartan-like Generators and the Pierce Decomposition Steven Duplij1, , Sergey Sinel’shchikov2 1 Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany.


2 Mathematics Division, B. I. Verkin Institute for Low Temperature Physics and Engineering,

National Academy of Sciences of Ukraine, 47 Lenin Ave, Kharkov 61103, Ukraine. E-mail: [email protected] Received: 9 April 2008 / Accepted: 19 June 2008 Published online: 15 October 2008 – © Springer-Verlag 2008

Dedicated to the memory of our colleague Leonid L. Vaksman (1951–2007) Abstract: Quantum bialgebras derivable from Uq (sl2 ) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.

1. Introduction The language of Hopf algebras [1,24] is among the principal tools of studying subjects associated to noncommutative spaces [5,18] and superspaces [6,13,23] appearing as quantization of commutative ones [12,25]. An important feature of supersymmetric algebraic structures is that their underlying algebras normally contain idempotents and other zero divisors [2,10,21]. Therefore, it is reasonable to render idempotents to some quantum algebras, to study their properties and the associated Pierce decompositions [20]. In this paper we introduce a new quantum algebra which admits an embedding of Uq (sl2 ) [9,14]. After adding some extra relations we obtain two worthwhile algebras that contain idempotents and von Neumann regular Cartan-like generators. One of the algebras has the Pierce decomposition which reduces to a direct sum of two ideals and can be treated as an extended version of the algebra with von Neumann regular antipode considered in [11,17], while another one appears to be a Hopf algebra in the sense of the standard definition [1]. We distinguish some special cases for which R-matrices of On leave of absence from: Theory Group, Nuclear Physics Laboratory, V. N. Karazin Kharkov National University, Svoboda Sq. 4, Kharkov 61077, Ukraine. E-mail: [email protected]; http://webusers.physics. umn.edu/~duplij

770

S. Duplij, S. Sinel’shchikov

simple form are available. This way both invertible and von Neumann regular R-matrices have been produced, the latter respecting the Pierce decomposition. 2. Preliminaries We start with recalling briefly some necessary notations and principal facts about Hopf algebras [1,4]. In our context an algebra U (alg) over C is a 4-tuple (C, A, µ, η), where A is a vector space, µ : A ⊗ A → A is a multiplication (alternatively denoted as de f

µ (a ⊗ b) = a · b), η : C → A is a unit so that 1 = η (1), 1 ∈A, 1 ∈ C. The multiplication is assumed to be associative µ ◦ (µ ⊗ id) = µ ◦ (id ⊗ µ) and the unit is characterized by the property µ ◦ (η ⊗ id) = µ ◦ (id ⊗ η) = id. An algebra map is a (alg) (alg) → U2 subject to ψ ◦ µ1 = µ2 ◦ (ψ ⊗ ψ) and ψ ◦ η1 = η2 . linear map ψ : U1 A coalgebra U (coalg) is a 4-tuple (C, C, ∆, ), where C is an underlying vector space, ∆ : C → C ⊗ C is a comultiplication with ∆ (A) = i Ai(1) ⊗ Ai(2) in the Sweedler notation, : C → C is a counit. These linear maps are subject to the following properties: coassociativity (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆, the counit property ( ⊗ id) ◦ (coalg) (coalg) → U2 ∆ = (id ⊗ ) ◦ ∆ = id. A coalgebra map is a linear map ϕ : U1 such that (ϕ ⊗ ϕ) ◦ ∆1 = ∆2 ◦ ϕ and 1 = 2 ◦ ϕ. A bialgebra U (bialg) is a 6-tuple (C, B, µ, η, ∆, ) which is an algebra and coalgebra simultaneously, with the compatibility conditions as follows: ∆ ◦ µ = (µ ⊗ µ) ◦ (id ⊗ τ ⊗id) ◦ (∆ ⊗ ∆), ∆ (1) = 1 ⊗ 1, ◦µ = µC ◦( ⊗), (1) = 1; here τ is the flip of tensor multiples, µC is the multiplication in the ground field. A Hopf algebra U (H op f ) is a bialgebra equipped with antipode, an antimorphism of algebra subject to the relation (S ⊗ id) ◦ ∆ = (id ⊗ S) ◦ ∆ = η ◦ . Let q ∈ C and q = ±1,0. We start with a definition of quantum universal enveloping (alg) algebra Uq (sl2 ) [8]. This is a unital associative algebra Uq (sl2 ) determined by its (Chevalley) generators k, k −1 , e, f , and the relations k −1 k = 1,

kk −1 = 1,

ke = q ek, k f = q 2

ef − f e =

−2

(1) f k,

(2)

− k −1

k . q − q −1 (H op f )

The standard Hopf algebra structure on Uq

(3) (sl2 ) is determined by

∆0 (k) = k ⊗ k,

(4)

∆0 (e) = 1 ⊗ e + e ⊗ k, ∆0 ( f ) = f ⊗ 1 + k −1

−1

−1

⊗ f,

S0 (k) = k , S0 (e) = −ek , S0 ( f ) = −k f, ε0 (k) = 1, ε0 (e) = ε0 ( f ) = 0. (alg)

(5) (6) (7)

The algebra Uq (sl2 ) is a domain, i.e. it has no zero divisors and, in particular, no idempotents [7,15]. A basis of the vector space Uq (sl2 ) is given by the monomim, n ≥ 0 [14]. We denote the Cartan subalgebra of Uq (sl2 ) by als k s em f n , where H0 1, k, k −1 . Our goal is to apply the Pierce decomposition to a suitably extended version of Uq (sl2 ). It is well known that there exists one-to-one correspondence between the central decompositions of unity on idempotents and decompositions of a module into a direct

Quantum Enveloping Algebras and the Pierce Decomposition

771

sum. Therefore we start with generalizing the Cartan subalgebra in Uq (sl2 ) towards the von Neumann regularity property [3,19,22]. 3. From the Standard Uq (sl 2 ) to U K + L Let us consider the generators K , K satisfying the relations K K K = K,

K K K = K,

(8)

which are normally referred to as von Neumann regularity [19]. Under the assumption of commutativity KK = KK

(9)

de f

we have an idempotent P = K K = K K subject to P K = K P = K, P 2 = P.

(10)

(11) The commutative algebra generated by K , K is not unital (we denote it by H K , K ), because unlike Uq (sl2 ) its relations do not anticipate unit explicitly, as in (1). Note that H K , K was considered as a Cartan-like part of the analog of the quantum enveloping algebra with von Neumann regular antipode Uqv = vslq (2) introduced by Duplij and Li [11,17]. The associated unital algebra derived by an exterior attachment of unit de f H 1, K , K = H K , K ⊕ C1 also appears in [11,17] as a part of Uqw = wslq (2). Observe that H 1, K , K contains one more idempotent (1 − P)2 = (1 − P). Therefore, we introduce another copy of the same algebra (we denote it by H L , L ) with generators L and L subject to similar relations as for K , K above L L L − L = 0,

L L L − L = 0.

(12)

Under the commutativity assumption LL = LL

(13)

de f

the idempotent Q = L L = L L satisfies QL = L Q = L,

(14)

Q = Q.

(15)

2

If there are no additional relations between K , K and L , L, the nonunital algebras H K , K and H L , L can form a free product only. On the other hand we merge together the unital algebras H 1, K , K and H 1, L , L so that their units are identified and add one more relation, the decomposition of unity P+Q=1

(16)

in order to produce the Pierce decomposition [20] of the resulting algebra H 1, K , K , L , L , which reduces to the direct product since Q P = P Q = 0. It follows from (10), (14) and (16) that K L = L K = L K = K L = K L = L K = 0.

(17)

772


The new (as compared to [11,17]) noninvertible generators L, L are introduced to justify the following Lemma 1. The sum a K + bL is invertible, and its inverse is a −1 K + b−1 L , where a, b ∈ R0. Proof. Reduces to a computation which involves (16) and (17) as (a K + bL) a −1 K + b−1 L = K K + L L = P + Q = 1.

(18)

This allows us to consider a two-parameter family of morphisms for the Cartan subal (a,b) gebra H : H0 1, k, k −1 → H 1, K , K , L , L given by k → a K + bL , (a,b)

Proposition 1. The map H

k −1 → a −1 K + b−1 L. (a,b)

is an embedding, i.e. ker H

(19)

= 0.

¯ (a,b) algebra H¯ 0 1, k, k −1 Proof. Use (19) to define a homomorphism H from the free generated by 1, k, k −1 into the free algebra H¯ 1, K , K , L , L generated by 1, K , K , ¯ (a,b) ¯ (a,b) In fact, if not, then L, L. We claim that H is an embedding. H annihilates some −1 nonzero element of H¯ 0 1, k, k . This element can be treated as a “noncommutative polynomial” in three indeterminates 1, k, k −1 . Because the linear change of variables (19) is non-degenerate, we obtain a nontrivial polynomial in 1, K , K , L, L, which cannot be zero in the free algebra H¯ 1, K , K , L , L . What remains is to observe that (a,b) H establishes one-to-one correspondence between the relations in H0 1, k, k −1 (a,b) and those induced on the image of H , which already implies our statement for the (a,b) morphism H between the quotient algebras H0 1, k, k −1 and H 1, K , K , L , L . Now we are in a position to add two more generators E and F, along with additional relations (a K + bL) E = q 2 E (a K + bL) , a −1 K + b−1 L E = q −2 E a −1 K + b−1 L ,

(20)

(a K + bL) F = q −2 F (a K + bL) , a −1 K + b−1 L F = q 2 F a −1 K + b−1 L , (a K + bL) − a −1 K + b−1 L EF − FE = , q − q −1

(22)

(21)

(23) (24) (alg)22

which together with (8)-(9) and (12)-(13) determine an algebra we denote by Ua K +bL , the indices 22 stand for the numbers of generators in the left (resp., right) hand sides of the relations between the Cartan-like generators (K , L) and E, F. This algebra corresponds to Uqw = wslq (2) introduced by Duplij and Li [11,17]. To be more precise, (alg)22

there exists an algebra homomorphism wslq (2) → Ua K +bL , which in the notation of [11] is given by K w → a K + bL ,

K w → a −1 K + b−1 L,

E w → E,

Fw → F.

(25)


773

As one can see from Lemma 1, together with (20) – (24), the image of this homomorphism is a copy of Uq (sl2 ), cf. [11, Prop. 1]. Next we present an analog of the algebra Uqv = vslq (2) as in [11]. This is an algebra (alg)22

having the same generators as Ua K +bL , and being subject to the relations (together with (8) – (9) and (12) – (13)), (26) (a K + bL) E a −1 K + b−1 L = q 2 E, a −1 K + b−1 L E (a K + bL) = q −2 E, (27) (28) (a K + bL) F a −1 K + b−1 L = q −2 F, a −1 K + b−1 L F (a K + bL) = q 2 F, (29) −1 (a K + bL) − a K + b−1 L EF − FE = , (30) q − q −1 (alg)31

which we denote Ua K +bL . This algebra corresponds to the algebra Uqv = vslq (2) [11] (alg)31

in the sense that there exists an algebra homomorphism vslq (2) → Ua K +bL . Again, this homomorphism, in the notation of [11], is given on the generators by (25), with the indices w being replaced by v. Another application of Lemma 1 allows one to observe that the image of this homomorphism is a copy of Uq (sl2 ), cf. [11, Prop. 1]. (a,b) Introduce an extension (a,b) of H to a morphism of Uq (sl2 ) with values in (alg)22

(alg)31

Ua K +bL and Ua K +bL as (a,b) :

k → a K + bL , k −1 → a −1 K + b−1 L, e → E, f → F. (alg)22

(alg)31

(31) (alg)22

Proposition 2. The algebras Ua K +bL and Ua K +bL are isomorphic to U K +L de f

(alg)22

(alg)31 de f

= Ua K +bL |a=1,b=1 and U K +L

(alg)31

= Ua K +bL |a=1,b=1 respectively. (alg)22,31

(alg)22,31

→ Ua K +bL is given by Proof. The desired isomorphism : U K +L K → a K , L → bL , K → a −1 K , L → b−1 L, E → E, F → F.

Therefore, we will not consider the parameters a and b below. 4. Splitting the Relations (alg)22

(alg)31

The idempotents P and Q are not central in U K +L and U K +L . By allowing certain misuse of terminology, we are going to ”split” the relations (20) – (24) and (26) – (30) in such a way that either P and Q become central, P E = E P, Q E = E Q,

(32)

P F = F P, Q F = F Q,

(33)

or satisfy the “twisting” conditions P E = E Q, Q E = E P,

(34)

P F = F Q, Q F = F P.

(35)

774


To be more precise, we are about to add the above relations in order to get the associated (alg)22 (alg)31 quotients of U K +L and U K +L . The “splitted” 22-algebras are given by the following lists of relations: (alg)22

(alg)22

U K ,L ,nor m

U K ,L ,twist

K K K = K, K K K = K, K K = K K, L L L = L , L L L = L, L L = L L, K K + L L = 1, K E = q 2 E K , L E = q 2 E L, K E = q −2 E K , L E = q −2 E L, K F = q −2 F K , L F = q −2 F L, K F = q 2 F K , L F = q 2 F L, (K + L) − K + L EF − FE = q − q −1

K K K = K, K K K = K, K K = K K, L L L = L , L L L = L, L L = L L, K K + L L = 1, K E = q 2 E L, L E = q 2 E K , K E = q −2 E L, L E = q −2 E K , K F = q −2 F L, L F = q −2 F K , K F = q 2 F L, L F = q 2F K , (K + L) − K + L EF − FE = q − q −1

(36)

and the ”splitted” 31-algebras are defined as follows: (alg)31

(alg)31

U K ,L ,nor m

U K ,L ,twist

K K K = K, K K K = K, K K = K K, L L L = L , L L L = L, L L = L L, K K + L L = 1, K E K = q 2 E K K , L E L = q 2 E L L, K E K = q −2 E K K , L E L = q −2 E L L, K F K = q −2 F K K , L F L = q −2 F L L, K F K = q 2 F K K , L F L = q 2 F L L, K−K K K (E F − F E) = , q − q −1 L−L L L (E F − F E) = q − q −1

K K K = K, K K K = K, K K = K K, L L L = L , L L L = L, L L = L L, K K + L L = 1, K E L = q 2 E L L, L E K = q 2 E K K , K E L = q −2 E L L, L E K = q −2 E K K , K F L = q −2 F L L, L F K = q −2 F K K , K F L = q 2 F L L, L F K = q 2 F K K , K−K K K (E F − F E) = , q − q −1 L−L L L (E F − F E) = q − q −1 (37)

Note that P = K K and Q = L L are not among the generators used in (36) and (37). The relations which appear in the tables form the (equivalent) translation in terms (alg)22 (alg)31 of the ”true” generators of the earlier relations for U K +L and U K +L , together with the ”splitting” relations (32) – (35). The procedure of deducing relations in tables from the original ”non-splitted” relations in most cases reduces to right and/or left multiplication by the idempotents P and Q with subsequent use of the ”annihilation rules” (17). Conversely, suppose that (36) and (37) are given. For example, let us start from the


775

relations in the left column of (37). To see that in this case P is central, one has, using (17), P E = K K E(P + Q) = K (K E K )K + K K (E L L) = K (q −2 E K K )K + K K (q −2 L E L) = q −2 K E K + 0 = E K K = E P. Of course, similar ideas work also in the rest of verifications. (alg)22 (alg)31 Proposition 3. We have the following isomorphisms: U K ,L ,nor m ∼ = U K ,L ,nor m , and (alg)22 (alg)31 ∼ . U =U K ,L ,twist

K ,L ,twist

Proof. A straightforward computation shows that, in both cases (normal and twisted), the ideals of relations in question coincide. For instance, the right multiplication of (alg)22 (alg)31 K E = q 2 E K by K in U K ,L ,nor m yields K E K = q 2 E P as in U K ,L ,nor m . Conversely, (alg)31

starting from the relation K E K = q 2 E P in U K ,L ,nor m we calculate K E = K (P E) = (alg)22 K (E P) = K E K K = q 2 E P K = q 2 E K as in U K ,L ,nor m . Multiplying the E F(alg)22

(alg)22

(alg)31

relations in U K ,L ,nor m , U K ,L ,twist by P and Q we obtain the E F-relations of U K ,L ,nor m , (alg)31

(alg)31

U K ,L ,twist , and conversely, summing up the last two E F-relations of U K ,L ,nor m and using (16), we obtain the E F-relations of second isomorphism.

(alg)22 U K ,L ,nor m .

Similar arguments establish the

(alg)22

(alg)22

Therefore, in what follows we consider the algebras U K ,L ,nor m , U K ,L ,twist (with the 22 superscript being discarded) only. Now we extend the morphism H to that taking values in the “splitted” algebras (alg) (alg) U K ,L ,nor m and U K ,L ,twist as follows: k → K + L, k −1 → K + L, : (38) e → E, f → F. Proposition 4. The map defined on the generators as above, admits an extension to (alg) (alg) a well defined morphism of algebras from Uq (sl2 ) to either U K ,L ,nor m or U K ,L ,twist , which is an embedding. Proof. Use an argument similar to that applied in the proof of Proposition 1. (alg)

(alg)

Corollary 1. Both algebras U K ,L ,nor m and U K ,L ,twist contain Uq (sl2 ) as a subalgebra. Proof. Follows from Proposition 4. (alg)

Note that the Pierce decomposition of U K ,L ,nor m is (alg)

(alg)

(alg)

U K ,L ,nor m = PU K ,L ,nor m P + QU K ,L ,nor m Q,

(39)

which reduces to a direct sum of the two ideals. This leads to (alg)

Proposition 5. U K ,L ,nor m is a direct sum of subalgebras with each summand being isomorphic to Uq (sl2 ). Proof. The desired isomorphism is given by K −→ k ⊕ 0, K −→ k −1 ⊕ 0, P E −→ e ⊕ 0, P F −→ f ⊕ 0, L −→ 0 ⊕ k, L −→ 0 ⊕ k

−1

, Q E −→ 0 ⊕ e, Q F −→ 0 ⊕ f,

(40) (41)

776


hence P −→ 1⊕0, Q −→ 0⊕1. This morphism splits as a direct sum of two morphisms each of the latter being, obviously, an isomorphism. In the “twisted” case the Pierce decomposition (alg)

(alg)

(alg)

(alg)

(alg)

U K ,L ,twist = PU K ,L ,twist P + PU K ,L ,twist Q + QU K ,L ,twist P + QU K ,L ,twist Q,

(42)

is nontrivial as all terms are nonzero, i.e. (42) is not a direct sum of ideals. (alg) (alg) Let us introduce a special automorphism of algebras U K ,L ,nor m and U K ,L ,twist , which will be denoted by the same letter ϒ. In either case, ϒ is given on the generators by E → E, F → F, K → L , K → L, L → K , L → K , 1 → 1,

(43)

and then extended to an endomorphism of the algebra in question. The very fact that it becomes this way a well defined linear map and then its bijectivity is established by observing that ϒ permutes the list of generators as well as the list of relations. Note that ϒ 2 = id. (alg)

Proposition 6. The Poincaré-Birkhoff-Witt basis of U K ,L ,nor m is given by the monomials

i P K i E j Fk ∪ K E j Fk i, j,k≥0 i>0, j,k≥0

i . (44) ∪ Q Li E j Fk ∪ L E j Fk i, j,k≥0

i>0, j,k≥0

(alg)

Proof. Since U K ,L ,nor m is a direct sum of two copies of Uq (sl2 ), the statement immediately follows from [14]. (alg)

In the case of U K ,L ,twist we have the decomposition into a direct sum of 4 vector subspaces (42). We present below a PBW basis which respects this decomposition. (alg)

Proposition 7. The Poincaré-Birkhoff-Witt basis of U K ,L ,twist is given by the monomials i P K i E j F k i, j,k≥0 ∪ K E j F k i>0, j,k≥0 ∪

j+k even

i

j

PK E F

k

⎢ ∪ ⎣ Q Li E j Fk

∪

i

j

QL E F

k

j+k even

i, j,k≥0 j+k odd

⎡

i

∪ K E F

i

i

j

k

i>0, j,k≥0 j+k odd

∪ L E j Fk

i, j,k≥0 j+k even

∪ L E F j

⎤

i, j,k≥0 j+k odd

k

⎥

i>0, j,k≥0 ⎦ j+k odd

i>0, j,k≥0 j+k even

.

(45)

Proof. It follows from (36) that the linear span of (45) is stable under multiplication by any of the generators K , K , L, L, E, F, which implies that this stability is also valid (alg) under multiplication by any element of U K ,L ,twist . Since P and Q are among the basis vectors, this linear span contains P + Q = 1, hence is just the entire algebra. To prove


777

the linear independence of (45) it suffices to prove that every part of this vector system which is inside a specific Pierce component, is linear independent. We now stick to the (alg) special case of the Pierce component P · U K ,L ,twist · P which is generated by the family of vectors i P K i E j F k i, j,k≥0 ∪ K E j F k i>0, j,k≥0 , (46) j+k even

j+k even

the part of the vector system (45) inside the first bracket. Consider a (finite) linear combination i αi jk P K i E j F k + βi jk K E j F k (47) i, j,k≥0 j+k even


which is non-trivial (not all αi jk and βi jk are zero). We are about to prove that (47) is non-zero. For that, we first use αi jk and βi jk to produce the associated non-trivial linear combination

αi jk k i e j f k +

i, j,k≥0 j+k even

βi jk k −i e j f k

(48)

i>0 , j,k≥0 j+k even

in Uq (sl2 ). Since the monomials involved form a PBW basis in Uq (sl2 ) [14], the linear combination (48) is non-zero. Now apply the map (38) to (48) to obtain i αi jk (K + L)i E j F k + βi jk K + L E j F k . (49) i, j,k≥0 j+k even


(alg)

As is an embedding by Proposition 4, we deduce that (49) is non-zero in U K ,L ,twist . Observe also that in the involved monomials j + k is even; it follows that the projections (alg) (alg) of (49) to the Pierce components P · U K ,L ,twist · Q and Q · U K ,L ,twist · P are both zero. (alg)

(alg)

Hence (49) is the sum of its projections to P · U K ,L ,twist · P and Q · U K ,L ,twist · Q, which are just i αi jk P K i E j F k + βi jk K E j F k i, j,k≥0 j+k even

and

i, j,k≥0 j+k even


αi jk Q L i E j F k +

i

βi jk L E j F k ,


respectively. It is easy to see that these are intertwined by the automorphism ϒ (43), which implies that these projections are simultaneously zero or non-zero. Of course, the second assumption is true, because their sum (49) is non-zero. In particular, i αi jk P K i E j F k + βi jk K E j F k i, j,k≥0 j+k even


is non-zero, which was to be proved. The proof of linear independence of all other subsystems of (45) (in brackets), related to other Pierce components, goes in a similar way. (alg)

(alg)

Let us consider the classical limit q → 1 for U K ,L ,nor m and U K ,L ,twist algebras.

778

S. Duplij, S. Sinel’shchikov (alg)

Proposition 8. The classical limit of U K ,L ,nor m is just a direct sum of two copies of classical limits for Uq (sl2 ) in the sense of [16]. Proof. This follows from Proposition 5. 5. Hopf Algebra Structure and von Neumann Regular Antipode To construct a bialgebra we need a counit on U K +L , to be denoted by ε. Since P and Q are idempotents in U K +L , one has ε (P) (ε (P) − 1) = 0 and ε (Q) (ε (Q) − 1) = 0, which implies that either ε (P) = 1, ε (Q) = 0 or ε (P) = 0, ε (Q) = 1. We assume the first choice. Then it follows from L = Q L that ε (L) = ε (Q L) = 0. Also it follows from (4) that ε(K + L) = 1, hence ε(K ) = 1. Elaborate the embedding defined in (19) and the standard relations (4), (5), (7) to transfer a coproduct onto the image of (31) as follows: ∆(K + L) = (K + L) ⊗ (K + L) , ∆ K+L = K+L ⊗ K+L ,

(50)

∆(E) = 1 ⊗ E + E ⊗ (K + L) , ∆(F) = F ⊗ 1 + K + L ⊗ F,

(52)

ε(E) = ε(F) = 0, ε(K + L) = 1, ε K + L = 1.

(54) (55)

(51) (53)

(56) (alg)

(alg)

To produce a comultiplication on the above algebras U K ,L ,nor m and U K ,L ,twist deter (alg) mined by (36), use (50)–(56) to define a coproduct ∆ first on Uq (sl2 ) (via (alg)

transferring from Uq (alg)

(alg)

(sl2 )) and then extend it to the entire algebras U K ,L ,nor m and

U K ,L ,twist as follows: (coalg)

(coalg)

U K ,L ,nor m

U K ,L ,twist

∆(K ) = K ⊗ K , ∆(K ) = K ⊗ K , ∆(L) = L ⊗ L + L ⊗ K + K ⊗ L , ∆(L) = L ⊗ L + L ⊗ K + K ⊗ L, ∆(E) = 1 ⊗ E + E ⊗ (K + L) , ∆(F) = F ⊗ 1 + K + L ⊗ F, ε(E) = ε(F) = 0, ε(K ) = 1, ε(K ) = 1, ε(L) = ε(L) = 0.

∆(K ) = K ⊗ K + L ⊗ L , ∆(K ) = K ⊗ K + L ⊗ L, ∆(L) = L ⊗ K + K ⊗ L , ∆(L) = L ⊗ K + K ⊗ L ∆(E) = 1 ⊗ E + E ⊗ (K + L) , ∆(F) = F ⊗ 1 + K + L ⊗ F, ε(E) = ε(F) = 0, ε(K ) = 1, ε(K ) = 1, ε(L) = ε(L) = 0.

(bialg)

(57)

(bialg)

The convolution on the bialgebras U K ,L ,nor m and U K ,L ,twist produced this way is defined by (A B) ≡ µ (A ⊗ B) ∆, where A,B are linear endomorphisms of the underlying vector space.

(58)


779

(bialg)

Let us first consider the bialgebra U K ,L ,nor m from the viewpoint of Hopf algebra structure. (bialg)

Proposition 9. The bialgebra U K ,L ,nor m has no conventional antipode S satisfying the standard Hopf algebra axiom S id = id S = η ◦ ε.

(59)

Proof. Since ε (P) = 1 and ∆(P) = P ⊗ P we have from (58) (S id) (P) = S (P) P = (id S) (P) = PS (P) = 1 · ε (P) = 1,

(60)

which is impossible since P is not invertible. (bialg)

Let us introduce an antimorphism T of U K ,L ,nor m as follows: T (K ) = K , T K = K , T (L) = L, T L = L , T (E) = −E K + L , T (F) = − (K + L) F.

(61) (62)

(bialg)

For U K ,L ,nor m we observe that

(T id) (K ) = (id T) (K ) = (T id) K = (id T) K = P, (T id) (L) = (id T) (L) = (T id) L = (id T) L = Q,

(64)

(T id) (E) = (id T) (E) = (T id) (F) = (id T) (F) = 0.

(65)

(63)

(bialg)

Proposition 10. The antimorphism T of U K ,L ,nor m is von Neumann regular id T id = id,

T id T = T.

(66)

Proof. First observe that, since a convolution of linear maps is again a linear map, it (bialg) (bialg) suffices to verify (66) separately on the direct summands PU K ,L ,nor m and QU K ,L ,nor m , (bialg)

associated to the central idempotents P and Q, respectively. We start with PU K ,L ,nor m , (bialg)

which is a sub-bialgebra. Denote by ϕ P : PU K ,L ,nor m → Uq (sl2 ) the isomorphism (40). Earlier it was introduced as an isomorphism of algebras (hence it intertwines the −1 −1 = µ0 = µUq (sl2 ) ), but now it follows from (57) products, ϕ P ◦ µ ◦ ϕ P ⊗ ϕ P and ∆(P) = P ⊗ P that ϕ P also intertwines the comultiplication (4)-(5) of Uq (sl2 ) (bialg) (bialg) and the restriction of the comultiplication ∆ of U K ,L ,nor m onto PU K ,L ,nor m , that is, (ϕ P ⊗ ϕ P ) ◦ ∆ ◦ ϕ −1 P = ∆0 . It follows that, given any two endomorphisms of the underlying vector space of (bialg) (bialg) U K ,L ,nor m which leave PU K ,L ,nor m invariant, then ϕ P sends the convolution of them (bialg)

(restricted to PU K ,L ,nor m ) to the convolution of the transferred maps on Uq (sl2 ). (bialg)

An obvious verification shows that both id and T leave PU K ,L ,nor m invariant, and then a computation shows that so do id T and T id. Specifically, one has (id T) (P X ) = (T id) (P X ) = ε0 (ϕ P (P X )) P (bialg)

for any X ∈ U K ,L ,nor m . This means that ϕ P establishes the equivalence of (66) on (bialg)

PU K ,L ,nor m and the von Neumann regularity conditions for the transfer of T via ϕ P on

780


Uq (sl2 ). An easy verification shows that this transfer is just S, the antipode of Uq (sl2 ). It is well known that S is also von Neumann regular, which finishes the proof of (66) (bialg) restricted to PU K ,L ,nor m . −1 : One can readily replace in the above argument ϕ P by the isomorphism Uq (sl2 ) → Uq (sl2 ), with being the embedding (38). This way we obtain (66) (bialg) restricted to Uq (sl2 ) . However, this argument is inapplicable to QU K ,L ,nor m , as the latter fails to be a sub-coalgebra. (bialg) Now observe that the projection of Uq (sl2 ) to the direct summand QU K ,L ,nor m (bialg) is just QU K ,L ,nor m . This is because the PBW basis k i e j f k j,k≥0 of Uq (sl2 ) transferred by is just i ∪ K + L E j Fk . (K + L)i E j F k i, j,k≥0

i>0, j,k≥0

(bialg)

These vectors project to QU K ,L ,nor m as i Q Li E j Fk ∪ L E j Fk i, j,k≥0

i>0, j,k≥0

,

(bialg)

(bialg)

which form a basis in QU K ,L ,nor m by Proposition 6. Thus, given any X ∈ U K ,L ,nor m , one can find x ∈ Uq (sl2 ) such that Q X = Q (x). In view of this, one has (id T id) (Q X ) = (id T id) ((1 − P) (x)) = (id T id) ( (x)) − (id T id) (P (x)) = (x) − P (x) = (1 − P) (x) = Q (x) = Q X, due to the above observations. Certainly, a similar computation is applicable to the sec(bialg) (bialg) ond part of (66), which completes its verification on QU K ,L ,nor m , hence on U K ,L ,nor m . Definition 1. We call the antimorphism T with property (66) a von Neumann regular antipode. Definition 2. We call a bialgebra with a von Neumann regular antipode a von Neumann-Hopf algebra. Remark 1. The standard Drinfeld-Jimbo algebra Uq (sl2 ) (which is a domain [14]) admits (bialg) no embedding of U K ,L ,nor m , because the latter contain zero divisors (e.g. (16)). (bialg)

Let us consider a possibility to produce a Hopf algebra structure on U K ,L ,twist . First we observe that the argument of the proof of Proposition 9 does not work in this case. Indeed, an application of (59) to P yields, instead of (60), the following relation: S (P) P + S (Q) Q = 1,

(67)

which does not contradict to noninvertibility of P and Q as in the context of (60). (bialg) Introduce an antimorphism S of U K ,L ,twist by the same formulas as (61)–(62), S (K ) = K , S K = K , S (L) = L, S L = L , (68) (69) S (E) = −E K + L , S (F) = − (K + L) F.


781

(bialg)

We have for U K ,L ,twist ,

(id S) (K ) = (S id) (K ) = (S id) K = (id S) K = 1, (id S) (L) = (S id) (L) = (S id) L = (id S) L = 0,

(70)

(id S) (E) = (S id) (E) = (S id) (F) = (id S) (F) = 0.

(72)

(71)

The proof of the following statement is basically due to [14, p.35]. Proposition 11. The relations (id S) (X ) = (S id) (X ) = ε (X ) · 1 are valid for any (bialg) X ∈ U K ,L ,twist . Proof. Note that X → ε(X )1 is a morphism of algebras. Hence, in view of an obvious induction argument, it suffices to verify that (id S) (X Y ) = (id S) (X ) · (id S) (Y ) and (S id) (X Y ) = (S id) (X ) · (S id) (Y ), with X being one of the generators K , K , L , L, E, F and Y arbitrary. We use the Sweedler notation ∆ (X ) = i X i ⊗ X i [24] to get S Y j S X i X i Y j . (S id) (X Y ) = ij

It follows from (70)–(72) that i S X i is a scalar multiple of 1, hence is central in (bialg) U K ,L ,twist , and we obtain S X i X i S Y j Y j (S id) (X Y ) =

X i

ij

⎞ ⎛ = S X i X i ⎝ S Y j Y j ⎠ = (S id) (X ) · (S id) (Y ) . i

j

Of course, a similar argument goes also for (id S). Thus, we have the following (H op f ) de f (bialg) Theorem 1. 1) U K ,L = U K ,L ,twist , S is a Hopf algebra; (v N −H op f ) de f (bialg) = U K ,L ,nor m , T is a von Neumann-Hopf algebra. 2) U K ,L 6. Structure of R-matrix and the Pierce Decomposition (v N −H op f )

(H op f )

and U K ,L . In Let us consider a version of the universal R-matrix for U K ,L order to avoid considerations related to formal series (the general context of R-matrices), we turn to quasi-cocommutative bialgebras [16]. Such bialgebras generate R-matrices of some simpler shape admitting (under some additional assumptions) an explicit formula to be described below. Definition 3. A bialgebra U (bialg) = (C, B, µ, η, ∆, ε) is called quasi-cocommutative, if there exists an invertible element R ∈ U (bialg) ⊗U (bialg) , called a universal R-matrix, such that ∆cop (b) = R∆ (b) R −1 , where

∆cop

is the opposite comultiplication in

∀b ∈ U (bialg) , U (bialg) .

(73)

782


The R-matrix of a braided bialgebra U (bialg) is subject to (∆ ⊗ id)(R) = R13 R23 , (id ⊗∆)(R) = R13 R12 , (74) where for R = i si ⊗ ti one has R12 = i si ⊗ ti ⊗ 1, etc. [9]. From now on we assume that q n = 1, which is a distinct case in the above context. (alg) Consider the two-sided ideal Isl2 in Uq (sl2 ) generated by {k n − 1, en , f n }, toge(alg) (alg) q ther with the associated quotient algebra U (sl2 ) = Uq (sl2 ) Isl2 . q(alg) (sl2 ) is Theorem 2 ([16, p.230]). The universal R-matrix of U ij = Am (q) · em k i ⊗ f m k j , R

(75)

0≤i, j,m≤n−1

1 (q − q −1 )m m(m−1) +2m(i− j)−2i j q 2 , n [m]! where [m]! = [1] [2] . . . [m], [m] = q m − q −m q − q −1 . ij

Am (q) =

(76)

(H op f )

Now we use (38) to obtain an analog of this theorem for U K ,L . In a similar way (H op f ) = U (H op f ) I (H op f ) , where the two-sided we consider the quotient algebra U (H op f )

ideal I K +L

K +L

K ,L

K +L

is generated by {K n + L n − 1, E n , F n }.

(H op f ) is given by Theorem 3. The universal R-matrix of U K ,L ij (H op f ) = R Am (q) · E m K i + L i ⊗ F m K j + L j . K +L

(77)

0≤i, j,m≤n−1

q(alg) (sl2 ) → U (H op f ) induced by (38) and : U Proof. In view of the morphism K +L (H op f ) = Theorem 2, it suffices (due to invertibility of R) to verify the relation ∆cop (b) R K +L (H op f ) ∆ (b) for b = K , K , because ∆ and ∆cop are morphisms of algebras. This claim R K +L reduces to the verification of (K ⊗ K + L ⊗ L) E m K i + L i ⊗ F m K j + L j (78) = E m K i + L i ⊗ F m K j + L j (K ⊗ K + L ⊗ L) , and

i j i j K ⊗ K + L ⊗ L Em K + L ⊗ Fm K + L i j i j = Em K + L ⊗ Fm K + L K⊗K+L⊗L ,

(79)

(H op f ) is into our picture, because R using (36). The relations (74) are transferred by K +L inside of the tensor square of the image of . Turn to writing down an explicit form for the universal R-matrix in the case of (v N −H op f ) (v N −H op f ) = U (v N −H op f ) U K ,L . Again we consider the quotient algebra U K +L K ,L (v N −H op f )

I K +L

(v N −H op f )

, where the two-sided ideal I K ,L

is generated by {K n + L n −1, E n, F n }.

Quantum Enveloping Algebras and the Pierce Decomposition (v N −H op f )

Theorem 4. The universal R-matrix of U K +L

is given by

ij Am (q) · E m K i + L i ⊗ F m K j + L j .

(v N −H op f ) = R K +L

783

(80)

0≤i, j,m≤n−1

Proof. Is the same as that of Theorem 3. Remark 2. In view of Theorem 2 the R-matrices we have introduced satisfy the YangBaxter equation by our construction. (v N −H op f )

Note that R is not submitted to the direct sum decomposition (39). Now we K +L present another notion of R-matrix which respects (39), but differs from that described in Definition 3 in the sense of being noninvertible. (bialg) = (C, B, µ, η, ∆, ε) is called near-quasiDefinition 4. A bialgebra U ∈ U (bialg) ⊗ U (bialg) , called a universal cocommutative, if there exists an element R near-R-matrix, such that = R∆ (b) , ∆cop (b) R

(bialg) , ∀b ∈ U

(81)

(bialg) and an element R † ∈ U (bialg) ⊗ where ∆cop is the opposite comultiplication in U (bialg) U is such that R † R † , = R, R † R R † = R R

(82)

† can be named the Moore-Penrose inverse for a near-R-matrix [19,22]. and R (bialg) is braided, if its near-R-matrix A near-quasi-cocommutative bialgebra U satisfies (74). (v N −H op f ) = U (v N −H op f ) I (v N −H op f ) , where the Consider the quotient algebra U K ,L K ,L K ,L (v N −H op f )

two-sided ideal I K ,L

is generated by {K n − P, L n − Q, E n , F n }. (v N −H op f )

Theorem 5. The universal R-matrix of U K ,L

is given by the sum

(v N −H op f ) + R (v N −H op f ) , (v N −H op f ) = R R K ,L PP QQ

(83)

where (v N −H op f ) = R PP

ij

(84)

ij

(85)

Am (q) · E m K i ⊗ F m K j ,

0≤i, j,m≤n−1

(v N −H op f ) = R QQ

Am (q) · E m L i ⊗ F m L j .

0≤i, j,m≤n−1

(v N −H op f ) can be presented in the form Remark 3. The universal near-R-matrix R K ,L (v N −H op f ) + (Q ⊗ Q) R (v N −H op f ) . (v N −H op f ) = (P ⊗ P) R R K ,L PP QQ

(86)

784

S. Duplij, S. Sinel’shchikov (v N −H op f )

Proof. Recall that U K ,L

admits the direct sum decomposition (39) with each (v N −H op f )

summand being isomorphic to Uq (sl2 ). After dividing out by the ideal I K ,L get (v N −H op f ) = PU (v N −H op f ) P I (v N −H op f ) ∩ PU (v N −H op f ) P U K ,L K ,L K ,L K ,L (v N −H op f ) (v N −H op f ) (v N −H op f ) + QU K ,L Q I K ,L ∩ QU K ,L Q .

we

(87)

q(alg) (sl2 ), Each of the summands of the right hand side of (87) is clearly isomorphic to U (alg) q and the isomorphisms in question take 1 ∈ U (sl2 ) to P and Q respectively. Now it follows from Theorem 2, that each of the terms of (86) satisfies the conditions of (v N −H op f ) . Also it follows from Definition 3 and (74), hence so does their sum R K ,L (v N −H op f )† ∈ U (v N −H op f ) ⊗ U (v N −H op f ) (v N −H op f )† , R Theorem 2, that there exist R K ,L K ,L PP QQ such that (v N −H op f ) R (v N −H op f )† R (v N −H op f )† = R (v N −H op f ) = P ⊗ P, R PP PP PP PP

(88)

(v N −H op f )† (v N −H op f ) R R QQ QQ

(89)

=

(v N −H op f )† R (v N −H op f ) R QQ QQ

= Q ⊗ Q,

hence the von Neumann regularity (82) is valid for (v N −H op f ) + R (v N −H op f ) , (v N −H op f ) = R R PP QQ (v N −H op f )

because R PP onal.

(v N −H op f )†

,R PP

(v N −H op f )

and R QQ

(v N −H op f )†

,R QQ

(90) are mutually orthog-

7. Conclusion Thus, we have introduced a couple of new bialgebras derived from Uq (sl2 ) which contain idempotents (hence some zero divisors). In some special cases explicit formulas for R-matrices are presented. We define near-R-matrices which satisfy the von Neumann regularity condition. In a similar way one can consider an analog of Uq (sln ) furnished by a suitable and more cumbersome family of idempotents. Also, it would be worthwhile to investigate supersymmetric versions of the presented structures. Hopefully, this approach will be able to facilitate a further research of bialgebras splitting into direct sums, which is a new way of generalizing the standard Drinfeld-Jimbo algebras. Acknowledgements. One of the authors (S.D.) is thankful to J. Cuntz, P. Etingof, L. Kauffman, U. Krähmer, G. Ch. Kurinnoj, B. V. Novikov, J. Okninski, S. A. Ovsienko, D. Radford, C. Ringel, J. Stasheff, E. Taft, T. Timmermann, S. L. Woronowicz for numerous and helpful discussions. Also he is grateful to the Alexander von Humboldt Foundation for valuable support and to M. Zirnbauer for kind hospitality at the Institute of Theoretical Physics, Cologne University, where this paper was finished. Both authors are indebted to L. L. Vaksman1 for stimulating communications related to the structure of quantum universal enveloping algebras. 1 Memorial Page: http://webusers.physics.umn.edu/~duplij/vaksman


785

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Communications in


An Infinite Genus Mapping Class Group and Stable Cohomology Louis Funar1 , Christophe Kapoudjian2 1 Institut Fourier BP 74, UMR 5582, University of Grenoble I,

38402 Saint-Martin-d’Hères cedex, France. E-mail: [email protected] 2 Laboratoire Emile Picard, UMR 5580, University of Toulouse III, 31062 Toulouse cedex 4, France. E-mail: [email protected] Received: 23 June 2005 / Revised: 25 January 2007 / Accepted: 20 September 2008 Published online: 17 February 2009 – © Springer-Verlag 2009

Abstract: We exhibit a finitely generated group M whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface S∞ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of M into the restricted symplectic group Spres (Hr ) of the real Hilbert space generated by the homology classes of non-separating circles on S∞ , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in H 2 (M, Z) is the pull-back of the Pressley-Segal class on the restricted linear group GLres (H) via the inclusion Spres (Hr ) ⊂ GLres (H). 1. Introduction 1.1. Statements of the main results. The tower of all extended mapping class groups was considered first by Moore and Seiberg ([19]) as part of the conformal field theory data. This object is actually a groupoid, which has been proved to be finitely presented (see [1,2,7,15]). When seeking for a group analog Penner ([24]) investigated a universal mapping class group which arises by means of a completion process and which is closely related to the group of homeomorphisms of the circle, but it seems to be infinitely generated. In [8], we introduced the universal mapping class group in genus zero B. The latter is an extension of Thompson’s group V (see [5]) by the infinite spherical pure mapping class group. We proved in [8] that the group B is finitely presented and we exhibited an explicit presentation. Our main difference with the previous attempts is that we consider groups acting on infinite surfaces with a prescribed behaviour at infinity that comes from actions on trees. L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.

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Following the same kind of approach, we propose a treatment of the arbitrary genus case by introducing a mapping class group M, called the asymptotic infinite genus mapping class group, that contains a large part of the mapping class groups of compact surfaces with boundary. More precisely, the group M contains all the pure mapping class groups PM(g,n ) of compact surfaces g,n of genus g with n boundary components, for any g ≥ 0 and n > 0. Its construction is roughly as follows. Let S denote the surface obtained by taking the boundary of the 3-dimensional thickening of the complete trivalent tree, and further let S∞ be the result of attaching a handle to each cylinder in S that corresponds to an edge of the tree (see Fig. 1). Then M is the group of mapping classes of those homeomorphisms of S∞ which preserve a certain rigid structure at infinity (see Definition 1.3 for the precise definition). This rigidity condition essentially implies that M induces a group of transformations on the set of ends of the tree, which is isomorphic to Thompson’s group V . The relation between both groups is enlightened by a short exact sequence 1 → PM → M → V → 1, where PM is the mapping class group of compactly supported homeomorphisms of S∞ . The latter is an infinitely generated group. Our first result is: Theorem 1.1. The group M is finitely generated. The interest in considering the group M, outside the framework of the topological quantum field theory where it can replace the duality groupoid, is the following homological property: Theorem 1.2. The rational homology of M is isomorphic to the stable rational homology of the (pure) mapping class groups. As a corollary of the argument of the proof (see Proposition 3.1), the group M is perfect, and H2 (M, Z) = Z. For a reason that will become clear in what follows, the generator of H 2 (M, Z) ∼ = Z is called the first universal Chern class of M, and is denoted c1 (M). Let Mg be the mapping class group of a closed surface g of genus g. We show that the standard representation ρg : Mg → Sp(2g, Z) in the symplectic group, deduced from the action of Mg on H1 (g , Z), extends to the infinite genus case, by replacing the finite dimensional setting by concepts of Hilbertian analysis. In particular, a key role is played by Shale’s restricted symplectic group Spres (Hr ) on the real Hilbert space Hr generated by the homology classes of non-separating closed curves of S∞ . We have then: Theorem 1.3. The action of M on H1 (S∞ , Z) induces a representation ρ : M → Spres (Hr ). The generator c1 of H 2 (Mg , Z) is called the first Chern class, since it may be obtained as follows (see, e.g., [20]). The group Sp(2g, Z) is contained in the symplectic group Sp(2g, R), whose maximal compact subgroup is the unitary group U (g). Thus, the first Chern class may be viewed in H 2 (BSp(2g, R), Z). It can be first pulled-back on H 2 (BSp(2g, R)δ , Z) = H 2 (Sp(2g, R), Z) and then on H 2 (Mg , Z) via ρg . This is the generator of H 2 (Mg , Z). Here BSp(2g, R)δ denotes the classifying space of the group Sp(2g, R) endowed with the discrete topology. The restricted symplectic group Spres (Hr ) has a well-known 2-cocycle, which measures the projectivity of the Berezin-Segal-Shale-Weil metaplectic representation in the bosonic Fock space (see [22], Chap. 6 and Notes, p. 171). Contrary to the finite dimension case, this cocycle is not directly related to the topology of Spres (Hr ), since the

Infinite Genus Mapping Class Group and Stable Cohomology

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latter is a contractible Banach-Lie group. However, Spres (Hr ) embeds into the restricted 0 (H) (see [25]), where H is the complexification linear group of Pressley-Segal GLres of Hr , which possesses a cohomology class of degree 2: the Pressley-Segal class 0 (H), C∗ ). The group GL0 (H) is homotopically equivalent to the P S ∈ H 2 (GLres res classifying space BU , where U = lim U (n, C), and the class P S does correspond n→∞

to the universal first Chern class. Its restriction on Spres (Hr ) is closely related to the Berezin-Segal-Shale-Weil cocycle, and reveals the topological origin of the latter. Via the composition of morphisms 0 M −→ Spres (Hr ) → GLres (H),

we then derive from P S an integral cohomology class on M (see Theorem 5.1 for a more precise statement): 0 (H), C∗ ) induces the first uniTheorem 1.4. The Pressley-Segal class P S ∈ H 2 (GLres 2 versal Chern class c1 (M) ∈ H (M, Z).

1.2. Definitions. 1.2.1. The infinite genus mapping class group M. Set M(g,n ) for the extended mapping class group of the n-holed orientable surface g,n of genus g, consisting of the isotopy classes of orientation-preserving homeomorphisms of g,n which respect a fixed parametrization of the boundary circles, allowing them to be permuted among themselves. We wish to construct a mapping class group, containing all mapping class groups M(g,n ). It seems impossible to construct such a group, but if one relaxes slightly our requirements then we could follow our previous method used for the genus zero case in [8]. The choice of the extra structure involved in the definitions below is important because the final result might depend on it. For instance, using the same planar punctured surface but different decompositions, one obtained in [9] two non-isomorphic braided PtolemyThompson groups. Definition 1.1 (The infinite genus surface S∞ ). Let T be the complete trivalent planar tree and S be the surface obtained by taking the boundary of the 3-dimensional thickening of T . By grafting an edge-loop (i.e. the graph obtained by attaching a loop to a boundary vertex of an edge) at the midpoint of each edge of T , one obtains the graph T∞ . The surface S∞ is the boundary of the 3-dimensional thickening of T∞ . The graph T (respectively T∞ ) is embedded in S (respectively S∞ ) as a cross-section of the fiber projection, as indicated on Fig. 1. Thus, S∞ is obtained by removing small disks from S centered at midpoints of edges of T and gluing back one holed tori 1,1 , called wrists which correspond to the thickening of edge-loops. It is convenient to assume that T is embedded in a horizontal plane, while the edgeloops are in vertical planes (see Fig. 1). Definition 1.2 (Pants decomposition of S∞ ). A pants decomposition of the surface S∞ is a maximal collection of distinct nontrivial simple closed curves on S∞ which are pairwise disjoint and non-isotopic. The complementary regions (which are 3-holed spheres)

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circles of pants decompositions graph drawn on the surface Fig. 1. The infinite genus surface S∞ with its canonical rigid structure

are called pairs of pants. By construction, S∞ is naturally equipped with a pants decomposition, which will be referred to below as the canonical (pants) decomposition, as shown in Fig. 1: • the wrists are decomposed using a meridian circle and the boundary circle of 1,1 . • there is one pair of pants for each edge, which has one boundary circle for attaching the wrists, and two circles to grip to the other type of pants. We call them edge pants. • there is one pair of pants for each vertex of the tree, called vertex pants. A pants decomposition is asymptotically trivial if outside a compact subsurface of S∞ , it coincides with the canonical pants decomposition. Definition 1.3. 1. A connected subsurface of S∞ is admissible if all its boundary circles are from vertex type pair of pants from the canonical decomposition and moreover, if one boundary circle from a vertex type pants is contained in then the entire pants is contained in . In particular, S∞ − has no compact components. 2. Let ϕ be a homeomorphism of S∞ . One says that ϕ is asymptotically rigid if the following conditions are fulfilled: • There exists an admissible subsurface g,n ⊂ S∞ such that ϕ(g,n ) is also admissible. • The complement S∞ − g,n is a union of n infinite surfaces. Then the restriction ϕ : S∞ − g,n → S∞ − ϕ(g,n ) is rigid, meaning that it maps the pants decomposition into the pants decomposition and maps T∞ ∩ (S∞ − g,n ) onto T∞ ∩ (S∞ − ϕ(g,n )). Such a surface g,n is called a support for ϕ. One denotes by M = M(S∞ ) the group of asymptotically rigid homeomorphisms of S∞ up to isotopy and call it the asymptotic mapping class group of infinite genus. In the same way one defines the asymptotic mapping class group M(S), denoted by B in [8]. Remark 1.1. In genus zero (i.e. for the surface S) a homeomorphism between two complements of admissible subsurfaces which maps the restrictions of the tree T one into


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the other is rigid, thus preserves the isotopy class of the pants decomposition. This is not anymore true in higher genus: the Dehn twist along a longitude preserves the edge-loop graph but it is not rigid, as a homeomorphism of the holed torus. Remark 1.2. Notice that, in general, rigid homeomorphisms ϕ do not have an invariant support i.e. an admissible g,n such that ϕ(g,n ) = g,n . Take for instance a homeomorphism which translates the wrists along a geodesic ray in T . Remark 1.3. Any admissible subsurface g,n ⊂ S∞ has n = g + 3. Moreover S∞ is the ascending union ∪∞ g=1 g,g+3 . Instead of the wrist 1,1 use a surface of higher genus g,1 and the same definitions as above. The admissible subsurfaces will be kg,k+3 . The asymptotic mapping class group obtained this way is finitely generated by small changes in the proof below. Remark 1.4. The surface S∞ contains infinitely many compact surfaces of type (g, n) with at least one boundary component. For any such compact subsurface g,n ⊂ S∞ , there is an obvious injective morphism i ∗ : PM(g,n ) → PM ⊂ M. However, the morphism i ∗ : M(g,n ) → M is not always defined. Indeed, it exists if and only if the n connected components of S∞ \g,n are homeomorphic to each other, by asymptotically rigid homeomorphisms. In particular, for any admissible subsurface g,n (hence n = g + 3), i ∗ extends to an injective morphism i ∗ : M(g,n ) → M defined by rigid extension of homeomorphisms of g,n to S∞ . 1.2.2. The group M and the Thompson groups Definition 1.4. 1. Let T be the planar trivalent tree. A partial tree automorphism of T is an isomorphism of graphs ϕ : T \τ1 → T \τ2 , where τ1 and τ2 are two finite trivalent subtrees of T (each vertex except the leaves are 3-valent). A connected component of T \τ1 or T \τ2 is a branch, that is, a rooted planar binary tree whose vertices are 3-valent, except the root, which is 2-valent. Each vertex of a branch has two descendant edges, and given an orientation to the plane, one may distinguish between the left and the right descendant edges. A partial automorphism ϕ : T \τ1 → T \τ2 is planar if it maps each branch of T \τ1 onto the corresponding branch of T \τ2 by respecting the left and right ordering of the edges. 2. Two planar partial automorphisms ϕ : T \τ1 → T \τ2 and ϕ : T \τ1 → T \τ2 are equivalent, which is denoted ϕ ∼ ϕ , if and only if there exists a third ϕ

: T \τ1

→ T \τ2

such that τ1 ∪ τ1 ⊂ τ1

, τ2 ∪ τ2 ⊂ τ2

and ϕ|T \τ1

= ϕ| T \τ

= ϕ

. One denotes 1 by [ϕ] the equivalence class of ϕ. 3. If ϕ and ϕ are planar partial automorphisms, one can find ϕ0 ∼ ϕ and ϕ0 ∼ ϕ such that the source of ϕ0 and the target of ϕ0 coincide. The product [ϕ] · [ϕ ] = [ϕ0 ◦ ϕ0 ] is well defined, as is easy to check. The set of equivalence classes of such automorphisms endowed with the above internal law, is a group with neutral element the class of idT . This is the Thompson group V . Remark 1.5. We warn the reader that our definition of the group V is different from the standard one (as given in [5]). Nevertheless, the present group V is isomorphic to the group denoted by the same letter in [5]. We introduce Thompson’s group T , the subgroup of V acting on the circle (see [11]), which will play a key role in the proofs.

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Definition 1.5 (Ptolemy-Thompson’s group T ). Choose a vertex v0 of T . Each g ∈ V may be represented by a planar partial automorphism ϕ : T \τ1 → T \τ2 such that v0 belongs to τ1 ∩ τ2 . Let D1 (respectively D2 ) be a disk containing τ1 (respectively τ2 ), whose boundary circle S1 (respectively S2 ) passes through the leaves of τ1 (respectively τ2 ), giving to them a cycling ordering. If ϕ preserves this cycling ordering, which amounts to saying that the bijection from the set of leaves of τ1 onto the set of leaves of τ2 can be extended to an orientation preserving homeomorphism from S1 onto S2 , then any other ϕ equivalent to ϕ also does, and one says that g itself is circular. The subset of circular elements of V is a subgroup, called the Ptolemy-Thompson group T . Proposition 1.1. Set PM for the inductive limit of the pure mapping class groups of admissible subsurfaces of S∞ . We have then the following exact sequences: 1 → PM → M → V → 1. Proof. Let ϕ be an asymptotically rigid homeomorphism of S∞ and g,n a support for ϕ. Then it maps T∞ ∩(S∞ −g,n ) onto T∞ ∩(S∞ −ϕ(g,n )), hence T ∩(S∞ −g,n ) onto T ∩ (S∞ − ϕ(g,n )) by forgetting the action on the edge-loops. This may be identified with a planar partial automorphism φ : T \τ1 → T \τ2 . The map [ϕ] ∈ M → [φ] ∈ V is a group epimorphism. The kernel is the subgroup of isotopy classes of homeomorphisms inducing the identity outside a support, and hence is the direct limit of the pure mapping class groups.

Remark 1.6. In [8] we prove the existence of a similar short exact sequence relating B to V , which splits over the Ptolemy-Thompson group T . It is worth noticing that the present extension of V is not split over T . 2. The Proof of Theorem 1.1 2.1. Specific elements of M. Recall that S∞ has a canonical pants decomposition, as shown in Fig. 1. We fix an admissible subsurface A = 1,4 which contains a central wrist and an admissible B = 0,3 ⊂ 1,4 which is not adjacent to the wrist. Let us consider now the elements of M described in the pictures below. Specifically: • Let γ be a circle contained inside B and parallel to the boundary curve labeled 3. Let t be the right Dehn twist around γ . This means that, given an outward orientation to the surface, t maps an arc crossing γ transversely to an arc which turns right as it approaches γ . The dashed arcs (also called seams) on the left-hand side picture figure out the boundary of the visible side of B, i.e. the part of B which is located on the side of the surface that contains the graph and which is represented on Fig. 1. Their images by t are represented on the right-hand side picture, 2

2

1

1

t

3

3

• π is the braiding, acting as a braid in M(0,3 ), with the support B. It rotates the circles 1 and 2 in the horizontal plane (spanned by the circles) counterclockwise.


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Assume that B is identified with the complex domain {|z| ≤ 7, |z − 3| ≥ 1, |z + 3| ≥ 1} ⊂ C. A specific homeomorphism in the mapping class of π is the composition of the counterclockwise rotation of 180 degrees around the origin—which exchanges the small boundary circles labeled 1 and 2 in the figure—with a map which rotates 180 degrees in the clockwise direction each boundary circle. The latter can be constructed as follows. Let A be an annulus in the plane, which we suppose for simplicity to be A = {1 ≤ |z| ≤ 2}. The homeomorphism D A,C acts as the counterclockwise rotation of 180 degrees on the boundary circle C and keeps the other boundary component pointwise fixed: √ z exp(π √−1(2 − |z|)), if C = {|z| = 1} . D A,C (z) = z exp(π −1(|z| − 1)), otherwise −1 −1 The map we wanted is D −1 A0 ,C0 D A1 ,C1 D A2 ,C2 , where A0 = {6 ≤ |z| ≤ 7}, C 0 = {|z| = 7}, A1 = {1 ≤ |z − 3| ≤ 2}, C1 = {|z − 3| = 1}, A2 = {1 ≤ |z + 3| ≤ 2}, and C2 = {|z + 3| = 1}. One has pictured also the images of the seams. 2

1

1

2

π

3

3

• β is the order 3 rotation in the vertical plane of the paper. It is the unique globally rigid mapping class which permutes counterclockwise and cyclically the three boundary circles of B. An invariant support for β is B. 2

1

1

3

β

3

2

• α is a twisted rotation of order 4 in the vertical plane which moves cyclically the labels of the boundary circles counterclockwise. Its support is a 4-holed torus A = 1,4 . 3

2

Q B′

Q

B′ B

B

3

4 rotate back the wrist

Q B′

4 rotate the support

B

4

1

3

1

2

1

2

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Let 0,5 be the 5-holed sphere consisting of the union of B with the edge pants Q near B and the next vertex pants B adjacent to Q. There are four boundary circles which are vertex type and one boundary circle which bounds a wrist. We perform first a rotation in R3 which preserves globally the pants decomposition and visible side, permutes counterclockwise and cyclically the four vertex type boundary circles of 0,5 and rotates the edge type circle according to one fourth twist. This rotation changes the position of the wrist 1,1 in R3 . We consider next the clockwise rotation of this wrist alone, of angle π2 around the vertical axis that meets the edge type circle in its center. This rotation restores the initial wrist position. The composition of the two partial rotations above is a homeomorphism of 1,4 that gives a well-defined element of M. • Let a1 , b1 be the meridian and longitude on the basic wrist in A. We denote by ta1 and tb1 the Dehn twists along these curves, and further, t0 states for the Dehn twist along the boundary circle of the wrist. Remark 2.1. It is worth noting that we have three types of Dehn twists: those along separating curves (conjugate either to t (the boundary on the vertex pants) or with t0 (the edge type pants)) and those along non-separating curves which are conjugate to the twist around such a curve on the wrist.

2.2. Generators for PM. Consider the following collection of simple curves drawn on S∞ :

Their description follows: 1. Choose, for each wrist, a longitude bi , which turns once along the wrist. 2. For each pair of wrists we choose a circle joining them as follows. For each wrist we have an arc going from the base point of its attaching circle to the longitude and back to the opposite point of the circle. Then join these two pairs of points by a pair of parallel arcs in the horizontal surface, asking that the arc which joins the two base points be a geodesic path in the tree T . We call them wrist-connecting loops. 3. Further we associate a loop to each pair consisting of a wrist and a vertex of the tree T . A vertex gives rise to a pair of pants in S∞ . Two of the boundary components of these pants correspond to the directions to move away from the wrist. Thus we can define again an arc on the pair of pants which joins a point p of the third circle (closest to the wrist), on the visible side of S∞ , to its opposite, on the hidden side, and separates the remaining two circles. Consider the loop resulting from gathering the following three kinds of arcs:


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(a) the arc on the wrist; (b) the arc on the pair of pants; (c) and a pair of parallel arcs which join them, asking that the arc which joins the point p to the base point of the wrist be a geodesic path of the tree T . We call them vertex connecting loops. 4. Consider the loops that come from the canonical pants decomposition of S by doubling them. We call them the horizontal pants decomposition loops. Lemma 2.1. The set H of Dehn twists along the meridians, the longitudes, the wrist connecting loops associated to edges, the vertex connecting loops and the horizontal pants decomposition loops generates PM. Proof. It suffices to consider the finite case of an admissible surface with boundary that contains g wrists and has g+3 boundary components. Then the lemma follows from [10], in which it is proved that the pure mapping class group of such a surface is generated by a set of Dehn twists Hg,n (with n = g + 3). It suffices to check that all the Dehn twists belonging to Hg,n also belong to the set H. Referring to the notations of [10], there are four types of Dehn twists in Hg,n : the αi ’s, the βi ’s, the γi j ’s and the δi s. The αi ’s are associated to wrist-connecting or vertex connecting loops, the βi ’s are associated to longitudes bi ’s, the γi j ’s are associated to wrist-connecting loops, except γ12 which is associated to a vertex-connecting loop, and finally, the δi ’s are associated to the circles of the boundary of the surface, hence of the pants decomposition (after doubling them) of S∞ . Therefore all of them belong to H. Remark also that in ([10], Fig. 1) the 1-handles are cyclically ordered and arranged on one side and then followed by all boundary components of the surface. However, we can arbitrarily permute the position of holes and 1-handles in the picture and keep the same system of generators.

2.3. The action of T on the generators of PM. 2.3.1. The groups T and T ∗ . Consider the subgroup T of M generated by the elements α and β. We will prove that the set of conjugacy classes for the action of T on H is finite by considering the action of T on some planar subsurface of S∞ . The surface obtained by puncturing (respectively deleting disjoint small open disks from) S at the midpoints of the edges is denoted by S ∗ (and respectively S • ). The 2-dimensional thickening in S of the embedded tree T is an infinite planar surface, which will be called the visible side of S, and will be denoted D. The intersection of D with S ∗ and S • is denoted D ∗ and D • , respectively. The elements α and β as defined above (i.e. as specific homeomorphisms, not only as mapping classes) keep invariant both S • and D • . If we crush the boundary circles to points then we obtain elements of M(D ∗ ), and there is a well defined homomorphism T → M(D ∗ ). We studied in [9] the asymptotic mapping class group M(D ∗ ) denoted by T ∗ there. Recall from [9] that: Proposition 2.1. The group T ∗ is generated by α and β. This implies that T → T ∗ is an epimorphism. The relation between the asymptotic mapping class groups T and T ∗ is made precise by the following:

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Lemma 2.2. We have an exact sequence 0 → Z∞ → T → T ∗ → 1, where the central factor Z∞ is the group of Dehn twists along attaching circles, normally generated by t0 . Proof. If an asymptotically rigid homeomorphism of S∞ preserving D • is isotopically trivial once the circles are crushed to points, then it is isotopic to a finite product of Dehn twists along those circles. Therefore, the kernel of T → T ∗ is contained in the subgroup denoted Z∞ . Observe that α 4 = t0 in T, so that t0 belongs to the kernel of T → T ∗ . Consequently, the kernel contains all the T-conjugates of t0 , hence Z∞ .

Thus if we understand the action of T ∗ on the isotopy classes of arcs embedded in then we can easily recover the action of T on homotopy classes of loops of S • , up to some twists along attaching circles. D∗

2.3.2. The action of T ∗ on the isotopy classes of arcs of D ∗ . The planar model of D ∗ is the punctured thick tree obtained from the binary tree by thickening in the plane and puncturing along midpoints of edges. The traces on D ∗ of the loops coming from the pants decomposition of S are arcs transversal to the edges. Thus D ∗ has a canonical decomposition into punctured hexagons. Each hexagon has three punctured sides coming from the arcs above, that we call separating side arcs. Moreover there are also three sides which are part of the boundary of D ∗ that we will call bounding side arcs. Notice that hexagons correspond to vertices of the binary tree, while separating side arcs. Further β is the rotation of order 3 supported on the hexagon B (image of the pants B) and α is the rotation of order 4 that is supported on the union of B with an adjacent hexagon.

Lemma 2.3. Let γ be an arc embedded in D ∗ that joins two punctures. Then there exists some element of T ∗ that sends γ in a prescribed arc joining the punctures 0 and 1. Proof. Recall from [9] that the infinite braid group associated to the punctures B∞ is contained in T ∗ . Further, there exists always a braid mapping class (supported in a compact subsurface of D ∗ ) sending the arc γ in the prescribed one.

Lemma 2.4. The group T ∗ acts transitively on the set of separating side arcs.


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Proof. The group T ∗ contains P S L 2 (Z), the group of orientation-preserving automorphisms of the tree T , generated by α 2 and β. It acts transitively on the set of edges of T , hence on the set of separating sides of the hexagons of D ∗ .

An arc joining a puncture belonging to a hexagon H to a bounding side of H is called standard if it is entirely contained in H . Lemma 2.5. For any arc joining a puncture to a bounding side arc of a hexagon, there exists some element of T ∗ sending it into a standard arc joining the puncture 0 to one of the bounding sides of its hexagon. Proof. As above, P S L(2, Z) ⊂ T ∗ also acts transitively on the set of all bounding sides. Thus we can use an element of T ∗ to send one end of our arc on a bounding side of the hexagon B. Next, one composes by a braid element in B∞ that moves the other endpoint of the arc onto the puncture 0 and then makes the arc isotopic to a standard arc.

Let ta1 and tb1 denote the Dehn twists along a meridian a1 and a longitude b1 on the wrist. Corollary 2.1. The elements α, β, t, ta1 , tb1 , π , a Dehn twist along one wrist connecting loop and a Dehn twist along a vertex connecting loop generate M. 3. The Rational Homology of M Theorem 3.1. The rational homology of M is isomorphic to the stable rational homology of the mapping class group: H∗ (M, Q) ∼ = H∗ (PM, Q). Proof. Recall first the theorem of stability, due to J. Harer (see [14]): Let R be a connected subsurface of genus g R of a connected compact surface S with at least one boundary component. Then the map Hn (PM R , Z) → Hn (PM S , Z) induced by the natural morphism PM R → PM S is an isomorphism if g R ≥ 2n + 1. The pure mapping class group PM is the inductive limit of the pure mapping class groups PM R , for all the compact subsurfaces R ⊂ S∞ . It follows that Hn (PM, Q) = lim Hn (PM R , Q) = Hn (PM R , Q) for any compact subsurface R ⊂ S∞ of genus →

R

g R ≥ 2n + 1. Therefore, the homology of PM is what is called the stable homology of the mapping class group. By Mumford’s conjecture proved in [18], H ∗ (PM, Q) is isomorphic to Q[κ1 , . . . , κi , . . .], where κi , the i th Miller-Morita-Mumford class, has degree 2i. Since H ∗ (PM, Q) = Hom(H∗ (PM, Q), Q), each Hn (PM, Q) is finite dimensional over Q. Write now the Lyndon-Hochschild-Serre spectral sequence in homology associated with 1 → PM → M → V → 1. The second term is E 2p,q = H p (V, Hq (PM, Q)). If we prove that V acts trivially on the finite dimensional Q-vector space Hq (PM, Q), and invoke a theorem of K. Brown ([4]) saying that V is rationally acyclic, then the only possibly non-trivial term of the 2 = H (PM, Q), and the proof is done. spectral sequence is E 0,n n Thus it remains to justify that V acts trivially on the homology groups Hq (PM, Q), for any integer q ≥ 0. This results from the fact that V is not linear, as we explain below.

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Indeed, if dimQ Hq (PM, Q) = N , then Aut(Hq (PM, Q)) ∼ = G L(N , Q). So, let ρ : V → G L(N , Q) be the representation resulting from the action of V on Hq (PM, Q). Since V is a simple group, ρ is either trivial or injective. Suppose it is injective, so that V is isomorphic to a finitely generated subgroup of S L(N , Q). Now each finitely generated subgroup of S L(N , K ) for any field K is residually finite. But V is not residually finite, since its unique normal subgroup of finite index is V itself. Therefore, ρ is trivial.

Proposition 3.1. The free universal mapping class group M is perfect, and H2 (M, Z) = Z. The generator of H 2 (M, Z) ∼ = Z is called the first universal Chern class of M, and is denoted c1 (M). Proof. Recall that the pure mapping class group of a surface of type (g, n) is perfect if g ≥ 3. Consequently, PM is perfect. Since V is perfect, M is perfect as well. The above spectral sequence may be written with integral coefficients. One obtains 2 = H (V, Z) = 0 (see [4]), E 2 = H (V, H (PM, Z)) = 0 since PM is perE 2,0 2 1 1 1,1 2 = H (V, H (PM, Z)). By Harer’s theorem ([13]) and stability ([14]), fect, and E 0,2 0 2 H2 (PM, Z) ∼ = Z. The action of V on Z = H2 (PM, Z) must be trivial, since V is sim2 = Z. Thus, the only non-trivial E ∞ term is E ∞ = E 2 = Z, ple, and it follows that E 0,2 0,2 0,2 and this implies H2 (M, Z) = Z.

4. The Symplectic Representation in Infinite Genus 4.1. Hilbert spaces and symplectic structure associated to S∞ . There is a natural intersection form ω : H1 (S∞ , R) × H1 (S∞ , R) → R on the homology of the infinite surface, but this is degenerate because it is obtained as a limit of intersection forms on surfaces with boundary. The M-module H1 (S∞ , R) is the direct sum of two submodules: H1 (S∞ , R) = H1 (S∞ , R)s ⊕ H1 (S∞ , R)ns , where H1 (S∞ , R)s is generated by the homology classes of separating circles of S∞ , while H1 (S∞ , R)ns is generated by the homology classes of non-separating circles of S∞ . The kernel ker ω of ω is H1 (S∞ , R)s , and the restriction of ω to H1 (S∞ , R)ns is a symplectic form. For each wrist torus occurring in the construction of S∞ (see Definition 1.1), we consider the meridian ak and the longitude bk , with intersection number ω(ak , bk ) = 1. Note that both collections {ak , k ∈ N} and {bk , k ∈ N} are invariant by the mapping class group T, since the generators α and β rigidly map a wrist onto a wrist. Moreover, these collections are almost invariant by M, meaning that for each g ∈ M, g({ak , k ∈ N}) (respectively g({bk , k ∈ N})) coincides up to isotopy with {ak , k ∈ N} (respectively {bk , k ∈ N}) for all but finitely many elements. a

2

a1

b4

b2 a

a4

b1

3 b3

a

b

5

5


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The classes {ak , bk , k ∈ N} form a symplectic basis for H1 (S∞ , R)ns . Each element of M acts on H1 (S∞ , R)ns by preserving the intersection form ω. In particular, there is a representation ρ : PM → Sp(2∞, R), where Sp(2∞, R) is the inductive limit of the symplectic groups Sp(2k, R), with respect to the natural inclusions Sp(2k, R) ⊂ Sp(2(k + 1), R). Note, though, that if g ∈ M is not in PM, it is not represented into Sp(2∞, R), but into a larger symplectic group, that we are defining below. One completes H1 (S∞ , R)ns as a real Hilbert space for which this basis orthonormal. Let Hr be this Hilbert space, and (. , .) denote its scalar product. Let J be the almost-complex structure induced by ω, i.e. the linear operator defined by ω(v, w) = (v, J w) for all v, w in Hr . We have J 2 = −1. Each linear operator on Hr decomposes into a J -linear part T1 and a J -antilinear part T2 , T = T1 + T2 , where T1 = T −2J T J and T2 = T + 2J T J . Recall that ([27]) the restricted symplectic group Spres (Hr ) is defined as the group of symplectic (i.e. ω preserving) bounded invertible operators T whose J -antilinear part T2 is a Hilbert-Schmidt operator. An operator T is called Hilbert-Schmidt if ||T ||2H S := 2 i ||T (ei )|| is finite, where (ei )i∈N is an orthonormal Hilbert basis. Theorem 4.1. The symplectic representation of the mapping class group PM extends to a representation ρˆ : M → Spres (Hr ) of M into the restricted symplectic group. 4.2. Proof of Theorem 4.1. Instead of a direct proof we will introduce the complexification of Hr to be used also in the next section. Let H = Hr ⊗R C. Extend ω and J by C-linearity, and (. , .) by sesquilinearity, and denote by ωC , JC and (. , .)C the extensions. Thus, (H, (. .)C ) is a complex Hilbert space. Let B be the indefinite hermitian form B(v, w) = √1−1 ωC (v, w), ¯ for all v, w in H, where w¯ is the complex-conjugate of w. Let Aut(H, ωC , B) be the group of bounded invertible operators of H which preserve ωC and B. The morphism φ : Sp(Hr ) −→ Aut(H, ωC , B), given by φ(T ) = T ⊗ 1C is an isomorphism (see [22]), since any T ∈ Aut(H, ωC , B) commutes with the complex conjugation and hence stabilizes Hr . √ Since JC2 = −id, H = H+ ⊕ H− , where H± = ker(J ± −1 · 1). Moreover, the direct sum is orthogonal. The complex conjugation interchanges H+ and H− . Let (ek )k∈N be an orthonormal basis of H+ and ( f k = ek )k∈N the conjugate basis of H− . According to ([22], 6.2) a symplectic operator T belongs to Spres (Hr ) if and only if the decomposition of φ(T) relative to the direct sum H+ ⊕ H− in the basis (ek )k∈N ∪ ( f k = , where ek )k∈N reads 1.

t − t

= 1 and t = t , where t T denotes the adjoint of T with respect

to (. , .)C . 2. : H− → H+ is a Hilbert-Schmidt operator.

We will apply this criterion for the action of M. Set √ √ 1 1 ek = √ (ak − −1 · bk ), f k = √ (ak + −1 · bk ), 2 2

800


Then (ek )k∈N is an orthonormal basis of H+ and ( f k )k∈N is the conjugate√orthonormal basis of H− . Moreover, ωC (ek , el ) = ωC ( f k , fl ) = 0, ωC (ek , fl ) = −1δkl , and B(ek , el ) = −B( f k , fl ) = δkl , B(ek , fl ) = 0 for all k, l. Consider now the action ρ(g) ˆ of g ∈ M on the M-invariant subspace Hr . We must check that (φ(ρ(g))) ˆ is a Hilbert-Schmidt operator. In fact, it is a finite rank operator. Let h,n be an admissible surface for g, that is g is the mapping class of a homeomorphism G so that G : S∞ \h,n → S∞ \ϕ(h,n ) is rigid. Any wrist torus Tk = 1,1 of S∞ \g,n is rigidly mapped by G onto another corresponding wrist torus Tσ (k) , for some infinite permutation σ . Therefore, for any such Tk , the associated matrices are such that φ(ρ(g))(e ˆ ˆ f k ) = f σ (k) . k ) = eσ (k) , φ(ρ(g))( In particular, for all but finitely many f k (i.e. excepting those corresponding to tori Tk ⊂ h,n ) we have φ(ρ(g))( ˆ f k ) ∈ H− . Now (φ(ρ(g))) ˆ : H− → H+ corresponds to the components of φ(ρ(g))( ˆ f k ) in H+ . This means that (φ(ρ(g))) ˆ has finite rank, and in particular, it is Hilbert-Schmidt. This proves that ρ(g) ˆ ∈ Spres (Hr ), as claimed. 5. The Universal First Chern Class 5.1. The Pressley-Segal extension. Let H be a polarized separable Hilbert space as above, that is, the orthogonal sum of two separable Hilbert spaces H = H+ ⊕ H− . The restricted linear group GLres (H) (see [25,27]) is the Banach-Lie group of operators in ab GL(H) whose block decomposition A = is such that b and c are Hilbert-Schmidt cd operators. Moreover, the invertibility of A implies that a is Fredholm in H+ , and has an index ind(a) ∈ Z. This gives a homomorphism ind : GLres (H) → Z, that induces an 0 (H) the connected component of the isomorphism π0 (GLres (H)) ∼ = Z. Denote by GLres 0 identity. Then GLres (H) is a perfect group (cf. [6], §5.4). Proposition 5.1. The restricted symplectic group Spres (Hr ) embeds into the restricted 0 (H). It is given the induced topology. linear group GLres Proof. Let be in Spres (Hr ). Since is a Hilbert-Schmidt operator, K = t is trace-class, hence compact. Then t = 1 + K ≥ 1, hence t ≥ 1 is injective, and the Fredholm alternative implies it is invertible. In particular, itself is invertible, and has null index.

Pressley-Segal’s extension of the restricted linear group. Let L1 (H+ ) denote the √ ideal of trace-class operators of H+ . It is a Banach algebra for the norm ||b||1 = T r ( b∗ b), where T r denotes the trace form. We say that an invertible operator q of H+ has a determinant if q − idH+ = Q is trace-class. Its determinant is the complex number +∞ det(q) = i=0 T r (∧i Q), where ∧i Q is the operator of the Hilbert space ∧i H+ induced by Q (cf. [28]). Denote by T the subgroup of G L(H+ ) consisting of operators which have a determinant, and by T1 the kernel of the morphism det : T → C∗ . Let E be the subgroup of GLres (H) × GL(H+ ) consisting of pairs (A, q) such that a − q is trace-class. Then ind(a) = ind(q + (a − q)) = ind(q) = 0, so that A belongs 0 (H). There is a short exact sequence to GLres i

p

0 1 → T −→ E −→ GLres (H) → 1


801

called the Pressley-Segal extension. Here p(A, q) = A, and i(q) = (1H , q). It induces the central extension 1→

T ∼ ∗ E 0 −→ GLres (H) → 1. = C −→ T1 T1

0 (H), C∗ ) is denoted by P S, and called The corresponding cohomology class in H 2 (GLres the Pressley-Segal class of the restricted linear group.

The Pressley-Segal class and the universal first Chern class. For G a Banach-Lie group, set E xt (G, C∗ ) for the set of equivalence classes of central extensions of G by C∗ , which are locally trivial fibrations (see [26]). Note that E xt (G, C∗ ) must not be confused 2 (G, C∗ ), since the latter only classifies with the group of continuous cohomology Hcont δ

the topologically split central extensions. One introduces two maps H 2 (G, C∗ ) ←− τ 2 (G, Z). The map δ associates to an extension E in E xt (G, C∗ ) E xt (G, C∗ ) −→ Htop its cohomology class in the Eilenberg-McLane cohomology of G. The map τ is the composition of E xt (G, C∗ ) −→ [G, BC∗ = K (Z, 2)], which sends an extension to the homotopy class of its classifying map G → BC∗ , with the isomorphism [G, K (Z, 2)] ∼ = 2 (G, Z). Htop 0 (H) and the central Pressley-Segal extenLet us apply this formalism to G = GLres 0 (H, C∗ )). Then δ(P S) = P S. The point sion, viewed as an element P S ∈ E xt (GLres 0 is that GLres (H) is a homotopic model of the classifying space BU . In fact E is contractible (see [25], 6.6.2) and T is homotopically equivalent to U (see [23]), hence the claim. It follows that the fibration P S corresponds to the universal first Chern class, that is, τ (P S) = c1 (BU ) ∈ H 2 (BU, Z). 0 (H), Sp (H ) and M. 5.2. Cocycles on GLres res r

Lemma 5.1. The class ι∗ (P S) in H 2 (Spres (Hr ), C∗ ) is represented by the cocycle C1 (g, g ) = det((g)(g )(gg )−1 ). 0 (H) consisting of operators A such that a is Proof. Let V be the open subset of GLres invertible. It is known that the central Pressley-Segal extensions splits over V, since it has the section σ : V → E, A → (A, a). In particular, there is a local cocycle for P S given by the formula ([25], 6.6.4): C(A, A ) = det(1 + aa a

−1 ), for A, A ∈ V, where a

is the first block of A · A . It suffices now to observe that Spres (Hr ) embeds into V.

In order to prove Proposition 5.2 below, we need the following result that contrasts sharply with the finite dimensional case: Lemma 5.2. The restricted symplectic group Spres (Hr ) is contractible. Proof. Denote by Z the set of symmetric Hilbert-Schmidt operators H− → H+ with norm < 1. Clearly, Z is a contractible subspace of the Banach space of Hilbert-Schmidt operators. The group Spres (Hr ) acts transitively and continuously (see [22], p. 177) on Z by means of g(S) = ((g)S + (g))((g)S + (g))−1 ∈ Z , for g ∈ Spres (Hr ), S ∈ Z .

802


0 such that is unitary in H+ . 0 Thus, it is isomorphic to U(H+ ). By a result of Kuiper ([17]), U(H+ ) is contractible. The claim is now a consequence of the contractibility of Spres (Hr )/U(H+ ) ∼

= Z.

The stabilizer of S = 0 is the group of matrices

Proposition 5.2. For each integer n ∈ Z, there is a well-defined continuous cocycle Cn defined on Spres (Hr ), with values in C∗ , such that 1 Cn (g, g ) = det ((g)(g )(gg )−1 ) n . Moreover, that

Cn |Cn |

may be lifted to a real cocycle ςn : Spres (Hr ) × Spres (Hr ) −→ R such Cn (g, g )

= e2iπ ςn (g,g ) , for all g, g ∈ Spres (Hr ). |Cn (g, g )|

The restriction ς1 of ς1 to Sp(2∞, R) defines an integral cohomology class [ς1 ] ∈ H 2 (Sp(2∞, R), Z). Proof. In fact, (g)−1 (g )−1 (gg ) = 1 + ((g )−1 (g)−1 (g)((g )). But, according to ([22], p. 168) we have ||−1 (g)(g)|| < 1 and ||(g )(g )

−1

|| < 1. 1

Thus, there is a non-ambiguous definition of (−1 (g)(gg )−1 (g )) n given by an absolutely convergent series. The existence of ςn is now an immediate consequence of the preceding lemma. det ((g)) The map : g ∈ Sp(2∞, R) → (g) = |det ((g))| is well-defined, so that the cocycle

(g, g ) ∈ Sp(2∞, R) × Sp(2∞, R) → e2iπ ς1 (g,g ) is the coboundary of . This proves that the cohomology class of ς1 restricted to Sp(2∞, R) is integral.

Remark 5.1. 1. The restrictions of the real cocycles ςn on the finite dimensional Lie group Sp(2g, R) are those constructed by Dupont-Guichardet-Wigner (see [12]). In fact, the authors of [12] proved that the cohomology class of the restriction of ς1 to Sp(2g, R) is integral, and is the image in H 2 (Sp(2g, R), R) of the generator of 2 (Sp(2g, R), Z) = Z, the second group of borelian cohomology of Sp(2g, R). Hbor They prove also that it is the image of the first Chern class c1 (BU (g, C)) by the composition of maps H 2 (BU (g, C), Z) ≈ H 2 (BSp(2g, R), Z) → H 2 (BSp(2g, R)δ , Z) ≈ H 2 (Sp(2g, R), Z) → H 2 (Sp(2g, R), R), where BSp(2g, R)δ is the classifying space of Sp(2g, R) as a discrete group. 2. The remark above implies that the map H ∗ (BU, Z) ≈ H ∗ (BSp(2∞, R), Z) → H ∗ (Sp(2∞, R), Z) sends the first universal Chern class c1 (BU ) onto [ς1 ] ∈ H 2 (Sp(2∞, R), Z). Further, the symplectic representation ρ : PM → Sp(2∞, R) maps [ς1 ] onto the generator c1 (PM) of H 2 (PM, Z).


803

3. According to ([22], Theorem 6.2.3), the Berezin-Segal-Shale-Weil cocycle is the complex conjugate of the cocycle C− 1 . 2

Theorem 5.1. Let [ςˆ1 ] ∈ H 2 (Spres (Hr ), R) be the cohomology class of ςˆ1 . The pullback of [ςˆ1 ] in H 2 (M, R) by the representation ρˆ of Theorem 1.3 is integral, and is the natural image of the generator c1 (M) of H 2 (M, Z) in H 2 (M, R). Proof. Let ι : PM → M and j : Sp(2∞, R) → Spres (Hr ) be the natural embeddings. Plainly, j ◦ ρ = ρˆ ◦ ι. Since j ∗ : H 2 (Spres (Hr ), R) → H 2 (Sp(2∞, R), R) maps [ςˆ1 ] onto [ς1 ], one has ι∗ (ρˆ ∗ [ςˆ1 ]) = ρ ∗ [ς1 ]. Let us denote by c¯1 (PM) (respectively c¯1 (M)) the image of c1 (PM) (respectively c1 (M)) in H 2 (PM, R) (respectively H 2 (M, R)). According to Remark 5.1, 2., ρ ∗ [ς1 ] = c¯1 (PM). By Proposition 3.1, ι∗ (c¯1 (PM)) = c¯1 (M), hence ρˆ ∗ [ςˆ1 ] = c¯1 (M).

Acknowledgements. The authors are indebted to Vlad Sergiescu for enlightening discussions and particularly for suggesting the existence of a connection between the first universal Chern class of M and the Pressley-Segal class. They are thankful to the referees for suggestions improving the exposition.

References 1. Bakalov, B., Kirillov, Jr. A.: Lectures on tensor categories and modular functors. A.M.S. University Lecture Series. 21, Providence, RI: Amer. Math. Soc., 2001 2. Bakalov, B., Kirillov, Jr. A.: On the Lego-Teichmüller game. Transform. Groups 5, 207–244 (2000) 3. Borel, A.: Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4) 7, 235–272 (1974) 4. Brown, K.S.: The geometry of finitely presented infinite simple groups. In: Algorithms and Classification in Combinatorial Group Theory, Baumslag, G., Miller, C.F. III, eds., MSRI Publications, Vol. 23. Berlin, Heidelberg, New-York: Springer-Verlag, 1992, pp. 121–136 5. Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompson’s groups. Enseign. Math. 42, 215–256 (1996) 6. Connes, A., Karoubi, M.: Caractère multiplicatif d’un module de Fredholm. K -Theory 2, 431–463 (1988) 7. Funar, L., Gelca, R.: On the groupoid of transformations of rigid structures on surfaces. J. Math. Sci. Univ. Tokyo 6, 599–646 (1999) 8. Funar, L., Kapoudjian, C.: On a universal mapping class group of genus zero. G.A.F.A. 14, 965–1012 (2004) 9. Funar, L., Kapoudjian, C.: The braided Ptolemy-Thompson group is finitely presented. Geom. Topol. 12, 475–530 (2008) 10. Gervais, S.: A finite presentation of the mapping class group of a punctured surface. Topology 40, 703–725 (2001) 11. Ghys, E., Sergiescu, V.: Sur un groupe remarquable de difféomorphismes du cercle. Comment. Math. Helve. 62, 185–239 (1987) 12. Guichardet, A., Wigner, D.: Sur la cohomologie réelle des groupes de Lie simples réels. Ann. Sci. École Norm. Sup. (4) 11, 277–292 (1978) 13. Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. Math. 72, 221–239 (1983) 14. Harer, J.: Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. 121, 215–249 (1983) 15. Hatcher, A., Lochak, P., Schneps, L.: On the Teichmüller tower of mapping class groups. J. Reine Angew. Math. 521, 1–24 (2000) 16. Hatcher, A., Thurston, W.: A presentation for the mapping class group of a closed orientable surface. Topology 19, 221–237 (1980) 17. Kuiper, N.: The homotopy type of the unitary group of Hilbert space. Topology 3, 19–30 (1965) 18. Madsen, I., Weiss, M.: The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math. (2) 165, 843–941 (2007) 19. Moore, G., Seiberg, N.: Classical and quantum field theory. Commun. Math. Phys. 123, 177–254 (1989) 20. Morita, S.: Structure of the mapping class group and symplectic representation theory. In: Essays on geometry and related topics, Ghys, E., De la Harpe, P., Jones, V., Sergiescu, V., eds., Vol. 2, Monogr. Enseign. Math., 38, Geneve: Univ. Geneve, 2001, pp. 577–596

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Communications in


From String Nets to Nonabelions Lukasz Fidkowski1,4 , Michael Freedman1 , Chetan Nayak1,2 , Kevin Walker1 , Zhenghan Wang1,3 1 Microsoft Station Q, University of California, Santa Barbara 93106-6105, USA.


2 Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA 3 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA 4 Department of Physics, Stanford University, Stanford, CA 94305, USA

Received: 23 January 2007 / Accepted: 12 December 2008 Published online: 14 February 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract: We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an S O(3)3 × S O(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.

1. Introduction In two dimensions, exotic quantum systems exist where interchange of identical quasiparticle excitations (often called anyons) can alter the wavefunction by a phase not equal to ±1 (as in the case of bosons and fermions). In fact, even non-abelian matrix operations, rather than just phases, are possible. The mathematical theory of anyons modular tensor categories - is extremely rich in examples, and existing physical theory provides a way of describing most of these examples as effective Chern-Simons gauge theories. In contrast, our knowledge is limited when it comes to identifying plausible solid state Hamiltonians from which a (2D) state of matter could emerge whose effective low energy description is, in fact, a Chern-Simons theory. The off-diagonal conductivity of fractional quantum Hall (FQHE) systems is tantamount to the equation of motion for a Chern-Simons Lagrangian, so Hall systems are the most developed source of such examples. The best-studied example among abelian states is the ν = 1/3 Laughlin state. The foremost candidate among non-Abelian states is the Pfaffian state [3], which is believed [4,5] to be realized at the ν = 5/2 fractional quantum hall plateau. Beyond

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L. Fidkowski, M. Freedman, C. Nayak, K. Walker, Z. Wang

ν = 5/2, more delicate plateaus at ν = 12/5, 4/7, etc. may also support nonabelions. However, in this paper we explore a quite distinct family of Hamiltonians. Because magnetic interactions in solids can be at energy scales as high as ∼ 103 Kelvin, it would be very exciting to find realistic families of spin Hamiltonians representing a nonabelian phase. (This has essentially been accomplished [6] for the simplest abelian phase, Z 2 gauge theory, although the corresponding experimental system has not been clearly identified.) This goal has been pursued for several years through the study of model Hamiltonians H acting in an effective Hilbert space H whose degrees of freedom are either unoriented loops [7], or, more recently, branching loops called “string nets” [1]. Such Hilbert spaces H are a kind of half-way house. Eventually, it will be necessary to understand how local spins can encode effective loops and nets, and some ideas on encoding nets are presented in Sect. 2. However, the premise of this paper is that we already have a Hilbert space H spanned by the simplest type of string net G, where the lines are unoriented and unlabeled, the nodes have valence 3 and lack internal states. Our goal then is to formulate, in the most general terms, what properties a Hamiltonian H : H → H should have in order to describe the simplest topological phase of string nets, the “doubled Fibonacci theory”, DFib, also sometimes denoted S O(3)3 × S O(3)3 . DFib is not only nonabelian but its braiding statistics is sufficiently rich as to serve as a basis for universal quantum computation [13]. It is thus an extremely attractive target phase. Following the microscopics of Sect. 2, we proceed in Sect. 3 to a derivation of DFib based on the concept of minimal degeneracy. To summarize our approach in a phrase: “nature abhors a degeneracy”. (Consider, for example, eigenvalue repulsion for random Hamiltonians). There is an irony here because topological phases are nothing else than a degenerate, yet stable, ground state for which no classical symmetry exists to be broken. From this viewpoint, we will see that building DFib (and other phases?) amounts to setting a trap for nature. By compelling a certain space V (D, n) of low energy modes (see Sec. 3 for a precise definition of V (D, n)) for small n to have dimension equal to d, unexpectedly small, we trap a class of Hamiltonians into an exponential growth √ of degeneracy: limn→∞ (dim(V (D, n)))1/n = τ = 1+2 5 . We present a rather surprising derivation of DFib from dimensional considerations alone; the F-matrix (or 6 jsymbol) derives from the assumption of unitarity and minimal dimension of “disk spaces”. Although the F-matrix obeys the pentagon equations we do not use the pentagon equation to find the F-matrix. The physical significance is that DFib should be a robust phase stabilized by a type of eigenvalue repulsion. Section 4 treats quasiparticle excitations. After DFib is derived in Sect. 5, a beautiful formula of Tutte (compare Ref. [2]) allows us to √make an exact connection to the Q = τ + 2 ≈ 3.618 state Potts model, where τ = 1+2 5 , the golden ratio. We find that the exactly solvable point in the DFib phase is the high temperature limit of the low temperature expansion of the τ + 2-state Potts model. This allows us to conjecture that the topological phase extends downward in “temperature” until the critical point is reached √ at log β = τ + 2 + 1. This suggests a one-parameter family of DFib-Hamiltonians whose ground state wave functions are not √ strictly topological but have a “length” or bond-fugacity, x, satisfying 0.345 ≈ 1/( τ + 2 + 1) ≤ x ≤ 1, implying a considerable stability within this phase. The approach in Sect. 5 and 6 complements the ideas presented in [1 and 2]. In [1], a finely tuned exactly solvable fixed point for DFib was produced; in this paper we focus instead on the minimum general requirements for the phase. In Sect. 5 we find an identity which is used in Sect. 6 to connect the statistical physics of DFib’s ground states to a critical Potts model, validating a key argument of [2].

From String Nets to Nonabelions

807

To recap, the paper is organized as follows. In Sect. 2, some ideas are presented for how string nets could emerge from microscopic models of quantum spins on a lattice. In Sect. 3, DFib is derived from dimensional considerations. In Sect. 4, we describe, in string net language, the quasiparticle excitations of DFib. In Sect. 5, string net wavefunctions and their squares are related to the chromatic polynomial. In Sect. 6 we show that the topological string net wave function has a “plasma analogy” to the τ + 2-state Potts model, and in Sect. 7 we discuss our conclusions. In the Appendix, Baxter’s hard hexagon model is used to extend a theorem of Tutte’s. 2. How to Construct a Net Hilbert Space H In this paper, we will be concerned with Hamiltonians H acting on Hilbert spaces H() of wave functions that assign complex-valued amplitudes to string nets (“nets” for short) on a surface . The surfaces which could be relevant to experimental systems are presumably planes with some number of punctures. However, it is quite profitable conceptually and for the purpose of numerical simulations to think about higher-genus surfaces as well. A net is what mathematicians call a trivalent graph; it has only simple branching and no “dead ends” (univalent vertices) except as defined by boundary conditions at the edge of the surface. According to this definition, (the Dirac function on the net in) Fig. 1 a) is not in the Hilbert space H but (those of) Fig. 1 b) and 1 c) are. In most of this paper, we will simply assume that the Hilbert spaces H() arise as the low-energy subspaces of the Hilbert spaces of a system of spins or electrons in a solid or ultra-cold atoms on an optical lattice. In such a formulation Fig. 1 a) could be thought of as a high, but finite, energy state of the spins, electrons in a solid or ultra-cold atoms. In this section, however, we will consider the question: from what kinds of lattice models do nets emerge in the low-energy description so that the Hilbert spaces H() are the low-energy subspaces? Three ideas A, B, and C are sketched for writing a spin Hamiltonian K : H¯ → H¯ on a large Hilbert space H¯ of microscopic degrees of freedom so that the ground state manifold H of K will be the “Hilbert space of nets” on which this paper is predicated. We emphasize that these spin Hamiltonians only reduce the Hilbert space down to a still rather large one supported on nets; they are not the full Hamiltonians describing the topological phase. In particular, their generic ground state wave functions do not satisfy isotopy invariance. We believe that they could be made to do so with the addition of extra terms to the hamiltonian, and this is the approach we take: first, cut the degrees of freedom to string nets, then make the string nets fluctuate appropriately to gain isotopy invariance. In the following, we accomplish the first part of this process. (a)

(b)

(c)

Fig. 1. a) is not in the Hilbert space H, but b) and c) are. In these figures, the outermost circle is the boundary of the system, where nets are allowed to terminate. The endpoint in the middle in (a) is a violation of the “no dead ends” condition

808


Fig. 2. An illegal pair of bonds which is energetically penalized by K p

All our Hamiltonians K break SU (2)-invariance and require fine tuning. Ideas A and B are conceptually very simple but both require a three-body interaction. Idea C is really an encryption of B into a 2-body interaction on higher spin (spin = 3/2) particles. A. H¯ = ⊗bonds C 2 , i.e. is a Hilbert space of spin = 1/2 particles living on the links of a trivalent graph such as the honeycomb. We interpret an Siz = 1/2 link as one on which the net lies. We take K = sites K s where K s projects onto the subspace of C 2 ⊗ C 2 ⊗ C 2 (i.e. of the Hilbert space of the three spins surrounding a lattice) of total 3 z i=1 Si = −1/2. This forbids dead ends. Unfortunately, the function total S z eigenvalue −3/2 −1/2 1/2 3/2

→ energy → 0, → nonzero →0 →0

(1)

is not quadratic (no parabola passes through (−3/2, 0), (−1/2, nonzero), (1/2, 0), and (3/2, 0). Hence, when K s is expanded in products of σ z i , i = 1, 2, 3, running over the bonds meeting s, it must contain a cubic σ z 1 σ z 2 σ z 3 term. B. An alternative is to put a spin = 1/2 particle at the sites of the honeycomb, so H = ⊗sites C 2 , C 2 = +, − . For each consecutive pair p of bonds, K has a term K p : K = p K p , where K p is a diagonal matrix all of whose entries are 0 except for that corresponding to the illegal pair of edges shown in Fig. 2). K penalizes both isolated +’s and +’s with exactly one + neighbor. The latter situation is shown in Fig. 3. Interpreting those bonds bounded by two + signs as the ones on which the net lies, we see that the zero modes of K are precisely the nets (trivalent graphs) within the honeycomb. Unfortunately K seems resolutely 3-body. B’. To set the stage for our final construction, it is helpful to reverse + and − spins on the index 2 Bravais lattice L within the honeycomb, honeycomb = L L . With this convention, K˜ = p centered on L K˜ p + p’ centered on L’ K˜ p where K˜ p projects to the highest total Sz eigenvalue, Sz = 3/2, and K˜ p projects to the lowest total Sz eigenvalue, Sz = −3/2. In other words, the penalized configurations are 3 consecutive pluses or 3 consecutive minuses. Also, we now draw bonds between plus sites of L and minus sites


809

Fig. 3. An up-spin which has only a single up-spin neighbor. This is energetically penalized by K = p K p

Fig. 4. Upon reversing the spins on one sublattice of the honeycomb lattice, K˜ p and K˜ p now penalize maximum and minimum S z eigenvalues respectively

Fig. 5. The bottom row represents the spin eigenvalues

of L (L and L labeled as black and red respectively in Fig. 4)(in color on line only). K˜ is still necessarily 3-body but at least it now has the form of the Klein Hamiltonian, see e.g. [8] C. The idea here is to take adjacent pairs of site spins from B and encrypt them as the state of a spin = 3/2 particle living on the bond b joining the two sites. We orient b from L to L. Let us set up an indeterminate bijection (Fig. 5). We now “simulate” K˜ on a

810


Fig. 6. The horizontal axis labels the i th spin and the vertical axis the j th spin

Hilbert space H¯ = ⊗bonds C 4 , the space of a spin = 3/2 particle on each bond. On sites l ∈ L (l ∈ L ) we must penalize t ⊗ t (q ⊗ q). Furthermore we must penalize inconsistent encryptions. For l ∈ L (l ∈ L ) the following pairs yield inconsistent site labels: q ⊗ r, r ⊗ q, q ⊗ t, t ⊗ q, s ⊗ r, r ⊗ s, s ⊗ t, and t ⊗ s (q ⊗ s, s ⊗ q, q ⊗ t, t ⊗ q, r ⊗ s, s ⊗ r, r ⊗ t, and t ⊗ r ). We again obtain the space of nets as zero modes H ⊂ H¯ by fixing K to be the following 2-body Hamiltonian: K = (t⊗t + q⊗r + r ⊗q + q⊗t + t⊗q + s⊗r + r ⊗s + s⊗t + t⊗s ) l∈L

+

(q⊗q + q⊗s + s⊗q + q⊗t + t⊗q + r ⊗s + s⊗r + r ⊗t + t⊗r ),

l ∈L

(2) where x⊗y denotes the projector onto the state x ⊗ y. We can write a Hamiltonian which effectively accomplishes such a projection in terms of the spins Siz (i is the bond index). For black sites H has terms 2 z 2 z Si + 1 + S j + 1 − 1/2 (3) Siz + S zj − 2 Siz + Sz j − 3 and for red sites

2 2 z − 1 + S j − 1 − 1/2 Siz + S zj + 2 Siz + S zj + 3 . Siz

The origin of these terms is illustrated in Fig. 6.

(4)


811

3. H, H, and V From now on, we will be concerned with the properties of the Hilbert spaces H() on a surface . If has boundary, ∂, we fix a boundary condition by specifying points where the nets must end. In the case where has connected boundary and the boundary condition consists of n points, we denote the Hilbert space by H(, n). We will be interested in “isotopy invariant” wave functions , whose value is independent of deformation. (In Sect. 6 we relax this condition to allow certain bond fugacities.) To avoid unnormalizable wave functions, these loops should really live on a lattice, as in the previous section. To produce invariant ’s we consider Hamiltonians H : H → H which contain fluctuations sufficient to enforce isotopy invariance (Fig. 7) on all low energy states. It should be remarked that it is not easy to set up such terms on a lattice; some fine tuning may be required (see [1] and [2] for a realization). Also there are questions of ergodicity - H must have sufficient fluctuations that crystals do not compete with the liquid condition described by Fig. 7. Nevertheless we start by assuming these problems solved: that we have H and a family {H } whose ground states V consist of isotopy invariant wave functions . The “axioms” we impose on the Hamiltonian H are implicit in the following conditions that we require its ground state manifolds V (, n) ⊂ H(, n) to satisfy. Axiom 1. H is gapped - this makes V (, n) sharply defined. Axiom 2. The following “minimal” dimensions on the 2-disk = D occur: (i) (ii) (iii) (iv) (v)

dim V (D, 0) = 1, dim V (D, 1) = 0, dim V (D, 2) = 1, dim V (D, 3) = 1, dim V (D, 4) takes the minimal value consistent with (i)-(iv) (to be computed below).

We make the further technical assumption that the constants a, b and c in Fig. 8 are neither 0 nor infinity. Axiom 2(ii) is the “no tadpole” axiom which says that although Fig. 1 b) is in H it has high energy. There is no low energy manifold V whatsoever when the boundary condition only allows tadpoles. If one thinks in terms of 1 + 1 dimensional physics, 2(ii) merely says the obvious: a single particle should not come out of the vacuum. If we nevertheless persisted in making dim (V (D, 1)) = 1 we would admit the very boring

Fig. 7. The condition of isotopy invariance. Wavefunctions assign a complex amplitude for any string net (which is the amplitude for this configuration to occur). These equations mean that wavefunctions in the low-energy Hilbert space assign the same amplitude to two string nets if one can be obtained from the other by smooth deformations

812


a

b

c

0 Fig. 8. We have stopped drawing the disk, but all diagrams above are nets in a 2-disk D with the endpoints on D.. Also we abuse notation to allow a net to also represent the ground state evaluation, g.s. (net)

case in which for all , dim(V ()) = 1 and V is spanned by the constant function on nets. Similarly 2(i) and 2(iii) are required in a 1 + 1 dimensional (unitary tensor category) context. Also, if either dimension is 0 then all V have dimension 0, by gluing formulae, so the entire theory collapses. Axiom 2(iv) does represent a choice. If we instead said dim (V (D, 3)) = 0, we would forbid our nets to branch. Here we know Z 2 gauge theory (i.e. the toric code) and the doubled semion theory — both abelian — can arise. Possibly higher doubled SU (2)k Chern-Simons theories might also arise from loop models, but entropy arguments [14] show that their ground state wave functions cannot be a simple Gibbs factor per loop [14]. So 2(iii) is not inevitable but represents our decision to set up whatever microscopics are necessary to build (H, H ) with branched nets occurring in V , i.e. to build a string net model. We now derive: Theorem 1. There is a unique theory (minimal category) V () compatible with 1 and 2 above. It satisfies dim (V (D, 4)) = 2 and is DFib, the doubled fibonacci category. Proof. By “Axiom 2” there are nonzero constants a, b, c ∈ C such that the conditions in Fig. 8 hold. Now consider the 4-point space V (D, 4). If its dimension is < 4 there must be relations among the 4 nets in the expression below (Fig. 9 a)), each thought of as an evaluation of the functional g.s. on V (D, 4). We find conditions on the coefficients by joining various outputs of R. This amounts to calculating consequent relations in the 1-dimensional spaces V (D, 2) and V (D, 3) which are implied by R. We work out, in part b) of Fig. 9 the implication of joining the upper outputs of R by an arc. By such arguments, we also have the three relations in fig. (10). Eliminating x and y, x = −ch − bi, y = −bh − ci, we obtain (b − c − ab)h + (−b − ac)i = 0, (−b − ac)i + (b − c − ab)h = 0.

(5)

Possible relations which can be imposed on string nets through the Hamiltonian correspond to solutions of this linear system. There can be at most two linearly independent solutions, and this case occurs exactly when the coefficients vanish: b = c + ab, b = −ac,


813

(a)

(b)

Fig. 9. If V (D, 4) < 4, then there must be some coefficients h, i, x, y such that the linear combination on the right-hand-side of (a) vanishes. By embedding these pictures within larger ones in such a way that the endpoints are connected as shown in (b) and using Fig. 8, we find relations satisfied by h, i, x, y

Fig. 10. Three other relations obtained by connecting the endpoints of R

(a)

(b)

Fig. 11. The 6 j symbols can be obtained from the h = 0, i = 1 and h = 1, i = a −1 relations

so −ac = c + √ (a 2 )c or a 2 = a + 1. We already see the golden ratio τ emerge: a = τ or −τ −1 , τ = 1+2 5 . It is well known that all relations among ground state string net amplitudes are determined by d-isotopy and local reconnection rules encoded in the 6 j symbol or F-matrix (Fig. 11) (see e.g. [2]). In order to calculate the 6 j-symbol, we set h = 0, i = 1 in R to 2 2 get the equation in Fig. 11 a). Unitarity requires | a −1 | + | b−1 | = 1 so a = τ and √ b = e2πiφ τ , where φ is an irrelevant phase (associated with identifying the simplest 3-point diagram with some unit vector in V (D, 3)) which we set to 1.

814


Similarly setting h = 1, i = a −1 (and hence x = 0) we find the second row of the 6 j-symbol (i.e. the F matrix) in Fig. 11 b) and thus:

−1 −1/2

τ τ

. F =

−1/2 −1 τ −τ

(6)

Thus, fixing V (D, 4) to have minimal dimension, 2, we find our constants specified: a = τ, b = τ 1/2 , c = −τ −1/2 , and the F-matrix as well. What we have obtained is the Turaev-Viro (or “doubled”) version of the unitary Fibonacci fusion category. From these rules – a, b, c, and F – all nets G on a sphere can be evaluated to a scalar G τ which is the “golden” quantum invariant. Similarly, with this data the entire unitary modular functor is specified on all surfaces with or without boundary: V ∼ = DFib. Remark. If V (D, 4) is allowed to have dimension 3, a generic solution, the Yamada polynomial, exists. It can be truncated to doubled S O(3)k - modular functors for k odd > 3 by imposing the correct dimension restriction on V (D, 2k). If V (D, 4) is allowed to be 4-dimensional, a relation in V (D, 5) realizes Kuperberg’s G 2 -spider which presumably admits further specializing relations which generate G 2 level k topological quantum field theories (TQFTs). 4. Deriving the Properties of DFib from V (, n) In the remaining three sections of this paper, we will discuss the properties of the ground states V (, n) ⊂ H(, n), thereby obtaining physical properties of the topological phase DFib. DFib is the product of two copies of opposite chirality of the Fibonacci theory, Fib, which the simplest 2 + 1 dimensional TQFT theory with nonabelian braiding rules. (Fib is also the simplest universal theory [13].) It arises as the “even sub-theory” of SU (2)3 (i.e. integer spins only) or SU (3)2 and also directly from G(2)1 . Fib has one non-trivial particle τ with fusion rule: τ ⊗ τ = 1 ⊕ τ.

(7)

Here we use the term particle to refer to a representation of the corresponding affine Lie algebra (SU (2)3 for the Fibonacci theory). This is because such representations can be associated with Wilson lines in the Chern-Simons picture. Particles arise in the usual way, when the three dimensional space under consideration is the product of a 2 dimensional Riemann surface (thought of as space) and the real line (thought of as time). Specializing to Wilson lines tracing out stationary trajectories, we obtain the particles discussed here,√and the degenerate ground states V (, n). Of course the quantum dimension of τ is 1+2 5 = τ . (In a slight abuse of notation, we use τ to denote both the particle and its quantum dimension, the golden ratio.) A discrete manifestation of this quantum dimension is that Fib(S 2 , n + 2), the Hilbert space for n + 2 τ -particles at fixed position on the 2-sphere, is Fib(n), the n th Fibonacci number. (Proof. Fuse two of the particles: the result will be either n + 1 or n τ ’s on S 2 depending on the fusion process outcome. This yields the famous recursion formula: dim Fib(S 2 , n + 2) = dim Fib(S 2 , n + 1) + dim Fib(S 2 , n) which defines Fibonacci numbers.) This gives the exponential growth referred to in the introduction.


815

Fib is a chiral theory with the following parameters:

1

1 τ

, S= √ τ + 2 τ −1

Sττ τ = e3πi/10 ,

−1 −1/2

τ τ

, F =

−1/2 −1 τ −τ

(8) (9) (10)

= exp(4 pi i / 5)

= exp(2 pi i / 5)

The theory V constructed in Sect. 3 is isomorphic to Fib∗ ⊗ Fib ∼ = End(Fib). Very briefly we explain this connection in the context of a closed surface (visualize = torus). Let G be a fine net on which “fills it” in the sense that all complementary regions are disks {δi }. According to Reshetikin-Turaev [16], labelings (by 1 or τ ) of the bonds of G consistent with Fib fusion rules span the Hilbert space for a very high ¯ (G¯ is a 3-D thickening of the net G). genus surface † which is the boundary of G, It can be seen that the fixed space under the F-matrix action on G-labelings can be obtained from projectors associated to {δi }. These disks determine projectors, or plaquette operators, onto the trivial particle type along a collection of “longitudes” on † (where δi intersects † ). Another way to implement these plaquette operators is to add ¯ G¯ {δ¯i } is homeomorphic to a product × I , surface {δ¯i } (thickenings of {δi } to G). cross interval. Adding the plaquet operators has cut Fib( † ) down to Fib(∂( × I )) = ¯ Fib( ) = Fib∗ () ⊗ Fib() = DFib(). From this point of view the double arises from the fact that surface × I has both an “inner” and “outer” boundary. It is nontrivial to align the various structures (e.g. particles) of DFib from the two perspectives: one as a theory of trivalent graphs on a surface (Sect. 3) and the other as a tensor product of a chiral theory and its dual. For this reason we are not content to merely state that the particle content of DFib is 1 ⊗ 1, 1 ⊗ τ, τ ⊗ 1, and τ ⊗ τ . Rather, we will give a direct string representation for these particles shortly. Remark. The expression of DFib (and more generally, th Turaev-Viro theories) through commuting local projectors was known to Kitaev and Kuperberg and made explicit for DFib in [1]. A conceptual understanding of the commutation relations is readily at hand from the preceding construction of × I . The “longitudes” on which we apply plaquette projectors are disjoint and thus commuting. The fusion rules that are enforced at disjoint vertices also commute. Finally vertex and plaquet terms commute because rules are preserved under the addition of an additional (“passive”) particle trajectory, labeled d in Fig. 12. We conclude this section by finding the 4 irreducible representations of the DFib annulus category. This is the linear C ∗ -category whose objects are finite point sets

816


c

cd

d a

b

a

b

Fig. 12. The addition of a passive arc to the (piece of a) string net on the right preserves the rules satisfied by nets

(a)

(b)

Fig. 13. a) 1 ∈ A0,0 . b) R ∈ A0,0

(boundary data) on a circle and whose morphisms are formal combinations of nets in the annulus, which obey the linear rules a, b, c, and F, and which mediate between the boundary data. The four irreducible category (or “Algebroid”) representations are detected as idempotents in A0,0 and A1,1 , the algebra under stacking of nets in annuli with either trivial or 1-point boundary data. A table which organizes the results and compares back to the Fib∗ ⊗ Fib picture is given below. The entries show the dimensions of the Hilbert space of (formal combinations of) nets on an annulus which start on the inner boundary with a given boundary condition (horizontal axis) and terminate near the outer boundary with a copy of a given idempotent (vertical axis). boundary conditions irreps (idempotents) 0 1 2 … e1 ∼ 1 0 >0 =1⊗1 e2 ∼ 1 1 >0 >0 =τ ⊗τ e3 ∼ 0 1 >0 =1⊗τ e4 ∼ 0 1 >0 =τ ⊗1 Let us start by finding the idempotents for A0,0 . A0,0 has the empty net as its identity and is generated by a single ring R. The a, b, c, F rules show: R 2 = 1 + R so A0,0 ∼ = C[R]/(R 2 = 1 + R). More generally, the idempotents in the algebra C[x]/P(x), P(x) = (x − a1 ) · · · (x − ak ), all roots distinct, are given by: ei =

− ai ) · · · (x − ak ) (x − ai ) · · · (x . (ai − a1 ) · · · (ai − ai ) · · · (ai − ak )

(11)

We get 1+τR ∼ ( = 1 ⊗ 1, the trivial particle), τ +2 1 + τ¯ R ∼ e2 = ( = τ ⊗ τ ), 2 + τ¯ e1 =

(12) (13)


(a)

817

(c)

(b)

Fig. 14. a) 1 ∈ A1,1 , b) T ∈ A1,1 , c) T −1 ∈ A1,1 (hint: apply F-matrix here)

Fig. 15. A compact representation of L = τ −1/2 1 + τ −3/2 (T + T −1 ), as may be seen by applying the F-matrix where indicated

Fig. 16. The relations depicted above can be obtained by applying the a, b, c, and F rules

τ¯ = −τ −1 . Using the a, b, c, and F rules the algebra A1,1 is seen to be generated by the identity 1, T , and T −1 (where T is defined in Fig. 14). There is an element L = τ −1/2 1 + τ −3/2 (T + T −1 ) which factors in a category sense through A0,0 . In fact, using the elementary rules in Fig. 8 we derive the useful identities in Fig. 16 and we find that L is equivalently represented as shown in Fig. 15. It follows that to find a new representation of the annulus category we should look for the idempotents in A1,1 /L. A1,1 /L ∼ = C[T ]/{T 2 + τ T + 1 = 0}. So, using (11) again and a little manipulation we find idempotents √ T − τ − 2τ −3 1 √ e˜3 = − τ −3

(14)

818


and

e˜4 =

T+

√ τ + τ −3 1 2

√

τ −3

(15)

in the quotient algebra. Also L 2 = (τ 1/2 + τ −3/2)L, so e L = L/τ 1/2 + τ −3/2 , so again using the identities in Fig. 16 we can find e3 = e˜3 (1 − e L ) and e4 = e˜4 (1 − e L ): √ τ +2+ τ −3 L, (16) e3 = e˜3 + √ 2(τ 1/2 + τ −3/2 ) τ − 3 √ −(τ + 2) − τ − 3 L. (17) e4 = e˜4 + √ 2(τ 1/2 + τ −3/2 ) τ − 3 5. Chromatic Polynomial and Yamada Polynomial The relation between the chromatic polynomial and ground state amplitudes of string nets has been studied before in [2]. In this section we use an identity for the chromatic polynomial (Tutte’s “golden ration theorem”) to deduce a simpler looking version of such a relation. The chromatic polynomial χGˆ (k) of a graph Gˆ at the positive integer k counts the number of k-colorings of the vertices of the graph (so that no two vertices connected by a bond are given the same color). χ obeys the famous “delete-contract” recursion relation: χGˆ (k) = χG−e ˆ (k) − χG/e ˆ (k).

(18)

This relation can be depicted graphically as shown in Fig. (17), in which we have suppressed χ (as we have consistently suppressed the wave function and written the relation out in terms of its pictorial argument). This is a local relation; there is no significance to the bit of Gˆ near the active edge e, which in the drawing is represented, purely for illustrative purposes, by three half-edges on top and bottom. χ (k) is completely fixed by (18) and the following conditions: that χ vanish on any graph in which the two ends of

(a)

(b)

(c) Fig. 17. (a) Graphical depiction of the “delete-contract” recursion relation. This relation together with the two depicted graphically in (b) and (c) and multiplicativity under disjoint union completely determine χGˆ (k)


819

Fig. 18. Jones-Wenzl projector

Fig. 19. The Yamada polynomial for the θ graph

a single bond are joined to the same vertex, that χsingle pt. (k) = k, and multiplicativity under disjoint union: χGˆ Hˆ = χGˆ χ Hˆ .

(19)

We will be interested in χGˆ evaluated at noninteger values as well as integral ones. We now turn to the Yamada polynomial defined for a net (trivalent graph) G lying in the plane (or 2-sphere); we denote it by G d , where d is the variable. To define G d , recall the 2-strand Jones-Wenzl projector, an idempotent familiar from the study of the Temperley-Lieb algebra T L d . Given G, G d is defined by labeling every arc of G by 2 as in (18) and then expanding to a weighted superposition of multi-loops. In each term, each loop contributes a numerical factor of d, G d = (coeff.)d # loops . (20) terms

For example, if G is a graph shaped like the Greek letter θ , we have the result shown in Fig. 19. As this example shows, the Yamada polynomial is actually a polynomial in d, d −1 . Theorem 2. If G is a net in the 2-sphere and Gˆ is the dual graph then: ˆ

G d = d −V (G) χGˆ (d 2 ),

(21)

V = number of vertices of Gˆ = number of faces of G. Proof. The above procedure for turning a net into a superposition of multi-loops may be generalized by declaring two local rules shown in Fig. 20. The first rule says a point is replaced by a circle with a possible numerical weight (vertex fugacity) and each arc

820


p

Fig. 20. Local rules for turning the dual graph of a net into a superposition of multi-loops. The ellipse represents an unknown combination

Fig. 21. A recoupling rule by which the dual of a trivalent graph can be converted into multi-loops. We will choose u and v (and also ρ) so that the procedure for making a trivalent graph into a multi-loop relates χGˆ (k) and G d

Fig. 22. Applying the rules in Fig. 20 to the graph on the left-hand side of the figure above yields the picture on the right

Fig. 23. Using the recoupling rule in Fig. 21, we can simplify the picture on the right-hand-side of Fig. 22

is replaced by two lines with some general recoupling. We express the fact that this recoupling is still a variable to be solved for by Fig. 21. Our goal is to find suitable values for ρ, u, and v so that χGˆ (k) comes out related to G d . The correct k will turn out to be d 2 . Let us look near a typical edge e of Gˆ and expand it using the rule in Fig. 22. With this expansion and Fig. 21 we obtain Fig. 23. Translating this back into graphs yields Fig. 24. Also, since graphs with an edge that connects a vertex to itself evaluate to zero, we have Fig. 25, implying v = −ud. Also, k = ρd. Comparing Fig. 24 with the chromatic relation (18) we find: v = −1, u = 1/ρ. Because we have v = −ud, we obtain u = 1/d so d = ρ. Finally, since k = ρd, k = d 2 . The mystery combination turns out to be a “sideways” P2 as shown in Fig. 26.


821

Fig. 24. A relation that must be satisfied by u, v, and ρ

Fig. 25. A graph with an edge that connects a vertex to itself evaluates to zero, from which we deduce a relation between u and v

Fig. 26. Solving for u, v, ρ, we see that the recoupling rule is just a “sideways” P2

Fig. 27. Applying the rules to a complete graph

To complete the proof, it remains to see the global geometry of how these P2 ’s hook together. We claim that they lie along (doubled) dual graph edges. It suffices to examine an example. We take Gˆ and G to be the complete graph C4 shown in Fig. 27. The

822

L. Fidkowski, M. Freedman, C. Nayak, K. Walker, Z. Wang ˆ

factor of ρ V (G) appearing with G d has been put on the right-hand-side in the theorem statement. It is well known that the Yamada polynomial and the unitary invariant G τ of Sect. 3 are closely related. Specifically, when d = τ , we should modify the Yamada polynomial G τ by a vertex fugacity to obtain G τ . This can be seen by checking that both quantities satisfy the elementary rules of Sect. 3 that determine them uniquely. From our example, shown in (19), we find θ τ = τ −1 while for the unitary theory θ τ = ab = τ 3/2 . To convert from the unnormalized (Kauffman) theory to the unitary theory (compare to [1]) one must multiply in a factor of τ 5/4 for each vertex. Thus: ˆ

G τ = (τ 5/4 )V (G) τ −V (G) χGˆ (τ 2 ).

(22)

ˆ + V (G) ˆ − E(G) ˆ and the fact that Gˆ is a triangulation, Using the Euler relation F(G) 3 implying E = 2 F, we find: ˆ = 2V (G) ˆ − 4. F(G)

(23)

Therefore: 3

ˆ

G τ = τ −5 τ 2 V (G) χGˆ (τ + 1).

(24)

We can now use Tutte’s “golden ratio theorem” ([T] and [L]) ([9]): For a planar trianˆ gulation G: (χGˆ (τ + 1))2 τ 3V (τ + 2)(τ −10 ) = χGˆ (τ + 2).

(25)

This formula allowed Tutte to conclude that the r.h.s. is positive, creating a curious analogue (and precursor) to the 4-color theorem. (Neither result has been shown to imply the other.) Now square (24) and substitute into (25); the result is a remarkable formula: (G τ )2 =

1 χ ˆ (τ + 2). τ +2 G

(26)

We have just proved the formula when Gˆ is a triangulation; in fact it holds more generally whenever G is a net. To establish this it suffices to check the formula when G is a single loop and to observe that the formula behaves well under disjoint union of disconnected components of G: the l.h.s. is obviously multiplicative and, it turns out, so is the r.h.s. The reason is that disjoint union of G 1 and G 2 corresponds to a 1-point union Gˆ 1 Gˆ 2 , and with the factor of 1/(τ + 2), the chromatic polynomial becomes multiplicative under 1-point unions. So we have proved: Theorem 3. Let G be a net in the plane or 2-sphere (possibly disconnected and possibly with circle components). Then 1 G τ 2 = χGˆ (τ + 2). (27) τ +2 Note: We would like to thank P. Fendley and E. Fradkin for a discussion of this identity. In their paper [2], a non-unitary normalization for G τ led to a vertex fugacity on the right-hand-side of (27), thereby obscuring the simplicity of (27) and the direct connection to the Potts model.


823

6. String Net Wavefunctions and the Potts Model Let us review the high- and low-temperature expansions of the Potts model. We begin by assuming a lattice with Q “spin values” σi = 0, 1, . . . , Q − 1 at each site i. The partition function Z (Q) is defined by: ⎛ ⎛ ⎞⎞ exp ⎝−β ⎝−J δσi ,σ j ⎠⎠ (28) Z = σ

=

i, j

exp β J δσi ,σ j ,

(29)

σ i, j

where the sum is over spin state configurations. Setting γ = eβ J − 1 and expanding in powers of γ we obtain the high-temperature expansion: Z= Qc γ b, (30) where the sum is over bond configurations; c is the number of clusters, and b is the number of bonds. From now on, we consider the ferromagnetic case J = 1. Note that the last sum is over the 2b distinct subsets, not the Q (# of sites) distinct colorings because a Fubini resummation has taken place. Also, note that isolated sites count as clusters in (30). This is the Fortuin-Kateleyn representation [17]. It√is known that for 0 < Q ≤ 4 the model is critical precisely at its self-dual point, γ = Q. The high temperature expansion has been used [2,7] to study loop gases with ground state wavefunction whose amplitude is d L for some real number d, where L is the number of loops. The square of such a wave function can be interpreted as a Gibbs weight (d 2 ) L providing a “plasma analogy” between topological ground states and the statisti2 cal physics of loop gases. √ We can easily see that a loop gas with Gibbs weight d per loop is critical if d ≤ 2: ∗

(d 2 ) L = (d 2 )c+c = (d 2 )2c+b = (d 4 )c (d 2 )b .

(31)

Here, c∗ is the number of dual clusters, i.e. the minimum number of occupied bonds which have to be cut in order to make each cluster tree-like (essentially the number of “voids” which are completely contained within clusters). The first equality follows from each loop being either the outer boundary of a cluster or a dual cluster. The second is due to the Euler relation c∗ = c + b + const. To map this squared wavefunction to the Potts model, we need d 4 = Q ≤ 4; note the edge fugacity for this loop model is automatically √ Q, placing the model at its self dual point. The low-temperature expansion of the Potts model is: Z (Q) = χGˆ (Q)(e−β J ) L , (32) G

where the sum over graphs G and L is the total length of the graph. We specialize to the case in which the dual lattice is trivalent (e.g. the triangular lattice, whose dual lattice is the honeycomb), so that the graphs are all trivalent. At the critical point (Q ≤ 4), this becomes L 1 Z (Q) = χGˆ (Q) √ , (33) Q+1 G √ using the condition for criticality eβc J − 1 = Q.

824


The (unnormalized) isotopy invariant wave function constructed in Sect. 3 satisfies (G) = G τ . Hence the corresponding statistical physics (of equal time correlators) is that of the normalized probability distribution: 2 1 G τ . (34) prob(G) = Z (Q = τ + 2, β = 0) Note that (using (27)) this is the high temperature limit of the low temperature expansion. This situation reminds us of the Toric code [18], where the weight on loops may be obtained by setting Q = 2, β = 0 (for Q = 2, branched nets have zero weight), so the toric code ground state wavefunction can also be understood in terms of the hightemperature limit of the low-temperature expansion of the Potts model. In both cases, we conjecture that for β < βc the wavefunction with isotopy invariance modified by: √

(G) = G τ x length(G)

(35)

for 1 ≥ x > 1/( Q + 1) (≈ 0.345 when Q = τ + 2, and ≈ 0.466 when Q = 2) will be the ground state for some gapped Hamiltonian in the corresponding topological phase DFib (or Toric code). Numerical work [19] already √ supports this conjecture in the Toric code case. Also, note that at β > βc (so x < 1/( Q+1)) an effective string tension prevents the nets from fluctuating. Thus, the system leaves the topological phase and enters a “geodesic phase” in which small nets dominate. Recoupling is now unlikely, so the necessary topological relations are not well enforced and geodesic continuation allows states on the torus to be fairly well guessed by measurement within a subdisk violating the disk axiom. Finally, (27) identifies the ground state of the gapped Levin-Wen [1] Hamiltonian: (G) = G τ as the β = 0 end point of the conjectured family. In ref. [2], it is conjectured that the unnormalized probability distribution on nets (G d )2 should somehow yield the same statistical physics as the Potts model at an “effective” Q eff satisfying: (Q eff − 1) = (d 2 − 1)2 .

(36)

If we set d = τ and adjust the G-vertex fugacity so as to replace G τ with G τ , as we did in Sect. 5, then the conjecture is precisely verified at Q = τ + 2, β = 0: (τ + 2 − 1) = (τ 2 − 1)2 .

(37)

We close this section with an observation whose importance, if any, we do not yet understand. The high temperature expansion weights loops while the low temperature expansion weights nets. But loops are nets, so we might ask: at what value of Q do the high and low temperature expansions weigh loops equally? In the low temperature expansion, √ loops have weights Q − 1 and in the high temperature expansion √ they have weight Q (from (31)). So loops are equally weighted when Q − 1 = Q, i.e. when Q = τ 2. 7. Conclusions DFib is among the simplest conceivable achiral particle theories. In some sense it rivals the toric code in simplicity (both have 4 particles), but is vastly richer (in fact, universal [13]) in its braiding. We have explored the path to this phase, taking a Hilbert space H of nets and an isotopy invariant Hamiltonian (possibly with a bond fugacity, see Sect. 6) H : H → H as our starting point. The exploration has been combinatorial in Sect. 3, algebraic in Sect. 4, and statistical in Sect. 6.


825

In Sect. 3, we saw that DFib emerges from minimizing certain degeneracies. This encourages us to believe the phase will ultimately be found in nature - as nature abhors a degeneracy. In Sect. 6, we establish that the nets G with (squared) topological weighting (G τ )2 (as usual, squaring the wavefunction to obtain a probability) are in a high temperature phase of the (τ + 2)-state Potts model (above criticality). This is also encouraging: the classical critical point looks as if it is the “plasma analogy” of a quantum critical point sitting at the entrance to the DFib phase. A parallel is explored between this situation and the Q = 2 Potts critical point which serves as an entrance to the toric code phase. Unresolved is what, more precisely, is required of H : H → H to be in the DFib phase. Is enforcement of the net G structure (encoded in the definition of H) plus strong dynamic fluctuation of G adequate? We do not know. Acknowledgement. We would like to thank Paul Fendley and Eduardo Fradkin for useful discussions. C.N. would like to acknowledge the support of the NSF under grant no. DMR-0411800 and the ARO under grant W911NF-04-1-0236 (C.N.). This research has been supported by the NSF under grants DMR-0130388 and DMR-0354772 (Z.W.). L.F. would like to acknowledge the support of the NSF under grant no. PHY -0244728. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Appendix A: The Chromatic Polynomial and Hard Hexagons In this appendix we extend a theorem of Tutte [9] which bounds the decay of the chromatic polynomial at τ + 2 of planar graphs. Specifically, we obtain a better sharp bound for graphs that consist of a large regions of hexagonal lattice (see Fig. 28). The ground state of the doubled Fibonacci theory DFib consists of a certain superposition of string net configurations on a hexagonal lattice. As in ref. [2] we may construct from it a classical statistical mechanical model of nets, with the Boltzmann weight of each net equal to the norm squared of its ground state amplitude. As discussed in Sect. 5, this statistical mechanical model is the (solvable) infinite-temperature limit of the Q = τ + 2 Potts model. We are interested in the rough quantitative behavior of the amplitudes for different types of graphs. In particular, we can consider a large chunk of hexagonal lattice, which is just the string net consisting of a large finite region whose bulk includes every available bond (see Fig. 28). We can ask about how its amplitude scales with the size of the region. Now, the Boltzmann weight of a string net is (up to overall scaling) just the chromatic polynomial at τ + 2 of the graph dual to the net, which can be interpreted

Fig. 28. Large chunk of hexagonal lattice

826


as the zero-temperature limit of the τ + 2 antiferromagnetic Potts model defined on the graph dual to the net. So for the large chunk configuration, we are just interested in the behavior of the zero-temperature free energy of the τ + 2 antiferromagnetic Potts model on the hexagonal lattice. This model is critical [11] and is expected to be described by a conformal field theory. On general grounds [10] we expect the free energy to scale like F = c0 A + c1 L + c2 log L ,

(A1)

where c2 is universal and related to the central charge of the CFT and c0 and c1 are non-universal (here A is the area, i.e. number of hexagons in the region, and L is the length of the boundary). It turns out that the model is exactly solvable and we can obtain an exact analytic expression for c0 . To do so, we first use the so-called shadow method [20] to evaluate G τ of a net. This method works as follows. We take the net to be located on the sphere and assume that the regions (or faces) that it bounds are simply connected and do not border themselves. Then the shadow method (applied to DFib) gives G τ as a sum over black and white colorings of the faces Gτ = FC E C −1 VC . (A2) colorings C

The colors black and white are identified with the two particle types of the theory, and dC(F) , (A3) FC = faces F

EC =

(C, E),

(A4)

edges E

VC =

Tetrahedron(C, V ).

(A5)

vertices V

Here dC(F) stands for the quantum dimension associated with the color of face F (1 for black, τ for white). (C, E) is the theta graph along whose three edges run particle types associated with the edge E (always the nontrivial particle) and the two faces which E borders. Tetrahedron (C, V ) is the tetrahedral graph with its six edges labeled by the particle types corresponding to the faces and edges adjoining V . Let v = # vertices, f = # faces, e = # edges. When applied to the net which consists of a large chunk of hexagonal lattice, the shadow method reduces to the following: the region containing the point at infinity is black, and the sum over colorings becomes a sum over colorings in which there are no adjacent black hexagons (and no black hexagons adjacent to the black outside region). Let us first compute the weight of the all white coloring. We ignore boundary effects. We have first of all v = 2 f and e = 3 f . Each edge contributes −1 = τ −3/2 , each vertex Tetrahedron = −τ , and each face the quantum dimension τ . The total weight of the all white coloring is thus e τ −3/2 (−τ )v (τ ) f = τ −3 f /2 . (A6) Now suppose we have a configuration with some black hexagons. Its weight is just that of the all white configuration multiplied by appropriate ratios of theta symbols, tetrahedron symbols, and quantum dimensions. More specifically, for each white hexagon that


827

one turns into a black hexagon, one must multiply the edge contribution by (τ/τ 3/2 )−6 , the vertex contribution by (τ 3/2 /(−τ ))6 , and the face contribution by τ −1 . The product of these is τ 5 , so that the weight of each such coloring is just τ 5#( black hexagons) times the weight of the all-white coloring. The sum over colorings now just yields the critical Hard Hexagon model, whose free energy per vertex was obtained by Baxter [12] (Eq. 10). We thus obtain f G τ = τ −3/2 κc , (A7) where

κc =

From (22) we surmise

√ 1/2 27(25 + 11 5) . 250

f χ (τ + 1) = G τ τ f −5/4v = τ −3 κc .

(A8)

(A9)

We note that this is sharper than Tutte’s bound [9] of const. τ − f on χ (τ + 1): 0.546 f < 0.618 f .

(A10)

Tutte’s bound is of course more general in that it applies to any net configuration on the sphere (whose regions are simply connected and don’t border on themselves). For completeness, we note from (26) and (A1) that c0 = τ −3/2 κc . References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Levin, M., Wen, X.G.: Phys. Rev. B. 71, 045110 (2005) Fendley, P., Fradkin, E.: Phys. Rev. B. 72, 024412 (2005) Moore, G., Read, N.: Nucl. Phys. B 360, 362 (1991) Morf, R.H.: Phys. Rev. Lett. 80, 1505 (1998) Rezayi, E.H., Haldane, F.D.M.: Phys. Rev. Lett. 84, 4685 (2000) Moessner, R., Sondhi, S.L.: Phys. Rev. Lett 86, 1881 (2001); Nayak, C., Shtengel, K.: Phys. Rev. B 64, 064422 (2001); Balents, L. et al.: Phys. Rev. B 65, 224412 (2002); Ioffe, L.B. et al.: Nature 415, 503 (2002); Motrunich, O.I., Senthil, T.: Phys. Rev. Lett. 89, 277004 (2002) Freedman, M., Nayak, C., Shtengel, K.: http://arXiv.org/list/cond-mat/0309120, 2003; Freedman, M., Nayak, C., Shtengel, K.: Phys. Rev. Lett. 94, 066401 (2005); Freedman, M., Nayak, C., Shtengel, K.: Phys. Rev. Lett. 94, 147205 (2005) Chayes, J.T., Chayes, L., Kivelson, S.A.: Commun. Math. Phys. 123, 53 (1989) Tutte, W.T.: On Chromatic Polynomials and the Golden Ratio. J. Combin. Th., Ser. B 9, 289–296 (1970) Cardy, J.: In: Les Houches 1988, Fields, Strings and Critical Phenomena. Brezin, E., Zinn-Justin, J. (eds.) London: Elsevier, 1989 Saleur, H.: Nucl. Phys. B 360, 219 (1991); Pasquier, V., Saleur, H.: Nucl. Phys. B, 330, 523 (1990) Baxter, R.: J. Phys. A: Math. Gen. 13, L61–L70 (1980) Freedman, M., Larsen, M., Wang, Z.: http://arXiv.org/list/quant-ph/0001108, 2000 Levin, M., Wen, X.G.: Phys. Rev. Lett. 96, 110405 (2006) Witten, E.: Commun. Math. Phys. 121, 351 (1989) Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. Berlin-New York: Walter de Gruyter, 1994 Fortuin, C.M., Kasteleyn, P.W.: Physica 57, 536 (1972) Kitaev, A.: Ann. Phys. 303, 2 (2003) Trebst, S., et al.: Phys. Rev. Lett. 98, 070602 (2007) Kauffman, L.H., Lins, S.L.: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Princeton, NJ: Princeton, University Press, 1994

Communicated by M. Aizenman


Communications in


A Rigorous Treatment of Energy Extraction from a Rotating Black Hole F. Finster1, , N. Kamran2, , J. Smoller3, , S.-T. Yau4, 1 NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany.


2 Department of Math. and Statistics, McGill University, Montréal, Québec, Canada H3A 2K6.


3 Mathematics Department, The University of Michigan, Ann Arbor, MI 48109, USA.


4 Mathematics Department, Harvard University, Cambridge, MA 02138, USA.

E-mail: [email protected] Received: 16 July 2007 / Revised: 22 May 2008 / Accepted: 10 November 2008 Published online: 7 February 2009 – © Springer-Verlag 2009

Abstract: The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou’s result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator.

1. Introduction and Statement of Results Near a rotating black hole there is the counter-intuitive effect that the physical energy of particles or waves need not be positive. As discovered by Roger Penrose [12], this effect can be used to extract energy from the black hole. In the process proposed by Penrose, a classical particle enters the so-called ergosphere, a region outside the event horizon, where it disintegrates into two particles. One particle of negative energy falls into the black hole, whereas the other particle has energy greater than the initial energy, and escapes to infinity. The wave analogue of the Penrose process is called superradiance; it was proposed in [16] and first studied in [13,14] and [4,5], see also [2,11,15]. In these papers, superradiance was considered only on the level of modes, i.e., by considering the reflection coefficients for the radial ODE obtained after separating out the time and angular dependence. A more convincing treatment of superradiance is to Research supported in part by the Deutsche Forschungsgemeinschaft.

Research supported by NSERC grant # RGPIN 105490-2004. Research supported in part by the Humboldt Foundation and the National Science Foundation, Grant

No. DMS-0603754. Research supported in part by the NSF, Grant No. 33-585-7510-2-30.

830

F. Finster, N. Kamran, J. Smoller, S.-T. Yau

consider the Cauchy problem for “wave packet” initial data, and in [1], this was carried out numerically for the scalar wave equation. In this paper we shall consider the Cauchy problem analytically, giving a mathematically rigorous treatment of superradiance for scalar waves. Our analysis is based on the integral representation for the wave propagator obtained in [8,9]. A rotating black hole is modeled by the Kerr metric, which in Boyer-Lindquist coordinates (t, r, ϑ, ϕ) with r > 0, 0 ≤ ϑ ≤ π , 0 ≤ ϕ < 2π , takes the form ds 2 = g jk d x j d x k =

(dt−a sin2 ϑ dϕ)2 −U U

sin2 ϑ dr2 + dϑ 2 − (a dt−(r2 + a 2 ) dϕ)2 . U (1.1)

Here M > 0 and a M > 0 denote the mass and the angular momentum of the black hole, respectively, and the functions U and are given by U (r, ϑ) = r2 + a 2 cos2 ϑ,

(r) = r2 − 2Mr + a 2 .

We consider only the non-extreme case M 2 > a 2 , where the function has two distinct roots. The largest root r1 = M +

M 2 − a2

defines the event horizon of the black hole. We restrict attention to the region r > r1 outside the event horizon where > 0. The metric (1.1) does not depend on t nor ϕ and is thus stationary and axisymmetric, admitting the two commuting Killing fields ∂t∂ and ∂ ∂ ∂ϕ . The ergosphere is defined to be the region where the Killing field ∂t is space-like, that is where r2 − 2Mr + a 2 cos2 ϑ < 0.

(1.2)

The ergosphere lies outside the event horizon r = r1 , and its boundary intersects the event horizon at the poles ϑ = 0, π . We now briefly review the Penrose process (for more details see [15]). The 4-momentum p j of a massive point particle is time-like and future-directed, and thus its energy p, ∂t∂ is positive whenever ∂t∂ is time-like. However, in the ergosphere the vector ∂t∂ becomes space-like, and hence the energy of the point particle need not be positive. Penrose considers a particle of positive energy which splits inside the ergosphere into two particles whose energies have opposite signs. By finely tuning the energy and momenta of these particles, one can arrange that the particle of negative energy crosses the event horizon and reduces the angular momentum of the black hole, whereas the other particle escapes to infinity, carrying (due to energy conservation) more energy than the original particle. In this way, one can extract energy from the black hole, at the cost of reducing its angular momentum. Christodoulou [3] showed that the infinitesimal changes of mass δ M and angular momentum δ(a M) of the black hole satisfy the inequalities δ(a M) ≤

r12 + a 2 δ M < 0, a

(1.3)

Rigorous Treatment of Energy Extraction from a Rotating Black Hole

831

and as a consequence he showed that it is not possible to reduce the mass of the black hole via the Penrose process below the irreducible mass Mirr defined by 2 = Mirr

1 2 4 M + M − (a M)2 . 2

(1.4)

We now recall the wave equation in the Kerr geometry and its separation; for more details see [8]. The scalar wave equation is ∂ 1 ij ∂ g i j ∇i ∇ j = √ − det g g = 0, (1.5) ∂x j − det g ∂ x i and in Boyer-Lindquist coordinates this becomes ∂ 1 ∂ 2 ∂ ∂ ∂ ∂ (r2 + a 2 ) + a sin2 ϑ − − + ∂r ∂r ∂t ∂ϕ ∂ cos ϑ ∂ cos ϑ ∂ 1 ∂ 2 2 a sin ϑ + = 0. − 2 ∂t ∂ϕ sin ϑ

(1.6)

Using the ansatz (t, r, ϑ, ϕ) = e−iωt−ikϕ R(r) (ϑ)

(1.7)

with ω ∈ R and k ∈ Z, the wave equation can be separated into both angular and radial ODEs, Rω,k Rλ = −λ Rλ ,

Aω,k λ = λ λ .

(1.8)

The angular operator Aω,k is also called the spheroidal wave operator. It has a purely discrete spectrum of non-degenerate eigenvalues 0 ≤ λ1 < λ2 , . . .. The corresponding eigenfunctions ω,k n are referred to as spheroidal wave functions. In order to bring the radial equation into a convenient form, we introduce a new radial function φ(r) by (1.9) φ(r) = r2 + a 2 R(r), and define the Regge-Wheeler variable u ∈ R by r2 + a 2 du = , dr

(1.10)

mapping the event horizon to u = −∞. The radial equation then takes the form of the Schrödinger equation d2 − 2 + V (u) φ(u) = 0 (1.11) du with the potential V (u) = − ω +

ak 2 r + a2

2 +

λn (ω) 1 2 2 2 + ∂ √ u r +a . (r2 + a 2 )2 r2 + a 2

(1.12)

832


Starobinsky [13] analyzes superradiance on the level of modes. Using the notation in [9], we can explain his ideas and results as follows. We fix the separation constants k > 0, ω and λn . Introducing the notation

= ω − ω0

ω0 = −

with

r12

ak , + a2

(1.13)

the potential (1.12) has the asymptotics lim V (u) = − 2 ,

u→−∞

lim V (u) = −ω2 .

u→∞

Thus there are fundamental solutions φ´ and φ` of (1.11) which behave asymptotically like plane waves, ´ ´ = 1, lim e−i u φ(u) = 0, (1.14) lim e−i u φ(u) u→−∞ u→−∞ ` ` = 1, lim eiωu φ(u) = 0, (1.15) lim eiωu φ(u) u→∞

u→∞

(see [9] for details). The solution φ = φ´ has, near the event horizon u = −∞, the asymptotics φ(u) = e−i u . Taking into account the factor e−iωt in the separation, this corresponds to a plane wave entering the black hole. A general solution φ can be expressed as a linear combination of φ` and its complex conjugate, ` ` φ(u) = A φ(u) + B φ(u).

(1.16)

` Equation (1.15) shows that the solution φ` behaves near infinity like φ(u) = e−iωu . Hence the corresponding time-dependent solution behaves like the plane wave e−iω(t+u) and corresponds to an incoming wave propagating from infinity. Likewise, the solution φ` corresponds to an outgoing wave propagating towards infinity. Computing the Wronskian of φ and φ near the event horizon and near infinity gives the relation |A|2 − |B|2 =

. ω

(1.17)

The quantities ω2 |A|2 and ω2 |B|2 have the interpretations as the energy flux of the incoming and outgoing waves, respectively. If the right side of (1.17) is positive, the outgoing flux is smaller than the incoming flux, and this corresponds to ordinary scattering. However, if the right side of (1.17) is negative, then the outgoing flux is larger than the incoming flux; this is termed superradiant scattering. According to (1.17), superradiant scattering appears precisely when ω and have opposite signs. Using (1.13), we see that superradiant scattering occurs if and only if ω lies in one of the following intervals: ω0 < ω < 0,

0 < ω < ω0 ,

(1.18)

depending on the sign of ω0 . The gain in energy is determined by the quotient of outgoing and incoming flux, R =

|B|2 . |A|2

(1.19)


833

Starobinsky [13] made an asymptotic expansion for R and found a relative gain of energy of about 5% for k = 1 and less than 1% for k ≥ 2. This was verified later numerically in [1]. Teukolsky and Press [14] made a similar mode analysis for higher spin and found numerically an energy gain of at most 4.4% for Maxwell (s = 1) and up to 138% for gravitational waves (s = 2). Our main result makes the above mode argument for the scalar wave equation mathematically precise in the setting of the Cauchy problem. To state our result, we combine and ∂t , as in [8], into a two-component vector = (, i∂t ). We always assume that the initial data is smooth and compactly supported outside the event horizon, 0 = (, i∂t )|t=0 ∈ C0∞ ((r1 , ∞) × S 2 )2 . The physical energy of the wave is then given by , where is the energy scalar product 2 2 2 ∞ 1 (r +a ) 2 2 = −a sin ϑ ∂t ∂t dr d(cos ϑ) r1 −1

1 a2 . (1.20) +∂r ∂r + sin2 ϑ∂cos ϑ ∂cos ϑ + ∂ ∂ − ϕ ϕ sin2 ϑ Provided that the limit t → ∞ exists, the energy radiated to infinity can be defined by E out = lim , t→∞

(1.21)

where χ is the characteristic function. Note that, according to the pointwise decay result in [9], we could replace the argument 2r1 by any radius greater than r1 . Moreover, again using pointwise local decay, the boundary term at r = 2r1 which arises when differentiating the characteristic function χ(2r1 ,∞) , vanishes in the limit t → ∞. We can now state our main theorem. Theorem 1.1. Consider the Cauchy problem for the scalar wave equation in the nonextreme Kerr geometry for initial data with compact support outside the event horizon, 0 = (, i∂t )|t=0 ∈ C0∞ ((r1 , ∞) × S 2 )2 , having energy . Fix k > 0, n ∈ N and ω ∈ (ω0 , 0). Then for any R > r1 and δ > 0 there is initial data 0 ∈ C0∞ ((R, ∞) × S 2 )2 such that the limit in (1.21) exists and E out ≤δ − R 0 0 with R as in (1.19). The same inequality holds for k < 0 and ω ∈ (0, ω0 ). We emphasize that we allow the initial data to be supported arbitrarily far away from the event horizon. This is important in order to avoid artificial initial data which would not correspond to an energy extraction mechanism. For example, if one allows the support of the initial data to intersect the ergosphere, one could take initial data close to a wave packet with zero total energy, in which case the quotient E out / could be made arbitrarily large. We also compute the energy E bh and the angular momentum Abh of the wave component crossing the event horizon of the black hole to obtain the following theorem.

834


Theorem 1.2. For any initial data 0 ∈ C0∞ ((r1 , ∞)× S 2 )2 , the quantities E bh and Abh defined by (2.8, 5.1) satisfy the inequality Abh ≤

r12 + a 2 E bh . a

This shows explicitly that Christodoulou’s estimates (1.3, 1.4) also hold for energy extraction by scalar waves, in agreement with Hawking’s area theorem [15]. 2. Absorption of Energy by the Black Hole Recall from [9] that given initial data 0 ∈ C0∞ (R × S 2 )2 , the solution of the Cauchy problem for the scalar wave equation (1.6) can be represented as (t, r, ϑ, ϕ) =

2 1 −ikϕ ∞ dω −iωt kωn a b e e tab kωn (r, ϑ), 2π ω −∞ n∈IN k∈Z

a,b=1

(2.1) where the sums and the integrals converge in L 2loc . In the above integral representation, the coefficients tab are defined by α α α t12 = t21 = −Im , t22 = 1 − Re , (2.2) t11 = 1 + Re , β β β and the complex coefficients α and β are defined by ´ φ` = α φ´ + β φ,

(2.3)

´ ` where φ(u) and φ(u) are the fundamental solutions of the radial equation having the asymptotics (1.14) and (1.15), respectively. We will refer to the complex coefficients α a (r, ϑ), a = 1, 2 are the and β as transmission coefficients. Finally the functions kωn solutions of the wave equation (1.6), with fixed quantum numbers k, ω, n, corresponding to the real-valued fundamental solutions of the radial equation given by ´ φ 1 = Re φ,

´ φ 2 = Im φ.

(2.4)

Here we shall always consider a fixed k-mode, and without loss of generality we assume that k > 0. For notational simplicity, from now on we omit the index k and the ϕ-dependence. Furthermore, we introduce the vector notation ωn 1 (r, ϑ) ωn ωn , 0 := , T = (tab )a,b=1,2 , (2.5) (r, ϑ) := 2ωn (r, ϑ) which allows us to write the integral representation (2.1) in the compact form 1 ∞ dω −iωt ωn e (r, ϑ), T ωn (t, r, ϑ) = 0 C2 . 2π n∈IN −∞ ω

(2.6)

We now introduce some basic quantities needed for the formulation of the superradiance property for wave packets. The total energy E tot of an initial data 0 ∈ C0∞ (R×S 2 )2 for the scalar wave equation (1.6) is defined by E tot = .

(2.7)


835

This energy is conserved throughout the time evolution of the scalar wave, meaning that if (t) denotes the solution of the Cauchy problem for the scalar wave equation, then = . Recall that the outgoing energy, which represents the energy radiated to infinity, has been defined in (1.21). Finally, the energy absorbed by the black hole is defined by E bh = lim . t→∞

(2.8)

We next derive useful expressions for these quantities in terms of the initial data and the transmission coefficients α and β. The expression for E tot is an immediate consequence of (2.6). Proposition 2.1. Choose a fixed k ∈ Z such that ω0 < 0. Then 1 ∞ dω ωn E tot = ωn 0 , T 0 C2 , 2π n∈IN −∞ ω where the series converges absolutely. We next want to compute E bh and E out . First, we need an argument which ensures that the sum over the angular momentum modes converges. Since E out = E tot − E bh and the convergence of the angular sum is already known for the total energy according to Proposition 2.1, it suffices to consider for example E out . In [10] it is shown that the outgoing energy is bounded uniformly in time, i.e. ≤ C()

for all t.

Moreover, it is also shown in [10] that the outgoing wave can be approximated by a finite number of angular momentum modes uniformly in time, i.e. for any δ > 0 there is N such that ≤ δ

for all t,

where 2 1 ∞ dω −iωt ωn ωn (t, r, ϑ, ϕ) = e tab a (r, ϑ) . 2π −∞ ω N

n≥N

a,b=1

This estimate will always allow us in what follows to interchange the limit t → ∞ with the sum over the angular momentum modes. To compute E bh , we make use of the following lemma, which generalizes [7, Lemma 9.1]. Lemma 2.2. For any Schwartz function f ∈ S(R × R) and any u 0 ∈ R, we define the four functions As,s , where s, s ∈ {−1, 1}, by ∞ ∞ u0 du dω dω e−i(ω−ω )t e−is u+is u f (ω, ω ). As,s = lim t→∞ −∞

Then

−∞

−∞

As,s = 2π δ1s δ1s

where δ1s denotes the Kronecker delta.

∞ −∞

f (ω, ω) dω,

836


Proof. Noting that − = ω − ω , the case s = s was proved in [7, Lemma 9.1]. Thus it remains to show that ∞ ∞ u0 du dω dω e−i(ω−ω )t e±i( + )u f (ω, ω ) = 0. lim t→∞ −∞

−∞

−∞

Noting

e−i(ω−ω )t e±i( + )u =

1 (∂ω +1)(∂ω +1)e−i(ω−ω )t±i( + )u , (−it ± iu+1)(it ± iu+1)

we can integrate the ω, ω -derivatives by parts. Applying Fubini’s theorem and estimating the resulting u-integral for t > 2|u 0 |, we find u 0 u0 − 1 − 1 1 2 2 ±i( + )u 2 2 = e (t+u) (t − u) du +1 +1 du −∞ (t ∓ u+i)(t ± u−i) −∞ u0 − 3 √ 1 4 (t+u)2 +1 ≤ 6 t− 2 du. −∞

Since the last integral is bounded uniformly in t, the entire expression tends to zero as t → ∞.

Proposition 2.3. Choose a fixed integer k > 0. Then the limit in (2.8) converges and 1 ∞ dω ωn E bh = ωn 0 , TPT 0 C2 , 2π n∈IN −∞ ω where P is the matrix

1 1 i . P = 2 −i 1

The above series is absolutely convergent. Proof. Substituting the integral representation (2.1) into (2.8), we find 2r1 ∞ ∞ 1 dω dω −i(ω−ω )t ω ,ω E bh = lim e dr E (r), 2 t→∞ (2π ) −∞ ω −∞ ω r1

(2.9)

where E

ω ,ω

(r) =

2

n,n ∈N a,b,c,d=1

ω n ωn tab tcd

S2

E(bω n , cωn ),

and E(bω n , cωn ) denotes the bilinear form corresponding to the energy density as given in (1.20). We next justify that we may interchange the limit t → ∞ with the sums over the angular momentum modes and that the series converge absolutely. According to [10, Theorem 8.1], the energy of the large angular momentum modes near infinity is small uniformly in time. Due to conservation of energy and the local decay [9], a similar result follows for the energy near the event horizon. Hence the integrals in (2.9) can be


837

approximated uniformly in time by a finite number of angular momentum modes. Thus in what follows we may restrict attention to fixed angular momentum modes n and n . Near the event horizon, we can expand the coefficient functions in the energy den sity E(bω n , cωn ) to obtain E(bω n , cωn ) r2 + a 2 2 2 γu ω n ∂ ωn − a k + ∂ ωb n ωn = (r12 + a 2 ) ω ω ωb n ωn u u c c c + Oω,ω (e ) b 2 2 r1 + a

a2k2 n ωn n n ωn ωn ω ω ω = n ω nω ω ω φb φc + ∂u φb ∂u φc − 2 2 2 φb φc + Oω,ω (eγ u ) (r1 +a ) with the fixed constant γ > 0 as in [9, Eq. (3.9)]. Here the error term can be bounded by |Oω,ω (eγ u )| ≤ C eγ u |φbω n | + |ω φbω n | + |∂u φbω n | |φaωn | + |ωφaωn | + |∂u φaωn | (2.10) independent of ω and ω . Using the Jost estimates of [9, Theorem 3.1],

with a constant C the fundamental solutions have the following expansion near the event horizon,

φ1ωn (u) = cos( u) + O˜ ω,ω (eγ u ), φ2ωn (u) = sin( u) + O˜ ω,ω (eγ u ). Here the error term is bounded by |O˜ ω,ω (eγ u )| ≤ C eγ u ,

|∂u O˜ ω,ω (eγ u )| ≤ C (1 + |ω|)eγ u ,

where the constant C is again uniform in ω (see [9, Eqs. (3.7, 3.10)]). This estimate also shows that the error term (2.10) is bounded by |Oω,ω (eγ u )| ≤ C eγ u (1 + |ω |)(1 + |ω|).

(2.11)

We thus conclude that r2

E(bω n , cωn ) = Oω,ω (eγ u ) 2 +a

+ n ω nω ω ω

ψbω

ψcω

+ ∂u ψbω ∂u ψcω

a2k2 ω ω − 2 ψ ψc , (r1 + a 2 )2 b

(2.12)

where Oω,ω (eγ u ) satisfies (2.11) (possibly with a new constant C) and ψ1ω (u) = cos( u), ψ2ω (u) = sin( u). Using the asymptotic formula (2.12) in (2.9) and writing the radial integral in terms of the Regge-Wheeler variable u, the factor /(r2 + a 2 ) drops out. Applying [10, Lemma 3.1], we can write E bh as u0 ∞ ∞ E bh = lim du dω dω e−i(ω−ω )t t→∞ −∞ −∞ −∞ ⎛ ⎞ 2 ×⎝ f b,c (ω, ω ) ψbω (u) ψcω (u) + g(ω, ω ) Oω,ω (eγ u )⎠, b,c=1

838


with Schwartz functions f b,c and g. To treat the error term, we first apply Fubini’s theorem to exchange the orders of integration. This gives u0 ∞ ∞ lim dω dω e−i(ω−ω )t g(ω, ω ) Oω,ω (eγ u ) du, t→∞ −∞

−∞

−∞

and using (2.11), we can apply the Riemann-Lebesgue lemma to conclude that this limit is zero. The term involving the f b,c can be treated by applying Lemma 2.2. Using the orthogonality of the spheroidal wave functions, we obtain

∞ 2k2 dω ωn TPT ωn 1 a 0 C2 ω2 + 2 − 2 , E bh = . 2π −∞ ω2 2 n∈IN 0 2 (r1 + a 2 )2 We finally apply the identity ω2 + 2 −

a2k2 = ω2 + 2 − ω02 = ω2 + ω2 − ωω0 = 2ω . (r12 + a 2 )2

Using the matrix identity T − TPT = TQT with Q := T−1 − P, a short calculation yields the following result. Proposition 2.4. Choose a fixed k ∈ Z such that ω0 < 0. Then the limit t → ∞ in (1.21) exists and 1 ∞ dω ωn E out = , TQT ωn 0 C2 , 2π n∈IN −∞ ω 0 where Q is the matrix

|α − β|2 i(α + β) (α − β) . Q = |α + β|2 2ω −i(α − β) (α + β)

3. Energy Propagation of Wave Packets near Infinity We fix ω˜ ∈ (ω0 , 0) and n˜ ∈ N and consider initial data 0 in the form of a linear combination of outgoing and incoming wave packets, η L (u) 1 1 −iωu ˜ iωu ˜ 0 = cin e + cout e , = n, ˜ ω˜ (ϑ) √ ω ˜ − ω˜ i∂t t=0 2 2 r +a (3.1) where L > 0,

u − L2 1 η L (u) = √ η , L L

with η ∈ C0∞ ((−1, 1)) being a smooth cut-off function, and where n, ˜ ω˜ (ϑ) is an eigenfunction of the angular operator A. In the next proposition we compute E tot asymptotically as L → ∞.


Proposition 3.1. For the initial data given by the wave packet (3.1), ∞ dω ` ωn ` ωn 1 lim E tot (0 ) = lim 0 , 0 C2 , L→∞ 4π n∈IN L→∞ −∞ ω2 where ` ωn 0

:=

` ωn , 0 > . ω ω

(3.9)

For clarity we point out that the L-dependence of g L comes from 0 (see (3.1)), as well 2 as from the phase factor eiωL . The corresponding contribution to E tot is obtained by integrating over ω, and thus the remaining task is to show that ∞ 2 e−2iωL g L (ω) dω = 0. lim L→∞ −∞

This is a direct consequence of the following two lemmas.

Lemma 3.2. Let f L ∈ L 1 (R), L > 0, be a family of functions with which converges in L 1 (R) as L → ∞. Then ∞ 2 lim e−2iωL f L (ω) dω = 0. L→∞ −∞


841

Proof. Let f = lim L→∞ f L in L 1 . Using the inequality ∞ ∞ −2iωL 2 −2iωL 2 + ≤ e f (ω) dω e f (ω) dω L −∞

−∞

∞

−∞

| f − f L |(ω) dω,

the first term tends to zero by the Riemann-Lebesgue lemma, while the second term tends to zero by hypothesis.

Lemma 3.3. For any n, n˜ and ω, ˜ the functions g L converge in L 1 (R) as L → ∞. Proof. In view of Lebesgue’s dominated convergence theorem, it suffices to show that (a) The pointwise limit lim L→∞ g L (ω) exists for all ω = 0, and (b) There is a constant C such that for all ω and L, |g L (ω)| ≤

C . ω2 + 1

To show (a), note that in the limit L → ∞, the support of 0 moves to infinity. Thus using the plane wave asymptotics of the Jost solutions ψ ωn together with the error estimates in the proof of Lemma 3.3 in [9], a straightforward calculation shows that both brackets in (3.9) converge for any fixed ω = 0. This proves (a). To prove (b), we first analyze the behavior near ω = 0. First note that the factors ω−1 in the brackets in (3.9) are compensated because the energy scalar product involves a factor of ω (see [8, Eq. (2.14)]). Thus the estimates for the Jost functions in Lemma 3.6 in [9] yield that the two brackets in (3.9) are bounded for small |ω|, uniformly in L. Next, the convexity argument in [9, Sect. 5] yields that the factor α/β is bounded near ω = 0. On any compact set which does not contain ω = 0, we know from [9, Sect. 7] that the factor α/β is a continuous function. Since the Jost solutions form holomorphic families and the estimates of [9, Lemma 3.3] hold, it follows that the integrand in (3.9) is continuous and thus bounded on any compact set, uniformly in L. To control the behavior for large |ω|, we first see from (3.5) that |α/β| < 1. Thus to finish the proof of (b) it suffices to show that the energy scalar products in (3.9) are bounded, uniformly in L. Indeed, we shall show that they have rapid decay in ω, uniformly in L, i.e. for any p ∈ N there is a constant c p such that cp ` ωn > + ≤ ∀ L , ∀ ω with |ω| > 1. (3.10) , (3.11) = 2l ≤ C |ω|l+1 , = + O(L −1 ). ` ωn , 0 > + O(L −1 ), φˆ R (ω) = S 2 ≤ ε for all ω ∈ Iε .

Now we can use the same argument as in the case n = n˜ to conclude the proof of (4.2).

5. Absorption of Angular Momentum by the Black Hole We first derive the expression for the angular momentum of the scalar wave. We recall from [8] that the Lagrangian is given by 2 1 2 ((r + a 2 )∂t + a∂ϕ ) 2 1 2 sin − sin2 ϑ |∂cos ϑ ϕ|2 − ϑ∂ + ∂ ) . (a t ϕ 2 sin ϑ

L = −|∂r |2 +

This Lagrangian is axisymmetric. Applying Noether’s theorem gives rise to the following conserved quantity: A[] =

∞

r1

dr

1 −1

2π

d(cos ϑ) 0

dϕ A, 2π

where A is given by ∂L A = Re ϕ ∂t

2 a ∂ϕ ∂ϕ (r + a 2 )2 2 2 . = Re − a sin ϑ ∂ϕ ∂t + ∂ϕ ∂ t + 2 r + a2 a sin2 ϑ The quantity A can be interpreted as the angular momentum of the wave , and A as the angular momentum density. Similar to (2.8), the angular momentum absorbed by the black hole is defined by Abh = lim

2r1

t→∞ r 1

dr

1

−1

2π

d(cos ϑ) 0

dϕ A[], 2π

(5.1)

provided that the limit exists. In the next proposition we compute Abh , again for a fixed k-mode. Proposition 5.1. Choose a fixed k ∈ Z such that ω0 < 0. Then the limit in (5.1) exists and 1 ∞ dω k ωn , TPT ωn Abh = 0 C2 , 2π n∈IN −∞ ω 0 where the sum converges absolutely.

846


Proof. As in the proof of Proposition 3.5 one sees that we may interchange the limit t → ∞ with the sum over the angular momentum modes, and that the resulting series converges absolutely. Thus it remains to consider a fixed angular momentum mode n. Substituting the integral representation (2.1) into the definition of Abh , we obtain 1 t→∞ (2π )2

Abh = lim

2r1

r1

dr

∞ −∞

dω ω

∞ −∞

dω −i(ω−ω )t ω ,ω A (r), e ω

where

Aω ,ω (r) =

2

ω n ωn tab tcd

n,n ∈N a,b,c,d=1

S2

A(bω n , cωn ),

and A(bω n , cωn ) denotes the bilinear form corresponding to the angular momentum density (similar to the bilinear form E(bω n , cωn ) appearing in the proof of Proposi tion 2.3). Near the event horizon, we can expand A(bω n , cωn ) to obtain 1 γu A(bω n , cωn ) = (r12 + a 2 ) k ( + ) ωb n ωn c + O(e ) r2 + a 2 2 1 = n ω nω k ( + ) φbω n φcωn + O(eγ u ). 2 Now we can proceed exactly as in the proof of Proposition 2.3.

Proof of Theorem 1.2. Without loss of generality we again restrict attention to the case k > 0, so that ω0 < 0. We set ρ(ω, n) =

1 ωn ωn 0 , TPT 0 C2 . 2π

The eigenvalues of the matrix TPT are computed to be zero and 1 + |α|2 /|β|2 . Thus ρ is non-negative. It follows that Abh

∞ k k = ρ dω ≤ ρ dω ω 2 ω 2 n∈N −∞ n∈N ω0 k ∞ 1 k k ∞ 1 ρ dω ≤ ρ dω = E bh , ≤ |ω0 | ω |ω | ω |ω 0 0| ω0 −∞ ∞

n∈N

and using (1.13) completes the proof.

n∈N

Acknowledgements. We would like to thank the Alexander-von-Humboldt Foundation and the Vielberth Foundation, Regensburg, for generous support. We also thank the referee his careful reading and helpful suggestions.


847

References 1. Andersson, N., Laguna, P., Papadopoulos, P.: Dynamics of scalar fields in the background of rotating black holes II: a note on superradiance. Phys. Rev. D58, 087503 (1998) 2. Chandrasekhar, S.: The Mathematical Theory of Black Holes. Oxford: Oxford University Press, 1983 3. Christodoulou, D.: Reversible and irreversible transformations in black hole physics. Phys. Rev. Lett. 25, 1596–1597 (1970) 4. Deruelle, N., Ruffini, R.: Quantum and classical relativistic energy states in stationary geometries. Phys. Lett. 52B, 437 (1974) 5. Deruelle, N., Ruffini, R.: Klein Paradox in a Kerr Geometry. Phys. Lett. 57B, 248 (1975) 6. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: The long-time dynamics of Dirac particles in the KerrNewman black hole geometry. Adv. Theor. Math. Phys. 7, 25–52 (2003) 7. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Decay rates and probability estimates for masssive Dirac particles in the Kerr-Newman black hole geometry. Commun. Math. Phys. 230, 201–244 (2002) 8. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: An integral spectral representation of the propagator for the wave equation in the Kerr geometry. Commun. Math. Phys. 260, 257–298 (2005) 9. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys. 264, 465–503 (2006), Erratum, Commun. Math. Phys. 280, 563–573 (2008) 10. Finster, F., Smoller, J.: A time independent energy estimate for outgoing scalar waves in the Kerr geometry. J. Hyperbolic Differ. Eq. 5, 221–255 (2008) 11. Frolov, V.P., Novikov, I.D.: Black Hole Physics. Basic Concepts and New Developments. Dordrecht: Kluwer Academic Publishers Group, 1998 12. Penrose, R.: Gravitational collapse: The role of general relativity. Rev. del Nuovo Cimento 1, 252–276 (1969) 13. Starobinsky, A.A.: Amplification of waves during reflection from a black hole. Sov. Phys. JETP 37, 28–32 (1973) 14. Teukolsky, S., Press, W.H.: Perturbations of a rotating black hole. III. Interaction of the hole with gravitational and electromagnetic radiation. Astrophys. J. 193, 443–461 (1974) 15. Wald, R.: General Relativity. Chicago, IL: University of Chicago Press, 1984 16. Zel’dovich, Ya.B.: Amplification of cylindrical electromagnetic waves from a rotating body. Sov. Phys. JETP 35, 1085–1087 (1972) Communicated by G. W. Gibbons


Communications in


Deformation of Curved BPS Domain Walls and Supersymmetric Flows on 2 d Kähler-Ricci Soliton Bobby E. Gunara, Freddy P. Zen Indonesia Center for Theoretical and Mathematical Physics (ICTMP), and Theoretical Physics Laboratory, Theoretical High Energy Physics and Instrumentation Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia. E-mail: [email protected]; [email protected] Received: 6 January 2008 / Accepted: 27 November 2008 Published online: 14 February 2009 – © Springer-Verlag 2009

Abstract: We consider some aspects of the curved BPS domain walls and their supersymmetric Lorentz invariant vacua of the four dimensional N = 1 supergravity coupled to a chiral multiplet. In particular, the scalar manifold can be viewed as a two dimensional Kähler-Ricci soliton generating a one-parameter family of Kähler manifolds evolved with respect to a real parameter, τ . This implies that all quantities describing the walls and their vacua indeed evolve with respect to τ . Then, the analysis on the eigenvalues of the first order expansion of BPS equations shows that in general the vacua related to the field theory on a curved background do not always exist. In order to verify their existence in the ultraviolet or infrared regions one has to perform the renormalization group analysis. Finally, we discuss in detail a simple model with a linear superpotential and the Kähler-Ricci soliton considered as the Rosenau solution. 1. Introduction Attempts to generalize the study of AdS/CFT correspondence [1] on curved spacetimes have been done, for example in the context of curved domain walls of five dimensional N = 2 supergravity [2,3]. In those papers the authors have constructed curved BPS domain walls and discussed their dual description in terms of the renormalization group (RG) flow described by a beta function. By putting the supersymmetric field theory studied in [4] on a curved four dimensional AdS background they have also demonstrated in a simple model that this holographic RG flow in the field theory on the curved spacetime has indeed a description in terms of curved BPS domain walls. So far, in the four dimensional N = 1 supergravity theory we only have the flat domain wall cases studied in some references, for example [5–9]. Therefore, the above results motivate us to apply the scenario to the case of the curved BPS domain walls of N = 1 supergravity in four dimensions. Our interest here is to study curved BPS domain walls together with their Lorentz invariant vacua in the context of the dynamical system and the RG flow analysis which is a generalization of the previous works [8,9].

850

B. E. Gunara, F. P. Zen

Particularly, we want to see how the general pattern looks in a simple model, namely the N = 1 supergravity coupled to a chiral multiplet whose scalar manifold can be regarded as a solution of the Kähler-Ricci flow equation [10,11].1 This geometric soliton generates a one-parameter family of scalar manifolds, i.e. Kähler manifolds, whose deformation parameter is τ ∈ R. In particular, this Kähler-Ricci soliton can be viewed as a volume deformation of a Kähler geometry for finite τ . 2 Thus, defining N = 1 supersymmetry on the Kähler-Ricci flow means that we deform it with respect to τ . As a direct implication of such treatment, all couplings such as the shifting quantities, the masses of the fields, and the scalar potential evolve with respect to τ since those quantities depend on this geometric soliton. Such behavior is generally inherited to all solitonic solutions such as the domain walls. So, the Lorentz invariant vacua do also possess such a property which can shortly be mentioned as follows. First, near the vacua the spacetime is in general non-Einsteinian, and then becomes a space of constant curvature (which is also non-Einsteinian) related to the divergences of the RG flow. Second, the Kähler-Ricci soliton indeed affects the nature of the vacua, mapping nondegenerate vacua to other degenerate vacua and vice versa. Moreover, in a model that admits a singular geometric evolution, the vacuum structure may have a parity pair of vacua in the sense that the vacua of the index λ turns into the other vacua of the index 2 − λ after hitting the singularity. This is an the example that also occurs in the general pattern for flat domain walls [8]. Finally, in order to have a consistent picture, the eigenvalues of the first order expansion of the BPS equation have to be real. This also shows that the above vacua do not always exist in general. By performing the RG flow analysis we can further verify the existence of such vacua in the infrared or ultraviolet regions, correspond to the field theory on the three dimensional curved background. The structure of this paper is as follows. In Sec. 2 we review the N = 1 supergravity on a two dimensional Kähler-Ricci soliton and introduce some quantities which are useful for our analysis. Then, the discussion is continued in Sec. 3 by addressing some aspects of the curved BPS domain walls on the two dimensional Kähler-Ricci solitons. Section 4 is assigned to the discussion of the nature of the supersymmetric Lorentz invariant vacua together with their deformation on the Kähler-Ricci soliton. We put the discussion about the deformation of the supersymmetric flow on curved spacetime in Section 5. A simple example is then given in Sec. 6. Finally, we summarize our results in Sec. 7.

2. 4d N = 1 Chiral Supergravity on 2d Kähler-Ricci Soliton In this section we provide a review of the four dimensional N = 1 supergravity coupled to a chiral multiplet in which the non-linear σ -model satisfies the Kähler-Ricci flow equation defined below, which in turn implies that our N = 1 supergravity is defined on a one-parameter family of Kähler manifolds deformed with respect to the real parameter τ [8]. The ingredients of the N = 1 theory are a gravitational multiplet and a chiral multiplet. The gravitational multiplet is composed of a vierbein eνa and a vector spinor 1 This Kähler-Ricci flow also appears as one loop approximation of the beta function of N = 2 supersymmetry in two and three dimensions, see for example [12,13]. In this case, the parameter τ is regarded as the energy scale of the theory. However, this is not the case in higher dimension, particularly in four dimensions. 2 This can be easily seen if the initial geometry is a Kähler-Einstein manifold. See Appendix C for details.

Deformation of Curved BPS Domain Walls and Supersymmetric Flows

851

ψν , where a = 0, . . . , 3 and ν = 0, . . . , 3 are the flat and the curved indices, respectively. The member of the chiral multiplet is a complex scalar z and a spin- 21 fermion χ. We then construct a general N = 1 chiral supergravity Lagrangian together with its supersymmetry transformation. This construction can be found, for example, in [14] 3 . Let us assemble the terms which are useful for our analysis. The bosonic part of the N = 1 supergravity Lagrangian has the form L N =1 = −

M P2 R + gz z¯ (z, z¯ ; τ ) ∂ν z ∂ ν z¯ − V (z, z¯ ; τ ) , 2

(2.1)

where M P is the Planck Mass, and by setting M P → +∞, the N = 1 global supersymmetric theory can be obtained. Next, R is the Ricci scalar of the four dimensional spacetime; the pair (z, z¯ ) spans a Hodge-Kähler manifold with metric gz z¯ (z, z¯ ; τ ) ≡ ∂z ∂z¯ K (z, z¯ ; τ ); and K (z, z¯ ; τ ) is a real function, called the Kähler potential. The scalar manifold satisfies the Kähler-Ricci flow equation4 ∂gz z¯ = −2Rz z¯ (τ ) = −2 ∂z ∂¯ z¯ lngz z¯ (τ ) , ∂τ

(2.2)

where τ is a real parameter related to the deformation of a Kähler surface mentioned in the previous section. The N = 1 scalar potential V (z, z¯ ; τ ) has the form 3 K (τ )/M P2 z z¯ ¯ ¯ ¯ V (z, z¯ ; τ ) = e (2.3) g (τ )∇z W ∇z¯ W − 2 W W , MP where W is a holomorphic superpotential and ∇z W ≡ (dW/dz) + (K z (τ )/M P2 )W . The Lagrangian (2.1) is invariant under the following supersymmetry transformations up to three-fermion terms 5 i i 2 δψ1ν = M P Dν 1 + e K (τ )/2M P W γν 1 + Q ν (τ ) 1 , 2 2M P δχ z = i∂ν z γ ν 1 + N z (τ ) 1 , = − Mi P (ψ¯ 1ν γ a 1 + ψ¯ ν1 γ a 1 ) ,

(2.4)

δeνa

δz = χ¯ z 1 , where N z (τ ) ≡ e K (τ )/2M P g z z¯ (τ )∇¯ z¯ W¯ , g z z¯ (τ ) = (gz z¯ (τ ))−1 , and the U (1) connection Q ν (τ ) ≡ − (K z (τ ) ∂ν z − K z¯ (τ ) ∂ν z¯ ). Here, we have also defined 1 ≡ 1 (x, τ ). The flow Eq. (2.2) implies that, for example, the scalar potential (2.3) and the shifting quantity N z do evolve with respect to τ as [8] 2

∂ N z (τ ) K τ (τ ) z K z¯ τ (τ ) K (τ )/2M 2 ¯ P W , = 2R zz (τ )N z (τ ) + N (τ ) + g z z¯ (τ ) e ∂τ 2M P2 M P2 (2.5) ∂ N z (τ ) ∂ V (τ ) ∂ Nz (τ ) z 3K τ (τ ) K (τ )/M 2 2 P = Nz (τ ) + N (τ ) − e |W | , ∂τ ∂τ ∂τ M P2 3 For an excellent review of N = 1 supergravity, see also for example [15,16]. 4 Classification of the special solutions of (2.2) using linearization method has been studied, for example,

in [17]. 5 The symbol D here is different from the one in reference [9]. D here is defined as D ≡ ∂ − 1 γ ωab . ν ν ν ν 4 ab ν

852


where R zz ≡ g z z¯ Rz z¯ . Note that as has been studied in [18,19] it is possible that such geometric flow (2.2) has a singular point at finite τ . For example, this situation can be directly observed in a model with Kähler-Einstein manifold as initial geometry. 6 In this paper we particularly consider the Rosenau solution of (2.2) which also admits such a property in Sec. 6. This special solution was first constructed in [20]. 3. Curved BPS Domain Walls on 2d Kähler-Ricci Soliton This section is assigned for the discussion of curved domain walls admitting partial Lorentz invariance. In particular, we consider curved BPS domain walls that maintain half of the supersymmetry of the parental theory. As a consequence, the background should be a three dimensional AdS spacetime. Let us now consider the ground states which partially break the Lorentz invariance, i.e. the domain walls. The starting point is to take the ansatz metric of the four dimensional spacetime as ds 2 = a 2 (u, τ ) gλ ν d x λ d x ν − du 2 ,

(3.1)

where λ, ν = 0, 1, 2, a(u, τ ) is the warped factor, and the parameter τ is related to the dynamics of the Kähler metric governed by (2.2). The metric gλ ν describes a three dimensional AdS spacetime. Therefore, the corresponding components of the Ricci tensor of the metric (3.1) are given by 2 3 a 2 gλ ν , Rλ ν = aa + 3 aa − a2 2 , (3.2) R33 = −3 aa + aa and the Ricci scalar has the form 2

a a 3 3 R=6 +2 − 2 , a a a

(3.3)

where a ≡ ∂a/∂u and 3 is the negative three dimensional cosmological constant. Writing the supersymmetry transformation (2.4) and setting ψ1ν = χ z = 0 on the background (3.1), leads to 1 i i 2 δψ1u = Du 1 + e K (τ )/2M P W γu 1 + Q u (τ ) 1 , MP 2 2M P 1 1 a K (τ )/2M P2 W 1 + i Q (τ ) , 1 M P δψ1ν = Dν 1 + 2 γν − a γ3 1 + ie 2M P ν

(3.4)

δχ z = i∂ν z γ ν 1 + N z (τ ) 1 . Supersymmetry further demands that the right-hand side of Eq. (3.4) vanish on the ground states. Then, on the three dimensional AdS spacetime there exists a Killing spinor [21] i Dν 1 + γˆν 1 = 0 , 2 6 See also Appendix C.

(3.5)


853

√ where ≡ − 3 /2. Here, γˆν means the gamma matrices in the three dimensional AdS spacetime, and therefore γν = a γˆν . Thus by taking z = z(u, τ ) the first equation in (3.4) shows that 1 depends on the spacetime coordinates, while the second equation gives a projection equation a

1 K (τ )/2M P2 γ3 1 = i e , W (z) − (3.6) a a which leads to 7

K (τ )/2M 2 a

P = ± e W (z) − . a a

(3.7)

So, the warped factor a is indeed τ dependent which is consistent with our ansatz (3.1). Next, since z = z(u, τ ) the third equation in (3.4) becomes simply z = ∓ 2eiθ(τ ) g z z¯ (τ ) ∂¯ z¯ W(τ ) ,

z¯ = ∓ 2e−iθ(τ ) g z z¯ (τ ) ∂z W(τ ) ,

where we have introduced the phase function θ (z, z¯ ; u, τ ) via 2 1 − e−K (τ )/2M P (aW )−1 , eiθ(τ ) = 2 1 − e−K (τ )/2M P (aW )−1

(3.8)

(3.9)

and the real function W(z, z¯ ; τ ) ≡ e K (τ )/2M P |W (z)| . 2

(3.10)

Note that at θ = 0 the flat domain wall case is regained, which corresponds to = 0. Thus, we have the gradient flow equations (3.8), called the BPS equations in a curved spacetime. Another supersymmetric flow related to the analysis is the renormalization group (RG) flow given by the beta functions β(τ ) ≡ a

∂z 2eiθ(τ ) g z z¯ (τ ) ∂¯ z¯ W(τ ) , = − K (τ )/2M P2 W (z) − /a ∂a e

¯ )≡a β(τ

∂ z¯ 2e−iθ(τ ) g z z¯ (τ ) ∂z W(τ ) , = − K (τ )/2M P2 ∂a W (z) − /a e

(3.11)

after using (3.7) and (3.8). These functions give a description of a conformal field theory (CFT) on the three dimensional AdS spacetime. Therefore, the scalars behave as coupling constants and the warped factor a can be viewed as an energy scale [4,7,22], and the scalar potential (2.3) can be written down as V (z, z¯ ; τ ) = 4 g z z¯ (τ ) ∂z W(τ ) ∂¯ z¯ W(τ ) −

3 W 2 (τ ) . M P2

(3.12)

To see the deformation of the domain walls clearly, we have to consider a case where the Kähler-Ricci flow has a singularity at finite τ = τ0 < ∞. Such a property may cause 7 This warped factor a is related to the c-function in the holographic correspondence [4,7].

854


a topological change of the scalar manifold. The simplest example of this case is when the initial manifold is a Kähler-Einstein manifold which has been studied in [8]. Here, our interest is only to look at another non-trivial solution of (2.2) where ⎧ ⎨ g1 (z, z¯ ; τ ) ; τ < τ0 , ; τ = τ0 , gz z¯ (τ ) = 0 (3.13) ⎩ −g (z, z¯ ; τ ) ; τ > τ , 2 0 with g1 (z, z¯ ; τ ) and g2 (z, z¯ ; τ ) are both positive definite functions. For the case at hand the N = 1 theory (2.1) and the walls described by (3.8) and (3.12) diverge at τ = τ0 . This singularity disconnects two different theories, namely for τ < τ0 we have N = 1 theory on a positive definite metric, while for τ > τ0 it becomes N = 1 theory on a negative definite metric. As we will see in Sec. 6 the Rosenau solution also has a similar property and produces a singularity at τ0 = 0. Concluding this section, we will look at the gradient flow Eq. (3.8). The critical points of Eq. (3.8) satisfy the following conditions ∂z W = ∂¯ z¯ W = 0,

(3.14)

which imply that the first derivative of the scalar potential (3.12) vanishes. In other words, there is a correspondence between the critical points of W(τ ) and the vacua of the N = 1 scalar potential V (τ ). 8 Moreover, the Kähler-Ricci flow, specifically the metric (3.13), causes an evolution of these critical points (vacua) which are characterized by the beta functions (3.12) splitting into two regions. If a → +∞ we have an ultraviolet (UV) region, while if a → 0 we have an infrared (IR) region . Several aspects of the vacuum and the supersymmetric flow deformation will be discussed in detail in Sec. 4 and Sec. 5 respectively. 4. Properties of the Supersymmetric Vacua Our attention in this section will be mainly drawn to the discussion of the supersymmetric vacua of the theory described by the scalar potential (3.12). As mentioned in the previous section these vacua are related to the critical points of the real function W(τ ) and changing with respect to τ . We firstly show that in general such critical points correspond to a four dimensional non-Einsteinian spacetime. Then, a second order analysis of the vacua related to the critical points of W(τ ) will be carried out. Note that the discussion here is incomplete since it does not involve a supersymmetric flow analysis which is provided in Sec. 5. A short review about critical points of surfaces is given in Appendix B which maybe useful for the analysis. Let us begin our discussion by mentioning that from (3.14) a critical point of the real function W(τ ), say p0 , is in general p0 ≡ (z 0 (τ ), z¯ 0 (τ )) due to the geometric flow (2.2). Such a point exists in the asymptotic regions, namely around u → ±∞. The form of the scalar potential (3.12) at p0 is V ( p0 ; τ ) = −

3 3 W 2 ( p0 ; τ ) ≡ − 2 W02 . 2 MP MP

(4.1)

For the case of the flat domain walls discussed in [9] Eq. (4.1) can be viewed as the cosmological constant of the spacetime at the vacuum. However, in general this is not 8 The vacuum of the scalar potential (2.3) means a Lorentz invariant vacuum (ground state). In the following we will mention the Lorentz invariant vacuum just as a vacuum or ground state.


855

the case. Therefore, we have to consider the behavior of the warped factor a near the vacua, which is related to the shape of the spacetime. Around p0 , the solution of Eq. (3.7) tends to 9 1/2 l

2 l2 −1 ∓W0 u ±W0 u A , (4.2) a(u, τ ) = 2 ± − e − A e 0 0 W0 W02 W04 where l ≡ e K ( p0 ;τ )/2M P ReW (z 0 ) and A0 = 0. Since a is real, then W0 > |l|/ . Moreover, in this case we have a /a = 0 near p0 . So, defining K ( p ;τ )/2M 2 a

0 P (4.3) = ± e W (z 0 ) − ≡ ± k , a a 2

with k ≡ k(u, τ ) ≥ 0, the Ricci tensor (3.2) and the Ricci scalar (3.3) become Rλ ν = ±k + 3k 2 + 2 2 e∓2 k du e±2 k du gλ ν , R33 = −3 ±k + k 2 , R = ±6k + 12k 2 + 6 2 e∓2

k du

,

(4.4)

respectively, which confirms that in general the spacetime is non-Einsteinian. 10 Let us consider some special cases as follows. For the = 0 case, we have a four dimensional AdS spacetime for k = 0 with k = 0 and the cosmological constant given by (4.1) appeared in the flat domain walls. Next, it is possible to have a case where k = 0. Here, the spacetime is a four dimensional non-Einsteinian space of constant curvature, where e K ( p0 ;τ )/2M P W (z 0 ) = 2

, a

(4.5)

or in other words, ImW (z 0 ) = 0 .

(4.6)

These facts tell us that the first order expansion of the beta function (3.12) would be ill defined. 11 Hence, this vacuum does not correspond to the CFT on a three dimensional AdS spacetime. Another singularity could occur at τ = τ0 which is caused by the divergence of the geometric flow (3.13). The detail of this aspect depends on the model in which both the form of the geometric flow and the superpotential are involved. Also, in this case some quantities would diverge. We give a simple model in Sec. 6. The last case is a static case in which W (z 0 ) = 0 and the U (1) connection vanishes. 12 This means that p does not depend on τ , but rather is determined by the holomorphic 0 superpotential W (z). In other words, p0 is a critical point of W (z). However, as we will see in the following, although p0 static, the second order analysis does depend on τ 9 In the limit of flat walls we have → 0 and A → ±∞. 0 10 It is important to notice that the analysis using supersymmetric flows, namely the gradient and the RG

flows, shows that such a spacetime does not always correspond to a CFT in three dimensional AdS, see the discussion in sec. 5. 11 See Eq. (5.9) and (5.10) in the next section for a detail. 12 Note that in the W (z ) = 0 case the warped factor a(u, τ ) in (4.2) becomes singular. 0

856


because the geometric flow described by the Kähler potential K (z, z¯ ; τ ) is involved in the analysis. An example of this situation is discussed in Sec. 6. The second part of the discussion is to study the properties of the critical points of the real function W(τ ) which is τ dependent in the second order analysis. Here, the eigenvalues of its Hessian matrix are also τ dependent and have the form 13 λW 1,2 (τ ) = where ∂z2 W0

2 e K ( p0 ;τ )/M P W¯ (¯z 0 ) ≡ 2W( p0 ; τ )

gz z¯ ( p0 ; τ ) W0 ± 2|∂z2 W0 | , M P2

(4.7)

d2W K zz ( p0 ; τ ) K z ( p0 ; τ ) dW (z 0 ) . (z 0 ) + W (z 0 ) + dz 2 dz M P2 M P2 (4.8)

Since the metric gz z¯ (τ ) satisfies (3.13), we split the discussion into two parts. First, in the interval τ < τ0 the metric is positive definite, namely gz z¯ ( p0 ; τ ) = g1 ( p0 ; τ ), and the possible cases for p0 are a local minimum if |∂z2 W0 |

1 g1 ( p0 ; τ )W0 . 2M P2

(4.10)

or a saddle if

Furthermore, p0 turns out to be degenerate when |∂z2 W0 | =

1 g1 ( p0 ; τ )W0 2M P2

(4.11)

holds. Second, for τ > τ0 we have a negative definite metric and gz z¯ ( p0 ; τ ) = −g2 ( p0 ; τ ). In this region the conditions (4.9) and (4.10) are modified by replacing g1 ( p0 ; τ ) with g2 ( p0 ; τ ) which further state that p0 is either a local maximum for |∂z2 W0 |

1 g2 ( p0 ; τ )W0 . 2M P2

(4.13)

or a saddle for

For the degenerate case, we use the same procedure on (4.11). Since the point p0 is dynamic with respect to τ , we can then summarize the above results as follows. A nondegenerate critical point can be changed into a degenerate critical point and vice versa by the special geometric flow (3.13). Moreover, this flow also affects the index of the critical points, namely the critical points of index λ turn to other critical points of index 2−λ after passing the singularity at τ = τ0 . In other words, this is a parity transformation 13 Note that since we have curved BPS domain walls, this eigenvalue has a restriction coming from the eigenvalues of the first order expansion of the gradient flow (3.8). Again, see sec. 5 for a detail.


857

of the Hessian matrix of W that maps a critical point in τ < τ0 to its parity partner in τ > τ0 , such as a local minimum to a local maximum and vice versa [8]. In the following we finally perform general analysis on the supersymmetric vacua of the scalar potential (3.12) and their relation to the critical points of W(τ ). As has been mentioned in the preceding section, the critical point p0 of W(τ ) defines a vacuum of the theory. At p0 the τ dependent eigenvalues of the Hessian matrix of the scalar potential (3.12) are given by. 14 gz z¯ ( p0 ; τ ) 2 W0 V z z¯ 2 2 λ1,2 (τ ) = −4 W0 − 2g ( p0 ; τ )|∂z W0 | ± 4 2 |∂z2 W0 | . (4.14) 4 MP MP The first step is to look for τ < τ0 . Local minimum of the scalar potential (3.12) exists if |∂z2 W0 | >

g1 ( p0 ; τ ) W0 . M P2

(4.15)

On the other side, local maximum is ensured by |∂z2 W0 |
τ0 , one gets a local maximum by replacing g1 ( p0 ; τ ) with g2 ( p0 ; τ ) in (4.15). By using the same procedure to (4.16) and (4.17), conditions for a local minimum and a saddle point are obtained, respectively. Again, we have degenerate vacua if g2 ( p0 ; τ ) W0 , 2M P2 g2 ( p0 ; τ ) |∂z2 W0 | = W0 . M P2

|∂z2 W0 | =

(4.19)

Here, some comments are in order. In the τ < τ0 region, the analysis of Eqs. (4.9)–(4.11) and (4.15)–(4.19) gives the same results as in [9]. The local minimum of W(τ ), given by (4.9), is mapped into the local maximum of the scalar potential (4.16). The other vacua, namely the local minimum, the saddle point, and the degenerate case, described 14 We have similar discussion as in footnote 13

858


by the second equality in (4.19), are coming from the saddle of W(τ ). Lastly, there is a case where both eigenvalues of the Hessian matrix of the scalar potential and the real function W(τ ) become singular. This means that this degenerate critical point of W(τ ) is mapped into the degenerate vacuum and such a case is called an intrinsic degenerate vacuum. For the τ > τ0 case, since the parity transformation exists, we have the following situations. The existence of the local minimum of the scalar potential is guaranteed by the local maximum of W(τ ) given by (4.12). The saddle of W(τ ) is mapped into the local maximum, the saddle point, and the degenerate vacua given by the second equation in (4.20) of the scalar potential. But still, the first equation in (4.20) describes intrinsic degenerate vacua. 5. Supersymmetric Flows On a Curved Spacetime This section provides the analysis of the supersymmetric flows, namely the gradient flow equations (3.8) and the RG flow described by the beta function (3.12) around the vacuum in the presence of the geometric soliton (3.13). As we will see, such a soliton affects the flows by a minus sign which is, in other words, the parity map mentioned above. First of all, we employ the dynamical system analysis on the gradient flows (3.8). A vacuum p0 is an equilibrium point of (3.8) if it is also a critical point of W(τ ). Around p0 the first order expansion of (3.8) gives the eigenvalues

1/2 2 (p ; τ) g W0 0 z z ¯ 1,2 = ∓ 2 cosθ0 (u, τ ) − 2g z z¯ ( p0 ; τ ) |∂z2 W0 |2 − W02 sin2 θ0 (u, τ ) , MP 4M P4 (5.1) where θ0 (u, τ ) ≡ θ ( p0 ; u, τ ) and the function θ (z, z¯ ; u, τ ) is defined in (3.9). In general the eigenvalues (5.1) are complex because the second term in the square root could be negative. Therefore in order to have a consistent model we simply set that they must have a real value in which 15 |gz z¯ ( p0 ; τ )| |∂z2 W0 | ≥ W0 |sinθ0 (u, τ )| (5.2) 2M P2 holds. This inequality gives a restriction of the critical points of the function W(τ ) and the vacua of the theory described by (4.7) and (4.14), respectively. In other words, in order to have some vacua related to a CFT on the curved spacetime, the condition (5.2) must be fulfilled 16 . For τ < τ0 and the cosθ0 (u, τ ) = 0 case we obtain that the stable nodes require |∂z2 W0 | >

g1 ( p0 ; τ ) W0 , 2M P2

(5.3)

while saddles are ensured by the condition g1 ( p0 ; τ ) g1 ( p0 ; τ ) W0 |sinθ0 (u, τ )| ≤ |∂z2 W0 | < W0 . 2M P2 2M P2

(5.4)

15 If we take the limit → 0, then sinθ ( p ; τ ) → 0. So, we regain the flat domain wall case with 0 |∂z2 W( p0 ; τ )| ≥ 0 [9]. 16 See also the discussion on the RG flow below.


859

Looking at (5.3), (4.15), and (4.17) we find that the dynamic of the walls described by (3.8) is stable, flowing along the local minimum and the stable direction of the saddles of the scalar potential (3.12). Along the local maximum the walls become unstable and the gradient flow is an unstable saddle. In other words, the dynamic of the walls is on an unstable curve flowing away from p0 . Moreover, in this linear analysis there is also a possibility of having p0 as a bifurcation point, namely one of the eigenvalues in (5.1) vanishes. Such a case occurs if the condition (4.11) holds and it takes place on the intrinsic degenerate vacua17 . These conclusions are similar to in the flat domain wall case [9]. For cosθ0 (u, τ ) = 0, only the stable nodes survive and hence we have stable walls. After crossing the singularity at τ = τ0 , i.e. the τ > τ0 case, we obtain the following inequality: |∂z2 W0 | >

g2 ( p0 ; τ ) W0 , 2M P2

(5.5)

describing unstable nodes, whereas saddles need g2 ( p0 ; τ ) g2 ( p0 ; τ ) W0 |sinθ0 (u, τ )| ≤ |∂z2 W0 | < W0 , 2M P2 2M P2

(5.6)

assuming cosθ0 (u, τ ) = 0. We have unstable walls flowing along the local maximum and the unstable direction of the saddles of the scalar potential (3.12). On the other hand, along local minima the walls become stable approaching p0 on the stable curve of the saddle flow. Again, a similar situation is obtained for a bifurcation point which requires |∂z2 W0 | =

g2 ( p0 ; τ ) W0 . 2M P2

(5.7)

If cosθ0 (u, τ ) = 0, then we have only unstable nodes which means that the model admits only unstable walls. Now let us perform an analysis on the RG flows described by the beta function (3.12) for finding out the nature of the vacuum p0 in the UV and IR regions. Our starting point is to expand the beta function (3.12) around p0 . We obtain the matrix ⎛ ⎞ ¯ ∂β/∂z( p0 ) ∂ β/∂z( p0 ) ⎠, U ≡ −⎝ (5.8) ¯ ∂β/∂ z¯ ( p0 ) ∂ β/∂ z¯ ( p0 ) whose eigenvalues are ⎛

1/2 ⎞ 2 (p ; τ) g W 0 0 −1 ⎝ ⎠, λU cosθ0 (u, τ )+2g z z¯ ( p0 ; τ ) |∂z2 W0 |2 − z z¯ 4 W02 sin2 θ0 (u, τ ) 1 =k M P2 4M P (5.9)

1/2 ⎞ gz2z¯ ( p0 ; τ ) 2 2 −1 ⎝W0 z z¯ 2 2 ⎠. λU cosθ (u, τ )−2g ( p ; τ ) |∂ W | − W0 sin θ0 (u, τ ) 0 0 0 z 2 =k M P2 4M P4 ⎛

(5.10) 17 To see the type of this fold bifurcation one has to check the higher order terms. At least one of these terms is non vanishing [23].

860


Similar to in the gradient flow, since the condition (5.2) is fulfilled, then both eigenvalues (5.9) and (5.10) are real. In the UV region where a → +∞, it demands that at least one of the above eigenvalues have to be positive along which the RG flow can depart from the vacuum. Let us first discuss the τ < τ0 region. In the cosθ0 (u, τ ) > 0 case, all possibilities of the vacua are allowed. But, for cosθ0 (u, τ ) ≤ 0 only stable nodes are permitted and therefore saddle, local minimum, and non-intrinsic stable degenerate vacua could exist. In other words, in this situation the gradient flow is only flowing along the stable direction of such vacua. In the τ > τ0 region we have the same result for the cosθ0 (u, τ ) > 0 case. However, for the cosθ0 (u, τ ) ≤ 0 case only unstable nodes are allowed which tell us the existence of unstable vacua such as saddle, local maximum, and non-intrinsic unstable degenerate vacua. In addition, the gradient flow is taking the unstable direction of the vacua and hence we have here an unstable situation. On the other side in the IR region, where a → 0, requires at least a negative eigenvalue of (5.8) which is the direction of the RG flow approaching the vacuum. In τ < τ0 and cosθ0 (u, τ ) ≥ 0 we find that only stable nodes survive and again, we have only a stable situation in which local maximum and intrinsic degenerate vacua are forbidden here. Conversely, we have all possible vacua for cosθ0 (u, τ ) < 0. Second, in τ > τ0 and cosθ0 (u, τ ) ≥ 0 it turns out that only unstable nodes exist, therefore the theory admits only the unstable vacua mentioned above. Lastly, everything is allowed for cosθ0 (u, τ ) < 0. 6. A Model with the Rosenau Soliton In this section we give an example in which the geometric flow is called the Rosenau soliton satisfying (3.13). This soliton was first studied in the context of fluid dynamics [20] and has been considered also by geometricians as a toy model in two dimensional complex surfaces. For a review see for example [24]. The Rosenau soliton has the form gz z¯ (τ ) = −

sinh(2bτ ) 4c2 , b cosh[2c(z + z¯ )] + cosh(2bτ )

whose corresponding U (1) connection is given by ic cosh[v + 2bτ ] Q(τ ) = − ln (dz − d z¯ ) , b coshv

(6.1)

(6.2)

where c ∈ R, b > 0, and v ≡ c(z + z¯ ) − bτ which diverges at τ = 0 or c = 0. It is easy to see that the metric (6.1) is invariant under parity transformation, c ↔ −c. We can further get its Kähler potential 2 cosh[v + 2bτ ] K (τ ) = − dv ln b coshv = 2bτ v + 2 Re dilog 1 + iev+2bτ − dilog 1 + iev , (6.3) where the dilogarithm function has been introduced [25]

x

dilog(x) ≡ 1

xk ln t dt = . 1−t k2 +∞

k=1

(6.4)


861

Looking at (6.1), we find that |sinh(2bτ )| 4c2 , b cosh[2c(z + z¯ )] + cosh(2bτ ) for all τ but not at singularity τ = 0 and c = 0. Now let us first choose for simplicity the holomorphic superpotential g1 (τ ) = g2 (τ ) =

W (z) = a0 + a1 z ,

(6.5)

(6.6)

with a0 , a1 ∈ R. So to find a supersymmetric vacuum one has to solve the condition (3.14), which in this model becomes 2c cosh[c(z 0 + z¯ 0 ) + bτ ] (a0 + a1 z 0 ) = 0 . a1 − ln (6.7) cosh[c(z 0 + z¯ 0 ) − bτ ] b M P2 Nondegeneracy requires W (z 0 ) = 0. We then obtain the solution of (6.7) as y =0, 0 tanh(2cx0 ) coth

ba1 M P2 4c(a0 +a1 x0 )

= coth(bτ ) ,

(6.8)

which follows that the imaginary part of W (z 0 ) vanishes. Thus, from (4.6) we find that this model admits only singular vacua which are not related to the CFT in three dimensions. In addition, it is easy to see that the origin belongs to this class of vacuum for a1 = 0 and a0 = 0 at which the U (1) connection disappears. The next case is to replace a1 by ia1 with a1 ∈ R in the superpotential (6.6). Then, we obtain a0 y0 = , a 1 bM 2 (6.9) tanh(2cx0 ) coth 4cxP0 = coth(bτ ) , in which the superpotential evaluated at p0 has the form W (z 0 ) = ia1 x0 .

(6.10)

Hence, the warped factor (4.2) simplifies to

∓W0 u a(u, τ ) = ± A0 e±W0 u − A−1 , e 0 W0 where W0 = e K (x0 ;τ )/2M P |a1 x0 | . 2

(6.11)

(6.12)

We close this section by providing the rest quantities related to the analysis of the eigenV U values, namely λW 1,2 , λ1,2 , 1,2 , and λ1,2 . These are W0 1 2 2 |∂z W0 | = 2M 2 gz z¯ ( p0 ; τ ) − M 2 |K z | ( p0 ; τ ) , P

P

−1/2 cosθ0 (u, τ ) = 1 + 2 (aW0 )−2 , −1/2 sinθ0 (u, τ ) = ± (aW0 )−1 1 + 2 (aW0 )−2 .

(6.13)

Therefore, we have a model where there may be a possibility of having vacua which correspond to the CFT. Moreover, in this model the singularities are at τ = 0 and at a1 = 0.

862


7. Conclusions We have studied the nature of the N = 1 supergravity curved BPS domain walls and their vacuum structure. In particular, we have considered a four dimensional N = 1 supergravity coupled to a chiral multiplet whose scalar manifold can be viewed as the Kähler-Ricci soliton satisfying the geometric flow equation (2.2). Some consequences emerge as follows. First of all, all couplings such as the scalar potential and the shifting quantity are evolving with respect to τ which are given in (2.5). Next, the warped factor, the BPS equations, and the beta function describing the domain walls and the three dimensional CFT, respectively, also depend on τ . The analysis on the vacua of the theory shows that in general the spacetime is nonEinsteinian, deformed with respect to τ whose Ricci tensor and Ricci scalar have the form provided in (4.4). This corresponds to the CFT on the three dimensional AdS spacetime which is ensured by the beta function (3.12). In this paper we just mentioned three special cases. First, for = 0 and k = 0 with k = 0 we regain four dimensional AdS spacetimes which appear in the flat domain walls. Second, there is a singularity at k = 0 and = 0 related to the RG flow analysis in which the spacetime has a constant curvature but non-Einsteinian. Note that the singularity caused by the geometric flow is trivial since some quantities in this case would become singular. Third, if W (z 0 ) = 0 and the U (1) connection vanishes, then we have a static case. For this case, the vacua are defined by the critical points of the superpotential W (z). Since our ground states generally depend on τ and additionally the theory admits the existence of a singular point at τ = τ0 described in (3.13), then we have to split the region in order to analyze the function W(τ ) and the scalar potential V (τ ). In the τ < τ0 region we reproduce the previous results in [9] as follows. Our analysis confirms that the deformation of the critical points of W(τ ) can only be in the following types, namely local minima, saddles, and degenerate critical points. The local minima verify the existence of local maximum vacua, whereas the saddles are mapped into local minimum, saddle, or non-intrinsic degenerate vacua. There is possibly a special situation where we have intrinsic degenerate vacua coming from the degenerate critical points of W(τ ). Hence, these also prove that the vacua certainly deform with respect to τ . For the τ > τ0 case, similar as mentioned above, W(τ ) also admits the evolution of its critical points which are local maxima, saddles, and degenerate critical points. In this case, however, the local maxima are mapped into local minimum vacua, while the saddles imply the existence of local maximum, saddle, or non-intrinsic degenerate vacua. In addition, intrinsic degenerate vacua could possibly exist. So, these results show that the geometric flow (3.13) indeed changes the Hessian matrix of the real function W(τ ) by a minus sign. Or in other words, the vacua of the index λ change to the other vacua of the index 2 − λ caused by Kähler-Ricci flow. The ground states of index 2 − λ in the τ > τ0 region are called the parity pair of those with the index λ in the τ < τ0 region. Furthermore, the analysis using the gradient and the RG flows shows that the above vacua do not always exist. First, the first order expansion of the BPS equations yields the condition (5.2) that ensures the existence of the vacua related to the three dimensional CFT. Then, in the interval τ < τ0 we have stable nodes flowing along the local minimum and the stable direction of the saddle vacua, whereas unstable saddles flow along the local maximum vacua. On the other side, for the τ > τ0 case, the gradient flow turns into the unstable nodes flowing along the local maximum and the unstable direction of the saddle vacua, while along the local minimum the gradient flow is flowing on the stable curve of the saddle flow which is called the stable saddles. In both regions there is a possibility of having a bifurcation point occurring near intrinsic degenerate vacua.


863

In order to check the existence of a ground state one has to carry out the analysis on the beta function. Particularly in the UV region and cosθ0 (u, τ ) ≤ 0, for τ < τ0 we have only the stable nodes which further imply that the theory admits only stable vacua, whereas for τ > τ0 the gradient flow turns into unstable nodes. On the other hand, in the IR region and cosθ0 (u, τ ) ≥ 0, for τ < τ0 again only the stable situation is allowed, while for τ > τ0 everything becomes unstable. Next, we have considered a simple model in which the superpotential is linear and the Kähler-Ricci soliton has a special form called the Rosenau solution. In the model where a0 , a1 ∈ R we find that the model has only singular ground states unrelated to the three dimensional CFT. However, if we set at least a1 ∈ C, we may obtain the vacua related to the CFT. Acknowledgement. We thank K. Yamamoto for the early stage of this work. We also acknowledge H. Alatas and A. N. Atmaja for useful discussions related to the topics of this paper. In addition, we particularly thank T. Mohaupt and M. Satriawan for carefully reading and correcting English grammar. We are grateful to the people at the theoretical astrophysics group and the elementary particle physics group of Hiroshima University for warmest hospitality where the early stage of this work was done. This work is supported by Riset KK ITB 2008 under contract No.: 035/K01.7/PL/2008 and ITB Alumni Association (HR IA-ITB) research project 2008 under contract No. 1241a/K01.7/PL/2008.

Appendix A. Convention and Notation The purpose of this appendix is to assemble our conventions in this paper. The spacetime metric is taken to have the signature (+, −, −, −) while the Christoffel symbol is given µ by νρ = 21 g µσ (∂ν gρσ + ∂ρ gνσ − ∂σ gνρ ), where gµν is the spacetime metric. The Rieµ µ µ µ µ mann curvature has the form R νρλ = ∂ρ νλ − ∂λ νρ + σνλ σρ − σνρ σ λ and the µ Ricci tensor is defined to be Rνλ = R νµλ . The following indices are given: ν, λ = 0, 1, 2,

label three dimensional curved spacetime indices,

a, b = 0, 1, 2,

label three dimensional flat spacetime indices,

µ, ν = 0, . . . , 3,

label four dimensional curved spacetime indices,

a, b = 0, . . . , 3,

label four dimensional flat spacetime indices.

B. Critical Point of A Function The structure and the logic of this section are similar to those in [9] which are useful for our analysis in the paper. Firstly, we consider any arbitrary (real) C ∞ -function f (z, z¯ ). A critical point p0 = (z 0 , z¯ 0 ) of f (z, z¯ ) satisfies ∂f ( p0 ) = 0 , ∂z

∂f ( p0 ) = 0 . ∂ z¯

(B.1)

864


The point p0 is said to be a non-degenerate critical point if the Hessian matrix of f (z, z¯ ) ⎞ ⎛ 2 ∂ f ∂2 f ∂z∂ z¯ ( p0 ) ∂z 2 ( p0 ) ⎟ ⎜ (B.2) Hf ≡ 2⎝ ⎠ 2 2 ∂ f ∂ f ( p0 ) ∂ z¯ ∂z ( p0 ) ∂ z¯ 2 is non-singular, i.e., det H f = 4

∂2 f ( p0 ) ∂z∂ z¯

2

∂2 f ∂2 f − 2 ( p0 ) 2 ( p0 ) = 0 . ∂z ∂ z¯

The eigenvalues of the Hessian matrix H f are given by 2 1 f λ1 = trH f + trH f − 4 detH f , 2 2 1 f trH f − trH f − 4 detH f . λ2 = 2

(B.3)

(B.4)

The eigenvalues defined in (B.4) can be used to classify the critical point p0 of the function f as follows: 1. 2. 3. 4.

f

f

If λ1 > 0 and λ2 > 0, then p0 is a local minimum describing a stable situation. f f If λ1 < 0 and λ2 < 0, then p0 is a local maximum describing an unstable situation. f f If λ1 > 0 and λ2 < 0 or vice versa, then p0 is a saddle point. If at least one of its eigenvalue vanishes, then p0 is said to be degenerate.

C. A Quick Review of the 2d Kähler-Ricci Flow This section is devoted to give a short review of the two dimensional Kähler-Ricci flow equation ∂gz z¯ = −2Rz z¯ (τ ) = −2 ∂z ∂¯ z¯ lngz z¯ (τ ) , ∂τ

(C.1)

discussed in Sec. 2 where τ ∈ R. Here, we particularly consider a simple case where the initial geometry at τ = 0 is a Kähler-Einstein surface satisfying Rz z¯ (x, 0) = 2 gz z¯ (x; 0) ,

(C.2)

where gz z¯ (x; 0) is an Einstein metric and 2 ∈ R. Then, we show that a Kähler-Ricci soliton can be viewed as a volume deformation of a Kähler geometry for finite τ . Let us simply choose that the constant 2 > 0 and the initial metric gz z¯ (x; 0) is definite positive. Taking the metric ansatz gz z¯ (x; τ ) = ρ(τ ) gz z¯ (x; 0) ,

(C.3)

and then the definition of Ricci tensor, we have Rz z¯ (x; τ ) = Rz z¯ (x; 0) = 2 gz z¯ (x; 0) .

(C.4)


865

Inserting (C.3) into (C.1), we get gz z¯ (x; τ ) = (1 − 2 2 τ ) gz z¯ (x; 0) .

(C.5)

This geometric soliton has a singularity at τ = 1/2 2 . After hitting the singularity, namely for τ > 1/2 2 , the geometry changes such that the metric is negative definite with a negative “cosmological” constant. This shows that the Kähler-Ricci flow interpolates two different N = 1 theories disconnected by the singularity at τ = 1/2 2 . Finally, we show that the Kähler-Ricci soliton can be viewed as a volume deformation of a Kähler manifold for finite τ . To be precise, the volume of the soliton (C.5) has the form detg(x; τ ) d 2 x = |1 − 2 2 τ | detg(x; 0) d 2 x . (C.6) For 0 ≤ τ < 1/2 2 , the geometry is diffeomorphic to the initial geometry endowed with a positive definite metric with 2 > 0, while for τ > 1/2 2 one has a geometry admitting a negative definite metric with negative “cosmological” constant. References 1. For a review see, for example: Aharony, O., Gubser, S.S., Maldacena, J., Ooguri, H., Oz, Y.: Large N Field Theories, String Theory and Gravity. Phys. Rep. 323, 183 (2000), and references therein 2. Cardoso, G.L., Dall’Agata, G., Lüst, D.: Curved BPS domain walls and RG flow in five dimensions. JHEP 0203, 044 (2002) 3. Cardoso, G.L., Lüst, D.: The holographic RG flow in a field theory on a curved background. JHEP 0209, 029 (2002) 4. Freedman, D.Z., Gubser, S.S., Pilch, K., Warner, N.P.: Renormalization group flows from holographysupersymmetry and a c-theorem. Adv. Theor. Math. Phys. 3, 363 (1999) 5. Cvetic, M., Griffies, S., Rey, S.J.: Static domain walls in N = 1 supergravity. Nucl. Phys. B 381, 301 (1992) 6. Cvetic, M., Soleng, H.H.: Supergravity domain walls. Phys. Rep. 282, 159 (1997) 7. Ceresole, A., Dall’Agata, G., Girvayets, A., Kallosh, R., Linde, A.: Domain walls, near-BPS bubbles and probabilities in the landscape. Phys. Rev. D74, 086010 (2006) 8. Gunara, B.E., Zen, F.P.: Kähler-Ricci Flow, Morse Theory, and Vacuum Structure Deformation of N = 1 Supersymmetry in Four Dimensions. Adv. Theor. Math. Phys 13, 217 (2009) 9. Gunara, B.E., Zen, F.P., Arianto: BPS Domain Walls and Vacuum Structure of N = 1 Supergravity Coupled To A Chiral Multiplet. J. Math. Phys. 48, 053505 (2007) 10. Cao, H.-D.: Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods p.1 in geometry. In: Chow, B., Gulliver, R., Levy, S., Sullivan, J. (eds.), Wellesley, MA: A K Peters, 1996 11. Cao, H.-D.: Limits of solutions to the Kähler-Ricci flow. J. Diff. Geom. 45, 257 (1997) 12. Higashijima, K., Itou, E.: Wilsonian Renormalization Group Approach to N = 2 Supersymmetric Sigma Models. Prog. Theor. Phys. 108, 737; A New Class of Conformal Field Theories with Anomalous Dimensions. Prog. Theor. Phys. 109, 751 (1997); Three Dimensional Nonlinear Sigma Models in the Wilsonian Renormalization Method, Prog. Theor. Phys. 110, 563 (2003) 13. Nitta, M.: Conformal Sigma Models with Anomalous Dimensions and Ricci Solitons. Mod. Phys. Lett. A20, 577 (2005) 14. Cremmer, E., Julia, B., Scherk, J., Ferrara, S., Girardello, L., van Nieuwenhuizen, P.: Spontaneous Symmetry Breaking and Higgs Effect in Supergravity Without Cosmological Constant. Nucl. Phys. B147, 105 (1979); Witten, E., Bagger, J.: Quantization of Newton’s Constant in Certain Supergravity Theories. Phys. Lett. B115, 202 (1982) 15. Wess, J., Bagger, J.: Supersymmetry and Supergravity. 2nd ed., Princeton, NJ: Princeton University Press, 1992 16. D’Auria, R., Ferrara, S.: On Fermion Masses, Gradient Flows and Potential in Supersymmetric Theories. JHEP 0105, 034 (2001); Andrianopoli, L., D’Auria, R., Ferrara, S.: Supersymmetry reduction of N -extended supergravities in four dimensions. JHEP 0203, 025 (2002) 17. Carstea, S.A., Visinescu, M.: Special solutions for Ricci flow equation in 2D using the linearization approach. Mod. Phys. Lett. A20, 2993 (2005)

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18. Topping, P.: Lectures on Ricci Flow. London Mathematical Society Lecture Note Series 325, Cambridge: Cambridge University Press, 2006, and references therein 19. Cao, H.-D., Zhu, X.-P.: Hamilton-Perelman’s Proof of the Poincaré and the Geometrization Conjecture. A revised version of the article which originally appeared in Asian J. Math. 10, 165, 2006. available at http://arxiv.org/list/math.DG/0612069, 2006 and references therein 20. Rosenau, P.: Fast and Superfast Diffusion Processes. Phys. Rev. Lett. 74, 1056 (1995) 21. De Wit, B., Herger, I.: Anti-de Sitter Supersymmetry. Lect. Notes Phys. 541, Berlin-Heidelberg-NewYork: Springer, 2000, pp. 79–100 22. Ceresole, A., Dall’Agata, G., Kallosh, R., van Proeyen, A.: Hypermultiplets, Domain Walls and Supersymmetric Attractor. Phys. Rev. D64, 104006 (2001) 23. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer-Verlag, New York, 1998 24. Chow, B.: Lectures on Ricci Flow, Lectures given at Clay Mathematics Institutes Summer School, Berkeley, 2005, and references therein, available at http://www.claymath.org/programs/summer_school/2005/ program.php#riccil-3, 2005 25. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. 9th ed., New York: Dover Publications, 1970 Communicated by G. W. Gibbons


Communications in


Fractional Moment Bounds and Disorder Relevance for Pinning Models Bernard Derrida1 , Giambattista Giacomin2 , Hubert Lacoin2 , Fabio Lucio Toninelli3 1 Laboratoire de Physique Statistique, Département de Physique,

École Normale Supérieure, 24, Rue Lhomond, 75231 Paris Cedex 05, France. E-mail: [email protected] 2 Université Paris Diderot (Paris 7) and Laboratoire de Probabilités et Modèles Aléatoires (CNRS U.M.R. 7599), U.F.R. Mathématiques, Case 7012, 2 Place Jussieu, 75251 Paris Cedex 05, France. E-mail: [email protected]; [email protected] 3 Laboratoire de Physique, ENS Lyon (CNRS U.M.R. 5672), 46 Allée d’Italie, 69364 Lyon Cedex 07, France. E-mail: [email protected]

Received: 11 January 2008 / Accepted: 24 November 2008 Published online: 17 February 2009 – © Springer-Verlag 2009

Abstract: We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K (·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K (n) = n −α−1 L(n), with α ≥ 0 and L(·) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α > 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open. The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power γ ∈ (0, 1) is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.

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B. Derrida, G. Giacomin, H. Lacoin, F. L. Toninelli

1. Introduction Pinning/wetting models with quenched disorder describe the random interaction between a directed polymer and a one-dimensional defect line. In absence of interaction, a typical polymer configuration is given by {(n, Sn )}n≥0 , where {Sn }n≥0 is a Markov Chain on some state space (for instance, = Zd for (1 + d)-dimensional directed polymers), and the initial condition S0 is some fixed element of which by convention we call 0. The defect line, on the other hand, is just {(n, 0)}n≥0 . The polymer-line interaction is introduced as follows: each time Sn = 0 (i.e., the polymer touches the line at step n) the polymer gets an energy reward/penalty n , which can be either positive or negative. In the situation we consider here, the n ’s are independent and identically distributed (IID) random variables, with positive or negative mean h and variance β 2 ≥ 0. Up to now, we have made no assumption on the Markov Chain. The physically most interesting case is the one where the distribution K (·) of the first return time, call it τ1 , of Sn to 0 has a power-law tail: K (n) := P(τ1 = n) ≈ n −α−1 , with α ≥ 0. This framework allows to cover various situations motivated by (bio)-physics: for instance, (1 + 1)-dimensional wetting models [12,17] (α = 1/2; in this case Sn ≥ 0, and the line represents an impenetrable wall), pinning of (1 + d)-dimensional directed polymers on a columnar defect (α = 1/2 if d = 1 and α = d/2 − 1 if d ≥ 2), and the Poland-Scheraga model of DNA denaturation (here, α 1.15 [27]). This is a very active field of research, and not only from the point of view of mathematical physics, see . e.g. [11] and references therein. We refer to [20, Ch. 1] and references therein for further discussion. The model undergoes a localization/delocalization phase transition: for any given value β of the disorder strength, if the average pinning intensity h exceeds some critical value h c (β) then the polymer typically stays tightly close to the defect line and the free energy is positive. On the contrary, for h < h c (β) the free energy vanishes and the polymer has only few contacts with the defect: entropic effects prevail. The annealed model, obtained by averaging the Boltzmann weight with respect to disorder, is exactly solvable, and near its critical point h ann c (β) one finds that the annealed free energy vanmax(1,1/α) [16]. In particular, the annealed phase transition is ishes like (h − h ann (β)) c first order for α > 1 and second order for α < 1, and it gets smoother and smoother as α approaches 0. A very natural and intriguing question is whether and how randomness affects critical properties. The scenario suggested by the Harris criterion [26] is the following: disorder should be irrelevant for α < 1/2, meaning that quenched critical point and critical exponents should coincide with the annealed ones if β is small enough, and relevant for α > 1/2: they should differ for every β > 0. In the marginal case α = 1/2, the Harris criterion gives no prediction and there is no general consensus on what to expect: renormalization-group considerations led Forgacs et al. [17] to predict that disorder is irrelevant (see also the recent [18]), while Derrida et al. [12] concluded for marginal relevance: quenched and annealed critical points should differ for every β > 0, even if the difference is zero at every perturbative order in β. The mathematical understanding of these questions witnessed remarkable progress recently, and we summarize here the state of the art (prior to the present contribution). (1) A lot is now known on the irrelevant-disorder regime. In particular, it was proven in [3] (see [28] for an alternative proof) that quenched and annealed critical points and critical exponents coincide for β small enough. Moreover, in [25] a small-disorder expansion of the free energy, worked out in [17], was rigorously justified. (2) In the strong-disorder regime, for which the Harris criterion makes no prediction, a few results were obtained recently. In particular, in [29] it was proven that

Disorder Relevance for Pinning Models

869

for any given α > 0 and, say, for Gaussian randomness, h c (β) = h ann c (β) for β large enough, and the asymptotic behavior of h c (β) for β → ∞ was computed. These results were obtained through upper bounds on fractional moments of the partition function. Let us mention by the way that the fractional moment method allowed also to compute exactly [29] the quenched critical point of a diluted wetting model (a model with a built-in strong-disorder limit); the same result was obtained in [8] via a rigorous implementation of renormalization-group ideas. Fractional moment methods have proven to be useful also for other classes of disordered models [1,2,9,15]. (3) The relevant-disorder regime is only partly understood. In [24] it was proven that the free-energy critical exponent differs from the quenched one whenever β > 0 and α > 1/2. However, the arguments in [24] do not imply the critical point shift. Nonetheless, the critical point shift issue has been recently solved for a hierarchical version of the model, introduced in [12]. The hierarchical model also depends on 2α/(2α−1) for the parameter α, and in [21] it was shown that h c (β) − h ann c (β) ≈ β β small (upper and lower bounds of the same order are proven). (4) In the marginal case α = 1/2 it was proven in [3,28] that the difference h c (β) − h ann c (β) vanishes faster than any power of β, for β → 0. Before discussing lower bounds on this difference, one has to be more precise on the tail behavior of K (n), the probability that the first return to zero of the Markov Chain {Sn }n occurs at n: if K (n) = n −(1+1/2) L(n) with L(·) slowly varying (say, a logarithm raised to a positive or negative power), then the two critical points coincide for β small [3,28] if L(·) diverges sufficiently fast at infinity so that ∞ n=1

1 < ∞. n L(n)2

(1.1)

The case of the (1 + 1)-dimensional wetting model [12] corresponds however to the case where L(·) behaves like a constant at infinity, and the result just mentioned does not apply. The case α = 1/2 is open also for the hierarchical model mentioned above. In the present work we prove that if α ∈ (1/2, 1) or α > 1 then quenched and 2α/(2α−1) for annealed critical points differ for every β > 0, and h c (β) − h ann c (β) ≈ β β 0 (cf. Theorem 2.3 for a more precise statement). In the case α = 1/2, while we do not prove that h c (β) = h ann c (β) in all cases in which condition (1.1) fails, we do prove such a result if the function L(·) vanishes sufficiently fast at infinity. Of course, h c (β) − h ann c (β) turns out to be exponentially small for β 0. We wish to emphasize that, although the Harris criterion is expected to be applicable to a large variety of disordered models, rigorous results are very rare: let us mention however [10,14]. Starting from the next section, we will forget the full Markov structure of the polymer, and retain only the fact that the set of points of contact with the defect line, τ := {n ≥ 0 : Sn = 0}, is a renewal process under the law P of the Markov Chain. 2. Model and Main Results Let τ := {τ0 , τ1 , . . .} be a renewal sequence started from τ0 = 0 and with inter-arrival law K (·), i.e., {τi − τi−1 }i∈N:={1,2,...} are IID integer-valued random variables with law

870


P(τ1 = n) = K (n) for every n ∈ N. We assume that recurrent) and that there exists α > 0 such that K (n) =

n∈N

K (n) = 1 (the renewal is

L(n) n 1+α

(2.1)

with L(·) a function that varies slowly at infinity, i.e., L : (0, ∞) → (0, ∞) is measurable and such that L(r x)/L(x) → 1 when x → ∞, for every r > 0. We refer to [6] for an extended treatment of slowly varying functions, recalling just that examples of L(x) include (log(1 + x))b , any b ∈ R, and any (positive, measurable) function admitting a positive limit at infinity (in this case we say that L(·) is trivial). Dwelling a bit more on nomenclature, x → x ρ L(x) is a regularly varying function of exponent ρ, so K (·) is just the restriction to the natural numbers of a regularly varying function of exponent −(1 + α). We let β ≥ 0, h ∈ R and ω := {ωn }n≥1 be a sequence of IID centered random variables with unit variance and finite exponential moments. The law of ω is denoted by P and the corresponding expectation by E. For a, b ∈ {0, 1, . . .} with a ≤ b we let Z a,b,ω be the partition function for the system on the interval {a, a + 1, . . . , b}, with zero boundary conditions at both endpoints: b (2.2) Z a,b,ω = E e n=a+1 (βωn +h)1{n∈τ } 1{b∈τ } a ∈ τ , where E denotes expectation with respect to the law P of the renewal. One may rewrite Z a,b,ω more explicitly as Z a,b,ω =

b−a

K (i j − i j−1 )e

h +β

j=1 ωi j

,

(2.3)

=1 i 0 =a 1) then L(1/ h) ∼ 1/E(τ1 ); (2) if α ∈ (0, 1), then L(1/ h) = Cα h −1/α Rα (h), where Cα is an explicit constant and Rα (·) is the function, unique up to asymptotic equivalence, that satisfies b0

Rα (bα L(1/b)) ∼ b. As a consequence of Theorem 2.1 and (2.7), the annealed critical point is simply given by ann h ann (β, h) = 0} = − log M(β). c (β) := sup{h : f

(2.10)

Via Jensen’s inequality one has immediately that f(β, h) ≤ fann (β, h) and as a consequence h c (β) ≥ h ann c (β), and the point of the present paper is to understand when this last inequality is strict. In this respect, let us recall that the following is known: if α ∈ (0, 1/2), then h c (β) = h ann c (β) for β small enough [3,28]. Also for α = 1/2 it has been shown that h c (β) = h ann c (β) if L(·) diverges sufficiently fast (see below). Moreover, assuming that P(ω1 > t) > 0 for every t > 0, one has that for every α > 0 and L(·) there exists β0 < ∞ such that h c (β) = h ann c (β) for β > β0 [29]: quenched and annealed critical points differ for strong disorder. The strategy we develop here addresses the complementary situations: α > 1/2 and small disorder (and also the case α = 1/2 as we shall see below). Our first result concerns the case α > 1: Theorem 2.2. Let α > 1. There exists a > 0 such that for every β ≤ 1, 2 h c (β) − h ann c (β) ≥ aβ .

Moreover, h c (β) > h ann c (β) for every β > 0.

(2.11)

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B. Derrida, G. Giacomin, H. Lacoin, F. L. Toninelli β0

2 Since h c (β) ≤ h c (0) = 0 and h ann c (β) ∼ −β /2, we conclude that the inequality (2.11) is, in a sense, of the optimal order in β. Note that h c (β) ≤ h c (0) is just a consequence of Jensen’s inequality: N E e n=1 (βωn +h)1{n∈τ } 1{N ∈τ } N Z N ,ω = Z N (h) E eh n=1 1{n∈τ } 1{N ∈τ } N

E 1{n∈τ } eh|τ ∩{1,...,N }| 1{N ∈τ }

ωn ≥ Z N (h) exp β , (2.12) h|τ ∩{1,...,N }| 1 E e {N ∈τ } n=1

from which f(β, h) ≥ f(0, h) and therefore h c (β) ≤ h c (0) immediately follows from E(ωn ) = 0. This can be made sharper in the sense that from the explicit bound in [20, Th. 5.2(1)] one directly extracts also that h c (β) ≤ −bβ 2 for a suitable b ∈ (0, 1/2) and every β ≤ 1, so that −h c (β)/β 2 ∈ (b, 1/2 −a). We recall also that the (strict) inequality h c (β) < h c (0) has been established in great generality in [4]. In the case α ∈ (1/2, 1) we have the following: Theorem 2.3. Let α ∈ (1/2, 1). For every ε > 0 there exists a(ε) > 0 such that (2α/(2α−1))+ε h c (β) − h ann , c (β) ≥ a(ε) β

for β ≤ 1. Moreover, h c (β) >

h ann c (β)

(2.13)

for every β > 0.

To appreciate this result, recall that in [3,28] it was proven that 2α/(2α−1) , h c (β) − h ann c (β) ≤ L(1/β)β

(2.14)

for some (rather explicit, cf. in particular [3]) slowly varying function L(·). Notably, L(·) is trivial if L(·) is. The conclusion of Theorem 2.3 can actually be strengthened and 2α/(2α−1) with L(·) ¯ ¯ we are able to replace the right-hand side of (2.13) with L(1/β)β ¯ does not match the bound in (2.14) another slowly varying function, but on one hand L(·) and on the other hand it is rather clear that it reflects more a limit of our technique than the actual behavior of the model; therefore, we decided to present the simpler argument leading to the slightly weaker result (2.13). The case α = 1/2 is the most delicate, and whether quenched and annealed critical points coincide or not crucially depends on the slowly varying function L(·). In [3,28] it was proven that, whenever 1 < ∞, (2.15) n L(n)2 n≥1

there exists β0 > 0 such that h c (β) = h ann c (β) for β ≤ β0 , and that when the same sum diverges then h c (β) − h ann (β) is bounded above by some function of β which vanishes c faster than any power for β 0. For instance, if L(·) is asymptotically constant then −c2 /β h c (β) − h ann , c (β) ≤ c1 e 2

(2.16)

for β ≤ 1. While we are not able to prove that quenched and annealed critical points differ as soon as condition (2.15) fails (in particular not when L(·) is asymptotically constant), our method can be pushed further to prove this if L(·) vanishes sufficiently fast at infinity:


873

Fig. 1. The decomposition of the partition function is simply obtained by fixing a value of k and summing over the values of the last contact (or renewal epoch) before N − k and the first after N − k. In the drawing the two contacts are respectively N − n and N − j and arcs of course identify steps between successive contacts

Theorem 2.4. Assume that for every n ∈ N, K (n) ≤ c

n −3/2 , (log n)η

(2.17)

for some c > 0 and η > 1/2. Then for every 0 < ε < η − 1/2 there exists a(ε) > 0 such that

1 h c (β) − h ann . (2.18) c (β) ≥ a(ε) exp − 1 β η−1/2−ε Moreover, h c (β) > h ann c (β) for every β > 0. 2.1. Fractional moment method. In order to introduce our basic idea and, effectively, start the proof, we need some additional notation. We fix some k ∈ N and we set for n∈N z n := eh+βωn .

(2.19)

Then, the following identity holds for N ≥ k: Z N ,ω =

N

Z N −n,ω

n=k

k−1

K (n − j) z N − j Z N − j,N ,ω .

(2.20)

j=0

This is simply obtained by decomposing the partition function (2.2) according to the value N − n of the last point of τ which does not exceed N − k (whence the condition 0 ≤ N − n ≤ N − k in the sum), and to the value N − j of the first point of τ to the right of N − k (so that N − k < N − j ≤ N ). It is important to notice that Z N − j,N ,ω has the same law as Z j,ω and that the three random variables Z N −n,ω , z N − j and Z N − j,N ,ω are independent, provided that n ≥ k and j < k. Let 0 < γ < 1 and A N := E[(Z N ,ω )γ ], with A0 := 1. Then, from (2.20) and using the elementary inequality γ

γ

(a1 + · · · + an )γ ≤ a1 + · · · + an ,

(2.21)

which holds for ai ≥ 0, one deduces k−1 N γ A N −n K (n − j)γ A j . A N ≤ E z1 n=k

The basic principle is the following:

j=0

(2.22)

874


Proposition 2.5. Fix β and h. If there exists k ∈ N and γ < 1 such that ∞ k−1 γ K (n − j)γ A j ≤ 1, ρ := E z 1

(2.23)

n=k j=0

then f(β, h) = 0. Moreover if ρ < 1 there exists C = C(ρ, γ , k, K (·)) > 0 such that A N ≤ C (K (N ))γ ,

(2.24)

for every N . Of course, in view of the results we want to prove, the main result of Proposition 2.5 is the first one. The second one, namely (2.24), is however of independent interest and may be used to obtain path estimates on the process (using for example the techniques in [23] and [20, Ch. 8]). Proof of Proposition 2.5. Let A¯ := max{A0 , A1 , . . . , Ak−1 }. From (2.22) it follows that for every N ≥ k A N ≤ ρ max{A0 , . . . , A N −k },

(2.25)

¯ The from which one sees by induction that, since ρ ≤ 1, for every n one has An ≤ A. statement f(β, h) = 0 follows then from Jensen’s inequality: f(β, h) = lim

N →∞

1 1 E log(Z N ,ω )γ ≤ lim log A N = 0. N →∞ N γ Nγ

In order to prove (2.24) we introduce γ γ E[z 1 ] k−1 j=0 K (n − j) A j , Q k (n) := 0

if n ≥ k, if n = 1, . . . k − 1.

(2.26)

(2.27)

Since ρ = n Q k (n), the assumption ρ < 1 tells us that Q k (·) is a sub-probability distribution and it becomes a probability distribution if we set, as we do, Q k (∞) := 1 − ρ. Therefore the renewal process τ with inter-arrival law Q k (·) is terminating, that is τ contains, almost surely, only a finite number of points. A particularity of terminating renewals with regularly varying inter-arrival distribution is the asymptotic equivalence, up to a multiplicative factor, of inter-arrival distribution and mass renewal function ([20, Th. A.4]), namely uN

N →∞

∼

1 Q k (N ), (1 − ρ)2

(2.28)

N τ ) and it satisfies the renewal equation u N = n=1 u N −n Q k (n) where u N := P(N ∈ for N ≥ 1 (and u 0 = 1). Since Q k (n) = 0 for n = 1, . . . , k − 1, for the same values of n we have u n = 0 too. Therefore the renewal equation may be rewritten, for N ≥ k, as uN =

N −k n=1

u N −n Q k (n) + Q k (N ).

(2.29)


875

N := A N 1 N ≥k then (2.22) implies that for N ≥ k, Let us observe now that if we set A N ≤ A

N −k

N −n Q k (n) + Pk (N ), with Pk (N ) := A

n=1

k−1

An Q k (N − n), (2.30)

n=0

and observe that Pk (N ) ≤ c Q k (N ), with c that depends on ρ, γ , k and K (·) (and on h and β, but these variables are kept fixed). Therefore N −k N A A N −n ≤ Q k (n) + Q k (N ), c c

(2.31)

n=1

for N ≥ k. By comparing (2.29) and (2.31), and by using (2.28) and Q k (N ) γ K (N )γ E[z 1 ] k−1 j=0 A j , one directly obtains (2.24).

N →∞

∼

2.2. Disorder relevance: sketch of the proof. Let us consider for instance the case α > 1, which is technically less involved than the others, but still fully representative of our strat2 egy. Take (β, h) such that β is small and h = h ann c (β) + , with = aβ . We are therefore considering the system inside the annealed localized phase, but close to the annealed critical point (at a distance from it), and we want to show that f(β, h) = 0. In view of Proposition 2.5, it is sufficient to show that ρ in (2.23) is sufficiently small, and we have the freedom to choose a suitable k. Specifically, we choose k to be of the order of the correlation length of the annealed system: k = 1/fann (β, h) = 1/f(0, ) ≈ const./(aβ 2 ), where the last estimate holds since the phase transition of the annealed system is first order for α > 1. Note that k diverges for β small. For the purpose of this informal discussion, assume that K (n) = c n −(1+α) , i.e., the slowly varying function L(·) is constant. The sum over n in the right-hand side of (2.23) is then immediately performed and (up to a multiplicative constant) one is left with estimating k−1 j=0

Aj . (k − j)(1+α)γ −1

(2.32)

One can choose γ < 1 such that (1 + α)γ − 1 > 1 and it is actually not difficult to show that sup j 1 case, here one can exploit the decay of P( j ∈ τ ) as j grows, while such a quantity converges to a positive constant if α > 1. Once again the case of j k can be dealt with by direct annealed estimates, while when one gets close to k a finer argument, direct generalization of the one used for the α > 1 case, is needed. 3. The Case α > 1 In order to avoid repetitions let us establish that, in this and the next sections, Ri , i = 1, 2, . . . denote (large) constants, L i (·) are slowly varying functions and Ci positive constants (not necessarily large). 2 Proof of Theorem 2.2. Fix β0 > 0 and let β ≤ β0 , h = h ann c (β) + aβ and γ < 1 sufficiently close to 1 so that

(1 + α)γ > 2.

(3.1)

It is sufficient to show that the sum in (2.23) can be made arbitrarily small (for some suitγ able choice of k) by choosing a small, since E[z 1 ] can be bounded above by a constant independent of a (for a small). √ We choose k = k(β) = 1/(aβ 2 ), so that β = 1/ ak(β). In order to avoid a plethora of ·, we will assume that k(β) is integer. Note that k(β) is large if β or a are small. First of all note that, thanks to Eqs. (A.21) and (A.24), the sum in the r.h.s. of (2.23) is bounded above by k(β)−1 j=0

L 1 (k(β) − j) A j . (k(β) − j)(1+α)γ −1

(3.2)

We split this sum as S1 + S2 :=

k(β)−1−R 1 j=0

L 1 (k(β) − j) A j + (k(β) − j)(1+α)γ −1

k(β)−1 j=k(β)−R1

L 1 (k(β) − j) A j . (3.3) (k(β) − j)(1+α)γ −1

To estimate S1 , note that by Jensen’s inequality A j ≤ (EZ j,ω )γ ≤ C1 with C1 a constant independent of j as long as j < k(β). Indeed, from (2.2) and the definition of the annealed critical point one sees that (recall (2.4)) 2 (3.4) EZ j,ω = Z j (aβ 2 ) = E eaβ |τ ∩{1,..., j}| 1{ j∈τ } ,


877

and the last term is clearly smaller than e. Therefore, using again (A.21) S1 ≤

L 2 (R1 ) (1+α)γ −2

R1

,

(3.5)

which can be made small with R1 large in view of the choice (3.1). As for S2 , one has S2 ≤ C2

max

k(β)−R1 ≤ j 0. If a is sufficiently small, for j ≤ k(β) = 1/(aβ 2 ) we have C3 C3 1 C3 β 1− √ ≤− (3.10) aβ 2 − √ ≤ √ . k(β) a 2k(β) a j As a consequence, E j,1/√ j (Z j,ω ) √ C3 C3 aβ 2 R1 /2 |τ ∩ {1, . . . , k(β)}| . ≤ e E exp − √ 2 ak(β) max

k(β)−R1 ≤ j 0, N n K (n) E(τ 1) n∈N n=1

(3.13)

almost surely (with respect to P) by the classical Renewal Theorem (or by the strong law of large numbers). The claim h c (β) > h ann c (β) for every β follows from the arbitrariness of β0 . 4. The Case 1/2 < α < 1 Proof of Theorem 2.3. To make things clear, we fix now ε > 0 small and 0 < γ < 1 such that γ (1 + α) + (1 − ε2 ) [1 − α + (ε/2)(α − 1/2)] > 2, (4.1) and γ (1 + α) + (1 − ε2 )(1 − α) > 2 − ε2 .

(4.2)

Moreover we take β ≤ β0 and 2α

ann 2α−1 (1+ε) . h = h ann c (β) + := h c (β) + aβ

(4.3)

We notice that it is crucial that (α − 1/2) > 0 for (4.1) to be satisfied. We will take ε sufficiently small (so that (4.1) and (4.2) can occur) and then, once ε and γ are fixed, a also small. We set moreover k(β) :=

1 f(0, )

(4.4)

and we notice that k(β) can be made large by choosing a small, uniformly for β ≤ β0 . As in the previous section, we assume for ease of notation that k(β) ∈ N (and we write just k for k(β)). Our aim is to show that f(β, h) = 0 if a is chosen sufficiently small in (4.3). We recall that, thanks to Proposition 2.5, the result is proven if we show that (3.2) is o(1) for k large. In order to estimate this sum, we need a couple of technical estimates which are proven at the end of this section (Lemma 4.2) and in Appendix 5 (Lemma 4.1). Lemma 4.1. Let α ∈ (0, 1). There exists a constant C4 such that for every 0 < h < 1 and every j ≤ 1/f(0, h), Z j (h) ≤

C4 . 1−α j L( j)

(4.5)

In view of Z j (h c (0)) = Z j (0) = P( j ∈ τ ) and (A.8), this means that as long as j ≤ 1/f(0, h) the partition function of the homogeneous model behaves essentially like in the (homogeneous) critical case.


879

Lemma 4.2. There exists ε0 > 0 such that, if ε ≤ ε0 (ε being the same one which appears in (4.3)), E j,1/√ j [Z j,ω ] ≤

C5 1−α+(ε/2)(α−1/2) j

(4.6)

for some constant C5 (depending on ε but not on β or a), uniformly in 0 ≤ β ≤ β0 and 2 in k (1−ε ) ≤ j < k. In order to bound above (3.2), we split it as 2

S3 + S4 :=

) k (1−ε

j=0

L 1 (k − j) A j + (k − j)(1+α)γ −1

k−1 2 j=k (1−ε ) +1

L 1 (k − j) A j . (k − j)(1+α)γ −1

(4.7)

For S3 we use simply A j ≤ (EZ j,ω )γ = [Z j ( )]γ and Lemma 4.1, together with (A.21) and (A.24): S3 ≤

1 L 3 (k) , k [(1+α)γ −1] k (1−ε2 )((1−α)γ −1)

(4.8)

where L 3 (·) can depend on ε but not on a. The second condition (4.2) imposed on γ guarantees that S3 is arbitrarily small for k large, i.e., for a√small. As for S4 , we use Lemma A.1 with N = j and λ = 1/ j to estimate A j (recall the definition in (A.1)). We get γ (4.9) A j ≤ E j,1/√ j (Z j,ω ) exp(cγ /(1 − γ )), √ 2 provided that 1/ j ≤ min(1, (1 − γ )/γ ), which is true for all j ≥ k 1−ε if a is small. 2 Then, provided we have chosen ε ≤ ε0 , Lemma 4.2 gives for every k (1−ε ) < j < k, Aj ≤

C6 . j [1−α+(ε/2)(α−1/2)]γ

(4.10)

Note that C6 is large for ε small (since from (4.1)–(4.2) it is clear that γ must be close to 1 for ε small) but it is independent of a. As a consequence, using (A.22), S4 ≤

max 2

k (1−ε ) ≤ j 0 such that, for every N ∈ N and γ ∈ (0, 1), E

Z N ,ω

γ

γ γ λ2 N , ≤ E N ,λ Z N ,ω exp c 1−γ

for |λ| ≤ min(1, (1 − γ )/γ ).

(A.2)


883

Proof. We have E

Z N ,ω

γ

= E N ,λ

Z N ,ω

γ

γ ≤ E N ,λ Z N ,ω

dP (ω) dP N ,λ

dP (ω) dP N ,λ

E N ,λ

1/(1−γ ) 1−γ

N

γ M(−λ)γ M (λγ /(1 − γ ))1−γ = E N ,λ Z N ,ω ,

(A.3)

where in the second step we have used Hölder inequality and the last step is a direct computation. The proof is complete once we observe that 0 ≤ log M(x) ≤ cx 2 for |x| ≤ 1 if c is the maximum of the second derivative of (1/2) log M(·) over [−1, 1].

A.2. Estimates on the renewal process. With the notation (2.4) one has Proposition A.2. Let α ∈ (0, 1) and r (·) be a function diverging at infinity and such that lim

N →∞

r (N )L(N ) = 0. Nα

(A.4)

For the homogeneous pinning model, Z N (−N −α L(N )r (N ))

N →∞

∼

N α−1 . L(N ) r (N )2

(A.5)

To prove this result we use: Proposition A.3 ([13, Theorems A & B]). Let α ∈ (0, 1). There exists a function σ (·) satisfying lim σ (x) = 0,

(A.6)

x→+∞

and such that for all n, N ∈ N, P(τn = N ) N ≤σ − 1 , n K (N ) a(n)

(A.7)

where a(·) is an asymptotic inverse of x → x α /L(x). Moreover, P(N ∈ τ )

N →∞

∼

α sin(π α) π

N α−1 . L(N )

(A.8)

We observe that by [6, Th. 1.5.12] we have that a(·) is regularly varying of exponent 1/α, in particular limn→∞ a(n)/n b = 0 if b > 1/α. We point out also that (A.8) has been first established for α ∈ (1/2, 1) in [19].

884


Proof of Proposition A.2. We put for simplicity of notation v(N ) := N α /L(N ). Decomposing Z N with respect to the cardinality of τ ∩ {1, . . . , N }, Z N (−r (N )/v(N )) =

N

P (|τ ∩ {1, . . . , N }| = n, N ∈ τ ) e−n r (N )/v(N )

n=1

=

N

P(τn = N )e−n r (N )/v(N )

n=1 √v(N ) r (N )

=

P(τn = N )e

−n

r (N ) v(N )

N

+

r (N )

P(τn = N )e−n v(N ) . (A.9)

) n= √v(N +1 r (N )

n=1

Observe now that one can rewrite the first term in the last line of (A.9) as (1 + o(1))K (N )

√ v(N )/ r (N )

n e−n r (N )/v(N ) ,

(A.10)

n=1

and o(1) is a quantity which vanishes for N → ∞ (this follows from Proposition A.3, which applies uniformly over all terms of the sum in view of lim N r (N ) = ∞). Thanks to condition (A.4), one can estimate this sum by an integral: √ v(N )/ r (N )

∞ v(N )2 v(N )2 (1 + o(1)) dx x e−x = (1 + o(1)). 2 r (N ) r (N )2 0 n=1 As for the second sum in (A.9), observing that n∈N P(τn = N ) = P(N ∈ τ ), we can bound it above by n e−n r (N )/v(N ) =

√ r (N )

P(N ∈ τ )e−

.

(A.11)

In view of (A.8), the last term is negligible with respect to N α−1 /(L(N ) r (N )2 ) and our result is proved. Proof of Lemma 4.1. Recalling the notation (2.4), point (2) of Theorem 2.1 (see in particular the definition of L(·)) and (A.8), we see that the result we are looking for follows if we can show that for every c > 0 there exists C23 = C23 (c) > 0 such that α E ec|τ ∩{1,...,N }|L(N )/N | N ∈ τ ] ≤ C23 , (A.12) uniformly in N . Let us assume that N /4 ∈ N; by Cauchy-Schwarz inequality the result follows if we can show that α E e2c|τ ∩{1,...,N /2}|L(N )/N | N ∈ τ ] ≤ C24 . (A.13) Let us define X N := max{n = 0, 1, . . . , N /2 : n ∈ τ } (last renewal epoch up to N /2). By the renewal property we have α E e2c|τ ∩{1,...,N /2}|L(N )/N | N ∈ τ ] =

N /2 n=0

α E e2c|τ ∩{1,...,N /2}|L(N )/N | X N = n] P (X N = n | N ∈ τ ). (A.14)


885

If we can show that for every n = 0, 1, . . . , N /2, P (X N = n | N ∈ τ ) ≤ C25 P (X N = n),

(A.15)

then we are reduced to proving (A.13) with E[·|N ∈ τ ] replaced by E[·]. Let us then observe that P (X N = n, N ∈ τ ) = P(n ∈ τ )P (τ1 > (N /2) − n, N − n ∈ τ ) N −n

= P(n ∈ τ )

P(τ1 = j)P (N − n − j ∈ τ ) . (A.16)

j=(N /2)−n+1

We are done if we can show that N −n

P(τ1 = j)P (N − n − j ∈ τ ) ≤ C26 P (N ∈ τ )

j=(N /2)−n+1

∞

P(τ1 = j),

j=(N /2)−n+1

(A.17) because the mass renewal function P(N ∈ τ ) cancels when we consider the conditioned probability and, recovering P(n ∈ τ ) from (A.16) we rebuild P(X N = n). We split the sum in the left-hand side of (A.17) in two terms. By using (A.8) (but just as upper bound) and the fact that the inter-arrival distribution is regularly varying we obtain N −n

P(τ1 = j)P (N − n − j ∈ τ )

j=(3N /4)−n+1

≤ C27

= C27

L(N ) N 1+α L(N ) N 1+α

N −n j=(3N /4)−n+1 N /4 j=1

1 (N − n − j + 1)1−α L(N − n − j + 1)

C28 1 ≤ . j 1−α L( j) N

(A.18)

Since the right-hand side of (A.17) is bounded below by 1/N times a suitable constant (of course if n is close to N /2 this quantity is sensibly larger) this first term of the splitting is under control. Now the other term: since the renewal function is regularly varying (3N /4)−n j=(N /2)−n+1

P(τ1 = j)P (N − n − j ∈ τ ) ≤ C29 P (N ∈ τ )

(3N /4)−n

P(τ1 = j),

j=(N /2)−n+1

(A.19) that gives what we wanted. It remains to show that (A.13) holds without conditioning. For this we use the asymp ∞ −1−α λ0 totic estimate − log E[exp(−λτ1 )] ∼ cα λα L(1/λ), with cα = 0 r (1 − exp(−r )) dr = (1 − α)/α, and the Markov inequality to get that if x > 0,

1 P |τ ∩ {1, . . . , N }|L(N )/N α > x = P (τn < N ) ≤ exp − cα λα L(1/λ)n + λN , 2 (A.20)

886


with n the integer part of x N α /L(N ) and λ ∈ (0, λ0 ) for some λ0 > 0. If one chooses λ = y/N , y a positive number, then for x ≥ 1 and N sufficiently large (depending on λ0 and y) we have that the quantity at the exponent in the right-most term in (A.20) is bounded above by −(cα /3)y α x + y. The proof is then complete if we select y such that (cα /3)y α > 2c (c appears in (A.13)) since ∞ if X is a non-negative random variable and q is a real number E[exp(q X )] = 1 + q 0 eq x P(X > x) dx. A.3. Some basic facts about slowly varying functions. We recall here some of the elementary properties of slowly varying functions which we repeatedly use, and we refer to [6] for a complete treatment of slow variation. The first two well-known facts are that, if U (·) is slowly varying at infinity, U (n) nm

N →∞

∼

U (N )

N 1−m , m−1

(A.21)

U (N )

N 1−m , 1−m

(A.22)

n≥N

if m > 1 and N U (n) n=1

nm

N →∞

∼

if m < 1 (cf. for instance [20, Sect. A.4]). The second two facts are that (cf. [6, Th. 1.5.3]) inf U (n)n m

n≥N

N →∞

∼

U (N ) N m ,

(A.23)

U (N ) N m ,

(A.24)

if m > 0, and sup U (n)n m

N →∞

∼

n≥N

if m < 0. Acknowledgements. G.G. and F.L.T. acknowledge the support of the ANR grant POLINTBIO. B.D. and F.L.T. acknowledge the support of the ANR grant LHMSHE. Note added in proof. After this work appeared in preprint form (arXiv:0712.2515 [math.PR]), several new results have been proven. In [5] it has been shown in particular that when L(·) is trivial, then ε in Theorem 2.3 can be chosen equal to zero, with a(0) > 0. The case α = 1 is also treated in [5]. The fractional moment method we have developed here may be adapted to deal with the α = 1 case too: this has been done in [7], where a related model is treated. Finally, the controversy concerning the case α = 1/2 and L(·) asymptotically constant has been solved in [22], where it was shown that h c (β) > h ann c (β) for every β > 0.

References 1. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) 2. Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. 224, 219–253 (2001) 3. Alexander, K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008)


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4. Alexander, K.S., Sidoravicius, V.: Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16, 636–669 (2006) 5. Alexander, K.S., Zygouras, N.: Quenched and annealed critical points in polymer pinning models. http:// arxiv.org/abs0805.1708V1[math.PR], 2008 6. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Cambridge: Cambridge University Press, 1987 7. Birkner, M., Sun, R.: Annealed vs quenched critical points for a random walk pinning model. http://arxiv. org/abs:0807.2752V1[math.PR], 2008 8. Bolthausen, E., Caravenna, F., de Tilière, B.: The quenched critical point of a diluted disordered polymer model. Stochastic Process. Appl. (to appear), http://arxiv.org/abs/0711.0141V2[math.PR], 2007 9. Buffet, E., Patrick, A., Pulé, J.V.: Directed polymers on trees: a martingale approach. J. Phys. A Math. Gen. 26, 1823–1834 (1993) 10. Chayes, J.T., Chayes, L., Fisher, D.S., Spencer, T.: Finite-size scaling and correlation lengths for disordered systems. Phys. Rev. Lett. 57, 2999–3002 (1986) 11. Coluzzi, B., Yeramian, E.: Numerical evidence for relevance of disorder in a Poland-Scheraga DNA denaturation model with self-avoidance: Scaling behavior of average quantities. Eur. Phys. J. B 56, 349–365 (2007) 12. Derrida, B., Hakim, V., Vannimenus, J.: Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992) 13. Doney, R.A.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Rel. Fields 107, 451–465 (1997) 14. von Dreifus, H.: Bounds on the critical exponents of disordered ferromagnetic models. Ann. Inst. H. Poincaré Phys. Théor. 55, 657–669 (1991) 15. Evans, M.R., Derrida, B.: Improved bounds for the transition temperature of directed polymers in a finite-dimensional random medium. J. Stat. Phys. 69, 427–437 (1992) 16. Fisher, M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–729 (1984) 17. Forgacs, G., Luck, J.M., Nieuwenhuizen, Th.M., Orland, H.: Wetting of a disordered substrate: exact critical behavior in two dimensions. Phys. Rev. Lett. 57, 2184–2187 (1986) 18. Gangardt, D.M., Nechaev, S.K.: Wetting transition on a one-dimensional disorder. J. Stat. Phys. 130, 483–502 (2008) 19. Garsia, A., Lamperti, J.: A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37, 221–234 (1963) 20. Giacomin, G.: Random Polymer Models. River Edge, NJ: Imperial College Press/World Scientific, 2007 21. Giacomin, G., Lacoin, H., Toninelli, F.L.: Hierarchical pinning models, quadratic maps and quenched disorder. Probab. Theory Rel. Fields (to appear), http://arxiv.org/abs/0711.4649V2[math.PR], 2007 22. Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models. http://arxiv. org/abs/0811.0723V1[math-ph], 2008 23. Giacomin, G., Toninelli, F.L.: Estimates on path delocalization for copolymers at selective interfaces. Probab. Theor. Rel. Fields 133, 464–482 (2005) 24. Giacomin, G., Toninelli, F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006) 25. Giacomin, G., Toninelli, F.L.: On the irrelevant disorder regime of pinning models. preprint (2007). http://arxiv.org/abs/0707.3340V1[math.PR] 26. Harris, A.B.: Effect of Random Defects on the Critical Behaviour of Ising Models. J. Phys. C 7, 1671–1692 (1974) 27. Kafri, Y., Mukamel, D., Peliti, L.: Why is the DNA denaturation transition first order? Phys. Rev. Lett. 85, 4988–4991 (2000) 28. Toninelli, F.L.: A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008) 29. Toninelli, F.L.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18, 1569–1587 (2008) Communicated by M. Aizenman


Communications in


Discrete Torsion Phases as Topological Actions Kiyonori Gomi1 , Yuji Terashima2 1 Department of Mathematics, University of Texas at Austin, 1 University Station C1200,

Austin, TX 78712-0257, USA. E-mail: [email protected]

2 Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama,

Meguro-Ku, Tokyo 152-8551, Japan. E-mail: [email protected] Received: 28 March 2008 / Accepted: 24 October 2008 Published online: 18 February 2009 – © Springer-Verlag 2009

Abstract: In this paper, we show that discrete torsion phases in string orbifold partition functions, and membrane discrete torsion phases, are topological actions on the simplicial manifolds associated to orbifold group actions. For this purpose, we introduce an integration theory of smooth Deligne cohomology on a general simplicial manifold, and prove that the integration induces a well-defined paring between the smooth Deligne cohomology and the singular cycles. 1. Introduction Discrete torsion phases are puzzling factors for twisted sectors in string orbifold partition functions introduced by C. Vafa [V]. In this paper, we show that discrete torsion phases are topological actions on the simplicial manifolds associated to the orbifold group actions, as Wess-Zumino terms are topological actions on the usual manifolds. Simplicial manifolds appear in many contexts. In particular, smooth groupoids give simplicial manifolds called their nerves. The convolution algebras of smooth groupoids are important examples of noncommutative spaces. Therefore, our result suggests that discrete torsion phases are topological actions on noncommutative spaces. Our key mathematical tool to get topological actions is an integration theory of Deligne cocycles on simplicial manifolds. This is a generalization of our previous result [GT] based on the works of J. L. Brylinski [B] and K. Gaw¸edzki [G]. K. Gaw¸edzki was the first to use integrations of Deligne cocycles on usual manifolds to describe Wess-Zumino terms (see also [A]). For a manifold with a finite group action, each group 2-cocycle gives a Deligne 2-cocycle on the simplicial manifold associated to the group action. We show that the discrete torsion phase in [V] is just the integration of the Deligne cocycle along a singular cycle on the simplicial manifold corresponding to a twisted sector. More generally, we show that the discrete torsion phase in E. Sharpe [S1], which is derived from his interpretation of a discrete torsion as a choice of the orbifold group action on a B-field, as well

890

K. Gomi, Y. Terashima

as the membrane discrete torsion phase in [S2] are just integrations of Deligne cocycles along singular cycles on the simplicial manifold corresponding to twisted sectors. Our construction is so general that, in order to get topological actions, it needs only simplicial manifolds with Deligne cocycle. In particular, the construction is applicable to the cases of Lie group actions in addition to the cases of finite group actions. In his stimulating paper [D], M. R. Douglas gave a new interpretation of discrete torsion by using a projective representation of the orbifold point group. It would be interesting to investigate a relation between his approach and ours. This paper is organized as follows: in Sect. 2, we recall the definition of the Deligne cohomology of a simplicial manifold. In Sect. 3, we show that, for any simplicial manifold, there exists an open cover compatible with singular cycles. Then, in Sect. 4, we define the integration of Deligne cocycles along singular cycles on simplicial manifolds, and prove our main theorem: the integration induces a well-defined pairing between the Deligne cohomology classes and the singular cycles on a simplicial manifold. This theorem, with its application to discrete torsion phases, was stated in Geometry of Quantization 2005, Tokyo.1 Finally, in Sect. 5, we show that the discrete torsion phases in [S1,S2 and V] arise as examples of our integration. 2. Deligne Cohomology on Simplicial Manifold A simplicial manifold X • is a sequence of smooth manifolds {X n }n≥0 equipped with the face maps εi : X n → X n−1 , (i = 0, . . . , n) and the degeneracy maps σi : X n−1 → X n , (i = 0, . . . , n − 1) satisfying the relations εi ε j = ε j−1 εi , (i < j), σi σ j = σ j+1 σi , (i ≤ j), ⎧ ⎨ σ j−1 εi , (i < j), id, (i = j, j + 1), εi σ j = ⎩ σ ε , (i > j + 1). j i−1 The smooth Deligne cohomology of X • is the hypercohomology of the complex of n • sheaves F D( p) = {F D( p) }n≥0 : d log

d

d

n ∗ 1 F D( p) : C X n → A X n → · · · → A X n → 0 → 0 → · · · , p

where C∗X n is the sheaf of smooth functions on X n with their values in non-zero complex q numbers, and A X n is the sheaf of smooth C-valued q-forms on X n . ˇ A convenient way to treat a hypercohomology is to use the Cech cohomology of an open cover of X • . An open cover U • = {U n }n≥0 of X • consists of open covers U n = {Uαn }α∈I n of X n such that their index sets I n form a simplicial set I • and the following relation of inclusions holds: εi (Uαn+1 ) ⊂ Uεni (α) . For a complex of sheaves F • = {F n }n≥0 : d

d

d

F n : F n,0 → F n,1 → F n,2 → · · · on X • , we have a triple complex 1 The transparency is available at http://www.f.waseda.jp/homma_yasushi/kori/2005/gq_iv.pdf

Discrete Torsion Phases as Topological Actions

891

C k,l,n =

α0 ,...,αk

F n,l (Uαn0 ···αk ).

ˇ The first differential δ : C k,l,n → C k+1,l,n is the usual Cech differential associated with the open cover U n , the second differential d : C k,l,n → C k,l+1,n comes from the differential in F n , and the third differential ε : C k,l,n → C k,l,n+1 is defined by n+1 (−1)i εi∗ . Then we have the cohomology H m (U • , F • ) of the associated total ε = i=0 k,l,n with its differential D. In our convention, complex C ∗ (U • , F • ) = ∗=k+l+n C the total differential D acts on the component C k,l,n by D = δ + (−1)k d + (−1)k+l ε. Now, the hypercohomology H m (X • , F • ) is given by the direct limit of the cohomology H m (U • , F • ) with respect to the direct system of open covers of X • . An open cover U • of X • will be called good when each U n is a good cover. If U • is good, then H m (X • , F • ) ∼ = H m (U • , F • ). 3. Singular Cycle on Simplicial Manifold 3.1. Definition and related notion. Let X • be a simplicial manifold. On each X n , we have the singular chain complex Sl (X n ) with its boundary operator ∂ : Sl (X n ) → Sl−1 (X n ). n If we define ε : Sl (X n ) → Sl (X n−1 ) by ε = i=0 (εi )∗ , then we get a double complex (Sl (X n ), ∂, ε). We write Cm (X • ) for the total complex of this double complex. We also write Z m (X • ) for the subgroup of cycles. Definition 3.1. A singular m-cochain on the simplicial manifold X • is defined to be an element in Cm (X • ). Similarly, a singular m-cycle on X • is defined to be an element in Z m (X • ). n • n n Let s • = m n=0 s be a singular m-cycle on X . Each chain s ∈ Sm−n (X ) can be expressed as: sn = (a n )r (s n )r , r

where (a n )r = ±1 and (s n )r : m−n → X n is a singular simplex. A choice of a triangulation K (m−n ) of the standard (m−n)-simplex m−n associates s n with another chain sˇ n ∈ Sm−n (X n ) given by (a n )r (s n )r |e . sˇ n = r

e∈K (m−n ) dime=m−n

m )} of We define a triangulation of s • to be a collection K = {K (1 ), . . . , K ( m n triangulations K (m−n ) of m−n such that the associated chain sˇ • = n=0 sˇ is • also a cycle on X . n • n For a triangulation K of s = m n=0 s ∈ Z m (X ), we define F (K ) to be the set of n n singular simplices (s )r |e constituting the associated chain sˇ . We also define: {(εi1 · · · εik )∗ τ | τ ∈ F n+k (K )}, F n (K ) = F n (K ) ∪ k≥1

which we call the set of ε-faces of K .

892


Definition 3.2. Let K be a triangulation of an m-cycle s ∈ Z m (X • ), and U • an open cover of X • . We write U n = {Uαnn }α n ∈I n for the open cover of X n constituting U • = {U n }. For k ≤ m, the triangulation K is said to be k-compatible with U • if there are maps φ n : F n (K ) → I n , (n ≤ k) satisfying • Image(τ ) ⊂ Uφnn (τ ) for all τ ∈ F n (K ), (n ≤ k), • εi (φ n+1 (τ )) = φ n ((εi )∗ τ )) for all τ ∈ F n (K ), (n ≤ k − 1). Such φ n as above will be called index maps. The triangulation K is simply said to be compatible with U • if it is m-compatible. Proposition 3.3. Any simplicial manifold X • has a good cover U • of X • such that: any singular cycle s admits a triangulation compatible with U • . This proposition is shown in the subsequent two subsections. 3.2. Construction of good cover. This subsection is devoted to constructing the good cover U • in Proposition 3.3. Definition 3.4. Let X • = {X n }n≥0 be a simplicial manifold. (a) For p ≥ 0, an open cover of {X n }n≤ p consists of open covers U n = {Uαnn }α n ∈I n of X n , (n = 0, . . . , p) and maps εi : I n → I n−1 , (n = 1, . . . , p, i = 0, . . . , n) such that: • εi (Uαnn ) ⊂ Uεn−1 n , i (α ) • εi ε j = ε j−1 εi for i < j. (b) An open cover {U n }n≤ p of {X n }n≤ p is said to be admissible if: n−1 n n n−1 for n = 1, . . . , p. • I = α n−1 ∈I n−1 I α0 , . . . , αn i

• For all α0n−1 , . . . , αnn−1 ∈ I n−1 , we have

n −1 n−1 U n−1 = εj j=0

αj

Uαnn .

α n ∈I n (α0n−1 ,...,αnn−1 )

• ε j (α n ) = α n−1 for all αn ∈ I n (α0n−1 , · · · , αnn−1 ) and α n−1 ∈ I n−1 . j j An admissible cover of X • is an open cover U • = {U n }n≥0 such that {U n }n≤ p is admissible for every p. Applying the first condition in Definition 3.4 (b) twice, we have:

In = I n α0n−1 , . . . , αnn−1 .

n−2 n−1 n−2 n−2 αi,n−2 α j ∈I n−1 α0, j ∈I j ,...,αn−1, j

The second condition in Definition 3.4 (a) and the third condition in Definition 3.4 (b) n−2 is a choice which does not satisfy the relation α n−2 = α n−2 , imply that: if αi,n−2 j ∈ I i, j j−1,i n−2 n−2 ∈ I n−1 (α0, (i < j), then I n (α0n−1 , . . . , αnn−1 ) = ∅ for any α n−1 j j , . . . , αn−1, j ). Hence the expression of I n reads: In = I n (α0n−1 , . . . , αnn−1 ). n−2 n−2 n−2 αi,n−2 α n−1 ∈I n−1 (α0, j ∈I j j ,...,αn−1, j ) n−2 n−2 αi, j =α j−1,i (i< j)


893

Proposition 3.5. Any simplicial manifold admits an admissible good cover. The proposition follows from Lemma 3.6 and 3.7. Lemma 3.6. Let X • be a simplicial manifold, and U 0 = {Uα00 }α 0 ∈I 0 a good cover of X 0 . Then there is a good cover U 1 of X 1 such that {U n }n≤1 is an admissible good cover of {X n }n≤1 . Proof. Because of the third relationship between the face maps and the degeneracy maps of X • described at the beginning of Sect. 2, the face maps εi : X 1 → X 0 are surjective. Hence the collection of open sets {ε0−1 (Uα00 ) ∩ ε1−1 (Uα00 )}α 0 ,α 0 ∈I 0 gives an open cover 0

0

1

1

of X 1 . We choose an open cover of ε0−1 (Uα00 ) ∩ ε1−1 (Uα00 ): 0 1

Uα11 ε0−1 Uα00 ∩ ε1−1 Uα00 = 0

1

α 1 ∈I 1 α00 ,α10

such that each Uα11 is geodesically convex with respect to a Riemannian metric on X 1 . Then {Uα11 }α 1 ∈I 1 is a good cover of X 1 , where

I1 = I 1 α00 , α10 .

α00 ,α10 ∈I 0 ×I 0

If we define εi : I 1 → I 0 , (i = 0, 1) to be εi (α 1 ) = αi0 for α 1 ∈ I 1 (α00 , α10 ), then

εi (Uα11 ) ⊂ Uε0 (α 1 ) clearly. i

Lemma 3.7. Let p be such that p ≥ 1. For an admissible good cover {U n }n≤ p of {X n }n≤ p , there is a good cover U p+1 such that {U n }n≤ p+1 is an admissible good cover of {X n }n≤ p+1 . Proof. We have the following open cover of X p+1 : ⎧ ⎫

⎬ p+1 ⎨ p Uα p ε−1 , j ⎩ ⎭ j j=0

p

p−1

p−1

p−1

p−1

p−1

where all α j ∈ I p (α0, j , α1, j , . . . , α p, j ) and all αi, j p−1

α j−1,i

∈ I p−1 such that αi, j

p+1 p U : for i < j are considered. We now take an open cover of j=0 ε−1 p j α

=

j

p+1 p −1 Uα p = εj j

j=0

p

p+1

p

α p+1 ∈I p+1 α0 ,...,α p+1

p+1

Uα p+1

such that each Uα p+1 is geodesically convex with respect to a Riemannian metric on p+1 X p+1 . Then U p+1 = Uα p+1 p+1 p+1 gives a good cover of X p+1 , where α ∈I

p p p+1 I = I p+1 α0 , . . . , α p+1 . p−1

αi, j ∈I p−1 p−1

p−1

αi, j =α j−1,i ,(i< j)

p p−1 p−1 α j ∈I n α0, j ,...,α p, j

894

K. Gomi, Y. Terashima p

p

p

For α p+1 ∈ I p+1 (α0 , . . . , α p+1 ) we put ε j (α p+1 ) = α j and define ε j : I p+1 → I p .

p+1 p Then we have εi Uα p+1 ⊂ Uε (α p+1 ) by construction. We also have εi ε j = ε j−1 εi for i

i < j. Hence {U n }n≤ p+1 is an admissible good cover. 3.3. Proof of Proposition 3.3. For a triangulation K = {K (1 ), . . . , K (m )} of a singular m-cycle s • on X • , we define a subdivision of K to be a triangulation Kˇ = { Kˇ (1 ), . . . , Kˇ (m )} of s • such that each Kˇ (k ) is a subdivision of K (k ). Notice that, if K is k-compatible with an open cover U, then so is its subdivision Kˇ . To see this fact, let φ n : F n (K ) → I n denote index maps of K . Then we get index maps φˇ n : F n ( Kˇ ) → I n of Kˇ by setting φˇ n (εi1 · · · εik (s n+k )r |eˇ ) = φ n (εi1 · · · εik (s n+k )r |e ), where e ∈ K (m−n−k ) is the simplex of the smallest dimension containing eˇ ∈ Kˇ (m−n−k ). Now Proposition 3.3 follows from Lemma 3.8 and 3.9. Lemma 3.8. Let U • be an admissible cover of X • . (a) Any singular cycle s • on X • admits a triangulation which is 0-compatible with U • . (b) Any triangulation of s • which is 0-compatible with U • has a subdivision which is 1-compatible with U • . Proof. For (a), let K be a triangulation of s • . Taking some barycentric subdivisions of the standard simplices simultaneously, we can get a subdivision Kˇ of K which is 0-compatible with U. For (b), if K is 0-compatible, then we have Image(εi τ ) ⊂ Uφ00 (ε τ )

for τ ∈ F 1 (K ), so that:

Image(τ ) ⊂ ε0−1 Uφ00 (ε

i

0τ )

∩ ε1−1 Uφ00 (ε

1τ )

=

α 1 ∈I 1 (φ 0 (ε0 τ ),φ 0 (ε1 τ ))

Uα11 .

Taking some subdivisions of the standard simplices, we can find a subdivision Kˇ of K such that: for each τ ∈ F 1 ( Kˇ ), there is φˇ 1 (τ ) ∈ I 1 (φ 0 (ε0 τ ), φ 0 (ε1 τ )) such that Image(τ ) ⊂ U 1ˇ 1 . Hence we have a map φˇ 1 : F 1 ( Kˇ ) → I 1 which makes Kˇ into φ (τ ) 1-compatible with U • .

Lemma 3.9. Let U • be an admissible cover of X • . For n ≥ 0, a triangulation K of an m-cycle s • which is (n + 1)-compatible with U • admits a subdivision which is (n + 2)compatible with U • . Proof. For τ ∈ F n+2 (K ), we have Image(ε j τ ) ⊂ Uφn+1 n+1 (ε Image(τ ) ⊂

n+2

Uφn+1 ε−1 n+1 (ε j

jτ)

jτ)

, so that:

.

j=0

If we define αi, j ∈ I n by αi, j = εi φ n+1 (ε j τ ) = φ n (εi ε j τ ), then αi, j = α j−1,i holds for i < j. Thus, by the definition of U • , we have the open cover: n+2 j=0

Uφn+1 ε−1 n+1 (ε j

jτ)

=

α n+2 ∈I n+2 (φ n+1 (ε0 τ ),...,φ n+1 (εn+2 τ ))

Uαn+2 n+2 .


895

n+2 ˇ ˇ ˇ n+2 We can

find a subdivision K of K such that: for τ ∈ F ( K ), there is φ (τ ) ∈ . Now we have I n+2 φˇ n+1 (ε0 τ ), . . . , φˇ n+1 (εn+2 τ ) such that Image(τ ) ⊂ U n+2 ˇ n+2 φ

(τ )

φˇ n+2 : F n+2 ( Kˇ ) → I n+2 making Kˇ into (n + 2)-compatible with U • .

4. Integration We introduce here a pairing, called integration, between smooth Deligne cohomology classes and singular cycles on X • . We then prove our main theorem about the integration. Let X • be a simplicial manifold, and U • = {U n }n≥0 an (admissible) open cover of • • • m m • X . Suppose that a Deligne m-cocycle c = m n=0 s ∈ Z (U , F D( p) ) and a singular • m-cycle s • = m n=0 ∈ Z m (X ) are given. Suppose also that there is a triangulation K of s • compatible with U • by the index maps φ n . We then define the pairing c• , s • ∈ C/Z as follows: m c , s = cn , s n , •

•

n=0

c , s = c , n

n

n

(a )r (s )r = n

r

cn , (s n )r =

m−n

n

m−n−i i=0 σ ∈Fli ((s n )r ) σ

(a n )r cn , (s n )r , r

θφi,m−n−i,n n (σ m−n )···φ n (σ m−n+i ) ,

where s n = r (a n )r (s n )r is the expression of the (m − n)-chain s n on X n by using singular simplices as before, Fli ((s n )r ) is the set of flags defined by p e ∈ K (m−n ), dime p = p, n n n , Fli ((s )r ) = ((s )r |em−n−i , . . . , (s )r |em−n ) em−n−i ⊂ · · · ⊂ em−n and cn = (θ m−n,0,n , . . . , θ 0,m−n,n ) is the expression of the Deligne cochain cn such that exp θαm−n,0,n ∈ C ∞ (Uαn0 ,...,αm−n , C∗ ) and θαi,m−n−i,n ∈ Am−n−i (Uαn0 ···αi ) for 0 < 0 ···αm−n 0 ···αm−n−i m − n − i ≤ min{m − n, p}. Theorem 4.1. The integration induces a well-defined homomorphism • • , : H m (X • , F D( p) ) ⊗Z Z m (X ) −→ C/Z.

Proof. We temporarily write s n , cn K ,φ for the pairing defined by using K and φ n : F n (K ) → I n . For another index map φ n : F n (K ) → I n of K , we obtain cn , s n K ,φ − cn , s n K ,φ = D n cn , s n φ,φ + cn , ∂s n φ,φ . In the above formula, D n cn , s n φ,φ is defined by m−n i n j (a )r (−1) r

i=0

σ

j=0

(Dc)i+1,m−n−i,n , φ(σ m−n )··φ(σ m−n− j )φ (σ m−n− j )··φ (σ m−n−i )

σ m−n−i

896


i,m−n−i,n + (−1)i+1 dθ i+1,m−n−i−1,n . The chain s n where we put (Dc)i+1,m−n−i,i r =n δθ n is expressed as s = r (a )r (s )r , and σ = (σ m−n−i , . . . , σ m−n ) runs through all the flags in Fli ((s n )r ). Similarly, cn , ∂s n φ,φ is

(bn )r

r

×

m−n−1 i i=0

σ m−n−1−i

σ

(−1) j

j=0

i+1,m−n−1−i,n θφ(σ m−n−1 )··φ(σ m−n−1− j )φ (σ m−n−1− j )··φ (σ m−n−1−i ) ,

where ∂s n is expressed as ∂s n = r (bn )r (∂s n )r , and σ = (σ m−n−1−i , . . . , σ m−n−1 ) belongs to Fli ((∂s n )r ). Since s • is a cycle, K is compatible with U • and c• is a cocycle, we have the following formula for n < m: cn , ∂s n φ,φ = cn , (−1)m−n εs n+1 φ,φ = (−1)m−n εcn , s n+1 φ,φ = −D n+1 cn+1 , s n+1 φ,φ , = δ + (−1)k d stands for the total differential on the Deligne complex C ∗ (U n , where D n n k,l,n on X n . Consequently, we have c• , s • • • F D( ) = K ,φ = c , s K ,φ and ∗=k+l F p) the pairing is independent of the choice of index maps. This result implies that the pairing is also independent of the choice of K , because: (i) we have cn , s n K ,φ = cn , s n Kˇ ,φˇ for a subdivision Kˇ of K with its index maps φˇ n : F n ( Kˇ ) → I n induced from φ n , and; (ii) any two triangulations of s • have a common subdivision. To complete the proof, we let m−1 n b be an (m −1)-cochain. We can show the formula D n bn , s n = bn , ∂s n b• = n=0 for each n. Hence we get Db• , s • = 0, since s • is a cycle and K is compatible with U •.

Note that there is a natural map from the smooth Deligne cohomology on X • to the group of (closed) differential forms on X 0 • F : H m (X • , F D(m) ) −→ Am+1 (X 0 ). • ), the closed form F([c• ]) is given For an m-cocycle c• = (θ i,m−n−i,n ) ∈ Z m (U • , F D(m) • 0,m,0 m • . Let H (X , C/Z) = Ker F be the subgroup in H m (X • , F • ) by F([c ])|Uα0 = dθα consisting of “flat Deligne classes”.

Proposition 4.2. The pairing induces a well-defined homomorphism , : H m (X • , C/Z) ⊗Z Hm (X • ) −→ C/Z. m+1 n n • Proof. Let c• = m n=0 c be a Deligne m-cocycle, t = n=0 t a singular (m + 1)chain, and s • = (∂ ± ε)t • its boundary. We can prove c0 , ∂t 0 = t 0 F(c• ) mod Z as well as cn , ∂t n = D n cn , t n for n > 0. Since c• is a cocycle, we have c• , s • = 0. This formula and our main theorem imply Proposition 4.2.


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5. Discrete Torsion Phases 5.1. In this section, we describe both the discrete torsion phases in [S1] and the membrane discrete torsion phases in [S2] as the integrations of Deligne cocycles on simplicial manifolds in Sect. 4. The simplicial manifold X • = • ×M associated to an action of a finite group on a smooth manifold M is given by X n = n ×M. The face maps are ⎧ ⎪ i =0 ⎨(g2 , . . . , gn , x) εi (g1 , . . . , gn , x) = (g1 , . . . , gi gi+1 , . . . , gn , x) i = 0, n ⎪ ⎩(g , . . . , g , g x) i = n. 1 n−1 n Each manifold X p = n ×M has an open cover U p = {(g1 , . . . , gn )×M} with its index set I p = p , where I 0 = {∗} consists of only one element. The sequence of the open covers U • = {U n }n≥0 gives rise to an open cover of X • = • ×M with the maps εi : I n → I n−1 defined by ⎧ ⎪ i =0 ⎨(g2 , . . . , gn ) εi (g1 , . . . , gn ) = (g1 , . . . , gi gi+1 , . . . , gn ) i = 0, n ⎪ ⎩(g , . . . , g ) i = n. 1 n−1 A set {B, (g), ω(g, h)}g,h∈ of differential forms B ∈ A2 (M), (g) ∈ A1 (M), ω(g, h) ∈ C ∞ (M, C∗ ), satisfying g ∗ B = B + d(g), (gh) = (h) + h ∗ (g) − d log ω(g, h), ω(g, hk)ω(h, k) = k ∗ ω(g, h)ω(gh, k), • ) on the simplicial manifold X • = • ×M. gives a Deligne cocycle c•B ∈ Z 2 (U • , F D(2) For non-trivial elements g, h in which commute with each other, we consider a smooth map T : [0, 1]×[0, 1] → M satisfying

gT (0, t2 ) = T (1, t2 ), hT (t1 , 0) = T (t1 , 1). Such maps are said to be in a twisted sector in the physics literature. The map T gives smooth maps v, v : 0 → × ×M, e, e : 1 → ×M, f : 2 → M, where v = {(g, h)}×T |(t1 ,t2 )=(0,0) , e = {g}×T |t1 =0 , f = T.

v = {(h, g)}×T |(t1 ,t2 )=(0,0) , e = {h}×T |t2 =0 ,

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From these maps with signs {−v, v , e, −e , f }, we get a cycle sT• ∈ Z 2 ( • ×M) by decomposing 1 ×1 into 2-simplices in a standard way. With the index map φ n given by φ 2 (v) = (g, h), φ 2 (v ) = (h, g), φ 1 (e) = g, φ 1 (e ) = h, φ 0 ( f ) = ∗, the integration of c•B along sT• is explicitly described as • • c B , sT = B+ (g) − (h) T

T |t1 =0

T |t2 =0

− log ω(g, h)(T (0, 0)) + log ω(h, g)(T (0, 0)). When B = 0, putting x = T (0, 0), we have

hx gx • • exp(c B , sT ) = exp (g) − (h) (ω(g, h)(x))−1 (ω(h, g)(x)). x

x

This is equal to the discrete torsion phase in [S1]. We remark that if B = 0 then the Deligne cocycle c•B is flat: F(c•B ) = 0. Therefore, in this case, the integration depends only on the homology class of the singular cycle sT• by Proposition 4.2. Moreover, when B = 0, (g) = 0, and ω(g, h) is constant on M, we have exp(c•B , sT• ) = (ω(g, h))−1 (ω(h, g)). This is equal to the discrete torsion phase in [V]. Similarly, a set {C, B(g), θ (g, h), ω(g, h, k)}g,h,k∈ of differential forms C ∈ A3 (M), B(g) ∈ A2 (M), θ (g, h) ∈ A1 (M), ω(g, h, k) ∈ C ∞ (M, C∗ ), satisfying g ∗ C = C + d B(g), B(gh) = B(h) + h ∗ B(g) + dθ (g, h), θ (g, hk) + θ (h, k) = k ∗ θ (g, h) + θ (gh, k) + d log ω(g, h, k), ω(g, h, kl)ω(gh, k, l) = ω(g, hk, l)ω(h, k, l)l ∗ ω(g, h, k), • ) on the simplicial manifold X • = • ×M. gives a Deligne cocycle cC• ∈ Z 3 (U • , F D(3) For elements g, h, k in which commute with each other, we consider a smooth map T : [0, 1]×[0, 1]×[0, 1] → M satisfying

gT (0, t2 , t3 ) = T (1, t2 , t3 ), hT (t1 , 0, t3 ) = T (t1 , 1, t3 ), kT (t1 , t2 , 0) = T (t1 , t2 , 1).


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The map T gives smooth maps v1 , . . . , v6 e1 , . . . , e6 f1 , f2 , f3 s

: : : :

0 → × × ×M, 1 → × ×M, 1 ×1 → ×M, 1 ×1 ×1 → M,

where v1 v3 v5 e1 e3 e5 f1 f3

= {(g, h, k)}×T |(t1 ,t2 ,t3 )=(0,0,0) , = {(k, h, g)}×T |(t1 ,t2 ,t3 )=(0,0,0) , = {(h, k, g)}×T |(t1 ,t2 ,t3 )=(0,0,0) , = {(g, h)}×T |(t1 ,t2 )=(0,0) , = {(g, k)}×T |(t1 ,t3 )=(0,0) , = {(h, k)}×T |(t2 ,t3 )=(0,0) , = {g}×T |t1 =0 , = {k}×T |t3 =0 ,

v2 = {(h, g, k)}×T |(t1 ,t2 ,t3 )=(0,0,0) , v4 = {(k, g, h)}×T |(t1 ,t2 ,t3 )=(0,0,0) , v6 = {(g, k, h)}×T |(t1 ,t2 ,t3 )=(0,0,0) , e2 = {(h, g)}×T |(t1 ,t2 )=(0,0) , e4 = {(k, g)}×T |(t1 ,t3 )=(0,0) , e6 = {(k, h)}×T |(t2 ,t3 )=(0,0) f 2 = {h}×T |t2 =0 , s = T.

These maps with signs {−v1 , v2 , v3 , −v4 , −v5 , v6 , −e1 , e2 , e3 , −e4 , −e5 , e6 , f 1 , − f 2 , f 3 , s} give a cycle sT• ∈ Z 3 ( • ×M) by decomposing 1 ×1 and 1 ×1 ×1 into 2 -simplices and 3-simplices, respectively. With the index map φ given by φ 3 (v1 ) = (g, h, k), φ 3 (v4 ) = (k, g, h), φ 2 (e1 ) = (g, h), φ 2 (e4 ) = (k, g), φ 1 ( f 1 ) = g, φ 0 (s) = ∗,

φ 3 (v2 ) = (h, g, k), φ 3 (v5 ) = (h, k, g), φ 2 (e2 ) = (h, g), φ 2 (e5 ) = (h, k), φ 1 ( f 2 ) = h,

φ 3 (v3 ) = (k, h, g), φ 3 (v6 ) = (g, k, h), φ 2 (e3 ) = (g, k), φ 2 (e6 ) = (k, h), φ 1 ( f 3 ) = k,

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the integration of cC• along sT• is explicitly described as cC• , sT• C+ = T

− −

T |t1 =0

T |(t1 ,t2 )=(0,0) T |(t1 ,t3 )=(0,0)

B(g) −

T |t2 =0

B(h) +

θ (g, h) +

T |(t1 ,t2 )=(0,0)

T |t3 =0

B(k)

θ (k, g) −

T |(t2 ,t3 )=(0,0)

θ (h, g) +

T |(t1 ,t3 )=(0,0)

θ (g, k)

θ (h, k) +

T |(t2 ,t2 )=(0,0)

θ (k, h)

− log ω(g, h, k)(T (0, 0, 0)) − log ω(h, g, k)(T (0, 0, 0)) + log ω(k, h, g)(T (0, 0, 0)) + log ω(k, g, h)(T (0, 0, 0)) + log ω(h, k, g)(T (0, 0, 0)) − log ω(g, k, h)(T (0, 0, 0)). When C = 0, putting x = T (0, 0, 0), we have ! exp(cC• , sT• )

= exp

T |t1 =0

B(g) −

× exp −

× exp −

kx

T |t2 =0

kx

T |t3 =0

x

hx

θ (g, k)

x

gx

θ (k, g) −

B(k)

θ (h, g) +

x

hx x

B(h) +

θ (g, h) +

x

"

gx

θ (h, k) +

θ (k, h)

x

×ω(g, h, k)(x)−1 ω(h, g, k)(x)−1 ω(k, h, g)(x) ×ω(k, g, h)(x)ω(h, k, g)(x)ω(g, k, h)(x)−1 , which is equal to the membrane discrete torsion phase in [S2]. If C = 0 then the Deligne cocycle cC• is flat, so that the integration depends only on the homology class of the singular cycle sT• . 5.2. Finally, we comment on a comparison between our approach and a traditional one. Traditionally, strings on an orbifold X/G are described by maps from a polyhedron # to X such that the edges are identified by the action of elements gi , h i ∈ G with i [gi , h i ] = 1:

Such a map corresponds to the following cycle in the simplicial manifold G • × X :


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Therefore, besides the discrete torsion phase, the rest of a string theory action on the orbifold, for example the energy term ∂φ i ∂φ j , d 2 σ G i j (φ)g ab a ∂σ ∂σ b make sense in this framework. Likewise, we can translate other traditional objects in terms of the simplicial manifold. The point is that G • × X is a concrete representative of E G ×G X , where E G is the universal G-bundle over the classifying space BG, which appears in the literature on strings on orbifolds. Acknowledgement. The authors would like to thank H. Kajiura and S. Terashima for helpful conversation. The authors are grateful to the referee for valuable comments. K.G. is supported by a JSPS Postdoctoral Fellowship for Research Abroad. Y.T. is supported in part by the Grants-in-Aid for Scientific Research, JSPS.

References [A] [B]

Alvarez, O.: Topological quantization and cohomology. Commun. Math. Phys. 100(2), 279–309 (1985) Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. Progress in Mathematics, 107. Boston, MA: Birkhäuser Boston, Inc., 1993 [D] Douglas, M.R.: D-branes and Discrete Torsion. http://arXiv.org/list/hep-th/9807235, 1998 [G] Gaw¸edzki, K.: Topological actions in two-dimensional quantum field theories. In: Nonperturbative quantum field theory, NATO Adv. Sci. Inst. Ser. B Phys., 185, New York: Plenum, 1988, pp. 101–141 [GT] Gomi, K., Terashima, Y.: Higher-dimensional parallel transports. Math. Res. Lett. 8(1-2), 25–33 (2001) [S1] Sharpe, E.: Discrete torsion. Phys. Rev. D (3) 68, no. 12, 126003, (2003) 20 pp [S2] Sharpe, E.: Analogues of discrete torsion for the M-theory three-form. Phys. Rev. D (3) 68(12), 126004, (2003) 12 pp [V] Vafa, C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys. B 273, 592–606 (1986)

Communicated by M. R. Douglas


Communications in


Genericity of Nondegeneracy for Light Rays in Stationary Spacetimes Roberto Giambò1 , Fabio Giannoni1 , Paolo Piccione2, 1 Dipartimento di Matematica e Informatica, Università di Camerino,

Via Madonna delle Carceri, 9, 62032 Camerino (MC), Italy. E-mail: [email protected]; [email protected]

2 Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010,

CEP 05508-900, São Paulo, SP, Brazil. E-mail: [email protected] Received: 10 April 2008 / Accepted: 5 November 2008 Published online: 18 February 2009 – © Springer-Verlag 2009

Abstract: Given a Lorentzian manifold (M, g), an event p and an observer U in M, then p and U are light conjugate if there exists a lightlike geodesic γ : [0, 1] → M joining p and U whose endpoints are conjugate along γ . Using functional analytical techniques, we prove that if one fixes p and U in a differentiable manifold M, then the set of stationary Lorentzian metrics in M for which p and U are not light conjugate is generic in a strong sense. The result is obtained by reduction to a Finsler geodesic problem via a second order Fermat principle for light rays, and using a transversality argument in an infinite dimensional Banach manifold setup. 1. Introduction Multiplicity of light rays from a light source to a receiver in a Lorentzian manifold models the so-called gravitational lensing effect in General Relativity. This is an active research field in both Physics and Geometry; a growing interest in this research area has been triggered in recent years by an increasing amount of observational material in Astrophysics. Some living reviews on the mathematical aspects (see [23]) as well as on the observational aspects (see [25]) of gravitational lensing are available on the web. Variational techniques apply to the light ray problem, which has a variational nature given by the Fermat principle. In particular, Morse theoretical results have been obtained in several contexts; Morse relations give lower estimates on the number of light rays issuing from a fixed event and terminating on a given observer. An essential assumption for the Morse theory of a functional f defined on a Hilbert manifold X is that all its critical points be (strongly) nondegenerate, in which case f is said to be a Morse function. In the light ray case, the functional to be studied is the so called arrival time (or departure time, according to the time orientation). The arrival time functional is defined on the set of all lightlike (future or past pointing) curves Current address: Department of Mathematics, University of Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain

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R. Giambò, F. Giannoni, P. Piccione

joining an event p and an observer U in a time oriented Lorentzian manifold (M, g). This is a Morse function precisely when p and U are not conjugate along any lightlike geodesic. If one fixes the ambient space (M, g), such non-conjugacy assumptions hold generically in the set of pairs ( p, U ), and this is established easily using the properties of the exponential map of (M, g). In this paper we address the question of establishing the genericity of the non-conjugacy property for a fixed pair ( p, U ) when the spacetime metric varies. This is no longer a finite dimensional problem, and it must be studied using an infinite dimensional Banach manifold approach. Before we get into the details of the result presented, let us discuss the original motivations that led us to this research. On one hand, it is clear that in view of physical applications, it is always desirable to have results that prove the stability of a theory by small perturbations of the data. Such data are possibly the outcome of an imperfect observation procedure in a physical experiment, or the result of some kind of theoretical approximation of the physical model. On the other hand, for the specific genericity problem discussed in the present paper, the main motivation comes from Morse theory. Unlike its finite dimensional counterpart, infinite dimensional Morse theory uses in an essential way several auxiliary data that do not belong properly to the variational problem itself. For instance, the choice of a specific Hilbert–Riemann structure in (an appropriate metric space completion of) the trial space, which does not belong specifically to the variational problem, is essential in order to guarantee: • the Palais–Smale condition, • the condition of transversality of the stable and unstable manifolds of critical points (Morse–Smale condition). As to the transversality of the stable and unstable manifolds, in many examples this is known to be generic in classes of variational problems having only nondegenerate critical points. Thus, it becomes an interesting question to establish the genericity of the nondegeneracy condition. For the Lorentzian geodesic variational problem, a recent result due to Abbondandolo and Majer (see [1,2]) shows that, under suitable technical assumptions, the homology of the Morse complex constructed using the dynamics of the gradient flow, is stable by uniformly (C 0 ) small perturbations of the Lorentzian metric tensor that preserve the non conjugacy property. An analogous result is likely to hold also in the case of light rays. This implies in particular that the Morse theory of light rays between a fixed event p and a fixed observer U is unchanged when the metric tensor varies in a connected open set containing a C 0 -dense subset of metrics for which p and U are non conjugate along any lightlike geodesic. With a result of this type at hand, clearly one would reduce the Morse theory for light rays to an analysis of the question in simple spacetimes, like for instance standard static products, avoiding the technical difficulties as encountered in [11–13,21,22] etc. Generic properties of geodesic flows have been studied mostly in the context of Riemannian geometry, see for instance [3,4,15,16]. Recently, B. White [26] has proved a genericity property of the nondegeneracy condition for minimal embeddings of higher dimensional submanifolds. Recently, the genericity of semi-Riemannian metrics for which two fixed endpoints are non conjugate has been proven in [8]. Nondegeneracy of (periodic) solutions of general flows on manifolds has been studied by several authors; a classical reference for this topic is the book [5]. As to the class of Hamiltonian systems, in which case the dynamics of the solutions may be different essentially in distinct energy levels, the nondegeneracy condition may fail to be generic, see [18]. As to the problem studied in this paper, the first observation is that the nonconjugacy property is easily stated in terms of transversality (Proposition 2.1, a manifold exten-

Genericity of Lightlike Nondegeneracy

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sion of the original result of B. White [26, Theorem 1.2]). Nevertheless, for the light ray problem there are two main obstructions to the classical genericity theory. First, the set of trial paths, i.e., the set of lightlike curves from a fixed event to a fixed observer, does not have the appropriate C 2 -regularity necessary to develop the theory. A second, more subtle, difficulty is the fact that, if one varies the spacetime metric, then it is the domain of the functional that changes, rather than the functional itself. This suggests that in order to study this problem one needs to restrict the class of spacetimes. More precisely, in order to maintain a fixed domain for the arrival time functional when the spacetime metric is changed, a natural assumption is that the spacetime admits a global splitting structure. In spacetime of this type there exists a bijection between lightlike curves and their projection onto the space component, which allows to keep the domain of the arrival time functional independent of the metric tensor. As an initial step for this research, we will consider Lorentzian manifolds (M, g) admitting a global splitting of the form M = M0 × R, with spacelike slices M0 × {t}, and with metric invariant by time translations ( p, t) → ( p, t + t0 ). These spacetimes form a class called standard stationary. Light rays in stationary spacetimes M0 ×R are associated naturally to Finsler geodesics in the base manifold M0 (see for instance the recent article [9] with full details on this topic), where the arrival time functional turns into the Finsler length functional. However, this does not solve the question of regularity of the functional. The Finlser metric on the base manifold M0 belongs to a special class of metrics called Randers metrics; Randers metrics are determined by the choice of a Riemannian metric g and a 1-form ω of norm less than 1 (see Sect. 3). For this class of metrics, the length functional, which is not smooth, can be replaced by a smooth functional Fg,ω (see (3.1)), as it was originally observed in [10]. A crucial point is then to establish that nondegenerate critical points of Fg,ω correspond to nondegenerate lightlike geodesics in M; this is proved by means of a second order variational principle (see Sect. 3.3). This principle says that elements in the kernel of the second variation of Fg,ω at a given critical point are the spatial component of a Jacobi field along the corresponding lightlike geodesic. The second order variational principle was originally stated and proved by A. Masiello, see [17], using an abstract argument. In this paper we reprove the result by an explicit computation of the index form and the differential equation satisfied by elements in its kernel, which is more suited for our purposes. A general remark on the existence of a second order Fermat principle is in order. The reader should recall that there exist other variational principles relating solutions of physical systems to geodesics in appropriate manifolds. Two classical examples are given by the Maupertuis principle and by the Kaluza–Klein principle. Maupertuis principle associates solutions of conservative dynamical systems in a fixed energy level to geodesics in conformal metrics. Kaluza–Klein principle associates trajectories in a relativistic electro-magnetic field of charged particles (having a fixed charge-to-mass ratio) to geodesics in a higher dimensional Lorentz manifold which is a principal fiber bundle over the spacetime. The interesting fact is that, in neither of these two cases, a second order principle holds, and thus nondegeneracy of solutions cannot be inferred from the nondegeneracy of the corresponding geodesic. In this sense, the Fermat principle for light rays is an exception. Using this second order principle, the genericity question is now studied in terms of the functionals Fg,ω ; these functionals are Fredholm (Proposition 3.2), and the genericity of the nondegeneracy condition for their critical point is established using an abstract criterion proved in [8,26] (see Proposition 2.1). This criterion involves the second mixed derivative of the functionals, computed explicitly in Sect. 4 (see (4.5), (4.6), (4.7)), and

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the existence of certain tensors on the manifold M0 whose value and covariant derivative have been assigned along a sufficiently small portion of an injective path. The existence of tensors with these data is established in a technical lemma (Lemma 4.1) proved by techniques of calculus with connections in affine manifolds. The main result of the paper, Theorem 4.4, states that, given a manifold M0 , the set of standard stationary metrics on M = M0 × R for which a fixed point ( p0 , 0) and a fixed observer U = { p1 } × R are joined only by nondegenerate lightlike geodesics is generic in a strong sense. More precisely, a standard stationary Lorentzian metric tensor on M0 × R is determined, up to an irrelevant conformal, by a Riemannian metric tensor g on the base manifold M0 and by the choice of a 1-form ω = g(δ, ·) on M0 (see (4.1)); here δ is a smooth vector field on M0 . We prove that, for fixed δ, the set of Riemannian metrics g for which the corresponding Lorentzian metric tensor has only nondegenerate lightlike geodesics between a fixed event ( p0 , 0) and a fixed observer U = { p1 } × R is generic. More surprisingly, we also prove that if one fixes g, then the set of δ’s for which the corresponding Lorentzian metric tensor has only nondegenerate lightlike geodesics between a fixed event ( p0 , 0) and a fixed observer U = { p1 } × R is generic. Let us conclude with some final remarks. First, we point out that we consider the case that the base manifold M0 is not necessarily compact. Note that in the noncompact case, there is no natural Banach space structure on the space of tensors of any type, and thus one needs to impose some restrictions on the growth at infinity of the metrics to be considered. This can be done in several (non canonical) ways. In order to preserve generality of our result, we axiomatize a few properties that characterize the type of Banach space of tensors that can be considered in the genericity result, with the introduction of the notion of an admissible triple of tensors (see Sect. 4.2). This is a very general notion, and it includes all possible types of Banach spaces of tensors with controlled C k -growth at infinity. Second, we remark that the genericity result is only proved in the case that the event ( p0 , 0) does not belong to the observer U = { p1 } × R, i.e., when p0 = p1 , although the result is very likely to be true also in this case. The proof presented in this paper fails in the case p0 = p1 due to a subtle technical point, which arises when one deals with lightlike geodesics whose spatial component is periodic. Given one such geodesic γ = (x, t) : [0, 1] → M0 × R with period T = 1/N , N ≥ 2, if there exists a nontrivial Jacobi field V = (ξ, σ ) along γ vanishing at the endpoints and with the property that N −1 i=0 ξt+i T = 0 for all t ∈ [0, T ], then the last part in the proof of Theorem 4.4 does not work. This suggests that nondegenericity of iterates should be dealt with by different (non variational) arguments, exactly as in the Riemannian case where the question of iterate closed geodesics has been treated in [4] by dynamical techniques. Finally, it is worth observing that an analogous degenericity result in the stationary Lorentzian manifold does not hold for geodesics of arbitrary causal character. In [8] an explicit example of a degenerate timelike geodesic in a standard static manifold whose degeneracy is preserved by arbitrary infinitesimal stationary perturbations of the metric tensor, is shown.

2. Notations and Preliminaries 2.1. Basic notations and references. Let M be a smooth Hausdorff paracompact manifold with dim(M) ≥ 2 and let ∇ be an arbitrarily fixed symmetric connection on T M. Given another (symmetric) connection ∇ on T M, there exists a (symmetric)


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(1, 2)-tensor on M defined by: ∇ = ∇ + , that will be called the Christoffel tensor of ∇ relative to ∇. An affine connection on a manifold M induces a connection on every vector bundle constructed by functorial construction on T M, like for instance all tensor bundles T M ∗ (r ) ⊗ T M (s) . These facts will be used in the proof of Lemma 4.1. Given a (semi-)Riemannian metric tensor g on M, we will denote by R the curvature tensor of the Levi–Civita connection of g, chosen with the sign convention: R(X, Y ) = [∇ X , ∇Y ] − ∇[X,Y ] . Finsler metrics in general do not determine compatible symmetric connections. A more general construction is obtained by considering the Chern connection associated to a Finsler metric, which is a connection defined on the vertical bundle over the tangent bundle of the Finsler manifold. In this paper we will consider a very special class of Finsler metrics, called Randers metrics (see Sect. 3), that are associated to the choice of a Riemannian metric tensor g and a 1-form ω. In this situation, it will be more convenient for the purposes of the present paper to use the Levi–Civita connection of g rather than Chern’s connection. Recommended references for all the background material in Lorentzian and semiRiemannian geometry are the textbooks [7,20]. Specific Finsler geometry techniques will be used only marginally in this paper; the basic reference for this topic is the book [6]. An extensive study of the relation between light rays in stationary spacetimes and geodesics in Randers metrics is carried out in the recent paper [9], which inspired parts of the present work. 2.2. Geodesics and Jacobi fields in stationary spacetimes. Let M be a differentiable manifold and g a Lorentzian metric tensor on M; (M, g) is said to be stationary if it admits a timelike Killing vector field. A Lorentzian manifold (M, g) is said to be standard stationary if M is given by a product M0 × R, where M0 is a differentiable manifold, and the metric tensor g is of the form: ¯ + gx (δ(x), v) r¯ + gx (δ(x), v) g(x,s) ((v, r ), (v, ¯ r¯ )) = gx (v, v) ¯ r − β(x)r r¯ , (2.1) where x ∈ M0 , s ∈ R, v, v¯ ∈ Tx M0 , r, r¯ ∈ Ts R ∼ = R, g is a Riemannian metric tensor on M0 , δ ∈ X(M0 ) is a smooth vector field on M0 , and β : M0 → R+ is a smooth positive function on M0 . The field Y = ∂s tangent to the lines {x0 } × R, x0 ∈ M0 , is a timelike Killing vector field in (M, g); an immediate computation shows that g(x,s) (Y, Y ) = −β(x) for all (x, s) ∈ M0 × R. We will endow (M, g) with the time orientation defined by the timelike vector field; Y ; thus, a causal curve (x, s) in M is future pointing when g (δ(x), x) ˙ − β s˙ is everywhere negative. Locally, every stationary Lorentzian metric tensor has the form (2.1). When the vector field δ in (2.1) vanishes identically on M0 , then the metric g is said to be standard static. Let ∇ be the Levi–Civita connection of the metric g in T M0 ; given a smooth map f 0 : M0 → R, denote by ∇ f 0 its gradient relative to the metric g and by H f0 (x) : Tx M0 → Tx M0 , x ∈ M0 , the Hessian of f 0 relative to g at the point x, which is the gx -symmetric linear operator on Tx M by H f0 (x)v = ∇v (∇ f 0 ), for all v ∈ Tx M0 . 0 given f If x is a critical point of f 0 , then gx H 0 (x)v, w = d2 f 0 (x)(v, w) is the standard second derivative of f 0 at x. Let us also recall the notion of Hessian of the smooth vector field δ, which is the (1, 2)-tensor on M0 : Hess(δ)(v, w) = ∇v ∇w˜ δ − ∇∇v w˜ δ,

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where v, w ∈ T M0 , and w is an arbitrary (local) extension of w to a vector field in M0 . We will need the symmetric part of the Hessian of δ, defined by: Hesss (δ)(v, w) =

1 2

[Hess(δ)(v, w) + Hess(δ)(w, v)] = Hess(δ)(v, w) − 21 R(v, w)δ,

for all v, w ∈ T M0 , where R is the curvature tensor of ∇. It will be convenient to introduce the following notation: by Hs (δ, v) : Tx M0 → M0 we will mean the linear operator defined by: Hs (δ, v)w = Hesss (δ)(v, w), for all x ∈ M0 and all v, w ∈ Tx M0 . Its g-adjoint, denoted by Hs (δ, v) , is defined by: g Hs (δ, v) w, z = g (Hs (δ, v)z, w) = g (Hesss (δ)(v, z), w), for all x ∈ M0 and all v, w, z ∈ Tx M0 . Moreover, the symbol Rs will denote the (1, 3)-tensor on M defined by Rs (a, b) : Tx M0 → Tx M0 : Rs (a, b)v =

1 2

(R(a, b)v + R(v, b)a),

for all x ∈ M0 and all a, b, v ∈ Tx M0 . Its g-adjoint Rs (a, b) is defined by: g Rs (a, b) v, w = g (Rs (a, b)w, v) for all x ∈ M0 and all a, b, v, w ∈ Tx M0 . A curve γ (t) = (x(t), s(t)) in M is a geodesic relative to the metric (2.1) if and only if its components x and s satisfy the system of differential equations: D dt x˙

+

D s δ) − s˙ (∇δ) (x) ˙ + 21 ∇β(x) s˙ 2 dt (˙

= 0,

d ˙ − β(x) s˙ ] = 0, [g (δ(x), x) dt (2.2)

D denotes covariant differentiation along x relative to the connection ∇, and (∇δ) where dt is the (1, 1)-tensor on M defined by g ((∇δ) (v), w) = g (∇w δ, w) for all v, w ∈ T M. Since Y is a Killing vector field, then for a geodesic γ = (x, s), the following quantity is constant:

cγ = g(γ˙ , Y ) = g(δ, x) ˙ − β(x)˙s ;

(2.3)

if γ is a future pointing causal geodesic, then cγ < 0. The second variation of the g-geodesic action functional at a geodesic γ (t) = (x(t), s(t)), t ∈ [0, 1], is given by: 1 D D ¯ Ig,δ,β (γ ) (ξ, σ ), (ξ , σ¯ ) = ˙ ξ¯ , x˙ + s˙ g Hesss (δ)(ξ, ξ¯ ), x˙ g dt ξ, dt ξ¯ + g R(ξ, x) 0 D¯ D ξ + σ¯ g ∇ξ δ, x˙ + σ g ∇ξ¯ δ, x˙ +˙s g ∇ξ δ, dt ξ + s˙ g ∇ξ¯ δ, dt D D ξ¯ , δ − σ¯ s˙ g (∇β(x), ξ ) − σ s˙ g ∇β(x), ξ¯ +σ¯ g dt ξ, δ + σ g dt

+˙s g R(ξ, x) ˙ ξ¯ , δ − 21 s˙ 2 g Hβ (x)ξ, ξ¯ − β(x)σ σ¯ dt , where ξ , ξ¯ are variational vector fields along x vanishing at the endpoints, and σ, σ¯ are smooth functions on [0, 1] vanishing at 0 and at 1. In the above formula and in the rest of the section we will denote by a dot the derivatives of the components x and s of the curve


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γ , and with a prime the derivatives of the component σ of the vector field V = (ξ, σ ) along γ . A pair V = (ξ, σ ) is a Jacobi field along the geodesic γ = (x, s) if it satisfies the second order linear system of differential equations: D D2 D s˙ ∇ξ δ − s˙ (∇δ) dt ξ − R(x, ˙ ξ ) x˙ − s˙ Hs (δ, ξ ) x˙ + dt ξ − σ (∇δ) (x) ˙ dt 2 D (σ δ) + σ s˙ ∇β(x) − s˙ Rs (ξ, x) ˙ δ + 21 s˙ 2 Hβ (x)ξ = 0, + dt

and d dt

D g ∇ξ δ, x˙ + g dt ξ, δ − s˙ g (∇β(x), ξ ) − β(x) σ = 0.

(2.4)

(2.5)

It is well known that lightlike geodesics and their conjugate points are invariant by conformal changes of the metric (see for instance [19, Theorem 2.36]). Thus, it will not be restrictive to assume in the remainder of the paper that β ≡ 1 in (2.1). Assume that γ = (x, s) is a future pointing lightlike geodesic, so that: g(x, ˙ x) ˙ + 2g(x, ˙ δ)˙s − s˙ 2 ≡ 0, s˙ > g(δ, x), ˙ and thus: s˙ = g(x, ˙ δ) +

g(x, ˙ x) ˙ + g(x, ˙ δ)2 = g(x, ˙ δ) − cγ .

(2.6)

(2.7)

Substituting (2.7) into (2.2) (with β ≡ 1) gives the following differential equation satisfied by the spatial part x of the lightlike geodesic γ = (x, s): D D (2.8) ˙ δ) δ) − g(x, ˙ δ)(∇δ) x˙ + cγ (∇δ) x˙ − (∇δ)x˙ = 0. dt x˙ + dt (g( x, Let V = (ξ, σ ) be a Jacobi field along γ ; from (2.4), (2.5) and (2.6), we obtain that ξ satisfies the following integro-differential equation: D2 D ξ − R(x, ˙ ξ )x˙ + (c0 − g(x, ˙ δ)) Hs (δ, ξ ) x˙ + (∇δ) dt ξ + Rs (x, ˙ δ)ξ 2 dt 1 D D A(x, ξ ) − − dt ˙ δ)) ∇ξ δ + dt A(x, ξ ) dt δ (c0 − g(x,

− A(x, ξ ) −

1

0

A(x, ξ ) dt (∇δ) x˙ = 0,

(2.9)

0

where A(x, ξ ) = g

D

dt ξ, δ

+ g x, ˙ ∇ξ δ .

(2.10)

2.3. An abstract genericity result for nondegenerate critical points. A subset of a metric space is said to be generic if it contains the intersection of a countable family of dense open subsets. Let us recall the following genericity result from [8,26]: Proposition 2.1. Let B be a Banach manifold, X a Hilbert manifold, A ⊂ B × X an open subset and F : A → R be a function of class C k , with k ≥ 2. Set Ab = {x ∈ H : (b, x) ∈ A}, let Fb : Ab → R be defined by Fb (x) = F(b, x) for all x ∈ Ab and set: C = (b0 , x0 ) ∈ A : x0 is a critical point of Fb0 .

910


Assume that for all (b0 , x0 ) ∈ C the following hypotheses are satisfied: (a) the Hessian d2 Fb0 (x0 ) = ∂∂ xF2 (b0 , x0 ) is Fredholm; (b) for all ξ ∈ Ker d2 Fb0 (x0 ) with ξ = 0 there exists β ∈ Tb0 B such that 2

∂2 F (b0 , x0 ) [(β, ξ )] = 0. ∂b ∂ x Then, denoting by : B × X → B the projection onto the first factor, the set of b ∈ (A) such that Fb is a Morse function is generic in (A). Proof. Condition (b) implies transversality of the map ∂∂ Fx : B × X → T X ∗ to the zero section of the cotangent bundle T X ∗ , thus C is an embedded submanifold of the product B × X . Together with assumption (a), it also implies that the projection |C : C → B is a nonlinear Fredholm operator of index 0, whose critical values are those b ∈ (A) for which Fb is not a Morse function. The result is then obtained as an application of the Sard–Smale theorem ([24]). See [26, Theorem 1.2] and [8, Sect. 3] for the details. 3. Light Rays and Randers Geodesic 3.1. Randers metrics. Let M0 be a differentiable manifold. A Randers metric on M0 is a pair (h, ω), where h is a Riemannian metric tensor on M0 and ω is a 1-form on M0 with ω p < 1 for all p ∈ M0 . Here, · is the norm of linear forms induced by h. Alternatively, a Randers metric can be described as a pair (h, X ), where X is a vector field on M0 with X p < 1 for all p ∈ M0 , the relation between X and ω being ω = h(X, ·). A Randers metric (h, ω) on M0 gives a Finsler metric f (h,ω) : T M0 → R obtained by setting: f (h,ω) (v) = h(v, v) + ω(v), v ∈ T M0 . Recall that a Finsler metric on M0 is a continuous function f : T M0 → [0, +∞[, smooth outside the zero section of T M0 , with f (v) = 0 for v = 0, which is positively homogeneous, and whose second derivatives in the directions of the vertical subbundle of T (T M0 ) are everywhere positive definite. A Finsler metric f defines a geometry on M0 in some aspects similar to the standard Riemannian geometry, but with interesting differences in several global properties. A Finsler geodesic is a smooth curve γ : [a, b] → M0 which (locally) minimizes its length L, defined by: b f (γ˙ (t)) dt. L(γ ) = a

Finsler geodesics are also stationary points of the functional: b f (γ˙ (t))2 dt Q(γ ) = a

defined in the space of paths γ joining two fixed endpoints. One should observe that neither L nor Q are functionals of class C 1 due to the lack of regularity of f . Stationary points of Q are parameterized by constant Finsler speed, i.e., f (γ˙ (t)) is constant along γ .


911

For the special class of Randers metrics, however, one can study geodesics in terms of critical points of a smooth functional. Namely, given a Randers metric (h, ω), the functional:

F(h,ω) (γ ) =

b

h(γ˙ , γ˙ ) dt

21

b

+

a

ω(γ˙ ) dt

(3.1)

a

is smooth, and its critical points (in the space of fixed endpoints paths) are geodesics relative to the Finsler metric f (h,ω) that are parameterized by constant Riemannian speed, i.e., h(γ˙ , γ˙ ) is constant along γ . 3.2. A first order Fermat principle. Let us now fix a Riemannian metric g and a smooth vector field δ on M0 ; the manifold M = M0 × R will be endowed with the stationary Lorentzian metric tensor (2.1) with β ≡ 1. Let p0 , p1 ∈ M0 be fixed, and consider the Hilbert manifold p0 , p1 (M0 ) consisting of all curves x : [0, 1] → M0 of Sobolev class H 1 and such that x(0) = p0 , x(1) = p1 . Consider the functional F : p0 , p1 (M0 ) → R defined by:

1

F(x) =

g(x, ˙ x) ˙ + g(x, ˙ δ) dt 2

21

0

1

+

g(x, ˙ δ) dt.

0

The reader will observe that F is the smooth functional (3.1) relative to the Randers metric (h, ω) defined by: h(v, w) = g(v, w) + g(δ p , v)g(δ p , w), ω p (v) = g(δ p , v),

(3.2)

for all p ∈ M0 and all v, w ∈ T p M0 . The Euler–Lagrange equation of F is given by: D dt x˙

+

D dt

˙ δ) δ) − g(x, ˙ δ)(∇δ) x˙ + cx (∇δ) x˙ − (∇δ)x˙ = 0 , (g(x,

(3.3)

where ˙ x) ˙ + g(x, ˙ δ)2 . cx (t) = − g(x,

(3.4)

If x is a critical point of F, the quantity cx is constant on [0, 1], and the following equality holds: 1 1 D D D g dt ξ, x˙ + g(x, ˙ δ) g dt ξ, δ + g x, ˙ ∇ξ δ dt = cx g dt ξ, δ + g x, ˙ ∇ξ δ dt 0

0

(3.5) for all vector fields ξ along x with ξ(0) = ξ(1) = 0. Remark. Observe that for every non constant critical point x of F the constant cx is strictly negative, which in particular implies that x is an immersion, i.e., x˙ never vanishes. Thus, when p0 = p1 , every critical point of F is an immersion. Comparing (3.3) with (2.8) gives the following result, originally proven in [10]:

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Proposition 3.1. (First order Fermat principle) The map γ = (x, s) → x gives a bijection from the set of future pointing lightlike geodesics in M = M0 × R from the event ( p0 , 0) and the observer U = { p1 }×R to the set of geodesics in M0 from p0 to p1 relative to the Randers metric (h, ω) given in (3.2). Its inverse is defined by x → γx = (x, sx ), where sx (t) = G x (t) − cx t and G x is the primitive of the function g(x, ˙ δ) satisfying G x (0) = 0. If x and γ are respectively a Randers geodesic and a lightlike geodesic related by the above bijection, then the constant cx corresponds to the constant cγ defined in (2.3). Moreover, the affine parameterization of γ corresponds to parameterization of x with constant Riemannian speed. Due to its geodesical nature, Eq. (3.3) shares many properties of a geodesic equation. For instance, the set of its solutions is invariant by affine reparameterizations, and this will be used systematically throughout (see for instance the proof of Lemma 4.3). 3.3. The second order Fermat principle. Let x ∈ x0 ,x1 (M0 ) be a critical point of F and let cx be the constant (3.4); the second variation of F at x is obtained by a direct computation, and it is given by:

1 1 D D d2 F(x) [ξ, η] = c1x g ∇ξ δ, x˙ + g dt ξ, δ dt g ∇η δ, x˙ + g dt η, δ dt − c1x

0

0

0

D D D ˙ x, ˙ η)+ g dt g dt ξ, dt η + g (R(ξ, x) ξ, δ + g ∇ξ δ, x˙ g dt η, δ + g ∇η δ, x˙ dt

1

D

D D ˙ δ)] g (Hesss (δ)(ξ, η), x) ˙ + g ∇ξ δ, dt η + g ∇η δ, dt ξ [cx − g(x, 0 ˙ δ) dt. +g (Rs (ξ, x)η,

+ c1x

1

(3.6)

Proposition 3.2. For all x ∈ x0 ,x1 (M0 ) critical point of F, the Hessian d2 F(x) is Fredholm. Proof. This is a standard argument. The first term in the second line of (3.6): 1 1 D D P[ξ, η] = − g dt ξ, dt η dt cx 0 is represented by a positive isomorphism of Tx x0 ,x1 (M0 ) (recall: cx < 0). The difference D[ξ, η] = d2 F(x) [ξ, η] − P[ξ, η] is represented by a compact operator of Tx x0 ,x1 (M0 ). Namely, the reader will observe that such bilinear form only contains at most one covariant derivative with respect to t of either ξ or η, i.e., each term of D[ξ, η] is continuous with respect to the C 0 -topology in one of its variables, and the claim follows from the fact that the inclusion H 1 → C 0 is compact. A straightforward computation using (3.6) shows that a vector field ξ ∈ Tx x0 ,x1 (M0 ) is in the kernel of d2 F(x) if and only if ξ is of class C 2 ; it vanishes at the endpoints ξ(0) = ξ(1) = 0, and it satisfies the integro-differential equation (2.9). Recalling (2.10), we have a bijective correspondence between vectors ξ in the kernel of d2 F(x) and Jacobi fields V = (ξ, σξ ) along the lightlike geodesic γ vanishing at the endpoints, where: t 1 A(x, ξ ) dr − t · A(x, ξ ) dr. σξ (t) = 0

We have therefore proven the following:

0


913

Proposition 3.3. (Second order Fermat principle) Let x ∈ p0 , p1 (M0 ) be a critical point of F, and let γ = (x, s) : [0, 1] → M = M0 × R be the corresponding future pointing lightlike geodesic. Then, γ (1) is conjugate to γ (0) along γ if and only if x is a degenerate critical point of F. Moreover, the multiplicity of γ (1) as a conjugate point along γ equals the dimension of the kernel of d2 F(x). The integro-differential equation (2.9) shares many properties of a standard Jacobi differential equation. For instance, all its solutions are defined globally. Moreover, the set of solutions of (2.9) along a periodic solution x of Eq. (3.3) with period T is invariant by T -translation. This fact will be used in formula (4.10).

4. Genericity of Lightlike Nondegeneracy 4.1. Tensors with arbitrarily prescribed value and covariant derivative along a curve. Let (M, ∇) be an affine manifold, i.e., M is a differentiable manifold and ∇ is a connection in the tangent bundle T M. Given nonnegative integers r, s, we will denote by T M ∗ (r ) ⊗ T M (s) the tensor product of r copies of T M ∗ and s copies of T M; sections of T M ∗ (r ) ⊗ T M (s) are called tensors of type (s, r ) on M. Let us show that one can construct globally defined tensors on M whose value and covariant derivative has been prescribed on a sufficiently short piece of curve. Lemma 4.1. Let (M, ∇) be an affine manifold, let γ : [a, b] → M be a smooth immersion, and let V be a smooth vector field along γ . Let t0 ∈ ]a, b[ be an instant at which Vt0 is not parallel to γ˙ (t0 ). Then, there exists an open interval I ⊂[a, b] containing t0 with the property that, given any pair of smooth sections H, K ∈ γ ∗ (T M ∗ (r ) ⊗ T M (s) ) over I , and given any open set U containing γ (I ), then there exists a globally defined (s, r )-tensor h on M with compact support contained in U , such that h γ (t) = Ht and ∇Vt h = K t for all t ∈ I . If both Ht and K t are symmetric or anti-symmetric relative to any of the variables for all t ∈ I , then the tensor h can be found to satisfy the same symmetries or anti-symmetries relative to the same variables. Proof. We choose an interval I containing t0 such that: • γ is injective on I ; • Vt is never parallel to γ˙ (t) for all t ∈ I ; • there exist: – a frame e1 , . . . , en of T M along γ | I , with e1 (t) = γ˙ (t) and e2 (t) = Vt for all t ∈ I; – a positive number ε and an open subset W containing γ (I ) which is the counterdomain of a system of Fermi coordinates: I ×]−ε, ε[n−1 (x1 , x2 , . . . , xn ) −→ expγ (x1 ) (x2 · e2 (x1 ) + · · · + xn · en (x1 )) ∈ W. Obviously, the open set W can be chosen arbitrarily small, i.e., contained in any prescribed open set U containing γ (I ). We will now define the desired tensor h as follows. First, we denote by d the connection in T M|W which is trivial in the Fermi coordinates (x1 , . . . , x2 ), and by d the Christoffel tensor of d relative to ∇. Given any (s, r )-tensor h with support in W , then its covariant derivatives relative to the connections ∇ and d

914


are related by the identity: dv h(α1 , . . . , αr , v1 , . . . , vs ) = ∇v h(α1 , . . . , αr , v1 , . . . , vs ) r h α1 , . . . , xd (v)∗ α j , . . . , αr , v1 , . . . , vs + −

j=1 s

h α1 , . . . , αr , v1 , . . . , xd (v)vk , . . . , vs ,

k=1

for all x ∈ W , all α1 , . . . , αr ∈ Tx M ∗ and all v, v1 , . .. , vs ∈ Tx M. ∈ γ ∗ (T M ∗ (r ) ⊗ T M (s) ) over the We will therefore define a smooth section K interval I by setting: t (α1 , . . . , αr , v1 , . . . , vr ) = K t (α1 , . . . , αr , v1 , . . . , vr ) K r Ht α1 , . . . , γd (t) (Vt )∗ α j , . . . , αr , v1 , . . . , vs − +

j=1 s

Ht α1 , . . . , αr , v1 , . . . , γd (t) (Vt )vk , . . . , vs ,

k=1

for all t ∈ I , all α1 , . . . , αr ∈ Tγ (t) M ∗ and all v1 , . . . , vs ∈ Tγ (t) M. The problem is now to determine

an (s, r )-tensor h with support contained in W , with h γ (t) = Ht and dVt h = d

∂ ∂ x2

h

coordinates (x1 , . . . , xn ), by setting:

γ (t)

t for all t ∈ I . This is an easy task, using =K

x1 , h(x1 , . . . , xn ) = Hx1 + x2 · β(x2 , . . . , xn ) K where β : W → R is a smooth nonnegative function with compact support, which is equal to one in a cylindrical neighborhood of I × 0n−1 (a segment on the x1 -axis) in I × ]−ε, ε[n−1 . Clearly, if both H and K have symmetries or anti-symmetries in any of the variables, then so does h.

4.2. Banach spaces of tensors. In order to state our genericity result, we need to introduce suitable Banach space structures on the set of tensors of class C k , k ≥ 2, on the manifold M0 . Since we are not assuming compactness for M0 , there is no canonical choice of such structure. We need two Banach spaces M and V satisfying the following axioms: • Elements of M are symmetric (0, 2) tensors of class C 2 on M0 ; M must contain all tensors of this type of class C ∞ and have compact support. • Elements of V are vector fields of class C 2 ; V must contain all vector fields of class C ∞ and have compact support. • Convergence in the norm of M and of V must imply C 2 -convergence on compact sets.


915

Moreover, we will need an open subset A of M whose elements are everywhere positive definite (0, 2) tensors, i.e., Riemannian metrics on M0 . For reference in the rest of the paper, we will call a triple (M, V, A) as above an admissible triple of tensors for the manifold M0 . Examples of the admissible triple of tensors (M, V, A) can be constructed as follows. Consider a fixed complete Riemannian metric g0 on M0 , and let ∇ 0 be a fixed symmetric connection on M0 , for instance the Levi–Civita connection of g0 . The metric g0 induces in a natural way a norm on each space (Tx M0∗ )(r ) ⊗(Tx M0 )(s) , and ∇ 0 induces a connection on every tensor bundle (T M0∗ )(r ) ⊗ (T M0 )(s) . Then, use these norms and connections to define M as the set of (0, 2)-symmetric tensors h of class C 2 on M0 such that: c0 (h) = sup h x < +∞, c1 (h) = sup ∇ 0 h x < +∞, x∈M0

x∈M0

c2 (h) = sup (∇ 0 )2 h x < +∞, x∈M0

and V the space of all vector fields V of class C 2 on M0 such that: d0 (V ) = sup Vx < +∞, d1 (V ) = sup ∇ 0 Vx < +∞, x∈M0

x∈M0

d2 (V ) = sup (∇ ) Vx < +∞. 0 2

x∈M0

A Banach space norm on M is given by: hM = max {c0 (h), c1 (h), c2 (h)} , and a Banach space norm on V is given by: V V = max {d0 (V ), d1 (V ), d2 (V )} . Completeness of the metrics · M and · V follows easily from the completeness of g. Clearly, M contains all symmetric (0, 2)-tensors of class C 2 with compact support, and V contains all vector fields of class C 2 with compact support. Moreover, convergence in these spaces implies C 2 -convergence on compact subsets of M0 . A typical open subset A of M consisting of everywhere positive definite tensors in given by: A = h ∈ M : inf λmin (h x ) > 0 , x∈M0

where λmin (h x ) is defined by: λmin (h x ) =

min

v∈Tx M0 gx (v,v)=1

h x (v, v).

Openness of A is proved easily using the fact that the map T −→ λmin (T ) = min T v, v ∈ R v,v=1

916


is Lipschitz continuous1 on the space of all symmetric operators T on a finite dimensional vector space with inner product ·, ·. Given an admissible triple of tensors (M, V, A) for M0 , then every element (g, δ) ∈ A defines a stationary Lorentzian metric tensor g g,δ on M0 × R by: g,δ

¯ + gx (δ(x), v) r¯ + gx (δ(x), v) g(x,s) ((v, r ), (v, ¯ r¯ )) = gx (v, v) ¯ r − r r¯ .

(4.1)

4.3. On the main result. We need a preliminary result concerning the distribution of instants along a lightlike geodesic at which the spatial component of a Jacobi field is parallel to the spatial component of the geodesic. Lemma 4.2. Let M = M0 × R be a standard stationary Lorentzian manifold endowed with the metric (2.1), let γ = (x, s) : [0, 1] → M0 × R be a lightlike geodesic, and let V = (ξ, σ ) be a Jacobi field along γ which is not everywhere parallel to γ˙ . Then, the set: ˙ T = {t ∈ [0, 1] : ξt is parallel to x(t)} is finite. Proof. We will show that the set T coincides with the set: {t ∈ [0, 1] : Vt is parallel to γ˙ (t)} ˙ which is finite (see [8, Lemma 2.5]). Assume that at some instant t one has ξt = λx(t) for some λ ∈ R and write σ = λ˙s + ρ. We will show that ρ = 0 as follows. First, observe that g(V, γ˙ ) ≡ 0, because g(V, γ˙ ) is an affine function vanishing at 0 and at 1. Thus, λg (x(t), ˙ x(t)) ˙ + 2λg (x(t), ˙ δ) s˙ + ρg (x(t), ˙ δ) − λ˙s (t)2 − s˙ (t)ρ = 0. Second, since γ is lightlike: s˙ = g(x, ˙ δ) ±

g(x, ˙ x) ˙ + g(x, ˙ δ)2 = g(x, ˙ δ) ∓ cx .

(4.2)

(4.3)

Substituting (4.3) into (4.2) gives: ρ · cx = 0, which yields ρ = 0. Here, we have cx = 0, because γ = (x, s) is a lightlike geodesic, and thus x is not constant. Note that Lemma 4.2 in particular applies when V is a nontrivial Jacobi field along γ such that V (0) = V (1) = 0; such a vector field is not everywhere parallel to γ˙ . As to the self-intersections of the spatial part of a lightlike geodesic, we have the following: Lemma 4.3. Let M = M0 × R be a standard stationary Lorentzian manifold endowed with the metric (2.1), let γ = (x, s) : [0, 1] → M0 × R be a lightlike geodesic. Then, either x is a periodic curve with period T < 1, or x has only a finite number of self-intersections. 1 The following inequality holds for all symmetric operators S and T : |λ min (T ) − λmin (S)| ≤ T − S.


917

Proof. Assume the existence of sequences (rn )n and (qn )n in [0, 1] with rn = qn and x(qn ) = x(rn ) for all n ∈ N. We can also assume that the limits lim rn = r and lim qn = q exist and, since x is an immersion and thus locally injective, that rn = r , qn = q for all n; clearly, x(q) = x(r ). Moreover, by the local injectivity, r = q, say r > q. For t near q, set y(t) = x(t + r − q); this is the spatial part of some lightlike geodesic in M with constant c y equal to cx . It is y(q) = x(r ) = x(q); moreover, qn = rn − r + q is a sequence tending to q with y(qn ) = x(rn ) = x(qn ) for all n. This implies that the tangent vectors y˙ (q) and x(q) ˙ are linearly dependent; since c y = cx , one deduces immediately that y˙ (q) = ±x(q), ˙ hence x(r ˙ ) = ±x(q). ˙ We claim that x(r ˙ ) = x(q). ˙ For, assume by contradiction x(r ˙ ) = −x(q); ˙ set w(t) = x(r − t) for t ∈ [q, r ]. Since w(r − q) = x(q) = x(r ) and w(r ˙ − q) = −x(q) ˙ = x(r ˙ ), by uniqueness we have x(r + q − t) = w(t − q) = x(t) for all t ∈ [q, r ]; computing the derivative at t = 21 (r + q) we have: −x˙

r +q 2

= x˙

r +q 2

⇒

x˙

r +q 2

= 0,

which is absurd, and proves that x(r ˙ ) = x(q). ˙ Again, a uniqueness argument shows that x has a periodic extension to R with period T = r − q; we need to show that T < 1. Assume by contradiction T = 1, i.e., q = 0 and r = 1. Consider x extended to the whole real line by periodicity; we have x(rn − 1) = x(rn ) = x(qn ) for all n ∈ N, which contradicts the local injectivity of x at 0. This concludes the proof. We will now set ourself the task of proving our main result on the genericity of the nondegeneracy condition for light rays. To this aim, we will use Proposition 2.1, whose assumption (b) involves the computation of a second mixed partial derivative, which is performed as follows. Let (M, V, A) be an admissible triple of tensors for M0 as defined in Sect. 4.2. Set g = (g, δ) ∈ A×V and consider the functional F : A×V× p0 , p1 (M0 ):

1

F(g, x) = 0

g(x, ˙ x) ˙ + g(x, ˙ δ) dt 2

21

+

1

g(x, ˙ δ) dt.

(4.4)

0

Let g0 = (g0 , δ0 ) ∈ A × V be fixed, and assume that x0 ∈ p0 , p1 (M0 ) is a critical point of F(g0 , ·). Let h = (h, ζ ) ∈ Tg0 A × Tδ0 V ∼ = M × V be an infinitesimal variationof g0 and ξ ∈ Tx0 p0 , p1 (M0 ) an infinitesimal variation of x0 . Then, setting

c0 = − g0 (x˙0 , x˙0 ) + g0 (x˙0 , δ0 )2 , the second mixed derivative computed as:

∂2 F ∂g ∂ x (x 0 , g0 ) [h, ξ ]

∂2 F (g0 , x0 ) [h, ξ ] ∂g ∂ x

1 D 1 D g0 x˙0 , dt ξ + g0 (x˙0 , δ0 ) g0 dt ξ, δ0 + g0 x˙0 , ∇ξ δ0 dt = 3 c0 0

1 · h(x˙0 , x˙0 ) + 2g0 (x˙0 , δ0 ) [h(x˙0 , δ0 ) + g0 (x˙0 , ζ )] dt 0

D 1 1 − ∇ξ h(x˙0 , x˙0 ) + 2h dt ξ, x˙0 2c0 0 D + 2 g0 dt ξ, δ0 + g0 x˙0 , ∇ξ δ0 [h(x˙0 , δ0 ) + g0 (x˙0 , ζ )] dt

is

(4.5)

918


D 1 1 − 2g0 (x˙0 , δ0 ) ∇ξ h(x˙0 , δ0 ) + h dt ξ, δ0 2c0 0 D + h x˙0 , ∇ξ δ0 + g0 dt ξ, ζ + g0 x˙0 , ∇ξ ζ dt 1 D D + ∇ξ h(x˙0 , δ0 ) + h dt ξ, δ0 + h x˙0 , ∇ξ δ0 + g0 dt ξ, ζ + g0 x˙0 , ∇ξ ζ dt. 0

Setting ζ = ∇ξ ζ ≡ 0 and h ≡ 0 in the right hand side of (4.5), we get: ∂2 F 1 (g0 , x0 ) [h, ξ ] = ∂g ∂ x 2c0

1 0

∇ξ h (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt;

(4.6)

while setting ζ ≡ 0 and h = ∇ξ h ≡ 0 in the right hand side of (4.5), we get: ∂2 F 1 (g0 , x0 ) [h, ξ ] = ∂g ∂ x c0

1 0

[c0 − g0 (x˙0 , δ0 )] g0 x˙0 , ∇ξ ζ dt.

(4.7)

Note that the quantity c0 − g0 (x˙0 , δ0 ) is strictly negative on [0, 1]. It should also be observed that the vector field W = 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 along x0 is never zero. Namely, if it were W = 0 at some instant, then at this instant it would be δ0 = λ · x˙0 for some (negative) real number λ, and therefore: 2 2 0 = g0 (W, x˙0 ) = 2 − g0 (x˙0 , x˙0 )(1 + λ ) − λ g0 (x˙0 , x˙0 ) − g0 (x˙0 , x˙0 ). However, since x0 is an immersion (Remark 3.2), the right hand side of the above formula is strictly negative, thus W never vanishes. We are finally ready for the statement and the proof of our main result. Theorem 4.4. Let M0 be a differentiable manifold, let p0 , p1 ∈ M0 be distinct points and let (M, V, A) be an admissible triple of tensors for M0 . Then, the set of all (g, δ) ∈ A such that the stationary Lorentzian metric g g,δ in M = M0 × R defined in (4.1) has only nondegenerate light rays from ( p0 , 0) to the observer U = {( p1 , s) : s ∈ R} is generic in A. More precisely, the following stronger result holds. For δ ∈ V, denote by Mδ ⊂ M the open set: Mδ = {g ∈ M : (g, δ) ∈ A}, and for g ∈ M, let Vg ⊂ V be the open set: Vg = {δ ∈ V : ∃ g ∈ M with (g, δ) ∈ A}. (1) For fixed g ∈ M, the set of δ ∈ Vg such that the metric tensor g g,δ defined in (4.1) has only nondegenerate light rays from the event ( p0 , 0) to the observer U = {( p1 , s) : s ∈ R} is generic in Vg. (2) For fixed δ ∈ V, the set of g ∈ Mδ such that the metric tensor g g,δ defined in (4.1) has only nondegenerate light rays from the event ( p0 , 0) to the observer U = {( p1 , s) : s ∈ R} is generic in Mδ .


919

Proof. The result will be proven as an application of Proposition 2.1, considering the function F : A × p0 , p1 (M0 ) → R defined by (4.4). By Proposition 3.3, nondegeneracy of all light rays from ( p0 , 0) to U for the metric tensor g g,δ is equivalent to the fact that the functional x → F g g,δ , x is Morse. Let g0 = (g0 , δ0 ) ∈ A be fixed, and let x0 ∈ p0 , p1 (M0 ) be a critical point of the functional x → F(g0 , x). The fact that the 2 second derivative ∂∂ xF2 (g0 , x0 ) is Fredholm, which is assumption (a) in Proposition 2.1, is proved in Proposition 3.2. As to assumption (b) in Proposition 2.1, in our geodesic setup this is translated into the following condition: given g0 = (g0 , δ0 ) ∈ A, a critical point x0 ∈ p0 , p1 (M0 ) of the functional x → F(g0 , x) and a non-trivial Jacobi field V = (ξ, σ ) along the corresponding lightlike geodesic γ = (x, s), with V (0) = V (1) = 0, then there should exist an element h = (h, ζ ) ∈ Tg0 A × Tδ0 V ∼ = M × V such that: ∂2 F (g0 , x0 ) [h, ξ ] = 0. (4.8) ∂g ∂ x More precisely, in order to prove statement (1) in the thesis, we need to show that inequality (4.8) holds for some choice of h of the form h = (0, ζ ) (i.e., h = 0), while statement (2) will be proved by showing that (4.8) will hold for some choice of h of the form h = (h, 0) (i.e., ζ = 0). With such choices of h, we will have to show that the right-hand side of formulas (4.6) and (4.7) are not zero. Since M (resp., V) contains all symmetric (0, 2)-tensors (resp., all vector fields) having compact support in M0 , it will suffice to search for smooth tensors h and vector fields ζ with the desired property and having compact support. Let us fix g0 , x0 and V = (ξ, σ ) as above; by the assumption that p0 = p1 , we have that x0 is not constant, and thus it is an immersion (Remark 3.2). We will distinguish two cases: when x0 is not periodic of period T < 1 and when x0 has period T < 1. Assume x0 not periodic of period T < 1. By Lemma 4.3, x0 has only a finite number of self-intersections. Using Lemma 4.2, we can find a non-empty open subset ]a, b[ ⊂ [0, 1] and an open subset U of M0 with the following properties: (i) x0−1 (U ) = ]a, b[; (ii) x0 |]a,b[ is an embedding; (iii) ξt is not parallel to x˙0 (t) for all t ∈ ]a, b[. As we have already observed, the vector field W = 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 along x0 never vanishes, and thus there exists a smooth field K of symmetric bilinear forms along x0 |[a,b] having compact support in ]a, b[ such that: b K t (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt = 0. a

Using (ii) and (iii), Lemma 4.1 tells us that there exists a globally defined smooth (0, 2)symmetric tensor h on M0 having compact support contained in U such that hx0 (t) = 0 and ∇ξt h = K t for all t ∈ ]a, b[. Then: 1 1 ∇ξ h (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt 2c0 0 b by (i) 1 = ∇ξ h (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt 2c0 a b 1 K t (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt = 0. = 2c0 a

920


Similarly, since [c0 − g0 (x˙0 , δ0 )] x˙0 never vanishes, there exists a smooth vector field K along x0 |[a,b] having compact support in ]a, b[ such that: b [c0 − g0 (x˙0 , δ0 )] g0 (x˙0 , K ) dt = 0. a

Another application of Lemma 4.1 gives us the existence of a globally defined smooth vector field ζ on M0 having compact support contained in U such that ζx0 (t) = 0 and ∇ξt ζ = K t for all t ∈ [a, b]. Thus: 1 1 [c0 − g0 (x˙0 , δ0 )] g0 x˙0 , ∇ξ ζ dt c0 0 b by (i) 1 = [c0 − g0 (x˙0 , δ0 )] g0 x˙0 , ∇ξ ζ dt c0 a 1 b [c0 − g0 (x˙0 , δ0 )] g0 (x˙0 , K ) dt = 0. c0 a This concludes the proof in the case that x0 is not periodic of period T < 1. Assume now that x0 is periodic, of period T < 1; for this case the proof goes along the same lines as the proof of [8, Prop. 4.3], that will be repeated here for the reader’s convenience. Consider the following numbers: t∗ = min {t > 0 : x0 (t) = q} , k∗ = max {k ∈ Z : kT < 1}, for which the following inequalities hold: ∗

≥ 1, 0 < t∗ < T, 1 = k∗ T + t∗ .

The curves x1 = x0 |[0,t∗ ] and x2 = x0 |[t∗ ,T ] join p1 and p2 (x2 with the opposite orientation), and the first part of the proof applies to both x1 and x2 . Thus, we can find open intervals I1 = [a1 , b1 ] ⊂ [0, t∗ ] and I2 = [a2 , b2 ] ⊂ [t∗ , T ] such that: (iv) t ∈ I1 , s ∈ ([0, t∗ ]\I1 ) ∪ [t∗ , T ] implies x0 (s) = x0 (t); (v) t ∈ I2 , s ∈ ([t∗ , T ]\I2 ) ∪ [0, t∗ ] implies x0 (s) = x0 (t). We can also find open subsets U1 , U2 ⊂ M, with x0 (Ii ) ⊂ Ui , i = 1, 2, satisfying: x0 (t) ∈ U1 ∩ x0 (I1 ) x0 (t) ∈ U2 ∩ x0 (I2 )

⇐⇒ ⇐⇒

∃ r ∈ {0, . . . , k∗ } such that t − r T ∈ I1 , ∃ r ∈ {0, . . . , k∗ − 1} such that t − r T ∈ I2 .

(4.9)

For j = 1, 2, consider the vector field η j along x j defined by: ηt1 =

k∗ r =0

ξt+r T , ηt2 =

k ∗ −1

ξt+r T .

(4.10)

r =0

Note that η1 and η2 are the spatial components of Jacobi fields V 1 and V 2 along a lightlike geodesics γ0 = (x0 , s0 ) in the stationary manifold M = M0 × R, and thus Lemma 4.2 applies in this situation. It is not the case that both η1 and η2 are everywhere parallel to x˙0 on I1 and I2 respectively, for otherwise from (4.10) one would conclude easily that ξ would be everywhere parallel to x˙0 , and this is not the case since V = (ξ, σ ) is a non trivial Jacobi field vanishing at 0 and 1 (Lemma 4.2). Assume that, say, η1 is not


921

everywhere parallel to x˙0 on I1 , i.e., by Lemma 4.2, there are only isolated values of t, where ηt1 is parallel to x˙0 (t); the other case is totally analogous. By reducing the size of I1 , we can assume that ηt1 is never a multiple of x˙0 (t) on I1 . Now, the first part of the proof can be repeated, by replacing the vector field ξ with η1 , as follows. Using Lemma 4.1, we can find a globally defined symmetric (0, 2)-tensor h on M0 , with compact support contained in U1 , with prescribed value hx0 (t) = 0 for t ∈ I1 , and covariant derivative ∇η1 h = K t along the curve x0 | I1 such that: t

b1

K t (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt = 0.

a1

The choice of such K is possible, due to the fact that both vectors [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 and x˙0 never vanish. For such a tensor h, we compute: 1 2c0

1

∇ξ h (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt

0

by (4.9)

=

k∗ b1 +r T 1 ∇ξ h (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt 2c0 a1 +r T r =0

=

k ∗ b1 1 ∇ξt+r T h (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt 2c0 a1 r =0

= =

1 2c0 1 2c0

b1

∇η1 h (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt t

a1 b1

K t (x˙0 , 2 [c0 − g0 (x˙0 , δ0 )] δ0 − x˙0 ) dt = 0.

a1

Similarly, by Lemma 4.1 one can find a globally defined vector field ζ on M0 , having compact support contained in U1 , with ζx0 (t) = 0 and with prescribed values ∇η1 ζ = K t t for all t ∈ I1 , where: b1 [c0 − g0 (x˙0 , δ0 )] g0 (x˙0 , K t ) dt = 0. a1

The choice of such K is possible, due to the fact that x˙0 never vanishes. For such a vector field ζ , one computes: 1 c0

1 0

[c0 − g0 (x˙0 , δ0 )] g0 x˙0 , ∇ξ ζ dt

by (4.9)

=

=

1 c0

k∗ b1 +r T 1 [c0 − g0 (x˙0 , δ0 )] g0 x˙0 , ∇ξ ζ dt c0 a1 +r T

r =0 k ∗ b1 r =0 a1

[c0 − g0 (x˙0 , δ0 )] g0 x˙0 , ∇ξt+r T ζ dt

922


1 = c0 =

1 c0

b1

a1 b1

[c0 − g0 (x˙0 , δ0 )] g0 x˙0 , ∇η1 ζ dt t

[c0 − g0 (x˙0 , δ0 )] g0 (x˙0 , K t ) dt = 0.

a1

The case in which η2 is not everywhere parallel to x˙0 is analogous, and this concludes the proof. A similar genericity result may be proved also in specific classes of stationary Lorentzian metric tensors, for instance the class of globally hyperbolic tensors. This can be done along the lines of an analogous result proved in [8, Sect. 4.5]. Acknowledgement. The authors thankfully acknowledge the help provided by E. Caponio, M. A. Javaloyes and A. Masiello during uncountable conversations.

References 1. Abbondandolo, A., Majer, P.: A Morse complex for infinite dimensional manifolds. I. Adv. Math. 197(2), 321–410 (2005) 2. Abbondandolo, A., Mejer, P.: A Morse complex for Lorentzian geodesics. Asian J. Math. 12, 299–320 (2008) 3. Abraham, R.: Bumpy metrics. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Providence, R.I.: Amer. Math. Soc., 1970, pp. 1–3 4. Anosov, D.V.: Generic properties of closed geodesics. Izv. Akad. Nauk SSSR Ser. Mat. 46(4), 675–709, 896 (1982) 5. Abraham, R., Robbin, J.: Transversal mappings and flows. New York: W. A. Benjamin, 1967 6. Bao, D., Chern, S.S., Shen, Z.: An introduction to Riemannian–Finsler geometry. Graduate Texts in Mathematics, New York: Springer-Verlag, 2000 7. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry. Vol. 202 of Monographs and Textbooks in Pure and Applied Mathematics, second ed., New York: Marcel Dekker Inc., 1996 8. Biliotti, L., Javaloyes, M.A., Piccione, P.: Genericity of nondegenerate critical points and Morse geodesic functionals. Preprint 2008, available at http://arXiv.org/abs/0804.3724v2, 2008 9. Caponio, E., Javaloyes, M.A., Masiello, A.: Variational properties of geodesics in non-reversible Finsler manifolds and applications. http://arXiv.org/abs/math/0702323v2, 2007 10. Fortunato, D., Giannoni, F., Masiello, A.: A Fermat principle for stationary space-times and applications to light rays. J. Geom. Phys. 15(2), 159–188 (1995) 11. Giannoni, F., Masiello, A., Piccione, P.: A Morse Theory for light rays in Stably Causal Lorentzian manifolds. Ann. Inst. H. Poincaré, Phys. Theor. 69, 359–412 (1998) 12. Giannoni, F., Masiello, A., Piccione, P.: A Morse Theory for massive particles and photons in General Relativity. J. Geom. Phy. 35, 1–34 (2000) 13. Giannoni, F., Masiello, A., Piccione, P.: The Fermat Principle in General Relativity and applications. J. Math. Phy. 43, 563–596 (2002) 14. Hasse, W., Perlick, V.: A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole. J. Math. Phys. 47(4), 042503 (2006) (17 pages) 15. Klingenberg, W.: Lectures on closed geodesics. Grundlehren der Mathematischen Wissenschaften, Vol. 230, Berlin: Springer-Verlag, 1978 16. Klingenberg, W., Takens, F.: Generic properties of geodesic flows. Math. Ann. 197, 323–334 (1972) 17. Masiello, A.: Variational Methods in Lorentzian Geometry. Pitman Research Notes in Mathematics, 309, London: Longman, 1994 18. Meyer, K.R., Palmore, J.: A generic phenomenon in conservative Hamiltonian systems. In: Global Analysis, Proc. Sympos. Pure Math., Vol. XIV, (Berkeley, Calif., 1968), Providence, RI: Amer. Math. Soc., 1970, pp. 185–189 19. Minguzzi, E., Sanchez, M.: The causal hierarchy of spacetimes. In: Recent developments in PseudoRiemannian geometry, Bawm, H., Alekseevsky, D. (eds.) Zurich: Eur. Math. Soc. Publ., 2008 20. O’Neill, B.: Semi-Riemannian geometry, Vol. 103 of Pure and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1983


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21. Perlick, V.: Infinite-dimensional Morse theory and Fermat’s principle in general relativity. I. J. Math. Phys. 36(12), 6915–6928 (1995) 22. Perlick, V.: Application of Morse theory to gravitational lensing. In: Current topics in mathematical cosmology (Potsdam, 1998), River Edge, NJ: World Sci. Publ., 1998, pp. 155–163 23. Perlick, V.: Gravitational Lensing from a Spacetime Perspective. Living Rev. Relativity 7, 9, available at http://www.livingreviews.org/lrr-2004-9, 2004 24. Smale, S.: An infinite dimensional version of Sard’s theorem. Amer. J. Math. 87, 861–866 (1965) 25. Wambsganss J.: Gravitational Lensing in Astronomy. Living Rev. Relativity 1, 12, available at http:// www.livingreviews.org/lrr-1998-12, 1998 26. White, B.: The space of minimal submanifolds for varying Riemannian metrics. Indiana Univ. Math. J. 40, 161–200 (1991) Communicated by G. W. Gibbons


Communications in


The Massless Higher-Loop Two-Point Function Francis Brown Institut de Mathematiques de Jussieu, UMR 7586, Université Pierre et Marie Curie-Paris 6, F-75005 Paris, France. E-mail: [email protected] Received: 10 April 2008 / Accepted: 10 November 2008 Published online: 7 February 2009 – © Springer-Verlag 2009

Abstract: We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with 6th roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph K 3,4 at one edge. 1. Introduction Let n 1 , . . . , nr ∈ N and suppose that nr ≥ 2. The multiple zeta value is the real number defined by the convergent nested sum: ζ (n 1 , . . . , nr ) =

0

1 m i−1 4 m i−2

for all i ≤ i 0 .

1 −1/2 0, 2 λ , p ∈ N be a complex-valued function such Corollary 2. Let f ∈ p ( p) that f (0) = 0 and max | f | ≤ C( p) λ 2 +1 for some constant C( p). Then for any ϕ0 , ϕ1 ∈ R, 1 1 λ−1/2 1 2 p+3 5 λ− 2 −1 −1 | f (t) − eiϕ1 t+iϕ0 |2 dt ≥ min 4− p− 2 , 4− 2 6 p C( p) p λ p holds. 9 0 (45) Cp

Proof. Let u = Re f and v = Im f . If max | f | ≥ 6, then at least one the expressions max |u|, max |v| is larger than or equal to 3. Without loss of generality we assume that max |u| ≥ 3 and apply Proposition 1 to the function u. If u satisfies (41), then there 1 1 exists an subinterval I ⊂ [0, 21 λ−1/2 ] of the length 3−1 4− p− 2 λ− 2 on which |u| ≥ 3/2. This implies 1 λ−1/2 2 1 1 | f (t) − eiϕ1 t+iϕ0 |2 dt ≥ 3−1 4− p− 2 λ− 2 . 0

If, on the other hand, u satisfies (42), then the length of the subinterval of [0, 21 λ−1/2 ], p−1

−1

−1− 1

on which |u| ≥ 3/2, is at least 3−1 4− 2 C( p) p λ 2 p , which gives 1 λ−1/2 2 p−1 − 1 −1− 1 | f (t) − eiϕ1 t+iϕ0 |2 dt ≥ 3−1 4− 2 C( p) p λ 2 p . 0

Assume now that max | f | < 6. The latter means that max |u| < 6 and max |v| < 6. Since u(0) = v(0) = 0, there exists a subinterval of [0, 21 λ−1/2 ], on which max{|u(t)|, |v(t)|} ≤ 1/3, which implies | f (t) − eiϕ1 t+iϕ0 |2 ≥ 1/4. Applying Proposition 1 to the functions u, v we find out that the length of this interval is bounded from below by 1 p+3 5 1 − 1 −1− 1 min 3−2 4− p− 2 λ− 2 , 3−2 4− 2 6 p C( p) p λ 2 p . This completes the proof.

972

H. Kovaˇrík, S. Vugalter, T. Weidl

With the above auxiliary results at hand, we can finally prove the following estimate, which will play a central role in the proof of Theorem 1 and 2. Proposition 2. Let f ∈ C p+1 [ω] be a complex valued function such that f (0, x2 ) = 0 for each x2 and ∂ p+1 f ∂ p f 1 p 1+ 2p ≤ β p+1 λ , p ≤ βp λ 2 + 2 p+1 2 ∂x ∂ x1 2 1 L (ω) L (ω) for some positive β p and β p+1 . Then the inequality 2 1 p 3 5 1 − 1 −1− 1p min 4− p− 2 λ−1 , 4− 2 − 2 6 2 p (β 2p+1 +β 2p ) 2 p λ f −ei(ξ1 x1 +ξ2 x2 +ϕ) 2 ≥ L (ω) 36 (46) holds true for all ξ1 , ξ2 , ϕ ∈ R. Proof. The measure of the set ⎧ √1 ⎨ 2 λ x2 ∈ [0, λ−1/2 ] : ⎩ 0

⎫ ∂ i f (x , x ) 2 ⎬ 3 1 2 2 i+ 2 d x ≤ 8 β λ , i ∈ { p, p + 1} 1 i ⎭ ∂ x1i

1

λ− 2 . For such x2 by Lemma 2, " ∂ p f (x , x ) √ 1 2 1+ 2p ≤ max 3 λ β 2p+1 + β 2p holds. p x1 ∂ x1

is obviously at least

1 4

Corollary 2 then implies the statement. 4.3. Proof of Theorem 1. Proof of Theorem 1. Fix λ > 0. Let λ j be the eigenvalues of the Dirichlet Laplacian on and let ψ j be the corresponding normalised eigenfunctions. For k ∈ N we define F(ξ ) =

k

|ψˆ j (ξ )|2 ,

j=1

where ψˆ j denotes the Fourier transform of ψ j . Moreover, we denote by F ∗ (|ξ |) the decreasing radial rearrangement of F(ξ ). Let ψ(x) = ci ψi (x), with |ci |2 ≤ V. λi ≤λ

µi ≤λ

For each j = 1,√. . . , n we choose on the middle part of p j several points tl such that dist(tl , tl+1 ) = 2 λ−1/2 for all l and denote by Tl the squares with the side 21 λ−1/2 constructed in the middle point between tl and tl+1 , see Fig. 2. We note that for each j the number of these squares is at least 1 1 Nj = √ lj λ2 . 3 2

Two-Dimensional Berezin-Li-Yau Inequalities with Correction Term

973

According to Corollary 1 for each l and p we have p+1 2 ∂ ψ A p ( p)V 2 p+2 λ , ≤ ∂ν p+1 2 4π L (Tl ) where ∂ψ ∂ν denotes the normal derivative of ψ. In view of Proposition 2 and Corollary 3 we get 2 1 p 3 5 1 − 1 −1− 1p min 4− p− 2 λ−1 , 4− 2 − 2 6 2 p (β 2p+1 +β 2p ) 2 p λ , (47) ≥ ψ−eiξ ·x 2 L (Tl ) 36 where A p ( p)V 2 . 4π

β 2p+1 =

We continue by estimating the sequence A p ( p). A direct inspection shows that A p ( p) ≤ c0 2( p+1) , c0 = 7 · 1022 . 2

1

This implies that (β 2p+1 + β 2p )− 2 ≥ 2−1/2 p=

#

(48)

√ −1/2 −1 −( p+1)2 /2 π c0 V 2 . Hence for

$ 3π − 12 c 2 log2 (V λ/c1 ) − 1, c1 = 2 0

we obtain 2 ψ − eiξ ·x 2

L (Tl )

≥

2−3 −1 c V 36 1

Vλ c1

−1− √

2 log2 (V λ/c1 )

.

Taking λ large enough such that dj , 3

λ−1/2 ≤

we make sure that the squares Tl lie inside and that they do not overlap each other. Summing this inequality for all l = 1, . . . , N j and all j = 1, . . . , n we thus arrive at 2 V − 4π 2 F ∗ (|ξ |) ≥ ψ − eiξ ·x 2 ≥ c2 V with c2 =

−1/2 2−3 √ c . 9 2 36 1

1 2

L (e )

Vλ c1

− 1 − √ 2

2 log2 (V λ/c1 )

n j=1

9 lj λ − 2 dj

(49)

This yields the following upper bound on F ∗ :

⎡ ⎤ − 1 − √ 2 n 2 log2 (V λ/c1 ) 1 V λ V 9 ⎣1−c2 V − 2 F ∗ (|ξ |) ≤ M( p, λ) := l j λ− 2 ⎦. 4π 2 c1 dj j=1

(50)

974


Now we use the minimiser (10) with V /4π 2 replaced by M( p, λ) to obtain

k

λj ≥

j=1

R2

F ∗ (|ξ |)|ξ |2 dξ ≥

λ2 V 2 . 8π 3 M( p, λ)

(51)

Employing the definition of M( p, λ) we then find out that k

λj ≥

λ2 V 2π

3

+ c2 c12 V − 2

Vλ c1

3−√ 2

2 log2 (V λ/c1 )

n

lj λ −

j=1

j=1

9 d 2j

.

(52)

Next we set λ = λk and note that inequality (2) yields 2π k ≤ λk . V

(53)

Since the right hand side of (52) is an increasing function of λ, we can use (53) to conclude that k

λj ≥

2π V

k 2 + 4 c3 k

3 √ 2 2 − log (2π k/c ) 2 1

n

lj k −

j=1

j=1

9V 2π d 2j

V −3/2 ,

(54)

where 5 1/4 2−3 c3 = √ (2π ) 4 c1 . 9 2 36

Finally, we combine inequalities (54) and (16) to get (21).

5. Proof for General Domains From now on we suppose that is a general domain satisfying Assumption A. To prove a Li-Yau type inequality with the correction term we cannot directly employ the approach invented for polygons, since ∂ is in general nowhere straight. However, we can extend by adding small “bumps” to certain parts of ∂, see Fig. 4. In this way we obtain an extended domain e whose boundary is in certain parts represented by straight lines. On these straight pieces of ∂e we will then employ the same strategy as in the case of polygons. Due to the monotonicity of eigenvalues, any lower bound on the sum of the eigenvalues on the extended domain gives also a lower bound on the sum of the eigenvalues on . On the other hand, we have to make sure that the volume of e is not much bigger than V , because otherwise it could destroy the effect of the correction term in (26) by decreasing the leading term. We will again split the exposition in several steps.


975

5.1. Step 1: Some geometrical remarks. Here we will show that ∂ ∩ j can be locally represented as a graph of a certain C 2 −smooth function. Let = {x1 (s), x2 (s)} be a part of the boundary of parametrised by its length s and such that x1 (s), x2 (s) ∈ C 2 (R+ ). Let κ0 :=

max

{x1 ,x2 }∈

|κ(x1 , x2 )|

be the maximal curvature of . We consider certain points A = {x1 (s ), x2 (s )} ∈ and B = {x1 (s ), x2 (s )} ∈ and choose a new system of coordinates (u, v) such that A = (0, 0) and the u−axes go along the line AB. Lemma 4. Assume that κ0 |s − s | ≤ π/4. Then the following statements hold true: (i) The part of connecting A and B can be written in the system of coordinates (u, v) as v = v(u), u ∈ [0, u 0 ], where u 0 = |AB|. Moreover, we have √ (55) max v(u) ≤ 2 κ0 u 20 . u∈[0,u 0 ]

(ii) The inequality 2−1/2 |s − s | ≤ |AB| ≤ |s − s |

(56)

holds. Proof. Let {u(s), v(s)} be the parametrisation of in the coordinates (u, v). By assumption we have |s −s | κ(s) ds ≤ π/4. (57) 0

This means that for any s ∈ [0, |s − s |] the angle between the tangent of at the point {u(s), v(s)} and the u−axes is less than or equal to π/4. Assume that there exists s1 , s2 ∈ [0, |s − s |] such that u(s1 ) = u(s2 ). Then there exists s3 ∈ [s1 , s2 ] such that the tangent of at {u(s3 ), v(s3 )} is orthogonal to the u−axes. The latter contradicts (57). This shows that the part of between A and B can be considered as the graph of the function v = v(u), u ∈ [0, u 0 ], v(0) = v(u 0 ) = 0 . This proves the first part of (i) and, in view of (57), it shows that |v (u)| ≤ 1 holds for any u ∈ [0, u 0 ]. It thus follows that u0 1/2 1 + |v (u)|2 du ≤ 21/2 u 0 , u 0 = |AB| ≤ |s − s | = 0

which implies (56). To prove (55) we note that v(u) is twice differentiable and therefore there exists some u 1 ∈ [0, u 0 ], such that v (u 1 ) = 0. Since |v (u)| = |κ(u)| (1 + |v (u)|2 )3/2 ≤ 23/2 κ0 , we obtain u |v (u)| du ≤ 23/2 κ0 u 0 ∀u ∈ [0, u 0 ]. |v (u)| ≤ u1

The last inequality together with the fact that v(0) = v(u 0 ) = 0 imply |v(u)| ≤

1 3/2 2 κ0 u 20 = 21/2 κ0 u 20 2

∀u ∈ [0, u 0 ] .

976


Fig. 3. Tiling of j

5.2. Step 2: Approximation of the boundary. Here we introduce a procedure that allows us to choose appropriate parts of ∂ ∩ j on which we will construct the additional “bumps”, see Fig. 4. Let j , j = 1 . . . m be the parts of boundary defined in Sect. 3 j with the end points A j , B j and the partition ai , i = 0, . . . , n j . We fix j ∈ {1, . . . , m} and take λ large enough, such that dj 3π π − 21 (58) λ ≤ min , if L( j ) > , √ 3 8 2 κj 8κ j and λ

− 21

d j L( j ) 3π , if L( j ) ≤ ≤ min . , √ 3 8κ j 3 2

j

(59)

j

Let us consider j ∩ (ai , ai+1 ) with 0 < i < n j . On this part of the boundary we choose √ several disjoint arcs (bl , bl ), see Fig. 3, such that each of them has the length 2 λ−1/2 and such that √ 1 j j j j s(bl , bl ) ≥ s(ai , ai+1 ), s(ai , ai+1 ) − s(bl , bl ) ≤ 2 λ−1/2 , 3 l

l

where s(a, b) denotes the arc-length between a and b. We now pick an l and connect bl and bl with a straight line and choose a local system of coordinates (y1 , y2 ) so that the y1 −axis goes along the straight line from bl to bl and j j j j π , which the origin is in bl , see Fig. 5. Notice that s(ai−1 , ai+1 ) = s(ai , ai+2 ) ≤ 2κ j according to Lemma 4 means that in the chosen coordinate system the boundary between j j ai−1 and ai+2 can be written explicitly as y2 = f (y1 ). Let y0 = dist(bl , bl ). In view of Lemma 4 we have √ 3 max | f (y1 )| ≤ 2 κ j y02 ≤ 2 2 κ j λ−1 . y1

Now we introduce

3 1 = (y1 , y2 ) : y1 ∈ [0, y0 ], y2 = 2 2 κ j λ−1

and

3 2 = (y1 , y2 ) : y1 ∈ [0, y0 ], y2 = −2 2 κ j λ−1 .


977

Fig. 4. Construction of the extended domain e . The thick line represents the boundary of e

Lemma 5. If λ > 6κ j /d j , then 1 ∩ ∂ = 2 ∩ ∂ = ∅ . j

j

Proof. Obviously 1 and 2 do not cross ∂ between ai−1 and ai+2 . On the other hand, for each point P = (y1P , y2P ), dist(P, (ai , ai+1 )) ≤ 23/2 κ j λ−1 holds. j

j

j j j j Since dist (ai , ai+1 ), ∂\(ai−1 , ai+2 ) ≥ d j , this implies dj j j > 0. dist P, ∂\(ai−1 , ai+2 ) ≥ d j − 23/2 κ j λ−1 > 2 The last lemma says that one of the sets 1 and 2 is inside and the other one is outside . Without loss of generality we assume that 1 is outside . 5.3. Step 3: Extended domain e . The extended domain e differs from if λ is large enough so that (58) respectively (59) is satisfied (otherwise it coincides with ). To define e we proceed as follows. For a fixed j ∈ {1, . . . , m}, fixed i ∈ {1, . . . , n j − 1} and fixed l, we consider the boundary between the points bl and bl . If it is a straight line, we do not change it. Otherwise we replace this piece of the boundary with the segment i , where i is such that i is outside , and connect the end points of 1 , b ) and b˜ ∈ (b , b ) with appropriate with the boundary at certain points b˜l ∈ (bl−1 l l l l+1 2 C functions, see Fig. 4. We choose these functions and the points b˜l , b˜l in such a way that the added area to is less than 3 times the area of the rectangle with the corners given by bl , bl and the end points of 1 . We then obtain a new region whose boundary, corresponding to the original piece j is again C 2 −smooth and which between the original boundary points bl and bl consists of a straight line, see Fig. 4. Repeating this procedure for all j , j = 1, . . . , m, all i ∈ {1, . . . , n j − 1} and all l we thus obtain a new domain e . As a next step we construct the squares Tl of the side 21 λ−1/2 between the points bl and √ bl centred in the middle, see Fig. 5. Note that, according to Lemma 4, |bl bl | ≥ λ−1/2 / 2. We have Lemma 6. The squares Tl do not overlap.

978


Fig. 5. The thick lines represent the square Tl . The dashed line represents the set 1

Proof. First we show that every Tl does not overlap any of the squares constructed on j j the part of the boundary different from the arch (ai−1 , ai+2 ). Indeed, each point of Tl has

distance to (bl , bl ) at most 21 λ−1/2 and the distance between (bl , bl ) and ∂\(ai−1 , ai+2 ) is at least d j . Since λ−1/2 < d j , see (58), the result follows. j j Consider now the arc (ai−1 , ai+2 ). This part can be written as y2 = f (y1 ) in the above introduced coordinate system. Consider the squares Tl1 and Tl2 with l1 = l2 . Let y11 be the y1 coordinate of the middle point between bl1 and bl 1 and let y12 be the y1 j

j

coordinate of the middle point between bl2 and bl 2 . Since | f (y1 )| ≤ 1 on (ai−1 , ai+2 ), we have |y11 − y12 | ≥ λ−1/2 . For all points (y1 , y2 ) ∈ Tl1 , |y1 − y11 | ≤ 41 λ−1/2 holds and j

√

for all points (y1 , y2 ) ∈ Tl2 , |y1 − y12 | ≤ we conclude that Tl1 ∩ Tl2 = ∅.

2 2

j

λ−1/2 holds. Collecting these inequalities

As a consequence of the last result we obtain estimates on the volume of e , which will be used in the proof of Theorem 2. Corollary 3. Let V e be the volume of the extended domain e . Then 3

V e ≤ V + 2 2 λ−1

m

κ j L( j ) .

(60)

j=1

Moreover, if λ ≥ 1 := 9 · 210 max κ 2j , j

then V e ≤ 2V .

(61)

Proof. Inequality (60) follows directly from the construction of e , since the area of the 3 added volume along j does not exceed 2 2 λ−1 κ j L( j ). As for the second inequality, we consider each pair bl , bl and note that for λ ≥ 9 · 210 κ 2j is the volume of the part added between the points b˜l and b˜l , see Fig. 4, bounded from above by 3

12 κ j λ− 2 ≤

1 −1 λ . 8


979

Fig. 6. Construction of the local coordinate system at the boundary of e

This follows from the choice of the points bl , see Sect. 5.3. On the other hand, for λ chosen as above we get 1 1 |Tl | = λ−1 . 2 8 Since Tl do not overlap, we obtain (61). |Tl ∩ | ≥

5.4. Proof of Theorem 2. Proof of Theorem 2. Fix λ > 0 and consider the extended domain e . Let µ j be the eigenvalues of the Dirichlet Laplacian on e and let φ j be the corresponding normalised eigenfunctions. For k ∈ N we define Fe (ξ ) =

k

|φˆ j (ξ )|2 ,

j=1

where φˆj denotes the Fourier transform of φ j . By Fe∗ (|ξ |) we denote the decreasing radial rearrangement of Fe (ξ ). Let ci φi (x), with |ci |2 ≤ V e , φ(x) = µi ≤λ

µi ≤λ

and let Tl be the sequence of squares constructed along j . For each j is the number of these squares at least 1 1 2 N j = √ L( j ) λ . 9 2 Next we take λ ≥ 1 so that V e ≤ 2V , see Corollary 3. According to Corollary 1 for each l and p we then have p+1 2 ∂ φ A p ( p)(V e )2 p+2 A p ( p)V 2 p+2 λ λ ≤ ≤ , ∂ν p+1 2 4π π L (Rn ) where ∂φ ∂ν denotes the normal derivative of φ. From Proposition 2 it follows that the inequality 2 1 p 3 5 1 − 1 −1− 1p min 4− p− 2 λ−1 , 4− 2 − 2 6 2 p (β 2p+1 + β 2p ) 2 p λ , ≥ φ − eiξ ·x 2 L (Tl ) 36

980


with β 2p+1 =

A p ( p)(V e )2 A p ( p)V 2 ≤ 4π π

holds for each l. Now we employ the arguments used in the proof of Theorem 1 in order to find an appropriate upper bound on Fe∗ . Since λ ≥ 1 we can use Corollary 3 to arrive at ⎤ ⎡ − 1 − √ 2 m 2 log2 (V λ/c1 ) 1 V λ V c 2 ⎣1 + Fe∗ (|ξ |) ≤ 23/2 V −1 κ j λ−1 − V − 2 L( j )⎦. 4π 2 2 c1 j=1

Note that for λ ≥ 2 := 22 c1 V −1 6

we have

Vλ c1

− 1 − √ 2

2 log2 (V λ/c1 )

≥

Vλ c1

− 3 4

, and therefore

Fe∗ (|ξ |) ≤ Me ( p, λ) ⎤ ⎡ − 1 − √ 2 m 2 log (V λ/c ) 1 Vλ V ⎣ c2 − 2 1 V 2 := L( j ) (λ − 3 ( j))⎦, 1− 4π 2 4 c1 j=1

where

3 ( j) := max 1 , 2 , c1−1 222 68 κ 4j V .

We now use again the Li-Yau type minimiser (10) with V /4π 2 replaced by Me ( p, λ) to obtain k

λj ≥

j=1

k

µj ≥

j=1

R2

Fe∗ (|ξ |)|ξ |2 dξ ≥

λ2 V 2 . 8π 3 Me ( p, λ)

As in the proof of Theorem 1 we set λ = λk and use the definition of Me ( p, λ) together with inequalities (58),(59) and (53) to obtain k

λj ≥

2π V

k 2 + c3 k

3 √ 2 2 − log (2π k/c ) 2 1

m j=1

L( j ) (k − k( j)) V −3/2 ,

j=1

where

2 V 9 128 κ j 6κ j max 3 ( j), 2 , k( j) := , . 2π π2 dj dj

Finally, we combine inequalities (62) and (16) to get (26). Acknowledgement. The support from the DFG grant WE 1964/2 is gratefully acknowledged.

(62)


981

References 1. Berezin, F.A.: Covariant and contravariant symbols of operators. Izv. Akad. Nauk SSSR Ser. Mat. 36, 1134–1167 (1972) 2. Freericks, J.K., Lieb, E.H., Ueltschi, D.: Segregation in the Falicov-Kimball model. Commun. Math. Phys. 227, 243–279 (2002) 3. Goldbaum, P.: Lower bound for the segregation in the Falicov-Kimball model. J. Phys. A. 36, 2227–2234 (2003) 4. Ivrii, V.: Microlocal analysis and precise spectral asymptotics. Springer Monographs in Mathematics. Berlin: Springer-Verlag, 1998 5. Ivrii, V.: The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary. (Russian) Funk. Anal. i Pril. 14(2), 25–34 (1980) 6. Kröger, P.: Estimates for sums of eigenvalues of the Laplacian. J. Funct. Anal. 126, 217–227 (1994) 7. Laptev, A.: Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J. Funct. Anal. 151, 531–545 (1997) 8. Laptev, A., Weidl, T.: Recent results on Lieb-Thirring inequalities. Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), Exp. No. XX, 14 pp., Nantes: Univ. Nantes, 2000 9. Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88, 309–318 (1983) 10. Lieb, E., Loss, M.: Analysis. 2nd Ed., Providence, RI: Amer. Math. Soc., 2001 11. Melas, A.D.: A lower bound for sums of eigenvalues of the Laplacian. Proc. Amer. Math. Soc. 131, 631–636 (2003) 12. Melrose, R.B.: Weyl’s conjecture for manifolds with concave boundary. In: Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Providence, RI: Amer. Math. Soc., 1980, pp. 257–274 13. Pólya, G.: On the eigenvalues of vibrating membranes. Proc. London Math. Soc. 11, 419–433 (1961) 14. Safarov, Yu., Vassiliev, D.: The asymptotic distribution of eigenvalues of partial differential operators. Translations of Mathematical Monographs, 155. Providence, RI: Amer. Math. Soc., 1997 15. Weidl, T.: Improved Berezin-Li-Yau inequalities with a remainder term. Preprint: arXiv: 0711.4925. To appear in Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. (2), to be published Jan 2009 16. Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71, 441–479 (1912) Communicated by B. Simon


Communications in


The Cauchy Two-Matrix Model M. Bertola1,2, , M. Gekhtman3, , J. Szmigielski4, 1 Centre de recherches mathématiques, Université de Montréal,

C.P. 6128, succ. centre ville, Montréal, Québec, Canada H3C 3J7. E-mail: [email protected]; [email protected]

2 Department of Mathematics and Statistics, Concordia University,

1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8

3 Department of Mathematics, 255 Hurley Hall, Notre Dame,

IN 46556-4618, USA. E-mail: [email protected]

4 Department of Mathematics and Statistics, University of Saskatchewan,

106 Wiggins Road, Saskatoon, Saskatchewan, S7N 5E6 Canada. E-mail: [email protected] Received: 14 April 2008 / Accepted: 17 November 2008 Published online: 10 February 2009 – © Springer-Verlag 2009

Abstract: We introduce a new class of two(multi)-matrix models of positive Hermitian matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann–Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitian matrix model is related to a hyperelliptic curve. Contents 1. 2. 3.

4. 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Biorthogonal Polynomials . . . . . . . . . . . . . . . . . 2.1 Christoffel–Darboux identities . . . . . . . . . . . . . . . . . 2.2 Riemann–Hilbert characterization of the integrable kernels . . Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Correlation functions: Christoffel–Darboux kernels . . . . . . 3.1.1 Correlation functions in terms of biorthogonal polynomials 3.2 A multi–matrix model . . . . . . . . . . . . . . . . . . . . . . Diagrammatic Expansion . . . . . . . . . . . . . . . . . . . . . . Large N Behaviour . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. 984 . 986 . 988 . 989 . 992 . 995 . 998 . 1000 . 1001 . 1002

Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Work supported in part by NSF Grant DMD-0400484. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC),

Grant. No. 138591-04.

984

M. Bertola, M. Gekhtman, J. Szmigielski

5.1 Continuum version: cubic spectral curve and solution problem . . . . . . . . . . . . . . . . . . . . . . . . . A. A Rectangular Mixed 3–Matrix Model with Ghost Fields . A.1 Integer h < N . . . . . . . . . . . . . . . . . . . . . . A.2 Half–integer h < N . . . . . . . . . . . . . . . . . . . A.3 Integer and half integer h > N . . . . . . . . . . . . . B. Relation to the O(1)–Model . . . . . . . . . . . . . . . . .

of the potential . . . . . . . . 1004 . . . . . . . . 1009 . . . . . . . . 1009 . . . . . . . . 1010 . . . . . . . . 1010 . . . . . . . . 1010

1. Introduction In the last two decades or so, the theory of matrix models have been an incredibly fertile ground for a fruitful interaction between theoretical physics, statistics, analysis, number theory and dynamical systems (see the classical book [36] and references therein). The interplay with analysis has been particularly beneficial for the Hermitian matrix model [15], largely due to the realization that the matrix model could be “solved” in terms of orthogonal polynomials. This allowed to translate questions about the spectrum of large random matrices into questions about the asymptotics of orthogonal polynomials, connecting the former with a very well developed area of analysis. The availability of Riemann–Hilbert methods [14,29] was the crucial ingredient in addressing questions of universality in the bulk and at the edge of the spectrum. It should be mentioned that this fortunate symbiosis relies on the following features: • the possibility of rewriting the matrix integral in terms of eigenvalues and the correlation functions in terms of suitable orthogonal polynomials; • the Riemann–Hilbert characterization of orthogonal polynomials [29]; • the Christoffel–Darboux formula, allowing to rewrite the kernel of the correlation functions in terms of only two orthogonal polynomials of degree N ; • the nonlinear steepest descent method [14] applied to the RH problem for orthogonal polynomials. There are by now several matrix models based on various ensembles of matrices. For some (Symplectic, Orthogonal), a connection with (skew-)orthogonal polynomials can be established [36,38] and some of the steps in the above list have been performed, but typically only for a certain restricted class of “potentials”. For others (like O(n) models, rectangular models) the methods used rely on symmetries of the integral and dynamical system approaches (the “loop equations”) [1,17–19,21] There are also the so–called multi-matrix-models which involve ensembles of several matrices (typically Hermitian) and one of the most studied among them is the Itzykson– Zuber–Harish-Chandra (IZHC) chain of matrices. In its simplest form, the two-matrix model, it amounts to the study of the spectral properties of a pair of Hermitian matrices of size N × N with a measure dµ(M1 , M2 ) = dM1 dM2 e−N Tr(V1 (M1 )+V2 (M2 )−M1 M2 ) ,

(1.1)

where the interaction term is e N TrM1 M2 . The first three items of the bulleted list above can be implemented at least for potentials whose derivative is a rational function [5– 7,20] while the last item is still not fully under control. One of the main reasons for this difficulty is that the size of the RHP depends on the potentials: for example if V j are polynomials of degree d j then there are two relevant RHPs of the same size. Clearly the question arises as to whether a Riemann–Hilbert method can be utilized for a more

The Cauchy Two-Matrix Model

985

general class of potentials, for example real-analytic, like in the case of the single-matrix model. The present paper is a part of a larger project initiated in [9], which puts forward a new multimatrix model (at the moment primarily two matrices) that can be completely solved along the lines described above. While the new model is very close in spirit to the IZHC model, the different interaction I (M1 , M2 ) =

1 det(M1 + M2 ) N

(1.2)

links the model to a new class of biorthogonal polynomials which were studied in [9] and termed “Cauchy biorthogonal polynomials”. These polynomials have several of the desirable features of classical orthogonal polynomials. We will fill in more details in Sect. 3 but here we just indicate that the model we want to study is defined on the space of pairs of positive Hermitian matrices of size N equipped with a measure dµ(M1 , M2 ) := dM1 dM2

α(M1 )β(M2 ) det(M1 + M2 ) N

(1.3)

for arbitrary measures α(x)dx, β(y)dy on R+ (to be understood in the formula above as conjugation-invariant measures on positive definite Hermitian matrices or, equivalently, measures on the spectra of Mi ’s). This immediately puts the problem at the same level of generality as the classical case. Corresponding to the above positive measure is the normalizing factor, customarily called the partition function, α(M1 )β(M2 ) Z N := dM1 dM2 (1.4) det(M1 + M2 ) N whose dependence on the measures α, β carries all relevant information about the model. We will show that this model is related to Cauchy biorthogonal polynomials α(x)dx β(y)dy = δmn pn (x)qm (y) (1.5) x+y R+ R+ defined and studied in [9] in relation with the spectral theory of the cubic string and the Degasperis-Procesi wave equation (see also [3,4,34,35]). Contrary to the biorthogonal polynomials of [7,8,20] the properties of these polynomials do not depend on the measures α, β and the main highlights are 1. 2. 3. 4. 5. 6.

they solve a four-term recurrence relation; their zeroes are positive and simple; their zeroes have the interlacing property; they possess Christoffel–Darboux identities relevant to matrix models; they can be characterized by a (pair of) Riemann–Hilbert problem(s) of size 3 × 3; the steepest descent method is fully applicable.

Our recent paper [9] contains a detailed discussion of all these points except the last one which will be addressed in a forthcoming publication. The paper is organized as follows; in Sect. 2 we review our previous results [9] on the Cauchy biorthogonal polynomials and the relevant formulæ and features needed in the following. In particular the Christoffel–Darboux identities (Sect. 2.1), the Riemann– Hilbert characterization in terms of 3 × 3 piecewise analytic matrices (Sect. 2.2). In

986


Prop. 2.7 we introduce the 3 × 3 matrix kernel in terms of the solution of the Riemann– Hilbert problems; this will be used in later sections to describe the spectral statistics of the two matrices. In Sect. 3 we introduce in detail the two–matrix model that we have outlined above and show how to reduce the study of its spectral statistics to Cauchy biorthogonal polynomials using a formula appeared in [26]. We also indicate (without details) how to deal with a similar model where R matrices are linked in a chain (Sect. 3.2). In Sect. 4 we show how a formal treatment of the partition function of the model for large sizes of the matrices can be used to extract combinatorial information for certain bi-colored ribbon graphs (as it has been done for the Hermitian one-matrix model in [12,33] and for the multi-matrix models with exponential coupling in [17]). In Sect. 5 we show that a saddle–point treatment of the partition function (see for example [16] for the one-matrix case) leads to a three–sheeted covering of the spectral plane (a trigonal curve); this (pseudo) algebraic curve plays the same rôle as the hyperelliptic curve in the Hermitian matrix model. The Appendices are devoted to the relation between our proposed model and other matrix models with rectangular matrices (and possibly with Grassmann entries, App. A) and a connection (App. B with the O(1) model of self-avoiding loops ([19] and references therein). Remark 1.1. For a particular case of measures α(x) = x a e−x and β(y) = y b e−y the corresponding biorthogonal polynomials appeared (in a somewhat disguised form) in the work [11]. As observed therein they are related to the classical Jacobi orthogonal polynomials for the weight x a+b dx on [0, 1]. We thank A. Borodin for pointing out this connection. 2. Cauchy Biorthogonal Polynomials 1 Let K (x, y) = x+y be the Cauchy kernel on R+ × R+ . It is known that it is totally positive in the sense of the classical definition [23,32]:

Definition 2.1. A totally positive kernel K (x, y) on I × J ⊂ R × R is a function such that for all n ∈ N and ordered n-tuples x1 < x2 < · · · < xn , y1 < y2 < · · · < yn we have the strict inequality det[K (xi , y j )] > 0.

(2.1)

It was shown in [9] that for any totally positive kernel K (x, y) (hence also for the Cauchy kernel) and for any pair of measures α(x)dx, β(y)dy supported in I, J ⊂ R+ respectively, the matrix of bimoments Ii j = x i y j K (x, y)α(x)dxβ(y)dy (2.2) I J

is a totally positive matrix, that is, every square submatrix has a positive determinant. This guarantees the existence of biorthogonal polynomials { p j (x), q j (y)} of exact degree j such that p j (x)qk (y)K (x, y)α(x)dxβ(y)dy = δ jk . (2.3) I J


987

The polynomials p j (x), q j (y) are defined uniquely up to a C× action p j → λ j p j (x), q j → λ1j q j and the ambiguity can be disposed of by requiring that their leading coefficients are the same and positive. With this understanding we have pn (x) = and qn = c1n y n + · · · , with the positive constant cn given by [9]

1 n cn x

+ ···

Dn+1 , Dn := det I jk 0≤ j,k≤n−1 , Dn := x j y k K (x, y)α(x)dxβ(y)dy.

cn = I jk

R+ R+

(2.4)

The determinantal expressions for the BOPs in terms of bimoments can be obtained using Cramer’s rule [9]. Proposition 2.1. (Thm. 4.5 in [9]) For any totally positive kernel K (x, y) the zeroes of the biorthogonal polynomials p j ’s (q j ’s) are simple, real and contained in the convex hull of the support of the measure α (β respectively). 1 In the case K (x, y) = x+y we named the corresponding polynomials { p j (x), q j (y)} Cauchy BOPs and proved, in addition,

Proposition 2.2. (Thm. 5.2 in [9]) The roots of adjacent polynomials in the sequences { p j (x)}, {q j (y)} are interlaced. Cauchy BOPs enjoy more structure than the BOPs associated to a generic kernel: they solve a four term recurrence relation of the form x(πn−1 pn (x)−πn pn−1 (x)) = an(−1) pn+1 (x)+an(0) pn (x) + an(1) pn−1 (x) + an(2) pn−2 (x), y(ηn−1 qn (y) − ηn qn−1 (y)) = bn(−1) qn+1 (y) + bn(0) qn (y) + bn(1) qn−1 (y) + bn(2) qn−2 (y), (2.5) where πn :=

R+

pn (x)α(x)dx ,

ηn :=

R+

qn (y)β(y)dy.

(2.6)

As proved in [9] πn , ηn are strictly positive. The other coefficients appearing in the recurrence relations are described in loc. cit. We also need to introduce certain auxiliary polynomials, qn , pn whose defining properties are 1. deg qn = n + 1, deg pn = n; (y) qn (y) qn (y) = qn+1 2. qn dβ = 0 or ηn+1 − ηn ; dαdβ qm (y) pn−1 − pn ) = pn . 3. pn (x) = δmn or η1n ( x+y

988


In addition pn , qn admit the determinantal representations: ⎡ ⎤ I00 . . . I0n+1 .. ⎢ .. ⎥ ⎢ . ⎥ . 1 ⎢ qn (y) = det ⎢ In−1 0 . . . In−1 n+1 ⎥ √ ⎥, ηn ηn+1 Dn Dn+2 ⎣ β ... β ⎦ 0 n+1 1 ... y n+1 ⎡ ⎤ I00 . . . I0 n 1 .. ⎥ ⎢ .. ⎢ . . ⎥ 1 ⎢ pn (x) = det ⎢ I n−1 ⎥ ⎥. . . . I x n−1 0 n−1 n Dn+1 ⎣ ⎦ n In0 . . . In n x β0 . . . βn 0

(2.7)

(2.8)

After lengthy manipulations one obtains several Christoffel–Darboux–like identities (CDIs) for Cauchy BOPs which play a crucial rôle in what follows and hence will be carefully described. 2.1. Christoffel–Darboux identities. Using recurrence coefficients featured in Eqs. (2.5) we define the following 3 × 3 matrices: ⎤ ⎡ ⎡ ⎤ (2) (2) bn+1 0 0 an+1 0 0 ⎥ ⎢ (−1) ⎢ (−1) ⎥ (0) (0) An (x) = ⎣ −bn−1 −bn + η x 0 ⎦ , Bn (y) = ⎣ −an−1 −an + π y 0 ⎦. n+1

0

−bn(−1)

n+1

0

0

−an(−1)

0 (2.9)

In addition we define the following integral transforms: qn (ζ )βdζ qn(1) (−x) (1) (2) qn (w) := ; qn (w) := α(x)dx, R+ w − ζ R+ w + x (1) pn (ξ )αdξ pn (−y) (1) (2) pn (z) := ; pn (z) := β(y)dy , z−ξ z+y R+ R+ and the following vectors

⎤ (µ) qn−2 (x) ⎥ ⎢ (µ) q n(µ) (x) = ⎣ qn−1 (x) ⎦ , (µ) qn (x) ⎡

(0)

(0)

⎤ (µ) pn−2 (x) ⎥ ⎢ (µ) (µ) p n (x) = ⎣ pn−1 (x) ⎦ , (µ) pn (x)

(2.10) (2.11)

⎡

(2.12)

where µ = 0, 1, 2 and qn ≡ qn , pn ≡ pn . For α(x), β(y) we define α (x) = α(−x) and β (y) = β(−y); next we define the Weyl functions (or Markov-functions) as 1 1 β(y)dy = −Wβ (−z), Wα ∗ (z) = α(x)dx = −Wα (−z), Wβ (z) = z−y z+x 1 α(x)β(y)dxdy, (2.13) Wα ∗ β (z) = − (z + x)(x + y) 1 Wβα ∗ (z) = α(x)β(y)dxdy. (z − y)(y + x)


989

Proposition 2.3. (Thm 7.3 in [9]) The following Christoffel–Darboux-like identities hold (z + w)

n−1

(µ)

n(µ) (w) · A(−w) · pn (ν) (z) − F(w, z)µ,ν q j (w) p (ν) j (z) = q

(2.14)

j=0

⎡

⎤ 0 0 1 ⎦ , (2.15) 1 Wβ ∗ (z) + Wβ (w) F(w, z) = ⎣0 1 Wα (z) + Wα ∗ (w) Wα ∗ (w)Wβ ∗ (z) + Wα ∗ β (w) + Wβ ∗ α (z) where the auxiliary vectors marked with a hat are characterized by a Riemann–Hilbert problem described below. Corollary 2.1. (Thm 7.4 in [9]) Evaluating (2.14) on the “antidiagonal” z = −w gives the perfect duality, q n(µ) (w) · A(−w) · pn (ν) (−w) = Jµν , ⎡ ⎤ 001 J := ⎣ 0 1 0 ⎦ . 100

(2.16)

Remark 2.1. The proposition above defines, in fact, 9 identities, but we will only need the 4 identities corresponding to µ, ν = 0, 1 (i.e. the principal submatrix of size 2 × 2).

2.2. Riemann–Hilbert characterization of the integrable kernels. The sums appearing on the left-hand side in Prop. 2.3 are all examples of a general framework of “integrable kernels” that were studied in great generality in [25,27,28,30,31]. Proposition 2.4 (Prop. 8.1 in [9]). Consider the Riemann–Hilbert problem (RHP) of finding a matrix (w) such that 1. (w) is analytic on C\(supp(β) ∪ supp(α )). 2. (w) satisfies the jump conditions ⎡

1 (w)+ = (w)− ⎣ 0 0 ⎡ 1 (w)+ = (w)− ⎣ 0 0

−2πiβ 1 0 0 1 0

⎤ 0 0⎦ , 1 ⎤

0 −2πiα ∗ ⎦ , 1

w ∈ supp(β) ⊂ R+ , (2.17) w ∈ supp(α ∗ ) ⊂ R−

3. Its asymptotic behavior at w = ∞ (w) = 0 is ⎡

⎤ wn 0 (w) = (1 + O(w−1 )) ⎣ 0 w −1 0 ⎦ . 0 0 w −n+1

(2.18)

990


Then such a (w) exists and is unique. Moreover (w) can equivalently be written as: ⎡ (0) (1) (2) ⎤ ⎡ ⎤ qn−1 qn−1 qn−1 cn ηn 0 0 ⎥ ⎢ 1 ⎢ ⎥ ⎢ (0) (1) (2) ⎥ 0 (2.19) (w) = ⎣ 0 ηn−1 ⎢qn−1 qn−1 qn−1 ⎥ , ⎦ ⎦ (−1)n−1 ηn−2 ⎣ 0 0 cn−2 (0) (1) (2) qn−2 qn−2 qn−2 or also ⎤⎡ 0 1 −cn ηn 0 1 0⎦ ⎢ (w) := ⎣0 ⎣ 0n n−2 (−1) 0 (−1)n−1 ηcn−2 1 ⎡

⎤ := Y (w) 0 cn 1 ⎥ (0) (1) (2) 0 [ q (w), q (w), q (w)], ⎦ ηn−1

cn−2

n

0

n

n

(2.20)

0

where the normalization constants ηn , cn have been introduced in (2.4, 2.6). Remark 2.2. The quantities referring to the letter q on the right hand side in Prop. 2.3 can be extracted from the RHP involving . The next proposition will achieve the same goal for the remaining quantities. Proposition 2.5. (Prop. 8.2 in [9]) Consider the Riemann–Hilbert problem (RHP) of finding a matrix (z) such that 1. (z) is analytic on C\(supp(α) ∪ supp(β )). 2. (z) satisfies the jump conditions ⎡ ⎤ 1 −2πiα(z) 0 1 0 ⎦ , z ∈ supp(α) ⊆ R+ (z)+ = (z)− ⎣ 0 0 0 1 ⎡ ⎤ 1 0 0 (z)+ = (z)− ⎣ 0 1 −2πiβ ∗ ⎦ , z ∈ supp(β ∗ ) ⊆ R− , (2.21) 0 0 1 3. Its asymptotic behavior at z = ∞ (z) = 0 is

⎤ ⎡ z n 0 1 ⎣ 0 1 0 ⎦. (z) = 1 + O z 0 0 z1n

(2.22)

Then such a (z) exists and is unique. Moreover (z) can equivalently be written as:

⎡

cn 0 (z) = ⎣ 0 −1 0 0

⎤⎡ ⎤ 0 p0,n p1,n p2,n 0 ⎦ ⎣ p0,n−1 p1,n−1 p2,n−1 ⎦ . (−1)n p0,n−1 p1,n−1 p2,n−1 cn−1

(2.23)

or also ⎡ (z) = ⎣

0 0

(−1)n cn−1 ηn−1

⎤⎡ 0 − ηcnn 1 −1 0 ⎦ ⎣ 0 0 0 0

−1 1 −1

(z) := Y ⎤ 0 (0) (z), (1) (z), (2) (z) . (2.24) 0⎦ p p p n n n 1

where the normalization constants ηn , cn have been introduced in (2.4, 2.6).


991 (0)

(1)

Remark 2.3. These polynomials pn (z) := pn (z) and the auxiliary functions pn (z), (2) pn (z) were introduced in [9] independently from a RHP formulation, but for all practical purposes the formulation above is sufficiently explicit. Finally the RHPs above allow us to reconstruct the ratio of two consecutive principal minors of the bimoment matrix (this will become relevant when discussing the partition function of the matrix model). Proposition 2.6. (Corollary 8.1 in [9]) If Dn is a leading principal n × n minor of the bimoment matrix I , then Dn 2,3 (w) 2 = cn−1 = (−1)n lim w 2n−1 > 0. w→∞ Dn−1 2,1 (w)

(2.25)

The CDIs in Prop. 2.3 can be written in a very simple form in terms of the solutions of the Riemann–Hilbert problems. To show this let us momentarily denote by Y (w) and (z) the matrices with columns given by the q n(µ) (w) and p (ν) Y n (z) respectively. Then (2.3) can be reformulated as ⎡ ⎤ n−1 (µ) (ν) (z + w)Hn (z, w) := (z + w)⎣ q j (w) p j (z)⎦ j=0

(z). + F(w, z) = Y t (w)A(−w) · Y µ,ν

(2.26) Now the perfect duality of Cor. 2.1 implies that (−w)−1 . Y (w)A(−w) = JY

(2.27)

(z) only by some Note that the solutions of the RHPs (w), (z) differ from Y (w), Y constant invertible left multipliers, and hence (z) = J · Y (−w)−1 · Y (z) = J · Y (w) · A(−w) · Y (−w)−1 · (z).

(2.28)

We collect this into the following proposition for later reference. Proposition 2.7. The matrix kernel Hn (z, w) ⎛ ⎞ n−1 (w, z) F µν (µ) ⎠ [Hn ]µν (z, w) := ⎝ q j (w) p (ν) j (z) + z+w

(2.29)

j=0

is given in terms of the solution of the Riemann–Hilbert problem (2.24)–(2.22) as Hn (z, w) :=

(z) J· (−w)−1 · . z+w

(2.30)

The relevance of the matrix kernel Hn (z, w) will become clear when we will discuss the spectral statistics of the matrix model.

992


3. Matrix Models Consider the vector space H N of Hermitian matrices of size N endowed with the U (N )–invariant Lebesgue measure, dM := d(Mi j )d (Mi j ) dMii . (3.1) i< j

j

Inside H N we consider the convex cone of positive definite matrices H+N := {M = M † , M > 0}

(3.2)

with the induced measure. Let α, β be two positive densities of finite mass supported on the positive real axis: α(M) will simply mean the product measure on the eigenvalues of M. Define now the measure on H+N × H+N = {(M1 , M2 )}, dµ(M1 , M2 ) := dM1 dM2

α(M1 )β(M2 ) . det(M1 + M2 ) N

(3.3)

Definition 3.1. For the positive finite–mass measure (3.3) we define the partition function as the integral Z N := dµ(M1 , M2 ). (3.4) H+N ×H+N

The resulting random matrix model falls into the general class of two–matrix–models although the “coupling” term is not the most common one (which is eT r (M1 M2 ) ). However these and much more general couplings have been considered relatively recently in [26]. As customary, we re-express the Lebesgue measures dMi in terms of the normalized Haar measure dUi of the unitary group U (N ) and the Lebesgue measure on the cone R+N , dM1 = G N dU

dx j 2 (X ) ,

dM2 = G N dV

dy j 2 (Y ) ,

(3.5)

j−1

where (X ) := det[xi ] is the Vandermonde determinant associated with the n-tuple X of ordered eigenvalues x1 ≤ · · · ≤ xn of M1 and (Y ) is defined in the same way relative to the matrix M2 . The constant G N is not of much relevance. It depends only on N but not on the densities. The resulting measure involves two copies of the unitary group and one of the integrals can be performed leading (up to a multiplicative constant depending on the normalizations of the Haar measures and the size N ) to the measure below on the spectra of M1 , M2 , dU α(X )dXβ(Y )dY, (3.6) dµ(X, Y ) = G 2N 2 (X ) 2 (Y ) † N U (N ) det(X + U Y U ) α(X ) :=

N j=1

α(x j ) , dX :=

N j=1

dx j

β(Y ) :=

N j=1

β(y j ) , dY :=

N j=1

dy j . (3.7)


993

At this point we need to compute the integral over U (N ); we can use the result on pp. 23–24 of [26] for the special example (A-28), det (1 − zai b j ) N −r −1 1≤i, j≤N † −r , (3.8) det(1 − z AU BU ) dU = C N ,r (A) (B) U (N ) N −1 k=1 k! C N ,r := , (3.9) N −1 N (N −1)/2 z k=1 (r − N + 1)k where (r − N + 1)k :=

k

(r − N + j)

(3.10)

j=1

is the Pochammer symbol. Setting z = −1, A = X −1 , B = Y in (3.8) we obtain y det (1 + xij ) N −r −1 dU 1≤i, j≤N . = C N ,r −1 U Y U † )r −1 ) (Y ) det(1 + X (X U (N ) Multiplying both sides by det(X )−r we then obtain det (xi + y j ) N −r −1 1≤i, j≤N dU N (N −1)/2 . = (−1) C N ,r † r (X ) (Y ) U (N ) det(X + U Y U )

(3.11)

(3.12)

The case of main relevance to us is r = N , which yields (C N ,N = (−1) N (N −1)/2 ) 1 det xi +y dU j . (3.13) = † N (X ) (Y ) U (N ) det(X + U Y U ) This shows that the measure dµ(M1 , M2 ) can be reduced to a measure on the spectra of the two matrices (we use the same symbol for the measure on the eigenvalues) dµ(X, Y ) = G 2N 2 (X ) 2 (Y )

det[K (xi , y j )] α(X )dXβ(Y )dY, (X ) (Y )

1 . K (x, y) = x+y

(3.14)

In fact we could have used the general formula (3.8) for any r := N + h; note, however, that for h integer and less than −1 Harnad–Orlov’s formula in the form presented above cannot be used, since the determinant in the numerator vanishes (for N ≥ −h) and so do some denominators in the definition of the constants C N ,r . In other words, one should take an appropriate limit and use de l’Hôpital’s rule. For any h ≥ 0 or any h ∈ / −N, tracing the steps above one could obtain general models where the reduced measures on the spectra have a form det(K h (xi , y j )) α(X )dXβ(Y )dY, (X ) (Y ) 1 K h (x, y) = , (x + y)1+h

dµh (X, Y ) = G 2N C N ,r 2 (X ) 2 (Y )

(3.15)

994


corresponding to the unreduced measure dµh (M1 , M2 ) = dM1 dM2

α(M1 )β(M2 ) . det(M1 + M2 ) N +h

(3.16)

It will be important for us in what follows that for any value of h ∈ −N the kernel K h is totally positive or at least sign regular on R+ × R+ . (Sign-regularity means that determinants in Def. 2.1 are nonzero and their sign depends only on their size.) Lemma 3.1. Consider the kernel K h =

1 (x+y)1+h

, restricted to (x, y) ∈ R+ × R+ . Then,

1. for 1 + h > 0, K h is totally positive, 2. for 1 + h ∈ R−\− N, K h is sign-regular. Proof. To prove this assertion define for 0 < y0 < · · · < yn functions ki (x) = 1 (x+yi )s , i = 0, . . . , n, s = 1+h . Denote for j = 1, 2, ..., c j (s) = s(s +1) . . . (s + j −1) and c0 (s) = 1. Then the Wronskian W (k0 , . . . , kn )(x) of k0 , . . . , kn is equal to det (−1) j c j (s)

1 (x + yi )s+ j

n = (−1) i, j=0

n(n−1) 2

n j=0

n c j (s) 1 det . (x + yi )s (x + yi ) j i, j=0 (3.17) n(n−1)

Thus, W (k0 , . . . , kn )(x) is a nonzero multiple of (−1) 2 (z 0 , . . . , z n ), where z i = 1 (x+yi ) and (z 0 , . . . , z n ) is the Vandermonde determinant constructed out of the z i ’s. Since z 0 > · · · > z n , we see that (−1)

n(n−1) 2

(z 0 , . . . , z n ) = (z n , . . . , z 0 ) > 0.

(3.18)

Thus W (k0 , . . . , kn )(x) is positive for any n if s > 0 since all the c j (s) are positive numbers; if s < 0 then — denoting by [s] the greatest integer less than s (hence negative) — we see that the sign of W is sign(W ) = (−1)−[s]n−

[s](1−[s]) 2

(3.19)

for any n ≥ 0 and x ≥ 0. Together these observations imply (Thm 2.3, ch. 2 of [32]), that K h is totally positive for s > 0 on R+ × R+ or at least sign regular for s ∈ R− \ −N. In App. A we will show that for h integer or half–integer, the model (3.16) is the reduction of a 3–matrix model, where a “ghost” Gaussian matrix A has been integrated out. Depending on the range h < N or h > N the matrix A consists of ordinary variables (bosons) or Grassmann variables. Remark 3.1. The fact that the measure dµh depends on N (in the determinant in the denominator) is a feature of the model rather than a problem; indeed, in studying the large N limit it is natural to make the strength of the interaction increase at the same rate as the size of the matrices. For example, in the “standard” two–matrix model one considers the interaction ecN T r (M1 M2 ) .


995

3.1. Correlation functions: Christoffel–Darboux kernels. In this section we will compute the correlation functions of the model. More precisely, Definition 3.2. The correlation functions of the model are defined as R(r,k) (x1 , . . . xr ; y1 , . . . , yk ) N ! rj=1 α(x j ) kj=1 β(y j ) := (N − r )!(N − k)!Z N × where Z N :=

1 N!

N

β(y j )dy j 2 (X ) 2 (Y )

j=k+1

N

α(x j )dx

=r +1

det[K (xi , y j )] , (X ) (Y )

(3.20)

(X ) (Y ) det[K (xi , y j )]i, j≤N α(X )β(Y )dX dY .

These functions allow one to compute the probability of having r eigenvalues of the first matrix and k eigenvalues of the second matrix in measurable sets of the real axis. The computations of these correlation functions in terms of biorthogonal functions reported below follows the general approach in [20 and 24]. Using the well–known formula for the Cauchy determinant, 1 (X ) (Y ) det , (3.21) = xi + y j i, j (x i + y j ) we obtain the measure (X )2 (Y )2 α(X )β(Y )dX dY. i, j (x i + y j )

(3.22)

We will be using the correlation functions only to compute expectations of spectral functions, namely functions of X, Y which are separately symmetric in the permutations of the x j ’s or y j ’s. Lemma 3.2. Suppose F(X ) is a symmetric function under the action of the symmetric group S N . Then N 1 1 F(X ) (X ) α(X )dX. F(X ) (X ) det[K (x j , y j )]α(X )dX = N! x j + yj j=1

Proof. Under the assumption F(X ) = F(X σ ) for any σ ∈ S N we have N 1 α(X )dX F(X ) (X ) x j + yj j=1

N 1 1 = α(X )dX F(X ) (X σ ) N! x + yj σ ∈S N j=1 σ ( j) ⎞ ⎛ N 1 = (σ ) K (xσ ( j) , y j )⎠ α(X )dX F(X ) (X ) ⎝ N! σ ∈S N j=1 1 = F(X ) (X ) det[K (x j , y j )]α(X )dX. N!

996


Corollary 3.1. Let F(X, Y ) be symmetric with respect to either set of variables X or Y . Then 1 F(X, Y ) (X ) (Y ) det[K (x j , y j )]α(X )β(Y )dX dY N! N 1 = F(X, Y ) (X ) (Y ) α(X )β(Y )dX dY. (3.23) x j + yj j=1

Since we are interested in the unordered spectrum of the matrices M1 , M2 , in view of the above Cor. 3.1 we will focus henceforth on the following unnormalized measure: d ν(X, Y ) := (X ) (Y )

N α(x j )β(y j ) α(X )β(Y ) dX dY. dX dY = (X ) (Y ) x j + yj det(X + Y ) j=1

(3.24) From the properties of the Vandermonde determinant we can write (X ) (Y ) = det pi (x j ) det qi (y j ) ,

(3.25)

where pi , qi are any monic polynomials of exact degree i in the respective variables (i = 0, . . . , N − 1). It is therefore natural [20] to choose the sets of monic polynomials { p j (x), q j (y)} j∈N α(x)β(y) to be biorthogonal with respect to the bi-measure x+y dxdy. This means that α(x)β(y) dxdy = ck 2 δk . pk (x) q (y) (3.26) x + y R+ R+ The constant ck was defined in (2.4, with K (x, y) = 1/(x + y)). The normalization constant for d ν is thus N −1 ZN = d ν(X, Y ) = N ! ck2 = N ! det I jk 0≤ j,k≤N −1 . (3.27) k=0

If we introduce the orthonormal biorthogonal polynomials pn :=

1 pn , cn

qn :=

1 qn , cn

(3.28)

then we can write the normalized measure as α(x j )β(y j ) N 1 det pi (x j ) det qi (y j ) d X d N Y. N! x j + yj N

ν(X, Y ) =

(3.29)

j=1

Notice now that the product of determinants in (3.29) is a determinant of the product of the two matrices indicated; hence det pi−1 (x j ) i, j≤N det q j−1 (yi ) i, j≤N = det K N (xi , y j ), (3.30) K N (x, y) :=

N −1 j=0

p j (x)q j (y).

(3.31)


997

The kernel K N (x, y) is a “reproducing” kernel K N (x, y) =

K N (x, z)K N (w, y)

α(w)β(z)dwdz , z+w

(3.32)

which follows immediately from the biorthogonality. In addition we have K N (x, y)

α(x)β(y)dxdy = N. x+y

(3.33)

Summarizing, in the new notation we have Proposition 3.1. The probability measure on X × Y induced by (3.3) is given by: 1 det[K N (xi , y j )] det[K (xi , y j )]α(X )β(Y )dX dY (N !)2

(3.34)

while the correlation functions are: R(r,s) (x1 , . . . xr ; y1 , . . . , ys ) r s j=1 α(x j ) j=1 β(y j ) = det[K N (xi , y j )] det[K (xi , y j )] (N − r )!(N − s)! ×

N

α(x j )dx j

=r +1

N

(3.35)

β(y j )dy j .

(3.36)

j=s+1

Example 3.1. Consider r = 1, s = 0. In this case, with the help of Lemma 3.2, we obtain: R(1,0) 1 α(x1 ) = (N − 1)!N ! = =

1 α(x1 ) (N − 1)! 1 α(x1 ) (N − 1)!

det[K N (xi , y j )] det[K (xi , y j )]

= α(x1 )

α(x )dx

=2

det[K N (xi , y j )]

N i=1

(σ )(σ )

N

β(y j )dy j

j=1

N N 1 α(x )dx β(y j )dy j xi + yi

=2

j=1

pσ (1)−1 (x1 ) . . . pσ (N )−1 (x N ),

σ,σ ∈S N

qσ (1)−1 (y1 ) . . . qσ (N )−1 (y N )

N

N i=1

N N 1 α(x )dx β(y j )dy j xi + yi

K N (x1 , y1 ) β(y1 )dy1 . x1 + y1

=2

j=1

(3.37)

998


3.1.1. Correlation functions in terms of biorthogonal polynomials In the paper by Eynard and Mehta [20] (generalized in [24]) they never used any specific information about the model they were considering (with the Itzykson–Zuber interaction) but only the fact that the matrices were coupled in a chain. We recall the relevant result here1 . Define H00 (x, y) := K N (x, y), H01 (x, x ) :=

β(y)dy α(x)dx , H , (3.38) (y, y ) := H00 (x, y ) 10 x +y x+y 1 α(z)dzβ(w)dw H11 (y, x) := − . H00 (z, w) (z + y)(x + w) x+y H00 (x, y)

Since H00 is a reproducing kernel H00 (x, y)H11 (y, x )β(y)dy = H01 (x, x ) − H01 (x, x ) = 0, and similarly

H11 (x, z)H00 (z, x )α(z)dz = H10 (x, x ) − H10 (x, x ) ≡ 0.

(3.39)

(3.40)

Integrating these two equations against (x + y )−1 α(x)dx (the first) or (x + y )−1 β(y )dy (the second) we find also H11 (y, x)H10 (y, y )β(y)dy = H11 (x, y)H01 (y, y )α(y)dy ≡ 0. (3.41) The correlation functions for r eigenvalues x1 , . . . , xr of M1 and s eigenvalues y1 , . . . , ys of M2 were computed in [20] and are given by R(r,s) (x1 , . . . , xr ; y1 , . . . , ys ) = ⎡ ⎢ × det ⎣

H01 (xi , x j )

H11 (yi , x j )

1≤i, j≤r

r

α(x j )

j=1

1≤i≤s, j≤r

β(yk )

k=1

H00 (xi , y j )

s

⎤

1≤i≤r,1≤ j≤s

H10 (yi , y j )

⎥ ⎦,

(3.42)

1≤i, j≤s

where 0 ≤ r ≤ N , 0 ≤ s ≤ N , 1 ≤ r + s, with the understanding that if either r or s is 0 then the corresponding blocks labeled by r, s respectively, are absent. Thus, for example R(1,0) (x1 ) = α(x1 )H0,1 (x1 , x1 ). This is a nontrivial result and perhaps one can get a bit of an insight by considering the special case r = s = N . We know that in this case R(N ,N ) (x1 , . . . x N ; y1 , . . . , y N ) = det[K N (xi , y j )] det[K (xi , y j )]α(X )β(Y ),

(3.43) (3.44)

so we have the following not at all obvious identity: 1 It could be extended to the chain of matrices (see below) but it becomes a bit cumbersome to describe.


999

Lemma 3.3.

⎡

⎢ det[K N (xi , y j )] det[K (xi , y j )] = det ⎣

H01 (xi , x j )

H11 (yi , x j )

i, j≤N

i, j≤N

H00 (xi , y j ) H10 (yi , y j )

i, j≤N

⎤ ⎥ ⎦.

i, j≤N

(3.45) Proof. It suffices to observe that the 2N × 2N matrix on the right-hand side has the following block structure: −P0 (x)T Q1 (−x) P0 (x)T Q0 (y) , (3.46) K= P1 (−y)T Q1 (−x) − K(x, y) −P1 (−y)Q0 (y) (1)

where P0 (x) = [ pi−1 (x j )]1≤i, j≤N , P1 (y) = [ pi−1 (y j )]1≤i, j≤N and, after exchanging pi s with qi s, we define the remaining symbols (see Eqs. (2.11) and (2.10) for definitions) accordingly. Moreover, K(x, y) = [K (xi , y j ]1≤i, j≤N . With this notation in place it is clear that K admits the following (Bruhat) decomposition: P0 (x)T 0 0 I −K(x, y) 0 . (3.47) K= −P1 (−y)T I I 0 −Q1 (−x) Q0 (y) Thus det K = det P0 (x)T det K(x, y) det Q0 (y) = det[K N (xi , y j )] det[K (xi , y j )].

A comparison of the definitions (3.38) with the entries of the Christoffel–Darboux identities of Prop. 2.3 shows that they are intimately related, in fact Hµ,ν (x, y) = (−)µ+ν

N −1 j=0

(µ) !

qj

" (ν) ! " δµ,1 δν,1 (−)µ x p j (−)ν y − x+y

(3.48)

and we recognize that (up to some signs) these are precisely the entries of the Hn kernel of Prop. 2.7: ! " Hµ,ν (x, y) = (−)µ+ν Hn,µ,ν (−)µ x, (−)ν y . (3.49) More explicitly we have Proposition 3.2. The kernels of the correlation functions are given in terms of the solution of the RHP in Prop. 2.5 as follows: −1 (x) 3,1 (−y) H00 (x, y) = H N ,00 (x, y) = , x+y −1 (−y ) 3,2 (−y) , H01 (y , y) = −H N ,01 (−y , y) = y − y (3.50) −1 (x ) (x) 2,1 H10 (x, x ) = −H N ,10 (x, −x ) = , x − x −1 (−y) 2,2 (x) H11 (y, x) = H N ,11 (−y, −x) = . x+y

1000


The potential relevance of these formulæ is that it allows one to compute the large N asymptotic behaviour of the correlation functions in terms of the asymptotic behaviour of only 3 consecutive biorthogonal polynomials and auxiliary functions associated with them that enter in the Riemann–Hilbert formulation of Props. 2.4, 2.5. The steepest–descent analysis of these problems will appear in a forthcoming paper. We expect that, after a complete description of the asymptotics of the BOPs is obtained, these formulæ can be used to address the issue of universality for this matrix model via the Riemann–Hilbert approach. Remark 3.2. Note also that the formula above is not symmetric inasmuch as the BOPs p, q play different roles in the Christoffel-Darboux theorem; we could however rewrite the theorem as N −1

p j (x)q j (y) =

t0 ,N (x)B N (−x)qˇ 0 ,N (y) p x+y

j=0

,

(3.51)

where B(z) was defined in (2.9). The auxiliary vector qˇ 0 ,N enters in a similar Riemann–Hilbert formulation with the rôles of the densities α and β interchanged.

3.2. A multi–matrix model. It is possible to extend the model (3.3) to a chain of matrices with the nearest neighbor interaction 1 . det(Mi + Mi+1 ) N +h i

(3.52)

The “strength” of the interaction (i.e. h j ’s) may depend on a site along the chain. Specif"×R ! and a collection of R positive densities α1 , . . . , α R ically, consider the space H+N on R+ . Define the finite mass measure dµ(M1 , . . . , M R ) =

R−1

=1

R 1 α (M )dM . det(M + M +1 ) N +h

(3.53)

=1

Using the Harnad–Orlov formula one we obtain, up to normalization constant, the following measure which we denote by the same symbol: dµ(X 1 , . . . , X R ) =

R

α (X ) (X ) dX 2

=1

R−1

det (x ,i + x +1, j )−1−h 1≤i, j≤N

=1

(X ) (X +1 )

.

(3.54) Following the same steps that led to the expression (3.24) as positive density for (S N )2 –invariant observables, we obtain the following reduced measure on the spectra (up to a normalization depending only on h ’s but not on the measures) d ν(X 1 , . . . , X R ) = (X 1 ) (X R )

R−1

=1

det(X + X +1 )−1−h

R

=1

α (X )dX . (3.55)


1001

In the case where all interactions are the same h = 0 we have d ν(X 1 , . . . , X R ) = (X 1 ) (X R )

R−1

=1

R 1 α (X )dX det(X + X +1 )

(3.56)

=1

which can be seen as a generalization of (3.24). This model too can be treated with the aid of biorthogonal polynomials that will now satisfy a R + 3 recurrence relation; note that the length of the recurrence relations does not depend on the “potentials”, or densities, α j . This is in sharp contrast with the usual multi-matrix model with interaction ecTr(M j M j+1 ) [7,20]. To characterize these biorthogonal polynomials, an (R +2)×(R +2) Riemann–Hilbert problem can be set up and the strong asymptotics can be dealt with, but the complexity of this problem definitely warrants a separate paper. 4. Diagrammatic Expansion In parallel with the N12 –expansion for the Hermitian matrix model and the IZHC two-matrix model, we would like to sketch the similar formal expansion of the model in terms of colored ribbon graphs. The weights α and β entering the definition (3.3) are assumed to be of the form α(M1 ) = e−N Tr(U (M1 )) , β(M2 ) = e−N Tr(V (M1 )) . We perform a shift and a rescaling of the matrices so that det(M1 + M2 ) → det(1 − ζ M1 − ηM2 ). Of course the values of ζ and η are suitably restricted to a neighborhood of the origin: however this restriction is irrelevant since the manipulations below are in the sense of formal power series. The procedure amounts to a perturbative Taylor–expansion around a Gaussian integral. In other words we will be considering a partition function in the form " ! 2 #∞ N 2

Z N := dM1 dM2 e− 2 Tr M1 +M2 +N =1 Tr(ζ M1 +ηM2 ) −N TrU p (M1 )−N TrV p (M2 ) , (4.1) where U p , V p are the perturbations of the Gaussian (including a quadratic term as well) which, for convenience, we parametrize as the following formal series: U p (x) := −

∞ uj −ζ j j x , j

V p (y) := −

j=1

We thus have $

⎛

Z N = exp N Tr ⎝

∞ u j −ζ j j=1

j

M1 j +

(4.2)

j=1

∞ v j −η j j=1

∞ vj − ηj j y . j

j

M2 j +

∞ 1%

=1

⎞' (ζ M1 + ηM2 ) ⎠ , &

(4.3) where the average is taken w.r.t. the underlying (uncoupled!) Gaussian measure & 1 N % 2 2 , dM dM exp − + M Tr M 1 2 1 2 (0) 2 ZN & N % (0) Z N := dM1 dM2 exp − Tr M12 + M22 . 2

(4.4)

1002


Note that, due to the shifts in u j , v j , the = 1 term in the third sum above cancels exactly against the shifts in the first two sums; similarly, only the term ζ2η Tr(M1 M2 ) remains in the quadratic part: ( % &) u2 v2 Z N = exp N Tr u 1 M1 + M1 2 + v1 M2 + M2 2 + ζ ηM1 M2 + · · · . (4.5) 2 2 Using Wick’s theorem for the evaluation of Gaussian integrals and the frequently used combinatorial interpretation (see [16] for an excellent introduction) one sees that the partition function is the sum over all possible bi-colored ribbon graphs respecting rules which we now specify • There are two colors for vertices/edges (say, red/blue); vertices are distinguished in monochromatic and bichromatic. • Each monochromatic vertex of valency j enters with a weight N u j /j (blue) or N v j /j (red). • Each edge (red or blue) enters with a weight N1 . k −k

• Each bichromatic vertex of valency enters with a weight ζ Nη , where 1 ≤ k ≤ −1 is the number% of&blue half-edges and − k the number of red ones and appears with a multiplicity k corresponding to all possible arrangements of k blue legs amongst

, up to cyclic reordering. In particular there is only one bichromatic bivalent vertex (up to automorphism) which enters with weight ζ η/N . In general, since each such vertex corresponds to a trace of the form Tr(M1 a1 M2 b1 M1 a2 . . . ) and in view of the cyclicity of the trace, there are precisely equivalent vertices obtained by cyclically permuting the matrices in the sum, which corresponds diagrammatically % & to a rota( −1)! 1 tion of the colors of the legs of the vertex. Hence there are in fact k = ( −k) ! k! inequivalent bicolored –valent vertices in each diagram contributing with a weight ζ k η −k /N . • Each connected Feynman diagram contributing to the perturbative sum has a power N F−E+V = N 2−2g , where g = g() is the genus of the surface over which the graph can be drawn. Summing up over all possible labeling of the bicolored fat-graphs leaves a factor |Aut ()| in the partition function (see [16]) and the result is hence ln Z N = Connected bichromatic fatgraphs

∞ −1 ∞ N 2−2g n j m j ( −k)r kr kη k, , u j vj ζ |Aut ()| j=1

(4.6)

=2 k=1

where n j = n j () is the number of blue vertices with valency j, m j = m j () the number of j–valent red vertices and rk = rk () is the number of –valent bichromatic vertices with k red (and hence − k blue) legs. 5. Large N Behaviour Consider two densities α(x), β(y) of the form 1

1

α(x) = α(x) = e− V (x) , β(y) = β(y) = e− U (y) ,

=

T , T > 0. (5.1) N


1003

Fig. 1. An example of a connected diagram contributing to the partition function

Here we have introduced a dependence on the small parameter on the measures (but we will not emphasize this dependence in the notation). From the experience amassed in the literature on the ordinary orthogonal polynomials we start by considering the heuristic “saddle–point” for the partition function Z N , namely the total mass of the reduced measure (3.24). The heuristics calls for a scaling approach where we send the size of the matrices N to infinity and the scaling parameter to zero as O(1/N ), namely that we vary the densities α, β by raising them to the power 1 . We thus have 1 α(X )β(Y )dX dY. (5.2) Z N = (X ) (Y ) det xi + y j i, j≤N Using 3.22 we have ZN ∝

* + N 1 (X )2 (Y )2 exp − U (xi ) + V (y j ) dX dY. i, j (x i + y j )

(5.3)

i=1

Taking

1 N2

times the logarithm of the integrand we have the expression

S(X, Y ) :=

N 1 1 1 U (x j ) + V (y j ) − 2 ln |x j − xk | − 2 ln |y j − yk | TN N N j=1

1 + 2 ln |x j + yk | N

j =k

j =k

j,k

= SU (X ) + SV (Y ) + J (X, Y ).

(5.4)

It is convenient –in order to deal with a more standard potential-theoretic problem– to map Y → −Y and define V (y) = V (−y) so that we can rewrite the action as S(X, Y ) = SU (X ) + SV (Y ) +

1 ln |x j − yk |, N2 j,k

(5.5)

1004


which describes the energy of a gas of 2N particles of charge +1 (for the x j ’s) and −1 (for the y j ’s) separated by an impenetrable, electrically neutral, partition confining the positively charged particles to the positive real axis under the potential U and the negatively charged particles to the negative axis under the potential V . The usual argument is that the configuration of the minimum contributes the most to the integral and –under suitable assumptions of regularity for the potential— one wants to show that the sequence of these minima configurations tends in-measure to some probability distributions. We define ρ=

N 1 δxk , N k=1

µ=

N 1 δy j , N

(5.6)

j=1

and rewrite the action as 1 S[ρ, µ] := U (x)ρ(x)dx − ρ(x)ρ(x ) ln |x − x |dxdx T R+ R+ R+ 1 + V (y)µ(y)dy − µ(y)µ(y ) ln |y − y |dydy T R− R− R− × ρ(x)µ(y) ln |x − y|dxdy. (5.7) R+ R−

5.1. Continuum version: cubic spectral curve and solution of the potential problem. We immediately rephrase the above minimization problem in a continuum version. In order to emphasize the symmetry of the problem it is convenient to denote U (x) by V1 (x) and V (y) = V (−y) by V2 (y) and denote the corresponding equilibrium densities by ρ1 (x), ρ2 (y). The main point of this section is Thm. 5.1 which states that the resolvents (Markov functions) of the equilibrium distributions are solutions of a cubic equation that defines a trigonal curve;2 this result is the analogue of the better-known result for the equilibrium measure appearing in the one–matrix model ([13] and references therein). The derivation that we present here is of a formal heuristic nature inasmuch as we discount several important issues about the regularity of the equilibrium measures. However this sort of manipulation is quite common and they can be obtained also from the loop equations as done in [19] for the O(n) model. Rewriting the functional in the new notation for the potentials we find 1 S[ρ1 , ρ2 ] := V1 (x)ρ1 (x)dx − ρ1 (x)ρ1 (x ) ln |x − x |dxdx T R+ R+ R+ 1 + V2 (y)ρ2 (y)dy − ρ2 (y)ρ2 (y ) ln |y − y |dydy T R− R− R− + ρ1 (x)ρ2 (y) ln |x − y|dxdy. (5.8) R+ R−

2 Namely a curve of the general form w 3 + Aw 2 + Bw + C = 0, with A, B, C smooth functions of the spectral parameter z.


1005

We will be studying the measures ρ1 , ρ2 that minimize the above functional; here we will assume their regularity (which shall be proved in a separate paper). Also, the potentials V1 , V2 will be assumed real analytic. More precisely, we make the following assumption on the nature of the potentials Assumption 5.1. The two potentials V1 (x), V2 (y) are restrictions to the positive/negative axis of two real–analytic functions such that V2 (y) • lim x→+∞ Vln1 (x) x = +∞, and similarly lim y→−∞ ln |y| = +∞; • the derivatives V j (x) have at most finitely many poles in a strip of finite width around R; • finally, V1 (x) > C1 | ln(x)| for some constant C1 > 0 and x > 0 and V2 (y) > C2 | ln |y| | for y < 0 and some constant C2 > 0.

It will be shown in a separate paper (see [2] for a very similar statement) that this assumption is sufficient to guarantee that the equilibrium densities exist, have compact support, are regular and their supports do not include the origin. In order to enforce the normalization of ρ1 , ρ2 –as customary– we will introduce in the action two suitable Lagrange multipliers S := S[γ1 , γ2 , ρ1 , ρ2 ] := S[ρ1 , ρ2 ] + γ+ ρ1 dx − 1 + γ− ρ2 dx − 1 . (5.9) The vanishing of the first variation of the functional S yields the equations V1 (z) ρ1 ln |z − x|dx + ρ2 ln |z − x|dx = γ+ , z ∈ Supp(ρ1 ), −2 T R+ R− V2 (z) −2 ρ2 ln |z − x|dx + ρ1 ln |z − x|dx = γ− , z ∈ Supp(ρ2 ). T R− R+ (5.10) Differentiating w.r.t. z yields 1 1 1 V1 (z) − 2P.V. ρ1 dx + ρ2 dx = 0, z ∈ Supp(ρ1 ), T z−x z−x (5.11) 1 1 1 V2 (z) − 2P.V. ρ2 dx + ρ1 dx = 0, z ∈ Supp(ρ2 ), T z−x z−x where P.V. indicates the Cauchy principal value. These equations are best written in terms of the resolvents (or Weyl functions) ρi (x)dx , z ∈ C \ Supp(ρi ). (5.12) Wi (z) := z−x Indeed, with the help of the Sokhotskyi–Plemelj formula, the equations above take a simpler form , W1,+ (z) + W1,− (z) = T1 V1 (z) + W2 (z), z ∈ Supp(ρ1 ) , (5.13) W2,+ (z) + W2,− (z) = T1 V2 (z) + W1 (z), z ∈ Supp(ρ2 ) Wi,+ − Wi,− = −2iπρi . (5.14)

1006


Fig. 2. A pictorial example of the three sheets of the Riemann surface of Y (x). In blue the support of ρ1 (right) and in green that of ρ2 (left)

Note that in the RHS of the system of equations, the resolvent of the other measure is evaluated at a regular point due to the fact that the two supports are disjoint. We define, partly motivated by [22], the shifted resolvents Y1 := −W1 +

2V1 + V2 , 3T

Y2 := W2 −

V1 + 2V2 . 3T

(5.15)

Observe that, in view of Assumption (5.1) , {Y1 , Y2 } have the same analytic structure as {W1 , W2 }, while the equations describing the jumps simplify to: Y1,+ + Y1,− = −Y2 (z), z ∈ Supp(ρ1 ) , (5.16) Y2,+ + Y2,− = −Y1 (z), z ∈ Supp(ρ2 ) Y2,+ − Y2,− = −2iπρ2 . (5.17) Y1,+ − Y1,− = 2iπρ1 , Using these equations one obtains by direct inspection that R(x) := Y1 2 + Y2 2 + Y1 Y2

(5.18)

has no jumps on either supports. Multiplying Eq. (5.18) on both sides by Y1 − Y2 one obtains R(z)(Y1 − Y2 ) = Y1 3 − Y2 3

(5.19)

Y1 3 − R(z)Y1 = Y2 3 − R(z)Y2 := D(z).

(5.20)

which can be rewritten as

The first expression may a priori have at most jumps across the support of ρ1 , while the second may have jumps only across the support of ρ2 : since the two supports are disjoint we conclude that D(x) is a regular function on the real axis. Remark 5.1. If we introduce the function Y0 := −Y1 − Y2 the jump relations (5.16, 5.17) can be rewritten (after a few straightforward manipulations) as Y0,± (z) = Y1,∓ (z) z ∈ Supp(ρ1 ), Y0,± (z) = Y2,∓ (z) z ∈ Supp(ρ2 ).

(5.21)


1007

This implies that we can think of the Y j (z)’s as the three branches (three sheets) of a Riemann surface realized as a triple cover of the x–plane branched at the endpoints of the spectral bands. The Riemann surface is depicted in Fig. 2. Theorem. 5.1 will be proving more formally this statement, realizing Y0 , Y1 , Y2 as branches of a cubic equation (which the reader can already guess from (5.20)). Theorem 5.1. Under Assumption (5.1), the two shifted resolvents satisfy the same cubic equation in the form E(y, z) := y 3 − R(z)y − D(z) = 0.

(5.22)

The coefficients R(x), D(x) are related to the equilibrium measures as follows: (V1 )2 + (V2 )2 + V1 V2 − R1 (z) − R2 (z), 3T 2 Vi (z) − Vi (x) 1 1 Ri (z) := ρi (x)dx = ∇i Vi , T z−x T 1 1 D(x) = U0 U1 U2 + U2 R1 + U1 R2 + ∇12 V1 − ∇21 V2 , T T R(z) =

(5.23) (5.24)

where the functions U0 , U1 , U2 and the operators ∇1 , ∇2 , ∇12 , ∇21 have been defined as 2V1 + V2 2V + V1 V − V1 , U2 := − 2 , U0 := 2 , 3T 3T 3T f (z) − f (x) ∇ j f (z) := dρ j (x) , z − x f (z) − f (x) dρ1 (x)dρ2 (y) , ∇12 f (z) := z−x x−y f (z) − f (x) dρ2 (y)dρ1 (x) . ∇21 f (z) := z−y y−x

U1 :=

(5.25)

(5.26) (5.27)

Furthermore the support of ρ1 , ρ2 must consist of a finite union of finite disjoint intervals. Proof. The statement about the support of the measures follows from the argument used in [13] that we sketch here. By results of Chapter 13 of [37]), Assumption 5.1 implies that the supports of the equilibrium measures are compact. Moreover, it follows that the branchpoints (and branchcuts) of the functions Y1,2,0 coincide with the supports of ρ1 , ρ2 . Since they solve a cubic algebraic equation, the branchpoints are determined as zeroes of the discriminant := 4R 3 − 27D 2 of (5.22). As suggested by the previous discussion, even without an explicit expression for R, D, it follows from Morera’s theorem that R, D are also real-analytic and so is . Thus cannot have infinitely many zeroes in a compact domain and so the number of endpoints of the supports is a-priori finite. The only part that is left to be proven are the formulæ for R, D. Computing the jump of W j 2 from Eqs. (5.14) one obtains 1 (5.28) V1 + W2 (W1 )2+ − (W1 )2− = −2iπρ1 T

1008


and a similar equation for W2 . This implies that ! " ρ1 (x) V1 (x)/T + W2 (x) 2 W1 (z) = dx z−x ! " ρ1 (x) V1 (z) − V1 (x) 1 ρ1 (x)ρ2 (y)dxdy = V1 (z)W1 (z) − dx + . T T (z − x) (z − x)(x − y) =:R1 (z)

(5.29) Note that R1 (z) is regular on the support of ρ1 since V1 is. Hence we have

=:W12

(x)ρ (y)dxdy ρ 1 1 2 , W1 2 (z) = V1 (z)W1 (z) − R1 (z) + T (z − x)(x − y) 1 ρ1 (x)ρ2 (y)dxdy W2 2 (z) = V2 (z)W2 (z) − R2 (z) + . T (z − y)(y − x)

(5.30) (5.31)

=:W21

Adding the two together and using the identity 1 1 1 + = (z − x)(x − y) (z − y)(y − x) (z − x)(z − y)

(5.32)

we obtain W1 2 + W2 2 =

1 1 V1 W1 + V2 W2 − R1 − R2 + W1 W2 , T T

(5.33)

which is precisely Eq. (5.18) when rewritten with W1 , W2 replaced by their expressions in terms of the Y1 , Y2 variables resulting from (5.15). The second formula for D(x) can be obtained by noticing that D(x) = Y0 Y1 Y2 . Explicitly, this reads (after rearranging the terms and using the definitions for the shifted resolvents (5.15) and the U j ’s (5.25), V D(x) = U0 U1 U2 − 2U0 W1 W2 − W12 W2 + W1 W22 − U2 W12 − 1 W1 T V (5.34) −U1 W22 − 2 W2 . T Substituting in the last two terms the expressions (5.30, 5.31) we obtain D(x) =U0 U1 U2 −2U0 W1 W2 −W12 W2 +W1 W22 +U2 R1 +U1 R2 −U2 W12 −U1 W21 . (5.35) Using W1 W2 = W12 + W21 (5.30, 5.31) we obtain D(x) = U0 U1 U2 − W12 W2 + W1 W22 + U2 R1 + U1 R2 +

V1 V W12 − 2 W21 . T T

(5.36)


1009

Now we can rewrite V1 W12 = ∇12 (V1 ) + V2 W21

=

∇21 (V2 ) +

V1 (x)ρ1 (x)ρ2 (y)dxdy , (z − x)(x − y)

(5.37)

V2 (y)ρ1 (x)ρ2 (y)dxdy , (z − y)(y − x)

(5.38)

so that we have obtained 1 1 D(x) = U0 U1 U2 + U2 R1 + U1 R2 + ∇12 V1 − ∇21 V2 + R, T T V1 (x) V2 (y) ρ1 (x)ρ2 (y)dxdy 1 + − W12 W2 + W1 W22 . R := T z−x z−y (x − y)

(5.39) (5.40)

We want to show that R ≡ 0: indeed it is clear that the only part of D(x) which may have jump-discontinuities is R, but we know that D(x) has no such discontinuities. Hence R has no discontinuities; on the other hand it is clear from its definition that R cannot have any other singularities and hence it is an entire function. Inspection shows that R(z) → 0 as z → ∞ and hence R must be identically vanishing by Liouville’s theorem. This concludes the proof of the theorem. The presence of a “spectral curve” will be one of the crucial ingredients for the large– degree asymptotic analysis of the biorthogonal polynomials using the Riemann–Hilbert formulation given in [9] and the Deift–Zhou nonlinear steepest descent method. Indeed we will show in a separate publication that the OPs are modeled by (spinorial) Baker– Akhiezer vectors (similarly to [10]) that naturally live on the three-sheeted covering specified by (5.22). A. A Rectangular Mixed 3–Matrix Model with Ghost Fields The model (3.16) (for any value of h) can be obtained from the following matrix models: A.1. Integer h < N . Consider the standard Lebesgue measure on the space Mat ((N − h) × N , C) of complex (N − h) × N matrices, viewed as a linear space: if A ∈ Mat ((N − h) × N , C) we will use a shorthand notation d Ad A† :=

N N −h

dAi j d Ai j

(A.1)

i=1 j=1

for the volume element. Let, as above, M1 , M2 ∈ H+N and consider the following normalizable measure on the space H+N × H+N × Mat (N − h, N , C), d µC (M1 , M2 , A) := dM1 dM2 d Ad A† α(M1 )β(M2 )e−Tr A(M1 +M2 )A . †

(A.2)

Since the measure is Gaussian in A one immediately sees that (up to inessential proportionality constants)

1010


Mat (N −h,N ,C)

d µC (M1 , M2 , A) ∝ dµh (M1 , M2 )

α(M1 )β(M2 ) , det(M1 + M2 ) N −h h ∈ {0, 1, . . . , N − 1}.

= dM1 dM2

(A.3)

A.2. Half–integer h < N . One could also consider a similar model where Mat (N − h, N , C) is replaced by Mat (2N − 2h, N , R), where h ∈ 21 Z and h < N obtaining then d A :=

N 2N −2h i=1

d Ai j ,

j=1

d µ (M , M , A) := dM1 dM2 d Aα(M1 )β(M2 )e−Tr A(M1 +M2 )A , R 1 2 d µR (M1 , M2 , A) ∝ dµ(M1 , M2 ) t

(A.4)

Mat (N −h,N ,R)

α(M1 )β(M2 ) . (A.5) det(M1 + M2 ) N −h Of course, if h is actually an integer then this case reduces to the previous one. = dM1 dM2

A.3. Integer and half integer h > N . For the sake of completeness we note that when h > N the determinant in the reduced measure (3.16) is a positive power in the numerator; this can be obtained from a Gaussian integral over Grassmann variables which can be obtained from “complex” anticommuting variables (for h integer) or “real” (for h ∈ 21 Z) much along the line of the previous (commuting variable) case. Clearly, the plethora of models is endless and each value of h can be studied with the aid of a different set of biorthogonal polynomial. We singled out the case h = 0, since it corresponds to the Cauchy BOPs that have many features in common with the much better–known orthogonal polynomials and yet seem to have a very rich but still tractable asymptotic theory. B. Relation to the O(1)–Model The O(n) matrix model [19] is a multimatrix model for a (positive) Hermitian matrix M and n Hermitian matrices A j with distribution ⎤⎞ ⎛ ⎡ n n (B.1) d A j exp ⎝−N Tr ⎣V (M) + M · A j 2 ⎦⎠ . dM j=1

j=1

With some manipulations, the integration over the Gaussian variables A j can be performed and the result written in terms of the eigenvalues z j of the matrix M, yielding a (unnormalized) measure over the space of eigenvalues given by N dz j e−N V (z j ) (Z )2 (Z )2 α(Z )dZ =: , Z := diag(z 1 , . . . , z N ). n n n i< j (z j + z i ) i< j (z j + z i ) z2

j=1

j

(B.2)


1011

We wish to show that we can specialize our Cauchy two-matrix-model so that, in principle, it reduces to a O(1) model. More precisely, we will show that the partition function of the Cauchy two–matrix model is the square of the partition function of the O(1) model for a particular choice of measures (see Prop. B.1) . We recall that from a diagrammatic standpoint the large N expansion of the O(n) model describes formally a gas of self-avoiding loops of n different colors (hence in our case of a single color) on random surfaces. Proposition B.1. The partition function of the Cauchy 2-matrix-model with N = 2k and β(x) = xα(x) is the square of the partition function of the O(1)-model of the same size. More precisely

2 1 (Z )2 α(Z )dZ n 2k k!N ! R2k i< j (z j + z i ) + 2k 1 j=1 y j (X ) (Y )α(X )α(Y )dX dY . = N ! R2k det(X + Y ) R2k + +

(B.3)

Remark B.1. We conjecture an analogous statement to hold for odd N . Proof. The matrix of moments for this choice of measures reads Ii j = x i y j+1 α(x) α(y) dxdy x+y and the partition function for the corresponding size–N Cauchy matrix model is by Eq. (3.27) (up to some combinatorial coefficients) the principal minor of this matrix det(I ) N (we use the subscript N to denote the principal minor of size N ). Such a minor coincides with the RHS of (B.3). We introduce the new skew symmetric matrix Mi j =

1 2

x i y j (y − x)

d2 α x+y

(B.4)

which is clearly skew–symmetric. Denoting by α = [ x j α(x)dx]tj=0,1,... the infinite vector of moments of the measure α(x)dx, we see that 1 I = M + αα t , 2

(B.5)

1 det(I N ) = det(M N ) + α t M N α, 2

(B.6)

and hence

where the tilde denotes the classical adjoint matrix of the principal minor of size N × N (the transposed of the matrix of cofactors). Since we are dealing with the case N = 2k even, the adjoint of a skew–symmetric matrix is skew-symmetric as well and hence the second term has to vanish. Thus we have det(I ) N = det M N = (P f (M N ))2 .

(B.7)

1012


We see that up to a multiplicative factor

k!2k Pf (M N ) =

(σ )

σ ∈S N

=

RN

k

1 , k!2k

Mσ2 j−1 σ2 j

j=1

⎡

k ⎢ α(x j )α(y j )dx j dy j ⎢ det ⎢ N ⎣ x + y j j R j=1

1 x1 .. .

y1 . . . y12 . . .

1 xk

yk yk2 .. .

x1N −1 y1N . . . xkN −1 ykN

⎤ ⎥ ⎥ ⎥. ⎦

We denote by Z the vector of size N with components Z = (z 1 , z 2 , . . . , z N ) = (x1 , y1 , x2 , y2 , . . . , xk , yk ).

(B.8)

With this notation in place we have k!2k Pf M N = =

RN

α(Z )dZ (Z )

k j=1

yj x j + yj

k 1 yj (x j + y ) (x j + x )(y j + y ) α(Z ) (Z )dZ . R N j< ≤N z j + z j=1 j = ≤k j< ≤k

(Z )

=:R(Z )

(B.9) The rational function (Z ) is invariant under permutations of the variables, whereas R(Z ) is not; however R(Z ) has the same total degree as (Z ), in particular its degree in y1 is N − 1. Symmetrizing under the integral sign the variables give: 1 k k!2 Pf M N = (Z σ ) (Z σ )R(Z σ )α(Z )dZ N! σ ∈S N 1 = (σ )R(Z σ )α(Z )dZ . (B.10) (Z ) (Z ) N! σ ∈S N

Now note that necessarily

(σ )R(Z σ ) = (Z ).

(B.11)

σ ∈S N

Indeed R(Z ) contains the monomial y1N −1 (with the coefficient 1), and has the same total degree in all variables; since the antisymmetrization must be divisible by (Z ) the assertion follows. We thus have 2 (Z ) 1 α(Z )dZ . (B.12) Pf M N = k 2 k!N ! R N j< ≤N (z j + z ) We see then that up to a proportionality constant Pf M N is the total integral of the measure (B.2) and thus the proposition is proved.


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The relationship between the two models does not seem to go much further in the sense that there is no direct and simple relationship between the correlation functions of the two models. It seems, however, that some connection should be present and is worth exploring. We leave it as an open problem to establish a connection between these two models on the level of the correlation functions.

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25. Harnad, J., Its, A.R.: Integrable Fredholm operators and dual isomonodromic deformations. Comm. Math. Phys. 226(3), 497–530 (2002) 26. Harnad, J., Orlov, A.Yu.: Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions. J. Phys. A 39(28), 8783–8809 (2006) 27. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. In Proceedings of the Conference on Yang-Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory, In: Barber, M.N., Pearce, P.A. (eds.) Singapore: World Scientific, 1990, pp. 303–338 28. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: The quantum correlation function as the τ function of classical differential equations. In: Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Berlin: Springer, 1993, pp. 407–417 29. Its, A.R., Kitaev, A.V., Fokas, A.S.: An isomonodromy approach to the theory of two-dimensional quantum gravity. Usp. Mat. Nauk., 45(6(276)), 135–136 (1990) 30. Izergin, A.G., Its, A.R., Korepin, V.E., Slavnov, N.A.: Integrable differential equations for temperature correlation functions of the Heisenberg XXO chain. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 205(Differentsialnaya Geom. Gruppy Li i Mekh. 13, 6–20, 179 (1993)) 31. Izergin, A.G., Its, A.R., Korepin, V.E., Slavnov, N.A.: The matrix Riemann-Hilbert problem and differential equations for correlation functions of the XXO Heisenberg chain. Algebra i Analiz 6(2), 138–151 (1994) 32. Karlin, S.: Total positivity. Vol. I. Stanford, Stanford University Press, 1968 33. Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157(4), 295–300 (1985) 34. Lundmark, H., Szmigielski, J.: Multi-peakon solutions of the Degasperis–Procesi equation. Inverse Problems 19, 1241–1245 (2003) 35. Lundmark, H., Szmigielski, J.: Degasperis-Procesi peakons and the discrete cubic string. IMRP Int. Math. Res. Pap. 2, 53–116 (2005) 36. Mehta, M.L.: Random matrices. Volume 142 of Pure and Applied Mathematics (Amsterdam). Third edition, Amsterdam: Elsevier/Academic Press, 2004 37. Saff, E.B., Totik, V.: Logarithmic potentials with external fields. Volume 316 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, 1997, Appendix B by Thomas Bloom 38. Widom, H.: On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Stat. Phys. 94(3–4), 347–363 (1999) Communicated by L. Takhtajan


Communications in


A Model of Heat Conduction P. Collet1 , J.-P. Eckmann2 1 Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique,

F-91128 Palaiseau Cedex, France

2 Département de Physique Théorique et Section de Mathématiques,

Université de Genève, CH-1211 Genève 4, Switzerland. E-mail: [email protected] Received: 18 April 2008 / Accepted: 8 August 2008 Published online: 21 November 2008 – © Springer-Verlag 2008

Abstract: In this paper, we first define a deterministic particle model for heat conduction. It consists of a chain of N identical subsystems, each of which contains a scatterer and with particles moving among these scatterers. Based on this model, we then derive heuristically, in the limit of N → ∞ and decreasing scattering cross-section, a Boltzmann equation for this limiting system. This derivation is obtained by a closure argument based on memory loss between collisions. We then prove that the Boltzmann equation has, for stochastic driving forces at the boundary, close to Maxwellians, a unique non-equilibrium steady state. 1. Introduction In this paper, we consider the problem of heat conduction for the continuum limit (N → ∞) of a particle model of a chain of N cells, each of which contains a (very simple) scatterer in its interior. Particles move between the cells, interacting with the scatterers, but not among themselves, similar to the model put forward in [4]. The paper begins with a detailed description of the scatterer model, which is one-dimensional, with the dynamics also in 1 dimension. Once the principles of interaction are laid down, we proceed to derive, in a heuristic way, under the usual closure assumptions, a Boltzmann equation for the limit system. (The reader who is interested only in the formulation of the Boltzmann equation can directly skip to Eq. (4.5).) It should be noted that in order to obtain a reasonable formal limit of the particle system, the scattering probabilities have to be chosen proportional to 1/N . Our main object of mathematical study is then this Boltzmann equation. In particular, starting from Sec. 5 we show how to formulate the heat conduction problem. Namely, heat conduction problems are usually described by prescribing the incoming fluxes on both ends of the system. We however reformulate this as a problem of prescribing what happens at one end only and then to find out what happens at the other. Using the inverse function theorem on a suitable space, we will be able to show that the problem is related

1016

P. Collet, J.-P. Eckmann

to a diffeomorphism in a suitable space (in fact a Banach cone) (starting from Sec. 7). This Banach cone allows for a wide range of initial conditions, and is basically only limited by conditions on the velocity distributions at infinite momenta. However, pure Maxwellians are not in this cone (but on its boundary), although they naturally form the equilibrium solutions. In a forthcoming paper with C. Mejía-Monasterio [1], we will illustrate some of the properties of both the particle model and its Boltzmann formulation, and also “justify” numerically the approximations which lead to the Boltzmann equation. 2. The Particle Model 2.1. One cell. To define the particle model, we begin by describing the scattering process in one cell. We begin with the description of one “cell”. The cell is 1-dimensional, of length 2L, and with particles entering on either side. These particles all have the same mass m, some velocity v and momentum p = mv. These particles do not interact among themselves. Note that v ∈ R and more precisely, v > 0 if the particle enters from the left, while v < 0 if it enters on the right side of the cell. In the center of the cell, we imagine a “scatterer” which is a point-like particle which can exchange energy and momentum with the particles, but does not change its own position. (This scatterer is to be thought of as a 1-dimensional variant of the rotating disks used in [4].) The scatterer has mass M and its “velocity” will be denoted by V . The collision rules are those of an elastic collision, where v˜ and V˜ denote quantities after the collision while v, V are those before the collision. In equations, v˜ = −v + (1 + )V, V˜ = (1 − )v + V, with =

M −m , M +m

µ≡

m 1− = . M 1+

(2.1)

Note that ∈ (−1, 1), since we assume m and M to be finite and non-zero. If v˜ > 0, we say that the particle leaves the cell to the right; if v˜ < 0, we say it leaves to the left. In particular, although everything is 1-dimensional, particles can cross the scatterers. For simplicity, we will assume > 0, that is, M > m. For the momenta, we get the analogous rules p˜ = −p + (1 − )P, P˜ = (1 + ) p + P. Note that the matrix

S =

− 1 − 1+

(2.2)

has determinant equal to −1 and furthermore S 2 = 1. We next formulate scattering in terms of probability densities (for momenta) for just one cell. We denote by g(t, P) the probability density that at time t the scatterer has momentum P (= M V ) and we will establish the equation for the time evolution of this function. To begin with, we assume that particles enter only from the left of the cell, with

A Model of Heat Conduction

1017

momentum distribution (in a neighboring cell or a bath) p → f L+ (t, p), where p = mv. Thus, there are, on average, p f L+ (t, p)d p/m particles entering the cell (per unit of time) from the left with momentum in [ p, p + d p].1 Note that f L+ has support on p ≥ 0 only, (indicated by the exponent “+”); it is the distribution of particles going to enter the cell from the left. Also note that the distribution of the momenta after collision, i.e., before leaving the cell, is in general not the same as f L+ . Denote by P(t) the stochastic process describing the momentum of the scatterer. We have for any interval (measurable set) of momenta A, for the probabilities P: P (P(t + dt) ∈ A) = P (P(t) ∈ A; no collision in [t, t + dt]) +P (P(t + dt) ∈ A; collisions occurred in [t, t + dt]). We assume for simplicity that with probability one, only one collision can occur in an interval [t, t + dt]. If there is a collision in [t, t + dt] with a particle of velocity v = p/m > 0, this particle must have left the boundary at time t − mp L with momentum p. Therefore, P (P(t) ∈ A; a collision occurred in [t, t + dt]) = dt d P˜ d p δ P˜ − P − (1 + ) p g(t, P) mp f L+ t − A

R+

m p L,

p .

This formulation neglects memory effects coming from the fact that a particle may have hit the scatterer, bounce out of the cell and reenter to hit again the scatterer. Similarly, P (P(t) ∈ A ; no collision occurred in [t, t + dt]) = 1 − dt d p mp f L+ t − mp L , p d P g(t, P). R+

A

We immediately deduce the evolution equation, ∂t g(t, P) = −g(t, P) d p mp f L+ t − R+ 1 p d p g t, P−(1+) + R+

m p L, p m

p

f L+ t −

m p L,

p .

(2.3)

Note that this equation preserves the integral of g over P, i.e., it preserves probability. This identity generalizes immediately to the inclusion of injection from the right, with distribution f R− having support in p < 0. One gets ∂t g(t, P) = −g(t, P) d p |mp| f L+ t− mp L , p + f R− t+ mp L , p R 1 p | p| − + m m f t− t+ d p g t, P−(1+) L , p + f L , p . (2.4) + L R m p p R In the stationary case, this leads to 1 p g(P) = d p g P−(1+) λ R

| p| m

f L+ ( p) + f R− ( p) ,

(2.5)

1 The factor p takes into account the probability of crossing the boundary of the cell (which is not the same as the probability of a particle with momentum p to be in the cell).

1018


where

λ=

R

dp

| p| m

f L+ ( p) + f R− ( p)

(2.6)

is the particle flux (see Sec. 4.1 below). It is important to note that the solution g of Eq. (2.5) only depends on the sum: f = f L+ + f R− , and thus, we can define a map f → g f , where g f is the (unique) solution of Eq. (2.5). It will be discussed in detail in Sec. 6. We can also compute the distribution of the momenta of the particles after collision. We have P ( p˜ ∈ A; a collision occurred in [t, t + dt]) = dt d p˜ d P δ ( p˜ + p − (1 − )P) g(t, P) |mp| f t − A

R

m | p| L ,

p .

This particle reaches the left or right boundary of the cell (according to the sign of p) ˜ after a time m L/| p| ˜ (assuming the scatterer is located in the center of the cell). Therefore, we have for the ejection distributions f L− (on the left) and f R+ (on the right): θ (− p) ˜ ˜ | p| ˜ − | p| f (t, p) ˜ = d p g t − |mp| L , p+p f t − |mp| L − |mp| L , p , m L 1− m ˜ ˜ 1− R and | p| ˜ + m f R (t,

p) ˜ =

θ (+ p) ˜ dp g t − 1− R

p+p ˜ m | p| ˜ L , 1−

| p| m

f t−

m | p| ˜ L

where θ is the Heaviside function. In the stationary case we get θ (− p) ˜ ˜ | p| ˜ − | p| ˜ = d p g p+p m f L ( p) 1− m f ( p), 1− R and | p| ˜ + ˜ m f R ( p)

=

θ (+ p) ˜ ˜ d p g p+p 1− 1− R

| p| m

f ( p).

−

m | p| L ,

p ,

(2.7)

(2.8)

Since g = g f is determined by the incoming distribution f in = f L+ + f R− (and is unique if we normalize the integral of g to 1) d P g(t, P) = 1, (2.9) R

we see that the outgoing distribution f out = f L− + f R+ is entirely determined by the incoming distribution. Note also that the flux is preserved: d p |mp| f in ( p) = d p |mp| f out ( p).


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2.2. Stationary solutions for one cell. Here, we will look for stationary states of the evolution equation (2.3), which have also the property that the ejected distributions are equal to the injected ones. It is almost obvious that Maxwellian fixed points can be found, but for completeness, we write down the formulas. The reader should note that the distributions f in and f out have singularities at p = 0. This reflects the well-known fact that slow particles need more time to leave the cell than fast ones. However F( p) ≡ mp f ( p) is a very nice function, and it is this function which appears in all the calculations of the fluxes, and stationary profiles. In this section we do the calculations with the quantity f . Starting from Sec. 4, we will use F. We impose the two incoming distributions f L+ ( p) = σ θ (+ p) |mp| e−βp

2 /(2m)

,

and f R− ( p) = σ θ (− p) |mp| e−βp

2 /(2m)

,

where σ is an arbitrary positive constant (related to λ in (2.6)) and θ is the Heaviside function. It is easy to verify, using Gaussian integration and the identity M = M2 +m(1+)2 , that the solution of Eq. (2.5) is given by β −β P 2 /(2M) β −β P 2 (1−)/((1+)2m) g(P) = = . e e 2π M 2π M Moreover, using the same identity several times one gets from Eqs. (2.7) and (2.8) for the exiting distributions f L− ( p) = σ θ (− p) |mp| e−βp

2 /(2m)

f R+ ( p) = σ θ (+ p) |mp| e−βp

.

,

and 2 /(2m)

Therefore, we see that the Maxwellian fixed points (divided by | p|) preserve both the distribution g of the scatterer, as well as the distributions of the particles. In fact, there are also non-Maxwellian fixed points of the form f L+ ( p) = σ θ (+ p) |mp| e−β( p−ma)

2 /(2m)

,

and f R− ( p) = σ θ (− p) |mp| e−β( p−ma)

2 /(2m)

.

It is easy to verify that the solution of Eq. (2.5) is now given by β −β(P−Ma)2 /(2M) e g(P) = . 2π M Moreover, f L− ( p) = σ θ (− p) |mp| e−β( p−ma)

2 /(2m)

f R+ ( p) = σ θ (+ p) |mp| e−β( p−ma)

.

,

and 2 /(2m)

The verification that this is a solution for any a ∈ R is again by Gaussian integration. Note that if a = 0 there is in fact a flux through the cell.

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3. N Cells The model generalizes immediately to the case of N cells which are arranged in a row, by requiring that the exit distributions of any given cell are equal to the entry distributions of the neighboring cells: the cells are numbered from 1 to N and we have the + , f − , f + , and f − for the particle and g for the scatterers, collections of functions f L,i i L,i R,i R,i i = 1, . . . , N . The equality of entrance and exit distributions is given by the identities + + , and f − = f − f L,i+1 = f R,i R,i L,i+1 for 1 ≤ i < N . The system is completely determined + and f − . Equations (2.5) generalize to by the two functions f L,1 R,N 1 p | p| − + f d p gi P−(1+) ( p) + f ( p) , gi (P) = L,i R,i m λ R

(3.1)

and similarly (2.7) and (2.8) lead to | p| ˜ − ˜ m f L,i ( p) | p| ˜ + ˜ m f R,i ( p)

= =

p+p ˜ θ(− p) ˜ | p| 1− R d p gi 1− m f i ( p), p+p ˜ θ(+ p) ˜ | p| 1− R d p gi 1− m f i ( p),

(3.2)

+ + f − . Clearly, the Gaussians of the previous section are still solutions where f i = f L,i R,i to the full equations for N contiguous cells. Here we have closed the model by assuming independence between the particles leaving and entering from the left (and from the right). In concrete systems this is not true since a particle can leave a cell to the left and re-bounce back into the original cell after just one collision with the scatterer in the neighboring cell, and, in such a situation there is too much memory to allow for full independence. It is possible to imagine several experimental arrangements for which independence is a very good approximation, see also [3,4] for discussions of such issues. One of them could be to imagine long channels between the scatterers where time decorrelation would produce independence. Note that “chaotic” channels may be more complicated since they can modify the distribution of left (right) traveling particles between two cells.

4. Continuous Space We are now ready to derive, non-rigorously, a continuum model. The cells are now replaced by a continuum, with a variable x ∈ [0, 1] replacing i/N , where i is the index of the i th cell. The relations we have derived so far will be generalized to this continuum formulation. So we have moving particles, of mass m and described by a time-dependent density f (t, p, x). The scatterers have mass M and their momentum distribution is called g(t, P, x). It is best to think that the continuous variable x ∈ [0, 1] replaces the discrete index i ∈ {0, . . . , N }. There is then an implicit rescaling of the form x ≈ i/N . Recall that the scatterers are fixed in space (although they have momentum) but that the particles will move in the domain [0, 1]. We first impose, for all x ∈ [0, 1], the normalization R

d P g(t, P, x) = 1,

(4.1)


1021

which is the generalization of Eq. (2.9). The particles again do not interact with each other, but only with the scatterers and, expressed in momenta, the matrix maps ( p, P) ˜ to ( p, ˜ P): p˜ − 1 − p p = ≡ S , 1+ P P P˜

(4.2)

as in Eq. (2.2). We now rewrite the problem in the form of a Boltzmann equation, which takes into ˜ related account this matrix, as well as the particle transport. One guesses, with ( p, ˜ P) to ( p, P) as above: p ∂t f (t, p, x) + ∂x f (t, p, x) m | p| | p| ˜ ˜ f (t, p, ˜ x) g(t, P, x) − f (t, p, x) g(t, P, x) , (4.3) = dP m m | p| ˜ ˜ x) − | p| f (t, p, x) g(t, P, x) . f (t, p, ˜ x) g(t, P, ∂t g(t, P, x) = d p m m The time independent version of the equation will be derived (non-rigorously) below from the model with a chain of cells. It is useful to introduce the function F(t, p, x) =

| p| f (t, p, x), m

and then Eq. (4.3) takes the form m∂t F(t, p, x) + p∂x F(t, p, x) ˜ x) − F(t, p, x) g(t, P, x) , = | p| d P F(t, p, ˜ x) g(t, P, ˜ x) − F(t, p, x) g(t, P, x) . ∂t g(t, P, x) = d p F(t, p, ˜ x) g(t, P,

(4.4)

Remark. One can also imitate a scattering cross section by introducing a factor γ ∈ [0, 1] in Eq. (4.4) (in the integrals) but this can be scaled away by a change of time and space scales. See also Sec. 9.3. We come now to the main equations whose solutions will be discussed in detail in the remainder of the paper. Equation (4.4) takes, for the stationary solution, the form ˜ x) − F( p, x) g(P, x) , (4.5a) sign( p)∂x F( p, x) = d P F( p, ˜ x) g( P, ˜ x) − F( p, x) g(P, x) . 0 = d p F( p, ˜ x) g( P, (4.5b) We will show that this equation has non-equilibrium solutions. Remark. Note that the model we have obtained here is not momentum-translation invariant, because of the term sign( p), except when the r.h.s. of the equation is 0.

1022


Derivation of (4.5). The derivation of (4.5) from (3.1–3.2) is based on the following formal limit: We replace the index i by the continuous variable x = i/N and set ε = ± ± 1/(2N ). We consider that f L,i is at (i − 21 )/N = x − ε, while f R,i is at x + ε. We have the correspondences, with θ± ( p) ≡ θ (± p): θ+ ( p)F( p, x − ε) =

| p| + m f L,i , | p| + m f R,i ,

θ+ ( p)F( p, x + ε) = g(P, x) = gi (P).

θ− ( p)F( p, x − ε) = θ− ( p)F( p, x + ε) =

| p| − m f L,i , | p| − m f R,i ,

To simplify momentarily the notation, let F− ( p, x) θ+ ( p) ≡ θ+ ( p)F( p, x − ε), F− ( p, x) θ− ( p) ≡ θ− ( p)F( p, x − ε), F+ ( p, x) θ+ ( p) ≡ θ+ ( p)F( p, x + ε), F+ ( p, x) θ− ( p) ≡ θ− ( p)F( p, x + ε). With these conventions, (3.1) becomes (setting λ = 1): 1 dq g P−(1+)q , x (F− (q, x)θ+ (q) + F+ (q, x)θ− (q)) , g(P, x) = R

(4.6)

which is equivalent to (4.5b). Similarly, Eq. (3.2) leads to p+q p) dq g , x F− ( p, x)θ− ( p) = θ(− (F− (q, x)θ+ (q) + F+ (q, x)θ− (q)) , 1− R 1− (4.7) p+q θ(+ p) F+ ( p, x)θ+ ( p) = 1− R dq g 1− , x (F− (q, x)θ+ (q) + F+ (q, x)θ− (q)) . Subtracting the first equation from the second in (4.7) leads to F+ ( p, x)θ+ ( p) − F− ( p, x)θ− ( p) sign( p) dq g p+q , x = (F− (q, x)θ+ (q) + F+ (q, x)θ− (q)) . 1− 1− R

(4.8)

On the other hand, since d p d P = d p˜ d P˜ we can, by (4.5b), impose the condition d P g(P, x) = 1, for all x. Then we have the trivial identity F− ( p, x)θ+ ( p)−F+ ( p, x)θ− ( p) = d P g(P, x) (F− ( p, x)θ+ ( p)−F+ ( p, x)θ− ( p)) . R

(4.9) Subtracting (4.9) from (4.8), we get for p > 0, 1 F+ ( p, x) − F− ( p, x) = , x dq g p+q (F− (q, x)θ+ (q) + F+ (q, x)θ− (q)) 1− 1− − dq g(q, x)F− ( p, x), (4.10)


1023

while for p < 0, 1 F+ ( p, x) − F− ( p, x) = − , x (F− (q, x)θ+ (q) + F+ (q, x)θ− (q)) dq g p+q 1− 1− + dq g(q, x)F+ ( p, x). (4.11) Note now that F− (q, x)θ+ (q) + F+ (q, x)θ− (q) = F(q, x − ε)θ+ (q) + F(q, x + ε)θ− (q) = F(q, x) + θ+ (q) (F(q, x − ε) − F(q, x)) +θ− (q) (F(q, x + ε) − F(q, x)) . Therefore, replacing the F± in the r.h.s. in (4.10) and (4.11) by F( p, x) is a higher order correction in ε, and we finally get sign( p) F( p, x + ε)−F( p, x − ε) = dq g p+q , x F(q, x)−g(q, x)F( p, x) . 1− 1− R A further change of integration variables leads to (4.5a), while (4.6) leads to (4.5b). (We have not taken into account another scaling of the scattering term by ε = 1/(2N ) which is needed to get the derivative. This corresponds to a particle model where the scattering cross-section is proportional to 1/N . (We will come back to this question in the discussion in Sect. 9.3 and in [1].) This ends the derivation of (4.5). The derivative term in Eq. (4.5) reflects the gradients which have to appear when the system is out of equilibrium. However, if the system is at equilibrium, the equivalence between Eq. (4.5) and Eqs. (2.5)–(2.8) immediately tells us that stationary solutions in the form of Gaussians (for F, not for f ) exist: F( p) = γ

β −βp2 /(2m) e , 2π m

g(P) =

β −β P 2 /(2M) e . 2π M

(4.12)

Furthermore, we have again translated versions of this fixed point, F( p) = γ

β −β( p−ma)2 /(2m) e , 2π m

g(P) =

β −β(P−Ma)2 /(2M) e , 2π M

(4.13)

because in this case, the r.h.s. of Eq. (4.5) is zero. 4.1. Flux. We can define various fluxes of the particles (recall that F( p, x) = | p| f ( p, x)/m): P = particle flux = d p sign( p) F( p, x), M = momentum activity = d p | p| F( p, x), 2 E = energy flux = d p sign(2mp) p F( p, x).

(4.14)

1024


Note that for the stationary Maxwellians of (4.13) these fluxes are equal to

2mπ erf(a βm/2), P = β

2m 3 π 2m −βma 2 /2 e erf(a βm/2), M = +a β β

2m 3 π 2am 2 −βma 2 /2 2 e + (1 + βma ) erf(a βm/2). E = 3 β β Also note that for a = 0 the quantity M does not vanish. This is because it measures the total outgoing flux, not the directed outgoing flux (which is of course 0 when a = 0). Lemma 4.1. For every stationary solution of (4.5) the 3 fluxes of (4.14) are independent of x ∈ [0, 1]. Proof. From (4.5a) we deduce that ˜ x) − F( p, x) g(P, x) , ∂x d p sign( p) F( p, x) = d p d P F( p, ˜ x) g( P, ˜ Similarly, multiplying (4.5a) by p and integrating which vanishes since d p d P = d p˜ d P. over p, we get ˜ x) − F( p, x) g(P, x) . ˜ x) g( P, ∂x d p | p| F( p, x) = d p d P p F( p, Multiplying (4.5b) by P and integrating over P, we get ˜ x) − F( p, x) g(P, x) . 0 = d p d P P F( p, ˜ x) g( P, Adding these two equations, we see that ˜ x) − F( p, x) g(P, x) . ˜ x) g( P, ∂x d p | p| F( p, x) = d p d P ( p + P) F( p, But this vanishes, since P + p = P˜ + p˜ by momentum conservation, and using again ˜ In a similar way, we first have d p d P = d p˜ d P. | p| p F( p, x) ∂x d p 2m p2 ˜ x) − F( p, x) g(P, x) . F( p, ˜ x) g( P, = dp dP 2m Finally, multiplying this time (4.5b) by P 2 /M, integrating over P, adding to the above equation and using energy conservation, we get | p| p F( p, x) ∂x d p 2m 2 P2 p ˜ x) − F( p, x) g(P, x) + F( p, ˜ x) g( P, = dp dP 2m 2M = 0. Thus, all three fluxes are independent of x, as asserted.


1025

5. Formulating the Heat-Conduction Problem Based on the stationarity equation (4.5), we now formulate the problem of heat conduction in mathematical terms. We imagine that the input of the problem is given by prescribing the incoming fluxes on both sides of the system. The system, in its stationary state, should then adapt all the other quantities, F and g, to this given input, which describes really the forcing of the system. In particular, the distribution of the outgoing fluxes will be entirely determined by the incoming fluxes. We now formulate this question in mathematical terms: the incoming fluxes are described by two functions F0 ( p) (defined for p ≥ 0) and F1 ( p) (defined for p ≤ 0). These are the incoming distributions on the left end (index 0) and the right end (index 1) of the system. In terms of these 2 functions, the problem of existence of a stationary state can be formulated as (recall that the rescaled system has length one): Is there a solution (F, g) of Eqs. (4.5) with the boundary conditions F( p, 0) = F0 ( p), ∀ p ≥ 0

and

F( p, 1) = F1 ( p), ∀ p ≤ 0.

(5.1)

Assume for a moment that, instead of the boundary conditions (5.1) we were given just F( p, 0), but now for all p ∈ R, not only for p > 0. Assume furthermore, that g( p, x) is determined by (4.5b). In that case, the relation (4.5) can be written as a dynamical system in the variable x: ∂x F(·, x) = X (F(·, x)).

(5.2)

Thus, if F(·, 0) is given, then, in principle, F(·, 1) is determined (uniquely) by Eq. (5.2), provided such a solution exists. We denote this map by Y0 : Y0 : F(·, 0) → F(·, 1). What is of interest for our problem is the restriction of the image of Y to functions of negative p only, since that corresponds to the incoming particles from the right side, and so we define (Y (F(·, 0))) ( p) ≡ θ− ( p) · (Y0 (F(·, 0))) ( p) = θ− ( p) · F( p, 1). Using this map Y , we will show that when F(·, 0) varies in a small neighborhood the map Y is invertible on its image. By taking inverses the problem of heat conduction for our model will be solved for small temperature and flux difference. Of course, this needs a careful study of the function space on which Y is supposed to act. This will be done below. To formulate the problem more precisely, we change notation, and let F0+ ( p) = θ+ ( p)F( p, 0),

F0− ( p) = θ− ( p)F( p, 0), F1+ ( p) = θ+ ( p)F( p, 1), F1− ( p) = θ− ( p)F( p, 1).

We assume now that F0+ is fixed once and for all and omit it from the notation. Then, we see that Y can be interpreted as a map which maps the function F0− to F1− , and we call this map .

1026


We will show below that for F0− in a small neighborhood D the map is 1-1 onto its image (D) and can therefore be inverted. For any Fˆ in (D), we can take ˆ and we will have solved the problem of existence of heat flux. F0− = −1 ( F), 6. The g Equation We start here with the study of existence of g for given F. Since (4.5b) does not couple different x, we can fix x. Equation (4.5b) is then equivalent to g = A F (g), where the operator A F acting on the function h is defined by ˜ x h P( p, P), ˜ x d p F p( ˜ p, P), ˜ x) = A F (h)( P, , d p F ( p, x) (provided the denominator does not vanish). Note that for fixed P˜ and p, we can solve the collision system (4.2) to find the corresponding P and p, ˜ namely P=

1 ˜ 1+ p P−

and

p˜ =

1− ˜ 1 P − p.

The action of the operator A F can then be rewritten as ˜ − (1 + ) p/, x −1 d p F( p, x)h P/ ˜ x) = A F (h)( P, . ds F(s, x)

(6.1)

In order to study this operator notice that it does not depend explicitly on x. It is convenient to study instead a family of operators indexed by functions ϕ of the momentum only. We define (assuming the integral of ϕ does not vanish) ˜ − (1 + ) p/ d p ϕ( p)ψ P/ ˜ = Lϕ ψ ( P) . d p ϕ( p) A final change of variables will be useful when we study Lϕ : p−q ψ(q) dq ϕ 1+ Lϕ ψ ( p) = . (1 + ) dq ϕ(q)

(6.2)

7. The Mathematical Setup and the Main Result Having formulated the problem of existence of the stationary solution in general, we now fix the mathematical framework in which we can prove this existence. This framework, while quite general, depends nevertheless on a certain number of technical assumptions which we formulate now. We fix once and for all the ratio µ = m/M of the masses, and assume, for definiteness, that µ ∈ (0, 1). It seems that this condition is not really necessary, and probably


1027

the condition m = M (and the masses non-zero) should work as well, but we have not pursued this. We next describe a condition on the incoming distribution, called F in the earlier sections. The basic idea, inspired from the equilibrium calculations, is that F( p, x) should be close to Freference ( p) = exp(−βp 2 /(2m)) ≡ exp(−αp 2 ), while the derived quantity g( p, x) should be close to greference ( p) = exp(−βp 2 /(2M)) ≡ exp(−µαp 2 ). Upon rescaling p, we may assume henceforth that α = 1. The operators of the earlier sections will now be described in spaces with weights Wν ( p) = exp(−νp 2 ), where we will choose ν = 1 for the F and ν = µ for the g. We recall that the operator L F in “flat” space is dq F p−q g(q) 1+ (L F g)( p) = . (1 + ) dq F(q) We then define the integral kernel in the space with weights exp(− p 2 ) for F and exp(−µp 2 ) for g, and write F( p) = e− p v( p) ,

g( p) = e−µp u( p).

2

2

Here, µ = m/M = (1 − )/(1 + ), as before. Expressed with u and v the operator L F takes the form (K v u) ( p) =

1 · (L v u) ( p), 2 (1 + ) dqe−q v(q)

where

(L v u)( p) =

and K ( p, q) = W1

dq v

p−q 1+

p−q 1+

(7.1)

K ( p, q)u(q),

· Wµ (q)/Wµ ( p).

A simple calculation shows that K ( p, q) = e−(p−q)

2 /(1+)2

.

(7.2)

Our task will be to understand under which conditions the linear operator L F has an eigenvalue 1. This will be done by showing that K v is quasi-compact. It is here that we were not able to give reasonable bounds on K ( p, q) in the case of different exponentials for p > 0 and p < 0, which represents different temperatures for ingoing and outgoing particles.

1028


7.1. Function spaces. We now define spaces which are adapted to the simultaneous requirement of functions being close to a Gaussian near | p| = ∞ and u and v having limits, and K v being quasi-compact. We define a space G1 of functions u with norm 2

u G1 = e−µp |u( p)|d p. Similarly, F1 is the space of functions v with norm 2

v F1 = e− p |v( p)|d p. Thus, the only difference is the absence of the factor µ = (1−)/(1+) in the exponent. We also define a smaller space G2 , contained in G1 , with the norm 2

u G2 = |du( p)| + e−µp |u( p)|d p, and the analogous space F2 contained in F1 with the norm 2

v F2 = |dv( p)| + e− p |v( p)|d p. Remark. To simplify notation we write |du( p)| instead of the variation norm. However, the “integration by parts” formula would hold with the “correct” definition of variation as well. Lemma 7.1. One has the inclusion G2 ⊂ L∞ , and more precisely 2 µ ∞

u L ≤ |du| + e e−µp |u( p)|d p ≤ eµ u G2 . Furthermore, if u ∈ G2 , then lim p→±∞ u( p) exists. The maps u → lim p→±∞ u( p) and u → d p exp(−µp 2 ) · u( p) are continuous functions from G2 to R. The unit ball of G2 is compact in G1 . Analogous statements hold for the spaces F2 (defined without the factor µ). Proof. The first statement is easy, but it will be convenient to have the explicit estimates. We have du, u(y) − u(x) = [x,y]

and therefore |u(x)| ≤

|du| +

1/2 −1/2

|u(y)|dy ≤

|du| + e

µ

e−µp |u( p)|d p. 2

The second statement follows at once since the functions in G2 are of bounded variation. ∞ For the last assertions, it follows from2 the inclusion in L that the unit ball of G2 is equi-integrable at infinity in L1 e−µp d p . Moreover, a set of uniformly bounded functions of uniformly bounded variation is compact in any L1 (K , d p) for any compact subset K of R (see [2], Helly’s selection principle).


1029

7.2. A cone in F2 . We will work in the space F2 but we will need a cone (of positive functions, with adequate decay) in this space, in order to prove quasi-compactness of Kv . We define a cone CF in F2 by the condition

(7.3) CF = v ∈ F2 , v ≥ 0 and Z · lim v( p) < 1 , p→±∞

where Z = Z (v) =

√ π d p e− p v( p) 2

.

(7.4)

Lemma 7.2. The cone CF has non empty interior (in F2 ) and is convex. Proof. By Lemma 7.1 the maps v → lim p→±∞ v( p) and v → exp(− p 2 )v( p)d p are continuous in F2 and hence the assertion follows.

Remark. Note that a function in the interior of the cone is necessarily bounded away from zero, since at infinity it must have a non-zero limit and in any compact set, if it is never zero, it is bounded away from zero. Remark. Note that the function v ≡ 1 (the Gaussian) is not in the cone CF . In fact, we 2 2 require that lim p→±∞ F( p)e p · e− p d p / F( p )d p < 1. 7.3. The main result. On the set CF , we consider now the spatial evolution equations (4.5) in the variables v and u v (which is the solution of K v u = u with K v defined in (7.1)): ∂x v( p, x)

˜ x) d P (W1 ·v)( p, ˜ x) (Wµ ·u v(·,x) )( P, (7.5) ×(W1 ·v)( p, x) (Wµ ·u v(·,x) )(P, x) = sign( p) d P (W1 ·v) (−p + (1 − )P, x) (Wµ ·u v(·,x) ) ((1 + ) p + P, x) 2 −sign( p) v( p, x) d P e−µP u v(·,x) (P, x),

= sign( p)

with initial condition v(·, x = 0) ∈ CF . We will give a more explicit variant in (8.9). Any solution of this equation is a function of p and x, and it is easy to verify that it satisfies Eq. (4.5a). Together with the definition of u v we have a complete solution of the nonlinear system (4.5). Here we assume of course that the r.h.s. of the above equation is well defined as a function, so that we can multiply by sign( p). Theorem 7.3. For any v0 ∈ CF , there are a number xv0 > 0 and a neighborhood Vv0 of v0 in CF such that the solution of (7.5) exists for any initial condition v0 = v( p, 0) ∈ Vv0 and for any x in the interval [0, xv0 ]. The function v0 → xv0 is continuous from CF to R+ compactified at infinity. We denote by x the semi-flow integrating (7.5). For any x ∈ [0, xv0 ], the map x : v → x (v) is a local diffeomorphism, i.e., a diffeomorphism on Vv0 .

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Note that this implies in particular that the probability densities for g(·, x) and F(·, x) remain positive for all x ∈ [0, xv0 ], which is of course crucial from the physics point of view. We will prove this in Sect. 8 (and in the Appendix). 8. Bound on the Operator K v and Proof of Theorem 7.3 These bounds are the crux of the matter. They actually show, that, under the conditions on F2 and the set CF , the operator K v is quasi-compact. In terms of the physical problem, this means that the scatterer is not heating up if the incoming fluxes are in F2 . The object of study is, for v ∈ CF , the operator 1 p−q K ( p, q)u(q)dq, v (K v u) ( p) = 2 1+ (1 + ) e−q v(q)dq and we are asking for a solution u of the equation K v (u) = u. Lemma 8.1. If v ≥ 0, K v is a positive (nonnegative) operator, and 2 −µp 2 dp e (K v u) ( p) = d p e−µp u( p) and

K v G1 = 1. Proof. Easy, compute and take absolute values. Alternately, consider that the probability is conserved in the original space.

Our main technical result is Proposition 8.2. For v ∈ CF , there exist a ζ , 0 ≤ ζ < 1 and an R > 0 (both depend on v continuously) such that for any u ∈ G2 one has the bound dK v (u) ≤ ζ du + R u G . 1 Proof. Since v will be fixed throughout the study of K v , it will be useful to introduce the abbreviation Q = Q v for the normalizing factor Q=

1

(1 + ) e−q v(q)dq 2

.

(8.1)

We will use a family of smooth cut-off functions χ L (L > 1) which are equal to 1 on [−L + 21 , L − 21 ] and which vanish on |q| > L + 21 . Let be a C ∞ function satisfying 0 ≤ ≤ 1, with (q) = 0 for q ≤ − 21 and (q) = 1 for q ≥ 21 . We define χ L by ⎧ 1 ⎪ ⎨(q + L) if q ≤ −L + 2 , 1 χ L (q) = 1 if − L + 2 ≤ q ≤ L − 21 , ⎪ ⎩(L − q) if q ≥ L − 1 . 2


1031

The functions χ L are C ∞ , satisfy 0 ≤ χ L ≤ 1 and χ L L∞ is independent of L. Let L 1 and L 2 be two positive numbers to be chosen large enough later on (depending on v). We will use the partition of unity 1 = χ L + χ L⊥ . (1)

Using this decomposition of unity with L = L 1 and L = L 2 , we write K v = K v + K v(2) + K v(3) with K v(1) u ( p) = Q dq v p−q K ( p, q)u(q) · χ L 1 (q), 1+ K ( p, q)u(q) · χ L⊥1 (q) χ L 2 ( p − q), K v(2) u ( p) = Q dq v p−q 1+ K ( p, q)u(q) · χ L⊥1 (q) χ L⊥2 ( p − q) . K v(3) u ( p) = Q dq v p−q 1+ We will now estimate the variation of the three operators separately. For the variation of the first term, we find 1 d K v(1) u ( p) = +Q dq dv p−q 1+ 1+ · K ( p, q)u(q)χ L 1 (q) ∂ p K ( p, q) · u(q)χ L 1 (q). +Q d p dq v p−q 1+ Using the explicit form of K ( p, q) (see Eq. (7.2)), and some easy bounds which we defer to the Appendix, we get the bound (1) ∞ d K v u ( p) ≤ O(1)Q v L + |dv|

1 L1+ 2 1 −L 1 − 2

|u(q)|dq

1 2 2 ≤ O(1)Q v L∞ + |dv| eµ(L 1 + 2 ) e−µq |u(q)|dq (8.2) ≤ const. u G1 · v F2 . (2)

The variation of K v leads to three terms: d K v(2) u ( p) K ( p, q)u(q) · χ L⊥1 (q) χ L 2 ( p − q) = Q d p dq v p−q 1+ ⊥ 1 +Q dq dv p−q 1+ 1+ · K ( p, q)u(q) · χ L 1 (q) χ L 2 ( p − q) ∂ p K ( p, q) · u(q) · χ L⊥1 (q) χ L 2 ( p − q) +Q d p dq v p−q 1+ := d J21 + d J22 + d J23 . In these terms, the variables p and q are in the domain D = {( p, q) ∈ R2 : | p − q| < L 2 +

1 2

and |q| > L 1 − 21 },

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and for L 1 = 3L 2 /(1 − 2 ) and L 2 sufficiently large we have from Lemma A.3: (8.3) K ( p, q) < exp −C1 (p − q)2 − C2 L 22 . Therefore, we get for d J21 : −C2 L 22

u ∞ v ∞ |d J21 | ≤ const.e

( p,q)∈D

d p dqe−C1 (p−q) . 2

(8.4)

The integral exists and is uniformly bounded in L 2 (since |p−q| → ∞ when |q| → ∞). The term d J22 is handled in a similar way and leads to the bound 2 (8.5) |d J22 | ≤ const.e−C2 L 2 u ∞ |dv|. The following identity is useful: ∂ p K ( p, q) = −∂q K ( p, q).

(8.6)

For the term d J23 we observe that from (8.6) one gets, upon integrating by parts, with the notation K ( p, q) · ∂q χ L⊥1 (q)χ L 2 ( p − q) d J23 = Q d p dq v p−q 1+ − ⊥ +Q d p dv p−q 1+ 1+ K ( p, q)u(q) · χ L 1 (q)χ L 2 ( p − q) K ( p, q)du(q) · χ L⊥1 (q)χ L 2 ( p − q) +Q d p v p−q 1+ := d J231 + d J232 + d J233 . All these terms are localized in the domain D. In d J231 there appears a derivative X = ∂q χ L⊥1 (q)χ L 2 ( p − q) = −χ L 1 (q)χ L 2 ( p − q) −χ L⊥1 (q)χ L 2 ( p − q) := X 1 + X 2 . The terms involving X 1 and X 2 can be bounded as d J21 and d J22 by observing that suppχ L 1 ⊂ {|q| < L 1 + 21 }, and similarly for X 2 . The terms d J232 and d J233 are bounded similarly. Together, these lead to a bound (2) dK (u) ≤ const.e−C2 L 22 v F u G . (8.7) v 2 2 Remark. Note that in this term, the norm u G2 appears with a small coefficient, while in (8.2) it was u G1 (with a large coefficient).


1033 (3)

Finally, we estimate the total variation of K v (u) and here, the nature of the set CF will be important. We have d K v(3) u ( p) ⊥ ⊥ 1 = Q dq dv p−q 1+ 1+ K ( p, q)u(q) · χ L 1 (q) χ L 2 ( p − q) K ( p, q)u(q) · χ L⊥1 (q) χ L 2 ( p − q) −Q d p dq v p−q 1+ ∂ p K ( p, q) · u(q) · χ L⊥1 (q) χ L⊥2 ( p − q) +Q d p dq v p−q 1+ := d J31 + d J32 + d J33 . The critical term is d J33 , but we first deal with the two others which are treated similar to earlier cases. For the first term we have by Lemma A.4 which tells us that K is exponentially bounded on D :

d J31 ≤ const. u L∞

1 |s|>(L 2 − 2 )/(1+)

|dv(s)|,

where D is the domain D = {( p, q) : |q| > L 1

and

| p − q| > L 2 }.

For the second term, we have, again by Lemma A.4 below, |d J32 | ≤ const. u L∞ v L∞ . The last term is more delicate, and uses the property Z · lim p→±∞ v( p) < 1 of the definition of the cone CF , Eq. (7.3). integrate by parts as before using (8.6) and get

− ⊥ ⊥ dv p−q 1+ 1+ K ( p, q)u(q) · χ L 1 (q) χ L 2 ( p − q) ⊥ K ( p, q)u(q) · ∂ χ ×Q d p dq v p−q (q)χ ( p − q) q L 1 L2 1+ K ( p, q)du(q) · χ L⊥1 (q) χ L⊥2 ( p − q) ×Qd p v p−q 1+

d J33 = Q d p

:= d J331 + d J332 + d J333 . The term d J331 is bounded like d J31 . In a similar way d J332 and d J32 are bounded by the same methods.

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The term d J333 makes use of the limit condition in CF . Consider the integral of |d J333 |. This leads to a bound and setting L 2 = (L 2 − 21 )/(1 + ): |d J333 ( p)| ≤ Q sup |v(s)| · |du(q)| |s|>L 2

d p K ( p, q) · χ L⊥1 (q) χ L⊥2 ( p − q) ⎛ ⎞ ⎜ ⎟ ≤ Q ⎝ sup d p K ( p, q)⎠ · sup |v(s)| |du| ·

|s|>L 2

1 |q|>L 1 − 2

=

√ π

dqe−q v(q) 2

(8.8)

· sup |v(s)| |s|>L 2

|du|

= Z · sup |v(s)| |s|>L 2

|du|,

where Z was defined in Eq. (7.4). Collecting all the estimates, we get 2 |dK v (u)| ≤ C e−µq |u(q)|dq + ζ (L 2 ) |du|, where ζ (L 2 ) = O(1)e

−C2 L 22

v F2 + O(1)

|s|>L 2

|dv(s)| + Z · sup |v(s)|. |s|>L 2

Since v belongs to CF , it follows that lim ζ (L 2 ) < 1,

L 2 →∞

and the lemma follows by taking L 2 large enough.

Proposition 8.3. For any v ∈ CF , the equation K v (u) = u has a solution in G2 . This solution can be chosen positive, it is then unique if we impose u G1 = 1. We call it u v . The map v → u v is differentiable. Proof. We apply the theorem of Ionescu-Tulcea and Marinescu [5] to prove the existence of u. Since for v > 0, the operator K v is positivity improving, it follows by a well known argument, see e.g., [7] that the peripheral spectrum consists only of the simple eigenvalue one and the eigenvector can be chosen positive. If normalized, it is then unique. Since the operator K v depends linearly and continuously on v (in F2 ), the last result follows by analytic perturbation theory (see [6]).

We next consider Eq. (7.5) for v: 2 2 2 p ∂x v( p) = sign( p) 1 e(1− )( p−q/(1+)) / v (1−)q− u v (q) dq 2 − v( p) e−µq u v (q) dq .

(8.9)


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Proposition 8.4. The r.h.s. of the equation for v is a C 1 vector field on F2 . Proof. This follows easily from the fact that the map v → u v is C 1 .

Theorem 8.5. Let v0 ∈ CF , and assume that v0 is bounded below away from zero and has nonzero limits at ±∞. Then there is a number s = s(v) > 0 such that the solution Eq. (8.9) with initial condition v0 exists in F2 and is nonnegative (moreover, it belongs to CF ). Proof. Follows at once from the previous proposition and the fact that v0 is in the interior of CF .

The proof of Theorem 7.3 is now completed by observing that the map : v0 → (v0 ) is indeed a local diffeomorphism, since it is given as the solution of an evolution equation.

9. Remarks and Discussion 9.1. The behavior of the solution at p = ∞. Consider the limit p → ∞ in the expression for K v . We need p − q = O(1) otherwise the Gaussian gives a negligible contribution. In other words, q ∼ p, and we are going to assume from now on that > 0 (the other case can be treated analogously). This implies p − q ∼ (1 − 2 ) p which also tends to infinity (the same infinity). Therefore, √ K v u(±∞) =

π v(±∞) u(±∞) . 2 e− p v( p)d p

In particular, if K v u = u and since we assumed

√ π v(±∞) e− p v( p)d p 2

= 1

we get u(±∞) = 0. For the v equation, we have for large p, q ∼ p(1 + ) and (1 − )q − p ∼ −2 p. Therefore (inverting limit and derivative) we get

√ ∂x v(±∞) = sign(±∞) π

1+ u(±∞) v(∓∞) − v(±∞) 1−

e

−µq 2

u(q)dq .

Note that the first term vanishes since u(±∞) = 0. Since the integral C(x) = u(q, x) is positive, we conclude that formally, ∂x v(±∞) = ∓v(±∞)C(x).

e−µq

2

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9.2. Essential spectrum. Conjecture. The essential spectrum of K v is the interval [0, σ (v)] with σ (v) = max

√ π v(±∞) e− p v( p)d p 2

.

If σ (v) < 1 we are looking for an eigenvalue 1 outside the essential spectrum, which is the case we have treated. If σ (v) > 1 we would be looking for an eigenvalue 1 inside the essential spectrum which would be a much more difficult task, since it may well not exist. Idea of proof. Similar to the above estimates, the operator K v should be written as something small plus something compact plus something whose essential spectrum can be computed. This last part is likely to be the limit operator at infinity.

9.3. Dependence on N . It should be noted that the equation for ∂x F has, in fact a scaling of the form N −1 ∂x F = O(1) + O(N −1 ). This means that in the main theorem (Theorem 7.3), the limit xv0 of x for which we have a result is quite probably bounded by a quantity of the form 1/(N · (v0 )), where (v0 ) measures the deviation of the initial condition v0 from a Gaussian. Thus, either xv0 is very small when N is large, or one has to take v0 very close to a Gaussian. Another way to look at this scaling is to introduce a scattering probability γ = b/N where b > 0 is a constant independent of N . In other words, a particle entering the array of cells from the left has for large N a probability e−b to traverse all the N cells (and leave on the right) without having experienced any scattering. This is analogous to a rarefied gas. It is easy to verify that Eq. (2.4) is modified by a factor b/N multiplying the right hand side, and hence Eq. (2.5) is unchanged. The stationary equations (4.5) become | p| ˜ − m f L (t,

b | p| ˜ p) ˜ = θ (− p) ˜ 1− f (t, p) ˜ N m ˜ b θ (− p) d p g t − m L/| p|, ˜ + N 1− R

p+p ˜ 1−

| p| m

f t−

m | p| ˜ L

−

m | p| L ,

p ,

| p| m

f t−

m | p| ˜ L

−

m | p| L ,

p .

and | p| ˜ + m f R (t,

b | p| ˜ p) ˜ = θ (+ p) ˜ 1− f (t, p) ˜ N m ˜ b θ (+ p) d p g t − m L/| p|, ˜ + N 1− R

p+p ˜ 1−

Equation (4.5a) follows as explained in Sect. 4 after a rescaling of space by a factor b.


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9.4. Discussion. The model presented in this paper has the nice property that one can control the existence of a solution out of equilibrium. In particular, this means that there is no heating up of the scatterers in the “chain”, when the system is out of equilibrium. The reader should note, however, that the initial condition at the boundary does not allow for different temperatures in the strict sense, only for different distributions at the ends. For example, a function of the form exp(−αp 2 ), if p > 0, F( p, 0) = exp(−α p 2 ), if p < 0, with α = α is not covered by Theorem 7.3. The reason for this failure is that we could not find an adequate analog of Lemma A.4 for initial conditions of this type, and therefore the bounds on the kernel K ( p, q) are not good enough. Acknowledgement. We thank Ph. Jacquet for a careful reading of the manuscript. This work was partially supported by the Fonds National Suisse.

A. Appendix: Bounds on K ( p, q) We study here the kernel K of (7.2), which equals K ( p, q) = e E( p,q) , with E( p, q) = µp 2 − µq 2 −

p−q 2 1+

= −(p − q)2 /(1 + )2 .

(A.1)

Lemma A.1. Assume |q| < L. There are constants C = C(L , ) and D = D(L , ) > 0 such that for all p, K ( p, q) < Ce−Dp , 2

(A.2)

and |∂ p K ( p, q)| < Ce−Dp . 2

(A.3)

Proof. Obvious.

Lemma A.2. Assume | p − q| < L. There are constants C = C(L , ) and D = D(L , ) > 0 such that for all q, K ( p, q) < Ce−Dq , 2

(A.4)

and |∂ p K ( p, q)| < Ce−Dq , 2

(A.5)

Proof. The proof is as in Lemma A.1, with the difference that now | p − q| < L.

1038


Lemma A.3. Consider the domain D defined by D = {( p, q) ∈ R2 : | p − q| < L 2 +

1 2

and |q| > L 1 − 21 },

(A.6)

with L1 =

3 L 2. 1 − 2

(A.7)

For fixed ∈ (0, 1) and sufficiently large L 2 there are positive constants C1 and C2 such that for ( p, q) ∈ D one has the bound K ( p, q) < exp −C1 (p − q)2 − C2 L 22 . Proof. From the definition of D and (1 − 2 )q = (p − q) − ( p − q), we find (for sufficiently large L 2 ): |p−q| ≥ 1−2 |q|−| p−q| ≥ 1−2 L 1 − 21 − L 2 + 21 > L 2 . (A.8) Using the form (p − q)2 > 41 (p − q)2 + 41 2 L 22 ,

(A.9)

the assertion follows immediately.

We next study the region D = {( p, q) : |q| > L 1

and

| p − q| > L 2 }.

(A.10)

In this region, we have the obvious bound Lemma A.4. For ( p, q) ∈ D , one has the bound 2 . E( p, q) = − p−q 1+

References 1. Collet, P., Eckmann, J.-P., Mejía-Monasterio, C.: ArXiv:0810.4464 2. Dunford, N., Schwartz, J.T.: Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space. With the assistance of William G. Bade and Robert G. Bartle, New York-London: Interscience Publishers John Wiley & Sons, 1963 3. Eckmann, J.-P., Mejía-Monasterio, C., Zabey, E.: Memory effects in nonequilibrium transport for deterministic Hamiltonian systems. J. Stat. Phys. 123, 1339–1360 (2006) 4. Eckmann, J.-P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1-D models. Commun. Math. Phys. 262, 237–267 (2006) 5. Ionescu Tulcea, C.T., Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 52, 140–147 (1950) 6. Kato, T.: Perturbation Theory for Linear Operators. Berlin: Springer-Verlag, 1984 (Second corrected printing of the second edition) 7. Schaefer, H.H., Wolff, M.P.: Topological vector spaces. second edition, Volume 3 of Graduate Texts in Mathematics, New York: Springer-Verlag, 1999 Communicated by A. Kupiainen


Communications in


Structure and f -Dependence of the A.C.I.M. for a Unimodal Map f of Misiurewicz Type David Ruelle1,2 1 Mathematics Department, Rutgers University, Piscataway, NJ 08854, USA 2 IHES, 35 route de Chartres, 91440 Bures sur Yvette, France. E-mail: [email protected]

Received: 20 April 2008 / Accepted: 27 May 2008 Published online: 16 September 2008 – © Springer-Verlag 2008

Abstract: By using a suitable Banach space on which we let the transfer operator act, we make a detailed study of the ergodic theory of a unimodal map f of the interval in the Misiurewicz case. We show in particular that the absolutely continuous invariant measure ρ can be written as the sum of 1/square root spikes along the critical orbit, plus a continuous background. We conclude by a discussion of the sense in which the map f → ρ may be differentiable. 0. Introduction This paper is part of an attempt to understand the smoothness of the map f → ρ, where (M, f ) is a differentiable dynamical system and ρ an SRB measure. [For a general introduction to the problems involved, see for instance [2,31]]. Smoothness has been established for uniformly hyperbolic systems (see [9,13,17,21,22]). In that case, one finds that the derivative of ρ with respect to f can be expressed in terms of the value at ω = 0 of a susceptibility function (eiω ) which is holomorphic when the complex frequency ω satisfies Im ω > 0, and meromorphic for Im ω > some negative constant. In the absence of uniform hyperbolicity, f → ρ need not be continuous. Consider then a family ( f κ )κ∈R . A theorem of H. Whitney [29] gives general conditions under which, if ρκ is defined on K ⊂ R, then κ → ρκ extends to a differentiable function of κ on R. Taking ρκ to be an SRB measure for f κ , this gives a reasonable meaning to the differentiability of κ → ρκ on K (as proposed in [24], see [11,20] for a different application of Whitney’s theorem), even though we start with a noncontinuous function κ → ρκ on R. Using Whitney’s theorem to study SRB states as proposed above is a delicate matter. A simple situation that one may try to analyze is when (M, f ) is a unimodal map of the interval and ρ an absolutely continuous invariant measure (a.c.i.m.). [From the vast literature on this subject, let us mention [6–8,12,15,28]]. A preliminary study of the Markovian case (i.e., when the critical orbit is finite, see [16,23]) shows that the

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D. Ruelle

susceptibility function (λ) has poles for |λ| < 1, but is holomorphic at λ = 1. This study suggests that in non-Markovian situations may have a natural boundary separating λ = 0 (around which has a natural expansion) and λ = 1 (corresponding to ω = 0). Misiurewicz [19] has studied a class of unimodal maps where the critical orbit stays away from the critical point, and he has proved the existence of an a.c.i.m. ρ for this class. This seems a good situation where one could study the dependence of ρ on f , as pointed out to the author by L.-S. Young. To be specific, let us consider the standard example of unimodal maps f c given by f c x = cx(1− x) on I = [0, 1]. It is known that for many values of c, f c has an a.c.i.m. ρ, and for a dense set it doesn’t. So, c → ρ cannot be differentiable in the usual sense. If c is restricted to a suitable set (say a set for which f c is Misiurewicz), it might be differentiable in the sense of Whitney interpolation. Our evidence is that c → ρ cannot be made differentiable in this way, but can nevertheless look differentiable numerically (and in experiments). An interesting feature will be the unexpected appearance of “acausal” singularities in the susceptibility function of unimodal maps. A desirable starting point to study the dependence of the a.c.i.m. ρ on f is to have an operator L on a Banach space A such that Lρ = ρ, and 1 is a simple isolated eigenvalue of L. The main content of the present paper is the construction of A and L with the desired properties. Specifically we write A = A1 ⊕A2 , where A2 consists of spikes, i.e., 1/square root singularities at points of the critical orbit, which are known to be present in ρ. We are thus able to prove that the a.c.i.m. ρ is the sum of a continuous background, and of the spikes (see Theorem 9, and Remarks 16). Note that the construction of an operator L with a spectral gap had been achieved earlier by G. Keller and T. Nowicki [18], and by L.-S. Young [30] (our construction, in a more restricted setting, leads to stronger results). We start studying the smoothness of the map f → ρ by an informal discussion in Sect. 17. Theorem 19 proves the differentiability along topological conjugacy classes (which are codimension 1) and relates the derivative to the value at λ = 1 of a modified susceptibility function (X, λ). [Following an idea of Baladi and Smania [5], it is plausible that differentiability in the sense of Whitney holds in directions tangent to a conjugacy class, see below]. Transversally to topological conjugacy classes the map f → ρ is continuous, but appears not to be differentiable. While this nondifferentiability is not rigorously proved, it seems to be an unavoidable consequence of the fact that the weight of the n th spike is roughly ∼ α n/2 (for some α ∈ (0, 1)) while its speed when f changes is ∼ α −n . [See Sect. 16(c). In fact, for a smooth family ( f κ ) restricted to values κ ∈ K such that f κ is in a suitable Misiurewicz class, the estimates just given for the weight and speed of the spikes suggest that κ → ρκ (A) for smooth A is 21 -Hölder, and nothing better, but we have not proved this]. Physically, let us remark that the spikes of high order n will be drowned in noise, so that discontinuities of the derivative of f → ρ will be invisible. Note that the susceptibility functions (λ), (X, λ) to be discussed may have singularities both for large |λ| and small |λ|. [The latter singularities do not occur for uniformly hyperbolic systems, but show up for the unimodal maps of the interval in the Markovian case, as we have mentioned above. A computer search of such singularities is of interest [10]]. A study similar to that of the present paper has been made (Baladi [3], Baladi and Smania [5]) for piecewise expanding maps of the interval. In that case it is found that f → ρ is not differentiable in general, but Baladi and Smania study the differentiability of f → ρ along directions tangent to topological conjugacy classes (horizontal

Structure and f -Dependence of A.C.I.M. for a Unimodal Misiurewicz Map f

1041

directions), not just for f restricted to a class. Note that our 1/square root spikes are replaced in the piecewise expanding case by jump discontinuities. This entails some serious differences, in particular, in the piecewise expanding case (λ) is holomorphic for |λ| < 1. 1. Setup Let I be a compact interval of R, and f : R → R be real-analytic such that f I ⊂ I . We assume that there is c in the interior of I such that f (c) = 0, f (x) > 0 for x < c, f (x) < 0 for x > c, and f (c) < 0. Replacing I by a possibly smaller interval, we assume that I = [a, b], where b = f c, a = f 2 c, and a < f a. We shall construct a horseshoe H ⊂ (a, b), i.e., a mixing compact hyperbolic set with a Markov partition for f . Following Misiurewicz [19] we shall assume that f a ∈ H . Note that the existence of H with a Markov partition is a weak condition, but the Misiurewicz condition f N c ∈ H for some N is a strong condition. Note also that f is not assumed to have a Markov partition on [a, b], which would mean that the critical point c is preperiodic ( f N c periodic for some N ). Our Misiurewicz condition is weaker than preperiodicity of c. Under natural conditions to be discussed below we shall study the existence of an a.c.i.m. ρ(x) d x for f , and its dependence on f . Studying the smoothness of f → ρ for unimodal maps f turns out to be a difficult problem. Our aim in the present paper will be to investigate new phenomena rather than to obtain very general results. In particular we make our life simpler by taking f to be real analytic rather than differentiable, and assume a Misiurewicz condition rather than Collet-Eckmann. Some other choices are made for the sake of simplicity, like f 3 c ∈ H rather than f N c ∈ H with N ≥ 3. Also we make a very geometric description of H in Sect. 2 in order to facilitate later discussion but , basically, Sects. 2–6 just say that H is a mixing hyperbolic set with a Markov partition. 2. Construction of the Set H(u1 ) Let u 1 ∈ [a, b] and define the closed set H (u 1 ) = {x ∈ [a, b] : f n x ≥ u 1 for all n ≥ 0}. We have thus f H (u 1 ) ⊂ H (u 1 ). Assuming that H (u 1 ) is nonempty, let v be its minimum element, then H (u 1 ) = H (v). [Since v ∈ H (u 1 ) we have v ≥ u 1 , hence H (v) ⊂ H (u 1 ). If H (u 1 ) contained an element w ∈ / H (v) we would have H (u 1 ) f k w < v for some k ≥ 0, in contradiction with the minimality of v]. Therefore we may (and shall) assume that H (u 1 ) u 1 . We shall also assume a < u 1 < c, f a (and f 2 u 1 = u 1 , which will later be replaced by a stronger condition). There is u 2 ∈ [a, b] such that f u 2 = u 1 and, since u 1 < f a, it follows that u 2 is unique and satisfies c c > u 1 and f u 2 ∈ H (u 1 )] and if x ∈ H (u 1 ) then x ≤ u 2 [because x > u 2 implies f x u 2 },

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D. Ruelle

then u 1 < V0 [because x ≤ u 1 implies f x ≤ f u 1 ∈ H (u 1 ) ≤ u 2 ] and V0 u 1 and f v1 = f v2 = u 2 ∈ H (u 1 )]. Our assumptions (H (u 1 ) u 1 , a < u 1 < c, f a and f 2 u 1 = u 1 ) and definitions give thus H (u 1 ) ⊂ [u 1 , v1 ] ∪ [v2 , u 2 ], f [u 1 , v1 ] ⊂ [u 1 , u 2 ], f [v2 , u 2 ] = [u 1 , u 2 ], and H (u 1 ) = {x ∈ [u 1 , u 2 ] : f n x ∈ / V0 for all n ≥ 0} = f H (u 1 ). Let us say that the open interval Vα ⊂ [u 1 , u 2 ] is of order n if f n maps homeomorphically Vα onto (v1 , v2 ) = V0 . We have thus H (u 1 ) = [u 1 , u 2 ]\ ∪ all Vα . By induction on n we shall see that [u 1 , u 2 ]\ ∪ the Vα of order ≤ n is composed of disjoint closed intervals J , such that f n J ⊂ [u 1 , v1 ] or [v2 , u 2 ] when n > 0, and the endpoints of f n J are u 1 , u 2 , v1 , v2 or an image of these points by f k with k ≤ n. Assume that the induction assumption holds for n (the case of n = 0 is trivial) and let J be as indicated. Since f n J ⊂ [u 1 , v1 ] or [v2 , u 2 ], f n+1 is monotone on J , and the endpoints of J are mapped by f n+1 outside of V0 [because u 1 , u 2 , v1 , v2 and their images by f are in H (u 1 ), hence ∈ / (v1 , v2 )]. The interval V0 is thus either inside of f n+1 J or disjoint from f n+1 J . Each Vα of order n +1 thus obtained is disjoint from other Vα of order ≤ n + 1, and the closed intervals J˜ in [u 1 , u 2 ]\ ∪ the Vα of order ≤ n + 1, are such that the endpoints of f n+1 J˜ are u 1 , u 2 , v1 , v2 or an image of these points by f k with k ≤ n + 1, in agreement with our induction assumption. We assume now that, for some N ≥ 0, we have f N +1 u 1 = u 1 (take N smallest with this property), and we assume also that ( f N +1 ) (u 1 ) > 0. [N = 0, 1 cannot occur, in particular f 2 u 1 = u 1 . Thus N ≥ 2, with f N u 1 = u 2 , f N −1 u 1 ∈ {v1 , v2 }. Furthermore, ( f N −1 ) (u 1 ) < 0 if f N −1 u 1 = v1 , and ( f N −1 ) (u 1 ) > 0 if f N −1 u 1 = v2 , i.e., f N −1 (u 1 +) = v1 − or v2 +]. Using the above assumption we now show that none of the intervals J in [u 1 , u 2 ]\ ∪ the Vα of order ≤ n is reduced to a point. We proceed by induction on n, assuming that f n J = [ f n x1 , f n x2 ], where f n x1 < f n x2 and f n x1 is of the form v2 , u 1 or f u 1 with ( f ) (u 1 ) > 0 while f n x2 is of the form v1 , u 2 or f u 2 with ( f ) (u 2 ) > 0. Therefore the lower limit of f n+1 J is of the form f m u 1 with ( f m ) (u 1 ) > 0 while the upper limit is of the form f m u 2 with ( f m ) (u 2 ) > 0. If f n+1 J ⊃ (v1 , v2 ) so that a new Vα of order n + 1 is created, the set f n+1 J \(v1 , v2 ) consists of two closed intervals, and one of them can be reduced to a point only if f m u 1 = v1 with ( f m ) (u 1 ) > 0 or if f m u 2 = v2 with ( f m ) (u 2 ) > 0. So, either f m+2 u 1 = u 1 with ( f m+2 ) (u 1 ) < 0, or f m+1 u 2 = u 2 with ( f m+1 ) (u 2 ) < 0 hence f m+1 u 1 = u 1 with ( f m+1 ) (u 1 ) < 0, in contradiction with our assumption that ( f N +1 ) (u 1 ) > 0.


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3. Consequences (No isolated points). H (u 1 ) is obtained from [u 1 , u 2 ] by taking away successively intervals Vα of increasing order. A given x ∈ H (u 1 ) will, at each step, belong to some small closed interval J , and the endpoints of J will not be removed in later steps, so that x cannot be an isolated point: H (u 1 ) has no isolated points. (Markov property). Our assumption f N +1 u 1 = u 1 implies that, for n = 1, . . . , N −1, the point f n u 1 is one of the endpoints of an interval Vα of order N −1−n, which we call VN −1−n . These open intervals Vk are disjoint, and their complement in [u 1 , u 2 ] consists of N intervals U1 , . . . , U N . Each Ui is closed, nonempty, and not reduced to a point. Furthermore, each Ui (for i = 1, . . . , N ) is mapped by f homeomorphically to a union of intervals U j and Vk : this is what we call Markov property. We impose now the following condition: 4. Hyperbolicity There are constants A > 0, α ∈ (0, 1) such that if x, f x, . . . , f n−1 x ∈ [u 1 , v1 ] ∪ [v2 , u 2 ], then d n −1 n d x f x < Aα . Note that in [19] hyperbolicity is automatic, because f is assumed to have a negative Schwarzian derivative. We label the intervals U1 , . . . , U N from left to right, so that u 1 is the lower endpoint of U1 , and u 2 the upper endpoint of U N . Define also an oriented graph with vertices U j

and edges U j → Uk when f U j ⊃ Uk . Write U j0 ⇒ U j if U j0 → U j1 → · · · → U j ,

and U j ⇒ Uk if U j ⇒ Uk for some > 0. 5. Lemma (mixing) r +3

(a) For each U j there is r ≥ 0 such that U j ⇒ U1 . s s s (b) If there is s > 0 such that U1 ⇒ U1 and U1 ⇒ U N , then U1 ⇒ Uk for k = 1, . . . , N . s (c) If there is s > 0 such that U j ⇒ Uk for all U j , Uk ∈ {U j : U1 ⇒ U j ⇒ U1 }, s then U j ⇒ Uk for all U j , Uk ∈ {U1 . . . , U N }, and we say that H (u 1 ) is mixing. (d) In particular if N + 1 is a prime, then H (u 1 ) is mixing. ˜ ˜ (e) Let u 1 < u˜ 1 < c, f a, and suppose that f N +1 u˜ 1 = u˜ 1 , ( f N +1 ) (u 1 ) > 0. Then if H (u 1 ) is mixing, so is H (u˜ 1 ). (a) The interval U j is contained in either [u 1 , v1 ] or [v2 , u 2 ]. Let the same hold for the successive images up to f r U j , but f r +1 U j c [hyperbolicity and the fact that U j r +1

is not reduced to a point imply that r is finite]. Then U j ⇒ Uk with Uk v1 or v2 , 2

r +3

hence Uk ⇒ U1 and U j ⇒ U1 . s (b) The U j such that U1 ⇒ U j form a set of consecutive intervals and, since this set contains U1 and U N by assumption, it contains all U j for j = 1, . . . , N .

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D. Ruelle s

s

s

(c) By assumption, U1 ⇒ U1 and U1 ⇒ U N , so that U1 ⇒ Uk for k = 1, . . . , N by (b). Therefore, {U j : U1 ⇒ U j ⇒ U1 } = {U1 , . . . , U N } by (a), and thus s U j ⇒ Uk for all U j , Uk ∈ {U1 . . . , U N }. (d) The transitive set {U j : U1 ⇒ U j ⇒ U1 } decomposes into n disjoint subsets 1

1

1

1

S0 , . . . , Sn−1 such that S0 ⇒ S1 ⇒ · · · ⇒ Sn−1 ⇒ S0 and there is s > 0 such that sn U j ⇒ Uk for all U j , Uk ∈ Sm , where m = 0, . . . , n −1. We may suppose that U1 ∈ S0 , and therefore if U(k) denotes the interval containing f k u 1 we have U(k) ∈ S(k) , where (k) = k(mod n). Therefore N +1 is a multiple of n, where n ≤ N < N +1. In particular, if s N + 1 is prime, then n = 1, and U j ⇒ Uk for all U j , Uk ∈ {U j : U1 ⇒ U j ⇒ U1 }, so that (c) can be applied. (e) Since H (u˜ 1 ) is a compact subset of H (u 1 ), without isolated points, the fact that H (u 1 ) is mixing implies that H (u˜ 1 ) is mixing. 6. Horseshoes Note that we have H (u 1 ) = {x ∈ [u 1 , u 2 ] : f n x ∈ / V0 for all n ≥ 0} = ∩n≥0 f −n ([u 1 , u 2 ]\V0 ). The sets Ui ∩ H (u 1 ) form a Markov partition of H (u 1 ), i.e., f (Ui ∩ H (u 1 )) is a finite union of sets U j ∩ H (u 1 ). A set H = H (u 1 ) as constructed in Sect. 2, with the hyperbolicity and mixing conditions will be called a horseshoe. A horseshoe is thus a mixing hyperbolic set with a Markov partition. Remember that the open interval Vα ⊂ [u 1 , u 2 ] is of order n if f n maps Vα homeomorphically onto V0 = (v1 , v2 ), and let |Vα | be the length of Vα . Hyperbolicity has the following consequence. 7. Lemma (a consequence of hyperbolicity) There are constants B > 0, β ∈ (0, 1) such that |Vα | ≤ Bβ n . α:order Vα =n

It suffices to prove that Lebesgue meas. ([u 1 , u 2 ]\ ∪ the Vα of order ≤ n) ≤ Gβ n [incidentally, this shows that H (u 1 ) has Lebesgue measure 0]. The above inequality is an immediate consequence of hyperbolicity: by a smooth conjugacy we may take A = 1 in Definition 4 of hyperbolicity, then if we remove the Vα of successive orders n = 1, 2, . . . we loose at each step a fraction > γ of the length for some γ ∈ (0, 1). ˜ 8. Remark (the set H) Starting from the horseshoe H = H (u 1 ) we can, by increasing u 1 to u˜ 1 such that u˜ 1 < c, f a, obtain a set H˜ = H (u˜ 1 ) ⊂ H such that u˜ 1 ∈ H˜ and the distance of H˜ to {u 1 , u 2 , v1 , v2 } is ≥ > 0. [In fact, using our hyperbolicity assumption we can arrange ˜ ˜ that there is N˜ such that f N +1 u˜ 1 = u˜ 1 , ( f N +1 ) (u˜ 1 ) > 0. In that case H˜ is mixing (Lemma 5(e)) and therefore again a horseshoe].


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9. Theorem Let H = H (u 1 ) be a horseshoe, suppose that f a = f 2 b ∈ H , and that { f n b : n ≥ 0} has a distance ≥ > 0 from {u 1 , u 2 , v1 , v2 }. Then f has a unique a.c.i.m. ρ(x) d x. Furthermore ρ(x) = φ(x) +

∞

Cn ψn (x).

n=0

The function φ is continuous on [a, b], with φ(a) = φ(b) = 0. For n ≥ 0 we shall choose wn ∈ {u 1 , u 2 , v1 , v2 } with (wn − c)(c − f n b) < 0 and let θn be the characteristic function of {x : (wn − x)(x − f n b) > 0}. Then, the above constants Cn and spikes ψn are defined by 1 Cn = φ(c)| f (c) f ( f k b)|−1/2 , 2 n−1 k=0

wn − x ψn (x) = · |x − f n b|−1/2 θn (x). wn − f n b [The condition that { f n b : n ≥ 0} has distance ≥ from {u 1 , u 2 , v1 , v2 } is achieved, according to Remark 8, by taking ≤ |u 1 − a|, |u 2 − b|, and f 2 b ∈ H˜ . Note also that ψn (c) = 0, so that φ(c) = ρ(c). Other choices of ψn can be useful, with the same singularity at f n b, but greater smoothness at wn and/or satisfying d x ψn (x) = 0]. 10. Analysis We analyze the problem before starting the proof. We may define a transfer operator L(1) on L 1 so that L(1) φ is the density of the image f ∗ (φ(x) d x) by f of φ(x) d x (in particular L(1) ρ = ρ). Near c we have y = f x = b − A(x − c)2 + h.o. 1/2 with A = − f (c)/2 √ > 0, hence x − c = ± ((b −∗ y)/A) + O(b − y). Therefore, writing U = ρ(c)/ A, the density (L(1) ρ)(x) of f (ρ(x)d x) has, near b, a singularity

√ U + O( b − x) √ (b − x) and, near a, a singularity √ U + O( x − a). − f (b)(x − a) To deal with the general case of the singularity at f n b, define sn = −sgn so that n−1 k=0

f ( f k b) = −sn U 2 Cn−2 .

n−1 k=0

f ( f k b),

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D. Ruelle

The density of f n∗ (ρ(x)d x) has then, near f n b, a singularity given when sn (x − f n b) > 0 by U + O( |x − f n b|) k n ( n−1 k=0 | f ( f b)|)|x − f b| U = + O( |x − f n b|) k −(x − f n b) n−1 k=0 f ( f b) Cn =√ sn (x − f n b)) + O( sn (x − f n b) and by 0 when sn (x − f n b) < 0. We let now w0 = u 2 and, for n ≥ 0, define wn+1 ∈ {u 1 , u 2 , v1 , v2 } inductively by: (wn+1 − c)( f n+1 b − c) > 0, (wn+1 − f n+1 b)( f wn − f n+1 b) > 0. We have thus w0 = u 2 , w1 = u 1 , and in general wn ∈ {u 1 , u 2 , v1 , v2 }, (wn − c)( f n b−c) > 0, sn (wn − f n b) > 0, |wn − f n b| ≥ . The above considerations show that the singularity expected near f n b for the density ρ(x) = (Ln(1) ρ)(x) is also represented by Cn x − f nb )· √ θn (x) wn − f n b sn (x − f n b) wn − x = Cn |x − f n b|−1/2 θn (x) = Cn ψn (x) wn − f n b

(1 −

in agreement with the claim of the theorem. 11. Lemma Write f (ψn (x)d x) = ψ˜ n+1 (x)d x, ψ˜ n+1 = | f ( f n b)|−1/2 ψn+1 + χn . Then, for n ≥ 0, the χn are continuous of bounded variation on [a, b], with χn (a) = b χn (b) = 0, and the Var χn = a |dχn /d x|d x are bounded uniformly with respect to n. Furthermore, if n ≥ 1 and Vα ⊂ suppχn , then χn |Vα extends to a holomorphic function χnα in a complex neighborhood Dα of the closure of Vα in R (further specified in Sect. 12), with the |χnα | uniformly bounded. The case n = 0 can be handled by inspection, and we shall assume n ≥ 1. We let ( f a, b) if f n b ∈ [a, c) In = , (a, b) if f n b ∈ (c, b) and define f n−1 : In → (a, b) to be the inverse of f restricted respectively to (a, c) or (c, b) in the two cases above. We have then ψn ( f n−1 x) . ψ˜ n+1 (x) = | f ( f n−1 x)|


1047

Since n ≥ 1, the region of interest f suppψn ∪ suppψn+1 is ⊂ [u 1 , u 2 ] ⊂ (a, b), and we have f n−1 x − f n b = (x − f n+1 b)An (x), where An is real analytic and An ( f n+1 b) = ( f ( f n b))−1 . Therefore we may write f ( f n b) (1 + (x − f n+1 b) A˜ n (x)), x − f n+1 b − f nb 1 1 = n (1 + (x − f n+1 b)) B˜ n (x), f ( f b) f ( f n−1 x) wn − f n−1 x = 1 + (x − f n+1 b)C˜ n (x), wn − f n b 1

f n−1 x

=

and since ψn ( f n−1 x)

=

wn − f n−1 x · | f −1 x θn ( f n−1 x) n wn − f n b

− f n b|−1/2 ,

we find

θn ( f n−1 x)| f ( f n b)|−1/2 1 + (x − f n+1 b) D˜ n (x) ψ˜ n+1 (x) = |x − f n+1 b| with D˜ n real analytic. Note that ψ˜ n+1 and | f ( f n b)|−1/2 ψn+1 have the same singularity at f n+1 b. It follows readily that ψ˜ n+1 − | f ( f n b)|−1/2 ψn+1 is a continuous function χn vanishing at the endpoints of its support, and bounded uniformly with respect to n. It is easy to see that Var χn is bounded uniformly in n. The extension of χn |Vα to holomorphic χnα in Dα is also handled readily (see Sect. 12 for the description of the Dα ). 12. The Operator L and the Space A We have f (ρ(x) d x) = (L(1) ρ)(x) d x, where the transfer operator L(1) on L 1 (a, b) is defined by L(1) ρ =

ρ ◦ f ±−1 ±

| f ◦ f ±−1 |

and we have denoted by f −−1 : [ f a, b] → [a, c]

and

f +−1 : [a, b] → [c, b]

the branches of the inverse of f . The invariance of ρ(x) d x under f is thus expressed by ρ = L(1) ρ.

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D. Ruelle

We shall look for a solution of this equation in a Banach space A defined below. Roughly speaking, A consists of functions φ+

∞

cn ψn ,

n=0

where the ψn are defined in the statement of Theorem 9, and φ : [a, b] → C is a less singular rest with certain analyticity properties. Remember that we may write [a, b] = H ∪ [a, u 1 ) ∪ (u 2 , b] ∪ the Vα of all orders ≥ 0. We have (see Remark 8) clos [a, u 1 ) ⊂ [a, u˜ 1 ), clos (u 2 , b] ⊂ (u˜ 2 , b], clos V0 ⊂ V˜0 , where u˜ 2 and V˜0 = (v˜ 1 , v˜ 2 ), are defined for H˜ as u 2 and V0 were defined for H . It is convenient to define V−1 = (u 2 , b] and V−2 = [a, u 1 ) (of order −1 and −2 respectively) so that [a, b] = H ∪ the Vα of all orders ≥ −2. We also define V˜−1 = (u˜ 2 , b], V˜−2 = [a, u˜ 1 ). We let now V˜α denote the unique interval in [a, b]\ H˜ such that Vα ⊂ V˜α . Note that the map Vα → V˜α is not injective! For each Vα of order ≥ 0 we may choose an open set Dα ⊂ C such that V˜α ⊃ Dα ∩ R ⊃ clos Vα and, if f Vβ = Vα of order ≥ 0, f Dβ ⊃ clos Dα [we have here denoted by clos Vα the closure of Vα in R, and by clos Dα the closure of Dα in C]. Let also Ra , Rb be two-sheeted Riemann surfaces, branched respectively at a, b, with natural projections πa , πb : Ra , Rb → C. We may choose open sets D−1 , D−2 ⊂ C such that, for α = −1, −2, V˜α ⊃ Dα ∩ {x ∈ R : a ≤ x ≤ b} ⊃ clos Vα and f extends to holomorphic maps f˜−1 : D0 → Rb , f˜−2 : ( f˜−1 D0 ) → Ra such that f˜−1 D0 ⊃ πb−1 clos D−1 , f˜−2 πb−1 D−1 ⊃ πa−1 clos D−2 . [We shall say that f˜−1 sends (v1 , c) to the upper sheet of Rb and (c, v2 ) to the lower sheet of Rb ; f˜−2 sends the upper (lower) sheet of Rb to the upper (lower) sheet of Ra ]. We come now to a precise definition of the complex Banach space A. We write A = A1 ⊕ A2 , where the elements of A1 are of the form (φα ) and the elements of A2 are of the form (cn ). Here the index set of the φα is the same as the index set of the intervals Vα (of order ≥ −2); the index n of the cn ∈ C takes the values 0, 1, . . . [the cn should not be confused with the critical point c]. We assume that φα is a holomorphic function in Dα when Vα is of order ≥ 0, while φ−1 , φ−2 are holomorphic on πb−1 D−1 , πa−1 D−2 and, for all α, ||φα || = supz∈Dα |φα (z)| < ∞. [We shall later consider a function φ : [a, b] → C such that φ|Vα = φα |Vα when Vα is of order ≥ 0. For x ∈ V−1 we shall require φ(x) = φ(x) = φ−1 (x + ) − φ−1 (x − ), where x + (x − ) is the preimage of x by πb on the upper (lower) sheet of πb−1 D−1 ; for x ∈ V−2 we shall require φ(x) = φ−2 (x) = φ−2 (x + ) − φ−2 (x − ) where x + (x − ) is the preimage of x by πa on the upper (lower) sheet of πa−1 D−2 . But at this point we


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discuss an operator L on A instead of the transfer operator L(1) acting on functions φ + n cn ψn .] Let γ , δ be such that 1 < γ < β −1 , 1 < δ < α −1/2 with β as in Lemma 7 and α as in the definition of hyperbolicity (Sect. 4). We write ||(φα )||1 = sup γ n |Vα |.||φα ||, ||(cn )||2 = sup δ n |cn | n≥−2

n≥0

α:order Vα =n

and, for = ((φα ), (cn )), we let |||| = ||(φα )||1 + ||(cn )||2 . We let then A1 , A2 be the Banach spaces of sequences (φα ), (cn ) as above , such that the norms ||(φα )||1 , ||(cn )||2 ˜ We first describe what contribution are finite. We shall define L on A such that L = . ˜ and then we shall check that this is a consistent description of each φα or cn gives to ˜ of A. an element (i) φβ ⇒ φˆ βα =

φβ ◦ ( f |Dβ )−1 in Dα if order β > 0 and | f |

f Vβ = Vα

[we have here denoted by | f | the holomorphic function ± f such that ± f > 0 for real argument, we shall use the same notation in (ii)-(vi) below].

φ0 1 −1 (ii) φ0 ⇒ cˆ0 = C0 φ0 (c) , φˆ −1 = ± ◦ f˜−1 − C0 φ0 (c)(± ψ0 ◦ πb ) in πb−1 D−1 , |f | 2 where the signs ± correspond to the upper/lower sheet of πb−1 D−1 . We claim that φˆ −1 is holomorphic in πb−1 D−1 as the difference of two meromorphic functions with a simple pole at the branch point b, with the same residue. To see this we uniformize πb−1 D−1 φ0 φ0 by the map u → b − u 2 . We have thus to express ± (c + x) = (c + x) in terms | f | f −1 2 ˜ ˜ of u where c + x = f −1 (b − u ) or u = b − f −1 (c + x) which gives a meromorphic √ function with a simple pole 1/2 Au. Since ±C0 φ0 (c)ψ0 (b − u 2 ) is meromorphic with the same simple pole, φˆ −1 is holomorphic in πb−1 D−1 . (iii) φ−1 ⇒ φˆ −2 = (iv) φ−2 ⇒ φˆ α =

φ−2 ◦ f −1 f

φ−1 ˜−1 ◦ f −2 | f |

in πa−1 D−2 .

in Dα if f (a, u 1 ) ⊃ Vα , 0 otherwise

[we have written φ−2 (x) = φ−2 (x + ) − φ−2 (x − ), where x + (x − ) is the preimage of x by πa on the upper (lower) sheet of πa−1 D−2 ].

ψ0 1 −1 −1/2 ˜ (v) c0 ⇒ cˆ1 = | f (b)|−1/2 c0 , χ0 = ± c0 ◦ π ◦ f − | f (b)| ψ ◦ π b 1 a −2 2 | f | in πa−1 D−2 where the sign ± corresponds to the upper/lower sheet of πa−1 D−2 . ψn n −1/2 −1 n −1/2 ◦ f n − | f ( f b)| (vi) cn ⇒ cˆn+1 = | f ( f b)| cn , χnα = cn ψn+1 | f |

in Dα if Vα ⊂ {x : θn ( f n−1 x) > 0} , 0 otherwise

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if n ≥ 1. We may now write ˜ = ((φ˜ α ), (cñ )), where (see (iii),(v)), φ˜ −2 = φˆ −2 + χ0 (see(ii)), φ˜ −1 = φˆ −1 φˆ βα + φˆ α + φ˜ α = χnα if order α ≥ 0 β: f Vβ =Vα

c˜0 = cˆ0 c˜1 = cˆ1 cñ = cˆn

(see (i),(iv),(vi)),

n≥1

(see (ii)), (see (v)), for n > 1

(see (vi)).

Note that, corresponding to the decomposition A = A1 ⊕ A2 , we have

L0 + L1 L2 , L= L3 L4 where L0 (φα ) = (

φˆ βα ),

β: f Vβ =Vα

L1 (φα ) = (φˆ α ), L2 (cn ) = (χ0 , ( χnα )α>−1 ), n≥1

L3 (φα ) = (cˆ0 , (0)n>0 ), L4 (cn ) = (0, (cˆn )n>0 ). Holomorphic functions in Dα are defined by (i),(iv),(vi) when order α ≥ 0, and in πb−1 D−1 , πa−1 D−2 by (ii),(iii),(v). Using Lemma 7, one sees that L0 , L1 are bounded A1 → A1 . Using Lemma 11, one sees that L3 is bounded A2 → A1 . It is also readily seen that L2 , L4 are bounded, so that L : A → A is bounded. 13. Theorem (structure of L). With our definitions and assumptions, the bounded operator L : A → A is a compact perturbation of L0 ⊕ L4 ; its essential spectral radius is ≤ max(γ −1 , δα 1/2 ). Since f a ∈ H˜ , we may assume that f (a, u 1 ) ⊃ Vα implies f (D−2 \negative reals) ⊃ clos Dα . Therefore, φ−2 → φˆ α |Dα is compact. For N positive integer, define the operator L N 1 such that L N 1 (φα ) =

φ−2 ◦ f −1 f

in Dα if f (a, u 1 ) ⊃ Vα and order α > N , 0 otherwise.

Then L1 is a perturbation of L N 1 by a compact operator and, using Lemma 7, we see that ||L N 1 (φα )||1 ≤ C sup γ n β n → 0 n>N

when N → ∞.


1051

We can write L2 = L N 2 + finite range, where L N 2 (cn ) = (0, 0, ( χnα )α≥0 ). n≥N

Using Lemma 11 we find a bound ||

n≥N

χnα || ≤ C δ N and, using Lemma 7,

||L N 2 ||A2 →A1 ≤ C δ N → 0

when N → ∞.

The operator L3 has one-dimensional range. Therefore L1 , L2 , L3 are compact operators, and the essential spectral radius of L is the max of the essential spectral radius of L0 on A1 and L4 on A2 . The spectral radius of L4 is ≤

||L4N ||1/N

≤ δ C sup N

N −1

≥0 k=0

1/N

|f (f

k+

b)|

−1/2

with limit < δα 1/2 when N → ∞.

The essential spectral radius of L0 is supn≥N γ n α:orderVα =n |Vα |.|| β: f Vβ =Vα φˆ βα || ≤ lim N →∞ supn≥N γ n+1 β:orderVβ =n+1 |Vβ |.||φβ || |Vα |.|| β: f Vβ =Vα φˆ βα || −1 = γ −1 . lim ≤γ orderVα →∞ β: f Vβ =Vα |Vβ |.||φβ || In fact, no eigenvalue of L0 can be > γ −1 , so the spectral radius of L0 acting on A1 is ≤ γ −1 . The essential spectral radius of L is thus ≤ max(γ −1 , δα 1/2 ). [Note also that when γ → β −1 , δ → 1, we have max(γ −1 , δα 1/2 ) → max(β, α 1/2 ).] 14. The Eigenvalue 1 of L Let the map : A1 → L 1 (a, b) be such that (φα )|(a, u 1 ) = φ−2 , (φα )|(u 2 , b) = φ−1 , and (φα )|Vβ = φβ if order β ≥ 0. We also define w : A → L 1 (a, b) by w((φα ), (cn )) = (φα ) + ∞ n=0 cn ψn and check readily that wL = L(1) w. If λ0 = 0 is an eigenvalue of L, and 0 = ((φα0 ), (cn0 )) is an eigenvector to this eigen0 = 0, φ 0 = 0, value, we have w0 = 0 [because w0 = 0 implies φ00 = 0, hence φ−1 −2 0 0 0 and (cn ) = 0; then (φα ) = 0, so φα = 0 when order α ≥ 0, i.e., 0 = 0]. Therefore |λ0 | a

λ0 w0 = L(1) (w0 ), b b |w0 | = |L(1) (w0 )| ≤ a

hence |λ0 | ≤ 1. If c00 = 0, then (cn0 ) = 0, and λ0 is thus that |λ0 | ≤ γ −1 (see Sect. 13). Therefore |λ0 |

a

b

L(1) |w0 | =

b

|w0 |,

a

an eigenvalue of L0 acting on A1 , so > γ −1 implies c00 = 0, c10 = 0, hence

1052

D. Ruelle

φ−1 + c0 ψ0 = 0, φ−2 + c1 ψ1 = 0. Note that, by analyticity, φ−2 + c1 ψ1 is nonzero almost everywhere in (a, u 1 ). The image f (a, u 1 ) contains some (small) interval Ui0 ∩ f −1 (Ui1 ∩ f −1 (Ui2 . . .)) on which the image of φ−2 + c1 ψ1 by L(1) does not vanish, and therefore (by mixing),

b

a

|L(1) w |
0]. (∗) also implies that if L = , then w 1 0 0 1 0 is proportional to w , hence φ0 is proportional to φ0 , hence is proportional to . Furthermore, the generalized eigenspace to the eigenvalue 1 contains only the multiples of [otherwise there would exist 1 such that Ln 1 = 1 + n0 , contradicting b 0 1 b 1 a wL = a w ]. We have proved the first part of the following 15. Proposition (a) Apart from the simple eigenvalue 1, the spectrum of L has radius < 1. The eigenvector 0 to the eigenvalue 1 (after multiplication by a suitable constant = 0) satisfies w0 ≥ 0. (b) Write 0 = ((φα0 ), (cn0 )) and (φα0 ) = φ 0 , then φ 0 is continuous, of bounded variation, and φ 0 (a) = φ 0 (b) = 0. The interval [u 1 , u 2 ] is divided into N closed intervals W1 , . . . , W N by the points f n u 1 for n = 1, . . . , N − 1. The intervals W1 , . . . , W N are ordered from left to right, by doubling the common endpoints we make the W j disjoint. Define γ 0 = (γ j0 ) Nj=1 by γ j0 = φ 0 |W j ∈ L 1 (W j ). Then, the equation 0 = L0 implies γ 0 = L∗ γ 0 + η

(*)

or γ j0 =

L jk γk0 + η j ,

k

where L = (L jk ) is a transfer operator defined as follows. Letting ( f −1 )k j : W j → Wk be such that f ◦ ( f −1 )k j is the identity on W j we write L jk γk =

γk ◦( f −1 )k j | f ◦( f −1 )k j |

0

if f Wk ⊃ W j otherwise


1053

[the term L∗ γ 0 in (∗) comes from (i) in Sect. 12]. We let ηj =

∞

η jn .

n=0

Here η j0 (x) =

0 (y) φ−2

f (y)

if f (a, u 1 ) ∩ W j contains more than one point, and y ∈ (a, u 1 ), f y = x ∈ W j ; we let η j0 (x) = 0 otherwise [this term comes from (iv) in Sect. 12]. For n ≥ 1, we let η jn = Cn χn |W j , where χn = (ψn /| f |) ◦ f n−1 − | f ( f n b)|−1/2 ψn+1 [this term comes from (vi) in Sect. 12]. Because f u 1 is one of the division points between the intervals W j , the function η j0 is continuous on W j ; the η jn for n ≥ 1 are also continuous. Furthermore, η j0 and the η jn for n ≥ 1 are uniformly of bounded variation. If H j denotes the Banach space of continuous functions of bounded variation on W j we have thus η j ∈ H j for j = 1, . . . , N . We shall now obtain an upper bound on the essential spectral radius of L∗ acting on H = ⊕1N H j by studying ||Ln∗ − Fn ||, where Fn has finite-dimensional range (we use here a simple case of an argument due to Baladi and Keller [4]). Define Win ···i0 = {x ∈ Win : f x ∈ Win−1 , . . . , f n x ∈ Wi0 } when f Wik ⊃ Wik−1 for k = n, . . . , 1. For η = (η j ) ∈ H, we let πn η = (π jn η j ), where π jn η j is a piecewise affine function on W j such that (π jn η j )(x) = η j (x) whenever x is an endpoint of W j or of an interval W jin−1 ···i0 , and is affine between all such endpoints. Then Fn = Ln∗ πn has finite rank (i.e., finite-dimensional range), and Ln∗ − Fn = Ln∗ (1 − πn ) maps H to H. Let Var γ = 1N Var j γ j , where Var j is the total variation on W j . Let also || · ||0 denote the sup-norm and || · || = max{Var ·, || · ||0 } be the bounded variation norm. We have Var(γ − πn γ ) ≤ 2Var γ , ||(γ − πn γ )|Win ···i0 ||0 ≤ Var γ i 0 ···i n

[the second inequality follows from the first because γ − πn γ vanishes at the endpoints of Win ···i0 ]. Since Ln∗ (1 − πn )γ vanishes at the endpoints of the W j , we have ||(Ln∗ − Fn )γ || = Var((Ln∗ − Fn )γ ) = Var ((γ − πn γ )in ◦ f˜in ···i0 )( f˜ ◦ f˜in ···i0 ) · · · ( f˜ ◦ f˜i1 i0 ), i 0 ···i n

where we have written f˜i ···i0 = ( f −1 )i i−1 ◦ · · · ( f −1 )i1 i0 and 1 f˜ = , |f |

1054

D. Ruelle

hence ||(Ln∗ − Fn )γ || ≤ =

Var[((γ − πn γ )in ◦ f˜in ···i0 )( f˜ ◦ f˜in ···i0 ) · · · ( f˜ ◦ f˜i1 i0 )]

i 0 ···i n

Var[((γ − πn γ )|Win ···i0 )

i 0 ···i n

n−1

( f˜ ◦ ( f |Win ···i0 ))].

=0

The right-hand side is bounded by a sum of n + 1 terms where Var is applied to (γ − πn γ )|Win ···i0 or a factor f˜ ◦ ( f |Win ···i0 )), and the other factors are bounded by their || · ||0 -norm. Thus, using the hyperbolicity condition of Sect. 4, we have ||(Ln∗ − Fn )γ || ≤ Var(γ −πn γ ).Aα n +

n−1 =0 i 0 ···i n

≤ 2 Aα Var γ + n A α n

2 n−1

Var f˜

||(γ −πn γ )|Win ···i0 ||0 .Aα .Var( f˜ |Win− ···i0 ).Aα n−−1

||(γ − πn γ )|Win ···i0 ||0

i 0 ···i n

≤ (2 A + n A2 α −1 Var f˜ )α n Var γ ≤ (2 A + n A2 α −1 Var f˜ )α n ||γ ||,

so that ||Ln∗ − Fn || ≤ (2 A + n A2 α −1 Var f˜ )α n , and therefore L∗ has essential spectral radius ≤ α < 1 on H. Suppose that there existed an eigenfunction γ ∈ H to the eigenvalue 1 of L∗ ; the fact that γ is continuous and = 0 on some W j would imply (Ln∗ |γ |)(x) d x < |γ |(x) d x [because, for some n, Ln∗ sends “mass” into V0 ]. But this is in contradiction with |γ |(x) d x = |Ln∗ γ |(x) d x ≤ (Ln∗ |γ |)(x) d x. Therefore, 1 cannot be an eigenvalue of L∗ , and there is γ = (1 − L∗ )−1 η ∈ H such that γ = L∗ γ + η. Since γ 0 satisfies the same equation in L 1 , we have γ 0 − γ = L∗ (γ 0 − γ ), hence γ 0 − γ = 0 by the same argument as above [|γ 0 − γ | is in L 1 , with “mass” in some Vα because H (u 1 ) has measure 0, and this is sent to V0 by Ln∗ for some n]. Thus γ 0 is continuous of bounded variation on the intervals W j for j = 1, . . . , N , and φ 0 has bounded variation on [a, b], with possible discontinuities only at f n u 1 for n = 0, . . . , N , and φ 0 (a) = φ 0 (b) = 0. We have L(1) φ 0 − c00 ψ0 +

∞

cn0 χn = φ 0 .

n=0

Therefore, hyperbolicity along the periodic orbit of u 1 shows that φ 0 cannot have discontinuities, and this proves part (b) of Proposition 15. This also concludes the proof of Theorem 9.


1055

16. Remarks (a) Theorem 9 shows that the density ρ(x) of the unique a.c.i.m. ρ(x) d x for f can be written as the sum of spikes ≈ |x − f n b|−1/2 θn (x) (where θn vanishes unless x > f n b or x < f n b) and a continuous background φ(x). In fact, one can also write ρ(x) as the sum of singular terms ≈ |x − f n b|−1/2 θn (x), |x − f n b|1/2 θn (x) and a background φ(x) which is now differentiable. This result is discussed in Appendix A. It seems clear that one could write ρ(x) as a sum of terms |x − f n b|k/2 θn (x) with k = −1, 1, . . . , 2−1 2 and a background φ(x) of class C , but we have not written a proof of this. (b) Let u ∈ (−∞, u 1 )∪(u 1 , v1 )∪(v2 , u 2 )∪(u 2 , ∞) and choose w ∈ {u 1 , u 2 , v1 , v2 } such that w is an endpoint of the interval containing u. If ±(w − u) > 0 and θ± is the characteristic function of {x : (w − x)(x − u) > 0} we define ψ(u±) (x) =

w−x · |x − u|−1/2 θ± (x) w−u

or a similar expression with the same singularity at u, greater smoothness at w, and/or ψ(u±) = 0. [Note that the ψn are of this form.] Claim: if u ∈ H˜ , there exists a unique (φα ) ∈ A1 such that φα = ψ(u±) |Vα for all α; furthermore ||(φα )||1 has a bound independent of u±. These results are proved in Appendix B (assuming γ < α −1/2 ). Note that if ((φα ), (cn )) ∈ A and c0 = c1 = 0, there is (φ˜ α ) ∈ A1 such that (φ˜ α ) = w((φα ), (cn )). It seems thus that we might have replaced A by A1 in our earlier discussions. However, separating the spikes (cn ) from the background (φα ) was needed in the spectral study of L. eigenvector 0 of L corresponding to the eigenvalue 1 (with w0 ≥ (c) The 0 0, w = 1) depends continuously on f . To make sense of this statement we may consider a one-parameter family ( f κ ) such that f 0 = f . We let Hκ , H˜ κ (hyperbolic sets) and A1κ (Banach space) reduce to H, H˜ and A1 when κ = 0. We restrict κ to a compact set K such that f κ3 cκ ∈ H˜ κ (where cκ is the critical point of f κ ). The intervals Vκα associated with Hκ can be mapped to the Vα associated with H , providing an identification ηκ : Aκ1 → A1 . There are natural definitions of Lκ : Aκ1 ⊕ A2 → Aκ1 ⊕ A2 and the 0 eigenvector 0κ reducing to L and 0 when κ = 0. We claim that κ → × κ = (ηκ , 1)κ is a continuous function K → A1 ⊕ A2 . This result is proved in Appendix C. It implies that, if A is smooth, κ → 0fκ , A is continuous on K . The weight of the n th spike is C0 nk=1 | f κ ( f κk−1 bκ )|−1/2 and its speed is n n n d n dbκ k−1 d fκ f κ bκ = + . f κ ( f κk−1 bκ ) f κ ( f κ bκ ) f κ∗ ( f κ−1 bκ ) with f κ∗ = dκ dκ dκ k=1

=1 k=+1

The weight may be roughly estimated as ∼ α n/2 and the speed as ∼ α −n for some α ∈ (0, 1), suggesting that κ → 0fκ , A is 21 -Hölder on K . 17. Informal Study of the Differentiability of f → 0f , A.

Writing 0f instead of 0 we want to study the change of 0f , A = d x (w0f ) (x)A(x) when f is replaced by fˆ close to f (and the critical orbit fˆk cˆ for k ≥ 3 is in the ˆ˜ ). Writing g = id − fˆ(c) perturbed hyperbolic set H ˆ + f (c), we see that fˆ is conjugate −1 ˆ With proper choice of the inverse to g ◦ fˆ ◦ g , which has maximum f (c) at g(c).

1056

D. Ruelle

f −1 we have f −1 ◦ (g ◦ fˆ ◦ g −1 ) = h close to id, hence g ◦ fˆ ◦ g −1 = f ◦ h and (h ◦ g) ◦ fˆ ◦ (h ◦ g)−1 = h ◦ f , i.e., fˆ is conjugate to h ◦ f and we may write 0ˆ , A = 0h◦ f , A ◦ h ◦ g. f

The differentiability of fˆ → A ◦ h ◦ g is trivial, and we concentrate on the study of h → 0h◦ f , A. Writing h = id + X , where X is analytic, we see that the change δ(w0f ) when f is replaced by (id + X ) ◦ f is, to first order in X , formally (1 − L)−1 D(−X 0f ), where D denotes differentiation. [The above formula is a standard first order perturbation calculation, and we have omitted the w map from our formula.] Writing 0f = ((φα0 ), (Cn )), we can identify D(−X ((φα0 ), 0)) with an element × of A (so that w× = D(X w((φα0 ), 0)) and d x w× (x) = 0, use Appendix A) which is easy to study, and we are left to analyze thesingular part D(−X (0, (Cn ))). To study this singular part we shall write (0, (Cn )) = ∞ n=0 C n ψ( f n b) , and use the equivalence ∼ modulo the elements of A. We extend the domain of definition of L so that Lψ(u) ∼ | f (u)|−1/2 ψ( f u) ,where we use the notation ψ(u±) of Section 16(b), but omit the ±, and we assume that ψ(u) = 0. We have thus D(−X (0, (Cn ))) ∼ −

∞

Cn X ( f n b)Dψ( f n b) ∼

n=0

=

∞

n−1

Cn X ( f n b)[

n=0

f ( f k b)]−1

k=0

(1 − λL)−1 D(−X (0, (Cn ))) ∼

(1 − λL)−1

∞

=

X ( f n b)[

n=0

d ψ(u) u= f n b du

∞

n−1

n=0

k=0

d d X ( f n b)[ f ( f k b)]−1 Ln C0 ψ(b) . ψ( f n b) ∼ db db

X ( f n b)[

n=0 n−1

Cn X ( f n b)

n=0

We may thus write (introducing

∞

∞

f ( f k b)]−1 λ−n

k=0

n−1

instead of (1 − L)−1 ) f ( f k b)]−1 λ−n

k=0

d (1 − λL)−1 (λL)n C0 ψ(b) db

d (1 − λL)−1 C0 ψ(b) − Z , db

where Z = ∼ =

∞

X ( f n b)[

n−1

n=0

k=0

∞

n−1

X ( f n b)[

n=0

k=0

∞

n−1

X ( f n b) ∞ n=0

X ( f n b)

n−1 d (λL) C0 ψ(b) db =0

f ( f k b)]−1

n−1

λ−n+ |

=0

λ−n+ [

=0

n=0

∼ −D

f ( f k b)]−1 λ−n

n−1

f ( f k b)]−1 |

=0

λ−n+ [

f ( f k b)|−1/2

d C0 ψ( f b) db

f ( f k b)|−1/2

d C0 ψ(u) u= f b du

k=0 −1 k=0

k= n−1

−1

n−1 k=

f ( f k b)]−1 C ψ


= −D = −D

∞ ∞

r −1

X ( f +r b)λ−r [

r =1 =0 ∞

∞

=0

r =1

1057

f ( f +k b)]−1 C ψ

k=0

C ψ

r −1

λ−r [

f ( f +k b)]−1 X ( f +r b).

k=0

We have thus an (informal) proof of the following result: For = 0, 1, . . . , define F (X ) =

∞ n=1

λ

−n

[

n−1

f ( f k+ b)]−1 X ( f n+ b),

k=0

which are holomorphic functions of λ when |λ| > α. Then the susceptibility function (λ) = (1 − λL)−1 D(−X 0f ), A has the form ∞

(λ) ∼ (X (b) + F0 (X ))

d (1 − λL)−1 C0 ψ(b) , A − F (X )C ψ , D A. db =0

d The derivative db (1 − λL)−1 C0 ψ(b) , A exists as a distribution, but is in principle a divergent quantity for given b. The corresponding term disappears however if X (b) + F0 (X ) = 0, and we are then left with a finite expression, meromorphic in λ for α < |λ| < min(β −1 , α −1/2 ) and holomorphic when α < |λ| ≤ 1. Note that in writing the equivalence ∼ we have omitted terms with the singularities of (1 − λL)−1 ; this explains the meromorphic contributions for |λ| > 1. The condition X (b) + F0 (X ) = 0 for λ = 1 is known as horizontality (see the discussion in Sect. 19 below).

18. A Modified Susceptibility Function (X, λ) At this point we extend the definition of the operator L to L∼ acting on a larger space. Remember that L was obtained from the transfer operator L(1) by separating the spikes ψn from the background in order to obtain better spectral properties. We now also introduce derivatives ψn of spikes, so that the transfer operator sends ψn to f ( f n b) ψ + a term in w(A1 + A2 ). | f ( f n b)|1/2 n+1 The coefficients of ψn form an element of A3 = {(Yn ) : ||(Yn )||3 = supn δ n |Yn | < ∞}. We define L∼ on A1 + A2 + A3 so that ⎛ ⎞ L0 + L1 L2 L5 L4 L6 ⎠ , L∼ = ⎝ L3 0 0 L7

1058

D. Ruelle

where we omit the explicit definition of L5 , L6 , and let Zn Z˜ n L7 n−1 = n−1 k 1/2 k 1/2 k=0 | f ( f b)| k=0 | f ( f b)| with Z˜ 0 = 0, Z˜ n = f ( f n−1 b)Z n−1 ⎛ 0 ⎝0 0

for n > 0. Since ⎞ 0 L5 0 L6 ⎠ L = 0 0 L7

we have L∼n = Ln +

n

Lk−1 (L5 + L6 )Ln−k + Ln7 7

k=1

and formally (1−λL∼ )−1 = (112 −λL)−1 + (13 −λL7 )−1 + (112 −λL)−1 λ(L5 + L6 )(13 − λL7 )−1 , where 112 and 13 denote the identity on A1 ⊕ A2 and A3 respectively. For λ close to 1, (13 − λL7 )−1 and thus (1 − λL∼ )−1 are not well defined. But there is a natural definition of a left inverse L−1 7L of L7 , where Z˜ n Zn −1 L7L n−1 = n−1 k 1/2 k 1/2 k=0 | f ( f b)| k=0 | f ( f b)| 1/2 /δ. This with Z˜ n = f ( f n b)−1 Z n+1 for n ≥ 0. The spectral radius of L−1 7L is thus ≤ α gives natural left inverses

(13 − λL7 )−1 L

=−

∞

λ−n L−n 7L

n=1

for |λ| >

α 1/2 /δ,

and

−1 −1 −1 (1−λL∼ )−1 + (13 −λL7 )−1 L = (112 −λL) L + (112 −λL) λ(L5 + L6 )(13 − λL7 ) L

when |λ| > α 1/2 /δ and (112 −λL)−1 exists. This gives a modified susceptibility function 0 L (λ) = (1 − λL∼ )−1 L D(−X f ), A

meromorphic in λ for α < |λ| < min(β −1 , α −1/2 ) and holomorphic for α < |λ| ≤ 1. Note that the A3 part of D(−X 0f ) is −X ( f n b) (Yn ) = 1 , n−1 k 1/2 1/2 k=0 | f ( f b)| 2 | f (c)| n≥0 where supn |X ( f n b)| < ∞. Therefore, for small |λ|, − nk=0 λk ( k=1 f ( f n− b))X ( f n−k b) −1 (13 − λL7 ) (Yn ) = n−1 k 1/2 1 1/2 k=0 | f ( f b)| 2 | f (c)| n≥0


1059

because the right-hand side is in A3 . Note that the right-hand side is also in A3 under the condition ∞

λ−n (

n=0

n−1

f ( f k b))−1 X ( f n b) = 0

(*)

k=0

because this condition implies −

n

λ−k (

k−1

f ( f b))−1 X ( f k b) =

=0

k=0

hence, multiplying by λn −

n

λn−k (

n−1

n−1 =0

λ−k (

k−1

k=n+1

=0

∞

k−1

f ( f b))−1 X ( f k b)

f ( f b),

f ( f b))X ( f k b) =

=k

k=0

∞

λn−k (

f ( f b))−1 X ( f k b)

=n

k=n+1

or −

n k=0

λ ( k

k =1

f (f

n−

b))X ( f

n−k

b) =

∞ k=1

λ

−k

(

k−1

f ( f n+ b))−1 X ( f n+k b)

=0

for each n, provided |λ| > α. We have proved that: Under the condition (∗), a resummation of the series defining (1 − λL∼ )−1 D(−X 0f ), A yields L (λ). It is then natural to define a modified susceptibility function (X, λ) by (X, λ) → (X, λ) = L (λ) on {(X, λ) : (∗) holds}. Note that the left-hand side of (∗) is equal to the quantity X (b) + F0 (X ) met in Sect. 17, and that (∗) with λ = 1 reduces to the horizontality condition. We shall conclude with a rigorous result agreeing in part with the informal study in Sect. 17, in part with a conjecture of Baladi [3], Baladi and Smania [5]. First we recall the definition of topological conjugacy. If I ⊂ X , I˜ ⊂ X˜ , we say that f : I → X , f˜ : I˜ → X˜ are topologically conjugate if there exists a homeomorphism h : X → X˜ such that h ◦ f = f˜ ◦ h on I . Topological conjugacy classes for unimodal maps have been analyzed in [1]; those called hybrid classes are Banach codimension 1 manifolds and their tangent vectors are characterized by a general horizontality condition. We shall be interested in the case where topological conjugacy classes are restricted to Misiurewicz diffeomorphisms, and might thus be called Misiurewicz classes (Misiurewicz classes are a special case of hybrid classes). We shall in fact not be concerned with the conjugacy h on I , but with a consequence of a special case of conjugacy (namely f κ3 c = ξκ f 3 c, corresponding to a Misiurewicz class). We shall also use the infinitesimal form of conjugacy, viz. horizontality. For a general discussion of conjugacy classes for unimodal maps, under somewhat different conditions than those used here, we must refer to [1], which also gives precise conditions under which a topological class contains a quadratic element f c (with f c x = cx(1 − x)).

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D. Ruelle

19. Theorem (differentiability along topological conjugacy classes) Let f κ = h κ ◦ f , where the h κ are real analytic, depend smoothly on κ, and f κ3 c = ξκ f 3 c identically in κ. [This last condition expresses that f κ belongs to a conjugacy class, and ξk : H → Hκ is the conjugacy defined in Appendix C.] Then, if A is smooth, κ → 0κ , A = d x (w0fk )(x)A(x) is continuously differentiable. Furthermore d 0fk , A |κ=0 = (X, 1) , dκ d where (X, λ) is defined in Sect. 18 with X = dκ h κ |κ=0 , and (X, λ) is holomorphic for α < |λ| ≤ 1, meromorphic for α < |λ| < min(β −1 , α −1/2 ). [The value κ = 0 plays no special role, and is chosen for notational simplicity in the formulation of the theorem.] Our notion of topological conjugacy class is a special case of that discussed in [1]. Note that ξ0 = id, and that ξκ depends differentiably on κ. Since f κ3 c = ξκ f 3 c and f κ ξκ = ξκ f on H , we have f κn c = ξκ f n c for n ≥ 3 and by differentiation (writing d ξ = dκ ξκ |κ=0 ): n n−1 [ f ( f c)]X ( f k c) = ξ ( f n c) k=1 =k

or n k−1 n−1 [ f ( f c)]−1 X ( f k c) = [ f ( f c)]−1 ξ ( f n c) k=1 =1

=1

and letting n → ∞: ∞ k−1 [ f ( f c)]−1 X ( f k c) = 0

∞ n−1 [ f ( f k b)]−1 X ( f n b) = 0.

or

k=1 =1

n=0 k=0

This is the horizontality condition derived much more generally in [1]. The proof of the theorem will be based on Appendices A, B, C, and use particularly the notation of Appendix C. We write 0fκ = 0κ and recall that the expression 0κ ,

Aκ =

dx

(wκ 0κ )(x)A(x)

=

α

Vκα

0 φκα A(x)d x

+

0 cκn

ψκn (x)A(x)d x

n

k depends explicitly on the intervals Vκα and the points k ≥ 1. We shall first f κ c for d 1 0 prove the existence of dκ κ , Aκ |κ=0 = limκ→0 κ (wκ 0κ − w0 )A and give an expression involving only the intervals Vα and the points f k c (corresponding to κ = 0). Then we shall transform the expression obtained to the form (X, 1). Since the map ξκ : H → Hκ depends smoothly on κ (in particular f κ ( f κk bκ ) = f κ (ξκ f k b) is continuous uniformly in k), it is easily seen that the operator L× κ defined in Appendix C now depends continuously and even differentiably on κ.


We may write 0κ , Aκ =

α

Vκα

0 φκα (x)A(x) d x +

1061

0 cκn

ψκn (x)A(x) d x

n

0 0 ), (cκn )), ((A|Vκα ), A)κ = ((φκα 0 = 0κ , ((A|Vκα ), 0)κ + 0κ , (0, (cκn ))κ .

For notational simplicity we study the derivative of this quantity at κ = 0 but the proof will show that the derivative depends continuously on κ. We have 1 0 κ , Aκ − 0 , A = I + I I, κ where 1 0 II = ψκn (x) − cn0 ψn (x)]A(x) d x [cκn κ n dc0 κn 0 d ψn (x) + cn ψκn (x)]A(x) d x → [ dκ dκ n κ=0

d [ dκ ψκn

d dκ |x

f κn bκ |−1/2 ;

is a distribution with singular part − integrating by part over x, and using f κn bκ = ξκ f n b for k ≥ 2, we see that the right-hand side makes sense, and is the limit of the left-hand side when κ → 0]. We also have 0κ , ((A|Vκα ), 0)κ = × κ , ((Aκα ), 0), −1 , so that where Aκα = (A|Vκα ) ◦ η˜ κα

× × Aκα − A0α κ − 0 , ((Aκα ), 0) + × ), 0), 0 , (( κ κ and the second term is readily seen to tend to a limit when κ → 0. In the first term × × 0 × remember that for κ = 0 we have × κ = 0 = , and Lκ = L0 = L. Also

I =

× × × × (1 − L)(× κ − 0 ) = (Lκ − L0 )κ ,

hence × × −1 × × × κ − 0 = (1 − L) (Lκ − L0 )κ .

Since (1 − L)−1 is bounded and κ → L× κ differentiable, we have

× × d κ − 0 0 , ((Aκα ), 0) → (1 − L)−1 ( L× κ κ=0 ) , ((A0α ), 0) κ dκ

when κ → 0, proving that κ → 0κ , A is differentiable. If we replace in the above calculation the Banach space A = A1 ⊕ A2 by d A = A1 ⊕ A2 as in Appendix A, we obtain an expression of dκ 0κ , Aκ |κ=0 that can be re-expressed in terms of the ψn , ψn and an element of A1 . We may thus write d ˜ A∼ , 0 , Aκ κ=0 = , dκ κ

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˜ ∈ A1 ⊕ A2 ⊕ A3 . The part ˜ 3 of ˜ in A3 is uniquely determined by where ˜ A∼ ; the calculation of II above shows that the n th component of ˜ 3 is A → , d n d n+1 f κ bκ cn0 = − f κ c cn0 − dκ dκ κ=0 κ=0 n−1 n+1 n n k 0 k =− X ( f c) f ( f c) cn = − X ( f b) f ( f b) cn0 , k=1

=k

k=0

=k

and as a result ˜ 3 = (−X ( f n b)Cn0 )n≥0 , (1 − L7 ) ˜ 3 = (1 − L7 )−1 (−X ( f n b)Cn0 )n≥0 . L

∗

˜ in A1 ⊕ A2 is not uniquely determined (because of the ambiguity The part of discussed in Appendix B); this part satisfies w∗ = 0. If L(1)κ is the transfer operator corresponding to f κ , we have L(1)κ wκ 0κ = wκ 0κ , hence (1 − L(1) )(wκ 0κ − w0 ) = (L(1)κ − L(1) )wκ 0κ . Therefore (using the fact that we may let L(1) act on A) we have 1 ˜ A∼ = lim (1 − L∼ ), A (1 − L(1) )(wκ 0κ − w0 ) κ→0 κ 1 1 0 = lim A (L(1)κ − L(1) )wκ κ = lim A (id∗ − h ∗−κ )wκ 0κ κ→0 κ→0 κ κ 1 ∗ ∗ 0 = lim A (h κ − id )w , κ→0 κ where h ∗ denotes the direct image of a measure (here a L1 function) under h, and the last equality uses the existence of a continuous derivative for κ → 0κ , Aκ . According to (0) (0) (1) (1) Appendix A we may write w0 as a sum of terms Cn ψn , Cn ψn , and a differentiable background. Corresponding to this we may identify limκ→0 κ1 (h ∗κ − id∗ )0 with a naturally defined element D(−X 0 ) of A1 ⊕A2 ⊕A3 , where D denotes differentiation. We write D(−X 0 ) = (D ∗ , D3 ) with D ∗ ∈ A1 ⊕ A2 , D3 ∈ A3 . Since the coefficient ˜ 3 . With ˜ = (∗ , ˜ 3) of ψn in D(−X 0 ) is −X ( f n b)cn0 , we have D3 = (1 − L7 ) we have thus ˜ 3 ), A∼ = D(−X 0 ), A∼ (1 − L∼ )(∗ , and

In particular

˜ 3 ), A. (1 − L)∗ , A = D(−X 0 ) − (1 − L∼ )(0, ˜ 3 )] = 0, and we may define w[D(−X 0 ) − (1 − L∼ )(0, ˜ 3 )] ∈ A. = (1 − L)−1 [D(−X 0 ) − (1 − L∼ )(0,

We have then (1 − L)(∗ − ), A = 0, hence w(1 − L)(∗ − ) = 0,


1063

hence w(∗ − ) = L(1) w(∗ − ) with

w(∗ − ) = 0, so that w(∗ − ) = 0, and

˜ 3 )], A ∗ , A = , A = (1 − L)−1 [D(−X 0 ) − (1 − L∼ )(0, −1 ∗ −1 ∗ ˜ = (1−L) [D + L5 + L6 )3 ], A = (1−L) [D + L5 + L6 )(1−L7 )−1 L D3 ], A −1 ∼ −1 ∗ ˜ 3 ), A∼ , = (1 − L ) (D , D3 ), A − (1 − L7 ) D3 , A = (X, 1) − (0, L

L

so that finally d ˜ A∼ = (X, 1) 0 , Aκ |κ=0 = , dκ κ as announced. Note that in [5], Baladi and Smania study (in the case of piecewise expanding maps) the more difficult problem of differentiability in horizontal directions (i.e., directions tangent to a topological class). It appears likely that this could be done here also (as conjectured in [5]) , but we have not tried to do so.

20. Discussion The codimension 1 condition X (b) + F0 (X ) = 0 for λ = 1 expresses that X is a horizontal perturbation, which means that it is tangent to a topological class of unimodal maps (see [1] and references given there). In our case, a family ( f κ ) is in a topological conjugacy class if f κ3 cκ = ξκ f 3 c in the notation of Appendix C. The informal result obtained in Sect. 17 and the formal proof of differentiability along a topological conjugacy class given by Theorem 19 support the conjecture by Baladi and Smania [5] that the map f → 0f , A is differentiable (in the sense of Whitney) in horizontal directions, i.e., along a curve tangent to a topological conjugacy class. Our Theorem 19 also relates the derivative along a topological conjugacy class to a naturally defined susceptibility function. It seems unlikely that a derivative (in the sense of Whitney) exists in nonhorizontal directions. Note however that if f → 0f , A is nondifferentiable, it will be in a mild way: the “nondifferentiable” contribution to (λ) is, as we saw above, proportional to d d (1 − λL)−1 ψ(b) , A ∼ λn Ln ψ(b) , A, db db n d Ln ψ(b) , A increases expowhere Ln ψ(b) , A decreases exponentially with n, while db nentially. Therefore, if one does not see the small scale fluctuations of b → (1 − λL)−1 ψ(b) , A, this function will seem differentiable. But the singularities with respect to λ (with |λ| < 1) may remain visible. In conclusion, a physicist may see singularities with respect to λ of a derivative (with respect to f or b) while this derivative may not exist for a mathematician.

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D. Ruelle

A. Appendix (Proof of Remark 16(a)) We return to the analysis in Sect. 10, and note that by an analytic change of variable x → ξ(x) we can get y = f x = b − ξ 2 [we have indeed b − y = A(x − c)2 (1 + 1 β(x).(x − c)) with β analytic, and we can take ξ = (x − c)A1/2 (1 + β(x).(x − c)) 2 ]. Write ρ(x) d x = ρ(ξ ˜ ) dξ (where ρ˜ is analytic near 0). The density of the image δ(y) dy by f of ρ(x) d x = ρ(ξ ˜ ) dξ is, near b, 1 ρ(y ˆ − b) δ(y) = √ (ρ( ˜ y − b) + ρ(− , ˜ y − b)) = √ 2 y−b y−b where ρˆ is analytic near 0. Therefore, near b, √ U δ(x) = √ + U b − x + . . . , b−x √ where U = ρ(c)/ A, and U is linear in ρ(c), ρ (c), ρ (c) with coefficients depending on the derivatives of f at c. Near a we find δ(x) = U | f (b)|−1/2 √

√ 1 3 + (U | f (b)|−3/2 − U f (b)| f (b)|−5/2 ) x − a. 4 x −a

n−1 k −1/2 k , we claim that near f n b Writing sn = −sgn n−1 k=0 f ( f b), tn = | k=0 f ( f b)| n we have a singularity given for sn (x − f b) < 0 by 0, and for sn (x − f n b) > 0 by δ(x) = √

n−1 f ( f k b) tn2 U tn 3 3 U t t − s ) sn (x − f n b). + (U n k+1 n 4 | f ( f k b)| tk2 sn (x − f n b) k=0

[To prove this we use induction on n, and the fact that, when f : x → y for x close to f n b we have: f ( f n b) sn+1 (y − f n+1 b) [1 − (y − f n+1 b)] | f ( f n b)| 2| f ( f n b)|2 f ( f n b) dy d x = n [1 − n 2 (y − f n+1 b)].] | f ( f b)| | f ( f b)|

sn (x − f n b) =

Define now

θn (x) x − f nb 2 ) ψn(0) (x) = 1 − ( , √ wn − f n b sn (x − f n b)

x − f nb 2 θ ) (x) sn (x − f n b) ψn(1) (x) = 1 − ( n wn − f n b

for sn (x − f n b) > 0, 0 otherwise. Then, the expected singularity of δ near f n b is given by f ( f k b) tn2 (1) 3 U tn ψn(0) + (U tn3 − U tn sk+1 k )ψ = Cn(0) ψn(0) + Cn(1) ψn(1) , 4 | f ( f b)| tk2 n n−1 k=0

Structure and f -Dependence of A.C.I.M. for a Unimodal Misiurewicz Map f (0)

1065

(1)

where C0 = U , C0 = U , and (0) = | f ( f n b)|−1/2 Cn(0) , Cn+1

3 (1) Cn+1 = | f ( f n b)|−3/2 Cn(1) − sn+1 | f ( f n b)|−5/2 f ( f n b)Cn(0) 4 f ( f n b) 3 = | f ( f n b)|−3/2 (Cn(1) − sn+1 n Cn(0) ). 4 | f ( f b)| Let (0) f (ψn(0) (x) d x) = ψ˜ n+1 (x) d x,

(1) f (ψn(1) (x) d x) = ψ˜ n+1 (x) d x,

and write 3 (0) (0) (1) ψ˜ n+1 = | f ( f n b)|−1/2 ψn+1 − sn+1 | f ( f n b)|−5/2 f ( f n b)ψn+1 + χn(0) 4 (1) (1) ψ˜ n+1 = | f ( f n b)|−3/2 ψn+1 + χn(1) . (0)

(0)

(1)

(1)

The density of f (Cn ψn (x) d x + Cn ψn (x) d x) is then (0) (0) (1) (1) Cn+1 ψn+1 + Cn+1 ψn+1 + Cn(0) χn(0) + Cn(1) χn(1) . (0)

(1)

The functions χn , χn have been constructed such that they and their first derivatives χn(0) , χn(1) have the properties of Lemma 11. Namely, χn(0) , χn(1) , χn(0) , χn(1) are continuous with bounded variation on [a, b] uniformly in n, they vanish at a, b, and if n ≥ 1 they extend to holomorphic functions on the appropriate Dα , with uniform bounds. , φ of φ , φ Let A1 ⊂ A1 consist of the (φα ) such that the derivatives φ−1 −1 −2 −2 (0)

(1)

vanish at πb−1 b and πa−1 a respectively. Let also A2 consist of the sequences (cn , cn ), (0) (1) with cn , cn ∈ C, n = 0, 1, . . . such that ||(cn(0) , cn(1) )||2 = sup δ n (|cn(0) | + |cn(1) |) < ∞. n≥0

If = ((φα ), (cn(0) , cn(1) )) ∈ A = A1 ⊕ A2 we let || || = ||(φα )||1 + ||(cn(0) , cn(1) )||2 , making A into a Banach space. We may now proceed as in Sect. 12, replacing A by A , and defining L : A → A in a way similar to L : A → A, but with (ii), (v), (vi) replaced as follows: φ0 1 (0) (0) (1) −1 (ii) φ0 ⇒ (cˆ0 , cˆ0 ) = (U, U ) , φˆ −1 = ± ◦ f˜−1 − U (± ψ0 ◦ πb ) − U |f | 2

−1 1 (1) ˆ (± 2 ψ0 ◦ πb ) , so that φ−1 is holomorphic in πb D−1 with vanishing derivative at πb−1 b.

3 (v) (c0(0), c0(1) ) ⇒ (cˆ1(0), cˆ1(1) ) = (| f (b)|−1/2 c0(0),| f (b)|−3/2 c0(1)− | f (b)|−5/2 f (b)c0(0) ), 4 (0) 1 (0) ψ 3 (0) (1) −1 −| f (b)|−1/2 ψ1 ◦ πa + | f (b)|−5/2 f (b)ψ1 ◦ πa) χ0 = ± c0 ( 0 ◦ πb ◦ f˜−2 2 |f | 4 (1) 1 (1) ψ0 (1) −1 −3/2 −1 ± c0 ( ◦ πb ◦ f˜−2 − | f (b)| ψ1 ◦ πa ) in πa D−2 . 2 |f |

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D. Ruelle

(vi) (cn(0) , cn(1) ) ⇒ 3 (0) (1) (0) (1) (0) (cˆn+1 , cˆn+1 ) = (| f ( f n b)|−1/2 cn , | f ( f n b)|−3/2 cn − sn+1 | f ( f n b)|−5/2 f ( f n b)cn ), 4 (0) 3 (0) ψn (0) (1) ◦ f n−1 − | f ( f n b)|−1/2 ψn+1 + sn+1 | f ( f n b)|−5/2 f ( f n b)|ψn+1 ] | f | 4 (1) (1) ψn (1) −1 −1 n −3/2 ψn+1 ] in Dα if Vα ⊂ {x : θn ( f n x) > 0}, 0 otherwise +cn [ ◦ f n − | f ( f b)| |f |

χnα = cn [

ifn ≥ 1.

We write then ˜ = ((φ˜ α ), (cñ(0) , cñ(1) )), L = where φ˜ −2 = φˆ −2 + χ0 , φ˜ −1 = φˆ −1 , φ˜ α = χnα φˆ βα + φˆ α + β: f Vβ =Vα

if order α ≥ 0,

n≥1

(cñ(0) , cñ(1) ) = (cˆn(0) , cˆn(1) )

for n ≥ 0

With the above definitions and assumptions we find, by analogy with Theorem 13, that L : A → A has essential spectral radius ≤ max(γ −1 , δα 1/2 ). There is (see Proposition 15) a simple eigenvalue 1, and the rest of the spectrum has radius < 1. It is 0(0) 0(1) convenient to denote by 0 = ((φα0 ), (cn , cn )) the eigenfunction to the eigenvalue 1. We find again that φ 0 = (φα0 ) is continuous, of bounded variation, and satisfies φ 0 (a) = φ 0 (b) = 0, but we can say more. Using the notation in the proof of Proposition 15, we have again γ j0 = L jk γk0 + η j k

0(0) (0) 0(1) (1) with η j = ∞ n=0 η jn , but now η jn = cn χn + cn χn |W j for n ≥ 1, so that the η j 0 have derivatives η j ∈ H j . The derivatives γ j of the γ j0 are measures satisfying γ j0 =

Ljk γk0 + η∗j .

k

The operator Ljk has the same form as L jk , but with an extra denominator f ◦ ( f −1 )k j , and therefore L∗ = (Ljk ) acting on measures has spectral radius ≤ α < 1. The term η∗j is 0 ∼ −1 ) |−1 the sum of ηj and a term k L∼ kj k j γk , where Lk j involves the derivative of | f ◦( f ∗ so that η j ∈ H j . The operator L∗ also maps H to H and, by the same argument as for L∗ , has essential spectral radius < 1 on H. Furthermore, 1 cannot be an eigenvalue since L∗ has spectral radius < 1 on measures. It follows that (γ 0 ) = (γ j0 ) = (1 − L∗ )−1 (η∗j ) ∈ H. Therefore, the derivative φ 0 of φ 0 may have discontinuities only on the orbit of


1067

u 1 , and hyperbolicity again shows that this cannot happen. In conclusion, φ 0 and its derivative φ 0 are both of bounded variation, continuous, and vanishing at a, b. A discussion similar to the above shows that the equation γ = (1 − L∗ )−1 η∗ also defines γ with finite norm in A1 , and this γ must coincide with (γ 0 ) as a measure. Therefore the family of derivatives (φα0 ) is an element of A1 . [For simplicity, we have 0 , φ 0 for the functions which, under application of , give the derivative of written φ−1 −2 0 0 .] φ−1 , φ−2 B. Appendix (Proof of Remark 16(b)) If u ∈ H˜ and ψ(u±) is defined as in Remark 16(b), we want to show that there is a unique (φα ) in A1 such that φα = ψ(u±) |Vα for all α. Furthermore ||(φα )||1 is bounded uniformly for u ∈ H˜ , provided we assume 1 < γ < min(β −1 , α −1/2 ). Note that uniqueness is automatic, and that φα = 0 unless order Vα > 0. Omitting the ± we let f (ψ( f n u) (x) d x) = [| f ( f n u)|−1/2 ψ( f n+1 u) (x) + χ( f n u) (x)] d x. k −1/2 χ n d x For n ≥ 0 there is a unique ωun such that f n+1 (ωun d x) = n−1 ( f u) k=0 | f ( f u)| k k u − c] × [supp f (ω (x) d x) − c] > 0 for 0 ≤ k ≤ n. Furthermore ψ(u) = and [ f un ∞ n u) ] denotes the element ω , where the sum restricted to each V is finite. If [χ un α ( f n=0 of A1 corresponding to χ( f n u) , we find that ||[χ( f n u) ]||1 is bounded uniformly in n and k −1/2 χ n by multiplying with u. Also note that we obtain ωun from n−1 ( f u) k=0 | f ( f u)| n−1 k | f ( f u)| (up to a factor bounded uniformly in n because of hyperbolicity) and k=0 n+1 n+1 (restricted to a small interval J such that f |J is invertible). We composing with f have thus ||[ωun ]||1 ≤ const γ n

n−1

| f ( f k u)|−1/2 ,

k=0

where [ωun ] is the element of A1 corresponding to ωun [This is because the replacement of |Vα | by |( f |J )−n−1 Vα | in the definition of ||·||1 is compensated up to a multiplicative k constant by the factor n−1 k=0 | f ( f u)|.] Thus ||[ωun ]||1 ≤ const (γ α 1/2 )n . Since γ < α −1/2 we find that n ||[ωun ]||1 < constant independent of u. Therefore, since (φα ) = n [ωun ], we see that ||(φα )||1 is bounded independently of u. C. Appendix (Proof of Remark 16(c)) We consider a one-parameter family ( f κ ) of maps, reducing to f = f 0 for κ = 0. We assume that (κ, x) → f κ x is real-analytic. For κ close to 0, f κ has a critical point cκ and maps [aκ , bκ ] to itself, with bκ = f κ cκ , aκ = f κ2 cκ . There is (by hyperbolicity of H with respect to f ) a homeomorphism ξκ : H → Hκ , where Hκ is an f κ -invariant hyperbolic set for f κ and f κ ◦ ξκ = ξκ ◦ f on H . We shall consider a compact set K of values of κ such that f κ aκ ∈ H˜ κ ; we let K 0, K of small diameter, and assume now κ ∈ K . We may in a natural way define a Banach space Aκ = Aκ1 ⊕ A2 and an operator

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D. Ruelle

Lκ : Aκ → Aκ associated with f κ so that Aκ , Lκ reduce to A, L for κ = 0. Note that, since κ ∈ K is close to 0, we may assume that the constants A, α in the definition (Sect. 4) of hyperbolicity, and the constants B, β (Sect. 7) are uniform in κ. Let ηκ,−2 be a biholomorphic map of the complex neighborhood D−2 of [a, u 1 ] to the complex neighborhood Dκ,−2 of the corresponding interval [aκ , u κ1 ], and lift ηκ,−2 to a holomorphic map η˜ κ,−2 : πa−1 D−2 → πa−1 Dκ,−2 . We also lift ηκ,−1 = f κ−1 ◦ ηκ,−2 ◦ f κ to −1 ◦ ηκ,−2 ◦ f˜, η˜ κ,−1 = f˜κ,−2

where the notation is that of Sect. 12, with obvious modification. We write −1 ◦ η˜ κ,−1 ◦ f˜−1 η˜ κ0 = f˜κ,−1

and η˜ κβ = ( f κ |Vκβ )−1 ◦ η˜ κα ◦ f |Vβ if order β > 0 and f Vβ = Vα . We have defined ηκα above for α = −1, −2, and we let ηκα = η˜ κα when order α ≥ 0. We introduce a map ηκ : Aκ1 → A1 by ) ηκ (φκα ) = ((φκα ◦ η˜ κα ).ηκα −1 so that L× κ = (ηκ , 1)Lκ (ηκ , 1) acts on A. Using the decomposition

Lκ0 + Lκ1 Lκ2 Lκ = Lκ3 Lκ4

as in Sect. 12, we define L × κ on A1 by 1 −1 −1 f κ ( f κk bk )|−1/2 ) L× κ (φα ) = ηκ (Lκ0 + Lκ1 )ηκ (φα ) + (ηκ φα )0 (cκ ).ηκ Lκ2 (| f κ (cκ ) 2 n−1

1 (cκ )−1 φ0 (cκ ).ηκ Lκ2 (| f κ (cκ ) = L0 (φα ) + ηκ Lκ1 ηκ−1 (φκ ) + ηκ0 2

k=0 n−1

f κ ( f κk bk )|−1/2 ).

k=0

−1 . L× κ is a compact perturbation of Lκ0 , and has therefore essential spectral radius ≤ γ If (φα ) is a (generalized) eigenfunction of L × to the eigenvalue µ, then κ

1 f κ ( f κk bk )|−1/2 )) ((φα ), ηκ0 (cκ )−1 φ0 (cκ ).(| f κ (cκ ) 2 n−1 k=0

is a (generalized) eigenfunction of L× κ to the same eigenvalue µ. We have thus a × multiplicity-preserving bijection of the eigenvalues µ of L × κ and Lκ when |µ| > −1 1/2 × max(γ , δα ). In particular, 1 is a simple eigenvalue of L κ for the values of κ considered (a compact neighborhood K of 0). ˆ The operator L × κ acting on A1 depends continuously on κ. [This is because φκα , χκ0 , χκnα depend continuously on κ (in particular, the χκnα for large n are uniformly small). Note however that L× κ does not depend continuously on κ because the continuity of f κ ( f κk bκ ) is not uniform in k.] There is > 0 such that L × κ has no eigenvalue µκ


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with |µκ − 1| < except the simple eigenvalue 1 [otherwise the continuity of κ → L × κ would imply that 1 has multiplicity > 1 for some κ]. Therefore, the 1-dimensional projection corresponding to the eigenvalue 1 of L × κ depends continuously on κ, and so does 0 × 0 the eigenvector with the the eigenvector × κ = (ηκ , 1)κ of Lκ , where κ denotes 0 eigenvalue 1 of Lκ normalized so that wκ κ ≥ 0 and wκ 0κ = 1, with the obvious definition of wκ (involving the spikes ψκn associated with f κ ). Note that a number of results have been obtained earlier on the continuous dependence of the a.c.i.m. ρ on parameters. I am indebted to Viviane Baladi for communicating the references [14,25,27], and also [26]. Acknowledgement. I am very indebted to Lai-Sang Young and Viviane Baladi for their help and advice in the elaboration of the present paper. L.-S. Young was most helpful in getting this study started, and V. Baladi in getting it finished. I am also thankful to the referees for valuable comments.

References 1. Avila, A., Lyubich, M., de Melo, W.: Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math. 154, 451–550 (2003) 2. Baladi, V.: Positive transfer operators and decay of correlations. Singapore: World Scientific, 2000 3. Baladi, V.: On the susceptibility function on piecewise expanding interval maps. Commun. Math. Phys. 275, 839–859 (2007) 4. Baladi, V., Keller, G.: Zeta functions and transfer operators for piecewise monotone transformations. Commun. Math. Phys. 127, 459–479 (1990) 5. Baladi, V., Smania, D.: Linear response formula for piecewise expanding unimodal maps. Nonlinearity 21, 677–711 (2008) 6. Benedicks, M., Carleson, L.: On iterations of 1 − ax 2 on (−1, 1). Ann. Math. 122, 1–25 (1985) 7. Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. 133, 73–169 (1991) 8. Benedicks, M., Young, L.-S.: Absolutely continuous invariant measures and random perturbation for certain one-dimensional maps. Ergod Th. Dynam Syst. 12, 13–37 (1992) 9. Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1, 301–322 (2007) 10. Cessac, B.: Does the complex susceptibility of the Hénon map have a pole in the upper half plane?. A Numerical Investigation Nonlinearity 20, 2883–2895 (2001) 11. Chierchia, L., Gallavotti, G.: Smooth prime integrals for quasi-integrable Hamiltonian systems. Nuovo Cim. 67B, 277–295 (1982) 12. Collet, P., Eckmann, J.-P.: Positive Lyapunov exponents and absolute continuity for maps of the interval. Ergod Th. Dynam Syst. 3, 13–46 (1981) 13. Dolgopyat, D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155, 389–449 (2004) 14. Freitas, J.M.: Continuity of SRB measure and entropy for Benedicks-Carleson quadratic maps. Nonlinearity 18, 831–854 (2005) 15. Jakobson, M.: Absolutely continuous invariant measures for certain maps of an interval. Commun. Math. Phys. 81, 39–88 (1981) 16. Jiang, Y., Ruelle, D.: Analyticity of the susceptibility function for unimodal Markovian maps of the interval. Nonlinearity 18, 2447–2453 (2005) 17. Katok, A., Knieper, G., Pollicott, M., Weiss, H.: Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98, 581–597 (1989) 18. Keller, G., Nowicki, T.: Spectral theory, zeta functions and the distribution of periodic points for ColletEckmann maps. Commun. Math. Phys. 149, 31–69 (1992) 19. Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math. IHES 53, 17–52 (1981) 20. Pöschel, J.: Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math. 35, 653–696 (1982) 21. Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997); Correction and complements. Commun. Math. Phys. 234, 185–190 (2003) 22. Ruelle, D.: Differentiation of SRB states for hyperbolic flows. Ergod. Th. Dynam. Syst. 28(2), 613–631 (2008)

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23. Ruelle, D.: Differentiating the absolutely continuous invariant measure of an interval map f with respect to f. Commun.Math. Phys. 258, 445–453 (2005) 24. Ruelle, D.: Application of hyperbolic dynamics to physics: some problems and conjectures. Bull. Amer. Math. Soc. (N.S.) 41, 275–278 (2004) 25. Rychlik, M., Sorets, E.: Regularity and other properties of absolutely continuous invariant measures for the quadratic family. Commun. Math. Phys. 150, 217–236 (1992) 26. Szewc, B.: The Perron-Frobenius operator in spaces of smooth functions on an interval. Ergod. Th. Dynam. Syst. 4, 613–643 (1984) 27. Tsujii, M.: On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps. Commun. Math. Phys. 177, 1–11 (1996) 28. Wang, Q.-D., Young, L.-S.: Nonuniformly expanding 1D maps. Commun. Math. Phys. 264, 255–282 (2006) 29. Whitney, H.: Analytic expansions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36, 63–89 (1934) 30. Young, L.-S.: Decay of correlations for quadratic maps. Commun. Math. Phys. 146, 123–138 (1992) 31. Young, L.-S.: What are SRB measures, and which dynamical systems have them?. J. Statistical Phys. 108, 733–754 (2002) Communicated by G. Gallavotti


Communications in


On the Crepant Resolution Conjecture in the Local Case Tom Coates Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK. E-mail: [email protected] Received: 21 April 2008 / Accepted: 10 October 2008 Published online: 3 February 2009 – © Springer-Verlag 2009

Abstract: In this paper we analyze four examples of birational transformations between local Calabi–Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov–Witten invariants, proving the Coates–Iritani–Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan’s original Crepant Resolution Conjecture should be modified, by including appropriate “quantum corrections”, and that there is no straightforward generalization of either Ruan’s original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds. 1. Introduction Suppose that X is an algebraic orbifold and that Y is an orbifold or algebraic variety which is birational to X . It is natural to try to understand the relationship between the quantum cohomology of X and that of Y. In this paper we analyze four examples of this situation — two crepant resolutions, a crepant partial resolution, and a flop — which together exhibit some of the range of phenomena which can occur. The spaces that we consider are local Calabi–Yau 3-folds. Our methods are based on mirror symmetry for toric orbifolds. The small quantum cohomology QC(X ) of an orbifold X is a family of algebras depending on so-called quantum parameters. It arises in string theory as the chiral ring of the topological A-model with target space X ; from this point of view the quantum parameters are co-ordinates on the stringy Kähler moduli space M of X . It is expected that the chiral rings form a family of algebras over the whole of M and that this family coincides with QC(X ) near the so-called large radius limit point 0X of M. Other target spaces Y which are birational to X are expected to correspond to other limit points 0Y

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of M; this suggests that if X and Y are birational then the relationship between QC(X ) and QC(Y) should involve analytic continuation in the quantum parameters. A precise mathematical formulation of this was given by Ruan in his influential Crepant Resolution Conjecture. To state it, we need to choose co-ordinates on (patches of) the stringy Kähler moduli space M. Suppose that Y → X is a crepant resolution or partial resolution of the coarse moduli space X of X . A choice of basis ϕ1 , . . . , ϕr for H 2 (Y; C) defines co-ordinates ti , 1 ≤ i ≤ r on H 2 (Y; C), and hence defines exponentiated flat co-ordinates qi = eti , 1 ≤ i ≤ r , on a neighbourhood of 0Y in M. Similarly a choice of basis φ1 , . . . , φs for H 2 (X ; C) defines exponentiated flat co-ordinates u i , 1 ≤ i ≤ s, near 0X in M. We take ϕ1 , . . . , ϕr to be primitive integer vectors on rays of the Kähler cone for Y and φ1 , . . . , φs to be primitive integer vectors on rays of the Kähler cone for X . Because Y → X is a (partial) resolution there is a natural embedding j : H 2 (X ; Q) → H 2 (Y; Q) which identifies the Kähler cone for X with a face of the Kähler cone for Y. We can therefore insist that j (φi ) = ri ϕi for some rational numbers ri , 1 ≤ i ≤ s. The presence of these rational numbers reflects the fact that the embedding j does not in general identify the integer lattices in H 2 (X ; Q) and H 2 (Y; Q). The variables q1 , . . . , qr and u 1 , . . . , u s thus defined are the quantum parameters: the parameters on which the small quantum cohomology algebras QC(Y) and respectively QC(X ) depend. Ruan’s Conjecture asserts that if Y → X is a crepant resolution then there are roots of unity ωi , 1 ≤ i ≤ r , and a choice of path of analytic continuation such that QC(X ) is isomorphic to the algebra obtained from QC(Y) by analytic continuation in the qi followed by the change of variables qi =

ωi u ri i 1 ≤ i ≤ s s < i ≤ r. ωi

(1)

One consequence of this is the Cohomological Crepant Resolution Conjecture (CCRC) [45], which asserts that the Chen–Ruan orbifold cohomology algebra of X is isomorphic to the algebra obtained from the small quantum cohomology of Y by analytic continuation in the qi followed by the change of variables qi =

0 1≤i ≤s ωi s < i ≤ r.

These conjectures have been verified in a number of examples [7,8,10–12,18,20,28,42, 46]. Recent progress in both mathematics and physics suggests, however, that Ruan’s Conjecture should be modified: that it is not an accurate reflection of the physical picture. In essence this is because when we identify the family of algebras over M (i.e. the chiral rings) with QC(X ) and QC(Y), we need to use exponentiated flat co-ordinates near 0X and 0Y . And even though the family of algebras near 0X is related to the family of algebras near 0Y by analytic continuation, the analytic continuation of exponentiated flat co-ordinates near 0X will not in general give exponentiated flat co-ordinates near 0Y . Thus we need also to analyze how the two co-ordinate systems are related. In some examples this has been done by Aganagic–Bouchard–Klemm [3] using ad hoc methods and by Coates–Iritani–Tseng [18] in a more systematic fashion; all of their examples satisfy the original Ruan Conjecture. One contribution of this paper is to give the first example (Example II below) of a crepant resolution where the change in exponentiated

On the Crepant Resolution Conjecture in the Local Case

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flat co-ordinate systems is sufficiently drastic that the original Ruan Conjecture probably fails. Our other examples suggest that this failure cannot easily be fixed, and that a different approach is needed. In recent joint work with Iritani and Tseng [18] we proposed1 such a different approach, giving a new conjectural picture of the relationship between the Gromov– Witten theory of X and that of Y. Our conjecture was phrased in terms of Givental’s symplectic formalism [23,33]. Genus-zero Gromov–Witten invariants of X (and respectively Y) are encoded in a Lagrangian submanifold-germ LX in a symplectic vector space HX (respectively LY ⊂ HY ). As LX and LY are germs of submanifolds it makes sense to analytically continue them, and we conjectured the existence of a linear symplectic isomorphism U : HX → HY satisfying some quite restrictive conditions such that after analytic continuation we have U(LX ) = LY . We also proved our conjecture when X is one of the weighted projective spaces P(1, 1, 2) or P(1, 1, 1, 3) and Y → X is a crepant resolution. Our conjecture has consequences for quantum cohomology: it implies both the Cohomological Crepant Resolution Conjecture and a modified version of Ruan’s Conjecture, each with the caveat that we must allow the quantities ωi to be arbitrary constants rather than roots of unity. (In the examples below the ωi turn out to be roots of unity and so the caveat disappears; Iritani has suggested an attractive conceptual reason for this to be true in general [37].) The modified version of Ruan’s Conjecture has an additional hypothesis, that X be semi-positive, and replaces the change of variables (1) by qi = f i (u 1 , . . . , u s ), where ωi u ri i + higher order terms in u 1 , . . . , u s 1 ≤ i ≤ s f i (u 1 , . . . , u s ) = ωi + higher order terms in u 1 , . . . , u s s < i ≤ r. Thus we get a “quantum corrected” version of Ruan’s original conjecture. In this paper we consider four examples: (I) the crepant resolution of X = C3 /Z3 , where Z3 acts with weights (1, 1, 1); (II) the crepant resolution of the canonical bundle X = K P(1,1,3) ; (III) a crepant partial resolution of X = C3 /Z5 , where Z5 acts with weights (1, 1, 3); (IV) a toric flop with X = OP(1,2) (−1)⊕3 and Y = OP2 (−1) ⊕ OP2 (−2). We prove the Coates–Iritani–Tseng/Ruan Crepant Resolution Conjecture in each case. This has implications as follows: Conjecture Example

CIT/Ruan

CCRC

original Ruan

modified Ruan

I II III IV

✓ ✓ ✓ ✓

✓ ✓ ? n/a

✓ ? ? n/a

✓ ✓ ? n/a

I expect that wherever there is a “?” in this table, the corresponding conjecture fails to hold, so that for example the original form of Ruan’s Conjecture fails in Example II and the modified form of Ruan’s Conjecture fails in Example III. It is difficult to prove these 1 Similar ideas occurred in unpublished work of Ruan; an expository account can be found in [24].

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assertions, as this would involve ruling out every possible choice of path of analytic continuation and all choices of roots of unity, but I know of no reason to expect these conjectures to hold. In forthcoming work, Iritani will prove our form of the Crepant Resolution Conjecture for all crepant birational transformations between toric Deligne–Mumford stacks. His method uses the full force of the mirror Landau–Ginzburg model, the variation of semi-infinite Hodge structure [5,37,38] associated to it, and the mirror theorem for toric Deligne–Mumford stacks [21]. Since all of our examples are included in his discussion, it is natural to ask: “What is the point of this paper?” The discussion here has quite modest goals, and is meant to illustrate four points. Firstly, these questions are not difficult. If X is a toric orbifold and Y → X is a crepant resolution then the relationship between the quantum cohomology of X and that of Y can be determined systematically, using well-understood methods from toric mirror symmetry. Secondly, our form of the Crepant Resolution Conjecture may also hold, without significant change, for more general crepant birational transformations: we see this here for a crepant partial resolution and a flop. Thirdly, the method of proof described here also applies without change to the more general crepant toric situation. Finally, it seems likely that no naïve modification of Ruan’s original conjecture holds true; we discuss this further in the next paragraph. Along the way, we will see two things which were perhaps already obvious: that Givental-style mirror theorems are well-adapted to the analysis of toric birational transformations, and that the methods of [18] are applicable to the (local) Calabi–Yau examples which are of greatest interest to physicists [3]. The original conjecture of Ruan has an attractive simplicity, and one might therefore ask whether our formulation of the Crepant Resolution Conjecture is unnecessarily complicated and whether some simpler statement holds [14]. The examples below constitute some evidence that the answer to these questions is “no”. In Example II below we see that quantum corrections to Ruan’s original conjecture are probably necessary, and in Example III the situation is even worse: there is probably not even a generalization of the Cohomological Crepant Resolution Conjecture to partial resolutions which involves only small (rather than big) quantum cohomology. This is related to the absence of a Divisor Equation for degree-two classes from the twisted sectors, and is discussed further in Sect. 6. A Note on the Bryan–Graber Conjecture. Jim Bryan and Tom Graber [11] have recently given a generalization of Ruan’s Crepant Resolution Conjecture which applies to big quantum cohomology, rather than just small quantum cohomology, under the assumption that the orbifold X involved satisfies a Hard Lefschetz condition on Chen–Ruan orbifold cohomology. We will not consider this here, as none of our examples satisfy the Hard Lefschetz condition. But as the conclusion of the Bryan–Graber Conjecture implies the original form of the Ruan Conjecture, Example II can be thought of as further evidence that the Bryan–Graber Conjecture probably fails to hold without the Hard Lefschetz assumption: see [18] for more on this. Conventions. We will assume that the reader is familiar with the Gromov–Witten theory of orbifolds. This theory is constructed in [1,2,15,16]; a rapid overview can be found in [22, Sect. 2]. We work in the algebraic category, so for us “orbifold” means “smooth algebraic Deligne–Mumford stack over C”. All of our examples are non-compact, but they carry the action of a torus T = C× such that the T -fixed locus is compact. We therefore work throughout with T -equivariant Gromov–Witten invariants, which in this setting behave much as the Gromov–Witten invariants of compact orbifolds (see e.g.


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[11]), and with T -equivariant Chen–Ruan orbifold cohomology. We always take the product of T -equivariant Chen–Ruan classes using the Chen–Ruan product; when we want to emphasize this, we will write the product as ∪ . The degree of a Chen–Ruan CR class always means its orbifold or age-shifted degree. An expository account of our Crepant Resolution Conjecture and its consequences can be found in [24]. The reader should take care when comparing the discussion in this paper with those in [11,24], as here we measure the degrees of orbifold curves using a basis of degree-two cohomology classes chosen as above, whereas there the authors use a so-called positive basis for H2 . Our choice of degree conventions fits well with toric geometry, and this will be important below, but we pay a price for our choice: the presence of the rational numbers ri described above. Outline of the Paper. In Sect. 2 we fix notation and give a precise description of the conjecture which we will prove. In Sect. 3 we collect various preparatory results, as well as describing how our conjecture implies versions of Ruan’s Conjecture and the Cohomological Crepant Resolution Conjecture. Examples I–IV are in Sects. 4–7 respectively. 2. Statement of the Conjecture In this section we give a precise statement of the conjecture that we will prove. Before we do so, we describe our general setup and fix notation. General setup. Let X be a Gorenstein orbifold with projective coarse moduli space X and let π : Y → X be a crepant resolution. Assume that X , X , and Y carry actions of a torus T = C× such that both π and the structure map X → X are T -equivariant and such that the T -fixed loci on X and Y are compact. Let C[λ] denote the T -equivariant cohomology of a point, where λ is the first Chern class of the line bundle O(1) → CP∞ , and • let C(λ) be its field of fractions. Write H (X ) := HCR,T (X ; C) ⊗ C(λ) for the localized T -equivariant Chen–Ruan orbifold cohomology of X , and H (Y ) := HT• (Y ; C) ⊗ C(λ) for the localized T -equivariant cohomology of Y . We work throughout over the field C(λ). The C(λ)-vector spaces H (X ) and H (Y ) carry non-degenerate inner products, given by i (α ∪ I β) j (α ∪ β) (α, β)X := and (α, β)Y := , IX T e(NIX T /IX ) Y T e(NY T /Y ) where I is the canonical involution on the inertia stack IX of X ; i : IX T → IX and j : Y T → Y are the inclusions of the T -fixed loci in IX and Y respectively; NIX T /IX and NY T /Y are the normal bundles to the T -fixed loci; and e is the T -equivariant Euler class. Note that the T -equivariant Euler classes are invertible over C(λ). The symplectic vector space. In what follows write Z for either X or Y , and write Z for the coarse moduli space of Z (i.e. for either X or Y ). Introduce the symplectic vector space HZ := H (Z) ⊗ C((z −1 )), the vector space, Z ( f, g) := Resz=0 ( f (−z), g(z))Z dz, the symplectic form,

+ := H (Z) ⊗ C[z], H− := z −1 H (Z) ⊗ C[[z −1 ]]. The polarization and set HZ Z + ⊕ H− identifies H with the cotangent bundle T H+ . We regard H HZ = HZ Z Z Z Z as a graded vector space where deg z = 2.

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Degrees and Novikov variables. Fix a basis ω1 , . . . , ωs for H 2 (X ; Q) consisting of primitive integer vectors on rays of the Kähler cone for X , and a basis ω1 , . . . , ωr for H 2 (Y ; Q) consisting of primitive integer vectors on rays of the Kähler cone for Y . Note that H 2 (X ; Q) is canonically isomorphic to H 2 (X ; Q), so we can regard ω1 , . . . , ωs as cohomology classes on X , and in our situation we can always insist that π ωi = ri ωi , 1 ≤ i ≤ s, for some rational numbers ri . We measure the degrees of orbifold curves using the bases ωi and ωi . Recall that a stable map f : C → Z from an orbifold curve to Z has a well-defined degree in the free part H2 (Z ; Z)free := H2 (Z ; Z)/H2 (Z ; Z)tors of H2 (Z ; Z); we write Eff(Z) ⊂ H2 (Z ; Z)free for the set of degrees of stable maps from orbifold curves to Z. Given an element d ∈ Eff(Z), set di = d, ωi if Z = X and di = d, ωi if Z = Y . Note that the di here are in general rational numbers. Define

d

d

Q d := Q d11 · · · Q ds s , where d ∈ Eff(X ) and Q d := Q 11 · · · Q r r , where d ∈ Eff(Y ). Here Q 1 , Q 2 , . . . are formal variables called Novikov variables; the number of Novikov variables associated with Z is b2 (Z ), the second Betti number of Z . Bases and Darboux co-ordinates. We fix C(λ)-bases φ0 , . . . , φ N and φ 0 , . . . , φ N for H (X ) such that (a) φ0 is the identity element 1X ∈ H (X ); (b) φ1 , φ2 , . . . , φs are lifts to T -equivariant cohomology of ω1 , ω2 , . . . , ωs ; (c) (φi , φ j )X = δi j ; and C(λ)-bases ϕ0 , . . . , ϕ N and ϕ 0 , . . . , ϕ N for H (Y ) such that (d) ϕ0 is the identity element 1Y ∈ H (Y ); (e) ϕ1 , ϕ2 , . . . , ϕr are lifts to T -equivariant cohomology of ω1 , ω2 , . . . , ωr ; (f) (ϕi , ϕ j )Y = δi j . Conditions (b) and (e) here will be useful below when we discuss the Divisor Equation. Write φi if Z = X φ i if Z = X and i = i = ϕi if Z = Y ϕ i if Z = Y . Then

qkα α z k +

k≥0

β

pl β (−z)−1−l

(2)

l≥0 β

gives a Darboux co-ordinate system {qkα , pl } on HZ ; here and henceforth we use the summation convention on Greek indices, summing repeated Greek (but not Roman) indices over the range 0, 1, . . . , N . Gromov–Witten invariants. We use correlator notation for T -equivariant Gromov– Witten invariants of Z, writing

α1 ψ i1 , . . . , αn ψ in

Z 0,n,d

=

n [Z0,n,d ]vir k=1

evk (αk ) · ψkik ,

(3)


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where α1 , . . . , αn are elements of H (Z) and i 1 , . . . , i n are non-negative integers. The cohomology classes ψ1 , . . . , ψn here are the first Chern classes of the universal cotangent line bundles on the moduli space Z0,n,d of genus-zero n-pointed stable maps to Z of degree d ∈ Eff(Z). The integral denotes the cap product with the T -equivariant virtual fundamental class of Z0,n,d : we discuss this further in the next paragraph. The right-hand

side of Eq. (3) is defined in Sect. 8.3 of [2] where it is denoted τi1 (α1 ), . . . , τin (αn ) 0,d ; our choice of notation allows compact expressions for many important quantities, such as Z 1

α m Z αψ for , 0,1,d z − ψ 0,1,d z m+1 m≥0

as correlators are multilinear in their entries. Twisted Gromov–Witten invariants. In most of the examples that we consider below, Z will be the total space of a concave vector bundle E over a compact orbifold (or manifold) B, and the T -action on Z will rotate the fibers of E and cover the trivial action on B. That E is concave means that H 0 (C, f E) = 0 for all stable maps f : C → B of non-zero degree. This implies that stable maps to E of non-zero degree all land in the zero section and so, for d = 0, the moduli space Z0,n,d coincides as a scheme with B0,n,d . The natural obstruction theories on Z0,n,d and B0,n,d differ, though, and the T -equivariant virtual fundamental classes satisfy [Z0,n,d ]vir = [B0,n,d ]vir ∩ e(Obs0,n,d ), where e is the T -equivariant Euler class and Obs0,n,d is the vector bundle over B0,n,d with fiber at a stable map f : C → B equal to H 1 (C, f E). Thus (· · · ) = (· · · ) ∪ e(Obs0,n,d ). [Z0,n,d ]vir

[B0,n,d ]vir

This means that Gromov–Witten invariants of Z coincide with twisted Gromov– Witten invariants [19,23] of B, where the twisting characteristic class is the inverse T -equivariant Euler class e−1 and the twisting bundle is E: this is explained in detail in [19]. Results of [19] allow us to compute these twisted Gromov–Witten invariants in terms of the ordinary Gromov–Witten invariants of B, a fact which we exploit repeatedly below. In the exceptional case d = 0, the moduli space Z0,n,d is non-compact and so we need to say what we mean by the integral in (3). Since Z0,n,d carries a T -action with compact fixed set, we can define the integral using the virtual localization formula of Graber–Pandharipande [34]; note that we could do this in the case d = 0, too, and this would reproduce the definition which we just gave. Gromov–Witten potentials. The genus-zero Gromov–Witten potential FZ0 is a generating function for certain genus-zero Gromov–Witten invariants of Z. It is a formal power series in variables τ a , 0 ≤ a ≤ N , and the Novikov variables Q i , 1 ≤ i ≤ b2 (Z ), defined by FZ0 =

n≥0 d∈Eff(Z )

n times Z Q d τ, τ, . . . , τ n!

0,n,d

,

(4)

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where τ = τ α α . Since correlators are multilinear, the expression τ, τ, . . . , τ Z 0,n,d expands into a polynomial in the variables τ a . The second summation here is over the set Eff(Z) of degrees of maps from orbifold curves to Z. 0 is a generating function for all genus-zero The genus-zero descendant potential FZ Gromov–Witten invariants of Z. It is a formal power series in variables tka , 0 ≤ a ≤ N , 0 ≤ k < ∞, and the Novikov variables Q i , 1 ≤ i ≤ b2 (Z ), defined by 0 FZ =

n≥0 0≤k1 ,...,kn 0, fI t I f I a formal power series in the variables t0a , 0 ≤ a ≤ b2 (Z ), I

and then analytically continue each f I . 2 These variables correspond to basis elements of H (Z) of degree 0 or 2.

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The Crepant Resolution Conjecture. We are now ready to state the conjecture. Conjecture 2.1. There is a degree-preserving C((z −1 ))-linear symplectic isomorphism U : HX → HY and a choice of analytic continuations of LX and LY such that U (LX ) = LY . Furthermore, U satisfies: −1 (a) U(1X ) =1Y + O(z ); (b) U ◦ ρ ∪ = (π ρ ∪) ◦ U for every untwisted degree-two class ρ ∈ H 2 (X ; C); + CR ⊕ HY− = HY . (c) U HX

This is a slight modification of a conjecture due to Coates, Iritani, and Tseng [18]; very similar ideas occurred, simultaneously and independently, in unpublished work of Ruan. An expository account of the conjecture and its consequences can be found in [24]. 3. General Theory In this section we describe various aspects of Givental’s symplectic formalism which we will need below, as well as stating some consequences of Conjecture 2.1. Big and small J -functions. Let τ = τ α α . The big J -function of Z is 1 Z big JZ (τ, z) := z + τ + τ, τ, . . . , τ, . n! z − ψ 0,n+1,d n≥0 d∈Eff(Z )

big

It is a formal family of elements of HZ — in other words, JZ is a formal power series in the variables τ a , 0 ≤ a ≤ N , which takes values in HZ . By writing out Eqs. (8) big defining LZ , it is easy to see that JZ (τ, −z) is the unique family of elements of LZ of the form −z + τ + O(z −1 ). Take Z = Y and restrict the parameter τ in the big J -function to the locus big τ = τ 1 ϕ1 + · · · + τ r ϕr . Then the Divisor Equation gives that JY (τ 1 ϕ1 + · · · + τ r ϕr , z) is equal to ⎛ ⎞ Y ϕ 1 r 1 r ed1 τ · · · edr τ ϕ ⎠. z eτ ϕ1 /z · · · eτ ϕr /z ⎝1 + (9) z(z − ψ) 0,1,d d∈Eff(Y )

Making the change of variables qi = eτ , 1 ≤ i ≤ r , we define the small J -function of Y to be ⎛ ⎞ Y ϕ ϕ /z ϕ /z q1d1 · · · qrdr ϕ ⎠. (10) JY (q, z) := z q1 1 · · · qr r ⎝1 + z(z − ψ) 0,1,d i

d∈Eff(Y )

In examples below we will see that this converges, in a domain where each |qi | is sufficiently small, to a multi-valued analytic function of q1 , . . . , qr which takes values in ϕ /z HY . The multi-valuedness comes from the factors qi i := exp(ϕi log(qi )/z). We have JY (q, −z) ∈ LY for all q in the domain of convergence of JY .


1081

Similarly, take Z = X and restrict the parameter τ in the big J -function to the locus τ = τ 1 φ1 + · · · + τ s φs . Then the Divisor Equation gives that big

JX (τ 1 φ1 + · · · + τ s φs , z) ⎛ = z eτ

1 φ /z 1

· · · eτ

s φ /z s

⎝1 +

e d1 τ · · · e ds τ 1

s

d∈Eff(X )

φ z(z − ψ)

X

⎞ φ ⎠.

0,1,d

Making the change of variables u i = eτ , 1 ≤ i ≤ s, we define the small J -function of X to be ⎛ ⎞ X φ φ /z φ /z JX (u, z) := z u 1 1 · · · u s s ⎝1 + u d11 · · · u ds s φ ⎠. (11) z(z − ψ) 0,1,d i

d∈Eff(X )

In the examples below this converges, in a domain where each |u i | is sufficiently small, to a multi-valued analytic function of u 1 , . . . , u s which takes values in HX . We have JX (u, −z) ∈ LX for all u in the domain of convergence of JX . Two consequences of Conjecture 2.1. Recall that the T -equivariant small quantum cohomology of X is a family of algebra structures on H (X ) parametrized by u 1 , . . . , u s , defined by

X φα • φ β = u d11 · · · u ds s φα , φβ , φ 0,3,d φ . (12) d∈Eff(X )

The T -equivariant small quantum cohomology of Y is a family of algebra structures on H (Y ) parametrized by q1 , . . . , qr , defined by

Y ϕα • ϕβ = q1d1 · · · qrdr ϕα , ϕβ , ϕ 0,3,d ϕ . (13) d∈Eff(Y )

For the remainder of this subsection, assume that: • Conjecture 2.1 holds; • the symplectic transformation U remains well-defined in the non-equivariant limit λ → 0; • X is semi-positive3 . Two consequences of Conjecture 2.1 are then as follows: these are proved4 in [24]. Define the class c ∈ H (Y ) by U(1X ) = 1Y − cz −1 + O(z −2 ), 3 The orbifold Z is semi-positive if and only if there does not exist d ∈ Eff(Z) such that

3 − dimC Z ≤ c1 (T Z), d < 0. All Fano and Calabi–Yau orbifolds are semi-positive, as are all orbifold curves, surfaces, and 3-folds. In particular, all the orbifolds that we consider in the examples below are semi-positive. 4 This is not, strictly speaking, true: the T -equivariant version of the Crepant Resolution Conjecture is not treated in [24]. It is straightforward to check, however, that the arguments given there also prove the results stated here. The key point is that U has a non-equivariant limit, and so only non-negative powers of λ can occur.

1082

T. Coates

and write c = c1 ϕ1 + · · · + cr ϕr + dλ, c1 , . . . , cr , d ∈ C;

(14)

such an equality exists because c has degree 2. Then: Corollary 3.1. The algebra obtained from the T -equivariant small quantum cohomology algebra of Y by analytic continuation5 in the parameters qs+1 , . . . , qr (if necessary) followed by the substitution qi =

0 1≤i ≤s i ec s < i ≤ r

is isomorphic to the T -equivariant Chen–Ruan orbifold cohomology algebra of X , via an isomorphism which sends α ∈ HT2 (X ; C) ⊂ H (X ) to π α ∈ H (Y ). This is a version of Ruan’s Cohomological Crepant Resolution Conjecture [45]. Define elements be ∈ H (Y ), 0 ≤ e ≤ N , by be = 0 if deg φe ≤ 2 and 1 U φe z 1− 2 deg φe = be + O(z −1 ) otherwise. Define power series f 1 , . . . , f r , g ∈ C[[u 1 , . . . , u s ]] by f 1 ϕ1 +· · ·+ f r ϕr +gλ =

N

X 1 1 (−1) 2 deg φe +1 φ e ψ 2 deg φe−2

0,1,d

d∈Eff(X ) e=r +1

u d11 · · · u ds s be ; (15)

such an equality exists because each class be has degree 2. Recall the definition of the rational numbers ri , 1 ≤ i ≤ s, from Sect. 2. Then: Corollary 3.2. The algebra obtained from the T -equivariant small quantum cohomology algebra of Y by analytic continuation in the parameters qs+1 , . . . , qr (if necessary) followed by the substitution qi =

i

i

ec + f u ri i 1 ≤ i ≤ s i i ec + f s >> >> >> >

X

K P(1,1,3) w w w ww w w {w w

(34)

induced by moving from Chamber I to Chamber II. In the next section we consider the crepant partial resolution 3 K P(1,1,3) C /Z5 III u III uu uu III u u I$ zuu 3 C /Z5 obtained by moving from Chamber II to Chamber III. We will not discuss Chamber IV at all. The T -action. The action of T = C× on C5 such that α ∈ T maps (x, y, z, u, v) −→ (x, y, z, u, αv) descends to give actions of T on X , X , and Y . The induced action on X is the canonical C× -action on the line bundle K P(1,1,3) → P(1, 1, 3); it covers the trivial action on P(1, 1, 3). The diagram (34) is T -equivariant. Bases for everything. We have r := rank H 2 (Y ; C) = 2, s := rank H 2 (X ; C) = 1. Let p1 , p2 ∈ H (Y ) denote the T -equivariant Poincaré-duals to the divisors {z = 0} and {x = 0} respectively, so that H (Y ) = C(λ)[ p1 , p2 ]/ p22 (λ + p2 − 2 p1 ), p1 ( p1 − 3 p2 ), p1 p22 .

1096

T. Coates

Set ϕ0 = 1, ϕ1 = p1 , ϕ2 = p2 , ϕ3 = p1 p2 , ϕ4 = p22 . * * Write the inertia stack IX of X as the disjoint union X0 X1/3 X2/3 , where X f isthe component of the inertia stack corresponding to the fixed locus of the ele ment 1, e2π i f ∈ (C× )2 . We have X0 = K P(1,1,3) and X1/3 = X2/3 = BZ3 . Define 1 f ∈ H (X ) to be the class which restricts to the unit class on the component X f and restricts to zero on the other components, and let p ∈ H (X ) denote the first Chern class of the line bundle O(1) → P(1, 1, 3), pulled back to K P(1,1,3) via the natural projection and then regarded as an element of Chen–Ruan cohomology via the inclusion X = X0 → IX . Set φ0 = 10 , φ1 = p, φ2 = p 2 , φ3 = 11/3 , φ4 = 12/3 , so that r1 = 13 . Step 1: A family of elements of LY . Consider 2 p2 1 + pz2 1 + pz1 1 + p1 −3 z IY (y1 , y2 , z) := z 2 p p −3 p p 1 + z1 + k 1 + 1 z 2 + k − 3l k,l≥0 1 + 2 + l z λ+ p2 −2 p1 1+ z y1k+ p1 /z y2l+ p2 /z . (35) × λ+ p2 −2 p1 1+ + l − 2k z

This series converges, in a region where |y1 | and |y2 | are sufficiently small, to a multivalued analytic function of (y1 , y2 ) which takes values in HY . We have: IY (y1 , y2 , z) =

k+ p1 /z l+ p2 /z y2 z m=l m=k 2 m=1 ( p2 + mz) m=1 ( p1 k,l≥0

×

y1

m≤0 (λ +

p2 − 2 p1 + mz)

m≤l−2k (λ +

p2 − 2 p1 + mz)

+ mz)

m≤0 ( p1

− 3 p2 + mz)

m≤k−3l ( p1

− 3 p2 + mz)

.

Note that all but finitely many terms in the infinite products here cancel. Proposition 5.1. IY (y1 , y2 , −z) ∈ LY for all (y1 , y2 ) in the domain of convergence of IY . Proof. The argument which proves Theorem 0.2 in [30] also proves the claim here. Theorem 0.2 as stated only applies to compact toric varieties, but the proof works for the non-compact toric variety Y as well. The reader who would prefer not to check this should wait for the full generality of [21].


1097

Step 2: IY determines LY . We have: Corollary 5.2. JY (q1 , q2 , z) = eλg(y1 ,y2 )/z IY (y1 , y2 , z), where q1 = y1 exp (2g(y1 , y2 ) − f (y1 , y2 )), q2 = y2 exp (3 f (y1 , y2 ) − g(y1 , y2 )), (−1)3l−k (3l − k − 1)! f (y1 , y2 ) = y1k y2l , (l!)2 k!(l − 2k)! 0 0 and ∈ HN red . The reduced model matrix (12) is closely related to fractional linear transformations, see for instance [16,24], [26, Chap. 10], and we introduce appropriate definitions in the present context. Definition 13. Let Ke be the multiplicity space of an edge e = (s, r ) so that Ke is a subspace of the network multiplicity Knet and let X ∈ B (Ke ). The feedback reduction of V ∈ M (h, K) through the edge e, with gain X , is the map Fe : M (h, K) × B (Ke ) → M (h, K Ke ) : (V, X ) → Fe (V, X ) , where in terms of block decomposition we have Fe (V, X )αβ Vαβ + Vαr X (1 − Vsr X )−1 Vsβ ,

(13)

where the indices are β ∈ {0} ∪ Pin \ {r0 } and α ∈ {0} ∪ Pout \ {s0 }. The domain is then the set of all pairs (V, X ) such that 1 − Vsr X is invertible in B (Ke ). In the special case of unit gain we write Fe (T ) Fe (T, 1). For fixed V, the map Fe (V, ·) is a non-commutative fractional linear, or Möbius, transformation. A B is a unitary operator on the direct sum H1 ⊕ Lemma 14 (Siegel). If S = C D H2 of two Hilbert spaces with A < 1, let T (X ) D + C X (1 − AX )−1 B with X ∈ Dom( S ) whenever X ≤ 1. For X, Y ∈ Dom( S ) we have the Siegel identities: −1

S (X )† S (Y ) − 1 = B † 1 − X † A† X † Y − 1 (1 − AY ) B,

S (X ) S (Y )† − 1 = C † (1 − X A)−1 X Y † − 1 1 − A† Y C † . A proof can be found in [24]. A B is unitary and 1− A is invertible, then D+C (1 − A)−1 B Corollary 15. If S = C D is unitary. For consistency, we would hope that the reduced model matrix belongs to the class of model matrices with multiplicity space one dimension lower. This we now show to be the case.

Quantum Feedback Networks: Hamiltonian Formulation

1123

Lemma 16. Let V be the model matrix determined by the operators (S, L, H). Then the reduced model matrix Vred obtained by eliminating the edge (r0 , s0 ) is determined by the operators Sred , Lred , Hred where −1 Ss0 r , Sred sr = Ssr + Ssr0 1 − Ss0 r0 −1 Ls0 , Lsred = Ls + Ssr0 1 − Ss0 r0 −1 red † Im Ls Ssr0 1 − Ss0 r0 Ls0 , H = H+ s∈Pout

for r ∈ Pin \{r0 } and s ∈ Pout \{s0 }. −1 red = red = Sred = S + S Proof. The identifications Vsr Ss0 r and Vs0 sr sr0 1 − Ss0 r0 sr −1 Lsred = Ls + Ssr0 1 − Ss0 r0 Ls0 are immediate. Unitarity of Sred follows from the red† red ≡ − above corollary to the Siegel identities. We next check that V0r s∈Pout \{s0 } Ls Sred sr . Here

Lsred† Sred sr =

s∈Pout \{s0 }

Ls† Sred sr +

s∈Pout \{s0 }

s∈Pout \{s0 }

−1 Ls†0 1 − S†s0 r0 S†sr0 Sred sr ,

and we use the simplification s∈Pout \{s0 }

S†sr0 Sred sr =

s∈Pout \{s0 }

−1 S†sr0 Ssr + S†sr0 Ssr0 1 − Ss0 r0 Ss0 r

−1 Ss0 r = −S†s0 r0 Ss0 r + (1 − S†s0 r0 Ss0 r0 ) 1 − Ss0 r0 −1 = −S†s0 r0 (1 − Ss0 r0 ) + (1 − S†s0 r0 Ss0 r0 ) 1 − Ss0 r0 Ss0 r −1 = (1 − S†s0 r0 ) 1 − Ss0 r0 Ss0 r

so that

Lsred† Sred sr =

s∈Pout \{s0 }

s∈Pout \{s0 }

−1 † Ls† Sred Ss0 r . sr + Ls0 1 − Ss0 r0

On the other hand −1 red V0r = V0r + V0r0 1 − Vs0 r0 Vs0 r −1 =− Ls† Ssr − Ls† Ssr0 1 − Ss0 r0 Ss0 r s∈Pout

=−

s∈Pout \{s0 }

=−

s∈Pout \{s0 }

s∈Pout

Ls† Sred sr

−1 − Ls†0 Ss0 r − Ls†0 Ss0 r0 1 − Ss0 r0 Ss0 r

−1 † Ls† Sred Ss0 r ≡ − sr − Ls0 1 − Ss0 r0

s∈Pout \{s0 }

Lsred† Sred sr .

1124

J. Gough, M. R. James

red† red red = − 1 red Finally we must check that V00 s∈Pout \{s0 } Ls Ls − iH . Let us use this as 2 the definition of Hred , then we have −1 1 −iHred = V00 + V0r0 1 − Vs0 r0 Vs0 0 + Lsred† Lsred 2 s∈Pout \{s0 } ⎛ ⎞ −1 1 † ⎠ = ⎝−iH − Ls Ls − Ls† Ssr0 1 − Ss0 r0 Ls0 2 +

1 2

s∈Pout

s∈Pout

Lsred† Lsred

s∈Pout \{s0 }

−1 1 1 = −iH − Ls†0 Ls0 − Ls† Ssr0 1 − Ss0 r0 Ls0 2 2 s∈Pout \{s0 } −1 −1 1 + Ls†0 1 − S†s0 r0 Ssr0 Ls − Ls†0 Ss0 r0 1 − Ss0 r0 Ls0 2 s∈Pout \{s0 } −1 −1 1 + Ls†0 1 − S†s0 r0 S†sr0 Ssr0 1 − Ss0 r0 Ls0 . 2 s∈Pout \{s0 }

We collected together several terms to get − 21 Ls†0 XLs0 where −1 −1 −1 1 − S†s0 r0 X = 1 + 2Ss0 r0 1 − Ss0 r0 − S†sr0 Ssr0 1 − Ss0 r0 s∈Pout \{s0 }

−1 −1 −1 1 − S†s0 r0 Ss0 r0 1 − Ss0 r0 = 1 + 2Ss0 r0 1 − Ss0 r0 − 1 − S†s0 r0 −1 −1 Ss0 r0 − S†s0 r0 1 − Ss0 r0 = 1 − S†s0 r0 . It follows that Hred = H + Im

s∈Pout \{s0 }

−1 −1 Ls† Ssr0 1 − Ss0 r0 Ls0 + Im Ls†0 1 − Ss0 r0 Ls0 .

−1 −1 Note that Im Ls†0 1 − Ss0 r0 Ls0 = Im Ls†0 Ss0 r0 1 − Ss0 r0 Ls0 we obtain H red as the self-adjoint operator given in the statement of the lemma. Lemma 17. Let e1 = (s1 , r1 ) and e2 = (s2 , r2 ) be a pair of edges in a network then Fe1 ◦ Fe2 = Fe2 ◦ Fe1 = Fe1 ⊕e2 .

(14)

Proof. Without loss of generality suppose that e1 = (1, 1) and e2 = (2, 2), then for α, β ∈ / {1, 2}, −1 (Fe2 ◦ Fe1 V)αβ = Vˆ αβ + Vˆ α2 1 − Vˆ 22 Vˆ 2β , where Vˆ αβ = Vαβ + Vα1 (1 − V11 )−1 V1β , and so we have (V , V ) (Fe2 ◦ Fe1 V)αβ = Vˆ αβ + α1 α2 Z

Vα1 Vα2


1125

with ! Z = ≡

1 1−V11

1 1 1 1 1 1−V11 V12 1−Vˆ 22 V21 1−V11 1−V11 V12 1−Vˆ 22 1 1 1 V21 1−V 11 1−Vˆ 22 1−Vˆ 22 −1 1 − V11 −V12 . −V21 1 − V22

+

"

The expression is clearly symmetric under interchange of 1 and 2, and corresponds to the double edge elimination. This implies that the order in which we apply zero time delay limits to eliminate multiple internal channels does not in fact matter, and can be combined simultaneously. In this manner, every quantum network may be reduced to a single Markovian component in a unique algebraically well-defined manner by eliminating the time delays in all internal channels by means of the map F = ◦e∈Eint Fe : M (h, Knet ) → M (h, Kext ) . If we decide to eliminate all internal channels by making the connections, inserting a gain matrix X , and taking the zero time-delay limit, then the resulting network will have model matrix F (V, X ) and of course has dimensions determined by the remaining (external) channels. The functorial property is that we are able to eliminate blocks of channels in one step, giving the same answer as if we performed the eliminations one-by-one. 5. Physical Applications We shall be interested in the situation where we eliminate all the internal channels (total multiplicity space Ki ) leaving only the external channels (total multiplicity space Ke ). With respect to the decomposition K = Kint ⊕ Kext , we write the operators (S, L, H) of the network as Li Sii Sie , L= . S= Sei See Le The feedback reduced model matrix may be conveniently expressed as F V, η−1 Vαβ + Vαi (η − Vii )−1 Viβ αβ

for α, β ∈ {0, e}, where η is the (unitary) adjacency matrix 1, if (s, r ) is an internal channel, ηsr = 0, otherwise. Here the “gain” η is the set of instructions as to which internal output port gets connected up to which internal input port. We, of course, have η = 1 if we match up the labels of the input and output ports according to the connections; however, it is computationally easier to work with a general labelling and just specify the adjacency matrix. The reduced

1126


model matrix Vred obtained by eliminating all the internal channels is determined by the red operators S , Lred , Hred given by Sred = See + Sei (η − Sii )−1 Sie , Lred = Le + Sei (η − Sii )−1 Li , Hred = H + Im L†j S ji (η − Sii )−1 Li . i=i,e

5.1. Quantum systems in feedforward. Our first application is to derive the formula for systems in series. This is the simplest nontrivial example of a quantum network.

Fig. 3. Systems in series

We begin by taking (S1 , L1 , H1 ) and (S2 , L2 , H2 ) to be the operators of the first and second system when considered as separate systems driven by independent noises. The model matrix for the network is then ⎛ ⎞ − j=1,2 ( 21 L†j L j + iH j ) −L1† S1 −L2† S2 V=⎝ L1 S1 0 ⎠ L2 0 S2 with respect to the labelling s = {0, s1 , s2 } for the outputs (rows) and r = {0, r1 , r2 } for the inputs (columns). We wish to reform a feedback reduction wherein we connect the systems via edge e = (s1 , r2 ) and take the zero time-delay limit along the edge. The resulting model should then be Markovian and its model matrix is given by Vseries = Fe V † (L , S ) − j=1,2 ( 21 L†j L j + iH j ) −L1† S1 + −L2 S2 (1 − 0)−1 1 1 = S2 L2 0 − j=1,2 ( 21 L†j L j + iH j ) − L2† S2 L1 −L1† S1 − L2† S2 S1 , = L2 + S2 L1 S2 S1 that is, the operators determining the reduced system are Sseries = S2 S1 , Lseries = L2 + S2 L1 ,

$ # Hseries = H1 + H2 + Im L2† S2 L1 . The Evans-Hudson maps associated with the feedforward system can be related to those of the individual systems via the identity (1)

(1)

(1) † (2) Lαβ (·) = Lαβ (·) + (Mµα ) Lµν (·) Mνβ .

(15)

The case most familiar to the quantum optics community is two cavity systems in cas√ √ √ cade. Here S 1 =S 2 = 1 and L i = γi ai (i = 1, 2) so we obtain L = γ1 a1 + γ2 a2 ,


1127

√ H = H1 + H2 + 2i1 γ1 γ2 a2† a1 − a1† a2 . This agrees with the calculations of Gardiner [11] for cascaded oscillators. Gardiner’s derivation [11] of the cascade rule is different from ours, but would extend to cover the gauge case due to (15). 5.1.1. The series product. The rule for determining the form of the model for systems in series has been previously given for the It¯o generator matrices where it was called the series product. It is related to the general question of how to “add” stochastic derivations in order to obtain a stochastic derivation [1]. We recall its definition and establish its basic properties. Definition 18. Let G1 and G2 be the It¯o generator matrices with the same multiplicity spaces, then the series product is defined to be G2 G1 G1 + G2 + G2 G1 . Lemma 19. The series product of two It¯o matrices G = G2 G1 is again an It¯o matrix and if the Gi have parameters (Si , Li , H) then G has parameters (Sseries , Lseries , Hseries ). If Mi is the Galilean matrix associated with Gi then the Galilean matrix associated with G = G2 G1 is M = M2 M1 . The series product is not symmetric, but is associative. Proof. One readily checks that the series product of two It¯o matrices satisfies the conditions to be an It¯o generating matrix. The specific form of the parameters is found by inspection. The associated Galilean transformation is M= = = =

1 + (G1 + G2 + G2 G1 ) 1 + G1 + G2 + G2 G1 (1 + G2 ) (1 + G1 ) M2 M 1 .

To prove associativity, let us construct the (1 + n + 1)-square matrix ⎛ ⎞ 1 −L† S − 21 L† L − iH ⎠ V = ⎝0 S L 0 0 1

(16)

from the model parameters (S, L, H). Then the series product corresponds to the ordinary matrix product Vseries = V2 V2 which is clearly associative. Associativity means that we can extend the result immediately to several systems cascaded in series. The easiest way to calculate the model matrix for several components in series is then by the ordinary matrix product of the augmented matrices Vseries = Vn · · · V2 V1 . In the lemma, the matrix V is of the type introduced by Belavkin [5] to efficiently cap⎛ ⎞ 001 ture the It¯o correction as an ordinary product. Let ζ = ⎝ 0 1 0 ⎠, then we obtain an 100 involution X = ζ X† ζ on the space of (1 + n + 1)-dimensional matrices. Then we are

1128


⎛

⎞ 1 B A considering precisely the class of -unitary matrices of the form V = ⎝ 0 D C ⎠, that 0 0 1 is V V = VV = 1. In particular, the product of two -unitaries is again a -unitary. Remark 20. Finally let us make the important remark that nowhere did we assume that the entries of G1 and G2 had to commute. This means that the series product can describe not only forward from one component system to an independent system, but also feedback into itself as well. This is captured in the picture below. 5.2. Beam Splitters. A simple beam splitter is a device performing physical superposiαβ tion of two input fields. It is described by a fixed unitary operator T = ∈ U (2): µν A1 αβ A˜ 1 = . ˜ µ ν A A2 2 This is a canonical transformation and the output fields satisfy the same canonical commutation relations as the inputs. The action of the beam splitter is depicted in the figure below. On the left we have a traditional view of the two inputs (A1 , A2 ) being split into two output fields ( A˜ 1 , A˜ 2 ). On the right we have our view of the beam splitter as being a component with two input ports and two output ports: we have sketched some internal detail to emphasize how the scattering (superimposing) of inputs; however we shall usually just draw this as a “black box” component in the following.

Fig. 4. Beam-splitter component

Our aim is to describe the effective Markov model for the feedback device sketched below where the feedback is achieved by means of a beam splitter. Here we have a component system, called the plant, in-loop and we assume that it is described by the parameters (S0 , L0 , H0 ). Markovianity here corresponds to the limit of instantaneous feedback.

Fig. 5. Feedback using a beam-splitter


1129

Fig. 6. Network representation

It is more convenient to view this as the network sketched below. Here we have the pair of internal edges (s2 , r3 ) and (s3 , r2 ). The model matrix for the network is ⎞ ⎛ − 21 L0† L0 − iH0 0 0 −L0† S0 ⎜ 0 T11 T12 0 ⎟ ⎟ V=⎜ ⎝ 0 T21 T22 0 ⎠ L0 0 0 S0 with respect to the labels (0, s1 , s2 , s3 ) for the rows and (0, r1 , r2 , r3 ) for the columns. Here we have T22 0 T21 , , Sie = Sii = 0 S0 0 Sei = (T12 , 0) , L0 , Li = 0

See = T11 , Le = 0, η =

01 . 10

Note that the adjacency matrix has row indices (s2 , s3 ) and columns indices (r2 , r3 ) labelling the internal ports, and that the edges are the off diagonals (s2 , r3 ) and (s3 , r3 ). We may perform the two eliminations simultaneously to obtain Sred

Lred

Hred

−1 T21 −T22 1 = T11 + 1 −S0 0 −1 = T11 + T12 S−1 T21 , 0 − T22 −1 (T 0) −T22 1 0 = 12 1 −S0 L0 (T12 0)

= T12 (1 − S0 T22 )−1 L0 , −1 † 0 −T22 1 = H0 + Im (0 L0 ) L0 1 −S0 = H0 + Im L0† (1 − S0 T22 )−1 L0 .

5.3. The Redheffer star product. An important feedback arrangement is shown in the figure below.

1130


Fig. 7. Composite System

!

A SA S11 12

"

We shall now derive this system taking component A to be described: , A SA S21 22 ! " " ! " B SB S33 L3B L1A 34 and B by , H , , H B . The operators of systems A are A B SB L2A S43 L4B 44 assumed to commute with those of B. We have two internal channels to eliminate which we can do in sequence, or simultaneously. We shall do the latter. here we have A A S11 0 S12 0 , Sei = , See = 0 SB 0 SB A 44 A 43 S21 0 S22 0 Sie = B , Sii = B , 0 S34 0 S33

!

and Le =

A L1A L2 01 , L , η = . = i 10 L4B L3B

The reduced operators are therefore S = ! = L = ! =

A 0 S11 B 0 S44

+

A 0 S12 B 0 S43

A −S22 1 B 1 −S33

−1

A 0 S21 B 0 S34

" A + S A S B 1 − S A S B −1 S A A 1 − S A S B −1 S B S11 S12 12 33 22 33 21 22 33 34 , B 1 − S A S B −1 S A B + S B 1 − S A S B −1 S A S B S43 S44 22 33 21 43 22 33 22 34

L1A L4B

+

A 0 S12 B 0 S43

A 1 −S22 B 1 −S33

−1

L2A L3B

" A S B 1 − S A S B −1 L A + S A 1 − S A S B −1 L B L1A + S12 22 33 2 12 22 33 3 , 33 B 1 − S A S B −1 L A + S B S A 1 − S A S B −1 L B L4B + S43 22 33 2 43 22 22 33 3


1131

−1 −1 B A B A A A H = H A + H B + Im L3B† 1 − S33 S22 L3B + L3B† 1 − S33 S22 S33 L2 −1 −1 A B A B A B +L2A† 1 − S22 S33 S22 L3 + L2A† 1 − S22 S33 L2A −1 −1 A B A A B A B B 1 − S33 1 − S33 +L1A† S12 S22 L3B + L1A† S12 S22 S33 L2 B +L4B† S43

−1 −1 % B† B A B A B A B 1 − S22 S33 S22 L3 + L4 S43 1 − S22 S33 L2A .

6. Topological Rules for Quantum Networks We wish to state algebraic rules for constructing the operators Snet , Lnet , Hnet covering all the examples of feedback networks studied so far. For a given network, we denote by ext the subset of output ports having external output. Pout the set of output ports and Pout With each i ∈ Pout there is an associated port operator L i . Similarly we have Pin and ext for the inputs. Pin For γ = (i, j) an ordered pair of ports, we set ⎧ 1, if j ∈ Pout and i ∈ Pin and there is a feedback connection ⎪ ⎪ ⎪ ⎪ ⎪ from j to i; ⎨ Sγ = Si j , if i ∈ Pout and j ∈ Pin and both i and j are in the same plant ⎪ ⎪ ⎪ ⎪ and where Si j is the scattering component from j to i; ⎪ ⎩ 0, otherwise. More generally, if γ is an ordered sequence (paths) of ports, alternating between input and output ports, we define S γ inductively. If γ is the concatenation of γ1 followed by γ2 then Sγ = Sγ2 Sγ1 . ext , Finally, we set (i, j) to be the set of all paths going from j to i. Then for i ∈ Pout ext j ∈ Pin , Sγ ; (Snet )i j = γ ∈(i, j)

(Lnet )i =

k∈Pout γ ∈(i,k)

Hnet = H0 +

Sγ Lk ;

(17) Ll† Sγ Lk ,

l,k∈Pout γ ∈(l,k)

where H0 is the sum of the individual component systems Hamiltonians. Acknowledgement. The authors would like to thank M. Yanagisawa for several useful comments in writing this paper, and to L. Bouten for suggesting the Trotter-Kato convergence technique in the main theorem. They are also grateful to the referee for bringing several important technical points to their attention.

1132


References 1. Accardi, L., Hudson, R.L.: Quantum stochastic flows and non-abelian cohomology. Quantum Probability V, Lecture Notes in Mathematics 1442, Berlin-Heidelberg-New York:Springer, 1990, pp. 54–69 2. Belavkin, V.P.: Optimization of Quantum Observation and Control. In: Proc. of 9th IFIP Conf on Optimizat Techn. Notes in Control and Inform Sci. 1, Warszawa: Springer-Verlag, 1979 3. Belavkin, V.P.: Theory of the Control of Observable Quantum Systems. Automatica and Remote Control 44(2), 178–188 (1983) 4. Belavkin, V.P.: Quantum continual measurements and a posteriori collapse on CCR. Commun. Math. Phys. 146, 611–635 (1992) 5. Belavkin, V.P.: On Quantum Ito Algebras and Their Decompositions. Lett. Math. Phys. 45, 131–145 (1998) 6. Bouten, L., Van Handel, R., James, M.R.: An introduction to quantum filtering. SIAM J. Control Optim. 46, 2199–2241 (2007) 7. Carmichael, H.J.: Quantum trajectory theory for cascaded open systems. Phys. Rev. Lett. 70(15), 2273–2276 (1993) 8. Caves, C.M.: Quantum limits on noise in linear amplifiers. Phys. Rev. D. 26, 1817 (1982) 9. Chebotarev, A.M.: Quantum stochastic differential equation is unitarily equivalent to a symmetric boundary problem in Fock space. Inf. Dim. Anal. Quantum Prob. 1, 175–199 (1998) 10. Davies, E.B.: One-parameter Semigroups. London: Academic Press Inc, 1980 11. Gardiner, C.W.: Driving a quantum system with the output field from another driven quantum system. Phys. Rev. Lett. 70(15), 2269–2272 (1993) 12. Gardiner, C.W., Collett, M.J.: Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. Phys. Rev. A 31(6), 3761–3774 (1985) 13. Gough, J.: Quantum flows as Markovian limit of emission, absorption and scattering interactions. Commun. Math. Phys. 254(2), 489–512 (2005) 14. Gough, J., Belavkin, V.P., Smolyanov, O.G.: Hamilton-Jacobi-Bellman equations for Quantum Filtering and Control. J. Opt. B: Quantum Semiclass. Opt. S237–244, Special issue on quantum control (2005) 15. Gough, J., James, M.R.: The series product and its application to feedforward and feedback networks. IEEE Trans. Automatic Control. http://arXiv.org/abs/07070048(v1) [quant-ph], 2007 16. Green, M., Limebeer, D.J.N.: Linear Robust Control. Englewood Cliffs, NJ: Prentice Hall, 1995 17. Gregoratti, M.: The Hamiltonian operator associated to some quantum stochastic differential equations. Commun. Math. Phys. 222, 181–200 (2001) 18. Hudson, R.L., Parthasarathy, K.R.: Quantum Ito’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984) 19. Lloyd, S.: Coherent quantum feedback. Phys. Rev. A 62, 022108 (2000) 20. Warszawski, P., Wiseman, H.M., Mabuchi, H.: Quantum trajectories for realistic detection. Phys. Rev. A 65, 023802 (2002) 21. Wiseman, H.: Quantum theory of continuous feedback. Phys. Rev. A 49(3), 2133–2150 (1994) 22. Yanagisawa, M., Kimura, H.: Transfer function approach to quantum control-part I: Dynamics of quantum feedback systems. IEEE Trans. Automatic Control 48, 2107–2120 (2003) 23. Yanagisawa, M., Kimura, H.: Transfer function approach to quantum control-part II: Control concepts and applications. IEEE Trans. Automatic Control 48, 2121–2132 (2003) 24. Young, N.: An Introduction to Hilbert Space. Cambridge Mathematical Textbooks, Cambridge: Cambridge Univ. Press, 1988 25. Yurke, B., Denker, J.S.: Quantum network theory. Phys. Rev. A 29(3), 1419–1437 (1984) 26. Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Englewood Cliffs, NJ: Prentice Hall, 1996 Communicated by A. Kupiainen


Communications in


Spectral Extrema and Lifshitz Tails for Non-Monotonous Alloy Type Models Frédéric Klopp1,2 , Shu Nakamura3 1 LAGA, U.M.R. 7539 C.N.R.S, Institut Galilée, Université de Paris-Nord,

99 Avenue J.-B. Clément, F-93430 Villetaneuse, France

2 Institut Universitaire de France, 103, Boulevard Saint-Michel, 75005 Paris, France.


3 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba,

Meguro-ku, Tokyo 153-8914, Japan. E-mail: [email protected] Received: 25 April 2008 / Accepted: 16 July 2008 Published online: 6 November 2008 – © Springer-Verlag 2008

Abstract: In the present note, we determine the ground state energy and study the existence of Lifshitz tails near this energy for some non monotonous alloy type models. Here, non monotonous means that the single site potential coming into the alloy random potential changes sign. In particular, the random operator is not a monotonous function of the random variables. Résumé: Cet article est consacré à la détermination de l’énergie de l’état fondamental et à l’étude de possibles asymptotiques de Lifshitz au voisinage de cette énergie pour certains modèles d’Anderson continus non monotones. Ici, non monotone signifie que le potentiel de simple site entrant dans la composition du potentiel aléatoire change de signe. En particulier, l’opérateur aléatoire n’est pas une fonction monotone des variables aléatoires. 0. Introduction and Results In this paper, we consider the continuous alloy type (or Anderson) random Schrödinger operator: Hω = − + Vω where Vω (x) = ωγ V (x − γ ) (0.1) γ ∈Zd

on Rd , d ≥ 1, where V is the site potential, and (ωγ )γ ∈Zd are the random coupling constants. Throughout this paper, we assume (H1) (1) V : Rd → R is L p (where p = 2 if d ≤ 3 and p > d/2 if d > 3), non-identically vanishing and supported in (−1/2, 1/2)d ; (2) (ωγ )γ are independent identically distributed (i.i.d.) random variables distributed in [a, b] (a < b) with essential infimum a and essential supremum b.

1134

F. Klopp, S. Nakamura

Let be the almost sure spectrum of Hω and E − = inf . When V has a fixed sign, it is well known that the E − = inf(σ (− + Vb )) if V ≤ 0 and E − = inf(σ (− + Va )) if V ≥ 0. Here, x is the constant vector x = (x)γ ∈Zd . Moreover, in this case, it is well known that the integrated density of states of the Hamiltonian (see e.g. (0.3)) admits a Lifshitz tail near E − , i.e., that the integrated density of states at energy E decays exponentially fast as E goes to E − from above. We refer to [5–7,9,11,20,22] for precise statements. In the present paper, we address the case when V changes sign, i.e., there may exist x+ = x− such that V (x− ) · V (x+ ) < 0.

(0.2)

The basic difficulty this property introduces is that the variations of the potential Vω as a function of ω are not monotonous. In the monotonous case, to get the minimum, one can simply minimize with respect to each of the random variables individually. In the non monotonous case, this uncoupling between the different random variables may fail. Our results concern reflection symmetric potentials since, as we will see, for these potentials we also have an analogous decoupling between the different random variables. Thus, we make the following symmetry assumption on V : (H2) V is reflection symmetric, i.e. for any σ = (σ1 , . . . , σd ) ∈ {0, 1}d and any x = (x1 , . . . , xd ) ∈ Rd , V (x1 , . . . , xd ) = V ((−1)σ1 x1 , . . . , (−1)σd xd ). We now consider the operator HλN = − + λV with Neumann boundary conditions on the cube [−1/2, 1/2]d . Its spectrum is discrete, and we let E − (λ) be its ground state energy. It is a simple eigenvalue and λ → E − (λ) is a real analytic concave function defined on R. We first observe: Proposition 0.1. Under the above assumptions (H1) and (H2), E − = inf(E − (a), E − (b)). For a and b sufficiently small, this result was proven in [17] without the assumption V (x)d x. The method used (H2) but with an additional assumption on the sign of Rd

by Najar relies on a small coupling constant expansion for the infimum of . These ideas were first used in [3] to treat other non-monotonous perturbations, in this case magnetic ones, of the Laplace operator. In [1], the authors study the minimum of the almost sure spectrumfor a random displacement model, i.e. the random potential is defined as Vω (x) = γ ∈Zd V (x − γ − ξγ ), where (ξγ )γ are i.i.d. random variables supported in a sufficiently small compact. We now turn to the results on Lifshitz tails. We denote by N (E) the integrated density of states of Hω , i.e., it is defined by the limit N (E) = lim

L→+∞

N #{eigenvalues of Hω,L ≤ E}

(2L + 1)d

,

(0.3)

N is the operator H restricted to the cube [−L −1/2, L +1/2]d with Neumann where Hω,L ω boundary conditions. This limit exists for a.e. ω and is independent of ω; it has been the object of a lot of studies and we refer to [20,23,24] for extensive reviews.

Spectral Extrema and Lifshitz Tails for Non-Monotonous Alloy Type Models

1135

We first give an upper bound on the integrated density of states. In the applications of the Lifshitz tails asymptotics, in particular, to localization, this side of the bound is the most important and also the difficult one to obtain. Theorem 0.1. Suppose assumptions (H1) and (H2) are satisfied. Assume moreover that E − (a) = E − (b).

(0.4)

Then d log | log N (E)| ≤ − − α+ , log(E − E − ) 2

lim sup + E→E −

(0.5)

where we have set c = a if E − (a) < E − (b) and c = b if E − (a) > E − (b), and log | log P({|c − ω0 | ≤ ε})| 1 ≥ 0. α+ = − lim inf 2 ε→0 log ε As will be clear from the proofs, we could also consider the model Hω = H0 + Vω , where Vω is as above and H0 = − + W, where W is a Zd -periodic potential that satisfies the symmetry assumption (H2). We now study a lower bound for the integrated density of states that we will prove in a more general case than the upper bound, i.e., we don’t need to assume (H2). The assumption is as follows: (HP) there exists ω P ∈ [a, b]Z that is periodic (i.e., for some L 0 ∈ N, for all γ ∈ Zd and β ∈ Zd , ωγP+L 0 β = ωγP ) such that inf = inf σ (Hω P ). d

Under this assumption, we have Theorem 0.2. Let Hω be defined as above, and assume (H1) and (HP) hold. Then −

log | log N (E)| d − α− ≤ lim inf , + log(E − E ) 2 E→E − −

(0.6)

where α− = − lim inf

log | log P({∀γ ∈ Zd /L 0 Zd ; |ωγP − ωγ | ≤ ε})| log ε

ε→0

.

Assume now that (H1) and (H2) hold, and hence, (HP) holds with ω P = a or ω P = b. Indeed, as we will see in the proof of Proposition 0.1 in Sect. 1, under assumption (H2), E − (a) is also the bottom of the spectrum of the periodic operator Ha . If we assume that α+ = α− = 0 and E − (a) = E − (b), we obtain the following corollary: Theorem 0.3. Assume that (H1) and (H2) and (0.4) hold and that α− = α+ = 0, then lim

+ E→E −

d log | log N (E)| =− . log(E − E − ) 2

Combining Theorem 0.1 with the Wegner estimates obtained in [4,10] and the multiscale analysis as developed in [2], we learn

1136


Theorem 0.4. Assume (H1), (H2) and (0.4) hold. Assume, moreover, that the common distribution of the random variables admits an absolutely continuous density. Then, the bottom edge of the spectrum of Hω exhibits complete localization in the sense of [2]. Lifshitz tails have already been proved for various non-monotonous random models, mainly models with random magnetic fields (see e.g. [3,15,18,19]). In the models we consider, we will now see that Lifshitz tails do not always appear. In a companion paper (see [14]), we study the case when E − (a) = E − (b). This requires techniques different from the ones used in the present paper and gets particularly interesting when the random variables are Bernoulli distributed. However, it is quite easy to see that, when E − (a) = E − (b), Lifshitz tails may fail; the density of states can even exhibit a van Hove singularity. Theorem 0.5. There exist potentials V and random variables (ωγ )γ satisfying (H1) and (H2) such that • E − (a) = E − (b) = 0. • There exists C > 0 such that, for E ≥ 0, one has 1 d/2 E ≤ N (E) ≤ C E d/2 . C This paper is constructed as follows. In Sect. 1, we determine the bottom of the almost sure spectrum, and prove Proposition 0.1. In Sect. 2, we prove our main theorems, Theorem 0.2 and Theorem 0.4. Finally, in Sect. 3, we prove Theorem 0.5. 1. Determining the Bottom of the Spectrum N , i.e., − + t V on We denote by t → E − (t) the ground state energy of the operator Ht,0 d [−1/2, 1/2] with Neumann boundary conditions. We first note that E − (t) is a concave function of t as, by the variational principle, it is the infimum of a family of affine functions of t. Hence, for t ∈ [a, b], we have E − (t) ≥ min(E − (a), E − (b)). Then, partitioning Rd into the cubes γ + [−1/2, 1/2]d for γ ∈ Zd , and, restricting Hω to each of these cubes with Neumann boundary conditions, we obtain HωNγ ,0 . Hω ≥ γ ∈Zd

So, we learn Hω ≥ min(E − (a), E − (b)). P , the operator H restricted to the cube [−L − 1/2, L + We let L ≥ 1, and consider Hω,L ω d 1/2] with periodic boundary conditions. Clearly, this operator depends only on finitely many random variables. We prove

Lemma 1.1. =

L≥1 ω admissible

P ), σ (Hω,L

where ω is called admissible if all the components of ω are in the support of the distribution of the random variables defining the alloy type random operator.


1137

This lemma is a variant of a standard characterization of the almost sure spectrum of an alloy type model. To prove Proposition 0.1, i.e., that E − = min(E − (a), E − (b)), it is hence enough to prove that, for any L sufficiently large, inf

d CL

ω∈[a,b]

P inf σ (Hω,L ) ≤ min(E − (a), E − (b)),

(1.1)

where C Ld = Zd ∩ [−L − 1/2, L + 1/2]d . To prove (1.1), we will use the assumption (H.2). For the sake of definiteness, let us assume E − (a) ≤ E − (b). N , say ψ, is simple and can be chosen uniquely as a norThe ground state of Ha,0 malized positive function. The reflection symmetry of the potential V guarantees that ψ is reflection symmetric. For γ ∈ Zd such that |γ | = |γ1 | + · · · + |γd | = 1, we can continue ψ to the γ + [−1/2, 1/2]d by reflection symmetry with respect to the common boundary of [−1/2, 1/2]d and γ + [−1/2, 1/2]d . As ψ is reflection symmetric, we continue this process of reflection with respect to the boundaries of the new cubes to obtain a continuation of ψ that is Zd -periodic, positive and reflection symmetric with respect to any plane that is common boundary to two cubes of the form γ +[−1/2, 1/2]d . P ψ = H P ψ = H N ψ = E (a)ψ. This Moreover, ψ satisfies, for any L ≥ 0, Ha,L − a,0 a,0 P proves that E − (a) ≥ inf σ (Ha,L ). Hence, (1.1) holds. This completes the proof of Proposition 0.1.

Proof of Lemma 1.1. Recall a well known characterization of the almost sure spectrum of an alloy type model in terms of periodic approximations (see e.g. [5,20]). Therefore, for L ≥ 1, define the LZd -periodic operator Hω,L = − + Vω,L , Vω,L (·) =

β∈L Zd

γ ∈Zd /(L Zd )

ωγ V (· − β − γ ).

(1.2)

Then, one has =

σ (Hω,L ),

L≥1 (ωγ )γ ∈Zd /(LZd ) admissible

where (ωγ )γ ∈Zd /(L Zd ) is admissible if all its components belong to the support of the random variables defining the alloy type model. P ) ⊂ σ (H Floquet theory (see e.g. [21]) guarantees that σ (Hω,L ω,L ). So, in order to prove Lemma 1.1, it is sufficient to prove that σ (Hω,L ) ⊂

n≥1

σ (HωPL ,n L )

(1.3)

for some well chosen admissible ω L . Consider ω L defined by ωγL+Lβ = ωγ for γ ∈ Zd /(LZd ) and β ∈ Zd . Clearly, Vω L = Vω,L ; hence, if E n (θ ) are the Floquet eigenvalues of Hω,L , the spectrum of the operator HωPL ,n L is the set {E n (2π γ /n); γ ∈ Zd /(n LZd )} (see e.g.[13]). The inclusion (1.3) follows from the continuity of the Floquet eigenvalues as functions of the Floquet parameter (see e.g Lemma 7.1 in [16]).

1138


2. Lifshitz Tails To fix ideas let us assume E − (a) < E − (b). The two bounds in Theorem 0.1 and Theorem 0.2 will be proved separately. 2.1. The upper bound. The upper bound on the integrated density of states, Theorem 0.1, will be an immediate consequence of the following result. Theorem 2.1. Suppose assumptions (H1) and (H2) are satisfied, and, that E − (a) < E − (b). Then, there exists C > 0 such that, for E ≥ E − (a), one has N (E) ≤ Nm (C(E − E − (a))), where Nm is the integrated density of states of the random operator Hωm = Ha − E − (a) + (ωγ − a)1[−1/2,1/2]d (x − γ )

(2.1)

(2.2)

γ ∈Zd

and Ha is defined above. The upper bound is then deduced from the same bound for the integrated density of states of Hωm which is standard, see e.g. [5,20,22] and references therein. Proof. We first note it is well known that, at E, a continuity point of N (E), the sequence N #{eigenvalues of Hω,L ≤ E} N N L (E) = E (2.3) (2L + 1)d is decreasing and converges to N (E) as L → +∞ (see e.g. [5,20]). So to prove Theorem 2.1, it suffices to prove that there exists C > 0 such that, for E real and L large, N LN (E) ≤ N Lm (C(E − E − (a))), N is replaced by H m , i.e., the restriction where N Lm (E) is defined by (2.3) where Hω,L ω,L of Hωm to [−L − 1/2, L + 1/2]d with Neumann boundary conditions. By the Rayleigh-Ritz principle ([21], Sect. XIII.1), this follows from the quadratic form inequality m N ≤ C(Hω,L − E − (a)). Hω,L

Under our assumptions on V , the form domain of both of these operators is H 1 ([−L − 1/2, L + 1/2]d ). Now, as for ψ ∈ H 1 ([−L − 1/2, L + 1/2]d ) and γ ∈ Zd ∩ [−L − 1/2, L +1/2]d , ψ1γ +[−1/2,1/2]d ∈ H 1 (γ +[−1/2, 1/2]d ), this inequality in turns follows from the inequalities ∀γ ∈ Zd ∩ +[−L − 1/2, L + 1/2]d , ∀ψ ∈ H 1 (γ + [−1/2, 1/2]d ), m N

Hω,L ψ, ψγ +[−1/2,1/2]d ≤ C (Hω,L − E − (a))ψ, ψγ +[−1/2,1/2]d .

Note that, here, the choice of the boundary condition is crucial: the form domain of the Neumann operator is the whole H 1 -space; moreover, the Neumann quadratic form does not involve boundary terms. Taking into account the structure of our random potentials, we see that this will follow from the operator inequality N N − E − (a)) + (t − a) ≤ C(Ht,0 − E − (a)), t ∈ [a, b], (Ha,0

(2.4)

N = − + t V on [−1/2, 1/2]d with Neumann boundary conditions as before. where Ht,0


1139

Lemma 2.1. Let H0 be self-adjoint on H a separable Hilbert space such that 0 = inf σ (H0 ). Let V1 be a closed symmetric operator relatively bounded with respect to H0 with bound 0. Set H1 = H0 + V1 and E 1 = inf σ (H1 ). Assume E 1 > 0. Then, there exists C > 0 such that, for t ∈ [0, 1], one has C(H0 + t V1 ) ≥ H0 + t. Lemma 2.1 applies to our case and, as a result, we obtain (2.4). This completes the proof of Theorem 2.1.

Proof of Lemma 2.1. Regular perturbation theory ensures that, for some β > 1, inf σ (H0 + βV1 ) = δ > 0. Then, for t ∈ [0, 1], we have H0 + t V1 = (1 − t/β)H0 + (t/β)(H0 + βV1 ) 1 ≥ (1 − 1/β)H0 + (t/β)δ ≥ (H0 + t), C where C −1 = min(1 − 1/β, δ/β).

2.2. The lower bound. We will use the techniques set up in [12,16]. We first recall two results from these papers which are valid in the generality of the present work. Consider a random Schrödinger operator of the form (0.1) where (HL1) V is a not identically vanishing, real valued, compactly supported function that is in L p (where p = 2 if d ≤ 3 and p > d/2 if d > 3); (HL2) the random variables (ωγ )γ ∈ are independent, identically distributed, non trivial and bounded. Clearly these two assumptions are consequences of assumption (H1). The existence of N (E), the integrated density of states defined by (0.3) is known ([20,23]). Theorem 2.2 ([12]). Assume (HL1) and (HL2) hold. Pick η0 > 0 and I ⊂ R, a compact interval. Then, there exists ν0 > 0 and ε0 > 0 such that, for 0 < ε < ε0 , E − ∈ I and n ≥ ε−ν0 , we have ν0 −η0

E(Nω,n (E − + ε/2) − Nω,n (E − − ε/2)) − e−(n ε ) ≤ N (E − + ε) − N (E − − ε) ≤ E(Nω,n (E − + 2ε) − Nω,n (E − − 2ε)) + exp −(nεν0 )−η0 ,

(2.5)

where Nω,n is the integrated density of states of the periodic operator Hω,n defined in (1.2). In [12], Theorem 2.2 is not stated in exactly the same form and under slightly stronger but unnecessary assumptions. The modifications necessary to obtain the form given here are simple and left to the reader. The second result we use is the following Lemma 2.2 ([12]). Consider a periodic Schrödinger operator of the form Hω P where ω P satisfies (HP). Let E − = inf σ (Hω P ). Then, for any α ∈ (0, 1) and any ε sufficiently small, there exists a function wε with the following properties:

1140


(1) wε is supported in a ball of center 0 and radius 2ε−(1+2α)/2 ; (2) 1 ≤ wε 2L 2 ≤ 2; (3) (Hω P − E − )wε 2L 2 ≤ Cε2(1+α) for some C > 0 (independent of ε). Though not formulated as a lemma, this result is proved in Sect. 2 of [12]. We now prove the lower bound (0.6). Pick α ∈ (0, 1) arbitrary. Let be the lattice with respect to which ω P is periodic. Pick n ∈ N∗ such that n ∼ ε−ν for ν > ν0 (ν0 given by Theorem 2.2), ν to be chosen sufficiently large. As wε constructed in Lemma 2.2 has compact support in the interior of C(0, n), it can be “periodized” to satisfy quasi-periodic boundary conditions; for θ ∈ Rd , we set wε,θ (·) =

e−iβθ wε (· + β).

β∈(2n+1)

Then, wε,θ satisfies wε,θ (x + β) = eiβθ wε,θ (x) for x ∈ Rd and β ∈ (2n + 1) , and we have wε,θ 2L 2 (C(0,n)) ≥ 1.

(2.6)

We define

α (ε) = γ ∈ ; for 1 ≤ j ≤ d, |γ j | ≤ ε−(1+3α)/2 . Since V satisfies (HL1), if we assume ωγ ≤ ε1+α for γ ∈ α (ε), then Vω,n (Hω P − E − − 1)−1 ≤ Cε1+α . This, (2.6) and point (3) of Lemma 2.2 imply that, for some C > 0 and ε sufficiently small, we have (Hω,n,θ − E − )wε,θ ≤ Cε1+α wε,θ . This proves that, for ε sufficiently small, if ωγ ≤ ε1+α for γ ∈ α (ε), then for all θ ∈ T∗n , Hω,n,θ has an eigenvalue in [E − − ε/2, E − + ε/2]. Hence, we learn that E(Nω,n (E − + ε/2) − Nω,n (E − − ε/2)) ≥

1 −d n P(ε, α), C

(2.7)

where P(ε, α) is the probability of the event {ω; ∀γ ∈ α (ε), ωγ ≤ ε1+α }. By assumption (HL2) and as n ∼ ε−ν , we have lim inf ε→0 ε>0

d log | log(P(ε, α))| ≥ − (1 + 3α) − α− , log ε 2

(2.8)

where α− is defined in Theorem 0.2. Plugging (2.8) and (2.7) into (2.5), since α > 0 is arbitrary, this completes the proof of Theorem 0.2.


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3. An Example where Lifshitz Tails Fail to Exist We now will construct an example of the type described in Theorem 0.5. Let ϕ ∈ C ∞ ((−1/2, 1/2)d ) be positive, reflection symmetric, constant near the boundary of [−1/2, 1/2]d and normalized in the cube. Let V = ϕ/ϕ; it satisfies (0.2) and assumption (H2). Moreover, ϕ is the positive normalized ground state of − + V on [−1/2, 1/2]d with Neumann boundary conditions, the associated ground state energy being 0. Let (ωγ )γ ∈Zd be Bernoulli random variables with support {0, 1}. Pick L ≥ 1 N acting on [−1/2, L − 1/2]d with and let ϕ L be the ground state of the operator Hω,L Neumann boundary conditions defined in Sect. 1. Then, this ground state can be described as follows: • in γ + [−1/2, 1/2]d , ϕ L (·) = L −d/2 ϕ(· − γ ) if ωγ = 1; • in γ + [−1/2, 1/2]d , ϕ L (·) = L −d/2 C0 if ωγ = 0, where the constant C0 is chosen to be equal to the constant value of ϕ near the boundary of [−1/2, 1/2]d . The thus constructed ground state is not normalized. We can now use the results of [8] to obtain a lower bound on the second eigenvalue N that is independent of ω. Indeed, the construction above shows that, for any ω, of Hω,L C0 ≤ L d/2 · min

x∈[−1/2,1/2]d

max

x∈[−1/2,L−1/2]d

ϕ(x) ≤ L d/2 ·

ϕ L (x) ≤

max

x∈[−1/2,1/2]d

min

x∈[−1/2,L−1/2]d

ϕ(x),

ϕ L (x) ≤ C0 .

N and the Neumann Laplacian on the same cube Then, Theorem 1.4 of [8] applied to Hω,L N is larger than cL −2 , where the constant guarantees that the second eigenvalue of Hω,L c does not depend on ω, nor on L. The standard upper bound for the integrated density of states by the normalized Neumann counting function (see e.g. [5,20,22]) yields, for any L ≥ 1,

N (E) ≤ E

N #{eigenvalues of Hω,L ≤ E}

Ld

.

If we pick L = (C 2 E)−1/2 for some C > 0 such that cC 2 ≥ 2, the second eigenvalue N is larger than 2E; thus for any realization of ω, we obtain of Hω,L N (E) ≤ L −d = C d E d/2 . The standard lower bound for the integrated density of states by the normalized Dirichlet counting function (see e.g. [5,20,22]) yields E

D #{eigenvalues of Hω,L ≤ E}

Ld

≤ N (E),

(3.1)

D is the Dirichlet restriction of H to [−1/2, L −1/2]d . Let ψ be the (positive where Hω,L ω L normalized) ground state of the Dirichlet Laplacian on [−1/2, L − 1/2]d . And define

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φ L = ψ L · ϕ L . This smooth function clearly satisfies Dirichlet boundary conditions and one computes, for some C > 1, D

Hω,L φ L , φ L = (− + Vω )φ L , φ L = (−ψ L )ϕ L , φ L − 2 ∇ψ L · ∇ϕ L , φ L C π2 = 2 2 φ L 2 + ϕ L |∇ψ L | 2 ≤ 2 φ L 2 , L L

(3.2)

where, in the final step, we integrated by parts and used the explicit form of the positive normalized ground state of the Dirichlet Laplacian. Hence, taking L = C E −1/2 , as C > 1, (3.2) ensures that the ground state energy of D Hω,L is less than E. Thus, from (3.1), we learn 1 d/2 E = L −d ≤ N (E). Cd Finally, combining the two estimates, we have proved that, for the above random model, the density of states exhibits a van Hove singularity at the bottom of the spectrum, that is, there exists C > 1 such that, for E ≥ 0, one has 1 d/2 E ≤ N (E) ≤ C E d/2 . C This completes the proof of Theorem 0.5.

Acknowledgement. FK would like to thank the University of Tokyo where part of this work was done. A part of this research was done when SN was invited to Univ. Paris 13 in 2007, and he would like to express thanks. SN is partially supported by JSPS Research Grant, Kiban (B) 17340033.

References 1. Baker, J., Loss, M., Stolz, G.: Minimizing the ground state energy of an electron in a randomly deformed lattice. Commun. Math. Phys. 283(2), 397–415 (2008). http://arxiv.org/abs/0707.3988vI[math-ph] 2. Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222(2), 415–448 (2001) 3. Ghribi, F.: Internal Lifshits tails for random magnetic Schrödinger operators. J. Funct. Anal. 248(2), 387– 427 (2007) 4. Hislop, P., Klopp, F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195, 12–47 (2002) 5. Kirsch, W.: Random Schrödinger operators. In: A. Jensen, H. Holden, eds, Schrödinger Operators, Sonderborg, Denmark 1988, Number 345 in Lecture Notes in Physics, Berlin: Springer-Verlag, 1989 6. Kirsch, W., Martinelli, F.: Large deviations and Lifshitz singularities of the integrated density of states of random hamiltonians. Commun. Math. Phys. 89, 27–40 (1983) 7. Kirsch, W.: Random Schrödinger operators and the density of states. In: Stochastic aspects of classical and quantum systems (Marseille, 1983), Volume 1109 of Lecture Notes in Math., Berlin: Springer, 1985, pp. 68–102 8. Kirsch, W., Simon, B.: Comparison theorems for the gap of Schrödinger operators. J. Funct. Anal. 75(2), 396–410 (1987) 9. Kirsch, W., Stollmann, P., Stolz, G.: Localization for random perturbations of periodic Schrödinger operators. Random Oper. Stochastic Eqs. 6(3), 241–268 (1998) 10. Klopp, F.: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167, 553–570 (1995) 11. Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators. Duke Math. J. 98(2), 335–396 (1999)


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12. Klopp, F.: Internal Lifshitz tails for Schrödinger operators with random potentials. J. Math. Phys. 43(6), 2948–2958 (2002) 13. Klopp, F.: Weak disorder localization and Lifshitz tails: continuous Hamiltonians. Ann. Henri Poincaré. 3(4), 711–737 (2002) 14. Klopp, F., Nakamura S.: In progress 15. Klopp, F., Nakamura, S., Nakano, F., Nomura, Y.: Anderson localization for 2D discrete Schrödinger operators with random magnetic fields. Ann. Henri Poincaré. 4(4), 795–811 (2003) 16. Klopp, F., Wolff, T.: Lifshitz tails for 2-dimensional random Schrödinger operators. J. Anal. Math. 88, 63–147 (2002) 17. Najar, H.: The spectrum minimum for random Schrödinger operators with indefinite sign potentials. J. Math. Phys. 47(1), 013515 (2006) 18. Nakamura, S.: Lifshitz tail for 2D discrete Schrödinger operator with random magnetic field. Ann. Henri Poincaré 1(5), 823–835 (2000) 19. Nakamura, S.: Lifshitz tail for Schrödinger operator with random magnetic field. Commun. Math. Phys. 214(3), 565–572 (2000) 20. Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. Volume 297 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Berlin: SpringerVerlag, 1992 21. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1978 22. Stollmann, P.: Caught by disorder. Volume 20 of Progress in Mathematical Physics. Boston, MA: Birkhäuser Boston Inc., 2001 23. Veselić, I.: Integrated density of states and Wegner estimates for random Schrödinger operators. In: Spectral theory of Schrödinger operators, Volume 340 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2004, pp 97–183 24. Veselić, I.: Existence and regularity properties of the integrated density of states of random Schrödinger operators. Lecture Notes in Math., 1917. Berlin: Springer-Verlag, 2008 Communicated by B. Simon


Communications in


A Construction of Frobenius Manifolds with Logarithmic Poles and Applications Thomas Reichelt Institut für Mathematik, Universität Mannheim, Mannheim, Germany. E-mail: [email protected] Received: 27 April 2008 / Accepted: 25 August 2008 Published online: 16 December 2008 – © Springer-Verlag 2008

Abstract: A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a partial generalization of the reconstruction theorem of Kontsevich and Manin for projective smooth varieties with convergent Gromov-Witten potential. A second application is a construction of Frobenius manifolds out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled.

Introduction In this paper we define the notion of a Frobenius manifold with logarithmic poles along a normal crossing divisor (logarithmic Frobenius manifold for short). Let M be a complex manifold and D a normal crossing divisor, then a logarithmic Frobenius manifold is a Frobenius manifold on M\D such that the identity field e and the Euler field E are logarithmic vector fields along D and the metric g and multiplication ◦ extend holomorphically with respect to logarithmic vector fields and the metric is nondegenerate with respect to them. If one restricts a logarithmic Frobenius manifold to an appropriate submanifold N one gets a logarithmic Frobenius type structure whose main ingredients are a vector bundle K := T M | N , a flat connection ∇ r , a Higgs field C which both have logarithmic poles along D ∩ N and a vector bundle endomorphism U which comes from multiplication with the Euler field. If some generation conditions are satisfied by the Higgs field and the endomorphism U the logarithmic Frobenius manifold can be uniquely reconstructed from the logarithmic Frobenius type structure (Theorem 1.12). This is a generalization of the corresponding theorem 4.5 of [HM]. The theorem of [HM] in turn generalizes a theorem of Malgrange [Mal] which treats the case N = { pt} and a result by Dubrovin [Du] which shows that a finite number of numbers suffices to reconstruct the whole

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Frobenius germ (M, 0) in the case N = { pt}, the multiplication ◦ being semisimple and U = E◦ having a simple spectrum. We give two applications of this unfolding result. The first is an application to quantum cohomology. We regard the even dimensional ∗ (X, C) as a manifold with coordinates t with respect to cohomology ring V = Heven i a homogeneous basis {Ti } and metric gi j = X Ti ∪ T j . The third derivatives of the Gromov-Witten potential provide us with the structure constants of a multiplication on the tangent bundle of V . As is well known these data give us a Frobenius manifold [Ma]. Let T1 , . . . , Tr be a basis of H 2 (X, C). To construct a logarithmic Frobenius manifold we do a coordinate change ϕ : ti → qi = eti for i = 1, . . . , r , where the other coordinates remain unchanged. The third derivatives of the Gromov-Witten potential are well defined on the image. We assume that they give holomorphic functions on a polydisc M around the large radius limit point p and degenerate at this point so that the corresponding multiplication becomes the cup product. The set {qi = 0} gives rise to a normal crossing divisor D. We show that the Frobenius manifold given on M\D has the demanded degeneration properties so that we retrieve a Frobenius manifold with loga∗ (X, C) be a minimal subspace rithmic poles along D on M. Now let W ⊂ V = Heven ∗ which generates Heven (X, C) with respect to the cup product. Then N = ϕ(W ) ∩ M is a submanifold of M and carries a logarithmic Frobenius type structure. We will use Theorem 1.12 to reconstruct the Frobenius manifold. In terms of Gromov-Witten invariants this means that they can be uniquely reconstructed from their values on the linear subspace V 2 × W l for l ≥ 1. In the case W = H 2 we recover the first reconstruction theorem of Kontsevich and Manin ([KM]) in the case of smooth projective varieties with h 2,0 = 0 and holomorphic Gromov-Witten potential. The second concerns variation of polarized Hodge structures. Here we prove that one can construct a logarithmic Frobenius type structure out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor D. If a certain condition called H 2 -generation holds, this logarithmic Frobenius type structure can be unfolded to give a logarithmic Frobenius manifold. The construction goes as follows: With a VPHS comes the associated Gauss-Manin connection ∇. One shows that the cohomology bundle decomposes into a direct sum of subbundles using the Hodge filtration F • and an opposite, ∇-flat filtration U• which is constructed out of the limiting PMHS. Because ∇ satisfies Griffiths-transversality one can show that ∇ maps O(F p ∩ U p ) to O(F p ∩ U p ) ⊕ O(F p−1 ∩ U p−1 ). Therefore one can split ∇ into a connection ∇ r and a Higgs field C. This Higgs field together with the polarization form will provide us with a logarithmic Frobenius type structure. This section is inspired by a paper of P. Deligne [De2]. Related to this section are papers of E. Cattani and J. Fernandez [CaFe] and of J. Fernandez and G. Pearlstein [FePe]. In the first they prove a correspondence between deformations of framed Frobenius modules and VPHS on (∗ )r with a limiting MHS of Hodge-Tate type and split over R. In the second paper they prove in the cases of weight k = 3, 4, 5 that a deformation of framed Frobenius modules with certain generation properties gives rise to a Frobenius manifold. From the results given here this follows for any weight k. 1. Unfoldings of Meromorphic Connections In this chapter we define Frobenius manifolds with logarithmic poles (Def. 1.4) and logarithmic Frobenius type structures (Def. 1.6). In Proposition 1.7 it is stated that every logarithmic Frobenius manifold gives rise to a logarithmic Frobenius type structure.

A Construction of Frobenius Manifolds

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To prove the construction theorem which is the primary goal of this chapter another structure has to be established, namely that of a (log D − tr TLEP (w))-structure (Def. 1.8). In Proposition 1.10 we show that there is a correspondence between logarithmic Frobenius type structures and (log D −tr TLEP (w))-structures. The latter has the advantage that the objects coming from a logarithmic Frobenius type structure are efficiently encoded in a vector bundle H over P1 × M with a flat connection ∇ and a pairing P. The connection is defined on C∗ × M\D having a pole of Poincare´ rank 1 along {0} × M and a logarithmic pole along {∞} × M ∪ P1 \{0} × D. In the case where the (log D − tr T L E P(w))-structure comes from a logarithmic Frobenius manifold the pair (H, ∇) is a natural generalization of the first structure connection of a Frobenius manifold (see e.g. [Du]). These structures enable us to prove the construction theorem for Frobenius manifolds with logarithmic poles by showing the existence of a universal unfolding of a (log D − tr T L E P(w))-structure. 1.1. Definitions. 1.1.1. Elementary sections. For the convenience of the reader we recall some definitions from [He1]. Let := {z ∈ C | | z |< 1} and ∗ := \{0}. Fix a holomorphic vector bundle H → (∗ )r × m−r of rank µ ≥ 1 with a flat connection ∇ on (∗ )r × m−r . We want to discuss special sections in H and extensions on r × m−r of the sheaf H of its holomorphic sections. Let Ti be the monodromy with respect to {z i = 0}. The monodromy determines the bundle uniquely up to isomorphism. Let Ti = Ti,s · Ti,u be the decomposition into semisimple and unipotent parts, N (i) := log(Ti,u ) the nilpotent part. A universal covering is e : Hr × m−r −→ (∗ )r × m−r , (ξ1 , . . . , ξr , zr +1 , . . . , z m ) → (e2πiξ1 , . . . , e2πiξr , zr +1 , . . . , z m ). The space of multivalued flat sections H ∞ is defined as H ∞ = { pr ◦ A : Hr × m−r → H | A is a global flat section of e∗ H }. All monodromies act on it. The indices of the simultaneous generalized eigenspace decomposition H ∞ = λ Hλ∞ can be considered as tuples λ = (λ1 , . . . , λr ) of eigenvalues for the monodromies T1 , . . . , Tr . ( j) Now choose A ∈ Hλ∞ and α = (α (1) , . . . , α (r ) ) with e−2πiα = λ j . The map Hr × m−r −→ H, r ξ = (ξ1 , . . . , ξr , zr +1 , . . . , z m ) → exp(2πi α ( j) ξ j ) exp(−ξ j N ( j) ) A(ξ ) j=1

is invariant with respect to any shift ξ j → ξ j + 1 and therefore induces a holomorphic section es(A, α) : (∗ )r × m−r → H, r exp(2πi α ( j) ξ j ) exp(−ξ j N ( j) ) A(ξ ) z = (z 1 , . . . , z m ) → j=1

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of the bundle H (with z = e(ξ )). It is called an elementary section and is usually denoted informally as z →

r

( j)

α ( j) − N2πi

zj

A.

j=1 ( j) α ( j) − N2πi It is nowhere vanishing if A = 0 because the twist with rj=1 z j is invertible. We now want to investigate extensions of H by elementary sections. A Zr -filtration (Pl )l∈Zr consists of subspaces Pl ⊂ H ∞ which are invariant with respect to all monodromies T1 , . . . , Tr and which satisfy for a suitable m > 0,

Pl = 0 Pl = H ∞ Pl ⊂ Pl Then Pl =

λ

if l j ≤ −m for some j, if l j ≥ m for all j, if l j ≤ l j for all j.

Pl,λ . A Zr -filtration (Pl ) induces r increasing exhaustive monodromy

invariant filtrations F• on H ∞ by j

j

F p :=

Pl .

l j=p

For λ j an eigenvalue of T j let α j be defined as e−2πiα j = λ j and −1 < Re α j ≤ 0. Set M := r × m−r . Proposition 1.1. Fix a Zr -filtration (Pl )l∈Zr . i of P . The sheaf 1. For any l ∈ ([−m, m] ∩ Z)r and any λ, choose generators Al,λ l,λ i L := O M · es(Al,λ , α + l) these l λ

i

is an O M -coherent extension of H with logarithmic pole along D. 2. The formula above yields a one-one correspondence between Zr -filtrations and O M -coherent extensions of H with logarithmic pole along D. 3. The sheaf L is a locally free O M -module if and only if Pl =

r

j

Fl j for all l

j=1 j

holds and the filtrations F• have a common splitting. Proof. See [EV], Appendix C and [He1], Chap. 8.3. Remark 1.2. A special case of Proposition 1.1 is the case when the filtrations on the simulj j taneous generalized eigenspaces Hλ∞ are all of type 0 = F p( j,λ)−1 ⊂ F p( j,λ) = Hλ∞ for some numbers p( j, λ) ∈ Z. The O M -locally free extension of H to M for the case when the real parts of all eigenvalues of all residue endomorphisms are contained in [0, 1) was called the canonical extension by Deligne.


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1.1.2. (Logarithmic) Frobenius manifolds. Before we can delve into logarithmic Frobenius manifolds, we state the definition of a Frobenius manifold. Let M be a complex manifold and TM its (holomorphic) tangent bundle. In this context a metric g on TM is a symmetric, C-bilinear, nondegenerate, holomorphic pairing on the fibers of the holomorphic bundle TM and a multiplication ◦ is a C-bilinear, holomorphic, commutative and associative multiplication on the fibers of TM. Definition 1.3 ([Du]). A Frobenius manifold is a tuple (M, ◦, e, E, g), where M is a manifold of dimension n ≥ 1 with metric g and multiplication ◦ on the tangent bundle, e is a global unit field and E is another global vector field, subject to the following conditions: 1. the metric is multiplication invariant, g(X ◦ Y, Z ) = g(X, Y ◦ Z ), 2. (potentiality) the (3,1)-tensor ∇◦ is symmetric (here ∇ is the Levi-Civita connection of the metric), 3. the connection ∇ is flat, 4. the unit field e is flat, ∇e = 0, 5. the Euler field E satisfies Lie E (◦) = 1 · ◦ and Lie E (g) = (2 − d) · g for some d ∈ C. Now we can give the definition of a logarithmic Frobenius manifold. Definition 1.4. Let M be a complex manifold and D a normal crossing divisor. Let M\D be a Frobenius manifold with e, E ∈ (M, Der M (log D)), ◦ ∈ (M, Der M (log D) ⊗ (A1M (log D) ⊗ S A1M (log D))), g ∈ (M, (A1M (log D) ⊗ S A1M (log D))), where g is nondegenerate on Der M (log D). Then M is a Frobenius manifold with logarithmic poles along D (logarithmic Frobenius manifold for short). Proposition 1.5. The induced Levi-Civita connection on M\D with respect to g has a logarithmic pole along D. Proof. Just compute the Christoffel symbols in aneighborhood U of the divisor with local coordinates (t1 , . . . , tn ) such that D ∩ U = ri=1 {ti = 0}. Define U := E◦ : Der M (log D) → Der M (log D) and V : Der M (log D) → Der M (log D) 2−d X → ∇ X E − X. 2

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1.2. Frobenius type structures with logarithmic poles. In this section we define Frobenius type structures with logarithmic poles. The advantage of these structures is that restricted to any submanifold they remain Frobenius type structures with logarithmic poles. Frobenius manifolds do not have this property in general because the multiplication ◦ is defined on the tangent bundle TM of M. Definition 1.6. A structure of Frobenius type with logarithmic poles along D is a tuple (K → (M, 0), D, ∇ r , C, U, V, g) with (M, 0) is a germ of a complex manifold, D is a normal crossing divisor with 0 ∈ D locally given by D = {z 1 · · · zr }, K −→ (M, 0) germ of a holomorphic vector bundle, ∇ r flat connection with logarithmic pole along D, C : O(K ) −→ O(K ) ⊗ A1M (log D) logarithmic Higgs field, U : O(K ) −→ O(K ) with [ U, C X ] = 0 ∀X ∈ Der M (log D), V : O(K ) −→ O(K ) ∇ r - flat, g : O(K ) × O(K ) −→ O M,0 symmetric, non-degenerate, ∇ r - flat, such that these data satisfy ∇ rX (CY ) − ∇Yr (C X ) − C[X,Y ] = 0, ∇ r (U) − [C, V] + C = 0, g(C X a, b) = g(a, C X b), g(Ua, b) = g(a, Ub), g(Va, b) = −g(a, Vb).

(1.1) (1.2) (1.3) (1.4) (1.5)

Proposition 1.7. Every Frobenius manifold with logarithmic poles gives rise to a Frobenius type structure with logarithmic poles. Proof. The proof is essentially the same as in [He2], Lemma 5.11, so we just define the objects: K := Der M (log D), ∇ r := ∇ g , U := E◦ : Der M (log D) −→ Der M (log D),

2−d X, 2 C : Der M (log D) −→ Der M (log D) ⊗ A1M (log D) , C X Y := −X ◦ Y.

V : Der M (log D) −→ Der M (log D) :=

X → ∇ X E −

1.3. Definition of (logD-trTLEP(w))-structures. Let (M, 0) be a germ of a complex manifold and D be a normal crossing divisor and let z be a coordinate on P1 . Definition 1.8. Fix w ∈ Z. A (log D − tr T L E P(w))-structure is a tuple ((M, 0), D, H, ∇, P) with H −→ P1 × (M, 0) a trivial holomorphic vector bundle with a flat connection on H|C∗ ×(M\D,0) with ∇ : O(H )(0,0) −→

1 1 (log Z 0 ) ⊗ O(H )(0,0) , z C×M,0


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where (Z 0 , (0, 0)) := ({0} × M ∪ C × D, (0, 0)) ⊂ (C × M, (0, 0)) and ∇ has a logarithmic pole along {∞} × (M, 0) ∪ (P1 \{0}) × (D, 0) ⊂ (P1 \{0}) × (M, 0). P is a (−1)w -symmetric, nondegenerate, ∇-flat pairing P : H(z,t) × H(−z,t) −→ C for (z, t) ∈ C∗ × (M\D, 0) on a representative of H such that the pairing extends to a nondegenerate, z-sesquilinear pairing P : O(H ) × O(H ) −→ z w OP1 ×(M,0) . If ((M, 0), D, H, ∇, P) is a (log D − tr T L E P(w))-structure and ϕ : (M , D , 0) → (M, D, 0) a holomorphic map of germs of manifolds then one can pullback H , ∇ and P with id × ϕ : (C, 0) × (M , 0) → (C, 0) × (M, 0). One easily sees that ϕ ∗ (H, ∇) and ϕ ∗ (P) gives a (log D − tr T L E P(w))-structure on (M , D , 0). Definition 1.9. Fix a (log D − tr T L E P(w))-structure ((M, 0), D, H, ∇, P). (a) An unfolding of it is a (log D − tr T L E P(w))-structure ((M × Cl , 0), D × Cl , H˜ , ˜ together with a fixed isomorphism ˜ P) ∇, ˜ H˜ , ∇, ˜ |(M×{0},0) , ˜ P) i : ((M, 0), D, H, ∇, P) → ((M × Cl , 0), D, ˜ where we denote D × Cl as D. l ˜ i) induces as second unfolding ˜ P, (b) One unfolding ((M × C , 0), D × Cl , H˜ , ∇, l ˜ ˜ ˜ ˜ ((M × C , 0), D , H , ∇ , P , i ) if there is an isomorphism j from the second unfolding to the pullback of the first unfolding by a map ˜ ϕ : ((M × Cl , 0), D˜ ) → ((M × Cl , 0), D),

which is the identity on (M × {0}, 0), and if i = j |(M×{0},0) ◦i . (c) An unfolding is universal if it induces any unfolding via a unique map ϕ. 1.4. A Correspondence. Proposition 1.10. There is a 1-1 correspondence between (log D − tr T L E P(w))structures and Frobenius type structures with logarithmic pole. Proof. Logarithmic Frobenius type structure → (logD-trTLEP(w))-structure. Let (K → (M, 0), D, ∇ r , C, U, V, g) be a Frobenius type structure with logarithmic pole. Set H := π ∗ K and let ϕz : H(z,t) → K t be the canonical projection. Extend ∇ r , C, U, V, g canonically on H. Define 1 w dz 1 r ∇ := ∇ + C + U − V + id z z 2 z and P : H(z,t) × H(−z,t) → C for (z, t) ∈ (C∗ , 0) × (M, 0), (a, b) → z w g(ϕz a, ϕ−z b). We have to show that (H, D, ∇, P) is a (log D − tr T L E P(w))-structure.

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H = π ∗ K is trivial because K is trivial. ∇ 2 = 0 results from a quick calculation. P is (−1)w -symmetric and nondegenerate because g is symmetric and nondegenerate. Let a, b ∈ O(H ). P is ∇-flat: 1 dP(a, b) = wz w−1 · g(a, b) + z w · dg(a, b), 1 w dz 1 w r U − V + id a, b , ∇ + C+ P(∇a, b) = z g z z 2 z 1 w dz 1 P(a, ∇b) = z w g a, ∇ r − C + − U − V + id b z z 2 z ⇒ ∇(P)(a, b) = d P(a, b) − P(∇a, b) − P(a, ∇b) = 0.

(1.6)

Because of the definition of ∇ above one sees easily that it has the correct pole order and that therefore (H, D, ∇, P) is a (log D − tr T L E P(w))-structure. (logD-trTLEP(w))-structure → Logarithmic Frobenius type structure. Let a (log D − tr T L E P(w))-structure ((M, 0), D, H, ∇, P) be given. Define K := H|{0}×(M,0) . Observe that K and H|{∞}×(M,0) are canonically isomorphic. The residual connection ∇ r es on H|{∞}×(M,0) is given by ∇ rXes [a] := [∇ X a] with X ∈ Der M (log D), a ∈ O(H ). The residue endomorphism V r es on H|{∞}×(M,0) is given by V r es [a] := [∇z˜ ∂z˜ a] a ∈ O(H ). Because of the canonical isomorphism these structures can be shifted to K . Let ∇ r , V be the shift of ∇ r es , V r es + w2 id respectively. Furthermore we set C := [z∇] : O(K ) −→ O(K ) ⊗ A1M (log D), C X [a] := [z∇ X a], U := [z∇z∂z ] : O(K ) −→ O(K ), g([a], [b]) := z −w P(a, b) mod zOC×M,0 . ∇ r is flat because ∇ is flat and has a logarithmic pole along D because ∇ has a logarithmic pole along C∗ × D. It follows from easy calculations that C is a logarithmic Higgs field, [U, C X ] = 0 and ∇ r (V) = 0. g is O M,0 -bilinear, symmetric and nondegenerate because P is OC×M z-sesquilinear, (−1)w -symmetric and nondegenerate. The conditions (1.3), (1.4), (1.5) and ∇ r (g) = 0 follow from ∇(P) = 0. Lift ∇ r , C, U, V canonically to H . Consider 1 w dz 1 r U − V + id . (1.7) := ∇ − ∇ + C + z z 2 z is OP1 ×M -linear. X for X ∈ π −1 Der M (log D) maps sections in π −1 π∗ O(H ) to sections with no pole along {0} × (M, 0) because of the definition of C and which vanish along {∞} × (M, 0) because of the definition of ∇ r . Therefore X = 0. The same holds for ∇z∂z , using the definitions of U and V. This shows = 0. 1 For better readability we suppress ϕ . ±z


1153

We show condition (1.1). Let a ∈ π −1 π∗ O(H ). ∇ X ∇Y − ∇Y ∇ X − ∇[X,Y ] = 0 holds with ∇ X = ∇ rX + 1z C X : 1 1 ∇ X ∇Y a = ∇ rX ∇Yr a + ∇ rX CY a + C X ∇Yr a + z z 1 1 ∇Y ∇ X a = ∇Yr ∇ rX a + ∇Yr C X a + CY ∇ rX a + z z 1 r ∇[X,Y ] a = ∇[X,Y ] a + C[X,Y ] a, z 1 r r ⇒ (∇ X (CY ) − ∇Y (C X ) − C[X,Y ] )a = 0. z

1 C X CY a, z2 1 CY C X a, z2

Condition 1.2 follows from a similar calculation. 1.5. The isomorphism case. In this section we prove a result which shows that if the Higgs field of a logarithmic Frobenius type structure provides an isomorphism between the fiber over 0 and the fiber over 0 of the tangent bundle and if a certain eigenvector condition is satisfied, then we can construct a logarithmic Frobenius manifold out of the logarithmic Frobenius type structure. This is a cornerstone for the general result which we prove in the next section. Proposition 1.11. Let (K → (M, 0), D, ∇ r , C, U, V, g) be a Frobenius type structure with logarithmic pole along D. Let ξ ∈ O(K ) be a section which is flat on M\D such that C : Der M (log D)0 −→ K 0 , X → −C X ξ |0 and Vξ = d2 ξ .

is an isomorphism

Then the Frobenius type structure corresponds to a germ of a Frobenius manifold with logarithmic pole along D ((M,0), D, ∇ g , ◦, e, E, g). ˜ Notice that, unlike in the corresponding result of [HM] where the vector ξ|0 could be extended to a flat section, we have to demand the existence of such a section ξ which is flat on M\D. The section ξ corresponds to the global unit field e under the isomorphism given above. Proof. We have an isomorphism v : Der M (log D) −→ O(K ) , so we can pullback the data of the Frobenius type structure ∇ g := v ∗ (∇ r ), g˜ := v ∗ (g), e := v −1 (ξ ),

E := v −1 (Uξ ).

∇ g is flat with ∇ g g˜ = 0 and ∇ g e = 0 and has a logarithmic pole along D. Torsion freeness follows from condition (1.1). Define the multiplication on Der M (log D), v(X ◦ Y ) := −C X v(Y ) = C X CY ξ .

1154

T. Reichelt

The potentiality follows easily from: 0 = ∇ rX (CY ) − ∇Yr (C X ) − C[X,Y ] g

g

⇔ −∇ X (Y ◦) + ∇Y (X ◦) + [X, Y ]◦ = 0 g

g

g

g

⇒ −∇ X (Y ◦ Z ) + Y ◦ ∇ X (Z ) + ∇Y (X ◦ Z ) − X ◦ ∇Y Z + [X, Y ] ◦ Z = 0, but this is equivalent to ∇ g (◦) symmetric (see [He1]). We denote the shift of V, U, C on Der M (log D) with the same letters. We want to show V = ∇•g E −

2−d 2−d g id = ∇ E − Lie E − id . 2 2

We need the following identities for the next calculation:

1 w ∇z∂z v(Y ) = U − V + id v(Y ) z 2 w 1 = v(E ◦ Y ) − v V − id Y , z 2 1 1 g ∇ X v(e) = v(∇ X e) − v(X ) = − v(X ). z z Therefore we have g

v(∇ X E) = ∇ rX v(E) 1 1 = ∇ X v(E) − C X v(E) = ∇ X Uv(e) + Uv(X ) z z d −w w e + ∇z∂z v(X ) + v V − id X = ∇ X z ∇z∂z v(e) + v 2 2 d −w w X + ∇z∂z v(X ) + v V − id X = −∇z∂z v(X ) + v(X ) − v 2 2 2−d id X =v V+ 2 2−d g id. ⇒ Lie E = ∇ E − V − 2 Now we can compute Lie E (g) and Lie E (◦): Lie E (g)(X, Y ) = Eg(X, Y ) − g(Lie E X, Y ) − g(X, Lie E Y ) g g = Eg(X, Y ) − g(∇ E X, Y ) − g(X, ∇ E Y ) + g(V X, Y ) + g(X, VY ) 2−d 2−d +g X, Y + g X, Y 2 2 = (2 − d)g(X, Y ).


1155 g

g

For the next calculation we are using ∇ X (CY ) − ∇Y (C X ) − C[X,Y ] = 0, ∇(U) − [C, V] + C = 0 and U = −C E : C Lie E (◦)(X,Y )−X ◦Y = C[E,X ◦Y ] − C X ◦[E,Y ] − C[E,X ]◦Y − C X ◦Y g

g

= ∇ E (C X ◦Y ) − ∇ X ◦Y (C E ) + C X C[E,Y ] + C[E,X ] CY + C X CY g

g

g

g

= −∇ E (C X CY ) − ∇ X ◦Y (C E ) + C X (∇ E (CY ) − ∇Y (C E )) g g + (∇ E (C X ) − ∇ X (C E ))CY + C X CY g g g = −∇ X ◦Y (C E ) − C X ∇Y (C E ) − ∇ X (C E )CY

+ C X CY = [C X ◦Y , V] − C X ◦Y + C X [CY , V] − C X CY + [C X , V]CY − C X CY + C X CY = 0. 1.6. The general case. Theorem 1.12. Let ((M, 0), D, K , ∇ r , C, U, V, g) be a Frobenius type structure with logarithmic pole along D, ξ ∈ O(K ) and ∇ r (ξ|M\D ) = 0 with (IC) C• ξ |0 : Der M (log D)0 −→ K 0 injective, (GC) ξ |0 and its images under iteration of the maps C X , X ∈ Der M (log D)0 and U : K 0 −→ K 0 generate K 0 , (EC) Vξ = d2 ξ for d ∈ C . ˜ 0), D, ˜ ◦, e, E, g, Then up to canonical isomorphism there exist unique data (( M, ˜ i, j) ˜ 0), D, ˜ ◦, e, E, g) ˜ i : with (( M, ˜ a Frobenius manifold with logarithmic pole along D, ˜ ˜ (M, 0) → ( M, 0) is an embedding such that i(M) ∩ D = i(D) and j : K → ˜ |i(M) is an isomorphism above i of germs of vector bundles which identifies Der M˜ (log D) the logarithmic Frobenius type structure on K with the natural logarithmic Frobenius ˜ |i(M) . type structure on Der M˜ (log D) Here we also have to demand the existence of a section ξ which is flat on M\D and fulfills (I C), (GC) and (EC) unlike in [HM] where they could extend a vector ξ to a flat section on M. First we give an overview of the proof. (A) Proposition 1.10 is used to get a (log D − tr TLEP (w))-structure. Then we prove Lemma 1.13 below. This lemma together with Lemma 1.15 gives us an unfolding of the (log D − tr T L E P(w))-structure which fulfills property (1.8) below for given f 1 , . . . , f n ∈ O M×Cl . (B) We investigate transformation properties of elementary sections with respect to pullback in the context of Lemma 1.13. This is needed in order to prove uniqueness of the constructed Frobenius manifold. (C) We choose an unfolding of the given (log D − tr T L E P(w))-structure so that the map ˜ −→ K 0 C˜• ξ |0 : Der M˜ (log D)

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T. Reichelt

is an isomorphism which is possible by Lemma 1.13. We prove that an unfolding with this property is a universal unfolding. But because two universal unfoldings are isomorphic, this shows that the property above characterizes the unfolding up to isomorphism. Now an application of Proposition 1.11 shows that this unfolding gives rise to a Frobenius manifold which is therefore also unique up to isomorphism. (A) Let ((M, 0), D, H, ∇, P) be a (log D − tr T L E P(w))-structure. We want to choose global sections (v1 , . . . , vn ) which are a basis of H so that the connection matrix has a convenient form for proving the existence of an unfolding. Let ∇ r es be the residue connection on H|{∞}×(M\D,0) . We choose the sections so that their restrictions to {∞} × (M, 0) are elementary sections with respect to the residue connection. We call such a choice of sections adapted to ((M, 0), D, H, ∇, P). Now let ((M × ˜ be an unfolding of the above structure. Then there exist unique Cl , 0), D × Cl , H˜ , ∇) ˜ so that (v˜1 |P1 ×M×{0} adapted sections (v˜1 , . . . , vñ ) of ((M × Cl , 0), D × Cl , H˜ , ∇) , . . . , vñ |P1 ×M×{0} ) = (v1 , . . . , vn ). We call these sections a canonical extension of (v1 , . . . , vn ). Recall the proof of Proposition 1.10. There we observed that K = H|{0}×(M,0) is canonically isomorphic to H|{∞}×(M,0) . Therefore it is possible to shift structure from one bundle to the other. This is the case for the residue connection ∇ r es which when shifted to K can be identified with ∇ r . Consider a global section v of H . Then its restriction to {0} × (M, 0) is ∇ r -flat if and only if its restriction to {∞} × (M, 0) is ∇ r es -flat. Lemma 1.13. Let w ∈ Z and ((M, 0), D, H, ∇, P) be the (log D − tr T L E P(w))structure which fulfills the conditions above. Choose adapted sections (v1 , . . . , vn ) of H such that v1|{0}×(M,0) = ξ . Choose l ∈ N and n functions f 1 , . . . , f n ∈ O(M×Cl ,0) with f i |(M×{0},0) = 0. Let (t1 , . . . tm , y1 , . . . , yl ) = (t, y) be coordinates on (M × Cl , 0). ˜ of (H, ∇) with Then there exists a unique unfolding ( H˜ → P1 × (M × Cl , 0), ∇) the following properties. H˜ is a trivial vector bundle with flat connection over (C∗ × ((M\D)×Cl , 0)) with logarithmic poles along {∞}×(M ×Cl , 0)∪P1 \{0}×(D×Cl , 0) and a pole of Poincaré rank 1 along {0} × (M × Cl , 0). Its restriction to P1 × (M × {0}, 0) is (H, ∇). Let (v˜1 , . . . , vñ ) be the canonical extensions of (v1 , . . . , vn ). Define ˜ U˜ as above. Then K˜ := H˜ |{0}×(M×Cl ,0) and C, C˜∂ yα v˜1 =

n ∂ fi v˜i for α = 1, . . . , l . ∂ yα

(1.8)

i=1

˜ were already constructed. The connection Proof. Suppose for a moment that ( H˜ , ∇) matrix with respect to the basis (v˜1 , . . . , vñ ), ˜ v˜1 , . . . , vñ ) = (v˜1 , . . . , vñ ) · , ∇( would take the form =

r

Ai

i=1

r m l dti 1 dti 1 1 1 1 + Ci + Ck dtk + Fα dyα + 2 U + V dz ti z ti z z z z i=1

k=r +1

α=1

with matrices Ai , Ci , C j , Fα , U, V ∈ M(n × n, O M×Cl ).


1157

˜ along {0, ∞} × This follows from v˜i elementary and from the pole orders of ( H˜ , ∇) (M × Cl , 0). In the following i, j ∈ {1, . . . , r }, k, l ∈ {r + 1, . . . m}. The flatness condition d + ∧ = 0 implies ∂ Ai = 0, ∂tk

(1.9)

∂ Ai = 0, ∂ yα

(1.10)

[Ai , A j ] = −ti

tj

∂Aj ∂ Ai + tj , ∂ti ∂t j

(1.11)

[Ci , C j ] = [Ci , Ck ] = [Ck , Cl ] = 0,

(1.12)

[Ci , Fα ] = [Ck , Fα ] = 0,

(1.13)

[Fα , Fβ ] = 0,

(1.14)

∂C j ∂Ci + [A j , Ci ] = ti + [Ai , C j ], ∂t j ∂ti

(1.15)

∂Ck ∂Ci = ti + [Ai , Ck ], ∂tk ∂ti

(1.16)

∂Cl ∂Ck = , ∂tk ∂tl

(1.17)

∂Ci ∂ Fα = ti + [Ai , Fα ], ∂ yα ∂ti

(1.18)

∂Ck ∂ Fα = , ∂ yα ∂tk

(1.19)

∂ Fβ ∂ Fα = , ∂ yβ ∂ yα

(1.20)

[U, C j ] = [U, Ck ] = 0,

(1.21)

[U, Fα ] = 0,

(1.22)

ti

ti

∂U = [V, Ci ] − Ci + [U, Ai ], ∂ti

(1.23)

∂U = [V, Ck ] − Ck , ∂tk

(1.24)

∂U = [V, Fα ] − Fα , ∂ yα

(1.25)

∂V = [V, Ai ], ∂ti

(1.26)

∂V = 0, ∂tk

(1.27)

1158

T. Reichelt

∂V = 0. ∂ yα

(1.28)

The condition (1.8) would mean (Fα )i1 =

∂ fi . ∂ yα

(1.29)

The proof consists of three parts. In Parts (I) and (II) we restrict to the case l = α = 1. In Part (I) we show inductively uniqueness and existence of matrices Ci , C j , F1 , U, V, W with (1.12)–(1.29) and with coefficients in O M,0 [[y1 ]]. In Part (II) their convergence will be proved with the Cauchy-Kovalevski theorem. In Part (III) the general case will be proved by induction in l. Part (I). Suppose l = α = 1 and y1 = y. Define for w ∈ Z≥0 , O M,0 [y]≤w :=

w

O M,0 · y k ,

k=0

O M,0 [y]>w := O M,0 [y] · y w+1 , O M,0 [[y]]>w := O M,0 [[y]] · y w+1 , and M(w) := M(n × n, O M,0 · y k ), M(> w) := M(n × n, O M,0 [y]>w ), M(≤ w) := M(n × n, O M,0 [y]≤w ).

(1.30)

Beginning of the induction for w = 0: The connection matrix (0) of (H, ∇) with respect to the basis v1 , . . . , vn takes the form r r m dti 1 (0) dti 1 (0) 1 1 (0) = Ai + Ci + C j dt j + 2 U (0) + V (0) dz ti z ti z z z i=1

i=1

(0)

j=r +1

(0)

with matrices Ai , Ci , C j , U (0) , V (0) ∈ M(0). The flatness condition d(0) + (0) ∧ (0) = 0 is equivalent to Eqs. (1.9),(1.11),(1.12),(1.15), (1.16), (1.17), (1.21), (1.23), (1.24), (1.26), (1.27) for w = 0. Equations (1.10), (1.13), (1.14), (1.18), (1.19), (1.20), (1.22), (1.25), (1.28) are empty. (k) (k) Induction hypothesis for w ∈ Z≥0 : Unique matrices Ci , C j , U (k) , V (k) ∈ M(k) for 0 ≤ k ≤ w and F1(k) ∈ M(k) for 0 ≤ k ≤ w − 1 are constructed such that the matrices (≤w)

Ci

:=

w

(k)

Ci

∈ M(≤ w)

k=0 (≤w)

(≤w−1)

and the analogously defined matrices C j , U (≤w) , V (≤w) , F1 and the matrices Ai satisfy (1.12), (1.15), (1.16), (1.17), (1.21), (1.23), (1.24), (1.26), (1.27) modulo M(> w), (1.13), (1.14), (1.18), (1.19), (1.22), (1.25),(1.28) modulo M(> w − 1) and (1.29) modulo O M,0 [[y]]>w−1 .


1159

Induction step from w to w + 1. It consists of three steps: 1. Construction of a matrix F1(w) ∈ M(w) such that the matrix F1(≤w) = F1(≤w−1) + (w) (≤w) (≤w) F1 together with the matrices Ci , C j , U (≤w) , V (≤w) satisfies (1.13), (1.22) mod M(> w) and (1.29) mod O M,0 [[y]]>w . (w+1) (w+1) , Ck , U (w+1) , V (w+1) ∈ M(w + 1) such that 2. Construction of matrices Ci (≤w+1) (≤w) (w+1) the matrices Ci = Ci + Ci and the analogously defined matrices (≤w+1) (≤w) (≤w+1) (≤w+1) ,U ,V and the matrix F1 satisfy (1.18), (1.19), (1.25), Cj (1.28) mod M(> w). 3. Proof of (1.12), (1.15), (1.16), (1.17), (1.21), (1.23),(1.24),(1.26), (1.27) mod M(> w + 1). (1) The matrices Ci(≤w) , Ck(≤w) and U (≤w) generate an algebra of commuting matrices in M(n × n, O M,0 [y]/O M,0 [y]>w ). Because of the generation condition (GC), the image of the column vector (1, 0, . . . , 0)tr under the action of this algebra is the whole space M(n × 1, O M,0 [y]/O M,0 [y]>w ). This shows two things: (≤w)

– This algebra contains matrices E i with the first column

∈ M(n × n, O M,0 [y]≤w ) for any i = 1, . . . , n

(≤w)

(E i

) j1 = δi j .

– Any matrix in M(n×n, O M,0 [y]≤w ) which commutes with the matrices Ci(≤w) , C (≤w) j and U (≤w) mod M(> w) is modulo M(> w) a linear combination of the matrices (≤w) Ei with coefficients in O M,0 [y]≤w . (≤w)

Therefore the matrix F1 (≤w)

F1

=

∈ M(≤ w) which is defined by

n ∂ fi i=1

∂y

(≤w)

· Ei

mod M(n × n, O M,0 [[y]]>w )

(1.31)

is the unique matrix which satisfies (1.13), (1.22) mod M(> w), (1.29) mod O M,0 [[y]]>w (≤w) (≤w−1) (w) (w) and F1 = F1 + F1 for some F1 ∈ M(w). (2) ∂Ci ∂ F1 = ti · + [A j , F1 ] mod M(> w) ∂y ∂ti (≤w+1)

⇒ Ci

constructed,

and so for the other matrices C (≤w+1) , U (≤w+1) , V (≤w+1) with the corresponding equaj tions. (3) Now we check that the derivatives by ∂∂y of the remaining equations hold modulo M(> w + 1). Because of the derivative we calculate modulo M(> w) and therefore we can use all equations modulo M(> w). For example one calculates, (1.12):

1160

∂ Ci , C j ∂y

T. Reichelt (1.18)

=

= (1.13)

=

(1.15)

=

∂ F1 ∂ F1 ti + [Ai , F1 ] , C j + Ci , t j + A j , F1 ∂ti ∂t j

∂ F1 ∂ F1 + Ci , A j , F1 ti , C j + [Ai , F1 ] , C j + Ci , t j ∂ti ∂t j

∂C j ∂Ci + F1 , C j , Ai + F1 , t j − F1 , Ci , A j − F1 , ti ∂ti ∂t j

0.

The other equations are similar or easier. This finishes the proof of the induction step from w to w + 1. It shows uniqueness and existence of matrices Ci , C j , F1 , U , V, W ∈ M(n × n, O M,0 [[y]]) with (1.12)–(1.29) and with restrictions (0) (0) ). (Ci , C j , U, V ) | y=0 = (Ci(0) , C (0) j ,U ,V Part (II). Now we have to show holomorphy of these matrices. We want to apply the Cauchy-Kovalevski theorem in the following form ([Fo](1.31),(1.40),(1.41); there the setting is real analytic, but proofs and statements hold also in the complex analytic setting): Given N ∈ N and matrices Hi , L ∈ M(N × N , C{t1 , . . . , tm , y, x1 , . . . , x N }) there exists a unique vector ∈ M(N × 1, C{t1 , . . . , tm , y}) with ∂ ∂ = Hi (t, y, ) + L(t, y, ), ∂y ∂ti

(1.32)

(t, 0) = 0.

(1.33)

m

i=1

We will construct a system (1.32)–(1.33) with N = (m + 3)n 2 such that it will be satisfied with the entries of the matrices Ci − Ci(0) , U − U (0) , V − V (0) as entries of . The system will be built from the following equations: (0)

∂Ci − Ci ∂y

= ti

∂ F1 + [Ai , F1 ], ∂ti

∂Ck − Ck(0) ∂ F1 = , ∂y ∂tk ∂U − U (0) = [V, F1 ] − F1 , ∂y ∂ V − V (0) = 0, ∂y

(1.34) (1.35) (1.36) (1.37)

and Eqs. (1.38), (1.39) with which one can express the entries of F1 as functions of the entries of . The commutative subalgebra of M(n × n, O M,0 [[y]]), which is generated by the matrices C1 , . . . , Cm , U , is a free O M,0 [[y]]-module of rank n. Choose monomials G ( j) , j = 1, . . . , n, in the matrices C1 , . . . , Cm , U which form an O M,0 [[y]]-basis ( j) of this module. Then the matrix (G i1 )i j of the first columns of the matrices G ( j) is invertible in M(n × n, O M,0 [[y]]). Equation (1.31) gives F1 =

n j=1

g j G ( j)

(1.38)


1161

with coefficients g j ∈ O M,0 [[y]] such that

∂ fn ∂ f1 ,..., ∂y ∂y

tr

( j)

= (G i1 ) · (g1 , . . . , gn )tr . (0)

(1.39)

(0)

Replacing the entries of the matrices Ci − Ci , Ck − Ck , U − U (0) , V − V (0) by indeterminates x1 , . . . x N , the coefficients of the matrices G ( j) become elements of C{t}[x1 , . . . , x N ], and the g j become elements of C{t1 , . . . , tm , y, x1 , . . . , x N }. One obtains from (1.34)–(1.39) a system (1.32), (1.33). Now the theorem of CauchyKovalevski shows Ci , Ck , F1 , U, V ∈ M(n × n, O M×C,0 ). This gives Lemma 1.13 in the case of l = 1. Part (III). By induction in l one obtains a slightly weaker version of the lemma, namely with formula (1.8) replaced by n ∂ fi ˜ C∂ yα v˜1 | yα+1 =···=yl =0 = v˜i | yα+1 =···=yl =0 ∂ yα i=1

for α = 1, . . . , l. This equation is equivalent to (1.29) with the same restrictions. But now one has a connection matrix as in the third line of the proof of Lemma 1.13. Flatness gives (1.9)–(1.28). Equation (1.20) together with the formula above gives the formula at the end of Lemma 1.13. It remains to prove that the pairing P extends to the unfolding. We need a lemma about the rigidity of logarithmic poles. Lemma 1.14. Let (L → (C∗ , 0) × (M, 0), ∇) be the germ of a holomorphic vector bundle with flat connection ∇, and let (L (0) → (C, 0) × {0}, ∇) be an extension of (L , ∇) |(C∗ ,0)×{0} with a logarithmic pole at 0. Then an extension (L → (C, 0) × (M, 0), ∇) of (L , ∇) with a logarithmic pole along {0} × (M, 0) exists with (L , ∇) |(C,0)×{0} = (L (0) , ∇). It is unique up to canonical isomorphism and it is isomorphic to the pullback ϕ ∗ (L (0) , ∇) where ϕ : (M, 0) → {0}. Proof. See [Sab] III.1.20 . ˜ be the unfolding of the (log D −tr TLEP Lemma 1.15. Let ((M ×Cl , 0), D ×Cl , H˜ , ∇) ˜ H˜ , (w))-structure (H, ∇, P) given above. Then P extends to O( H˜ ) and ((M×Cl , 0), D, ˜ ˜ ∇, P) is a log D − tr T L E P(w)-structure. Proof. It is sufficient to consider an unfolding in one parameter y. For some representative of H˜ the pairing P extends to a ∇-flat pairing on the restriction to (C∗ , 0)×(M\D × C, 0). We have to show that it takes values on O( H˜ ) in z w OP1 ×M×C . A priori the values are in OC∗ ×M\D×C . Recall that ∇ has a logarithmic pole along (C∗ , 0) × (D, 0). ˜ |C∗ ×M\D ×C is Because of the rigidity of logarithmic poles the extension to ( H˜ , ∇) sing trivial, therefore P takes values in OC∗ ×M\Dsing ×C . A codimension 2 argument shows that P takes values OC∗ ×M×C . The same reasoning shows that on P1 \{0} × M × C the pairing P takes values in z w OP1 \{0}×M×C . It remains to prove the corresponding property on C × M × C. Denote n := r k H and let (z, t1 , . . . , tm , y) be coordinates on

1162

T. Reichelt

(C × M × C, 0). Choose an OC×M×C,0 -basis (v˜1 , . . . , vñ ) of O( H˜ ) with connection matrix r m 1 ˇ 1 ˇ dti 1 ˇ 1 + Ci Ck dtk + Fdy = + Uˇ dz z ti z z z i=1

i=r +1

ˇ Uˇ ∈ M(n × n, OC×M×C,0 ) and the matrix with matrices Cˇ i , Cˇ k , F, R := (P(v˜i , v˜ j )) ∈ M(n × n, OC∗ ×M×C,0 ). Flatness and z-sesquilinearity of the pairing give d R(z, t, y) = tr (z, t, y)R(z, t, y) + R(z, t, y)(−z, t, y), that means, ∂ R(z, t, y) ∂z ∂ R(z, t, y) ti ∂ti ∂ R(z, t, y) ∂tk ∂ R(z, t, y) ∂y z

= 1z Uˇ tr (z, t, y)R(z, t, y) − 1z R(z, t, y)Uˇ (−z, t, y),

(1.40)

= 1z Cˇ itr (z, t, y)R(z, t, y) − 1z R(z, t, y)Cˇ i (−z, t, y),

(1.41)

= 1z Cˇ ktr (z, t, y)R(z, t, y) − 1z R(z, t, y)Cˇ k (−z, t, y),

(1.42)

ˇ = 1z Fˇ tr (z, t, y)R(z, t, y) − 1z R(z, t, y) F(−z, t, y).

(1.43)

Write R as a power series R(z, t, y) =

∞

R (l) (z, t) with R (l) ∈ M(n × n, OC∗

×M,0

· yl )

l=0

and define R (≤l) (z, t, y) :=

l

R ( j) (z, t),

j=0 (l)

(l)

ˇ Uˇ , with Cˇ , Cˇ , Fˇ (l) , Uˇ (l) ∈ M(n ×n, OC×M,0 · y l ). Then analogously for Cˇ i , Cˇ k , F, i k (0) w R ∈ M(n × n, z OC×M,0 ) because (H, ∇, P) is a (log D − tr T L E P(w))-structure. Induction hypothesis for k ∈ Z≥0 : R (≤l) ∈ M(n × n, z w OC×M×C,0 ). Induction step from l to l + 1. Recall the definition of M(> l) in (1.30). Equations (1.40), (1.41) and (1.42) show that one has modulo M(> l) Cˇ i(≤l)tr (0, t, y)[z −w R (≤l) (z, t, y)] |z=0 ≡ [z −w R (≤l) (z, t, y)] |z=0 Cˇ i(≤l) (0, t, y), Cˇ k(≤l)tr (0, t, y)[z −w R (≤l) (z, t, y)] |z=0 (≤l) ≡ [z −w R (≤l) (z, t, y)] |z=0 Cˇ k (0, t, y),

Uˇ (≤l)tr (0, t, y)[z −w R (≤l) (z, t, y)] |z=0 ≡ [z −w R (≤l) (z, t, y)] |z=0 Uˇ (≤l) (0, t, y).


1163

Because of the generation condition (GC) the matrix Fˇ (≤l) (0, t, y) is an element of the commutative subalgebra of M(n × n, O M,0 [y])/M(> l) which is generated by (≤l) (≤l) Cˇ 1 , . . . , Cˇ m , Uˇ (≤l) . Therefore modulo M(> l) Fˇ (≤l)tr (0, t, y)[z −w R (≤l) (z, t, y)] |z=0 ≡ [z −w R (≤l) (z, t, y)] |z=0 Fˇ (≤l) (0, t, y). This together with (1.43) completes the induction step. (B) Transformation properties. In this section we want to clarify the transformation properties of the global sections v1 , . . . , vn , which were defined in Lemma 1.13, under pullback by a map ϕ : M × Cl → M × Cl of a given log(D) − tr TLEP (w)-structure. Furthermore the meaning of the base change matrix B between the elementary sections s j around the divisor { 1z · t1 · · · · · tr = 0} and the global sections v j of an adapted basis will be investigated. Without loss of generality we can assume that B = I + B (1) z −1 + B (2) z −2 + · · · . Let rank H = n and s1 , . . . , sn be elementary sections as above. There exists α j , (1) (r ) β j , . . . , β j ∈ C with j ∈ {1, . . . , n}, Aj ∈ H

(1) (r ) −2πiβ j −2πiα j −2πiβ j e , ,e ,...,e

∞

s j = zα j −

N (0) 2πi

r

β

(k)

tk j

(k)

− N2πi

Aj,

k=1

and for k ∈ {1, . . . , r } M (0) := Monodromy around z = 0, M

(k)

:= Monodromy around tk = 0,

N (0) = log(unipotent part of M (0) ), N (k) = log(unipotent part of M (k) ). (k)

(k)

(k)

(k)

With respect to the basis A1 , . . . , An we have matrices Mmat , Ms,mat , Mu,mat , Nmat with ⎞ ⎛ (k) e−2πiβ1 ⎟ ⎜ (k) .. ⎟ Ms,mat = ⎜ . ⎠ ⎝ (k) e−2πiβn (0)

for k ∈ {0, 1, . . . , r } with β j general the matrix

:= α j . Observe that these matrices commute. But in ⎛

⎜ (k) βmat = ⎝

⎞

(k)

β1

..

.

⎟ ⎠ βn(k)

1164

T. Reichelt (k)

(k)

does not commute with Mu,mat and Nmat . Set ⎛ ⎜ := ⎜ ⎝

z α1

r

β

⎞

(k)

1 k=1 tk

..

. z αn

r

(k)

⎟ ⎟. ⎠

βn k=1 tk

We have ∇tk ∂tk s j = 0 ∇tk ∂tk s j = tk ∂tk (z

(0) α j − N2πi

= β (k) j · sj +

−1 2πi

r l=1 n

(l)

i=1

for k ∈ {r + 1, . . . , m},

(l)

β −N tl j 2πi

Aj)

for k ∈ {1, . . . , r }

(k) ii−1 (Nmat )i j j j si .

This means in matrix notation (for k ∈ {1, . . . , r }) −1 −1 (k) (k) tk ∂tk (s1 , . . . , sn ) = (s1 , . . . , sn ) · βmat + Nmat . 2πi

(1.44)

We have (k)

(k)

(−1 Nmat )i j = ii−1 (Nmat )i j j j (k)

= (Nmat )i j · z α j −αi

r

(l)

(l)

β j −βi

tl

.

l=1

Because (s1 , . . . , sn ) is a holomorphic basis of H |(P1 \{0})×M we have the following (k) restrictions on the entries of Nmat : (k) (l) (Nmat )i j = 0 ⇒ α j ≤ αi and β (l) j ≥ βi . (k)

Now we decompose Nmat in terms of powers of z −1 . Write (k,l) (k) = Nmat , Nmat l≥0

(k,l) )i j = (Nmat

0 α j − αi = −l . (k) (Nmat )i j α j − αi = −l

As above we have a basis v1 , . . . , vn of global sections such that ∞ −k (k) (v1 , . . . , vn ) = (s1 , . . . , sn ) · z B (t) . k=0

Set αmat

⎛ α1 ⎜ .. := ⎝ .

⎞ ⎟ ⎠. αn


1165 (0)

(k)

(k)

It follows that there are formulas for Ai , Ci , U, V in terms of , Nmat , Nmat , βmat , αmat and B (1) . Define ⎞ ⎛ (k) β r tk 1 k=1 ⎟ ⎜ .. ⎟. ˜ := ⎜ . ⎠ ⎝ (k) r βn k=1 tk We have for example −1 −1 (i,0) Nmat i ∈ {1, . . . , r }, 2πi ∂ (1) −1 −1 (i,1) ˜ + ti ˜ Nmat Ci = [Ai , B (1) ] + B i ∈ {1, . . . , r }, 2πi ∂ti ∂ B (1) j ∈ {r + 1, . . . , m}, Cj = ∂t j (i) Ai = βmat +

Fα =

∂ B (1) α ∈ {1, . . . , l}. ∂ yα

But we will not use these formulas in the following. Recall that an elementary section is utterly dependent on the chosen coordinates cf. [He1]. Assume that we have given two unfoldings ( H˜ → P1 × (M × Cl , 0)) with coordinates yα on Cl , ( H˜ → P1 × (M × Cl , 0)) with coordinates yα on Cl . We now want to examine the properties of a map

ϕ : (M × Cl , D × Cl , 0) → (M × Cl , D × Cl , 0) ˜ with ( H˜ , ∇˜ ) ϕ ∗ ( H˜ , ∇) and ϕ | M×{0} = id. Therefore ϕ has the following form: ϕ = (ϕ1 , . . . , ϕr , ϕr +1 , . . . , ϕm , ϕm+1 , . . . , ϕm+l )

= (t1 · eu 1 (t,y ) , . . . , tr · eur (t,y ) , ϕr +1 , . . . , ϕm , ϕm+1 , . . . , ϕm+l ).

(1.45)

But this has the following effect on the elementary sections (here we identify ϕ ∗ (A j ) with Aj ) r r (k) N (k) (k) N (k) ) N (0) β j − 2πi (β − )u (t,y tk · e j 2πi k ϕ ∗ s j = z α j − 2πi Aj = =

k=1 r

e

(k) (k) (β j − N2πi )·u k (t,y )

k=1 r β (k) ·u k j

e

k=1

·

r

k=1

(k)

e

k=1

s j

− N2πi ·u k

s j ,

1166

T. Reichelt

which is in matrix notation ∗

ϕ (s1 , . . . , sn ) =

r

e

k=1

=

(k)

− N2πi ·u k

(s1 , . . . , sn ) ·

(s1 , . . . , sn )

·

r k=1

r

e

(k)

(k)

eβmat ·u k

1 (− 2πi −1 Nmat )u k

k=1

·

r

e

(k)

βmat ·u k

.

k=1

We now want to expand the matrix 1

−1 ·N (k) ·)u k mat

e(− 2πi

(k,l)

in powers of z −1 . We use the matrices Nmat defined above. This yields (k) (k,l) ˜ ˜ −1 Nmat · · z −l . −1 · Nmat · = l≥0 (k,l) in general do not commute. Therefore we Observe that for fixed k the matrices Nmat have the following expansion:

e

(k) 1 −1 ·Nmat · u k − 2πi (k,0)

˜ −1 · e− 2πi Nmat ·u k · ˜ = z0 · −1 −1 (k,1) ˜ ˜ + z −1 · Nmat · · uk 2πi 1 −1 2 −1 (k,0) (k,1) (k,1) (k,0) ˜ ˜ + ··· + · Nmat · Nmat + Nmat Nmat · 2! 2πi 1

+ z −2 · [. . .] + z −3 · [. . .] + · · · . Now set 1 =

=

r k=1 r

(k,0)

˜ −1 · e− 2πi Nmat 1

·u k

˜ ·

and

(k)

eβmat ·u k .

k=1

Let v˜1 , . . . , vñ respectively v˜1 , . . . , vñ be the canonical extensions of v1 , . . . , vn . We then have ∞ z −k B˜ (k) with B˜ (0) = I (v˜1 , . . . , vñ ) = (s1 , . . . , sn ) · k=0

and analogously for H˜ → P1 × (M × Cl , 0). Now we are able to examine how the global sections (v˜1 , . . . , vñ ) behave under a pullback (where we identify the flat sections Aj of H˜ and the sections ϕ ∗ (A j ) of ϕ ∗ ( H˜ )):


ϕ ∗ (v˜1 , . . . , vñ ) =ϕ

∗

(s1 , . . . , sn ) ·

= (s1 , . . . , sn ) ·

∞

1167

z

−k

B˜ (k)

k=0 r − 1 −1 N (k) u k mat 2πi

e

k=1

=:

(s1 , . . . , sn ) ·

·

r

e

(k) βmat ·u k

∞ z −k ( B˜ (k) ◦ ϕ) ·

k=1

z · 1 + z 0

−1

· 2 + z

−2

· [. . .] + . . . · ·

k=0 ∞

z

−k

˜ (k)

(B

◦ ϕ)

k=0

= (s1 , . . . , sn ) · 1 · + z −1 · 2 · + 1 · · ( B˜ (1) ◦ ϕ) + z −2 · [. . .] + · · · , and therefore ϕ ∗ (v˜1 , . . . , vñ ) · −1 · 1−1 = (s1 , . . . , sn ) · 1 + z −1 · 2 · 1−1 + 1 · · ( B˜ 1 ◦ ϕ) · −1 · 1−1 + · · · . Now the matrix 2 above is defined implicitly and its form is extremely complicated. But we do not need its explicit form in the following. Recall the definition of the global sections v˜1 , . . . , vñ , which are a canonical extension of an adapted basis v1 , . . . , vn . They were defined as elementary sections of the residue connection ∇˜ r es on H˜ |{∞}×(M×Cl ,0) when restricted to {∞} × (M × Cl , 0) and (v˜1 , . . . , vñ ) |P1 ×(M×{0},0) = (v1 , . . . , vn ). Now recall the correspondence between logarithmic Frobenius type structures and (log D − tr TLEP (w))-structures. Given a (log D − tr TLEP (w))-structure we set K := H |{0}×M and used the canonical isomorphism between H |{∞}×M and K to shift the residue connection of ∇ on {∞} × M to a connection on K . We therefore have (v˜1 , . . . vñ ) |{∞}×M×Cl = (s1 , . . . , sn ) |{∞}×M×Cl ,

(v˜1 , . . . vñ ) |{∞}×M×Cl = (s1 , . . . , sn ) |{∞}×M×Cl , but this yields ϕ ∗ (v˜1 , . . . , vñ ) · −1 · 1−1 = (v˜1 , . . . , vñ ) .

(1.46)

This formula is the crucial ingredient in proving the universality of an unfolding. (C) We now prove Theorem 1.12. Proof. Let ((M, 0), D, K , ∇ r , C, U, V, g) be a Frobenius type structure with logarithmic pole along D as in Theorem 1.12; let w ∈ Z and ((M, 0), D, H, ∇, P) be the corresponding (log D − tr T L E P(w))-structure. Because of the lemma and condition ˜ H˜ , ∇, ˜ exists such that for the corresponding ˜ P) (IC) an unfolding ((M × Cl , 0), D, l ˜ U˜ , V, ˜ g) ˜ ˜ Frobenius type structure ((M × C , 0), D, K , ∇˜ r , C, ˜ ˜ 0 −→ K˜ 0 C˜• ξ : Der M×Cl (log D) is an isomorphism. Now we are in the isomorphism case which shows that the constructed Frobenius type structure corresponds to a Frobenius manifold. It remains to prove uniqueness of this Frobenius manifold. As mentioned above this is equivalent to the universality of the unfolding. Therefore we now prove universality

1168

T. Reichelt

of the unfolding. Consider a second unfolding ((M × Cl , 0), D˜ , H˜ , ∇˜ , P˜ ). The two ) ( connection matrices of the corresponding (log D˜ − tr TLEP(w))-structures are

=

r i=1

=

r i=1

Ai

r m l dti 1 dti 1 1 1 1 + Ci + C j dt j + Fα dyα + 2 U + V dz , ti z ti z z z z i=1

j=r +1

α=1

r m l dti 1 dti 1 1 1 1 Ai + Ci + C j dt j + Fα dyα + 2 U + V dz, ti z ti z z z z

i=1

j=r +1

α=1

where is with respect to adapted sections (v˜1 , . . . , vñ ) and is with respect to adapted sections (v˜1 , . . . , vñ ), which are both canonical extensions of adapted sections (v1 , . . . , vn ). ˜ with ϕ|M×{0} = id such We want to find a map ϕ : (M × Cl , D˜ ) → (M × Cl , D) that ˜ = ∇˜ . ϕ ∗ (∇)

(1.47)

Now the discussion in (B) shows that the ϕ ∗ (v˜1 , . . . , vñ ) |{∞}×M×Cl are not elementary ˜ on {∞} × M × Cl in general. But we know sections of the residue connection of ϕ ∗ (∇) from formula (1.46) that ϕ ∗ (v˜1 , . . . , vñ ) · −1 · 1−1 |{∞}×M×Cl ˜ Now set are elementary sections of the residue connection of ϕ ∗ (∇). := −1 · 1−1 . Therefore if we identify (ϕ ∗ (v˜1 ), . . . , ϕ ∗ (vñ ))· with (v˜1 , . . . , vñ ) the following relation must be satisfied if we want (1.47): = −1 d + −1 ϕ ∗ () .

(1.48)

The reason that we have to use this matrix is that the sections (v˜1 , . . . , vñ ) were defined using elementary sections (with respect to the residue connection). But elementary sections are coordinate dependent, therefore we can not expect equality of (ϕ ∗ (v˜1 ), . . . , ϕ ∗ (vñ )) and (v˜1 , . . . , vñ ), unless the first i components of ϕ are the identity. We want to use the uniqueness statement in Lemma 1.13 that an unfolding is determined by the first columns of the Fα in the connection matrix. Therefore if we want to ˜ must satisfy ϕ | M×{0} = id and the first have (1.47), ϕ : (M × Cl , D˜ ) → (M × Cl , D) columns of the Fα must be equal. Now observe that if we have ϕ | M×{0} = id then we have r r dti dti −1 −1 ∗ d + ϕ Ai Ai | |{∞}×M×Cl = l , ti ti {∞}×M×C i=1

i=1

because (v˜1 , . . . , vñ ) and ϕ ∗ (v˜1 , . . . , vñ ) · are both elementary sections on {∞} × M × Cl which coincide on {∞} × M × {0} and therefore coincide on all of {∞} × M × Cl .


1169

We have l equations coming from (1.48), and using (1.45), Fα =

r

−1 (Ci ◦ ϕ)

i=1

m l ∂ϕ j −1 ∂ϕm+β ∂u i −1 + (C ◦ ϕ) + (Fβ ◦ ϕ) . j ∂ yα ∂ yα ∂ yβ j=r +1

β=1

We consider the first columns of −1 · Ci · and −1 · Fα · respectively and denote these with an upper I . The matrices S = (C1I , . . . , CmI , F1I , . . . , FlI ) , and S˜ which is defined by (( −1 · (C1 ◦ ϕ) · ) I . . . ( −1 · (Cm ◦ ϕ) · ) I ( −1 · (F1 ◦ ϕ) · ) I . . . ( −1 · (Fl ◦ ϕ) · ) I ) are locally invertible around {0} × {0} because | M×{0} = I , therefore S˜ | M×{0} = S | M×{0} and S is invertible in {0} × {0} by construction. We get the following system of equations: ∂u 1 ∂u r ∂ϕr +1 ∂ϕm ∂ϕm+1 ∂ϕm+l tr S˜ −1 (Fα ) I = , . . . , , , . . . , , , . . . , . ∂ yα ∂ yα ∂ yα ∂ yα ∂ yα ∂ yα Now ϕ is determined on M × {0} (the identity) and we can solve for ϕ with the theorem of Cauchy-Kovalevski by using induction in α. The beginning of the induction is clear: ( −1 d + −1 ϕ ∗ ()) | M×{0} = | M×{0} because | M×{0} = 1 and ϕ | M×{0} = id. Now assume that we have constructed ϕ such that ( −1 d + −1 ϕ ∗ ()) | M×{yα =···=yl =0} = | M×{yα =···=yl =0} . We use the equation S˜ −1 (Fα ) I | yα+1 =···=yl =0 =

∂u 1 ∂u r ∂ϕr +1 ∂ϕm+l ,..., , ,...,, ∂ yα ∂ yα ∂ yα ∂ yα

tr | yα+1 =···=yl =0

to construct an extension of ϕ from M × {yα = · · · = yl = 0} to M × {yα+1 = · · · = yl = 0} by using the theorem of Cauchy-Kovalevski. From the construction of the extension of ϕ it follows that the first column of Fα and the first column of the corresponding matrix in −1 d + −1 ϕ ∗ () coincide on M × {yα+1 = · · · = yl = 0}. The uniqueness statement of Lemma 1.13 in the case l = 1 gives ( −1 d + −1 ϕ ∗ ()) | M×{yα+1 =···=yl =0} = | M×{yα+1 =···=yl =0} . which was to be shown. It remains to prove uniqueness of ϕ. But the condition ϕ | M×{0} = id and the equations tr ˜S −1 (Fα ) I = ∂u 1 , . . . , ∂u r , ∂ϕr +1 , . . . , ∂ϕm , ∂ϕm+1 , . . . , ∂ϕm+l ∂ yα ∂ yα ∂ yα ∂ yα ∂ yα ∂ yα show that ϕ is unique.

1170

T. Reichelt

2. Logarithmic Frobenius Manifolds and Quantum Cohomology In this section we show how to construct a Frobenius manifold out of quantum cohomology where we are following [Ma]. With the definition of a logarithmic Frobenius manifold at hand we show how this construction extends for smooth, projective varieties with h 2,0 = 0 to give us logarithmic Frobenius manifolds. This construction enables us to prove a partial generalization of the first Reconstruction theorem in [KM]. 2.1. Quantum Cohomology and Frobenius manifolds. We now define quantum cohomology on the even dimensional cohomology ring. That means we define a potential ∗ (X, C) which we use to define a quantum deformation of the cup product. Let on Heven T0 = 1 ∈ H 0 (X, C), T1 , . . . , Tr be a basis of H 2 (X, C) and let T r +1 , . . . , Tm be a m ∗ (X, C). Put γ = basis for the other cohomology groups lying in Heven i=0 ti Ti . In this section we only consider the free parts of H2 (X, Z) and H 2 (X, Z) and denote them by the same letters. Definition 2.1. Let X be a smooth projective variety. Then the Gromov-Witten potential is the formal sum (γ ) =

∞

n=0 β∈H2 (X,Z)

1 = 6

γ + 3

X

1 I0,n,β (γ n ) n!

∞

n=0 β∈H2 (X,Z) β =0

= class

1 I0,n,β (γ n ) n!

+ quantum ,

(2.1)

where we set I0,n,β = 0 for n ≤ 2. There are two problems with this definition. The first is, if for fixed n the sum β∈H2 (X,Z)

1 I0,n,β (γ n ) n!

(2.2)

does converge. Usually one introduces the Novikov ring as the coefficient ring to split the contribution of the different β. But here we will assume that (2.2) does converge. The second problem is if the whole sum (2.1) gives a well-defined function at least in some domain of H ∗ (X, C). Assumption. We assume in the rest of the paper that the individual summands in the Gromov-Witten potential are convergent. We assume also that the Gromov-Witten potential gives a well defined holomorphic function in the domain B := B0 ×

r m ∗ {Re(ti ) < ri } × B j ⊂ Heven (X, C), i=1

where B0 resp. for all i.

m j=r +1

j=r +1

B j are (poly)discs in H 0 resp. H 4 ⊕ · · · ⊕ H 2dim X and ri ∈ R


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Remark 2.2. Write γ = γ0 + δ with δ ∈ H 2 (X, C). We use the divisor axiom and the fundamental class axiom to rewrite the expression for the potential: 1 quantum (γ0 + δ) = I0,n,β (γ0⊗i ⊗ δ ⊗k ) i!k! i,k≥0 β =0

=

e(β,δ) I0,i,β (γ0⊗i ) i! i≥0 β =0

=

r +1 e(β,δ) I0,i,β (Tr +1 ⊗ . . . ⊗ Tmm )

j

j

jr +1 ,..., jm β =0

j

j

r +1 . . . tmm tr +1 . jr +1 ! . . . jm !

Definition 2.3. Let be the Gromov-Witten potential for a smooth projective variety X. Then define ∂ 3 Ti ∗ T j = T k, ∂ti ∂t j ∂tk k where g kl is as above and T k = l g kl Tl . Extending this linearly gives the big quantum ∗ (X, C). product on the cohomology Heven Lemma 2.4. For all i, j, k, we have ∞

∂ 3 = ∂ti ∂t j ∂tk

n=0 β∈H2 (X,Z)

=

1 I0,n+3,β (Ti , T j , Tk , γ n ) n!

r +1 e(β,δ) I0,i+3,β (Ti ⊗ T j ⊗ Tk ⊗ Tr +1 ⊗ · · · ⊗ Tmm )

j

jr +1 ,..., jm β =0

j

j

j

r +1 . . . tmm tr +1 jr +1 ! . . . jm !

Ti ∪ T j ∪ Tk .

+ X

Proof. See [CK], Lemma 8.2.3. It is a well known fact that ∗ is commutative and associative, with unit T0 . For a proof see [CK]. 2.1.1. Construction of Frobenius manifold. Because of the assumption on the GromovWitten potential stated above we will get a Frobenius manifold on the open subset ∗ (X, C). We consider now B ⊂ H ∗ (X, C) as a manifold with global B ⊂ Heven even coordinates ti with respect to a basis {Ti } as above. At each point the tangent space is ∗ (X, C). Denote the global vector fields as ∂ and define canonically isomorphic to Heven ti a metric g(∂ti , ∂t j ) := gi j = Ti ∪ T j . X

Observe that the metric is constant so that the induced Levi-Civita connection is ∇ = d with respect to the sections ∂ti . Denote by deg(∂ti ) the degree of ∂ti in H ∗ (X, C). We define the Euler vector field as deg(∂ti ) E := r j ∂t j , 1− ti ∂ti + 2 i

deg(∂t j )=2

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T. Reichelt

where the r j are defined as c1 (X ) =

r j Tj .

degT j =2

Proposition 2.5. Let E be as above. It holds a) E quantum = (3 − dim X ) quantum , b) (E − E(0)) class = (3 − dim X ) class , deg∂ti + deg∂t j + deg∂tk i jk . c) E i jk = (3 − dim X ) i jk − 3 − 2 Proof. Recall that quantum =

j j e(β,δ) t r +1 . . . tmm jr +1 j I0,i,β (Tr +1 . ⊗ · · · ⊗ Tmm ) r +1 i! jr +1 ! . . . jm !

jr +1 ,..., jm β =0

We apply E to that term. The E(0) term acts only upon e(β,δ) and multiplies it by j j (c1 (X ), β). The E − E(0) partmultiplies any monomial ta11 . . . tann in non-divisorial deg(Tai ) coordinates by i 1 − . Observe that the Gromov-Witten invariants contrib2 ute only for such β which satisfy 1 deg(Tai ) = dim X + 2 n

i=1

β

c1 (X ) + n − 3.

Hence every non-vanishing term is an eigenvector of E with eigenvalue 3 − dim X . b) is clear. deg∂ c) We are using [E, ∂tm ] = − 1 − 2 tm ∂tm : E i jk = E∂ti ∂t j ∂tk

deg∂ti + deg∂t j + deg∂tk i jk . = ∂ti ∂t j ∂tk E − 3 − 2

The statement follows with E = E quantum + (E − E(0)) class + E(0) class = (3 − dim X ) quantum + (3 − dim X ) class + E(0) class and the observation that ∂ti ∂t j ∂tk E(0) class = 0 because class is only cubic in the tm . We now have to check that this defines the structure of a Frobenius manifold. The properties (1)–(4) in Definition 1.3 are easy to show. We are using the Einstein sum convention. (5) Lie E (∗) = 1 · ∗ and Lie E g = (2 − d) · g with d = dim X .


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Here we use Proposition 2.10c. Lie E (∗)(∂ti , ∂t j ) = [E, ∂ti ∗ ∂t j ] − [E, ∂ti ] ∗ ∂t j − ∂ti ∗ [E, ∂t j ] deg(∂t j ) deg(∂ti ) lk = [E, i jk g ∂tl ] + 1 − ∂ti ∗ ∂t j + 1 − ∂ti ∗ ∂t j 2 2 deg(∂tk ) deg(∂tl ) kl i jk g ∂tl − 1 − i jk g kl ∂tl = (3 − dim X )∂ti ∗ ∂t j − 1 − 2 2 = ∂ti ∗ ∂t j , Lie E (g)(∂ti , ∂t j ) = Eg(∂ti , ∂t j ) − g([E, ∂ti ], ∂t j ) − g(∂ti , [E, ∂t j ]) deg(∂t j ) deg(∂ti ) =0+ 1− gi j + 1 − gi j 2 2 = (2 − dim X )g(∂ti , ∂t j ). 2.1.2. Logarithmic Frobenius manifolds. Above we constructed a Frobenius manifold ∗ (X, C) but this description has a little drawback. The on an open subset B of Heven point where the multiplication is simplest is not included. We now consider smooth, projective varieties X with h 2,0 (X ) = 0. Therefore we can choose a homogeneous basis T0 ∈ H 0 , T1 , . . . , Tr ∈ H 2 , Tr +1 ∈ H 4 , . . . , Tm ∈ H 2dim X such that T1 , . . . , Tr lies in the Kähler cone. If we look at quantum =

j j e(β,δ) t r +1 . . . tmm jr +1 j I0,i,β (Tr +1 , ⊗ · · · ⊗ Tmm ) r +1 i! jr +1 ! . . . jm !

jr +1 ,..., jm β =0

we see that the quantum potential is periodic with respect to δ but observe that class is not. Consider the map ϕ : B → M ∗ := B0 ×

r

{0 < |qi | < eri } ×

i=1

m

Bj ,

j=r +1

ϕ : ti → qi = eti i ∈ {1, . . . , r } , ϕ : t j → t j j ∈ {0, r + 1, . . . , m} , ϕ∗ : ∂ti → qi ∂qi . The potential quantum is well-defined on the image but class is not. But if we only consider the third derivatives of the potential, ∂ 3 class is a constant and because of that well-defined on the image. We want to investigate the limitqi → 0 for some i. Therefore we have to look at the term e(β,δ) more closely. Let δ = ri=1 ti Ti , then we have e(β,δ) =

r i

(β,Ti )

qi

.

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T. Reichelt

Now for all effective β we have (β, Ti ) ≥ 0 but only these β occur in the Gromov-Witten potential. Therefore the limit is defined. In the limit qi → 0 ∀i all terms in the quantum potential vanish because β = 0 and the Ti , i ∈ {1, . . . , r } are a basis of H 2 (X ) = H 1,1 (X ). Set M = B0 ×

r

{0 ≤ |qi | < 1 + } ×

i=1

m

Bj .

j=r +1

It is now clear that quantum , being a power series in the qi for i ∈ {1, . . . , r }, is convergent on M. As a consequence we have defined a Frobenius manifold structure on M ∗ and the quantum multiplication degenerates at qi = 0 ∀i to the usual cup product. It rests to examine if this defines a logarithmic Frobenius manifold. Note that the divisor D = M\M ∗ is normal crossing and that the ϕ∗ (∂ti ) are a basis of Der M (log D). Because of ∂ti → qi ∂qi we have m r deg(∂ti ) 1− ti ∂ti + E = t 0 ∂t 0 + r j q j ∂q j 2 i=r +1

j=1

so e, E ∈ Der M (log D). Because i jk is defined for qi → 0 and ϕ is locally biholomorphic we can define a multiplication on Der M (log D), ϕ∗ (∂ti ) ∗ ϕ∗ (∂t j ) := ϕ∗ (∂ti ∗ ∂t j ). Because g(∂ti , ∂t j ) is constant, we can define g(ϕ∗ (∂ti ), ϕ∗ (∂t j )) := g(∂ti , ∂t j ). We extend both the multiplication and the metric g O M -linearly. It follows that M carries the structure of a logarithmic Frobenius manifold. We call the point p defined by {t0 = q1 = · · · = qr = tr +1 = · · · tm = 0} the large radius limit point. 2.2. The first Reconstruction Theorem. We want to give a geometric interpretation and partial generalization of the first Reconstruction Theorem of Kontsevich and Manin given in [KM]. For the convenience of the reader we restate the theorem. Theorem 2.6. Let X be a smooth projective variety with the property that H ∗ (X, Q) is generated by H 2 (X, Q). Also assume that we know the Gromov-Witten invariants I0,3,β (α1 , α2 , α3 ) for all β ∈ H2 (X, Z) with deg α3 = 2. Then we can determine all tree-level Gromov-Witten classes I0,n,β (α1 , . . . , αn ) for all β ∈ H2 (X, Z). In [KM] the additional condition β c1 (X ) ≤ dim X + 1 is demanded. This can be explained as follows. In order to be non-zero the classes must fulfill deg(αi ) = 3 2( β c1 (X )+dim X ) but we have i=1 deg(αi ) ≤ 4 dim X +2 (assuming deg(α3 ) = 2). Therefore this condition is automatically fulfilled. There is the following partial generalization of the theorem: Theorem 2.7. Let X be a smooth projective variety with h 2,0 = 0.Let δ1 , . . . , δu u ∗ (X, C) with respect to the cup product. Set be generators of Heven k=1 C · δk = ∗ W ⊂ Heven (X, C). Assume that we know the Gromov-Witten invariants I0,n,β (α1 , α2 , α3 , . . . , αn ) for all β ∈ H2 (X, Z) with α3 , . . . , αn ∈ W and for all n ≥ 3. Then we can determine all tree-level Gromov-Witten classes I0,n,β (α1 , . . . , αn ) for all β ∈ H2 (X, Z).


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Proof. We first remark that the tree-level Gromov-Witten classes can be uniquely reconstructed from the tree-level Gromov-Witten invariants. This follows from [KM] Proposition 2.5.2. Therefore it remains to prove that we can reconstruct the corresponding Gromov-Witten invariants. Recall the map ϕ : B → M ∗ . and the logarithmic Frobenius manifold constructed on M. In Proposition 1.7 we proved that the latter gives rise to a logarithmic Frobenius type structure on M. Restrict this logarithmic Frobenius type structure to the submanifold N = ϕ(W ∩ B). The Higgs field on N (which comes from the quantum multiplication restricted to N ) incorporates only Gromov-Witten invariants of the form I0,n,β (α1 , . . . , αn ) with α3 , . . . , αn ∈ W (cf. Definition 2.3 and Lemma 2.4). Let ξ = ϕ∗ (∂t0 ) = ∂t0 . Then it is clear that (I C) is sat∗ (X, C) isfied. The condition (GC) holds because the δ1 , . . . , δu are generators of Heven and ∗ is simply the cup product at p (the large radius limit point). Recall the definition of E in the quantum cohomology case and that of V. We see that ∇∂t0 E = ∂t0 . So (EC) is satisfied. Now Theorem (1.12) states that we have a universal unfolding which is equivalent to a Frobenius manifold. So we have reconstructed the big quantum product. To extract the Gromov-Witten invariants we recall the following formula: Ti ∗ T j =

k

∂ 3 Tk . ∂ti ∂t j ∂tk

∂3

Therefore we know ∂ti ∂t j ∂tk for all i, j, k. If we integrate this three times we get . Now recall the definition of : ∞ 1 I0,n,β (γ n ) . (γ ) = n! n=0 β∈H2 (X,Z)

Notice that because of the integration we have an ambiguity in the terms for n ≤ 2 but with the definition above (I0,n,β = 0 for n ≤ 2, β = 0) this ambiguity is fixed. Now we want to extract the single Gromov-Witten invariants out of the potential. Recall Remark 2.2 where we saw that j j t r +1 . . . tmm jr +1 j quantum (γ0 + δ) = . e(β,δ) I0,i,β (Tr +1 ⊗ . . . ⊗ Tmm ) r +1 jr +1 ! . . . jm ! jr +1 ,..., jm β =0

Because of our assumption that the potential is holomorphic, we can differentiate with n respect to ti , ji times such that i=r +1 ji = n to get the term n ∂ quantum jr +1 j (δ, 0) = e(β,δ) I0,n,β (Tr +1 ⊗ · · · ⊗ Tmm ). j j m r +1 ∂ tr +1 . . . ∂ tm β =0

Choose a basis T1 , . . . Tr in H 2 (X, Z) and a basis S1 , . . . Sr in H2 (X, Z) such that (Si , T j ) = δi j , this is possible due to Poincaré duality. Restrict the term above to the R-vector space spanned by i T1 , . . . , i Tr and let t = (t1 , . . . , tr ) be coordinates on that space. Then the term above gets a multi-dimensional Fourier series jr +1 j ei l·t I0,n,β (Tr +1 ⊗ . . . ⊗ Tmm ). l∈Zd \{0}

Now the single coefficients can be obtained by integration.

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T. Reichelt

∗ (X, C) is H 2 -generated, then W = H 2 . Let us compare the two theorems. If Heven Thus the corresponding Higgs field incorporates only Gromov-Witten invariants of type I0,3,β (Ti ⊗T j ⊗Tk ) with deg(Tk ) = 2 and the other restrictions mentioned above (here we use the divisor axiom). Now Theorem (2.7) generalizes Theorem (2.6) for smooth projective varieties with h 2,0 = 0.

3. Hodge Asymptotics In this section we prove the existence of a filtration U• opposite to the Hodge filtration F • of a polarized variation of Hodge structures which degenerates along a normal crossing divisor (Prop. 3.11). This is used in Theorem 3.19 to show that the above data give rise to a logarithmic Frobenius manifold if the limiting PMHS is Hodge-Tate, split over R and an additional generation condition is satisfied.

3.1. Generalities. First we recall some definitions. Definition 3.1. A mixed Hodge structure is a triple (HZ , W• , F • ), where HZ is a lattice, W• is a finite increasing filtration on HQ = HZ ⊗ Q, and F • is a finite decreasing filtration on H = HZ ⊗C, such that the induced filtration on the quotient GrkW H = Wk /Wk−1 defines a Hodge structure of weight k. Here F p GrkW H is the image of F p ∩Wk in GrkW H , F p GrkW H = (Wk ∩ F p + Wk−1 )/Wk−1 . F • is called the Hodge filtration and W• the weight filtration. Let H be as above and HR := HZ ⊗ R. Let S be a nondegenerate (−1)w -symmetric pairing on H with real values on HR . Let N be a nilpotent endomorphism of HR and of H which is an infinitesimal isometry of S. One obtains a weight filtration W• in the following way: Lemma 3.2. Let (H, HR , S, N , w) as above. (a) There exists a unique finite increasing filtration W• on HR such that N (Wl ) ⊂ Wl−2 W −→ Gr W is an isomorphism. and such that N l : Grw+l w−l (b) The filtration satisfies S(Wl , Wl ) = 0 for l + l < w. W for (c) A nondegenerate (−1)w+l -symmetric bilinear form Sl is well defined on Grw+l ˜ for a, b ∈ Gr W with representatives a, l ≥ 0 by Sl (a, b) := S(a, ˜ N l b) ˜ b˜ ∈ Ww+l . w+l W (d) The primitive subspace Pw+l ⊂ Grw+l is defined by W W −→ Grw−l−2 ) Pw+l := ker (N l+1 : Grw+l

for l ≥ 0 and by Pw+l := 0 for l < 0. Then W = Grw+l

N i Pw+l+2i i≥0

and this decomposition is orthogonal with respect to Sl if l ≥ 0. Proof. See [Sch] Lemma 6.4.


1177

Definition 3.3. A polarized mixed Hodge structure of weight w consists of data (H, HR , S, N , w), W• as above and an exhaustive decreasing Hodge filtration F • on H with the following properties. 1. F • GrkW gives a pure Hodge structure of weight k, that means, GrkW = F p GrkW ⊕ F k+1− p GrkW . 2. N (F p ) ⊂ F p−1 , i.e. N is a (−1, −1)-morphism of mixed Hodge structures. 3. S(F p , F w+1− p ) = 0. 4. The pure Hodge structure F • Pw+l of weight w + l on Pw+l is polarized by Sl , that means Sl (F p Pw+l , F w+l+1− p Pw+l ) = 0, i p−(w+l− p) Sl (a, a) > 0 for a ∈ F p Pw+l ∩ F w+l− p Pw+l − {0} . A PMHS comes equipped with a generalized Hodge decomposition, namely Deligne’s I p,q . Lemma 3.4. For a PMHS as above define I p,q := (F p ∩ W p+q ) ∩ (F q ∩ W p+q +

F q− j ∩ W p+q− j−1 ),

j>0 p,q I0

Then

:= ker (N

p+q−w+1

:I

p,q

−→ I

w−q−1,w− p−1

).

F p = i,q:i≥ p I i,q , Wl = p+q≤l I p,q , j p+ j,q+ j , N (I p,q ) ⊂ I p−1,q−1 , I p,q = j≥0 N I0 S(I p,q , I r,s ) = 0 p,q S(N i I0 ,

N

j r,s I0 )

=0

for (r, s) = (w − p, w − q), for (r, s, i + j) = (q, p, p + q − w),

I q, p I p,q

mod W p+q−2 ,

q, p I0

mod W p+q−2 .

p,q I0

Proof. See [De1], (1.2.8), [CaK] or [He3], 2.3. Definition 3.5. A variation of Hodge structures of weight n is a quadruple (HZ , ∇, M, F • ) where M is a complex manifold, HZ is a local system with coefficient group Zh for some h, and ∇ a flat holomorphic connection on the holomorphic vector bundle H = HZ ⊗ O M such that the sections of HZ are ∇-flat. F • is a decreasing filtration of holomorphic vector bundles F p ⊂ H. These objects have to satisfy the following two conditions: – for each point t ∈ M the filtration F • induces a filtration Ft• on the fibre HZ ⊗ C at t ∈ M constituting an HS of weight n, – ∇(F p ) ⊂ 1M ⊗O M F p−1 for each p.

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T. Reichelt

Moreover, if there is a flat, non-degenerate bilinear form S : HZ ×HZ −→ Q M defining a polarized HS for every t ∈ M, then the VHS (HZ , ∇, M, F • ) is said to be polarized. n Now assume that we have given a manifold rM = , a normal crossing divisor Z ∗ r n−r on M such that M\Z = ( ) × , Z = j=1 D j and a polarized VHS on M\Z . We want to construct a Frobenius manifold which encodes these data. First we construct a filtration U• which is opposite to the Hodge filtration F • . This is done using Schmid’s limiting MHS and Deligne’s I p,q ’s. Then following [HM] Chap. 5 we construct a Frobenius manifold out of the latter. Now let H → M\Z be the flat vector bundle with connection ∇ coming from a polarized VHS. Let T j be the monodromies of ∇ with respect to D j for j = 1, . . . , r . Let N j := log(T j,u ) be the logarithm of the unipotent part of T j . Observe that both T j and N j act on the fibers of H and all Ti and N j commute with each other. Let

τ : U −→ M\Z (z, w) = (z 1 , . . . , zr , wr +1 , . . . , wn ) → (e2πi z 1 , . . . , e2πi zr , wr +1 , . . . , wn ) = (t, w) be the universal covering. We fix a flat isomorphism for the rest of the paper ρ∞ : τ ∗ H H ∞ × U . For each (z, w) ∈ U we get an isomorphism ρ(z,w) : Hτ (z,w) −→ H ∞ . Now we have ρ(z 1 ,...,z j +1,...,zr ,w) ◦ T j = ρ(z,w) . The monodromies T j act on H ∞ through the maps ρ(z,w) ◦ T j ◦ (ρ(z,w) )−1 : H ∞ → H ∞ which are actually independent of (z, w). Therefore we denote them by T j , too. The same applies to the N j . The Deligne extension. For the rest of the chapter assume that all T j are unipotent. Define for (z, w) ∈ U the map: σ(z,w) : Hτ (z,w) → Hτ (z,w) ,

A →

r

−N j /(2πi)

tj

(A) .

j=1

σ(z,w) also acts on H ∞ . Now the maps σ(z,w) map a multi-valued ∇-flat section A on H to an elementary section on H : r

−N j /(2πi)

tj

(A) .

j=1

It holds: −1 ρ(z,w) ◦ σ(z,w) = ρ(z 1 ,...,z j +1,...,zr ,w) ◦ σ(z−11 ,...,z j +1,...,zr ,w) .


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Therefore we can define −1 : Hτ (z,w) → H ∞ . χτ (z,w) := ρ(z,w) ◦ σ(z,w)

The χτ (z,w) map an elementary section to a constant value in H ∞ . ˜ Let ∇˜ be the connection for which the elementary sections are ∇-flat: ∇˜ := ∇ +

r N j dt j . 2πi t j j=1

The elementary sections provide an extension of the vector bundle H → M\Z to a vector bundle H˜ → M on M, the Deligne extension. The connection ∇˜ gives us a ˜ trivialization of H˜ → M. The maps χ(t,w) provide an identification of the ∇-flat bundle ∞ H˜ → M with H × M, which we will use in the following discussion. Fix (z, w) ∈ U and consider the isomorphism ρ(z,w) : Hτ (z,w) → H ∞ . Then we can shift the polarized HS on Hτ (z,w) to H ∞ . We denote the obtained reference polarized HS by (H ∞ , HR∞ , S, F0• ). The space p Dˇ := {filtrations F • ⊂ H ∞ | dim F p = dim F0 , S(F p , F w+1− p ) = 0}

is a complex homogeneous space and a projective manifold, and the subspace D := {F • ∈ Dˇ | F • is a part of a polarized Hodge structure} is an open submanifold and a real homogeneous space. It is a classifying space for polarized Hodge structures with fixed Hodge numbers. • Now let F • = (t,w)∈M\Z F(t,w) . The monodromies give rise to a representation of the fundamental group φ : π1 (M\Z ) −→ Gl(H ∞ ). Set = φ(π1 (M\Z )) = φ(Zr ). We have the following commutative diagram U ⏐ ⏐ τ"

˜

−−−−→

D ⏐ ⏐ "

M\Z −−−−→ \D with ˜ : (z, w) → ρ(z,w) (Fτ•(z,w) ) , and where is Griffiths’ period mapping for the polarized VHS in question. With respect to elementary sections we get the map : M\Z −→ Dˇ • (t, w) → χ(t,w) (F(t,w) ). The fact that takes values in Dˇ rather than D, reflects the fact that these sections are not real.

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ˇ the Definition 3.6. A pair (F • , N ) is said to give rise to a nilpotent orbit if F • ∈ D, endomorphism N of HR is nilpotent and an infinitesimal isometry with N (F p ) ⊂ F p−1 , and e z N F • ∈ D for I mz > b , b ∈ R. Then the set {e z N F • | z ∈ C} is called a nilpotent orbit. Theorem 3.7. (Nilpotent Orbit Theorem) The map extends holomorphically to n . For (w) ∈ n−r , we have a(w) = (0, w) ∈ Dˇ . For any given number η with 0 < η < 1, there exist constants α, β ≥ 0, such that under the restrictions I m z i ≥ α, 1 ≤ i ≤ r, and | w j |≤ η, r + 1 ≤ j ≤ n, l the point exp i=1 z i Ni ◦ a(w) lies in D and satisfies the inequality d exp

l

z i Ni

˜ ◦ a(w), (z, w) ≤

i=1

r

β I m(z i )

i=1

l

exp(−2π I m(z i ));

i=1

here d denotes a G R invariant Riemannian metric on D. Finally the mapping l (z, w) → exp z i Ni ◦ a(w) i=1

is horizontal. Proof. See [Sch] Theorem 4.12. • . We denote the point (0, 0) as Flim

r • , Remark 3.8. Observe that the pair (Flim j=1 a j N j ) gives rise to a nilpotent orbit, for any a j > 0, j = 1, . . . , r . Schmid’s Nilpotent Orbit Theorem yields ⎛ ⎞ r exp ⎝ z j N j ⎠ ψ(0, 0) ∈ D for I m(z j ) > α. j=1

Then

⎛ exp ⎝z

r

⎞ λ j N j ⎠ ψ(0, 0) (λ j > 0)

j=1

is a nilpotent orbit, because for I m(z) > b = max j

α λj

the above expression lies in D.


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To build a PMHS on (0, 0) we need a nilpotent endomorphism to define a weight filtration but the question is which of the Ni we should use. The following result of Cattani and Kaplan provides an answer. Theorem 3.9. Let C=

⎧ r ⎨ ⎩

j=1

⎫ ⎬

λj Nj | λj > 0 ∀ j . ⎭

C is called the monodromy cone. Then each N ∈ C defines the same monodromy weight filtration. Proof. See, e.g. [CaK] Theorem 3.3. We need a criterion for which N the tuple (H, HR , S, F • , N ) is a PMHS. Theorem 3.10. 1. The tuple (H, HR , S, F • , N ) is a PMHS of weight w ⇔ the pair (F • , N ) gives rise to a nilpotent orbit. 2. If (H, HR , S, F • , N ) is a PMHS with I q, p = I p,q , then e z N F • ∈ D for I mz > 0 . Proof. ‘⇐’ is [Sch], Theorem 6.16. It is a consequence of the S L 2 -orbit theorem. ‘⇒’ is [CaKSch], Cor. 3.13. The special case (b) is proved in [CaKSch], Lemma 3.12. Now we are finally able to prove the first step in the construction of a logarithmic Frobenius manifold out of a variation of Hodge structures. Namely the construction of a filtration U• which is opposite to the filtration F • in vicinity of (0, 0). Proposition 3.11. Assume as above that we have given M = n , M\Z = (∗ )r ×n−r , Z = rj=1 D j , and a polarized VHS on a ∇-flat bundle H → M\Z with unipotent monodromies. Let H˜ → M be the Deligne extension of H → M\Z . ˜ 1. The bundle H˜ → M comes equipped with a canonical flat connection ∇˜ and a ∇-flat ∞ ˜ isomorphism H → H × M. • on 2. The Hodge subbundles of H → M\Z extend to H˜ → M. The filtration Flim H˜ 0 H ∞ is part of a PMHS. For it one has Deligne’s I p,q . Then the filtration U• • on H ∞ with U p := i,q:i≤ p I i,q is opposite to all F(t,w) for (t, w) ∈ M close to 0, ˜ and it is ∇-flat as well as ∇-flat. • is part of a Proof. Part (1) follows from what has been said above. For part (2), Flim • is clear. PMHS because of Remark 3.8 and Theorem 3.10. That U• is opposite to Flim ∞ ˜ We are using the ∇-flat identification H˜ → H × M mentioned above to construct ˜ subbundles U p on H˜ . Because being opposite is an open extensions of the U p to ∇-flat • property, all F(t,w) are opposite to U• for (t, w) near 0. For N ∈ C we have because of Lemma 3.4,

N : I p,q −→ I p−1,q−1 . Because these N build an open cone in rj=1 R · N j all N j are linear combinations of such N . Therefore we have for all N j , N j : I p,q −→ I p−1,q−1 , and therefore N j : U p → U p−1 . Because U• is increasing U p is N j -invariant for all N dt N j . Because we have ∇˜ = ∇ + rj=1 2πij t j j we conclude that U• is also ∇-invariant.

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3.2. Construction of Frobenius manifolds. Section 3.2 is a generalization of chapter 5 of [HM]. Lemma 3.12. (a) The structures (α) and (β) are equivalent: (α) A logarithmic Frobenius type structure ((M, 0), D, K , ∇ r , C, U, V, g) together with an integer w such that U = 0 and V is semisimple with eigenvalues in w2 + Z, (β) A tuple ((M, 0), D, K , ∇, F • , U• , w, S). Here K → (M, 0) is a germ of a holomorphic vector bundle; ∇ is a flat connection with logarithmic poles along D; F • is a decreasing filtration by germs of holomorphic subbundles F p ⊂ K , p ∈ Z, which satisfies Griffiths transversality 1 ∇ : O(F p ) −→ Ω M (log D) ⊗ O(F p−1 );

U• is an increasing filtration by subbundles U p ⊂ K which are flat outside D such that K = F p ⊕ U p−1 =

F q ∩ Uq ;

(3.1)

q

w ∈ Z, and S is a ∇-flat, (−1)w -symmetric, nondegenerate pairing on K with S(F p , F w+1− p ) = 0, S(U p , Uw−1− p ) = 0.

(3.2) (3.3)

(b) One passes from (α) to (β) by defining ∇ := ∇ r + C,

w id : K −→ K , ker V − q − F := 2 q≥ p w id : K −→ K , U p := ker V − q − 2 q≤ p p

S(a, b) := (−1) p g(a, b) for a ∈ O(F p ∩ U p ), b ∈ O(K ).

(3.4) (3.5) (3.6) (3.7)

Proof. First we prove part (b). The connection ∇ r and the logarithmic Higgs field C are 1 (log D) ⊗ O(K ). Then the flatness ∇ r means (∇ r )2 = 0, the Higgs maps O(K ) → Ω M 2 field C satisfies C = 0, and the potentiality condition means ∇ r (C) = ∇ r ◦C+C◦∇ r = 0. Therefore ∇ 2 = (∇ r + C)2 = 0, the connection ∇ is flat and has logarithmic poles along D. The filtrations F • and U• obviously satisfy (3.1) and w id : K −→ K . F p ∩ U p = ker V − p − 2 The connection ∇ r maps O(F p ∩ U p ) to itself because V is ∇ r -flat. Because U = 0 we have [C, V] = C, which is equivalent to C X , X ∈ Der M (log D) mapping O(F p ∩ U p ) to O(F p−1 ∩ U p−1 ). Therefore U• is ∇-flat and F • satisfies Griffiths transversality. Obviously ∇ r and C have logarithmic poles along D. The condition (1.5) says that for a ∈ O(F p ∩ U p ), b ∈ O(F q ∩ Uq ), S(a, b) = (−1) p g(a, b) = 0 if p + q = w . Therefore S is (−1)w -symmetric and satisfies (3.2) and (3.3). To show that it is ∇-flat we have to choose a representant of ((M, 0), D, K , ∇ r ) which we denote by the same


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letters. Now let U ⊂ M\D open and simply connected. For X ∈ Der M (log D)(U ) and ∇ r -flat sections a ∈ O(F p ∩ U p )(U ), b ∈ O(F q ∩ Uq )(U ) we have (∇ X S)(a, b) = X S(a, b) − S(∇ rX a + C X a, b) − S(a, ∇ rX b + C X b) = 0 − (−1) p+1 g(C X a, b) − (−1) p g(a, C X b) = 0. This shows (b). To pass from (β) to (α) we notice that V is uniquely determined by w F p ∩ U p = ker V − p − id : K −→ K 2 and g(a, b), a ∈ O(F p ∩ U p ), b ∈ O(K ) is defined by (3.7). We can decompose ∇ into ∇ r mapping O(F p ∩ U p ) to itself and C X for X ∈ Der M (log D) mapping O(F p ∩ U p ) to O(F p−1 ∩ U p−1 ) respectively. This is possible because U• is flat and F • satisfies Griffiths transversality. Now we decompose ∇ 2 into its components. Thus we have (∇ r )2 = 0, ∇ r ◦ C + C ◦ ∇ r = 0 and C 2 = 0. The second is equivalent to condition (1.1) in the definition of logarithmic Frobenius structure and the third shows that C is a Higgs field with logarithmic poles. Condition (1.2) is satisfied because U = 0 and [C, V] = C per construction of C. The condition g(C X a, b) = g(a, C X b) follows from the very same calculation above. g(Va, b) = −g(a, Vb) follows right from the definitions. Remark 3.13. If one adds in (β) a real structure with suitable conditions one obtains a germ of a variation of polarized Hodge structures of weight w. Definition 3.14. (a) A germ of an H 2 -generated variation of filtrations of weight w, which has logarithmic poles at D, is a tuple ((M, 0), D, K , ∇, F • , w) with the following properties: w is an integer, K → (M, 0) is a germ of a holomorphic vector bundle with flat connection ∇ with logarithmic poles along D and a variation of filtrations F • which satisfies Griffiths transversality and 0 = F w+1 ⊂ F w ⊂ · · · ⊂ K , r kF w = 1, r kF w−1 = 1 + dim M ≥ 2 . Griffiths transversality and flatness of ∇ give a Higgs field C on the graded bundle p p+1 with commuting endomorphisms p F /F C X = [∇ X ] : O(F p /F p+1 ) −→ O(F p−1 /F p ) for X ∈ Der M (log D) . H2 -generation condition: The whole module p O(F p /F p+1 ) is generated by O(F w ) and its images under iterations of the maps C X , X ∈ Der M (log D). (b) A pairing and an opposite filtration for an H 2 -generated variation of filtrations ((M, 0), D, K , ∇, F • ) of weight w are a pairing S and a filtration U• as in Lemma (3.12). Definition 3.15. An H 2 -generated germ of a logarithmic Frobenius manifold of weight w ∈ N≥1 is a germ ((M, 0), D, ◦, e, E, g) of a logarithmic Frobenius manifold with the properties (I) and (II) below and with E |t=0 = 0.

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Let ∇ g be the Levi-Civita connection of ((M, 0), g). It has logarithmic poles along D by Proposition 1.5. The endomorphism ∇ g E : Der M (log D)0 → Der M (log D)0 , g X → ∇ X E acts on the space of ∇ g -flat logarithmic vector fields. In particular e ∈ ker (∇ g E − id). (I) It acts semisimply with eigenvalues {1, 0, . . . , −(w − 1)}. It turns out that then the multiplication on the algebra Der M (log D)0 respects the grading w

Der M (log D)0 =

ker (∇ g E − (1 − p)id : Der M (log D)0 −→ Der M (log D)0 ) p=0

and that Lie E (g) = (2 − w) · g. (II) H 2 -generation condition: The algebra Der M (log D)0 is generated by ker (∇ g E : Der M (log D)0 −→ Der M (log D)0 ). Theorem 3.16. There is a one-to-one correspondence between the structures in (α), (β) and (γ ). (α) A logarithmic Frobenius type structure ((M, 0), D, K , ∇ r , C, U, V, g) with U = 0 and with a fixed section ξ ∈ O(K )0 which is ∇ r -flat on M\D and which satisfies the conditions (IC),(GC) and (EC) in Theorem (1.12). (β) A germ of an H 2 -generated variation of filtrations with logarithmic pole along D((M, 0), D, K , ∇, F • , w, S, U• ) of weight w ∈ N≥1 with pairing and opposite filtration and with a fixed generator ξ ∈ O(F w ) such that ∇ξ ∈ O(Uw−1 ) ⊗ Der M (log D). ˜ 0), D, ˜ ◦, e, E, g) (γ ) An H 2 -generated germ of a logarithmic Frobenius manifold (( M, ˜ of weight w ∈ N≥1 . One passes from (α) to (β) by Lemma (3.12) and from (α) to (γ ) by Theorem (1.12). One passes from (γ ) to (α) by defining M := {t ∈ M˜ | E |t = 0}, D := D˜ ∩ (M, 0) , ˜ |(M,0) K := Der M˜ (log D) with the canonical logarithmic Frobenius type structure, and ξ := e |(M,0) . The eigenvector condition (EC) is Vξ = w2 ξ . In contrast to the corresponding theorem in [HM] where they could simply extend ξ |0 ∇ r -flat, we have to demand the existence of a section ξ which is flat on M\D. Proof. (α) ⇒ (β). We have to show that the conditions in Lemma 3.12 (α) are satisfied. For that we have to show that for some w ∈ N≥1 the endomorphism V + w2 id is semisimple with eigenvalues in {0, 1, . . . , w}. For the conditions on F • in Definition 3.14 we have to show Vξ = w2 ξ . The eigenvector condition says Vξ = d2 ξ for some d ∈ C. Condition (1.2) reads as [C, V] = C. This together with the generation condition )(GC) shows * that V acts semisimply on K 0 with eigenvalues in d2 + Z≤0 and that ker V − d2 id = C · ξ . The ) ) * * injectivity condition (IC) implies dim ker V − d2 − 1 id = dim M > 0. Condition


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* ) * ) (1.5) says that the eigenvalues of V are in d2 + Z≤0 ∩ − d2 + Z≤0 . Therefore d ∈ N. Define w := d. The conditions (IC) and (GC) show that one passes from (α) to (β) by Lemma (3.12). ∇ξ | M\D ∈ O(Uw−1 ) follows from ∇ r ξ | M\D = 0. (β) ⇒ (α). Lemma (3.12) gives a Frobenius type structure. It holds ∇ r ξ | M\D = 0 because ∇ξ ∈ O(Uw−1 ) ⊗ Der M (log D). We have to show (IC),(GC) and (EC). (IC) follows from the condition r kF w−1 = 1 + dim M and from the H 2 -generation condition. (GC) follows from the H 2 -generation condition. * Finally (EC)*followsw from the ) ) w p characterization q of V as F ∩U p = ker V − p − 2 id : K −→ K , r kF = 1 and K = q F ∩ Uq . (α) ⇒ (γ ). Theorem (1.12) applied to (α) gives a logarithmic Frobenius manifold ˜ 0), D, ◦, e, E, g), ˜ 0) and an isomorphism of (( M, ˜ an embedding i : (M, 0) → ( M, ˜ |i(M) . It maps V to the logarithmic Frobenius type structures j : K → Der M˜ (log D) 2−d g ˜ restriction of ∇ E − 2 id on Der M˜ (log D) |i(M) . Therefore one obtains an H 2 -generated germ of a Frobenius manifold of weight w. In suitable flat coordinates the Euler field of this Frobenius manifold takes the form E=

dim M˜ i=1

di ti

∂ ∂ti

with di ∈ {1, 0, . . . , −(w − 1)} and

d − 1 id = dim M. (i | di = 0) = dim ker (∇ g E) = dim ker V − 2

Observe that for logarithmic coordinates di = 0, because U = 0 on M. Therefore i(M) = {t ∈ M˜ | E |t = 0}. (γ ) ⇒ (α). One passes from (γ ) to (α) as described above. (IC) is fulfilled because ξ = e. (GC) is because the Frobenius manifold is H 2 -generated. (EC) is also from the definition.

3.3. Application to VPHS. In this section we apply the machinery developed above to VPHS and investigate some properties of the resulting Frobenius type structures. Proposition 3.17. Assume as above that we have given M = n , M\Z = (∗ )r ×n−r , r Z = j=1 D j , and a VPHS on a ∇-flat bundle H → M\Z with unipotent monodromies. Let H˜ → M be the Deligne extension of H → M\Z . Then one can construct a logarithmic Frobenius type structure and a (log D − tr T L E P(w))-structure out of this data. Proof. Set K := H˜ . Construct an increasing filtration by ∇-flat subbundles U p with the help of Lemma 3.11, which is opposite to the filtration F• . Now Lemma 3.4 gives us S(I p,q , I r,s ) = 0 for (r, s) = (w − p, w − q) and because U p = i,q:i≤ p I i,q we see that condition (3.3) (S(U p , Uw−1− p ) = 0) is fulfilled. Now we have data as in Lemma 3.12(a)(β) and therefore a logarithmic Frobenius type structure ((M, 0), Z , K , ∇ r , C, U, V, g, w) with U = 0 and V semisimple with eigenvalues in w2 +Z. Lemma 1.10 provides a (log D − tr T L E P(w))-structure.

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Proposition 3.18. 1. The connection ∇ r has trivial monodromy. 2. It can be extended holomorphically from H → M\Z to K → M. Proof. Recall the proof of Lemma 3.12 part (β) → (α). There the map ∇ : O(F p ∩ U p ) → (O(F p ∩ U p ) ⊕ O(F p−1 ∩ U p−1 )) ⊗ Der M (log D), was decomposed into ∇ r mapping to O(F p ∩ U p ) ⊗ Der M (log D) and C mapping to O(F p−1 ∩ U p−1 ) ⊗ Der M (log D) respectively. A different characterization of ∇ r and C would be the following: All U p are ∇-flat subbundles of K . Now ∇ induces flat connections ∇ (U ) on the quotients U p /U p−1 . Because F • and U• are opposite we have canonical isomorphisms F p ∩ U p U p /U p−1 and therefore U p /U p−1 p

F p ∩ Up = K . p

∇ (U )

goes over to ∇ r and C := ∇ − ∇ r . Because of T j = e N j Under this isomorphism and N j : U p → U p−1 , T j induces the identity on U p /U p−1 . Therefore also ∇ r has trivial monodromy. This shows (1). For (2) recall that U p is the Deligne extension of its restriction U p | M\Z on M\Z with respect to ∇. Therefore U p /U p−1 is the Deligne extension with respect to ∇ (U ) of its restriction to M\Z . Because of that and (1), ∇ (U ) has an extension to M. This shows (2). We will now give conditions on the PMHS on H˜ 0 so that the logarithmic Frobenius type structure constructed in Proposition 3.17 gives rise to a logarithmic Frobenius manifold. Theorem 3.19. Assume that we have given M = n and M\Z = (∗ )n and a VPHS on a ∇-flat bundle H → M\Z with unipotent monodromies Ti . Let H˜ → M be the • be part of the limiting MHS on H ˜ 0 with Deligne extension and Flim w+1 w ⊂ Flim ⊂ . . . ⊂ K, 0 = Flim w−1 w = 1, dim Flim = 1 + dim M. and dim Flim

Let Ni be the logarithm of the unipotent part of Ti acting on H˜ 0 . If the vector space p p+1 w p Flim /Flim is generated by Flim and its images under iterations of the linear maps Ni , then we can construct an H 2 -generated germ of a logarithmic Frobenius manifold out of these data. w . Recall Proposition 3.17 where we constructed a Proof. Pick a nonzero vector v ∈ Flim logarithmic Frobenius type structure ((M, 0), Z , K , ∇ r , C, U, V, g, w) out of the VPHS. Because of Proposition 3.18 we can extend this vector v to a ∇ r -flat section ξ ∈ O(K ). • and v ∈ F w give the conditions (I C) and (GC) in Theorem 1.12 The conditions on Flim lim resp. Theorem 3.16 with respect to the section ξ ∈ O(K ). The condition (EC) follows from the construction of the logarithmic Frobenius type structure in Lemma 3.12 and the existence of an opposite filtration U• on H˜ 0 = K 0 (cf. 3.11). • it follows that Remark 3.20. From the generation conditions and the properties of Flim the PMHS on H˜ 0 is Hodge-Tate and split over R. This can be easily seen by choosing v to be real and by noticing that every N j is a (−1, −1)-morphism defined on H˜ R,0 .


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References [CaK] [CaKSch] [CaFe] [CK] [De1] [De2] [Du] [EV] [FePe] [Fo] [He1] [He2] [He3] [HM] [KM] [Mal] [Ma] [Sab] [Sch]

Cattani, E., Kaplan, A.: Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structures. Invent. Math. 67, 101–115 (1982) Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. Math. 123, 457–535 (1986) Cattani, E., Fernandez, J.: Frobenius modules and Hodge asymptotics. Commun. Math. Phys. 238, 489–504 (2003) Cox, D.A., Katz, S.: Mirror symmetry and Algebraic Geometry. Mathematical Surveys and monographs, vol. 68, Providence, RI: Amer. Math. Soc., 1999 Deligne, P.: Théorie de Hodge. II. Publ. Math. I.H.E.S. 40, 5–57 (1971) Deligne, P.: Local behaviour of Hodge structures at infinity. In: Mirror Symmetry II (eds. Greene, B., Yau, S.-T.), Studies in Advanced Mathematics, volume 1, Providence, RI: AMS/IP, 1997, pp 683–699 Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and quantum groups (Montecatini, Terme 1993), (Francoviglia, M., Greco, S., eds.). Lecture Notes in Math. 1620, Berlin-Heidelberg-New York: Springer Verlag 1996, pp 120–348 Esnault, H., Viehweg, E.: Logarithmic de Rham complexes and vanishing theorems. Invent. Math. 86, 161–194 (1986) Fernandez, J., Pearlstein, G.: Opposite filtrations, variations of Hodge structure, and Frobenius modules. In: Frobenius manifolds, eds. Hertling, C., Marcolli, M., Aspects of Mathematics E 36, Wiesbaden: Vieweg, 2004, pp 19–34 Folland, G.: Introduction to partial differential equations. 2nd ed., Princeton, NJ: Princeton University Press, 1995 Hertling, C.: Frobenius manifolds and moduli spaces for singularities. Cambridge Tracts Math. 151, Cambridge: Cambridge University Press, 2002 Hertling, C.: tt* geometry, Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. 555, 77–161 (2003) Hertling, C.: Classifying spaces and moduli spaces for polarized mixed Hodge structures and for Brieskorn lattices. Compos. Math. 116, 1–37 (1999) Hertling, C., Manin, Yu.: Unfoldings of meromorphic connections and a construction of Frobenius manifolds. In: Frobenius manifolds, eds. Hertling, C., Marcolli, M., Aspects of Mathematics E36, Wiesbaden: Vieweg, 2004, pp 113–144 Kontsevich, M., Manin, Yu.: Gromov-Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994) Malgrange, B.: Deformations of differential systems, ii. J. Ramanujan Math. Soc. 1, 3–15 (1986) Manin, Yu.: Three constructions of Frobenius manifolds: a comparitive study. Asian J. Math 3, 179–220 (1999) Sabbah, C.: Déformation isomonodromique et variétés de Frobenius, une introduction. Centre des Mathematiques, Ecole Polytechnique, U.M.R. 7640 du C.N.R.S., no. 2000-05, 251 pages Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)

Communicated by N. A. Nekrasov

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