Communications in Mathematical Physics - Volume 301

Commun. Math. Phys. 301, 1–21 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1174-9 Communications in Mathe...

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Commun. Math. Phys. 301, 1–21 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1174-9

Communications in

Mathematical Physics

Quantum Symmetries and Exceptional Collections Robert L. Karp Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA. E-mail: [email protected] Received: 14 December 2008 / Accepted: 24 February 2010 Published online: 20 November 2010 – © Springer-Verlag 2010

Abstract: We study the interplay between discrete quantum symmetries at certain points in the moduli space of Calabi-Yau compactifications, and the associated identities that the geometric realization of D-brane monodromies must satisfy. We show that in a wide class of examples, both local and compact, the monodromy identities in question always follow from a single mathematical statement. One of the simplest examples is the Z5 symmetry at the Gepner point of the quintic, and the associated D-brane monodromy identity.

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . Monodromies as Autoequivalences . . . . . . . . 2.1 Fourier-Mukai functors . . . . . . . . . . . . 2.2 Monodromies in general . . . . . . . . . . . 2.3 Exceptional collections and autoequivalences Examples . . . . . . . . . . . . . . . . . . . . . . 3.1 C 2 /Z3 . . . . . . . . . . . . . . . . . . . . . 3.2 C 3 /Z5 . . . . . . . . . . . . . . . . . . . . . 3.3 A compact example: P49,6,1,1,1 [18] . . . . . . 3.3.1 Monodromy around the orbifold point. . . 3.3.2 Monodromy around the P2 point. . . . . 3.3.3 Monodromy around the LG point. . . . . Discussion . . . . . . . . . . . . . . . . . . . . .

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2 4 4 5 7 8 8 11 13 15 15 17 19

Present address: Department of Systems Biology, Harvard Medical School, 200 Longwood Avenue, Boston, MA 02115, USA.

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1. Introduction Studying B-type topological D-Branes using the derived category allows us to go beyond the picture of D-branes as vector bundles over submanifolds, and opens the window toward understanding various α -corrections. At the same time this technology is very efficient at studying certain problems, like the superpotential [1,2], which seem hard by traditional boundary conformal field theory (CFT) techniques. From the point of view of strings in string theory, the appearance of the derived category is intriguing, but D-branes mandate the categorical approach [3,4]. In particular, B-type topological D-branes are objects in the bounded derived category of coherent sheaves. The A-type D-branes have a very different description, involving the derived Fukaya category. Mirror symmetry exchanges the A and B branes, and naturally leads to Kontsevich’s homological mirror symmetry (HMS) conjecture. For a detailed exposition of these ideas we refer the reader to the recent book [5], or the review articles [6,7]. The fact that B-type D-branes undergo monodromy as one moves in the moduli space of complexified Kahler forms is expressed quite naturally in this language. This is in fact a surprisingly rich area, where the interplay between abstract mathematics (autoequivalences of derived categories) and string theory (discrete symmetries in CFT’s) is particularly evident. The main motivation of the present paper is to further our understanding in this area. To motivate our result we need to start with mirror symmetry in its pre-HMS phase. In this form mirror symmetry is an isomorphism between the (complexified) Kahler moduli space M K (X ) of a Calabi-Yau variety X and the moduli space of complex deformations Mc ( X ) of its mirror X . For the precise definitions we refer to the book by Cox and Katz [8]. The complexified Kahler moduli space M K (X ) is an intricate object, but for X a hypersurface in a toric variety it has a rich combinatorial structure and is relatively wellunderstood. In particular, the fundamental group of M K (X ) in general is non-trivial, and one can talk about various monodromy representations. More concretely, there are two types of boundary divisors in M K (X ): “large radius divisor” and the “discriminant”. Both are reducible in general. At a large radius divisor certain cycles of X (or X itself), viewed as a Kahler manifold, acquire infinite volume. The discriminant is somewhat harder to describe. The original definition is that the CFT associated to a string probing X becomes singular at such a point in moduli space. Generically this happens because some D-brane (or several of them, even infinitely many) becomes massless, and therefore the effective CFT description provided by the strings fails. A consequence of this fact is that, by using the mirror map isomorphism of the moduli spaces, as one approaches the discriminant in M K (X ) one is moving in Mc ( X ) to a point where the mirror X is developing a singularity. Armed with this picture of M K (X ), we can fix a basepoint O, and look at loops in M K (X ) based at O. Traversing such loops the D-branes will undergo monodromies, similarly to the BPS particles in Seiberg-Witten theory, and for topological B-branes this leads to non-trivial functors D(X ) → D(X ),1 which are in fact equivalences. Therefore we arrive at a group homomorphism, the monodromy representation, first suggested by Kontsevich:2 μ : π1 (M K (X )) −→ Aut(D(X )). 1 D(X ) will always denote the bounded derived category of the the variety (or smooth stack) X . 2 Kontsevich’s ideas were generalized by Horja and Morrison. We refer to [9] for more details on the history

of this topic.

Quantum Symmetries and Exceptional Collections

3

At present writing very little is known about μ. The question at hand is: given a pointed loop in M K (X ), what is the associated autoequivalence in D(X )? Progress in this direction was made in [9], where this question is answered for the EZ-degenerations introduced in [10]. It is clear now that given a presentation of π1 (M K (X )) where we know the images under μ of the generators, the relations in the presentation will determine interesting identities in Aut(D(X )). In particular, whenever one is at a point in moduli space which is an orbifold of some sort (like a Landau-Ginzburg orbifold), the moduli space locally is an orbifold itself. The fact that moduli spaces are in general stacks rather than varieties, precisely because of the appearance of additional automorphisms at different points, complicates matters a bit, as we will see in the example of the next paragraph. But it is clear that there are loops encircling the orbifold point in moduli space which are finite order. Therefore the associated monodromy operator in Aut(D(X )) has to satisfy an analogous relation. Understanding these relations in Aut(D(X )) is the goal of this paper. We will find that in a broad range of examples of “toric” Calabi-Yau varieties, both local and compact, finite orderness always follows from general statements concerning Seidel-Thomas twist functors and complete exceptional collections (Prop. 2.8 and Prop. 2.9 are two special cases). For illustration, let us look at the example of the quintic 3-fold in P4 . In this case the compactification M K (X ) of the Kahler moduli space M K (X ) is isomorphic to P1 . M K (X ) is also isomorphic to Mc ( X ), the complex structure moduli space of the mirror. In either of these moduli spaces we have three distinguished points: 1. PL V is the large volume limit point in M K (X ). It also corresponds to the large complex structure limit point (with maximally unipotent monodromy) in Mc ( X ). 2. P0 is the conifold point. Here the D6-brane wrapping X becomes massless, and therefore the effective CFT description breaks down. Alternatively, the mirror family X develops rational double points, in physics language conifolds, and is singular. 3. PLG is the Gepner point, and is a Landau-Ginzburg (LG) orbifold. At this point in moduli space the mirror X has the Fermat form, and has an additional Z5 automorphisms. In both formulations we see that at this point the moduli space has a stacky Z5 structure.3 Let M P denote the monodromy associated to a loop around the point P. Since PL V and P0 are the only limit points of M K (X ), and the compactification of this is isomorphic to P1 (see [8]), with π1 (P1 − {2 points}) = Z, one would want to conclude, incorrectly, that M PL V and M P0 are related. But as we discussed, PLG is a stacky point in the moduli space, with finite stabilizer, and so, at best, the 5th power of M PLG ∼ = M PL V ◦ M P0 is the identity. This was proposed by Kontsevich, who checked it in K-theory. Later Aspinwall [6] realized that in fact M P5 LG ∼ = (−)[2].

(1)

The authors of [11] observed that the exceptional collection OP4 , OP4 (1), . . . , OP4 (4) and its dual collection {kP4 (k)}4k=0 (kP4 is the k th wedge power of the holomorphic cotangent bundle) were implicit in Aspinwall’s proof, and this was the aspect of the 3 Mathematically the compactified moduli space M (X ) ∼ M ( = = P1 is only a coarse moduli space, c X) ∼ K while the moduli stack is P1 (5, 1). As a scheme P1 (5, 1) ∼ = P1 , but not as a stack.

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proof that gave a handle for generalizations. This observation allowed [11] to show that (1) generalizes to Calabi-Yau hypersurfaces in weighted projective spaces of arbitrary dimensions. More precisely, for the genericCalabi-Yau hypersurface X in weighted pro wi ∼ jective space Pnw0 ...wn , one has that M PLG = (−)[2], where M PLG = TO X ◦ LO X (1) in the notation of Sect. 2.2. The proof constructs a Beilinson resolution of the diagonal for Pnw0 ...wn using the full exceptional collection O, O(1), . . . , O( wi − 1) and its dual. Canonaco generalized this approach [12], and shows that given a full exceptional collection on a smooth stack, it leads to a Beilinson type resolution. This resolution is then used to prove Prop. 2.8 and Prop. 2.9. The aim of this paper is to understand the identities stemming from quantum symmetries in the case when M K (X ) is higher dimensional. The recurring feature in our investigations will be the existence of exceptional collections, either on divisors in the Calabi-Yau, or in the ambient space. Exceptional collections have appeared in the physics literature before [13,14]. They were first applied in the context of Landau-Ginzburg models in [15–17]. They reappeared in the AdS/CFT literature [18,19] in the context of quiver gauge theories [20]. Their role in determining the gauge theory living on D-branes placed at Calabi-Yau singularities was clarified in [21,22]. In this paper we show that they can also be used to establish the monodromy identities. The organization of the paper is as follows. In Sect. 2 we briefly review some of the Fourier-Mukai technology, which is used throughout the paper. In Sect. 2.2 we consider three very different examples with two dimensional moduli space, and show that in every instance, the monodromy identities that follow from the emergence of additional cyclic symmetries in moduli space always follow from Prop. 2.8 and Prop. 2.9. We conclude the paper with a discussion of how general our results are, and an outlook to possible generalizations. 2. Monodromies as Autoequivalences We start this section with a brief review of Fourier-Mukai functors. Then we express the various monodromy actions on D-branes in terms of Fourier-Mukai equivalences. 2.1. Fourier-Mukai functors. For the convenience of the reader we review some of the key notions concerning Fourier-Mukai functors, and at the same time specify the conventions used. We will make extensive use of this technology in the rest of the paper. Our notation follows [23]. Given two non-singular proper algebraic varieties (or smooth Deligne-Mumford stacks), X 1 and X 2 , an object K ∈ D(X 1 × X 2 ) determines a functor of triangulated categories K : D(X 1 ) → D(X 2 ) by the formula4 L K (A) := R p2∗ K ⊗ p1∗ (A) , 4 R p is the total right derived functor of p , i.e., it is an exact functor from D(X ) to D(X ). Similarly, 2∗ 2∗ L

⊗ is the total left derived functor of ⊗. Most of the time these decorations will be omitted.


5

where pi : X × X → X is projection to the i th factor: X1 × X2 p1

p2

X 2.

X1

The object K ∈ D(X 1 × X 2 ) is called the kernel of the Fourier-Mukai functor K . It is convenient to introduce the external tensor product of two objects A ∈ D(X 1 ) and B ∈ D(X 2 ) by the formula L

A B = p2∗ A ⊗ p1∗ B. The importance of Fourier-Mukai functors when dealing with derived categories stems from the following theorem of Orlov (Theorem 2.18 in [24]), later generalized for smooth quotient stacks associated to normal projective varieties [25]). Theorem 2.1. Let X 1 and X 2 be smooth projective varieties. Suppose that F : D(X 1 ) → D(X 2 ) is an equivalence of triangulated categories. Then there exists an object K ∈ D(X 1 × X 2 ), unique up to isomorphism, such that the functors F and K are isomorphic. The first question to ask is how to compose Fourier-Mukai (FM) functors. Accordingly, let X 1 X 2 and X 3 be three non-singular varieties, while let F ∈ D(X 1 × X 2 ) and G ∈ D(X 2 × X 3 ) be two kernels. Let pi j : X 1 × X 2 × X 3 → X i × X j be the projection map. A well-known fact is the following: Proposition 2.2. The composition of the functors F and G is given by the formula L ∗ ∗ G ◦ F H , where H = R p13∗ p23 (G) ⊗ p12 (F) . Proposition 2.2 shows that composing two FM functors gives another FM functor, with a simple kernel. Now we have all the technical tools ready to study the monodromy actions of physical interest. 2.2. Monodromies in general. As discussed in the Introduction, the moduli space of CFT’s contains the moduli space of Ricci-flat Kahler metrics. This, in turn, at least locally has a product structure, with the moduli space of Kahler forms being one of the factors. This is the moduli space of interest to us. This space is a priori non-compact, and its compactification consists of two different types of boundary divisors. First we have the large volume divisors. These correspond to certain cycles being given infinite volume. The second type of boundary divisors are the irreducible components of the discriminant. In this case the CFT becomes singular. Generically this happens because some D-brane (or several of them, even infinitely many) becomes massless at that point, and therefore the effective CFT description breaks down. For the quintic this breakdown happens at the well known conifold point. The monodromy actions around the above divisors are understood to some extent. An extensive treatment of monodromies in terms of Fourier-Mukai functors was given in [9]. We will review now what is known.

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Large volume monodromies are shifts in the B field: “B → B + 1”. If the Kahler cone is higher dimensional, then we need to be more precise, and specify a two-form, or equivalently a divisor D. Then the monodromy becomes B → B + D. We will have more to say about the specific D’s soon. The simplest physical effect of this monodromy on a D-brane is to shift its charge, and this translates in the Chan-Paton language into tensoring with the line bundle O X (D). This observation readily extends to the derived category: Proposition 2.3. The large radius monodromy associated to the divisor D is L

L D (B) = B ⊗ O X (D),

for all B ∈ D(X ).

Furthermore, this is a Fourier-Mukai functor L , with kernel L = δ∗ O X (D), where δ : X → X × X is the diagonal embedding. Now we turn our attention to the conifold-type monodromies. For this we need to introduce the Fourier-Mukai functor with kernel Cone (A∨ A → O ), where O = δ∗ O X , and for A ∈ D(X ) its derived dual is A∨ = RHomD(X ) (A, O X ). By Lemma 3.2 of [26], for any B ∈ D(X ): L ∼ Cone(A∨ A→O ) (B) = Cone HomD(X ) (A, B) ⊗ A −→ B . Since the functor Cone(A∨ A→O ) will play a crucial role, we introduce a notation for it: L TA := Cone(A∨ A→O ) , TA (B) = Cone HomD(X ) (A, B) ⊗ A −→ B . (2) The functor TA is sometimes referred to as the Seidel-Thomas twist functor. Returning to conifold-type monodromies, we have the following conjecture from [9]: Conjecture 2.4. If we loop around a component of the discriminant locus associated with a single D-brane A becoming massless, then this results in a relabeling of D-branes by applying TA . The question of when is TA an autoequivalence has a simple answer. For this we need the following definition: Definition 2.5. Let X be smooth projective Calabi-Yau variety (stack) of dimension n. An object E in D(X ) is called n-spherical if ExtrD(X ) (E, E) ∼ = Hr (S n , C), that is C for r = 0, n and zero otherwise. One of the main results of [26] is the following: Theorem 2.6 (Prop. 2.10 in [26]). If the object E ∈ D(X ) is n-spherical, then the functor TE is an autoequivalence. Let us mention at this point that in the rest of the paper whenever we have an expression involving the functor TE , then E will always be spherical.


7

2.3. Exceptional collections and autoequivalences. Exceptional collections play a surprisingly central role concerning the monodromy identities that we are to discuss. First we recall some facts about them, then we will list two propositions from [12] which constitute the technical backbone of this paper. Definition 2.7. Consider the bounded derived category of coherent sheaves D(X ) on the algebraic variety (smooth stack) X . 1. An object E ∈ D(X ) is called exceptional if Extq (E, E) = 0 for q = 0 and Ext 0 (E, E) = C. 2. An exceptional collection (E1 , E2 , . . . , En ) in D(X ) is an ordered collection of exceptional objects such that Extq (Ei , E j ) = 0,

for all q, whenever i > j.

3. An exceptional collection (E1 , E2 , . . . , En ) in D(X ) is complete or full if it generates D(X ). In other words, the minimal full triangulated subcategory containing all the n objects {Ei }i=1 is D(X ) itself. Full subcategory in the last sentence is meant in the usual categorical sense. For more details see, e.g., [27]. The existence of a full and strong exceptional collection for a given variety X constrains the structure of X considerably. In particular, no smooth projective (and therefore compact) Calabi-Yau variety admits such a collection; the obstruction comes from Serre duality. The exceptional collections we are interested in are constructed on an exceptional divisor or the ambient space, i.e., where the Calabi-Yau is embedded, rather than on X itself. Now we turn to two propositions that will be used repeatedly in the remainder of this paper. As we will see, in all the examples considered, the monodromy identities dictated by CFT can always be explained using these two statements. It is also gratifying to remark that these statements were motivated by precisely the kind of identities that they are now used to prove. More precisely, these statements were formulated by Alberto Canonaco as a consequence of trying to prove with the author the conjectures substantiated in [28] and [23]. Let X and Y be smooth varieties/stacks, with canonical bundles ω X resp. ωY . Also assume that we have a full exceptional collection in D(Y ) : E1 , E2 , . . . , Em . There are two cases to consider. The first is when X is a Calabi-Yau hypersurface in Y . In this case we have that Proposition 2.8 (Corollary 5.2. in [12]). If i : X → Y is the inclusion of a hypersurface such that ωY ∼ = OY (−X ), then ∼ LO (−X )[2] . Ti ∗ E0 ◦ · · · ◦ Ti ∗ Em = X In the second case Y is a hypersurface in X , where X is Calabi-Yau; and we have an analogous statement. Proposition 2.9 (Corollary 5.6. in [12]). If j : Y → X is the inclusion of a hypersurface such that j ∗ ω X ∼ = OY , then T j∗ E0 ◦ · · · ◦ T j∗ Em ∼ = LO X (Y ) . Note that if X is Calabi-Yau, then the condition j ∗ ω X ∼ = OY is automatically satisfied.

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3. Examples 3.1. C 2 /Z3 . In this section we focus on the C2 /Z3 geometric orbifold and the associated CFT. In this case there is only one supersymmetric Z3 action (z 1 , z 2 ) → (ωz 1 , ω2 z 2 ),

ω3 = 1.

First let us review some of the findings and notations of [28], then generalize them. The crepant resolution of the singularity, denoted by X , has a reducible exceptional divisor. The toric fan of the resolved space X is shown in Fig. 1. Let Ci denote the divisor associated to the vertex vi , which in this case is a curve. The exceptional locus of the blow-up consists of the divisors C3 and C4 , both −2 curves. The curves C3 and C4 are the generators of the Mori cone of effective curves. The Kahler cone is dual to the Mori cone, and both are two dimensional. The Poincaré duals of the curves Ci are denoted by Di . Since we are in two complex dimensions, an irreducible divisor is a curve. This leads to potential confusion. To avoid it, the reader should remember that Ci lives in the second homology H2 (X, Z), while Di lives in the second cohomology H2 (X, Z). Now let us look at the moduli space of complexified Kahler forms. The point-set spanned by the rays of the toric fan, A = {v1 , . . . , v4 }, admits four triangulations, i.e., in the language of the gauged linear sigma model we have four phases. The secondary fan is depicted in Fig. 2, which is the toric fan of the Kahler moduli space. The four phases are the completely resolved smooth phase; the two phases where one of the P1 ’s has been blown up to partially resolve the Z3 fixed point to a Z2 fixed point; and finally the Z3 orbifold phase. The orbifold points in the moduli space are themselves singular points. This fact is related to the quantum symmetry of an orbifold theory. For either of the Z2 points, one has a C2 /Z2 singularity with weights (1, −1), while for the Z3 point the moduli space locally is of the form C2 /Z3 , with weights (1, 2). The four maximal cones of Fig. 2, C1 ,…,C4 , correspond to the four distinguished phase points. The four edges correspond to curves in the moduli space, denoted L1 , . . . , L4 . These are all weighted projective lines, all isomorphic to P1 as varieties, but not as stacks. The four curves connecting the different phase points are sketched in Fig. 3, together with the discriminant locus of singular CFT’s. The discriminant 0 intersects the four lines transversely. We depicted this fact in Fig. 3 using short segments.

Fig. 1. The toric fan for the resolution of the C2 /Z3 singularity


9

Fig. 2. The phase structure of the C2 /Z3 model

Fig. 3. The moduli space of the C2 /Z3 model

When talking about monodromy there are two cases to be considered. One can loop around a divisor, i.e., real codimension two objects; or one can loop around a point inside a complex curve. Of course the two notions are not unrelated. Our interest will be the second type of monodromy: looping around a point inside a curve.5 It was shown in [28] that monodromy around the Z2 point inside L1 is MZ2 = Ti∗ OC3 ◦ L D2 ,

(3)

while monodromy around the Z2 point inside L2 is MZ2 = Ti∗ OC4 ◦ L D1 .

(4)

Here i : C3 → X resp. j : C4 → X are the embedding maps, while C3 and C4 are the exceptional divisors defined at the beginning of the subsection. Finally, monodromy 5 As explained by many authors, since L is not part of the moduli space, rather it is only a compactification i divisor, one cannot consider loops inside it. Instead, the loops in question are infinitesimally close to being in Li .

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R. L. Karp

inside L3 around the Z3 point is given by MZ3 = T j∗ OC4 ◦ MZ2 = T j∗ OC4 ◦ Ti∗ OC3 ◦ L D2 .

(5)

Using an approach analogous to the one deployed in [11], [28] showed that (MZ2 )2 = L D1

(MZ2 )2 = L D2 .

(6)

These identities were reproved in [12] using Proposition 2.8. [28] also conjectured that (MZ3 ) 3 ∼ = idD(X ) ,

(7)

and checked this statement at the level of Chern characters (idD(X ) is the identity functor of D(X )). Here we provide a proof for (7), which is different from the one in [12] by being more closely tied with the physics of the example, and which will pave the way for a similar result presented in the sequel, which of course are not proven in [12]. Canonaco’s proof can be found on p. 14 of [12], after the proof of Corollary 5.6. We start by recalling Lemma 8.21 from [29], which shows how to conjugate a Seidel-Thomas twist functor by an autoequivalence. Lemma 3.1. If F ∈ D(X ) and G is an autoequivalence of D(X ), then G ◦ TF ∼ = TG(F ) ◦ G. First we simplify (MZ3 )2 using Lemma 3.1:6 MZ23 = MZ3 ◦ (TOC4 ◦ MZ2 ) ∼ = TMZ3 (OC4 ) ◦ MZ3 ◦ MZ2 = TMZ3 (OC4 ) ◦ TOC4 ◦ MZ22 . As it was shown in Sect. 4.2 of [28], MZ3 (OC4 ) = OC3 ; while (6) shows that MZ22 ∼ = L D1 , therefore MZ23 ∼ = TOC3 ◦ TOC4 ◦ L D1 = TOC3 ◦ MZ2 ,

(8)

where MZ2 was defined in (4). Now observe that MZ3 = TOC4 ◦ MZ2 = MZ2 ◦ L−1 D1 ◦ MZ2 .

(9)

Therefore (8) and (9) imply that MZ33 ∼ = TOC3 ◦ (MZ2 ) 2 ◦ L−1 D1 ◦ MZ2 . But (MZ2 ) 2 ∼ = L D2 by (6), while TOC3 ◦ L D2 = MZ2 . Thus ∼ MZ33 ∼ = MZ2 ◦ L−1 D1 ◦ MZ2 = idD(X ) , where in the last relation we used (6) again. The idea that one has to take away from this proof is the concept, rather than the manipulations. One starts out with MZ3 , and then rewrites its third power in such a way that the already established identities in (6) can be used. Note also that both identities in (6) were needed, along with two of the three fractional branes, OC3 and OC4 , and the fact that the Z3 monodromy permutes the fractional branes, i.e., MZ3 (OC4 ) = OC3 from [28]. The same philosophy will guide the proof of the next subsection. 6 For brevity we omit i and j from T i ∗ OC and T j∗ OC . 3

4


11

Fig. 4. The toric fan for the resolution of the C3 /Z5 singularity

3.2. C 3 /Z5 . Let us now turn to the example of C 3 /Z5 , as treated in [23]. First we review the relevant toric geometry of C3 /Z5 , then the moduli space of complexified Kahler forms. Throughout, we follow closely the notation of [23]. Once again, there is a unique supersymmetric Z5 action, i.e., Z5 ⊂ SL(3, Z): (z 1 , z 2 , z 3 ) → (ωz 1 , ωz 2 , ω3 z 3 ),

ω5 = 1.

The toric fan of the resolved space X is the cone over Fig. 4. We denote the divisor associated to vi by Di . The exceptional locus of the blow-up is reducible, with two irreducible components corresponding to v4 and v5 . Toric geometry immediately tells us that the divisor D4 is a P2 , while D5 is the Hirzebruch surface F3 . Let f be the fiber of F3 (e.g., the cone generated by v1 and v5 ), and h its −3 section (the cone generated by v4 and v5 ). At the same time, h is the hyperplane class of P2 , while f does not intersect P2 . The curves h and f are the generators of the Mori cone of effective curves. Shrinking h also shrinks the divisors D4 , and hence gives a Type II contraction, while shrinking f collapses the Hirzebruch surface F3 onto its base, giving a Type III degeneration. There are again four phases. The secondary fan is depicted in Fig. 5. The four phases are as follows: the completely resolved smooth phase; the two phases where one of the compact divisors D4 or D5 has been blown up to partially resolve the Z5 fixed point; and finally the Z5 orbifold phase. The phase corresponding to the cone C2 can be reached from the smooth phase C1 by blowing down the divisor D5 . This creates a line of Z2 singularities in the Calabi-Yau. We will refer to this phase as the Z2 phase. Similarly, the phase C3 is reached by blowing down the divisor D4 , and creates a C3 /Z3 singularity. We call this the Z3 phase. The orbifold points in the moduli space are themselves singular points. The Z2 point is a C2 /Z2 singularity with weights (1, −1), while the Z3 point for the moduli space locally is of the form C2 /Z3 , with weights (1,2). We have already seen that the four maximal cones C1 ,…, C4 in Fig. 5 correspond to the four distinguished phase points. Similarly, the four rays correspond to curves in the moduli space, and once again are weighted projective lines: L1 , . . . , L4 . The discriminant intersects L1 tangentially, while it is transverse to the other Li ’s. We depicted this fact in Fig. 6 using a parabola and resp. short segments.

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Fig. 5. The phase structure of the C3 /Z5 model

Fig. 6. The moduli space of the C3 /Z5 model

It was shown in [23] that monodromy around the Z2 point inside L1 is7 MZ2 = T j∗ O D5 (− f ) ◦ T j∗ O D5 ◦ L D2 ,

(10)

and monodromy around the Z3 point inside L2 is MZ3 = Ti∗ O D4 ◦ L D1 ,

(11)

while monodromy inside L3 around the Z5 point is given by MZ5 = Ti∗ O D4 ◦ MZ2 .

(12)

In [23] it was conjectured that (MZ2 )2 = L D1

(MZ3 )3 = L D2 ,

(13)

and the statements were checked at the level of Chern characters. These identities were proved in [12] using Proposition 2.8, and full exceptional collections on F3 resp. P2 . [23] also conjectured that 7 Once again, i and j are the embeddings, and will be omitted later on.


13

Proposition 3.2. MZ55 ∼ = idD(X ) . Proof. We prove the statement using (13), and the philosophy of the previous subsection. For this we first rewrite MZ25 in a more convenient form using Lemma 3.1: MZ25 = MZ5 ◦ (TO D4 ◦ MZ2 ) ∼ = TMZ5 (O D4 ) ◦ MZ5 ◦ MZ2 = TMZ5 (O D4 ) ◦ TO D4 ◦ MZ22 . (14) But MZ22 ∼ = L D1 by (13), while TO D4 ◦ L D1 = MZ3 . Furthermore, Sect. 4.1.3 of [23] proves that MZ5 (O D4 ) = O D5 ( f ). (As shown in [23], both are fractional branes, and this is just the statement that the Z5 monodromy permutes the fractional branes.) Therefore MZ25 ∼ = TO D5 ( f ) ◦ MZ3 .

(15)

A short computation shows that MZ3 (O D5 ( f )) = O D5 (2 f ). Using this fact, (15) and Lemma 3.1, we have that MZ45 ∼ = TO D5 ( f ) ◦ TO D5 (2 f ) ◦ MZ23 . On the other hand it is easy to verify that −1 TO D5 ( f ) ◦ TO D5 (2 f ) ∼ = L D2 1 ◦ MZ2 ◦ L−2 D1 ◦ L D2 .

Consequently, −1 2 MZ55 = (TO D4 ◦ MZ2 ) ◦ (L D2 1 ◦ MZ2 ◦ L−2 D1 ◦ L D2 ) ◦ MZ3 −1 2 ∼ = TO D4 ◦ (MZ2 ◦ L D2 1 ◦ MZ2 ) ◦ (L−2 D1 ◦ L D2 ) ◦ MZ3 ∼ = TO ◦ L 3 ◦ (L−2 ◦ L−1 ) ◦ M 2 D4

= (TO D4

D1

D1 D2 −1 2 ◦ L D1 ) ◦ L D ◦ MZ 3 2

Z3

2 ∼ = MZ3 ◦ L−1 D2 ◦ MZ3 = idD(X ) .

To go from line one to line two we used the fact that composition of Fourier-Mukai functors is associative. To go to line three we used the first relation in (13). In the last line we used (11), and the second relation in (13). 3.3. A compact example: P49,6,1,1,1 [18]. The degree 18 hypersurface in the weighted projective space P49,6,1,1,1 is a well studied example of a Calabi-Yau 3-fold with two dimensional complexified Kahler moduli space. We will follow the notation of [30], where the relevant identities that are the subject of this paper were partly checked at K-theory level. We will lift these identities to the derived category, and prove them. This example is particularly interesting, since both Proposition 2.8 and 2.9 will be needed in the proof, while the examples studied so far used only one of them per example. First we review the geometry of the blown-up P49,6,1,1,1 , which we call Z , as the details will be important in the sequel. Z is particularly easy to describe torically: its fan has the same rays, v1 , . . . , v5 , as P49,6,1,1,1 , and one additional ray corresponding to the blow-up, v6 : v1 = (1, 0, 0, 0), v2 = (0, 1, 0, 0), v3 = (0, 0, 1, 0), v4 = (0, 0, 0, 1), v5 = (−9, −6, −1, −1), v6 = (−3, −2, 0, 0).

(16)

14

R. L. Karp

Fig. 7. The moduli space of the P49,6,1,1,1 [18] model

The weighted projective space P49,6,1,1,1 has a curve of Z3 singularities, located at the points [z 1 , z 2 , 0, 0, 0], where [z 1 , . . . , z 5 ] are homogeneous coordinates on P49,6,1,1,1 . Locally these are of the form C3 /Z3 . We blow up this curve by introducing the ray v6 , which satisfies the additional relation: v3 + v4 + v5 − 3v6 = 0.

(17)

Let di denote the divisor corresponding to the ray vi . It is clear from (16) that we have the following linear equivalence relations: d1 ∼ 3h, d2 ∼ 2h, d3 ∼ d4 ∼ d5 ∼ l, d6 ∼ e, e ∼ h − 3l.

(18)

The generic degree 18 hypersurface in P49,6,1,1,1 intersects the curve of Z3 singularities in one point, and hence is singular. Blowing up P49,6,1,1,1 resolves the singularity of the hypersurface as well. This is achieved torically by considering a generic anti-canonical hypersurface X in Z , the blow-up of P49,6,1,1,1 discussed before. It follows from (16) that in the notation of (18) −K Z = 6h, thus X is a generic element in the linear system |6h|. The exceptional divisor e of Z intersects X in a P2 , which we call E, and X is elliptically fibered over E. Let H resp. L denote the restriction of the divisors h resp. l of Z to X . As a consequence of (18) and the fact that −K Z = 6h, on X we have the linear equivalence relations E ∼ H − 3L ,

X ∼ 6H.

(19)

This model has four phases (depicted in Fig. 7):8 1. the smooth Calabi-Yau phase; 2. an orbifold phase, whose limit point has the orbifold singularity C3 /Z3 , but the Calabi-Yau has infinite volume; 3. a P2 phase, where the elliptic fibration X collapses onto its base P2 . In the limit, this elliptic fiber has zero area and the P2 has infinite volume; 4. a Landau-Ginzburg (LG) phase with the Gepner point as the limit point. The discriminant locus of singular CFT’s is reducible, with irreducible components 0 and 1 . These intersect the four lines of interest in the way depicted in Fig. 7: the 8 See Sect. 4 of [30] for more details.


15

intersections at L1 and L3 are simple transverse; 0 and 1 meet L4 at the same point; while L2 and 0 meet at third order. We have three interesting monodromy identities to consider: at the orbifold point, at the P2 point, and at the LG point. We start with the orbifold point. 3.3.1. Monodromy around the orbifold point. It was argued in [30] that monodromy inside L1 around the orbifold point is given by M1 = Ti∗ O E ◦ L L ,

(20)

where i is the embedding of the exceptional divisor i : E → X . Guided by the Z3 quantum symmetry of the C3 /Z3 orbifold, which is created by blowing down E, [30] shows that

ch (M1 )3 (O X ) = e H . Now we will show that indeed Proposition 3.3. (M1 )3 ∼ = LH . Proof. We start by rewriting (M1 )3 in a more convenient form using the definition (20) and Lemma 3.1: (M1 )3 = (Ti∗ O E ◦ L L ) ◦ (Ti∗ O E ◦ L L ) ◦ (Ti∗ O E ◦ L L ) ∼ = Ti∗ O E ◦ TL L (i∗ O E ) ◦ TL2 (i∗ O E ) ◦ L3L . L

(21)

But L L (i ∗ O E ) = i ∗ (O E (i ∗ L)), and it is easy to see, particularly in the toric presentation, that the divisor L restricts to the hyperplane divisor on E ∼ = P2 , i.e., L L (i ∗ O E ) = 2 i ∗ O E (1); and similarly L L (i ∗ O E ) = i ∗ O E (2). But O, O(1), O(2) is a full exceptional collection on P2 . Using Prop. 2.9, Eq. (21) becomes (M1 )3 ∼ = LO X (E) ◦ L3L = L E+3L = L H . In the last equality we used the linear equivalence (19).

3.3.2. Monodromy around the P2 point. It was shown in [30] that monodromy inside L2 around the P2 point is given by M2 = L H ◦ L−2 L ◦ TO X ◦ L L ◦ TO X ◦ L L ◦ TO X .

(22)

Passing from the Calabi-Yau point to the P2 point represents collapsing a large radius elliptic fiber to an LG orbifold theory, which is a Z6 -orbifold, and thus has a Z6 quantum symmetry. This motivated [30] to prove that

ch (M2 )6 (O X ) = 1. Lifting this to the derived category, we have the following: Proposition 3.4. (M2 )6 ∼ = (−)[2].

16

R. L. Karp

Proof. We first rewrite M2 using the definition (22) and Lemma 3.1: M2 = L H ◦ TO X (−2L) ◦ TO X (−L) ◦ TO X . Then once again using Lemma 3.1, (M2 )6 = L H ◦ TO X (−2L) ◦ TO X (−L) ◦ TO X ◦ TO X (H −2L) ◦ TO X (H −L) ◦ TO X (H ) 5 ◦ · · · ◦ TO X (5H −2L) ◦ TO X (5H −L) ◦ TO X (5H ) ◦ L H . (23) Recall that Z denotes the resolved weighted projective space P49,6,1,1,1 , i.e., the ambient space where X is embedded as a smooth hypersurface, as discussed at the beginning of Sect. 3.3, and also recall the toric divisors l and h on Z from (18). We have the following lemma Lemma 3.5. O Z (−2l), O Z (−l), O Z , O Z (h − 2l), O Z (h − l), O Z (h), . . . , O Z (5h − 2l), O Z (5h − l), O Z (5h) is a full exceptional collection on Z . Proof of the lemma. First note that both P49,6,1,1,1 and its blow-up Z are singular toric varieties, and hence we need to work with them as smooth stacks. For convenience we tensor each element of the collection by O Z (2l), which of course is irrelevant for exceptionality. The task now is to prove that O Z , O Z (l), O Z (2l), O Z (h), O Z (h + l), O Z (h + 2l), . . . , O Z (5h), O Z (5h + l), O Z (5h + 2l)

(24)

is exceptional. The obvious route is to compute the Ext groups by standard toric techniques available for line bundles on toric varieties, a straightforward but tedious work. Instead, we will use a recent result by Kawamata [31] which we paraphrase for the convenience of the reader. Section 5 of [31] considers a toric divisorial contraction φ : X → Y, with exceptional divisor E, where X and Y are toric stacks. φ : X → Y is also known as the blow-up map. n Let Ei be the prime divisors on Y corresponding to the rays {vi }i=1 of the toric fan, n define prime which in turn define the toric stack Y. Since X is a blow-up, the rays {vi }i=1 divisors on X as well, which we call Di . Of course, X has one more prime divisor, the exceptional divisor Dn+1 = E, corresponding to the additional ray vn+1 of the blow-up. The contraction morphism is described by an equation a1 v1 + · · · + an+1 vn+1 = 0,

(25)

for integers ai . In general there is no morphism of stacks X → Y, but there is still a fully faithful functor : D(Y) → D(X ). Kawamata proves the following isomorphism: n n n −1 OY ( ki Ei ) ∼ ki Di + Nk E , where Nk = ai ki . = OX an+1 i=1

i=1

i=1

(26) x is the integer part of the rational number x (the floor function).


17

Returning to our problem, it is known that O, O(1), . . . O(17)

(27)

is a full exceptional collection on the toric stack P49,6,1,1,1 . We will now show that (24) is

the image of (27) under . The exceptionality of the sequence (24) then follows immediately, since is a fully faithful functor. For this we use (26) with the role of (25) being played by (17): (OP4

9,6,1,1,1

1 (i)) ∼ = O Z (il + Ni e), where Ni = i. 3

(28)

Running through the index set 0, 1, . . . , 17, and using the linear equivalence e ∼ h − 3l from (18) completes the proof of exceptionality. The fact that (24) is full follows from Theorem 5.2(1) of [31]. Returning to the proof of Prop. 3.4, in particular Eq. (23), first recall that L and H are the restrictions of the toric divisors l and h on Z . Therefore, O X (α H + β L) = j ∗ O Z (αh + βl), for all α, β ∈ Z, where j : X → Z is the embedding. Using Lemma 2.8 and Lemma 3.5, Eq. (23) becomes (M2 )6 ∼ = L H ◦ LO X (−X )[2] ◦ L5H = L6H −X [2] = (−)[2]. In the last equality we used again the linear equivalence (19).

3.3.3. Monodromy around the LG point. Monodromy inside L3 around the LG point was shown to be [30] M3 = TO X ◦ M1 .

(29)

The fact that the LG point is a Z18 orbifold motivated [30] to prove that

ch (M3 )18 (O X ) = 1. But in fact more is true: Proposition 3.6. (M3 )18 ∼ = (−)[2]. Proof. Using the definition (29) and Lemma 3.1 we have that (M3 )3 = (TO X ◦ M1 ) ◦ (TO X ◦ M1 ) ◦ (TO X ◦ M1 ) ∼ = TO X ◦ TM1 (O X ) ◦ TM2 (O X ) ◦ M31 . 1

Using Prop. 3.3, we have that (M3 )3 ∼ = TO X ◦ TM1 (O X ) ◦ TM2 (O X ) ◦ L H . 1

(30)

To proceed, we need to compute M1 (O X ) and M21 (O X ). Let’s start with M1 (O X ). From its definition in (20), M1 (O X ) = Ti∗ O E (O X (L)). From the definition (2) it’s clear that we need to compute HomD(X ) (i ∗ O E , O X (L)). For later convenience we compute HomD(X ) (i ∗ O E , O X (k L)) for all k ∈ Z.

18

R. L. Karp

Lemma 3.7. HomaD(X ) (i ∗ O E , O X (k L)) = Ha (P2 , OP2 (k−3)) for all a ∈ Z, and k ∈ Z. Proof of the lemma. First observe that i ∗ O E = Ri ∗ O E , and use the fact that i ! is the right adjoint functor of Ri ∗ : HomD(X ) (i ∗ O E , O X (L)) = HomD(E) (O E , i ! O X (L)).

(31)

On the other hand, for i : E → X an embedding of a divisor, and X a Calabi-Yau, L

i ! O X (k L) = Li ∗ O X (k L) ⊗ ω E ∼ = O E (k − 3)

(32)

(see, e.g., [10] or the Appendix of [2] for some properties of i ! ). Since E ∼ = P2 , (31) and (32) imply that HomD(X ) (i ∗ O E , O X (k L)) = H∗ (P2 , OP2 (k − 3)). Using the lemma, and the fact that Hi (P2 , OP2 (−2)) = Hi (P2 , OP2 (−1)) = 0 for all i ∈ Z, we immediately see that M1 (O X ) = O X (L), M21 (O X ) = O X (2L). Thus (30) becomes (M3 )3 ∼ = TO X ◦ TO X (L) ◦ TO X (2L) ◦ L H . Repeatedly using Lemma 3.1 (by moving L H to the right) gives (M3 )18 ∼ = TO X ◦ TO X (L) ◦ TO X (2L) ◦ TO X (H ) ◦ TO X (H +L) ◦ TO X (H +2L) 6 ◦ · · · ◦ TO X (5H ) ◦ TO X (5H +L) ◦ TO X (5H +2L) ◦ L H . But this is the exceptional collection (24) in the proof of Lemma 3.5, and therefore Lemma 2.8 gives that ∼ LO (−X )[2] ◦ L6 = L6H −X [2] = (−)[2]. (M3 )18 = H X Let us mention that the direct approach used in proving Prop. 3.6 does not work for proving (7) or Prop. 3.2. The reason is that we have sheaves supported on different divisors in those cases, and the analogs of the steps in the proof of Prop. 3.6 do not lead to an exceptional collection, and we are led to use a cleverer approach.


19

4. Discussion In the light of our results, the reader will naturally ask the question of how generic are these type of results? On the physics side, given a Calabi-Yau compactification, we expect discrete symmetries to arise at various points in moduli space, and hence the moduli space locally is an orbifold. Therefore the fundamental group of the moduli space is non-trivial. Choosing a presentation, the generators of this group will satisfy certain relations. D-branes, through their monodromy, translate these relations into relations between the associated autoequivalences. As a result, we expect interesting identities between autoequivalences for a general Calabi-Yau compactification. The next question is how often do these identities follow from the technique of exceptional collections developed here? Obviously, we cannot expect to have an exceptional collection for every instance. In this paper we looked at three examples with two dimensional moduli space, two local and one compact, and proved a total of nine identities. Every one of them followed from the existence of an exceptional collection (either on a divisor, or in the ambient space). The author also studied the compact models P41,1,2,2,2 and P41,1,2,8,12 , with two resp. three dimensional Kahler moduli space, and obtained similar results. On the other hand, most known Calabi-Yau varieties are subvarieties in toric varieties (hypersurfaces and complete intersections). Kawamata proves (Theorem 1.1. of [25]) that if X is a projective toric variety with at most quotient singularities, then the associated smooth Deligne-Mumford stack X has a complete exceptional collection consisting of sheaves. This suggests that for a large class of Calabi-Yau varieties at least part of the monodromy identities should indeed follow from the exceptional collection techniques. It would be interesting to study examples where the ambient space does not have an exceptional collection. Unfortunately general statements are beyond reach at this point, even in the toric case. The first obstacle is that in order to write down the identities we need a detailed understanding of the singularities of the moduli space, the discriminant loci, and its intersections with the large radius divisors. All of these are hard to get, but computing the discriminant in the general case seems impossible. There are two ways to approach the problem: 1. write down the equations enforcing that the mirror is singular, and use elimination theory, 2. use Horn parametrization, and then elimination theory (see, e.g., [28]). Using Groebner basis for the elimination part, both approaches give the same answer, but as the dimension of the moduli space increases, today’s computers are unable to solve the elimination problem. Even if we knew the discriminant, its intersections with the large radius divisors is very diverse (we already saw evidence in our examples), hence writing down general statements seems impossible. Therefore, the best one can do is to prove the identities case by case, as we have done in this paper. On the other hand, this is not an unsatisfactory state of affairs, since this is the best one can do for the proof of mirror symmetry as well: prove it case by case. One family of examples where we can make general statements is the case of hypersurfaces in weighted projective spaces. Physics-wise the relevant case is the 3-fold, and it is meaningless to work on 6-folds and higher, but mathematically the statement holds for all weights and all dimensions. The weighted projective space has only quotient singularities, and also has a strong and full exceptional collection when viewed as a stack.

20

R. L. Karp

As discussed in the Introduction, the relevant identity was proven in full generality in [11]. There are two future directions that our technique seems suitable to tackle. The first is to extend our results to complete intersections in toric varieties. The second is the connection with Horja’s EZ-transformations [10]. The conjectured autoequivalences were proved only in the case when the Calabi-Yau is a fibration over the projective space of dimension d, Pd [9]. The proof used the so-called “Verdier 9-diagram”, and had no mention of exceptional collections. Canonaco [12] pointed out that the same result also follows from the exceptional collection approach. This opens up the possibility of proving other cases of Horja’s EZ-conjecture. Acknowledgements. It is a pleasure to thank Paul Aspinwall, Tom Bridgeland, Mike Douglas, Alastair King and Eric Sharpe for useful conversations. I am especially indebted to Alberto Canonaco for collaboration on a related project. This work was partially supported by NSF grant PHY-0755614.

References 1. Aspinwall, P.S., Katz, S.: Computation of superpotentials for D-Branes. Commun. Math. Phys. 264, 227–253 (2006) 2. Diaconescu, D.-E., Garcia-Raboso, A., Karp, R.L., Sinha, K.: D-Brane Superpotentials in Calabi-Yau Orientifolds. Adv. Theor. Math. Phys. 11, 471–516 (2007) 3. Douglas, M.R.: D-branes, categories and N = 1 supersymmetry. J. Math. Phys. 42, 2818–2843 (2001) 4. Sharpe, E.R.: D-branes, derived categories, and Grothendieck groups. Nucl. Phys. B 561, 433–450 (1999) 5. Aspinwall, P.S., Bridgeland, T., Craw, A., Douglas, M.R., Gross, M., Kapustin, A., Moore, G., Segal, G., Szendroi, B., Wilson, P.: Dirichlet Branes and Mirror Symmetry. Providence, RI: Amer. Math. Soc. Clay Math. Inst. Vol. 4, 2009 6. Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. In: Recent Trends in String Theory. River Edge, NJ: World Scientific, 2004, pp. 1–152 7. Sharpe, E.: Derived categories and stacks in physics. http://arxiv.org.abs/hep-th/0608056v2, 2006 8. Cox, D.A., Katz, S.: Mirror symmetry and algebraic geometry, Vol. 68 of Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc., 1999 9. Aspinwall, P.S., Karp, R.L., Horja, R.P.: Massless D-branes on Calabi-Yau threefolds and monodromy. Commun. Math. Phys. 259, 45–69 (2005) 10. Horja, R.P.: Derived category automorphisms from mirror symmetry. Duke Math. J. 127(1), 1–34 (2005) 11. Canonaco, A., Karp, R.L.: Derived autoequivalences and a weighted Beilinson resolution. J. Geom. Phys. 58, 743–760 (2008) 12. Canonaco, A.: Exceptional sequences and derived autoequivalences. http://arxiv.org.abs/0801.0173v1 [math.A6], 2008 13. Zaslow, E.: Solitons and helices: The Search for a math physics bridge. Commun. Math. Phys. 175, 337– 376 (1996) 14. Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. http://arxiv.org.abs/hep-th0005247v2, 2000 15. Govindarajan, S., Jayaraman, T.: D-branes, exceptional sheaves and quivers on Calabi-Yau manifolds: From Mukai to McKay. Nucl. Phys. B 600, 457–486 (2001) 16. Tomasiello, A.: D-branes on Calabi-Yau manifolds and helices. JHEP 02, 008 (2001) 17. Mayr, P.: Phases of supersymmetric D-branes on Kaehler manifolds and the McKay correspondence. JHEP 01, 018 (2001) 18. Cachazo, F., Fiol, B., Intriligator, K.A., Katz, S., Vafa, C.: A geometric unification of dualities. Nucl. Phys. B 628, 3–78 (2002) 19. Wijnholt, M.: Large volume perspective on branes at singularities. Adv. Theor. Math. Phys. 7, 1117– 1153 (2004) 20. Douglas, M.R., Moore, G.W.: D-branes, Quivers, and ALE Instantons. http://arxiv.org.abs/hep-th/ 9603167v1, 1996 21. Herzog, C.P., Karp, R.L.: Exceptional collections and D-branes probing toric singularities. JHEP 02, 061 (2006) 22. Herzog, C.P., Karp, R.L.: On the geometry of quiver gauge theories (Stacking exceptional collections). Adv. Theor. Math. Phys. 13, 1–38 (2010) 23. Karp, R.L.: On the Cn /Zm fractional branes. J. Math. Phys. 50, 022304 (2009)


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24. Orlov, D.O.: Equivalences of derived categories and K 3 surfaces. J. Math. Sci. (New York) 84(5), 1361– 1381 (1997) 25. Kawamata, Y.: Equivalences of derived categories of sheaves on smooth stacks. Amer. J. Math. 126(5), 1057–1083 (2004) 26. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001) 27. Rudakov, A.N., et al.: Helices and vector bundles. Vol. 148 of London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press, 1990 28. Karp, R.L.: C2 /Zn fractional branes and monodromy. Commun. Math. Phys. 270, 163–196 (2007) 29. Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. Oxford: The Clarendon Press Oxford University Press, 2006 30. Aspinwall, P.S.: Some navigation rules for D-brane monodromy. J. Math. Phys. 42, 5534–5552 (2001) 31. Kawamata, Y.: Derived categories of toric varieties. Michigan Math. J. 54(3), 517–535 (2006) Communicated by A. Kapustin

Commun. Math. Phys. 301, 23–35 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1148-y

Communications in


Blowup Criterion for Viscous Baratropic Flows with Vacuum States Xiangdi Huang1,4 , Jing Li2,3,4 , Zhouping Xin4 1 Department of Mathematics, University of Science and Technology of China, Hefei 230026,

P. R. China. E-mail: [email protected]

2 Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100190, P. R. China.

E-mail: [email protected]

3 Hua Loo-Keng Key Laboratory of Mathematics, Academia Sinica, Beijing 100190, P. R. China 4 The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong.

E-mail: [email protected] Received: 16 February 2009 / Accepted: 19 June 2010 Published online: 9 October 2010 – © Springer-Verlag 2010

Abstract: We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases. 1. Introduction The time evolution of the density and the velocity of a general viscous compressible barotropic fluid occupying a domain ⊂ R 3 is governed by the compressible Navier-Stokes equations ∂t ρ + div(ρu) = 0, (1.1) ∂t (ρu) + div(ρu ⊗ u) − μu − (μ + λ)∇(divu) + ∇ P(ρ) = 0, where ρ, u, P denote the density, velocity and pressure respectively. The equation of state is given by P(ρ) = aρ γ (a > 0, γ > 1), μ and λ are the shear viscosity and the bulk viscosity coefficients respectively. They satisfy the following physical restrictions: 2 μ > 0, λ + μ ≥ 0. 3

(1.2)

24

X. Huang, J. Li, Z. Xin

Equations (1.1) will be studied with initial conditions: (ρ, u)(x, 0) = (ρ0 , u 0 )(x),

(1.3)

and three types of boundary conditions: 1) Cauchy problem: = R 3 and (in some weak sense) ρ, u vanish at infinity;

(1.4)

2) Dirichlet problem: in this case, is a bounded smooth domain in R 3 , and u = 0 on ∂;

(1.5)

3) Navier-slip boundary condition: in this case, is a bounded smooth domain in R 3 , and u · n = 0, curlu × n = 0 on ∂,

(1.6)

where n = (n 1 , n 2 , n 3 ) is the unit outward normal to ∂. The first condition in (1.6) is the non-penetration boundary condition, while the second one is also known in the form (D(u) · n)τ = −κτ u τ ,

(1.7)

where D(u) is the deformation tensor: D(u) =

1 (∇u + ∇u t ), 2

(1.8)

and κτ is the corresponding principal curvature of ∂. Condition (1.7) implies the tangential component of D(u) · n vanishes on flat portions of the boundary ∂. Note that ∇u can be decomposed as ∇u = D(u) + S(u),

(1.9)

where D(u) is the deformation tensor defined by (1.8) and S(u) =

1 ∇u − ∇u t , 2

(1.10)

is known as the rigid body rotation tensor. The tensors D(u) and S(u) are respectively the symmetric and skew-symmetric parts of ∇u. There are huge amounts of literature on the large time existence and behavior of solutions to (1.1). The one-dimensional problem has been studied extensively by many people, see [15,21,28,29] and the references therein. The multidimensional problem (1.1) was investigated by Matsumura-Nishida [23], who proved global existence of smooth solutions for initial data close to a non-vacuum equilibrium, and later by Hoff [15–17] for discontinuous initial data. For the existence of solutions for arbitrary data in three-dimension, the major breakthrough is due to Lions [22], where he established global existence of weak solutions for the whole space, periodic domains or bounded domains with Dirichlet boundary conditions provided γ ≥ 95 . The restriction on γ is improved to γ > 23 by Feireisl [12–14]. The main restriction on initial data is that the initial energy is finite, so that the density is allowed to vanish.

Blowup Criterion for Viscous Baratropic Flows with Vacuum States

25

However, the regularity and uniqueness of such weak solutions remains completely open. Desjardins ([10]) treated local existence of weak solutions with higher regularity. It should be noted that one would not expect too much regularity of Lions’s weak solutions in general because of the results of Xin ([32]), who showed that there is no global smooth solution (ρ, u) to the Cauchy problem for (1.1) with a nontrivial compactly supported initial density at least in 1-D. Xin’s blowup result ([32]) raises the questions of the mechanism of blowup and structure of possible singularities: what kinds of singularities will form in finite time? What is the main mechanism for possible breakdown of smooth solutions for the 3-D barotropic compressible Navier-Stokes equations? Before stating the main result, we explain the notations and conventions used throughout this paper. We denote f dx = f d x.

For 1 ≤ r ≤ ∞, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows: ⎧ 1 () : ∇ k u r < ∞}, ⎪ L r = L r (), D k,r = {u ∈ L loc L ⎪ ⎪ ⎨W k,r = L r ∩ D k,r , H k = W k,2 , D k = D k,2 ,

⎪ D01 = u ∈ L 6 : ∇u L 2 < ∞, and (1.4) or (1.5) or (1.6) holds , ⎪ ⎪ ⎩ H 1 = L 2 ∩ D 1 , u

k D k,r = ∇ u L r . 0 0 We begin with the local existence of strong (or classical) solutions. In the absence of vacuum, the local existence and uniqueness of classical solutions are known in [24,30]. In the case where the initial density need not be positive and may vanish in an open set, the existence and uniqueness of local strong (or classical) solutions are proved recently in [4–7,27]. Before stating their local existence results, we first give the definition of strong solutions. Definition 1.1 (Strong solutions). (ρ, u) is called a strong solution to (1.1) in ×(0, T ), if for some q0 ∈ (3, 6], 0 ≤ ρ ∈ C([0, T ], W 1,q0 ), ρt ∈ C([0, T ], L q0 ), u ∈ C([0, T ], D01 ∩ D 2 ) ∩ L 2 (0, T ; D 2,q0 ) ∞

u t ∈ L (0, T ; L ) ∩ L (0, T ; 2

2

(1.11)

D01 ),

and (ρ, u) satisfies (1.1) a.e. in × (0, T ). In particular, Cho et al. [4] proved the following result. Theorem 1.1. If the initial data ρ0 and u 0 satisfy 0 ≤ ρ0 ∈ L 1 ∩ W 1,q˜ , u 0 ∈ D01 ∩ D 2 ,

(1.12)

for some q˜ ∈ (3, ∞) and the compatibility condition: 1/2

− μu 0 − (λ + μ)∇divu 0 + ∇ P(ρ0 ) = ρ0 g for some g ∈ L 2 ,

(1.13)

then there exists a positive time T1 ∈ (0, ∞) and a unique strong solution (ρ, u) to the initial boundary value problem (1.1) (1.3) together with (1.4) or (1.5) or (1.6) in

26


× (0, T1 ]. Furthermore, the following blow-up criterion holds: if T ∗ is the maximal time of existence of the strong solution (ρ, u) and T ∗ < ∞, then sup ( ρ H 1 ∩W 1,q0 + u D 1 ) = ∞,

t→T ∗

0

(1.14)

with q = min(6, q). ˜ There are several works ([8,11,18,20]) trying to establish blowup criteria for the strong (smooth) solutions to the barotropic compressible Navier-Stokes equations (1.1). In particular, it is proved in [11] for three dimensions, if 7μ > 9λ, then

T 4 lim ∗ sup ρ L ∞ + ( ρ W 1,q0 + ∇ρ L 2 )dt = ∞, T →T

0

0≤t≤T

where T ∗ < ∞ is the maximal time of existence of a strong solution and q0 > 3 is a constant. Later, we [18–20] first establish a blowup criterion, analogous to the Beale-KatoMajda criterion [1] for the ideal incompressible flows, for the strong and classical solutions to the barotropic compressible flows in three-dimension: T lim ∗

∇u L ∞ dt = ∞, (1.15) T →T

0

under a stringent condition on the viscous coefficients: 7μ > λ.

(1.16)

It should be noted that for ideal incompressible flows, Beale-Kato-Majda [1] established a well-known blowup criterion for the 3-dimensional incompressible Euler equations that a solution remains smooth if T

S(u) L ∞ dt (1.17) 0

is bounded, where S(u) is the rigid body rotation tensor defined by (1.10). Later, Ponce [25] rephrased the Beale-Kato-Majda’s theorem in terms of deformation tensor D(u), that is, the same results in [1] hold if T

D(u) L ∞ dt (1.18) 0

remains bounded. Moreover, as pointed out by Constantin[9], the solution is smooth if and only if T

((∇u)ξ ) · ξ L ∞ dt (1.19) 0

is bounded, where ξ is the unit vector in the direction of vorticity curlu. All these facts in [1,9,25] show that the solution becomes smooth when either the skew-symmetric or symmetric part of ∇u is controlled. The aim of this paper is to improve all the previous blowup criterion results for the barotropic compressible Navier-Stokes equations by removing the stringent condition (1.16), and allowing initial vacuum states, and furthermore, instead of (1.15), describing the blowup mechanism in terms of the deformation tensor D(u). Our main result can be stated as follows:


27

Theorem 1.2. Let (ρ, u) be a strong solution of the initial boundary value problem (1.1) (1.3) together with (1.4) or (1.5) or (1.6) satisfying (1.11). Assume that the initial data (ρ0 , u 0 ) satisfies (1.12) and (1.13). If T ∗ < ∞ is the maximal time of existence, then T lim ∗

D(u) L ∞ dt = ∞, (1.20) T →T

0

where D(u) is the deformation tensor defined by (1.8). A few remarks are in order: Remark 1.1. Theorem 1.2 also holds for classical solutions to the compressible flows with initial vacuum. In addition, Theorem 1.2 holds for all μ and λ satisfying the physical restrictions (1.2), which removed the condition (1.16) which is essential in the analysis in [18–20]. Remark 1.2. In 1998, Xin [32] gave a life span estimate of classical solutions to the compactly supported initial density of the Cauchy problem (1.1) (1.3) (1.4) at least in 1-D. However, it’s unclear which quantity becomes infinite as the critical time approaches. Theorem 1.2 shows that singularity can develop only if the size of the deformation tensor becomes arbitrarily large. Remark 1.3. Theorem 1.2 gives a counterpart of Ponce’s result in [25] for the ideal incompressible flows. Remark 1.4. We will extend our results to the viscous heat-conductive flows in a forthcoming paper. We now comment on the analysis of this paper. Note that in all previous works [18–20], the assumption (1.16) played an important role in their analysis in order to obtain an improved energy estimate which is essential not only for bounding the L 2 norm of the convection term F = ρu t + ρu · ∇u, but also for improving the regularity of the solutions. Their method also depends on the L ∞ -norm of ∇u instead of D(u). It is thus difficult to adapt their analysis here. To proceed, some new ideas are needed. The key step in proving Theorem 1.2 is to derive the L 2 -estimate on gradients of both the density ρ and the velocity u. Observe that there are two main difficulties here: one is due to the possible vacuum states, the other is the strong nonlinearities of convection terms. In order to overcome these difficulties, we will use the simple observation that the momentum equations (1.1)2 become “more” diffusive near vacuum if divided on both sides by ρ as long as ρ remains bounded above which is guaranteed by the boundedness of the temporal integral of the super-norm in space of the deformation tensor. Thus a new energy estimate by using the effective stress tensor will lead to a prior estimates on the L 2 -norms of gradients of both the density and the velocity. The detail of the proof of Theorem 1.2 is given in Sect. 2. 2. Proof of Theorem 1.2 Let (ρ, u) be a strong solution to the problem (1.1)–(1.2) as described in Theorem 1.2. First, the standard energy estimate yields T γ 1/2 2 sup ρ u(t) L 2 + ρ L γ +

∇u 2L 2 dt ≤ C0 , 0 ≤ T < T ∗ . (2.1) 0≤t≤T

0

28


To prove the theorem, we assume otherwise that T lim ∗

D(u) L ∞ dt ≤ M0 < ∞. T →T

(2.2)

0

Then (2.2), together with (1.1)1 , immediately yields the following L ∞ bound of the density ρ. Indeed, one has Lemma 2.1. Assume that T

divu L ∞ dt ≤ C, 0 ≤ T < T ∗ .

0

Then sup ρ L ∞ ≤ C, 0 ≤ T < T ∗ .

(2.3)

0≤t≤T

Here and after, C will denote a generic constant depending only on C0 , M0 , T , the initial data and the domain . Proof. It follows from (1.1)1 that for any p ≥ γ , ∂t (ρ p ) + div(ρ p u) + ( p − 1)ρ p divu = 0.

(2.4)

Integrating (2.4) over leads to d ρ p d x ≤ ( p − 1) divu L ∞ ρ p d x, dt that is, ∂t ρ L p ≤

p−1

divu L ∞ ρ L p , p

which implies immediately

ρ L p (t) ≤ C, with C independent of p, so Lemma 2.1 follows immediately. The key estimates on ∇ρ and ∇u will be given in the following lemma. Lemma 2.2. Under the condition (2.2), it holds that for any T < T ∗ , T

∇u 2H 1 dt ≤ C. sup ∇ρ 2L 2 + ∇u 2L 2 + 0≤t≤T

(2.5)

0

To prove Lemma 2.2, we need the following lemma (see [3]), which gives the estimate of ∇u by divu and curlu. Lemma 2.3. Let u ∈ H s () be a vector-valued function satisfying u · n|∂ = 0, where n is the unit outer normal of ∂. Then

u H s ≤ C( divu H s−1 + curlu H s−1 + u H s−1 ), for s ≥ 1 and the constant C depends only on s and .

(2.6)


29

Proof of Lemma 2.2. Multiplying ρ −1 (μu + (μ + λ)∇divu − ∇ P) on both sides of the momentum equations (1.1)2 , integrating the resulting equation over , one has after integration by parts d μ μ+λ |∇u|2 + (divu)2 d x + ρ −1 (μu + (μ + λ)∇divu − ∇ P)2 d x dt 2 2 = −μ u · ∇u · ∇ × curlud x + (2μ + λ) u · ∇u · ∇divud x − u · ∇u · ∇ Pd x − u t · ∇ Pd x, (2.7)

due to u = ∇divu − ∇ × curlu. When u satisfies boundary condition (1.4) or (1.5), we deduce from the standard L 2 -theory of elliptic system that

∇ 2 u 2L 2 − C ∇ P 2L 2 ≤ C μu + (μ + λ)∇divu 2L 2 − C ∇ P 2L 2 + C ∇u 2L 2 ≤ C μu + (μ + λ)∇divu − ∇ P 2L 2 + C ∇u 2L 2 ≤ C ρ −1 (μu + (μ + λ)∇divu − ∇ P)2 d x + C ∇u 2L 2 ,

(2.8)

due to ρ −1 ≥ C −1 > 0. Lemma 2.3 yields that (2.8) also holds for u satisfying boundary condition (1.6) due to the following simple fact by (1.6):

(2μ + λ)∇divu 2L 2 + μ∇ × curlu 2L 2 = μu + (μ + λ)∇divu 2L 2 . Next, we shall treat each term on the righthand side of (2.7) under the boundary condition (1.6), since the estimate here is more subtle than the other cases, (1.4) or (1.5), due to the effect of boundary. Using (1.6) and the facts that u × curlu = 21 ∇(|∇u|2 ) − u · ∇u and ∇ × (a × b) = (b · ∇)a − (a · ∇)b + (divb)a − (diva)b, one gets after integration by parts and direct computations that (u · ∇)u · ∇ × curlud x = curlu · ∇ × ((u · ∇)u) d x = curlu · ∇ × (u × curlu)d x 1 2 = |curlu| divud x − curlu · D(u) · curlud x 2 ≤ C ∇u 2L 2 D(u) L ∞ , and

u · ∇u · ∇divud x 1 i j t 3 = u ∂i u n j divud S − ∇u : ∇u divud x + (divu) d x 2 ∂ ≤ ε ∇ 2 u 2L 2 + C(ε) ∇u 2L 2 ∇u 2L 2 + D(u) L ∞ ,

(2.9)

(2.10)

30


due to the following simple fact: i j i i j = u ∂ u n divud S u ∂ (u · n)divud S − u u ∂ n divud S i j i i j ∂ ∂ ∂ = u i u j ∂i n j divud S ∂

≤ C u 2L 4 (∂) divu L 2 (∂) ≤ C ∇u 2L 2 ∇u H 1 ≤ C(ε) ∇u 4L 2 + ε ∇ 2 u 2L 2 + C(ε), where (1.6) and the Poincaré type inequality and the Ehrling inequality have been used. Similarly, u · ∇u · ∇ Pd x i j i j i = u ∂i u n j Pd S − ∂ j u ∂i u Pd x − u ∂i divu Pd x ∂ i j i j 2 = u u ∂i n j Pd S + ∂ j u ∂i u Pd x − (divu) Pd x − u · ∇ Pdivud x ∂ ≤ C ∇u 2L 2 + u · ∇ Pdivud x ≤ C ∇u 2L 2 + u L 6 divu L 3 ∇ P L 2 5

1

≤ C ∇u 2L 2 + C ∇u L3 2 D(u) L3 ∞ ∇ρ L 2 ≤ C ∇ρ 2L 2 ∇u 2L 2 + C ∇u 2L 2 ( D(u) L ∞ + 1) + C,

(2.11)

which yields also that − u t · ∇ Pd x d = Pt divud x Pdivud x − dt d = Pdivud x + u · ∇ Pdivud x + (γ − 1) P(divu)2 d x dt d ≤ Pdivud x + C ∇u 2L 2 ( D(u) L ∞ + 1) dt + C ∇ρ 2L 2 ∇u 2L 2 + C.

(2.12)

Substituting (2.8)–(2.12) into (2.7) gives that for ε suitably small, μ μ+λ d 2 2 |∇u| + (divu) − Pdivu d x + C0 ∇ 2 u 2L 2 dt 2 2 ≤ C ∇ρ 2L 2 + ∇u 2L 2 ∇u 2L 2 + D(u) L ∞ + 1 + C.

(2.13)


31

It remains to bound the L 2 -norm of ∇ρ. To this end, one can differentiate (1.1)1 and then multiply the resulting equation by 2∇ρ to get ∂t |∇ρ|2 + div(|∇ρ|2 u) + |∇ρ|2 divu = −2(∇ρ)t ∇u∇ρ − 2ρ∇ρ · ∇divu = −2(∇ρ)t D(u)∇ρ − 2ρ∇ρ · ∇divu.

(2.14)

Integrating (2.14) over yields d

∇ρ 2L 2 ≤ C ∇ρ 2L 2 D(u) L ∞ + ε ∇ 2 u 2L 2 + C(ε) ∇ρ 2L 2 . dt

(2.15)

Adding (2.15) to (2.13), we deduce, after choosing ε suitably small and using Gronwall’s inequality, that (2.5) holds. The proof of Lemma 2.2 is completed. We next improve the regularity of the density ρ and the velocity u. We start with some bounds on derivatives of u based on the above estimates. Lemma 2.4. Under the condition (2.2), it holds that 1/2 sup ρ u t (t) L 2 + ∇u H 1 +

0

0≤t≤T

T

∇u t 2L 2 dt ≤ C, 0 ≤ T < T ∗ .

(2.16)

Proof. Differentiating the momentum equations (1.1)2 with respect to t yields ρu tt + ρu · ∇u t − μu t − (μ + λ)∇divu t = −∇ Pt − ρt u t − ρu t · ∇u − ρt u · ∇u. (2.17) Taking the inner product of the above equation with u t in L 2 and integrating by parts, one gets 1 d 2 μ|∇u t |2 + (λ + μ)(divu t )2 d x ρu t d x + 2 dt = Pt divu t d x − ρ(u t · ∇u) · u t d x − ρu · ∇ |u t |2 + u · ∇u · u t d x ≤ C (|u||∇ρ| + |∇u|) |∇u t |d x + C ρ|u t |2 |∇u| + ρ|u||u t ||∇u t | d x +C |u||u t ||∇u|2 + |u|2 |u t ||∇ 2 u| + |u|2 ||∇u||∇u t | d x =

3

Ii .

(2.18)

i=1

Noticing that by (2.5) and Sobolev’s inequality, one has I1 ≤ C u L ∞ ∇ρ L 2 + ∇u L 2 ∇u t L 2 ≤ C ∇u t L 2 ∇u H 1 ≤ ε ∇u t 2L 2 + C(ε) ∇u 2H 1 .

(2.19)

32


Similarly, I2 ≤ C ρ 1/2 u t L 2 u t L 6 ∇u L 3 + C u L ∞ ρ 1/2 u t L 2 ∇u t L 2 ≤ C ρ 1/2 u t L 2 ∇u t L 2 ∇u H 1 ≤ ε ∇u t 2L 2 + C(ε) ρ 1/2 u t 2L 2 ∇u 2H 1 ,

(2.20)

and I3 ≤ C u L 6 u t L 6 ∇u 2L 3 + C u 2 L 3 u t L 6 ∇ 2 u L 2 + C ∇u t L 2 ∇u L 6 u 2 L 3 ≤ C ∇u t L 2 ∇u L 2 ∇u L 6 + ∇ 2 u L 2 ≤ ε ∇u t 2L 2 + C(ε) ∇u 2H 1 .

(2.21)

We conclude from (2.18)–(2.21) that 1 d 2 ρu t d x + μ |∇u t |2 d x 2 dt ≤ 6ε ∇u t 2L 2 + C(ε) ∇u 2H 1 + C(ε) ρ 1/2 u t 2L 2 ∇u 2H 1 , which, together with Gronwall’s inequality, implies that for ε suitably small, T sup ρ 1/2 u t (t) 2L 2 +

∇u t 2L 2 dt ≤ C, 0 ≤ T < T ∗ ,

(2.22)

0

0≤t≤T

due to the fact that 1

1

1

ρ0 (x) 2 u t (x, t = 0) = ρ02 u 0 · ∇u 0 (x) − ρ02 g ∈ L 2 , which comes from the compatibility condition (1.13). Moreover, since u satisfies μ u + (μ + λ)∇divu = ρu t + ρu · ∇u + ∇ P, (1.4) or (1.5) or (1.6) holds, similar to (2.8), one has 1

∇ 2 u L 2 ≤ C( ρ 2 u t L 2 + u L ∞ ∇u L 2 + ∇ P L 2 ) 1/2

≤ C + C ∇ 2 u L 2 . Hence, sup ∇u H 1 ≤ C.

0≤T 0, if n = 0.

Quantization of the Lie Bialgebra of String Topology

41

Since A is quasi-isomorphic to C ∗ (M), it follows by a well-known result of Jones (see [19, Theorem A] or [11, Theorem 1.5.1, Corollary 1.5.2]) that the equivariant chain 1 complex C∗S (L M) is quasi-isomorphic to (Hoch∗ (V )[u], b + u −1 B). Let Hoch∗ (V ) = C ⊕

∞

V ⊗ C[1]⊗n

n=1

be the reduced Hochschild complex. Then the reduced equivariant chain complex of L M and (Hoch∗ (V )[u], b + u −1 B) are quasi-isomorphic. On the other hand, note that the normalized and reduced Hochschild complexes are non-negatively graded, so the bicomplex (Hoch∗ (V )[u], b + u −1 B) lies in the first quadrant. It follows by a standard argument using filtrations (see [21, Proposition 2.2.14]) that the map (Hoch∗ (V )[u], b + u −1 B) → CC∗ (C)[1] B(α), if n = 0, α ⊗ u n → 0, if n > 0, is a quasi-isomorphism. The gives the desired result.

Remark 6. The equivariant homology of L M is isomorphic to the direct sum of the reduced equivariant homology of L M with k[u] (where deg(u) = 2). Since V is simply-connected, we may replace the direct product in the definition of Hoch∗ (C) by direct sum. 3. Lie Bialgebra 3.1. Construction of the Lie bialgebra. In this section, C is the coaugmentation coideal of a counital coaugmented simply-connected DG open Frobenius algebra V (of degree m). One can also take C to be V itself. We shall write ± for signs determined by the usual Koszul convention. Definition 7 (Lie coalgebra). Let L be a vector space over k. A skew-symmetric map δ : L → L ⊗ L defines a Lie coalgebra structure on L if (τ 2 + τ + id) ◦ (δ ⊗ id) ◦ δ = 0 : L → L ⊗ L ⊗ L ,

(6)

where τ is the permutation a ⊗ b ⊗ c → ±c ⊗ a ⊗ b, for a, b, c ∈ L. The map δ is called the cobracket and (6) is called the co-Jacobi identity. Definition 8 (Lie bialgebra). Suppose (L , { , }) is a Lie algebra and (L , δ) is a Lie coalgebra. The triple (L , { , }, δ) defines a Lie bialgebra on L if the following identity, called the Drinfeld compatibility, holds: δ{a, b} = {a, δ(b)} + {δ(a), b}.

(7)

If moreover, { , } ◦ δ : L → L vanishes identically, the Lie bialgebra (L , { , }, δ) is called involutive.

42

X. Chen, F. Eshmatov, W. L. Gan

Let L := (CC∗ (C)[1]) [m − 2] = CC∗ (C)[m − 1], where CC∗ (C) is defined in the previous section. We shall write elements of L in the form N ([a1 | · · · |an ]), where ai ∈ C[1] for i = 1, . . . , n. Define on L the following two operators: { , } : L ⊗ L → L by {α, β} :=

±ε(ai · b j )N ([ai+1 | · · · |an |a1 | · · · |ai−1 |b j+1 | · · · |bm |b1 | · · · |b j−1 ]) (8)

i, j

and δ : L → L ⊗ L by δ(α) :=

±ε(ai · a j )N ([a1 | · · · |ai−1 |a j+1 | · · · |an ]) ∧ N ([ai+1 | · · · |a j−1 ]), (9)

i< j

for any homogeneous α = N ([a1 | · · · |an ]), β = N ([b1 | · · · |bm ]) ∈ L, where in the above ε is the augmentation, and in (9), a ∧ b means a ⊗ b − b ⊗ a, and will also be written as a ⊗ b − Alt. Theorem 9. Let L be as above. Then (L , { , }, δ) forms an involutive DG Lie bialgebra. The Lie bracket { , } is of degree 0, and the Lie cobracket δ is of degree 2(2 − m). The rest of this section is devoted to the proof of Theorem 9. The proof is divided into several steps.

3.2. Proof of the DG Lie algebra. The product on V is graded commutative, hence if we shift the degree of C down by 1, the induced pairing ε(a ·b) : C[1]⊗C[1] → k is graded skew-symmetric. Therefore the bracket { , } defined by (8) is graded skew-symmetric. We now show the Jacobi identity: for any α = N ([a1 | · · · |an ]), β = N ([b1 | · · · |bm ]), γ = N ([c1 | · · · |c p ]) ∈ L, {{α, β}, γ } = ±ε(ai b j )ε(ak cl )N ([a1 | · · · |b j+1 | · · · |b j−1 | · · · |cl+1 | · · · |cl−1 | · · · |an ]) (10) i, j,k,l

+

±ε(ai b j )ε(bk cl )N ([a1 | · · · |b j+1 | · · · |cl+1 | · · · |cl−1 | · · · |b j−1 | · · · |an ]).

i, j,k,l

(11) Similarly, we have {{β, γ }, α} = ±ε(b j cl )ε(bk ai )N ([b1 | · · · |cl+1 | · · · |cl−1 | · · · |ai+1 | · · · |ai−1 | · · · |bm ]) (12) i, j,k,l

+

±ε(b j cl )ε(ck ai )N ([b1 | · · · |cl+1 | · · · |ai+1 | · · · |ai−1 | · · · |cl−1 | · · · |bm ]),

i, j,k,l

(13)


43

and {{α, β}, γ } ±ε(cl ai )ε(ck b j )N ([c1 | · · · |ai+1 | · · · |ai−1 | · · · |b j+1 | · · · |b j−1 | · · · |c p ]) (14) = i, j,k,l

+

±ε(cl ai )ε(ak b j )N ([c1 | · · · |ai+1 | · · · |b j+1 | · · · |b j−1 | · · · |ai−1 | · · · |c p ]).

i, j,k,l

(15) Note that by the cyclic invariance of N , (10) cancels with (15), so do (11) with (12) and (13) with (14). This proves the Jacobi identity. We next show that b respects the bracket. It is easy to see that the bracket thus defined commutes with the internal differential, hence we only check that it commutes with the external differential. For any α = N ([a1 | · · · |an ]), β = N ([b1 | · · · |bm ]), b(α) =

n i=1

N ([a1 | · · · |ai |ai | · · · |an ]), and b(β) =

n

N ([b1 | · · · |bj |bj | · · · |bm ]).

j=1

Therefore, {b(α), β} = ±ε(ak bl )N ([ak+1 | · · · |an |a1 | · · · |ai |ai | · · · |ak−1 |bl+1 | · · · |bm |b1 | · · · |bl−1 ]) i,k,l

+

(16) ±ε(ai bl )N ([ai | · · · |an |a1 | · · · |ai−1 |bl+1 | · · · |bm |b1 | · · · |bl−1 ])

(17)

±ε(ai bl )N ([ai+1 | · · · |an |a1 | · · · |ai−1 |ai |bl+1 | · · · |bm |b1 | · · · |bl−1 ]),

(18)

i,l

+

i,l

{α, b(β)} ±ε(ak bl )N ([ak+1 | · · · |an |a1 | · · · |ak−1 |bl+1 | · · · |bj |bj | · · · |bm |b1 | · · · |bl−1 ]) = j,k,l

+

(19) ±ε(ak bj )N ([ak+1 | · · · |an |a1 | · · · |ak−1 |bj | · · · |bm |b1 | · · · |b j−1 ])

(20)

k, j

+

±ε(ak bj )N ([ak+1 | · · · |an |a1 | · · · |ak−1 |b j+1 | · · · |bm |b1 | · · · |b j−1 |bj ]), (21)

k, j

while b{α, β} = ±ε(ak bl )N ([ak+1 | · · · |an |a1 | · · · |ai |ai | · · · |ak−1 |bl+1 | · · · |bm |b1 | · · · |bl−1 ]) i,k,l

+

(22) ±ε(ak bl )N ([ak+1 | · · · |an |a1 | · · · |ak−1 |bl+1 | · · · |bj |bj | · · · |bm |b1 | · · · |bl−1 ]).

j,k,l

(23)

44


Note that (17) and (21) cancel, so do (18) and (20). The remaining terms of {b(α), β} + {α, b(β)} are identical to (22) + (23), which is exactly b{α, β}. 3.3. Proof of the DG Lie coalgebra. The cobracket is skew-symmetric. The co-Jacobi identity holds due by a similar computation as the Jacobi identity, and so we leave its verification to the reader. Next, we show that b respects the cobracket. As before, we check that the external differential commutes with the cobracket: by definition, δ N ([a1 | · · · |an ]) = ±ε(ai a j )N ([a1 | · · · |ai−1 |a j+1 | · · · |an ]) ⊗ N ([ai+1 | · · · |a j−1 ]) − Alt, i< j

hence b(δ N ([a1 | · · · |an ]) = ±ε(ai a j )N ([a1 | · · · |ak |ak | · · · |ai−1 |a j+1 | · · · |an ]) i< j,k

⊗ N ([ai+1 | · · · |a j−1 ]) − Alt ±ε(ai a j )N ([a1 | · · · |ai−1 |a j+1 | · · · |an ]) +

(24)

i< j,l

while δ

⊗ N ([ai+1 | · · · |al |al | · · · |a j−1 ]) − Alt,

k

(25)

N ([a1 | · · · |ak |ak | · · · |an ]) has not only (24) and (25), but also

±ε(ai ak )N (a1 | · · · |ai−1 |ak | · · · |an ])

i,k

⊗ N ([ai+1 | · · · |ak−1 ]) − Alt + ±ε(ai ak )N ([a1 | · · · |ai−1 |ak+1 | · · · |an ])

(26)

i,k

⊗ N ([ai+1 | · · · |ak−1 |ak ]) − Alt + ±ε(ai a j )N ([a1 | · · · |ak−1 |a j+1 | · · · |an ])

(27)

k, j

⊗ N ([ak |αk+1 | · · · |a j−1 ]) − Alt ±ε(ak a j )N ([a1 | · · · |ak |a j+1 | · · · |an ]) +

(28)

k, j

⊗ N ([ak+1 | · · · |a j−1 ]) − Alt.

(29)

Since C is an open Frobenius algebra, by the module compatibility (5), (26) cancels with (29), and (27) cancels with (28), and hence b commutes with the cobracket. 3.4. Proof of the Drinfeld compatibility. Let α = N ([a1 | · · · |an ]) and β N ([b1 | · · · |bm ]), and write δ(α) = α (1) ⊗α (2) and δ(β) = β (1) ⊗β (2) .

=


45

We have {α, β} =

±ε(ai b j )N ([ai+1 | · · · |ai−1 |b j+1 | · · · |b j−1 ])

i, j

and δ{α, β} ±ε(ai b j )ε(ak al )N ([ak+1 | · · · |al−1 ]) = i, j,k,l

⊗ N ([al+1 | · · · |ai−1 |b j+1 | · · · |b j−1 |ai+1 | · · · |ak−1 ]) + ±ε(ai b j )ε(ak bl )N ([ak+1 | · · · |ai−1 |b j+1 | · · · |bl−1 ]) i, j,k,l

⊗ N ([bl+1 | · · · |b j−1 |ai+1 | · · · |ak−1 ]) ±ε(ai b j )ε(al ak )N ([ak+1 | · · · |ai−1 |b j+1 | · · · |b j−1 |ai+1 | · · · |al−1 ]) + i, j,k,l

⊗ N ([al+1 | · · · |ak−1 ]) + ±ε(ai b j )ε(bk bl )N ([bk+1 | · · · |bl−1 ]) i, j,k,l

⊗ N ([bl+1 | · · · |b j−1 |ai+1 | · · · |ai−1 |b j+1 | · · · |bk−1 ]) ±ε(ai b j )ε(bk al )N ([bk+1 | · · · |b j−1 |ai+1 | · · · |al−1 ]) + i, j,k,l

⊗ N ([al+1 | · · · |ai−1 |b j+1 | · · · |bk−1 ]) ±ε(ai b j )ε(bl bk )N ([bk+1 | · · · |b j−1 |ai+1 | · · · |ai−1 |b j+1 | · · · |bl−1 ]) + i, j,k,l

⊗ N ([bl+1 | · · · |bk−1 ]). In the above, the second summation and the fifth summation cancel with each other. The first summation is equal to α (1) ⊗{α (2) , β}; the third summation is equal to {α (1) , β}⊗α (2) ; the fourth summation is equal to β (1) ⊗{α, β (2) }; and the sixth summation is equal to {α, β (1) }⊗β (2) . Thus we obtain the Drinfeld compatibility. 3.5. Proof of the involutivity. Let α = N ([a1 | · · · |an ]), then δ(α) =

±ε(ai a j )N ([a1 | · · · |ai−1 |a j+1 | · · · |an ]) ⊗ N ([ai+1 | · · · |a j−1 ])

i< j

−

±ε(ai a j )N ([ai+1 | · · · |a j−1 ]) ⊗ N ([a1 | · · · |ai−1 |a j+1 | · · · |an ]).

i< j

By a similar argument as above, one checks that { , } ◦ δ = 0 holds identically. The above constructions and proofs, except for the compatibilities of the differential with the Lie bracket and cobracket, are similar to the proof of the Lie bialgebras of Turaev [24], Chas-Sullivan [9], Hamilton [16] and Schedler [23].

46


4. Quantization of the Lie Bialgebra 4.1. Construction of the Hopf algebra. In this section, we construct a DG Hopf algebra which quantizes the DG Lie bialgebra of Sect. 3. We follow Schedler [23] closely. We will also define a new differential in Definition 12 below which is not present in [23]. Definition 10 (Quantization). Let h be a formal parameter. Suppose A is a Hopf algebra over k[h]. We say A quantizes the Lie bialgebra (L , { , }, δ) if there is a Hopf algebra isomorphism ∼ =

φ : A/ h A −→ U (L), where U (L) is the universal enveloping algebra of L, such that for any x0 ∈ L, and any x ∈ A, φ(x) = x0 , 1 ((x) − op (x)) ≡ δ(x0 ) h

mod h,

where op is the opposite comultiplication of A. Definition 11. Let C H := C ⊗k k[μ, μ−1 ], where μ is a formal variable (of degree 0). We shall write an element a ⊗ μu ∈ C H as (a, u) and call u ∈ Z the height of (a, u). ⊗n ∗ (C H ) be the graded vector space ∞ Let Hoch n=0 C H ⊗ C H [1] , and denote by ∗ (C H ) the subspace of cyclically invariant elements in Hoch ∗ (C H ). Let CC

∗ (C H )[m − 1]. ∗ (C H )[1] [m − 2] = CC L H := CC There is a canonical projection L H → L by forgetting the heights in L H (recall L is given in §3.1). Let S L H be the symmetric algebra of L H . Definition 12. Define a differential b on S L H so that on the homogeneous components it is given by

b N ([(a1,1 , h 1,1 )| · · · |(a1, p1 , h 1, p1 )]) • · · · • N ([(an,1 , h n,1 )| · · · |(an, pn , h n, pn )]) (30) pi n := − ± · · · • N ([(ai,1 , h i,1 )| · · · |(dai, j , h i, j )| · · · |(ai, pi , h i, pi )]) • · · · (31)

+

i=1 j=1 pi n

±N ([(a1,1 , h 1,1 )| · · · |(a1, p1 , h 1, p1 )]) • · · ·

(32)

i=1 j=1 (a(i, j) )

h i,1 )| · · · |(ai, j , h i, j )|(ai, j , h i, j + 1)| · · · |(ai, pi , h i, pi )]) • · · · , · · · • N ([(ai,1 , (33) where in (32) and (33), for all (i , j ) = (i, j), hi , j if h i , j ≤ h i, j hi , j = h i , j + 1 if h i , j > h i, j .

(34)


47

Remark 13. In the above definition, we assign ai, j and ai, j with heights h i, j and h i, j + 1 respectively, and raise all heights h i , j greater than h i, j to h i, j + 1. The coassociativity of C implies that b2 = 0. Let h be a formal parameter of degree 2(m−2). The differential b on S L H extends to a differential on the k[h]-module S L H [h]. Consider the subcomplex of S L H [h] which is spanned by elements whose homogeneous components are of the form: N [(a1,1 , h 1,1 )| · · · |(a1, p1 , h 1, p1 )]) • · · · • N ([(ak,1 , h k,1 )| · · · |(ak, pk , h k, pk )]),

(35)

where all the h i, j are distinct. Denote this subcomplex by S L H [h]. Let A˜ be the quotient module of S L H [h] defined by identifying any element of the form (35) with other elements obtained by replacing h i, j with any h˜ i, j satisfying h i, j < h i , j if and only if h˜ i, j < h˜ i , j . Let B˜ be the submodule of A˜ generated by elements of the following form: X − X i, j,i , j − X i, j,i , j , where i = i , h i, j < h i , j , and (i , j ) with h i, j < h i , j < h i , j ; X − X i, j,i, j − h X i, j,i, j , where h i, j < h i, j , and (i , j ) with h i, j < h i , j < h i, j ,

(36) (37)

where the X and X terms are defined as follows: if i = i , X i, j,i , j is the same as X except that the heights h i, j and h i , j are interchanged, while X i, j,i , j replaces the components N ([(ai,1 , h i,1 )| · · · |(ai, pi , h i, pi )]) and N ([(ai ,1 , h i ,1 )| · · · |(ai , pi , h i , pi )]) by ±ε(ai, j ai , j )N ([(ai, j+1 , h i, j+1 )| · · · |(ai, j−1 , h i, j−1 )|(ai , j +1 , h i , j +1 ) ×| · · · |(ai , j −1 , h i , j −1 )]); similarly, X i, j,i, j is the same as X but with the heights h i, j and h i, j interchanged, while X i, j,i, j is given by replacing the component with the following two components: ±ε(ai, j ai, j )N ([(ai, j +1 , h i, j +1 )| · · · |(ai, j−1 , h i, j−1 )]) • N ([(ai, j+1 , h i, j+1 ) ×| · · · |(ai, j −1 , h i, j −1 )]). Lemma 14. Let A˜ and B˜ be as above. Then A˜ is a chain complex and B˜ is a subcomplex ˜ of A. ˜ and so A˜ is a chain complex. We have to Proof. It is clear that b is well-defined on A, ˜ ˜ check that B is a subcomplex of A. The equivalence relation (36) only involves operations on two components in the elements of S L H [h], so without loss of generality, we may assume X = N ([(a1 , 2h 1 )| · · · |(an , 2h n )]) • N ([(b1 , 2g1 )| · · · |(bm , 2gm )]). Suppose that in X , the heights 2h i and 2g j satisfy condition (36). Then X − X − X = ±N ([(a1 , 2h 1 )| · · · |(an , 2h n )]) • N ([(b1 , 2g1 )| · · · |(bm , 2gm )]) ∓N ([(a1 , 2h 1 )| · · · |(ai , 2g j )| · · · |(an , 2h n )]) • N ([(b1 , 2g1 ) ×| · · · |(b j , 2h i )| · · · |(bm , 2gm )]) ∓ε(ai b j )N ([(ai+1 , 2h i+1 )| · · · |(ai−1 , 2h i−1 )|(b j+1 , 2g j+1 )| · · · |(b j−1 , 2g j−1 )]).

48


Therefore b(X ) =

±N ([· · · |(ak , 2h k )|(ak , 2h k + 1)| · · · ]) ⊗ N ([· · · |(bl , 2gl )| · · · ])

(38)

k =i

+ +

±N ([· · · |(ai , 2h i )|(ai , 2h i + 1)| · · · ]) ⊗ N ([· · · |(bl , 2gl )| · · · ])

(39)

±N ([· · · |(ak , 2h k )| · · · ]) ⊗ N ([· · · |(bl , 2gl )|(bl , 2gl + 1)| · · · ])

(40)

l = j

+ ±N ([· · · |(ak , 2h k )| · · · ] ⊗ N ([· · · |(bj , 2g j )|(bj , 2g j + 1)| · · · ]), b(X ) = ±N ([· · · |(ak , 2h k )|(ak , 2h k + 1)| · · · ]) ⊗ N ([· · · |(b j , 2h i )| · · · ])

(41) (42)

k =i

+ +

±N ([· · · |(ai , 2g j )|(ai , 2g j +1)| · · · ])⊗ N ([(· · · |(b j , 2h i )| · · · )])

(43)

±N ([· · · |(ai , 2g j )| · · · ]) ⊗ N ([· · · |(bl , 2gl )|(bl , 2gl + 1)| · · · ])

(44)

l = j

+ ±N ([· · · |(ai , 2g j )| · · · ]) ⊗ N ([· · · |(bj , 2h i )|(bj , 2h i + 1)| · · · ]), b(X ) = ±ε(ai b j )N ([· · · |(ak , 2h k )|(ak , 2h k + 1)| · · · ])

(45) (46)

k =i

+

±ε(ai b j )N ([· · · |(bl , 2gl )|(bl , 2gl + 1)| · · · ]).

(47)

l = j

˜ It is plain that both (38) − (42) − (46) and (40) − (44) − (47) are contained in B. ˜ To see that (39) + (41) − (43) − (45) is also contained in B, we introduce the following interpolating terms: ±N ([· · · |(ai , 2h i )|(ai , 2g j )| · · · ]) ⊗ N ([· · · |(b j , 2h i + 1)| · · · ]), (48) ±ε(ai b j )N ([(ai+1 , 2h i+1 )| · · · |(ai , 2h i )|(b j+1 , 2g j+1 )| · · · |(b j−1 , 2g j−1 )]), (49) ±N ([· · · |(ai , 2h i )|(ai , 2g j + 1)| · · · ]) ⊗ N ([· · · |(b j , 2g j )| · · · ]), (50) ±ε(ai b j )N ([(ai , 2g j +1)| · · · |(ai−1 , 2h i−1 )|(b j+1 , 2g j+1 )| · · · |(b j−1 , 2g j−1 )]),

(51) ±N ([· · · |(ai , 2g j )| · · · ]) ⊗

N ([· · · |(bj , 2h i )|(bj , 2g j

+ 1)| · · · ]),

(52)

±ε(ai bj )N ([(ai+1 , 2h i+1 )| · · · |(ai−1 , 2h i−1 )|(bj , 2g j + 1)| · · · |(b j−1 , 2g j−1 )]), (53) ±N ([· · · |(ai , 2h i + 1)| · · · ]) ⊗

N ([· · · |(bj , 2h i )|(bj , 2g j )| · · · ]),

±ε(ai bj )N ([(ai+1 , 2h i+1 )| · · · |(ai−1 , 2h i−1 )|(b j+1 , 2g j+1 )| · · · |(bj , 2h i )]).

One has ˜ (39) − (48) − (49) ∈ B, ˜ (50) − (43) − (51) ∈ B,

˜ (41) − (52) − (53) ∈ B, ˜ (54) − (45) − (55) ∈ B.

(54) (55)


49

Moreover, (48) = (50), (52) = (54), (49) = −(53), (51) = −(55). ˜ Hence, (39) + (41) − (43) − (45) ∈ B. By a similar argument, the subspace spanned by elements of the form X − X − h X ˜ in (37) is also stable under b, and therefore B˜ is a subcomplex of A.

˜ B. ˜ There is a DG Hopf algebra structure on A, which quanTheorem 15. Let A = A/ tizes the DG Lie bialgebra (L , { , }, δ) of Theorem 9. Moreover, A is isomorphic to U (L)[h] as k[h]-modules. The proof of the above theorem is given in the following subsections. 4.2. Proof of DG algebra. For any two elements X, X ∈ A, define the product of X and X as follows: suppose X, X are both represented by elements of the form (35); let X be the element which is the same as X but with the corresponding heights h i , j (X ) replaced by h i , j (X )+C, where C = 1+maxi, j,i , j (h i, j (X )−h i , j (X )). Here, h i, j (X ) are the heights in X and similarly h i , j (X ) are the heights in X . Thus, X is obtained from X by shifting the heights of the latter such that its heights are larger than those of X . The product of X and X is defined to be X • X . It is easy to see that this is well-defined, and commutes with the boundary b. 4.3. Proof of DG coalgebra. For an element X in the form of (35), let P := PX := {(i, j) | 1 ≤ i ≤ k, 1 ≤ j ≤ pi }. If (i, j) ∈ P, we let

(i, j) + (0, 1) =

(i, j + 1) if j < pi , (i, 1)

if j = pi .

Let n be an integer greater than or equal to 2. Now let I be any subset of P such that #I is even, and let φ : I → I be an involutive, fixed point-free map, where by being involutive we mean φ 2 = id. We call (I, φ, f ) an n-labeling of X if f : P → {1, 2, . . . , n} is a map such that:

f (i, j) =

f ((i, j) + (0, 1)),

if (i, j) ∈ / I;

f (φ(i, j) + (0, 1)),

if (i, j) ∈ I,

(56)

and f (i, j) > f (φ(i, j)) if and only if h i, j > h φ(i, j) , for (i, j) ∈ I. For an n-labeling (I, φ, f ), let q : P → P be given by (i, j) + (0, 1), if (i, j) ∈ / I, (i, j) → φ(i, j) + (0, 1), otherwise,

(57)

50


and define g : P\I → P\I by the following: for (i, j) ∈ P\I , let g(i, j) be the first element not in I under the iterations of the map q. Since q is a permutation of the finite set P, g is well-defined. Suppose the orbits of P under iterations of q is {Q 1 , . . . , Q w }. Then f descends to a map f : {Q 1 , . . . , Q w } → {1, . . . , n}, where f (Q m ) = f (i, j) for any (i, j) ∈ Q m , 1 ≤ m ≤ w. Similarly, suppose the orbits of P\I under iterations of g is {P1 , . . . , Pl }. Then f descends to a map f¯ : {P1 , . . . , Pl } → {1, . . . , n}, where f¯(Pm ) = f (i, j) for any (i, j) ∈ Pm , 1 ≤ m ≤ l. Suppose Pm (1 ≤ m ≤ l) is the orbit of (i, j) under g; then we define an element X m ∈ L H by X m = N ([(ai, j , h i, j )|(ag(i, j) , h g(i, j) )| · · · ]). (i)

Let 1 ≤ i ≤ n. Now define an element X (I,φ, f ) in A by ⎧ if f −1 (i) = ∅, ⎨1 (i) X (I,φ, f ) = 0 if #( f¯−1 (i)) < #( f −1 (i)), ⎩ −1 X i1 • · · · • X ir if #( f¯ (i)) = #( f −1 (i)) and f¯−1 (i) = {Pi1 , . . . , Pir }. The n-fold coproduct of X is defined by (1) (n) n (X ) := ε(I,φ, f ) h (I,φ, f ) X (I,φ, f ) ⊗ · · · ⊗ X (I,φ, f ) , I,φ, f

where ε(I,φ, f ) =

ε(ai, j · aφ(i, j) )

{(i, j)∈I | f (i, j)< f (φ(i, j))}

and h (I,φ, f ) = h (#I −2k+2l)/4 . Define := 2 . The following lemma yields the DG coalgebra on A: Lemma 16. Let n and be defined as above. Then (i) n is well-defined. (ii) is coassociative. (iii) b commutes with . Proof.

˜ then (i) To check that it is well-defined, one has to verify that if X˜ ∈ B, n−1 id ⊗i−1 ⊗ B˜ ⊗ id ⊗n−i . The proof of this is completely similar n−1 ( X˜ ) ∈ i=1

to [23, §3.5] (by considering a quiver in [23] with just one vertex); we omit the details. (ii) The proof is similar to [23, §3.7]. We have ( ⊗ id) ◦ = 3 = (id ⊗ ) ◦ . As explained in [23, §3.7], one can group the labelings 1 and 2 in 3 into labeling 1 and consider 1 and 3; this gives the first identity. Similarly, grouping the labelings 2 and 3 together into 2 and considering 1 and 2 gives the second identity. (iii) The proof is by a direct verification similar to the proof for the DG Lie coalgebra.


51

4.4. The Hopf identity. The proof is similar to [23, §3.8]. For any X, Y ∈ A, (X Y ) = 2 (X Y ) (X Y ) ⊗ (X Y ) = 2-labelings of X Y

=

2-labelings of X

X Y ⊗ X Y +

(X Y ) ⊗ (X Y ) .

φ(I ∩PX )∩PY =∅

and of Y

The last summation is over all 2-labelings (I, φ, f ) of X Y such that φ(I ∩ PX )∩ PY = ∅. However, in the product X Y , the heights of Y are all greater than that of X , and if the set φ(I ∩ PX ) ∩ PY is nonempty, then by (56) and (57), one has f (i, j) − f ((i, j) + (0, 1)) 0= (i, j)∈PX

=

f (i, j) − f ((i, j) + (0, 1)) < 0,

{(i, j)∈I ∩PX | φ(i, j)∈PY }

a contradiction. Hence, (X Y ) = (X )(Y ). Remark 17. The antipode map S : A → A is defined by replacing the heights h i, j in X by −h i, j , and then multiplying by (−1)#(PX ) . See [23, §3.9]. 4.5. Proof of the quantization. Let BC be a basis for C. Let B L := {N ([a1 | · · · |an ]) ∈ L | ai ∈ BC for all i}. Then B L is a basis for L. Let S L be the symmetric algebra for L and B S L := {x1 • · · · • xk ∈ S L | xi ∈ B L for all i}. Then B S L is a basis for S L. Suppose x ∈ B S L is the element N ([a1,1 | · · · |a1, p1 ]) • · · · • N ([ak,1 | · · · |ak, pk ]), where ai, j ∈ BC for all i, j. Then we fix an element Y (x) ∈ A˜ of the form (35) where the sequence h 1,1 , . . . , h k, pk is a permutation of 1, 2, . . . , # PY (x) . Let Y¯ (x) := Y (x)+B˜ ∈ A. Theorem 18. The set B A := {Y¯ (x) ∈ A | x ∈ B S L } is a basis for A over k[h]. We refer the reader to [23, Corollary 4.2] for the proof of Theorem 18. (An alternative proof can also be given following the proof of the usual PBW Theorem for universal enveloping algebras of Lie algebras given, for example, in [5, §9.2].) It follows from Theorem 18 and the PBW Theorem for L that A is isomorphic to U (L)[h] as k[h]-modules.

52


Note that any element of L has a canonical lifting to an element of A/ h A since by (37) the heights do not matter. Thus, there is a natural map L → A/ h A. By (36), this map induces a homomorphism ι : U (L) → A/ h A. It follows by Theorem 18 that ι is an isomorphism of DG Hopf algebras. Let x = N ([a1 | · · · |an ]) ∈ L, and assume without loss of generality that its lifting X ∈ A is represented by N ([(a1 , 1)| · · · |(an , n)]). From the definition of , we have (X ) = 1 ⊗ X + X ⊗ 1 +h ±ε(ai a j )N ([(a1 , 1)| · · · |(ai−1 , i − 1)| · · · |(a j+1 , j + 1)| · · · |(an , n)]) i< j

⊗N ([(ai+1 , i + 1)| · · · |(a j−1 , j − 1)]) + higher order terms. It follows that δ(x) ≡

1 ((X ) − op (X )) h

mod h.

4.6. Proof of Theorem 1. (i) If (L , d) is a DG Lie algebra, then its universal enveloping U (L) with the induced differential is a DG Hopf algebra; denote the induced differential by b. We have H∗ (U (L), b) is the enveloping algebra of H∗ (L , d) (see Quillen [22, Appendix, Proposition 2.1]). Therefore, by Theorems 9 and 15, H∗ (A, b) quantizes the Lie bialgebra H C∗ (C)[m − 1]. (ii) This is immediate from Theorem 5 and Theorem 1 (i) . Acknowledgements. The first author would like to thank Professor Yongbin Ruan for his encouragement during the preparation of this paper. The third author was partially supported by NSF grant DMS-0726154. 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.

References 1. Abbaspour, H., Tradler, T., Zeinalian, M.: Algebraic string bracket as a Poisson bracket. http://arxiv.org/ abs/0807.2351v3 [math.AT], 2008 2. Abbaspour, H., Zeinalian, M.: String bracket and flat connections. Alg. Geom. Top. 7, 197–231 (2007) 3. Andersen, J.E., Mattes, J., Reshetikhin, N.: The Poisson structure on the moduli space of flat connections and chord diagrams. Topology 35(4), 1069–1083 (1996) 4. Andersen, J.E., Mattes, J., Reshetikhin, N.: Quantization of the algebra of chord diagrams. Math. Proc. Camb. Phils. Soc. 124(3), 451–467 (1998) 5. Carter, R.: Lie Algebras of Finite and Affine Type. Cambridge: Cambridge Univ. Press, 2005 6. Cattaneo, A., Fröhlich, J., Pedrini, B.: Topological field theory interpretation of string topology. Commun. Math. Phys. 240(3), 397–421 (2003) 7. Cattaneo, A., Rossi, C.: Higher-dimensional B F theories in the Batalin-Vilkovisky formalism: the BV action and generalized Wilson loops. Commun. Math. Phys. 221(3), 591–657 (2001) 8. Chas, M., Sullivan, D.: String topology, http://arXiv.org/abs/math/9911159v1 [math.GT], 1999 9. Chas, M., Sullivan, D.: Closed string operators in topology leading to Lie bialgebras and higher string algebra. In: The legacy of Niels Henrik Abel, Berlin: Springer, 2004, pp. 771–784 10. Chen, K.-T.: Iterated path integrals. Bull. AMS, 83(5), 831–879 (1977) 11. Cohen, R., Voronov, A.: Notes on string topology. In: String Topology and Cyclic Homology, Adv. Courses in Math: CRM, Barcelona, Basel-Boston: Birkhauser, pp. 1–95, 2006 12. Felix, Y., Thomas, J.-C.: Rational BV-algebra in String Topology. http://arXiv.org/abs/0705.4194v1 [math.AT], 2007


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13. Getzler, E., Jones, J.D.S., Petrack, S.: Differential forms on loop spaces and the cyclic bar complex. Topology 30, 339–371 (1991) 14. Ginzburg, V.: Non-commutative symplectic geometry, quiver varieties, and operads. Math. Res. Lett. 8(3), 377–400 (2001) 15. Goldman, W.: Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85(2), 263–302 (1986) 16. Hamilton, A.: Noncommutative geometry and compactifications of the moduli space of curves. http:// arXiv.org/abs/0710.4603v1 [math.QA], 2007 17. Hamilton, A., Lazarev, A.: Symplectic A∞ -algebras and string topology operations. http://arXiv.org/abs/ 0707.4003v2 [math.QA], 2007 18. Hess, K., Parent, P.-E., Scott, J.: CoHochschild homology of chain coalgebras. J. Pure Appl. Alg. 213(4), 536–556 (2009) 19. Jones, J.D.S.: Cyclic homology and equivariant homology. Invent. Math. 87, 403–423 (1987) 20. Lambrechts, P., Stanley, D.: Poincaré duality and commutative differential graded algebras. Ann. Sci. l’École Nor. Sup. 41, 495–509 (2008) 21. Loday, J.-S.: Cyclic homology. Second edition. Grundlehren der Mathematischen Wissenschaften, 301. Berlin: Springer-Verlag, 1998 22. Quillen, D.: Rational homotopy theory. Ann. of Math. 90(2), 205–295 (1969) 23. Schedler, T.: A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver. Int. Math. Res. Notices 2005(12), 725–760 (2005) 24. Turaev, V.G.: Skein quantization of Poisson algebras of loops on surfaces. Ann. Sci. École Norm. Sup. (4) 24(6), 635–704 (1991) Communicated by Y. Kawahigashi


Communications in


Poisson-Lie Interpretation of Trigonometric Ruijsenaars Duality L. Fehér1,2 , C. Klimˇcík3 1 Department of Theoretical Physics, MTA KFKI RMKI, P.O.B. 49, 1525 Budapest, Hungary 2 Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, H-6720 Szeged,

Hungary. E-mail: [email protected]

3 Institut de Mathématiques de Luminy, 163, Avenue de Luminy, 13288 Marseille, France.

E-mail: [email protected] Received: 26 June 2009 / Accepted: 17 June 2010 Published online: 13 October 2010 – © Springer-Verlag 2010

Abstract: A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented. The phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard Heisenberg double of U (n). The collections of commuting Hamiltonians of the systems in duality are shown to descend from two families of ‘free’ Hamiltonians on the double which are dual to each other in a Poisson-Lie sense. Our results give rise to a major simplification of Ruijsenaars’ proof of the crucial symplectomorphism property of the duality map.

1. Introduction In 1986 Ruijsenaars and Schneider [32] introduced a remarkable deformation of the non-relativistic integrable many-body systems due to Calogero [3], Sutherland [34] and Moser [22]. The deformation corresponds to a passage from Galilei to Poincaré invariance, and for this reason the deformed systems can be called relativistic Calogero systems. The family of Calogero type systems is very important both from the physical and from the mathematical point of view, and has been the subject of intense studies ever since its inception. See, e.g., the reviews [8,31,35]. When constructing action-angle maps for the classical, non-elliptic An−1 systems, Ruijsenaars [27,29,30] discovered an intriguing relation that arranges the Calogero type systems into ‘dual pairs’. The main feature of the duality between system (i) and system (ii) is the fact that the action variables of system (i) are the particle-position variables of system (ii), and vice versa. The simplest example is provided by the non-relativistic rational Calogero system, whose self-duality was already noticed by Kazhdan, Kostant, and Sternberg [16] when treating the system by symplectic reduction of T ∗ u(n) u(n) × u(n), and thereby relating the symmetry between the u(n) factors to the self-duality property. In his papers Ruijsenaars hinted at the possibility that there

56

L. Fehér, C. Klimˇcík

might exist an analogous geometric picture behind the duality in the other cases, too, leaving this as a problem for future investigation. Later Gorsky and Nekrasov [15] and their coworkers [13,14,23] introduced new ideas and conjectures in the area of Ruijsenaars’ duality. By generalizing [16], they proposed to interpret the duality in general in terms of symplectic reduction of two distinguished families of commuting Hamiltonians living on a suitable higher dimensional phase space. Upon a single reduction of this phase space described using two alternative gauge slices, i.e., two alternative models of the reduced phase space, those two families of commuting Hamiltonians may reduce to the two respective sets of action variables of the mutually dual systems. They mainly focused on reductions of infinite-dimensional phase spaces aiming to relate the Calogero type many-body systems to field theories. We wish to stress that also relatively simple finite-dimensional phase spaces can be considered for which these ideas work fully as expected. For example, we worked out in [10] the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider systems [27] by reducing the cotangent bundle of the group G L(n, C). The key problem in the reduction approach is to find for each particular case of Ruijsenaars’ duality two distinguished families of commuting Hamiltonians on an appropriate higher-dimensional phase space and to find a way how to reduce those families. In the present paper, we solve this problem for one of the structurally most interesting and technically most involved cases of the duality that links together two particular real forms of the complex trigonometric Ruijsenaars-Schneider system [30]. The first real form is the original trigonometric Ruijsenaars-Schneider system [32] characterized by the Lax matrix L and symplectic form ω, 1 4 e pk sinh( x2 ) sinh2 2x L j,k (q, p) = 1 + sinh(iq j − iqk + x2 ) sin2 (q j − qm ) m= j ×

ω=

x 2

1 4

,

(1.1)

dpk ∧ dqk , 0 ≤ qk < π, q1 > q2 > · · · > qn , pk ∈ R,

(1.2)

1+

m=k n

sinh2

sin2 (qk − qm )

k=1

and the other real form is the Ruijsenaars dual of (1.1-2) that can be locally characterized by the Lax matrix Lˆ and symplectic form ω, ˆ 1 4 eiqˆk sinh(− x2 ) sinh2 2x ˆ p) ˆ = Lˆ j,k (q, 1 − sinh( pˆ j − pˆ k − x2 ) sinh2 ( pˆ j − pˆ m ) m= j ×

m=k

ωˆ =

n k=1

1−

sinh2

x 2

1

sinh2 ( pˆ k − pˆ m )

d pˆ k ∧ d qˆk , 0 ≤ qˆk < 2π,

4

, pˆ j − pˆ j+1 >

(1.3) |x| ( j = 1, . . . , n − 1). 2 (1.4)

Poisson-Lie Interpretation of Trigonometric Ruijsenaars Duality

57

The Lax matrices are generating functions for Hamiltonians in involution. The ‘main Hamiltonians’ from which these systems derive their names read 1 tr L(q, p) + L(q, p)−1 2 1 n 2 sinh2 2x = (cosh p j ) 1+ sin2 (q j − qm ) j=1 m= j

HRS (q, p) =

(1.5)

and 1 ˆ ˆ q, Hˆ RS (q, q, ˆ p) ˆ + L( ˆ p) ˆ −1 ˆ p) ˆ = tr L( 2 1 n 2 sinh2 2x = (cos qˆ j ) . 1− sinh2 ( pˆ j − pˆ m ) j=1 m= j

(1.6)

Here x is a real, non-zero coupling constant and the ‘velocity of light’ has been set to unity. The variables (q, p) and (q, ˆ p) ˆ provide one-to-one set theoretic parametrizations of the phase spaces P and Pˆ of the original and the dual Ruijsenaars-Schneider systems, respectively. As manifolds, P is the cotangent bundle of the configuration space consisting of unordered n-tuples of distinct points on the circle U (1), which corresponds to n indistinguishable particles moving on the circle, and Pˆ is an open submanifold of the cotangent bundle of the torus Tn = U (1)×n . In what follows (P, ω, L) denotes the collection of the Ruijsenaars-Schneider Hamiltonian systems defined by the spectralˆ ω, ˆ invariants of L (1.1), and similarly for ( P, ˆ L). The duality between the real forms (1.1-2) and (1.3-4) was previously studied in [30] by the ‘direct’ approach, or, in other words, by an approach that did not use the methods of symplectic reduction. In particular, Ruijsenaars presented integration algorithms for the Hamiltonian flows of the original and dual systems and proved that the flows of the original system are complete on (P, ω) but the flows of the dual system are not ˆ ω). complete on ( P, ˆ He also pointed out that this singular behaviour of the dual system ˆ ω) is related to the fact that the injective action-angle map from ( P, ˆ into (P, ω) is not surjective but has only a dense open image. He then introduced an extension ( Pˆc , ωˆ c ) ˆ ω) of the phase space ( P, ˆ in order to achieve bijectivity of the corresponding extension of the action-angle map (alias the duality map).1 Remarkably, the dual flows turned out to be complete on the extended phase space ( Pˆc , ωˆ c ), which can be therefore referred to ˆ ω). as the completion of ( P, ˆ Our interest in the real forms of the complex trigonometric Ruijsenaars-Schneider system was inspired by a conjecture that Gorsky and Nekrasov raised in [15]. They derived yet another trigonometric real form (called the IIIb system in [30]) from gauged WZW theory, and conjectured that it should be possible to derive the same system also from an appropriate finite-dimensional Heisenberg double. Although, as it stands, this conjecture remains still open, we recently succeeded to prove its ‘extrapolation’ to the trigonometric real form (1.1-2). Indeed, in our paper [11], we obtained the collection (P, ω, L) of the original Ruijsenaars-Schneider Hamiltonian systems by a reduction of a 1 Our notations P, P, ˆ Pˆc correspond, respectively, to , , ˆ ˆ in [30]. Further notational correspondence is given in footnotes 3 and 4 in the text.

58


certain family of ‘free’ Hamiltonians on the Heisenberg double of the standard compact Poisson-Lie group U (n). The ‘free’ Hamiltonians were constructed as the pull-backs of the dressing-invariant functions on the dual Poisson-Lie group B(n) by the Iwasawa map R . (See Sect. 2.1 for the definitions of the Iwasawa maps and of the pertinent actions of U (n).) Their basic properties are explicit integrability and invariance with respect to the so-called quasi-adjoint action of U (n) on the double. Our present investigation relies on the existence of a dual family of ‘free’ Hamiltonians that have the same properties. In fact, the members of the dual family can be constructed by pulling-back to the double the conjugation-invariant functions on the Poisson-Lie group U (n) by means of the dual Iwasawa map R . As the main technical result of the present paper, we shall obtain the ˆ ω, ˆ of the dual Ruijsenaars-Schneider Hamiltonian completion of the collection ( P, ˆ L) systems by reducing the dual family of free Hamiltonians on the Heisenberg double of U (n). Notice that the first occurrence of the term ‘dual’ in the previous sentence refers to Ruijsenaars’ duality of integrable systems and the second occurrence refers to duality in the sense of Poisson-Lie groups. Said in other words, the duality between the two trigonometric Ruijsenaars-Schneider systems described in [30] can be interpreted as a remnant of the ‘geometric democracy’ between the Poisson-Lie group U (n) and the corresponding dual Poisson-Lie group B(n) that survives the symplectic reduction. We believe that our results are related to the quantum group aspects of the standard trigonometric Ruijsenaars-Schneider system [8,21,24,28] through the general correspondence between quantum groups and Poisson-Lie groups. Both the original and the dual families of free Hamiltonian systems on the Heisenberg double are Poisson-Lie symmetric with respect to the quasi-adjoint action of U (n) on the double. A decisive step in our work [11] was the choice of a suitable value ν(x) of the Poisson-Lie moment map of this quasi-adjoint symmetry such that the original Ruijsenaars-Schneider phase space (P, ω) could be identified with the constraint-manifold Fν(x) := −1 (ν(x)) factorized by the gauge group given by the isotropy group G ν(x) < U (n). In other words, (P, ω) was identified in [11] with the reduced phase space arising from the symplectic reduction of the Heisenberg double by the quasiadjoint Poisson-Lie symmetry at the value ν(x) of the moment map . In this paper, we ˆ ω) shall demonstrate that the completion ( Pˆc , ωˆ c ) of the dual phase space ( P, ˆ is also symplectomorphic to the same reduced phase space Fν (x)/G ν (x). Therefore we can view (P, ω) and ( Pˆc , ωˆ c ) as two distinct models of a single reduced phase space. Moreover, as was already mentioned, we shall interpret the two collections of commuting Hamiltonians associated with the dual pair of Ruijsenaars-Schneider systems as reductions of two commutative families of free Hamiltonians on the Heisenberg double. Our results thus fit the geometric interpretation of Ruijsenaars’ duality advocated by Gorsky and his collaborators for example in [13]. Speaking generally, the symplectic reduction approach often represents useful technical streamlining or simplification with respect to direct methods. This feature occurs also in our particular case. For instance, the free Hamiltonian systems on the Heisenberg double are themselves integrable and, as we shall see, their very simple Lax matrices reduce to the relatively complicated Lax matrices of the Ruijsenaars-Schneider systems. In addition, we can recover the integration algorithms for the original and dual Ruijsenaars-Schneider flows [30] simply by projecting the ‘free’ flows on the respective models of the reduced phase space Fν(x) /G ν(x) , and these projected flows are automatically complete. However, there is one aspect of the duality story where the reduction approach yields more than technical streamlining or simplification and, in fact, represents a major technical advantage with respect to the direct approach. Indeed, in the direct


59

approach it was very difficult to prove the crucial symplectomorphism property of the extended action-angle map between the phase space (P, ω) and the extended dual phase space ( Pˆc , ωˆ c ). To realize the difficulties, the reader may consult the ‘hyperbolic proof’ published in [27] and its ‘trigonometric analytic continuation’ presented in [30]. In these references, a sophisticated web of non-trivial steps had been used combining scattering theory with demanding analysis and intricate analytic continuation arguments. However, from the point of view of the reduction approach the extended action-angle map is automatically a symplectomorphism, since it is easily recognized to be the composition of the symplectomorhism relating (P, ω) with Fν(x) /G ν(x) and of the symplectomorphism relating Fν(x) /G ν(x) with ( Pˆc , ωˆ c ). To render justice to the direct methods, we note that so far considerably more examples of the duality were thoroughly investigated in the direct approach [27,29,30] than in the symplectic reduction approach. We hope, however, that the attractive features mentioned above supply sufficient motivation to further develop the reduction approach, along the lines discussed at the end of this article. The rest of the paper is organized as follows. In Sect. 2, we first review the geometry of the standard Heisenberg double of U (n) and recall the definition of the quasi-adjoint action of U (n) on it. We then describe the two families of free Hamiltonian systems on the double that are Poisson-Lie symmetric with respect to the quasi-adjoint action. We do not claim any new results in this section, although we could not find in the literature any previous detailed treatment of the free flows as given by our Propositions 2.1 and 2.2. The reduction of the two families of free Hamiltonians to the respective two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented in Sect. 3. After a summary of required notions, Theorems 3.1 and 3.2 state in a strengthened form the result of [11], where the first family was reduced to the system (P, ω, L). Then the new Theorems 3.3 and 3.4 are formulated, which claim that the same symplectic reduction ˆ ω, ˆ Section 4 is applied to the second family yields the completion of the system ( P, ˆ L). devoted to the proofs of Theorems 3.1 and 3.2. Although many results described in this section were obtained already in our letter [11], we still add here important new material. Namely, we pay special attention to the Weyl group covering of the phase space P of the original Ruijsenaars-Schneider system, since this enables us to recover the geometric meaning of the standard coordinates on P from the reduction. Indeed, in [11], we identified the Ruijsenaars-Schneider system with the reduced system on Fν(x) /G ν(x) by means of a somewhat mysterious coordinate transformation that was just ‘cooked up’ to do the job. Here we explain the geometrical origin of this transformation. Theorems 3.3 and 3.4 represent the main results obtained in this paper and their proof occupies much of Sect. 5. As a by-product, the integration algorithm for the dual flows is also treated at the end of Sect. 5. Further discussion, including comparison with [30], is offered in Sect. 6. Finally, Appendix A contains an alternative proof of the fact that Fν(x) is an embedded submanifold of the Heisenberg double, and Appendix B is devoted to the subtle topology of the configuration space of n indistinguishable point-particles moving on the circle. 2. Free Systems on the Heisenberg Double Let us recall that a Poisson-Lie group is a Lie group, G, equipped with a Poisson bracket, {., .}G , such that the multiplication map G × G → G is Poisson. It is an important concept that makes it possible to generalize the usual notion of symmetry for Hamiltonian

60


systems. Namely, if the Poisson-Lie group G acts on a symplectic manifold M in a Poisson way (i.e. the action map G × M → M is Poisson) and, moreover, if the Hamiltonian H is G-invariant, then one says that the system (M, H ) is G Poisson-Lie symmetric [33]. For our purpose we shall focus on the group G = U (n) equipped with its standard Poisson-Lie structure, although all results collected in this section hold true, with minor modifications, for any compact reductive group G. In Subsect. 2.1, we summarize the necessary information concerning the structure of the Heisenberg double of the dual pair of Poisson-Lie groups U (n) and B(n) and also recall the notion of the quasi-adjoint action of U (n) on the double. Then, in Subsect. 2.2, we describe the two families of U (n) Poisson-Lie symmetric Hamiltonian systems that will descend upon symplectic reduction to the trigonometric Ruijsenaars-Schneider system (1.1) and its dual. These systems turn out to be explicitly integrable, and for this reason we call the underlying Hamiltonians ‘free’ Hamiltonians.

2.1. Recall of the Heisenberg double and the quasi-adjoint action. The Heisenberg double is a Poisson manifold (D, {., .}+ ) that Semenov-Tian-Shansky [33] associated with any Poisson-Lie group (G, {., .}G ). The manifold D is itself a Lie group, the Drinfeld double of G, and its Poisson bracket {., .}+ can be expressed in terms of the factorizable r -matrix of Lie(D). One can also directly define the double (D, {., .}+ ) and then recover from it the Poisson-Lie group (G, {., .}G ) and its dual. Consider the real Lie group D := G L(n, C) and endow the corresponding real Lie algebra D := gl(n, C) with the non-degenerate, invariant ‘scalar product’ (X, Y )D := tr (X Y ),

∀X, Y ∈ D.

(2.1)

Here z stands for the imaginary part of the complex number z. Let B := B(n) be the subgroup of D formed by the upper-triangular matrices having positive entries along the diagonal, and denote G := U (n). As a real vector space, we have the direct sum decomposition D = G + B,

(2.2)

where G := Lie(G) = u(n) and B := Lie(B) are isotropic subalgebras mutually dual to each other with respect to the pairing provided by (., .)D . In other words, we have a so-called Manin triple in our hands, and thus the real Lie group D carries two natural Poisson structures {., .}± . Here we need only the structure {., .}+ , and to define it we introduce the projection operators πG : D → G and πB : D → B associated with the splitting (2.2). Furthermore, for any real function ∈ C ∞ (D) introduce the left- and right gradients ∇ L ,R ∈ C ∞ (D, D) by d (es X K esY ) = (X, ∇ L (K ))D +(Y, ∇ R (K ))D , ds s=0

∀X, Y ∈ D, ∀K ∈ D. (2.3)

By using the factorizable r -matrix of D, ρ :=

1 (πG − πB ), 2

(2.4)


61

for any 1 , 2 ∈ C ∞ (D) one has the Poisson bracket { 1 , 2 }+ (K ) = ∇ R 1 (K ), ρ(∇ R 2 (K )) + ∇ L 1 (K ), ρ(∇ L 2 (K )) . D

D

(2.5) The Poisson manifold (D, {., .}+ ) is called the Heisenberg double of G. By the Iwasawa decomposition, each element K ∈ D has the unique representations and K = g L b−1 with b L ,R ∈ B, g L ,R ∈ G. K = b L g −1 R R

(2.6)

As a result of the global character of this decomposition, our (D, {., .}+ ) is actually symplectic. The underlying symplectic form, called ω+ , was found by Alekseev and Malkin [1]. In fact, with the help of the Iwasawa maps L ,R : D → B and L ,R : D → G defined by using (2.6) as L ,R (K ) := b L ,R and L ,R (K ) := g L ,R ,

(2.7)

one has ω+ =

1 1 −1 −1 −1 tr (d L −1 L ∧ d L L ) + tr (d R R ∧ d R R ). 2 2

(2.8)

Having described the Heisenberg double, next we recall how the Poisson bracket {., .}+ induces Poisson-Lie structures on the groups B and G. As a preparation, define the left- and right derivatives d L ,R f ∈ C ∞ (B, G) for a real function f ∈ C ∞ (B) by the equality d s X sY L R f (e be ) = X, d f (b) + Y, d f (b) , ∀X, Y ∈ B, ∀b ∈ B. D D ds s=0 (2.9) For a real function φ ∈ C ∞ (G) define d L ,R φ ∈ C ∞ (G, B) similarly, d φ(es X gesY ) = X, d L φ(g) + Y, d R φ(g) , ∀X, Y ∈ G, ∀g ∈ G. D D ds s=0 (2.10) By using the negative-definite scalar product on G furnished by

X, Y G := tr (X Y ),

∀X, Y ∈ G,

introduce also D L ,R φ ∈ C ∞ (G, G) by d φ(es X gesY ) = X, D L φ(g)G + Y, D R φ(g)G , ds s=0

(2.11)

∀X, Y ∈ G, ∀g ∈ G. (2.12)

Defining R i ∈ End(G) as R i (X ) = πG (−iX ),

∀X ∈ G,

(2.13)

62


which is actually nothing but the standard r -matrix of G, it is easy to check that the two types of derivatives over G are related by d L φ = iD L φ + R i (D L φ),

d R φ = iD R φ + R i (D R φ),

(2.14)

which in particular implies that d L ,R φ = πB (iD L ,R φ),

∀φ ∈ C ∞ (G).

(2.15)

After fixing all the above notations, we are ready to write down explicit formulas for the Poisson-Lie structures {., .} B on B and {., .}G on G. In fact, it can be shown that the algebras of functions C ∞ (B) and C ∞ (G) pulled-back, respectively, by the Iwasawa maps R and R form two Poisson subalgebras of C ∞ (D). Because of the surjectivity of the Iwasawa maps in (2.7), the bracket {., .}+ on D thus induces two brackets {., .} B and {., .}G by the following prescriptions: {∗R f 1 , ∗R f 2 }+ = ∗R { f 1 , f 2 } B and {∗R φ1 , ∗R φ2 }+ = ∗R {φ1 , φ2 }G .

(2.16)

The brackets {., .} B and {., .}G induced in this way are just the Poisson-Lie brackets on B and on G, respectively. The explicit form of the induced Poisson bracket on B reads { f 1 , f 2 } B (b) = − b−1 (d L f 1 (b))b, d R f 2 (b) , ∀ f 1 , f 2 ∈ C ∞ (B), ∀b ∈ B. D

(2.17) This is obtained from (2.5) and (2.6) using that if = ∗R f for some f ∈ C ∞ (B), then ∇ L (K ) = −g L d R f (b R ) g −1 L ,

∇ R (K ) = −b R d R f (b R ) b−1 R .

(2.18)

The induced Poisson-Lie structure on G permits the analogous formula {φ1 , φ2 }G (g) = g −1 (d L φ1 (g))g, d R φ2 (g) , ∀φ1 , φ2 ∈ C ∞ (G), ∀g ∈ G, D

(2.19) which can be conveniently rewritten as {φ1 , φ2 }G (g) = D R φ1 (g), R i (D R φ2 (g))G − D L φ1 (g), R i (D L φ2 (g))G

(2.20)

by virtue of (2.15). In the last formula only G features explicitly, while (2.17) and (2.19) are formulated relying on conjugations defined in the group D. The Poisson bracket {., .}+ closes also on ∗L C ∞ (B) and on ∗L C ∞ (G). In fact, by using L and L (2.7) one obtains the same Poisson-Lie structures on B and on G as by means of (2.16). Another important fact is that the elements of ∗R C ∞ (B) (respectively ∗R C ∞ (G)) commute with the elements of ∗L C ∞ (B) (respectively ∗L C ∞ (G)) with respect to {., .}+ . Finally, we recall from [18] the so-called quasi-adjoint Poisson action of G on (D, ω+ ). The corresponding action map G × D → D sends (g, K ) to g K defined by g K := g K R (g L (K )), ∀g ∈ G, ∀K ∈ D.

(2.21)


63

The composition property, g1 (g2 K ) = (g1 g2 ) K for all g1 , g2 ∈ G and K ∈ D, can be checked by using (2.6) and (2.7). The G-action admits the equivariant Poisson-Lie moment map : D → B given by (K ) = L (K ) R (K ),

∀K ∈ D.

(2.22)

This means that enjoys the following two properties. First, X D [ ] = (X, { , }+ −1 )D , ∀X ∈ G, ∀ ∈ C ∞ (D), where (X D [ ])(K ) :=

d sX ds (e

(2.23)

K )|s=0 for all K ∈ D. Second,

(g K ) = Dressg ((K )), ∀g ∈ G, ∀K ∈ D,

(2.24)

where we use the dressing action of G on B that operates as Dressg (b) := L (gb), ∀g ∈ G, ∀b ∈ B.

(2.25)

The quasi-adjoint action (2.21) was found in [18] by the following method. One first observes that the map defined by (2.22) satisfies (2.24) and it generates via (2.23) an infinitesimal action of G on D. These statements are implied [20] by the fact that : D → B is a Poisson map, which is obvious. The resulting infinitesimal action was then integrated, and the G-action obtained in this way automatically has the Poisson property, i.e., the action map G × D → D is a Poisson map. Remark 2.1. It follows from (2.21) that, like for the ordinary adjoint action, the central U (1) subgroup of G = U (n) acts trivially, and the factor group U (n)/U (1) acts effectively. It is also easy to check (see Appendix A) that the map (2.22) takes its values in the Poisson-Lie subgroup S B of B consisting of the elements of determinant one. Consider the projection π D : D → D¯ := D/G L(1, C),

(2.26)

where G L(1, C) denotes the center of D = G L(n, C), and define subgroups of D¯ by G¯ := π D (G) and B¯ := π D (B) = π D (S B).

(2.27)

The groups G¯ and B¯ sit in D¯ in quite the same way as G and B sit in D, and therefore are dual to each other in the Poisson-Lie sense. In fact, D¯ carries a symplectic structure inherited from (D, ω+ ), whereby it can be regarded as the Heisenberg double constituted ¯ The projection π D gives rise to natural isomorphisms by G¯ and B. U (n)/U (1) G¯

and

¯ S B B.

(2.28)

By using these identifications, the map : D → S B given by (2.22) yields the Poisson-Lie moment map for the quasi-adjoint action of U (n)/U (1) on D.

64


2.2. Integration algorithms for the free flows. Our principal aim now is to present the flows of two families of commuting Hamiltonians on the Heisenberg double that are invariant with respect to the quasi-adjoint action of G. The Hamiltonians of our interest form the rings H and Hˆ defined by H := ∗R C ∞ (B)c

and

Hˆ := ∗R C ∞ (G)G .

(2.29)

Here C ∞ (B)c denotes the center of the Poisson-Lie structure on C ∞ (B) and C ∞ (G)G contains the functions on G that are invariant with respect to the standard adjoint action of G on G. It is obvious that all elements of H mutually Poisson commute which each other. The Poisson commutativity of Hˆ also takes places, because D L φ = D R φ for any φ ∈ C ∞ (G)G , and hence (2.20) implies that {φ1 , φ2 }G = 0 for any φ1 , φ2 ∈ C ∞ (G)G . Note, however, that the elements of C ∞ (G)G do not lie in the center of the Poisson-Lie structure on C ∞ (G) in general. It is well known and is readily seen from (2.17) and (2.25) that C ∞ (B)c = C ∞ (B)G

(2.30)

with the dressing-invariant functions on the right-hand side. Moreover, if inv : B → B is the (anti-Poisson) inversion map, then one can verify that inv∗ stabilizes C ∞ (B)c and ∗R f = ∗L ( f ◦ inv),

∀ f ∈ C ∞ (B)c .

(2.31)

In particular, it follows that H := ∗R C ∞ (B)c = ∗L C ∞ (B)c .

(2.32)

To obtain another useful characterization of H, consider the diffeomorphism, P, from B to the space of Hermitian positive definite matrices, P, defined by P(b) := bb† ,

∀b ∈ B.

(2.33)

The map P intertwines the dressing action (2.25) on B with the ordinary conjugation action on P, P(Dressg b) = gP(b)g −1 ,

∀g ∈ G, ∀b ∈ B,

(2.34)

whereby the elements of C ∞ (B)c correspond to the spectral-invariants C ∞ (P)G on P. It is of crucial importance that all Hamiltonians in H and in Hˆ are invariant under the quasi-adjoint action (2.21) on the double. This holds since for all g ∈ G and K ∈ D one has L (g K ) = Dressg ( L (K )), R (g K ) = R (g L (K ))−1 R (K ) R (g L (K )).

(2.35)

The first relation and (2.30), (2.32) imply the G invariance of the elements of H, which also follows directly from R (g K ) = Dress R (g L (K ))−1 ( R (K )). Now we are ready to show that the evolution equations of the Hamiltonian systems ˆ (D, ω+ , H ) and (D, ω+ , Hˆ ) can be explicitly solved for any H ∈ H and any Hˆ ∈ H. We start with H.


65

Proposition 2.1. The flow induced on the Heisenberg double D by the Hamiltonian H = ∗R f with an arbitrary f ∈ C ∞ (B)c is given by

(2.36) K (t) = g L (0) exp −td R f (b R (0)) b−1 R (0). In terms of the Iwasawa decompositions (2.6), the flow can be written as

g L (t) = g L (0) exp −td R f (b R (0)) , b R (t) = b R (0),

g R (t) = exp −td L f (b L (0)) g R (0), b L (t) = b L (0).

(2.37) (2.38)

Proof. Equation (2.5) entails that the Hamiltonian vector field, V , generated by an arbitrary ∈ C ∞ (D) has the form V (K ) = Kρ(∇ R (K )) + ρ(∇ L (K ))K . Notice from (2.17) that πB b d R f (b) b−1 = 0,

∀b ∈ B, ∀ f ∈ C ∞ (B)c .

By combining these formulae and (2.18), we obtain ∗ ∞ c V (K ) = −g L d R f (b R ) b−1 R , if = R f, f ∈ C (B) .

(2.39)

(2.40)

(2.41)

At the same time, since R : D → B is a Poisson map, we also have V (b R ) = 0. It follows immediately that the flow is given by (2.36), which obviously translates into (2.37). The alternative formula (2.38) can be established using that b L (t) = b L (0) by (2.32), and that d L f (b) = b(d R f (b))b−1 for any f ∈ C ∞ (B)c . Proposition 2.1 says that the Poisson-Lie momenta b L , b R are constants of motion and both ‘position-like’ variables g L , g R follow Killing geodesics on G. In a special case, this statement first appeared in [38]. Now we turn to the flows associated with the conjugation-invariant functions on G. We begin by treating this problem on the Poisson-Lie group G. Lemma 2.1. For φ ∈ C ∞ (G)G , consider the Hamiltonian evolution equation on (G, {., .}G ), g˙ := {g, φ}G = [g, R i (Dφ(g))],

(2.42)

where we denote D L φ = D R φ simply by Dφ. Taking an arbitrary initial value g(0), let the curves β(t) ∈ B and γ (t) ∈ G be the (unique, smooth) solutions of the factorization problem eitDφ(g(0)) = β(t)γ (t).

(2.43)

Then the solution of (2.42) with the initial value g(0) is given by g(t) = γ (t)g(0)γ (t)−1 .

(2.44)

66


Proof. By taking the derivative of (2.44), we obtain g(t) ˙ = [γ˙ (t)γ (t)−1 , g(t)].

(2.45)

On the other hand, by taking the derivative of (2.43) we obtain ˙ β(t)γ (t)(iDφ(g(0))) = β(t)γ (t) + β(t)γ˙ (t).

(2.46)

˙ + γ˙ (t)γ (t)−1 , iγ (t)Dφ(g(0))γ (t)−1 = iDφ(g(t)) = β(t)−1 β(t)

(2.47)

This implies

where the first equality follows from the invariance property of φ. We see from (2.47) that γ˙ (t)γ (t)−1 = πG (iDφ(g(t))) = −R i (Dφ(g(t))), which concludes the proof.

(2.48)

Proposition 2.2. The flow K (t) = b L (t)g R (t)−1 induced on the Heisenberg double by the Hamiltonian Hˆ = ∗R φ with an arbitrary φ ∈ C ∞ (G)G is given by g R (t) = γ (t)g R (0)γ (t)−1 ,

b L (t) = b L (0)β(t)

(2.49)

in terms of the solutions γ (t) ∈ G, β(t) ∈ B of the factorization problem eitDφ(g R (0)) = β(t)γ (t). Proof. Consider first an arbitrary ‘collective Hamiltonian’ on D of the type = ∗R φ,

φ ∈ C ∞ (G).

(2.50)

In this case one obtains directly from the definitions that ∇ L (K ) = −b L (d R φ(g R ))b−1 L ,

∇ R (K ) = −g R (d R φ(g R ))g −1 R .

(2.51)

With the aid of these relations, the evolution equation K˙ = V (K ) (2.39) can be spelled out as g˙ R = g R R i (D R φ(g R )) − R i (D L φ(g R ))g R , b˙ L = b L (d R φ(g R )) = b L πB (iD R φ(g R )).

(2.52) (2.53)

Besides (2.51), we also used (2.13) and (2.14) to get this system of equations. Notice that (2.52) is just the Hamiltonian evolution equation generated by φ on (G, {., .}G ). If we now assume that φ ∈ C ∞ (G)G , then the desired solution of the last system of equations is easily found with the aid of (2.44) and (2.47). This yields the flow as claimed in (2.49). Corollary 2.1. The integral curve of the Hamiltonian Hˆ = ∗R φ, φ ∈ C ∞ (G)G satisfies K (t)K † (t) = b L (t)b L (t)† = b L (0)e2itDφ(g R (0)) b L (0)† .

(2.54)

Proof. Combine (2.49) with b L (0)β(t)(b L (0)β(t))† = b L (0)β(t)γ (t)(b L (0)β(t)γ (t))† .


67

We finish this section with a few comments. First we notice from the statement below (2.34) that C ∞ (B)c is functionally generated by the (not all independent) invariants f k (b) :=

1 tr (bb† )k , 2k

∀k ∈ Z∗ .

(2.55)

It is also clear that C ∞ (G)G is generated by the functions φk (g) :=

1 1 tr (g k + g −k ), φ−k (g) := tr (g k − g −k ), ∀k ∈ Z+ . 2k 2ki

(2.56)

The flows of the generators Hk := ∗R f k of H and Hˆ k := ∗R φk of Hˆ can be written down explicitly using that d R f k (b) = i(b† b)k , d L f k (b) = i(bb† )k , ∀k ∈ Z∗ , 1 1 Dφk (g) = (g k − g −k ), Dφ−k (g) = (g k + g −k ), ∀k ∈ Z+ . 2 2i

(2.57) (2.58)

For later reference, we record that if one considers a real linear combination φ := k=0 μk φk , and writes it with some analytic function χ in the form φ(g) = tr (χ (g)) + c.c., then one has Dφ(g) = gχ (g) − (gχ (g))† .

(2.59)

We note from the above that the Hamiltonians Hk and Hˆ k , and more generally the elements of the rings H and Hˆ (2.29) that they generate, are spectral-invariants of the respective matrix functions L and Lˆ on the Heisenberg double D defined by L : K → b R b†R

and

Lˆ : K → g R

(2.60)

with the Iwasawa decompositions in (2.6). For our purpose, it will be fruitful to view L and Lˆ as ‘unreduced Lax matrices’ and the following convention will also prove to be very convenient. Definition 2.1. By using the previous notations (2.29) and (2.60), we define the collections of Hamiltonian systems (D, ω+ , L) := { (D, ω+ , H ) | H ∈ H }, ˆ := { (D, ω+ , Hˆ ) | Hˆ ∈ Hˆ }, (D, ω+ , L)

(2.61)

and henceforth refer to these collections as the ‘canonical free systems’. Any member of the above collections is integrable in the obvious sense that one can directly write down its Hamiltonian flow, as given by Propositions 2.1 and 2.2. In the rest of the paper we shall study the symplectic reduction of the canonical free systems, i.e., the simultaneous reduction of all the Hamiltonian systems that constitute them. The main advantage of our, somewhat non-standard, notation (2.61) is that it suggests that ˆ instead of separately reducing the one should directly reduce the Lax matrices L and L, Hamiltonians that they generate. In this respect, observe from (2.35) and the sentence afterwards that the quasi-adjoint action (2.21) operates by similarity transformations on L and on Lˆ in (2.60). Since, as generators of commuting Hamiltonians, any two Lax matrices that are related by a similarity transformation are equivalent, one can indeed

68


use the quasi-adjoint action to reduce the (equivalence classes of the) Lax matrices L ˆ The usefulness of this point of view will become clear in Sect. 3. and L. Finally, for clarity, let us remark that the elements of the Abelian subalgebra ∗L C ∞ (G)G of the Poisson algebra C ∞ (D) are in general not invariant under the quasiadjoint action (2.21). Indeed L (g K ) = g L (K ) L ( R (K )−1 R (g L (K )))

(2.62)

is not conjugate to L (K ) in general. However, the elements of ∗L C ∞ (G)G are actually invariant with respect to the alternative quasi-adjoint action [18] of G on D that can be associated with the ‘flipped moment map’ := R L : D → B. (The elements of H (2.32) are invariant under both quasi-adjoint actions.) Since = L R can be converted into by inversion on the group D, it is sufficient to consider only the quasi-adjoint action given by (2.21). 3. Reduction of the Canonical Free Systems Our goal here is to present the results that permit the identification of a certain symplectic ˆ of Definition 2.1 with reduction of the canonical free systems (D, ω+ , L) and (D, ω+ , L) the Ruijsenaars-Schneider system (P, ω, L) (1.1) and with a natural extension of the ˆ ω, ˆ (1.3), respectively. For this purpose we must take D = G L(n, C), dual system ( P, ˆ L) but it is worth stressing that the preliminaries described in Sect. 2 remain valid in a more general context. We first review the necessary theoretical background concerning symplectic reduction based on Poisson-Lie symmetry with an equivariant moment map [20]. (The notations ¯ B, ¯ used in this overview anticipate the application studied later on.) Consider a G, symplectic manifold M acted upon smoothly and effectively by a compact Poisson-Lie group G¯ in such a way that the action map : G¯ × M → M is Poisson and choose a ¯ The -preimage Fν regular value ν ∈ B¯ in the image of the moment map2 : M → B. of the point ν is then an embedded submanifold of M. The maximal subgroup G¯ ν < G¯ which leaves Fν invariant is called the gauge group. If G¯ ν acts freely on Fν then, as is well known, there exists a unique manifold structure on the space of orbits, Fν /G¯ ν , such that the canonical projection π : Fν → Fν /G¯ ν

(3.1)

is a smooth submersion, i.e., Fν (Fν /G¯ ν , G¯ ν , π ) is a principal fiber bundle. Let us note that at every point m ∈ Fν the tangent space Tm Fν has the vertical subspace Vm Fν ⊂ Tm Fν

(3.2)

generated by the infinitesimal action of G¯ ν . Smooth functions on Fν /G¯ ν correspond to smooth G¯ ν -invariant functions on Fν . The manifold structure on Fν /G¯ ν is constructed with the aid of local cross sections for the G¯ ν action [7]. In fact, it turns out that Fν /G¯ ν is a symplectic manifold. It is referred to as the reduced symplectic manifold (or reduced phase space) and the symplectic form ν on it is uniquely determined by the requirement | Fν = π ∗ ν .

(3.3)

2 The moment map is a Poisson map into the Poisson-Lie group B¯ dual to G. ¯ It generates the G¯ action via the Poisson bracket on M by X M [ f ] = X, { f, } M −1 , where X M is the vector field on M corresponding ¯ f ∈ C ∞ (M), and ., . is the dual pairing. to X ∈ Lie(G),


69

Here | Fν denotes the pull-back of the original symplectic form of M on the submanifold Fν ⊂ M. ¯ ¯ The Hamiltonian flow induced by any G-invariant function H ∈ C ∞ (M)G preserves ¯ Fν , and the restricted flow is projectable from Fν to Fν /G ν . The projection gives the flow associated with the reduced Hamiltonian H ν ∈ C ∞ (Fν /G¯ ν ) (for which H | Fν = H ν ◦π ) ¯ Hamiltoby means of ν . It further follows that the involutivity of a set of G-invariant ¯ ∞ G nians {H j } ⊂ C (M) is inherited by the corresponding set of reduced Hamiltonians {H jν } ⊂ C ∞ (Fν /G¯ ν ). In concrete examples one wishes to exhibit models of the reduced symplectic manifold (Fν /G¯ ν , ν ). In principle, any symplectic manifold that is globally symplectomorphic to (Fν /G¯ ν , ν ) can serve as a model, but the real aim is to construct models as explicitly as possible. The simplest situation occurs when the manifold Fν is a trivial G¯ ν -bundle. Then the reduced symplectic manifold (Fν /G¯ ν , ν ) can be modelled by any global cross section of the G¯ ν action on Fν . To be more precise, we present the following (standard) result. ¯ ω) Lemma 3.1. Suppose that ( P, ¯ is a symplectic manifold and J : P¯ → Fν is a smooth injective map such that 1. J ∗ (| Fν ) = ω, ¯ 2. the image S := {J (y) | y ∈ P¯ } intersects every G¯ ν -orbit in Fν exactly in one point. ¯ ω) Then the map π ◦ J : P¯ → Fν /G¯ ν is a symplectic diffeomorphism, and ( P, ¯ can thus serve as a model of the reduced phase space (Fν /G¯ ν , ν ). Proof. The procedure of symplectic reduction rests on the fact that Ker m (| Fν ) = Vm Fν ,

∀m ∈ Fν ,

(3.4)

where Ker m (| Fν ) is the annihilator of | Fν in Tm Fν . This equation and J ∗ (| Fν ) = ω¯ imply that the tangent (derivative) map Ty J ≡ (DJ )(y) : Ty P¯ → TJ (y) Fν is injective and ¯ = {0}, VJ (y) Fν ∩ Ty J (Ty P)

¯ ∀y ∈ P.

(3.5)

¯ Ty J (Y ) ∈ VJ (y) Fν for a non-zero Y ∈ Ty P¯ Indeed, the hypothesis that, at some y ∈ P, ¯ However, this is excluded by the would entail that J ∗ (| Fν )(Y, Z ) = 0 for all Z ∈ Ty P. ∗ non-degeneracy of J (| Fν ). The injectivity of Ty J means that the map J : P¯ → Fν is an immersion. Next, being the composition of two smooth maps, π◦J : P¯ → Fν /G¯ ν is smooth and we see from (3.5) that it is also an immersion. Therefore, since π ◦ J is injective and surjective by assumption, it must be a submersion, which in particular requires that ¯ TJ (y) Fν = VJ (y) Fν ⊕ Ty J (Ty P),

¯ ∀y ∈ P.

(3.6)

It is well known that the one-to-one smooth submersions are precisely the diffeomorphisms. Hence we conclude that the map π ◦J is a diffeomorphism. Finally, using (3.3), we obtain ¯ (π ◦ J )∗ ν = J ∗ (π ∗ ν ) = J ∗ (| Fν ) = ω.

(3.7)

70


Remark 3.1. On account of (3.6), the smooth one-to-one map ◦(id G¯ ν , J ) : G¯ ν × P¯ → Fν is again a submersion. This implies that the map ◦ (id G¯ ν , J ) is a diffeomorphism. ¯ Hence the map J is an embedding and the restriction of the projection π to S = J ( P) yields a diffeomorphism, π S : S → Fν /G¯ ν . Moreover, if we define σ : Fν /G¯ ν → Fν by σ (b) := J (y) for the unique y ∈ P¯ such that b = π(J (y)), then σ is a smooth global section of the bundle π : Fν → Fν /G¯ ν in the usual sense, i.e., σ ∈ C ∞ (Fν /G¯ ν , Fν ) and π ◦ σ = id Fν /G¯ ν . This follows by noting that σ = ι S ◦ π S−1 , where ι S : S → Fν is the tautological injection. Remark 3.2. We call the map J of Lemma 3.1 a global cross section of the G¯ ν action on Fν . The same term can be used to refer to the image S of J , too, but here we adopt the more widespread terminology that refers to S as a global gauge slice. Note that (S, | S ) can be also thought of as a model of the reduced phase space, since it is ¯ ω) symplectomorphic to ( P, ¯ by J . From now on we take the unreduced symplectic manifold to be the Heisenberg double, (M, ) := (G L(n, C), ω+ ),

(3.8)

and identify the effectively acting symmetry group G¯ with U (n) divided by its center. As explained in Remark 2.1, G¯ acts by the quasi-adjoint action (2.21) according to ([g], K ) := g K ,

¯ ∀K ∈ G L(n, C), ∀[g] ∈ G,

(3.9)

where g ∈ U (n) is an arbitrary representative of [g] ∈ U (n)/U (1). In Definition 2.1 of Sect. 2, we introduced two families of G¯ Poisson-Lie symmetric Hamiltonian systems, ˆ Since these canonical free systems namely (G L(n, C), ω+ , L) and (G L(n, C), ω+ , L). have the same symplectic structure ω+ and they are Poisson-Lie symmetric with respect ¯ we can consider their simultaneous symplectic reduction based to the same action of G, on a moment map value ν. In our letter [11], we identified the (parameter-dependent) value ν(x) ∈ B¯ such that the symplectic reduction of the system (G L(n, C), ω+ , L) gives the original Ruijsenaars-Schneider system (P, ω, L) (1.1). An enhanced formulation of this result is given by Theorems 3.1 and 3.2 below. Later on, we shall formulate ˆ Theorems 3.3 and 3.4, which claim that the symplectic reduction of (G L(n, C), ω+ , L) with respect to the same ν(x) gives an extension of the dual Ruijsenaars-Schneider ˆ ω, ˆ (1.3). system ( P, ˆ L) In order to formulate precisely the above mentioned theorems, first we have to recall some notations and results from our previous paper [11]. Thus let x be a real non-zero parameter and denote by ν(x) the element of the group B¯ S B < B (remember (2.28)) such that ν(x)kk = 1, ∀k,

ν(x)kl = (1 − e−x )e

(l−k)x 2

, ∀k < l.

(3.10)

There holds the equation

enx − 1 v(x)v(x)† , ν(x)ν(x)† = e−x 1n + n

where the components of the real column vector v(x) read n(e x − 1) − kx vk (x) = e 2 , ∀k = 1, . . . , n. 1 − e−nx

(3.11)

(3.12)


71

The isotropy subgroup G ν(x) < U (n) is the direct product G v(x) ×U (1), where G v(x) < U (n) is the isotropy group of v(x) ∈ Cn and U (1) is the center of U (n). The corresponding subgroup G¯ ν(x) of the effective symmetry group G¯ = U (n)/U (1) admits the natural isomorphisms G¯ ν(x) G ν(x) /U (1) G v(x) ,

(3.13)

and therefore we can identify G¯ ν(x) with G v(x) . Let : G L(n, C) → B¯ S B (2.28) be the moment map of the quasi-adjoint action (3.9) of G¯ = U (n)/U (1). With ν(x) in (3.10), consider the symplectic reduction of the Heisenberg double (G L(n, C), ω+ ) defined by imposing the constraint (K ) := L (K ) R (K ) = ν(x),

K ∈ G L(n, C).

(3.14)

The associated gauge group is given by G¯ ν(x) (3.13), and we have the following basic lemma. Lemma 3.2 ([11]). The set Fν(x) consisting of the solutions of the moment map constraint (3.14) is an embedded submanifold of G L(n, C) and the compact group G¯ ν(x) acts freely on it. Lemma 3.2 was proven in [11] by finding explicitly all solutions of the moment map constraint (3.14). For completeness, in Appendix A we offer an alternative proof of the fact that Fν(x) is an embedded submanifold by showing that the element ν(x) is a regular value of the moment map . It follows from the lemma that the orbit space Fν(x) /G¯ ν(x) is a smooth manifold: the base of the principal fiber bundle with total space ˆ the Fν(x) and structure group G¯ ν(x) . For any H ∈ H (2.29) (respectively Hˆ ∈ H), reduction of the Hamiltonian system (G L(n, C), ω+ , H ) yields a system with complete Hamiltonian flow on the reduced phase space, which can be constructed by projecting the original flow given by Proposition 2.1 (respectively Proposition 2.2). Before giving a characterization of the reduction of the canonical free system (G L(n, C), ω+ , L) of Definition 2.1, next we present a convenient description of the phase space (P, ω) of the original Ruijsenaars-Schneider system (1.1). Let T0n denote the regular part of the standard maximal torus Tn < G = U (n). The symmetric group S(n) acts freely on T0n , any σ ∈ S(n) acts by permuting the diagonal entries of the elements T = diag(T1 , . . . , Tn ) ∈ T0n . Thus the corresponding space of S(n)-orbits Q(n) := T0n /S(n)

(3.15)

is a smooth manifold. This is the set of unordered n-tuples of pairwise distinct points of the circle S 1 = U (1). The phase space of the trigonometric Ruijsenaars-Schneider system, when interpreted as a many-body system of indistinguishable particles, is in fact given by (P, ω) := (T ∗ Q(n), T ∗ Q(n) ),

(3.16)

where T ∗ Q(n) is the canonical symplectic form of the cotangent bundle T ∗ Q(n). Note that S(n) acts freely on T ∗ T0n , too, by the cotangent lift of the S(n)-action on T0n . If we use the realization T ∗ T0n T0n × Rn = {(e2iq , p)},

(3.17)

72


then this S(n)-action operates by the simultaneous permutations of the entries of e2iq and p. We identify p with the diagonal matrix p ≡ diag( p1 , . . . , pn ) and choose the normalization of the natural symplectic form T ∗ T0n as T ∗ T0n ≡

n

dpk ∧ dqk =

k=1

1 tr (dp ∧ e−2iq de2iq ). 2

(3.18)

The point is that T0n is a covering space of Q(n), and T ∗ T0n is a symplectic covering space of T ∗ Q(n). To state this formally, we have π1∗ (T ∗ Q(n) ) = T ∗ T0n ,

(3.19)

π1 : T ∗ T0n → (T ∗ T0n )/S(n) ≡ T ∗ (T0n /S(n)) ≡ T ∗ Q(n)

(3.20)

where

is the natural submersion. Therefore one can identify the Poisson algebra of the smooth functions on T ∗ Q(n) with the Poisson algebra of the smooth S(n)-invariant functions on T ∗ T0n . For example, one may regard the Lax matrix L(q, p) (1.1) as a function on T ∗ T0n , with its spectral invariants (symmetric functions) giving the commuting Hamiltonians on T ∗ Q(n). Actually Q(n) is a rather non-trivial manifold, and the ensuing technical complications are avoided if one works with the S(n)-invariant functions on T ∗ T0n . Theorem 3.1. Consider the smooth map I˜ : T ∗ T0n → G L(n, C) defined by the formula ˜ 2iq , p)kk = e− I(e

pk 2

−2iqk

mk

1+

sinh2

x 2

sin2 (qk − qm )

1 4

, (3.21)

x l−k x 2iql e 2 e −e− 2 e2iqk+m 2iq 2iq 2iq ˜ ˜ ˜ I(e , p)kl = 0, k >l, I(e , p)kl = I(e , p)ll , k C¯ x := { pˆ ∈ Rn | pˆl − pˆl+1

(3.26)

Remark 3.4. We shall refer to the elements of Cx as the interior elements of C¯ x . Note that C¯ 0 is the standard Weyl chamber associated with gl(n, C) and the phase space of ˆ ω, ˆ is Pˆ = Tn × Cx . We often identify an element of Rn with a corthe system ( P, ˆ L) responding n × n diagonal matrix. For example, qˆ in (1.4) parametrizes Tn by eiqˆ and we may use pˆ diag( pˆ 1 , . . . , pˆ n ). Definition 3.2. Consider the phase space Pˆ = Tn × Cx (1.4) of the dual RuijsenaarsSchneider system and the symplectic manifold Pˆc := Cn−1 × C× , where C× denotes the complex plane without the origin and the symplectic form ωˆ c on Pˆc is defined by ωˆ c :=

n−1 id Z ∧ d Z¯ idz j ∧ d z¯ j , + sign(x) 2 Z¯ Z j=1

Z ∈ C× , z ∈ Cn−1 .

(3.27)

Define the smooth injective map Zx : Pˆ → Pˆc by 1

z j (x, q, ˆ p) ˆ = ( pˆ j − pˆ j+1 − x/2) 2

n

e−iqˆk ,

j = 1, . . . , n − 1,

k= j+1

Z (x, q, ˆ p) ˆ = e− pˆ1

n

(3.28)

e−iqˆk , x > 0,

k=1

z j (x, q, ˆ p) ˆ = ( pˆ j − pˆ j+1 + x/2)

1 2

j

e−iqˆk ,

j = 1, . . . , n − 1,

k=1

Z (x, q, ˆ p) ˆ = e− pˆn

n k=1

e−iqˆk , x < 0.

(3.29)

74


ˆ ω) Remark 3.5. One can check that Zx is a symplectic embedding of ( P, ˆ into ( Pˆc , ωˆ c ), i.e., Zx∗ ωˆ c = ωˆ

(3.30)

ˆ conwith ωˆ in (1.4). The Zx -image of Pˆ in Pˆc is a dense open submanifold; Pˆc \Zx ( P) sists of the points for which one or more of the complex coordinates z j is equal to zero. It is important to note that the same embedding of Pˆ into Pˆc was also used in [30].3 This fact, together with the requirements forced on us by the technical analysis in Sect. 5, motivated Definition 3.2. Definition 3.3. By introducing the ‘special index’ a := n for x > 0 and a := 1 for x < 0, define the n × n orthogonal matrix κ L (x) by vi (x)v j (x) va (x) vi (x) κ L (x)aa = √ , κ L (x)i j = δi j − , κ L (x)ia = −κ L (x)ai = √ , i, j = a, √ n n + nva (x) n (3.31) where v(x) is given by (3.12). Then consider z ∈ Cn−1 and introduce the smooth functions k−1 |x| x sinh ( l= j z l z¯ l + (k − j) 2 − 2 ) , 1 ≤ j < k ≤ n, (3.32) Q jk (x, z) = k−1 sinh ( l= j zl z¯l + (k − j) |x| 2 ) with Q jk (x, z) := Q k j (−x, z) for j > k. By using the above notations and J (y) := sinh y ˆ y for y = 0, J (0) := 1, define the smooth n × n matrix function ζ (x, z) as sinh x2 ζˆ (x, z)aa = Q al (x, z), ζˆ (x, z)a j = −ζˆ (x, z) ja , j = a, (3.33) sinh nx 2 l=a sinh x2 z j J (z j z¯ j ) ζˆ (x, z) jn = Q jl (x, z), x > 0, j = n, nx sinh 2 sinh (z z¯ + x ) j j

ζˆ (x, z) j1 =

2

l= j, j+1

(3.34) sinh x2 sinh nx 2

z j−1 J (z j−1 z¯ j−1 ) sinh (z j−1 z¯ j−1 − x2 )

Q jl (x, z), x < 0,

j = 1,

l= j−1, j

(3.35) ζˆ (x, z) jk = δ jk

ζˆ (x, z) ja ζˆ (x, z)ak + , 1 + ζˆ (x, z)aa

j, k = a.

(3.36)

Next, define the smooth n × n matrix function θˆ (x, z) for x > 0 as nx ˆ ˆ 2 )sign(k − j − 1)ζ (x, z) jn ζ (−x, z)1k , k = j + 1, max(k, j)−1 sinh ( l=min(k, j) zl z¯l + |k − j − 1| x2 ) − sinh x2 θˆ (x, z) j, j+1 = Q jl (x, z)Q j+1,l (−x, z), x sinh (z j z¯ j + 2 ) l= j, j+1

θˆ (x, z) jk =

(sinh

(3.37) (3.38)

3 One can see this from Eq. (1.73) in [30], where the completion of the dual phase space was formulated ˆ in terms of a covering space of P.


75

and for x < 0 as θˆ (x, z) = θˆ (−x, z)† .

(3.39)

Finally, let (x, z, Z ) be the diagonal matrix function on Pˆc given for x > 0 by the components 1 (x, z, Z ) = Z , j (x, z, Z ) = |Z | exp (

j−1 l=1

x zl z¯l + ( j − 1) ), 2

j = 2, . . . , n, (3.40)

and for x < 0 by the components n (x, z, Z ) = Z ,

j (x, z, Z ) = |Z | exp (−

n−1 l= j

x zl z¯l +(n − j) ), 2

j = 1, . . . , n−1. (3.41)

Remark 3.6. The origin of the above formulae will become clear in Sect. 5, and there ˆ z) and ζˆ (x, z) for all values of we shall demonstrate the unitarity of the matrices θ(x, their arguments. Recall Remark 3.2 concerning our terminology for a global cross section. Theorem 3.3. The symplectic manifold ( Pˆc , ωˆ c ) (3.27) is a model of the reduced phase space (Fν(x) /G¯ ν(x) , ν(x) ). With the notations introduced in Definition 3.3, the map Iˆ : Pˆc → Fν(x) given by ˆ I(z, Z ) := κ L (x)ζˆ (x, z)−1 (x, z, Z )θˆ (x, z)−1 (3.42) is a global cross section, and π ◦ Iˆ : Pˆc → Fν(x) /G¯ ν(x) is a symplectomorphism. Theorem 3.4. By using the symplectic embedding Zx : Pˆ → Pˆc introduced in Definition 3.2 and Lˆ defined in (2.60), the composition map Lˆ ◦ Iˆ ◦ Zx gives (up to an inessential similarity transformation) the Lax matrix Lˆ (1.3) of the dual Ruijsenaarsˆ ω, ˆ Schneider system ( P, ˆ L). To sum up, Theorems 3.1 and 3.2 state that the original trigonometric RuijsenaarsSchneider system (P, ω, L) is exactly the result of the symplectic reduction of the canonical free system (G L(n, C), ω+ , L). Theorems 3.3 and 3.4 affirm that the reducˆ gives a certain integrable tion of the other canonical free system (G L(n, C), ω+ , L) ˆ ˆ ˆ system ( Pc , ωˆ c , L ◦ I) which can be viewed (due to the non-surjectivity of the map ˆ ω, ˆ ˆ L). Zx : Pˆ → Pˆc ) as an extension of the dual Ruijsenaars-Schneider system ( P, Moreover, together with Remark 3.5, they ensure that the extended dual Ruijsenaarsˆ coincides with the ‘minimal completion’ of the system Schneider system ( Pˆc , ωˆ c , Lˆ ◦ I) ˆ ω, ˆ constructed by Ruijsenaars by means of the direct method [30]. The guiding ( P, ˆ L) principle behind his extension of Pˆ was the aim to obtain a bijective correspondence between the phase spaces of the dual pair of systems. At the same time, the extension

76


ˆ It is pleasgave rise to the completion of the dual flows, which are not complete on P. ing that Ruijsenaars’ minimal completion comes about naturally from the symplectic reduction. In this framework the geometric origin of the duality symplectomorphism between (P, ω) and ( Pˆc , ωˆ c ), which has been established in [30] by a complicated web of arguments, becomes transparent: any two models of the reduced phase space are naturally symplectomorphic to each other. The natural symplectomorphism maps to each other those points of the two different models that correspond to the same point of the reduced phase space. We shall further discuss in Sect. 6 why the geometrically induced symplectomorphism between (P, ω) and ( Pˆc , ωˆ c ) is the same as the duality map (alias ‘action-angle map’) constructed in [30]. 4. Proofs of Theorems 3.1 and 3.2 Many ingredients of the proofs that follow were already given in our previous paper [11] but here we present all this material in a more complete and natural way. In particular, we shall explain how the non-trivial topology of the configuration space of indistinguishable particles on the circle (see also Appendix B) is reflected in the symplectic reduction and why this aspect of the story explains the geometric origin of the important formula (3.13) of [11]. We recall that the somewhat complicated formula (3.13) of [11] (which appears as (4.19) below) relates the group theoretically simplest coordinates on the reduced phase space with the cotangent-bundle coordinates in which the Ruijsenaars-Schneider Hamiltonians are usually expressed. Let A < B denote the subgroup of diagonal matrices with positive real entries and N < B the subgroup of upper-triangular matrices with unit diagonal. Define the smooth function N : T0n → N by the formula x l−k x e 2 Tl − e− 2 Tk+m N (T )kl = , Tl − Tk+m−1

∀k < l,

(4.1)

m=1

and introduce the subset S˜ ⊂ G L(n, C) as follows: S˜ := {N (T )aT −1 | a ∈ A, T ∈ T0n }.

(4.2)

Lemma 4.1. The set S˜ lies in the constraint-manifold Fν(x) ⊂ G L(n, C) and it intersects every orbit of the gauge group G¯ ν(x) acting on Fν(x) . Every K ∈ Fν(x) with R (K ) ∈ Tn ˜ The map belongs to S. T0n × A → Fν(x) , (T, a) → N (T )aT −1

(4.3)

is an embedding, and the corresponding pull-back of the form ω+ is the symplectic form ω S˜ = tr (T −1 dT ∧ a −1 da).

(4.4)

Proof. The statement is just a reformulation of part of Theorem 1 of [11]. The fact that S˜ is an embedded submanifold of G L(n, C) is obvious from the Iwasawa decomposition, which also implies by Lemma 3.2 that S˜ is an embedded submanifold of Fν(x) .


77

Lemma 4.2. For every fixed K ∈ S˜ and permutation σ ∈ S(n) there exists a unique element [g(K , σ )] ∈ G¯ ν(x) for which ([g(K , σ )], K ) ∈ S˜ and R (([g(K , σ )], K )) = σ ( R (K )),

(4.5)

where denotes the action (3.9) and σ (T ) is obtained by permuting the entries of any T ∈ T0n . All gauge transformations that map K ∈ S˜ to S˜ are of the above type, and the formula ˜ σ (K ) := (σ, K ) := ([g(K , σ )], K ) : S(n) × S˜ → S,

(4.6)

˜ which preserves the symplectic form ω ˜ . defines a smooth, free action of S(n) on S, S Proof. This can be extracted from [11], too, and thus we can be brief here. First, for fixed K ∈ S˜ and σ ∈ S(n) there cannot exist two gauge transformations subject to (4.5), since the action of G¯ ν(x) is free on Fν(x) and two different elements K 1 , K 2 ∈ S˜ satisfying R (K 1 ) = R (K 2 ) are never gauge equivalent [11]. Second, because of the second relation in (2.35) and the surjectivity of the map g K : U (n) → U (n) given by g K (η) = R (η L (K )) for each K , it is clear that for any K ∈ S˜ and σ ∈ S(n) there exists some η ∈ G for which R (η K ) = σ ( R (K )).

(4.7)

This implies that (Dressη (ν(x))) j j = (η K ) j j = 1,

∀ j,

(4.8)

where we used both the equivariance of the moment map as well the formula (2.22), which shows that (K ) j j = 1 if K = bT −1 for some b ∈ B and T ∈ Tn . Next, it is not difficult to see (e.g. from the proof of Lemma 1 in [11]) that for any element Dressη (ν(x)) with unit diagonal there exists some τ ∈ Tn for which Dressη (ν(x)) = Dressτ (ν(x)). Then it follows from (4.7) that R (τ −1 η K ) = σ ( R (K )) ∈ Tn . Consequently, [g(K , σ )] := [τ −1 η] ∈ G¯ ν(x) is the required element. Each gauge transformation that maps K ∈ S˜ to S˜ is associated with some σ ∈ S(n) according to (4.5), since these gauge transformations act by some permutation on R (K ) (again because of (2.35)). It is clear from the established uniqueness property that (4.6) ˜ This action preserves ω ˜ since it is defines indeed a smooth, free action of S(n) on S. S given by gauge transformations (the gauge transformations preserve ω+ | Fν (x) , and ω S˜ is ˜ the pull-back of ω+ | Fν (x) on S). The following important formula was found by first making a detailed inspection in the n = 2 case, and then generalizing the result for arbitrary n. Lemma 4.3. The action of the transposition σk,k+1 ∈ S(n) on N (T )aT −1 ∈ S˜ is given explicitly by the formula ˆ k,k+1 (T )−1 , σk,k+1 (N (T )aT −1 ) = N (σk,k+1 (T ))aσ

(4.9)

/ {k, k + 1} and where aˆ j = a j if j ∈ 1 2 sinh2 2x ak with Wk (T ) := 1+ 2 , T = e2iq . aˆ k = ak+1 Wk (T ), aˆ k+1 = Wk (T ) sin (qk −qk+1 ) (4.10)

78


˜ For any γ ∈ (0, π ) define the Proof. Fix 1 ≤ k ≤ (n − 1) and K = N (T )aT −1 ∈ S. matrix g(γ ) ∈ SU (2) by

x x α β g(γ ) := , α := cos γ + i sin γ tanh , β := i sin γ /cosh . (4.11) β α¯ 2 2 Then introduce the element gk (γ ) ∈ U (n) by gk (γ ) := diag(, . . . , , g(γ ), , . . . , ), := eiγ ,

(4.12)

where the first string of ’s occupies the first (k −1)-entries along the diagonal. Introduce similarly the matrix χk (γ ) ∈ U (n) by

0 i χk (γ ) := diag(, . . . , , χ , , . . . , ) with χ := . (4.13) i 0 It is readily verified that the vector v(x) (3.12) is an eigenvector of gk (γ ), gk (γ )v(x) = eiγ v(x),

(4.14)

which implies that gk (γ ) ν(x) = ν(x), i.e., [gk (γ )] belongs to the gauge group G¯ ν(x) . It is also straightforward to check that if γ is determined by the equality cot γ = (tanh x/2) cot(qk − qk+1 ),

T j = e2iq j (∀ j = 1, . . . , n),

(4.15)

then the following Iwasawa decomposition is valid: gk (γ )N (T )a = bχk (γ ) with b ∈ B, b j, j = aˆ j (∀ j = 1, . . . , n),

(4.16)

where aˆ ∈ A is as claimed by the lemma. To finish the proof, we notice from (4.16) that R (gk (γ ) L (K )) = χk (γ )−1 , and this allows us to calculate (cf. (2.35)) that R (gk (γ ) K ) = σk,k+1 (T ).

(4.17)

˜ Therefore the By the second sentence of Lemma 4.1, this implies that gk (γ ) K ∈ S. element [g(K , σk,k+1 )] of Lemma 4.2 is provided by gk (γ ) with γ in (4.15). Since any K ∈ S˜ is determined by R (K ) and the diagonal part of L (K ), it follows that b in (4.16) is given by b = N (σk,k+1 (T ))a, ˆ which can be checked also by direct calculation. Lemma 4.4. The image of the map I˜ defined in Theorem 3.1 is the submanifold S˜ defined in (4.2). The corresponding map I˜ : T ∗ T0n → S˜ is an S(n)-equivariant symplectic diffeomorphism. Here, we refer to the S(n)-action on (T ∗ T0n , T ∗ T0n ) obtained ˜ ω˜) as the cotangent lift of the permutation action on T0n and to the S(n)-action on ( S, S described in Lemmas 4.2 and 4.3. Proof. It follows from the definition of I˜ in Theorem 3.1 and from Eq. (4.1) that we have the equality ˜ 2iq , p) = N (e2iq )a(e2iq , p)e−2iq I(e

(4.18)


79

with the function a = diag(a1 , . . . , an ) given by a j (e

2iq

, p) := e

−

pj 2

1+

m< j

×

1+

m> j

sinh2

− 1

x 2

4

sin2 (q j − qm ) sinh2

1

x 2

4

sin2 (q j − qm )

,

j = 1, . . . , n.

(4.19)

By taking into account the identification (3.17) and the definition (4.2), this ensures the validity of the first sentence of the lemma. One can see from the formula of I˜ or directly from Lemma 4.1 that I˜ : T ∗ T0n → S˜ is a diffeomorphism. The symplectic property I˜ ∗ ω S˜ = T ∗ T0n (with (3.18) and (4.4)) can be established directly. In fact, the properties mentioned so far would hold also if one replaced [1 + (sinh2 2x )/ sin2 (q j − qm )] in (4.19) by any positive even function W (q j − qm ). It is sufficient to confirm the equivariance property of I˜ for the transpositions σ := σk,k+1 ∈ S(n), k = 1, . . . , n − 1.

(4.20)

From (4.18) we obtain ˜ (e2iq ), σ ( p)) = N (σ (e2iq ))a(σ (e2iq ), σ ( p))(σ (e2iq ))−1 . I(σ

(4.21)

/ {k, k +1} For any fixed k, it is easily checked that a j (σ (e2iq ), σ ( p)) = a j (e2iq , p) if j ∈ and ak (σ (e2iq ), σ ( p)) = ak+1 (e2iq , p)Wk (e2iq ), ak+1 (σ (e2iq ), σ ( p)) =

ak (e2iq , p) Wk (e2iq ) (4.22)

with the same function Wk as in (4.10). The comparison of (4.21) with (4.9) shows that the proof is complete. Our lemmas explain the geometric picture behind Theorem 3.1, which is now easy to prove. Proof of Theorem 3.1. Lemmas 4.1 and 4.2 give rise to the identification of symplectic manifolds ˜ S/S(n) Fν(x) /G¯ ν(x) .

(4.23)

˜ Here, S/S(n) is the space of orbits of the S(n)-action given by Lemma 4.2, its symplectic ˜ while (Fν(x) /G¯ ν(x) , ν(x) ) is the reduced phase space of form descends from ω S˜ on S, interest. Moreover, we constructed the following commutative diagram of maps: T ∗ T0n π1 ↓ T ∗ Q(n)

I˜

−→

S˜ ↓ π S˜

I ˜ −→ S/S(n)

(4.24)

80


The map I is well-defined by this diagram and is a diffeomorphism, because of Lemma 4.4. (To compare with (3.24), note that π S˜ ◦ I˜ = π ◦ I˜ since the image of I˜ is S˜ ⊂ Fν (x).) We also established the relation (3.19) as well as π ∗˜ (ν(x) ) = ω S˜ (by (4.23)) and S I˜ ∗ (ω ˜ ) = T ∗ T0 (by Lemma 4.4). These relations and the fact that π1 and π ˜ are local S

n

diffeomorphisms imply that I ∗ (ν(x) ) = T ∗ Q(n) .

S

˜

Proof of Theorem 3.2. Denote by L S the restriction of the unreduced Lax matrix L (2.60) to S˜ (4.2). The definition directly yields the formula ˜

L S = T a −1 N (T )−1 (N (T )−1 )† a −1 T −1 ,

(4.25)

where T , a and N (T ) are understood as evaluation functions on S˜ T0n × A. Next, we remark that the restriction of the moment map constraint (3.14) to S˜ is equivalent to the relation

enx − 1 ˜ v(x)v(x)† , (4.26) N (T )aL S (N (T )a)† = ν(x)ν(x)† = e−x 1n + n which can be rewritten as x x x ˜ ˜ −1 , e 2 L Sjk − e− 2 T j−1 L Sjk Tk = 2U j U¯k a −1 j ak sinh 2

if we define

U j :=

x

e− 2 (enx − 1) 2n sinh x2

1 2

N (T )−1 v(x)

j

:= η j |U j |.

(4.27)

(4.28) ˜

Here, T = diag(T1 , . . . , Tn ) and a = diag(a1 , . . . , an ). By solving (4.27) for L S we arrive at ˜ L Sjk

= ηj

−1 2a −1 j |U j | ak |Uk | sinh

x 2

e 2 − e− 2 T j−1 Tk x

x

ηk−1 .

(4.29)

With the inverse N (T )−1 displayed in [11], it is also straightforward to calculate that 1 2 sinh2 2x |U j | = , (4.30) 1+ sin2 (q j − qm ) m> j where we use the parametrization Tk = e2iqk for all k = 1, . . . , n. Next, let us parametrize a ∈ A according to (4.19) and insert also (4.30) into (4.29). Then we obtain 1 1 p j + pk 4 4 γ j γk−1 e 2 sinh x2 sinh2 2x sinh2 2x ˜S L jk = (4.31) 1+ 2 1+ 2 x sinh( 2 + iq j − iqk ) sin (q j −qm ) sin (qk −qs ) m= j ˜

s=k

with γ j := η j eiq j . Hence L S is conjugate to the standard Ruijsenaars-Schneider Lax matrix L in (1.1). Recall from (4.18) that the map I˜ : T ∗ T0n → Fν(x) in (3.21), (3.22) was obtained from the map in (4.3) by using the identification T ∗ T0n T0n × Rn (3.17) and the parametrization (4.19) of a j . Therefore the foregoing arguments prove Theorem 3.2.


81

Remark 4.1. The statement of Theorem 3.2 was also obtained in [11], but there the details of the proof were omitted for lack of space. The ‘useful substitution’ (4.19) was introduced in [11] (Eq. (3.13) in loc. cit.) just on the basis that it converts the expression T a −1 N (T )−1 (N (T )−1 )† a −1 T −1 (4.25) into a conjugate of the standard Lax matrix L(q, p) (1.1). The deeper geometric meaning of this substitution is now revealed by ˜ Lemma 4.4 above. Note that L S transforms by conjugation under the ‘residual gauge transformations’ of Lemma 4.2, and this corresponds to the fact that L(q, p), viewed as a function on T ∗ T0n (3.17), transforms by conjugation under the natural S(n)-action. Remark 4.2. The fact that T ∗ Q(n) appears as a factor space (4.24), and not directly as a global gauge slice, reflects the fact that Q(n) is not a submanifold of T0n (see also Appendix B). Remark 4.3. It is worth pointing out that the consequence (4.27) of the moment map constraint is essentially identical to the ‘commutation relation of the Lax matrix’ that played an important rôle in the analysis presented in [30]. 5. Proofs of Theorems 3.3 and 3.4 The proofs of Theorems 3.3 and 3.4 will be based on a series of preliminary lemmas. From now on we adopt the identification pˆ diag( pˆ 1 , . . . , pˆ n ) for any pˆ ∈ C¯ 0 (3.26). Lemma 5.1. Every element K of the group G L(n, C) can be decomposed as K = k L (e− pˆ k −1 R ),

k L , k R ∈ U (n),

pˆ ∈ C¯ 0 .

(5.1)

Moreover, if τ is any element of the maximal torus Tn pˆ i+1 , ∀i = 1, . . . , n − 1). Proof. The statement of the lemma is a direct consequence of the standard Cartan decomposition of the elements of G L(n, C). Indeed, it is well known that every element K of the group G L(n, C) can be decomposed as K = η L e− pˆ η−1 R , η L , η R ∈ U (n),

pˆ ∈ C¯ 0 .

(5.2)

For each K , the diagonal matrix pˆ in the standard Cartan decomposition (5.2) is defined unambiguously. Moreover, simultaneous right multiplication of the pair η L , η R by an element τ of the maximal torus Tn gives an equally good pair η L τ, η R τ and this is the maximal possible ambiguity of the decomposition (5.2) if pˆ is regular. From (5.2) and the definition (2.21) of the quasi-adjoint action, we obtain K = k L (e− pˆ k −1 R ),

k L , k R ∈ U (n),

(5.3)

where k L = ηL ,

k R = R (η L e− pˆ )η R .

(5.4)

The proof is finished by noting that (η L τ, η R τ ) corresponds by (5.4) to (k L τ, τ −1 k R τ ) for all τ ∈ Tn .

82


Lemma 5.2. If K = k L (e− pˆ k −1 R ) (5.1) is a solution of the moment map constraint (3.14) then pˆ ∈ C¯ x , as defined in (3.26). Proof. By using the formula L (K ) = R (K −1 ), ∀K ∈ G L(n, C),

(5.5)

we can rewrite the moment map constraint (3.14) as − pˆ − pˆ (k L (e− pˆ k −1 ) L (−1 )k R e pˆ ) = ν(x), R )) = L (k L e R (k L e

(5.6)

or, equivalently, as − pˆ −1 † − pˆ −1 (k L (e− pˆ k −1 k R )) = k L e− pˆ k R e2 pˆ k −1 k L = ν(x)ν(x)† . (5.7) R ))(k L (e R e

By means of (3.11), the last equality can be further rewritten as

nx 2 pˆ −1 2 pˆ −x −x e − 1 pˆ −1 † pˆ kRe kR = e e + e e k L v(x)v(x) k L e . n

(5.8)

The equality of the characteristic polynomials of the matrices on the two sides of (5.8) gives ⎞ ⎛ nx 2 pˆ j 2 pˆ j −x −x e − 1 2 p ˆ 2 2 p ˆ −x ⎝e j |w j | (e − λ) = (e − λ) + e (e k − λ)⎠ , n j

j

j

k= j

(5.9) where w := k −1 L v(x)

(5.10)

and λ is a complex variable. To derive (5.9), we used the identity det(1n + uy † ) = 1 + y † u,

(5.11)

which is valid for arbitrary n-component column vectors u and y. Suppose that (5.8) holds for some regular K , i.e., pˆ 1 > pˆ 2 > · · · > pˆ n . We can then evaluate the polynomials on both sides of (5.9) at the n different values λ = e2 pˆ j −x , j = 1, . . . , n. This yields |w j |2 = n

1 − e−x 1 − e2 pˆ j −2 pˆk −x , 1 − e−nx 1 − e2 pˆ j −2 pˆk k= j

j = 1, . . . , n.

(5.12)

Consider first the case x > 0. On account of pˆ 1 > pˆ 2 > · · · > pˆ n , we find from (5.12) for each j = 1, . . . , n − 1 the following inequality k> j

(e2 pˆ j −2 pˆk −x − 1) =

1 − e2 pˆ j −2 pˆk 1−e−nx |w j |2 2 pˆ j −2 pˆk (e −1) ≥ 0. 1−e−x n 1 − e2 pˆ j−2 pˆk −x k> j k< j (5.13)


83

Now we prove by induction that pˆl − pˆl+1 −

x ≥ 0, 2

∀l = 1, . . . , n − 1.

(5.14)

First of all, for j = n − 1, the inequality (5.13) gives immediately (5.14) for l = n − 1. It is easy to see that if (5.14) holds for l = j + 1, j + 2, . . . , n − 1, then it holds also for l = j. Indeed, this follows from 0 ≤ (e2 pˆ j −2 pˆk −x −1) = (e2 pˆ j −2 pˆ j+1 −x − 1) (e2 pˆ j −2 pˆk−1 +2 pˆk−1 −2 pˆk −x − 1). k> j

k> j+1

(5.15) The case x < 0 is very similar. The point of departure is the following inequality

(1 − e2 pˆ j −2 pˆk −x ) =

k< j

e−nx − 1 |w j |2 (1 − e2 pˆ j −2 pˆk ) e−x − 1 n k< j

×

e2 pˆ j −2 pˆk − 1 k> j

e2 pˆ j −2 pˆk −x − 1

≥ 0, ∀ j = 2, . . . , n. (5.16)

In this case we prove by induction that pˆl−1 − pˆl +

x ≥ 0, 2

∀l = 2, . . . , n.

(5.17)

Now (5.16) for j = 2 gives (5.17) for l = 2. By using 0 ≤ (1−e2 pˆ j −2 pˆk −x ) = (1 − e2 pˆ j −2 pˆ j−1 −x ) (1 − e2 pˆ j −2 pˆk+1 +2 pˆk+1 −2 pˆk −x ), k< j

k< j−1

(5.18) one sees that if (5.17) holds for l = 2, . . . , j − 1, then it holds also for l = j. So far we have proved Lemma 5.2 for the regular solutions of the moment map constraint (3.14), i.e., for those K (5.1) for which pˆ 1 > pˆ 2 > · · · > pˆ n . We remark that such regular solutions exist. As an example, consider K = k L (e− pˆ k −1 R ) with pˆ such that pˆl − pˆl+1 =

|x| , 2

∀l = 1, . . . , n − 1.

(5.19)

In fact, a solution is then provided by k L := κ L (x) (given by (3.31)) and k R := κ R (x), where, for x > 0, κ R (x)n1 = κ R (x)i,i+1 = 1, i = 1, . . . , n − 1,

κ R (x)i j = 0 otherwise,

(5.20)

and for x < 0, κ R (x) := κ R (−x)−1 .

(5.21)

To finish the proof, it remains to treat the case of non-regular solutions of (3.14), for which two or more pˆ j ’s are equal to each other. Suppose that such a non-regular solution, K 0 , exists. Note that the space of solutions of (3.14), Fν(x) , is connected, since

84


it is the total space of a principal fiber bundle with connected structure group and connected base, as follows from Lemma 3.2 and Theorem 3.1. Then take a regular solution, K 1 , e.g. the one exhibited above, and connect K 0 with K 1 by a continuous path K s , s ∈ [0, 1], in Fν(x) . Now the diagonal matrix p(s) ˆ in the modified Cartan decomposition

(5.1) of K s (or, in other words, the spectrum of the element K s K s† ) varies continuously with s. However, this is not possible because the set of the non-regular elements of C¯ 0 is disconnected from C¯ x for x = 0. Hence, non-regular solutions of the constraint (3.14) do not exist. Lemma 5.3. If K = k L (e− pˆ k −1 R ) (5.1) is a solution of the moment map constraint (3.14) then the matrix k R must have the form k R = δl θ (x, p)δ ˆ r.

(5.22)

Here δl , δr are some diagonal unitary matrices and θ (x, p) ˆ is the real orthogonal matrix defined for every pˆ ∈ C¯ x by θ(x, p) ˆ jk

1 sinh( pˆ j − pˆ m − x ) sinh( pˆ k − pˆ m + x ) 2 sinh x2 2 2 := , sinh( pˆ j − pˆ m ) sinh( pˆ k − pˆ m ) sinh pˆ k − pˆ j m= j,k

θ(x, p) ˆ j j :=

m= j

sinh( pˆ j − pˆ m − x2 ) sinh( pˆ j − pˆ m + x2 )

j = k,

(5.23)

1 2

.

sinh2 ( pˆ j − pˆ m )

Proof. Consider the following variant of the moment map constraint (5.8): x x x ξ(x)ξ(x)† , k R (x)e2 pˆ e 2 k R (x)−1 = e2 pˆ e− 2 + 2 sinh 2

(5.24)

(5.25)

where the vector ξ(x) is defined as ξ(x) :=

enx − 1 pˆ −1 e k L v(x). n(e x − 1)

(5.26)

Here, our notation emphasizes the dependence of k R and ξ on x while their dependence on pˆ remains tacit. Observe from the comparison of (5.10), (5.12) and (5.26) that |ξ(x) j |2 = e2 pˆ j

e x − e2 pˆ j −2 pˆk k= j

1 − e2 pˆ j −2 pˆk

.

(5.27)

For any given pˆ ∈ C¯ x and ξ(x) subject to (5.27), the constraint (5.25) admits a solution for k R (x), since the characteristic polynomials of the matrices on the two sides of (5.25) are equal. The solution k R (x) can be chosen to be unitary because the matrix on the right-hand side of (5.25) is Hermitian. Suppose that a pair (k R (x), ξ(x)) ∈ U (n) × Cn satisfies (5.25), at some fixed pˆ ∈ C¯ x . Then all pairs satisfying (5.25) can be obtained by replacing the given solution by (δl k R (x)δr , δl ξ(x)) with arbitrary δl , δr ∈ Tn , since (5.25) fixes the vector ξ(x) up to phases, according to (5.27), and the eigenvalues of x e2 pˆ e 2 are distinct.


85

Let us rearrange the constraint (5.25) as

x x x k R (x)−1 e2 pˆ e− 2 k R (x) = e2 pˆ e 2 − 2 sinh k R (x)−1 ξ(x)ξ(x)† k R (x). 2 Let us also consider (5.25) for x → −x, x x x ξ(−x)ξ(−x)† . k R (−x)e2 pˆ e− 2 k R (−x)−1 = e2 pˆ e 2 − 2 sinh 2 By comparing (5.28) and (5.29), we conclude that (k R (x)−1 ξ(x)) j = eiη j ξ(−x) j ,

(5.28)

(5.29)

(5.30)

where the eiη j are some phases. Indeed, this follows from the fact that the constraint (5.29) determines the components of ξ(−x) up to phases. Let us multiply the constraint (5.25) from the right by k R (x). This gives x x x k R (x)e2 pˆ e 2 − e2 pˆ e− 2 k R (x) = 2 sinh ξ(x)ξ(x)† k R (x). (5.31) 2 Next, by spelling out (5.31) in components and taking into account (5.30), we obtain x x x e2 pˆl + 2 − e2 pˆ j − 2 k R (x) jl = 2 sinh ξ(x) j ξ(−x)l† e−iηl . (5.32) 2 For any pˆ ∈ Cx , we can deduce from (5.32) that † x e−iηl ξ(x) j ξ(−x)l k R (x) jl = 2 sinh x x . 2 e2 pˆl + 2 − e2 pˆ j − 2

(5.33)

Now we note that if pˆ ∈ Cx , then θ (x, p), ˆ defined by (5.23) and (5.24), can be also rewritten as x |ξ(x) j ||ξ(−x)l | (5.34) θ (x, p) ˆ jl = 2 sinh x x , 2 e2 pˆl + 2 − e2 pˆ j − 2 where the absolute values of the components of the vectors ξ(±x) are given by (5.27). It is clear from (5.33) and (5.34) that each unitary solution k R (x) of (5.25) verifies ˆ r with some δl , δr ∈ Tn , k R (x) = δl θ (x, p)δ

∀ pˆ ∈ Cx ,

(5.35)

which proves (5.22) for pˆ ∈ Cx . Moreover, the formula in (5.35) entails that the real matrix θ (x, p) ˆ must be itself unitary, and hence orthogonal, and it must also satisfy (5.25) for all pˆ ∈ Cx . By the continuity of θ (x, p) ˆ as a function of pˆ ∈ C¯ x , it then follows that θ (x, p) ˆ must be an orthogonal matrix for all pˆ ∈ C¯ x , and there must exist ˜ also a vector, say ξ (x, p), ˆ such that x x x ˆ −1 = e2 pˆ e− 2 +2 sinh θ (x, p)e ˆ 2 pˆ e 2 θ (x, p) ξ˜ (x, p) ˆ ξ˜ (x, p) ˆ † , ∀ pˆ ∈ C¯ x . (5.36) 2 x

x

ˆ −1 − e2 pˆ e− 2 )/(2 sinh x2 ) is a rank-one proIndeed, the fact that (θ (x, p)e ˆ 2 pˆ e 2 θ (x, p) jector for all pˆ ∈ Cx implies that the same statement holds also at the boundary of C¯ x . (The rank cannot decrease at the boundary, since the vectors ξ(x) satisfying (5.25) cannot vanish at any pˆ ∈ C¯ x .) Thus we have shown that θ (x, p) ˆ is unitary and solves (5.25) for all pˆ ∈ C¯ x . By the remarks given after (5.27), this guarantees the validity of (5.22) at every pˆ ∈ C¯ x . We note in passing that, since the vector ξ˜ (x, p) ˆ is determined by (5.36) up to an overall phase at any fixed p, ˆ and θ (x, p) ˆ is real and continuous, the components of ξ˜ (x, p) ˆ can be chosen to be real, continuous functions of pˆ ∈ C¯ x .

86


Lemma 5.4. If K = k L (e− pˆ k −1 R ) (5.1) is a solution of the moment map constraint (3.14) then the matrix k L must have the form k L = hκ L (x)ζ (x, p) ˆ −1 δ −1 .

(5.37)

Here h ∈ G v(x) (3.13), δ is some diagonal unitary matrix, the matrix κ L (x) is given by (3.31), and ζ (x, p) ˆ is the real orthogonal matrix defined for every pˆ ∈ C¯ x by r (x, p) ˆ i r (x, p) ˆ j , 1 + r (x, p) ˆ a = r (x, p) ˆ i , i, j = a,

ζ (x, p) ˆ aa = r (x, p) ˆ a , ζ (x, p) ˆ i j = δi j − ζ (x, p) ˆ ia = −ζ (x, p) ˆ ai

where a = n for x > 0, a = 1 for x < 0 and, for all x = 0, 1 − e−x 1 − e2 pˆ j −2 pˆk −x r (x, p) ˆ j := , 1 − e−nx 1 − e2 pˆ j −2 pˆk

j = 1, . . . , n.

(5.38)

(5.39)

k= j

Proof. First, let us show that the above real matrix ζ (x, p) ˆ is orthogonal. For this, note that |r (x, p) ˆ j |2 = 1. (5.40) j

This can be deduced from the comparison of (5.39) and (5.12) by using (5.9) for λ = 0. One can then easily check that the columns of the matrix ζ (x, p) ˆ form an orthonormal system. We know from Eqs. (5.10), (5.12) and (5.39) that there exists δ ∈ Tn such that √ ˆ (5.41) k −1 L v(x) = nδr (x, p), where the vector r (x, p) ˆ is defined by its components (5.39). On the other hand, the formula (3.31) leads immediately to ⎛ ⎞ ⎛√ ⎞ 0 n ⎜ . ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ κ L (x)−1 v(x) = ⎜ ⎟ for x > 0 and κ L (x)−1 v(x) = ⎜ ⎟ for x < 0. ⎜ . ⎟ ⎜ . ⎟ ⎝ 0 ⎠ ⎝ . ⎠ √ 0 n (5.42) For all x, we then obtain √ nr (x, p). ˆ

(5.43)

ˆ L (x)−1 v(x), δ −1 k −1 L v(x) = ζ (x, p)κ

(5.44)

ζ (x, p)κ ˆ L (x)−1 v(x) = By combining this with (5.41), we get

which implies the existence of an element h from the isotropy group G v(x) of the vector v(x) (3.12) such that (5.37) holds.


87

Lemma 5.5. For each τ = diag(τ1 , . . . , τn ) ∈ Tn set τ(x) := diag(τ2 , . . . , τn , 1) if x > 0,

τ(x) := diag(1, τ1 , . . . , τn−1 ) if x < 0. (5.45)

Using the previous notations, define the map K x : G v(x) × Tn × C¯ x → G L(n, C) by −1 ˆ := hκ L (x)τ(x) ζ (x, p) ˆ −1 e− pˆ τ τ(x) θ (x, p) ˆ −1 , K x (h, τ, p) (5.46) ∀(h, τ, p) ˆ ∈ G v(x) × Tn × C¯ x . Then the image of the map K x coincides with the submanifold Fν(x) = −1 (ν(x)) of G L(n, C). Proof. Consider h = 1n , τ = 1n and pˆ ∈ C¯ x . We first wish to show that K x (1n , 1n , p) ˆ solves the constraint (3.14). This statement is equivalent to x x x (x, p)(x, ˆ p) ˆ †, ˆ −1 = e2 pˆ e− 2 + 2 sinh (5.47) θ (x, p)e ˆ 2 pˆ e 2 θ (x, p) 2 where the real vector (x, p) ˆ is defined as enx − 1 pˆ e r (x, p), ˆ (x, p) ˆ := ex − 1

∀ pˆ ∈ C¯ x ,

(5.48)

with r (x, p) ˆ in (5.39). This is so simply because (5.43) holds, and the moment map constraint is (5.25) with (5.26). Let us recall from the proof of Lemma 5.3 that there exists a real vector ξ˜ (x, p) ˆ that satisfies (5.36). We also know that ξ˜ (x, p) ˆ verifies |ξ˜ (x, p) ˆ j | = (x, p) ˆ j for all j, and is determined by (5.36) up to an overall sign. Notice from (5.27) that ξ˜ (x, p) ˆ n = 0 holds for x > 0 and ξ˜ (x, p) ˆ 1 = 0 holds for x < 0, at each pˆ ∈ C¯ x . This fact allows us to fix the sign ambiguity of ξ˜ (x, p) ˆ by requiring that ξ˜ (x, p) ˆ n = (x, p) ˆ n for x > 0 and ˜ξ (x, p) ˆ 1 = (x, p) ˆ 1 for x < 0. Now we are going to prove that the unique vector ξ˜ (x, p) ˆ specified above actually satisfies ξ˜ (x, p) ˆ = (x, p), ˆ

∀ pˆ ∈ C¯ x ,

(5.49)

which converts (5.36) into (5.47). We start by noting from Eqs. (5.23) and (5.24) that θ (−x, p) ˆ = θ (x, p) ˆ −1 .

(5.50)

Equations (5.36) and (5.50) together imply x

θ (−x, p)e ˆ 2 pˆ e− 2 θ (−x, p) ˆ −1 x x θ (x, p) ˆ −1 ξ˜ (x, p) ˆ ξ˜ (x, p) ˆ † θ (x, p). = e2 pˆ e 2 − 2 sinh ˆ 2

(5.51)

This entails ˆ = ±ξ˜ (−x, p), ˆ θ (x, p) ˆ −1 ξ˜ (x, p)

(5.52)

88


where the overall sign will be determined soon. Let us multiply Eq. (5.36) from the right by θ (x, p). ˆ This gives x x x ˆ = 2 sinh ˆ (5.53) θ (x, p)e ˆ 2 pˆ e 2 − e2 pˆ e− 2 θ (x, p) ξ˜ (x, p) ˆ ξ˜ (x, p) ˆ † θ (x, p). 2 Using Eqs. (5.52) and (5.53), we then find easily ˆ j ξ˜ (−x, p) ˆ l x ξ˜ (x, p) θ (x, p) ˆ jl = ±2 sinh x x . 2 p ˆ + 2 p ˆ − l j 2 e 2 −e 2 On the other hand, we know from (5.34) that ˆ l ˆ j (−x, p) x (x, p) θ (x, p) ˆ jl = 2 sinh . 2 pˆl + x2 2 pˆ j − x2 2 e −e

(5.54)

(5.55)

Let us restrict pˆ to Cx in the above two equations (although both (5.54) and (5.55) extend from Cx to C¯ x by continuity). Comparing Eqs. (5.54) and (5.55) for j = n, l = 1 and j = 1, l = n fixes the positive sign in (5.52) and (5.54). Comparing Eqs. (5.54) and (5.55) for j = n, l arbitrary and l = 1, j arbitrary gives then ξ˜ (x, p) ˆ = (x, p) ˆ for ¯ ˜ all pˆ ∈ Cx . By the continuity of ξ (x, p) ˆ and (x, p) ˆ at every pˆ ∈ Cx , the claim of Eq. (5.49) then holds everywhere. In the above we have proved that K x (1n , 1n , p) ˆ solves the moment map constraint (3.14). Notice from (5.42) and (5.45) that κ L (x)τ(x) κ L (x)−1 ∈ G v(x) ,

∀τ ∈ Tn .

(5.56)

It follows that whenever k L , pˆ and k R solve (5.8), then also hκ L (x)τ(x) κ L (x)−1 k L , pˆ and ˆ k R τ −1 τ(x) solve (5.8) for every h ∈ G v(x) and τ ∈ Tn . This means that also K x (h, τ, p) solves the moment map constraint (3.14). Let us now show that all solutions of the moment map constraint (3.14) are of the form K x (h, τ, p). ˆ Using Lemmas 5.2, 5.3 and 5.4, we know that the most general solution of ¯ the moment map constraint must be of the form k L (e− pˆ k −1 R ), where pˆ ∈ Cx and ˆ r, k R = δl θ (x, p)δ

k L = hκ L (x)ζ (x, p) ˆ −1 δ −1

(5.57)

with arbitrary h ∈ G v(x) and certain δl , δr , δ ∈ Tn . The substitution of (5.57) into the moment map constraint (5.8) gives θ (x, p)e ˆ 2 pˆ θ (x, p) ˆ −1 enx − 1 pˆ −1 e δl δζ (x, p)κ = e2 pˆ e−x + e−x ˆ L (x)−1 v(x)v(x)† κ L (x)ζ (x, p) ˆ −1 δ −1 δl e pˆ . n (5.58) On the other hand, from Eqs. (5.38), (5.42), (5.47) and (5.48), we deduce ˆ −1 θ (x, p)e ˆ 2 pˆ θ (x, p) enx − 1 pˆ = e2 pˆ e−x + e−x ˆ L (x)−1 v(x)v(x)† κ L (x)ζ (x, p) ˆ −1 e pˆ . e ζ (x, p)κ n

(5.59)

The compatibility of Eqs. (5.58) and (5.59) requires that δl−1 δζ (x, p)κ ˆ L (x)−1 v(x) = γ ζ (x, p)κ ˆ L (x)−1 v(x)

(5.60)


89

with some phase γ ∈ U (1). This then implies that ˆ L (x)−1 = γ ζ (x, p)κ ˆ L (x)−1 h −1 δl−1 δζ (x, p)κ

(5.61)

with some h ∈ G v(x) . By taking the inverse of this equation, we obtain ˆ −1 δ −1 δl = h γ −1 κ L (x)ζ (x, p) ˆ −1 . κ L (x)ζ (x, p)

(5.62)

Finally, we conclude

−1 −1 − pˆ −1 −1 −1 e ) = hκ (x)ζ (x, p) ˆ δ δ θ (x, p) ˆ δ k L (e− pˆ k −1 L r R l = K x (h , ℵ(x, δl δr ), p), ˆ

(5.63)

where −1 h = hh κ L (x)ℵ(x, δl δr )−1 ∈ G v(x) (x) κ L (x)

(5.64)

with ℵ(x, τ ) j :=

n

τk−1 , x > 0 and ℵ(x, τ ) j :=

k= j

j

τk−1 , x < 0.

(5.65)

k=1

The proof is complete by the last equality in (5.63). To derive this, in addition to (5.62) we also employed the identity −1 ℵ(x, τ )(ℵ(x, τ ))−1 , (x) = τ

∀τ ∈ Tn ,

which is satisfied by the bijection ℵ(x, ·) : Tn → Tn defined by (5.65).

(5.66)

Recall the following lemma proved in [17] (Lemma 5.1 therein). Lemma 5.6 ([17]). Let A < G L(n, C) be the subgroup of real diagonal matrices with positive entries. Consider the three maps ι, ι L ,R : U (n) × A × U (n) → G L(n, C) defined by −1 ι(η L , a, η R ) := η L aη−1 R , ι L (η L , a, η R ) := η L a, ι R (η L , a, η R ) := aη R . (5.67)

Then the ι-pullback and ι L ,R -pullbacks of the symplectic form ω+ given by (2.8) are related as ι∗ ω+ = ι∗L ω+ + ι∗R ω+ .

(5.68)

Lemma 5.7. Using the notations (5.46) and (5.65), define the smooth map k x from the phase space Pˆ = Tn × Cx of the dual trigonometric Ruijsenaars-Schneider system into G L(n, C) by k x (q, ˆ p) ˆ := K x (1n , ℵ(x, eiqˆ ), p). ˆ

(5.69)

k x∗ ω+ = ω, ˆ

(5.70)

Then it holds

where ωˆ is the symplectic form on Pˆ given by (1.4).

90


Proof. Define the map η : Tn × Cx → U (n) × A × U (n) by η(τ, p) ˆ := (η L ( p), ˆ e− pˆ , η R (τ, p)), ˆ

∀τ ∈ Tn , ∀ pˆ ∈ Cx ,

(5.71)

where ˆ := κ L (x)ζ (x, p) ˆ −1 , η R (τ, p) ˆ := R (κ L (x)ζ (x, p) ˆ −1 e− pˆ )−1 θ (x, p)τ ˆ (x) τ −1 . η L ( p) (5.72) It follows from Eqs. (2.21), (5.46) and (5.67) that K x (1n , τ, p) ˆ = (κ L (x)τ(x) κ L (x)−1 ) ι(η(τ, p)). ˆ

(5.73)

Now we wish to calculate the pull-back K x (1n )∗ ω+ where the map K x (1n ) : Tn ×Cx → G L(n, C) is defined by K x (1n )(τ, p) ˆ := K x (1n , τ, p). ˆ The factor κ L (x)τ(x) κ L (x)−1 does not contribute to this pull-back, since K x (1n ) takes its values in Fν(x) by Lemma 4.5 and κ L (x)τ(x) κ L (x)−1 acts by gauge transformations on Fν(x) because of (5.56). By combining (5.73) with Lemma 4.6, we then obtain K x (1n )∗ ω+ = η∗ ι∗ ω+ = η∗ ι∗L ω+ + η∗ ι∗R ω+ .

(5.74)

η∗ ι∗L ω+ = 0.

(5.75)

Next, notice that

ˆ − pˆ is real, the images of the Iwasawa maps L , This is because the element η L ( p)e R , R , L are therefore real, too, and hence the imaginary part of the expression under the trace in (2.8) vanishes in this case. In order to find η∗ ι∗ ω+ = η∗ ι∗R ω+ , we have to calculate −1 −1 −1 −1 −1 , L (aη−1 L (aη−1 R ) = a, R (aη R ) = ρτ(x) τ R ) = τ τ(x) μ, R (aη R ) = ρa μ,

(5.76) where −1 ˆ τ(x) , ρ := η R (τ, p)τ

μ := L (e− pˆ ρ −1 ).

(5.77)

By using the formula (2.8), we compute directly η∗ ι∗R ω+ =

1 −1 −1 tr d pˆ ∧ (dτ(x) τ(x) − dτ τ −1 − τ τ(x) dμμ−1 τ(x) τ −1 ) 2 1 ˆ −1 + ρe pˆ dμμ−1 e− pˆ ρ −1 ) ∧ (dρρ −1 + tr (dρρ −1 + ρd pρ 2

−1 + ρ(dτ(x) τ(x) − dτ τ −1 )ρ −1 ) .

(5.78)

By eliminating the terms that vanish due to the fact that the imaginary part of a real number vanishes, it remains 1 −1 η∗ ι∗R ω+ = tr (2d pˆ + ρ −1 dρ + dμμ−1 ) ∧ (dτ(x) τ(x) − dτ τ −1 ) . (5.79) 2


91

Finally, we note that both ρ and μ are real orthogonal matrices, hence both ρ −1 dρ and dμμ−1 are 1-forms taking values in the space of antisymmetric matrices. Since the trace of the product of an antisymmetric matrix with a diagonal matrix vanishes, we arrive at −1 K x (1n )∗ ω+ = η∗ ι∗R ω+ = tr d pˆ ∧ τ τ(x) d(τ(x) τ −1 ) . (5.80) With the help of the relation k x (q, ˆ p) ˆ = K x (1n )(ℵ(x, eiqˆ ), p) ˆ and (5.66), we conclude from (5.80) that k x∗ ω+ = d pˆ j ∧ d qˆ j = ω. ˆ (5.81) j

Proof of Theorem 3.3. First of all we recall that the dual Ruijsenaars-Schneider phase space Pˆ is identical to Tn × Cx as a manifold (cf. Remark 3.4). The map Zx : Pˆ → Pˆc , viewed as the map Zx : Tn × Cx → Pˆc , can be extended to a surjective map Z˜ x : Tn × C¯ x → Pˆc by using the same formulae (3.29) and (3.30) that define Zx . In correspondence with the Cartesian product Pˆc = Cn−1 × C× , we denote the components of this smooth map Z˜ x by z˜ and Z˜ . Then it is straightforward to check that the following ˆ ∈ Tn × C¯ x : identities hold for all (eiqˆ , p) −1 ζˆ (x, z˜ (x, q, ˆ p)) ˆ = τ(x) ζ (x, p)τ ˆ (x) ,

(5.82)

−1 ˆ τ˜(x) , θˆ (x, z˜ (x, q, ˆ p)) ˆ = τ(x) θ (x, p)

(5.83)

(x, z˜ (x, q, ˆ p), ˆ Z˜ (x, q, ˆ p)) ˆ =

−1 − pˆ τ τ˜(x) e ,

(5.84)

where τ = ℵ(x, eiqˆ ) as in (5.65), τ(x) is given by (5.45), and for every τ ∈ Tn we employ τ˜(x) := diag(1, τ2 , . . . , τn ) if x > 0,

τ˜(x) := diag(τ1 , . . . , τn−1 , 1) if x < 0. (5.85)

Recall that the orthogonal matrices θ (x, p), ˆ ζ (x, p) ˆ were defined in Lemma 5.3 and 5.4, respectively, and the matrices ζˆ (x, z) and θˆ (x, z) were introduced in Definition 3.3. The surjectivity of the map Z˜ x : Tn × C¯ x → Cn−1 × C× now implies the unitarity of the matrices ζˆ (x, z) and θˆ (x, z) for every x = 0 and z ∈ Cn−1 . The above identities give rise to the relation (Iˆ ◦ Z˜ x )(q, ˆ p) ˆ = K x (1n , ℵ(x, eiqˆ ), p), ˆ

∀eiqˆ ∈ Tn , ∀ pˆ ∈ C¯ x ,

(5.86)

where the map Iˆ was defined in Eq. (3.42) and K x in Eq. (5.46). This implies by Lemma 5.5 and the surjectivity of the maps Z˜ x , ℵ(x, ·) that every point of the conˆ straint-manifold Fν(x) = −1 (ν(x)) can be written as h I(z, Z ) with some h ∈ G v(x) and some (z, Z ) ∈ Cn−1 × C× . In particular, the image of Iˆ intersects every gauge orbit in Fν(x) . Next, we wish to show that Iˆ ∗ ω+ = ωˆ c .

(5.87)

92


Referring to (5.69), the restriction of the relation (5.86) to Tn × Cx can be expressed as k x = Iˆ ◦ Zx .

(5.88)

Zx∗ ωˆ c = ωˆ = k x∗ ω+ = Zx∗ Iˆ ∗ ω+ .

(5.89)

By Eqs. (3.30) and (5.70), we have

We recall from Remark 3.5 that the map Zx yields a diffeomorphism between Pˆ and the dense open submanifold Pˆc0 of Pˆc given by Pˆc0 := {(z, Z ) ∈ Cn−1 × C× | z j = 0, ∀ j = 1, . . . , n − 1}.

(5.90)

We thus see from Eq. (5.89) that the pull-back Iˆ ∗ ω+ and the form ωˆ c coincide everywhere on the dense open subset Pˆc0 ⊂ Pˆc . Therefore, by smoothness, they coincide everywhere on Pˆc . For h ∈ G v(x) and (z, Z ), (z , Z ) ∈ Pˆc , we now prove the following implication: ˆ ˆ , Z ) h I(z, Z ) = I(z

⇒ h = 1n , z = z ,

Z = Z .

(5.91)

We present the argument in detail for the case x > 0 and leave to the reader the analoˆ gous case x < 0. The assumption on the left-hand side of (5.91) entails that h I(z, Z) − p ˆ ˆ and I(z , Z ) have the same matrix e in the modified Cartan decomposition (5.1). By means of the formulae (3.40), (3.41) and (3.42), this gives (x, z, |Z |) = (x, z , |Z |),

(5.92)

whence |Z | = |Z |,

|z i | = |z i |, ∀i = 1, . . . , n − 1.

(5.93) z

Define the diagonal matrix ϒ = diag(ϒ1 , . . . , ϒn−1 , 1) by setting ϒ j := z jj if |z j | = |z j | = 0 and ϒ j := 1 otherwise. The inspection of the formulae (3.36), (3.37) and (3.38) together with Eq. (5.93) leads to the following equalities: ϒ ζˆ (x, z)ϒ −1 = ζˆ (x, z ),

ˆ z ), ϒ θˆ (x, z)ϒˆ −1 = θ(x,

(5.94)

where ϒˆ := diag(1, ϒ1 , . . . , ϒn−1 ). Furthermore, the equality ˆ , Z )I(z ˆ , Z )† ˆ ˆ (h I(z, Z ))(h I(z, Z ))† = I(z

(5.95)

implies hκ L (x)ζˆ (x, z)−1 (x, z, |Z |)2 ζˆ (x, z)κ L (x)−1 h −1 = κ L (x)ζˆ (x, z )−1 (x, z, |Z |)2 ζˆ (x, z )κ L (x)−1 .

(5.96)

This gives ζˆ (x, z)κ L (x)−1 h −1 κ L (x) = ζˆ (x, z ),

(5.97)


93

for some = diag(1 , . . . , n ) ∈ Tn and, consequently, ζˆ (x, z)κ L (x)−1 v(x) = ζˆ (x, z)κ L (x)−1 h −1 κ L (x)κ L (x)−1 v(x) = ζˆ (x, z )κ L (x)−1 v(x).

(5.98)

By using Eqs. (5.42), (3.34) and the first equation of (5.94), we find from (5.98) that n = 1 and −1 j = ϒ j ∀ j ∈ {1, . . . , n − 1}

for which z j = 0.

(5.99)

By multiplying Eq. (5.97) by −1 from the right and by using the first equation of (5.94), we then arrive at ζˆ (x, z)κ L (x)−1 h −1 κ L (x) −1 = ζˆ (x, z),

(5.100)

κ L (x)−1 hκ L (x) = −1 .

(5.101)

and thus

Now we return to the equation on the left-hand side of the implication (5.91) and, by using Eqs. (3.42), (5.94) and (5.101), we rewrite it as

hκ L (x)ζˆ (x, z)−1 (x, z, Z )θˆ (x, z)−1

= κ L (x) −1 ζˆ (x, z)−1 (x, z, Z )θˆ (x, z)−1

= κ L (x) −1 ζˆ (x, z)−1 (x, z, Z ) −1 θˆ (x, z)−1

ˆ z )−1 . (5.102) = κ L (x)ζˆ (x, z )−1 (x, z , Z )θ(x, We remark that −1 ζˆ (x, z)−1 = ϒ ζˆ (x, z)−1 ϒ −1 = ζˆ (x, z )−1 , and hence from Eq. (5.102) we obtain (x, z , Z )θˆ (x, z )−1 = (x, z, Z ) −1 θˆ (x, z)−1 .

(5.103)

Taking into account Eq. (5.94), this yields ϒ θˆ (x, z)ϒˆ −1 (x, z , Z )−1 = −1 θˆ (x, z)(x, z, Z )−1 .

(5.104)

The ( j, j +1) entries of the matrices on the two sides of Eq. (5.104) never vanish. Together with (5.99), the equality of these entries implies that = 1n = ϒ −1 . We conclude that h = 1n by (5.101) and z = z by the definition of ϒ. Similarly, the (n, 1) entry never vanishes in (5.104), and this gives Z = Z . The implication (5.91) is therefore proven. It is clear from its formula (3.42) and Lemma 3.2 that Iˆ : Pˆc → Fν(x) is a smooth map. We see from the implication (5.91) that the map Iˆ is injective and its image intersects every gauge orbit at most in one point. Since we have shown also that the image of Iˆ intersects every gauge orbit, we conclude that condition 2) of Lemma 3.1 is satisfied. Equation (5.87) guarantees that condition 1) of the same lemma holds. In conclusion, Iˆ is a global cross section and ( Pˆc , ωˆ c ) is a model of the reduced phase space.

94


Proof of Theorem 3.4. We first observe from (5.86) and the definition of Lˆ (2.60) that (Lˆ ◦ Iˆ ◦ Zx )(q, ˆ p) ˆ = Lˆ (K x (1n , ℵ(x, eiqˆ ), p)) ˆ ˆ = R (K x (1n , ℵ(x, eiqˆ ), p)), ˆ ∀(q, ˆ p) ˆ ∈ P. (5.105) Using the notation X ∼ Y to signify that the matrices X and Y are similar, we then conclude from (2.35) and the formula (5.46) of K x that iqˆ R (K x (1n , ℵ(x, eiqˆ ), p)) ˆ ∼ R (e− pˆ (ℵ(x, eiqˆ ))−1 ˆ −1 ) = θ (x, p)e ˆ iqˆ . (x) ℵ(x, e )θ (x, p)

(5.106) To obtain the last equality, we used the identity (5.66). The similarity transformation that appears in (5.106) is by a unitary matrix that one can find explicitly from the above. On the other hand, one can directly check with the aid of the formula of θ (x, p) ˆ given in Lemma 5.3 that the dual Ruijsenaars-Schneider Lax matrix (1.3) can be rewritten as ˆ q, ˆ (x, p)e L( ˆ p) ˆ = δθ ˆ iqˆ δˆ−1

(5.107)

with δˆ = diag(δˆ1 , δˆ2 , . . . , δˆn ), δˆ j =

1 sinh ( pˆ j − pˆ m + x ) 4 2 , ∀ j = 1, . . . , n. sinh ( pˆ j − pˆ m − x2 )

m= j

(5.108) ˆ q, ˆ p) ˆ and L( ˆ p) ˆ are both similar to the matrix Therefore the matrices (Lˆ ◦ Iˆ ◦ Zx )(q, i q ˆ θ (x, p)e ˆ . The implication (5.91) also confirms that G v(x) acts freely on Fν(x) , as was already shown in [11]. Here, some further clarifying remarks are in order, which will be referred to in Sect. 6. 0 be the image of G Remark 5.1. Let Fν(x) v(x) × Tn × Cx by the map K x (5.46). It follows 0 from the above that Fν(x) is a dense open submanifold of Fν(x) , which is stable under the 0 /G gauge group G v(x) and is diffeomorphic to G v(x) × Tn × Cx by K x . Hence Fν(x) v(x) is a dense open submanifold of the full reduced phase space Fν(x) /G v(x) . The phase ˆ ω) space ( P, ˆ (1.4) is a model of this dense open submanifold. In fact, a corresponding 0 (5.69) cross section (in the sense of Remark 3.2) is provided by the map k x : Pˆ → Fν(x) that satisfies k x = Iˆ ◦ Zx (5.88) with the symplectic embedding Zx : Pˆ → Pˆc given by Definition 3.2.

Remark 5.2. We can define the Rn -valued smooth (even real-analytic) G v(x) -invariant function πˆ on Fν(x) by setting πˆ (K x (h, τ, p)) ˆ := p. ˆ More directly, πˆ diag(πˆ 1 , . . . , πˆ n ) can be characterized by the property ˆ ) ∈ C¯ 0 , K K † ∼ e−2πˆ (K ) with π(K

(5.109)

where, as before, ∼ denotes similarity of matrices. Clearly, πˆ induces a smooth function on the full reduced phase space and it has C¯ x as its range. From the perspective of the completed dual Ruijsenaars-Schneider system, πˆ can be viewed as a C¯ x -valued globally ˆ well-defined ‘position variable’ (which coincides with pˆ on the phase space P).


95

We explained in [11] how Theorems 3.1 and 3.2 imply the known integration algorithm for the time development of the position variable q along the commuting flows of the system (P, ω, L). Now we present the analogous result about the ‘dual position variable’ πˆ (5.109) along the commuting flows of the dual system characterized by Theorems 3.3 and 3.4. Remark 5.3. Take a Hamiltonian Hˆ := ∗R φ, φ ∈ C ∞ (G)G and regard it as a function on the model ( Pˆc , ωˆ c ) of the reduced phase space. Choose also an initial value (z(0), Z (0)) ∈ Pˆc . Directly from the definitions, the associated initial value πˆ (0) := ˆ πˆ (I(z(0), Z (0))) is subject to e−πˆ (0) = L (0 ) with 0 := (x, z(0), Z (0)).

(5.110)

By combining Theorem 3.3 with Corollary 2.1, it is easy to obtain the following result:

ˆ z(0)) R (0 )) . e−2πˆ (t) ∼ e−2πˆ (0) exp 2itDφ(θ(x, (5.111) ¯ Since (5.109) defines πˆ also as a G-invariant function on the double, (5.111) folˆ lows immediately from (2.54) by replacing the initial value I(z(0), Z (0)) (3.42) by ¯ (x, z(0), Z (0))θˆ (x, z(0))−1 . Indeed, the respective solutions K (t) (2.49) are G-related and yield the same πˆ (t). If (z(0), Z (0)) ∈ ( Pˆc \ Pˆc0 ) (see (5.90)), then the flow (5.111) does not stay in this lower dimensional closed submanifold in general, which explains by Theorem 3.4 ˆ ω). why the commuting dual flows are not complete on ( P, ˆ On the other hand, if 0 ˆ ˆ p(0)) ˆ is the corresponding initial value in Pˆ = Zx−1 ( Pˆc0 ), (z(0), Z (0)) ∈ Pc and (q(0), then (5.111) can be rewritten as

ˆ iq(0) ˆ e−2πˆ (t) ∼ e−2 p(0) exp 2itDφ(θ (x, p(0))e ˆ ) . (5.112) To further elaborate, let us consider φ(g) = tr (χ (g)) + c.c. with some complex power series χ , for which Dφ(g) − (ψ(g))† with ψ(z) := zχ (z) as was mentioned ∞= ψ(g) k ¯ k ˜ ¯ in (2.59). For ψ(z) = k=1 ψk z , let us define ψ(z) := ∞ k=1 ψk z , where ψk is the complex conjugate of ψk . Then, by using (5.107), we can rewrite (5.112) equivalently as

ˆ ˆ q(0), ˜ Lˆ −1 )) with Lˆ 0 := L( e−2πˆ (t) ∼ e−2 p(0) exp 2it (ψ( Lˆ 0 )− ψ( ˆ p(0)). ˆ 0 (5.113) These formulas for the flows were obtained previously in [30]. Our geometric picture renders their derivation essentially obvious. 6. Discussion In this section, we summarize our construction in terms of diagrams of maps and explain the connection with the related results in [30]. Then we conclude and comment on open problems.

96


Our main arguments were concerned with the following diagram: P R

I

−→ Fν(x) /G v(x)

↓ Pˆc

↓

(6.1)

id

π ◦Iˆ

−→ Fν(x) /G v(x)

The maps I and π ◦ Iˆ are symplectomorphisms as claimed by Theorems 3.1 and 3.3. The map R is defined as the symplectomorphism that makes this diagram commute. ˆ ω) We can add the phase space ( P, ˆ (1.4) to (6.1) by using the embedding Zx of Definition 3.2 and restrictions to the relevant dense open submanifolds. This leads to the second diagram: I0

id

0 /G P 0 −→ P 0 −→ Fν(x) v(x)

R0

Pˆ

↓

R0c

↓

Zx −→ Pˆc0

↓

id

(6.2)

π ◦Iˆ 0

0 /G −→ Fν(x) v(x)

Here, P 0 ⊂ P and Pˆc0 ⊂ Pˆc are the dense open submanifolds symplectomorphic to the 0 /G ˆ (5.109) dense open submanifold Fν(x) v(x) of the reduced phase space, for which π varies in Cx (3.26). (See also (5.90) and Remark 5.1.) Whenever appropriate, the maps in (6.2) are obtained as the restrictions of those in (6.1). The map R0 is defined by the commutativity of the diagram (6.2). The symplectic forms ω (1.2) on P, ωˆ c (3.27) on Pˆc , and the reduced symplectic form on Fν(x) /G v(x) induce symplectic forms on the respective dense open submanifolds. Then all the maps I 0 , π ◦ Iˆ 0 , R0c , Zx and R0 are symplectomorphisms. The maps R and R0 reproduce the ‘complete’ and the ‘restricted’ duality symplectomorphisms (alias action-angle maps) originally obtained by Ruijsenaars in [30] by means of direct arguments. In order to confirm this, we now present a useful consequence of our construction. Lemma 6.1. Consider a point (q, p) ∈ P 0 and its image (q, ˆ p) ˆ ∈ Pˆ by the symplectomorphism R0 defined by (6.2): R0 (q, p) = (q, ˆ p), ˆ

(6.3)

where (q, p) parametrizes some point (e2iq , p) of the covering space T ∗ T0n (3.17) of ˆ q, P (3.16). Then the Lax matrices L(q, p) (1.1) and L( ˆ p) ˆ (1.3) satisfy the similarity relations ˆ q, L(q, p) ∼ e2 pˆ and L( ˆ p) ˆ ∼ e2iq . (6.4) ˜ 2iq , p). The Proof. First, let us recall from Theorem 3.2 that L(q, p) ∼ (L ◦ I)(e 2iq ˜ 2iq , p)−1 × ˜ , p) = I(e definition (2.60) of the unreduced Lax matrix L gives (L ◦ I)(e ˜ 2iq , p)−1 )† , and then we obtain (I(e ˜ 2iq , p)−1 (I(e ˜ 2iq , p)−1 )† ∼ (I(e ˜ 2iq , p)I(e ˜ 2iq , p)† )−1 ∼ I(e −1 ∼ ((Iˆ ◦ Zx )(q, ˆ p))(( ˆ Iˆ ◦ Zx )(q, ˆ p)) ˆ † ∼ e2 pˆ .

(6.5)


97

˜ 2iq , p) and (Iˆ ◦ Zx )(q, Here we used that, because of (6.3), I(e ˆ p) ˆ must lie on the 0 same gauge orbit in Fν(x) , which implies the second similarity relation in (6.5) on account of the first relation in (2.35). The last relation in (6.5) follows from (5.109), ˆ Thus the first claim of (6.4) is proved. To prove ˆ p)) ˆ = pˆ holds on P. since πˆ (Iˆ ◦ Zx (q, the second claim of (6.4), we notice from Theorem 3.4 together with the definition of Lˆ in (2.60) and the second relation in (2.35) that ˜ 2iq , p)) = e2iq . ˆ q, L( ˆ p) ˆ ∼ (Lˆ ◦ Iˆ ◦ Zx )(q, ˆ p) ˆ ∼ R (I(e

(6.6)

˜ 2iq , p). The last equality above is a direct consequence of the formula (4.18) of I(e

We see from (6.4) that our symplectomorphism R0 between the restricted phase spaces P 0 and Pˆ converts the action variables of the original system (1.1) into the particle-coordinates of the dual system (1.3), and vice versa. These relations completely characterize the map R0 and also hold for the restricted duality symplectomorphism constructed in [30]. Therefore the latter is indeed reproduced by our geometric construction. The complete duality symplectomorphism R of diagram (6.1) is also the same as the one obtained in [30],4 simply because the embedding Zx : Pˆ → Pˆc occurs in [30] as well and its image is dense in Pˆc . To conclude, we have shown in this paper that the duality symplectomorphism ˆ ω, ˆ (1.3) is between the system (P, ω, L) (1.1) and the completion of the system ( P, ˆ L) nothing but the geometrically natural map between two models of the reduced phase space associated with a symplectic reduction of the Heisenberg double of ˆ ω) U (n). The character of the completion of the phase space ( P, ˆ is thereby illuminated: ˆ ( P, ω) ˆ represents a dense open submanifold of the full reduced phase space wherein the reduced flows inherited from the free flows on the double are naturally complete. We have also seen that the reduction turns the two Abelian algebras H and Hˆ (2.29) (spanned by the unreduced free Hamiltonians) into the Abelian algebras of the action and the particle-position variables of the system (P, ω, L), respectively, and the rôle of these algebras is exchanged when viewed from the perspective of the dual system. In addition, we obtained the Ruijsenaars-Schneider Lax matrices as well as the integration algorithms for their flows as easy by-products. The present article, together with the previous one [10], leave little doubt that all cases of the duality studied in [27,29,30] must permit analogous interpretation in terms of symplectic reduction of finite-dimensional integrable systems of group theoretic origin.5 The details are not trivial and we plan to return to other examples elsewhere. It is proper to mention at this point that although we expect that all cases of Ruijsenaars’ duality can be treated by the reduction method, the direct approach also has its own advantages. For instance, it appears that the detailed analyses of the scattering behaviour performed in [27,29,30] cannot be simplified by group theoretic means. It can be surmised from results in [12,26] that the classical trigonometric BCn systems of van Diejen [36] must admit a derivation based on a reduction of the Heisenberg double of U (2n), which should produce the so far missing Lax matrices for these systems. We plan to elaborate this, building on the description of the free systems given in Sect. 2 of the present paper. The results collected there might prove to be useful in other 4 To provide a short dictionary: our maps R0 and R reproduce the maps and that feature in diagrams (1.67) and (1.74) in [30], respectively. The content of our Remark 5.3 is consistent with Theorem 5.7 in [30]. 5 The question of possible dualities in the elliptic case is wide open. For the reduction approach to elliptic systems, see for example [2] and references therein.

98


studies in the reduction approach too, for example to obtain spin Ruijsenaars-Schneider systems. Finally, it is pertinent to remark that the quantum mechanical analogue of Ruijsenaars’ duality is the so-called bispectral property [6] and it is known that the quantum ˆ ω, ˆ form a bispectral pair [28]. mechanical variants of the systems (P, ω, L) and ( P, ˆ L) This follows from explicit inspection of the eigenfunctions of the respective Hamiltonian operators. The eigenfunctions are provided by Macdonald polynomials [4,28,37], which also have close connections to quantum groups [9,21,24] and to double affine Hecke algebras [5,25]. It would be interesting to understand the bispectral property, at least for certain (integer) values of the coupling constant, in terms of a quantum analogue of our classical reduction. Acknowledgements. We wish to thank Ian Marshall for useful comments on the manuscript. L.F. was partially supported by the Hungarian Scientific Research Fund (OTKA) under the grant K 77400.

A. Regularity of the Moment Map Value ν(x) The purpose of this Appendix is to demonstrate that ν(x) (3.10) is a regular value of the Poisson-Lie moment map associated with the quasi-adjoint action on G L(n, C). We first recall from [11] that every element K ∈ G L(n, C) can be represented as K = g (bT −1 ) with g ∈ G = U (n), T ∈ Tn , b ∈ B.

(A.1)

It was also shown in [11] that every element of the constraint-manifold Fν(x) = {K ∈ G L(n, C) | (K ) = ν(x)}

(A.2)

K = g (N (T )aT −1 ) with g ∈ G ν(x) , T ∈ C, a ∈ A.

(A.3)

can be written as

Here C is a subset of the regular elements T0n ⊂ Tn , such that C intersects every orbit of the permutation group S(n) in T0n precisely in one point. A < B is the group of diagonal matrices with positive real entries, and N (T ) belongs to the subgroup N < B of upper-triangular matrices with unit diagonal, given explicitly by the formula (4.1). The formula (2.22) of gives (bT −1 ) = bT b−1 T −1 ∈ N ,

∀T ∈ Tn , ∀b ∈ B.

(A.4)

By using this together with the U (n) equivariance of and (A.1), we easily obtain that det (K ) = 1 for all K ∈ G L(n, C). In other words, takes its values in the normal subgroup S B of B consisting of elements of unit determinant. Proposition A.1. The constant ν(x) (3.10) is a regular value of the Poisson-Lie moment map : G L(n, C) → S B. Proof. By the equivariance property of and (A.3), it is sufficient to show that the derivative (tangent) map λ K := T (K ) : TK G L(n, C) → Tν(x) S B

(A.5)


99

is surjective at every point of Fν(x) of the form K = N (T )aT −1 with T ∈ C, a ∈ A.

(A.6)

Now the tangent space to S B at any of its elements can be identified, as a vector space, with sB: the space of upper triangular, traceless matrices with real diagonal entries. This simply follows from the structure of S B as a matrix group. Take an arbitrary Y ∈ sB and consider the tangent vector Y K ∈ TK G L(n, C),

(A.7)

which is the velocity of the curve eY t K at t = 0. At K of the form (A.6), the formula (A.4) leads to λ K (Y K ) = Y ν(x) − ν(x)(T Y T −1 ) ∈ Lie(N ) ⊂ sB,

∀Y ∈ sB.

(A.8)

This equation implies that for any Z ∈ Lie(N ) there exists a unique Y ∈ Lie(N ) for which λ K (Y K ) = Z . To verify this, consider the principal gradation of gl(n, C) by ‘heights’ and write accordingly Y = Y1 +Y2 +· · ·+Yn−1 ,

Z = Z 1 + Z 2 +· · ·+ Z n−1 , ν(x) = 1n +ν1 +ν2 +· · ·+νn−1 . (A.9)

First, the grade 1 part of Y ν(x) − ν(x)(T Y T −1 ) = Z

(A.10)

Y1 − T Y1 T −1 = Z 1 ,

(A.11)

reads

which determines Y1 uniquely in terms of Z 1 and T since T is regular. Second, the grade 2 part of (A.10) reads Y2 − T Y2 T −1 + (Y1 ν1 − ν1 T Y1 T −1 ) = Z 2 ,

(A.12)

which determines Y2 uniquely in terms of Z 1 , Z 2 and T . Obviously, this argument can be continued increasing the grade until the top grade, (n − 1), is reached. We have seen that the image of λ K (A.5), at K in (A.6), contains Lie(N ). Denoting by sA the space of diagonal elements of sB, we have sB = sA + Lie(N ),

(A.13)

and it remains to show that the sA-projections of the elements in the image of λ K span sA. For W ∈ u(n), denote by W K the velocity of the curve et W K at t = 0. Because the moment map is equivariant, we obtain λ K (W K ) = dressW ν(x) ≡ ν(x)(ν −1 (x)W ν(x))B ,

(A.14)

where we use the unique decomposition X = X B + X u(n) ,

∀X ∈ gl(n, C).

(A.15)

100


Note from (3.10) that x E j, j+1 ν(x) = 1n + (2 sinh )I1 + · · · , where I1 = 2 n−1

(A.16)

j=1

and the dots stand for the higher grade parts of ν(x). Let us now take an element W ∈ u(n) of the form W = W−1 − (W−1 ) , †

W−1 =

n−1

ξk E k+1,k

(A.17)

k=1

with some real numbers ξk . Then we easily obtain x [I1 , W−1 ]+non-zero grade contributions. ν −1 (x)W ν(x) = − 2 sinh B 2 (A.18) Therefore, from (A.13), in this case we get the sA-projection n−1 x πs A (λ K (W K )) = −2 sinh ξk (E k,k − E k+1,k+1 ). 2

(A.19)

k=1

These elements span sA. Hence we conclude that λ K (A.5) is surjective.

Remark 7.1. It was shown in [11] (p. 135) that the ‘constituents’ T ∈ C, a ∈ A and g ∈ G ν(x) that appear in (A.3) are uniquely determined by K ∈ Fν(x) up to the obvious freedom of multiplying g by an element from the central U (1) subgroup of U (n). This means, in particular, that G¯ ν(x) (3.13) acts freely on Fν(x) . To avoid possible confusion, we point out that the set C cannot be taken as a globally smooth submanifold of T0n . This follows from the non-triviality of the S(n)-bundle T0n → Q(n); see the subsequent appendix.

B. The Structure of the Configuration Space Q(n) In this Appendix we deal with some features of the non-trivial manifold, Q(n) = T0n /S(n),

(B.1)

which is the configuration space of n indistinguishable non-coinciding ‘point-particles’ moving on the circle. We shall expound the following statements: 1. The principal S(n)-bundle T0n → Q(n) is topologically non-trivial. 2. The manifold Q(n) is orientable if and only if n is odd. 3. Q(n) cannot be separated into the Cartesian product of a ‘center of mass circle’ and a ‘space of relative positions’.


101

Note that the first statement can be extracted also from [30] and the second one generalizes the result of Leinaas and Myrheim [19] who showed that Q(2) is the open Möbius strip. To begin, we remark that the elements of the manifold T0n can be viewed as the configurations of n distinguished non-coinciding points on a circle. We can label those points by integers 1, . . . , n according to their registration on the diagonal in the matrix realization of T0n . Consider an element E of T0n and its S(n)-action image σ (E) under any permutation σ ∈ S(n) that changes the cyclic order of the distinguished points on the circle. It is evident that we cannot smoothly connect the element E with its image σ (E) without violating the condition of non-coincidence. This means that the element E and its image σ (E) live respectively in two different connected components of Tn0 . On the other hand, it is not difficult to see that the permutations respecting the cyclic order (i.e. the cyclic permutations) can be realized smoothly, which means that the element E and its image under the cyclic permutation live on the same connected component in T0n . Thus the connected components of T0n correspond to the coset space S(n)/Zn . In particular, this implies that the principal S(n)-bundle T0n → Q(n) is topologically non-trivial, since if it were trivial then the element E and its image σ (E) would live on different connected components for each permutation σ not equal to the identity. Consider the connected component K of T0n characterized by the requirement that the order of the labels 1 < 2 < · · · < n conforms with the cyclic order in which the distinguished points appear on the circle. We know from the preceding paragraph that the manifold K is a topologically non-trivial Zn -covering of the manifold Q(n). In fact, the non-triviality of the Zn -bundle K → Q(n) follows from the connectedness of K , which also implies that Q(n) is connected. Now we remark that K can be identified as the Cartesian product of the unit circle, on which the distinguished points live, and of the (n − 1)-dimensional open simplex Simpn−1 given by Simpn−1 := {δ ∈ Rn−1 | δ j > 0,

n−1

δ j < 2π }.

(B.2)

j=1

Indeed, the point z on the unit circle corresponds to the position of the distinguished point 1, the number 0 < δ1 < 2π is the angle between the points 1 and 2, δ2 between the points 2 and 3 etc. The generator of the cyclic group Zn is the cyclic permutation (123 . . . n) → (n12 . . . n − 1). It is very easy to find the action of on K = U (1) × Simpn−1 : (z, δ1 , δ2 , . . . , δn−2 , δn−1 ) = (zeiδ1 , δ2 , δ3 , . . . ., δn−1 , 2π −

n−1

δ j ).

(B.3)

j=1

A simple calculation of the Jacobian gives the transformation law of the volume form on K under the -transformation: ∗ (iz −1 dz ∧ dδ1 ∧ . . . ∧ dδn−1 ) = (−1)n+1 iz −1 dz ∧ dδ1 ∧ . . . ∧ dδn−1 . (B.4) We observe that for n odd the volume form is Zn invariant and therefore it descends to an everywhere non-vanishing n-form on the quotient K /Zn = Q(n). This means that for n odd Q(n) is orientable. Suppose now that there is also an everywhere non-vanishing n-form α on Q(n) for n even. Its pull-back α˜ to K must be given by a formula α˜ = f (z, δ1 , . . . , δn−1 )iz −1 dz ∧ dδ1 ∧ . . . ∧ dδn−1 ,

(B.5)

102


where f (z, δ1 , . . . , δn−1 ) is a smooth and everywhere non-vanishing real function on K . On the other hand, it must hold also that ˜ = α, ˜ ∗ (α)

(B.6)

because otherwise α˜ would not be the pull-back of α from Q(n). For n even, the condition (B.6) says that f (zeiδ1 , δ2 , δ3 , . . . ., δn−1 , 2π −

n−1

δ j ) = − f (z, δ1 , δ2 , . . . , δn−2 , δn−1 ). (B.7)

j=1

Since f changes sign if we move from a point on K to its image and f is smooth, it must vanish somewhere on K , which is in contradiction with the orientability of Q(n) for n even. So far we have proved the first two statements displayed at the beginning. For the last statement, we notice that the permutation action (B.3) of Zn on K commutes with the obvious action of U (1) on K given by u(z, δ1 , δ2 , . . . , δn−2 , δn−1 ) = (uz, δ1 , δ2 , . . . , δn−2 , δn−1 ), u ∈ U (1).

(B.8)

Therefore this U (1) action descends to Q(n). In specific systems, e.g. in the trigonometric Ruijsenaars-Schneider and Sutherland systems, the U (1) action on Q(n) just described can be interpreted as global rotation symmetry. Suppose that one tries to separate Q(n) into the Cartesian product of a ‘center of mass circle’ and some manifold of ‘relative positions’, say R(n − 1). A reasonable definition of such separation requires the existence of a product representation, Q(n) U (1) × R(n − 1), such that U (1) acts only on U (1) and not on R(n − 1). One might contemplate actions Au of the global rotations on U (1) × R(n − 1) defined by Au (w, r ) := (u k w, r ), ∀(w, r ) ∈ U (1) × R(n − 1),

(B.9)

where k is a fixed, non-zero integer. The choice k = 1 might appear the most natural, while k = n corresponds to taking the hypothetical center of mass as the product of the n points on the unit circle. However, we now show that neither of these separations of Q(n) exists. Indeed, if we consider a point (z, δ1 , . . . , δn−1 ) in K such that δ j = 2π n for every j, then we obtain e

2π i n

(z,

2π i 2π 2π 2π 2π 2π 2π 2π 2π 2π , ,..., ) = (ze n , , ,..., ) = (z, , ,..., ). n n n n n n n n n (B.10) 2π i

Hence the action of e n ∈ U (1) leaves invariant the point of Q(n) covered by 2π 2π (z, 2π n , n , . . . , n ) ∈ K . On the other hand, the isotropy group of the generic elements of Q(n) is trivial under the natural U (1) action. In contrast, under the ‘separated action’ (B.9) all points have the same isotropy group, Zk . This contradiction implies the validity of our third statement.


103

References 1. Alekseev, A.Yu., Malkin, A.Z.: Symplectic structures associated to Lie-Poisson groups. Commun. Math. Phys. 162, 147–174 (1994) 2. Arutyunov, G.E., Chekhov, L.O., Frolov, S.A.: Quantum dynamical R-matrices. In: Moscow Seminar in Mathematical Physics, AMS Transl. Ser. 2, Vol. 191, Providence, RI: Amer. Math. Soc., 1999, pp. 1–32 3. Calogero, F.: Solution of the one-dimensional N -body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971) 4. Chalykh, O.: Macdonald polynomials and algebraic integrability. Adv. Math. 166, 193–259 (2002) 5. Cherednik, I.: Double Affine Hecke Algebras. Cambridge: Cambridge University Press, 2005 6. Duistermaat, J.J., Grünbaum, F.A.: Differential equations in the spectral parameter. Commun. Math. Phys. 103, 177–240 (1986) 7. Duistermaat, J.J., Kolk, J.A.C.: Lie Groups. Berlin-Heidelberg-New York: Springer, 2000 8. Etingof, P.: Calogero-Moser Systems and Representation Theory. Zurich: European Mathematical Society, 2007 9. Etingof, P.I., Kirillov, A.A. Jr.: Macdonald’s polynomials and representations of quantum groups. Math. Res. Lett. 1, 279–294 (1994) 10. Fehér, L., Klimˇcík, C.: On the duality between the hyperbolic Sutherland and the rational RuijsenaarsSchneider models. J. Phys. A: Math. Theor. 42, 185202 (2009) 11. Fehér, L., Klimˇcík, C.: Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction. Lett. Math. Phys. 87, 125–138 (2009) 12. Fehér, L., Pusztai, B.G.: A class of Calogero type reductions of free motion on a simple Lie group. Lett. Math. Phys. 79, 263–277 (2007) 13. Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V.: Duality in integrable systems and gauge theories. JHEP 07, 028 (2000) 14. Fock, V.V., Rosly, A.A.: Poisson structure on moduli of flat connections on Riemann surfaces and the r -matrix. In: Moscow Seminar in Mathematical Physics, AMS Transl. Ser. 2, Vol. 191, Providence, RI: Amer. Math. Soc., 1999, pp. 67–86 15. Gorsky, A., Nekrasov, N.: Relativistic Calogero-Moser model as gauged WZW theory. Nucl. Phys. B 436, 582–608 (1995) 16. Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian group actions and dynamical systems of Calogero type. Comm. Pure Appl. Math. XXXI, 481–507 (1978) 17. Klimˇcík, C.: Quasitriangular WZW model. Rev. Math. Phys. 16, 679–808 (2004) 18. Klimˇcík, C.: On moment maps associated to a twisted Heisenberg double. Rev. Math. Phys. 18, 781–821 (2006) 19. Leinaas, J.M., Myrheim, J.: On the theory of identical particles. Nuovo Cim. B 37, 1–23 (1977) 20. Lu, J.-H.: Moment maps and reduction of Poisson actions. In: Proc. Sem. Sud-Rhodanien de Geometrie à Berkeley (1989), Springer-Verlag MSRI Publ., Vol. 20, Berlin-Heidelberg-New York: Springer, 1991, pp. 209–226 21. Mimachi, K.: Macdonald’s operator from the center of the quantized universal enveloping algebra Uq (gl(N )). Int. Math. Res. Notices IMRN 1994 no.10, 415–424 (1994) 22. Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220 (1975) 23. Nekrasov, N.: Infinite-dimensional algebras, many-body systems and gauge theories. In: Moscow Seminar in Mathematical Physics, AMS Transl. Ser. 2, Vol. 191, Providence, RI: Amer. Math. Soc., 1999, pp. 263–299 24. Noumi, M.: Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123, 16–77 (1996) 25. Oblomkov, A.A.: Double affine Hecke algebras and Calogero-Moser spaces. Represent. Theory 8, 243–266 (2004) 26. Oblomkov, A.A., Stokman, J.V.: Vector valued spherical functions and Macdonald-Koornwinder polynomials. Compos. Math. 141, 1310–1350 (2005) 27. Ruijsenaars, S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems I. The pure soliton case. Commun. Math. Phys. 115, 127–165 (1988) 28. Ruijsenaars, S.N.M.: Finite-dimensional soliton systems. In: Integrable and Superintegrable Systems, ed. Kupershmidt, B. Singapore: World Scientific, 1990, pp. 165–206 29. Ruijsenaars, S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems II. Solitons, antisolitons and their bound states. Publ. RIMS 30, 865–1008 (1994) 30. Ruijsenaars, S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems III. Sutherland type systems and their duals. Publ. RIMS 31, 247–353 (1995) 31. Ruijsenaars, S.N.M.: Systems of Calogero-Moser type. In: Proceedings of the 1994 CRM–Banff Summer School ‘Particles and Fields’, Berlin-Heidelberg-New York: Springer, 1999, pp. 251–352

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Communications in


A Frequency Localized Maximum Principle Applied to the 2D Quasi-Geostrophic Equation Henggeng Wang1 , Zhifei Zhang2 1 School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018,

P. R. China. E-mail: [email protected]

2 School of Mathematical Science, Peking University, Beijing 100871, P. R. China.

E-mail: [email protected] Received: 5 August 2009 / Accepted: 31 July 2010 Published online: 8 October 2010 – © Springer-Verlag 2010

Abstract: In this paper, we prove a maximum principle for a frequency localized transport-diffusion equation. As an application, we prove the local well-posedness of the 1−α , and global supercritical quasi-geostrophic equation in the critical Besov spaces B˚ ∞,q 0 well-posedness of the critical quasi-geostrophic equation in B˚ ∞,q for all 1 ≤ q < ∞. s s ˚ Here B∞,q is the closure of the Schwartz functions in the norm of B∞,q . 1. Introduction In this paper, we consider the two dimensional quasi-geostrophic equation ∂t θ + α θ + v · ∇θ = 0 in (0, T ) × R2 , θ (0) = θ 0 in R2 ,

(1.1)

where α ∈ (0, 1], the real-valued function θ (t, x) denotes the potential temperature of the fluid. The velocity v of the fluid is determined by the Riesz transforms of θ : v = (−∂2 −1 θ, ∂1 −1 θ ) = (−R2 θ, R1 θ ). A fractional power of the Laplacian α is defined by α f (ξ ) = |ξ |α f (ξ ),

where f denotes the Fourier transform of f . The quasi-geostrophic equation is an important model in geophysical fluid dynamics, which is a special case of the general quasi-geostrophic approximation for atmospheric and oceanic fluid flow with small Rossby and Ekman numbers. We may refer to [9,19] for more details about its background in geophysics. The cases α > 1, α = 1 and α < 1 are called subcritical, critical and supercritical respectively.

106

H. Wang, Z. Zhang

Due to the deep analogy between the Eq. (1.1) with α = 1 and the 3D Navier-Stokes equations, there are many mathematicians devoted to the study of the quasi-geostrophic equation. In the subcritical case, Constantin and Wu [10] proved the global existence of the smooth solution. In the critical case, Constantin, Cordoba, and Wu[8] proved the global existence of smooth solution for small periodic initial data. Recently, Kiselev, Nazarov and Volberg [18] proved the same result for arbitrary smooth periodic initial data. Caffarelli and Vasseur [3] established the global regularity of weak solutions associated to the initial data in L 2 (R2 ). In the supercritical case, whether smooth solutions of (1.1) develop the singularity in finite time is still an open problem. We may refer to [11–13] and references therein for some relevant results. In this paper, we are concerned with the well-posedness problem of (1.1) in the critdef

ical spaces. It is easy to find that if θ is a solution of (1.1), then θλ = λα−1 θ (λα t, λx) is also a solution. A functional space X is said to be critical if θλ X = θ X for any 2/ p+1−α λ > 0. Obviously, the homogeneous Besov space B˙ p,q (R2 ) is a critical space for any p, q ∈ [1, ∞]. Let us review some known well-posedness results. Chae and Lee [4] 1−α . proved the global well-posedness for small initial data in the critical Besov spaces B˙ 2,1 Cordoba-Cordoba [14], Ning [17] studied the well-posedness in the Sobolev spaces H s , s ≥ 1 − α, α ∈ [0, 1]. Wu [22,23] established the well-posedness in the Besov spaces B sp,q , s > 1 − α, p = 2 N . Recently, by establishing the generalized Bernstein’s inequality, Chen, Miao and Zhang [7] proved the global well-posedness for small initial 2/ p+1−α data in the critical Besov space B p,q (R2 ) for (α, p, q) ∈ (0, 1] × [2, ∞) × [1, ∞), and the local well-posedness for large initial data. By obtaining some regularization effects, Hmidi and Keraani [16] also proved the similar well-posedness results including s 0 the well-posedness in the limiting space B∞,1 ∩ B˙∞,1 for s ≥ 1 − α. Very recently, Abidi and Hmidi [1] proved the global well-posedness for the critical quasi-geostrophic 0 (R2 ). Here B˙ 0 (R2 ) is the closure of equation with large initial data belonging to B˙∞,1 ∞,1 0 (R2 ). the Schwartz functions in the norm of the homogenous Besov space B˙ ∞,1 This paper is devoted to the well-posedness of (1.1) in the inhomogeneous critical 1−α for 0 < α ≤ 1, 1 ≤ q ≤ ∞, which have a certain vanishing property Besov spaces B˚ ∞,q at infinity. That is, 1−α 1−α B˚ ∞,q is the closure of the Schwartz functions in the norm of B∞,q .

The key point of the proof is to establish the following maximum principle for the frequency localized transport-diffusion equation. Theorem 1.1 (Localized Maximum Principle). Let θ, v, f be smooth functions with j θ (t) ∈ C0 (R2 ) for t > 0 and j ≥ 0, and let θ be a solution of the equation ∂t j θ + v · ∇ j θ + α j θ = f. Then there exists a positive constant c independent of θ, v, f, j such that for a.e. t > 0, ∂t j θ ∞ + c2 jα j θ ∞ ≤ f ∞ . Here C0 (R2 ) denotes the set of the continuous function vanishing at infinity, and j is the frequency localization operator (see Sect. 2).

Maximum Principle and the 2D Quasi-Geostrophic Equation

107

Remark 1.2. By using the generalized Bernstein’s inequality, Chen, Miao and Zhang [7] established the following inequality: ∂t j θ p + c p 2 jα j θ p ≤ f p , for 2 ≤ p < ∞, here c p → 0 as p → ∞. However, it seems difficult to adapt the method of [7] to the case when p = ∞. As an application of Theorem 1.1, we obtain the well-posedness results of (1.1) by means of the Fourier localization technique and Bony’s decomposition. It should s be pointed out that since the Riesz transform Ri is not bounded in B∞,q , we need to use a subtle commutator estimate (Lemma 6.2) and the fact that Ri −1 θ ∈ C0 (R2 ) if 1−α (see Remark 2.8) in order to deal with the velocity v. θ ∈ B˚ ∞,q Theorem 1.3. Let 0 < α ≤ 1, 1 ≤ q ≤ ∞. Assume that the initial data θ 0 belongs to 1−α . Then there hold B˚ ∞,q (a) If q < ∞, there exist T > 0 and a unique solution θ to (1.1) satisfying def 1−α 1 θ ∈ E T = C([0, T ]; B˚ ∞,q )∩ L 1T B∞,q .

Moreover, let T ∗ be the maximal existence time of θ , then there exists an absolute constant η > 0 such that if T ∗ < ∞, then α

lim inf (T ∗ − t) 2−α ∇θ (t) B∞,∞ ≥ η. 0 ∗ t→T

(b) If q = ∞ and θ 0 B∞,∞ 1−α ≤ 0 for small enough 0 , then there exist T > 0 and a solution θ to (1.1) satisfying def

1−α 1 θ ∈ E T = L ∞ ([0, T ]; B∞,∞ )∩ L 1T B∞,∞ .

Moreover, the solution is unique under the following extra assumption: sup lim θ

t 1, then the solution θ has the higher regularity, i.e. s θ ∈ C([0, T ]; B˚ ∞,q ),

for any T < T ∗ . Remark 1.4. For 1 ≤ q < ∞, there holds 1 lim 2 j (1−α) ω j (t) 2 j θ 0 ∞ t→0

q

= 0, ω j (t) = 1 − e−c2

jα t

,

which ensures that the estimates we obtained are uniform in the short time. However, it seems impossible for q = ∞. This is the reason why we assume that the initial data is small in Theorem 1.3 (b).

108

H. Wang, Z. Zhang

By using the argument of modulus of continuity introduced by Kiselev, Nazarov and Volberg [18], we also establish the global well-posedness for the critical quasigeostrophic equation. Theorem 1.5. If α = 1 and 1 ≤ q < ∞, then the solution θ obtained in Theorem 1.3 is global in time. Remark 1.6. Due to the inclusion relation: 2

+1−α

p B p,q

1−α B˚ ∞,q for 1 ≤ p < ∞, 1 ≤ q ≤ ∞,

Theorem 1.3 is an improvement of the well-posedness results given by [7] and [16]. Throughout this paper, C stands for a positive constant which may be different from line to line. We denote by · p the norm of the Lebesgue space L p (R2 ). 2. Preliminaries First of all, we introduce the Littlewood-Paley decomposition. Choose two nonnegative radial functions χ , ϕ ∈ S(R2 ), supported respectively in B = {ξ ∈ R2 , |ξ | ≤ 4/3} and C = {ξ ∈ R2 , 3/4 ≤ |ξ | ≤ 8/3} such that ϕ(2− j ξ ) = 1, ξ ∈ R2 . χ (ξ ) + j≥0

Let h = F −1 ϕ and h˜ = F −1 χ . The frequency localization operators j and S j are defined by j f = ϕ(2− j D) f = 22 j h(2 j y) f (x − y)dy for j ≥ 0, 2 R −j ˜ j y) f (x − y)dy, and S j f = χ (2 D) f = k f = 2 2 j h(2 R2

−1≤k≤ j−1

−1 f = S0 f, j f = 0 for j ≤ −2. With our choice of ϕ, it is easy to verify that j k f = 0 if | j − k| ≥ 2 and j (Sk−1 f k f ) = 0 if | j − k| ≥ 5.

(2.1)

In the sequel, we will constantly use Bony’s decomposition from [2]: uv = Tu v + Tv u + R(u, v), with Tu v =

j

S j−1 u j v,

R(u, v) =

(2.2)

j u j v.

| j − j|≤1

With the introduction of j , let us recall the definition of Besov space, see [21].


109

Definition 2.1. Let s ∈ R, 1 ≤ p, q ≤ ∞, the inhomogeneous Besov space B sp,q is defined by B sp,q = f ∈ S (Rn ) : f B sp,q < ∞ . Here

def f B sp,q = 2 js j f p q .

We denote by

B˚ sp,q

the completion of the Schwartz functions under the norm · B sp,q . ρ

Besides the usual space-time space L T B sp,q , we also need the Chemin-Lerner-Besov ρ space L T B sp,q , which is defined as the set of all distributions f satisfying def js ρ ρ s 2 f = f p j L L q < ∞. L B p,q T

T

From Minkowski’s inequality, it is easy to find that L rt (B sp,q ) ⊆ L rt (B sp,q ) if r ≤ q and L rt (B sp,q ) ⊆ L rt (B sp,q ) if r ≥ q. s σ Lemma 2.2. Let 1 ≤ q < ∞, σ ∈ R. If f ∈ B˚ ∞,q ∩ B∞,q for some s < σ , then σ ˚ f ∈ B∞,q .

Proof. Take χ (x) ∈ C0∞ with χ (x) = 0 for |x| ≤ 1, and let χ M (x) = χ (x/M). Thanks s σ to f ∈ B˚ ∞,q , it is easy to verify that S N f ∈ B˚ ∞,q and for fixed N , σ −→ 0, (1 − χ M (x))S N f B∞,q

σ as M → +∞. On the other hand, since f ∈ B∞,q and q < ∞, there holds σ (1 − S N ) f B∞,q −→ 0,

as N → +∞. Thus, σ σ σ χ M (x)S N f − f B∞,q ≤ (1 − S N ) f B∞,q + (1 − χ M (x))S N f B∞,q −→ 0,

by letting M → +∞, then N → +∞. The lemma is proved. Lemma 2.3 [5]. Let 1 ≤ p ≤ q ≤ +∞. Assume that f ∈ L p (R2 ), then for any γ ∈ (N ∪ {0})2 , there exists a constant C independent of f , j such that 1

1

j|γ |+2 j ( p − q ) supp fˆ ⊆ {|ξ | ≤ A0 2 j } ⇒ ∂ γ f q ≤ C2 f p, j j − j|γ | ˆ supp f ⊆ {A1 2 ≤ |ξ | ≤ A2 2 } ⇒ f p ≤ C2 sup ∂ β f p . |β|=|γ |

Next we recall a result of the fractional integral operator. Lemma 2.4 [14]. Let 0 ≤ α < 2, and θ ∈ S(R2 ). Then there holds θ (x) − θ (y) α θ (x) = cα dy, 2+α R2 |x − y| with cα > 0.

110

H. Wang, Z. Zhang

Following the proof of Lemma 2.3 in [24], we can obtain Lemma 2.5. Let m(ξ ) be a homogeneous function of degree α with α > 0. Define def

K (x) =

m(ξ )χ (ξ )ei x·ξ dξ,

where χ ∈ C0∞ (R2 ) with suppχ ⊂ {ξ : |ξ | ≤ 2}. Then there holds |K (x)| ≤ C(1 + |x|)−2−α . f ⊂ {ξ : |ξ | ≤ 2}, we have In particular, if f ∈ L p (R2 ) with supp m(D) f p ≤ C f p , for any 1 ≤ p ≤ ∞. Remark 2.6. Thanks to Lemma 2.5, we have s s+α , α f B∞,q ≤ C f B∞,q

for any α > 0, s ∈ R, q ∈ [1, ∞]. Let us conclude this section by recalling the BMO and VMO spaces. A locally integrable function f is said to belong to BMO if the inequality 1 |B|

| f (x) − f B |d x ≤ A B

holds for all balls B; here f B = |B|−1

B

f d x. It is well-known that

Ri : L ∞ → BMO but Ri : L ∞ → L ∞ . Let VMO be the closure of C0 (R2 ) in the norm of BMO. We have the following properties about VMO space (see [20], p. 180).

Proposition 2.7. (1) Let ∈ S(R2 ), R2 d x = 1. Suppose that f ∈ B M O, then f ∈ V M O if and only if f ∗t ∈ C0 (R2 ) for all t > 0, and f − f ∗t B M O → 0 as t → 0. (2) If f ∈ C0 (R2 ), then Ri f ∈ VMO for i = 1, 2. s Remark 2.8. If f ∈ B˚ ∞,q for all s ∈ R, q ∈ [1, ∞], then Ri −1 f ∈ C0 (R2 )∩C ∞ (R2 ). s Indeed, since f ∈ B˚ ∞,q , −1 f ∈ C0 (R2 ). Thanks to Proposition 2.7 (2), Ri −1 f ∈ VMO, which together with Proposition 2.7 (1) implies Ri −1 f ∈ C0 (R2 ). On the other hand, Ri −1 f ∈ C ∞ (R2 ) is a direct consequence of Lemma 2.3.


111

3. A Frequency Localized Maximum Principle We consider the transport-diffusion equation ∂t θ + α θ + v · ∇θ = f.

(3.1)

We have the following parabolic maximum principle. Proposition 3.1 (Maximum principle). Let v be a smooth vector field and f be a smooth function. Assume that θ (t, x) is a solution of Eq. (3.1) with θ (t) ∈ C0 (R2 ). Then there holds t θ (t)∞ − θ (0)∞ ≤ f (τ )∞ dτ. 0

In the case when v is a divergence free vector, the inequality was proved in [14] by using the positivity lemma. Here we generalize it to a general vector v. To prove Proposition 3.1, we need the following classical lemma. Lemma 3.2. Let f (t, x) be a smooth function on [0, +∞) × R2 with f (t) ∈ C0 (R2 ) for d all t ≥ 0. Then dt f (t)∞ exists for a.e. t ≥ 0 and d f (t)∞ = (∂t f )(t, xt )sign( f (t, xt )), dt here f (t, xt ) = ± f (t)∞ and sign x is a sign function of x. Proof. Given t ≥ 0, for all h ∈ R such that t + h ≥ 0, we have | f (t + h)∞ − f (t)∞ | ≤ f (t + h) − f (t)∞ ≤

sup

τ ∈[0,t+1]

∂τ f (τ )∞ + 2 sup f (τ )∞ |h|. τ ≥0

def

d f (t)∞ Thus, g(t) = f (t)∞ is a Lipschitz function in t ≥ 0, and then g (t) = dt exists for a.e. t ≥ 0. For every h > 0, since f (t, x) is a smooth function with f (t) ∈ C0 (R2 ), there always exists a point xt+h ∈ R2 such that | f (t + h, x)| reaches its maximum at xt+h , that is g(t + h) = | f (t + h, xt+h )|. Then we can find a sequence h n → 0 such that xt+h n → xt and g(t) = | f (t, xt )|. Assume that f (t, xt ) > 0 (The case when f (t, xt ) ≤ 0 can be similarly proved) and g (t) exists. Thus, for n big enough,

f (t + h n ) L ∞ − f (t) L ∞ f (t + h n , xt+h n ) − f (t, xt ) = hn hn f (t + h n , xt+h n ) − f (t, xt+h n ) ≤ −→ ∂t f (t, xt ) hn as n → ∞, which implies that d f (t)∞ ≤ ∂t f (t, xt ). dt

(3.2)

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H. Wang, Z. Zhang

On the other hand, we have f (t + h n ) L ∞ − f (t) L ∞ f (t + h n , xt ) − f (t, xt ) ≥ → ∂t f (t, xt ) hn hn as n → ∞, which implies that d f (t)∞ ≥ ∂t f (t, xt ), dt from which and (3.2), we conclude that d f (t)∞ = ∂t f (t, xt ). dt This finishes the proof of Lemma 3.2. Proof of Proposition 3.1. Due to θ ∈ C0 (R2 ), without loss of generality, we may assume that there exists a point xt ∈ R2 such that θ (t, xt ) = θ (t)∞ . Then ∇θ (t, xt ) = 0. By Lemma 2.4, we have θ (xt ) − θ (y) α θ (xt ) = Cα dy ≥ 0. 2+α 2 R |x t − y| Thus, it follows from Lemma 3.2 that ∂t θ (t)∞ = (∂t θ )(t, xt ) = −v(t, xt ) · ∇θ (t, xt ) − α θ (t, xt ) + f (t, xt ) ≤ f (t)∞ , which implies Proposition 3.1. Next, we consider the frequency localized transport-diffusion equation ∂t j θ + v · ∇ j θ + α j θ = f.

(3.3)

The following maximum principle plays a key role in the proof of Theorem 1.3. Theorem 3.3 (Localized maximum principle). Let θ, v, f be smooth functions with j θ (t) ∈ C0 (R2 ) for t > 0 and j ≥ 0, and let θ be a solution of (3.3). Then there exists a positive constant c independent of θ, v, f, j such that for a.e. t > 0, ∂t j θ L ∞ + c2 jα j θ L ∞ ≤ f L ∞ . The proof of Theorem 3.3 is based on the following localized positivity lemma. Lemma 3.4. Let

8 3 def 2 g ⊂ {ξ : ≤ |ξ | ≤ } . A = g ∈ C0 (R ) : g L ∞ = 1, supp 4 3 Suppose that g ∈ A and |g(x0 )| = g L ∞ = 1 for some x0 ∈ R2 . Then there exists a constant c independent of g such that sign(g(x0 ))α g(x0 ) ≥ c.


113

Proof. We prove the lemma by the contradiction argument. Assume that there exist {gn } ⊂ A and xn ∈ R2 such that |gn (xn )| = 1,

cn = α gn (xn ) → 0.

Let τh be a translation operator defined as τh g(x) = g(x + h). It is easy to verify that τh α g(x) = α τh g(x). For the sequence {τxn gn }, there exist a subsequence {τxnk gn k } and g ∈ L ∞ such that |τxnk gn k (0)| = 1 and τxnk gn k g in L ∞ . Since supp F(τxnk gn k )(ξ ) ⊂ {ξ : we have supp g ⊂ {ξ :

3 4

3 8 ≤ |ξ | ≤ }, 4 3

≤ |ξ | ≤ 83 } and for any x ∈ R2 ,

τxnk gn k (x) = F −1 (ψ) ∗ (τxnk gn k )(x) → g(x). Here ψ ∈ C0∞ (R2 ) satisfies ψ(ξ ) = 1 in {ξ : 43 ≤ |ξ | ≤ 83 }. Moreover, g ∈ C ∞ ∩ L ∞ , |g(0)| = 1, and g L ∞ = 1. In addition, g ≡ ±1. Otherwise, g = F −1 (ψ) ∗ g ≡ 0 since R2 F −1 (ψ)(x)d x = ψ(0) = 0. By Lemma 2.4 and the Fatou Lemma, we find that g(0) − g(y) α sign(g(0)) g(0) = Cα sign(g(0)) dy |y|2+α R2 τ g (0) − τ g (y) xnk n k xnk n k ≤ Cα sign(g(0)) lim dy = sign(g(0)) lim cn k = 0. 2+α |y| k→∞ R2 k→∞ Since g ≡ ±1, the above inequality contradicts the fact:

α

sign(g(0)) g(0) = sign(g(0))

R2

g(0) − g(y) dy > 0. |y|2+α

This completes the proof of Lemma 3.4. def

Proof of Theorem 3.3. Since θ j = j θ ∈ C0 (R2 ) for j ≥ 0, there exists a point xt, j ∈ R2 so that |θ j (t, xt, j )| = θ j (t)∞ > 0. By using a scaling argument and Lemma 3.4, there exists a positive constant c independent of j and θ such that sign(θ j (xt, j ))α θ j (xt, j ) ≥ c2 jα θ j ∞ . Then we get by Lemma 3.2 and (3.3) that ∂t θ j (t)∞ = sign(θ j (xt, j ))(∂t θ j )(t, xt, j ) = sign(θ j (xt, j )) −v(t, xt, j ) · ∇θ j (t, xt, j ) − α θ j (t, xt, j ) + f (t, xt, j ) ≤ −c2 jα θ j (t)∞ + f (t)∞ , where we used ∇θ j (t, xt, j ) = 0 in the last inequality.

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H. Wang, Z. Zhang

4. Proof of Theorem 1.3 In this section, we will prove Theorem 1.3. To construct the approximate solution, we need the following lemma. 0 ˚ s+3 ˚s Lemma 4.1. Let v = (−R2 f, R1 f ) with f ∈ L∞ T B∞,q , and θ ∈ B∞,q for s ≥ 3. Then the transport-diffusion equation ∂t θ + α θ + v · ∇θ = 0, (4.1) θ (0) = θ 0 , s has a unique solution θ ∈ C([0, T ]; B˚ ∞,q ). s , we can choose θ 0,n ∈ S(R2 ) such that as n tends to ∞, Proof. Since θ 0 ∈ B˚ ∞,q s θ 0,n −→ θ 0 in B∞,q .

Then we solve (4.1) with the initial data θ 0,n and obtain a sequence of solutions θ n ∈ s s C([0, T ]; H s+2 ). Since H s+2 → B˚ ∞,q , θ ∈ C([0, T ]; B˚ ∞,q ). On the other hand, from Step 1 in the proof of Theorem 1.3 (especially (4.2) and (4.5)), we can infer that t 0,n s θ n ≤ θ + 2 js [ j , v] · ∇θ n ∞ q dτ, s ∞ B∞,q L t B∞,q 0

from which and Lemma 6.2 (6.5), we infer that t 0,n s s s θ n ∇θ n L ∞ f B∞,q dτ ≤ θ + C + ∇v L ∞ θ n B∞,q s ∞ B∞,q L t B∞,q 0

n s ∞ ∞ f ∞ s ≤ θ 0,n B∞,q + Ct∇θ n L ∞ . s L t B∞,q + Ct∇v L t L ∞ θ L∞ t L t B∞,q s Here we also used f L ∞ ≤ f . Noticing that s L∞ t B∞,q t B∞,q

n ∞ ≤ Cθ ∞ s ∞ ≤ C f ∞ s ∇θ n L ∞ L t B∞,q , ∇v L ∞ L t B∞,q , t L t L

we obtain s ≤ θ 0,n B∞,q + Ct f θ n , θ n s s s L∞ L∞ L∞ t B∞,q t B∞,q t B∞,q

which implies s ). θ n −→ θ in C([0, T ]; B∞,q s Thus θ ∈ C([0, T ]; B˚ ∞,q ).

Proof of Theorem 1.3. We divide the proof into several steps. Step 1. A priori estimates. We assume that θ (t, x) ∈ C0 (R2 ) is a smooth solution of def

(1.1). Set θ j = j θ for j ≥ −1, then θ j satisfies ∂t θ j + α θ j + v · ∇θ j = −[ j , v] · ∇θ.


115

First of all, we get by Proposition 3.1 that 0 θ−1 (t)∞ ≤ θ−1 ∞ +

t

0

[ −1 , v] · ∇θ L ∞ dτ.

(4.2)

We infer from Lemma 6.2 (6.4) that

t

[ −1 , v] · ∇θ L ∞ dτ ≤ 2 j (1−α) [ − j , v] · ∇θ L 1 L ∞ q t

0

≤ C∇v ≤ Cθ 2

−α 2 L 2t B∞,q 1− α

L 2t B∞,q2

∇θ

−α 2 L 2t B∞,q

.

Thus we deduce that 0 ∞ + Cθ 2 θ−1 (t)∞ ≤ θ−1

1− α

L 2t B∞,q2

.

(4.3)

Thanks to Theorem 3.3, we get for j ≥ 0, ∂t θ j ∞ + c2 jα θ j ∞ ≤ [ j , v] · ∇θ ∞ , from which and Gronwall’s inequality, we infer that θ j (t)∞ ≤ e−c2

jα t

θ 0j ∞ + e−c2

jα t

∗ [ j , v] · ∇θ ∞ .

(4.4)

Here the convolution is defined as f ∗ g(t) =

t

f (t − s)g(s)ds.

0

Using (4.4) and the Young’s inequality, we get for 1 ≤ r ≤ ∞ and j ≥ 0, α

1

2 j r θ j L rt L ∞ ≤ Cω j (t) r θ 0j L ∞ + C[ j , v] · ∇θ L 1 L ∞ , t

def

(4.5)

where ω j (t) = 1 − e−c2 t . Multiplying by 2 j (1−α) on both sides of (4.5), and then summing over j, we obtain jα

⎛ jq(1− α ) q ⎝ r θ j r 2

Lt L

⎞1 q

⎠ ∞

j≥0 1

≤ C2 j (1−α) ω j (t) r θ 0j L ∞ q + C2 j (1−α) [ j , v] · ∇θ L 1 L ∞ q . t

While due to Lemma 6.2 and Remark 2.6, we have 2 j (1−α) [ j , v] · ∇θ L 1 L ∞ q ≤ Cθ 2 t

1− α

L 2t B∞,q2

,

(4.6)

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H. Wang, Z. Zhang

which together with (4.3) and (4.6) gives for 1 ≤ r ≤ ∞, θ r

1−α/r

L t B∞,q

θ

1− α

L 2t B∞,q2

1

1

2 ≤ C(1 + t r )θ 0 B∞,q 1−α + C(1 + t r )θ

1− α

L 2t B∞,q2

1

,

(4.7)

1

≤ t 2 −1 θ 0 L ∞ + C2 j (1−α) ω j (t) 2 θ 0j L ∞ q 1

+ C(1 + t 2 )θ 2

1− α

L 2t B∞,q2

.

(4.8)

Step 2. Approximate solutions and uniform estimates. We construct the approximate solutions of (1.1) by solving the following linear system: ⎧ (k+1) + α θ (k+1) + v (k) · ∇θ (k+1) = 0, ⎨ ∂t θ (k) (4.9) v = (−R2 θ (k) , R1 θ (k) ), ⎩ (k+1) (0) = Sk+4 θ (0) , θ s0 for all k ≥ 0. Here we set (θ (0) , v (0) ) = ( −1 θ 0 , −1 v 0 ). Since θ (k) (0) ∈ B˚ ∞,q for any s0 (for example, we take s0 ≥ 3), we infer from Lemma 4.1 that the solution s0 ) for k ≥ 0. Then a similar argument as leading to (4.7) and θ (k) ∈ C([0, +∞); B˚ ∞,q (4.8) ensures that

θ (k+1) r

1−α/r

L t B∞,q

1

≤ C(1 + t r )θ 0 B∞,q 1−α 1

+ C(1 + t r )θ (k) θ (k+1)

1− α

L 2t B∞,q2

1− α

L 2t B∞,q2

1− α

L 2t B∞,q2

1

,

(4.10)

1

≤ t 2 −1 θ 0 L ∞ + C2 j (1−α) ω j (t) 2 θ 0j L ∞ q 1

+ C(1 + t 2 )θ (k)

1− α

L 2t B∞,q2

Noticing that for 1 ≤ q < ∞, 1 lim 2 j (1−α) ω j (t) 2 j θ 0 ∞ t→0

θ (k+1)

θ (k+1)

q

1− α

L 2t B∞,q2

.

(4.11)

= 0,

and θ 0 B∞,q 1−α ≤ 0 for q = ∞, then by a standard induction argument, (4.10)-(4.11) ensure that there exists T > 0 such that 2 θ (k) 2 1− α2 ≤ C 0 , θ (k) r 1−α/r ≤ C θ 0 B∞,q 1−α + 0 , (4.12) L t B∞,q

L t B∞,q

for k ≥ 0 and 0 ≤ t ≤ T . Step 3. Convergence and existence. In this step, we assume that 0 ≤ t ≤ T ≤ 1. Set δθ (k) = θ (k+1) − θ (k) , δv (k) = v (k+1) − v (k) . Then (δθ k , δv k ) satisfies ∂t δθ (k) + α δθ (k) + v (k) · ∇δθ (k) + δv (k) · ∇θ (k+1) = 0, δθ (k) (0) = k+3 θ 0 .

(4.13)


117

Taking j on both sides of (4.13), we obtain ∂t j δθ (k) + α j δθ (k) + v (k) · ∇ j δθ (k) = −[ j , v (k) ] · ∇δθ (k) − j (δv (k) · ∇θ (k+1) ). Noticing that −1 k+3 θ 0 = 0, we get by Proposition 3.1 that −1 δθ (k) (t)∞ ≤

t 0

[ −1 , v (k) ] · ∇δθ (k) + −1 (δv (k) · ∇θ (k+1) )∞ dτ. (4.14)

Exactly as in the proof of (4.6), we can obtain ⎛ ⎝

⎞1 q

2

jq( α2 −s)

j≥0

(k) q δθ j L 2 L ∞ ⎠ t

− js −s + C 2 ≤ C k+3 θ 0 B∞,q [ j , v (k) ] · ∇δθ (k) L 1 L ∞ q t − js (k) (k+1) + C 2 j (δv · ∇θ ) L 1 L ∞ q . t

(4.15)

From (4.14) and (4.15), we deduce that δθ (k)

α −s

2 L 2t B∞,q

− js −s + C 2 ≤ C k+3 θ 0 B∞,q [ j , v (k) ] · ∇δθ (k) L 1 L ∞ q t − js (k) (k+1) + C 2 j (δv · ∇θ ) L 1 L ∞ q . t

Taking s such that

α 2

(4.16)

< s < 1, we get by Lemma 6.2 and Remark 2.6 that

− js 2 [ j , v (k) ] · ∇δθ (k) L 1t L ∞

≤ Cθ (k)

q

1− α

L 2t B∞,q2

δθ (k)

α −s

2 L 2t B∞,q

,

(4.17)

and by Lemma 6.1 we have − js 2 j (δv (k) · ∇θ (k+1) ) L 1 L ∞ t

(k)

(k+1)

q

≤ Cδv 2 α2 −s θ 2 1− α2 L t B∞,q L t B∞,q (k) ∞ ≤ C −1 δv L t L ∞ + δθ (k)

α −s 2 L 2t B∞,q

θ (k+1)

1− α

L 2t B∞,q2

.

(4.18)

From (4.16)–(4.18), we infer that δθ (k)

α −s

2 L 2t B∞,q

≤ C2−k(1+s−α) θ 0 B∞,q 1−α +C

k+1 m=k

θ (m)

1− α L 2t B∞,q2

(k) ∞ + δθ −1 δv (k) L ∞ t L

α −s 2 L 2t B∞,q

. (4.19)

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H. Wang, Z. Zhang

∞ . In order to do this, we take Ri −1 (i = 1, 2) It remains to estimate −1 δv (k) L ∞ t L on both sides of (4.13) to obtain

∂t Ri −1 δθ (k) + v (k) · ∇ Ri −1 δθ (k) + α Ri −1 δθ (k)

= −[Ri −1 , v (k) ] · ∇δθ (k) − Ri −1 (δv (k) · ∇θ (k+1) ),

from which and the fact that Ri −1 θ (k) ∈ C0 (R2 ), we get by Proposition 3.1 that −1 δv (k) (t)∞ ≤

2

t

[Ri −1 , v (k) ] · ∇δθ (k) ∞ dτ

i=1 0 2 t

+

0

i=1

Ri −1 (δv (k) · ∇θ (k+1) )∞ dτ.

(4.20)

Firstly, from Lemma 6.3 and s < 1, we infer that t [Ri −1 , v (k) ] · ∇δθ (k) ∞ dτ 0 t ≤C −1 δθ (k) ∞ j θ (k) ∞ + j θ (k) ∞ j δθ (k) ∞ dτ 0

≤ Cδθ

| j− j |≤6

j>4 (k)

α −s

2 L 2t B∞,q

θ

(k)

1− α

L 2t B∞,q2

.

(4.21)

Secondly, from Lemma 2.5 and Bony’s decomposition (2.2), we can deduce that Ri −1 (δv (k) · ∇θ (k+1) )∞ = Ri ∇ −1 ·(δv (k) θ (k+1) )∞ ≤ C −1 (δv (k) θ (k+1) )∞ ≤ C −1 (Tδv (k) θ (k+1) )∞ + −1 (Tθ (k+1) δv (k) )∞ + −1 R(θ (k+1) , δv (k) )∞ S j−1 δv (k) ∞ j θ (k+1) ∞ + S j−1 θ (k+1) ∞ j δv (k) ∞ ≤C j≤4

+C

j θ (k+1) ∞ j δv (k) ∞ ,

| j− j |≤1

which implies that t Ri −1 (δv (k) · ∇θ (k+1) )∞ dτ 0

≤ Cδv (k) 2 α2 −s θ (k+1) 2 1− α2 L t B∞,q L t B∞,q (k) ∞ ≤ C −1 δv L t L ∞ + δθ (k)

α −s 2 L 2t B∞,q

θ (k+1)

1− α

L 2t B∞,q2

.

(4.22)

Summing up (4.20)-(4.22), we obtain −1 δv (k) (t)∞ ≤ C

k+1 m=k

θ (m)

1− α L 2t B∞,q2

(k) ∞ + δθ −1 δv (k) L ∞ t L

α −s 2 L 2t B∞,q

,


119

which together with (4.19) gives δθ (k)

α −s

2 L 2t B∞,q

k+1

+C

θ

+ −1 δv (k) (t)∞ ≤ C2−k(1+s−α) θ 0 B∞,q 1−α (m)

m=k

1− α

L 2t B∞,q2

(k) ∞ + δθ −1 δv (k) L ∞ t L

α −s

2 L 2t B∞,q

.

(4.23)

The above inequality and (4.12) ensure that θ (k) and −1 v (k) are two Cauchy sequences α 2 −s ∞ in L 2 (0, T ; B∞,q ) × L∞ T L . Thus, there exists a limit θ (t, x) such that as n → ∞, α

2 −s θ (k) −→ θ in L 2 (0, T ; B∞,q ),

−1 v (k) −→ −1 v

(4.24)

∞ L∞ T L ,

in

with v = (−R2 θ, R1 θ ). Then by (4.12) and interpolation, we get for any σ < 1 − α2 , σ θ (k) −→ θ in L 2 (0, T ; B∞,q ),

(4.25)

1− α

σ ) for 1 ≤ r ≤ ∞ with and θ ∈ L r (0, T ; B∞,qr ) ∩ L 2 (0, T ; B∞,q

θ

1− α L rt B∞,qr

2 θ ≤ C θ 0 B∞,q 1−α + 0 ,

1− α

L 2t B∞,q2

≤ C 0 .

(4.26)

With (4.25) and (4.26), a standard limit argument will ensure that (θ (t, x), v(t, x)) satisfies (1.1) in the sense of distribution. 1−α ) for 1 ≤ q < ∞. Thanks to the definition of Next, we prove θ ∈ C([0, T ], B˚ ∞,q Besov spaces, we have θ (t) − θ (t ) B∞,q 1−α ≤

+

j 0 be arbitrarily small. Since θ ∈ L∞ t B∞,q , there exists N > 0 such that

q 1/q 2 j (1−α) θ j (t) − θ j (t )∞ < .

(4.27)

j≥N

On the other hand, from Eq. (1.1), we obtain q 1/q 2 j (1−α) θ j (t) − θ j (t )∞ jN s 2− j (s−1) θ B∞,q .

122

H. Wang, Z. Zhang log(e+θ L∞ Bs

Then, taking N such that 2−N (s−1) θ ≤ 1 (i.e., N = s L∞ t B∞,q obtain t t (∇v∞ + ∇θ ∞ )dτ ≤ 2 j j θ ∞ dτ + t 0

j≤N

t

s log 2

∞,q

)

), we

0

≤ Cθ L1 B1 t

∞,q

ln(e + θ ) + t. s L∞ t B∞,q

Thus we have s ln(e + θ ) ≤ ln(e + θ 0 B∞,q ) + Cθ s L∞ L1 B1 t B∞,q t

∞,q

ln(e + θ ) + Ct. s L∞ t B∞,q

1 , we can choose t small enough such that Since q < ∞ and θ ∈ L 1t B∞,q

θ L1 B1 t

∞,q

≤

1 . 2C

s ∞ s Thus, θ ∈ L∞ t B∞,q . A standard continuity argument concludes that θ ∈ L T ∗ B∞,q . s Then θ ∈ C([0, T ∗ ); B˚ ∞,q ) from the proof at the end of Step 3. This completes the proof of Theorem 1.3. s Remark 4.2. If the initial data θ 0 ∈ B˚ ∞,q for s > 1 − α, there exist T0 > 0 and a unique solution θ of (1.1) such that s s+α θ ∈ C([0, T0 ]; B˚ ∞,q )∩ L 1T0 B∞,q .

This remark can be deduced by following the proof of the existence and uniqueness part of Theorem 1.3. Let us conclude this section by establishing the higher regularity of the local solution of (1.1). Theorem 4.3. Assume that θ is a solution of (1.1) on [0, T ∗ ) , as stated in Theorem 1.3 s (a). Then θ (t) ∈ C([δ, T ∗ ); B˚ ∞,q ) for any δ ∈ (0, T ∗ ) and s > 0. 1 , thus θ ∈ L 1 B 1− for any > 0, Proof. Thanks to Theorem 1.3 (a), θ ∈ L 1T ∗ B∞,q T ∗ ∞,q which together with Lemma 2.2 implies that for any ε < T ∗ , there exists t0 ∈ (0, ε) such 1− . Then from Remark 4.2 and the uniqueness of the solution, we infer that θ (t0 ) ∈ B˚ ∞,q 1 1− +α ) for some T > t . Repeating the above argument again, we that θ ∈ L (t0 , T0 ; B∞,q 0 0 can conclude that there exists t0 ∈ (0, δ) such that the local solution θ (t0 ) obtained in α 1+ Theorem 1.3 belongs to B˚ ∞,q2 . Theorem 1.3 (c) will ensure that the solution θ satisfies α 1+ θ ∈ C([t0 , T ∗ ); B˚ ∞,q2 ). For any β ∈ R+ , let us consider the equation

∂t [(t − t0 )β θ ] + α (t − t0 )β θ + v · ∇(t − t0 )β θ = β(t − t0 )β−1 θ, (t − t0 )β θ |t=t0 = 0.


123

In a similar way as (4.29), we get for t0 ≤ t < T ∗ , (t − t0 )θ 1 ) L ∞ (t0 ,t;B∞,q t ≤ 2 j [ j , v] · (τ − t0 )∇θ ∞ q dτ + Cθ 1−α L ∞ B∞,q t

t0

≤C

t

t0

(∇v∞ + ∇θ ∞ ) (τ − t0 )θ 1−α , 1 ) dτ + Cθ L ∞ (t0 ,τ ;B∞,q L ∞ B∞,q t

from which and Gronwall’s inequality, we infer that (t − t0 )θ 1−α e 1 ) ≤ Cθ L ∞ (t0 ,t;B∞,q L ∞ B∞,q

C

t

≤ Cθ 1−α e L ∞ B∞,q

C

t

t0 (∇v∞ +∇θ∞ )dτ

t t0

θ B 1

∞,1

t

dτ

,

where we used in the last inequality ∇v∞ ≤ C∇v B 0

≤ Cθ B 1 .

∞,1

∞,1

Then we can get by an induction argument that (t − t0 )n+1 θ n+1 ≤ C(n + 1)e L ∞ B∞,q t

≤ Cn e

C(n+2)

C

t

t t0

t0

θ B 1

∞,1

θ B 1

∞,1

dτ

dτ

(t − t0 )n θ n L∞ t B∞,q

θ 1−α , L ∞ B∞,q t

β from which and interpolation, we can deduce that (t − t0 )β θ ∈ L ∞ (t0 , t; B˚ ∞,q ) for any β β ∈ R+ and t ∈ [t0 , T ∗ ). Thus θ ∈ C([δ, T ∗ ); B˚ ∞,q ).

5. Proof of Theorem 1.5 This section is devoted to the global well-posedness of (1.1) with the critical dissipation. We shall use the argument of modulus of continuity introduced by Kiselev, Nazarov and Volberg[18]. Here we will only sketch the proof, see [1] for more details. 0 )∩ Let T ∗ be the maximal existence time of the solution in the space C([0, T ∗ ); B˚ ∞,q 1 ). Let λ be a real positive number determined later and T ∈ (0, T ∗ ). L 1 (0, T ∗ ; B∞,q 1 We define the set def I = T ∈ [T1 , T ∗ ); ∀t ∈ [T1 , T ], x = y ∈ R2 , |θ (t, x) − θ (t, y)| < ωλ (|x − y|) , where ω : R+ → R+ is strictly non-decreasing, concave, ω(0) = 0, ω (0) < +∞, limξ →0+ ω (ξ ) = −∞ and ωλ (|x − y|) = ω(λ|x − y|). First of all, if we take λ=

ω−1 (3θ 0 ∞ ) ∇θ (T1 )∞ , 2θ 0 ∞

then we can deduce from the maximum principal and the properties of ω that T1 ∈ I , thus the set I is non-empty. From the construction, the set I is an interval of the form [T1 , T∗ ). Now there are three possibilities:

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The first case is T∗ = T ∗ and in this case we have the necessary T ∗ = +∞ because the Lipschitz norm of θ does not blow up. The second case is T∗ ∈ I which is impossible. In fact, using the higher regularity of θ (see Theorem 4.3), we can infer that there exists a δ > 0 such that for all t ∈ [T∗ , T∗ +δ], |θ (t, x) − θ (t, y)| < ωλ (|x − y|). Thus, we obtain that T∗ + δ ∈ I which contradicts the fact that T∗ is maximal. The last case is T∗ ∈ / I . By the time continuity of θ , there exists x = y such that θ (T∗ , x) − θ (T∗ , y) = ωλ (ξ ), with ξ = |x − y|. As in [18], we choose the continuous function ω as follows: when 0 ≤ ξ ≤ δ, ω(ξ ) = ξ − ξ 3/2 , γ , when ξ > δ, ω (ξ ) = ξ(4+log(ξ/δ)) here δ, γ are small enough and satisfy δ > γ > 0. With this choice, it is shown in [18] that f (T∗ ) < 0, where f (t) = θ (t, x) − θ (t, y). Thus, this scenario can not occur since f (t) ≤ f (T∗ ), ∀t ∈ [0, T∗ ]. This completes the proof of Theorem 1.5. Acknowledgements. The authors thank the referee for invaluable comments and suggestions which helped improve the paper greatly. The authors would like to thank Professor Miao Changxing and Professor Taoufik Hmidi for helpful suggestions. Z. Zhang was partially supported by NSF of China under Grant 10990013, 11071007, and SRF for ROCS, SEM.

6. Appendix In this Appendix, we shall present the product estimate and commutator estimate in the inhomogeneous Besov spaces. We can refer to Lecture notes [6,15] for a more detailed introduction. Lemma 6.1. Let 1 ≤ q ≤ ∞, r1 = r11 + r12 ≤ 1, ρ1 , ρ2 < 0, ρ1 + ρ2 + 1 > 0, and v be a solenoidal vector field. Then there holds v · ∇θ L r B ρ1 +ρ2 ≤ Cv L r1 B ρ1 θ L r2 B 1+ρ2 . t

∞,q

t

∞,q

t

∞,q

Proof. Using Bony’s decomposition (2.2), we write v · ∇θ = Tvi ∂i θ + T∂i θ v i + R(v i , ∂i θ ). Thanks to (2.1), we have j (Tvi ∂i θ ) =

| j − j|≤4

j (S j −1 v i ∂i j θ ).


125

From Lemma 2.3 and ρ1 < 0, it follows that j (Tvi ∂i θ )∞ ≤ C 2j k v∞ j θ ∞ | j − j|≤4

k≤ j −2

ρ1 ≤ C2 j (1−ρ1 ) v B∞,q

Similarly, we have

j (T∂i θ v i ) =

j θ ∞ .

(6.1)

| j − j|≤4

j (S j −1 (∂i θ ) j v i ).

| j − j|≤4

From ρ2 < 0, Lemma 2.3 applied gives j (T∂i θ v i )∞ ≤ C

2k k θ ∞ j v∞

| j − j|≤4 k≤ j −2

≤ C2− jρ2 θ B 1+ρ2 ∞,q

Since divv = 0, we have

j R(v i , ∂i θ ) =

j v∞ .

(6.2)

| j − j|≤4

∂i j ( j v i j θ ),

j , j ≥ j−3;| j − j |≤1

from which and Lemma 2.3, it follows that j R(v i , ∂i θ )∞ ≤ C

2 j j v∞ j θ ∞

j , j ≥ j−3;| j − j |≤1 ρ1 ≤ C2 j v B∞,q

2− j ρ1 j θ ∞ .

(6.3)

j ≥ j−3

Summing up (6.1)–(6.3), we obtain v · ∇θ L r B ρ1 +ρ2 ≤ Cv L r1 B ρ1 θ L r2 B 1+ρ2 . t

∞,q

t

∞,q

t

∞,q

Indeed, we have by the Young’s inequality for the series and ρ1 + ρ2 + 1 > 0 that ρ1 2 j (ρ1 +ρ2 ) j R(v i , ∂i θ )∞ q ≤ Cv B∞,q 2(ρ1+ρ2 +1)( j−j ) 2 j (ρ2 +1) j θ ∞ q j ≥ j−3

≤ Cv

ρ1 B∞,q

θ B 1+ρ2 . ∞,q

This finishes the proof of Lemma 6.1. Lemma 6.2. Let 1 ≤ q ≤ ∞, r1 = r11 + r12 ≤ 1, ρ1 , ρ2 < 1, ρ1 + ρ2 > 0, ρ1 > 0, and v be a solenoidal vector field. Then there holds {2 j (ρ1 +ρ2 −1) [ j , v] · ∇θ L rt L ∞ }q ≤ C∇v r ρ −1 ∇θ r2 ρ2 −1 . L 1B 1 L B t

∞,q

t

∞,q

Moreover, if v = (−R2 f, R1 f ), we have for all s > 0, s s . + ∇v∞ θ B∞,q 2 js [ j , v · ∇]θ ∞ q ≤ C ∇θ ∞ f B∞,q

(6.4)

(6.5)

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H. Wang, Z. Zhang

Proof. We firstly prove (6.4). Using Bony’s decomposition (2.2), we write [ j , v · ∇]θ = [ j , Tvi ]∂i θ + j (T∂i θ v i ) − T∂i j θ v i + j R(v i , ∂i θ ) − R(v i , ∂i j θ ) def

= I + I I + I I I.

Firstly, in view of the definition of j , we write I =

j (Sk−1 v · ∇ k θ ) − Sk−1 v · j ∇ k θ k

=

2

|k− j|≤4

=

h(2 j (x − y))(Sk−1 v(y) − Sk−1 v(x)) · ∇ k θ (y)dy

2j

22 j

1

h(2 j y)

y · ∇ Sk−1 v(x + τ y)dτ · ∇ k θ (y + x)dy,

0

|k− j|≤4

which implies I ∞ ≤ C2− j

∇ Sk−1 v∞ ∇ k θ L ∞ ,

(6.6)

|k− j|≤4

from which and ρ1 < 1, it follows that j (ρ1 +ρ2 −1) I L rt L ∞ } q {2 j (ρ +ρ −2) 1 2 ≤ C {2 ∇ Sk−1 v L r1 L ∞ ∇ k θ L r2 L ∞ } t t |k− j|≤4

q

≤ C∇v r ρ −1 ∇θ r2 ρ2 −1 . L 1B 1 L B t

∞,q

t

(6.7)

∞,q

Similarly, we have I I ∞

i i j (Sk−1 ∂i θ k v ) − Sk−1 j ∂i θ k v = k ∞ i i ≤ j (Sk−1 ∂i θ k v ) − Sk−1 j ∂i θ k v |k− j|≤4 ∞ + Sk−1 j ∂i θ k v i k> j+4 ∞ −j ≤ C2 ∇ k v∞ ∇ Sk−1 θ ∞ + C∇ j θ ∞ k v∞ , |k− j|≤4

k> j+4


127

from which and the assumption that ρ2 < 1 and ρ1 > 0, we infer that j (ρ1 +ρ2 −1) I I L rt L ∞ } q {2 j (ρ +ρ −2) 1 2 ≤ C {2 ∇ k v L r1 L ∞ ∇ Sk−1 θ L r2 L ∞ t t q |k− j|≤4 jρ 1 + C∇θ k v L r1 L ∞ } r ρ2 −1 {2 L t 2 B∞,q t q k> j+4

≤ C∇v r ρ −1 ∇θ r2 ρ2 −1 , L 1B 1 L B t

∞,q

t

(6.8)

∞,q

where we used Lemma 2.3 in the last inequality. Since div v = 0, we can rewrite I I I as III = j ( k v · k ∇θ ) − k v · j k ∇θ |k− j|≤4;|k−k |≤1

+

div j ( k v k θ ).

k> j+4;|k−k |≤1

Then a similar proof of (6.6) gives I I I ∞ ≤ C2− j

∇ k v∞ ∇ k θ ∞

|k− j|≤4;|k−k |≤1

+ C2 j

k v∞ k θ ∞ ,

k> j+4;|k−k |≤1

which together with Lemma 2.3 and ρ1 + ρ2 > 0 implies that {2 j (ρ1 +ρ2 −1) I I I L rt L ∞ }q ≤ C∇v r ρ −1 ∇θ r2 ρ2 −1 . L 1B 1 L B t

∞,q

t

∞,q

(6.9)

Summing up (6.7)–(6.9), we conclude the proof of (6.4). Since the proof of (6.5) is completely similar, here we omit it. Let us close this section by the following commutator estimate. Lemma 6.3. Let v = (−R2 f, R1 f ). Then there holds [Rk −1 , v · ∇]θ ∞ ≤ C −1 θ ∞ j f ∞ + C j>4

j θ ∞ j f ∞ .

| j− j |≤6

Proof. Using Bony’s decomposition (2.2), we write [Rk −1 , v · ∇]θ = [Rk −1 , Tvi ]∂i θ + Rk −1 (T∂i θ v i ) − T∂i Rk −1 θ v i + Rk −1 R(v i , ∂i θ ) − R v i , ∂i Rk −1 θ def

= I + I I + I I I.

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We denote by K (x) the kernel of the operator ∇ −1 Rk . From Lemma 2.5, we have |K (x)| ≤ C(1 + |x|)−3 . Given 0 < δ < 1, thanks to div v = 0, we have I ∞ = K (x − y) · (S j−1 v(y) − S j−1 v(x)) j θ (y)dy j≤4

∞

δ |S j−1 v(y) − S j−1 v(x)| = | j θ (y)|dy |K (x − y)||x − y| δ |x − y| j≤4

≤

|K (x)||x|δ d x sup x= y

j≤4

≤C

∞

|S j−1 v(y) − S j−1 v(x)| j θ ∞ |x − y|δ

S j−1 f ∞ j θ ∞ .

(6.10)

j≤4

Here in the last inequality we used the fact that for j ≤ 3, sup

x= y

|S j v(y) − S j v(x)| |S j f (y) − S j f (x)| ≤ C sup |x − y|δ |x − y|δ x= y ≤ CS j f C δ ≤ C2 jδ S j f ∞ ≤ CS j f ∞ , (6.11)

which follows from the boundedness of Rk on the homogeneous Hölder spaces and Lemma 2.3. Similarly, from div v = 0 and the fact that j v∞ ≤ C j f ∞ for j ≥ 0, we deduce that I I ∞ ≤ K (x − y) · ( j v(y) − j v(x))S j−1 θ (y)dy ∞

j≤4

+

S j−1 (∇ Rk −1 )θ ∞ j v∞

j>4

≤C

S j−1 f ∞ j θ ∞ + C −1 θ ∞

j≤4

and I I I ∞

j f ∞ ,

(6.12)

j>4

= v(y) − v(x)) θ (y)dy K (x − y) · ( j j j j≤4;| j− j |≤1 ∞ + divRk −1 ( j v j θ )∞ j>4;| j− j |≤1

≤C

j f ∞ j θ ∞ .

| j− j |≤1

Summing up (6.10)–(6.13), we conclude the proof of this lemma.

(6.13)


129

References 1. Abidi, H., Hmidi, T.: On the global well-posedness of the critical quasi-geostrophic equation. SIAM J. Math. Anal. 40, 167–185 (2008) 2. Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246 (1981) 3. Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171(3), 1903–1930 (2010) 4. Chae, D., Lee, J.: Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys. 233, 297–311 (2003) 5. Chemin, J.Y.: Perfect Incompressibe Fluids, New York: Oxford University Press, 1998 6. Chemin, J.Y.: Localization in Fourier space and Navier-Stokes system. In: Phase Space Analysis of Partial Differential Equations, Proceedings 2004, CRM Series, Pisa, pp. 53–136 7. Chen, Q., Miao, C., Zhang, Z.: A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation. Commun. Math. Phys. 271, 821–838 (2007) 8. Constantin, P., Córdoba, D., Wu, J.: On the critical dissipative quasi-geostrophic thermal active scalar. Indiana Univ. Math. J. 50, 97–107 (2001) 9. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) 10. Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999) 11. Constantin, P., Wu, J.: Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Annal. L’Inst. H. Poincaré Analyse Non Linéaire 25, 1103–1110 (2008) 12. Córdoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998) 13. Córdoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Amer. Math. Soc. 15, 665–670 (2002) 14. Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004) 15. Danchin, R.: Fourier analysis methods for PDEs, http://perso-math.univ-mlv.fr/users/danchin.raphael/ courschine.pdf, 2005 16. Hmidi, T., Keraani, S.: Global solutions of the super-critical 2D Q-G equation in Besov spaces. Adv. in Math. 214, 618–638 (2007) 17. Ju, N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251, 365–376 (2004) 18. Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167, 445–453 (2007) 19. Pedlosky, J.: Geophysical Fluid Dynamics, New York: Springer, 1987 20. Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton Univ Press, 1993 21. Triebel, H.: Theory of Function Spaces. Monograph in Mathematics, Vol. 78, Basel: Birkhauser Verlag, 1983 22. Wu, J.: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36, 1014–1030 (2004) 23. Wu, J.: The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity 18, 139–154 (2005) 24. Zhang, P., Zhang, Z.: On the local well-posedness for the dumbbell model of polymeric fluids. J. Part. Diff. Eqs. 21, 234–252 (2008) Communicated by P. Constantin


Communications in


A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory G. I. Lehrer1 , Hechun Zhang2 , R. B. Zhang1 1 School of Mathematics and Statistics, University of Sydney, Sydney, Australia.

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

2 Department of Mathematical Sciences, Tsinghua University, Beijing, China.

E-mail: [email protected] Received: 11 August 2009 / Accepted: 10 July 2010 Published online: 16 October 2010 – © Springer-Verlag 2010

Abstract: We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry. Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalities on Module Algebras over Quantum Groups . . . . . . . . 2.1 Module algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Braided symmetric algebras . . . . . . . . . . . . . . . . . . . . . 2.3 A quantum analogue of the symmetric algebra . . . . . . . . . . . 2.4 The quantum trace and invariants . . . . . . . . . . . . . . . . . . Invariant Theory of the Quantum Even Orthogonal Groups Uq (so2n ) . 3.1 The natural module for Uq (so2n ) . . . . . . . . . . . . . . . . . . 3.2 The braided symmetric algebra of the natural Uq (so2n )-module . . 3.3 A Uq (so2n ) module algebra: Sq (V )⊗m with twisted multiplication 3.4 Noncommutative FFT of invariant theory . . . . . . . . . . . . . . Invariant Theory of the Quantum Odd Orthogonal Groups Uq (so2n+1 ) .

. . . . . . . . . . . .

. . . . . . . . . . . .

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132 134 134 137 138 139 140 140 142 144 149 150

132

G. I. Lehrer, H. Zhang, R. B. Zhang

4.1 Braided symmetric algebra of the natural Uq (so2n+1 )-module . 4.2 A Uq (so2n+1 )-module algebra and the FFT of invariant theory 5. Invariant Theory for the Quantum Symplectic Groups Uq (sp2n ) . . 5.1 Braided symmetric algebra of the natural Uq (sp2n )-module . . 5.2 A Uq (sp2n )-module algebra and the FFT of invariant theory . . 6. Invariant Theory for the Quantum General Linear Group . . . . . . 6.1 Algebra of functions on the quantum general linear group . . . 6.2 Invariant theory for the quantum general linear group . . . . . 6.3 Proof of the FFT for the quantum general linear group . . . . . 6.4 The braided exterior algebra . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

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. . . . . . . . . . . .

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151 153 154 155 156 158 159 164 166 169 172 173

1. Introduction The fundamental theorems of classical invariant theory may be formulated in three equivalent ways. Given a module V for a reductive group G over a field k, the first formulation provides a complete description of EndkG (V ⊗r ); the second does the same thing for the linear space of G-invariant multilinear functions f : W → k, where W = ⊕r V ⊕s V ∗ , i.e. multilinear functions which are constant on G-orbits of W . Thirdly, the commutative algebra S(W ∗ ) of polynomial functions on W , is naturally a G-module, and one describes the subalgebra of invariant functions. In each case the ‘first fundamental theorem’ (FFT) provides generators (suitably defined) for the space of invariants, while the ‘second fundamental theorem’ (SFT) describes all relations among the generators. Although equivalent in principle, the statements of the fundamental theorems in their three formulations are not equally straightforward. For example, the SFT for orthogonal and symplectic groups in the first context would require a hitherto unknown description of an ideal in the Brauer algebras (cf. [19]). A typical case addressed by the fundamental theorems is when G is one of the classical groups over the complex numbers C, and the G-module V is the natural module. In these cases, the standard polynomial (third) form of the first fundamental theorem of classical invariant theory [32] yields a finite set of algebra generators for the subalgebra of invariants. In the first formulation given above, the fundamental theorems are often referred to as (generalised) Schur-Weyl-Brauer duality. In this work we shall consider quantum analogues of the FFT in its third formulation. That is, we shall consider the action of the quantum groups corresponding to the classical Lie algebras on non-commutative analogues of the S(W ∗ ) above, and provide a finite set of generators of the (also generally non-commutative–see below) subalgebra of invariants. Note that the concept of “generators” of course depends on the structure of the algebra on which the quantum group acts, and that is a crucial aspect of this work. There is some work in this direction in the literature. A type of Howe duality between quantum general linear groups was constructed in [36], which implies the JimboSchur-Weyl duality between the quantum general linear group and the Hecke algebra. Also the paper [30] investigated quantum analogues of polynomial invariants for the symplectic Lie algebra. The works [10,11] have elements in common with our work, as does [4]. The work [22] provides a general context for Hopf algebra actions on noncommutative algebras. However, to our knowledge, there is no systematic treatment of the invariant theoretic aspects of the subject.

Noncommutative Classical Invariant Theory

133

There is a considerable literature on endomorphism algebras of tensor powers of quantum group modules. In particular, when g is a classical Lie algebra and V is the natural module for the quantum group Uq (g) (a slight enlargement of Uq (g) is more convenient when g = so2n ), the algebra EndUq (g) (V ⊗r ) at generic q is known to be a quotient of the Hecke algebra of type A if g = gln and a quotient of a Birman-WenzlMurakami algebra in the other cases [9,16–18,26]. The duality between the quantum general linear group and Hecke algebra is known to be valid even when q is a root of unity [9]. In [18], the algebras EndUq (g) (V ⊗r ),where V is the 7-dimensional irreducible module for Uq (G 2 ) or any finite dimensional irreducible module for Uq (sl2 ) were described in terms of representations of the Artin braid group. The endomorphism algebras for tensor powers of certain irreducible representations of the E series of quantum groups were studied by Wenzl [31]. The general framework for developing a noncommutative analogue of classical invariant theory for quantum groups is well established in the setting of arbitrary Hopf algebras [22] (see also [4]). Let Uq (g) be a quantum group, and let A be a module algebra over Uq (g) in the sense of [22, §4.1]; that is, A is an associative algebra whose underlying vector space is a Uq (g)-module, and whose algebraic structure is preserved by the quantum group action. Then the subspace AUq (g) of invariants in A is a subalgebra. Our aim is to describe the algebraic structure of AUq (g) . Corresponding to each classical Lie algebra g, we construct an associative algebra, which is a module algebra for the quantum group Uq (g) and reduces in the limit q → 1 to the polynomial algebra over a direct sum of copies of the natural g-module (and its dual if g is type A). Since the usual tensor product of Uq (g) module algebras is not generally a module algebra, this construction makes essential use of the braiding of the quantum group given by the universal R-matrix, which does not appear in the setting of general Hopf algebras [22]. The question of a module algebra structure on tensor products has also been studied in [3,37] and will be discussed below. We then study the subspace of quantum group invariants in the module algebra. We shall show that the subspace of invariants always forms a subalgebra of the module algebra, and find a set of generators for the subalgebra of invariants together with the commutation relations obeyed by these generators. This result may be regarded as the FFT of the noncommutative analogue of classical invariant theory for quantum groups associated with the classical Lie algebras, in the third of the three formulations described above. In all the cases studied, we find that the subalgebras of invariants are finitely generated. We prove this by explicit analysis of the subalgebras of invariants using the diagrammatical method of [27,28]. We should point out that since a quantum group Uq (g) is a non-cocommutative Hopf algebra, the relevant module algebras are noncommutative, and so also are their subalgebras of invariants. For this reason, most of the techniques in classical invariant theory, based on commutative algebra, do not apply here. In particular the general proofs of finite generation by Hilbert, Weyl and Nagata in the commutative context do not generalise to the quantum group setting. Returning to the problem of constructing appropriate quantum analogues of polynomial algebras over quantum group modules, we note that it has long been known that there is no good quantum analogue of the coordinate ring of the 4-dimensional irreducible Uq (sl2 )-module [29]. Recently Berenstein and Zwicknagl [3,37] have carried out a systematic study of the braided symmetric algebras, which might be construed as quantum analogues of coordinate rings. They found that almost all the braided symmetric algebras are ‘smaller’ than the corresponding polynomial algebras and thus are not

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suitable quantum analogues (e.g. in the sense of [29, Definition 2]) of the latter. That is, they are not flat deformations of the coordinate rings. The same is true even of the braided symmetric algebras on direct sums of copies of the natural module of quantum groups of type B, C and D. This may be a partial explanation of the lack of results available on quantum analogues of classical invariant theory in the third (polynomial function) formulation of the three described above. Our construction of quantum analogues of polynomial algebras will follow an approach suggested in [5], and the resulting algebras are not braided symmetric algebras in the sense of [3,37]. There is a connection between the present work and noncommutative geometry [6]. The relationship between quantum groups and noncommutative geometry is well described in [21]. This suggests that quantum groups play much the same role in noncommutative geometry as that played by Lie groups in ordinary differential geometry. Indeed a noncommutative space X is specified by its associative algebra A(X ) of functions [6]. If A(X ) is a module algebra over some quantum group Uq (g), then we may regard X as having a Uq (g) symmetry algebra. Typical examples of such noncommutative geometries are the quantum homogeneous spaces studied in [12]. In order to understand the noncommutative space X , it is useful to determine the Uq (g)-modules structure of A(X ), in particular, the subalgebra of invariants. We shall realise this in Lemma 3.6 and Corollary 4.3, for the quantum sphere. This paper is organised as follows. Section 2 discusses generalities concerning module algebras for quantum groups. The concept of “generator” is based on the well known Theorem 2.3, which defines a twisted multiplication on the tensor product of any two module algebras by using the universal R-matrix of the quantum group to endow the tensor product with the structure of a module algebra. Sections 3 and 4 treat the invariant theory of the orthogonal quantum groups. The main results are summarised in Theorems 3.11 and 4.7, which provide a noncommutative FFT’s of invariant theory for the quantum orthogonal groups. The subalgebra of invariants is noncommutative in the quantum setting, and we describe the commutation relations among the (finite set of) generators of the subalgebra of invariants in Lemmas 3.9 and 4.6. Section 5 treats the invariant theory of the symplectic quantum groups. The noncommutative FFT for the quantum symplectic groups is Theorem 5.6, and the commutation relations among the generators of the noncommutative subalgebra of invariants are given in Proposition 5.5. In Sect. 6, we study the invariant theory of the quantum general linear group. The techniques used in this section are similar to those of [12] but rather different from those in the previous sections. Theorem 6.10 provides a noncommutative FFT, and Lemma 6.9 describes the commutation relations among the generators of the subalgebra of invariants. In Theorem 6.16 we prove a quantum analogue of the skew (GLm , GLn ) duality [14, Theorem 4.1.1] of Howe. Finally, we note that an alternative approach to the questions addressed here may be to start with the study of endomorphisms of tensor powers (the first formulation), and translate versions of the first and second fundamental theorems to the other two settings. One might even hope to apply methods such as those in the appendix of [2] to move from type A to the other classical types. This strategy seems to provide a route towards a SFT in the quantum setting, and we intend to return to this theme in a future work. 2. Generalities on Module Algebras over Quantum Groups 2.1. Module algebras. For a simple complex Lie algebra g, we denote by Uq (g) the 1 associated quantum group over the field of rational functions K = C(q 2 ), where q is an


135

indeterminate. The algebra Uq (g) has a standard presentation with generators ei , f i , ki±1 (1 ≤ i ≤ rank(g)) and the usual relations (see, e.g. [15, Ch. 4], whose conventions we do not follow precisely, but which is a good general reference). It is well known that Uq (g) is a Hopf algebra. Write , and S for the co-multiplication, co-unit and the antipode respectively. We use the following convention in defining co-multiplication: (ei ) = ei ⊗ ki + 1 ⊗ ei , ( f i ) = f i ⊗ 1 + ki−1 ⊗ f i , (ki ) = ki ⊗ ki . If g is a semi-simple Lie algebra, or a direct sum of general linear algebras, we denote by Uq (g) the tensor product of the quantum groups of the components of g. The definition of the quantum general linear group Uq (gln ) will be given in Sect. 6.1. An important structural property of a quantum group is its braiding, namely, the existence of a universal R-matrix. We may think of R as an invertible element in some appropriate completion of Uq (g) ⊗ Uq (g), whose action on the tensor product of any two finite dimensional Uq -modules of type (1, . . . , 1) is well defined. It satisfies the following relations: R(x) = (x)R, ∀x ∈ Uq (g), ( ⊗ id)R = R13 R23 , (id ⊗ )R = R13 R12 ,

(2.1) (2.2)

where is the opposite co-multiplication. Here the subscripts of R13 etc. have the usual meaning as in [8]. It follows from (2.2) that R satisfies the celebrated Yang-Baxter equation R12 R13 R23 = R23 R13 R12 .

(2.3)

Remark 2.1. Note that if V1 , V2 are Uq -modules and P : V1 ⊗ V2 → V2 ⊗ V1 interchanges the factors of a tensor, then in terms of the operator Rˇ := P R, (2.1) reads ˇ ˇ and (2.3) becomes the braid relation Rˇ 1 Rˇ 2 Rˇ 1 = Rˇ 2 Rˇ 1 Rˇ 2 . R(x) = (x) R, Because of the Hopf algebra structure, the tensor product V1 ⊗K V2 of any two Uq (g)modules V1 and V2 becomes a Uq (g)-module. In view of 2.1, the R-matrix (which is Rˇ

invertible) defines an isomorphism of Uq (g)-modules, V1 ⊗ V2 −→V2 ⊗ V1 . We shall make extensive use of the notion of module algebras over a Hopf algebra [22, §4.1]. An associative algebra (A, μ) with multiplication μ and identity element 1 is a Uq (g)-module algebra if A is a Uq (g)-module such that the Uq (g)-action preserves the algebraic structure of A, that is, xμ(a ⊗ b) = μ(x(1) (a) ⊗ x(2) (b)), x(1) = (x)1, (x)

for all a, b ∈ A and x ∈ Uq (g). We shall sometimes use the term Uq (g)-algebra as a synonym for Uq (g)-module algebra. Let AUq (g) = {t ∈ A | x(t) = (x)t, ∀x ∈ Uq (g)} be the subspace of Uq (g)invariants in the module algebra A. The following result is well known. Lemma 2.2. Let A be a Uq (g)-algebra. Then the subspace AUq (g) of invariants is a subalgebra of A.

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Proof. Using Sweedler’s notation, write the co-multiplication in Uq (g) as (x) = Uq (g) , (x) x (1) ⊗ x (2) . Then for all t, t ∈ A x(1) (t)x(2) (t ) = (x(1) )(x(2) )tt = (x)tt , ∀x ∈ Uq (g). x(tt ) = Thus tt ∈ AUq (g) .

We remark that the above result is valid for arbitrary Hopf algebras. A module algebra A is called locally finite if Uq (g)a is finite dimensional for any a ∈ A, and is said to be of type (1, 1, . . . , 1) if A is of type (1, 1, . . . , 1) as a Uq (g)module. Recall that locally finite Uq (g)-modules are semi-simple (see e.g., [1]). The next result is well known, cf. [5] or [22]. Its significance in our context is that it defines what is meant by “generator”. Theorem 2.3. Let (A, μ A ) and (B, μ B ) be locally finite Uq (g)-module algebras of type (1, 1, . . . , 1). Then A ⊗ B acquires a Uq (g)-module algebra structure when endowed with the following multiplication: μ A,B = (μ A ⊗ μ B )(id A ⊗ P R ⊗ id B ),

(2.4)

where R is the universal R-matrix of Uq (g), and P : B ⊗ A −→ A⊗ B is the permutation map defined by a ⊗ b → b ⊗ a.

We omit the proof. Remark 2.4. Henceforth all Uq (g)-modules will be assumed to be locally finite of type (1, 1, . . . , 1), as will all Uq (g)-module algebras. The next two results give the basic properties of the tensor product construction above. The first asserts the associativity of the tensor product on the category of Uq (g)-algebras. Lemma 2.5. Let A, B and C be three Uq (g)-algebras. The module algebras (A ⊗ B ⊗ C, μ A⊗B,C ) and (A ⊗ B ⊗ C, μ A,B⊗C ) are equal.

Remark 2.6. It follows from Lemma 2.5 that given Uq (g)-algebras Ai (i = 1, 2, . . . , k), one can iterate the construction of Theorem 2.3 to obtain an unambiguous Uq (g)-module algebra structure on A1 ⊗ A2 ⊗ · · · ⊗ Ak . The next statement sets out the key properties of homomorphisms of Uq (g)algebras in the tensor category. Proposition 2.7. Let α : A → A and β : B → B be homomorphisms of Uq (g)algebras. Then (1) The kernel and image of α (and β) are Uq (g)-algebras. (2) The map α ⊗ β : A ⊗ B−→A ⊗ B is a homomorphism of Uq (g)-algebras. Proof. The first assertion is clear. To verify the second, we need to show that α ⊗ β respects the multiplication, and the Uq (g)-action. Take elements a1 , a2 ∈ A and b1 , b2 ∈ B, and write the R-matrix R as R = r ⊗ st , with rt , st ∈ Uq (g). Then α ⊗ β(a1 ⊗ b1 .a2 ⊗ b2 ) is equal to t t t α(a1 ).st α(a2 ) ⊗ rt β(b1 ).β(b2 ). It is straightforward to check that this is equal to the product α(a1 ) ⊗ β(b1 ).α(a2 ) ⊗ β(b2 ) in A ⊗ B , which shows that α ⊗ β respects algebra multiplication. Finally, if u ∈ Uq (g) and (u) = i u i ⊗u i , then for elements a ∈ A and b ∈ B, we have α ⊗ β(u.a ⊗ b) = i α(u i a) ⊗ β(u i b) = i u i α(a) ⊗ u i β(b) = u.α(a) ⊗ β(b). Hence α ⊗ β respects the Uq -action.


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2.2. Braided symmetric algebras. The first step in our construction of the noncommutative analogues of coordinate rings with which we shall work is to define, for each finite dimensional Uq (g)-module V , a braided symmetric algebra Sq (V ); these algebras have been considered previously, e.g. in [3, §2.1]. Write R V V ∈ G L(V ⊗ V, K) for the R-matrix associated to Uq (g). Let P : V ⊗ V −→ V ⊗ V , be the permutation map v ⊗ w → w ⊗ v, and define Rˇ := P R V,V . Then Rˇ ∈ EndUq (g) (V ⊗ V ), and ˇ Rˇ ⊗ idV ) = (idV ⊗ R)( ˇ Rˇ ⊗ idV )(idV ⊗ R). ˇ ( Rˇ ⊗ idV )(idV ⊗ R)( It follows from [18, Theorem 6.2] that Rˇ 2 acts on each isotypic component of V ⊗ V as a scalar of the form q m , where m ∈ Z. Hence Rˇ is semisimple on V ⊗ V , and has characteristic polynomial of the form (cf. [18, (6.10)]) k+ i=1

(+)

t − q χi

k−

(−)

t + q χi

,

j=1

where k± are positive integers and χi(+) and χi(−) are rational numbers (in fact ∈ 21 Z). Define submodules a2 and s2 of V ⊗ V by a2 =

k+ i=1

(+)

Rˇ − q χi

(V ⊗ V ), s2 =

k−

(−)

Rˇ + q χi

(V ⊗ V ).

(2.5)

j=1

Let T (V ) be the tensor algebra of V . This is a Uq (g)-module in the obvious way, and the Uq (g) action preserves the algebraic structure of T (V ); that is, T (V ) is a Uq (g) module algebra. Let Iq be the two-sided ideal of T (V ) generated by a2 ; since Rˇ commutes with Uq (g), this is stable under Uq (g). Define the braided symmetric algebra Sq (V ) of the Uq (g)-module V by Sq (V ) = T (V )/Iq . Let τ : T (V ) −→ Sq (V ) be the natural surjection. Since the ideal Iq is a Uq -subalgebra of T (V ), it follows from Proposition 2.7 that Sq (V ) is a module algebra over Uq (g). Note that the two-sided ideal a2 of T (V ) is a Z+ -graded submodule of the Z+ graded module T (V ). Hence the quotient Sq (V ) is Z+ -graded, and we shall denote by Sq (V )n its degree n homogeneous component. The symmetric algebra S(V ) of V over K is also Z+ -graded. Following [3], Sq (V ) is called a flat deformation of S(V ) if dim Sq (V )n = dim S(V )n for all n. This happens if and only if dim Sq (V )3 = dim S(V )3 by a result of Drinfeld. In this case, we shall also say that the Uq (g)-module V itself is flat. The results of [3,37] show that flat modules of quantum groups are extremely rare. Other than the natural modules for the quantum groups associated with the classical Lie algebras, there are hardly any modules which are flat. Even in the case of Uq (sl2 ), all the irreducible modules of dimensions greater than 3 are not flat! To develop an analogue of classical invariant theory for quantum groups, we require quantum analogues of polynomial algebras on direct sums of copies of a given module. However even when a module is flat, the direct sum of several copies of it is usually not flat. For example the natural module V of Uq (son ) is known to be flat [37] (see also below). Let ⊕m V denote the direct sum of m copies of V . It is easily verified that Sq (⊕m V ) is not flat if m > 1, and thus is not a suitable quantum analogue of S(⊕m V ).

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2.3. A quantum analogue of the symmetric algebra. To construct a quantum analogue of S(⊕k V ), we shall use Theorem 2.3. Given a module V for a quantum group Uq (g), we have the braided symmetric algebra Sq (V ) as above. Iterating the construction of Theorem 2.3, we obtain the Uq (g)-module algebra Am (V ) := (Sq (V )⊗m , μ S ).

(2.6)

Whenever there is no danger of confusion, we shall drop the V from the notation and simply write Am for Am (V ). This is a Zm + -graded Uq -algebra, with homogeneous component (Am )d of multi-degree d = (d1 , d2 , . . . , dm ) given by (Am )d = Sq (V )d1 ⊗ Sq (V )d2 ⊗ · · · ⊗ Sq (V )dm . If V is flat, then dim Sq (V )k = dim S(V )k for all k. In this case, dim

(Am )d = dim S(⊕m V )k , |d| =

|d|=k

m

di .

i=1

This proves the following statement. Theorem 2.8. If V is a flat Uq (g)-module, the Uq (g)-module algebra Am (V ) defined by (2.6) is a flat deformation (in the sense of [3]) of the symmetric algebra S(⊕m V ) of the direct sum of m-copies of V . Let {vs | s = 1, 2, . . . , dim V } be a basis for the Uq (g)-module V . Since the natural map τ : T (V ) −→ Sq (V ) maps T (V )1 = V injectively into Sq (V ), we also denote by vs the image of this basis element in Sq (V ). We introduce the following elements of Am : X ia := 1 · · ⊗ 1 ⊗va ⊗ 1 ⊗ · · · ⊗ 1 . ⊗ · i−1

(2.7)

m−i

Then for all i < j, X ja X ib =

2n

Rˇ a a,b b X ia X jb ,

(2.8)

a ,b =1

ˇ where Rˇ aa ,bb are the entries of the R-matrix. The same construction yields the Zm + -graded Uq (g)-module algebra Tm (V ) := (T (V )⊗m , μT ).

(2.9)

The next statement is an immediate consequence of Proposition 2.7. Lemma 2.9. The natural map τ ⊗m : Tm −→ Am is a surjection of Zm + -graded Uq (g)-algebras. (i)

(i)

It is useful to describe the map τ ⊗m explicitly. For a(i) = (a1 , . . . , adi ) (i ∈ [1, m]), let t[a(i) ] = va (i) ⊗ · · · ⊗ va (i) ∈ T (V )di . Write X [a(i) ] = X ia (i) · · · X ia (i) ∈ Sq (V )di . 1

di

1

di

Then τ ⊗m (t[a(1) ] · · · t[a(m) ]) = X [a(1) ] · · · X [a(m) ], where the right-hand side belongs to the homogeneous component of Am of multi-degree (d1 , d2 , . . . , dm ).


139

2.4. The quantum trace and invariants. It will be useful to keep in mind the structural context of the computations which appear below. Denote Uq (g) by Uq . For any two Uq -modules N , M, we have a canonical ∼ Uq -module isomorphism ξ : N ⊗ M ∗ −→HomK (M, N ), where ξ(n ⊗ φ) : m →

φ, by u. f (m) = mn, and the action of u ∈ Uq on the right side is given u . f (S(u )m) for f ∈ Hom (M, N ), where (u) = u (1) (2) K (u) (u) (1) ⊗ u (2) . The map ξ induces an isomorphism ∼

(N ⊗ M ∗ )Uq −→HomK (M, N )Uq = HomUq (M, N ).

(2.10)

Now taking M = V ∗ and N = V in (2.10), we obtain an isomorphism ξ0 : (V ⊗ ∗ Uq (V , V ). Write K = K 2ρ , where 2ρ is the sum of the positive roots

∼ V ∗∗ )Uq −→Hom

∼

(in the notation of [15]). There is an isomorphism of Uq -modules ε K : V −→V ∗∗ , where ε K (v) = ε(K v), and ε(w) is evaluation at w ∈ V . Composing these maps, we obtain an isomorphism f : (V ⊗ V )Uq −→(V ⊗ V ∗∗ )Uq −→HomUq (V ∗ , V ).

(2.11)

If V is simple, the dimension of the right side of (2.11) is 0 or 1; in this case any non-zero element T ∈ (V ⊗ V )Uq thus gives rise to an isomorphism of Uq -modules fT : V ∗ → V . Next, note that the map τ1 : V ∗ ⊗ V → K given by φ ⊗ v → φ, v is a homomorphism of Uq -modules. Applying this with V replaced by V ∗ , we see that the quantum trace τq : End(V ) ∼ = V ⊗ V ∗ → K defined by τq (ξ(v ⊗ φ)) = φ, K v is also a Uq -module map. Now given two Uq -modules M, N , we have an isomorphism of Uq -modules D : ∼ ∗ M ⊗ N ∗ −→(M ⊗ N )∗ , defined by D = κ ◦ R, where R is the R-matrix, and κ is defined by κ(φ ⊗ ψ), m ⊗ n = φ, m ψ, n. This is because for u ∈ Uq , ◦ S(u) = S ⊗ S ◦ (u), where is the opposite comultiplication to . It follows that τ1 ∈ ((V ∗ ⊗ V )∗ )Uq (V ⊗ V ∗ )Uq and similarly that τq ∈ ((V ⊗ ∗ V )∗ )Uq (V ∗ ⊗ V )Uq Invariants in terms of basis elements. We shall make explicit the above identifications in terms of bases of V and V ∗ . Let V be a Uq -module, and let v1 , . . . , vn be a basis of weight vectors, with wt(vi ) = ∗ λi ; we assume the λi are pairwise distinct. Let v1∗ , . . . , vn∗ be the dual basis of V ; then wt(vi∗ ) = −λi . Under the isomorphism (2.10), idV corresponds to γ := i vi ⊗ vi∗ ∈ V ⊗ V ∗ . Since the former is Uq -invariant, we have vi ⊗ vi∗ ∈ (V ⊗ V ∗ )Uq. (2.12) γ = i

Now the map ε K : V → V ∗∗ takes vr to q (2ρ,λr ) vr∗∗ , where (vr∗∗ ) is the basis of V ∗∗ dual to the basis (vr∗ ) of V ∗ . Applying ε K ⊗ idV ∗ to γ and reinterpreting in terms of V (rather than V ∗ ), we see (taking into account that vr∗ has weight −λr ) that γq = q −(2ρ,λi ) vi∗ ⊗ vi ∈ (V ∗ ⊗ V )Uq. (2.13) i

If V is simple then (V ⊗V ∗ )Uq (V ∗ ⊗V )Uq is one-dimensional. Hence the element γq of (2.13) is a non-zero scalar multiple of the quantum trace τq .

140


Now assume that V is simple and self dual. Then whenever λr is a weight of V , so is −λr , and we write v−r for the basis element with weight −λr . Since any element T ∈ (V ⊗ V )Uq has weight zero, it must be of the form T =

n

cr vr ⊗ v−r ,

(2.14)

r =1

for elements cr ∈ K. The corresponding Uq -isomorphism f T : V ∗ → V (2.11) is readily seen to satisfy f T (vi∗ ) = c−i q (2ρ,λi ) v−i .

(2.15)

Since f T is an isomorphism, it follows that ci = 0 for all i, and the inverse f T−1 : V → V ∗ is described similarly. We may now apply idV ⊗ f T−1 and f T−1 ⊗ idV to T , to obtain invariant elements of V ⊗ V ∗ and V ∗ ⊗ V respectively. The latter yields the element γq of (2.13), while the −1 −(2ρ,λi ) former yields i ci c−i q vi ⊗ vi∗ . Comparing with γ (2.13) shows that −1 −(2ρ,λi ) q is independent of i. ci c−i

(2.16)

It is evident that if zero is a weight of V , then the implied constant is 1. In the next three sections, we shall study the algebras Am associated with the natural modules of the quantum orthogonal and symplectic groups. 3. Invariant Theory of the Quantum Even Orthogonal Groups Uq (so2n ) Let i (i = 1, 2, . . . , n) be an orthonormal basis of the weight lattice of the Lie algebra so2n ; a set of simple roots may then be taken to be i − i+1 1 ≤ i < n together with n−1 + n . In this section, Uq will denote the quantum group Uq (so2n ). 3.1. The natural module for Uq (so2n ). We realise the group S O2n (C) as the subgroup 0 In , where of S L 2n (C) preserving the bilinear form defined by the matrix J := In 0 In is the n × n identity matrix. Then g = so2n is the subalgebra of sl2n (C) consisting of matrices satisfying X t = −J X J . Accordingly, there is a Cartan subalgebra consisting of diagonal matrices. Now the natural representation of so2n is minuscule. Thus it lifts to the natural representation of the quantum group Uq in such a way that matrices for the Chevalley generators remain the same (see, e.g. [34]); this is exploited in the description below. The natural module V for Uq has highest weight 1 and weights ±i , i = 1, 2, . . . , n. It therefore has a basis {va | a ∈ [1, n]∪[−n, −1]}, where [1, n] = {1, 2, . . . , n}, [−n, −1] = {−n, −(n−1), . . . , −1}, and va has weight sgn(a)|a| , where sgn(a) = a/|a|. Let E ab be the matrix units in End(V ) relative to this basis, defined by E ab vc = δbc va .

(3.1)


141

Then we have the following explicit formulae for the natural representation π : Uq −→ End(V ) of Uq relative to the above basis: π( f i ) = E i+1,i − E −i,−i−1 , i < n, π(ei ) = E i,i+1 − E −i−1,−i , π(en ) = E n−1,−n − E n,−n+1 , π( f n ) = E −n,n−1 − E −n+1,n , π(ki ) = 1 + (q − 1)(E ii + E −i−1,−i−1 ) + (q −1 − 1)(E i+1,i+1 + E −i,−i ), i < n, π(kn ) = 1 + (q − 1)(E n−1,n−1 + E nn ) + (q −1 − 1)(E −n+1,−n+1 + E −n,−n ). Note that the subalgebra of Uq generated by ei , f i , ki±1 (1 ≤ i < n) is isomorphic to Uq (sln ), and the vectors vi (1 ≤ i ≤ n) span its natural module. The v−i (1 ≤ i ≤ n) span its dual. The tensor product V ⊗ V is the direct sum of three distinct irreducible submodules L 0 , L 21 and L 1 +2 with highest weights 0, 21 and 1 + 2 respectively. The following explicit bases for these irreducible summands will be useful later. (1) A basis for L 0 (note that this is an element of the form of T in (2.14)): n q n−i vi ⊗ v−i + q i−n v−i ⊗ vi .

(3.2)

i=1

(2) A basis for L 21 : vi ⊗ vi , v−i ⊗ v−i , 1 ≤ i ≤ n, vi ⊗ v j + qv j ⊗ vi , v− j ⊗ v−i + qv−i ⊗ v− j , i < j ≤ n, vi ⊗ v− j + qv− j ⊗ vi , i = j,

(3.3)

q −1 vi ⊗ v−i + qv−i ⊗ vi − (vi+1 ⊗ v−i−1 + v−i−1 ⊗ vi+1 ), i ≤ n − 1. (3) A basis for L 1 +2 : vi ⊗ v j − q −1 v j ⊗ vi , v− j ⊗ v−i − q −1 v−i ⊗ v− j , i < j ≤ n, vi ⊗ v− j − q −1 v− j ⊗ vi , i = j, vi ⊗ v−i − v−i ⊗ vi − (qvi+1 ⊗ v−i−1 − q −1 v−i−1 ⊗ vi+1 ), i < n − 1, (3.4) vn−1 ⊗ v1−n − v1−n ⊗ vn−1 − (qvn ⊗ v−n − q −1 v−n ⊗ vn ), vn−1 ⊗ v1−n − v1−n ⊗ vn−1 + (q −1 vn ⊗ v−n − qv−n ⊗ vn ). Let Ps , Pa and P0 be the idempotent projections mapping V ⊗ V onto the irreducible submodules with highest weights 21 , 1 + 2 and 0 respectively. Then the R-matrix of Uq acting on V ⊗ V is given by Rˇ = q Ps − q −1 Pa + q 1−2n P0 .

(3.5)

In Sect. 3.3, we shall need the R-matrix in the following slightly more general situation for the proof of Lemma 3.7. Let V1 and V2 be two isomorphic copies of the natural (1,2) module. Denote by Pμ the idempotent projection from V1 ⊗ V2 onto its irreducible submodule with highest weight μ, where μ = 21 , 1 + 2 or 0. Similarly, define the idempotents Pμ(2,1) in End(V2 ⊗ V1 ). Then the R-matrix Rˇ : V1 ⊗ V2 −→ V2 ⊗ V1

(3.6)

142


is the (unique) Uq -linear map satisfying the following relations: (1,2) (2,1) ˇ (1,2) = q P2 , R P1 +1 = −q −1 P(2,1) , Rˇ P0(1,2) = q 1−2n P0(2,1) . Rˇ P2 1 +2 1 1

(3.7)

(α)

In particular, if we denote by {vb | b ∈ [−n, −1] ∪ [1, n]} the standard basis for n (β) (β) (α) (α) (α,β) n−i i−n Vα (α = 1, 2), and let T = i=1 q vi ⊗ v−i + q v−i ⊗ vi , we have ˇ (1,2) = q 1−2n T (2,1) . RT

(3.8)

3.2. The braided symmetric algebra of the natural Uq (so2n )-module. Let T (V ) be the

⊗k . The submodule tensor algebra of the natural Uq -module V . Then T (V ) = ∞ k=0 V ∼ Pa (V ⊗ V ) = L 1 +2 generates a graded two-sided ideal Iq of T (V ) which is also graded since it is generated by homogeneous elements. Hence the braided symmetric algebra Sq (V ) = T (V )/Iq of V inherits a Z+ -grading from T (V ). It is evident, given the basis of L 1 +2 constructed above, that the following is a presentation of Sq (V ). Here we abuse notation by writing v j for the image in Sq (V ) of v j ∈ T (V ). Lemma 3.1. The braided symmetric algebra Sq (V ) of the natural Uq (so2n )-module V is generated by {va | a ∈ [−n, −1] ∪ [1, n]} subject to the following relations: vi v j − q −1 v j vi = 0, 1 ≤ i < j ≤ n, v− j v−i − q −1 v−i v− j = 0, 1 ≤ i < j ≤ n, vi v− j − q −1 v− j vi = 0, i = j, vi v−i − v−i vi = qvi+1 v−i−1 − q vn v−n − v−n vn = 0.

(3.9) −1

v−i−1 vi+1 , 1 ≤ i ≤ n − 1,

Note that the last two sets of relations may be written as (+)

v−i vi = vi v−i − (q − q −1 )q i+1−n φi+1 , i ∈ [1, n],

(3.10)

where φi(+) is the quadratic element nk=i q n−k vk v−k . Set φi(−) = nk=i q k−n v−k vk . An easy computation using (3.10) shows that φi(+) = q 2n−2 φi(−) for all i. Using results of [37] or by direct calculation, one sees that the ordered monomials in vi and v−i (i ∈ [1, n]) form a basis of Sq (V ). That is, Sq (V ) is a flat deformation of the symmetric algebra of V in the sense of [3]. In summary, we have Theorem 3.2. (1) The braided symmetric algebra Sq (V ) of V is a Z+ -graded module algebra over Uq (so2n ). (2) The ordered monomials in vi , v−i (i ∈ [1, n]) form a basis of Sq (V ). (+)

(−)

Let := φ1 + φ1 . Then (+)

= q 1−n (q n−1 + q 1−n )φ1 . Proposition 3.3. (1) We have ∈ Sq (V )Uq (so2n ) . (2) The element belongs to the centre of Sq (V ).

(3.11)


143

Proof. The first statement is immediate because is the image in Sq (V ) of the basis element in (3.2) of L 0 := (V ⊗ V )Uq . (+) In view of (3.11), to prove (2), it clearly suffices to show that φ1 is central. Consider v j φ1(+) : we have (+)

v j φ1 = =

j−1

(+)

q n−i vi v−i v j + q n− j v j v j v− j + q −2 φ j+1 v j

i=1 (+) φ1 v j

(+)

+ (q −2 − 1)φ j+1 v j + q n− j v j (v j v− j − v− j v j ).

By (3.10), we have (+)

(+)

q n− j v j (v j v− j − v− j v j ) = (q 2 − 1)v j φ j+1 = −(q −2 − 1)φ j+1 v j. (+)

It follows that v j commutes with φ1 for all j ≥ 0. Similarly one shows that v− j commutes with φ1(+) for all j, and the proposition follows.

Let Sq (V )Uq (so2n ) be the space of Uq -invariants in Sq (V ); these form a subalgebra by Lemma 2.2. Proposition 3.4. The subalgebra Sq (V )Uq (so2n ) of invariants is generated by and is isomorphic to the polynomial algebra C[]. Proof. Denote by Sq (V )k the homogeneous subspace of degree k in Sq (V ). Let Sq (V )k be the Uq -submodule of Sq (V )k generated by (v1 )k , that is Sq (V )k = Uq · (v1 )k . Then Sq (V )k is isomorphic to the irreducible Uq -module with highest weight k1 , and thus 2n + k − 1 2n + k − 3 has dimension − . Taking into account the weights of Uq k k−2 occurring in Sq (V )k−2 , one sees that Sq (V )k ∩ Sq (V )k−2 = 0. Now dim Sq (V )k = 2n + k − 1 = dim Sq (V )k + dim Sq (V )k−2 , and recalling that is invariant, we have k the Uq -module decomposition Sq (V )k = Sq (V )k ⊕ Sq (V )k−2 . U

U

It follows that Sq (V )k q = Sq (V )k q ⊕ (Sq (V )k−2 )Uq . Since Sq (V )k is a nontrivial U

irreducible Uq -module for all k > 0, Sq (V )k q = 0, and since is invariant,

Sq (V )k

Uq

U = Sq (V )k−2 q .

U (so ) k It follows that Sq (V )k q 2n is spanned by 2 for even k and is 0 for odd k.

The results of this section may be applied to the construction of a quantum sphere Sq2n−1 with manifest quantum orthogonal group symmetry, that is, with an action of Uq (so2n ). Since is a central element in Sq (V ) by Proposition 3.3 (2), the left ideal generated by − 1 coincides with the right, and hence is a two-sided ideal, which we denote by − 1. Then the quantum sphere Sq2n−1 is defined by Sq2n−1 = Sq (V )/ − 1.

(3.12)

144


Remark 3.5. Quantum spheres have been defined in other contexts. In [24], C ∗ algebras are used to define a quantum analogue of the 2-sphere, while in [23] there is a holomorphic analogue of a quantum homogeneous space. In neither case is the Uq (g) symmetry evident, as in our definition. Moreover odd dimensional quantum spheres have no realisation as quantum homogeneous spaces. From the proof of Proposition 3.4, we have the following result, analogous to its classical (q = 1) counterpart. Lemma 3.6. The quantum sphere Sq2n−1 is a Uq (so2n )-module algebra whose decom position as Uq (so2n )-module is Sq2n−1 = ∞ k=0 L k1 . 3.3. A Uq (so2n ) module algebra: Sq (V )⊗m with twisted multiplication. Let Tm = (T (V )⊗m , μT ) be the Uq -module algebra with Uq = Uq (so2n ), where multiplication μT is defined by iterating (2.4). We shall construct the quantum analogue of the coordinate ring of ⊕m V as a module algebra Am = (Sq (V )⊗m , μ S ) over Uq by repeatedly using (2.4). It will be convenient to relabel the standard basis elements of V as va with a ∈ [1, 2n], where va is identified with va−2n−1 if a > n. The following result describes the algebraic structure of Am . For i ∈ [1, m] and a ∈ [1, 2n], let X ia be the image in Am of 1 ⊗ · · · ⊗ 1 ⊗ va ⊗ 1 ⊗ · · · ⊗ 1, where va is the i th factor. Lemma 3.7. The Uq (so2n )-module algebra Am is generated by X ia with i ∈ [1, m] and a ∈ [1, 2n], subject to the following relations: (1) for fixed i, the elements X ia obey the relations (3.9) with va = X ia ; (2) for i < j in [1, m] and a, b ∈ [1, 2n]: X ja X ia = q X ia X ja , ∀a, X jb X ia = X ia X jb + (q − q −1 )X ib X ja , a < b = 2n + 1 − a, X ja X ib = X ib X ja , a < b = 2n + 1 − a, (i, j)

X jt X i,2n+1−t = q X i,2n+1−t X jt − (q − q −1 )q n−t ψt −1

,

X j,2n+1−t X it = q X it X j,2n+1−t − (q − q )X i,2n+1−t X jt (i, j) + (q − q −1 )q t−n ψ¯ t+1 − (i, j) , t ∈ [1, n], where (for all i, j ∈ [1, m]) (i, j) ψt

=

t

q k−n X i,2n+1−k X jk ,

k=1 (i, j) ψ¯ t =

(i, j)

=

t

q n−k X ik X j,2n+1−k ,

k=1 (i, j) ψn

(i, j)

+ ψ¯ n

.

(3.13)

(3.14)


145

Proof. For fixed i, the elements X ia (i ∈ [1, 2n]) generate a subalgebra isomorphic to Sq (V ) in Am , whence (1). Part (2) is obtained from (2.8) by straightforward but tedious calculation. We indicate the main steps. First, using (3.7), we rewrite relation (2.8) for the present case in the following form (for all i < j): X ja X ia = q X ia X ja , ∀a, X ja X ib + q X jb X ia = q X ia X jb + q X ib X ja , a < b = 2n + 1 − a, q −1 X jt X i,2n+1−t + q X j,2n+1−t X it − X j,t+1 X i,2n−t + X j,2n−t X i,t = q q −1 X it X j,2n+1−t + q X i,2n+1−t X jt

(3.15)

− q X i,t+1 X j,2n−t + X i,2n−t X j,t , 1 ≤ t < n; X ja X ib − q −1 X jb X ia = −q −1 X ia X jb − q −1 X ib X ja , a < b = 2n + 1 − a, X jt X i,2n+1−t − X j,2n+1−t X it − q X j,t+1 X i,2n−t − q −1 X j,2n−t X i,t = −q −1 X it X j,2n+1−t − X i,2n+1−t X jt + q −1 q X i,t+1 X j,2n−t − q −1 X i,2n−t X j,t , 1 ≤ t < n,

(3.16)

X jn X i,n+1 − X j,n+1 X in = −q −1 X in X j,n+1 − X i,n+1 X jn ; ( j,i) = q 1−2n (i, j) .

(3.17)

The first relation of (3.13) is just the first relation of (3.15), and the other relations of (3.13) are obtained by combining the second relation of (3.15) with the first relation in (3.16) (that is, the relations with a < b = 2n + 1 − a). The relations (3.14) are obtained from the third relation of (3.15), the second and third relations of (3.16) and the relation (3.17) by a rather lengthy sequence of routine manipulations. The fact that these relations suffice is a consequence of the flat nature of Sq (V ), which implies that there is a linear isomorphism Am −→S(V )⊗m .

The elements (i, j) will play an important role in the study of the subalgebra of Uq -invariants in Am , and we now study their properties. Lemma 3.8. (1) For all i, j ∈ [1, m], the elements (i, j) are Uq -invariant.

146


(2) The elements (i, j) satisfy (3.17) as well as the following relations: X ka (i,i) − (i,i) X ka = 0, for all i, k,

X ka (i, j) − (i, j) X ka = 0, k > i, j or k < i, j, X ka (i, j) − (i, j) X ka = (q − q −1 ) X ia (k, j) − (i,k) X ja , i < k < j, (3.18) (i,i) (i, j) X ia − q −1 X ia (i, j) = (q − q −1 )ψ¯ n X ja , i < j, ( j, j)

X ja (i, j) − q −1 (i, j) X ja = (q − q −1 )X ia ψ¯ n

, i < j.

Proof. Observe that the natural map T (V ) → Sq (V ) restricts to an injection of V into Sq (V ). Hence for i < j, (i, j) may be thought of as the basis element (3.2) of (V ⊗ V )Uq , where the two copies of V are in the i th and j th factors of ⊗m Sq (V ). Taking Proposition 3.3 into account, it follows that (i, j) is invariant if i ≤ j. By Eq. (3.17), (1) follows. The k = i case of the first relation of (3.18) follows from Proposition 3.3. Write the universal R-matrix of Uq as R = t αt ⊗ βt . If k > i, j, we have βt ( (i, j) )αt (X ka ). X ka (i, j) = t

Since (i, j) is Uq -invariant by part (1), the right side is equal to (βt )( (i, j) )αt (X ka ) = (i, j) X ka , t

proving the second relation of (3.18) for k > i, j. The case k < i, j is similar. Taking i = j, we obtain the case k = i of the first relation of (3.18). To prove the third relation of (3.18), it suffices to consider the case i = 1, k = 2 and j = 3. Since Sq (V )1 ∼ = T (V )1 ∼ = V , we have canonical Uq -module isomorphisms V ⊗ V ⊗ V −→ (T3 (V ))(1,1,1) −→ (A3 )(1,1,1) , where the second map is the restriction of τ ⊗3 to (T3 (V ))(1,1,1) . Denote the first map by ι(1,1,1) . Our strategy is to deduce the third relation of (3.18) from appropriate relations in V ⊗ V ⊗ V . This will be done using the diagrammatical method of Reshetikhin-Turaev (see [27,28]), which is equivalent to working in the B M W -algebra, to describe homomorphisms between tensor powers of V . Recall that their functor sends tangle diagrams to Uq -module homomorphisms. In our case, we shall colour all the components of any tangle diagram with the module V , so that the images of tangle diagrams under the Reshetikhin-Turaev functor are Uq -maps between tensor powers of V . Because the module V is self dual, there is no need to orient the tangle diagrams. We shall identify tangle diagrams with the corresponding Uq -module homomorphisms; the relations we seek will arise from relations among diagrams whose images lie in V ⊗3 . Write the basis element (3.2) of L 0 ⊂ V ⊗ V defined in Sect. 3.1 as : K −→ a,b C ab va ⊗ vb with Cab ∈ K; then in terms of diagrams, we have V ⊗ V, 1 → a,b Cab va ⊗ vb . Furthermore, we have the following skein relation: . (3.19) − = (q − q −1 ) −


147

Let us denote D+ := D0 := Then D0 (va ) = va ⊗ D+ (va ) =

: V −→ V ⊗3 ,

D− :=

: V −→ V ⊗3 ,

: V −→ V ⊗3 ,

D0 :=

: V −→ V ⊗3 .

D0 (va ) =

Cbd vb ⊗ vd ,

b,d

Cbd vb ⊗ vd ⊗ va ,

b,d

ˇ a ⊗ vb ) ⊗ vd , Cbd R(v

D− (va ) =

b,d

ˇ d ⊗ va ), Cbd vb ⊗ R(v

b,d

from which we obtain τ ⊗3 ◦ ι(1,1,1) ◦ D+ (va ) = X 2a (1,3) ,

τ ⊗3 ◦ ι(1,1,1) ◦ D− (va ) = (1,3) X 2a ,

τ ⊗3 ◦ ι(1,1,1) ◦ D0 (va ) = X 1a (2,3) ,

τ ⊗3 ◦ ι(1,1,1) ◦ D0 (va ) = (1,2) X 3a .

Evaluating both sides of (3.19) at va , then applying the map τ ⊗3 ◦ ι(1,1,1) to the resulting elements of V ⊗3 , we obtain the third relation of (3.18) for i = 1, k = 2 and j = 3. The fourth and fifth relations of (3.18) can be proved in much the same way, and we shall consider the fourth relation only. We may assume that i = 1 and j = 2. Note that the map τ ⊗2 : T2 (V ) −→ A2 restricts to an isomorphism (Ps + P0 )T (V )2 ⊗ T (V )1 ∼ = Sq (V )2 ⊗ Sq (V )1 . Denote by ι(2,1) the canonical Uq -isomorphism from (V ⊗ V ) ⊗ V to T (V )2 ⊗ T (V )1 . Let F := ((Ps + P0 ) ⊗ id)D0 = (Ps ⊗ id)D0 +

1 D , dimq V 0

B := ((Ps + P0 ) ⊗ id)D− = q −1 (Ps ⊗ id)D0 +

q 2n−1 D , dimq V 0

where dimq V = [n]q (q n−1 + q 1−n ) is the quantum dimension of V . Then F − q B = −q n

q n − q −n D0 : V ⊗3 −→ V ⊗3 . dimq V

Note that τ ⊗2 ◦ ι(2,1) ◦ F(va ) = X 1a (1,2) ,

τ ⊗2 ◦ ι(2,1) ◦ B(va ) = (1,2) X 1a ,

and τ ⊗2 ◦ ι(2,1) ◦ D0 (va ) = (1,1) X 2a . Thus for i = 1 and j = 2, we have q (i, j) X ia − X ia (i, j) = q n

q n − q −n (i,i) X ja . dimq V

This leads to the desired result taking into account that (i,i) =

q 1−n dimq V (i,i) ψ¯ n , [n]q

which is implied by (3.11). This completes the proof of the lemma.

(3.20)

148


The following relations are easy consequences of part (2) of Lemma 3.8: Lemma 3.9. (i,i) ( j,k) − ( j,k) (i,i) = 0, for all i, j, k, (i,i)

(i,k) (i, j) − q −1 (i, j) (i,k) = (q − q −1 )ψ¯ n

( j,k) , k = i, j; i < j,

( j, j) ( j,k) (i, j) − q −1 (i, j) ( j,k) = (q − q −1 )ψ¯ n (i,k) , k = i, j; i < j,

(3.21)

(i, j) (k,l) − (k,l) (i, j) = 0, k < i < j < l, (i, j) (k,l) − (k,l) (i, j) = (q − q −1 ) (i,k) ( j,l) − (i,l) (k, j) , i < k < j i, j or k < i, j, X ka (i, j) − (i, j) X ka = (q − q −1 ) X ia (k, j) − (i,k) X ja , i < k < j, (4.6) (i, j) X ia − q −1 (i, j) X ia = (q − q −1 )ϕ (i,i) X ja , i < j, X ja (i, j) − q −1 (i, j) X ja = (q − q −1 )ϕ ( j, j) X ia , i < j, where ϕ (i,i) :=

q 2n − q −1 (i,i) 1 − q −1 ¯ n(i,i) + = ψ (X i,n+1 )2 . q − q −1 q − q −1

It follows from part (2) of Lemma 4.5 that the elements (i, j) also satisfy the following relations: Lemma 4.6. (i,i) ( j,k) − ( j,k) (i,i) = 0, for all i, j, k, (i,k) (i, j) − q −1 (i, j) (i,k) = (q − q −1 )ϕ (i,i) ( j,k) , k = i, j; i < j, ( j,k) (i, j) − q −1 (i, j) ( j,k) = (q − q −1 )ϕ (i,i) (i,k) , k = i, j; i < j,

(4.7)

(i, j) (k,l) − (k,l) (i, j) = 0, k < i < j < l, (i, j) (k,l) − (k,l) (i, j) = (q − q −1 ) (i,k) ( j,l) − (i,l) (k, j) , i < k < j l, then (i, j) (k,l) = (k,l) (i, j) . < k and j < l, then (k,l) (i, j) − (i, j) (k,l) = (q − q −1 ) (i,l) (k, j) + (i,k) ( j,l) .

The following result is the quantum analogue of the first fundamental theorem of invariant theory for Uq (sp2n ). Its proof is a straightforward adaptation of that of Theorem 3.11. U (sp )

Theorem 5.6. The subalgebra Amq 2n of Uq (sp2n )-invariants in Am is generated by the elements (i, j) (i < j) and the identity. 6. Invariant Theory for the Quantum General Linear Group In this section, we study the invariant theory of the quantum general linear group. Our first requirement is a quantum analogue of S(⊕k V ⊕l V ∗ ), where V is the natural gln -module and V ∗ is its dual. There are various ways this could be approached, all of which turn out to be equivalent. One way is to follow the construction of Sect. 2 to first construct the braided symmetric algebras Sq (V ) and Sq (V ∗ ) over Uq (gln ), and then to form the module algebras Sq (V )⊗k and Sq (V ∗ )⊗l . From these, one constructs the module algebra Sq (V )⊗k ⊗ Sq (V ∗ )⊗l using Theorem 2.3, and this is a quantum analogue of the symmetric algebra over S(⊕k V ⊕l V ∗ ). Another possibility is to directly construct quantum analogues of the symmetric algebras of ⊕k V and of ⊕l V ∗ . In this case, we need to consider the Lie algebras gln ×glk and gln × gll . Let g = gln × glk , and let W = V ⊗ V (k) , where V and V (k) are respectively the natural modules for Uq (gln ) and Uq (glk ). Then W is an irreducible Uq (g)-module, and following §2 we may form the braided symmetric algebra Sq (W ). It is known [37] that W is one of the extremely rare modules such that Sq (W ) is flat. That is, Sq (W ) is (linearly) isomorphic to Sq (V )⊗k . We may similarly consider the irreducible Uq (g)module W = V ∗ ⊗ V (l) , where now g = gln ⊗ gll , and construct Sq (W ). Then using Theorem 2.3 we obtain the module algebra Sq (W ) ⊗ Sq (W ) over Uq (gln ). The above two constructions are essentially equivalent, and are equivalent to a third construction, which is conceptually simpler, and is the one we shall use in the present work. We now discuss this third construction.


159

6.1. Algebra of functions on the quantum general linear group. The quantum general linear group Uq (gl N ) is generated by K b±1 (1 ≤ b ≤ N ), ea and f a (1 ≤ a < N ). The defining relations are essentially the same as those for Uq (sl N ) except for those involving the elements K b±1 , which are given by K b ea K b−1 = (1 + (q − 1)δba + (q −1 − 1)δb,a+1 )ea , K b f a K b−1 = (1 + (q − 1)δb,a+1 + (q −1 − 1)δba ) f a , ea f b − f b ea = δab

−1 K a K a+1 − K a+1 K a−1 . q − q −1

Let {va | 1 ≤ a ≤ N } be the standard basis of weight vectors of the natural module V (N ) for Uq (gl N ), with wt(vi ) = i , and let π : Uq (gl N ) −→ End(V (N ) ) be the usual representation of Uq (gl N ) relative to this basis. Define elements tab (1 ≤ a, b ≤ N ) of the dual vector space Uq (gl N )∗ of Uq (gl N ) by ta b , x = π(x)a b (the (a, b) entry of the matrix π(x)), ∀x ∈ Uq (gl N ), and call these the cooordinate functions of the representation π . Now the dual space of Uq (gl N ) has a natural algebraic structure with multiplication defined as follows. For any t, t ∈ Uq (gl N )∗ , t t , x =

t (x) ⊗ t , x (1) ⊗ x (2) , for all x ∈ Uq (gl N ). Following an idea of [35], we consider the subalgebra M N of Uq (gl N )∗ generated by the elements tab (a, b ∈ [1, N ]). Then M N has a presentation with generators tab (a, b ∈ [1, N ]) and relations Raa ,bb ta c tb d = tbb taa Ra c,b d , (6.1) a ,b

a ,b

where the Raa ,bb are the entries of the R-matrix, which in our case has the form R = 1 ⊗ 1 + (q − 1)

N

E aa ⊗ E aa + (q − q −1 )

a=1

E ab ⊗ E ba .

(6.2)

a s,

R

t

(tab ) =

tab , b ≤ t, qtab , b > t,

we have ta1 b1 ta2 b2 · · · tak bk ∈ Ms,t if and only if 1 ≤ ai ≤ s and 1 ≤ b ≤ t for all i. Such elements obviously form a basis of Ms,t . This shows that the subalgebra Ms,t is generated by the elements ti j with 1 ≤ i ≤ s and 1 ≤ j ≤ t. The relations (6.6) follow from (6.1) and the explicit form of the R-matrix (6.2).

Remark 6.3. Let V (t) be the natural module for Uq (glt ) and let Sq (V (t) ) be the braided symmetric algebra over V (t) . Consider the algebra Sq (V (t) )⊗s discussed above. The relations (6.6) show that this algebra is isomorphic to Ms,t , as was pointed out in [5]. The next result is the quantum Howe duality (cf. [36]) applied to the subalgebra Ms,t of M N . Theorem 6.4. (1) The subalgebra Ms,t of M N is a L(Uq (gls )) ⊗ R(Uq (glt ))-module algebra, and Ms,t ∼ =

λ

(s) ∗

Lλ

(t)

⊗ Lλ ,

(6.7)

where the summation is over all partitions of length ≤ min(s, t). ˜ q (gls )) ⊗ R(Uq (glt )) and (2) The subalgebra Ms,t is a module algebra over L(U Ms,t ∼ =

λ

(s)

(t)

Lλ ⊗ Lλ

˜ q (gls )) ⊗ R(Uq (glt ))-module, as L(U

(6.8)

where the range of the summation in λ is the same as in part (1). Proof. Consider the following assertions, which we shall prove shortly. For all partitions λ,

(N ) Lλ

t

(N ) ∗ Lλ

s

L (t) λ , if the length of λ is ≤ t, 0, otherwise, ∗ (s) Lλ , if the length of λ is ≤ s, = 0, otherwise. =

(6.9)

Granted (6.9), part (1) of the theorem follows immediately. Part (2) then follows from the second statement in part (1). We turn to the proof of Eq. (6.9), for which we provide details, since similar arguments will be used later. Consider the subalgebras Uq (lt ) and Uq (pt ) of Uq (gl N ), defined as follows. • Uq (lt ) is generated by K a±1 (for all a) and eb , f b (b = t); • Uq (pt ) is generated by the elements of Uq (lt ) and et .

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G. I. Lehrer, H. Zhang, R. B. Zhang (N )

The irreducible Uq (gl N )-module L λ with highest weight λ = (λ1 , λ2 , . . . , λ N ) may (N ),0 be the irreducible Uq (pt )-module with highest be constructed as follows. Let L λ ),0 . weight λ, and construct the generalised Verma module Vλ = Uq (gl N ) ⊗Uq (pt ) L (N λ (N )

Then the quotient of Vλ by its unique maximal submodule is isomorphic to L λ . (N ),0 If μ is a weight of the generalised Verma module but not of L λ , then it follows from the definition of Vλ that μ = λ − i≤t = − < . Define X jk θ () = (−1)

i

k

t=1 at

a=1

+ (q − q

−1

q−

i

a=1 ak

[θ (< )(X jk θ (> ))

)θ (< + 1n ik − 1il )(X jl θ (> ))].

Note that the degree of θ (> ) is smaller than that of θ (), so that by induction, the action of X jk and X jl on it are already defined. Therefore X jk θ (> ) and X jl θ (> ) are well defined elements in K[θ ]. The construction guarantees that the defining relations of q (V ) are satisfied. But clearly X ()(1) = θ (), and the linear independence of the elements X () follows from that of the θ () in the Grassmann algebra K[θ ].

The following result is a generalisation to the quantum group setting of [14, Theorem 4.1.1] known as ‘skew (G L m , G L n ) duality’ in the terminology of Howe. Theorem 6.16. Let V (m) and V (n) be the natural modules for Uq (glm ) and Uq (gln ) respectively, and set V = V (m) ⊗ V (n) . Then as a Uq (glm ) ⊗ Uq (gln )-module, the braided exterior algebra q (V ) decomposes into a multiplicity free direct sum of irreducibles as follows. (m) (n) L λ ⊗ L λ , q (V ) = λ

λ

where denotes the conjugate of the partition λ, and the sum is over all partitions λ = (λ1 , λ2 , . . . , λm ) such that λ1 ≤ n. The proof of the theorem will make use of the following result. Lemma 6.17. For any partition λ = (λ1 , λ2 , . . . , λm ) with λ1 ≤ n, let λ := X 11 . . . X 1λ1 X 21 . . . X 2λ2 . . . X m1 . . . X mλm ∈ q (V ). Then (1) The element λ is a highest weight vector with respect to the actions of Uq (glm ) and of Uq (gln ). (2) The Uq (glm ) weight of λ is λ, while the Uq (gln ) weight is λ .

172


Proof. It is understood that if λi = 0, then λ contains no X it as a factor. This also implies that there is no X jt factor in λ for all j ≥ i. If λi+1 = 0, then the Chevalley generator ei ∈ Uq (glm ) acts on λ as 0. Assume λi+1 > 0. Then ei λ is a linear combination of terms of the form ν X i1 . . . X iλi X i+1,1 . . . X i+1,t−1 X it X i+1,t+1 . . . X i+1,λi+1 . . . X m1 . . . X mλm , where ν = (λ1 , . . . , λi−1 , 0, . . . , 0) and t ≤ λi+1 . We prove that this is equal to zero by proving the more general result that for all t such that λi ≥ t > s, X i1 X i2 . . . X iλi X i+1,1 X i+1,2 . . . X i+1,s X i,t = 0.

(6.24)

If s = 0, then using the second relation of (6.21) we can shift X it to the left to arrive at the expression (−q)λi −t X i1 X i2 . . . X it X it X i,t+1 . . . X iλi , which is zero by the first relation of (6.21). We now use induction on s. By (6.20), we may write X i+1,s X i,t = −X i,t X i+1,s + (q − q −1 )X is X i+1,t . Thus X i1 X i2 . . . X iλi X i+1,1 X i+1,2 . . . X i+1,s X i,t = X i1 X i2 . . . X iλi X i+1,1 X i+1,2 . . . X i+1,s−1 (−X i,t X i+1,s + (q − q −1 )X is X i+1,t ), from which (6.24) follows by the induction hypothesis. The fact that λ is also a Uq (gln ) highest weight vector is proved similarly, by applying the Chevalley generators ei ∈ Uq (gln ) to λ and using the first and second relations of (6.21). The second part of the lemma is obvious.

Proof of Theorem 6.16. For any Z≥0 -graded module M, let M≤k denote the sum of the

(m) (n) graded components of degree at most k. It follows from Lemma 6.17 that λ L λ ⊗ L λ m n is a Uq (g)-submodule of q (V ), where g = glm × gln . Let (C ⊗ C ) be the exterior algebra of Cm ⊗ Cn over C, and denote by |λ| the size of the partition λ, that is, the sum of its parts (recall we always assume that λ1 ≤ n). Then by skew (G L m , G L n ) duality [14, Theorem 4.1.1], we have (m) (n) dimK L λ ⊗ L λ = dimC (Cm ⊗ Cn )≤k |λ|≤k

since irreducible modules of g and of Uq (g) with the same highest weight have the same dimension. By Proposition 6.15, dimC (Cm ⊗ Cn )≤k = dimK q (V )≤k for all k. Hence (m) (n) dimK L λ ⊗ L λ , ∀k. dimK q (V )≤k = |λ|≤k

This completes the proof.

Acknowledgement. The authors thank the Australian Research Council and National Science Foundation of China for their financial support.


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Communications in


Harmonic Functions and Instanton Moduli Spaces on the Multi-Taub–NUT Space Gábor Etesi, Szilárd Szabó Department of Geometry, Mathematical Institute, Faculty of Science, Budapest University of Technology and Economics, Egry J. u. 1, H ép., H-1111 Budapest, Hungary. E-mail: [email protected], [email protected] Received: 25 August 2009 / Accepted: 27 July 2010 Published online: 6 October 2010 – © Springer-Verlag 2010

Abstract: Explicit construction of the basic SU(2) anti-instantons over the multi-Taub– NUT geometry via the classical conformal rescaling method is exhibited. These antiinstantons satisfy the so-called weak holonomy condition at infinity with respect to the trivial flat connection and decay rapidly. The resulting unit energy anti-instantons have trivial holonomy at infinity. We also fully describe their unframed moduli space and find that it is a five dimensional space admitting a singular disk-fibration over R3 . On the way, we work out in detail the twistor space of the multi-Taub–NUT geometry together with its real structure and transform our anti-instantons into holomorphic vector bundles over the twistor space. In this picture we are able to demonstrate that our construction is complete in the sense that we have constructed a full connected component of the moduli space of solutions of the above type. We also prove that anti-instantons with arbitrary high integer energy exist on the multi-Taub–NUT space. 1. Introduction The aim of this paper is to construct the most relevant anti-instanton moduli space over the multi-Taub–NUT geometry by elementary means. An important class of non-compact but complete four dimensional geometries is the collection of the so-called asymptotically locally flat (ALF) spaces including several mathematically as well as physically important examples. The flat R3 × S 1 plays a role in finite temperature Yang–Mills theories, the Euclidean Schwarzschild space [20] deals with quantum gravity and Hawking radiation. If the metric is additionally hyper-Kähler then the space is also called an ALF gravitational instanton (in the narrow sense). The flat R3 × S 1 is a straightforward example; non-trivial ones for this restricted class are provided by the multi-Taub–NUT (or Ak ALF or ALF Gibbons–Hawking) spaces [15] which carry supersymmetric solutions of string theory and supergravity models, the

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G. Etesi, Sz. Szabó

Atiyah–Hitchin manifold (and its universal double cover) [3] describing the 2-monopole moduli space over R3 and last but not least the recently constructed Dk ALF spaces [9]. The ALF asymptotics is a natural generalization of the well-known ALE (asymptotically locally Euclidean) one including the multi-Eguchi–Hanson geometries [15]. Instanton theory over these later spaces possessing several phenomena related with noncompactness (e.g. existence of four dimensional moduli spaces isometric to the original space) is well-known due to the important paper of Kronheimer and Nakajima [27] in which a full ADHM construction was established. The existence of this construction is in some sense not surprising because the original ADHM construction was designed for the flat R4 and all ALE spaces arise by an algebro-geometric deformation of the flat quotients C2 / , where ⊂ SU(2) are various discrete subgroups [26]. The natural question arises: what about instanton theory over ALF spaces? Unlike the ALE geometries, these are essentially non-flat spaces in the sense above, therefore one may expect that instanton theory somewhat deviates from that over the flat R4 . Some general questions have been answered recently [13] and these investigations pointed out that in spite of their more transcendental nature, open Riemannian spaces with ALF asymptotics rather resemble compact four manifolds at least from the point of view of instanton theory. To test this interesting observation more carefully in this paper we work out the simplest moduli space over the multi-Taub–NUT geometry: we will find that this moduli space is five dimensional, furthermore its part containing concentrated instantons looks like a “collar” of the original manifold, supporting the analogy with the compact case. We note that this apparent compactness is related to the existence of a smooth compactification of ALF spaces motivated by L 2 cohomology theory [19]. The paper is organized as follows. In Sect. 2 we quickly summarize two important tools for solving the self-duality equations over an anti-half-conformally flat space: the conformal rescaling method of Jackiw–Nohl–Rebbi [4,25] as well as the Atiyah–Ward correspondence [5]. Via conformal rescaling one constructs instantons out of positive harmonic functions with at most pointlike singularities and appropriate decay toward infinity while the Atiyah–Ward correspondence establishes a link between instantons and holomorphic vector bundles. Since these methods meet in Penrose twistor theory we shall also briefly outline it here. Then we introduce ALF spaces. Referring to recent results on instanton theory over these spaces [13] we precisely define the class of anti-instantons which are expected to form nice moduli spaces. These solutions must obey two conditions: the so-called weak holonomy condition with respect to some smooth flat connection and the rapid decay condition (cf. Definition 2.1 here). The former condition, albeit looks like an analytical one, in fact deals with the topology of infinity of the ALF space only [13, Theorem 2.3] meanwhile the latter one controls the fall-off properties of an anti-instanton, hence is indeed analytical in its nature. In Sect. 3, taking probably the most relevant ALF example namely the multi-Taub– NUT series, following Anderson–Kronheimer–LeBrun [1] and LeBrun [28] we review this geometry with special attention to the description of its generic complex structures as algebraic surfaces in C3 . We also identify a finite number of distinguished complex structures that can be realized as blowups of these algebraic surfaces. The group S 1 acts on these spaces via isometries and this action has isolated fixed points, called NUTs. Then in Sect. 4 we come to our main results. We shall prove via conformal rescaling that over the multi-Taub–NUT space (MV , gV ) unframed L 2 moduli spaces of SU(2) anti-instantons obeying both the aforementioned weak holonomy condition with respect to the trivial flat connection ∇ as well as the rapid decay condition are not empty.

Instantons over the Multi-Taub–NUT Space

177

Since the input in this construction is a positive harmonic function, the so-called nonparabolic manifolds [17,21,34] are the key concept here which admit an abundance of positive minimal Green functions G(·, y) with arbitrarily prescribed pointlike singularity y ∈ MV (cf. Definition 4.1 here). Non-parabolicity deals with the volume growth toward infinity and is closely related to the ALF property of our space as it was observed by Minerbe [29, Theorem 0.1]. These functions provide us with a five real paremeter family f y,λ = 1 + λG(·, y) of positive harmonic functions, hence that of non-gauge equivalent irreducible SU(2)anti-self-dual connections of unit energy on the positive chiral spinor bundle—parameterized by a point y ∈ MV and a “concentration parameter” λ ∈ (0, +∞). These solutions obey the weak holonomy condition with respect to the trivial flat connection as well as decay rapidly. This way we obtain a familiar “collar” for the original manifold (cf. Theorem 4.1 here). Since we know from [13, Theorem 3.2] that the unit energy unframed moduli space of these irreducible solutions is a smooth five dimensional manifold it follows that this collar in fact realizes an open subset of it (or one of its connected components) and describes the regime of concentrated unit energy anti-instantons with trivial holonomy at infinity. Then we proceed further and construct additional anti-instantons in the “centerless” direction i.e., when the concentration parameter λ becomes infinite. These solutions just correspond to pure Green functions. However in this limit anti-instantons corresponding to the NUTs are reducible to U(1), consequently it is natural to talk about (a connected (1, ) including these reducible solutions component of) the unframed moduli space M as well. Therefore we will find in particular that this extended moduli space is not a manifold anymore but rather admits usual singularities in the reducible points, and moreover that our moduli space itself admits a singular fibration (1, ) −→ R3 :M whose generic fibers are 2 dimensional open while the singular ones are 1 dimensional semi-open balls (cf. Theorem 4.2 here). The aforementioned anti-instantons with infinite concentration parameter λ = +∞ are represented in the generic case by the centers of these open 2-balls meanwhile in the singular case by the closed end of these semiopen 1-balls. For a given generic fiber the λ = const. < +∞ circles around the center describe anti-instantons “centered” about points in (MV , gV ) along the same isometry orbit and with fixed finite concentration parameter. The singular fibers correspond to the fixed points of the isometric S 1 -action, hence in a given singular fiber there is a unique anti-instanton with λ = const. ≤ +∞. In other words, all “centerless” anti-instantons stemming from pure Green functions with pointlike singularities along a given isometry orbit are gauge equivalent. We will also be able to demonstrate that similar moduli spaces with arbitrary high energy and trivial holonomy at infinity are non-empty according to previous expectations. For a comparison we also construct the moduli space of rapidly decaying unit energy SU(2) instantons with trivial holonomy in infinity over the flat R3 × S 1 (cf. Theorem 4.4 here). In the last section, Sect. 5 we round things off by proving, as already mentioned, that our construction is complete in the sense that at least a whole connected component of the basic moduli space emerges this way. This will require, however, quite tedious work.

178


To this end first we provide a detailed construction of the twistor space of the multiTaub–NUT geometry together with its real structure à la Hitchin [6,22,24]. Then we transform our anti-instantons constructed out of harmonic functions into holomorphic vector bundles on the twistor space in the spirit of the Atiyah–Ward correspondence [5]. More precisely, by the aid of Atiyah [2] we work out a smooth compactification of the twistor space which makes it possible to identify the harmonic functions used to construct anti-instantons with elements of certain Ext groups of the compactified twistor space (cf. Theorem 5.1 here). A novelty of our compactification compared to Atiyah’s is that our resulting Ext groups have the correct dimension (cf. Lemma 5.5) i.e., they include the constant harmonic functions as well. (Note that without compactification the Ext groups would be infinite dimensional.) These Ext groups can be used to obtain holomorphic bundles on the twistor space. In particular gluing data of the bundles corresponding to minimal Green functions can be worked out explicitly [2] and one can conclude that the isometric S 1 -action induces an isomorphism of these bundles, therefore the associated “centerless” anti-instantons indeed must be gauge equivalent. This provides us with the picture of the moduli space mentioned above. Working with the twistor space also can be viewed as a step toward establishing a general ADHM-like instanton factory [8] which can be used to replace the conformal rescaling method; this is certainly necessary since our technique is not capable to grasp all higher energy moduli spaces. We conjecture that in fact we have obtained the full moduli space of unit energy, rapidly decaying anti-instantons with trivial holonomy at infinity in this paper, i.e. the basic moduli space is connected. Finally for a comparison we make a comment here on other gravitational instantons of ALG and ALH type introduced by Cherkis and Kapustin. By their definition, these geometries are parabolic, hence one cannot expect the existence of positive minimal Green functions over them and accordingly, a canonical five parameter family of antiinstantons. Indeed, the simplest ALG space is the flat R2 × T 2 and it is known that the framed unit energy anti-instanton moduli space here is one dimensional only [7]. Hence in the ALE-ALH hierarchy, apparently ALF spaces represent the closest analogues of compact four-manifolds from an instanton theoretic viewpoint. Analogously, in this hierarchy ALF spaces are the only ones whose L 2 cohomological compactification [19] is smooth. 2. Constructing Instantons In this section we quickly summarize two methods to obtain instantons over an antiself-dual Riemannian manifold: the conformal rescaling method and the Atiyah–Ward correspondence. These two approaches meet each other in twistor theory, therefore we also review the basic facts about it here. Let us begin with the conformal rescaling method which was first used by Jackiw, Nohl and Rebbi to obtain instantons over the flat R4 [25]. Remember that an SU(2) (anti-)self-dual connection ∇ A over a four dimensional oriented Riemannian manifold (M, g) is a smooth SU(2) connection on any SU(2) rank two complex vector bundle E over M whose curvature satisfies the (anti-)self-duality equations ∗g FA = ±FA . The gauge equivalence class [∇ A ] of this connection is called an (anti-)instanton if additionally its energy is finite: FA L 2 (M) < +∞. Note that this assumption on the energy is non-trivial only if (M, g) is non-compact but complete.


179

Let (M, g) be an oriented Riemannian spin four-manifold, possibly non-compact but complete. Take the Levi–Civita connection ∇ of the metric and lift it to get the corresponding spin connection ∇ s . Referring to the splitting so(4) = su(2)+ ⊕ su(2)− the projected connection ∇ A := p + (∇ s ), with p + : so(4) → su(2)+ being the canonical projection, is an SU(2)+ connection on the SU(2)+ rank two complex vector bundle + (the positive chiral spinor bundle). Assume (M, g) is moreover anti-half-conformally flat, i.e. Wg+ = 0. Picking up an everywhere smooth and positive function f on M, one can consider the conformally rescaled metric g˜ := f 2 g. The new space (M, g) ˜ still satisfies Wg˜+ = 0 by conformal invariance of the Weyl-tensor. Its scalar curvature however transforms as f 3 sg˜ = 6g f + f sg , where g refers to the scalar Laplacian on (M, g). If this new scalar curvature sg˜ happens to vanish over M, i.e. the rescaling function satisfies the equation g f +

sg f = 0, 6

(1)

then the corresponding projected connection ∇ A˜ , constructed like ∇ A before, satisfies the anti-self-duality equations ∗g˜ FA˜ = −FA˜ over (M, g) ˜ by a variant of the Atiyah–Hitchin–Singer theorem [4, Prop. 2.2]. However taking into account the conformal invariance of self-duality, we obtain that ∇ A˜ in fact satisfies the original SU(2)+ anti-self-duality equations ∗g FA˜ = −FA˜ over (M, g). Hence we have constructed a solution of the anti-self-duality equation via the conformal rescaling method. This anti-self-dual connection locally can be constructed as follows. Let U ⊂ M be a coordinate ball and let (ξ 0 , . . . , ξ 3 ) denote the orthonormal frame field which diagonalizes g|U . In this gauge write ∇|U = d + ω for the Levi–Civita connection. The rescaled metric f 2 g has a corresponding local frame field ( f ξ 0 , . . . , f ξ 3 ) and U = d + ω. ˜ The connection 1-form ω˜ can be calculated by Levi–Civita connection ∇| the aid of the Cartan equation d( f ξ i ) + ω˜ ij ∧ ( f ξ j ) = 0. An easy computation shows that ω˜ ij = ωij + ((d log f )ξi )ξ j − ((d log f )ξ j )ξ i , where (ξ0 , . . . , ξ3 ) denotes the dual vector field on T U (with respect to the original to the spin bundle, then metric g|U ) and (d log f )ξi = g|U (grad(log f ), ξi ). Lifting ∇ projecting it to the positive side (using the ’t Hooft matrices, cf. [12]) and exploiting the identification su(2)+ ∼ = Im H given by (σ1 , σ2 , σ3 ) → (i, −j, −k) we obtain in this induced gauge 1 A˜ = A + Im((d log f )ξ ) 2

(2)

with A := p + (ωs ) being the projection of the original spin connection ∇ s |U = d + ωs and the quaternion-valued 0-form d log f and 1-form ξ in the second term are defined respectively as follows: d log f := d log f (−ξ0 + ξ1 i + ξ2 j + ξ3 k),

ξ := ξ 0 + ξ 1 i + ξ 2 j + ξ 3 k.

Writing a|U := 21 Im((d log f )ξ ) for the only term involving f , it is clear that it extends over M as an Im H-valued 1-form i.e., one finds a ∈ C ∞ (M; 1 M ⊗ End + )

(3)

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and it is insensitive for an overall rescaling of f by a non-zero real constant. This implies that from the point of view of our anti-instanton factory, we can regard two conformal scaling functions identical if they differ by an overall rescaling only. Moreover note that the new connection takes the form ∇ A˜ = ∇ A+a i.e., this whole conformal rescaling procedure can be regarded as an anti-self-dual perturbation of the original connection. For an explicit application of this method we refer to [25] over the flat R4 and to [12] in the case of the multi-Taub–NUT space. If (M, g) is compact then this machinery provides us with an anti-instanton on + . If the space is non-compact but complete and ∇ A has finite energy (which can be achieved if the metric decays sufficiently fast to the flat one at infinity as we have seen), then if the perturbation a also decays fast, i.e. f is sufficiently bounded at infinity, then ∇ A˜ has still finite energy and consequently we obtain an anti-instanton on the same bundle + over (M, g). An important particular case is when the original space (M, g) has already vanishing scalar curvature. In this case (1) cuts down to the ordinary Laplace equation g f = 0 and the original connection ∇ A is already anti-self-dual corresponding to the trivial solution f = 1. However it is clear by the maximum principle that in many situations there are no non-trivial everywhere smooth bounded solutions to (1); therefore this method is apparently vacuous. Fortunately we can modify f to possess mild singularities over M. At this level of generality by a “mild singularity” we mean the following: (i) There are finitely many points {y1 , . . . , yk } ⊂ M such that f diverges in yi for all i = 1, 2, . . . , k; (ii) The local energy of the rescaled connection around each point is finite, i.e. for Ui a neighbourhood of yi one has FA˜ L 2 (Ui \{yi }) < +∞ for all i = 1, 2, . . . , k. (In terms of f this condition simply controls its behaviour about yi .) Under these circumstances Uhlenbeck’s theorem [33] allows us to remove the singularities from ∇ A˜ introduced by f . Therefore if f is a positive solution to (1) with such singularities then the resulting anti-self-dual connection will be smooth again and corresponds to a non-trivial anti-instanton over a bundle E if M is compact. This SU(2)+ bundle may be no more isomorphic to + as a consequence of the allowed singularities of the scaling function. In fact its topological type is determined by the singularity structure of f . In the case of a non-compact complete base geometry, if in addition f is sufficiently bounded at infinity, then the resulting anti-self-dual connection has finite energy, i.e. describes an anti-instanton on the same bundle + , regardless how many singularities f possesses. This is because in fact over a non-compact four-manifold all SU(2)+ bundles, including + , are isomorphic to the trivial one. The significance of the singularity structure of scaling functions in instanton theory therefore becomes transparent. We move on and recall twistor theory of anti-half-conformally flat geometries and in particular the Atiyah–Ward construction [5] which is another powerful way to solve the (anti-)self-duality equations at least in principle. Let us recall the general theory [6, Chap. 13]. Let (M, g) be an oriented Riemannian four-manifold and consider the projectivization Z := P( + ) of the positive chiral spinor bundle + on M (this projectivization is well-defined even if M is not spin). Clearly,


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Z admits a fiber bundle structure p : Z → M with CP 1 ’s as fibers. A fiber Fx over x ∈ M can be viewed as the parameter space of orthogonal complex structures on Tx M whose induced orientations agree with that of M. Using the Levi–Civita connection on (M, g) we can split the tangent space Tz Z at any point z ∈ Z ; writing z = (x, p) with x ∈ M and p ∈ Fx , one endows the horizontal subspace H(x, p) ∼ = Tx M of T(x, p) Z with the complex structure J p given by p ∈ Fx itself, meanwhile the vertical subspace V(x, p) ∼ = T p Fx of T(x, p) Z is equipped with the standard complex structure I coming from Fx ∼ = CP 1 . Therefore T(x, p) Z carries a complex structure (J p , I ) and Z as a real six-manifold possesses an almost complex structure. A basic theorem of Penrose [32] or Atiyah, Hitchin and Singer [4, Theorem 4.1, 6, Theorem 13.46] states that this complex structure is integrable, i.e. Z is a complex three-manifold if and only if (M, g) is antihalf-conformally flat. The space Z is called the twistor space of (M, g). The complex structure in fact depends on the confomal class of the metric only. The basic holomorphic structure of Z looks as follows [6, Chap. 13]: (i) For all y ∈ M the fibers Fy ⊂ Z represent holomorphic lines Y ⊂ Z with normal bundles isomorphic to H ⊕ H, where H is the usual line bundle H with c1 (H ), [Y ] = 1 on Y ∼ = CP 1 . (ii) The antipodal map on Y induces a real structure on Z , i.e. an anti-holomorphic involution τ : Z → Z satisfying τ 2 = Id Z . (iii) The space of all holomorphic lines in Z near the fibers Y form a locally complete family which is a complex manifold M C of complex dimension four and the fibers Y among them are distinguished by the property that the real structure fixes them, hence are called real lines. A remarkable property of twistor spaces is that they allow one to solve certain linear or non-linear field equations over (M, g). These equations are of great importance in physics. We shall also quickly review these constructions. 1. The linear field equation relevant to us here is the conformal scalar Laplace equation (1). We shall denote by H Z the holomorphic line bundle on the whole Z such that for any fiber the restriction H Z |Y is the standard line bundle H on Y ∼ = CP 1 and by k th H Z its k tensor product. For notational convenience write O Z (k) for the sheaf O(H Zk ). Recall (cf. e.g. [23]) that complex-valued real analytic functions satisfying (1) on open subsets (U, g|U ) ⊂ (M, g) are in one-to-one correspondence with elements of the sheaf cohomology group H 1 ( p −1 (U ); O p−1 (U ) (−2)), i.e. there is a natural isomorphism T : H 1 ( p −1 (U ); O p−1 (U ) (−2)) ∼ = Ker g |U called the Penrose transform: given an element ϕ ∈ H 1 ( p −1 (U ); O p−1 (U ) (−2)) and a real line Y ⊂ p −1 (U ) one can take the restriction ϕ|Y ∈ H 1 (Y ; OY (−2)) ∼ = C. This gives rise to a complex-valued function f (y) := ϕ|Y , consequently one has a map T : H 1 ( p −1 (U ); O p−1 (U ) (−2)) −→ C ∞ (U, C). It turns out that f is a solution to (1) on U ⊂ M and all local solutions arise this way. 2. The non-linear (anti-)self-duality equations of Yang–Mills theory, also can be treated in twistor theory (cf. e.g. [4]): solutions of the SU(2) anti-self-duality equations over (M, g) with approriate orientation can be converted into certain holomorphic vector bundles on Z as follows. There is a one-to-one correspondence of gauge equivalence classes of anti-self-dual SU(2) connections on a rank two complex vector bundle E on M and isomorphism classes of rank two complex vector bundles F on Z such that

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(i) F is holomorphic; (ii) For any real line Y ⊂ Z the restricted bundle F|Y is holomorphically trivial; (iii) There exists a conjugate linear map τ˜ : F → F lying over the real structure τ : Z → Z and satisfies τ˜ 2 = −Id F . If ∇ A is irreducible and Z (that is, M) is compact then F is a stable holomorphic bundle on Z and in this case τ˜ is unique up to scalar multiplication. This construction, called the Atiyah–Ward correspondence, goes as follows. Take an SU(2) vector bundle E and an anti-self-dual connection on it over (M, g). Let F := p ∗ E be a C ∞ bundle; by anti-self-duality the curvature of the lifted connection p ∗ ∇ A will be of (1, 1)-type hence F will be holomorphic. One checks that properties (ii) and (iii) are also satisfied. Conversely, take a holomorphic bundle F satisfying (ii) and (iii). Define E y to be the space of holomorphic sections over F|Y (this space is two complex dimensional via (ii)). Property (iii) induces a symplectic structure on F, consequently we can take the corresponding unique unitary connection ∇ on F. By uniquenss and (ii) this connection is the trivial connection on each F|Y , hence is of the form p ∗ ∇ A with a connection ∇ A on E. Since the curvature of this connection is of (1, 1)-type with respect to all complex structures on Ty M for all y ∈ M, it follows that it is anti-self-dual. 3. Another non-linear equation, namely the (Riemannian) anti-self-dual vacuum Einstein equations of general relativity can also be naturally adjusted to twistor theory, known as Penrose’ non-linear graviton construction [22,32]. Assume here that (M, g) is moreover simply connected and Ricci-flat, i.e. simply connected and anti-selfdual. Then it follows that the induced connection on + is the trivial flat one and Z can be retracted onto its particular fiber Fy by parallel transport. Therefore topologically Z∼ = M × S 2 in this case. The associated twistor space of (M, g) has in addition to the basic holomorphic properties above the following: ∼ CP 1 with (i) There is a holomorphic fibration π : Z → Y over a particular line Y = k ∗ k ∼ fibers being the original space M. Consequently H Z = π H for all k ∈ Z; (ii) Since the complex normal bundle NY of each Y is isomorphic to H ⊕ H it follows that the canonical bundle K Z of Z is isomorphic to π ∗ H −4 . Conversely, it can be shown [6, Chap. 13] that a complex 3-space Z having these properties encodes the conformal class of an anti-self-dual simply connected four-space (M, g). Indeed, the complexified space M C carries a natural conformal structure by declaring two points Y , Y ∈ M C to be null-separated if and only if for the corresponding lines Y ∩ Y = ∅ in Z . Restricting this conformal structure to the real lines parameterized by M ⊂ M C , one comes up with the confomal class of an anti-self-dual structure (M, g). The particular metric emerges via an appropriate isomorphism K Z ∼ = π ∗ H −4 . We have encountered two techniques for constructing anti-instantons over an antihalf-conformally flat space. The link between them is an observation of Atiyah [2] which relates solutions to (1) to holomorphic vector bundles on the twistor space. We will work out this in detail in Sect. 5. In the rest of this section we introduce a special class of four-manifolds: suppose from now on that (M, g) is a four dimensional asymptotically locally flat (ALF) space as it was defined in [13]. By a recent powerful theorem of Minerbe [29, Theorem 0.1] these four dimensional geometries at least in the connected, geodesically complete, hyper-Kähler case can also be characterized by requiring their curvature to be L 2 with respect to the r4 measure Vol (B dVg as well as prescribing their asymptotical volume growth to be 4 g r (x)) O(r ν ) with 3 ≤ ν < 4.


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Framed moduli spaces of certain “admissible” SU(2) (anti-)instantons over these geometries have been investigated recently [13]. By “admissible” we mean the following. Topologically, an ALF space (M, g) (with a single end) admits a decomposition M = K ∪ W, where K is a compact interior space and W is an end or neck homeomorphic to N × R+ , where π N : N → B∞ is a connected, compact, oriented three-manifold fibered over a compact Riemann surface B∞ with circle fibers F ∼ = S 1 . Note that for any x ∈ M and sufficiently large R > 0, W is homeomorphic to M\B R4 (x), where B R4 (x) is a geodesic ball about x ∈ M with radius R > 0. Definition 2.1. Let (M, g) be an ALF four-manifold. Take an arbitrary smooth, finite energy SU(2) connection ∇ A on a (necessarily trivial) SU(2) vector bundle E over M. This connection is said to be admissible if it satisfies two conditions (cf. [13]): (i) The first is called the weak holonomy condition and says that to ∇ A there exists a smooth flat SU(2) connection ∇ |W on E|W along the end W ⊂ M and a constant c1 = c1 (g) > 0, independent of R > 0, such that in any smooth gauge on M\B R4 (x) the inequality

A − L 2

1,

M\B R4 (x)

≤ c1 FA L 2 M\B 4 (x) R

M\B R4 (x);

holds along the neck (ii) The second condition requires ∇ A to decay rapidly at infinity i.e., √ R FA L 2 M\B 4 (x) = 0. lim R

R→∞

The first condition ensures us that the connection has a well-defined holonomy at infinity given by ∇ |W ; however note that if the infinity is not simply connected then this holonomy can in principle be non-trivial.1 The second condition regulates how fast the finite energy connection decays to this flat connection and also can be reformulated (cf. [13]) by saying that FA belongs to a weighted Sobolev space with weight δ = 21 , i.e. FA ∈ L 21 (M; 2 M ⊗ End E). Note that this second condition is stronger 2

than the finite energy requirement on ∇ A . It was demonstrated in [13] that irreducible SU(2) anti-instantons satisfying conditions (i) and (ii) with a fixed limiting flat connection ∇ |W and energy e form framed moduli spaces M (e, ) which are smooth finite dimensional manifolds. Their dimensions have also been calculated by the aid of a dimension formula. Now take an ALF space (M, g) with a fixed orientation whose curvature satisfies Wg+ = 0 and sg = 0, therefore the Atiyah–Hitchin–Singer theorem applies and the projected Levi–Civita connection ∇ A is an anti-self-dual connection on + . By an appropriate curvature decay imposed on the metric (which is incorporated into the precise definition of an ALF space, cf. [13]) this connection has not only finite energy but also decays rapidly as in (ii) of Definition 2.1; moreover if additionally it satisfies the weak holonomy condition (i) in Definition 2.1, then it gives rise to an anti-instanton on + in the sense above. Performing a conformal rescaling of the metric we obtain another anti-self-dual connection ∇ A˜ on + . Since it is of the form ∇ A+a with a as in (3) except finitely many points, its curvature reads as FA˜ = FA+a = FA +d A a +a ∧a, consequently for its energy FA˜ L 2 (M) ≤ FA L 2 (M) + d A a L 2 (M) + c2 a 2L 2 1 2

1 2

1 2

1 In this paper the case of the trivial holonomy will be considered only.

(M) 1 2 ,1,

(4)

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holds with a constant c2 = c2 (g) > 0. We conclude that the regularity and decay properties of the perturbed connection is determined by the perturbation term. More precisely, the conditions 2 A˜ ∈ L loc,1, (M; 1 M ⊗ End + ), A˜

a ∈ L 21 ,1, (M; 1 M ⊗ End + ), 2

2 FA˜ ∈ L loc (M; − M ⊗ End + )

(5)

ensure us that a affects neither the asymptotics of ∇ A or its fall-off and any pointlike singularities are removable. Therefore if ∇ A is smooth and obeys both the weak holonomy and the rapid decay conditions above, then ∇ A˜ = ∇ A+a will be also smooth, decays rapidly and converges to the same limiting flat connection. More precisely one can show that the energy e˜ of ∇ A˜ can differ from that of ∇ A by a non-negative integer only [13]. That is, [∇ A ] ∈ M (e, ) implies [∇ A˜ ] ∈ M (e, ˜ ) and e˜ − e ∈ {0, 1, 2, ...}. In fact as one expects, the difference e˜ − e is governed by the singularity structure of the perturbation, i.e. of the scaling function f in (1). In the next sections we will apply the conformal rescaling technique and twistor theory to understand the moduli space over the multi-Taub–NUT geometries of unit energy SU(2) anti-instantons decaying rapidly and obeying the weak holonomy condition with respect to the trivial connection ∇ . 3. Review of the Multi-Taub–NUT Geometry In this section we take a closer look at the multi-Taub–NUT spaces following [1,28]. Consider s > 0 distinct points q1 , . . . , qs ∈ R3 . We will construct a four-manifold MV and a smooth map π : MV → R3 such that π −1 (q j ) is a point for all j = 1, 2, . . . , s but π −1 (x) ∼ = S 1 for all other points x ∈ R3 \{q1 , . . . , qs }. To begin with, let π : UV → R3 \{q1 , . . . , qs } be the principal S 1 bundle whose Chern class is −1 when restricted to a small sphere of radius r j < min |q j − qk | around q j . Thus π −1 (Br3j (q j )) is diffeomork= j

phic to the punctured ball B 4j \{0} ⊂ R4 in a manner such that the S 1 action becomes the action of S 1 ⊂ C on C2 ∼ = R4 by scalar multiplication. We then define MV := (UV B14 · · · Bs4 )/ ∼, where ∼ means that B 4j \{0} is identified with π −1 (Br3j (q j )). The map π : UV → R3 \{q1 , . . . , qs } extends to a smooth map π : MV → R3 . Note that there is an S 1 action on MV whose fixed points with index −1 are exactly the q j ’s—called NUTs. If i j is a straight line segment joining qi with q j , then π −1 (i j ) ⊂ MV is a smoothly embedded 2-sphere whose self-intersection number is −2. The 2-spheres π −1 ( j, j+1 ) with j = 1, 2, . . . , s − 1 are attached together according to the As−1 Dynkin diagram and generate the singular cohomology group H 2 (MV , Z) ∼ = Zs−1 . Take a real number c > 0 and let V : R3 \{q1 , . . . , qs } → R+ be defined by 1 1 . V (x) := c + 2 |x − q j | s

j=1

In this paper we will suppose c = 1. Then V = 0 that is, it is a positive har1 monic function on R3 \{q1 , . . . , qs }. The cohomology class [ 2π ∗3 dV ] ∈ H 2 (UV , Z)


185

is the first Chern class of the bundle π : UV → R3 \{q1 , . . . , qs }. Consequently there is a connection ∇ on π : UV → R3 \{q1 , . . . , qs } whose curvature is ∗3 dV . More precisely, let ω ∈ C ∞ ( 1 (R3 \{q1 , . . . , qs })) be a real valued connection 1-form on R3 \{q1 , . . . , qs } so that ∗3 dV = 1i Fω = dω. The form ω is unique up to gauge transformation since R3 \{q1 , . . . , qs } is simply connected. If we introduce a coordiante system (x 1 , x 2 , x 3 , τ ), where x i are Cartesian coordinates on R3 and τ ∈ [0, 2π ) parameterizes the circles on UV , then the multi-Taub–NUT metric gV |UV on UV is defined to be ds 2 = V π ∗ ((dx 1 )2 + (dx 2 )2 + (dx 3 )2 ) +

1 (dτ + π ∗ ω)2 . V

(6)

The space (UV , gV |UV ) extends smoothly across the fixed points of the S 1 action: write 1 V (x) = V j (x) + f (x) with V j (x) := 2|x−q and f a smooth function around q j . Let j| ω j be the connection whose curvature is ∗3 dV j , then we obtain a decomposition ds 2 = V j π ∗ ((dx 1 )2 + (dx 2 )2 + (dx 3 )2 ) +

1 (dτ + π ∗ ω j )2 + h, Vj

where h is a smooth symmetric bilinear form around p j induced by f . Moreover the first term is just the 1-Eguchi–Hanson metric on R4 which is known to be isometric to the standard flat metric on R4 , hence extends over p j . We come, therefore, up with a Riemannian manifold (MV , gV ) with an isometric S 1 action whose fixed points with index −1 are π −1 (q j ). It is moreover complete with a single ALF end, is anti-self-dual since dω = ∗3 dV, that is, anti-half-conformally flat and Ricci-flat. Since MV is simply connected, it follows that it is hyper-Kähler as well, i.e. there is an entire 2-sphere’s worth of complex structures for which gV is a Kähler metric. To display these complex structures explicitly at least on UV , let e1 , e2 , e3 be an orthonormal basis in R3 . Consider these as constant vector fields on R3 and let eˆ1 , eˆ2 , eˆ3 be their horizontal lifts to UV via the connection, as well as let eˆ0 be the generator of the S 1 -action. Then √ 1 1 1 (7) V eˆ0 , √ eˆ1 , √ eˆ2 , √ eˆ3 (ξ0 , ξ1 , ξ2 , ξ3 ) := V V V is an orthonormal frame on (UV , gV |UV ) which we define to be oriented. Relative to this frame we take the almost complex structure ⎛ ⎞ 0 −1 0 0 ⎜1 0 0 0 ⎟ . Je1 := ⎝ 0 0 0 −1 ⎠ 0 0 1 0 It is parallel, hence is integrable, and it depends on the choice of e1 . Of course it extends over the whole (MV , gV ) which we continue to denote by Je1 . By the aid of this picture we see that all possible orthogonal complex structures with compatible orientation on (MV , gV ) are parameterized by a CP 1 and the situation can be described as a holomorphic fibration: the fiber over e1 ∈ CP 1 is the complex manifold (MV , Je1 ) in accord with the general theory outlined in Sect. 2. Moreover if the direction of any of the straight line segments i j coincides with e1 , then the corresponding 2-sphere π −1 (i j ) will represent a holomorphic curve with self-intersection number −2 in (MV , Je1 ). For a generic complex structure however, there are no holomorphic projective lines.

186


These complex structures on the whole (MV , gV ) can be described in a rather explicit algebraic way as follows (cf. [1,28]). Lemma 3.1. Fix a generic configuration of points q1 , . . . , qs in R3 and consider the associated space (MV , gV ). Take a direction e1 in R3 . Assume that e1 is not parallel with any line segments i j joining qi with q j . Then (MV , Je1 ) is biholomorphic to the algebraic surface X ⊂ C3 given by x y − (z − p1 ) . . . (z − ps ) = 0

(8)

with (x, y, z) ∈ C3 and fixed distinct complex numbers p1 , . . . , ps ∈ C. The S 1 -action on X is given by (x, y, z, τ ) → (xeiτ , ye−iτ , z) with fixed points (0, 0, p j ) ∈ X of index −1 for all j = 1, . . . , s. Under this biholomorphism the fixed point πV−1 (q j ) ∈ MV of the S 1 -action on MV correspond to the fixed point (0, 0, p j ) ∈ X with the same index for all j = 1, . . . , s, consequently with this S 1 -action on X we obtain 1 an isomorphism of complex s manifolds with an S -action. There exist at most 2 2 = s(s − 1) different directions when e1 is parallel with any i j . In this case (MV , Je1 ) is biholomorphic to the complex surface X which arises by taking pi = p j for the fixed pair (i, j) in (8) and blowing up the resulting singular surface X ∗ ⊂ C3 in (0, 0, p j ) ∈ X ∗ . As a real oriented four-manifold X is diffeomorphic to X . Proof. By genericity we assume that the straight line segments i j ⊂ R3 with i, j = 1, 2, . . . , s are mutually non-parallel. First take a direction e1 in R3 which is not parallel with any i j . Let (e2 e3 ) ⊂ R3 be the oriented plane passing through the origin, perpendicular to e1 and identify it with the complex plane with its standard orientation: (e2 e3 ) ∼ =C such that e2 → 1 and e3 → i. Let pe1 : R3 → C be the projection along e1 which induces Pe1 := pe1 ◦π : MV → C, the projection from MV . Then Pe1 is a holomorphic map from (MV , Je1 ) to C. For a point a ∈ MV , define a complex coordinate by z := Pe1 (a) ∈ C. In particular for a NUT q j ∈ R3 put p j := Pe1 (π −1 (q j )) = pe1 (q j ) ∈ C for all j = 1, . . . , s and also call them NUTs. Next we analyze the preimage Pe−1 (z) ⊂ MV . If z = p j ∈ C, then Pe−1 (z) is 1 1 holomorphically isomorphic to an infinite cylinder C∗ := {(x, y) ∈ C2 | x y = 1} in (MV , Je1 ), since the directions eˆ0 and eˆ1 span a holomorphic line for the complex structure Je1 . On the other hand Pe−1 ( p j ) is homeomorphic to two complex affine lines 1 intersecting in one point. In particular we have a model like {(x, y) ∈ C2 | x y = 0} for Pe−1 ( p j ). Since the complex structure on the fibers is independent of z, the above two 1 cases can be managed together into a global equation by writing x y = c(z), where c(z) is a polynomial in the variable z on C which vanishes exactly in the points p1 , . . . , ps ∈ C. Therefore we put c(z) := (z − p1 ) . . . (z − ps ) to obtain the desired biholomorphism between (MV , Je1 ) and X given by (8). The statement about the S 1 -action is clear. If e1 happens to be parallel with any i j connecting qi ∈ R3 and q j ∈ R3 , then we find pe1 (qi ) = pi = pe1 (q j ) = p j ∈ C, hence two roots coincide in (8). However now Pe−1 ( pi ) = Pe−1 ( p j ) consists of not only two complex affine lines as above but in 1 1 addition the projective line π −1 (i j ) and the two complex affine lines do not intersect


187

each other, rather each individual line hits π −1 (i j ) in one point respectively. These two points are distinct. The situation can be modeled on {(x, y, [u : v]) ∈ C2 × CP 1 | x y = 0, xv = yu}. Therefore to obtain the resulting smooth complex surface we have to take a singular curve X ∗ ⊂ C3 represented by (8) with pi = p j and blow it up at (0, 0, pi ) = (0, 0, p j ) ∈ X ∗ to obtain X. It readily follows that as a real oriented four-manifold, X is diffeomorphic to X since both are just the original MV equipped with different complex structures. Remark. Incidentally we note that the above definition of the coordinate z depends on the identification of the oriented plane perpendicular to e1 with C, i.e. on the choice of a vector e2 in the tangent space of CP 1 at e1 . For all such choices, the above procedure yields a complex number. As T CP 1 ∼ = H 2 , this means that globally over CP 1 the coordinate z must be regarded as a section of the line bundle H 2 . This fact will play an important role in constructing the twistor space, cf. Sect. 5. 4. Moduli Spaces with Trivial Holonomy We already know that a multi-Taub–NUT space (MV , gV ) is anti-half-conformally flat and Ricci-flat. Consequently for its curvature Wg+V = 0 and sgV = 0 holds, and the Atiyah–Hitchin–Singer theorem applies and we can construct SU(2)+ (anti-)self-dual connections on the (trivial) positive chiral spinor bundle + . If we fix an orientation coming from any compex structure in the hyper-Kähler family, these connections will be anti-self-dual. As we have seen in Sect. 2, this approach amounts to find positive solutions of the Laplace equation gV f = 0 with a finite number of mild singularities and appropriate bounds at infinity over the original manifold (MV , gV ). This program partially was carried out in [10,12] and will be stretched to its limits here. An abundance of such harmonic functions are provided by minimal positive Green functions therefore we make a short digression on them following [17,21,34]. Definition 4.1. Let (M, g) be a non-compact complete Riemannian manifold of dimension m ≥ 2 and y ∈ M be a point. A function G(·, y) : M\{y} → R is called the minimal positive Green function concentrated at y ∈ M if it has the following properties: (i) It satisfies g G(·, y) = δ y in the sense of distributions on (M, g). (ii) Let r := d(·, y) denote the distance from y on (M, g). If r → +∞ then G(·, y) tends to zero and if r → 0 then G(·, y) is O(r 2−m ) and |∇ y G(·, y)|g is O(r 1−m ) for m ≥ 3 as well as G(·, y) is O(log r ) and |∇ y G(·, y)|g is O(r −1 ) for m = 2. (iii) Moreover 0 < G(·, y) and G(·, y) ≤ G (·, y) for any other positive Green function with the same singularity. This function if exists is obviously unique and is characterized by these properties. One can try to construct them as follows [21, p. 229]. Take y ∈ M and M an open subset with smooth boundary and compact such that y ∈ . Let G (·, y) be the unique Green function on satisfying g G (·, y) = δ y and G (·, y)|∂ = 0. By the maximum principle we can assume that G (·, y) is positive on \{y} and, if we extend G (·, y) by zero outside , then G (·, y) ≤ G (·, y) whenever ⊆ . Consequently setting

188


G(·, y) := sup G (·, y)

it follows that if this function exists then it is the minimal positive Green function with singularity in y ∈ M. One can prove [21, Theorem 8.1] (also cf. [17,34]) that minimal positive Green functions with singularity y ∈ M constructed this way either exist over M for all y or do not exist at all. In the former case (M, g) is called non-parabolic, while in the later case it is called parabolic. Therefore we should be able to decide whether or not a Riemannian manifold is parabolic. For our purposes the easiest way is to recall a result of Varopoulos [34] which states that if the Ricci curvature of a non-compact complete Riemannian manifold (M, g) with m ≥ 2 is non-negative and for some point x ∈ M with its geodesic ball of radius r, +∞ 1

r dr < +∞ Volg (Br4 (x))

holds, then (M, g) is non-parabolic. With this in mind for the Ricci-flat multi-Taub–NUT space (MV , gV ) we find that VolgV (Br4 (x)) ∼

8π 2 3 r , 3

demonstrating that it is non-parabolic (and we can also see that it is indeed ALF); consequently for all y ∈ MV an associated minimal positive Green function G(·, y) exists. In fact they were explicitly constructed by Page [31] and the simplest of them also appeared in [12]. Making use of these functions we obtain plenty of positive harmonic functions of the form f y,λ (x) := 1 + λG(x, y)

(9)

with a real constant λ ∈ (0, +∞). We claim that these functions have the required properties and can be used to construct finite-energy smooth SU(2)+ anti-instantons over the multi-Taub–NUT space. To check this, we construct the associated five parameter family ∇ A˜ y,λ of non-gauge equivalent anti-self-dual SU(2)+ connections on + with respect to the orientation on (MV , gV ) coming from the complex structures and demonstrate first that they are smooth around y ∈ MV , hence everywhere, and secondly that they are admissible, hence in particular have finite energy. Lemma 4.1. Fix a point y ∈ MV and a number λ ∈ (0, +∞). Take a geodesic ball Bε4 (y) ⊂ MV around y and write Bε∗ (y) := Bε4 (y)\{y} for the punctured ball. Consider the orthonormal frame (7) on (Bε4 (y), gV ). In this gauge plugging (9) into (2) write ∇ A˜ y,λ | Bε∗ (y) = d + A˜ y,λ for the resulting anti-self-dual connection. There exists an L 2

1, A˜ y,λ

gauge transformation γ : Bε∗ (y) → SU(2)+ such that the

gauge transformed potential A˜ y,λ := γ −1 A˜ y,λ γ +γ −1 dγ extends smoothly over Bε4 (y), that is the SU(2)+ connection ∇ A˜ y,λ | Bε4 (y) is smooth on + | Bε4 (y) . Consequently the anti-self-dual connetion ∇ A˜ y,λ is smooth everywhere on + .


189

Proof. Take an orthonormal frame (ξ 0 , ξ 1 , ξ 2 , ξ 3 ) over Bε4 (y) as in (7) with its associated complex structure Je1 on MV . Since the multi-Taub–NUT metric is Kähler with respect to this complex structure, there exists a complex coordinate system (z 1 , z 2 ) on Bε4 (y) centered in y such that gV | Bε4 (y) = (δi j + O(r 2 ))dz i dz j , i.e. the metric osculates the flat metric in second order. Take a neighbourhood y ∈ U ⊂ MV and identify it with H such that y ∈ MV is mapped into y := y 0 + y 1 i+ y 2 j+ y 3 k ∈ H. Then Bε4 (y) ⊂ U becomes a quaternionic ball Bε4 (y) ⊂ H centered about y. Introduce a quaternionic coordinate x ∈ H via x := y + z 1 + z 2 j on it. To simplify notation also write gV for the pullback metric on Bε4 (y). There is a bounded positive function a such that r (x) = a(x)|x − y|, hence setting z 1 = x 0 + x 1 i and z 2 = x 2 + x 3 i, we find in this real coordinate system that j

ξ i (x) = (δ ij + O(|x − y|2 )) dx j and ξi (x) = (δi + O(|x − y|2 )

∂ . ∂x j

Similarly for the harmonic function in (9) we obtain λ + O(|x − y|−1 ) and |x − y|2 i x − yi i −2 d f y,λ (x) = −2λ δ + O(|x − y| ) dx j . |x − y|4 j f y,λ (x) = 1 +

Inserting these expansions into (3) we obtain ay,λ = By,λ + by,λ for the perturbation where the singular Euclidean term looks like By,λ (x) = Im

λ(x − y) dx |x − y|2 (λ + |x − y|2 )

and by,λ is of O(1), hence regular in Bε4 (y). Regarding the spin connection ∇ A , note that the SU(2)+ bundle + is trivial over MV . The metric is anti-self-dual hence the spin connection is flat and since MV is simply connected it is just the trivial flat connection. Writing ∇ A | Bε4 (y) = d + A we simply find A = 0 in the natural gauge (7). Putting all of these into (2) we eventually come up in the gauge (7) we use with an expansion of the vector potential as follows: A˜ y,λ = By,λ + by,λ , whose only singular term is By,λ but still A˜ y,λ ∈ L 2

1, A˜ y,λ

( 1 Bε∗ (y) ⊗ End + | Bε∗ (y) ).

Calculating the curvature we find FA˜ y,λ = FBy,λ + d By,λ by,λ + by,λ ∧ by,λ , that is, FA˜ y,λ (x) =

λ dx ∧ dx + d By,λ (x) by,λ (x) + by,λ (x) ∧ by,λ (x), (λ + |x − y|2 )2

(10)

190


demonstrating that the curvature FBy,λ of the Euclidean term as well as by,λ ∧ by,λ are smooth in the ball Bε4 (y), meanwhile d By,λ by,λ is singular. Let ·, · denote the pointwise scalar product and | · | the induced pointwise norm on su(2)+ -valued 2-forms associated to the Killing norm on su(2)+ and the metric on 2 MV . This gives rise to the scalar product (·, ·) L 2 and norm · L 2 on the punctured ball. Separating the regular and singular terms we obtain FA˜ y,λ 2L 2 ( B ∗ (y)) = FBy,λ + by,λ ∧ by,λ 2L 2 ( B ∗ (y)) + d By,λ by,λ 2L 2 ( B ∗ (y)) ε ε ε +2 FBy,λ + by,λ ∧ by,λ , d By,λ by,λ L 2 ( B ∗ (y)) . ε

Since the dual of the volume form ∗(dVgV | Bε (y) (x)) with respect to the flat Hodge star ∗ on H is O(|x − y|3 ), we find on the one hand that the regular energy term is of course finite: ε

FBy,λ

+ by,λ ∧ by,λ 2L 2 ( B ∗ (y)) ε

|FBy,λ + by,λ ∧ by,λ | dVgV ≤ c3

=

r 3 dr < +∞.

2

Bε∗ (y)

0

The singular term |d By,λ (x) by,λ (x)|, on the other hand, is O(|x − y|−1 ), consequently ε

d By,λ by,λ 2L 2 ( B ∗ (y)) ε

|d By,λ by,λ | dVgV ≤ c4

=

2

Bε∗ (y)

r −2 r 3 dr < +∞.

0

Finally, the cross term also satisfies the estimate (here and only here | · | is the ordinary pointwise absolute value on reals) FBy,λ + by,λ ∧ by,λ , d By,λ by,λ L 2 ( B ∗ (y)) ε FB + by,λ ∧ by,λ , d B by,λ dVg ≤ y,λ y,λ V Bε∗ (y)

ε ≤ c5

r −1r 3 dr < +∞.

0

We conclude that the last two conditions in (5) hold, hence the apparent singularity in ∇ A˜ y,λ at y can be removed by a gauge transformation via Uhlenbeck’s theorem [33], providing us with a smooth anti-self-dual SU(2)+ connection on the whole positive spinor bundle + . Proceeding further we demonstrate that our solutions are anti-instantons, i.e. have finite energy. To carry out this we construct an asymptotic expansion for (9) to see explicitly its fall-off. Intuitively [10], a harmonic function with a fixed singularity on (MV , gV ) asymptotically looks like pulling the singular point of the function as well as all the NUTs in MV together. Therefore our strategy will be as follows: we construct an expansion of a harmonic function whose singularity coincides with that of a singular multi-Taub–NUT space. After finding its asymptotic expansion we show that the decay


191

rate of the leading term remains unchanged if we perturb this singular space into the original smooth one. In light of Lemma 3.1, this collapsed space arises when all the roots of the right-hand side in (8) coincide: p1 = · · · = ps = 0, hence the resulting space X ∗ ⊂ C3 is given by x y − z s = 0. On the other hand, if (u, v) ∈ C2 , then the polynomials u s , v s and uv are invariant under the standard action of Zs ⊂ SU(2) on C2 , and putting x := u s , y := v s , z := uv, then they also satisfy x y − z s = 0, i.e. X ∗ is a model in C3 for the As−1 -singularity MV∗ := C2 /Zs . Removing the origin {0} ∈ C2 the resulting smooth space W ∗ := C2 \{0}/Zs is the neck at infinity and is topologically a lens space: W ∗ ∼ = L(s, −1) × R+ . Here 3 ∼ L(s, −1) = S /Zs denotes the usual lens space of (s, −1)-type with its orientation inherited from the complex structure on W ∗ . Regarding W ∗ as a complex line bundle of Chern class −s over S 2 , let us denote by (r, τ ) polar coordinates on the fibres centered in the singular origin of C2 /Zs , and by (, φ) the usual spherical coordinates on the base.2 We simply get V (r ) = 1 +

s s , α(, φ) = cos dφ, 2r 2

consequently the multi-Taub–NUT metric (6) reduces to a singular metric gV∗ on MV∗ whose shape on W ∗ is ds 2 =

2r + s s 2 2r (dr 2 + r 2 d2 + r 2 sin2 dφ 2 ) + (dτ + cos dφ)2 2r 4 2r + s

with parameters 0 < r < +∞, 0 ≤ τ
0 and r 0 and r >> 1 we obtain sl

l −lr − 2 −1 K− r (1 + O(r −1 )) and K lj (r ) = elr r j−1 (r ) = e

sl 2 −1

(1 + O(r −1 )).

Separating variables therefore forces us to put the pointlike singularity of f ∗ into the origin of MV∗ , showing our approach is consistent. To summarize, the general solution formally looks like f ∗ (r, τ, φ, ) =

j ∞

2j

[s ] k,l l l λk,l K (r ) + μ K (r ) Y jk,l (τ, φ, ) − j−1 j j j

(13)

j=0 k=− j l=−[ 2 j ] s

with the only singularity in the origin. Because Y jk,l = Y j−k,−l , a real basis for the solutions is given by 1 k,l 1 k,l (Y + Y j−k,−l ) and (Y − Y j−k,−l ). 2 j 2i j −l −l l l Moreover from (12) we obtain that K − j−1 = K − j−1 and K j = K j , consequently −k,−l −k,−l if we suppose λk,l as well as μk,l , then we come up with formal j = λj j = μj real solutions in (13). In addition asymptotically bounded solutions from (13) arise by setting μk,l j = 0 for j > 0.

Taking into account that Y00,0 is a constant, the leading j = 0 term shows that the asymptotic shape of asymptotically bounded real harmonic functions looks like 0,0 f ∗ (r ) = μ + λr −1 + O(r −2 ) (we put simply μ0,0 0 = μ and λ0 = λ, both real) on the ∗ ∗ singular space (MV , gV ). Now we ask ourselves how this picture changes over the original non-singular space (MV , gV ) for the Green function G(·, y) with singularity at y ∈ MV . For this purpose, we use results of [30], where a careful study of analytic properties of the Laplace operator on perturbations of ALF spaces has been carried out. If W ∗ and W are the necks of MV∗ and MV respectively, then we suppose W ∗ ∼ = W and use the same coordinate system along them. It is clear that for α = 2 we have gV |W = gV∗ |W ∗ + O(r −α )h with some O(1) symmetric tensor field h. Let χ be a cut-off function supported in a compact neighborhood of y and identically equal to 1 in a smaller neighborhood of y. We will try to look for the Green function in the form G(·, y) = χ G (4) (·, y) + u,

194


where G (4) (·, y) stands for the Green function on R4 and u is some locally L 2 function. Then, u must satisfy the equation gV u = f

(14)

for some prescribed compactly supported smooth function f only depending on χ and G (4) (·, y). In particular, f belongs to any weighted Sobolev space L 2δ (MV ; R). Setting δ = 2 it follows from [30, Cor. 2] that Eq. (14) can be solved with u ∈ L 22 (MV ; R). Put now δ := − 21 + ε for any ε > 0, and denote the interval (δ , δ) by I . Notice that the only critical weights in I are δ0 = 23 and 2 − δ0 = 21 . It then follows from [30, Prop. 4 and Cor. 1] that up to perturbations in L 2 1 (MV ; R) and L 2 3 (MV ; R) with some − 2 +ε

− 2 +ε ,2

ε > 0, u is equal to a linear combination of solutions −

0 (r ) = r ν0 = r −1 K 00 (r ) = r ν0 = 1 and K −1 +

of the non-perturbed Laplacian. Notice that the corresponding spherical harmonic Y00,0 = √ s 4π

is constant and that the decay corresponding to L 2

moreover the decay corresponding to L 2 (9) asymptotically also looks like

− 23 +ε ,2

− 21 +ε

(MV ; R) is r ν with ν = −2+ε,

(MV ; R) is even stronger. Consequently

f y,λ (r ) = 1 + λr −1 + O(r −2+ε )

(15)

and the leading term is of course positive for λ > 0. One can also immediately read off this from the explicit Green function of Page [31, Eq. (18)]. By the aid of these observations we obtain Lemma 4.2. The smooth anti-self-dual connection ∇ A˜ y,λ with y ∈ MV and λ ∈ (0, +∞) is subject to both the weak holonomy condition with respect to the trivial flat connection ∇ and the rapid decay condition of Definition 2.1, consequently it has not only finite energy but is moreover admissible, i.e. its gauge equivalence class gives rise to an admissible anti-instanton on + . The energy of this anti-instanton is of one unit. Proof. We already have seen that the spin connection on + is gauge equivalent to the trivial flat connection ∇ . Consider the neck W ⊂ MV of the multi-Taub–NUT manifold and using the gauge (7) along W write ∇ A˜ y,λ |W = d + A˜ y,λ with A˜ y,λ = a y,λ as in (2), where a y,λ is the perturbation term involving (9). The asymptotic shape of (9) looks like (15) in light of our previous calculations. This ensures us that if λ is finite and positive, |d log f λ |gV as well as |∇ (d log f λ )|gV are O(r −2 ) along W . Consequently, since 3 |d log f y,λ |gV |a y,λ | = 2 in the gauge (7), we also obtain that both |a y,λ | and |d a y,λ | are O(r −2 ) along W for all y ∈ MV and 0 < λ < +∞. We see that the first condition in (5) is also satisfied. Inserting this and F = 0 into (4) we can see that our connection can be viewed as a rapidly decaying perturbation of the trivial connection, hence satisfies both the weak holonomy condition with respect to the trivial flat connection as well as the rapid decay condition.


195

It follows [13, Theorem 4.2] that in this situation the energy of the connection ∇ A˜ y,λ is integer, hence it must be one, because the harmonic function in (9) has only one singularity. Remark. Before summarizing our findings we may raise the question: what about the higher overtones with j > 0 in the harmonic expansion (13)? The radial function r 2 = |u|2 + |v|2 in C2 is invariant under Zs , consequently we can talk about lens spaces of radius r centered at the origin of C2 /Zs . Take such a lense space with its metric g inherited from the round metric of S 3 . By orthogonality the spherical harmonics obey for j > 0 that 1 1 k,l −k,−l Yj + Yj dVg = Y jk,l − Y j−k,−l dVg = 0, 2 2i L(s,−1)

L(s,−1)

hence these real functions must change sign somewhere along any lense space showing that in fact all individual terms in (13) with j > 0 vanish on three dimensional subsets of MV∗ . Additionally we know from the asymptotic formulae for the radial functions l l K− j−1 and K j that they behave very differently for small or large r ’s, consequently the aforementioned three dimensional zero sets of the individual higher overtones cannot be removed by forming particular linear combinations of them. Consequently these harmonic functions represent “non-mild” singularities, hence cannot be used to construct anti-instantons. Let ∇ |W be a smooth flat SU(2)+ connection on + |W . Then following [13] if ∞ C0 (MV ; End + ) is the space of smooth, compactly supported endomorphisms of the positive chiral spin bundle, then define the L 2 gauge group G + as the L 22, completion of the space

{γ − 1 ∈ C0∞ (MV ; End + ) | γ − 1 L 2

2, (M V )

< +∞, γ ∈ C ∞ (MV ; Aut + ) a.e.}.

Let us denote by M (e, ) the framed moduli space consisting of pairs ([∇ A ], ), the G + gauge equivalence classes of smooth irreducible SU(2)+ anti-self-dual connections on + of energy e < +∞, decaying rapidly and obeying the weak holonomy condition with respect to ∇ |W and a fixed gauge at infinity preserved by G + . Associated to this (e, ) consisting of the G + equivalence space one has the unframed moduli space M classes [∇ A ] only. In other words it is formed by dividing via the group G + ∼ = G + × SU(2)+

(16)

of gauge transformations tending to any fixed element γ0 ∈ SU(2)+ (not to the identity only) at infinity. (k, ) corresponding to In particular we can consider the spaces M (k, ) and M the trivial flat connection ∇ ; in this case the energy is an integer k ∈ N, cf. [13, Theorem 4.2]. It was demonstrated in [13, Theorem 3.2] that these spaces are smooth manifolds and [13, Theorem 4.2] ensures us that dim M (k, ) = 8k and therefore (k, ) = 8k − 3. dim M Putting together our findings so far as well as noting that all of our anti-instantons are irreducible [12, Theorem 5.1] we obtain the following

196


Theorem 4.1. Let (MV , gV ) be the multi-Taub–NUT space with s > 0 NUTs equipped with an orientation coming from any of its complex strucures in the hyper-Kähler family. Then there exists a 5 real parameter family of unframed, non-gauge equivalent, irreducible, smooth SU(2)+ anti-instantons {[∇ A˜ y,λ ] | (y, λ) ∈ MV × (0, +∞)} on the positive chiral spinor bundle + which satisfies the weak holonomy condition with respect to the trivial flat connection and also satisfies the rapid decay condition. All these anti-instantons have unit energy and provide us with an open subset of (one (1, ). connected component of) the unframed moduli space M Theorem 4.1 can be regarded as a familiar “collar theorem” for the 1-anti-instanton moduli space over the multi-Taub–NUT space. The patient reader may write down these solutions explicitly by inserting the explicit Green functions [31, Eq. (18)] of Page into (9) and then by the aid of the gauge (7) construct the corresponding connection 1-form via (2). However the quite complicated result is not informative. Rather we turn attention to a puzzling feature of the description provided by Theorem 4.1. This is the limit λ → +∞. In this case (9), up to reparametrization, remains meaningful and provides us with the positive minimal Green function G(·, y). It readily follows from the proofs of Lemmata 4.1 and 4.2 that the resulting anti-self-dual connection ∇ A˜ y,+∞ gives rise to an unframed anti-instanton [∇ A˜ y,+∞ ] on + obeying the weak holonomy condition with respect to ∇ and the rapid decay condition, hence these solutions are expected to complete the moduli space. Indeed, for a fixed y ∈ MV taking the limit λ → +∞ (especially in (10)) the proof of Lemma 4.1 continues to hold, hence the corresponding anti-self-dual connection is smooth everywhere on + . Additionally, in light of (15) the asymptotics of G(·, y) looks like 1/r, therefore the proof of Lemma 4.2 also can be repeated if we notice that this time |a y | and |d a y | (MV ; 1 MV ⊗ End + ), but it follows from a are O(r −1 ) only. Hence a y ∈ L 21 2 ,1,

direct calculation that the corresponding curvature still belongs to the weighted Sobolev space: FA˜ y ∈ L 21 (MV ; 2 MV ⊗ End + ). We conclude that [∇ A˜ y,+∞ ] provides us with 2

an admissible unframed anti-instanton with unit energy on the positive spin bundle; more precisely it decays rapidly and one can also check that the weak holonomy condition with respect to the same trivial flat connection ∇ is fulfilled. If y ∈ MV is not a NUT then the corresponding anti-instanton is irreducible, otherwise reducible to U(1) and these Abelian solutions span the full L 2 cohomology of the multi-Taub–NUT geometry as it was stated in [12, Theorem 5.1]. Therefore if y ∈ MV is not a NUT then these limiting irreducible solutions are expected to give the completion of the collar MV × (0, +∞) to the full unframed moduli space in the unconcentrated or “centerless” regime—or at least one connected component of the moduli space emerges this way. Since it is natural to include the reducible (1, ) as well, for simplicity we shall consolutions corresponding to the NUTs into M (1, ) tinue to denote (one connected component of) this extended moduli space by M from now on and note that it is not a manifold anymore: usual conical singularities appear around the reducible points as we know from the general theory but will also see shortly. We also conjecturially note that in our opinion not only a connected component but the whole moduli space of unit energy admissible anti-instantons with trivial holonomy arises this way.


197

Theorem 4.2. Consider the multi-Taub–NUT space (MV , gV ) with s > 0 NUTs p1 , . . . , ps ∈ MV , equipped with the natural orientation as above. Then for (one connected component of) the unframed moduli space of unit energy SU(2)+ admissible anti-instantons decaying rapidly to the trivial flat connection on + we find (1, ) ∼ M = (MV × (0, +∞])/ ∼,

(17)

where the equivalence relation ∼ means that MV ×{+∞} is pinched into R3 by collapsing the S 1 -isometry orbits of (MV , gV ). Consequently there exists a singular fibration (1, ) −→ R3 :M

(18)

with generic fibers homeomorphic to the open 2-ball B 2 and as many as s singular fibers homeomorphic to the semi-open 1-ball (0, +∞]. Therefore (one connected component of) the moduli space is contractible and in particular is orientable. (1, ) with i = 1, 2, . . . , s represent reducThe images of the points ( pi , +∞) in M ible anti-instantons and M (1, ) around these points looks like a cone over CP 2 or 2 CP (depending on the orientation). Proof. Although we stated it already in Theorem 4.1, first of all we remark that from the expression (10) of the curvature one can immediately read off that both y ∈ MV and λ ∈ (0, +∞) are gauge-invariant parameters, i.e. anti-instantons corresponding to different values of (y, λ) cannot be gauge equivalent. Next, in Sect. 5 we will prove that if y ∈ MV and eiτ y is its image by an element of the S 1 isometry group of (MV , gV ), then the corresponding two anti-instantons with (y, λ = +∞) and (eiτ y, λ = +∞) are actually gauge equivalent unit energy, rapidly decaying anti-instantons with trivial holonomy at infinity. In addition to these, we know from [13, Theorem 3.2] that away from the reducible points the moduli space is a finite dimensional, possibly non-connected manifold without boundary. Consequently we are forced to look at our moduli space as follows. Fix a point y ∈ MV different from any NUT and consider the S 1 isometry orbit through y: it is parameterized by the cyclic coordinate τ ∈ [0, 2π ) of the local coordinate system on (MV , gV ) introduced in Sect. 3; as well as take the concentration parameter λ ∈ (0, +∞). Then by Theorem 4.1, associated to any y ∈ MV , we obtain a subset of the moduli space parameterized by (τ, λ). We regard this as a punctured open B 2 such that the points with λ = 0 are supposed to represent the boundary while λ = +∞ is the center of this (τ, λ)-ball. In this picture we interpret the existence of smooth solutions with λ = +∞ as filling in the punctured (τ, λ)-ball over y ∈ MV with the anti-instanton [∇ A˜ y,+∞ ] corresponding to the limit λ → +∞. In other words, this anti-instanton represents the center of the unpunctured (τ, λ)-ball that consequently will be homeomorphic to an open B 2 over any y ∈ MV different from the NUTs. Since the NUTs p1 , . . . , ps ∈ MV are the only fixed points of the isometry group action on (MV , gV ) it follows that over these points the moduli space is parameterized by λ ∈ (0, +∞] only, hence is a semi-open B 1 . In this case the “center” is the endpoint λ = +∞. Therefore completing our collar MV × (0, +∞) with these limiting solutions, the moduli space takes the shape as in (17), where the equivalence relation ∼ means that MV × {+∞} is collapsed into R3 by shrinking all the isometry orbits of (MV , gV ) into

198


single points respectively and admits a singular fibration (18) whose generic fiber is an open B 2 and with s pieces of singular fibers homeomorphic to a semi-open B 1 . In (1, ) is contractible, hence orientable. particular M (1, ) We already know that the only reducible solutions are the images ri ∈ M of the points ( pi , +∞) with i = 1, 2, . . . , s. We will study the structure of the moduli space around these points. Take a NUT pi ∈ MV and consider its neighbourhood (1, ). We will construct a decomposition ∂U (ri ) = X 1 ∪ X 2 of its boundU (ri ) ⊂ M (1, ) is ary as follows. It is clear that if y is not a NUT and the image of (y, +∞) in M 3 2 5 m then U (m) ∼ = B , an open 5-ball. Let us decompose its closure as U (m) ∼ = B ×B , 2 3 where B , is the aforementioned (τ, λ)-disk and B is a ball around the image of y in 2 3 R3 . Then ∂U (m) ∼ = (S 2 × B ) ∪ (B × S 1 ) which is of course an S 4 glued together 2 3 from S 2 × B and B × S 1 along their common boundary S 2 × S 1 . Now let us move 2 2 y toward a particular NUT pi ; then one of the B ’s in the S 2 × B component of the above decomposition, namely that one which is moved exactly into the position of the 1 NUT, gets shrink to a B in a manner such that meanwhile its radius (measured by λ) is kept constant, its circumference (measured by τ ) vanishes3 . Put 2 X1 ∼ = S 2 × B / ∼1

corresponding to this collapsed space. We obviously find for the singular cohomology that Z if j = 0, 2 H j (X 1 ; Z) ∼ = 0 otherwise. 3

Since this movement also collapses an S 1 , the boundary in the other component B × S 1 2 of the previously collapsed B , we also get a corresponding collapsed space 3 X2 ∼ = B × S 1 / ∼2 .

Since the generator of π1 (B × S 1 ) ∼ = Z is just killed out in X 2 , we also obtain Z if j = 0 H j (X 2 ; Z) ∼ = 0 otherwise. 3

It is also clear that X 1 ∩ X 2 is nothing else than a 3-sphere with exactly two antipodal points pinched consequently Z if j = 0, 3 H j (X 1 ∩ X 2 ; Z) ∼ = 0 otherwise. Recalling the Mayer–Vietoris sequence of the decomposition ∂U (ri ) = X 1 ∪ X 2 which looks like . . . → H j ∂U (ri ); Z → H j (X 1 ; Z) ⊕ H j (X 2 ; Z) → H j (X 1 ∩ X 2 ; Z) → H j+1 ∂U (ri ); Z → . . . 3 This way of collapsing B 2 into B 1 looks like closing an umbrella.


199

we end up with H ∂U (ri ); Z ∼ = j

Z 0

if j = 0, 2, 4 otherwise.

Moreover we know that ∂U (ri ) is simply connected and is smooth. However from all of 2 these we obtain via Freedman’s classification [14] that ∂U (ri ) ∼ = CP 2 or CP depend(1, ). Therefore the unframed moduli space about a ing on the orientation put onto M reducible point looks like a cone over one of these spaces as stated. We make a comment on the higher-energy moduli spaces including anti-instantons with trivial holonomy in infinity. It is obvious that instead of the functions (9) we can use harmonic functions with a finite number of isolated singularities yielding (k, ) are non-empty for all Theorem 4.3. The moduli spaces M (k, ), hence M k ∈ N. For sake of completeness without proof we exhibit here the degenerate s = 0 case as well, i.e. the flat R3 × S 1 whose associated instantons also referred to as calorons. This space is flat hence conformal rescaling works. Using coordinates (x 1 , x 2 , x 3 , τ ) = (x, τ ) on R3 × S 1 we have an obvious five paramerer family of positive harmonic functions (9) with Green functions in a rather explicit form [31] G((x, τx ), (y, τ y )) = =

tanh |x − y|

1 16π 2 |x

− y| 1 −

cos(τx −τ y ) cosh |x−y|

+∞ 1 1 . 2 2 4π |x − y| + (τx − τ y + 2π k)2 k=−∞

Making use of [13, Theorem 2.3] to describe the holonomy issue in this situation as well as [13, Theorem 4.1] which provides us with the dimension moreover taking into account that there are no reducible SU(2)± instantons on this space we state Theorem 4.4. Take the flat R3 × S 1 with an arbitrary fixed orientation. Then any nontrivial SU(2)± instanton on ± is irreducible, moreover the rapidly decaying ones obey the weak holonomy condition with respect to some flat connection ∇ on the neck and their energies are always non-negative integers k ∈ N. The corresponding framed moduli (k, ) = 8k − 3 spaces, if not empty, have dimensions dim M (k, ) = 8k while dim M for the unframed ones. Assume the asymptotic flat connection is the trivial one ∇ and let M (1, ) be the corresponding framed moduli space. Then for (one connected component of) the unframed space we find (1, ) ∼ M = (R3 × S 1 × (0, +∞])/ ∼ ∼ = R3 × B 2 (k, ), are also moreover the higher energy framed moduli spaces M (k, ), hence M non-empty with k ∈ N arbitrary.

200


5. Completion of the Moduli Space In this closing section our aim is to prove that conformal rescaling gives rise to at least one connected component of unit energy, rapidly decaying anti-instantons obeying the weak holonomy condition with respect to the trivial flat connection (however we note again that in our opinion this moduli space is connected). Our starting point is the construction of the twistor space of the multi-Taub–NUT geometry by the aid of Hitchin [6,22,24] and also a certain compactification of it. Then following Atiyah [2] this enables us to identify the harmonic functions used so far with elements of certain Ext groups on the compactified twistor space. These Ext groups also provide us with twisted holomorphic vector bundles over the original non-compact twistor space and the corresponding untwisted ones represent our anti-instantons in a new form in the spirit of the Atiyah–Ward correspondence. To get these twisted vector bundles more explicitly we investigate the real structure of the twistor space leading to the multi-Taub–NUT geometry and also construct the corresponding real twistor lines; our twisted vector bundles then, referring to Serre’s method, can be constructed via sections vanishing along these real lines. The transition functions of these twisted bundles corresponding to pure Green functions, i.e., in the λ → +∞ limit can be constructed rather explicitly. We will find that the twisted bundles corresponding to G(·, y) and G(·, eiτ y) are actually isomorphic, hence the corresponding anti-instantons must be gauge equivalent verifying our picture on the moduli space presented in Theorem 4.2. To begin with, we claim that there exist nice compactifications of the complex manifolds underlying a given a multi-Taub–NUT space. Lemma 5.1. Consider any multi-Taub–NUT space (MV , gV ) with s NUTs and let X denote the smooth complex surface constructed in Lemma 3.1 given by picking one generic complex structure in the hyper-Kähler family. Also let X be the smooth complex surface belonging to any exceptional complex structure, as constructed in Lemma 3.1. Then both X and X admit compactifications X and X respectively which are smooth compact rational surfaces. These spaces arise by attaching s + 3 lines to the finite parts in a suitable way. Proof. Fix a generic configuration of s points q1 , . . . , qs ∈ R3 and consider the corresponding multi-Taub–NUT space (MV , gV ). Pick a direction e1 in R3 not parallel with any straight line segment i j connecting qi and q j and put the corresponding (generic) integrable complex structure Je1 onto MV from the hyper-Kähler family. We know from Lemma 3.1 that (MV , Je1 ) is biholomorphic to the algebraic surface X ⊂ C3 given by (8) and there are s(s −1) exceptional directions to choose for e1 so that the corresponding complex manifolds are not biholomorphic to X but rather to its blowup denoted by X in agreement with the notations of Lemma 3.1. First consider the case of X . Introducing homogeneous coordinates ([x : u], [y : v], [z : w]) ∈ CPx1 × CPy1 × CPz1 , Eq. (8) can be made homogeneous of degrees (1, 1, s) respectively as follows: x yw s − uv

s

(z − wpi ) = 0.

(19)

i=1 ∗

Let us denote the resulting compact complex surface in CPx1 × CPy1 × CPz1 by X . We ∗ claim that X is smooth except one point. Indeed, let 0x := [0 : 1] and ∞x := [1 : 0]


201

denote the origin and the infinity of CPx1 respectively and in the same fashion introduce the notation 0 y , ∞ y ∈ CPy1 and 0z , ∞z ∈ CPz1 . Then in the compactification C3 ⊂ CPx1 × CPy1 × CPz1 infinity is represented by the bouquet ∞ := ({∞x } × CPy1 × CPz1 ) ∨ (CPx1 × {∞ y } × CPy1 ) ∨ (CPx1 × CPy1 × {∞z }), ∗

∗

and one obtains that X ∩ ∞ consists of four lines, that is X = X (1 ∪ 2 ∪ 3 ∪ 4 ) with 1 := {∞x } × {0 y } × CPz1 , 2 := {∞x } × CPy1 × {∞z }, 3 := CPx1 × {∞ y } × {∞z }, 4 := {0x } × {∞ y } × CPz1 . These lines are not disjoint, they intersect each other as follows: 1 ∩ 2 = (∞x , 0 y , ∞z ), 2 ∩ 3 = (∞x , ∞ y , ∞z ), 3 ∩ 4 = (0x , ∞ y , ∞z ). By calculating the gradient of the left hand side in (19) we see that the only point where ∗ ∗ X is singular is the intersection 2 ∩ 3 , where X locally looks like uv −

s

w(1 − wpi )−1 = 0,

(20)

i=1

i.e. it possesses a classical As−1 quotient singularity. Carrying out the same procedure for the exceptional singular surface X ∗ ⊂ C3 ∗ with the notations of Lemma 3.1, we obtain a similar compactification X ∗ in CPx1 × 1 1 CPy × CPz . The only difference is that in addition to the As−1 singularity at infinity ∗ as above, X ∗ possess more rational double points in the finite part in each point where [ pi : 1] = [ p j : 1]. We proceed further and desingularize these singular compactifications. In light of Lemma 3.1 we simply do this by replacing them with their non-singular blowups. Again ∗ consider first the generic case X and let us denote the resulting smooth space by X . ∗ Clearly, in this case we have to just remove the As−1 -singularity at infinity from X , therefore we have to attach 4 + s − 1 = s + 3 lines in an approriate way: X = X (1 ∪ 2 ∪ 1 ∪ · · · ∪ s−1 ∪ 3 ∪ 4 )

(21)

∗

to obtain X . Denoting by X the desingularization of X ∗ , it stems exactly the same way from the smooth X , the desingularization of X ∗ ⊂ C3 . Finally we prove that both of these smooth compactifications are rational surfaces. ∗ ∗ Indeed, consider first the general case when X is smooth except at infinity. Then X is given by Eq. (19) for distinct values of p j ’s. Consider the map ∗

X −→ CPx1 × CPz1 , ([x : u], [y : v], [z : w]) −→ ([x : u], [z : w]). Over a given ([x : u], [z : w]) ∈ CPx1 × CPz1 there exists a unique [y : v] ∈ CPy1 such ∗ that ([x : u], [y : v], [z : w]) ∈ X , except in the case xw s = u (z − wp j ) = 0 when any [y : v] is a solution. It readily follows that these exceptional points are precisely {(0x , [ p1 : 1]), . . . , (0x , [ ps : 1]), (∞x , ∞z )} ∈ CPx1 × CPz1 .

(22)

202

G. Etesi, Sz. Szabó ∗

Furthermore, all the points of X are non-singular except the isolated, hence codimension 2 surface singularity of type As−1 at (∞x , ∞ y , ∞z ) ∈ CPx1 × CPy1 × CPz1 which ∗ is mapped onto (∞x , ∞z ) in CPx1 ×CPz1 . It follows that X is isomorphic to the blowup of CPx1 × CPz1 in the points (22) outside of a codimension 2 subset. The same argument ∗ works for the special case of X ∗ possessing a further rational double point, if we add ∗ ∗ this singular point to the codimension 2 subset above. Therefore both X and X ∗ are ∗ ∗ rational. Since both X and X arise by blowing further up X and X ∗ respectively in finitely many points, we conclude that the non-singular compact spaces X and X are also rational. Remark. It is worth comparing the complex compactification (21) with the simple smooth real compactification of MV in [11,13] motivated by L 2 -cohomology theory of ALF spaces [19]. We proceed further and smoothly compactify the twistor space of a multi-Taub–NUT space in two steps. Lemma 5.2. The first approximation of the twistor space (what we denote here by) Z of any multi-Taub–NUT space (MV , gV ) with s NUTs admits a smooth complex compactification (what we denote here by) Z . This compactification arises by adding finitely many various Hirzebruch surfaces to this Z in a suitable way. Proof. First we construct a sort of approximation of the twistor space of (MV , gV ) and then modify it into the true twistor space in the next lemma. Being (MV , gV ) simply connected with vanishing Ricci curvature, its true twistor space admits a holomorphic fibration over CP 1 , consequently we would like to regard Eq. (19) as a set of equations parameterized by a projective line. Therefore we proceed as follows (cf. [6, pp. 393–395] and [22]). Consider the holomorphic fiber bundle π : P(H k ⊕ H 0 ) ⊕ P(H l ⊕ H 0 ) ⊕ P(H m ⊕ H 0 ) −→ CP 1

(23)

whose fibers are Fx × Fy × Fy with Fi ∼ = CPi1 (i = x, y, z) as before. Take the subbundle k 0 1 ∼ π x : P(H ⊕ H ) → CP with Fx = CPx1 . Referring to the isomorphism ⊗H k

P(H 0 ⊕ H −k ) −→ P(H k ⊕ H 0 ), take the canonical section ∞x := (1, 0) of P(H 0 ⊕ H −k ) and regard its image as a divisor which we also denote by ∞x ; one then has an associated line bundle L x := [∞x ] and corresponding sheaf Orel,x (1) on P(H 0 ⊕ H −k ). This line bundle has a canonical section u ∈ Orel,x (1) satisfying (u) = ∞x . We also have a canonical section 0x := (0, 1) of P(H k ⊕ H 0 ) which gives rise to the bundle L x ⊗ π ∗x H k and a canonical section x which is an element of Orel,x (1) ⊗ O P(H k ⊕H 0 ) (k), where O P(H k ⊕H 0 ) (k) is the associated sheaf of π ∗x H k . Take the fibration π y,z : P(H k ⊕ H 0 ) ⊕ P(H l ⊕ H 0 ) ⊕ P(H m ⊕ H 0 ) −→ P(H k ⊕ H 0 ) whose fibers are Fy × Fz . Pulling back x, u we therefore obtain elements π ∗y,z x ∈ π ∗y,z Orel,x (1) ⊗ π ∗y,z O P(H k ⊕H 0 ) (k), π ∗y,z u ∈ π ∗y,z Orel,x (1).


203

In the same fashion we construct π ∗x,z y ∈ π ∗x,z Orel,y (1) ⊗ π ∗x,z O P(H l ⊕H 0 ) (l), π ∗x,z v ∈ π ∗x,z Orel,y (1), and finally π ∗x,y z ∈ π ∗x,y Orel,z (1) ⊗ π ∗x,y O P(H m ⊕H 0 ) (m), π ∗x,y w ∈ π ∗x,y Orel,z (1). For simplicity we shall denote them as x, u, y, v, z, w respectively and observe that they provide us with coordinates on the total space of the fibration (23). The situation gets simplified somewhat by virtue of the following observation. It is not difficult to check that the bundle π ∗y,z (π ∗x H k ) on the total space of (23) depends only on k and not on the way we pulled it back from CP 1 . Consequently π ∗y,z (π ∗x H k ) ∼ = π ∗x,y (π ∗z H k ) ∼ = π ∗x,z (π ∗y H k ) ∼ = π∗ Hk, and the same is true for the other bundles H l and H m . Write W := P(H k ⊕ H 0 ) ⊕ P(H l ⊕ H 0 ) ⊕ P(H m ⊕ H 0 ) for the total space of (23) and put L a0 ,a1 ,a2 ,a3 := π ∗ H a0 ⊗ π ∗y,z L ax1 ⊗ π ∗x,z L ay2 ⊗ π ∗x,y L az 3 with ai ∈ Z; we will also use the convenient notation for the corresponding sheaf: OW (a0 , a1 , a2 , a3 ) := OW (a0 ) ⊗ π ∗y,z Orel,x (a1 ) ⊗ π ∗x,z Orel,y (a2 ) ⊗ π ∗x,y Orel,z (a3 ) (therefore OW (a0 ) = OW (a0 , 0, 0, 0)). Then x ∈ OW (k, 1, 0, 0) and u ∈ OW (0, 1, 0, 0), etc. and if we regard pi ∈ OW (m, 0, 0, 0) we obtain x yw s ∈ OW (k + l, 1, 1, s) and uv

s

(z − wpi ) ∈ OW (sm, 1, 1, s),

i=1

consequently setting k + l = sm, we end up with a well-defined map P : W −→ L sm,1,1,s , where formally P is given by the left-hand side of (19) and a singular model ∗ for the first approximation of the compactified twistor space Z ⊂ W is provided by the hypersurface P = 0, hence restricting (23) it admits a holomorphic fibration ∗ π : Z −→ CP 1 . To simplify the notation of Lemma 5.1 from now on we shall denote the fibers ∗ ∗ π −1 ([a : b]) of this fibration as X [a:b] for all [a : b] ∈ CP 1 . Note that Z has an obvious singularity along the line which is the section passing through the singu∗ ∗ lar points (∞x , ∞ y , ∞z )[a:b] ∈ X [a:b] for an [a : b] ∈ CP 1 . However Z has more singularities, namely the points (0, 0, pi ([a : b])), where exactly two roots coincide: pi ([a : b]) = p j ([a : b]). (We may assume that the configuration is generic, hence no more than two roots coincide at the same time.) Hence these points represent rational ∗ double points in Z . Therefore the number of such bad points is m · 2s = m · 21 s(s − 1), since pi ∈ OW (m, 0, 0, 0). The geometric origin of these bad points is clear: they just correspond to the special complex structures analyzed in Lemma 3.1 under which (MV , Je1 ) is not biholomorphic to (8). Since the number of these complex structures is

204


s(s − 1) as we have seen, we have to set m = 2 yielding k + l = 2s, and since x and y play symmetric roles, we take k = l = s in (23). ∗ We desingularize Z to obtain a smooth compact complex 3-manifold Z whose finite part Z ⊂ Z provides us with a non-singular model for the approximated twistor space of (MV , gV ). In light of Lemmata 3.1 and 5.1 we simply do this by replacing the fibers ∗ ∗ X [a:b] of π : Z −→ CP 1 with their non-singular blowups for all [a : b] ∈ CP 1 . Let us denote the resulting smooth fiber by X [a:b] . Since these fibers are diffeomorphic to each other via Lemma 3.1, we end up with a smooth holomorphic fibration; we denote it as π : Z −→ CP 1 . Finally, referring to (21) we conclude that the compactified smooth Z arises by adding s + 3 Hirzebruch surfaces to the smooth twistor space Z of (MV , gV ) as follows: Z = Z (S1 ∪ S2 ∪ S1 ∪ · · · ∪ Ss−1 ∪ S3 ∪ S4 ). ∗

Two of them, S1 and S4 stemming from the lines 1 and 4 in X , are biholomorphic to P(H 2 ⊕ H 0 ), while S2 and S3 given by 2 and 3 are biholomorphic to P(H s ⊕ H 0 ). To determine the type of the exceptional ones, notice that the local resolution of the singularity (20) is given by defining the new variables u s := w −s u, vs := v, ws := w, whence the equation of the proper transform becomes u s vs −

s (1 − ws pi )−1 = 0. i=1

This means that we add a family of projective lines parametrized by CP 1 with sections ws . As ws is untwisted in the base direction, this implies that the Hirzebruch surfaces S1 , . . . , Ss−1 we add through this process are trivial CP 1 × CP 1 ’s. Remark. The reader may recognize that this “first approximation” of the twistor space of the multi-Taub–NUT space is nothing other than the twistor space of the corresponding ALE Gibbons–Hawking space [22]. Lemma 5.3. The true twistor space Z of any multi-Taub–NUT space (MV , gV ) with s NUTs also admits a smooth complex compactification Z . It also arises by adding s + 3 Hirzebruch surfaces to Z in a suitable way. Proof. The idea is to twist (cf. [6, pp. 393-395] and [24]) the natural algebraic model constructed in Lemma 5.2 with a section of a certain line bundle L c which lives on the total space of the line bundle πz : H 2 → CP 1 , where this base is the same as that of (23). For the definition of L c , let this CP 1 be covered by the affine open sets Ua and Ub with coordinates [a : b] satisfying a = 0 and b = 0 respectively and write Ua = πz−1 (Ua ) and Ub = πz−1 (Ub ) for the open subsets of H 2 covering the affine subsets of CP 1 , moreoverdenote by z anypoint in the total space of H 2 . Therefore we obtain coordinate systems ( 1 : ab , z) and ( ab : 1 , z) on Ua and Ub respectively. Define the holomorphic line bundle L c with a complex parameter c by the transition function b

g : Ua ∩ Ub −→ C∗ , g(z, [a : b]) := e−cz a . It is clear that L 0 is just the trivial bundle over H 2 and L −c is canonically isomorphic to (L c )∗ .


205

Next we construct a section of L c . For this we exploit a nice geometric description of H 2 ∼ = T CP 1 by identifying it with the space of oriented affine lines in R3 (with a fixed origin) as follows. If ⊂ R3 is an oriented affine line and (t) ∈ is its point, t ∈ R, then there clearly exist unique vectors u, v with |u| = 1 and u · v = 0 such that (t) = v + tu. Regarding u as a point on S 2 and v a vector in its tangent space we obtain the isomorphism. There is a natural real structure on the space of oriented real lines, namely (u, v) → (−u, v), which gives rise to the antipodal map on CP 1 inside H 2 as a zero section. Consequently we also obtain a map from the space of holomorphic sections into that of anti-holomorphic sections r : H 0 (CP 1 ; O(2)) −→ H 0 (CP 1 ; O(2)).

(24)

A section is called real if it is invariant under this map. If z ∈ H 2 then there is a unique oriented affine line z ⊂ R3 corresponding to it with a distinguished point z (0) ∈ R3 (depending on the choice of origin in R3 ). This point can be characterized by the set of all lines in R3 intersecting it, which provides us with a real section in H 0 (CP 1 ; O(2)) ∼ = C3 , hence is of the form z([a : b]) = 2 2 αa + 2βab − αb with α ∈ C and β ∈ R. Therefore we have two ways of looking at the same object: we can regard z either as a point on the total space z ∈ H 2 or as a real section αa 2 + 2βab − αb2 ∈ H 0 (CP 1 ; O(2)). Of course, quite tautologically speaking, the section z passes through the point z corresponding to it. Regarding z as a real section z([a : b]) = αa 2 + 2βab − αb2 now, this homogeneous polynomial is canonically identified with the single-valued polynomial in the affine coordinate ab over Ub : z|Ub

a b

=α

a 2 b

+ 2β

a b

− α.

Now we define a nowhere vanishing section ηc of L c by ! a ec(α ( b )+β ) c η (z, [a : b]) := a g(z, [a : b])ec(α ( b )+β )

on Ub ; on Ua .

It is well-defined since it looks like exp c α ab + β on Ub , i.e. where b = 0 and looks b like exp c α a − β on Ua , i.e. where a = 0. c Next we claim that the pair (L c , ηc ) can be extended uniquely to a pair (L , ηc ), c where L is a line bundle over π z : P(H 2 ⊕ H 0 ) → CP 1 which restricts to L c over the finite part as before, moreover c ηc ∈ H 0 P(H 2 ⊕ H 0 ) ; E P(H 2 ⊕H 0 ) (L ) is a section of it which also gives back ηc on L c . In order to construct these extensions we proceed as follows. First, notice that there is a covering of P(H 2 ⊕ H 0 ) by four affine charts: the open sets Ub and Ua of the total space of H 2 defined above and additional charts Vb and Va covering the section at infinity (w = 0). Explicitly, let (z, a) and (z , b) be standard coordinates on Ub and Ua respectively with b = a −1 and z = za −2 , then coordinates on Vb and Va can be chosen to be (w, a) and (w, b) with w = z −1 . Since we want to extend the bundle L c , c the transition function of L from the chart Ub to Ua must be that of L c , i.e. equal to

206


g = exp(−cz/a). On the other hand, the requirement that ηc should extend to a section c of L fixes the transition function between the charts Ub and Vb to be exp(−cαa − cβ). Similarly, the transition between Ua and Va is constrained to be exp(cαb + cβ). From the cocycle-relation we then deduce that the restriction to the total space of H 2 (i.e. the open part of P(H 2 ⊕ H 0 ) away from the section at infinity) of the transition between Vb and Va must be identically 1. In particular, this transition function extends to infinity c and the set of transition functions just described yields the line bundle L with a section c η we were looking for. Then referring to the notations of Lemma 5.2 (and knowing already that k = l = s and m = 2 as before) consider the fibration π x,y : W −→ P(H 2 ⊕ H 0 ) c

and use it to pull back both L and ηc over W . As an extension of the construction in Lemma 5.2 set L a0 ,a1 ,a2 ,a3 ,c := L a0 ,a1 ,a2 ,a3 ⊗ π ∗x,y L

c

with ai ∈ Z and c ∈ C; we will also use the convenient notation for the corresponding sheaf: c EW (a0 , a1 , a2 , a3 , c) := OW (a0 , a1 , a2 , a3 ) ⊗ EW π ∗x,y L . ±c If—again in analogy with Lemma 5.2—we denote π ∗x,y η±c ∈ EW π ∗x,y L simply as η±c then we have x · ηc ∈ EW (k, 1, 0, 0, c) and y · η−c ∈ EW (k, 1, 0, 0, −c), etc. and can regard them as twisted coordinates on W . Making use of these new twisted ∗ coordinates in Lemma 5.2 we define the true compactified singular twistor space Z of 2s,1,1,s,0 the multi-Taub–NUT geometry by the equation P c = 0, where P c : W → L looks like P c (x, u, y, v, z, w) := (x · η )(y · η c

−c

)w − (u · η )(v · η s

c

−c

)

s

(z − wpi ),

i=1

which simply gives back (19). Therefore we come up with a singular holomorphic fibration ∗

π : Z −→ CP 1 .

(25)

∗

Carrying out desingularization of Z as before we obtain finally that there is a smooth complex fibration π : Z −→ CP 1

(26)

whose fibers X [a:b] are biholomorphic to the rational surfaces constructed in Lemma 5.1. This is the true compact non-singular twistor space of (MV , gV ). ∗ For clarity we remark that from now on, the notations Z , Z , Z ∗ and Z will denote the corresponding true twistor spaces of the multi-Taub–NUT geometry (and we will call them simply as the twistor space of the corresponding kind). Now we are in a position to read off certain cohomology groups of the compactified twistor space.


207

Lemma 5.4. The middle sheaf cohomology groups of the compactified smooth twistor space Z constructed in Lemma 5.3 satisfy H 1 (Z ; O Z (−2)) ∼ = C and H 2 (Z ; O Z (−2)) ∼ = 0. Proof. First note that in light of Lemma 5.1 all the fibers X [a:b] with [a : b] ∈ CP 1 of (26) are rational. Next, notice that O Z (−2) = π ∗ O(−2) is trivial along the fibers of (26). Rationality implies in particular that for all [a : b] and 1 ≤ q ≤ 2 one has H q X [a:b] ; (π | X [a:b] )∗ O(−2) ∼ = H q X [a:b] ; O X [a:b] = 0, so R q π ∗ π ∗ O(−2) is the zero sheaf. On the other hand R 0 π ∗ π ∗ O(−2) ∼ = O(−2) by the projection formula, consequently H 0 CP 1 ; R 0 π ∗ π ∗ O(−2) = 0 and H 1 CP 1 ; R 0 π ∗ π ∗ O(−2) = C. The second level of the Leray sequence associated to (26) and the sheaf π ∗ O(−2) spectral p,q ∗ p 1 q is given by E 2 = H CP ; R π ∗ π O(−2) . Taking into account our calculations so far, the only non-trivial part is a one-dimensional space in bidegree (1, 0): 0 0 q 0 0 0 C p

implying that the spectral sequence collapses at this step and the statement follows.

Keeping these results in mind now we recall the notion of the “global Ext” groups Ext k (V ; R, S ) over a complex manifold V with two coherent sheaves R and S on it. The background material can be found in [16]. These objects are appropriate generalizations of the sheaf cohomology groups over complex manifolds. For instance if R = OV then we know that Ext k (V ; OV , S ) ∼ = H k (V ; S ).

(27)

If V is compact n dimensional then there is a generalization of the Serre duality theorem for coherent sheaves, called the Grothendieck duality theorem: H k (V ; S ) ∼ = (Ext n−k (V ; S , KV ))∗ ,

(28)

and in the same fashion if S = O(E) is locally free then the Serre duality theorem implies that (Extn−k (V ; O(E), KV ))∗ ∼ = H n−k (V ; O(E ∗ ) ⊗ KV ).

(29)

We will be particularly interested in Ext 1 (V ; R, S ) which classifies extensions of coherent sheaves over V : 0 −→ S −→ E −→ R −→ 0. Following Atiyah [2] we obtain:

208


Lemma 5.5. Let JY denote the ideal sheaf of holomorphic functions on Z vanishing on a fixed real line Y ⊂ Z ⊂ Z . Then Ext 1 (Z ; JY , O Z (−2)) ∼ = C2 .

Proof. Note that OY (−2) ∼ = KY , the canonical sheaf of Y . Let j : Y → Z be the inclusion and denote by j∗ KY the extension of KY by zero over Z . Hence j∗ KY is a coherent sheaf on Z . Applying (28) on Z in the last step we obtain H k (Y ; KY ) ∼ = H k (Z ; j∗ KY ) ∼ = (Ext 3−k (Z ; j∗ KY , K Z ))∗ , and additionally we know that Ext 3−k (Z ; j∗ KY , K Z ) ∼ = Ext 3−k (Z ; j∗ OY ⊗ π ∗ KY , K Z ) ∼ = Ext 3−k (Z ; j∗ OY , π ∗ KY∗ ⊗ K Z ). We already know that (π ∗ K Y∗ ⊗ K Z )|Y is non-canonically isomorphic to H −2 , therefore O Z (−2) is an extension of (π ∗ KY∗ ⊗ K Z )|Y over the whole Z that is, Ext 3−k (Z ; j∗ OY , π ∗ KY∗ ⊗ K Z ) ∼ = Ext 3−k (Z ; j∗ OY , O Z (−2)). Therefore, taking into account that H 2 (Y ; KY ) ∼ = 0 and H 1 (Y ; KY ) ∼ = C, we conclude that Ext 1 (Z ; j∗ OY , O Z (−2)) ∼ = 0 and Ext 2 (Z ; j∗ OY , O Z (−2)) ∼ = C. Referring to (27) we know Exti (Z ; O Z , O Z (−2)) ∼ = H i (Z ; O Z (−2)) consequently Lemma 5.4 additionally yields Ext 1 (Z ; O Z , O Z (−2)) ∼ = C and Ext 2 (Z ; O Z , O Z (−2)) ∼ = 0. Now consider the short exact sequence of coherent sheaves over Z : 0 −→ JY −→ O Z −→ j∗ OY −→ 0. A segment of the associated long exact sequence of the global Ext groups with O Z (−2) looks like . . . −→ Ext 1 (Z ; j∗ OY , O Z (−2)) −→ Ext 1 (Z ; O Z , O Z (−2)) −→ δ

−→ Ext 1 (Z ; JY , O Z (−2)) −→ Ext 2 (Z ; j∗ OY , O Z (−2)) −→ −→ Ext 2 (Z ; O Z , O Z (−2)) −→ . . . which gives 0 −→ C −→ Ext 1 (Z ; JY , O Z (−2)) −→ C −→ 0, providing the result.

(30)


209

After these algebro-geometric preliminaries the time has come to relate our considerations so far with anti-instantons over a multi-Taub–NUT space. First we establish a link with harmonic functions. The embedding i : Z \Y ⊂ Z together with (27) implies a homomorphism i ∗ : Ext 1 (Z ; JY , O Z (−2)) −→ Ext 1 (Z \Y ; JY , O Z \Y (−2)) ∼ = H 1 (Z \Y ; O Z \Y (−2)). However for a fixed real line Y ⊂ Z representing the point y ∈ MV via Penrose transform we have an isomorphism T : H 1 (Z \Y ; O Z \Y (−2)) ∼ = Ker gV | MV \{y} . Consequently one can think of the elements of Ext1 (Z ; JY , O Z (−2)) as harmonic functions. Therefore Lemma 5.5 provides us with a distinguished 2-parameter family of complex-valued harmonic functions on MV \{y}. These functions can easily be found by observing that the positive minimal Green function with singularity in y ∈ MV of Defintion 4.1 provides us with a splitting of Ext 1 (Z ; JY , O Z (−2)) ∼ = C2 as a vector space into two one dimensional summands. More precisely (30) gives 0 −→ Ext 1 (Z ; O Z , O Z (−2)) δ

−→ Ext 1 (Z ; JY , O Z (−2)) −→ Ext 2 (Z ; j∗ OY , O Z (−2)) −→ 0, and by [2, Theorem 1] the Green function defines a map Ext 2 (Z ; j∗ OY , O Z (−2)) −→ Ext 1 (Z ; JY , O Z (−2)), so that all elements of the middle term canonically can be written in the form α + β with the first factor α coming from Ext 1 (Z ; O Z , O Z (−2)), which is independent of Y, hence corresponds to a constant harmonic function μ ∈ C on (MV , gV ) while β = λG(·, y) with λ ∈ C. Consequently we obtain the following (also cf. [2]): Theorem 5.1. Fix a point y ∈ MV of the multi-Taub–NUT space (MV , gV ) and consider the associated real line Y ⊂ Z and the group Ext1 (Z ; JY , O Z (−2)) ∼ = C2 . Then any element of this group can canonically be written in the form μ + λG(·, y), where G(·, y) is the unique minimal positive Green function concentrated at y ∈ MV and μ, λ ∈ C are constants. Remark. Notice that in fact the above theorem holds for any anti-half-conformally flat non-parabolic manifold whose twistor space admits a compactification satisfying Lemma 5.4. One can then see that the positive harmonic functions of (9)—from which our antiinstantons [∇ A˜ y,λ ] stem—can also be regarded as elements of Ext 1 (Z ; JY , O Z (−2)), hence they describe extensions of two coherent sheaves 0 −→ O Z (−2) −→ F (−1) −→ JY −→ 0. Restricting this to Z , the resulting sheaves F (−1) will be locally free, hence provide us with rank 2 holomorphic vector bundles F(−1) over Z . Our aim is now to understand those twisted vector bundles together with their canonical sections among the aforementioned vector bundles over Z which are associated to

210


pure Green functions. We want to show that for two points y, eiτ y ∈ MV on the same S 1 -orbit, these twisted vector bundles are isomorphic. For this purpose we have to construct the real structure on Z which gives rise to the multi-Taub–NUT geometry. It is sufficient to work over the open twistor space Z ⊂ Z , because we are only interested in the real lines lying over points y ∈ MV at finite distance. Therefore we take a closer look at this space. Restricting our construction in Lemma 5.3 and in particular the singular fibration (25) to the finite part, we obtain a model for the non-compact singular twistor space π : Z ∗ → CP 1 as the hypersurface Pc = 0, where Pc : (H s ⊗ L c ) ⊕ (H s ⊗ L −c ) ⊕ H 2 → H 2s takes the shape Pc (x, y, z) := (x · ηc )(y · η−c ) −

s

(z − p j ).

(31)

j=1

(For clarity we remark that in (31) and from now on without introducing extra notation all the bundles H k over CP 1 will be pulled back to H 2 .) But first we make a digression on the choice k = l = s and m = 2 in Lemmata 5.2 and 5.3 and note that it is also dictated by the requirement that the normal bundle of a real line Y ⊂ Z must be isomorphic to H ⊕ H and Z must possess a non-trivial real structure leading to the multi-Taub–NUT geometry. Indeed, to describe the normal bundle of a real line Y ⊂ Z is equivalent to describe that of a generic real line Y ⊂ Z ∗ . Since it is a section of the bundle π : Z ∗ → CP 1 its normal bundle is the restriction of the tangent bundle of H s ⊕ H s ⊕ H 2 to a generic (nonsingular) line, i.e. the kernel of the map (Px , Py , Pz ) : T (H s ⊕ H s ⊕ H 2 ) → T H 2s . This shows that c1 (NY ), [Y ] = s + s + 2 − 2s = 2, moreover the type of the normal bundle is stable under small perturbations yielding NY ∼ = H ⊕ H . An important consequence is that since K Z |Y ∼ = K Y ⊗ 2 NY∗ , we therefore find K Z ∼ = π ∗ H −4 for the canonical bundle of Z as we already noted in Sect. 2. Notice that this relation does not extend to the compactification Z . Indeed, for every real line Y at finite distance the relation NY ∼ = H ⊕ H continues to hold, but because the fibers of the map (26) become compact, in K Z there will be a twist coming from the canonical bundle of X as well. This explains the asymmetry between the middle cohomology groups in Lemma 5.4. Next we move to the construction of the real structure on Z following [10] — originally due to [6,22,24] — which gives rise to the multi-Taub–NUT geometry. As we have seen in Sect. 2 the real structure on our twistor space must be induced by the antipodal map on CP 1 . Therefore, assuming c ∈ R, we use (24) to construct induced maps rc : H 0 H 2 ; E H 2 (H s ⊗ L c ) −→ H 0 H 2 ; E H 2 (H s ⊗ L −c ) satisfying rc2 = (−1)s Id. The real structure τc : Z ∗ → Z ∗ on the twistor space given by Pc = 0 in (31) is defined to be τc (x, y, z) := (−1)s rc (y · η−c ), rc (x · ηc ), −r (z) . (32) A twistor line is called real if it is invariant under this map. We claim that the space Z ∗ constructed in Lemma 5.3 together with this τc is indeed (a singular model of) the twistor space of the multi-Taub–NUT geometry. The simplest way of demonstrating this would be the derivation of the multi-Taub–NUT metric (6) via Penrose’ non-linear graviton construction outlined in Sect. 2. However, after a considerable hesitation we decided not


211

to present this calculation here because it is quite long, moreover it is already available in the literature for a long time ([6, pp. 393–395], [22,24]). We rather find the corresponding real lines by a factorization method ([6, pp. 393– 395] and [22]). Using the notations and constructions of Lemma 5.3 if we encode the NUT q j ∈ R3 as the real section of H 2 whose shape over Ub ⊂ CP 1 looks like 2 p j ([ ab : 1])|Ub = α j ab + 2β j ab − α j as well as write a general point in R3 as 2 ζ ([ ab : 1])|Ub = α ab + 2β ab − α, then the roots ρ j , σ j ∈ C ⊂ CP 1 of the equation ζ − p j = 0 are

ρ j := σ j :=

−(β − β j ) − −(β − β j ) +

" "

(β − β j )2 + |α − α j |2

α − αj (β − β j )2 + |α − α j |2

α − αj

,

.

The real lines appear by simply factorizing in a τc -invariant way Eq. (31) and imposing the reality condition (this already has been done for ζ ). They are of the form ⎧ s c ⎨ ξ(ζ, [a : b]) = Aη (ζ, [a : b]) j=1 (a − ρ j b) υ(ζ, [a : b]) = Bη−c (ζ, [a : b]) sj=1 (a − σ j b) ⎩ ζ ([a : b]) = αa 2 + 2βab − αb2 . To see that this is indeed a τc -invariant factorization, the only non-trivial fact we have to check is that η−c goes to ηc if c ∈ R and vice versa on both charts Ua and Ub . However for example for [a : b] ∈ Ua we have [−b : a] ∈ Ub , and η−c (ζ, [−b : a]) = exp(c(α( ab ) − β)) which is just ηc (ζ, [a : b]) written on the chart Ua . The coefficients A, B here are arbitrary constants satisfying AB = (α − αi ), the leading coefficient of (ζ − pi ). As this latter number is non-zero for α = α j , say B can be expressed in terms of A and α. But the reality constraint moreover implies

|A|2 =

s

(β − β j ) +

"

(β − β j )2 + |α − α j |2 .

j=1

Hence the set of these real lines is locally parameterized4 by argA ∈ S 1 and (Re α, Im α, β) ∈ R3 and provides us with a local chart of MV as it was constructed in Sect. 3. The choice c = 0 gives rise to the ALE As−1 geometries while c > 0 real provides us with the ALF As−1 i.e., the multi-Taub–NUT spaces. Hence from now on we will restrict to c = 1 in accord with previous sections. A given real line will be denoted as Yα,β,A . The next step is to express the transition matrices for thevector bundle Fα,β,A (−1) on Z as well as its section sα,β,A ∈ H 0 Z ; O Z (Fα,β,A (−1)) , associated to the canonical element G(·, yα,β,A ) ∈ Ext 1 (Z \Yα,β,A ; JYα,β,A , O Z \Yα,β,A (−2)), 4 Note that the way of assigning the roots of ζ − p to ξ and υ is well-defined only up to the action of the j Galois group of the problem which explains that we have a local parameterization only.

212


representing the Green function. Here, we follow again [2] and first work over Z ∗ . Set f := x · η1 − ξ, g := y · η−1 − υ, h := z − ζ. By definition, the real twistor line Yα,β,A is given by the equations f = g = h = 0. Define open subsets Uξ and Uυ by ξ = 0 and υ = 0 respectively. Then, restricted to Uξ ∩ Z ∗ the real line is given by the complete intersection f = h = 0 as well as restricted to Uυ ∩ Z ∗ it is the complete intersection g = h = 0. Furthermore, for all generic ζ the polynomials ζ − pi and ζ − p j do not have common roots for i = j, so Uξ and Uυ cover Z ∗ . We restrict to such choices α, β, A, but by continuity our results continue to hold in the general case too. Let us now set θ :=

xy − ξυ (x · η1 )(y · η−1 ) − ξ υ = , h h

a polynomial in a, b. Straightforward verification yields the identities f =−

θ h y · η−1 x · η1 g + h and h = − g + h. υ υ υ υ

Obviously f, g and h are well-defined sections, vanishing on Yα,β,A , of H s ⊗ L 1 , H s ⊗ L −1 and H 2 respectively. Hence, we can define a rank 2 vector bundle Fα,β,A (−1) on Z ∗ by gluing the sections f and h of the bundle (H s ⊗ L 1 ) ⊕ H 2 restricted to Uξ with the sections g and h of the bundle (H s ⊗ L −1 ) ⊕ H 2 on Uυ using the gluing matrix 1 −x · η1 θ (33) Mα,β,A := −h y · η−1 υ on Uξ ∩ Uυ . Moreover, we get a section sα,β,A of this vector bundle by setting it to be equal to ( f, h) on Uξ and to (g, h) on Uυ . This section vanishes precisely on Yα,β,A . This bundle provides us with a Green function, hence is positive [17,21,34] moreover it is the minimal positive one since from [2, Eq. 7.11] one can check that it converges to zero at infinity. Recall also that S 1 acts on MV as identified with the space of real sections by multiplication on the component A, or equivalently in view of the identity AB = (α − αi ), by inverse multiplication on B. Our aim is to show that if we consider the twistor line associated to a point (α, β, eiτ A), then the vector bundle F(−1) constructed for this point will be isomorphic to the one corresponding to (α, β, A). But the gluing matrix (33) for these former data is related to that of the latter ones via Mα,β,eiτ A = e−iτ Mα,β,A . It follows that the change of trivializations ( f, h) → (eiτ f, eiτ h) on Uξ induces an isomorphism between the bundles Fα,β,A (−1) and Fα,β,eiτ A (−1).


213

As we have seen we obtain Z from the singular space Z ∗ by blowing it up in finitely many points. Carrying out this procedure and then pulling back the bundles just constructed we come up with isomorphic vector bundles Fα,β,A (−1) and Fα,β,eiτ A (−1) over the regular (and non-compact) twistor space Z . Consider the untwisted vector bundle Fα,β,A = Fα,β,A (−1) ⊗ H . First of all, it is clear that it is a holomorphic rank 2 vector bundle over Z . Secondly, the restriction of Fα,β,A to any real line in Z is holomorphically trivial. This follows since the bundle we have constructed just corresponds to the Green function and this function is everywhere positive as we already noted above. Thirdly, the anti-linear map (32) induces a real structure on Fα,β,A (−1) over τ1 simply because τ1 switches H s ⊗ L 1 and H s ⊗ L −1 while it maps H 2 to H 2 . Therefore, since over CP 1 the bundle H carries a symplectic structure [18], the untwisted bundle Fα,β,A = Fα,β,A (−1) ⊗ H also carries a symplectic structure τ˜1 : Fα,β,A → Fα,β,A lying over τ1 . Since the same conclusions are also true for Fα,β,eiτ A , it then follows that the corresponding untwisted vector bundles satisfy the three conditions of the Atiyah–Ward correspondence (cf. the summary of the Atiyah–Ward correspondence in Sect. 2), hence they indeed give rise to anti-instantons. Since Fα,β,A and Fα,β,eiτ A are isomorphic, we therefore conclude that the limiting anti-instantons are gauge equivalent, i.e. [∇ A˜ y,+∞ ] = [∇ A˜ iτ ] as desired. e y,+∞

6. Conclusion In this paper we considered SU(2) anti-instanton moduli spaces over the multi-Taub– NUT spaces containing solutions with integer energy. As a consequence of the general theory [13] in principle anti-instantons with the same nice properties but with fractional energy also may exist over these spaces. The tantalizing question therefore arises whether or not these other moduli spaces are empty. Acknowledgements. The second author would like to thank the Alfréd Rényi Institute of Mathematics, where he had a scholarship during the early stages of this article, and Vincent Minerbe for useful discussions. Both authors were partially supported by OTKA grant No. NK81203 (Hungary).

References 1. Anderson, M.T., Kronheimer, P.B., LeBrun, C.: Complete Ricci-flat manifolds with infinite topological type. Commun. Math. Phys. 125, 637–642 (1989) 2. Atiyah, M.F.: Green’s functions for self-dual four-manifolds. In: Mathematical analysis and applications, Part A, Advances in mathematics supplementary studies 7A, New York: Academic Press, pp. 129–158, 1981 3. Atiyah, M.F., Hitchin, N.J.: The geometry and dynamics of magnetic monopoles. Princeton, NJ: Princeton University Press, 1988 4. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London A362, 425–461 (1978) 5. Atiyah, M.F., Ward, R.: Instantons and algebraic geometry. Commun. Math. Phys. 55, 117–124 (1977) 6. Besse, A.L.: Einstein Manifolds. Berlin: Springer-Verlag, 1987 7. Biquard, O., Jardim, M.: Asymptotic behaviour and moduli space of double periodic instantons. J. Eur. Math. Soc. 3, 335–375 (2001) 8. Cherkis, S.: Moduli spaces of instantons on the Taub–NUT space. Commun. Math. Phys. 290, 719–736 (2009) 9. Cherkis, S., Hitchin, N.J.: Gravitational instantons of type Dk . Commun. Math. Phys. 260, 299–317 (2005)

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10. Etesi, G.: Classification of ’t Hooft instantons over multi-centered gravitational instantons. Nucl. Phys. B662, 511–530 (2003) 11. Etesi, G.: The topology of asymptotically locally flat gravitational instantons. Phys. Lett. B641, 461– 465 (2006) 12. Etesi, G., Hausel, T.: On Yang–Mills-instantons over multi-centered gravitational instantons. Comm. Math. Phys. 235, 275–288 (2003) 13. Etesi, G., Jardim, M.: Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons. Commun. Math. Phys. 280, 285–313 (2008), Erratum: ibid. 288, 799–800 (2009) 14. Freedman, M.H.: The topology of four-manifolds. J. Diff. Geom. 17, 357–454 (1982) 15. Gibbons, G.W., Hawking, S.W.: Gravitational multi-instantons. Phys. Lett. B78, 430–432 (1978) 16. Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry. New York: John Wiley & Sons, Inc., 1984 17. Grigor’yan, A.A.: The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds (Russian). Math. Sb. (S.N.) 128, 354–363, 446(170) (1985) 18. Hartshorne, R.: Stable vector bundles and instantons. Commun. Math. Phys. 59, 1–15 (1978) 19. Hausel, T., Hunsicker, E., Mazzeo, R.: Hodge cohomology of gravitational instantons. Duke Math. J. 122, 485–548 (2004) 20. Hawking, S.W.: Gravitational instantons. Phys. Lett. A60, 81–83 (1977) 21. Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes 5, Providence, RI: Amer. Math. Soc., 2000 22. Hitchin, N.J.: Polygons and gravitons. Math. Proc. Camb. Phil. Soc. 85, 465–476 (1979) 23. Hitchin, N.J.: Linear field equations on self-dual spaces. Proc. Roy. Soc. London A370, 173–191 (1980) 24. Hitchin, N.J.: Twistor construction of Einstein metrics. In: Global Riemannian geometry (Durham, 1983), Ellis Horwood Ser.: Math. Appl., Chichester: Horwood, 1984, pp. 115–125 25. Jackiw, R., Nohl, C., Rebbi, C.: Conformal properties of pseudo-particle configurations. Phys. Rev. D15, 1642–1646 (1977) 26. Kronheimer, P.B.: The construction of ALE spaces as hyper-Kähler quotients. J. Diff. Geom. 29, 665– 683 (1989) 27. Kronheimer, P.B., Nakajima, H.: Yang–Mills instantons on ALE gravitational instantons. Math. Ann. 288(2), 263–607 (1990) 28. LeBrun, C.: Complete Ricci-flat Kähler metrics on Cn need not be flat. Proc. Symp. Pure Math. 52(2), 297– 304 (1991) 29. Minerbe, V.: On some asymptotically flat manifolds with non-maximal volume growth. http://arXiv.org/ abs/0709.1084v1[math.DG], 2007 30. Minerbe, V.: A mass for ALF manifolds. Commun. Math. Phys. 289, 925–955 (2009) 31. Page, D.N.: Green’s function for gravitational multi-instantons. Phys. Lett. B85, 369–372 (1979) 32. Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31–52 (1976) 33. Uhlenbeck, K.K.: Removable singularities in Yang–Mills fields. Commun. Math. Phys. 83, 11–29 (1982) 34. Varopoulos, N.: Potential theory and diffusion on Riemannian manifolds. In: Conference on harmonic analysis in honor of Antoni Zygmund (Chicago, Ill. 1981), Vol. I, II, Wadsworth Math. Ser., Belmont, CA: Wadsworth, 1983, pp. 821–837 Communicated by N.A. Nekrasov


Communications in


Betti Numbers of a Class of Barely G 2 Manifolds Sergey Grigorian Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, D-14476 Golm, Germany. E-mail: [email protected] Received: 8 October 2009 / Accepted: 2 August 2010 Published online: 23 October 2010 – © Springer-Verlag 2010

Abstract: We calculate explicitly the Betti numbers of a class of barely G 2 manifolds that is, G 2 manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces from which they inherit all their (1, 1)-cohomology and the involutions are free acting. 1. Introduction One of the key concepts in String and M-theory is the concept of compactification - here the full 10- or 11-dimensional spacetime is considered to be of the form M4 × X, where M4 is the “large” 4-dimensional visible spacetime, while X is the “small” compact 6- or 7-dimensional Riemannian manifold. Due to considerations of supersymmetry, these compact manifolds have to satisfy certain conditions which place restrictions on the geometry. In the case of String theory, the 6-dimensional manifolds have to be Calabi-Yau manifolds - that is Kähler manifolds with vanishing first Chern class. The existence of Ricci-flat Kähler metrics for these manifolds has been proven by Yau in 1978 [1]. One of the first examples of a Calabi-Yau 3-fold (6 real dimensions) was the quintic - a degree 5 hypersurface in CP4 . Later, Candelas et al. [2] found the first large class of Calabi-Yau manifolds - the Complete Intersection Calabi-Yau (CICY) manifolds, which are given by intersections of hypersurfaces in products of complex projective spaces. We review the details in Sect. 3. Since then even larger classes of Calabi-Yau manifolds have been constructed - such as Weighted Complete Intersection manifolds [3], and complete intersection manifolds in toric varieties [4]. So overall there is a very large pool of examples of Calabi-Yau manifolds, and it is in fact still an open question whether the number of topologically distinct Calabi-Yau 3-folds is finite or not. One of the great discoveries in the study of Calabi-Yau manifolds is Mirror Symmetry [5,6]. This symmetry first appeared in String Theory where evidence was found that

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conformal field theories (CFTs)related to compactifications on a Calabi-Yau manifold with Hodge numbers h 1,1 ,h 2,1 are equivalent to CFTs on a Calabi-Yau manifold with Hodge numbers h 2,1 , h 1,1 . Mirror symmetry is currently a powerful tool both for calculations in String Theory and in the study of the Calabi-Yau manifolds and their moduli spaces. However if we go one dimension higher, and look at compactifications of M-theory, a natural analogue of a Calabi-Yau manifold in this setting is a 7-dimensional manifold with G 2 holonomy. These manifolds are also Ricci-flat, but being odd-dimensional they are real manifolds. The first examples of G 2 manifolds have been constructed by Joyce in [7]. While some work has been done both on the physical aspects of G 2 compactifications (for example [8–11] among others) and on the structure and properties of the moduli space (for example [7,12–15] among others), still very little is known about the overall structure of G 2 moduli spaces. One of the problems is that there are relatively few examples of G 2 manifolds, and for the ones that are known it is hard to do any calculations, because the examples are not very explicit. However there is a conjectured method of constructing G 2 manifolds from Calabi-Yau manifolds, which could potentially yield many new examples of G 2 manifolds. Here we take a Calabi-Yau 3-fold Y and let Z = (Y × S 1 )/σˆ , where σˆ acts as antiholomorphic involution on Y and acts as z −→ −z on the S 1 . In general, the result will have singularities, and it is still an unresolved question how to systematically resolve these singularities to obtain a smooth manifold with G 2 holonomy. This construction has been suggested by Joyce in [7,16]. A more basic approach is to only consider involutions without fixed points, so that the resulting manifold Z is smooth. Manifolds belonging to this class have been called barely G 2 manifolds in [8]. Such manifolds do not have the full G 2 holonomy, but rather only Z2 SU (3). However, they do share many of the same properties as full G 2 manifolds, so for many purposes they can play the same role as genuine G 2 manifolds [8,17]. In particular, if we consider a specific class of of Calabi-Yau manifolds, such as CICY manifolds, we can construct a corresponding class of barely G 2 manifolds rather explicitly. This is what we focus on in this paper. We first give an overview of G 2 manifolds and CICY manifolds, and then describe the algorithm that was used to systematically calculate the Betti numbers of the barely G 2 manifolds corresponding to the independent CICY manifolds. In order to work out how the involution acts on 2-forms, we need to know the structure of the second cohomology of the CICY manifold, and for this reason we limit our attention to those CICY manifolds which inherit all of their second cohomology from the ambient product space. 2. G 2 Manifolds 2.1. Basics. We will first review the basics of manifolds with G 2 holonomy. The 14-dimensional exceptional Lie group G 2 ⊂ S O (7) is precisely the group of automorphisms of imaginary octonions, so it preserves the octonionic structure constants [18]. 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)

These precisely give the structure constants of the octonions, so G 2 preserves ϕ0 . Since G 2 preserves the standard Euclidean metric g0 on R7 , it preserves the Hodge star, and hence the dual 4-form ∗ϕ0 , which is given by ∗ ϕ0 = e4567 + e2367 + e2345 + e1357 − e1346 − e1256 − e1247 .

(2.2)

Betti Numbers of a Class of Barely G 2 Manifolds

217

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 positive if 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. It is shown in [16] 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. It turns out that the holonomy group H ol (X, g) ⊆ G 2 if and only if X has a torsionfree G 2 structure [16]. In this case, the invariant 3-form ϕ satisfies dϕ = d ∗ ϕ = 0

(2.3)

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. The holonomy group is precisely G 2 if and only if the fundamental group π1 (X ) is finite. In particular, if H ol (X, g) = G 2 , the first Betti number b1 vanishes. The reverse is however not true in general. 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

(2.4)

for a spinor ξ . As lucidly explained in [10], condition (2.4) 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. Hence eleven-dimensional supergravity compactified on a G 2 holonomy manifold gives rise to a N = 1 effective theory. From [10,11] and [9] we know that the deformations of the G 2 3-form ϕ give b3 real moduli which combine with the deformations of the supergravity 3-form C to give b3 complex moduli. Together with modes of the gravitino, this gives b3 chiral multiplets. Decomposition of the C-field also gives b2 abelian gauge fields, which again combine with gravitino modes to give b2 vector multiplets. The structure of the moduli space has been studied in detail in [15]. Examples of compact G 2 manifolds have been first constructed by Joyce [7] as resolutions of orbifolds T 7 / for a discrete group . There is taken to be a finite group of diffeomorphisms of T 7 preserving the flat G 2 -structure on T 7 . The resulting orbifold will have a singular set coming from the fixed point of the action of , and these singularities are resolved by gluing ALE spaces with holonomy SU (2) or SU (3).

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2.2. G 2 manifolds from Calabi-Yau manifolds. A simple way to construct a manifold with a torsion-free G 2 structure is to consider X = Y × S 1 , where Y 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 ,

(2.5) (2.6)

where θ is the coordinate on S 1 , ω is the Kähler form on Y and is the holomorphic 3-form on Y . This then defines a torsion-free G 2 structure, with ∗ϕ =

1 ω ∧ ω − dθ ∧ Im . 2

(2.7)

However, the holonomy of X in this case is SU (3) ⊂ G 2 . From the Künneth formula we get the following relations between the Betti numbers of X and the Hodge numbers of Y : b1 = 1, b2 = h 1,1 , b3 = h 1,1 + 2 h 2,1 + 1 . In [7] and [16], Joyce describes a possible construction of a smooth manifold with holonomy equal to G 2 from a Calabi-Yau manifold Y . So suppose Y is a Calabi-Yau 3-fold as above. Then suppose σ : Y −→ Y is an antiholomorphic isometric involution on Y , that is, χ preserves the metric on Y and satisfies σ 2 = 1, σ ∗ (ω) = −ω, ¯ σ ∗ ( ) = .

(2.8a) (2.8b)

Such an involution σ is known as a real structure on Y . Define now a quotient given by (2.9) Z = Y × S 1 /σˆ , where σˆ :Y × S 1 −→ Y × S 1 is defined by σˆ (y, θ ) = (σ (y) , −θ ). The 3-form ϕ defined on Y × S 1 by (2.6) is invariant under the action of σˆ and hence provides Z with a G 2 structure. Similarly, the dual 4-form ∗ϕ given by (2.7) is also invariant. Generically, the action of σ on Y will have a non-empty fixed point set N , which is in fact a special Lagrangian submanifold on Y [16]. This gives rise to orbifold singularities on Z . The singular set is two copies of N . It is conjectured that if there exists a non-vanishing harmonic 1-form on N , then it is possible to resolve each singular point using an ALE 4-manifold with holonomy SU (2) in order to obtain a smooth manifold with holonomy G 2 . The precise details of the proof of this conjecture are not yet available however. We will therefore consider only free-acting involutions, that is those without fixed points. Manifolds defined by (2.9) with a freely acting involution were called barely G 2 manifolds by Harvey and Moore in [8]. The cohomology of barely G 2 manifolds is expressed in terms of the cohomology of the underlying Calabi-Yau manifold Y : H 2 (Z ) = H 2 (Y )+ , H 3 (Z ) = H 2 (Y )− ⊕ H 3 (Y )+ .

(2.10a)


219

Here the superscripts ± refer to the ± eigenspaces of σ ∗ . Thus H 2 (Y )+ refers to twoforms on Y which are invariant under the action of involution σ and correspondingly H 2 (Y )− refers to two-forms which are odd under σ . Wedging an odd two-form on Y with dθ gives an invariant 3-form on Y × S 1 , and hence these forms, together with the invariant 3-forms H 3 (Y )+ on Y , give the three-forms on the quotient space Z . Also note that H 1 (Z ) vanishes, since the 1-form on S 1 is odd under σˆ . Consider the action of σ on H 3 (Y ). It sends H 3,0 (Y ) to H 0,3 (Y ) and H 2,1 (Y ) to 1,2 H (Y ). Therefore the positive and negative eigenspaces are of equal dimension, so dim H 3 (Y )+ = h 2,1 + 1. Therefore the Betti numbers of Z in terms of Hodge numbers of Y are b1 = 0, b2 = h +1,1 , b = 3

h− 1,1

(2.11a)

+ h 2,1 + 1.

Hence in order to construct barely G 2 manifolds we need to be able to find involutions of Calabi-Yau manifolds and determine the action of the involution on H 1,1 (Y ). A relatively large class of Calabi-Yau manifolds for which this is not hard to do are the complete intersection Calabi-Yau manifolds. We review the properties of these manifolds in the next section. 3. Complete Intersection Calabi-Yau Manifolds 3.1. Basics. Complete intersection Calabi-Yau (CICY) manifolds were the first major class of Calabi-Yau manifolds which was discovered by Candelas et al. in [2]. Such a manifold M is defined as a complete intersection of K hypersurfaces in a product of m complex projective spaces W = CPn 1 × · · · × CPn m . Each hypersurface is defined as the zero set of a homogeneous holomorphic polynomial f a z μr = 0 a = 1, . . . , K . (3.12) Each such polynomial is homogeneous of degree qar with respect to the homogeneous coordinates of CPnr . By complete intersection it is meant that the K -form = df1 ∧ ··· ∧ df K does not vanish on M. This condition ensures that the resulting manifold is defined globally. In order for M to be a 3-fold, we obviously need K =

m

n i − 3.

(3.13)

i=1

The standard notation for a CICY manifold is a m × (K + 1) array of the form [ n q],

(3.14)

where n is a column m-vector whose entries nr are the dimensions of the CPnr factors, and q is a m × K matrix with entries qar which give the degrees of the polynomials in the coordinates of each of the CPnr factor. Each such array defining a CICY is known as a configuration matrix, while an equivalence class of configuration matrices under

220

S. Grigorian

permutation of all rows and all columns belonging to q is called a configuration. Clearly each such permutation defines exactly the same manifold. As it was shown in [2], Chern classes can be computed directly from the defining quantities n and q. In particular, we immediately get the condition for a vanishing first Chern class: nr + 1 =

K

qar ∀r.

(3.15)

a=1

That is, the sum of entries of in each row of q must equal the dimension of the corresponding CPnr factors. This is hence precisely the condition for the complete intersection manifold to be Calabi-Yau. Moreover from the expressions for Chern classes, an expression for the Euler number is also obtained. This is given by ⎞ ⎡⎛ m ⎤ m K

c3r st xr xs xt ⎠ · qbu xu ⎦ , (3.16) χ E (M) = ⎣⎝ r,s,t=1

where c3r st

b=1

u=1

coefficient of

m

r =1 (xr )

nr

K 1 r st r s t = qa qa qa (nr + 1) δ − 3 a=1

and δr st = 1 for r = s = t and vanishes otherwise. Varying the coefficients of polynomials in a CICY configuration generally corresponds to complex structure deformations, but as it was shown in [19], there is no one to one correspondence. So it is said that each configuration corresponds to a partial deformation class. There are also various identities which relate different configurations, so not all configurations are independent. There are however 7868 independent configurations. A method for calculating Hodge numbers of the CICY manifolds has been found by Green and Hübsch in [19], and in [20] Green, Hübsch and Lütken calculated the Hodge numbers for each of the 7868 configurations. They found there were 265 unique pairs of Hodge numbers. Unfortunately, the original data with the CICY Hodge numbers has been lost, and the original computer code by Hübsch has been written in a curious mix of C and Pascal so the original code had to be rewritten in standard C in order to be able to recompile the list of Hodge numbers for CICY manifolds, which is necessary to be able to calculate the Betti numbers of corresponding barely G 2 manifolds. 3.2. Involutions. Antiholomorphic involutions of projective spaces have been classified in [17], and here we briefly review their results. First consider involutions of a single projective space CPn . Suppose we have homogeneous coordinates (z 0 , z 1 , . . . , z n ) on CPn , then we can represent an anti-holomorphic involution σ by a matrix M which acts as z i −→ Mi j z¯ j .

(3.17)

Without loss of generality we fix det M = 1 since multiplication by any non-zero complex number still gives the same involution. Moreover, involutions which differ only by a holomorphic change of basis can be regarded to be the same.


221

Also σ 2 = 1 must be true projectively, so we get M M¯ = λI.

(3.18)

Taking the determinant of (3.18), we find that λn+1 = 1, and taking the trace we see that λ is real. Thus λ = 1 for n even and λ = ±1 for n odd. The involution σ is required to be an isometry - that is, it must preserve the standard Fubini-Study metric of CPn . Together with previous restrictions on M, this gives the condition M M † = I.

(3.19)

Combining (3.18) and (3.19), we see that for λ = 1 these equations imply that M is symmetric, and for λ = −1 that M is antisymmetric. Moreover, due to (3.18), the real and imaginary parts of M commute, and so can be simultaneously brought into a canonical form - diagonal for λ = 1 and block-diagonal for λ = −1. Another change of basis can be used to normalize the coefficients. Hence we get two distinct antiholomorphic involutions: A : (z 0 , z 1 , . . . , z n ) −→ (¯z 0 , z¯ 1 , . . . , z¯ n ), B : (z 0 , z 1 , . . . , z n−1 , z n ) −→ (−¯z 1 , z¯ 0 , . . . , −¯z n , z¯ n−1 ).

(3.20a)

The involution A corresponds to λ = +1 and is defined for n both odd and even, whereas the involution B corresponds to λ = −1 and is only defined for n odd. An important difference between the two involutions is that A has a fixed point set {z i = z¯ i }, whereas B acts freely without any fixed points. So far we considered antiholomorphic involutions of a single projective space, but in general we are interested in products of projective spaces, so we should also consider involutions which mix different factors. As pointed out in [17], the only possibility for this is to exchange two identical projective factors CPn , giving another involution C: C : ({yi } ; {z i }) −→ ({¯z i } ; { y¯i }).

(3.21)

This involution clearly has a fixed point set {yi = z¯ i }. Now that we have antiholomorphic involutions of projective spaces, we can use these to construct barely G 2 manifolds from CICY manifolds, as in (2.9). In general we must either have an involution acting on each projective factor - either involutions A or B on single factors or involution C on a pair of identical projective factors. Given a CICY configuration matrix, we will denote the resulting barely G 2 manifold by the same configuration matrix, but indicating in the first column of the configuration matrix which involutions are acting on each projective factor. These actions will be

denoted by n, ¯ nˆ and

n for involutions A, B and C, respectively. For example, consider n

the configuration matrix: 1 ⎢ ⎢1 ⎢ ⎢1 ⎢ ⎣2 3 ⎡

0 0 0 1 1

0 0 0 1 1

0 1 1 1 0

0 1 1 0 1

⎤ 2 1,39 0⎥ ⎥ 0⎥ 0⎦ 1

(3.22)

222

S. Grigorian

This denotes the barely G 2 manifolds constructed from CICY with the same configuration matrix but with involution A acting on the CP2 and CP3 factors, involution B acting on the first remaining CP1 factor and involution C acting on the remaining CP1 × CP1 . The superscripts (1, 39) give the Betti numbers b2 and b3 of the resulting 7-manifold. Note that since this example includes the action of involution B which has no fixed points, the full involution acting on the whole CICY is also free, so the resulting space is a smooth barely G 2 manifold. When the projective space involution restricts to the complete intersection space, conditions are imposed on the coefficients of the defining homogeneous equations. Thus the involutions must be compatible with the defining equations, and this may not always be possible. In particular, the invariance of the defining equations under the involution implies that the transformed equations must be equivalent to the original equations. Let us use the configuration matrix (3.22) to demonstrate this. Let u i , vi , wi for i = 0, 1 be the homogeneous coordinates on the CP1 spaces, let y j for j = 0, 1, 2 be coordinates on CP2 and z k for k = 0, 1, 2, 3 be the homogeneous coordinates on the CP3 factor. Then the original defining equations are ⎧ f 1 (y, z) = f 2 (y, z) = 0, ⎨ g1 (v, w, y) = g2 (v, w, z) = 0, (3.23) ⎩ h (u, z) = 0, where the f i and gi are polynomials homogeneous of degree 1 in their variable and h is a polynomial which is homogeneous of degree 2 in u i and of degree 1 in z k . Under the involution presented in (3.22), after taking the complex conjugates, these equations become ⎧ f¯1 (y, z) = f¯2 (y, z) = 0, ⎨ g¯ 1 (w, v, y) = g¯2 (w, v, z) = 0, (3.24) ⎩ h¯ u, ˆ z = 0, where uˆ 2k = −u 2k+1 and uˆ 2k+1 = u 2k . Then for some complex numbers λ1 , λ2 and λ3 we must have g1 (v, w, y) = λ1 g¯ 1 (w, v, y), g2 (v, w, z) = λ2 g¯ 2 (w, v, z), h (u, z) = λ3 h¯ u, ˆ z , and for some matrix M in G L (2, C) we must have f 1 (y, z) f¯ (y, z) = M ¯1 . and f 2 (y, z) f 2 (y, z)

(3.25a) (3.25b) (3.25c)

(3.26)

For consistency in (3.25a) and (3.25b), we find that λ1 λ¯ 1 = 1 and λ2 λ¯ 2 = 1. Without loss of generality, we can set λ1 = λ2 = 1. From (3.25c), we have ˆˆ z = λ3 λ¯ 3 h (u, z). (3.27) ˆ z = λ3 λ¯ 3 h u, h (u, z) = λ3 h¯ u, Here we have used the fact that h (u, z) is of degree 2 in u i , so even though uˆˆ = −u, the minus sign cancels, and we get λ3 λ¯ 3 = 1. So we can set λ3 = 1 without loss of generality. In order for (3.26) to be consistent, we find that we must have M M¯ = I, but


223

M = I satisfies this condition and so fulfills the consistency criteria. We can see that all these conditions on the coefficients of the defining polynomials halve the number of possible choices for the coefficients. This also shows that not all combinations of involutions are possible. In particular, suppose if we wanted a B involution to act on the CP3 factor. Then since zˆˆ = −z, and h (u, z) is of degree 1 in z, from (3.27) we would get that λ3 λ¯ 3 = −1, which is clearly not possible. Also, the C involution is not always possible - the configuration must be invariant under the interchange of factors. In order to construct all possible barely G 2 manifolds from CICY manifolds, we must be able to find all possible involutions of a given CICY configuration. Since we want freely acting involutions, we only consider those combinations of involutions which contain a B involution. The overall strategy is the following. We first find all possible combinations of C involutions, and then for each such combination we find the possible B involutions. The remaining factors which do not have any involutions acting on them get an A involution. Suppose we have a configuration matrix with m rows and K columns - that is we have K hypersurfaces in a product of m projective factors. Let the coordinates be labelled by x 1 , . . . , x m , and let the homogeneous polynomials be f 1 , . . . , f K . So the intersection of hypersurfaces is given by f 1 = f 2 = . . . = f K = 0.

(3.28)

We want to check whether a C involution is possible on the first two factors. For this we assume that the two factors are of the same dimension, as this is a basic necessary condition for a C involution. Then we have to make sure that after the interchange of x 1 and x 2 the new set of homogeneous equations is equivalent to (3.28). This is true if and only if under the interchange of x 1 and x 2 the polynomials remain the same up to a change of ordering. In terms of the configuration matrix this means that under the interchange of two rows the matrix remains invariant up to a permutation of the columns. For more than one C involution acting on the same configuration matrix, we thus require that under the full set of row interchanges the matrix remains invariant up to a permutation of the columns. To find all the possible C involutions for a given configuration matrix we do an exhaustive search of all possibilities. First we find all the possible combinations of pairs of rows that correspond to projective factors of equal dimensions. Then for each such combination of pairs we check if under the interchange of rows in each pair the configuration matrix stays invariant up to a reordering of columns. If this is true, then it is possible to have C involutions acting on each of these pairs of rows. This procedure then gives us the full set C = {C1 , . . . , C N } of all possible combinations of C involutions acting on the configuration matrix. Now given all the possible C involutions on a configuration matrix, for each such combination Ci ∈ C, we need to find the possible B involutions. Suppose we have a configuration matrix as before, and we want to check whether a B involution is possible on the first projective factor. The basic necessary condition is that the dimension of this projective factor is odd. Then we need to make sure that the new set of homogeneous equations is equivalent to the old set. Let I be the set of columns which have non-zero entries in the first row - or equivalently, the set of polynomials that involve x 1 . First suppose that all columns in I are distinct. Then for each i ∈ I we require f i z 1 , . . . = λi f¯i zˆ 1 , . . .

(3.29)

224

S. Grigorian

for some constant λi ∈ C. As in (3.27), we then have the consistency requirement (3.30) f i z 1 , . . . = λi f¯i zˆ 1 , . . . = λi λ¯ i f i zˆˆ 1 , . . . . However, zˆˆ 1 = −z 1 , but f i is homogeneous of degree q 1i in z 1 , so f i zˆˆ 1 , . . . = 1 (−1)qi f i z 1 , . . . . Hence in order for (3.30) to be consistent, qi1 needs to be even for each i. If this is true, then we can have a B involution on the first projective factor. More generally, however, suppose that we have some identical columns in I. In particular assume that columns k1 , . . . , kr ∈ I are all identical, and that the remaining columns in I are distinct from these. These columns correspond to polynomials which have the same degrees in projective space coordinates. We can have an involution B if and only if f k1 = f k2 = . . . = f kr = 0 ⇐⇒ fˆk1 = fˆk2 = . . . = fˆkr = 0. So for some matrix M ∈ G L (r, C) we must have ⎛ ⎛ ⎞ ⎞ f k1 z 1 , . . . f¯k1 zˆ 1 , . . . ⎝ ⎠ ⎝ ⎠ . . . = M . . . . f kr z 1 , . . . f¯kr zˆ 1 , . . .

(3.31)

From (3.31) we have the consistency condition ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ 1 ˆˆ 1 , . . . z f k f k1 z 1 , . . . f k1 z , . . . ⎟ ⎜ 1 Q ⎟ ¯ ⎝ ¯ ⎜ ⎠ ⎠ ⎝ . . . ⎠ = (−1) M M . .1. = M M ⎝ . .1. , (3.32) 1 f kr z , . . . f kr z , . . . f kr zˆˆ , . . . where Q = qk11 + . . . + qk1r . If r is even, then we can always find a block-diagonal real matrix M such that M M¯ = M 2 = −I , so in this case the condition (3.32) is alwaysconsistent, independent of the parity of Q. For example for r = 2 we could set 0 1 . However if r is odd, then it is not possible to find a matrix which M = −1 0 satisfies M M¯ = −I , so we then cannot have Q odd. To find all possible B involutions, we again proceed with an exhaustive search. We look for all possible combinations of B involutions for each combination of C involutions Ci ∈ C. First we find the set R of all possible combinations of rows such that the dimensions of the corresponding projective factors are odd, and such that these rows do not have a C involution from Ci acting on them. Given a combination R ∈ R, we want to check if it is possible to have a B involution acting on each row in R. We look for the set I of columns which have a non-zero entry in at least one of the rows in R. The set I is then split into maximal subsets of identical columns. For each such subset we evaluate Q as above, and if for some subset of size r , r Q is odd, then the consistency condition (3.32) is not fulfilled, and so the combination of rows R does not admit a B involution. The above algorithm has been implemented in the programming language C. After running the algorithm, for each configuration matrix in the original list of 7868 CICY configurations we find the possible combinations of C-involutions, and for each combination of C-involution all the possible combinations of B involutions. Since we are


225

interested in manifolds with free-acting involutions, we are only concerned with those configuration that admit a B-involution. It turns out that a total of 4652 configurations do admit a B-involution, out of which 153 have unique pairs of Hodge numbers. The Hodge pairs for which there exist configurations that admit B involutions are listed in (3.33): h 1,1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 19

h 2,1 65, 73, 89 50 + 2k for k 31 + 2k for k 26 + 2k for k 25 + 2k for k 24 + 2k for k 23 + 2k for k 22 + 2k for k 21 + 2k for k 20 + 2k for k 19 + 2k for k 18 + 2k for k 17 + 2k for k 16 + 2k for k 15, 21 20 19

= 0, . . . , 13, 18 = 0, 2, 3, . . . , 17, 19, 22 = 0, 1, . . . , 19, 21 = 0, 1, . . . , 18 = 0, 1, . . . , 13, 15 = 0, 1, . . . , 10, 12, 13 = 0, . . . , 11 = 0, . . . , 9 = 0, . . . , 7 = 0, . . . , 6 = 0, . . . , 3, 5 = 0, . . . , 4 = 0, 1, 3

(3.33)

As we can see there is a clear pattern - all these pairs of Hodge numbers have an even sum. In fact the only pairs of Hodge numbers that have an even sum but do not admit any B involutions are (2, 46) , (2, 64) , (3, 27) and (3, 33). 4. Barely G 2 Manifolds 4.1. Betti numbers. Now that we have found the CICY involutions, we can calculate the Betti numbers of the corresponding barely G 2 manifolds. Thus we need to find the harmonic forms on these manifolds. As we know from Sect. 2.1, for this we only need to determine the stabilizer of the involution σ acting on the H 1,1 (Y ) of a CICY Y . In general, we can expect part of the cohomology group to come from H 1,1 (W ) (where W is the product of projective factors) and some of it may come from the embedding of the hypersurface. In fact, from [21] we have K E a∗ , ker j : H 1,1 (W ) −→ H 1,1 (Y ) = H 1 Y, (4.1) a=1

where E a are the line bundles over W , the sections of which correspond to the polynomials Pa . The rank of this cohomology group can easily be calculated for CICY manifolds [19,22]. However, whenever the rank is non-zero, the configuration matrix can be reduced to an equivalent one for which the rank does indeed vanish [20, Cor. 2]. The simplest example of such a reduction is that a homogeneous hypersurface of degree 1 in CP1 × CP1 is again CP1 . So in fact, the map j from H 1,1 (W ) to H 1,1 (Y ) may be taken to be injective. It turns out that all of the 7868 CICY configurations in [2] satisfy

226

S. Grigorian

this. There could still however be some elements of H 1,1 (Y ) that do not come from H 1,1 (W ). The cycles corresponding to these cohomology classes are called vanishing cycles. However if h 1,1 (Y ) = h 1,1 (W ), then j is in fact an isomorphism. We restrict our attention to this particular case, because otherwise we cannot say how the cohomology classes that correspond to vanishing cycles behave under the involution. Since W is a product of complex projective factors, we have in fact that h 1,1 = m, the number of complex projective factors in the given CICY. Then the harmonic (1, 1)forms on Y are simply the pullbacks of the Kähler forms J1 , . . . , Jm on the corresponding complex projective factors. In the list of CICYs by Candelas et al., 4874 configurations satisfy this criterion, while the rest do not. The class of CICYs for which this holds have been referred to as favourable by Candelas and He [23]. Now suppose we have some involutions acting on Y × S 1 . First let us consider the case when there are no C involutions. In this case, no projective factors are mixed, and each of the Kähler forms is odd under the involution. Hence in this case, h − 1,1 = h 1,1 and h +1,1 = 0. From (2.11), we thus have on the 7-dimensional quotient space that b2 = 0 and b3 = h 11 + h 2,1 + 1. Now consider the case when we have one C involution acting on Y . Without loss of generality assume that the C involution acts on the first two projective factors. Then J1 + J2 is odd, while J1 − J2 is even under this involution. The remaining Kähler forms + remain odd as before. So in this case, h − 1,1 = h 1,1 − 1 and h 1,1 = 1, and so b2 = 1 and b3 = h 1,11 + h 2,1 . When we have multiple C involutions, b2 correspondingly is equal to the number of C involutions: b2 = n c , b3 = h 1,1 + h 2,1 + 1 − n c ,

(4.2a) (4.2b)

where n C is the number of C involutions acting on the base CICY manifold. After doing all the calculations we find the following pairs of Betti numbers of the barely G 2 manifolds: b2 0 1 2 3 4 5

b3 31 + 2k 30 + 2k 29 + 2k 28 + 2k 27 + 2k 26

for k = 0, . . . , 22, 24, 29, 30 for k = 0, . . . , 19, 21 for k = 0, . . . , 10, 12, 13, 15 for k = 0, . . . , 7, 9, 10 for k = 0, 1, 2, 3

(4.3)

Thus we have a total of 76 distinct pairs of Betti numbers. All of these pairs have odd b2 + b3 , and while most of Joyce’s examples of G 2 holonomy manifolds have b2 + b3 ≡ 3 mod 4, here we have a mix between b2 + b3 ≡ 1 mod 4 and b2 + b3 ≡ 3 mod 4. 5. Concluding Remarks We have obtained the Betti numbers of barely G 2 manifolds obtained from Complete Intersection Calabi-Yau manifolds. This gives a class of manifolds that have an explicit description. One of the ways to use these examples is to try and understand the moduli spaces. On one hand we know the structure of the moduli space of the underlying CICY manifolds, but on the other hand, previous general results about the structure of G 2 moduli spaces [14,15] could be applied to these specific cases. In particular, quantities


227

like the Yukawa couplings and curvature could be calculated for these examples. This should then give a relationship between the corresponding Calabi-Yau quantities and the G 2 quantities. This could then lead to much better understanding of G 2 moduli spaces and their relationship to Calabi-Yau moduli spaces. Another direction could be to construct barely G 2 manifolds from some larger class of Calabi-Yau manifolds. In particular it is interesting to see what is the relationship between manifolds constructed from Calabi-Yau mirror pairs, and whether this could shed some light on possible G 2 mirror symmetry. Acknowledgements. I would like to thank Tristan Hübsch for the useful correspondence about CICY Hodge number, Rahil Baber for the help with programming, and the anonymous referee for very helpful remarks.

References 1. Yau, S.-T.: On the Ricci curvature of a compact Kaehler manifold and the complex monge-ampère equation. I. Comm. Pure Appl. Math. 31, 339–411 (1978) 2. Candelas P., Dale A.M., Lutken C.A., Schimmrigk R.: Complete Intersection Calabi-Yau Manifolds. Nucl. Phys. B298, 493 (1988) 3. Greene, B.R., Roan, S.S., Yau, S.-T.: Geometric singularities and spectra of Landau-Ginzburg models. Commun. Math. Phys. 142, 245–260 (1991) 4. Batyrev, V.V., Borisov, L.A.: On Calabi-Yau Complete Intersections in Toric Varieties. http://arXiv.org/ abs/alg-geom/9412017v1, 1994 5. Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B479, 243–259 (1996) 6. Hori, K., et.al.: Mirror symmetry. Providence, RI: Amer. Math. Soc., 2003 7. Joyce, D.D.: Compact Riemannian 7-manifolds with holonomy G 2 . I, II. J. Diff. Geom. 43(2), 291–328, 329–375 (1996) 8. Harvey, J.A., Moore, G.W.: Superpotentials and membrane instantons. http://arXiv.org/abs/hep-th/ 9907026v1, 1999 9. Gutowski, J., Papadopoulos, G.: Moduli spaces and brane solitons for M theory compactifications on holonomy G(2) manifolds. Nucl. Phys. B615, 237–265 (2001) 10. Acharya, B.S., Gukov, S.: M theory and Singularities of Exceptional Holonomy Manifolds. Phys. Rept. 392, 121–189 (2004) 11. Beasley, C., Witten, E.: A note on fluxes and superpotentials in M-theory compactifications on manifolds of G(2) holonomy. JHEP 0207, 046 (2002) 12. Lee, J.-H., Leung, N.C.: Geometric structures on G(2) and Spin(7)-manifolds. http://arXiv.org/abs/math/ 0202045v2 [math.DG], 2007 13. Karigiannis, S.: Flows of G 2 Structure, I. Quart. J. Math. 60, 487–522 (2009) 14. Karigiannis, S., Leung, N.C.: Hodge theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps. Proc. London Math. Soc. 99(3), 297–325 (2009) 15. Grigorian, S., Yau, S.-T.: Local geometry of the G2 moduli space. Commun. Math. Phys. 287, 459–488 (2009) 16. Joyce, D.D.: Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford: Oxford University Press, 2000 17. Partouche, H., Pioline, B.: Rolling among G(2) vacua. JHEP 0103, 005 (2001) 18. Baez, J.: The Octonions. Bull. Amer. Math. Soc. (N.S.) 39, 145–205 (2002) 19. Green, P., Hubsch, T.: Polynomial deformations and cohomology of Calabi-Yau manifolds. Commun. Math. Phys. 113, 505 (1987) 20. Green, P.S., Hubsch, T., Lutken, C.A.: All Hodge Numbers of All Complete Intersection Calabi-Yau Manifolds. Class. Quant. Grav. 6, 105–124 (1989) 21. Green, P.S., Hubsch, T.: (1, 1)3 Couplings in Calabi-Yau Threefolds. Class. Quant. Grav. 6, 311 (1989) 22. Hubsch, T.: Calabi-Yau manifolds: A Bestiary for physicists. Singapore: World Scientific, 1992 23. He, A.-M., Candelas, P.: On the number of complete intersection Calabi-Yau manifolds. Commun. Math. Phys. 135, 193–200 (1990) Communicated by N.A. Nekrasov

Commun. Math. Phys. 301, 229–283 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1149-x

Communications in


Diffusive Behavior for Randomly Kicked Newtonian Particles in a Spatially Periodic Medium Jeremy Clark1 , Christian Maes2 1 Department of Mathematics, University of Helsinki, Gustaf Hällströmin katu 2b,

00014 Helsinki, Finland. E-mail: [email protected]

2 Instituut voor Theoretische Fysica, K.U. Leuven, Celestijnenlaan 200d, 3001 Heverlee, Belgium

Received: 15 October 2009 / Accepted: 25 June 2010 Published online: 27 October 2010 – © Springer-Verlag 2010

Abstract: We prove a central limit theorem for the momentum distribution of a particle undergoing an unbiased spatially periodic random forcing at exponentially distributed times without friction. The start is a linear Boltzmann equation for the phase space density, where the average energy of the particle grows linearly in time. Rescaling time, the momentum converges to a Brownian motion, and the position is its time-integral showing superdiffusive scaling with time t 3/2 . The analysis has two parts: (1) to show that the particle spends most of its time at high energy, where the spatial environment is practically invisible; (2) to treat the low energy incursions where the motion is dominated by the deterministic force, with potential drift but where symmetry arguments cancel the ballistic behavior. 1. Introduction Recent times show a great renewed interest in obtaining diffusive behavior from microscopically defined dynamics. The motivation is much older, to derive, as Fourier, Navier or Boltzmann first did in their ways and times, irreversible and dissipative behavior starting from the reversible microscopic laws. The limiting behavior is often associated to a conserved quantity like energy in classical mechanics and the challenge is then to express the (energy) current in terms of gradients of the (energy) density itself. Obviously, for the sharpness of the limit, some scaling must be done, combined with typicality arguments on the level of the initial or boundary conditions. For example, more recently the search for a rigorous derivation of Fourier’s law of heat conduction was relaunched in [2] and many attempts and models have been taken up after that. More modestly one starts half-way with an effective description on the level of singleparticle dynamics. The one-particle phase space density then really refers to a cloud of weakly interacting particles brought in contact with some environment, and the conserved quantity is simply the number of particles. In the present paper we study the diffusive scaling limit of a massive particle in a one-dimensional periodic potential to which

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we add random forcing. The latter is not derived from first principles, but has various physical motivations. Heuristically, the forcing corresponds to the random collisions with an effectively infinite temperature granular bath. The granular structure is in the discrete kicks the particle undergoes at random exponentially distributed times. Depending on its present position, the distribution of the momentum kicks differs. Together with the potential, that specifies the spatial inhomogeneity and makes the problem nontrivial. The spatial heterogeneity brings us to a second motivation of the present work: the study of active particles where flights of ballistic motion are interrupted by spatially depending re-orientations, or the statistical characterization of particle trajectories in active heterogeneous fluids, see e.g. [15]. Our present work adds a rigorous result establishing under what conditions diffusive behavior for the momentum and superdiffusive behavior for the position get realized. Finally, the present contribution fits in the long tradition of proofs of the central limit theorem (invariance principle, and its modifications) for additive functionals, with position as the integral of the momenta and momenta as the integral of the forces. Much of all that for studies of interacting particle systems started from the pioneering work in [12]. In the spirit of the present work the papers [4,5] added very important symmetry considerations, making it possible to apply the work to strongly dependent variables. The fact that these arguments avoided the use of mixing assumptions or strong enough decay of time-correlations appears like an important lesson for today’s pursuit of diffusive behavior mentioned at the very beginning of this Introduction. Indeed, one often emphasizes that strong enough chaoticity assumptions are needed in the mathematical control of the transition from the microscopic reversible laws to macroscopic irreversible behavior. One then refers for example to the problem of obtaining regular transport properties via well-controlled Green-Kubo expressions where some temporal decay certainly seems necessary. These Green-Kubo relations are however needed only for very special observables, and not for all possible even microscopically defined quantities. It is therefore very welcome if symmetry considerations can help to establish diffusive behavior for certain classes of functionals that share symmetry properties with the relevant observables of statistical mechanics. The present paper is not starting from the microscopic classical mechanical world, but it does deal with the problem of exploiting symmetry to cancel ballistic behavior. Other more recent work on the central limit theorem that shares important ambitions with the present study includes [10,9,1,6]. For the study of fluctuations in Markov processes with an overview of central limit results, we refer to the recent book [13]. The next section introduces the model, the results and the main strategy of the proof. The momentum variable is not autonomous since it is coupled to the position of the particle. Its changes come from two sources, the momentum jumps by the external Poisson noise and the acceleration due to the presence of the potential. That is translated in the structure of the argument. The idea is to obtain a martingale central limit theorem for the momentum jumps while the effect of the potential should vanish in the long time limit. Section 3 establishes that, most of the time, the particle’s energy grows linearly with time. That is sufficient to show in Sect. 4 that the absolute value of the momentum process converges in distribution to the absolute value of a Brownian motion. Next, in Sect. 5, follow the estimates characterizing the motion at high energy, where the (bounded) potential has very little effect. The low energy motion is discussed under Sect. 6. There the drift due to the potential gets controlled by symmetry

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arguments. The combination of high and low energy estimates yields the main result of Sect. 7. 2. Main Result 2.1. Informal description. Consider a one-dimensional classical particle whose position and momentum (X t , K t ) evolve deterministically with the Hamiltonian H (x, k) = 21 k 2 + V (x) for some bounded periodic potential 0 ≤ V (x) ≤ V¯ except at Poisson times at which the particle may receive a momentum kick from the environment. That is, independent of its current momentum k and at the rate jx (v) when its current position is x, the particle receives a momentum jump v. On the level of the phase space densities, the dynamics we consider is then governed by the linear Boltzmann equation dV ∂ Pt d ∂ Pt Pt (x, k) = −k (x, k) + (x) (x, k) + dv jx (v)(Pt (x, k −v)− Pt (x, k)) dt ∂x dx ∂k R (2.1) for the phase space probability density Pt (x, k) ∈ L 1 (R2 ) for the particle at each time t ≥ 0. The rates jx have the same periodicity as the potential V . It is however probabilistically simpler to imagine a universal Poisson clock having rate R > 0, such that when the alarm rings, a (biased) coin is tossed to decide whether or not a momentum kick will occur. The probability of the coin and the distribution of the momentum jump v are respectively 0 ≤ κ(a) ≤ 1 and Pa (v), where a = X t mod 1 is the position (modulo the period 1 of the potential V ) at the Poisson time t. We assume that the momentum jumps are symmetric Pa (−v) = Pa (v) and that there is a uniform lower bound for the coin probabilities 0 < ν ≤ κ(a). Then, in (2.1), jx (v) = R κ(x)Px (v). Our main result is to show, under certain technical conditions, that the normalized s 3 1 variables (t − 2 X s t , t − 2 K s t ), s ≥ 0 approach the process ( 0 dr Br , Bs ) in distribution, where Bs is Brownian motion whose diffusion constant σ depends on the spatial average 1 of the periodic noise σ = 0 da R dv ja (v) v 2 . There is clearly no energy relaxation in (2.1), since, no matter where you start, the time-derivative of the expected energy satisfies d k2 d E [E t ] = d x dk ( + V (x)) Pt (x, k) dt dt R2 2 1 2 Pt (x, k), d x dk dv jx (v) v = 2 R2 R

(2.2)

and thus the mean energy grows linearly as t t inf sup dv ja (v) v 2 ≤ E [E t ] ≤ E [E 0 ] + dv ja (v) v 2 . E [E 0 ] + 2 a∈[0, 1] R 2 a∈[0, 1] R Moreover, as we will show, by time t not only the average but also the typical energy is of order t. Since the potential V (x) is bounded, the absolute value of the momentum 1 is then |k| ∝ t 2 . As a consequence of having high momentum, the particle will pass through one period of the potential (periodic cell) much faster than the time scale of the Poisson clock governing the noise. The particle then effectively “feels” a spatial average

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˜ of the noise in which the averaged distribution of a jump P(v) and the averaged Poisson ˜ rate are R, 1 1 da κ(a) Pa (v) ˜ =R ˜ P(v) = 0 1 and R da κ(a). 0 0 da κ(a) Moreover, at very high momentum, the force field by the potential can only displace the momentum by relatively negligible values. The effective dynamics at high energy is thus d ∂ Pt ˜ ˜ (Pt (x, k − v) − Pt (x, k)). (2.3) Pt (x, k) = −k (x, k) + R dv P(v) dt ∂x R This dynamics is translation invariant and so the momentum process has now become s 3 1 a Markov process. Showing that (t − 2 X st , t − 2 K st ) converges to ( 0 dr Br , Bs ) is then straightforward. Another way of expressing this result on the level of single-time marginals is to consider the rescaled density t 2 Pt (t 3/2 x, t 1/2 k) at time t. Its limit t ↑ ∞ is Gaussian P∞ (x, k), √ 3 − 6 (x− k )2 − 1 k 2 2 2σ P∞ (x, k) = , e σ πσ 1 and σ = 0 d x R dv jx (v) v 2 . The coupling between position x and momentum k results from the correlation between the Brownian motion (for the momentum) and its time-integral (for the position). 2.2. Strategy of proof. The process (X t , K t ) is Markovian over the set of rightcontinuous paths (having left limits) from t ∈ R+ to R2 , bounded over finite time intervals. Since the position variable is an integral of the momentum, the proof that s 3 1 (t − 2 X st , t − 2 K st ) converges to ( 0 dr Br , Bs ) for a Brownian motion Bs is implied by showing that the momentum component converges to a Brownian motion. The momentum process can be written as st dV (X r ), dr (2.4) K st = K 0 + Mst + dx 0 where Mt is the martingale of jumps, Mt = ts vs over the jump times 0 ≤ s ≤ t in Poisson process at rate R and vs is the actual momentum kick. On the other hand, the t dV 0 dr d x (X r ) is the net drift due to the conservative force up to time t. The analysis splits into two semi-independent parts corresponding to the two last 1 terms in (2.4). First we show that the momentum jump part t − 2 Mst converges to a Brownian motion. That requires establishing a martingale central limit theorem. Because of the inhomogeneity in the momentum jumps we need to prove that there is asymptotic regularity in the variances of the momentum jumps (quadratic variation process). That is realized because the particle spends most of its time at high energy where translation invariance is recovered. We call that the high energy analysis. 1 st Secondly, for the low energy analysis we show that the drift process t − 2 0 dr ddVx (X r ) makes a vanishing contribution for large times; in other words the variance of the time


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integral of the drift converges to zero. Indeed note again that the final process, obtained after scaling, does not depend on the potential V . The superdiffusion of the position makes the position process almost deterministic and the random kicks become rare compared to the fast movement of the particle. Mathematically, in Sect. 6 we use that periods of low energy are well separated by times of high energy. There is therefore some independence between the low energy incursions. Moreover symmetry arguments constrain the gained momentum in each such incursion to have zero expectation. 2.3. Main theorem. Our main mathematical result is a central limit theorem for the momentum process. We give here the precise statement. Assumptions List 2.1.

I. There exists 0 < r1 such that for all a ∈ [0, 1], r1 ≤ R dv ja (v) v 2 . II. There is ρ > 0 such that for all a ∈ [0, 1], R dv Pa (v) v 4 ≤ ρ. III. ja (v) = ja (−v) IV. V¯ > V (x) ≥ 0 is bounded and has a bounded derivative.

The first three assumptions are on the rate of momentum jumps. They should be symmetric, allow spreading but still have a fourth momentum. For the Hamiltonian part, both the potential and the force is bounded. The assumptions of List 2.1 are designed to be the minimal assumptions for Sect. 3, and most results from later sections require both List 2.1 and some of the assumptions from List 2.2. Assumptions List 2.2. i. There exists C and η > 0 such that for all a ∈ [0, 1] and v, w ∈ R with |v|−|w| ≥ 0, Pa (w) ≤ C e−η(|v|−|w|) Pa (v). ii. There exists a μ such that for all a ∈ [0, 1], sup (Pa (v))

v∈R

−1

(1 + |v|)

−1

dPa (w) ≤ μ. sup |w−v|≤1 dv

. iii. There exists a reflection R on the torus such that V (R(x)) = V (x) and j R(a) (v) = ja (v) for a ∈ [0, 1] and v ∈ R. Condition (i) implies that the Laplace transform of Pa is finite in a neighborhood around zero and thus that the fourth moment as in (II) and all other moments are finite. In later sections, r1 , r2 , ν will be defined as 2 r1 = inf dv ja (v) v , r2 = sup dv ja (v) v 2 , ν = inf κa . a∈[0,1] R

a∈[0,1] R

a∈[0,1]

Since ja (v) = Rκa Pa (v), the condition (I) and (II) with Jensen’s inequality imply that 1 1 0 < r1 R−1 ρ − 2 ≤ ν. Also by (II) and Jensen’s inequality, r2 < Rρ 2 < ∞. (III) and (iii) are the symmetries that we assume for the dynamics. (III) says that for every point a ∈ [0, 1] in the periodic cell, the rate of kicks by a momentum v occurs with the same rate as kicks by a momentum −v. (iv) specifies that in addition to the periodicity of the dynamics, there is also a spatial reflection symmetry. The combination

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of these symmetries forms a “momentum time-reversal symmetry” which is used in Sect. 6.2. Pa (ii) is a technical assumption so that the values of the derivative | ddv (w) for w in a neighborhood around v cannot be too large compared to the value Pa (v). The constraint becomes more flexible at large |v|, where the ratio is allowed to increase as |v|. This condition effectively forbids densities with tails that vanish faster than a Gaussian density G(v) in which dG (v) dv

G(v)

∝ |v|.

The support of Pa (v) for each a cannot be finite for instance. Avoiding this decay of P(v) as |v| → ∞ is not essential to the analysis, but generalizing the condition (ii) (for instance by replacing (1 + |v|) by (1 + |v|)m ) requires making other conditions more complicated. Theorem 2.3 (Main result). Assume List 2.1, List 2.2, and that the initial joint phase space distribution P0 (x, k) ∈ L 1 (R2 ) has finite second moments. In the limit t → s 3 1 ∞, (t − 2 X st , t − 2 K st ) converges in distribution to ( 0 dr Br , Bs ), where Bs is Brownian 1 motion with diffusion constant σ = 0 da R dv ja (v) v 2 . 3. A Martingale Central Limit Theorem In this section, we prove that the typical energy for the particle is on the order of t. That implies a regularity in the momentum process, at least concerning its absolute value and for the quadratic variation of the momentum jumps. The net result is a martingale central limit theorem for the martingale part in (2.4). Note that we always assume the natural filtration Ft specifying the Markov process up to time t. Theorem 3.1. Assume List 2.1 and (i)–(ii) of List 2.2 and that the initial joint distribu1 tion P0 (x, k) has finite second moments. Then t − 2 Mst converges in distribution to a Brownian motion Bs with diffusion constant σ . The proof follows in Sect. 7. It will be built on the lemmas below and in the next section. The following lemma relies on a martingale central limit theorem [7] and on having some bounds for the time-change of that central limit theorem. The result applies to more general class of martingales, but we develop it here to the martingale process Mt =

Nt

wn S(K tn− ),

n=1

where tn are the Poisson time for the underlying Poisson clock Nt with rate R, S(K t − ) is the left limit up to time t for the sign of the momentum, and wn = Mtn − Mtn− . Note that wn is zero if at the Poisson time tn there is no momentum jump, and it is equal to the momentum jump if it does happen.


Lemma 3.2. For T (t) = , δ > 0,

1 0

235

1

dr χ (|t − 2 Mr t | > ), there exists a C > 0 such that for all

lim inf Pr t→∞

T (t)

1

r2 ≥1−δ ≥1−C 2 . r1 δ 1

1

1

The same result remains true with |t − 2 Mst | replaced by t − 2 Ms t − inf 0≤u≤s t − 2 Mu t . Proof. We start from the lower bound:

Pr T (t) ≤ 1 − δ = 1 − Pr

1 0

1 dr χ t − 2 |Mst | ≤ ≥ δ

1 1 − 21

ds χ (|t Ms t | ≤ ) ≥ 1− E δ 0 1 1 1 ds Pr |t − 2 Ms t | ≤ . = 1− δ 0

(3.1)

1 (t) Define B˜ u = t − 2 Mτ u t , where τu is the hitting time 1

M st ≥ u , τu = inf s ≥ 0 r1 t

and M t is the predictable quadratic variation of M up to time t. In our situation, M t has the form t

M t = dr dv j X r (v) v 2 . 0

R

(t) By the martingale central limit theorem, B˜ u converges to a Brownian motion in the uniform metric. The Lindberg condition is guaranteed by the boundedness of the fourth moment of single momentum jumps in (II) of List 2.1 and the fact that the jump times occur according to a Poisson clock (having rate R). 1 We also have t − 2 Mst = B˜ R(t)s , where Rs = r11 t M st , since τu and Rs are inverses of one another. By (II) of List 2.1 and the discussion following it, R dv ja (v) v 2 ranges between the values 0 < r1 ≤ r2 for a ∈ [0, 1]. It follows that r2 r1 ≤ τu ≤ u and s ≤ Rs ≤ s . (3.2) u r2 r1

By (3.2), Rs has the range s ≤ Rs ≤ 0

1

1 −2

ds χ |t Mst | ≤ =

0

1

r2 r1 s,

and thus

(t) ˜ ds χ | B Rs | ≤ ≤

r2 r1

0

du χ | B˜ u(t) | ≤ . (t)

Taking the expectation of the right-hand side and using the fact that B˜ s approaches a Brownian motion, we have that 2 r 2 r2 r2 − 2rx s 1 r1 r1 r1 e (t) (t) . du χ | B˜ u | ≤ = du Pr | B˜ u | ≤ −→ du dx √ E 2πr1 s 0 0 0 −

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By a change of variables, the right side is bounded by a constant multiple of r1−1r22 . With (3.1) this proves the result. 1 1 (t) To generalize the result to Hs = t − 2 Ms t − inf 0≤a≤s t − 2 Ma t , we make the same 1 (t) (t) (t) time-change τ , to define a process Z u = Hτu . Since B˜ u = t − 2 Mτ u converges to a (t)

Brownian motion, it will follow that Z u converges to the absolute value of a Brownian motion. Indeed, the function f : L ∞ ([0, 1]) → L ∞ ([0, 1]) defined by f (xs ) = xs + sup0≤r ≤s − xr (read the supremum as an essential supremum) satisfies f (xs ) − ˜ (t) f (ys ) ∞ ≤ 2 x s − ys ∞ . Thus the convergence of Bu to a Brownian motion Bu (t) implies that f B˜ u converges in distribution to f (Bu ). However, by the basic result for Brownian motion [11], f (Bu ) is equal in distribution to |Bu |. Thus we can apply the same reasoning as above to get the result. The lemma below will be used in the proof of Lemma 3.4. Its proof follows from basic calculus but requires consideration of several cases. Lemma 3.3. For V¯ = supx∈R V (x) < ∞, there exists a c > 0 such that 1 1 1 2 2 1 1 2 2 2 2 − |k + w| + V |k − w| + V 2 w S(k) − 2 2 1 1 1 2 2 2 1 1 1 |k + w|2 + V |k − w|2 + V k2 + V ≤ 2 w 2 1|w|>J + c J + −2 2 2 2 (3.3) 1 for all k, w ∈ R, 0 ≤ V ≤ V¯ , and J ≥ V 2 . 1

We now apply Lemma 3.2 to attain a similar inequality with |Ms | replaced by E s2 . 1 1 Our analysis is based on the fact that E t2 = 21 K t2 + V (X t ) 2 is a submartingale. The 1 1 submartingale property of E t2 follows since f (k) = 21 k 2 + V 2 is a convex function. Since the jumps occur with symmetric probabilities Pa (v) = Pa (−v), we can write 1

1

E t2 − E 02 = Mt + At in terms of a martingale part Mt and a stochastically increasing part At as 1 1 N 2 2 1 1 t 1 2 2 |K tn− + wn | + V (X tn ) − |K tn− − wn | + V (X tn ) Mt = , (3.4) 2 2 2 n=1

N

1 t At = 2 n=1

1 |K − + wn |2 + V (X tn ) 2 tn

1 |K − |2 + V (X tn ) −2 2 tn

1

2

+

1 |K − − wn |2 + V (X tn ) 2 tn

1 2

1 2

,

(3.5)

where tn , wn for n = 1, . . . , Nt are the Poisson times and their corresponding momentum jumps (when they occur), respectively. The processes Mt , At form a Doob-Meyer

Diffusive Behavior for Randomly Kicked Newtonian Particles 1

237

1

decomposition for E t2 − E 02 , however At is not a predictable process so the decomposition is not in the unique sense. Lemma 3.4 (Energy Lemma). Assume List 2.1. Define T ,(t)V = There exists a constant C such that lim inf Pr t→∞

(t) T , V

1 0

1

1

ds χ (|t − 2 E st2 | > ).

1

r2 ≥1−δ ≥1−C 2 . r1 δ

Proof. Define the martingale Mt and the increasing process A t as Mt =

Nt

wn S(K tn− ) and A t = sup −Ms , 0≤s≤t

n=1

where S(K t − ) is the left limit up to time t for the sign of the momentum. Also define √ 1 1 1 (t) (t) G s as the difference G s = 2t − 2 E st − t − 2 Mst − t − 2 A st . In general, we have that √ (t) Pr T (t) > 1 − δ, sup G (t) (3.6) s ≤ ( 2 − 1) ≤ Pr T ,V > 1 − δ , 0≤s≤1

−2 −2

as in Lemma 3.2 for the where T (t) is defined process t1 Mst + t Ast . We will prove √ − below that Pr sup0≤s≤1 G (t) s > ( 2 − 1) = O(t 4 ). In that case, by applying the inclusion-exclusion principle to the left side of (3.6), then (3.6) can be written 1 Pr T (t) > 1 − δ + O(t − 4 ) ≤ Pr T ,(t)V > 1 − δ . 1

1

We can then apply Lemma 3.2 to the left-side to complete the proof. √ −1 Now we work towards establishing Pr sup0≤s≤1 G (t) s > ( 2 − 1) = O(t 4 ). Consider the martingale Mt − Mt . The square of the jumps of Mt − Mt can be bounded by the jumps of At plus an extra term through the inequality from Lemma 3.3: 1 1 √ 2 2 1 1 2 2 2 |k + w| + V (x) − |k − w| + V (x) 2 w S(k) − 2 2 1 1 2 2 1 1 2 2 2 |k + w| + V (x) + |k − w| + V (x) ≤ 2 v 1|w|>J + c J 2 2 1 2 1 2 k + V (x) −2 2 r 2 1 4 for all J > 1, k, w, x ∈ R. Define the process Q r = N n=1 wn χ (|wn | ≥ t ).

By (3.7), the quadratic variation process [M − M]t has the bound 1

[M − M]t ≤ c t 4 At + 2Q r .

(3.7)

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This will allow us to bound E (Mt − Mt )2 = E [M − M]t . By the fact that At is the 1

1

increasing part of the Doob-Meyer decomposition for E t2 − E 02 and Jensen’s inequality we have 1 1 1 1 1 1 1 2 E[At ] = E[E t2 − E 02 ] ≤ (E[E t ]) 2 ≤ E[E 0 ] + 2−1r2 t ∼ 2− 2 r22 t 2 . Also we have ⎤ ⎡ Nt

1 1 1 wn2 χ (|wn | ≥ t 4 )⎦ ≤ E[Nt ] ρ t − 2 = R ρt 2 , E [Q t ] = E ⎣ n=1

where we have used that the fourth moments of the jumps are bounded as E wn4 ≤ ρ, since 1 1 1 1 E wn2 χ (|wn | ≥ t 4 ) ≤ t − 2 E wn4 χ (|wn | ≥ t 4 ) ≤ t − 2 ρ. Thus, using the above and Doob’s maximal inequality E

sup t −1 |Mst − Mst |2 ≤ 4t −1 E (Mt − Mt )2 = 4E [M − M]t

0≤s≤1

3

1

1

1

1

1

≤ 4 c t − 4 E [At ] + 8t −1 E [Q t ] ≤ 4 c 2− 2 r22 t − 4 + 8Rρ t − 2 = O(t − 4 ). (3.8) 1

1

Next we show that t − 2 Ast is typically bounded from below by t − 2 A st in the sense E

−1

2 1 1 − sup (t 2 Ast − t 2 Ast )1 A st >Ast = O(t − 4 ).

0≤s≤1

We can write

√ − 1 21 2t 2 E st as

1 √ −1 1 √ 1 1 1 1 1 2t 2 E st2 = t − 2 Mst + 2t − 2 E 02 + t − 2 Mst − t − 2 Mst + t − 2 Ast ,

(3.9)

where since the left side is positive for all a, we must have 1

1

−t − 2 Mat χ(t − 2 Mat ≤ 0) ≤

√

1 1 1 1 1 1

2t − 2 E 02 + t − 2 Mat − t − 2 Mat + t − 2 Aat χ(t − 2 Mat ≤ 0).

Taking the supremum in a up to s ≤ 1 of both sides, 1 √ −1 1 1 1 1 1

+ t 2 Ast , t − 2 A st = sup −t − 2 Mat ≤ sup 2t − 2 E 02 + t − 2 Mat − t − 2 Mat

0≤a≤s

0≤a≤s


239

where we have used that M0 = 0 for the left side and that Ar is increasing for the 1 1 right side. Subtracting t − 2 Ast and taking E sup0≤s≤1 | · |2 2 of both sides 1 2 21 1 E sup (t − 2 A st − t − 2 Ast ) 1 A >A ≤

√

2t

− 21

st

st

0≤s≤1

1 2

E0 + E

sup t

−1

Mst −

0≤s≤1

2 Mst

1 2

1

= O(t − 8 ).

(3.10)

Observe that G (t) s < 0 implies Ast − A st ≤

√

1

2 E 02 − 0 ∧ (Mst − Mst ).

(3.11)

Finally, by the triangle inequality, (3.8), (3.10), and (3.11), 1 2 E

2 sup |G (t) s | 1G (t)

In the following lemma, we set L st = Mst + A st , where Mst and A st are defined as in the proof of Lemma 3.4. The Doob-Meyer decompositions in the lemma are not unique, since the increasing parts are not constrained to be predictable. Lemma 3.5. Consider the submartingales E t and L 2t . 1. E t admits a Doob-Meyer decomposition as a sum of martingale and increasing parts: Mt =

Nt

N

wn K tn−

and At = E 0 +

n=1

1 t 2 wn . 2 n=1

2. L 2t admits a Doob-Meyer decomposition as a sum of martingale and increasing parts M¯ t = Nt αn and A¯ t = Nt βn respectively, where n=1

n=1

αn = 2wn L tn−1 1 αn = S(K tn− )S(wn )(L tn−1 + |wn |)2 2 βn = wn2 1 βn = wn2 + (|wn | − L tn−1 )2 2

for |L tn−1 | ≥ |wn |, for |L tn−1 | ≤ |wn |, for |L tn−1 | ≥ |wn |, for |L tn−1 | ≤ |wn |.

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3. In the limit t → ∞,

E

sup t

1 2Ast − A¯ st = O(t − 4 ).

−1

0≤s≤1

Proof. Part (1): E t can be rewritten as Et = E0 +

Nt

n=1

1 wn K tn− + wn2 , 2

(3.13)

which follows by an inductive expansion using the conservation of energy between momentum jumps: E tn =

1 1 1 (K tn− + wn )2 + V (X tn− ) = K t2− + V (X tn− ) + wn K tn− + wn2 n 2 2 2 1 2 1 = K tn−1 + V (X tn−1 ) + wn K tn− + wn2 . 2 2

The middle term onthe right of (3.13) is a martingale by the symmetry of the jump rates (III) of List 2.1: E wn Ftn− = 0. Part (2): To find an expression for L t , we apply an inductive argument as with E t except that the analysis breaks into two cases. Expanding L 2tn when |L tn−1 | > |wn | is the easy case since L 2tn = L 2tn−1 + 2vn S(K tn− ) + wn2 . Again wn S(K tn− ) is the martingale contribution, since E wn2 Ftn− = 0. Expanding L 2tn in the case that |L tn−1 | ≤ |wn | , then L tn = 21 1 + S(K tn− )S(wn ) (L tn−1 + |wn |), and 1 1 S(K tn− )S(wn )(L tn−1 + |wn |)2 + wn2 + (|wn | − L tn−1 )2 . 2 2 The first term on the right has mean zero since E S(wn ) Ftn− , |wn | = 0. L 2tn − L 2tn−1 =

Part (3): By Part (1) and Part (2), t −1 ( A¯ st − 2Ast ) = −2t −1 E 0 + t −1

Nst

2 1 |wn | − L tn−1 χ (|wn | ≥ L tn−1 ). 2 n=1

Since |wn | ≤ J , the sum above is bounded by Nst

(|wn | − L tn−1 )2 χ (|L tn−1 | ≤ |wn |)

n=1 1

≤ t4

Nst Nst

1 (|wn | − L tn−1 )χ (|L tn−1 | ≤ |wn |) + |wn |2 χ (|wn | ≥ t 4 ). n=1

n=1

(3.14)


241

By the estimates in Lemma 3.4, ⎤ ⎡ ⎤ ⎡ Nst Nt

1 1 1 |wn |2 χ (|wn | ≥ t 4 )⎦ = E ⎣t −1 |wn |2 χ (|wn | ≥ t 4 )⎦ = O(t − 2 ). E ⎣ sup t −1 0≤s≤1

n=1

n=1

The first term on the right-side is closely related to Ar , since Ar can be written sup − 0≤s≤r

Nr

wn S(K tn− ) =

Ar

Nr

= (|wn | − L tn−1 )

n=1

n=1

×χ |L tn−1 | ≤ |wn |, S(K tn− )S(wn ) = −1 ,

since increases in A t occur |L tn−1 | ≤ |wn | and the jump wn has sign such that S(K tn− ) S(wn ) = −1. In fact, the conditional expectation of a single term from the sum with respect to the information up to time tn− and the size |wn | of the n th jump is E (|wn | − L tn−1 )χ |L tn−1 | ≤ |wn |, S(K tn− )S(wn ) = −1 Ftn− , |wn | =

1 (|wn | − L tn−1 )χ (|L tn−1 | ≤ |wn |), 2

since wn = −|wn | and wn = |wn | have equal probability. Thus, ⎤ ⎡ Nt

3 3 3 (|wn | − L tn−1 )χ (|L tn−1 | ≤ |wn |)⎦ = t − 4 E A t = t − 4 E Mt + A t E ⎣t − 4

n=1 1 2

=E 2 t

− 43

1 2

Et − t

− 41

(t) G1

1 2

≤2 t

− 43

1 2

E Et

3

+ O(t − 8 )

1 3 1 3 1 3 1 3 1 1 ≤ 2 2 t − 4 E [E t ] 2 + O(t − 8 ) ≤ 2 2 t − 4 (E[E 0 ] + r2 t) 2 + O(t − 8 ) = O(t − 4 ), 2 (3.15)

where the second equality is because Mr is a mean zero martingale, the third (t) equality is from the definition of G r , and the first inequality uses the result 1 1 (t) E sup0≤s≤1 |G s |2 1G (t) 0, let τ ∈ [0, 1] be the first time that Hs(t) reaches above a and put τ = 1 if that event does not occur. By (4.7), (t) (t) (t) (t) E τ = a Pr sup Hs ≥ a + E H1 χ sup Hs < a , 0≤s≤1

which implies that sup a Pr

Hs(t)

sup

a∈R+

0≤s≤1

0≤s≤1

≥ a ≤ E τ(t) + E

sup 0≤s≤1

1

|Hs(t) |1 H (t) 0 and αt = supa∈R+ a Pr[Yt ≥ a] 2 . If E Yt2 ≤ y2 for s all t, then for any m > αt > 0, αt m ∞ ∞ da Pr [Yt ≥ a] E [Yt ] = da Pr [Yt ≥ a] = + + 0

≤ αt +

1 αt

αt

0

m

αt

da a Pr [Yt ≥ a] +

1 m

m ∞

da a Pr [Yt ≥ a] ≤ αt + mαt +

m

y2 . 2m

a Pr[Yt ≥ a] for the first the second. We can pick m large to make the last term on the right-side small, and then pick t large enough so that with (4.8) αt (1 + m) is small. To finish the argument we just need to show that E Yt2 can be uniformly bounded The second inequality aboveuses that αt2 is the supremum of ∞ integral and the relation 21 E Yt2 = 0 da a Pr [Yt ≥ a] for

(t)

1

1

1

1

3

1

1

(t)

in t. Since G τ is the difference of 2 2 t − 2 E st2 and t − 2 L st , and 2 2 t − 2 E s2 − G s is their sum, E

Yt2

1 2

1

< 4E

sup t 0≤s≤1

−2

2

2 E st

+2E

1 sup t

−2

2

|L st |

4

.

0≤s≤1

To attain a value y, the terms on the right can be bounded by standard calculations using the Doob-Meyer decompositions for Er and |L r |2 from Lemma 3.5 and using Doob’s maximal inequality. 5. Estimates at High Energy In this section, we provide estimates that are useful for understanding the dynamics when the particle has high energy, which by Lemma 3.4, is the majority of the time. The estimates are based on the idea that the particle will feel a spatially averaged noise and that the momentum is too high to be shifted through the action of the force generated by the potential. The following elementary bound is used many times in this section and later sections. It follows from the conservation of energy and the quadratic formula. It basically says 1 1 that if the initial momentum k0 has |k0 | V¯ 2 = supa∈[0,1] V (x) 2 , then the future momenta ks , as determined by the Hamiltonian evolution, will stay close to k0 .


245

Lemma 5.1. Let (xt , kt ) evolve according to the Hamiltonian H (x, k) = 21 k 2 + V (x), for positive potential bounded by V¯ . If the initial momentum has |k0 |2 > 4V¯ , then the displacements in momentum kt − k0 and kt − ks have bounds |kt − k0 |
4V¯ , the momentum kt will not change signs at any time. By the conservation of energy, 2 1 1 k0 + (kt − k0 ) − k02 = −V (xt ) + V (x0 ). 2 2 Using the quadratic formula and that kt , k0 have the same sign, 1 2 |kt − k0 | = |k0 | − k02 + 2V (x0 ) − 2V (xt ) 1 2V (x0 )−2V (xt ) − 1 2 V¯ 2 ≤ . da k02 + a < 2 0 |k0 | √ − 1 Since 21 k02 + a 2 ≤ 2|k0 |−1 < 2|k0 |−1 for a ≤ 21 k02 , by the triangle inequality, we can bound the difference |kt − ks |. 1 1 ˜ As before P(v) = 0 da κ(a) κ¯ Pa (v), where κ¯ = 0 da κ(a). By our assumptions inf 0≤a≤1 κ(a) = ν > 0. Lemma 5.2. Assume List 2.1 and (i)–(ii) of List 2.2. Starting from the point (x, k) with |k|2 V¯ , let r(x, k) ∈ L 1 ([0, 1]) and r˜(x, k) ∈ L 1 ([0, 1]) be the probability density for the torus position of the particle at the first Poisson time and at the time of the first momentum jump respectively. Let P(x, k) ∈ L 1 (R) be the density for the first momentum jump. Let T(x, k),t ∈ L 1 ([0, 1]) be the density of time that a determinstic trajectory starting from (x, k) and evolving according to the Hamiltonian H (x, k) = k 2 + V (x) spends at a torus point over a time interval [0, t]. We have the following bounds: R 1. supa∈[0, 1] r(x, k) (a) − 1 ≤ 2|k| + O( |k|1 2 ), −2 κ¯ − 1 ≤ 2R|k|ν + O( |k|1 2 ), 2. supa∈[0, 1] r˜(x,k) (a) κ(a) −2 P (v) 3. supv∈R (x,˜ k) − 1 ≤ 2R|k|ν + O( |k|1 2 ). P(v) 2t 4. supa∈[0, 1] T(x, k),t (a) − t ≤ |k| . Proof. Part (1): Let (xs , ks ) ∈ R2 be the position and momentum for a particle beginning at (x, k) and evolving over a time period s for Hamiltonian H (x, k) = 21 k 2 + V (x). Notice that r(x, k) (a) can be written as r(x, k) (a) =

∞

|k(x,k) (a)|−1 R e−R rn (a) ,

n=1

where s = r1 (a), r2 (a), . . . are the periodic sequence of times for which xs mod(1) = a, 1 and k(x,k) (a) = s(k) (H (x, k) − V (a)) 2 is their momentum at the point. These times will exist for every a ∈ [0, 1] as long as H (x, k) > V¯ .

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If 4V¯ ≤ k 2 , then |ks − k| ≤ 2 V¯ |k|−1 by Lemma 5.1. Thus for large momentum 1 |k| (V¯ ) 2 , ks is nearly constant and the hit times rn (a) will be close to the sequence of times s = sn (a) at which x + sk mod(1) = a for a time period at least on the order of 1 1 . When |k| is t 2 . The period τ such that rn (a) − rn−1 (a) = τ should thus be close to |k| 1 2 −1 ¯ large enough so that |ks − k| ≤ 2 V |k| < |k|, then clearly τ ≤ , and |k|

2

|τ −

1 |k| 1 τ 1 |≤ ds k − |k| |k| 0 0

1 τ |k| 1 6V¯ ds k ≤ ds |ks − k| + ds |ks − k| < 3 . |k| |k| 0 0

(5.1) The difference between the first crossing-times |r1 (a) − s1 (a)| of the point a can be similarly bounded. Using the triangle inequality ∞

r(x,k) (a) − 1 ≤ r(x,k) (a) − 1 R e−R rn (a) |k| n=1

∞ ∞ 1

1

+ R e−R rn (a) − R e−R sn (a) |k| |k| n=1

n=1

∞ 1

2R 1 + + O( 2 ), R e−R sn (a) − 1 ≤ |k| |k| |k|

(5.2)

n=1

where the last inequality follows by further computation using the inequalities above. For instance, we can bound the first term on the right as ∞

−R r (a) 1 k Rτ 2 V¯ 1 r(x,k) − 1 e R e−R rn (a) ≤ 1 − ≤ 2. − R τ |k| k(x,k) (a) |k|τ 1 − e k n=1

κ¯ Part (2): We will bound supa∈[0, 1] r˜(x,k) (a) κ(a) − 1 by invoking Part (1). This will involve breaking down an expression for r˜(x,k) (a). Since there are a random number of Poisson times before the time of the first momentum jump, the expression will have a series of integrals whose n th term corresponds to a momentum jump occurs at the (n + 1)th Poisson time, ∞

r˜(x, k) (a) = κ(a) n=0

(R + )n

ds1 . . . dsn Rn e−R S(n) nm=1 1 − κ(x S(m) )r(x S(n) , k S(n) ) (a), (5.3)

and S(m) = s1 + · · · + sm . 1 Let |k| > 4V¯ 2 so that with two applications of Lemma 5.1, sups, t≥0 |kt − ks | ≤ |k|−1 2 V¯ . In particular, for any time S(n), we can apply Part (1) to the difference


247

|r(x S(n) , k S(n) ) (a) − 1| to get ∞

κ¯ − κ¯ ds1 . . . dsn Rn e−R S(n) nm=1 1 − κ(x S(m) ) ˜r(x, k) (a) + n κ(a) n=0 (R )

∞ 2R 1 ≤ ds1 . . . dsn Rn e−R S(n) nm=1 1 − κ(x S(m) ) + O( 2 ) + n |k| |k| n=1 (R )

∞ 2R 2R 1 1 n −1 ≤ (1 − ν) = ν (1 − ν) (5.4) + O( 2 ) + O( 2 ) , |k| |k| |k| |k| n=1

where the second inequality follows since 1−κ(x S(m) ) ≤ 1−ν for all m and Rn e−R S(n) defines on (R+ )n . a probability measure n − R ¯ n in If (R+ )n ds1 . . . dsn R e S(n) nm=1 1 − κ(x S(m) ) were replaced by (1 − κ) ∞ ¯ n = κ −1 would make the difference the left-side of (5.4), then identity n=0 (1 − κ) zero. Using a telescoping sum and the definition of r(x,k) (a),

¯ n ds1 . . . dsn Rn e−R S(n) nm=1 1 − κ(x S(m) ) − (1 − κ)

(R + )n n

≤

(1 − κ)n−m

m=0

× 0

1

(R)m−1

ds1 . . . dsm−1 Rm−1 e−R S(m−1) rm−1 =1 1 − κ(x S(r ) )

da 1 − κ(a) r(x S(m−1) , k S(m−1) ) (a) − 1.

Again by Part (1), since Rm−1 e−R S(m−1) is a probability measure on (R+ )m−1 , and by the bounds 1 − κ(a), 1 − κ¯ ≤ 1 − ν, we can estimate the right-side above by n(1 − ν)

n

2R 1 + O( 2 ) . |k| |k|

Putting everything together κ¯ sup r˜(x,k) (a) − 1 ≤ κ(a) a∈[0, 1]

2R 1 + O( 2 ) |k| |k|

∞

(n + 1)(1 − ν)n ,

n=0

and the sum of the series is ν −2 . Part (3): Now we study the probability density P(x, k) (v) for the next momentum jump. We can write the density for the next momentum jump as P(x, k) (v) =

1 0

da r˜(x, k) (a) Pa (v),

(5.5)

248

J. Clark, C. Maes

We then have a bound using 1 κ(a) P(x, k) (v) 1 κ¯ − 1 Pa (v) sup − 1 ≤ sup da r˜(x, k) (a) ˜ ˜ κ(a) κ¯ P(v) 0 v∈R v∈R P(v) 1 1 κ¯ κ¯ da Pa (v) ≤ sup r˜(x,k) (a) − 1 sup ˜ κ(a) κ(a) a∈[0, 1] |v|≤J P(v) 0 κ ¯ (5.6) − 1, = sup r˜(x,k) (a) κ(a) a∈[0, 1] ˜ where in the last equality we have used the definition of P(v). Applying Part (2) then we get the bound. Part (4): The density T(x,k),t can be written T(x,k),t (a) = n (x,k) (a, t)k(x,k) (a), where k(x,k) (a) is defined as in Part (1), and n (x,k) (a, t) is the number of times the particle passes over the torus point a over a time period t when starting from (x, k). The result follows by bounding the errors for k(x,k) (a) ∼ k and |k|−1 n (x,k) (a, t) ∼ t which is similar to Part (1). The following lemma bounds the contribution of the cumulative drift over periods of high-energy. Define τ (t) to be the time of the next to last momentum jump and put it equal to zero if two jumps have not occurred. Lemma 5.3. Assume List 2.1. In the limit t → ∞, for 0 < β < 21 , E

sup t −1+2β

0≤r ≤t

0

r

2 dV (X s ) χ |K τ (s) | > t β ds dx

1 2

−→ 0.

Proof. It is convenient to split the total integral into a sum of integrals over the periods between Poisson times (at which there may be a momentum jump breaking the conservation of energy) which include only the even and odd terms respectively. Let t1 , . . . , tNr be the Poisson times up to a time r .

r 0

N2r

t2n+1 dV dV β (X s ) χ |K τ (s) | > t ≈ (X s ) ds χ |K t2n−1 | ≥ t β ds dx dx t2n n=1

N2r

+

n=0

χ |K t2n | ≥ t β

t2n+2

ds t2n+1

dV (X s ), dx (5.7)

where on the right side, we have neglected the integral from t2 Nr to r , which will 2 be small, and we define t−1 = 0. We will focus on the sum with interval starting at even numbered times [t2n , t2n+1 ]. First, we will show that the terms in the sum can be replaced by the same expressions conditioned on the information up to time tn−1 . These conditional expectations can be uniformly bounded by an argument following at the end


249

of the proof. The strategy is to invoke Lemma 3.4 to guarantee that most of the terms 1 in the sum have |K t2n−1 | on the order of t 2 rather than just |K t2n | ≥ t β . This lowers the bounds available for the conditional expectations of those terms. The following is a martingale: N2r

Yr =

n=0

t2n+1

t2n+1 1 dV dV χ |K t2n−1 | ≥ t β ds ds (X s ) − E (X s ) Ft2n−1 . 2 dx dx t2n t2n

(5.8) However, the second moment for a single term from the sum is bounded by t2n+1

t2n+1

1 dV dV 2 (X s ) − E (X s ) Ft2n−1 ds ds E χ |K t2n−1 | ≥ t β 2 dx dx t2n t2n

t2n+1 2 dV ≤E (X s ) |K t2n−1 | ≥ t β ≤ 16V¯ 2 t −2β ds d x t2n

2 dV 1 2β −2 β (5.9) (a) Pr |K t2n | ≤ t |K t2n−1 | ≥ t , +2R sup 2 0≤s≤1 d x where we have considered separate bounds for the event that |K t2n | ≥ 21 t β or |K t2n | < 1 β 1 β 5.1 to bound the drift by 4V¯ t −β , 2 t . When |K t2n | ≥ 2 t , then we can apply Lemma dV 1 β and when |K t2n | < 2 t then we use that the forces d x (a) are bounded and that the difference between two Poisson times has an exponential distribution with second moment 1 2R−2 . Finally, since the force can only change the momentum by at most V¯ 2 14 t β over any time interval, only a large momentum jump can send |K t2n | below 21 t β . However, since the fourth moments of Pa (w) are less than ρ,

1 β 1 β β β Pr |K t2n | ≤ t |K t2n−1 | ≥ t ≤ Pr |wn | ≥ t |K t2n−1 | ≥ t 2 4 ∞ 44 ρ ≤ sup dw Pa (w) ≤ 4β , t a∈[0,1] 41 t β where the last inequality is Chebyshev’s and thus the right side is O(t −4β ), which make the right term on the left side of (5.9) negligible compared to the left term. Consider again the variance of a single term in (5.8). By Doob’s maximal inequality, 2 2 E sup Yr ≤ 4 E Yt ≤ 16V¯ 2 t −2β + O(t −4β ) E[Nt ] = 16V¯ 2 Rt 1−2β +O(t 1−4β ). 0≤r ≤t

t Thus we can focus on bounding the expressions E t2n2n+1 ds

dV dx

(X s ) Ft2n−1 when

|K t2n−1 | ≥ t β . The end result of the analysis below will be to show that there is a constant c such that for all sufficiently large t β 1, t2n+1 dV β (X s ) Ft2n−1 ≤ c t −2β . ds (5.10) E χ |K t2n−1 | ≥ t dx t2n

250

J. Clark, C. Maes 1

Applying the above inequality (also with t β replaced by t 2 ), N2t

E χ |K t2n−1 | ≥ t β

t2n+1

ds

t2n

n=1

≤ c Nt,

−2 −1

t

dV (X s ) Ft2n−1 dx

+ c Nt − Nt, t −2β , 1

where Nr, is the number of terms with |K t2n−1 | ≥ t 2 up to time r . By the triangle inequality ⎡ N2t

21 t2n+1 dV 2 ⎢ −1+2β

β E ⎣t E χ |K t2n−1 | ≥ t ds (X s ) Ft2n−1 dx t2n n=1

2 21 2 21 1 ≤ 2 2 c R −2 t −1+2β + c t −1 E Nt − Nt, − γ (t) + c t −1 E γ (t) , (5.11) where it was used that Nt,

1 1 2 2 ≤ Nt , E Nt ≤ 2 2 tR, and γ (t) is defined as

γ (t) = R

Nr

1 (tn − tn−1 )χ (|K tn−1 | ≤ t 2 ). n=1

The difference Nr − Nt, − γ (r ) is a martingale since tn − tn−1 are exponentially distributed with mean R−1 . The variance of the martingale satisfies ⎡ ⎤ Nt

2 1 |R(tn − tn−1 ) − 1|2 χ (|K tn−1 | ≤ t 2 )⎦ ≤ E[Nt ]. E Nt − Nt, − γ (t) = E ⎣ n=1 1

Thus the middle term on the right-side of (5.11) is O(t 2 −2β ). γ (t) is less than the amount 1

1 2

of time r ∈ [0, t] the particle spends with t − 2 Er ≤ . In other terms, t −1 γ (t) ≤ 1 − (t) (t) T , V , where T , V is defined as in Lemma 3.4. Thus by Lemma 3.4, Pr t −1 γ (t) ≥ δ ≤ 1

r2 C r21 δ .

Since t −1 γ (t) is bounded by 1, 1

2 21 r2 E t −1 γ (t) ≤ δ Pr t −1 γ (t) < δ + Pr t −1 γ (t) ≥ δ ≤ δ + C 2 . r1 δ Thus we can pick δ to make the first term small and then pick to make the second term small. We now turn to showing (5.10). By the Markov property

t2n+1 dV β E χ |K t F (X | ≥ t ds ) s t2n−1 2n−1 dx t2n t2n+1 dV (X s ) (X t2n−1 , K t2n−1 ) = (x, k) , ≤ d x dk Pω (x, k) χ |k| ≥ t β E ds dx t2n

(5.12) where Pω (x, k) is the distribution (X t2n−1 , K t2n−1 ) conditioned on ω ∈ Ft2n−1 .


251

Assuming that |K t2n−1 | ≥ t β and K t2n ≥ 21 t β , t2n+1 dV V (X t2n+1 ) − V (X t2n ) (X s ) − ds dx K t2n−1 t2n t2n+1 t2n+1 dV 1 dV ≤ (X s ) − (X s ) K s ds ds dx K t2n t2n dx t2n 1 1 + V (X t2n+1 ) − V (X t2n ) − K t2n K t2n−1 dV < 2t −2β V¯ sup (x) (t2n+1 − t2n ) + 2t −2β V¯ |K t2n − K t2n−1 |, (5.13) 0≤x≤1 d x t where we have used the identity r ds ddVx (X s ) K s = V (X t ) − V (X r ). The only thing random in the final bound is the difference tn+1 − tn , which is an exponential random variable with mean R−1 and the difference |K t2n − K t2n−1 | which has variance less than r2 R . Thus t2n+1 dV (X s ) Ft2n−1 ds E χ |K t2n−1 | ≥ t β dx t2n

1

dV r2 (x) + 2 t −2β V¯ 21 sup 0≤x≤1 d x R2 V (X )−V (X t2n ) t 2n+1 β + d x dk Pω (x, k) χ |k| ≥ t E (X t2n−1 , K t2n−1 ) = (x, k) , K t2n−1

< O(t −4β ) + 2 t −2β RV¯

O(t −4β )

(5.14) < 21 t β ,

corresponds to the unlikely event that Pr K t2n where ω ∈ F2n−1 and which we have treated above following (5.9). 1 Adding and subtracting the spatial average of the potential, 0 da V (a) = V in the expectation above, V (X t2n+1 ) − V (X t2n ) E (X t2n−1 , K t2n−1 ) K t2n−1 1 ≤ d x dk P(X t , K t2n−1 ) (x, k) E V (X t2n+1 ) − V (X t2n , K t2n ) = (x, k) 2n−1 |K t2n−1 | R2 + E V (X t2n ) − V (X t2n−1 , K t2n−1 ) , (5.15)

where P(X t2n−1 , K t2n−1 ) is the probability density for (X t2n , K t2n ) given (X t2n−1 , K t2n−1 ). Finally, we can work with quantities that allow more explicit expressions, ∞ 1 E V (X t2n ) (X tn−1 , K tn−1 ) = dt R e−Rt V (xt ) = da r(x0 , k0 ) (a) V (a), 0

0

where xt is the position at time t for the particle evolving according to the dynamics from the initial point (x0 , k0 ) = (X tn−1 , K tn−1 ), and r(x, k) ∈ L 1 ([0, 1]) is defined as in Lemma 5.2, 1 E V (X t ) (X t , K t ) − V ≤ V¯ da r(x0 , k0 ) (a) − 1. 2n n−1 n−1 0

252

J. Clark, C. Maes

β −β −2β ). By Part (1) of Lemma 5.2, when |K tn−1 | ≥ t , then |r(x0 , k0 ) (a)−1| ≤ 4R t +O(t A similar analysis bounds the term E V (X t2n+1 ) − V (X t2n , K t2n ) = (x, k) . Thus with the factor of |K tn−1 |−1 on the right side of (5.15), then (5.15) is O(t −2β ), which completes the proof.

6. Bounding the Momentum Drift t In general, we have that K t = K 0 + Mt + 0 dr ddVx (X r ). In this section, we develop tools st for controlling the cumulative drift 0 dr ddVx (X r ). The end result, under the assumption of the symmetry (iii) of 2.2, is that st 1 d V (6.1) (X s ) −→ 0. ds E sup t − 2 dx 0≤s≤1 0 1

Thus on the scale t 2 of a central limit theorem for K t , the drift term vanishes. By Lemma 3.4, the particle spends most of the time at “high energy” (in Lemma 3.4 this meant ∝ t), where the contribution to the total drift over any given finite time interval is small. However, the particle is also making occasional shorter incursions to “low energy” where the contribution may be larger over a finite interval. In this section, “low 1 energy” roughly means below t 4 . In order to bound (6.1), the analysis is split into parts treating the drift at high and low energies respectively. From this section onwards, we contract the position degree of freedom to a single periodic cell x ∈ [0, 1]. This clearly does not affect the statistics for the drift process (6.1). Thus the dynamics satisfies the same linear Boltzmann equation (2.1) as before but with periodic boundary conditions; the derivatives in position at the boundaries of the interval are symmetric. We now define what we mean by low energy incursions. They are limited by starting 1 and ending times. Define the hitting time θ0 = min{s ∈ [0, t] |K s | ≥ t 4 }. For j ≥ 1 define the sequences of hitting times σ j , θ j : 1 σ j = min{s ∈ [0, ∞), Ms − Ms − = 0 s > θ j−1 , |K s | < t 4 }, 1 θ j = min{s ∈ [0, ∞) s > σ j , |K s | > 2 t 4 }.

(6.2) (6.3)

Notice that θ0 is defined differently than θ j for j ≥ 1. We refer to [σ j , θ j ] as the time period of the j th incursion. In the lemma below, we give a bound on the expected number of incursions NY () over a time interval [0, t], and show that the time periods of incursions θ j − σ j have finite first moments. The time periods between incursions σ j+1 − θ j can be shown to be almost surely finite. This follows from an argument using Lévy’s zero-one law and Theorem 2.3, but showing σ j+1 − θ j to be finite is not required to prove Theorem 2.3. In any case, bounds on σ j+1 − θ j are intrinsically less important to us, since the challenge is to get estimates for the low-energy part of the walk. Lemma 6.1. Assume 2.1 and (i) of List 2.2. 1

1. Given σ j < ∞, the difference θ j − σ j has expectation O(t 2 ).


253

2. Let > 0. For large enough t, the expectation for the number of incursions in the interval [0, t] is bounded as 1

1

1

E[NY (t)] ≤ 2 r22 t 4 . Proof. Part (1): Let us construct the stopping time θT = (θ j − σ j ) ∧ T for some bound T > 1 and set σ j = 0, r2 r2 E [θT ] ≤ E M θT = E [M]θT , r1 r1 where the equality follows since [M]t − M t is a martingale, and the inequality is a d consequence of r1 ≤ dt M t ≤ r2 . r 2 −1 Lemma 3.5 states that 2−1 N n=1 wn = 2 [M]r differs from E r − E 0 by a martingale, and by the Optional Sampling Theorem the expectation of the martingale part is zero at time θT so E [M]θT = 2E E θT − E 0 . 1

1

Let D be the size of the over-jump of the boundary −2t 4 or 2t 4 when θT < T , and V¯ be the max of the potential,

1 1 1 2 ¯ 4 E E θT − E 0 ≤ E (2t + D) + V θT < T + (2t 2 + V¯ ) Pr [θT = T ] 2 1

1

≤ 2t 2 + O(1) < 3 2 . By Lemma B.4, there are universal bounds determined by C and η on all the moments of D, E D 2 < ρ2 (C, η). Using that |x + y|2 ≤ 2 x 2 + 2 y 2 , Pr [θ > T ] ≤ 1, and that 1 V¯ t 4 , 1 r2 1 E[θT ] < 6 t 2 + O(1) = O(t 2 ). r1 Finally, by taking the limit T → ∞, we get a bound for the second moment of θ j − σ j : 1 r2 1 E θ j − σ j = lim sup E[θT ] ≤ 6 t 2 + O(1) = O(t 2 ). r 1 T →∞ Part (2): By Part (1) each incursion ends. Thus for each count of NY (t) there is a dis1 1 tinct up-crossing in which |K s | begins below t 4 and ends up above 2t 4 . However, for 1

large values of momentum √1 |K s | ≈ E s2 , we bound NY by the number of up-cross2 1 1 1 1 1 1 −1 ings Ut 2 t 4 , 2 t 4 ; E s (ω) that E s2 makes between 2−1 t 4 and 2t 4 . Since E s2 is a submartingale, we can apply the submartingale up-crossing inequality [3] to attain 1 1 E[NY (t)] ≤ E Ut 2−1 t 4 , 2 t 4 ; E s (ω) ≤ √ ≈

1

E[E t2 ] 1

1

2 t 4 − 2−1 t 4

1 1 1 2 21 1 1 r2 2 t 4 < r22 2 t 4 . 3

254

J. Clark, C. Maes

The basic idea for our analysis is the following: • For large t, there is an asymptotic independence between the events during a single incursion and all events up to the end of a previous incursion. • Events during the incursion, which occur far enough after the starting time of the incursion are independent of the initial state of the incursion. It is convenient in many places to have an effective bound on the size of the momentum jumps that are likely to occur in the interval [0, t]. We thus consider the statistics for our model conditioned on the event 1

{|vn | ≤ t 40 for all n such that tn ≤ t}, 1

where a jump greater than t 40 is considered to be large. The lemma below shows that 1 the probability that there is a jump above t 40 over the interval [0, t] decays superpolynomially, and that for dealing with the drift over up to time t, we can neglect the 1 possibility of large jumps. The choice of 40 involves constraints from Proposition B.6. Lemma 6.2. Assume (i) of List 2.2, then the probability of a momentum jump vn with η

1

1

|vn | ≥ t 40 over the interval [0, t] is O(t e− 2 t 40 ), for η as in (i) of List 2.2. Moreover, the difference in the quantity

1 E sup t − 2 0≤s≤1

0

st

2 dV (X s ) ds dx

1 2

, 1

for the dynamics conditioned not to make jumps greater than t 40 and the unconditioned η

1

dynamics is O(t 2 e− 2 t 40 ) Proof. Assumption (i) of List 2.2 implies that the Laplace transforms for single momentum jumps are uniformly bounded by C(1 − e−η−q )−1 . Using Chebyshev’s inequality 1 we can bound the probability that any jump isgreater than t 40 . On the other hand, the d V integral of the drift can be at most t supa∈[0,1] d x (a). We refer to Appendix B for a discussion of boundary crossing distributions and the definition of the boundary crossing density φ∞ : [0, 1] × R+ → R+ . If H is a random variable, then Pr[H = y], for dummy variable y ∈ R, refers to the distributional measure of H or its probability density if it exists. For a signed measure μ on R, then

μ 1 = |μ|(R), where |μ| is the absolute value of the measure. Most of the random variables in this article (e.g. energy and momentum jumps) have well-defined densities. The following lemma states that incursions beginning at points (X σ j , K σ j ) for s1 K σ j ∈ 1

1

1

[t 4 − t 40 , t 4 ] for fixed s1 = ± all have approximately the same probabilities for ending the incursion in the positive or negative direction. Lemma 6.3. Assume List 2.1 and (i)–(ii) of List 2.2. Consider the dynamics condi1 tioned not to have jumps greater than t 40 . Let s1 , s2 ∈ {+, −}, and (x, s1 k) ∈ [0, 1] × 1 1 1 [t 4 −t 40 , t 4 ]. There are constants ρs1 , s2 (t) such that as t → ∞, Pr s2 K θ > 0 (X σ , s1 K σ ) = (x, k) j j j − 1 −→ 0. sup ρ (t) s , s (x,k) 1 2

Diffusive Behavior for Randomly Kicked Newtonian Particles (a, v)

(a, v)

Proof. Fix s1 = s2 = +. Define φ↑, t , φ↓, t 1 4

1 40

255

to be the boundary crossing distributions

1 4

1

1

above and below the set S = [t − t , t + t 40 ] starting from (a, t 4 − v). Using the Markov property and that H is a function of the process after the time τ , 1 Pr K θ j > 0 (X σ j , K σ j ) = (a, t 4 − v) 1 1 (a, v) = dq dp φ↑, t (q, p) Pr K θ j > 0 (X τ , K τ ) = (q, t 4 + t 20 + p) [0,1]×R+ 1 1 (a, v) 4 20 (X + φ↓, (q, p) Pr K > 0 , K ) = (q, t − t − p)

1 . (6.4) θj τ τ t 1

Thus for (a, v), (a , v ) ∈ [0, 1] × [0, t 40 ], 1 sup Pr H = y (X σ j , K σ j ) = (a, t 4 − v) a, a , v, v

1 −Pr H = y (X σ j , K σ j ) = (a , t 4 − v ) 1

≤

sup

a, a , v, v

(a, v)

φ↑, t

(a, v)

(a ,v )

− φ↑, t

1 +

sup

a, a , v, v

(a, v)

φ↓ t

(a ,v )

− φ↓, t

1 .

(6.5)

(a, v)

By Proposition B.7, φ↑, t and φ↓, t

converge uniformly to φ∞ in L 1 . Thus the diam 1 eter D(t) of the set of possible values for Pr K θ j > 0 (X τ , K τ ) = (a, t 4 − v) as a function of (a, v) shrinks to zero as t → ∞. Let us define 1 1 ρ+, + (t) = Pr K θ j > 0 (X σ j , K σ j ) = (x, k) + Dt2 + , t

for any choice of (x, k), where t −1 is merely to ensure that ρ+, + (t) is non-zero, and the − 21

square root is introduced so that Dt Dt of the lemma.

1

= Dt2 → 0. Then we will have the conclusion

6.1. Bounding the drift over an incursion . In this section, we define incursions to have end times ς j which are different but related to the end times to θ j . Define the sequence of times ς j : 1 ς j = min{s ∈ [0, t], Ms − Ms − = 0 s > σ j , inf |K r | > t 4 }. (6.6) s 0, s2 K ς j > 0) dr (X r ), dx σj where j and m are related through n

χ (s1 K σi > 0) }. j = min{n ≥ 0 m =

(6.7)

i=1

Ys1 ,s2 (m) is equal to the drift for the m th incursion that begins with momentum having sign s1 provided that the incursion ends with sign s2 . Naturally, if there is no m th incursion with sign s1 , then we set Ys1 ,s2 (m) = 0. For s ∈ {±}, we also define Ns (r ) to be the NY (r ) χ (sK σi > 0). number Ns (r ) = i=1 Define the constant 1 cs1 , s2 (t) = t − 4

×

1

[0,1]×(0, t 40 )

ς

ds 0

da dv φ(a, v) E

1

(a, s1 t 4 −s1 v)

dV (X s )χ (s2 K ς > 0) , dx

(6.8)

where ς is defined analogously to the ς j ’s as the last time that there is a momentum 1 1 1 1 jump inside (−t 4 , t 4 ) before exiting the larger interval (−2 t 4 , 2 t 4 ): 1 ς = min{s ∈ [0, θ ), Ms − Ms − = 0 inf |K r | > t 4 }, 1 θ = min{s ∈ [0, ∞) |K s | > 2 t 4 }.

s 0) for any fixed (a, v) ∈ [0, 1] × R+ as t → ∞, so the density φ(a, v) appearing in the definition of cs1 ,s2 is not important (except as a matter of convenience). The main purpose of the following proposition is to establish Part (3) which says that the sum of the Ys1 , s2 (m)’s can be replaced by the number Ns1 (st) multiplied by the constant cs1 ,s2 (t). Proposition 6.4. Assume List 2.1 and (i)–(ii) of 2.2. 1. For large enough t, then for all j and ω ∈ Fσ j , 1

E[Y j2 |Fσ j ] 2 < 5. 2. Let cs1 ,s2 (t) be defined as in (6.8) and j and m be related by (6.7). As t → ∞, we have the L 2 () convergence, 2 21 E E[Ys1 , s2 (m)|Fθ j−1 ] − cs1 ,s2 (t) −→ 0.


257

3. As t → ∞, there is L 2 () convergence, ⎤ ⎡ s1 (st) N

1 t − 4 E ⎣ sup Ys1 , s2 (m) − Ns1 (st) cs1 ,s2 (t)⎦ −→ 0. 0≤s≤1

m=1

Proof. 1 Part (1): By Lemma 6.2, we can take the jumps to be bounded by t 40 , although we will not employ this till the end of the proof. Set σ = σ j , ς = ς j , and θ = θ j . By the triangle inequality, 1 1 1 1 1 2 E[Y j2 |Fσ ] 2 ≤ t − 4 E[(Mς − − Mσ )2 |Fσ ] 2 + t − 4 E (K ς − − K σ )2 |Fσ . 1 1 The times σ and ς are defined such that |K σ |, |K ς − | ≤ t 4 . Thus E (K ς − − K σ )2 |Fσ 2 1

≤ 2t4. As in the proof of Part (1) of Lemma 6.1 define the stopping time θT = θ ∧ T and the capped time ςT = ς ∧ T . By Doob’s maximal inequality and ςT ≤ θT , E M

ςT−

1 2 2 − Mσ |Fσ ≤E

2 sup Mr − Mσ |Fσ

1 2

σ ≤r ≤θT

1 2 1 2 ≤ 2E MθT − Mσ |Fσ = 2E [M]θT − [M]σ |Fσ 2 , (6.9) where [M]t is the quadratic variation and of the martingale Mt up to time t, and the last equality follows from the optional sampling theorem. By Lemma 3.5, At in the increasing part in a Doob-Meyer decomposition for E t , 2At − 2As = [M]t − [M]s . Applying the above for the time interval from σ up to stopping time θT , the right-hand side of (6.9), is bounded by 1

2E[AθT − Aσ |Fσ ] = 2E[E θT − E σ |Fσ ] ≤ 4t 2 + O(1), where the equality is another use of the optional sampling theorem. The inequality comes from the proof of Part (1) of Lemma 6.1. Part (2): Let us take s1 , s2 = +. By Lemma 6.2, we can take jumps to be bounded by 1 t 40 . By the Markov property and by the definition of Y+, + (m),

ς 1 dV (X s ) χ (K ς > 0) E Y+, + (m)|Fθ j−1 = E(X θ j−1 ,K θ j−1 ) t − 4 ds dx 0

= 1 da dv φt (a , v ) E 1 (a , t 4 −v ) [0,1]×[0,t 40 ]

ς 1 dV (X s ) χ (K ς > 0) , × t− 4 ds (6.10) dx 0

258

J. Clark, C. Maes 1

where φt (a, v) is the joint distribution of (X σ j , −K σ j + t 4 ) given (X θ j−1 , K θ j−1 ). We attach a subscript (a, v) to the symbol φt to indicate the ending point (X θ j−1 , K θ j−1 ) = 1

(a, t 4 − v) of the last previous incursion. We have that

2 E c+, + (m) − E Y+,+ (m) Fθ j−1

ς 2 dV 1 (X s ) χ ( K σ > 0) ≤ sup E 1 t − 4 ds dx 0 (a , v ) (a , t 4 −v ) (a,v) ×E φt, (a, v) − φ∞ 1 1 (X θ j−1 , t 4 −K θ j−1 )

≤ 25 r2

sup

(a, v)∈[0,1]×R+

φt, (a, v) − φ∞ 1 ,

(6.11)

where the first inequality follows from the definition of c+,+ and (6.10) to which we apply Jensen’s inequality over the measure determined by φt, (a, v) (a , v )−φ∞ (a , v )| da dv

and finally Hölder’s inequalityto pull the supremum outside the integral. The second 1 ς 2 inequality follows since E(x,k) t − 4 0 ds ddVx (X s ) is smaller than 25r2 for (x, k) ∈ 1

[0, 1] × [0, t 20 ] by the same argument as in Part (1). Finally by Proposition B.9, φt, (a, v) converges to φ∞ in L 1 uniformly for (a, v) ∈ 1 1 [0, 1] × [0, 2t 4 ] (which includes [0, 1] × [0, t 40 ]) as t → ∞. Part (3): Again we invoke Lemma 6.2, to work with the process conditioned to have 1 jumps bounded by t 40 . Let Fθ j−1 be the σ -algebra of all information known up to the end of the last incursion θ j−1 , and j and m are related by (6.7). For a random process 1 X s , 0 ≤ s ≤ 1, define X s p,∞ for p ≥ 1 as E sup0≤s≤1 |X s | p p . By the triangle inequality and by Jensen’s inequality for the first term on the right, N

s (st) 1 Y (m) − c (t) s ,s s ,s 1 2 1 2 m=1

N

s (st) 1 Ys1 ,s2 (m) − E Ys1 ,s2 (m)|Fθ j−1 ≤ j=1 1,∞ 2,∞ N

s1 (st) − N + E Y (m)|F (st) c (t) s ,s s s ,s θ 1 2 1 1 2 j−1 j=1

.

1,∞

(6.12) Since the information of previous incursions is contained in Fθ j−1 , the sum of the differences Ys1 , s2 (m) − E Ys1 , s2 (m)|Fθ j−1 up to m = Nst is a martingale. By Doob’s inequality and Lemma A.1, Ns1 (st)

Ys1 , s2 (m) − E Ys1 ,s2 (m)|Fθ j−1 m=1

2,∞

⎡ ⎤1 s1 (st) N

2 2 ≤ E ⎣ Ys1 ,s2 (m) − E Ys1 ,s2 (m)|Fθ j−1 ⎦ m=1


259

1 2 1 2 ≤ E Ns1 (st) 2 sup E Ys1 ,s2 (m) − E[Ys1 ,s2 (m)|Fθ j−1 ] m ≤ Ns1 (st) m

1 2

≤ r2 t

1 8

sup

m,ω∈Fσ j

1 1 1 2 2 E Ys1 , s2 (m) Fσ j ≤ 5r22 t 8 ,

(6.13)

where the last inequality uses Part (1), and the third inequality uses Part (2) of Lemma 6.1, and the following: 1 2 2 E Ys1 , s2 (m) − E[Ys1 , s2 (m)|Fθ j−1 ] m ≤ Ns1 (st) 1 2 2 ≤ sup E Ys1 , s2 (m) − E[Ys1 , s2 (m)|Fθ j−1 ] Fσ j m,ω∈Fσ j

≤

sup

m,ω∈Fσ j

1 2 2 E Ys1 , s2 (m) Fσ j .

(6.14)

Now we can work on the second term on the right of (6.12). By the triangle inequality and conditioning that m ≤ Ns1 (st) for the terms in the sum as above, ⎤ ⎡ s1 (st) N

E ⎣ E Ys1 , s2 (m)|Fθ j−1 − Ns1 (st)cs1 , s2 (t)⎦ m=1

⎡

≤ E⎣

Ns1 (st)

⎤ ⎦. 1 ,s2 (m)|Fθ j−1 ] − cs1 , s2 (t) m ≤ Ns1 (st)

E E[Ys

m=1

However, we will split the terms in the sum on the right-side into the two groups m ∈ [Ns1 (st − t) + 1, Ns1 (st)] and m ∈ [0, Ns1 (st − t)] for some 0 < 1 and in particular < s. For m ∈ [Ns1 (st − t) + 1, Ns1 (st)], ⎡ ⎤ Ns1 (st)

E⎣ E E[Ys1 , s2 (m)|Fθ j−1 ] − cs1 , s2 (t)m ≤ Ns1 (st) ⎦ m=Ns1 (st−t)+1

1 1 1 ≤ 2E Ns1 (st) − Ns1 (st − t) sup E |Ys1 , s2 (m)|Fσ j ≤ 5 r22 2 t 4 ,

ω∈Fσ j

(m)|Fθ j−1 ] and cs1 , s2 (t) are convex comwhere the first inequality follows since E[Y s1 , s2 binations of values E |Ys1 , s2 (m)|Fσ j for different ω ∈ Fσ j . The second inequality employs Part (1) and then Part (2) of Lemma 6.1 for an interval of length t. For the sum of the terms with m ∈ [0, Ns1 (st − t)], we need to better understand the expressions E E[Ys1 , s2 (m)|Fθ j−1 ] − cs1 , s2 (t)m ≤ Ns1 (st) , (6.15) and, in particular, how the information m ≤ Ns1 (st) will change the expectation. If m ≤ Ns1 (st), then it will already be known at time θ j−1 ≤ (s − )t, that m − 1 ≤ Ns1 (st). However, it was shown in the beginning of the proof of Corollary B.9 that the

260

J. Clark, C. Maes 1

1

probability of a jump into the region [−t 4 , t 4 ] (which is the beginning of an incur1 1 sion) after starting in any point (x, k) with |k| ∈ [2t 4 , 4t 4 ] occurs with probability 1 approaching one for t → ∞. Since, we have assumed jumps bounded by t 40 , the point 1 1 1 (X θ j−1 , K θ j−1 ) will have |K θ j−1 | ≤ [2t 4 , 2t 4 + t 40 ]. Thus knowing m ≤ Ns1 (st) will add little is known at time θ j . With this consideration, we can give a crude upper bound for the expression (6.15) by doubling the unconditioned value of the expectation of |Ys1 , s2 (m)|Fθ j−1 ] − cs1 , s2 (t)|: E E[Ys1 , s2 (m)|Fθ j−1 ] − cs1 ,s2 (t)Ns1 (st) − Ns1 (θ j−1 ) > 0 (6.16) < 2E E[Ys1 ,s2 (m)|Fθ j−1 ] − cs1 ,s2 (t) , where the inequality is due to the event Ns1 (st) − Ns1 (θ j−1 ) > 0 having probability close to one by Corollary B.9. Finally, ⎤ ⎡ Ns1 (st−t)

1 − E ⎣t 4 E E[Ys1 ,s2 (m)|Fθ j−1 ] − cs1 , s2 (t) ⎦ m=1

< 2t

− 41

E Ns1 (st − t) sup E E[Ys1 , s2 (m)|Fθ j−1 ] − cs1 , s2 (t) , m

which goes to zero by Part (2) of this proposition and by Part (2) of Lemma 6.1. 6.2. The torus reflection symmetry. Now we move on to results which are specific to having a reflection symmetry on the torus such that the potential V (x) and jump densities jx (v) satisfy V (x) = V (R(x)) and jx (v) = j R(x) (v). A consequence of this symmetry along with the symmetric jump rates ja (v) = ja (−v) between positive and negative momenta is that a specific phase space trajectory (xs , ks ), s ∈ [0, t] from (x0 , k0 ) to (xt , kt ) will occur with the same density as a trajectory (R(xs ), −ks ) from (R(x0 ), −k0 ) to (R(xt ), −kt ). This combines with the time reversal symmetry of the model to yield a “time reversal in momentum”. Original trajectory (xs , ks ) Time reversal symmetry (xt−s , −kt−s )

Torus reflection (R(xs ), −ks ) Time rev. with torus reflection (R(xt−s ), kt−s )

The significance of the symmetries for the study of the drift is that ddVx (x) = − ddVx (R(xs )). Thus for the trajectories (xs , ks ) and (R(xt−s ), kt−s ) over the interval [0, t], t t dV dV (xs ) = − (R(xt−s )). ds ds dx dx 0 0 This will provide a basis for showing that there is no systematic bias for the contributions of the incursions.


261 ρ

(t)

Lemma 6.5. Assume List 2.1 and List 2.2. For the momentum-capped dynamics ρ+,− − −,+ (t) 1 tends to zero. Proof. By the torus reflection symmetry of the dynamics, the probability density of 1 1 going from (a, t 4 − v) to (a , −2t 4 − v ) from the initial time to time θ is the same as 1 1 the probability density of going from (R(a), −t 4 + v) to (R(a ), 2t 4 + v ). Thus Pr

1

(a, t 4 −v)

[K θ < 0] = Pr

By the triangle inequality, ρ+,− (t) ρ+,− (t) − 1 ≤ − ρ−,+ (t) ρ−,+ (t) Pr

1

(R(a), −t 4 +v)

[K θ > 0] .

ρ+,− (t) + Pr [K θ > 0] 1

(R(a), −t 4 +v)

ρ+,− (t) − 1, [K θ < 0] 1

(a, t 4 −v)

(6.17) By Proposition 6.3, [K < 0] [K > 0] 1 Pr 1 Pr (a, t 4 −v) θ (a, −t 4 +v) θ − 1, − 1 −→ 0, ρ+,− ρ−,+ 1

for all (a, v) ∈ [0, 1] × [0, t 40 ]. Thus (6.17) goes to zero. The following lemma constructs a specific joint distribution φt∗ (a, v; a , v ) for the first entrance coordinates (a, v) and last exit coordinates (a , v ) for the set S = {(x, k) ∈ 1 1 [0, 1]×R||k| ≤ t 4 } for trajectories conditioned to begin with s1 K 0 > t 4 and to end with 1 s2 K θ > t 4 . The symmetry (6.18) will play a key role in the proof of Proposition 6.7. Lemma 6.6 (Equilibrium first-entrance/last-exit distribution). Assume List 2.1 and List 1 2.2. Consider the dynamics conditioned to have jumps capped by t 40 . For large enough 1 t, there exists a unique joint density φt∗ (a, v; a , v ) with support in ([0, 1] × [0, t 40 ])2 such that the marginals ∗ ∗

φF ,t (a, v) = da dv φt∗ (a, v; a , v ) and φL ,t (a , v ) [0,1]×R+ da dvφt∗ (a, v; a , v ), = [0,1]×R+

satisfy the relations ∗

φL ,t (a , v ) = ∗

φF ,t (a , v ) =

[0,1]×R+ [0,1]×R+

∗ da dvφF ,t (a, v)Pr ∗ da dvφL ,t (a, v)Pr

1 (a,s1 t 4

−s1 v)

(R) 1 (a,s1 t 4

−s1 v)

1 (X ς , s2 K ς ) = (a , t 4 − v )s2 K ς > 0 , 1 (X ς , s2 K ς ) = (a , t 4 − v )s2 K ς > 0 ,

(R)

where Pr(x,k) refers to the law of the time-reversed dynamics starting from the point (x, k) ∈ [0, 1] × R. Moreover, φt∗ has the symmetry φt∗ (a, v; a , v ) = φt∗ (R(a ), v ; R(a), v).

(6.18)

262

J. Clark, C. Maes

Proof. Set s1 = s2 = +. We pick a number in (1, 2), say, 23 . In the event that K σ > 0, define ς to be the last time the particle has a jump with momentum < 1

3 2

1

t 4 before it

continues on to reach 2 t 2 (at time θ ), 3 1 ς = inf{s ∈ [0, θ ], Ms − Ms − = 0 inf |K r | > t 4 }. s ς . Consider the two maps F , R : L 1 ([0, 1] × R+ ), F (ϕ)(a, v) = da dv ϕ(a, v)Pr 3 1 (a, 2 t 4 −v)

1 3 1 × (X ς , K ς ) = (a , t 4 − v ) K ς > 0, inf K r < t 4 , 0≤r ≤θ 2 (R) R (ϕ)(a, v) = da dv ϕ(a, v)Pr 1 (a, 3 t 4 −v)

2

1 1 3

4 4 × (X ς , K ς ) = (a , t − v ) K ς > 0, inf K r < t , 0≤r ≤θ 2

(R)

where Pr(x,k) refers to the statistics for the time-reversed Markov dynamics starting from the point (x, k). F and R send probability densities to probability densities, and for large enough t, we claim that F , R are contractive on differences of densities. Consider the hit ting time τ = inf{s ∈ [0, θ ], Ms − Ms − = 0 K s ∈ / S} for the set S = {(x, k) ∈ 1

1

[0, 1] × R||k − 23 t 4 | < t 20 }. Since τ < ς , by the Markov property, F (ϕ)(a, v) = da dv ϕ(a , v ) + [0,1]×R 1 1 3 1 3 1 (a ,v ) (a ,v ) × d x dk ψ↑,t (x, k − t 4 − t 20 ) + ψ↓,t (x, −k + t 4 − t 20 ) 2 2 [0,1]×R+

1 3 1 (6.20) × Pr(x,k) (X ς , K ς ) = (a, t 4 − v) K ς > 0, inf K r < t 4 , 0≤r ≤θ 2

(a ,v ) (a ,v ) ψ↑,t and ψ↓,t are the boundary crossing densities for the set S for paths start1

1

1

ing from (a , 23 t 4 − v ) and conditioned to reach below t 4 before going above t 4 . Using (6.20), for two probability densities ϕ1 , ϕ2 ,

F (ϕ1 − ϕ2 ) 1 ≤ ϕ1 − ϕ2 1 (a,v) (a ,v ) (a,v) (a ,v )

ψ↑,t − ψ↑,t 1 + φ↓,t − φ↓,t 1 . (6.21) × sup 1

a, a ∈[0,1],v v ∈[0, t 40 ]

However, by Proposition B.8 we have that (a,v)

sup a∈[0,1],v v ∈[0, t

1 40 ]

ψ↑,t

1 − φ∞ 1 −→ 0 2

and

(a,v)

sup a∈[0,1],v∈[0,t

1 40 ]

φ↓,t

1 − φ∞ 1 −→ 0. 2


263

Thus for large enough t there exists a constant 0 < λ < 1 such that

F (ϕ1 − ϕ2 ) 1 ≤ λ ϕ1 − ϕ2 1 , for any two probability densities ϕ1 , ϕ2 ∈ L 1 ([0, 1] × R+ ). For large enough t, the constant λ > 0 can be made arbitrarily small. By symmetry, the same proof holds for R . For t large enough for the strict contractive property above, we can now construct special densities by defining the limits πtF = lim ( R F )n (P) and πtR = lim ( F R )n (P), n→∞

n→∞

where the limit is independent of the probability density P. We have constructed equilibrium states in which we can do computations using the Markov property for both the original dynamics and the time-reversed dynamics. We thus define the first entrance distribution ∗ φF (a, v) = da dv πtF (a , v ) ,t + [0,1]×R

1

41

4 × Pr 1 (X τ , K τ ) = (a , t − v ) K θ > 0 inf K r < t , (a,t 4 −v)

0≤r ≤θ

(6.22) and finally

φt∗ (a, v; a , v )

as the product

∗ φt∗ (a, v; a , v ) = φF ,t (a, v)Pr

(a,t

1 4 −v)

1 (X ς , K ς ) = (a , t 4 − v ) K ς > 0 .

∗ (a, v) with The first relation in the statement of the lemma, which determines φL ,t ∗ (a, v) using the forward dynamics, follows immediately from the definition of φ ∗ . φF t ,t ∗ (a , v ) is determined as the last exit time of S = {(x, k) ∈ In the above formula, φL ,t 1

1

[0, 1] × R | |k| ≤ t 4 }. However, starting from the points (q, 23 t 4 − p) with distribution ∗ (a, v) is the first entrance distriπtR (q, p) in the time-reversed dynamics, then φL ,t bution for set S. The second relation then follows from the Markov property for the time-reversed dynamics (R) ∗

∗

14

K φF (X (a, v) = da , dv φ (a , v )Pr , K ) = (a , t − v ) > 0 . ς ς ς 1 ,t L,t (a,t 4 −v)

(6.23) Proposition 6.7 (Antisymmetry of constants). Assume List 2.1 and List 2.2. The constants c+,+ (t), c−,− (t), c+,− (t) + c−,+ (t) tend to zero for large times. Proof. Let us fix s1 = s2 = +. By Lemma 6.2, we can take the dynamics conditioned 1 to make jumps capped by t 40 . Define the functional : L 1 [0, 1] × R+ → R,

ς dV − 41 t (X (ϕ) = da dvϕ(a, v)E ds )χ (K > 0) . 1 s ς 1 (a,t 4 −v) dx [0,1]×[0,t 40 ] 0 (6.24)

264

J. Clark, C. Maes

∗ then By our comments above c+,+ ≈ (φ∞ ), and when ϕ = φF ,t ∗ ) = Pr[K > 0] (φF θ ,t

1

(a ,t 4 −v )

da dv da dv φt∗ (a, v; a , v )E

1

(a,t 4 −v)

1

t− 4

ς ds 0

dV (X s ) , dx

(6.25) (x ,k )

where E(x,k) is the expectation conditioned on the trajectories that begin at (x, k) and have a last exit (x , k ) at the time ς . Note the anti-symmetry

ς ς 1 1 dV dV (a ,t 4 −v ) − 14 (R(a),t 4 −v) − 41 t t (X s ) = −E (X s ) , E 1 ds ds 1 dx dx (a,t 4 −v) (R(a ),t 4 −v ) 0 0 (6.26) which is due to the “time-reversal in momentum” mentioned at the beginning of the section in which for every trajectory (xt , kt ) on the interval [0, t], there is a backwards trajectory with a torus-reflected position (R(xt−s ), kt−s ) which occurs in the forward dynamics with the same “probability” as a fraction of the trajectories that begin at (x0 , k0 ) and (xt , kt ) respectively. Even though the final time ς is not deterministic, the two trajectories are still weighted equally in the expectations (6.26). This follows since ς is a hitting time for the time-reversed Markov process (for when the momentum first 1 jumps below t 4 ). Due to (6.26) and the symmetry (6.18) of φt∗ (a, v; a , v ), it follows that (6.25) is zero. We now focus on showing that due to a dynamical loss of memory of the initial conditions over the time interval [0, σ ] , the values of (ϕ) for any ϕ with support 1 1 in [0, 1] × [0, t 40 ] are close. In particular, (ϕ) for ϕ = φ¯ ∞ = φ∞ χ (|v| ≤ t 40 ) or ∗ are close, which would prove the result. ϕ = φF ,t 1 1 Define τ to be the exit time for the set S = {(x, k) ∈ [0, 1] × R|k − t 4 | < t 20 }. Thus for a probability density ϕ,

τ ∧σ dV − 41 (ϕ) = t (X s )χ K ς > 0 da dvϕ(a, v)E 1 ds (a,t 4 −v) dx [0,1]×R+ 0 1 1 1 1 ϕ ϕ + d x dk φ↑,t (x, k − t 4 − t 20 ) + φ↓,t (x, −k + t 4 − t 20 )

σ dV − 41 (X s )χ K ς > 0 , ds (6.27) × E(x,k) t dx 0 (q, p) ϕ ϕ where φ↑,t = dq dpϕ(q, p)φ↑,t and an analogous definition for φ↓,t . We argue that the first term on the right-side of (6.27) tends to zero for large t. Since 1 the momentum is greater than 21 t 4 up to time τ , then by Lemma 5.1,

τ ∧σ 4V¯ dV − 41 E 1 t (X s ) ≤ 1 E 1 ds [Nτ ] , (a,t 4 −v) dx 0 t 4 (a,t 4 −v) where Nτ is the number of momentum jumps up to time τ . In the proof of Proposi1 tion B.6, it was shown that E(x,k) [N ] = O(t 10 ). Thus the drift up to time τ ∧ σ vanishes 3 for large t vanishes as O(t − 20 ).


265

Thus (φ) − φ ∗ ≤ O(t − 203 ) + φ ϕ | ¯ − φ ϕ |ϕ=φ ∗ 1 + φ ϕ | ¯ − φ ϕ |ϕ=φ ∗ 1 ϕ= φ ϕ= φ F ,t ↑,t ↑,t ↓,t ↓, t ∞ ∞ F ,t F ,t

− 1 σ dV ds × sup E(x,k) t 4 (X s )χ K ς > 0 . dx 0 (x,k) ϕ

ϕ

ϕ

However, by Proposition B.7, φ↑,t , φ↓,t → ϕ∞ , since by definition φ↑,t is a convex (q, p)

1

combination of φ↑,t for (a, p) with p ∈ [0, t 40 ]. ϕ ϕ ϕ ϕ φ↑,t |ϕ=φ∞ , φ↑,t |ϕ=φF∗ ,t , φ↓,t |ϕ=φ∞ , and φ↓,t |ϕ=φF∗ ,t tend to φ∞ in L 1 . Moreover, by the same argument as for Part (1) of Proposition 6.4,

1 sup E(x,k) t − 4

(x,k)

σ

ds 0

1 2 dV 2 (X s ) χ (K θ > 0) ≤ 5. dx

∗ (a, v) converges to zero for t → ∞ which proves We then have that (φ) − φF ,t the result. That shows the c+,+ case. The other cases for c−,− , and c+,− + c−,+ are similar. Theorem 6.8. Assume List 2.1 and List 2.2. In the limit t → ∞, E

1 sup t − 2

0≤s≤1

0

st

dV (X r ) −→ 0. dr dx

(6.28)

1 st Proof. Our basic idea is to break the integral t − 2 0 dr ddVx (X r ) into parts corresponding to where |K r | is high and low energy respectively. The low energy parts are controlled ς by our study of the random variables Yn = σnn dr ddVx (X r ) and the high energy parts will be controlled by Lemma 5.3,

1

t− 2

st

dr 0

st 1 1 dV dV (X r ) + t − 2 (X r )χ (|K τ (r ) | > t 4 ) dr d x d x 0 0 ςn N

Y (st) 1 1 dV (X r )χ (|K τ (r ) | > t 4 ) Yn − dr + t− 4 d x σn n=1 θ NY (st) d V (X r ), −χ ∃( j) : st ∈ [σ j , ς j ] dr (6.29) dx st

1 dV (X r ) = t − 2 dx

θ0

dr

where τ (r ) is the next to last jump time before time r as in Lemma 5.3, and the last term corresponds removing an overlap due to a last incomplete incursion which begins before. For the first term on the right side of (6.29), an argument analogous to Part (1) of Proposition 6.4 gives the bound,

E 0

θ0

2 dV (X r ) dr dx

21

1

≤ 5t 4 ,

(6.30)

266

J. Clark, C. Maes

and thus that term is negligible. In this case, the end time γ is a hitting time, which makes the argument easier. The last term has the same bound. By Lemma 5.3,

1 sup t − 2

E

0≤s≤1

st 0

1 2 dV (X r )χ (|K τ (r ) | > t 4 ) dr dx

1 2

converges to zero. The same argument as in the proof of Lemma 5.3 shows that ⎡

Y (st) 1 N

E ⎣ sup t − 2

0≤s≤1

n=1

ςn σn

⎤1 2 2 1 dV dr (X r )χ |K τ (r ) | > t 4 ⎦ −→ 0, dx

since it includes even less terms. 1 NY (st)−1 We are left with the sequence t − 4 n=1 Yn , which we can write as 1

t− 4

N

+ (st)

1

Y+,+ (m) + Y+,− (m) + t − 4

m=1

N− (st)

Y−,− (m) + Y−,+ (m).

m=1 1

By Part (4) of Proposition 6.4 these sums can be approximated by t − 4 Ns1 (st)cs1 ,s2 (t): ⎤ ⎡ s1 (st) 1 N

1 Ys1 ,s2 (m) − t − 4 Ns1 (st)cs1 ,s2 (t)⎦ −→ 0. E ⎣ sup t − 4 0≤s≤1

j=1

For the sequences with s1 = s2 = s, 1 1 −1 E sup t 4 Ns,s (st)cs,s (t) = E t − 4 Ns,s (t)cs,s (t) ≤ r 2 |cs,s (t)|, 0≤s≤1

2

1

where the inequality uses Part (2) of Lemma 6.1. By Proposition 6.7, cs, s (t) converges to zero. The cases of (s1 , s2 ) = (+, −) and (s1 , s2 ) = (−, +) must be treated together. We −1 1 1 will take a step backward and approximate t − 4 Ns1 (st)cs1 , s2 (t) with t − 4 ρs1 ,s2 (t) Ns1 (st) cs1 ,s2 (t)χ (s2 K ς j > 0). By the triangle inequality n=1 ⎤ Ns1 (st) −1 −1

1 E ⎣ sup t 4 ρs1 ,s2 (t) cs1 ,s2 (t)χ (s2 K ς j > 0) − t − 4 Ns1 (st)cs1 ,s2 (t)⎦ ⎡

0≤s≤1

≤t

− 14

n=1

⎡ ⎤ Ns1 (st)

−1 cs ,s (t)E ⎣ sup ρs1 ,s2 (t) χ (s2 K ς j > 0) − Pr[s2 K σ j > 0]⎦ 1 2 0≤s≤1

+t

− 14

n=1

⎤

−1 ρs1 ,s2 (t) cs1 ,s2 (t)E ⎣ sup Pr[s2 K ς j > 0] − ρs1 ,s2 (t)⎦ . ⎡

Ns1 (st)

0≤s≤1

n=1

(6.31)


267

Ns1 (st) The sum n=1 χ (s2 K ς j > 0)−Pr[s2 K ς j > 0] is a martingale, so by Doob’s maximal inequality and by Lemma A.1, ⎡ ⎤1 s1 (st) N

2 2 E ⎣ sup χ (s2 K ς j > 0) − Pr[s2 K ς j > 0] ⎦ 0≤s≤1

n=1

⎡

⎤1 s1 (t) N

2 2 ≤ 2E ⎣ χ (s2 K ς j > 0) − Pr[s2 K ς j > 0] ⎦ n=1

1 2 1 2 ≤ sup E χ (s2 K ς j > 0) − Pr[s2 K ς j > 0] j ≤ Ns1 (st) E Ns1 (t) 2 j

1 ≤ sup Pr[s2 K ς j > 0| j ≤ Ns1 (st)] − Pr[s2 K ς j > 0]r2 t 8 ,

(6.32)

j,s

where the last inequality follows since Ns1 (t) < NY (t) and from Part (2) of Proposition 6.1 and by an explicit calculation for the expectation of the indicator in 1 the variance-type formula. Since the factor of t − 4 in (6.31) over-powers the factor −1 1 going to zero. Since the event j ≤ Ns1 (st) is adapted t 8 , the only worry is ρs1 ,s2 (t) to the information known up to time σ j , −1 ρs1 ,s2 (t) sup Pr[s2 K ς j > 0| j ≤ Ns1 (st)] − Pr[s2 K ς j > 0] j

≤

Pr[s2 K ς > 0|Fσ ] Pr[s2 K ς > 0] j j j − . ρ (t) ρ (t) s1 ,s2 s1 ,s2 j, ω∈Fσ j sup

(6.33) By adding and subtracting 1 in the expression on the right-side, then by two applications of Lemma 6.3, which is permitted by our assumption on the boundedness of the jumps, shows that the above goes to zero. For the application of Lemma 6.3, note that by the definitions θ j and ς j that s2 K ς j > 0 is equivalent to s2 K θ j > 0. Also note that Pr[s2 K ς j > 0] is a convex combination of the probabilities Pr[s2 K ς j > 0|(X σ j , K σ j )]. 1

Due to the decay of t − 8 , we had only needed this term to be bounded, but we apply these principles again below. For the second term in (6.31), ⎤ ⎡ Ns1 (st) Pr[s2 K ς > 0]

Pr s K > 0 1 1 2 ςj j t − 4 E ⎣ sup − 1⎦ ≤ t − 4 sup − 1 E[NY ]. ρ (t) ρ (t) s1 ,s2 s1 ,s2 0≤s≤1 j n=1

1

1

We apply Part (2) of Lemma 6.1 to show that t − 4 E[NY ] ≤ r22 is bounded and Lemma 6.3 to show that the ratio of probabilities converges to one. Now we just need to bound N− (st) + (st) −1 N

−1

1 1 t − 4 ρ+,− (t) c+,− (t)χ (K ς j < 0) + t − 4 ρ−,+ (t) c−,+ (t)χ (K ς j > 0). n=1

n=1

(6.34)

268

J. Clark, C. Maes

Using Lemma 6.5 and the same techniques above, we can replace ρ−,+ (t) by ρ+,− (t). 1 1 More critically, since the number of up-crossings from below −2 t 4 to above 2 t 4 can 1 differ by at most one from the number of down-crossings from above 2 t 4 to below 1 −2 t 4 ,

−1

⎤ N

− (st) + (st) 1 N

1 E ⎣ sup t − 4 c+,− (t)χ (K ς j < 0) + t − 4 c−,+ (t)χ (K ς j > 0)⎦

⎛

ρ+,− (t)

 0

n=1

⎞

ρ+,− (t)

⎠

1 × t − 4 |c+,− (t) + c−,+ (t)|E [NY (t)] + c+,− (t) ∧ c−,+ (t) . 1

As above E [NY (t)] = O(t 4 ), and by Lemmas 6.7 and 6.3, sup j

Pr K ς j > 0 ρ+,− (t)

+

Pr K ς j > 0 ρ+,− (t)

< 4.

Finally, |c+,− (t) + c−,+ (t)| converges to zero by Lemma 6.7, which finishes the proof. 7. Proof of Main Results Proof of Theorem 3.1. Denote the initial distribution P0 (x, k) as μ. We wish to apply the martingale central limit theorem. The Lindberg condition follows easily, since jumps occur at Poisson rate R and have finite fourth moments by (II) of List 2.1. Weak convergence with respect to the uniform metric then follows by convergence in probability of −1 the predictable quadratic variation σ s for every s. Without losing generality, −1 t M stto we take s = 1 and show Eμ t M t − σ → 0. 1 1 (t) Define S ,δ to be the event that 0 dsχ t − 2 |K st | > > 1 − δ. The same state1

1

1

1

ment as in Lemma 3.4 holds for t − 2 E st replaced by t√− 2 |K st |, since Er2 ≈ 2− 2 |K r | r (t) when |K r | 1. Thus for some C, Pr[S ,δ ] ≥ 1 − C r12 δ for large enough t. It will 1 r be convenient to observe that R 0 ds χ t − 2 |K r | ≤ is close to the number nr, of 1

Poisson times tm such that |K tm− | ≤ t 2 up to time r ≤ t. This is a law of large numbers 1 r following from the difference nr, − R 0 ds χ t − 2 |K s | ≤ being a martingale with Eμ

t

−1

1

n t, − R

dsχ t

− 21

|K st | ≤

1 2 2

1

≤ t − 2 R.

(7.1)

0

Using conditional expectations and the triangle inequality over a telescoping sum determined by the Poisson times tm , m = 1, . . . , Nt , we get the first inequality below.


269

N

1 t M t Eμ E(X tm ,K tm ) M tm+1 − M tm − σ (tm+1 − tm ) − σ ≤ Eμ t t m=0 N 1 1 1 t 2 (t) (t) < 2 2 r2 1 − Pr S ,δ + Eμ χ(S ,δ ) E(X tm , K tm ) M tm+1 − M tm − σ (tm+1 − tm ) . t m=0

(7.2) The second inequality follows by using the brute upperbound for the complement of the event S ,δ : N 1 t Nt , E(X tm ,K tm ) M tm+1 − M tm − σ (tm+1 − tm ) < r2 t tR

(7.3)

m=0

which holds since M r always increases at rates ranging between r1 and r2 , and thus E(X tm ,K tm ) M tm+1 − M tm − σ (tm+1 − tm ) ≤ |r2 − σ | ∨ |r1 − σ |R−1 < r2 R−1 . (7.4) (t) To bound t −1 R−1r2 E 1 − χ (S ,δ ) Nt , we apply the Cauchy-Schwarz inequality to (t)

1

1

attain 2 2 tR(1 − Pr[S ,δ ]) 2 .

(t) , by the discussion in the second paragraph of this proof, we expect In the event S ,δ 1

that a fraction of at least (1 − δ) of the times tn have |K tn− | ≥ t 2 . For large t, it will be easy to see that the terms corresponding to momentum jumps K tn − K tn− = vn 1

1

with |vn | ≥ 21 t 2 will be negligible, and we can assume |K tn | > 21 t 2 . Using that the differences tn+1 − tn are exponentially distributed with expectation R−1 , then for 1 (x, k) = (X tn , K tn ), |k| ≥ 2−1 t 2 , E(x,k) M tn+1 − M tn − σ (tn+1 − tn ) ∞ 1 = dτ Re−Rτ da T(x,k),τ (a) − τ dv ja (v)v 2 R 0 0 ∞ −1 −Rτ −1 −1 − 21 ≤ r2 |k| dτ Rτ e ≤ 2r2 R t , (7.5) 0

where r2 = sup0≤a≤1 R dv ja (v)v 2 and T(x,k),r ∈ L 1 ([0, 1]) is as defined in Lemma 5.2 which has been applied to get the second inequality. Our recipe for bounding (7.2) is the following: 1. Pick δ so that r2 δ √ 1, 1 r 2. Pick so that r2 (C r12 δ ) 2 1, 1

3. Pick t so that r2 −1 t − 2 1.

√ r (t) By the observations in the second paragraph of this proof, 1 − Pr S ,δ ≤ C r12 δ , and hence the first term on the right side of (7.2) is small. 1 1 Let n be the number of Poisson times with |K tn− | > 2−1 t 2 and |K tn | ≤ 2−1 t 2 up 1

to time t, and n

be the number of times with |K tn | ≥ 2−1 t 2 .

270

J. Clark, C. Maes 1

Applying the bounds (7.5) and (7.4) depending on whether or not |K tn | ≥ 2−1 t 2 , then we get the first inequality below: ⎤ ⎡ Nt

(t) 1 Eμ ⎣χ (S ,δ ) E(X tm ,K tm ) M tm+1 − M tm − σ (tm+1 − tm ) ⎦ t m=0 3 1 (t) < Eμ χ (S ,δ ) (n ,t + n )R−1r2 t −1 + n

2r2 R−1 −1 t − 2 ≤ r2 δ + O(t − 2 ) 1. (7.6) Now we look at the expectation of the terms n ,t , n , and n

to reach the second inequality. Since n

is smaller than the total number of Poisson times Nt , Eμ [n

] ≤ Rt, which t 1 makes its contribution in (7.6) O(t − 2 ). The number n ,t is smaller than N n=1 χ (|vn | ≥ 1

2−1 t 2 ), so for the distribution Pa (v) of a momentum jump conditioned to occur at a torus point a ∈ [0, 1], we can write ⎤ ⎡ Nt

1 t −1 Eμ n ,t ≤ t −1 Eμ ⎣ χ (|vn | ≥ 2−1 t 2 )⎦ n=1

≤ 2t

−1

Eμ [Nt ] sup

0≤a≤1

∞

1 2−1 t 2

dv Pa (v) = O(t −1 ).

The decay for the above expression follows by Chebyshev’s inequality and by the uniformly bounded second moments for Pa , a ∈ [0, 1]. To bound the t −1 n ,t , we will finally be able to use the definition of the event S ,δ in the second inequality below:

1 1 (t) (t) Eμ χ (S ,δ )t −1 n ,t ≤ REμ χ (S ,δ ) dsχ t − 2 |K st | ≤ 0

+ O(t

− 21

1

) ≤ Rδ + O(t − 2 ).

Finally, the first inequality follows from (7.1). 1

Proof of Theorem 4.1. By Lemma 4.2, t − 2 |K st | obeys the equation s a 1 S(K r t ) d Mr(t) + sup − S(K r t ) d Mr(t) + Es(t) t − 2 |K st | = 0≤a≤s

0

1

= t− 2

Nst

0

1

wn S(K tn− ) + t − 2 sup −

n=1

0≤s≤1

Nst

wn S(K tn− ) + Es(t) ,

(7.7)

n=1

(t) (t) where the error Es vanishes in the norm E sup0≤s≤1 Es . The martingale Mr = Nr n=1 wn S(K tn− ) has the same quadratic variation as the martingale of momentum jumps Mr , [M]r = [M ]r =

Nr

n=1

wn2 .


271

Thus the predictable quadratic variations M r and M are alsoequal. However, by the proof of Theorem 3.1, Eμ t −1 M st − σ s → 0 for s ∈ [0, 1], and thus Mr converges to a Brownian motion. By the same argument as at the end of 1 the proof of Lemma 3.2, we have that t − 2 |K st | converges to the absolute value of a Brownian motion. 1

Proof of Main Result. It is sufficient to prove that t − 2 K s t converges to a Brownian motion, since by a change of integration variable, st s 3 1 − 23 − 23 dr K r = t − 2 X 0 + dr t − 2 K r t . t Xs t = t X0 + 0 − 23

− 21

0

s 1 K s t ) converges in distribution to ( 0 dr Br , Bs ) if t − 2 K s t con-

Hence (t X s t , t verges to Bs . t Recall that K t = K 0 + Mt + 0 ds

(X s ). By Lemma 6.8, the first moment of s dV dr (X r ) dx 0

dV dx

1 sup t − 2 0≤s≤t

1

converges to zero. By Theorem 3.1, t − 2 Mst converges to a Brownian motion with 1 diffusion constant σ . Hence t − 2 K st converges to a Brownian motion. Acknowledgements. We thank Wojciech De Roeck, Karel Netoˇcný and Frank Redig for useful discussions. J. C. acknowledges support from the Belgian Interuniversity Attraction Pole P6/02 and the Marie Curie Research. Training Network project MRTN-CT- 2006-035651, Acronym CODY, of the European Commission.

A. A Martingale Lemma Below we formalize a simple lemma for a martingale Mt which makes jumps at discrete times. For a basic introduction to martingale theory, see [3]. t Lemma A.1. Let Mt = N n=1 X n be a right-continuous martingale adapted to a filtration Ft , and making jumps X n (ω), ω ∈ Ftn at discrete times according to some random counter Nt , where tn = inf{s ∈ R+ Ns = n}. Assume also that E [Nt ] < ∞ for every t and that E X n2 < ∞ for every n. Then E Mt2 ≤ E [Nt ] sup E X n2 n ≤ Nt , n

where E X n2 n ≤ Nt is defined as zero for n such that Pr[n ≤ N(t)] = 0.

Proof. By orthogonality of martingale increments, N ∞ ∞ t

2 2 2 E Mt = E Xn = E X n χ (n ≤ Nt ) = Pr[n ≤ Nt ]E X n2 n ≤ Nt n=1

≤

∞

n=1

n=1

n=1

Pr[n ≤ Nt ] sup E X n2 n ≤ Nt ≤ E [Nt ] sup E X n2 n ≤ Nt ,

n

where the inequality is Hölder’s.

n

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B. Boundary Crossing Distributions Consider symmetric independent and identically distributed random variables X 1 , X 2 , . . . with mean zero and density ˜ P(v) =

1

da 0

κ(a) Pa (v). κ¯

n Construct the random walk Yn = m=1 X m . We will refer to this as the so-called “averaged random walk” Let L ≥ 0. Our interest in this section is to understand the probability density π L (a, b) =

∞

Pr[Yn − L = a, X n = b, Yr < L for r < n], a, b ≤ R+ . (B.1)

n=1

In the present appendix we use the notation Pr[·] for the induced density by the random walk, in its obvious meaning for continuous densities. To be clear, π L describes the distribution of jumps in excess over point L for the random walk Yn on the first time that it passes over that L. For L = 0, we write D(v, w) = π L=0 (v, w), which is in fact the “successive record increment” distribution. Indeed, define Rn = sup0≤m≤n Ym , the record in the positive direction for the random walk up to time n, and let τm be the time of the m th record. ∞Then, the increments Rτm − Rτm−1 are i.i.d. random variables with density D(a) = 0 db D(a, b). We use this fact in the proof of the following proposition. In particular

E Rτm − Rτm−1 =

∞ ∞

da db a D(a, b). 0

0

The topic of “record distributions” has a much wider scope and has a long history in extreme value statistics, see e.g. [8] for some pioneering contribution. Lemma B.1. Assume (III) of List 2.1 and (i)–(ii) of List 2.2. 1. The Laplace transform ϕ(q) of D(v) satisfies ϕ(q) ≤

C 1 − e−η−q

for q > −η.

2. D(v) is bounded and continuous. 3. The following is a probability density on R+ × R+ : ∞

π∞ (v, w) = ∞ v∞ 0

0

d x D(x, w) d x d y x D(x, y)

,

which is positive for all v < w. Proof. Part (1) follows, since D(v) inherits the property (i) of List 2.2.


273

Proposition B.2. Assume (III) of List 2.1 and (i)–(ii) of List 2.2. Let the random walk Yn , π L (v, w), and D(v, w) be defined as above. In the limit of large L, we have the L 1 (R+ × R+ ) convergence ∞ d x D(x, w) π L (v, w) −→ π∞ (v, w) = ∞ v∞ . 0 0 d x d y x D(x, y) Moreover, the difference between π∞ and π L converges exponentially and monotonically. Corollary B.3. Let Yn be a random walk as above. For L > 0 and d ∈ R with |d| ≤ 21 L define S = (−d −L , L) ⊂ R, and let π↑,L (v, w), π↓,L (v, w) be the probability densities on R+ × R+ defined as π↑,L (v, w) =

∞

Pr X n = w, Yn − L = v, Ym ∈ S for 0 ≤ m < n 1Yn >0 , (B.2)

n=1

π↓,L (v, w) =

∞

Pr −X n = w, Yn + L + d = −v, Ym ∈ S for m < n 1Yn −η.

Let P(v1 , . . . , vn ) be the joint probability distribution for the increments v j = K t j − K t j−1 , where t j is the time of the j th momentum jump. Lemma B.5. Let us make assumptions (i)–(ii) from List 2.2. Fix β, γ > 0. Let the dynamics begin at a point (x, k) and v1 , . . . , vn ∈ [−2t γ , 2 t γ ] be such that |k+ rm=0 vr | ≥ 21 t β for all 0 ≤ m ≤ n. Then P(v , · · · , v ) γ −β 1 n − 1 < c n t γ −β ec n t , ˜ ˜ P(v1 ) · · · P(vn ) where c = 8V¯ μ. Proof. We start with the one-jump case. The only difference between Lemma B.5 and Lemma 5.2 is that here the increment v is the sum of the first momentum kick ànd the drift up to the time of the kick. Let (xs , ks ) be on the trajectory determined by the Hamiltonian H (x, k) = 21 k 2 +V (x) starting from (x, k) ∈ [0, 1] × R. For fixed starting point (x, k), s = ks − k = (xs )


275

is a function of xs . For the one-jump case, we can use the density r˜(x,k) ∈ L 1 ([0, 1]) from the proof of Lemma 5.2, to write

1

P(v) = 0

dar˜(x,k) (a)Pa (v − (a)) .

1 Define the density P (v) = 0 da κ(a) κ¯ Pa (v − (a)). Following the same line of reasoning as (5.6) in Lemma 5.2, 1 P(v) 1 κ(a) − 1 ≤ sup

Pa (v − (a)) sup

da r˜(x,k) (a) − κ¯ v∈R P (v) v∈R P (v) 0 1 1 κ¯ κ(a) −1

Pa (v − (a)) da ≤ sup r˜(x,k) (a) κ(a) P (v) κ¯ 0≤a≤1 0 κ¯ − 1. = sup r˜(x,k) (a) κ(a) a∈[0,1] Since, |k| ≥ 21 t β , then by Part (2) of Lemma 5.2, κ¯ − 1 ≤ ν −2 4Rt −β + O(t −2β ). sup r˜(x,k) (a) κ(a) a∈[0,1] Moreover, 1 P (v) 1 κ(a) Pa (v − (a)) − Pa (v) −1 ≤ da ˜ ˜ κ¯ P(v) P(v) 0 1 1 dP 1 κ(a) a = |(a)| da ds (v + s(a)) ˜ κ¯ dv P(v) 0 0 μ4V¯ (1 + |v|)t −β 1 κ(a) ≤ Pa (v) = μ4V¯ (1 + |v|)t −β . (B.6) da ˜ κ ¯ P(v) 0 For the second inequality we have used Lemma 5.1 to get sup0≤s≤1 |(a)| ≤ 4V¯ t −β . Pa In particular, |(a)| < 1 allows us to use (ii) of List 2.2 to bound | ddv (v + s(a)) | by μ(1 + |v|)Pa (v). Since 1 + |v| ≤ 1 + 2 t γ < 4t γ for large enough t, we have P (v) − 1 < 16V¯ μt γ −β , sup ˜ v∈R P(v) P(v) P (v) P (v) P(v) − 1 − 1 ≤

+ − 1 ˜ ˜ ˜ P (v) P(v) P(v) P(v) γ −β ¯ ≤ (16V μt ) 1 + 4R ν −2 t −β + O(t −2β ) + 4Rν −2 t −β + O(t −2β ) < 20 V¯ μt γ −β = ct γ −β , where the last inequality is for t large enough and c = 20 V¯ μ.

(B.7)

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To extend to arbitrary n, n P(v , . . . , v )

P(v1 , . . . , vm−1 ) 1 n P(v1 , . . . , vm ) − 1 ≤ − . ˜ n) ˜ m) ˜ m−1 ) ˜ 1 ) . . . P(v ˜ 1 ) . . . P(v ˜ 1 ) . . . P(v P(v P(v P(v m=1

(B.8)

Moreover, we can write the summand as

m−1 P(v1 , . . . , vm ) P(v1 , . . . , vr ) # − 1 . ˜ m) ˜ r) P(v1 , . . . , vm−1 ) P(v P(v1 , . . . , vr −1 ) P(v r =1

The ratios appearing in the above expression can be written in terms of an expectation: P(X tm−1 ,K tm−1 ) (vm ) P(v1 , . . . , vm ) =E K tr − K tr −1 = vr for r < m , ˜ m) ˜ m) P(v1 , . . . , vm−1 ) P(v P(v (B.9) where P(X tm−1 ,K tm−1 ) (vm ) is the one jump distribution starting from (x, k) = (X tm−1 , K tm−1 ). The right side of (B.9) is thus an average of the one-jump case over different starting points (x, k) = (X tm−1 , K tm−1 ) and thus we can apply the n = 1 case. We thus conclude P(v , . . . , v ) γ −β 1 n − 1 < c n t γ −β (1 + c t γ −β )n < c n t γ −β ec n t . ˜ n) ˜ 1 ) . . . P(v P(v For notational convenience, we will take the initial point (x, k) to have positive momentum k > 0 for the propositions and lemmas below. Proposition B.6. Assume (i)–(ii) from List 2.2. Let (X s , K s ) evolve from an initial point (x, k) with k ≥ t β . Define S = {y ∈ R|k − t γ − d < y < k + t γ } for 0 < γ < 14 β and |d| ≤ 21 t γ , and the densities π↑,t (v, w) = π↓,t (v, w) =

∞

n=1 ∞

Pr[K tn − K tn−1 = w, K tn − k − t γ = v ≥ 0, K tm ∈ S∀(m < n)]1 K tn >S , Pr[K tn − K tn−1 = w, K tn − k + t γ + d = −v ≤ 0, K tm ∈ S∀(m < n)]1 K tn <S .

n=1

In the limit t → ∞, there is L 1 (R+ × R+ ) convergence π↑,t −→ p↑ π∞ , π↓,t −→ p↓ π∞ , where p↑ and p↓ are the exit probabilities for the averaged random walk, and the convergence is uniform for all |d| ≤ 21 t γ .


277

Proof. Define the sets Sn ( p, w) = {(v1 , . . . , vn ) ∈ Rn |vn = w, p = k +

n

vr ∈ / S, k +

r =1

m

vr ∈ S for m < n},

r =1

and define Sn ( p, w) so that Sn ( p, w) = Sn ( p, w) ∩ {|vm | ≤ 2t γ for m ≤ n} when n ≤ t 3γ and Sn ( p, w) = ∅ when n > t 3γ . Then π↑,t (v, w), π↓,t (v, w) satisfy π↑,t ( p − k − t γ , w) + π↓,t (k − t γ − d − p, w) ∞

= dv1 · · · dvn P(v1 , . . . , vn ), n=1 Sn ( p,w)

(B.10)

where P(v1 , . . . , vn ) are the joint densities from Lemma B.5. The sum of π↑,t and π↓, t forms a probability measure, since, by an analogous argument as in the proof of Lemma 6.1, the exit time for S is almost surely finite (and has expectation ∝ t 2β ). We will argue that the right side is close in L 1 (R+ × R+ ) as a function of ( p, w) to the same ˜ 1 ) · · · P(v ˜ n ): expression with P(v1 , . . . , vn ) replaced by P(v (0)

(0)

π↑,t ( p − k − t γ , w) + π↓, t (k − t γ − d − p, w) =

∞

n=1 Sn ( p,w)

˜ n ). ˜ 1 ) · · · P(v dv1 · · · dvn P(v

This expression corresponds to the averaged random walk. We can then apply Lemma B.3 (0) (0) for an ordinary random walk to get the convergence of π↑,t and π↓,t to p↑ π∞ and p↓ π∞ . m 1 By definition of Sn ( p, w), Sn ( p, w), K + n=1 vm > 2 t β for all 0 ≤ m < n. First, notice that by Lemma B.5, ∞

n=1

≤

˜ n ) P(v1 , . . . , vn ) − 1 ˜ 1 ) · · · P(v dv1 · · · dvn P(v ˜ n) ˜ 1 ) · · · P(v P(v Sn ( p,w)

3γ t

n=1

Sn ( p,w)

˜ n ) c n t γ −β ec n t γ −β . ˜ 1 ) · · · P(v dv1 · · · dvn P(v

(B.11)

The L1 (R+ × R+ ) norm of the right-side of (B.11) (by integrating over p, w) is bounded 3γ by E q (N0 ∧ t 3γ )eq(N0 ∧t ) q=c t γ −β , where N0 is the number of steps that a random ˜ walk with jumps having density ct−1 P(v)1 |v|≤t γ , for normalization constant ct , takes to leave S starting from k. The right-side of (B.11) vanishes as 3γ 4γ −β ≈ c t 4γ −β . E q (N0 ∧ t 3γ ) eq (N0 ∧t ) q=c t γ −β ≤ c t 4γ −β ec t Now we need to show that not much probability was lost by replacing Sn ( p, w) with (0) (0) Sn ( p, w). Since both π↑,t + π↓,t and π↑,t + π↓,t are probability measures, it is enough to show that the probability of the event {|vm | > 2t γ for some m ≤ N or N > t 3γ } is small for the random walk. First, note that the expected number of steps N to leave S

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J. Clark, C. Maes

for the averaged random walk will be smaller than the expected number of the steps N0 for the capped random walk, Pr[N > t 3γ ] ≤ Pr[N0 > t 3γ ]

and

E[N ] ≤ E[N0 ].

This follows since the first jump of size greater than 2t γ will immediately leave S, and the corresponding capped trajectory may have to continue on for more steps before leaving S. Now we argue that E[N0 ] = O(t 2γ ). Let us set k = 0. Define the stopping time N0,T = N0 ∧ T . Reasoning as in the proof of Lemma 6.1, then

E N0,T

⎡ = ζ −1 E ⎣

N0,T

⎤ vn2 ⎦ = ζ −1 E

2 ≤ ζ −1 (3 t γ + d)2 < 16 ζ −1 t 2γ , v1 + · · · + v N0,T

n=1

where the expectations are with respect to the statistics for the capped random walk, and 2. ˜ ζ = ct−1 |v|≤t γ dv P(v)v In the second equality, we used that v1 + · · · + v N0,T is either inside [−t γ − d, t γ ] when T < N0 or has jumped out this interval with a jump smaller than 2t γ . The bound on the right is independent of T , so E [N0 ] = lim sup E N0,T ≤ 16 ζ −1 t 2γ . T →∞

By Chebyshev’s inequality, Pr[N0 > t 3γ ] ≤ t −3γ E [N0 ] ≤ 16ζ −1 t −γ −→ 0, since ζ converges to σ for large t. We still need to show that Pr [|vn | > 2t γ for some n ≤ N ] is small, Pr |vn | > 2t γ for some n ≤ N ≤ E [N ] Pr |vn | ≥ 2t γ . By our remarks above E [N ] ≤ E [N0 ] = O(t 2γ ). Using Chebyshev’s inequality, and the bound on the fourth moment of a single momentum jump by ρ, 16t 4γ Pr |vn | ≥ 2t γ ≤ E vn4 ≤ ρ. Putting the above inequalities together, Pr |vn | ≥ 2t γ , for some n ≤ N ≤ E[N0 ] Pr |vn | ≥ 2t γ < ρζ −1 t −2γ , hence the event that the last jump vn is greater than 2t γ is negligible for large t. We have shown that π↑,t , π↓,t converge to π∞ in L 1 (R+ × R+ ) at t → ∞. The convergence of π↑,t and π↑,t to π∞,t is uniform over |d| ≤ 21 t γ by the uniformity in Lemma B.3 and by the uniformity in the bounds above.


279

Proposition B.7. Assume (i)–(ii) from List 2.2. Let (X s , K s ) evolve from an initial point (x, k) with k ≥ t β . Define S = {y ∈ R|k − t γ − d < y < k + t γ } for 0 < γ < 41 β, and the densities φ↑,t (a, v) =

∞

K tn − k − t γ = v ≥ 0, K tm ∈ S for 0 ≤ m < n]1 K tn >S ,

n=1

φ↓,t (a, v) =

∞

Pr[X tn = a, K tn − k + t γ + d = −v ≤ 0,

n=1

for K tm ∈ S 0 ≤ m < n]1 K tn <S . In the limit t → ∞, there is L 1 ([0, 1] × R+ ) convergence, φ↑,t −→ p↑ φ∞ (a, v), φ↓,t −→ p↓ φ∞ (a, v), where p↑ and p↓ are probabilities that the averaged random walk exits above or below S respectively. Moreover, the convergence is uniform for |d| ≤ 21 t γ . Proof. Define the joint density ↑,t (a, v, w) for the position a, increment v for the over-jump of the boundary of S, and the size of the jump w which exits above S: ↑,t (a, v, w) =

∞

Pr[X tn = a, K tn = w, K tn − k − t γ = v ≥ 0,

n=1

K tm ∈ S

for 0 ≤ m < n]1 K tn >S .

The definition for ↓,t (a, v, w) is analogous. Let primed densities be normalized (e.g. −1 −1

↑,t and π↑,t = p↑,t π↑,t , where p↑,t is the probability of leaving S from ↑,t = p↑ the top). We will show that ↑,t converges in L 1 to ∞ (a, v, w) = π∞ (v, w)

κ(a) κ¯ Pa (w)

˜ P(w)

.

(B.12)

(a, v) = Since φ↑,t R+ dw ↑,t (a, v, w), and p↑,t converges to the probability that the random walk exits in the up direction by Proposition B.6, this would prove the result. In particular, since |d| ≤ 21 t γ , neither of the probabilities p∞ (↑, t) or p∞ (↓, t) will be close to zero. Also by the proof of B.6, the probability that the final momentum increment w = K t N − K t N −1 is greater than 2t γ , where t N is the time of the last momentum jump leaving S and t N −1 is the time of previous momentum jump, decays as O(t −2γ ). This is true also for the random walk case. ∞ χ (w > 2t γ ) 1 and ↑,t χ (w > 2t γ ) 1 thus vanish for large t. (t) Define g 1 = g χ (w ≤ 2t γ , v < w) 1 . We placed the constraint v < w in

(v, w) is strictly positive and, in the indicator so that by Part (3) of Lemma B.1, π∞

(v, w) is also strictly positive for v < w < 2t γ , since particular, we can divide by it. π↑,t there is a non-zero density for jumping from k to k + t γ − w + v in one jump and then to k + t γ + v on a second jump.

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We now focus on showing that

(a, v, w) ↑,t

(v, w) π↑,t

∞

−

(t) ↑,t 1

tends to zero:

= E Pr X t N = a K t N = t β + t γ + v, K t N − K t N −1 = w, X t N −1 , (B.13) 1

(v, w) =

since π↑,t 0 da ↑,t (a, v, w). We may write the conditional probability density in the expectation above as Pr X t N = a K t N = t β + t γ + v, K t N − K t N −1 = w, X t N −1 Pa (w − (a)) r˜(X t N −1 ,K t N −1 ) (a) = . (B.14) P(X t N −1 ,K t N −1 ) (w)

(a) is a drift term which was defined in Lemma B.5. P(x,k) (w) are defined as in the proof of Lemma B.5 as the difference in momentum (sum of one momentum jump plus some drift) between a starting time with state (x, k) and next time jump time. r˜(x,k) (a) is a defined as in Lemma 5.2 as the distribution for the position of the particle at the next momentum jump. (t) (t) Analogously to g 1 define the semi-norm g ∞ = gχ (|w| ≤ 2t γ , v < w) ∞ . Putting together (B.12)–(B.14),

˜ P(w) κ¯ ↑,t π∞ (t) − 1

− 1 ∞ ≤ E sup r˜(X t N −1 ,K t N −1 ) (a) π↑,t ∞ κ(a) P(X t N −1 ,K t N −1 ) (w) w≥2t γ ,a

˜ P(w) κ¯ ≤ sup r˜(x,y) (a) E −1 κ(a) P(X t N −1 ,K t N −1 ) (w) y≥t β ,a,x

κ¯ + E sup r˜(X t N −1 ,K t N −1 ) (a) − 1 κ(a) a < 1 + 4Rν −2 t −β + O(t −2β ) 2ct γ −β + 4Rν −2 t −β + O(t −2β ) = O(t γ −2β ), where in the strict inequality we have used Part (2) Lemma 5.2 and Lemma B.5: P(x,k) (w) κ¯ 1 − 1 ≤ 8Rν −2 t −β , sup − 1 ≤ 2ct γ −β , for|k| ≥ t β sup r˜(x,k) (a) ˜ κ(a) 2 P(w) 0≤a≤1 w∈R +

(where we doubled the constant factor in front of the higher order term to get rid of the lower), and we used that | b1 − 1| ≤ 2 b for b in a small neighborhood around one. By adding and subtracting by

↑,t

−

(t)

↑,∞ 1

π↑,∞ ,

↑,t π↑,t

and the triangle inequality

(t)

(t) ↑,t ↑,∞ ↑,t

π↑,t − π↑,∞ ≤ −

π↑,∞ +

π↑,t π↑,t π↑,∞ 1 1 (t)

(t) ↑,∞ ↑,t ↑,t

≤ π↑,∞

1 − + π↑,t − π↑,∞

1 , π↑,t π↑,t π↑,∞ ∞

∞


281

π↑,∞

1 = 1 and so with the analysis above the left term on the second line tends to (t) ↑,t

is zero. For the right term, π↑,t − π↑,∞ 1 vanishes by Proposition B.6 and

π↑,t ∞ ∞ (t) bounded by π∞ plus a small number by the analysis above. ∞

Proposition B.8 states the same results as in Proposition B.7 when there is some conditioning on the future of the particle’s trajectories. For a particle starting with momentum in (t β , 2t β ), let θ↑ , θ↓ , and τ be the first times the particle has a momentum landing above 2t β , below −2 t β and below t β respectively. By the same argument as in Lemma 6.1, θ↑ ∧ θ↓ has finite expectation. We omit the proof here, since the techniques are similar to those used before. Proposition B.8. Assume (i)–(ii) from List 2.2. Consider the dynamics conditioned to γ have momentum jumps bounded by t 2 . Let (X s , K s ) evolve from an initial point (x, k) γ with 23 t β − t 2 ≤ k ≤ 23 t β . Let S = {y ∈ R|k − t γ < y < k + t γ } for 0 < γ < 41 β. The boundary crossing densities ψ↑,t , ψ↓, t for trajectories that are conditioned so that τ < θ↑ < θ↓ , have L 1 ([0, 1] × R+ ) convergence, ψ↑,t −→

1 1 φ∞ , ψ↓,t −→ φ∞ . 2 2

The convergence is uniform for the allowed range of k. The same statements hold for ψ↑,t , ψ↓,t for trajectories which are conditioned so that θ↓ < θ↑ . In the next corollary, like Proposition B.2, there is only one boundary. However, we place an optional time constraint t, ∈ (0, ∞] here on the amount of time the particle is allowed to have before reaching the boundary. Corollary B.9. Assume (i)–(ii) from List 2.2. Let (X s , K s ) evolve according to the dynamics from some initial point (x, k) for 2t β ≤ k ≤ 4 t β , 41 ≤ β < 21 . Define the density φt (a, v) =

∞

Pr[X tn = a, t β − K tn = v ≥ 0, K tm > t β for 0 ≤ m < n, tn < t].

n=1

In the limit t → ∞, φt (a, v) → φ∞ (a, v). The convergence is uniform over all starting points (x, k). Proof. First, we will argue that the probability pt = φt 1 that the particle jumps below t β in the interval [0, t] approaches one as t → ∞ for ∈ R+ :

− 21 β− 21 pt = Pr inf t K r ≥ t 0≤r ≤t −1 r dV 1 β− 21 − 21 − ≥ Pr t + t k + sup t 2 ds (X s ) ≤ sup −t 2 Mr , dx 0≤r ≤ t 0≤r ≤t 0 (B.15)

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r

ds ddVx (X s ). By Chebyshev’s inequality and that k ≤ 4 t β , −1 r dV β− 21 − 21 2 + t k + sup t ds Pr t (X s ) ≥ dx 0≤r ≤ t 0 1 r dV 2 −2 −2 β− 21 ≤ E 5t (X s ) . + sup t ds (B.16) dx 0≤r ≤t 0

since K r = k + Mr +

0

The right-side converges to zero by Lemma 5.3, and thus the probability that the supremum of the drift is greater than any finite 1 tends to zero. 1 By (3.1), t − 2 Mst , s ∈ [0, 1] converges to a Brownian motion Bs . Since the supremum over an interval [0, 1] is a uniformly continuous functional on elements L ∞ [0, 1], 1 sup0≤s≤st −t − 2 Mst converges in distribution to sup0≤s≤1 −Bs . In the large t limit, 1

Pr[ sup −t − 2 Mst > 2 ] −→ Pr[ sup −Bs > 2 ]. 0≤s≤t

0≤s≤

Thus we can pick so that Pr[sup0≤s≤1 −Bs > 2 ] is close to one, and then pick t so 1

that both Pr[sup0≤s≤ t −t − 2 Mst > 2 ] is close to Pr[sup0≤s≤ −Bs > 2 ] and (B.16) is close to zero. By the inclusion exclusion principle pt ≥ 1−2 Pr[ sup 0≤s≤ t

1 1 1 −t − 2 Mst > 2 ] ∧ Pr t β− 2 + t − 2 k +

1 sup t − 2

0≤r ≤ t

dV ds (X s ) ≥ , dx 0

r

and so pt can be made arbitrarily close to 1. Now we continue with showing L 1 convergence of φt to φ∞ . Since (1 − φt ) is negligible, it is sufficient to show φt → (1 − φt )φ∞ . Let γ < 13 β. We construct a sequence of hitting times σ j = min{s ∈ [0, ∞)s > θ j−1 , K s < t β + t γ }, θ j = min{s ∈ [0, ∞)s > σ j , K s > t β + 2t γ }, where we set θ0 = 0. Let N be the number of σ j ’s in the interval [0, t] before the first time t N that K t N jumps below t β . The process will usually have its first jump below t β from a point with momentum in the interval [t β , t β + 21 t γ ]. In other words, Pr |K t N −1 | > t γ is small. If, as defined in Lemma B.4, π↓ (v) is the distribution for the over-jump for the lower boundary of the set S = [t β + t γ , ∞) starting from some point in S, then by Lemma B.4 R+ dvπ↓ (v)v 2 < J . By Chebyshev’s inequality,

1 1 J Pr |K t N −1 | > t β + t γ ≤ dvπ↓ (v)χ (|v| ≥ t γ ) ≤ t −2γ . + 2 2 4 R The above comments allow us to write the following: ⎤ ⎡ φ (X σ j ,K σ j ) ∞

1 ↓, t ⎥ ⎢ γ γ χ N = j; K σ j − t β ≥ t γ φt − E ⎣ ⎦ ≤ Pr |K t N −1 | > t ≤ J t , 2 p(X σ j , K σ j ) j=1 1


283

(X σ j ,K σ j )

where φ↓,t

is the lower boundary crossing distribution as in Proposition B.7 start (X σ ,K σ ) ing from the point (X σ j , K σ j ), and da dv φ↓,t j j (a, v) = p(X σ j , K σ j ) is the probability of exiting the domain [t β , t β + 2t γ ) at the lower boundary. The difference d between the distance to the upper and lower boundaries 2t β and t β + t 2γ respectively is d = 2(t β + t γ − K σ j ): ⎡ ⎤ (X σ j ,K σ j ) ∞ φ↓,t 1 γ ⎢

⎥ β χ N = j; K σ j − t ≥ t E ⎣ ⎦ − pt π∞ 2 p(X σ j , K σ j ) j=1 1 φ (a,y) φ (a,y) ∞

↓, t ↓,t

≤ Pr χ (N = j) sup sup p(a, y) − π∞ ≤ p(a, y) − π∞ . 1 1 y−t β ≥ t γ ,a y−t β ≥ t γ ,a n=1 2

1

2

1

By Proposition B.7 p(a, y), y ≥ t β + 21 t γ converge uniformly to the probabilities p↓ for the averaged random walk to exit in the down direction. p↓ are bounded away from zero, since the ratio of the distance to the lower boundary to the upper boundary is less (a,y) than or equal to 3. Moreover, by Proposition B.9 φ↓, t converges uniformly to p↓ π∞ . References 1. Basile, G., Bovier, A.: Convergence of a kinetic equation to a fractional diffusion equation. http://arxiv. org/abs/0909.3385v1 [math.PR], 2009 2. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s Law: a Challenge for Theorists, Mathematical Physics 2000, Edited by A. Fokas, A. Grigoryan, T. Kibble, B. Zegarlinsky, London: Imeprial College Press, 2000, pp. 128–151 3. Chung, K.L.: A Course in Probability Theory. New York: Academic Press, 1976 4. De Masi, A., Ferrari, P.A., Goldstein, S., Wick, D.W.: An Invariance Principle for Reversible Markov Processes. Applications to Random Motions in Random Environments. J. Stat. Phys. 55, 767–855 (1989) 5. Goldstein, S.: Antisymmetric functionals of reversible Markov processes. Ann. Inst. Henri Poincaré 31, 177–190 (1995) 6. Hairer, M., Pavliotis, G.A.: From ballistic to diffusive behavior in periodic potentials. J. Stat. Phys. 131(1), 175–202 (2008) 7. Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. London-NewYork: Academic Press, 1980 8. Husak, D.V.: On the joint distribution of the time and value of the first overjump for homogeneous processes with independent increments. Teor. Ver. Primen. 14, 15–23 (1969) 9. Jara, M., Komorowski, T.: Non-Markovian limits of additive functionals of Markov processes. http:// arxiv.org/abs/0905.2163v1 [math.PR], 2009 10. Jara, M., Komorowski, T., Olla, S.: Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19(6), 2270–2300 (2009) 11. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Berlin-Heidelberg-NewYork: Springer-Verlag, 1988 12. Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104, 1–19 (1986) 13. Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes. See webpage http://w3.impa. br/~landim/notas.html, 2008 14. Pollard, D.: Convergence of Stochastic Processes. Berlin-Heidelberg-NewYork: Springer-Verlag, 1984 15. Weihs, D., Teitell, M.A., Mason, T.G.: Simulations of complex particle transport in heterogeneous active liquids. Microfluid Nanofluid 3, 227–237 (2007) Communicated by G. Gallavotti


Communications in


Transverse Laplacians for Substitution Tilings Antoine Julien1 , Jean Savinien1,2 1 Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43 blvd du 11 novembre 1918,

F-69622 Villeurbanne Cedex, France. E-mail: [email protected]; [email protected]

2 SFB 701, Universität Bielefeld, Bielefeld, Germany

Received: 26 October 2009 / Accepted: 14 June 2010 Published online: 22 October 2010 – © Springer-Verlag 2010

Abstract: Pearson and Bellissard recently built a spectral triple – the data of Riemannian noncommutative geometry – for ultrametric Cantor sets. They derived a family of Laplace–Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular self-similar Cantor sets. We use Bratteli diagrams to encode the self-similarity, and Cuntz–Krieger algebras to implement it. We show that the abscissa of convergence of the ζ -function of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace–Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels. Contents 1. 2. 3. 4. 5. 6.

Introduction and Summary of the Results . . . Weighted Bratteli Diagrams and Substitutions Spectral Triple, ζ -Function, and Complexity . Laplace–Beltrami Operator . . . . . . . . . . Cuntz–Krieger Algebras and Applications . . Examples . . . . . . . . . . . . . . . . . . . .

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1. Introduction and Summary of the Results In a recent article [39], Pearson and Bellissard defined a spectral triple – the data of Riemannian noncommutative geometry (NCG) [13] – for ultrametric Cantor sets. They used a construction due to Michon [38]: any ultrametric Cantor set (C, d) can be represented isometrically as the set of infinite paths on a weighted rooted tree. The tree Work supported by the NSF grants no. DMS-0300398 and no. DMS-0600956.

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A. Julien, J. Savinien

defines the topology, and the weights encode the distance. The spectral triple is then given in terms of combinatorial data on the tree. With this spectral triple, they could define several objects, including a ζ -function, a measure μ, and a one-parameter family of operators on L 2 (C, μ), which were interpreted as Laplace–Beltrami operators. They showed the abscissa of convergence of the ζ -function to be a fractal dimension of the Cantor set (the upper box dimension). The goal of Pearson and Bellissard was to build a spectral triple for the transversals of tiling dynamical systems. This opened a new, geometrical approach to the theory of tilings. Until now, all the operator-algebraic machinery used to study tilings and tiling spaces was coming from noncommutative topology. Striking applications of noncommutative topology were the studies of the K -theory for tiling C ∗ -algebras [3,5,29,30], and namely [2,15,18] for computations applied to substitution tilings. A follow-up of this study was the gap-labeling theorems for Schrödinger operators [4,6,8,28,44] – a problem already appearing in some of the previously cited articles. Other problems include cyclic cohomology and index theorems (and applications to the quantum Hall effect) [7,9,31–34]…. These problems were tackled using mainly topological techniques. The construction of a spectral triple is a proposition for bringing geometry into play. In this article, we revisit the construction of Pearson and Bellissard for the transversals of some tiling spaces. For this purpose, we use the formalism of Bratteli diagrams instead of Michon trees. This approach is equivalent, and applies in general to any ultrametric Cantor set. But in some cases, the diagram conveniently encodes the self-similarity, and is very well suited to handle explicit computations. For this reason, we will focus on diagrams arising from substitution tilings, a class of tilings which is now quite well studied [2,35,40,42]. Bratteli diagrams were introduced in the seventies for the classification of AFalgebras [10]. They were used by Veršik to encode measurable Z-actions, as a tool to approach some dynamical systems by a sequence of periodic dynamical systems [45]. They were then adapted in the topological setting to encode Z-actions on the Cantor set [15,23,25]. Then, Bratteli diagrams were used to represent the orbit equivalence relation arising from an action of Z2 [21] (and recently Zd [22]) on a Cantor set. Yet, it is not well understood how the dynamics itself should be represented on the diagram. The case we will look at is when the Cantor set is the transversal of a tiling space, and the action is related to the translations. The idea of parameterizing tilings combinatorially dates back to the work of Grünbaum and Shephard in the seventies, and the picture of a Bratteli diagram can be found explicitly in the book of Connes [12] for the Penrose tiling. However, it took time to generalize these ideas, and to understand the topological and dynamical underlying questions. In the self-similar case (for example when the Cantor set is the transversal of a substitution tiling space), the diagram only depends on an adjacency (or Abelianization) matrix. There is a natural C ∗ -algebra associated with this matrix, called a Cuntz– Krieger algebra [14]. Its generators implement recursion relations and therefore provide a method of computation for the eigenvalues of the Laplace–Beltrami operators. While Bratteli diagrams are suited to facilitate computations for any self-similar Cantor set, we focus on diagrams associated with substitution tilings. Indeed, a tiling space comes with a convenient distance, which is encoded (up to Lipschitz-equivalence) in a natural way by weights on the diagram. The Laplace–Beltrami operators we build here are similar to two other types of constructions found in the literature. First, the C ∗ -algebras in the spectral triples presented here are commutative AF-algebras. Christensen and Ivan built a family of spectral triples

Transverse Laplacians for Substitution Tilings

287

for general AF-algebras in [11], and their Dirac operators are, in the commutative case, related (and in some case identical) to our Laplacians, see Sect. 4.2. Second, those Laplacians come from Dirichlet forms and are generators of Markov semi-groups. They can also be constructed with purely probabilistic means. There is a rich literature on the subject of stochastic processes on homogeneous or regular trees, when additional algebraic structure is at a disposal [16,37]. For non-compact totally disconnected spaces, similar (jump) processes were obtained in [1]. Laplacians and stochastic processes are used in physics to model diffusion phenomena. There is a peculiar type of atomic diffusion which is expected to occur in quasicrystals due to structural effects (flip-flops) and not thermodynamics, and might become thus predominant at low temperature. Quasicrystals are often modeled by substitution tilings, and we believe the Laplacians studied in this paper to be good candidates to describe this phenomenon. Results of the Paper. Let B be a weighted Bratteli diagram (Definition 2.2, and 2.9), and (∂B, dw ) the ultrametric Cantor set of infinite paths in B (Proposition 2.10). The Dixmier trace associated with the spectral triple gives a measure μDi x on ∂B. The construction of Pearson–Bellissard gives a family of Laplace–Beltrami operators Δs , s ∈ R, on L 2 (∂B, dμDix ), see Definition 4.2. For all s, Δs is a nonpositive, selfadjoint, and unbounded operator. For a path γ , we denote by [γ ] the clopen set of infinite paths with prefix γ , and by χγ its characteristic function. And we let ext1 (γ ) denote the set of ordered pairs of edges that extend γ one generation further. The operator Δs was shown to have pure point spectrum in [39]. In this paper, we determine all its eigen elements explicitly. Theorem 4.3. The eigenspaces of Δs are given by the subspaces 1 1 χγ ·a − χγ ·b : (a, b) ∈ ext1 (γ ) Eγ = μDix [γ · a] μDix [γ · b] for any finite path γ in B. We have dim E γ = n γ − 1, where n γ is the number of edges in B extending γ one generation further. The associated eigenvalues λγ are also calculated explicitly, see Eq. (4.3). An eigenvector of Δs is simply a weighted sum of the characteristic functions of two paths of the same lengths that agree apart from their last edge, see Fig. 2 for an example. The transversal Ξ to a substitution tiling space (Ω, ω) of Rd , can be described by a stationary Bratteli diagram, like the Fibonacci diagram shown in Fig. 1. There is a “natural map”, called the Robinson map in [29], ψ : ∂B → Ξ , which under some technical conditions (primitivity, recognizability, and border forcing) is a homeomorphism (Theorem 2.22). We endow Ξ with the combinatorial metric dΞ : two tilings are ε-close if they agree on a ball of radius 1/ε around the origin. Let A be the Abelianization matrix of the substitution, and Λ P F its Perron–Frobenius eigenvalue. We denote by w(γ ) the weight of a finite path γ in B (Definition 2.9). Theorem 2.25. If there are constants c+ > c− > 0 such that −n/d

c− Λ P F

−n/d

≤ w(γ ) ≤ c+ Λ P F

for all paths γ of lengths n, then the homeomorphism ψ : ∂B → Ξ is (dw –dΞ ) bi-Lipschitz.

288

A. Julien, J. Savinien •

•

•

•

•

•

◦

Fig. 1. A self-similar Bratteli diagram associated with the matrix

1 1 1 0

(root on the left)

Fig. 2. Example of eigenvectors of Δs for the Fibonacci diagram

In our case of substitution tiling spaces, we have a ζ -function which is given as in [39] by ζ (s) =

w(γ )s ,

γ ∈Π

where Π is the set of finite paths in B. It is proven in [39] that, when it exists, the abscissa of convergence s0 of ζ is the upper box dimension of the Cantor set. For self-similar Cantor sets it always exists and is finite. Theorem 3.8. For a weighted Bratteli diagram associated with a substitution tiling space of dimension d, the abscissa of convergence of the ζ -function is s0 = d. We also have an interpretation of s0 which is not topological. We link s0 to the exponent of the complexity function. This function p, associated with a tiling, counts the number of distinct patches: p(n) is the number of patches of radius n (up to translation). We present two results.


289

Theorem 3.14. For the transversal of any minimal aperiodic tiling space with a welldefined complexity function, the box dimension, when it exists, is given by the following limit: ln( p(n)) . dim Ξ = lim sup ln(n) n→+∞ And we deduce the following. Corollary 3.16. Let Ξ be the transversal of a substitution tiling of dimension d, with complexity function p. Then there exists a function ν such that: p(n) = n ν(n) ,

with lim ν(n) = d. n→+∞

With the above choice of weights we can compute the Dixmier trace μDi x . Furthermore, there is a uniquely ergodic measure on (Ω, Rd ), which was first described erg by Solomyak [42]. It restricts to a measure μΞ on the transversal, and we have the following. erg

Theorem 3.9. With the above weights one has ψ∗ (μDi x ) = μΞ . As B is stationary, the sets of edges between two generations (excluding the root) are isomorphic. Let us denote by E this set, and by E0 the set of edges linking to the root. Thanks to the self-similar structure we can define affine maps u e , e ∈ E, that act on the eigenvalues of Δs as follows (see Sect. 5.1) (d+2−s)/d

u e (λη ) = λUe η = Λ P F

λη + βe ,

where Ue η is an extension of the path η (see Definition 5.2), and βe a constant that only depends on e. Those maps Ue , and u e , e ∈ E, implement two faithful ∗-representations of a Cuntz–Krieger algebra associated with the matrix A (Lemma 5.2). If γ = (ε, e1 , e2 , . . . en ), ε ∈ E0 , ei ∈ E, is a path of length n, let u γ be the map u e1 ◦ u e2 ◦ · · · u en . Let us denote by λε , ε ∈ E0 , the eigenvalues of Δs corresponding to paths of length 1. For any other eigenvalue λγ , there is a (unique) λε such that λγ = u γ (λε ) = Λns λε +

n

j−1

Λs

βe j ,

j=1 (d+2−s)/d

. That is, the Cuntz–Krieger algebra allows to calculate explicwhere Λs = Λ P F itly the full spectrum of Δs from the finite data of the λε , ε ∈ E0 , and βe , e ∈ E – which are immediate to compute, see Sect. 5.1. For instance, for the Fibonacci diagram (Fig. 1) and s = s0 = d, there are only two such √ u e maps, namely u a (x) = x · φ 2 − φ, and u b (x) = x · φ 2 + φ, where φ = (1 + 5)/2 is the golden mean. The eigenvalues of Δd are all of the form p + qφ 2 for integers p, q ∈ Z. They can be represented as points ( p, q) in the plane; these points stay within a bounded distance to the line directed by the Perron–Frobenius eigenvector of the Abelianization matrix, see Fig. 3. This is an example of a general result, valid for hyperbolic substitutions, see Theorem 5.9. We treat further examples in Sect. 6, in particular the Thue–Morse and Penrose tilings. The name Laplace–Beltrami operator for Δs can be justified by the following two results. For s = s0 = d, in analogy with the Laplacian on a compact d-manifold, Δd satisfies the classical Weyl asymptotics, and the trace of its heat kernel follows the leading term of the classical Seeley expansion. Let N (λ) be the number of eigenvalues of Δd of modulus less than λ.

290


Fig. 3. Distribution of the eigenvalues for the Fibonacci diagram

Theorem 5.7. There are constants 0 < c− < c+ such that as λ → +∞ one has c− λd/2 ≤ N (λ) ≤ c+ λd/2 . Theorem 5.8. There are constants 0 < c− < c+ such that as t ↓ 0 one has c− t −d/2 ≤ Tr etΔd ≤ c+ t −d/2 . 2. Weighted Bratteli Diagrams and Substitutions We first give the general definitions of Cantor sets, Bratteli diagrams, and substitution tilings. We then describe how to associate diagrams to tilings. Definition 2.1. A Cantor set is a compact, Hausdorff, metrizable topological space, which is totally disconnected and has no isolated points. An ultrametric d on a topological space X is a metric which satisfies this strong triangle inequality: ∀x, y, z ∈ X, d(x, y) ≤ max{d(x, z), d(y, z)}. 2.1. Bratteli diagrams. To a substitutive system, one can naturally associate a combinatorial object named a Bratteli diagram. These diagrams were first used in the theory of C ∗ -algebras, to classify AF-algebras. Then, it was mainly used to encode the dynamics of a minimal action of Z on a Cantor set. Definition 2.2. A Bratteli diagram is an oriented graph defined as follows: B = (V, Etot , r, s), where V is the set of vertices, Etot is the set of directed edges, and r, s are functions Etot → V (range and source), which define adjacency. We have a partition of V and Etot in finite sets: Vn ; Etot = En , V= n≥0

n≥0


291

V0 is a single element called the root and noted ◦. The edges of En have their source in Vn and range in Vn+1 , that is: r : En → Vn+1 , s : En → Vn . We ask that s −1 (v) = ∅ for all v ∈ V, and r −1 (v) = ∅ for all v ∈ V\V0 . Remark 2.3. If, for all v ∈ V\{◦}, r −1 (v) is a single edge, then B is a tree. In that sense, the formalism of Bratteli diagrams includes the case of trees, and so is a generalization of the case studied in [39]. However, our goal is to restrict to self-similar diagrams, for which computations are easier. Definition 2.4. A path γ of length n ∈ N ∪ {+∞} in a Bratteli diagram is an element (ε0 , e1 , e2 , . . .) ∈

n−1

Ei ,

i=0

which satisfies: for all 0 ≤ i < n, r (ei ) = s(ei+1 ). We call Πn the set of paths of length n < +∞, Π the set of all finite paths, and ∂B the set of infinite paths. The function r naturally extends to Π : if γ = (ε0 , . . . , en ), then r (γ ) := r (en ). In addition to the definition above, we ask that a Bratteli diagram satisfies the following condition. Hypothesis 2.5. For all v ∈ V, there are at least two distinct infinite paths through v. Remark 2.6. If, for all n ≥ 1 and (v, v ) ∈ Vn × Vn+1 , there is at most one edge from v to v , we can simply encode the path by vertices: the following map is an homeomorphism onto its image: (ε0 , e1 , . . .) −→ (ε0 , r (e1 ), . . .). Definition 2.7. Given two finite or infinite paths γ and γ , we note γ ∧ γ the (possibly empty) longest common prefix of γ and γ . The set ∂B is called the boundary of B. It has a natural topology inherited from the +∞ Ei , which makes it a compact and totally disconnected set. product topology on i=0 A basis of neighborhoods is given by the following sets: [γ ] = {x ∈ ∂B ; γ is a prefix of x}. Hypothesis 2.5 is the required condition to make sure that there are no isolated points. This implies the following. Proposition 2.8. With this topology, ∂B is a Cantor set.

292


Definition 2.9. A weight on ∂B is a function w : V → R∗+ , satisfying the following conditions: (i) w(◦) = 1; (ii) sup{w(v) ; v ∈ Vn } tends to 0 when n tends to infinity; (iii) ∀e ∈ Etot , w(s(e)) > w(r (e)). A weight extends naturally on paths: by definition, w(γ ) := w(r (γ )). Proposition 2.10. We define a function dw on (∂B)2 by: w (r (x ∧ y)) if x = y; dw (x, y) = 0 otherwise. It is an ultrametric on ∂B, which is compatible with the topology defined above, and so (∂B, dw ) is an ultrametric Cantor set. A case of interest is when the Bratteli diagram is self-similar. For a diagram, selfsimilarity means that all the Vn are isomorphic, and all En are isomorphic for n ≥ 1, and r, s commute with these identifications. We will focus on self-similarity when the diagram is associated with a substitution. 2.2. Substitution tilings. Let us give the definition of a tiling of Rd . The tilings we are interested in are constructed from a prototile set and an inflation and substitution rule on the prototiles. The notion of substitution dates back to the sixties. For example the self-similarity of the Penrose tiling is well-known. However, systematic formalization of substitution tilings and properties of their associated tiling space was done in [35,43] in the nineties. In this section, we follow the description by Anderson and Putnam [2] with some minor changes. In particular, some non trivial facts are cited along the text. We do not claim to cite the original authors for all of these. The reader can refer to the reviews [41] and [19]. By tile, we mean a compact subset of Rd , homeomorphic to the closed unit ball. A tile is punctured if it has a distinguished point in its interior. A prototile is the equivalence class under translation of a tile. Let A be a given set of punctured prototiles. All tilings will be made from these tiles. By abuse of notation, we may consider an element t ∈ A as a representant of a tile, when it is not relevant to make precise the position of the tile in Rd . A patch p on A is a finite set of tiles which have disjoint interior. We call A∗ the set of patches modulo translation. A partial tiling is an infinite set of tiles with disjoint interior, and the union of which is connected, and a tiling T is a partial tiling which covers Rd , which means: t = Rd . Supp(T ) := t∈T

A substitution rule is a map ω which maps tiles to patches, and such that for all tiles p, Supp(ω( p)) = λ Supp( p) for some λ > 1, and ω( p + x) = ω( p) + λx for all x ∈ Rd . See Fig. 4 for an example of dimension 2. The factor λ inflates the tiles p, which is then cut into pieces; these pieces have their translational classes in A. The map ω extends to patches, partial tilings and tilings. Since we will represent tilings spaces by diagrams, we have a specific interest for the combinatorics of the substitution.


293

Fig. 4. A process of inflation and substitution (chair tiling). A whole tiling of R2 can be obtained as a fixed point of this map

Definition 2.11. The Abelianization matrix of a given substitution ω is the integer-valued matrix Aω = (a pq ) p,q∈A (or simply A) defined by: ∀ p, q ∈ A, a pq = number of distinct translates of p included in ω(q). Remark that, since ω( p + x) = ω( p) + λx, the coefficient a p,q only depends on the classes of p and q modulo translation. We now define the tiling space associated with ω. Definition 2.12. The tiling space Ω is the set of all tilings T such that for all patch p of T , p is also a subpatch of some ωn (t) (n ∈ N, t ∈ A). We make the following assumptions on ω: Hypothesis 2.13. (i) The matrix A is primitive: it has non-negative entries, and some power An has positive entries. (ii) Finite local complexity (FLC): for all R > 0, the set of all patches p ⊂ T which can be included in a ball of radius R is finite up to translation, for all T ∈ Ω. Just as every tile has a distinguished point inside it, Ω has a distinguished subset. This is the Cantor set which will be associated with a Bratteli diagram. Definition 2.14. Let Ξ be the subset of Ω of all tilings T such that 0 is the puncture of one of the tiles of T . This set is called the canonical transversal of Ω. Note that, by this definition, the substitution ω extends to a map Ω → Ω. We will assume in the following that ω : Ω → Ω is one-to-one. This condition is equivalent (Solomyak [43]) to the fact that no tiling in Ω has any period. Furthermore, it implies that Ω is not empty, and ω : Ω → Ω is onto. The set of interest for us is Ξ . Its topology is given as follows. For any patch p, define the following subset of Ξ : U p = {T ∈ Ξ ; p ⊂ T }. Note that U p can be empty. Nevertheless, the family of all the U p ’s is a basis for a topology on Ξ . Let us now define a distance d on Ω: d(T, T ) = inf > 0 ; ∃x, y ∈ B(0, ) such that B(0, 1/) ⊂ Supp (T − x) ∩ (T − y) ∪ {1} .

294


Two tilings are d-close when, up to a small translation, they agree on a large ball around the origin. This distance, when restricted to Ξ , is compatible with the topology defined above. With this topology, Ξ is a Cantor set. Furthermore, the map ω : Ω → Ω is a homeomorphism, and the dynamical system (Ω, Rd ) given by translation is continuous and uniquely ergodic [42]. Substitution tiling spaces are minimal, which means that every Rd -orbit is dense in Ω. Combinatorially, this is equivalent to the fact that these tilings are repetitive: for all R > 0, there is a bound ρ R , such that every patch of size R appears in the tiling within range ρ R to any tile. We need an additional assumption – the border forcing condition. It is required in order to give a good representation of Ξ as the boundary of a Bratteli diagram. Definition 2.15. Let p ∈ L be a patch. The maximal unambiguous extension of p is the following patch, obtained as an intersection of tilings: {T ∈ Ω ; p ⊂ T } . Ext( p) = So any tiling which contains p also contains Ext( p). The patch Ext( p) is called the empire of the patch p by some authors [24]. Definition 2.16. Assume there is some m ∈ N and some > 0 such that for all t ∈ A, Ext ωm (t) contains an -neighborhood of ωm (t). Then ω is said to force its border. It is always possible to give labels to the tiles, in order to change a non-border forcing substitution into a border-forcing one. See [2]. Example 2.17. Let us give an example in dimension one. Consider the substitution defined on symbols by: a → baa, b → ba. It has√a geometric realization, with a being associated with an interval of length φ = (1 + 5)/2, and b with an interval of length 1. Then the map above is a substitution in the sense of our definitions, with λ = φ 2 . The transversal Ξ is naturally identified with a subset of {a, b}Z (the subshift associated with ω). The map ω satisfies the border forcing condition. Indeed, if p is a patch of a tiling T (which we identify symbolically with a finite word on {a, b} in a bi-infinite word), let x, y ∈ {a, b} be the letters preceding and following p in T respectively. Then ω(x) ends by an a and ω(y) begins by a b. Therefore, ω(T ) is always followed by b and preceded by a. This proves that ω forces its border. 2.3. Bratteli diagrams associated with substitution tilings. We show how to associate a Bratteli diagram to a substitution, and identify the transversal with the boundary of the diagram. It is clear that these two are homeomorphic, being Cantor sets. We will give an explicit and somehow natural homeomorphism. Let ω be a primitive, FLC and border forcing substitution, and A = (ai j )i, j∈A its Abelianization matrix. Let λ be the expansion factor associated with ω. Let Ξ be the transversal of the tiling space associated with ω. The diagram B associated with the substitution is defined as follows.


295

Definition 2.18. The diagram associated with ω is B = (V, Etot , r, s), where: ∀n ≥ 1, Vn = A × {n}, for all n ≥ 1, for all i, j ∈ A, there are exactly ai j edges of En from (i, n) to ( j, n + 1), and for all v ∈ V1 there is exactly one edge εv from the root ◦ to v. We call E the set of “models” of edges from one generation to another. All En are copies of E (for example with identification En = E × {n}). Since the models for the vertices are the elements of A, we do not use a specific name. The maps’ source and range can be restricted as maps E → A. When the “depth” of a vertex is not important, we will sometimes consider r, s as functions valued in A. Remark 2.19. Combinatorially, the diagram only depends on the Abelianization matrix of ω. It is indeed possible to associate a diagram to a primitive matrix A with integer coefficients. We would have similar definitions, with Vn = I × {n}, with I the index set of the matrix. Proposition 2.20. This is a Bratteli diagram in the sense of Definition 2.2, except in the case A = (1). Example 2.21. Figure 1 is an example of the self-similar diagram associated with the Fibonacci substitution: a → ab; b → a. There is a correspondence between the paths on B and the transversal Ξ . It depends on a choice on the edges which remembers the geometry of the substitution: each e ∈ E from a ∈ A to b ∈ A corresponds to a different occurrence of a in ω(b). This correspondence is a homeomorphism ϕ : Ξ → ∂B, called the Robinson map, as defined in [29]. We first give the definition of ϕ, and we will then give a condition for the weight function w so that ϕ is bi-Lipschitz. To construct ϕ, start with T ∈ Ξ . Let t ∈ T be the tile containing 0, and [t] ∈ A the corresponding prototile. Then, the first edge of ϕ(T ) is the edge from ◦ to ([t], 1) ∈ V1 . Assume that the prefix of length n of ϕ(T ) is already constructed and ends at vertex ([t ], n), where t is the tile of ω−(n−1) (T ) which contains 0. Let t be the tile containing 0 in ω−n (T ); then the (n + 1)th edge of ϕ(T ) is the edge corresponding to the inclusion of t in ω(t ). This edge ends at ([t ], n + 1). By induction we construct ϕ(T ). Theorem 2.22 (Theorem 4 in [29]). The function ϕ : Ξ → ∂B is a homeomorphism. We give explicitly the inverse for φ as follows. Let us define ψ : Π → A∗ which is increasing in the sense that if a path is a prefix of another, then the patch associated to the first is included in the patch associated to the second. The image of an infinite path will then be defined as the union of the images of its prefixes. If (ε) is a path of length one, then ψ(ε) is defined as the tile r (ε) with puncture at the origin. Now, given a path γ of length n, assume its image by ψ is well defined, and is some translate of ωn−1 (r (γ )), with the origin at the puncture of one of its tiles. Consider the path γ .e, with e ∈ En such that s(e) = r (γ ). Then, e encodes an inclusion of s(e) inside ω(r (e)). It means that it encodes an inclusion of ωn−1 (r (γ )) inside ωn (r (e)). The patch ψ(γ .e) is defined as the translate of ωn (r (e)) such that the inclusion ψ(γ ) ⊂ ψ(γ .e) is the inclusion defined by the edge e. Remark that if x ∈ ∂B, ψ(x) does not a priori define more than a partial tiling. The fact that it corresponds to a unique tiling results from the border forcing condition. Then the map ψ, extended to infinite paths, is the inverse of ϕ.

296


Continuity can be proved directly, but can also be seen as a consequence of Theorem 2.25, which we prove later. Notation. Remember the definition of the empire of a patch (Definition 2.15). Given γ ∈ Πn , define the following object: {ψ(x) ∈ Ω ; x ∈ [γ ]} . Ext(ψ(γ )) = This is a maximal unambiguous extension of ψ(γ ) given that this patch appears as the n-substitute of some tile. It does not need to be the same as Ext(ψ(γ )). Lemma 2.23. There exists C1 , C2 positive constants, such that for all path γ of length n in B, xt (ψ(γ )) , B(0, C1 λn ) ⊂ Supp E B(0, C2 λn ) ⊂ Supp (Ext (ψ(γ ))), where λ is the expansion factor of ω. Proof. Let k be the smallest number such that for all v ∈ V, there exists two distinct paths of length k starting from v. For example, k = 1 when there are two elements of Etot starting from v, for all v, and k = 2 in the case of Fibonacci, pictured in Fig. 1. Let C be the maximum diameter of the tiles. Let γ be a path of length n. Then, one can find two distinct paths of length n + k with the same prefix γ ; call them γ1 and γ2 . Remember that ψ(γi ) is some translate of ωn+k−1 (ti ) for some tile ti , and 0 ∈ Supp (ψ(γ )). So ψ(γ1 ) and ψ(γ2 ) differ within range Cλn+k−1 . Therefore, with C1 = Cλk−1 , the second statement of the lemma holds. For the first inclusion, let m be the exponent for which ω satisfies the border forcing m (t)) condition, and γ a path of length m. Let t ∈ A be the range of γ . Then Ext(ω covers an -neighborhood of Supp(ωm (t)). It means that, since ψ(γ ) is a translate of ωm (t) which contains 0, B(0, ) ⊂ E xt(ψ(γ )). Similarly, for all path of length m + k, )). B(0, λk ) ⊂ Ext(ψ(γ With n = m + k and C1 = /λm , one has the result. For n < m, the inequality still holds (up to a reduction of ). Remark 2.24. We can simplify the diagram in the case where a symmetry group G ⊂ On (Rd ) acts freely on A, such that ∀g ∈ G, ∀ p ∈ A, ω(g · p) = g · ω( p). With this, it is possible to extend the action of G to the edges of Etot . It induces naturally an action of G on ∂B. The group G also acts on Ω by isometries, and these two actions are conjugate by ψ. , One can “fold” the diagram by taking a quotient as follows. Define: B = (V , Etot r , s ), where V0 = V, E0 = E0 × G, and ∀n ≥ 1, Vn = Vn /G and En = En /G.


297

Furthermore, for [e]G ∈ En /G, r ([e]G ) := [r (e)]G and s ([e]G ) := [s(e)]G . These definitions do not depend on the choice of the representant e. For (e, g) ∈ E0 = E0 × G, define r ((e, g)) := [r (e)]G and s ((e, g)) := ◦. One can check that these two diagrams are “the same” in the sense that the tree structure of their respective sets of paths of finite length is the same. In terms of substitution, the image by ψ of a path γ = (ε0 , e1 , . . . , en−1 ) is the image of ωn (r (γ )) under some element of Rd G. The translation part is given by the truncated path (e1 , . . . , en−1 ). The rotation part is encoded in the first edge ε0 ∈ E0 : ε0 = (ε , g), where ε brings no additional information, but g corresponds to the choice of an orientation for the patch. There is still an action of G on ∂B , defined by: g · ((ε0 , h), e1 , . . .) = ((ε0 , gh), e1 , . . .). 2.4. Weights and metric. We gave in Theorem 2.22 an explicit homeomorphism ϕ between the boundary of the Bratteli diagram, and the transversal of the tiling space. Since we are interested in metric properties of the Cantor set, we now show that a correct choice of weights on the vertices of the Bratteli diagram gives a metric on ∂B which is Lipschitz equivalent to the usual metric on Ξ (the equivalence being induced by ϕ). Theorem 2.25. Let B be the Bratteli diagram associated with a substitution ω. Let λ be the inflation factor of ω. We make the following assumption on the weight function w: ∀n ≥ 1, ∀v ∈ V, w(v, n + 1) =

1 w(v, n). λ

(2.1)

Then the function ϕ : (Ξ, d) → (∂B, dw ) defined in Proposition 2.22 is a bi-Lipschitz homeomorphism. Proof. It is enough to show that there are two constants m, M > 0, such that for all γ ∈ Π, m≤

diam([γ ]) ≤ M. diam(ϕ −1 ([γ ]))

Since the [γ ] are a basis for the topology of ∂B, this will prove the result. Let γ ∈ Πn . By Lemma 2.23, any two tilings in ϕ −1 ([γ ]) coincide on a ball of radius at least C1 λn . Therefore, diam(ϕ −1 ([γ ])) ≤

1 −n λ . C1

On the other hand, it is possible to find two tilings in ϕ −1 ([γ ]) which disagree on a ball of radius C2 λn . Therefore, diam(ϕ −1 ([γ ])) ≥

1 −n λ . C2

And by definition of the weights, min{w(v, 1); v ∈ V}λ−n+1 ≤ diam([γ ]) ≤ max{w(v, 1); v ∈ V}λ−n+1 . Together with the previous two inequalities, this proves the result.

298


3. Spectral Triple, ζ -Function, and Complexity 3.1. Spectral triple. Let B be a weighted Bratteli diagram, and let (∂B, d) be the ultrametric Cantor set of infinite rooted paths in B. Pearson and Bellissard built in [39] a spectral triple for (∂B, d) when B is a tree (that is for any vertex v ∈ V\{◦}, the fiber r −1 (v) contains a single point). In our setting, their construction is adapted as follows. A choice function on B is a map Π → ∂B × ∂B τ : such that d (τ+ (γ ), τ− (γ )) = diam[γ ], (3.1) γ → (τ+ (γ ), τ− (γ )) and we denote by E the set of choice functions on B. Let CLip (∂B) be the pre-C ∗ algebra of Lipschitz continuous functions on (∂B, d). Given a choice τ ∈ E we define a faithful ∗-representation πτ of CLip (∂B) by bounded operators on the Hilbert space H = l 2 (Π ) ⊗ C2 as f (τ (γ )) 0 + . (3.2) πτ ( f ) = 0 f (τ− (γ )) γ ∈Π

This notation means that for all ξ ∈ H and all γ ∈ Π , 0 f (τ+ (γ )) · ξ(γ ). (πτ ( f ) · ξ ) (γ ) = 0 f (τ− (γ )) A Dirac operator D on H is given by D=

γ ∈Π

1 01 , diam[γ ] 1 0

(3.3)

that is D is a self–adjoint unbounded operator such that (D 2 + 1)−1 is compact, and the commutator f (τ+ (γ )) − f (τ− (γ )) 0 −1 , (3.4) [D, πτ ( f )] = 1 0 diam[γ ] γ ∈Π

is bounded for all f ∈ CLip (∂B). Finally a grading operator is given by Γ = 1l 2 (Π) ⊗ 1 0 , and satisfies Γ 2 = Γ ∗ = Γ , and commutes with πτ and anticommutes with 0 −1 D. The following is Proposition 8 in [39]. Proposition 3.1. CLip (∂B), H, πτ , D, Γ is an even spectral triple for all τ ∈ E. In [39] the set of choice functions E is considered an analogue of a tangent bundle over ∂B, so that the above commutator is interpreted as the directional derivative of f along the choice τ . The metric on ∂B is then recovered from the spectral triple by using the Connes formula, i.e. by taking the supremum over all directional derivatives. Theorem 3.2 (Thm. 1 in [39]). The following holds: d(x, y) = sup | f (x) − f (y)| ; f ∈ CLip (∂B), sup [D, πτ ( f )] ≤ 1 . τ ∈E

(3.5)


299

Definition 3.3. The ζ -function associated with the spectral triple is given by: ζ (s) =

1 −s Tr |D| = diam[γ ]s , 2

(3.6)

γ ∈Π

and we will denote by s0 ∈ R its abscissa of convergence, when it exists. We now assume that the weight system on B is such that s0 ∈ R. Definition 3.4. The Dixmier trace of a function f ∈ CLip (∂B) is given by the following limit, when it exists: Tr |D|−s πτ ( f ) . μ( f ) = lim (3.7) s↓s0 Tr |D|−s It is a linear form, and actually defines a probability measure on ∂B and does not depend on the choice τ ∈ E. Remark 3.5. In their article, Pearson and Bellissard make an assumption on the spectral triple (which they call ζ -regularity). This condition ensures that the above limit exists (Theorem 3 in [39]). In the following, we will prove directly that the limit defining the Dixmier trace exists for self-similar weighted Bratteli diagrams. Proving that the spectral triple is ζ -regular in this case would use the same methods. Furthermore, when f = χγ is a characteristic function of a cylinder [γ ], the limit above can be rewritten: s η∈Πγ w(r (η)) μ([γ ]) := μ(χγ ) = lim , (3.8) s s↓s0 η∈Π w(r (η)) where Πγ stands for the set of paths in Π with prefix γ . 3.2. The measure on ∂B. Let (Ω, ω) be a substitution tiling space, A be the Abelianization matrix of ω, and B be the associated Bratteli diagram, as in the previous section (Definition 2.18). We assume that B comes together with a weight function w, which satisfies the properties given in Definition 2.9, and adapted to the substitution ω. In particular, it satisfies w(a, n + 1) = λ−1 w(a, n) for all a ∈ A and n ∈ N. As we assumed A to be primitive, it has a so-called Perron–Frobenius eigenvalue, denoted Λ P F , which satisfies the following (see for example [26]): (i) (ii) (iii) (iv)

Λ P F is strictly greater than 0, and equals the spectral radius of A; For all other eigenvalues ν of A, |ν| < Λ P F ; The right and left eigenvectors, v R and v L , have strictly positive coordinates; If v R and v L are normalized so that v R , v L = 1, then: An = v L v tR ; n→+∞ Λn PF lim

(v) Any eigenvector of A with non-negative coordinates corresponds to the eigenvalue ΛP F .

300


A classical result about linear dynamical systems together with the properties above gives the following result, which will be needed later. Lemma 3.6. Let A be a primitive matrix with Perron–Frobenius eigenvalue Λ P F . Let p PM be its minimal polynomial: PM = (X − Λ P F ) i=1 (X − αi )m(i) . Then, the coefficients of An are given by: [A ]ab = n

cab ΛnP F

+

p

(a,b)

Pi

(n)αin ,

i=1

where the Pi ’s are polynomials of degree m(i), and cab > 0. Note that the coefficients (a, b) of An gives the number of paths of length n in the diagram between some vertex (a, k) ∈ Vk and (b, n + k). This lemma states that this number is equivalent to cab ΛnP F when n is large. Proof. We have cab > 0, as it is the (a, b) entry of the matrix v L v tR defined above. The rest is classical, and results from the Jordan decomposition of the matrix A. We assume that Ω is a d-dimensional tiling space. Since ω expands the distances by a factor λ, the volumes of the tiles are dilated by λd . This gives the following result: Proposition 3.7. Let Λ P F be the Perron–Frobenius eigenvalue of the matrix A. Then Λ P F = λd . In particular, Λ P F > 1. Theorem 3.8. The ζ -function for the weighted Bratteli diagram (∂B, w) has abscissa of convergence s0 = d. Proof. We have ζ (s) =

w(γ )s .

γ ∈Π

The w(γ ) only depends on r (γ ). Furthermore, w(a, n) tends to zero like λn = n quantity 1/d Λ P F when n tends to infinity. So we have: n∈N

ns/d

m s Λ P F Card(Πn ) ≤ ζ (s) ≤

ns/d

M s Λ P F Card(Πn ),

n∈N

where m (resp. M) is the minimum (resp. the maximum) of the w(a, 1), a ∈ A. Now, since Card(Πn ) grows like ΛnP F up to a constant (see Lemma 3.6), we have the result. Theorem 3.9. The measure μ given by the Dixmier trace (Definition (3.4)) is well defined, and given as follows. Let v = (va )a∈A be the (right) eigenvector for A, normalized such that e∈E0 vr (e) = 1. For all γ ∈ Π , let (a, n) := r (γ ) ∈ Vn . Then: μ([γ ]) = va Λ−n+1 PF . In particular, in the case of a substitution tiling, ψ∗ (μ) is the measure given by the frequencies of the patches, and therefore is the restriction to Ξ of the unique ergodic measure on (Ω, Rd ) (see [42]).


301

Proof. Let γ = (ε0 , e1 , . . . , en−1 ) ∈ Πn , and (a, n) := r (γ ). Let Πγ be the subset of Π of all paths which have γ as a prefix. Define: s η∈Πγ w(η) . f (s) = s η∈Π w(η) Then μ([γ ]) = lims↓d f (s), when this limit exists. The terms of the sum above can be grouped together: if η is a path of Πγ , then the quantity w(η)s only depends on the length of η, say n + k (k ≥ 0), and on r (η) = b ∈ A. Then, if we call N (a, b; k) the number of paths of length k from a to b, we can group the sum and write: N (a, b; k)w(b, n + k)s k≥0 b∈A f (s) = . 1 + k≥0 ε∈E0 b∈A N (r (ε), b; k)w(b, k)s Now, since w(b, n)s = λ(−n+1)s w(b, 1)s , we can write: N (a, b; k)w(b, n + k)s = λ(−n−k+1)s E at Ak W (s), b∈A

where E a is the vector (δa (b))b∈A , and W is the continuous vector-valued function s → (w((b, 1))s )b∈A . Similarly, N (r (ε), b; k)w(b, k)s = λ(−k+1)s E Et0 Ak W (s), ε∈E0 b∈A

where E E is the sum over ε ∈ E0 of all Er (ε) . Now, by Lemma 3.6, we have: p

E at Ak W (s) = ca (s)ΛnP F +

Pi (n, s)αin ,

i=1

where the Pi ’s are polynomial in n for s fixed, with Λ P F > |αi | for all i, and ca (s) > 0. Furthermore, the Pi are continuous in s. Similarly, E Et0 Ak W (s) = cE (s)ΛnP F +

p

Q i (n, s)αin ,

i=1

and since E E0 is a linear combination of the E a ’s, cr (ε) . cE =

(3.9)

ε∈E0

Then, we write: f (s) = λ =

−ns

k≥0 ca (Λ P F /λ

λs +

s )n

+

p k≥0

i=1

p

s n k≥0 cE (Λ P F /λ ) + k≥0 s )n + R (s) c (Λ /λ a P F 1 k≥0 λ−ns s . λ + k≥0 cE (Λ P F /λs )n + R2 (s)

Pi (n, s)(αi /λs )n

i=1

Q i (n, s)(αi /λs )n

302


Note that the expression above is defined a priori for s > d, but Ri (i = 1, 2) is defined and continuous for s ≥ d (the continuity results from the absolute convergence of the sum). The remaining sums above can be computed explicitly, and we have: f (s) = λ−ns

ca (s) + (1 − (Λ P F /λs )n ) R1 (s) . cE + (1 − (Λ P F /λs )n ) (R2 (s) + λs )

Then it is now clear that this expression is continuous when s tends to d, and has limit Λ−n P F ca (d)/cE (d). Let u a be defined as ca /cE for all a. Let us show that u = v. First, show that (u a )a∈A (or equivalently, (ca )a∈A ) is an eigenvector of A associated with Λ P F . We have: ca (d) = lim

n→+∞

E a An W (d) ΛnP F

= E a L W (d), where L = x y t , with x (resp. y) an eigenvector of A (resp. of At ) associated with Λ P F , and x, y = 1. So ca is the a-coordinate of L W (d) = y, W (d) x, and so is an eigenvector of A associated to Λ P F . Equation (3.9) now proves that u has the good normalization, and that u = v. 3.3. Complexity and box counting dimension. Definition 3.10. Let (X, d) be a compact metric space. Then, the box counting dimension is defined as the following limit, when it exists: dim(X, d) = lim − t→0

ln(Nt ) , ln(t)

where Nt is the minimal number of balls of radius t needed to cover X . Theorem 3.11 (Thm. 2 in [39]). Given an ultrametric Cantor set and its associated ζ -function, let s0 be the abscissa of convergence of ζ . Then: s0 = dim(X, d). This dimension can be linked to complexity for aperiodic repetitive tilings. Definition 3.12. The complexity of a tiling T is a function p which associates to n the number of patches of size n which have a puncture at the origin. Formally: p(n) = Card (T − x) ∩ B(0, n) ; x ∈ Rd and T − x ∈ Ξ , where (T − x) ∩ B(0, n) is a shorthand for the set of all tiles of T − x which are included in the ball of radius n centered at the origin. Notice that when T is repetitive (for example when T is a substitution tiling) the complexity is the same for all the tilings which are in the same tiling space. Remark 3.13. The definition of a patch actually depends on the norm chosen. Asymptotically, a change in the norm gives the same behavior for the complexity function. Notice that in the definition above, a patch of size n is simply the set of all tiles included in a ball of size n. It is not related to the hierarchical structure of the tiling which could be given by a substitution.


303

Theorem 3.14. Let Ω be a minimal tiling space, and Ξ its canonical transversal, endowed with the metric defined in Sect. 2.2. Let p be the associated complexity function. Then, for this metric, the box dimension is given by: ln( p(n)) . n→+∞ ln(n)

dim(Ω, d) = lim

Proof. Let Nt be the number of balls of diameter smaller than t needed to cover Ξ . Let us first prove that for all n ∈ N, p(n) = N1/n . Let L(n) be the set of all patches of size n, so that p(n) = Card(L(n)). Then, since all the tilings of Ξ have some patch of size n at the origin, the set:

Uq ; q ∈ L(n) is a cover of Ξ by sets of diameter smaller than 1/n. So p(n) ≥ N1/n . To prove the equality, assume we have a covering of Ξ by open sets {Vi ; i ∈ I }, with Card(I ) < p(n), and diam(Vi ) ≤ 1/n for all n. Then, in every Vi , we can find some set of the form Uq , q a patch. This allows us to associate some patch q(i) to all i ∈ I . We claim that for all such q(i), B(0, n) ⊂ Ext(q(i)), where Ext(q) is the maximal unambiguous extension of p, as defined in 2.15. Indeed, if it were not the case, diam(Uq ) would be smaller than n. Therefore, by restriction, to each i ∈ I , we can associate a patch q (i) of size n, such that Uq (i) ⊂ Vi . Since the {Vi }i∈I cover Ξ , all patches of size n are obtained this way, and p(n) ≤ Card(I ). Since this holds for all cover, p(n) ≤ N1/n . Now, Nt is of course an increasing function of t. Therefore, Nt−1 ≤ p ([1/t]) ≤ Nt , and so: −

p ([1/t]) Nt Nt−1 ≤ ≤− . ln(t) ln([1/t]) ln(t − 1)

Letting t tend to zero proves the theorem. Corollary 3.15. Let Ξ be the transversal of a minimal aperiodic tiling space, with a complexity function which satisfies C1 n α ≤ p(n) ≤ C2 n α for C1 , C2 , α > 0. Then: dim(Ξ, d) = α. Let’s now consider a tiling space Ω associated to a substitution ω. Let B be the weighted Bratteli diagram associated with it. We proved in Sect. 3.2 that for a Bratteli diagram associated with a substitution tiling of dimension d, the abscissa of convergence is exactly d. It is furthermore true that the box dimension of the transversal Ξ with the usual metric is d; it results from the invariance of the box dimension under bi-Lipschitz equivalence, which is proved in [17, Ch. 2.1]. Therefore, we can deduce the following result:

304


Corollary 3.16. Let Ω be a substitution tiling space satisfying our conditions. Then there exists a function ν such that: p(n) = n ν(n) , with lim ν(n) = d. n→+∞

Equivalently, for all > 0, there exists C1 , C2 > 0 such that for all n large enough, C1 n d− ≤ p(n) ≤ C2 n d+ . This result is actually weaker than what we can actually expect: in fact, there exist C1 , C2 > 0 such that C1 n d ≤ p(n) ≤ C2 n d . The upper bound was proved by Hansen and Robinson for self-affine tilings (see [41]). The lower bound can be proved by direct analysis for substitution tilings. It would also result from the conjecture that any d-dimensional tiling with low complexity (which means p(n)/n d tends to zero) has at least one period (see [36]). However, it is still interesting to see how the apparently abstract fact that the abscissa of convergence s0 equals the dimension gives in fact a result on complexity. 4. Laplace–Beltrami Operator We introduce here the operators of Pearson and Bellissard, revisited in our context of Bratteli diagram, and give their explicit spectrum. We then compare those operators with the Dirac operators proposed by Christensen and Ivan in [11] for AF-algebras. 4.1. The Pearson–Bellissard operators. Let B be a weighted Bratteli diagram. The Dixmier trace (3.7) induces a probability measure ν on the set E of choice functions (see [39] Sect. 7.2.). The following is [39] Theorem 4. Proposition 4.1. For all s ∈ R the bilinear form on L 2 (∂B, dμ) given by 1 Q s ( f, g) = Tr |D|−s [D, πτ ( f )]∗ [D, πτ (g)] dν(τ ), 2 E

(4.1)

with dense domain Dom Q s = χγ : γ ∈ Π , is a closable Dirichlet form. The classical theory of Dirichlet forms [20] allows to identify Q s ( f, g) with f, Δs g for a non-positive definite self-adjoint operator Δs on L 2 (∂B, dμ) which is the generator of a Markov semi-group. We have Dom Q s ⊂ Dom Δs ⊂ Dom Q˜ s where Q˜ s is the smallest closed extension of Q s . The following is taken from [39] Sect. 8.3. Theorem 4.2. The operator Δs is self-adjoint and has pure point spectrum. Following Pearson and Bellissard we can calculate Δs explicitly on characteristic functions of cylinders. For a path η ∈ Π let us denote by ext1 (η) the set of ordered pairs of distinct edges (e, e ) which can extend η one generation further, |γ |−1

1 (μ[γk ] − μ[γk+1 ])χγ − μ[γ ](χγk − χγk+1 ) G s (γk ) k=0 1 μ[η · e]μ[η · e ]. with G s (η) = diam[η]2−s 2

Δ s χγ = −

(e,e )∈ext1 (η)

(4.2a) (4.2b)


305

Note that the term χγk − χγk+1 is the characteristic function of all the paths which coincide with γ up to generation k and differ afterwards, i.e. all paths which split from γ at generation k. And μ[γk ] − μ[γk+1 ] is the measure of this set. We now state the main theorem which gives explicitly the full spectrum of Δs . Theorem 4.3. The spectrum of Δs is given by the following. (i) 0 is a single eigenvalue with eigenspace 1 = χ∂ B . 1 1 (ii) λ0 = G s1(◦) is eigenvalue with eigenspace E 0 = μ[ε] χε − μ[ε ] χε : ε, ε ∈ E0 , ε = ε of dimension dim E 0 = n 0 − 1, where n 0 is the cardinality of E0 . (iii) For γ ∈ Π , λγ =

|γ |−1 k=0

μ[γ ] μ[γk+1 ] − μ[γk ] − G s (γk ) G s (γ )

(4.3)

is eigenvalue with eigenspace Eγ =

1 1 : (e, e ) ∈ ext1 (γ ) χγ ·e − χ γ ·e μ[γ · e] μ[γ · e ]

(4.4)

of dimension dim E γ = n γ − 1, where n γ is the number of edges extending γ one generation further. Proof. The formula for the eigenvalues is calculated easily noticing that Δs χγ ·e and Δs χγ ·e only differ by the last term in the sum in Eq. (4.2a). The spectrum of Δs is always the closure of its set of eigenvalues (whatever its domain may be). Hence we do not miss any of it by restricting to characteristic functions. We now show that all the eigenvalues of Δs are exactly given by the λγ . It suffices to check that the restriction of Δs to Πn has exactly dim Πn eigenvalues (counting multiplicity). Notice that χγ is the sum of χγ ·e over all edges e extending γ one generation further. Hence an eigenfunction in E γ for γ ∈ Πk can be written as a linear combination of characteristic functions ofpaths in Πm , for any m > k. The number of eigenvalues λγ for γ ∈ Πn−1 is γ ∈Πn−1 dim E γ = γ ∈Πn−1 (n γ − 1) = dim Πn − dim Πn−1 . So the number of eigenvalues λγ for γ ∈ Πk for all 1 ≤ k ≤ n − 1 is n−1 k=1 (dim Πk+1 − dim Πk ) = dim Πn − dim Π1 . And counting 0 and λ0 adds up (dim Π1 − 1) + 1 to make the count match. Remark 4.4. As noted in Remark 2.3, our formalism with Bratteli diagrams includes as a special case the approach of Pearson–Bellissard for weighted Cantorian Michon trees. Hence Theorem 4.3 gives also the spectrum and eigenvectors of their Laplace–Beltrami operators. The eigenvectors (4.4) are very simple to picture. Given a path γ , and two extensions (a, b) ∈ ext1 (γ ), an eigenvector is the difference of their characteristic functions weighted by their measures. See Fig. 2 in Sect. 1 for an example for the Fibonacci diagram. Remark 4.5. One can easily see that the eigenspaces of Δs are pairwise orthogonal. Upon orthonormalizing E 0 and the E γ , one gets an orthonormal basis of L 2 (∂B, dμ) made of eigenvectors of Δs .

306


4.2. Comparison with the Dirac operator of Christensen and Ivan. Christensen and Ivan introduced in [11] a family of spectral triple for an AF-algebra A = limn An with a faith− → ful state τ . We recall briefly the construction of their Dirac operators. They considered the GNS representation associated with τ , on the Hilbert space Hτ that is the completion of A under the dot product a, b = τ (b∗ a). If [a] denotes the image of a ∈ A in Hτ , the representation is given by πτ (a)[b] = [ab]. The projection A → An becomes Hτ → Hn = {πτ (a)[1] : a ∈ An }. There is an orthogonal projection Pn : [An+1 ] → [An ], with kernel the finite dimensional vector space ker Pn = [An ]⊥ ∩ [An+1 ]. We let Q n be the projection Hτ → ker Pn . Given a sequence of positive real numbers λ = (λn )n∈N , Christensen and Ivan defined the following self-adjoint operator: Dλ = λn Q n . (4.5) n∈N

Under some growth conditions for the sequence λ, they showed that Dλ is a suitable Dirac operator, and that the metric it induces on the space of states generates the weak-∗ topology. Now let A = C(∂B) and An = χγ : γ ∈ Πn . An element χγ ∈ An can be written χγ ·e , where the sum runs over the edges that extend γ one generation further. The dot product in H = L 2 (∂B, dμ) reads on A χγ ·e , χγ · f = μ[γ · e] δe f , and the completion of A under , is H here of course. To simplify notation we identify χγ with its image in H. The orthogonal complement of χγ in [An+1 ] is generated by elements of the form χγ ·e /μ[γ · e] − χγ · f /μ[γ · f ], for (e, f ) ∈ ext1 (γ ), i.e. it is exactly the eigenspace E γ of Δs given in Theorem 4.2. Therefore ker Pn corresponds in this case to the direct sum of the E γ for γ ∈ Πn . Thus Pn is the sum of the spectral projections for eigenvalues associated with paths γ ∈ Πn . In the case where λγ = λn for all γ ∈ Πn (as in the Thue–Morse example in Sect. 6.2) then Δs is exactly the Dirac operator of Christensen and Ivan Dλ given in Eq. (4.5). 5. Cuntz–Krieger Algebras and Applications We now consider stationary Bratteli diagrams. We use the self-similar structure to further characterize the operator Δs and its spectrum. 5.1. Cuntz–Krieger algebras. Let B be a stationary Bratteli diagram. Let A be its Abelianization matrix. Let us denote by E0 the set of edges ε linking the root to generation 1, and by E the set of edges linking two generations (excluding the root). Let A˜ = a˜ e f e, f ∈E be the square matrix with entries a˜ e f = 1 if e can be composed with (or followed down the diagram by) f and a˜ e f = 0 otherwise. There is an associated ˜ which is “dual” to B in the sense that Bratteli diagram B˜ with Abelianization matrix A, its vertices correspond to the edges of B and its edges to the adjacencies of edges in B. Note that, because the entries of A˜ are zeros or ones, all the edges in B˜ are simple. Furthermore, since the matrix A is primitive, it is easily checked that A˜ is primitive as well. Remark 5.1. If B has only simple edges, we can simply take B˜ = B and A˜ = A.


307

Fig. 5. Example of some Cuntz–Krieger operators acting on finite paths

The Cuntz-Krieger algebra O A˜ , is the C ∗ -algebra generated by the partial isometries Ue , e ∈ E (on a separable, complex, and infinite dimensional Hilbert space H) that satisfy the following relations: O A˜ = C ∗ Ue , Ue∗ , e ∈ E | Ue Ue∗ , Ue∗ Ue ∈ P(H), Ue∗ Ue = A˜ e f U f U ∗f , f ∈E

(5.1) where P(H) denotes the set of projections in H: p ∈ P(H) ⇐⇒ p 2 = p ∗ = p. By abuse of notation we write the basis elements of l 2 (Π \Π1 ) as γ ∈ Π \Π1 . We define a representation of O A˜ on l 2 (Π \Π1 ) as follows:

(ε , e, e1 , e2 , . . .) 0 (ε , e2 , e3 , . . .) π0 (Ue∗ )(ε, e1 , e2 , . . .) = 0

π0 (Ue )(ε, e1 , e2 , . . .) =

if Aee1 = 1, otherwise if e1 = e, otherwise

(5.2a) (5.2b)

where ε ∈ E0 stands for the (possibly) new edges linking to the root, as illustrated in the case of the Penrose substitution below. The orientation of ε however is taken to be the same as that of ε: if ε = (s(e1 ), g) then we have ε = (s(e), g) (see Remark 2.24). In other words we require that ψ(ε , e, e1 , . . .) = ω ◦ ψ(ε, e1 , . . .) + x for some x ∈ Rd , and where ψ : ∂B → Ξ is the homeomorphism of Proposition 2.22. See Fig. 5 for some examples. Lemma 5.2. π0 is a faithful ∗-representation of O A˜ on l 2 (Π \Π1 ).

308


Proof. This follows from the unicity of the algebra O A˜ for a primitive matrix A˜ [14, Theorem 2.18]. We now define a representation on L 2 (∂B, dμ). To simplify notation we will simply write Ue for π0 (Ue ). First on DomΔs ⊂ L 2 (∂B, dμ) we let: (∗) χU (∗) γ if Ue γ = 0 (∗) e (5.3) π(Ue )χγ = 0 otherwise for paths in Πn≥2 and by linearity for shorter paths, using the relations χγ = e χγ ·e (the sum running over all edges e extending the path one generation further). If ϕ ∈ E γ , for γ ∈ Πn≥2 , is an eigenfunction of Δs , then we see from Eq. (5.3) that π(Ue )ϕ, if not zero, is another eigenfunction. Namely, we have EUe γ if Ue γ = 0 (5.4) π(Ue ) E γ = {0} otherwise. By Remark 4.5 the E γ give a basis of L 2 (∂B, dμ) made of eigen elements of Δs . As for Lemma 5.2 we have the following. Lemma 5.3. π is a faithful ∗-representation of OA˜ on L 2 (∂B, dμ). This allows to define another representation on the set of eigenvalues of Δs as follows. The Hilbert space we consider is the l 2 space of pairs (λγ , γ ) together with the element (0, 0). We simply write λγ and 0 for such elements. We set λUe γ if Ue γ = 0 u e (λγ ) = (5.5) 0 else, and one can easily check that this defines a faithful ∗-representation of OA˜ . Those maps u e are calculated as shown below. Lemma 5.4. Let Λ P F be the Perron–Frobenius eigenvalue of A, and consider a path γ = (ε, e1 , e2 , . . .) ∈ Πn≥2 . Then, if Ue γ = 0, we have μ[ε ] − μ[◦] μ[ε e] − μ[ε ] μ[ε] − μ[◦] (d+2−s)/d λγ − + + . u e (λγ ) = Λ P F G s (◦) G s (◦) G s (ε ) (5.6) Proof. By Eq. (4.3) we have λUe γ = −

|U e γ |−1 k=0

μ[Ue γ ] μ[(Ue γ )k ] − μ[(Ue γ )k+1 ] − . G s ((Ue γ )k ) G s (Ue γ )

(5.7)

The terms for k = 0, 1, in the above sum give the last two terms in Eq. (5.6). For (s−d)/d−2 all k ≥ 2, μ[(Ue γ )k = Λ−1 G s (γk−1 ) (see P F μ[γk−1 ], and G s ((Ue γ )k ) = Λ P F Theorem 3.9 for the rescaling of the measures). Hence the rest of the sum, over k = (d−s+2)/d 2, . . . |Ue γ | − 1 (and the last term) in Eq. (5.7), rescales by a factor Λ P F to the sum over k = 1, . . . |γ | − 1 (and the last term) for the eigenvalue λγ (Eq. (4.3)). We then add the contribution of the root, i.e. the term for k = 0, to get Eq. (5.6).


309

Note that the maps u e are affine, with constant terms, written βe , that only depend on e. We will write from now on (d+2−s)/d

u e (λγ ) = Λs λγ + βe ,

with Λs = Λ P F

.

(5.8)

The eigen elements of Δs corresponding to E 0 , and E ε , ε ∈ E0 , are immediate to calculate explicitly from Eqs. (4.3) and (4.4). We can therefore calculate explicitly all other eigen elements by action of OA˜ on those corresponding to the E ε , ε ∈ E0 . We summarize this in the following. Proposition 5.5. For γ = (ε, e1 , e2 , . . . en ) ∈ Πn≥1 , set Uγ = Ue1 Ue2 · · · Uen and u γ = u e1 ◦ u e2 ◦ · · · u en . For any γ ∈ Πn≥1 , we have E γ = Uγ E ε , and λγ = u γ (λε ) = Λns λε +

n

j−1

Λs

βe j .

(5.9)

j=1

5.2. Bounded case. We consider here the case s > d + 2. We show that Δs is bounded and characterize the boundary of its spectrum. Proposition 5.6. For s > d + 2, Δs is a bounded, and we have Δs B(L 2 (∂ B,dμ)) ≤ c

1 (d+2−s)/d 1 − ΛP F

,

with c = max maxε∈E0 |λε |, maxe∈E |βe | . Proof. By Proposition 5.5, Eq. (5.9), we see that for γ ∈ Πn we have n j Λs . |λγ | ≤ max max |λε |, max |βe | ε∈E0

e∈E

j=0

(d−s+2)/d

< 1, therefore the above geometric sum From Eq. (5.8) we have Λs = Λ P F converges, and is bounded for all n by its sum 1/(1 − Λs ). We define the ω-spectrum of Δs as Spω (Δs ) =

Sp(Δs )\Sp Δs |Πn .

n∈N

In our case here, this is the boundary of the (pure point) spectrum of Δs . Under some conditions on A and the βe , e ∈ E, and for s > d + 2 large enough, one can show that Spω (Δs ) is homeomorphic to Ξ , and that this homeomorphism is Hölder [27].

310


5.3. Weyl asymptotics. The following theorem justifies calling Δs a Laplace–Beltrami operator. Indeed, for s = s0 = d, Theorem 5.7 shows that the number of eigenvalues of Δd of modulus less than λ behaves like λd/2 when λ → ∞, which is the classical Weyl asymptotics for the Laplacian on a compact d-manifold.

Theorem 5.7. Let Ns (λ) = Card λ eigenvalue of Δs : |λ | ≤ λ . For s < d + 2, we have the following Weyl asymptotics: c− λd/(d−s+2) ≤ Ns (λ) ≤ c+ λd/(d−s+2) ,

(5.10)

as λ → +∞, for some constants 0 < c− < c+ . Proof. By Proposition 5.5, Eq. (5.9), there exist constants x± , y± > 0, such that for all γ ∈ Πn we have x− Λns + y− ≤ |λγ | ≤ x+ Λns + y+ . For s < d + 2, Λs > 1 (see Eq. (5.8)), so there is an integer k > 0 (independent of n) such that x+ Λns + y+ ≤ + y− . Hence for all γ ∈ Πl , l ≤ n, we have |λγ | ≤ x+ Λns + y+ , and for all x− Λn+k s + y− . Therefore we get the inequalities: γ ∈ Πl , l ≥ n + k, we have |λγ | ≥ x− Λn+k s Card Πn ≤ N (x+ Λns + y+ ) ≤ Card Πn+k . There are constants c1 > c2 > 0 such that for all l ∈ N, c1 ΛlP F ≤ Card Πl ≤ c2 ΛnP F , so that we get c1 ΛnP F ≤ N (x+ Λns + y+ ) ≤ n c2 Λn+k P F = c3 Λ P F . We substitute Λs from Eq. (5.8) to complete the proof. 5.4. Seeley equivalent. For the case s = s0 = d we give an equivalent to the trace Tr etΔd , for s = s0 = d, as t ↓ 0. The behavior of Tr etΔd like t −d/2 as t ↓ 0 is in accordance with the leading term of the classical Seeley expansion for the heat kernel on a compact d–manifold. Theorem 5.8. There exist constants c+ ≥ c− > 0, such that as t ↓ 0, c− t −d/2 ≤ Tr etΔd ≤ c+ t −d/2 .

(5.11)

Proof. Let Pγ be the spectral projection (onto E γ ) for γ ∈ Πn≥1 , and P0 that on E 0 . The trace reads ∞ etλγ Tr (Pγ ). Tr etΔs = 1 + eλ0 t Tr (P0 ) +

(5.12)

n=1 γ ∈Πn

Now Tr (Pγ ) = n γ − 1, with n γ the number of possible extensions of γ one generation further (see Eq. (4.4) in Theorem 4.3), and Tr (P0 ) = n 0 − 1. Since the Bratteli diagram of the substitution is stationary, the integers n γ are bounded, so there are p− , p+ > 0, such that for all γ ∈ Π we have: p− ≤ Tr (Pγ ) ≤ p+ .

(5.13)

By Eq. (5.9), the λε , βe , being bounded, there exists λ− , λ+ > 0, such that for all γ ∈ Π we have: nd/2

nd/2

λ− Λ P F ≤ |λγ | ≤ λ+ Λ P F .

(5.14)

The cardinality of Πn grows like ΛnP F so there are π− , π+ > 0, such that for all n ≥ 0 we have: π− ΛnP F ≤ |Πn | ≤ π+ ΛnP F .

(5.15)


311

We substitute inequalities (5.13), (5.14), and (5.15) into Eq. (5.12) to get 1 + p− π−

∞

∞ nd/2 nd/2 ΛnP F e−tλ+ Λ P F ≤ Tr etΔs ≤ 1 + p+ π+ ΛnP F e−tλ− Λ P F . (5.16)

n=0

n=0

Set Nt = d log(1/t)/(2 log(Λ P F )), and split the above sums into two parts: the sum over n < Nt , and the remainder. For the finite sum we have: N t −1

nd/2

ΛnP F e−tλ± Λ P F = t −d/2 Λ−Nt

n=0

N t −1

(n−Nt )d/2

ΛnP F e−λ± Λ P F

,

(5.17)

n=0 (n−Nt )d/2

where we have used Λ P tF = 1/t. With e−λ± ≤ e−λ± Λ P F ≤ 1, the above sum Nt −1 n on the right-hand side of (5.17) is bounded by the geometric series n=0 ΛP F = Nt −d/2 −N t (Λ P F − 1)/(Λ P F − 1). Multiplying by t Λ we get the inequalities: N d/2

≤ t −d/2 c−

N t −1

nd/2

ΛnP F e−tλ± Λ P F ≤ t −d/2 c+ ,

(5.18)

n=0 , c > 0, and t small enough. for some constants c− + For the remainder of the sums in (5.16) we have ∞ n=Nt

nd/2

ΛnP F e−tλ± Λ P F = Λ NP tF

∞

md/2

ΛmP F e−λ± Λ P F = t −d/2 c± ,

(5.19)

m=0

where c±

> 0 is the sum of the absolutely convergent series. We put together inequalities (5.18) and Eq. (5.19) into inequalities (5.16) to complete the proof. 5.5. Eigenvalues distribution. We now restrict to the case s = s0 = d, so that Λd = 2/d Λ P F in Eq. (5.8). In dimension 2, and for s = s0 = 2, the weights are not involved in the formula which define the Laplace operator. In dimension 1, we assume that the weights are taken equal to the measures: diam[γ ] = μ[γ ]. In particular (and it is the important point if one wants to generalize), all the coefficients in the formula defining the Laplacian belong to the same algebraic field, which is the splitting field for the characteristic polynomial of the matrix A. Let us consider F the splitting field of the characteristic polynomial of A over Q. It is clear that Λ P F and the coordinates of the associated eigenvectors belong to F. From the above assumption and using Eqs. (4.2a), (4.2b), (4.3), and (5.6), we see that the λε , ε ∈ E0 , and the βe , e ∈ E, belong to F. Now remark that F is just a finite dimensional Q-vector space. Therefore we can choose a basis and represent its elements by vectors in Qr for some r . Now, the linear map which is multiplication by Λ P F or Λ2P F can be represented by a matrix in this basis, 2/d which we call C. This matrix has coefficients in Q. Notice that necessarily, Λ P F is an eigenvalue of C, and so the characteristic polynomial of C is divided by the minimal 2/d polynomial of Λ P F . The affine maps βe can be represented in matrix form, and Eq. (5.8) becomes here (5.20) u e λγ = C λγ + β e ,

312


where β e is the matrix of βe in the basis of F we just chose. The general expression of an eigenvalue given in Proposition 5.5, Eq. (5.9), takes here the form λγ = C n λε +

n

C j−1 β e j .

(5.21)

j=1

This identification of F and Qr allows to describe the distribution of eigenvalues in a geometric way. We now assume that Λ P F is Pisot, i.e. it is real and greater than 1, and all its algebraic conjugates μ satisfy |μ| < 1. It implies that the matrix C has a single eigenvalue of 2/d modulus greater than 1, which is Λ P F , and all other eigenvalues are smaller than 1 in modulus. Let us denote by V the eigenspace (of dimension 1) of the matrix C associated with 2/d Λ P F . We can characterize the distribution of the eigenvalues of Δd as follows. Theorem 5.9. Let distV be a distance (given by a norm) on Qr ∼ = F. There exist a constant M > 0 such that for any eigenvalue λγ of Δd , one has distV λγ , V ≤ M, (5.22) Proof. One can decompose the space F as a direct sum of V on the one hand, and the sum of all other characteristic spaces of C on the other hand, which we call V . With this 2/d decomposition, C is block-diagonal, with Λ P F as the first diagonal coefficient. Denote by C the other block (this is the matrix of the application induced by C on V ). Choose some Euclidean norm arbitrarily, but such that V and V are orthogonal. Furthermore, it is possible to choose the restriction of the norm on V so that the induced matrix norm satisfies C < 1. The distance distV λγ , V equals the norm of the projection of λγ onto V . Call P the projection to V with respect to V . Using the fact that C and P commute (PC = C P = PC P = C ), we have ⎛ ⎞ n j−1 ⎝C n λε + ⎠ P distV λγ , V = C β ej j=1 ≤ (C )n .λε +

n

(C ) j−1 .Pβ e j .

j=1

Now, remark that there are only finitely many βe and finitely many λε , so they can be bounded uniformly by a constant. Furthermore, (C )n ≤ (C )n , which is a geometric sequence decreasing to zero. So it is enough to choose: M = max (Pβ e ) (C )n . e

n≥0

Remark 5.10. (i) Up to rescalling the basis of the space F by an integer large enough, it is possible to express the eigenvalues as integer column vectors. (ii) It is possible to generalize Theorem 5.9 in dimension d > 2. The required property is that the weights must belong to a suitable algebraic field. (iii) If C is no longer Pisot but strictly hyperbolic (i.e. has no eigenvalue of modulus 1), then the above result still holds but with V the unstable space (the span of the eigenvectors with eigenvalues |μ| > 1).


313

6. Examples We illustrate here the results of Sects. 4 and 5 for the classic examples of the Thue–Morse, Fibonacci, Ammann–A2, and Penrose tilings. 6.1. The Fibonacci diagram. The Bratteli diagram for the uncollared Fibonacci substia → ab tution reads b → a α2 = α3 = α4 = = == == == == = = == == == == ◦? ?? === === ?? == == ?? = = ? 2 α α3 α

B

where the √ top vertices are of “type b” and the bottom ones of “type a”, where α = 1/φ = ( 5 − 1)/2 is the inverse of the golden mean, and the term α n at a vertex is the measure of the cylinder of infinite paths through that vertex. Note that this substitution does not force the border, so that ∂B is not the transversal of the Fibonacci tiling space. For illustration purposes it is however worth carrying this example in details. We treat the “real” Fibonacci tiling together with the Penrose tiling in Sect. 6.3. Since the Bratteli diagram has only simple edges, as noted in Remark 2.6, the paths can be indexed by the vertices they go through. The paths in Π1 are thus written a and b, and the paths in Π2 are written aa, ab, and ba (note that these are not orthonormal bases for the dot product given by the Dixmier trace , so that the Laplacians written below will not be symmetric). The restrictions of the Laplace operator (4.2) for s = s0 = 1 to Π1 and Π2 are given below together with their eigenelements: 1 − φ 2 −1 + φ 2 , with eigenelements Δ|Π1 = φ2 −φ 2 1 1 ), and (1 − 2φ 2 , ), (0, 1 1 − φ2 ⎡

Δ|Π2

⎤ 2 − 3φ 2 −1 + 2φ 2 −1 + φ 2 = ⎣ −1 + 3φ 2 2 − 4φ 2 −1 + φ 2 ⎦ , with eigen elements −1 + φ 2 1 −φ 2 ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ 1 1 1 (0, ⎣ 1 ⎦), (1 − 2φ 2 , ⎣ 1 ⎦), (3 − 6φ 2 , ⎣ 1 − φ 2 ⎦). 1 1 − φ2 0

Using the identities χb = χba , and χa = χaa + χab , we see that the first two eigenvectors of Δ|Π2 are exactly those of Δ|Π1 expressed in Π2 . Note that 1 − φ 2 = −φ and that the above eigenvectors of Δ are χa + χb , χa − φχb , and χaa − φχab . And all other eigenvectors are given by χγ aa − φχγ ab for γ ∈ Π .

314


Since the Bratteli diagram hasonly simple edges, as noted in Remark 5.1, we can 1 1 take B˜ = B and A˜ = A = . The action of the two Cuntz–Krieger operators Ua 1 0 and Ub on the eigenvalues of Δ as in Eq. (5.5) is given in here by u a λγ = φ 2 λγ + 1 − φ 2 , u b λγ = φ 2 λγ − 1 + φ 2 , (6.1) if γ is compatible with their action, and u a (λγ ) = 0 or u b (λγ ) = 0 otherwise. 0 1 , and the operators Over the ring Z ⊕ φ 2 Z, the companion matrix of A is 1 1 (6.1) become the affine maps 0 1 0 1 −1 −1 u a λγ = λ − λ + , u b λγ = , 1 1 γ 1 1 γ 1 1 when γ is compatible with the corresponding action. Figure 3 illustrates Theorem 5.9 that characterizes the repartition of the eigenvalues of −Δ as point of integer coordinates that stay within a bounded strip to the PerronFroebenius eigenline of A (slope φ) in Z ⊕ φ 2 Z. Note that the repartition of points in the strip is not “homogeneous”, i.e. the number of points within √a distance r to the origin is not linear in r , but rather follows the Weyl asymptotics in r (Theorem 5.7). 6.2. The dyadic Cantor set and the Thue–Morse tiling. Those examples have enough symmetries to allow easy and direct calculations (without using the operators of the Cuntz–Krieger algebra). The Bratteli diagram B of the dyadic Cantor set is the dyadic odometer, ◦

B

0 1

1 2

0 1

1 4

0 1

1 8

and its associated diagram for its Cuntz-Krieger algebra B˜ is the Bratteli diagram of the 0 → 01 (uncollared) Thue-Morse substitution : 1 → 10

B˜

◦; ;; ;; ;; ;

1 2

1 2

99 99 99 99 99 9 999 99 99

1 4

1 4

99 99 99 99 99 9 999 99 99

1 8

1 8

where the term 21n at a vertex is the measure of the cylinder of infinite paths through that vertex. The top vertices are of “type 0”, the bottom ones of “type 1”. We label the paths in B by sequences of 0’s and 1’s labeling the edges they go through from the root: γ ∈ Πn is written γ = (ε1 , . . . , εn ). The Laplacian on B commutes with the following operators: χ(ε1 ,...,εi +1,...εn ) if 1 ≤ i ≤ n, τi χ(ε1 ,...,εn ) = (6.2) χ(ε1 ,...,εn ) otherwise,


315

where the addition is taken mod 2. The operators τi commute with each other and square up to the identity. One can therefore choose an eigenbasis for Δ made of eigen elements of the τi ’s: that is Haar functions on the dyadic Cantor set ∂B. We recover this way the example treated in [39] and we refer the reader there for the details. ˜ we can also index paths by sequences of 0’s and 1’s For the Thue–Morse diagram B, labeling the vertices they go through from the root. The Laplacian is also commuting with the operators τ˜i defined like the τi defined in Eq. (6.2). A basis of eigenvectors of Δ for s = s0 = 1 is given by the constant function χ∂ B˜ (with eigenvalue 0), and the functions ϕn,γ = χγ − τñ χγ , γ ∈ Π˜ n , for n ∈ N, with eigenvalues λn = − 23 7 · 4n−1 − 1 of degeneracy 2n−1 = Card Πn . The eigenvalues satisfy the induction formula λn+1 = 4λ ' n − 2. ' 1 6 6 4 The Weyl asymptotics of Theorem 5.7 reads here 2 7 λ + 10 ≤ N (λ) ≤ 7 7λ + 7.

6.3. The Penrose tiling. The Fibonacci, Penrose, and Ammann–A2 [24] tilings have formally the “same” substitution on prototiles modulo their symmetry groups, with Abelianization matrix 2 1 . A= 1 1 The Penrose and Ammann–A2 substitutions force the border. And for the Fibonacci tiling, one considers the conjugate substitution a → baa, b → ba, which is primitive, recognizable, and forces the border as noted in Example 2.17. Those three substitu2 tion tilings √ have the same Perron–Frobenius eigenvalue, namely Λ P F = φ , where φ = (1 + 5)/2 is the golden mean. In conclusion, the transversals of those tiling spaces can be described by the set of infinite paths in the same Bratteli diagram B illustrated below: ◦= == == = εa ==

α2 |G|

εb

B

α |G|

e5

;; ;; ;; e4 ; ;; ;; ;; e3 ;; ;; e2 e1

α4 |G|

α3 |G|

;; ;; ;; ;; ;; ;;; ;; ;;

α6 |G|

α5 |G|

√ where α = 1/φ = ( 5 − 1)/2 is the inverse of the golden mean, and the term α n /|G| at a vertex is the measure of the set of infinite paths through that vertex, and where G is the symmetry group of the tiling introduced in Sect. 2.24, and |G| the cardinality of G. That is G = {1} is the trivial group (so |G| = 1) for the Fibonacci tiling, G = C2 × C2 for the Ammann–A2 tiling (symmetries of “vertical and horizontal” reflections, |G| = 4), and G = D10 for the Penrose tiling (10-fold rotational symmetries, and reflections, |G| = 20).

316


Let us denote by Ui , i = 1, . . . 5, the generators of the Cuntz–Krieger algebra (5.1) ˜ The induced action on the eigenvalues associated with the Abelianization matrix of B. of Δ = Δs0 as in Eq. (5.5) reads here for Penrose and Ammann–A2 (for Fibonacci, 2/d Λ P F = φ 4 has to replace φ 2 in the following equations): μ[εa ] − μ(◦) μ[εa e1 ] − μ[a] u 1 λγ = φ 2 λγ + (1 − φ 2 ) + , G(◦) G(εa ) μ[εa ] − μ(◦) μ[εa e2 ] − μ[a] + , u 2 λγ = φ 2 λγ + (1 − φ 2 ) G(◦) G(εa ) μ[εa ] − μ(◦) μ[εb ] − μ(◦) μ[εb e3 ] − μ[a] + + , u 3 λγ = φ 2 λγ + −φ 2 G(◦) G(◦) G(εb ) μ[εb ] − μ(◦) μ[εa ] − μ(◦) μ[εa e4 ] − μ[a] + + , u 4 λγ = φ 2 λγ + −φ 2 G(◦) G(◦) G(εa ) μ[εb ] − μ(◦) μ[εb e5 ] − μ[εb ] + , u 5 λγ = φ 2 λγ + (1 − φ 2 ) G(◦) G(εb ) if λγ is compatible with the operators. Here G = G s0 as in Eq. (4.2b), so we have G(◦) =

α α α 2 α 2 α 2 α 2 α α |G|(|G|−1) ( |G| ) + ( |G| ) +|G|2 |G| |G| , G(εa ) = 2 ( |G| ) + ( |G| ) +4 |G| |G| and 2

2

2

3

3

4

α α G(εb ) = 2 |G| |G| . The eigen elements of Δ|Π2 are 0 for χ∂ B , λ0 for χεa − φχεb , λεa for χεa e − φχεa f , e, f ∈ ext1 (εa ), and λεb for χεb e − φχεb f , e, f ∈ ext1 (εb ), where 3

4

−2|G| |G| + 1 − 4φ 2 , λ0 = |G|2 − 10|G| + 5 μ[εb ] μ[εb ] − μ[◦] − . λεb = G(◦) G(εb )

λεa =

μ[εa ] μ[εa ] − μ[◦] − , G(◦) G(εa )

Acknowledgements. The authors were funded by the NSF grants of Jean Bellissard no. DMS-0300398 and no. DMS-0600956, while visiting the Georgia Institute of Technology for the Spring term 2009. It is a pleasure to thank J. Bellissard for his many supportive, insightful, and enthusiastic remarks on this work. The authors would like to thank Ian Putnam for useful discussions which lead in particular to the study of the eigenvalue distribution in Sect. 5.5. The authors would like to thank Johannes Kellendonk for useful discussions, in particular for explaining to us the relation between the Pearson–Bellissard operators and the Dirac operators of Christensen and Ivan given in Sect. 4.2. The authors would like to thank Marcy Barge for inviting them to Montana State University. J.S. also acknowledges financial support from the SFB 701, Universität Bielefeld, and would like to thank Michael Baake for his generous invitations.

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6. Bellissard, J., Kellendonk, J., Legrand, A.: Gap-labelling for three-dimensional aperiodic solids. C. R. Acad. Sci. Paris Sér. I Math. 332(6), 521–525 (2001) 7. Bellissard, J., van Elst, A., Schulz-Baldes, H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10), 5373–5451 (1994) 8. Benameur, M.-T., Oyono-Oyono, H.: Gap-labelling for quasi-crystals (proving a conjecture by J. Bellissard). In: Operator algebras and mathematical physics (Constan¸ta, 2001). Bucharest: Theta, 2003, pp. 11–22 9. Benameur, M.-T., Oyono-Oyono, H.: Index theory for quasi-crystals. I. Computation of the gap-label group. J. Funct. Anal. 252(1), 137–170 (2007) 10. Bratteli, O.: Inductive limits of finite dimensional C ∗ -algebras. Trans. Amer. Math. Soc. 171, 195–234 (1972) 11. Christensen, E., Ivan, C.: Spectral triples for AF C ∗ -algebras and metrics on the cantor set. J. Operator Theory 56(1), 17–46 (2006) 12. Connes, A.: Géométrie non commutative. Paris: InterEditions, 1990 13. Connes, A.: Noncommutative geometry. San Diego, CA: Academic Press Inc., 1994 14. Cuntz, J., Krieger, W.: A class of C ∗ -algebras and topological Markov chains. Invent. Math. 56(3), 251–268 (1980) 15. Durand, F., Host, B., Skau, C.: Substitutional dynamical systems, Bratteli diagrams and dimension groups. Erg. The. Dyn. Syst. 19(4), 953–993 (1999) 16. Evans, S.N.: Local properties of Lévy processes on a totally disconnected group. J. Theoret. Probab. 2(2), 209–259 (1989) 17. Falconer, K.: Fractal geometry. Chichester: John Wiley & Sons Ltd., 1990 18. Forrest, A.H.: K -groups associated with substitution minimal systems. Israel J. Math. 98, 101–139 (1997) 19. Frank, N.P.: A primer of substitution tilings of the Euclidean plane. Expo. Math. 26(4), 295–326 (2008) 20. Fukushima, M.: Dirichlet forms and Markov processes. Volume 23 of North-Holland Mathematical Library. Amsterdam: North-Holland Publishing Co., 1980 21. Giordano, T., Matui, H., Putnam, I.F., Skau, C.F.: Orbit equivalence for Cantor minimal Z2 -systems. J. Amer. Math. Soc. 21(3), 863–892 (2008) 22. Giordano, T., Matui, H., Putnam, I.F., Skau, C.F.: Orbit equivalence for Cantor minimal Zd -systems. Invent. Math. 179(1), 119–158 (2010) 23. Giordano, T., Putnam, I.F., Skau, C.F.: Topological orbit equivalence and C ∗ -crossed products. J. Reine Angew. Math. 469, 51–111 (1995) 24. Grünbaum, B., Shephard, G.C.: Tilings and patterns. A Series of Books in the Mathematical Sciences. New York: W. H. Freeman and Company, 1989 25. Herman, R.H., Putnam, I.F., Skau, C.F.: Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3(6), 827–864 (1992) 26. Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge: Cambridge University Press, 1994. corrected reprint of the 1991 original 27. Julien, A., Savinien, J.: Embedding of self-similar ultrametric Cantor sets. Preprint available at http:// arxiv.org/abs/1008.0264v1 [math.GN], 2010 28. Kaminker, J., Putnam, I.: A proof of the gap labeling conjecture. Michigan Math. J. 51(3), 537–546 (2003) 29. Kellendonk, J.: Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7(7), 1133–1180 (1995) 30. Kellendonk, J.: The local structure of tilings and their integer group of coinvariants. Commun. Math. Phys. 187(1), 115–157 (1997) 31. Kellendonk, J.: Gap labelling and the pressure on the boundary. Commun. Math. Phys. 258(3), 751–768 (2005) 32. Kellendonk, J., Richter, T., Schulz-Baldes, H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14(1), 87–119 (2002) 33. Kellendonk, J., Schulz-Baldes, H.: Boundary maps for C ∗ -crossed products with R with an application to the quantum Hall effect. Commun. Math. Phys. 249(3), 611–637 (2004) 34. Kellendonk, J., Schulz-Baldes, H.: Quantization of edge currents for continuous magnetic operators. J. Funct. Anal. 209(2), 388–413 (2004) 35. Kenyon, R.: The construction of self-similar tilings. Geom. Funct. Anal. 6(3), 471–488 (1996) 36. Lagarias, J.C., Pleasants, P.A.B.: Repetitive Delone sets and quasicrystals. Erg. Th. Dyn. Syst. 23(3), 831– 867 (2003) 37. Marchal, P.: Stable processes on the boundary of a regular tree. Ann. Probab. 29(4), 1591–1611 (2001) 38. Michon, G.: Les cantors réguliers. C. R. Acad. Sci. Paris Sér. I Math. 300(19), 673–675 (1985) 39. Pearson, J.C., Bellissard, J.V.: Noncommutative riemannian geometry and diffusion on ultrametric cantor sets. J. Noncommut. Geom. 3(3), 447–481 (2009) 40. Queffélec, M.: Substitution dynamical systems—spectral analysis. Volume 1294 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1987

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Communications in


Optimal Convergence Rates of Classical Solutions for Vlasov-Poisson-Boltzmann System Tong Yang1 , Hongjun Yu2 1 Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong.


2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P. R. China.

E-mail: [email protected] Received: 28 October 2009 / Accepted: 17 April 2010 Published online: 13 October 2010 – © Springer-Verlag 2010

Dedicated to Professor Ling Hsiao on the Occasion of Her 70th Birthday Abstract: The dynamics of charged dilute particles can be modeled by the two species Vlasov-Poisson-Boltzmann system when the particles interact through collisions in the self-induced electric field. By constructing the compensating function for multi-species particle system, the optimal time decay of global classical solutions to this system near a global Maxwellian is obtained through a refined energy method. 1. Introduction Consider the Vlasov-Poisson-Boltzmann system for two species of particles ∂t F+ + v · ∇x F+ + ∇x φ · ∇v F+ = Q(F+ , F+ ) + Q(F+ , F− ), ∂t F− + v · ∇x F− − ∇x φ · ∇v F− = Q(F− , F+ ) + Q(F− , F− ), {F+ − F− }dv, φ =

(1.1)

R3

with initial data F± (0, x, v) = F0,± (x, v). Here F± (t, x, v) represent the number density functions for ions (+) and electrons (−) respectively at time t ≥ 0, with spatial coordinate x = (x1 , x2 , x3 ) ∈ R3 and velocity v = (v1 , v2 , v3 ) ∈ R3 . And the normalized Boltzmann collision operator is given by, cf. [2], |(u − v) · ω|{g1 (v )g2 (u ) − g1 (v)g2 (u)}dudω, Q(g1 , g2 ) = R3 ×S2

where v = v − [(v − u) · ω]ω, u = u + [(v − u) · ω]ω. Even though the global existence of classical solutions to this system is known, cf. [19], the optimal convergence rates to the equilibrium state have not been obtained yet. The purpose of this paper is to obtain the optimal convergence rates for this system when

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the solution is a small perturbation of an equilibrium state, that is, a global Maxwellian. 2 Without loss of generality, we take the global Maxwellian to be μ(v) = e−|v| . Then we define the perturbation f ± (t, x, v) around this Maxwellian by √ F± = μ + μ f ± . Then the normalized vector-valued Vlasov equation for the perturbation f (t, x, v) = [ f + (t, x, v), f − (t, x, v)] takes the form √ [∂t + v · ∇x + q∇x φ · ∇v ] f − 2∇x φ · v μq1 + L f = q∇x φ · v f + ( f, f ), √ φ = μ{ f + − f − }dv,

(1.2) (1.3)

R3

with initial data f (0, x, v) = f 0 (x, v). From now on, [a, b] represents a column vector with elements a and b, q1 = [1, −1], and the diagonal matrix q is given by diag(1, −1). For any given g = [g+ , g− ], the linearized collision operator in (1.2) is defined by Lg = [L + g, L − g], where 1 1 √ √ L ± g = −2 √ Q( μg± , μ) − √ Q(μ, μ{g± + g∓ }). μ μ As usual, cf. [7], L can be decomposed as Lg = ν(v)g − K g, where ν(v) = |(u − v) · ω|μ(u)dudω R3 ×S2

(1.4)

is the collision frequency. For the hard sphere model, there is a constant C > 1 such that ν(v) satisfies 1 (1 + |v|) ≤ ν(v) ≤ C(1 + |v|). C

(1.5)

For g = [g+ , g− ] and h = [h + , h − ], the nonlinear collision operator is defined by (g, h) = [+ (g, h), − (g, h)], with 1 1 √ √ √ √ ± (g, h) = √ Q( μg± , μh ± ) + √ Q( μg± , μh ∓ ). μ μ It is well known that the linearized collision operator L is non-negative and self-adjoint. And for fixed (t, x), the null space of L is given by √ √ √ √ √ √ N = span{[ μ, 0], [0, μ], [vi μ, vi μ], [|v|2 μ, |v|2 μ]} (i = 1, 2, 3). (1.6) Define P as the orthogonal projection in L 2 (R3 ) to the null space N . Any function g(t, x, v) = [g+ (t, x, v), g− (t, x, v)] can be decomposed into

Optimal Convergence Rates of Classical Solutions for Vlasov-Poisson-Boltzmann System

g(t, x, v) = Pg(t, x, v) + (I − P)g(t, x, v),

321

(1.7)

where (t, x) is taken as a parameter. With this decomposition, Pg and (I − P)g correspond to the macroscopic and microscopic components of the function g respectively. Writing P f = [P f + , P f − ] according to the basis in (1.6) yields 3 2 √ bi (t, x)vi + c(t, x)|v| μ. (1.8) P f ± ≡ a± (t, x) + i=1

In this paper, the following notations will be needed. For α = [α0 , α1 , α2 , α3 ] and β = [β1 , β2 , β3 ], denote ∂βα ≡ ∂tα0 ∂xα11 ∂xα22 ∂xα33 ∂vβ11 ∂vβ22 ∂vβ33 . If each component of β is not greater than the corresponding one of β, we use the stanβ¯ dard notation β ≤ β. And β < β means that β ≤ β and |β| < |β|. Cβ is the usual binomial coefficient. To discuss the optimal time decay of solutions to (1.2) and (1.3), the space Z q = L 2 (Rv3 ; L q (Rx3 )) will be used and its norm is defined by f Zq =

R3

21

2 R3

| f (x, v)|q d x

q

dv

.

In the following, ·, · to denote the standard L 2 inner product in Rv3 for a pair of functions g1 (v), g2 (v) ∈ L 2 (Rv3 ; R2 ), and (·, ·) for the one in Rx3 × Rv3 for a pair of functions g1 (x, v), g2 (x, v) ∈ L 2 (Rx3 × Rv3 ; R2 ). | · |2 denotes the L 2 norm in Rv3 , and · denotes the L 2 norms in Rx3 × Rv3 or Rx3 without any ambiguity. In addition, for q ∈ [1, ∞], L q (R3 ) denotes the usual Lebesgue space on R3 with the norm · L q , and H s (R3 ) stands for the standard Sobolev space on R3 with index s. And C denotes a generic positive constant which may vary from line to line. Corresponding to the linearized operator L, the dissipation rate is given in the following norms: |g|ν = |ν 1/2 g|2 ,

gν = ν 1/2 g,

(1.9)

that is, there exists a constant δ > 0 such that Lg, g ≥ δ|(I − P)g|2ν .

(1.10)

To obtain the global existence and the time convergence rate of classical solutions, a key step is to derive an energy estimate so that the a priori estimate can be closed. For this, we will need to introduce the weight w = 1 + |v| for convergence estimates. With this, the following instant energy functional for any l ≥ 0 and a solution f (t, x, v), denoted by El (t) will be used, which is equivalent to E˜l (t) with the standard L 2 norm on the solution and the electric field: wl ∂βα f (t)2 + ∇x φ(t)2 . (1.11) El (t) ∼ E l (t) = |α|+|β|≤N

Note that the initial value of E l (t) at t = 0 is in fact defined only by the initial data f 0 because

322

T. Yang, H. Yu

E l (0) =

|α|+|β|≤N

wl ∂βα f 0 2 + ∇x φ0 2 ,

where by the Poisson equation (1.3), we have ∇x φ0 (x) = ∇x φ(0, x) = ∇x −1 f 0,+ − f 0,− , Correspondingly, the dissipation rate Dl (t) satisfies

l (t) = Dl (t) ∼ D ∂ α P f (t)2 + |α|+|β|≤N

1≤|α|≤N

√

μ .

wl ∂βα (I − P) f (t)2ν .

(1.12)

Throughout this paper, the Sobolev index is taken to be N ≥ 4 for applying the Sobolev imbedding for the estimation on the nonlinear term. With the above preparation, the main result can be stated as follows. √ Theorem 1.1. Let the initial data be F0,± (x, v) = μ+ μ f 0,± (x, v) ≥ 0. For any given index l ≥ 0, if the initial data satisfies El (0) ≤ ε for some small constant ε > 0, then the Vlasov-Poisson-Boltzmann system √ (1.2)–(1.3) has a unique global classical solution f (t, x, v) with F± (t, x, v) = μ + μ f ± (t, x, v) ≥ 0, which satisfies t Dl (s)ds ≤ CEl (0). (1.13) El (t) + 0

If we further assume l ≥ 1, and for some small constant ε1 > 0, f 0 Z 1 + ∇x φ0 L 1 (Rx3 ) ≤ ε1 ,

(1.14)

we have the following time convergence rate estimates: 1≤|α|≤N

f (t)2 = P f (t)2 + (I − P) f (t)2 ≤ C(1 + t)−3/2 , (1.15) α 2 l α 2 2 −5/2 ∂ P f (t) + w ∂β (I − P) f (t) + ∇x φ(t) ≤ C(1 + t) . |α|+|β|≤N

(1.16) Remark 1.2. As in [4,14,20,22,23], to obtain the convergence rate of the solutions to the Boltzmann equation, the weight w is needed to estimate the collision operator and the external force governed by the Poisson equation. The above time convergence rates are optimal in the sense that they are the same as those for the linearized system. We now review some works related to the study in this paper. First of all, the global existence of the renormalized solutions with large initial data to the Vlasov-PoissonBoltzmann system was proved in [15] and this result was later generalized to the case with boundary in [18]. It was shown in [10] that the one species Vlasov-Poisson-Boltzmann system near Maxwellian admits a unique global classical solution in periodic box. Global solutions to this system near a global Maxwellian in the whole space was obtained in [24,25]. On the other hand, the global smooth solutions to two species Vlasov-MaxwellBoltzmann system near a global Maxwellian were constructed in [11] in the spatially periodic case and this result was later generalized to the whole space in [19]. For the time convergence rate estimates, it was shown to be exponential in [11] for the gas in a periodic box. In the whole space, the authors in [25,27] applied the techniques used in [5,16,17,26] to obtain the time decay rate in the · L 2 -norm of the order O(t −1/2 ).


323

And the time asymptotic behavior of the renormalized solutions with extra regularity assumptions was studied in [6] without any convergence rate. Although the two species Vlasov-Poisson-Boltzmann system near a global Maxwellian in periodic box was proved to be exponentially decay in time, cf. [11], the optimal time decay for the case in whole space has remained open. One reason is that for the spatially periodic solution, the Poincaré inequality can be applied together with the celebrated H-theorem on the dissipation rate and such argument does not hold for the whole space. On the other hand, a method was introduced in [4] by combining the spectrum analysis and energy method for the study on the optimal time decay rate for the Boltzmann equation with the external force. However, this method is still difficult to apply to the study of Vlasov-Poisson-Boltzmann system. Thus, despite time decay of solutions to the linearized Vlasov-Poisson-Boltzmann system studied in [8,9], the optimal convergence rate for the nonlinear case is unknown. Recently the authors in [22,23] studied the nonlinear stability and the optimal time decay of the solutions for the relativistic kinetic equations and the kinetic equations with the external force near an equilibrium by combining Kawashima’s compensating function method and the macro-micro decomposition of the solution. The main observation is that the uniform macroscopic energy estimate can be obtained by using Kawashima’s compensating function method [14]. And the advantage is not only to obtain the global existence of the solution, but also to obtain the optimal time decay to the equilibrium by avoiding the study on the complicated spectrum property of the linearized operator as in [1,20]. Motivated by [4,11,22], in this paper, we consider the optimal time decay of the two species Vlasov-Poisson-Boltzmann system near a global Maxwellian in the whole space. As a main ingredient in the analysis, we introduce a way to construct the compensating function introduced by Kawashima for one species particles to the two-species particles. In fact, the method can also be applied to the Boltzmann equation for multi-species particles. It is known that for the Boltzmann equation modeling the one species particles, there exist the well known thirteen moments when one analyzes the convection term. Accordingly, for the two species system, since the space of the collision invariants is spanned by six 2-dimensional vector functions, the corresponding vector moments consist of a subspace with seventeen dimensions. Moreover, in order to take care of the external force governed by the Poisson equation in the Vlasov-Poisson-Boltzmann system, we need to carefully choose the base for this subspace. In particular, the space of the collision invariants is a linear combination of the first nine functions in the base as shown later. Notice that without the external force, the choice of the base can be different so that the first six functions in the base can be chosen as the collision invariants. In fact, we can expect this analytic technique can be used to study some other behavior of gases with mixed particles. The rest of the paper will be organized as follows. In the next section, we recall Kawashima’s compensating function and construct the compensating function for the linearized Vlasov-Poisson-Boltzmann system. Some refined energy estimates will be given in Sect. 3 and the optimal time convergence rate estimates will be obtained in the last section. 2. Compensating Function In this section, we will construct the compensating function for the Vlasov-PoissonBoltzmann system (1.2) and (1.3). Even though the basic idea on the compensating

324

T. Yang, H. Yu

function follows from the work by [7,14] for the one particle Boltzmann equation, there are some new ideas and new phenomena for the multi-species particles through the construction. Moreover, the appearance of the electric field ∇x φ also reveals its effect on the property of the compensating function as one can see below. Let v · ξ (ξ ∈ R3 ) be the symbol of the streaming operator v · ∇x . Note that v · ξ is

as the subspace of L 2 (R3 ) spanned by the a linear operator in L 2 (R3 ) from N into W seventeen functions ϕ j , j = 1, 2, . . . , 17, defined in the following. That is,

= span{ϕ j | j = 1, . . . , 17}, W where for j = 1, 2, 3, √ √ √ √ ϕ1 = [ μ, 0]; ϕ2 = [0, μ]; ϕ j+2 = [v j μ, 0]; ϕ j+5 = [0, v j μ]; √ √ √ √ √ √ ϕ j+8 = [v 2j μ, v 2j μ]; ϕ12 = [v1 v2 μ, v1 v2 μ]; ϕ13 = [v2 v3 μ, v2 v3 μ]; √ √ √ √ ϕ14 = [v3 v1 μ, v3 v1 μ]; ϕ j+14 = [v j |v|2 μ, v j |v|2 μ]. Notice that the null space of the linearized operator L is spanned by ϕ1 , ϕ2 , ϕ j+2 + ϕ j+5 and 3j=1 ϕ j+8 . Thus, we can rewrite the seventeen functions as: √ √ √ √ ψ1 = [ μ, 0]; ψ2 = [0, μ]; ψ j+2 = [v j μ, 0]; ψ j+5 = [0, v j μ]; √ √ √ √ √ √ ψ9 = [|v|2 μ, |v|2 μ]; ψ10 = [v22 μ, v22 μ]; ψ11 = [v32 μ, v32 μ]; √ √ √ √ ψ12 = [v1 v2 μ, v1 v2 μ]; ψ13 = [v2 v3 μ, v2 v3 μ]; √ √ √ √ ψ14 = [v3 v1 μ, v3 v1 μ]; ψ j+14 = [v j |v|2 μ, v j |v|2 μ]. Notice that the basis {ψ j | j = 1, . . . 17} is the same as the basis {ϕ j | j = 1, . . . 17} except for the difference between ϕ9 and ψ9 . By a standard Gram-Schmidt procedure, the above seventeen functions can be made as a set of pairwise orthonormal vectors: √ 2 √ e1 = [0, μ]; e j+2 = 3/4 [v j μ, 0]; 3/4 π π π √ 2 1 2 1 3 3 √ √ √ = 3/4 [0, v j μ]; e9 = √ [(|v|2 − ) μ, (|v|2 − ) μ]; 3/4 π 2 2 2 3π 3 1 √ 1 √ c2 j π −3/4 [ v 2j − μ, v 2j − μ]; e10 = 2 2 √ [ μ, 0]; e2 = 3/4

1

e j+5

1

√

j=1

e11 =

3 j=1

2π −3/4 √ √ −3/4 2 1 √ 2 1 √ c3 j π [ vj − μ, v j − μ]; e12 = √ [v1 v2 μ, v1 v2 μ]; 2 2 2

e13 =

2π −3/4 2π −3/4 √ √ √ √ √ [v2 v3 μ, v2 v3 μ]; e14 = √ [v3 v1 μ, v3 v1 μ]; 2 2 1 2 5 √ 5 √ e15 = √ √ π −3/4 [v1 (|v|2 − ) μ, v1 (|v|2 − ) μ], 2 2 2 5


325

1 2 5 √ 5 √ e16 = √ √ π −3/4 [v2 (|v|2 − ) μ, v2 (|v|2 − ) μ], 2 2 2 5 1 2 5 √ 5 √ e17 = √ √ π −3/4 [v3 (|v|2 − ) μ, v3 (|v|2 − ) μ]. 2 2 2 5 Here, the constant vectors (ci1 , ci2 , ci3 ), i = 2, 3, which satisfy the condition 3 j=1 ci j = 0, will be defined later. It is obvious that the null space of the linearized operator is a six-dimensional space N = span{e1 , e2 , e3 + e6 , e4 + e7 , e5 + e8 , e9 }.

: Let P0 be the orthogonal projection from L 2 (Rv3 ) onto W P0 f =

17 f, ek ek . k=1

Consider the linearized Vlasov-Poisson-Boltzmann system √ [∂t + v · ∇x + L] f = g + 2∇x φ · v μq1 ,

(2.1)

with initial data f (0, x, v) = f 0 (x, v) and a source term g = [g+ , g− ]. Set Wk = f, ek , k = 1, . . . , 17, and W = [W1 , . . . , W17 ]T . Then we have by using (2.1) that ∂t W +

V j ∂x j W + L W = g + R,

j

where V j ( j = 1, 2, 3) and L are the symmetric matrices defined by L = { L[el ], ek }17 k,l=1 , V (ξ ) =

3

V j ξ j = { (v · ξ )ek , el }17 k,l=1 ,

j=1

and g is the vector component g, ek . Here R denotes the remaining term which contains either the factor (I − P0 ) f or ∇x φ(t, x). In the following, we will use Rz to denote the real part of z ∈ C. For ξ ∈ R3 , we know that V (ξ ) = {((v · ξ )ek , el )}17 k,l=1 . Thus V (ξ ) is symmetric and can be written as V (ξ ) =

V11 (ξ )9×9

V12 (ξ )9×8

V21 (ξ )8×9

V22 (ξ )8×8

T = V , V T = V and V = V T . Notice that V11 11 21 22 12 22

.

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T. Yang, H. Yu

More precisely, we have ⎛ 0 0 ⎜ ⎜0 0 ⎜ ⎜ξ ⎜ √1 0 ⎜ 2 ⎜ ⎜ √ξ2 0 ⎜ 2 ⎜ ⎜ξ 0 V11 (ξ ) = ⎜ √32 ⎜ ⎜ ξ √1 ⎜0 2 ⎜ ⎜ ξ2 √ ⎜0 2 ⎜ ⎜ ξ3 √ ⎜0 2 ⎝ 0 0 and

⎛ 0 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎜ ⎜ ⎜0 ⎜ V21 (ξ ) = ⎜ ⎜0 ⎜ ⎜ ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎝ 0

ξ2 √ 2

ξ3 √ 2

0

0

0

0

0

0

0

ξ1 √ 2

ξ2 √ 2

ξ3 √ 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

ξ1 √ 6

ξ2 √ 6

ξ3 √ 6

ξ1 √ 6

ξ2 √ 6

ξ3 √ 6

0

0

c√ 22 ξ2 2 c√ 32 ξ2 2 ξ1 2 ξ3 2

0

ξ3 2

0

ξ2 2 ξ1 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0 0

c√ 21 ξ1 2 c√ 31 ξ1 2 ξ2 2

ξ1 √ 2

c√ 23 ξ3 2 c√ 33 ξ3 2

0

c√ 21 ξ1 2 c√ 31 ξ1 2 ξ2 2

0

c√ 22 ξ2 2 c√ 32 ξ2 2 ξ1 2 ξ3 2

c√ 23 ξ3 2 c√ 33 ξ3 2

ξ3 2

0

ξ2 2 ξ1 2

0

0

⎞

⎟ 0⎟ ⎟ ⎟ ξ1 ⎟ √ 6⎟ ⎟ ξ 2 √ ⎟ 6⎟ ⎟ ξ 3 √ ⎟ ⎟, 6⎟ ⎟ ξ √1 ⎟ 6⎟ ξ2 ⎟ √ ⎟ 6⎟ ξ3 ⎟ √ ⎟ 6⎠ 0

0

⎞

⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟. 0 ⎟ ⎟ ⎟ 1 5 ⎟ 2 6 ξ1 ⎟ ⎟ ⎟ 1 5 ⎟ 2 6 ξ2 ⎟ ⎠ 1 5 2 6 ξ2 0

Set W I = [W1 , W2 , . . . , W9 ]T , W I I = [W10 , W11 , . . . , W17 ]T . We have the following lemma about the property of the matrix V (ξ ). Lemma 2.1. There exist three 17 × 17 real constant entry skew symmetric matrices R j ( j = 1, 2, 3) such that for R(ω) =

3

R jωj,

j=1

we have for all ω ∈ S2 , R R(ω)V (ω)W, W ≥ c1 (|W1 |2 +|W2 |2 +|W9 |2 +|W3 +W6 |2 +|W4 +W7 |2 +|W5 +W8 |2 ) − C1 (|W3 − W6 |2 + |W4 − W7 |2 + |W5 − W8 |2 ) − C2 |W I I |2 , for some positive constants c1 and C1 . Here , is the inner product on C17 .


327

Proof. Let

3

V12 (ξ )9×8 . 08×8

α R 11 (ξ )9×9 R(ξ ) = R ξj = −V21 (ξ )8×9 j=1 j

Here α > 0 is a constant to be specified later and ⎛ 0 0 ξ1 ξ2 ξ3 ⎜ 0 0 0 0 0 ⎜ ⎜−ξ1 0 0 0 0 ⎜ 0 0 0 0 ⎜−ξ2 ⎜ 0 0 0 0 R 11 (ξ ) = ⎜−ξ3 ⎜ 0 −ξ1 0 0 0 ⎜ ⎜ 0 −ξ2 0 0 0 ⎜ ⎝ 0 −ξ3 0 0 0 0 0 0 0 0

0 ξ1 0 0 0 0 0 0 0

0 ξ2 0 0 0 0 0 0 0

0 ξ3 0 0 0 0 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ . 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

Then each R j is 17 × 17 real skew symmetric with constant entries. Set V11 (ξ )9×9 V12 (ξ )9×8 V (ξ ) = , V21 (ξ )8×9 V22 (ξ )8×8 and

α R 11 V11 + V12 V21 U (ξ ) = R(ξ )V (ξ ) = −V21 V11

α R 11 V12 + V12 V22 . −V21 V12

We have

(α R 11 V11 + V12 V21 )W I + (α R 11 V12 + V12 V22 )W I I , R(ω)V (ω)W = −V21 V11 W I − V21 V12 W I I

so that R(ω)V (ω)W, W = (α R 11 V11 + V12 V21 )W I , W I + (α R 11 V12 + V12 V22 )W I I , W I − V21 V11 W I , W I I − V21 V12 W I I , W I I . Consider U11 (ξ ) = (α R 11 V11 + V21 V12 )(ξ ). Notice that

R 11 V11

⎛ |ξ |2 ⎜ ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ 1 ⎜ 0 = √ ⎜ 2⎜ ⎜ 0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 0

0

0

0

0

0

0

0

|ξ |2 0 0 0 0 0 0 0

0 −ξ12 −ξ1 ξ2 −ξ1 ξ3 0 0 0 0

0 −ξ1 ξ2 −ξ22 −ξ2 ξ3 0 0 0 0

0 −ξ1 ξ3 −ξ2 ξ3 −ξ32 0 0 0 0

0 0 0 0 −ξ12 −ξ1 ξ2 −ξ1 ξ3 0

0 0 0 0 −ξ1 ξ2 −ξ22 −ξ2 ξ3 0

0 0 0 0 −ξ1 ξ3 −ξ1 ξ3 −ξ32 0

1

3

|ξ |2

⎞ ⎟

1 2⎟ 3 |ξ | ⎟ ⎟

0 0 0 0 0 0 0

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

328

T. Yang, H. Yu

It follows that √

2R R 11 (ω)V11 (ω)W I , W I = R W1 (W1 +

1 W9 ) + W2 (W2 + 3

1 W9 ) 3

+ W3 (−ω12 W3 − ω1 ω2 W4 − ω1 ω3 W5 ) + W4 (−ω1 ω2 W3 − ω22 W4 − ω2 ω3 W5 ) + W5 (−ω12 W3 − ω2 ω3 W4 − ω32 W5 ) + W6 (−ω12 W6 − ω1 ω2 W7 − ω1 ω3 W8 ) + W7 (−ω1 ω2 W6 − ω22 W7 − ω2 ω3 W 8) + W8 (−ω12 W6 − ω2 ω3 W7 − ω32 W8 ) ≥ c2 (|W1 |2 + |W2 |2 ) − C2

9

|Wk |2 .

k=3

Therefore, R R 11 (ω)V11 (ω)W I , W I

≥ c3 (|W1 |2 + |W2 |2 ) − C4 |W9 |2 + (W3 + W6 )2 + (W5 + W7 )2 + (W5 + W8 )2 −C4 (W3 − W6 )2 + (W4 − W7 )2 + (W5 − W8 )2 .

Next, we write V12 (ω)V21 (ω)W I , W I = |V21 (ω)W I |2 , and direct calculation gives ⎞ ⎛ 1 √ (c21 ω1 (W3 + W6 ) + c22 ω2 (W4 + W7 ) + c23 ω3 (W5 + W8 )) ⎟ ⎜ 12 ⎜ √ (c31 ω1 (W3 + W6 ) + c32 ω2 (W4 + W7 ) + c33 ω3 (W5 + W8 ))⎟ ⎟ ⎜ 2 ω2 ω1 ⎟ ⎜ ⎟ ⎜ 2 (W3 + W6 ) + 2 (W4 + W7 ) ⎟ ⎜ ω3 ω2 (W4 + W7 ) + 2 (W5 + W8 ) ⎟ ⎜ 2 ⎟ ⎜ ω3 ω1 V21 (ω)W I = ⎜ ⎟. (W + W ) + (W + W ) 3 6 8 5 2 2 ⎟ ⎜ ⎟ ⎜ 1 5 ω W ⎟ ⎜ 1 9 26 ⎟ ⎜ ⎟ ⎜ 1 5 ⎟ ⎜ ω2 W9 2 6 ⎠ ⎝ 1 5 ω W 2 6 3 9 Note the set 1 1 c2 = [ √ , − √ , 0], 2 2 2 1 1 [ , , −1]. c3 = 3 2 2 To simplify the calculation, set X 1 = W3 + W6 , X 2 = W4 + W7 , X 3 = W5 + W8 .


329

Then for the sum of the squares of the first entries in V21 (ω)W I , we have 1 1 |c21 ω1 X 1 + c22 ω2 X 2 + c23 ω2 X 3 |2 + |c31 ω1 X 1 + c32 ω2 X 2 + c33 ω3 X 3 |2 2 2 1 1 = |ω1 X 1 − ω2 X 2 |2 + |ω1 X 1 + ω2 X 2 − 2ω3 X 3 |2 4 12 1 2 1 2 2 = R [ω1 |X 1 | + ω2 |X 2 |2 − 2ω1 ω2 X 1 X 2 ] + R [ω12 |X 1 |2 4 12 + ω22 |X 2 |2 + 4ω32 |X 3 |2 + 2ω1 ω2 X 1 X 2 − 4ω1 ω3 X 1 X 3 − 4ω2 ω3 X 2 X 3 ] 1 1 = (ω12 |X 1 |2 + ω22 |X 2 |2 + ω32 |X 3 |2 )− R(ω1 ω2 X 1 X 2 +ω2 ω3 X 2 X 3 +ω3 ω1 X 3 X 1 ). 3 3 2 Hence, |V21 (ω)W I | is equal to 1 R(ω12 |X 1 |2 + ω22 |X 2 |2 + ω32 |X 3 |2 − ω1 ω2 X 1 X 2 − ω2 ω3 X 2 X 3 − ω3 ω1 X 3 X 1 ) 3 1 + R(ω22 |X 1 |2 + ω12 |X 2 |2 + 2ω1 ω2 X 1 X 2 ) 4 1 + R(ω32 |X 2 |2 + ω22 |X 3 |2 + 2ω1 ω2 X 2 X 3 ) 4 1 5 + R(ω32 |X 1 |2 + ω12 |X 3 |2 + 2ω1 ω3 X 1 X 3 ) + |W9 |2 4 24 |X 1 |2 |X 2 |2 |X 3 |2 1 1 1 = + ω12 |X 1 |2 + + ω22 |X 2 |2 + + ω32 |X 3 |2 4 12 4 12 4 12 1 1 1 5 ω1 ω2 X 1 X 2 + ω2 ω3 X 2 X 3 + ω3 ω1 X 3 X 1 + |W9 |2 +R 6 6 6 24 1 1 1 ≥ |X i |2 + |W9 |2 + R[ω12 |X 1 |2 + ω22 |X 2 |2 + ω32 |X 3 |2 6 6 12 3

j=1

+ ω1 ω2 X 1 X 2 + ω2 ω3 X 2 X 3 + ω3 ω1 X 3 X 1 ] 3 1 1 1 |X i |2 + |W9 |2 + |ω1 X 1 + ω2 X 2 + ω3 X 3 |2 = 6 6 12 j=1

1 ≥ (|W3 + W6 |2 + |W4 + W7 |2 + |W5 + W8 |2 + |W9 |2 ). 6 Thus, for any α > 0, we have αR R 11 (ω)V11 (ω)W I , W I + |V21 (ω)W I |2 ≥ α c1 (|W1 |2 + |W2 |2 ) − C4 (|W9 |2 + |W3 + W6 |2 + |W4 + W7 |2 + |W5 + W8 |2 ) 1 + (|W3 + W6 |2 + |W4 + W7 |2 + |W5 + W8 |2 + |W9 |2 ) 6 − αC4 (|W3 − W6 |2 + |W4 − W7 |2 + |W5 − W8 |2 ). Take α =

1 12C4

to get

R U11 (ω)W I , W I ≥ c(|W1 |2 +|W2 |2 +|W3 +W6 |2 +|W4 +W7 |2 +|W5 +W8 |2 +|W9 |2 ) − αC4 (|W3 − W6 |2 + |W4 − W7 |2 + |W5 − W8 |2 ).

330

T. Yang, H. Yu

By noticing that the moduli of the second term and the third terms in R(ω)V (ω)W, W are not larger than c|W I ||W I I | ≤ |W I |2 + c |W I I |2 for any > 0, and the one for the last term is of the order of |W I I |2 , the conclusion in the lemma then follows. We are now ready to construct the compensating function for (2.1). We set R(ω) in Lemma 2.1 as R(ω) = {ri j (ω)}i,17j=1 . For ω ∈ S2 and some constant λ > 0 defined later, set the compensating function S(ω) to be S(ω) f =

17

λrk (ω) f, e ek ,

(2.2)

k,=1

for a function f ∈ L 2 . The following lemma is similar to the corresponding one in [9,14] for the compensating function of one species particles. Note that the dissipation rate |(I − P)h|2ν will be included in the lower bound estimate. Lemma 2.2. The compensating function S(ω) defined in (2.2) enjoys the following properties: (i) S(·) is C ∞ on S2 with values in the space of bounded linear operators on L 2 (R3 ), and S(−ω) = −S(ω) for all ω ∈ S2 . (ii) i S(ω) is self-adjoint on L 2 (R3 ) for all ω ∈ S2 . (iii) There exist constants λ > 0 and c0 > 0 such that for all f ∈ L 2 (R3 ) and ω ∈ S2 , R S(ω)(v · ω) f, f + L f, f ≥ c0 (|P f |22 + |(I − P) f |2ν ). Proof. Parts (i)–(ii) follow immediately from (2.2) and the definition of R(ω) in Lemma 2.1 because R(ω) is skew-symmetric. For part (iii), from (2.2), we have S(ω)(v · ω) f, f ≡

17

λrk (ω) (v · ω) f, e f, ek .

k,=1

Set f = P0 f + (I − P0 ) f . Then for W j = f, e j , we have (v · ω) f, e = (v · ω)P0 f, e + (v · ω)(I − P0 ) f, e =

17 j=1

Wj

3

e j , ω p v p e + (v · ω)(I − P0 ) f, e .

p=1

Notice that 17 j=1

Wj

3

e j , ω p v p e =

p=1

17 j=1

=

17 j=1

Wj

3

ω p (V p ) j

p=1

W j (V (ω)) j =

17 j=1

W j (V (ω))j .


331

Thus, we have R S(ω)(v · ω) f, f = Rλ

17

rk (ω)

17

Vj (ω)W j W k

j=1

k,=1

+ Rλ

17

rk (ω) (I − P0 ) f, (v · ω)e W k

k,=1

= Rλ R(ω)V (ω)W, W + Rλ

17

rk (ω) (I − P0 ) f, (v · ω)e f, ek

k,=1

≥ λ[c1 (|W1 |2 +|W2 |2 +|W9 |2 +|W3 +W6 |2 +|W4 +W7 |2 +|W5 +W8 |2 ) − C1 (|W3 −W6 |2 +|W4 −W7 |2 +|W5 −W8 |2 )−C2 |(I−P) f |2ν ] + Rλ

17

rk (ω) (I − P0 ) f, (v · ω)e f, ek ,

(2.3)

k,=1

where we have used Lemma 2.1 and the exponential decay property of ek in v. Since | (v · ω)(I − P0 ) f, e | ≤ c5 |(I − P0 ) f |2 ≤ c5 |(I − P) f |ν , and | f |22 ≤ |(I − P) f |2ν + |P f |22 , we have R S(ω)(v · ω) f, f ≥ λc1 (|W1 |2 + |W2 |2 + |W9 |2 + |W3 + W6 |2 + |W4 + W7 |2 + |W5 + W8 |2 )−C1 λ(|W3 − W6 |2 +|W4 − W7 |2 +|W5 − W8 |2 ) − C3 λ|(I − P) f |2ν − ε|P f |22 , (2.4) where ε > 0 is a small constant. Since P0 f =

17

W j e j , W j = f, e j ,

j=1

and N = span{e1 , e2 , e3 + e6 , e4 + e7 , e5 + e8 , e9 }, we have P f = W1 e1 + W2 e2 + W9 e9 W3 + W6 W4 + W7 W5 + W8 (e3 + e6 ) + (e4 + e7 ) + (e5 + e8 ). + 2 2 2

(2.5)

332

T. Yang, H. Yu

Clearly, P0 f − P f =

17

Wjej +

j=10

+

W3 − W6 W6 − W3 e3 + e6 2 2

W4 − W7 W7 − W4 W5 − W8 W8 − W5 e4 + e7 + e5 + e8 . (2.6) 2 2 2 2

Since the linearized operator satisfies, L f, f ≥ δ0 |(I − P) f |2ν ,

for some δ0 > 0,

thus we obtain L f, f = L(I − P) f, (I − P) f = L(I − P0 ) f + L(P0 − P) f, (I − P0 ) f + (P0 − P) f = L(I − P0 ) f, (I − P0 ) f + 2 L(I − P0 ) f, (P0 − P) f + L(P0 − P) f, (P0 − P) f ≥ δ|(I − P0 ) f |2ν − C |(I − P0 ) f |2ν − |(P − P0 ) f |2ν + δ|(P − P0 ) f |2ν ≥ −C|(I − P) f |2ν + (δ − )|(P − P0 ) f |2ν , where ε > 0 is a small constant. Note that |(P− P0 ) f |2ν

∼ |(P −

P0 ) f |22

∼

17

|W j |2 +|W3 −W6 |2 +|W4 −W7 |2 +|W5 −W8 |2 .

j=10

Thus, there exists some positive constant C such that δ (1 + C ) L f, f ≥ C|(I − P) f |2ν + |(P − P0 ) f |2ν 2 ≥ C|(I − P) f |2ν + C {|W3 − W6 |2 + |W4 − W7 |2 + |W5 − W8 |2 }. In fact, we can also obtain L f, f ≥ C4 |(I − P) f |2ν + C5 {|W3 − W6 |2 + |W4 − W7 |2 + |W5 − W8 |2 }. (2.7) From (2.4) and (2.5), we have R S(ω)(v · ω) f, f ≥ C1 λ|P f |22 − C2 λ|(I − P) f |2ν − C1 λ(|W3 − W6 |2 + |W4 − W7 |2 + |W5 − W8 |2 ). (2.8) If the constant λ > 0 is chosen to be small enough, a linear combination of (2.7) and (2.8) yields that R S(ω)(v · ω) f, f + L f, f ≥ C3 |P f |22 + C4 |(I − P) f |2ν . And this completes the proof of the lemma.


333

By using the compensating function S(ω), we will derive some energy estimate on the linearized Vlasov-Poisson-Boltzmann system. Set ω = ξ/|ξ | and take the Fourier transform in x of (2.1) to have √ ∂t (2.9) f + i|ξ |(v · ω) f +L f = g + 2 ∇x φ · v μq1 . By taking the inner product of (2.9) with f , we have 1 2 √ f, f = R g, f + 2R iξ φˆ · v μ, ( fˆ+ − fˆ− ) . (2.10) ∂t | f |2 + L 2 √ √ We take the L 2 inner product of (2.9) with [ μ, 0] and [0, μ] respectively to get √ √ √ f + , μ + i|ξ |(v · ω) f + , μ = g+ , μ , ∂t √ √ √ ∂t f − , μ + i|ξ |(v · ω) f − , μ = g− , μ . We easily see that √ √ √ ∂t fˆ+ − fˆ− , μ + i|ξ |(v · ω)( fˆ+ − fˆ− ), μ = gˆ + − gˆ − , μ .

(2.11)

Notice that the Fourier transform of the Poisson equation (1.3) gives √ ˆ = μ( f + − fˆ− )dv. φ R3

This and (2.11) imply that √ √ ∂t | ∇x φ|2 + 2 i|ξ |(v · ω)( fˆ+ − fˆ− ), μ φ = 2 gˆ + − gˆ − , μ φ . By (2.10) and the above equation, we can obtain 1 2 2 √ f | + |∇x φ| ) + L f, f = R g, f + 2R gˆ + − gˆ − , μ φ . ∂t ( | 2 2

(2.12)

Applying −i|ξ |S(ω) to (2.9) gives −i|ξ |S(ω)∂t f ) − i|ξ |S(ω)L f f + |ξ |2 S(ω)((v · ω) √ = −i|ξ |S(ω) g − 2i|ξ |S(ω) ∇x φ · v μq1 .

(2.13)

The inner product of (2.13) with f yields f, f f, f + |ξ |2 R S(ω)(v · ω) R −i|ξ |S(ω)∂t √ = |ξ |R i S(ω)L f, f − i S(ω) g, f −2R i|ξ |S(ω) ∇x φ · v μq1 , f .

(2.14)

Since i S(ω) is self-adjoint, the first term is just − 21 ∂t [|ξ | i S(ω) f, f ]. By multiplying 2 (1 + |ξ | ) to (2.12), and by adding κ times (2.14), we have 1 2 2 κ|ξ | f |2 + |∇x φ| ) − i S(ω) f, f + (1 + |ξ |2 − κ|ξ |2 ) L ∂t (1 + |ξ |2 )( | f, f 2 2 f, f + L f, f } + κ|ξ |2 {R S(ω)(v · ω) √ 2 + κ|ξ |R i S(ω)L = (1 + |ξ | ) R f , g + 2R gˆ + − gˆ − , μ φ f, f − i S(ω) g, f √ − 2κ R i|ξ |S(ω) ∇x φ · v μq1 , f . (2.15)

334

T. Yang, H. Yu

For the second term on the left-hand side of (2.15), when 0 < κ < 1, we have (1 + |ξ |2 − κ|ξ |2 ) L f, f ≥ (1 − κ)(1 + |ξ |2 ) · δ0 |(I − P) f |2ν . And by Lemma 2.2, the third term on the left-hand side of (2.15) is bounded by κ|ξ |2 {R S(ω)(v · ω) f, f + L f, f } ≥ κ|ξ |2 · c0 (|P f |22 + |(I − P) f |ν ). Notice that S(ω)L f =

17

λrk (ω) L f, e ek =

k,=1

17

λrk (ω) L[(I − P) f ], e ek ,

k,=1

and since ν(v) ≤ C(1 + |v|) and K is bounded from L 2 to itself, we have | L[(I − P) f ], e | ≤ C|(I − P) f |ν , where we have used Lg = ν(v)g − K g and the exponential decay of e (v) in v. The second term on the right-hand side of (2.15) is dominated by Cκ|ξ | | i S(ω)L f, f | + | i S(ω) g, f | ⎫ ⎧ 17 ⎬ ⎨ f |2 + | g , e || f , ek | ≤ Cκ|ξ | |(I − P) f |ν | ⎭ ⎩ k,=1

≤ 4Cκ|(I − P) f |2ν +

κ 2 2 |ξ | | f |2 + Cκ | g , e |2 . 2 17

=1

We have from (2.2) and the decomposition (1.7) that 17 √ √ − i|ξ |S(ω) ∇x φ · v μq1 , f = − iλ|ξ |rk, (ω) ∇x φ · v μq1 , e f, ek k,=1

=−

8 9

√ iλ|ξ |rk, (ω) ∇x φ · v μq1 , e P f , ek

k=1 =3

−

8 17

√ iλ|ξ |rk, (ω) ∇x φ · v μq1 , e (I − P) f , ek .

k=1 =3

√ √ By noticing that q1 = [1, −1] and [v μ, −v μ] is orthogonal to all the e with = 1, 2 2 3 2 and = 9, . . . , 17 in L (Rv ; R ), for the first part in the above equation, we can obtain 9 8

√ iλ|ξ |rk, (ω) ∇x φ · v μq1 , e P f , ek

k=1 =3

=

9 5

iλ|ξ |rk, (ω)( ∇x φ)−2 P f , ek

k=1 =3 9 8

−

iλ|ξ |rk, (ω)( ∇x φ)−5 P f , ek ,

k=1 =6

where ( ∇x φ)i denotes the i th component of ∇x φ.

(2.16)


335

Since rk, (ω) = 0, when 2 ≤ k ≤ 9 and 3 ≤ ≤ 5, or k = 1, 3 ≤ k ≤ 9 and 6 ≤ ≤ 8, we have 9 8

√ iλ|ξ |rk, (ω) ∇x φ · v μq1 , e P f , ek = iλ|ξ |ω · ∇x φ(W1 − W2 )

k=1 =3

2, = λ|φ| √ √ where we have used W1 = f + , μ and W2 = f − , μ together with the Poisson equation (1.3) in the last equality. For the second part in the same equation, we have $ $ √ $ $ ∇x φ · v μq1 , e (I − P) f , ek $ $|ξ |rk, (ω) 1 ≤ C|(I − P) f |2ν + |ξ |2 | ∇x φ|2 2 1 2 ≤ C|(I − P) f |2ν + |φ| . 2 By choosing κ > 0 small enough and by combining the above estimates, we know that there exist constants δ1 , δ2 > 0 such that κ 2 1 2 2 f |2ν ∂t (1 + |ξ | )( | f |2 + |∇x φ| ) − |ξ | i S(ω) f , f + δ1 (1 + |ξ |2 )|(I − P) 2 2 √ 2 ≤ (1 + |ξ |2 ) R + δ2 |ξ |2 |P f |22 + δ2 |φ| f , g + 2R gˆ + − gˆ − , μ φ +C

17

| g , e |2 .

(2.17)

=1

This is a main estimate obtained by the compensating function, which will be used later. For the study on the optimal convergence rates, we need the following decay estimates on the solution operator of the linearized Vlasov-Poisson-Boltzmann system: √ ∂t f + v · ∇x f − 2∇x φ · v μq1 + L f = 0, (2.18) √ μ{ f + − f − }dv, (2.19) φ = R3

with f (0, x, v) = f 0 (x, v). Note that the solution to this system can be written as f (t, x, v) = U (t, 0) f 0 (x, v),

(2.20)

with U (t, s) being the solution operator which has the following decay estimates: Lemma 2.6. Let k ≥ k1 ≥ 0, f 0 ∈ H N ∩ Z q and ∇x φ0 (x) ∈ H N ∩ L q . If f (t, x, v) ∈ C 0 ([0, ∞); H N ) ∩ C 1 ([0, ∞); H N −1 ) is a solution of (2.18), we have ∇xk U (t, 0) f 0 ≤ C(1 + t)−σq,m (∇xk1 f 0 Z q + ∇xk1 ∇x φ0 L q + ∇xk f 0 + ∇xk ∇x φ0 ), for any integer m = k − k1 ≥ 0, where q ∈ [1, 2] and 1 m 3 1 − + . σq,m = 2 q 2 2

336

T. Yang, H. Yu

Proof. By setting g = 0 in (2.17), we have 1 2 2 κ ∂t [(1 + |ξ |2 )( | f |2ν f |2 + |∇x φ| ) − |ξ | i S(ω) f, f ] + δ1 (1 + |ξ |2 )|(I − P) 2 2 +δ2 |ξ |2 |P f |22 + δ2 |ξ |2 | ∇x φ|2 ≤ 0. (2.21) Hence, there is a δ > 0 such that ∂t E[ f ] + δ

|ξ |2 E[ f ] ≤ 0, 1 + |ξ |2

(2.22)

where E[ f ] =

1 κ|ξ | | f (t, ξ, v)|22 + | i S(ω) f, f . ∇x φ(t, ξ )|2 − 2 2(1 + |ξ |2 )

Since κ > 0 is small enough and S(ω) is a compensating function as a bounded operator, it is clear that 1 ˆ | f (t, ξ, v)|22 + | ∇x φ(t, ξ )|2 ≤ E[ f ] ≤ | fˆ(t, ξ, v)|22 + | ∇x φ(t, ξ )|2 . 4

(2.23)

From (2.22) and (2.23), we have | fˆ(t, ξ, v)|22 ≤ ce

−δt

|ξ |2 1+|ξ |2

(| fˆ0 (ξ )|22 + | ∇x φ 0 (ξ )|2 ).

Multiplying this by |ξ |2k and integrating over ξ give k 2 ∇x f = |ξ |2k | fˆ(t, ξ, ·)|22 dξ R3

≤c Set

R3

|ξ |2k e

I0 =

|ξ |≤1

|ξ |2k e

−δt

+

|ξ |≥1

|ξ |2k e

−δt

|ξ |2 1+|ξ |2

|ξ |2 1+|ξ |2

−δt

(| fˆ0 (ξ )|22 + | ∇x φ 0 (ξ )|2 )dξ.

(2.24)

(| fˆ0 (ξ )|22 + | ∇x φ 0 (ξ )|2 )dξ

|ξ |2 1+|ξ |2

(| fˆ0 (ξ )|22 + | ∇x φ 0 (ξ )|2 )dξ.

(2.25)

For the first term in I0 , we have from Hölder inequality that $ $ $ $ |ξ |2 |ξ |2 $ $ 2k −δt 1+|ξ |2 ˆ 2 |ξ | e | f 0 (ξ )|2 dξ $ ≤ |ξ α−α |2 e−δt 2 |ξ α fˆ0 (ξ )|22 dξ $ $ |ξ |≤1 $ |ξ |≤1 1/ p 1/q 2 2q 2 p m −δp t |ξ2| α ˆ ≤ |ξ | e dξ |ξ f 0 (ξ )|2 dξ , |ξ |≤1

|ξ |≤1

where |α| = k, |α | = k1 , m = |α − α | and p ∈ [1, ∞) with p1 + |ξ |2 |ξ |2 p m e−δp t 2 dξ ≤ C(1 + t)−3/2− p m . |ξ |≤1

1 q

= 1. We see that


337

By the Hausdorff-Young inequality and the inequality |h| L 2v L qx ≤ |h L qx | L 2v for q ≤ 2, cf. [1], we obtain

|ξ |≤1

2q |ξ α fˆ0 (ξ )|2 dξ

1/q

=

R3

2q |ξ α fˆ0 (ξ )|2 dξ

1/q

1 1 + = 1. q 2q

≤ c∂ α f 0 2Z q , Thus, we have |ξ |≤1

|ξ |2k e

−δt

|ξ |2 1+|ξ |2

(| fˆ0 (ξ )|22 + | ∇x φ 0 (ξ )|2 )dξ

≤ c(1 + t)−3/2 p −m (∂ α f 0 2Z q + ∂ α ∇x φ0 2L q ). For the second term in I0 , we directly have

2

|ξ | 2k −δt 1+|ξ |2

|ξ |≥1

|ξ | e

| fˆ0 (ξ )|22 dξ ≤ ce

−δt 2

∂ α f 0 2 .

Hence, one has

I0 ≤ C(1 + t)−3/2 p −m (∇xk1 f 0 2Z q + ∇xk1 ∇x φ0 2L q ) + ce

−δt 2

(∂ α f 0 2 + ∂ α ∇x φ0 2 )

≤ C(1 + t)−2σq,m (∇xk1 f 0 2Z q + ∇xk1 ∇x φ0 2L q + ∇xk f 0 2 + ∇xk ∇x φ0 2 ). Therefore, Lemma 2.6 follows from the above estimates on I0 . And this completes the proof of the lemma. 3. Energy Estimates In this section, we will establish the basic energy estimates in order to obtain the global existence and the optimal time decay of solution with weight. To this end, we recall the following estimate of collision operator, which is from [11,13]. Lemma 3.1. Let functions f (v), g(v) and h(v) ∈ Cc∞ (R3 ; R2 ). For any l ≥ 0, we have 1 w2l ∂βα L f, ∂βα f ≥ |wl ∂βα f |2ν − C|∂ α f |2ν , 2 $ $ $ 2l $ [|wl ∂β1 f |ν |wl ∂β2 g|2 $ w ∂β ( f, g), ∂β h $ ≤ C

(3.1)

β1 +β2 ≤β

% % % %

R

% % % % % ( f, g)hdv % %+% 3

R3

+ |wl ∂β1 g|ν |wl ∂β2 f |2 ]|wl ∂β h|ν , (3.2) % 1/2 % 3 (g, f )hdv % |g(x, v)|2 dv f . % ≤ C sup |ν h| sup x,v

x

R3

(3.3) For the weighted estimate on the nonlinear collision operator, we have the following lemma.

338

T. Yang, H. Yu

Lemma 3.2. Let l ≥ 0 and |α| + |β| ≤ N . Then there is some constant Cη > 0 for any η > 0 such that $ $ $ $ 2l α

l (t). (3.4) $ w ∂β ( f, f ), ∂βα (I − P) f $ ≤ ηwl ∂βα (I − P) f 2ν + Cη E l (t)D Proof. By using the decomposition (1.7), we have ( f, f ) = (P f, P f )+(P f, (I − P) f )+((I − P) f, P f ) + ((I − P) f, (I − P) f ). By (3.2), we obtain $ $ $ 2l α $ $ w ∂β (P f, P f ), ∂βα (I − P) f $ 1 ≤C [|wl ∂βα11 P f |2 |wl ∂βα−α P f |ν 2 R3

1 +|wl ∂βα11 P f |ν |wl ∂βα−α P f |2 ]|wl ∂βα (I − P) f |ν d x. 2

(3.5)

Here, the summation is over α1 ≤ α and β1 + β2 ≤ β. Notice that there exists a constant C > 0 such that |g|2ν ≥ C|w 1/2 g|22 ,

|wl ∂β P f |2ν ≤ C|P f |22 .

We only consider the first term in (3.5) because the second term can be estimated similarly. If |α1 | + |β1 | ≤ N /2, for any η > 0, we have 1 |wl ∂βα11 P f |2 |wl ∂βα−α P f |ν |wl ∂βα (I − P) f |ν d x 2 3 R 1 wl ∇x ∂ α ∂βα11 P f 2 wl ∂βα−α P f 2ν ≤ ηwl ∂βα (I − P) f 2ν + Cη 2 |α |≤1

≤ ηwl ∂βα (I − P) f 2ν + Cη

∇x ∂ α ∂ α1 P f 2 ∂ α−α1 P f 2 ,

|α |≤1

which is bounded by the right-hand side of (3.4). The case |α1 | + |β1 | ≥ N /2 is similar. By applying Lemma 3.1, we have $ $ $ 2l α $ $ w ∂β (P f, (I − P) f ), ∂βα (I − P) f $ 1 ≤C [|wl ∂βα11 P f |2 |wl ∂βα−α (I − P) f |ν 2 R3 l α1 1 +|w ∂β1 (I − P) f |2 |wl ∂βα−α P f |ν ]|wl ∂βα (I − P) f |ν d x. 2

Again, we only consider the first term because the second term can be estimated similarly. If |α1 | + |β1 | ≤ N /2, by using the Sobolev inequality, we obtain 1 |wl ∂βα11 P f |2 |wl ∂βα−α (I − P) f |ν |wl ∂βα (I − P) f |ν d x 2 3 R 1 wl ∂ α ∂βα11 P f 2 wl ∂βα−α (I − P) f 2ν ≤ ηwl ∂βα (I − P) f 2ν + Cη 2 |α |≤2


|α |≤2

1 ∂ α ∂ α1 P f 2 wl ∂βα−α (I − P) f 2ν . 2


339

If |α − α1 | + |β2 | ≤ N /2, we also have

1 |wl ∂βα11 P f |2 |wl ∂βα−α (I − P) f |ν |wl ∂βα (I − P) f |ν d x 2 1 wl ∂ α ∂βα−α (I − P) f 2ν wl ∂βα11 P f 2 ≤ ηwl ∂βα (I − P) f 2ν + Cη 2

R3

|α |≤2


|α |≤2

1 wl ∂ α ∂βα−α (I − P) f 2ν ∂ α1 P f 2 . 2

Both are bounded by the right-hand side of (3.4). Since the other terms can be estimated similarly, we complete the proof of this lemma. Next we will estimate the electric field ∇x φ in terms of f (t, x, v) through the system (1.2) and (1.3). Lemma 3.3. Let f (t, x, v) be the classical solution to the system (1.2) and (1.3). There exists ε > 0 such that if

∂ α f 2 +

|α|≤N

∂ α ∇x φ ≤ ε.

(3.6)

|α|≤N

Then there is a constant C > 0 such that

∂ α ∇x φ2 ≤ C

|α|≤N

∂ α P f 2 + C

1≤|α |≤N

∂ α (I − P) f 2 .

|α |≤N

α 2 Proof. √ We take the ∂ derivative of (1.2) and take the L inner product of this with [v μ, 0] to get

√ √ ∂t ∂ α P f + , v μ + v · ∇x ∂ α P f + , v μ − 2∂ α ∇x φ √ = − {∂t + v · ∇x + L + }∂ α (I − P) f + , v μ √ + ∂ α (∇x φ · v f + − ∇x φ · ∇v f + + + ( f, f )), v μ .

(3.7)

We first estimate the right-hand side of (3.7). Let |α| ≤ N −1. Since ν(v) ≤ C(1+|v|) and K is bounded from L 2 to itself, we have $ $ $ {∂t + v · ∇x + L + }∂ α (I − P) f + , v √μ $2 ≤ C{|∂t ∂ α (I − P) f |22 + |∇x ∂ α (I − P) f |22 + |∂ α (I − P) f |22 }, √ where we have used the exponential decay of v μ.

(3.8)

340

T. Yang, H. Yu

We now come to the second part of (3.7). For this, we have $ $ α $ ∂ (∇x φ · v f + − ∇x φ · ∇v f + ), v √μ $2 d x R3

$ $2 $ $ √ √ $ $ α1 α1 α−α1 α1 α−α1 Cα ( ∂ ∇x φ · v∂ f + , v μ + ∂ ∇x φ∂ f + , ∇v (v μ) )$ d x = $ $ R3 $α1 ≤α ≤C Cαα1 |∂ α1 ∇x φ|2 |∂ α−α1 f + |22 d x.

α1 ≤α

R3

If |α − α1 | ≤ N /2 and N ≥ 4, the Sobolev inequality implies that |∂ α1 ∇x φ|2 |∂ α−α1 f + |22 d x ≤ C ∇x ∂ α f + 2 ∂ α1 ∇x φ2 . R3

|α |≤N −1

Similarly, if |α1 | ≤ N /2, we have |∂ α1 ∇x φ|2 |∂ α−α1 f + |22 d x ≤ C R3

|α |≤N −1

For the collision term, we have $ $ α $ ∂ + ( f, f ), v √μ $2 d x ≤ Cαα1 R3

∇x ∂ α ∇x φ2 ∂ α−α1 f + 2 .

R3

α1 ≤α

$ $ $+ (∂ α1 f, ∂ α−α1 f ), v √μ $2 d x.

Without loss of generality, we assume |α1 | ≤ N /2. Then from (3.3), we have $ $ α $ ∂ + ( f, f ), v √μ $2 d x ≤ C sup |∂ α1 f |2 dv ∂ α−α1 f 2 x R3 R3 ≤C ∇x ∂ α1 ∂ α f 2 ∂ α−α1 f 2 . |α |≤1

Therefore, by using (3.2), we have the estimate on the collision term: $ $ α $ ∂ + ( f, f ), v √μ $2 d x ≤ Cε ∂ α P f 2 + Cε ∂ α (I − P) f 2 . R3

1≤|α |≤N

|α |≤N

For |α| ≤ N − 1, it is obvious that $ $ $ ∂t ∂ α P f + , v √μ + v · ∇x ∂ α P f + , v √μ $ ≤ C

∂ α P f .

1≤|α |≤N

Therefore, (3.7) implies the following estimate: ∂ α ∇x φ2 ≤ C ∂ α P f 2 + C ∂ α (I − P) f 2 . |α|≤N −1

1≤|α |≤N

(3.9)

|α |≤N

2 Next, √ we shall estimate √ the case when |α| = N . We take the L inner product of (1.2) with [ μ, 0] and [0, μ] respectively to get √ √ ∂t f + , μ + v · ∇x f + , μ = 0, √ √ ∂t f − , μ + v · ∇x f − , μ = 0.


341

By the Poisson equation (1.3) and the above two equations, we have √ √ ∂t φ = ∂t μ( f + − f − )dv = −∇x · v μ( f + − f − )dv. R3

R3

It follows that ∂t ∂ α φ = −∇x ·

R3

√ v μ(∂ α f + − ∂ α f − )dv.

If |α| = N − 1 and ∂ α = ∂tα , we can multiply the above equation by ∂t ∂ α φ to have ∂ α ∇x φ ≤ C ∂ α f . |α|=N

|α|=N −1

On the other hand, when ∂ α = ∂xα , it is clear that ∂ α ∇x φ2 ≤ C ∂ α ∇x φ2 ≤ C |α|=N

|α|=N −2

Then by (3.9), we obtain ∂ α ∇x φ2 ≤ C

∂ α f 2 .

|α|=N −1

∂ α P f 2 + C

1≤|α |≤N

|α|≤N

∂ α (I − P) f 2 ,

|α |≤N

which is the desired estimate. This completes the proof of the lemma. The following lemma is about the estimate on the macroscopic part which is based on (2.17) obtained by the compensating function. Lemma 3.4. For the system (1.2) and (1.3), we have the following two estimates: ⎤ ⎡ d ⎣ 1 α 2 ( ∂ f + ∂ α ∇x φ2 ) − κ (1 + |ξ |2 ) N −2 |ξ |3 i S(ω) f, f dξ ⎦ 3 dt 2 R 1≤|α|≤N + δ1 (I − P)∂ α f 2ν + δ2 P∂ α f 2 1≤|α|≤N

1≤|α|≤N

l (t), ≤ C∇x P f + C E l (t)D 2

and

(3.10)

⎤ ⎡ d ⎣ 1 α 2 α 2 2 N −1 ( ∂ f + ∂ ∇x φ ) − κ (1 + |ξ | ) |ξ | i S(ω) fˆ, fˆ dξ ⎦ 3 dt 2 R |α|≤N

l (t), + δ1 (I − P)∂ α f 2ν + δ2 P∂ α f 2 ≤ C E l (t)D (3.11) |α|≤N

1≤|α|≤N

where κ > 0 is a small constant. Proof. We prove the above estimate by considering two cases.

342

T. Yang, H. Yu

Case 1. The case when α = [0, α1 , α2 , α3 ]. By multiplying (2.17) by (1 + |ξ |2 ) N −2 |ξ |2 and integrating it over ξ , we have ⎤ ⎡ d ⎣ 1 α 2 κ ( ∂ f + ∂ α ∇x φ2 ) − (1 + |ξ |2 ) N −2 |ξ |3 i S(ω) f, f dξ ⎦ dt 2 2 R3 1≤|α|≤N + δ1 (I − P)∂ α f 2ν + δ2 P∂ α f 2 1≤|α|≤N

+ δ2 ≤

R3

R3

2≤|α|≤N

2 dξ (1 + |ξ |2 ) N −2 |ξ |2 |φ|

(1 + |ξ |2 ) N −1 |ξ |2 R f , g dξ + C

+2 R3

(1 + |ξ |2 ) N −1 |ξ |2 R gˆ + − gˆ − ,

17 3 =1 R

√

(1 + |ξ |2 ) N −2 |ξ |2 | g , e |2 dξ

dξ. μ φ

The Poisson equation implies that $ $ $ $ $ (1 + |ξ |2 ) N −1 |ξ |2 |φ| 2 dξ $ = $ $ R3

(3.12)

∂ α φ2 .

1≤|α|≤N −1

Now for the source term g, we know that it is equal to q∇x φ · v f + ( f, f ) −q∇x φ · ∇v f from the Vlasov-Poisson-Boltzmann system. Firstly, consider the last √ term on the√right-hand side of (3.12). By recalling that ( f, f ) is orthogonal to [ μ, 0] and [0, μ] and √ √ √ ∇x φ · v f − ∇x φ · ∇v f, μ = ∇x φ · v f, μ + ∇x φ f, ∇v ( μ) = 0, we have

R3

(1 + |ξ |2 ) N −2 |ξ |2 R gˆ + − gˆ − ,

√

dξ = 0. μ φ

For the second term on the right-hand side of (3.12), we get 2 N −2 2 2 (1 + |ξ | ) |ξ | | g , e | dξ ≤ C | ∂ α g, e |2 d x. R3

3 1≤|α|≤N −1 R

Plug in the expression of g into the last term of the above inequality. For the term related to the first term in g, we obtain |q ∂ α (∇x φ · v f ), e |2 d x = Cαα1 |q ∇x ∂ α1 φ · v∂ α−α1 f ), e |2 d x R3

α1 ≤α

≤C

R3

3 α1 ≤α R

|∇x ∂ α1 φ|2 |∂ α−α1 f |22 d x.

Now if |α1 | ≤ (|α| − 1)/2, (3.13) is bounded by ∇x ∂ α +α1 φ2 ∂ α−α1 f 2 ≤ C |α |≤2

1≤|α|≤N −1

∂ α ∇x φ2

1≤|α|≤N

∂ α f 2 .

(3.13)


343

On the other hand, if |α − α1 | ≤ (|α| − 1)/2, (3.13) is bounded by ∇x ∂ α ∂ α−α1 f 2 ∇x ∂ α1 φ2 ≤ C ∂ α ∇x φ2 ∂ α f 2 . |α |≤1

1≤|α|≤N −1

1≤|α|≤N

l (t), we have Thus, by using Lemma 3.3 and the definition of E l (t) and D

l (t). |q ∂ α (∇x φ · v f ), e |2 d x ≤ C E l (t)D R3

A similar argument works for the term related to the last term of g which gives the following estimate:

l (t). |q ∂ α (∇x φ · ∇v f ), e |2 d x ≤ C E l (t)D R3

Now for the estimation on the second term in (3.12), it remains to consider the term related to the nonlinear operator in g. For this, by using Lemma 3.1, we have | ∂ α ( f, f ), e |2 d x ≤ C [|∂ α1 f |22 |∂ α−α1 f |2ν + |∂ α1 f |2ν |∂ α−α1 f |22 ]d x R3

3 α1 ≤α R

≤C

|α |≤1

+C

∇x ∂ α1 +α f 22 ∂ α−α1 f 2ν

|α |≤1

∇x ∂ α1 +α f 2ν ∂ α−α1 f 22 ,

where we have assumed |α1 | ≤ (|α| − 1)/2 with |α| ≥ 1. It is easy to show that

l (t). | ∂ α ( f, f ), e |2 d x ≤ C E l (t)D R3

For the first term on the right-hand side of (3.12), we firstly have $ $ $ $ $ (1 + |ξ |2 ) N −1 |ξ |2 R $ f , g

dξ $ 3 $ R ≤ Cαα1 | ∂ α1 ∇x φ · v∂ α−α1 f, ∂ α f |d x 1≤|α|≤N α1 ≤α

R3

α1

+ R3

| ∂ ∇x φ · ∇v ∂

α−α1

α

f, ∂ f |d x +

R3

| (∂

α1

f, ∂

α−α1

α

f ), ∂ f |d x . (3.14)

For the first integral in (3.14), we have | (∂ α1 ∇x φ · v∂ α−α1 f, ∂ α f |d x ≤ ε∂ α f 2ν + C∂ α1 ∇x φ · |∂ α−α1 f |ν 2 R3 ≤ ε∂ α f 2ν + C ∂ α ∇x φ2 ∂ α f 2ν 1≤|α|≤N

≤ ε∂

α

f 2ν

+C

|α|≤N

1≤|α|≤N

α

∂ P f

2

1≤|α|≤N

where we have used the Poisson equation (1.3).

∂ α f 2ν ,

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T. Yang, H. Yu

The second integral of (3.14) vanishes when |α1 | = 0. When |α1 | = 0, we have | ∂ α1 ∇x φ · ∇v ∂ α−α1 f, ∂ α f |d x R3

≤ ε∂ α f 2ν + C∂ α1 ∇x φ · |∇v ∂ α−α1 f |2 2 ≤ ε∂ α f 2ν + C ∂ α P f 2 ∇v ∂ α f 2ν . |α|≤N

1≤|α|≤N −1

Since ( f, g), h = ( f, g), (I − P)h , we obtain α1 α−α1 α | (∂ f, ∂ f ), ∂ f |d x = R3

R3

| (∂ α1 f, ∂ α−α1 f ), (I − P)∂ α f |d x

l (t). ≤ ε(I − P)∂ α f 2ν + C E l (t)D

Note that ∂ α f 2ν ≤ (I − P)∂ α f 2ν + P∂ α f 2 . By taking ε > 0 small enough and by summarizing the above estimates on (3.12), we have ⎤ ⎡ d ⎣ 1 κ ( ∂ α f 2 + ∂ α ∇x φ2 ) − (1 + |ξ |2 ) N −2 |ξ |3 i S(ω) f, f dξ ⎦ dt 2 2 R3 1≤|α|≤N + δ1 (I − P)∂ α f 2ν + δ2 P∂ α f 2 + δ2 ∂ α φ2 1≤|α|≤N

1≤|α|≤N −1

2≤|α|≤N

l (t). ≤ C∇x P f + C E l (t)D 2

(3.15)

Case 2. The case α = [α0 , α1 , α2 , α3 ] with α0 = 0 and 1 ≤ |α| ≤ N . Taking ∂ α on (1.2) yields √ {∂t + v · ∇x + q∇x φ · ∇v }∂ α f − 2∂ α ∇x φ · v μq1 + L∂ α f =− Cαα1 ∂ α1 ∇x φ · ∇v ∂ α−α1 f + Cαα1 {q∂ α1 ∇x φ · v∂ α−α1 f α1 =0 α1

+(∂

α1 ≤α

f, ∂

α−α1

f )}.

By the Poisson equation (1.3), we obtain d √ −2 ∂ α ∇x φ · v μq1 , ∂ α f = ∂ α ∇x φ2 . dt Since L∂ α f, ∂ α f ≥ δ0 (I − P)∂ α f 2ν ,

(3.16)


345

by taking the inner product of ∂ α f with Eq. (3.16) and by summing over 1 ≤ |α| ≤ N , we can show that there exists a constant C > 0 such that ⎤ ⎡ d ⎣ 1 ( ∂ α f 2 + ∂ α ∇x φ2 )⎦ + δ1 (I − P)∂ α f 2ν dt 2 1≤|α|≤N 1≤|α|≤N

l (t). ≤ Cε ∂ α P f 2 + C E l (t)D (3.17) 1≤|α|≤N

In order to estimate ∂ α P f 2 , we rewrite (3.16) into √ ∂t ∂ α P f + ∂t (I − P)∂ α f + v · ∇x ∂ α f + q∇x φ · ∇v ∂ α f − 2∂ α ∇x φ · v μq1 + L∂ α f =− Cαα1 ∂ α1 ∇x φ · ∇v ∂ α−α1 f + Cαα1 {q∂ α1 ∇x φ · v∂ α−α1 f α1 =0 α1

+ (∂

α1 ≤α

f, ∂

α−α1

f )}.

(3.18)

Note that √ ∂t ∂ α P f, ∂t (I − P)∂ α f − 2∂ α ∇x φ · v μq1 + L∂ α f + (∂ α1 f, ∂ α−α1 f ) = 0. Direct calculation implies that for |α| ≤ N − 1,

l (t). ∂t ∂ α P f 2 ≤ C∇x ∂ α P f 2 + C∇x ∂ α (I − P) f 2 + C E l (t)D

(3.19)

Now by combining Cases 1 and 2, that is, by a suitable linear combination of (3.15), (3.17) and (3.19), we have the desired estimate (3.10). Finally, the estimate (3.11) can be proved similarly. In fact, by multiplying (2.11) by (1 + |ξ |2 ) N −1 and integrating it over ξ , we have ⎤ ⎡ κ d ⎣ 1 α 2 ( ∂ f + ∂ α ∇x φ2 ) − (1 + |ξ |2 ) N −1 |ξ | i S(ω) f, f dξ ⎦ dt 2 2 R3 |α|≤N 2 dξ + δ1 (I − P)∂ α f 2ν + δ2 P∂ α f 2 + δ2 (1 + |ξ |2 ) N −1 |φ| |α|≤N

≤

R3

(1 + |ξ |2 ) N R h, g dξ + Cε

17 =1

+2 R3

R3

1≤|α|≤N

(1 + |ξ |2 ) N R gˆ + − gˆ − ,

√

R3

(1 + |ξ |2 ) N −1 | g , e |2 dξ

dξ. μ φ

(3.20)

A similar argument as the one for (3.12) and Lemma 3.3 lead to the proof when ∂ α contains only spatial differentiation. And then the case with time differentiation can be obtained as the above Case 2. This completes the proof of the lemma. For the time decay rate estimate, we also need the following weighted energy estimate.

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T. Yang, H. Yu

Lemma 3.5. For any l ≥ 0, we have ⎤ ⎡ d ⎣ wl ∂ α f 2 + wl (I − P) f 2 ⎦ + wl (I − P)∂ α f 2ν dt 1≤|α|≤N |α|≤N α 2 α

l (t). ≤C ∂ P f + C (I − P)∂ f 2ν + C E l (t)D |α|≤N

1≤|α|≤N

Proof. We apply ∂ α on Eq. (1.2) with 1 ≤ |α| ≤ N to have

√ {∂t + v · ∇x }∂ α f + q∇x φ · ∇v ∂ α f − 2∂ α ∇x φ · v μq1 + L∂ α f =− Cαα1 ∂ α1 ∇x φ · ∇v ∂ α−α1 f α1 =0

+

α1 ≤α

Cαα1 {q∂ α1 ∇x φ · v∂ α−α1 f + (∂ α1 f, ∂ α−α1 f )}.

(3.21)

In the L 2 inner product of (3.21) with w 2l ∂ α f , denoted by I, the first term on the d left-hand side is equal to 21 dt wl ∂ α f 2 . The other terms can be estimated as follows. Lemma 3.1 implies that w 2l L∂ α f, ∂ α f ≥ 21 wl ∂ α f 2ν − C∂ α f 2ν . By using (3.6) and integration by parts, we have | q∇x φ · ∇v ∂ α f, w 2l ∂ α f | ≤ Cεwl ∂ α f 2ν . By Lemma 3.3, the third term on the left-hand side of I is bounded by √ | 2∂ α ∇x φ · v μq1 , w 2l ∂ α f | ≤ εwl ∂ α f 2ν + Cε ∂ α ∇x φ2 ≤ εwl ∂ α f 2ν + Cε ∂ α P f 2 + Cε ∂ α (I − P) f 2 . 1≤|α |≤N

|α |≤N

For the first term on the right-hand side of I, we apply the Sobolev inequality to get | ∂ α1 ∇x φ · ∇v ∂ α−α1 f, w 2l ∂ α f | ≤ εwl ∂ α f 2ν + Cε ∂ α1 ∇x φ · |wl ∇v ∂ α−α1 f |2 2

l (t). ≤ εwl ∂ α f 2ν + Cε E l (t)D Similarly, we have

l (t). | ∂ α1 ∇x φ · v∂ α−α1 f, w 2l ∂ α f | ≤ εwl ∂ α f 2ν + Cε E l (t)D For the last term on the right-hand side of I, one has from Lemma 3.2 that

l (t). | (∂ α1 f, ∂ α−α1 f ), w 2l ∂ α f | ≤ εwl ∂ α f 2ν + Cε E l (t)D Note that ∂ α f 2ν ≤ (I − P)∂ α f 2ν + P∂ α f 2 . By combining the above estimates and taking ε > 0 small enough, we have from I that d wl ∂ α f 2 + wl (I − P)∂ α f 2ν dt 1≤|α|≤N 1≤|α|≤N α 2

l (t). (3.22) ≤C ∂ P f + C (I − P)∂ α f 2ν + C E l (t)D 1≤|α|≤N

|α|≤N


347

We now rewrite (1.2) as √ {∂t + v · ∇x }(I − P) f + q∇x φ · ∇v f − 2∇x φ · v μq1 + L f = −{∂t + v · ∇x }P f + q∇x φ · v f + ( f, f ).

(3.23)

If we take the inner product of (3.23) with w 2l (I − P) f , the standard energy estimate leads to d wl (I − P) f 2 + wl (I − P) f 2ν dt

l (t). ≤C ∂ α P f 2 + C (I − P)∂ α f 2ν + C E l (t)D

(3.24)

|α|≤N

1≤|α|≤N

Finally, a suitable linear combination of (3.22) and (3.24) gives the desired estimate and this completes the proof of the lemma. We now turn to consider the differentiation of the solution in the velocity variable with weight. For this, we have the following lemma. Lemma 3.6. For any l ≥ 0, we have d l α 2 l α 2 w ∂β (I − P) f + w ∂β (I − P) f ν dt 1≤|β|,|α|+|β|≤N

l (t). (3.25) ∂ α P f 2 + C wl (I − P)∂ α f 2ν + C E l (t)D ≤C |α|≤N

1≤|α|≤N

Proof. By applying ∂βα on (3.23) with |α| + |β| ≤ N and |β| ≥ 1, we obtain √ {∂t + v · ∇x }∂βα (I − P) f − 2∂ α ∇x φ · ∂β (v μ)q1 + ∂βα L f β α + Cβ 1 ∂β1 v · ∇x ∂β−β (I − P) f 1 |β1 |=1

+

|α1 |=0

Cαα1 q∂ α1 ∇x φ · ∇v ∂βα−α1 f + q∇x φ · ∇v ∂βα (I − P) f

=

β

α1 ≤α,β1 ≤β

−

|β1 |=1

α−α1 Cαα1 Cβ 1 q∂ α1 ∇x φ · ∂β1 v∂β−β f + ∂βα ( f, f ) 1

β

α Cβ 1 ∂β1 v · ∇x ∂β−β P f − {∂t + v · ∇x + q∇x φ · ∇v }∂βα P f. (3.26) 1

We take the inner product of (3.26) with w 2l ∂βα (I − P) f and denote it by J . The first d term on the left-hand side of J is 21 dt wl ∂βα (I − P) f 2 . By using Hölder inequality, the second term on the left-hand side of J is bounded by √ ∂ α ∇x φ2 . | ∂ α ∇x φ · ∂β (v μ)q1 , w 2l ∂βα (I − P) f | ≤ εwl ∂βα (I − P) f 2 + Cε |α|≤N

By Lemma 3.1, the third term on the left-hand side of J satisfies ∂βα L f, w 2l ∂βα (I − P) f ≥

1 l α w ∂β (I − P) f 2ν − C∂ α (I − P) f 2ν . 2

348

T. Yang, H. Yu

Similarly, the fourth term on the left-hand side of J is bounded by α wl ∇x ∂β−β (I − P) f 2 . εwl ∂βα (I − P) f 2 + Cε 1 |β1 |=1

For the fifth term on the left-hand side of J , we have | q∂ α1 ∇x φ · ∇v ∂βα−α1 f, w 2l ∂βα (I − P) f | ≤ εwl ∂βα (I − P) f 2 + Cε ∂ α1 ∇x φ · |wl ∇v ∂βα−α1 f |2 2

l (t). ≤ εwl ∂βα (I − P) f 2 + Cε E l (t)D Similarly, the first term on the right-hand side of J has the following estimate: α−α1 | q∂ α1 ∇x φ · ∂β1 v∂β−β f, w 2l ∂βα (I − P) f | 1

l (t). ≤ εwl ∂βα (I − P) f 2 + Cε E l (t)D For the term in J involving the collision operator, we can apply Lemma 3.2 to obtain

l (t). | ∂βα ( f, f ), w 2l ∂βα (I − P) f | ≤ εwl ∂βα (I − P) f 2 + Cε E l (t)D By using the expression of P f , we know that the last two terms in J are bounded by

l (t). ∂ α P f 2 + εwl ∂βα (I − P) f 2 + Cε E l (t)D C 1≤|α|≤N

By combining the above inequalities and taking ε > 0 small enough, we have from Lemma 3.2 that d wl ∂βα (I − P) f 2 + wl ∂βα (I − P) f 2ν ≤ C ∂ α P f 2 dt 1≤|α|≤N α 2 l α

l (t). (3.27) + C∂ (I − P) f ν + C w ∇x ∂β−β1 (I − P) f 2 + Cε E l (t)D |β1 |=1

For any 0 < |β| ≤ N and any l ≥ 0, a suitable summation of (3.27) over |α| + |β| ≤ N gives (3.25). And this completes the proof of the lemma. 4. Optimal Time Decay Rate In this section, we shall derive a refined energy estimate for proving the global existence and the optimal convergence rates. For the periodic initial data, the local existence result without weight function was given in [11]. By a simple modification of their argument, we have the following local existence result with weight function for the system (1.2)–(1.3) in the whole space. We omit the proof for brevity. Lemma 4.1. For any l ≥ 0, if El (0) ≤ ε, there exists T ∗ = T ∗ (ε) > 0 such that there is a unique solution f (t, x, v) to the system (1.2)–(1.3) in [0, T ∗ ) × R3 × R3 satisfying t wl ∂ α f (s)2ν ds ≤ CEl (0). El (t) + |α|≤N

0

√ √ Moreover, if F0,± (x, v) = μ+ μ f 0,± (x, v) ≥ 0, F± (t, x, v) = μ+ μ f ± (t, x, v) ≥ 0.


349

We are now ready to prove Theorem 1.1 and we divide the proof into two parts, namely, the one for existence and the one optimal decay for clear presentation. Proof of global existence. By Lemma 3.4, we have ⎤ ⎡ d ⎣ 1 ( ∂ α f 2 + ∂ α ∇x φ2 ) − κ (1 + |ξ |2 ) N −1 |ξ | i S(ω) fˆ, fˆ dξ ⎦ 3 dt 2 R |α|≤N

l (t), + δ1 (I − P)∂ α f 2ν + δ2 P∂ α f 2 ≤ C E l (t)D (4.1) |α|≤N

1≤|α|≤N

where κ > 0 is a small constant. From Lemma 3.5, we have ⎤ ⎡ d ⎣ wl ∂ α f 2 + wl (I − P) f 2 ⎦ + wl (I − P)∂ α f 2ν dt 1≤|α|≤N |α|≤N α 2 α

l (t), ≤C ∂ P f + (I − P)∂ f 2ν + C E l (t)D

(4.2)

|α|≤N

1≤|α|≤N

and by Lemma 3.6, we have d wl ∂βα (I − P) f 2 + wl ∂βα (I − P) f 2ν dt 1≤|β|,|α|+|β|≤N

l (t). ∂ α P f 2 + C wl (I − P)∂ α f 2ν + C E l (t)D ≤C

(4.3)

|α|≤N

1≤|α|≤N

A suitable linear combination of (4.1), (4.2) and (4.3) yields ⎧ d ⎨ 1 α 2 α 2 ( ∂ f + ∂ ∇x φ ) − κ (1 + |ξ |2 ) N −1 |ξ | i S(ω) fˆ, fˆ dξ dt ⎩ 2 R3 |α|≤N

+

wl ∂ α f 2 + wl (I − P) f 2 +

1≤|α|≤N

+

⎧ ⎨ ⎩

+

|α|≤N

wl ∂βα (I − P) f 2

1≤|β|,|α|+|β|≤N

∂ α P f 2 +

|α|≤N

1≤|α|≤N

∂ α (I − P) f 2ν +

⎭

wl ∂ α (I − P) f 2ν

wl ∂βα (I − P) f 2ν

1≤|β|,|α|+|β|≤N

⎫ ⎬ ⎭

l (t). ≤ C E l (t)D

(4.4)

On the other hand, the boundedness of the operator S(ω) implies $ $ $ $ α 2 $ (1 + |ξ |2 ) N −1 |ξ | i S(ω) fˆ, fˆ dξ $ ≤ C ∂ f + C $ $ R3

⎫ ⎬

|α|≤N

|α|≤N −1

∂ α f 2 .

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T. Yang, H. Yu

Therefore, we can define two functionals which are equivalent to those defined in (1.11) and (1.12) respectively as α 2 ∂ f − κ (1 + |ξ |2 ) N −1 |ξ | i S(ω) fˆ, fˆ dξ El (t) = R3

|α|≤N

+

1≤|α|≤N

Dl (t) =

wl ∂ α f 2 + wl (I − P) f 2 + α

∂ P f + 2

|α|≤N

1≤|α|≤N

wl ∂βα (I − P) f 2 ,

1≤|β|,|α|+|β|≤N

∂

α

(I − P) f 2ν

+

|α|+|β|≤N

wl ∂βα (I − P) f 2ν .

With these notations, we have d

l (t) ≤ CEl (t)Dl (t). El (t) + Dl (t) ≤ C E l (t)D dt

(4.5)

If we assume El (0) ≤ ε for a sufficiently small constant ε > 0, based on the local existence stated in Lemma 4.1, the global existence follows from the standard continuity argument. The following proof on the optimal decay estimate is motivated by [4] on the Boltzmann equation by using the approach of combination of spectrum analysis and energy method. Proof of optimal convergence rates. By Lemma 3.4, we have ⎤ ⎡ d ⎣ 1 ( ∂ α f 2 + ∂ α ∇x φ2 ) − κ (1 + |ξ |2 ) N −2 |ξ |3 i S(ω) f, f dξ ⎦ 3 dt 2 R 1≤|α|≤N + δ1 (I − P)∂ α f 2ν + δ2 P∂ α f 2 1≤|α|≤N

1≤|α|≤N

l (t), ≤ C∇x P f + C E l (t)D 2

(4.6)

where κ > 0 is a small constant. From Lemma 3.5, we have ⎤ ⎡ d ⎣ wl ∂ α f 2 + wl (I − P) f 2 ⎦ + wl (I − P)∂ α f 2ν dt 1≤|α|≤N |α|≤N α 2 α

l (t). ≤C ∂ P f + (I − P)∂ f 2ν + C E l (t)D 1≤|α|≤N

Lemma 3.6 implies that d wl ∂βα (I − P) f 2 + wl ∂βα (I − P) f 2ν dt 1≤|β|,|α|+|β|≤N

l (t). ∂ α P f 2 + C wl (I − P)∂ α f 2ν + C E l (t)D ≤C 1≤|α|≤N

(4.7)

|α|≤N

|α|≤N

(4.8)


351

In addition, the standard energy estimate gives d (I − P) f 2 + (I − P) f 2ν dt

l (t). ≤ C∇x P f 2 + C∇x (I − P) f 2ν + C E l (t)D

(4.9)

A suitable linear combination of (4.6), (4.7), (4.8) and (4.9) yields ⎧ d ⎨ 1 α 2 ( ∂ f + ∂ α ∇x φ2 ) − κ (1 + |ξ |2 ) N −2 |ξ |3 i S(ω) f, f dξ 3 dt ⎩ 2 R 1≤|α|≤N

+

wl ∂ α f 2 + wl (I−P) f 2 +

1≤|α|≤N

+

∂ α P f 2 +

|α|≤N

1≤|α|≤N

+

wl ∂βα (I−P) f 2 +(I−P) f 2

1≤|β|,|α|+|β|≤N

⎧ ⎨ ⎩

wl ∂ α (I − P) f 2ν +

wl ∂βα (I − P) f 2ν

1≤|β|,|α|+|β|≤N

⎫ ⎬

|α|≤N

∂ α (I − P) f 2ν

⎭

l (t) + C∇x P f 2 . ≤ C E l (t)D

On the other hand, the boundedness of the operator S(ω) implies that $ $ $ $ $ (1 + |ξ |2 ) N −2 |ξ |3 i S(ω) $≤C f , f

dξ $ 3 $ R

∂ α f 2 + C

∂ α f 2 .

1≤|α|≤N −1

1≤|α|≤N

Therefore, we can define an energy functional Hl (t) by

Hl (t) =

1≤|α|≤N

+

1 ( ∂ α f 2 + ∂ α ∇x φ2 ) − κ 2

R3

(1 + |ξ |2 ) N −2 |ξ |3 i S(ω) f, f dξ

wl ∂ α f 2 + wl (I − P) f 2

1≤|α|≤N

+

wl ∂βα (I − P) f 2 + (I − P) f 2 ,

1≤|β|,|α|+|β|≤N

which is bounded by CDl (t) from Lemma 3.3. Thus, we obtain d

l (t) + C∇x P f 2 ≤ CEl (t)Dl (t) + C∇x P f 2 . Hl (t) + Dl (t) ≤ C E l (t)D dt

⎫ ⎬ ⎭

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T. Yang, H. Yu

For El (t) < ε for some ε > 0 small enough, we have d Hl (t) + Dl (t) ≤ C∇x P f 2 , dt which gives from the fact that Hl (t) ≤ CDl (t), d Hl (t) + cHl (t) ≤ C∇x P f 2 , dt

(4.10)

for some positive constant c. On the other hand, if we set G = q∇x φ · v f + ( f, f ) − q∇x φ · ∇v f, then system (1.2) can be written as

∂t f + v · ∇x f + L f = G, φ =

R3

√

μ{ f + − f − }dv.

With (2.20), the solution to system (1.2) can be written in the mild form t U (t, s)G(s)ds. f (t) = U (t, 0) f 0 + 0

By using Lemma 2.6, we can obtain ∇x P f (t) ≤ ∇x f (t) ≤ Cλ0 (1 + t)−5/4 t +C (1 + t − s)−5/4 (G(s) Z 1 + ∇x G(s))ds, 0

where λ0 = f 0 Z 1 + ∇x φ0 L 1 + f 0 + ∇x φ0 . If l ≥ 1, by using the classical estimate on the collision operator, cf. Lemma 3.2 in [4], ( f, g) Z 1 ≤ C f wg + Cgw f , we have ( f, f ) Z 1 ≤ C f w f

* * ≤ Cw(I − P) f 2 + CP f 2 ≤ C El (t) Hl (t) + CP f 2 ,

∇x ( f, f )2 ≤ (∇x f, f )2 + ( f, ∇x f )2 ≤ CEl (t)Hl (t). Then by using Hölder inequality and Lemma 3.3, we have G(s) Z 1 ≤ ( f, f ) Z 1 + ∇x φ · v f Z 1 + ∇x φ · ∇v f Z 1 * * ≤ C El (t) Hl (s) + CP f 2 + C∇x φ(∇v f + v f ) * * √ * ≤ C El (t) Hl (s) + CP f 2 ≤ C ε Hl (s) + CP f 2 , where we have used the fact that El (t) ≤ ε. Similarly, for l ≥ 1, it holds that * * √ * ∇x G(s) ≤ C El (t) Hl (s) ≤ C ε Hl (s).


Define M(t) = sup

+

, (1 + s)5/2 Hl (s) ,

M0 (t) = sup

0≤s≤t

+ , (1 + s)3/2 f (s)2 .

353

(4.11)

0≤s≤t

Notice that M(t) and M0 (t) are non-decreasing. When l ≥ 1, we have √ * G(s) Z 1 + ∇x G(s) ≤ C ε Hl (s) + CP f (s)2 * √ ≤ C ε(1 + s)−5/4 M(t) + C(1 + s)−3/2 M0 (t), for any 0 ≤ s ≤ t. With this, we have ∇x P f (t) ≤ ∇x f (t) ≤ Cλ0 (1 + t)−5/4 t √ * + C( ε M(t) + M0 (t)) (1 + t − s)−5/4 (1 + s)−5/4 ds 0 √ * (4.12) ≤ C(1 + t)−5/4 (λ0 + ε M(t) + M0 (t)). On the other hand, by Gronwall inequality, (4.10) gives t −ct e−c(t−s) ∇x P f (s)2 ds. Hl (t) ≤ e Hl (0) + C 0

Then (4.12) yields Hl (t) ≤ e

−ct

Hl (0) + C 0

t

e−c(t−s) (1 + s)−5/2 ds(λ20 + εM(t) + M02 (t))

≤ C(1 + t)−5/2 (Hl (0) + λ20 + εM(t) + M02 (t)). Hence, for any t ≥ 0, M(t) = sup 0≤s≤t

+

, (1 + s)5/2 Hl (s) ≤ C(Hl (0) + λ20 + εM(t) + M02 (t)).

That is, when ε > 0 is small enough, one has M(t) ≤ C(Hl (0) + λ20 + M02 (t)). By (4.11), this gives Hl (t) ≤ C(1 + t)−5/2 (Hl (0) + λ20 + M02 (t)).

(4.13)

By using Lemma 2.6, it holds that f (t) ≤ C(1 + t)−3/4 ( f 0 Z 1 + f 0 + ∇x φ0 + ∇x φ0 L 1 ) t (1 + t − s)−3/4 (G(s) Z 1 + G(s))ds. +C 0

Since G(s) ≤ ( f, f ) + ∇x φ · v f + ∇x φ · ∇v f * √ * √ * ≤ C ε Hl (t) + C El (t)∇x φ ≤ C ε Hl (t),

(4.14)

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we have √ * G(s) Z 1 + G(s) ≤ C ε Hl (s) + P f 2 * √ ≤ C( ε + λ0 + Hl (0) + M0 (s))(1 + s)−5/4 + (1 + s)−3/2 M0 (s).

(4.15)

Finally, by plugging (4.15) into (4.14), we have f (t) ≤ C(1 + t)−3/4 ( f 0 Z 1 + f 0 + ∇x φ0 + ∇x φ0 L 1 ) t * √ (1 + t − s)−3/4 (1 + s)−5/4 ds( ε + λ0 + Hl (0) + M0 (t)) +C 0 √ ≤ C(1 + t)−3/4 ( f 0 Z 1 + f 0 + ∇x φ0 + ∇x φ0 L 1 + ε + λ0 * + Hl (0) + M0 (t)). (4.16) Notice that the constants ε and ε1 can be taken to be sufficiently small and f 0 Z 1 + f 0 + ∇x φ0 + ∇x φ0 L 1 +

√

ε + λ0 +

*

√ Hl (0) ≤ C( ε + ε1 ).

From (4.16), we obtain *

√ M0 (t) ≤ C( ε + ε1 ) + C M0 (t).

Thus we have M0 (t) ≤ C, which implies f (t) ≤ C(1 + t)−3/4 .

(4.17)

From (4.13) and (4.17), we obtain Hl (t) ≤ C(1 + t)−5/2 .

(4.18)

Since Hl (t) ∼

1≤|α|≤N

∂ α P f (t)2 +

|α|+|β|≤N

wl ∂βα (I − P) f (t)2 ,

(1.16) is proved by Lemma 3.3, and this completes the proof of Theorem 1.1. Acknowledgements. The authors would like to thank the referee for the valuable comments to revise the paper and for pointing out the reference [3]. The research of the first author was supported by the General Research Fund of Hong Kong CityU #103607, and the NNSFC Grant 10871082. The research of the second author was supported in part by FANEDD, NCET, the NNSFC Grant 11071085, Huo Ying Dong Foundation 121002 and The Project-sponsored by SRF for ROCS, SEM.


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References 1. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences 106, New York: Springer-Verlag 1994 2. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge: Cambridge Univ. Press, 1952 3. Duan, R.-J., Strain, R.M.: Optimal Time decay of the Vlasov-Poisson-Boltzmann system in R3 . Arch. Rat. Mech. Anal., doi:10.1007/s00205-010-0318-6, 2010 4. Duan, R.-J., Ukai, S., Yang, T., Zhao, H.-J.: Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications. Commun. Math. Phys. 277, 189–236 (2008) 5. Deckelnick, K.: Decay estimates for the compressible Navier-Stokes equations in unbounded domains. Math. Z. 209(1), 115–130 (1992) 6. Desvillettes, L., Dolbeault, J.: On long time asymptotics of the Vlasov-Poisson-Boltzmann equation. Comm. P.D.E. 16(2–3), 451–489 (1991) 7. Glassey, R.T.: The Cauchy Problem in Kinetic Theory. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1996 8. Glassey, R.T., Strauss, W.A.: Decay of the linearized Boltzmann-Vlasov system. Transport Theory Stat. Phys. 28(2), 135–156 (1999) 9. Glassey, R.T., Strauss, W.A.: Perturbation of essential spectra of evolution operators and the VlasovPoisson-Boltzmann system. Disc. Contin. Dynam. Syst. 5(3), 457–472 (1999) 10. Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9), 1104–1135 (2002) 11. Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153(3), 593–630 (2003) 12. Guo, Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4), 1081–1094 (2004) 13. Guo, Y.: Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm. Pure Appl. Math. 59(5), 626–687 (2006) 14. Kawashima, S.: The Boltzmann equation and thirteen moments. Japan J. Appl. Math. 7, 301–320 (1990) 15. Lions, P.L.: On kinetic equations. In: Proceedings of the International Congress of Mathematicians, Kyoto: Math. Soc. Japan, 1991, pp. 1173–1185 16. Liu, T.-P., Yang, T., Yu, S.-H.: Energy method for the Boltzmann equation. Physica D 188(3–4), 178–192 (2004) 17. Liu, T.-P., Yu, S.-H.: Boltzmann equation: micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004) 18. Mischler, S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210(2), 447–466 (2000) 19. Strain, R.M.: The Vlasov-Maxwell-Boltzmann system in the whole space. Commun. Math. Phys. 268(2), 543–567 (2006) 20. Ukai, S.: On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation. Proc. Japan. Acad. 50, 179–184 (1974) 21. Villani, C.: A survey of mathematical topics in kinetic theory. In: Handbook of Mathematical Fluid Dynamics, Friedlander, S., Serre, D. eds., Elsevier Sience, 2002, pp 71–305 22. Yang, T., Yu, H.-J.: Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space. J. Diff. Eqs. 248(6), 1518–1560 (2010) 23. Yang, T., Yu, H.-J.: Optimal convergence rates of Landau equation with external forcing in the whole space. Acta Math. Sci. Ser. B Engl. Ed. 9(4), 1035–1062 (2009) 24. Yang, T., Yu, H.-J., Zhao, H.-J.: Cauchy Problem for the Vlasov-Poisson-Boltzmann system. Arch. Rat. Mech. Anal. 182(3), 415–470 (2006) 25. Yang, T., Zhao, H.-J.: Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 268(3), 569–605 (2006) 26. Yang, T., Zhao, H.-J.: A new energy method for the Boltzmann equation. J. Math. Phys. 47 (5), 053301, 19 pp (2006) 27. Zhang, M.: Stability of the Vlasov-Poisson-Boltzmann system in R3 . J. Diff. Eqs. 247(7), 2027–2073 (2009) Communicated by P. Constantin


Communications in


Spanning Forest Polynomials and the Transcendental Weight of Feynman Graphs Francis Brown1 , Karen Yeats2, 1 Institut de Math. de Jussieu Paris, UMR 7586, Univ. Pierre et Marie Curie-Paris 6,

F-75005 Paris, France. E-mail: [email protected]

2 Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby,

BC V5A 156, Canada. E-mail: [email protected] Received: 29 October 2009 / Accepted: 21 May 2010 Published online: 14 October 2010 – © Springer-Verlag 2010

Abstract: We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in φ 4 theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a list of subgraphs which induce a drop in the transcendental weight. 1. Introduction It is well-known since the work of Broadhurst and Kreimer [3] that single-scale massless Feynman integral calculations in perturbative quantum field theories give rise empirically to multiple zeta values. In particular, there is a map from primitive graphs in φ 4 theory at low loop orders to linear combinations of multiple zeta values. Currently there is no way to predict this map without intensive numerical analysis. In this paper, we consider the most simple invariant of multiple zeta values: their transcendental weight. Conjecturally, there should exist no linear relations between multiple zetas of different weights, and hence one expects there to be a grading on the ring of MZVs. Thus the transcendental weight of the multiple zeta value ζ (n 1 , . . . , nr ) is n 1 +· · ·+nr and coincides with the number of integrations in its standard iterated integral representation (see Definition 30). On the other hand, the perturbative expansion in massless φ 4 theory is also graded by the number of loops. The surprising fact is that these gradings do not quite coincide. In the generic case, the transcendental weight of a graph is equal to twice its loop number minus 3. This is the case for the left and middle graphs in the examples below:

Supported by an NSERC discovery grant.

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But the non-planar graph on the right has 5 loops and hence its expected weight should be 2 × 5 − 3 = 7; yet it has weight 6. In other words, a weight drop can occur. The goal of this paper is to understand combinatorially why such a weight drop arises. Note that we are only considering the numbers coming from the contribution to the β-function for 4-point graphs in massless φ 4 with no divergent subgraphs. However, these results are of more general interest for two reasons. First, this is the case where the interesting patterns of transcendental numbers first appear in practice. Second, our results concern the behaviour of the denominators as we integrate; much the same story will hold for integrals with more complicated numerator structure but the same denominators. Our main result describes some operations on graphs in φ 4 theory under which the weight is preserved. As a corollary, we produce some infinite families of graphs which are of maximal weight, and other families which have a weight drop. These two classes should contribute to φ 4 theory in a quite different way. In order to obtain physical predictions, one must sum a large number of diagrams in a given quantum field theory at each loop order, and it is known that not all graphs contribute equally to the final sum. Indeed, some can even be discarded altogether. We believe that the notion of weight drop may shed some light on this phenomenon. Remark 1. It is likely that most primitive graphs in φ 4 theory do not evaluate to multiple zetas at very high loop orders, but the residues are always periods in the sense of [6]. Therefore the general conjectural picture is that there should exist a large pro-algebraic (‘motivic’) Galois group which acts on the set of all periods, and should in particular equip the perturbative expansion of a quantum field theory with a lot of extra structure. The notion of weight is the first non-trivial piece of information that such a theory would provide. 1.1. Outline. Let G be a primitive graph in φ 4 theory with eG edges. Its residue is defined by the formula IG =

eG 1 dαi δ(ακ = 1), G2 i=1

where αi is the Schwinger coordinate of each edge, and IG does not depend on the choice of κ. The graph polynomial G is defined by G =

T ⊆G e∈T /

αe ,

Weight of Feynman Graphs

359

where T ranges over all spanning trees of G. In order to understand the integral IG , I,J , where one is naturally led [4] to consider auxilliary (or ‘Dodgson’) polynomials G,K I, J, K are subsets of edges of G satisfying |I | = |J |. In the first part of this paper, we introduce spanning forest polynomials GP associated to any partition P of a subset of vertices of G. These are sums over families of trees whose leaves contain the vertices I,J in each partition of P. We show that every polynomial G,K can be written as a linear I,J . combination of GP , which in particular gives a formula for the signs in G,K Next, in §3 we study some algebraic identities between spanning forest polynomials, and give a universal formula for the graph polynomial of any 3-vertex connected graph G in terms of the GP . The graph polynomial for any such graph is the graph polynomial of the single graph K 2,3 which has 6 edges, which are decorated by spanning forest polynomials. Thus to prove a statement about any 3-vertex connected graph, it suffices to prove it in this single case. In §4 we recall some elementary properties of hyperlogarithms and give a sufficient condition for a graph to have a weight drop. This can be phrased in terms of higher graph invariants [4], which in some special cases can be computed in terms of spanning forest polynomials. Using the properties of the GP proved previously, we show that any 2-vertex connected graph always has a weight drop. We then show that the operation of splitting a triangle and moving some of its outer edges preserves the difference between the expected and the actual weights. This operation is pictured below: given any graph G containing seven edges in the configuration shown on the left (where A,B,C,D denote the rest of the graph), one can replace it with the smaller graph G on the right, and G has a weight drop if and only if G does.

These two results alone suffice to explain almost all of the known weight-drop graphs in φ 4 theory up to 8 loops. In §4.6, we seek a classification of weight-preserving, or weight-dropping operations. Using the universal formula for 3-vertex connected graphs, it suffices to write down all the local minors of 3-vertex connected graphs at k edges, for small values of k, and compute their graph polynomials. From this we deduce some new families of weight-preserving operations which are not attainable by splitting triangles. Remark 2. One way to circumvent the transcendence conjectures for periods is to replace them with mixed Hodge structures. Following [1], one defines the graph hypersurface X G ⊂ PeG −1 to be the zero locus of the graph polynomial G , which is singular in general. The differential form which defines IG gives a cohomology class ωG ∈ H DeGR−1 (PeG −1 \X G ), e G i ∧. . .∧dαeG . The cohomology (−1)i αi ∧dα1 . . . dα where ωG is the class of ψG−2 i=1 group on the right carries a mixed Hodge structure, so the natural question one can ask,

360

F. Brown, K. Yeats

is where the class ωG sits with respect to the Hodge and weight filtrations. One should in fact consider a certain relative cohomology group, constructed in [1], which takes into account the domain of integration also. In this context, the weight always makes sense, but it is currently not known how to carry this out save for a few examples of graphs. Therefore, in the face of the geometric difficulties of this problem, we have instead focused on the combinatorial aspects of the weight drop, which seems to be a necessary prerequisite before tackling the Hodge side. It came as a surprise to us quite how intricate this first step already is. 1.2. Background. Let G be a connected multigraph, with self-loops1 allowed. Let E(G), V (G) denote the set of edges and vertices of G, and let eG = |E(G)|, vG = |V (G)|. To each edge e of G, we associate a Schwinger parameter αe . The graph polynomial of G is defined by G = αe ∈ Z[αe , e ∈ E(G)], T ⊆G e∈E(T / )

where the sum is over all spanning trees T of G. Definition 3. Choose an orientation on the edges of G, and for every edge e and vertex v of G, define an incidence matrix: ⎧ if the edge e begins at v, ⎨ 1, (EG )e,v = −1, if the edge e ends at v, ⎩ 0, otherwise. Let A be the diagonal matrix with entries αe , for e ∈ E(G), and set A EG

, MG = −EGT 0 where the first eG rows (resp. columns) are indexed by the set of edges of G, and the remaining vG rows (resp. columns) are indexed by the set of vertices of G, in some order.

G has zero determinant. The matrix M Definition 4. Choose any vertex of G and let MG denote the minor obtained by deleting

G . the corresponding row and column of M The matrix MG is not well-defined, but one can show that G = det(MG ) is the graph polynomial of G. This motivates the following: Definition 5. Let I, J, K be subsets of the set of edges of G which satisfy |I | = |J |. Let MG (I, J ) K denote the matrix obtained from MG by removing the rows (resp. columns) indexed by the set I (resp. J ) and setting αe = 0 for all e ∈ K . Set I,J G,K = det MG (I, J ) K . 1 Or tadpoles.


361

∅,∅ I,J It is clear that G,∅ = G . The polynomials G,K are well-defined up to sign. The following results are proved in [4].

Proposition 6. Let e ∈ E(G) such that e ∈ / I ∪ J ∪ K . Let G\e denote the graph obtained by deleting the edge e (and removing any isolated vertices) and let G/e denote the graph obtained by contracting e (i.e., deleting e and identifying its endpoints).2 Then I,J I ∪e,J ∪e G\e,K = G,K , I,J I,J = G,K G/e,K ∪e . I,J Since G,K is linear in the Schwinger parameters this implies that I,J I,J I,J = G\e,K αe + G/e,K . G,K

Corollary 7. Let G, I, J, K be as above. Then

I,J G,K = GI ,J ,∅ ,

where G = G\(I ∩ J )/(K \(I ∩ J )), and I ∩ J = ∅. In other words, by passing to a minor of G, we can assume that I ∩ J = K = ∅. Now let U ⊂ E(G) be a set of |U | = h 1 (G) = eG − vG + 1 edges. Define EG (U ) to be a square (vG − 1) × (vG − 1) matrix obtained from EG by removing the rows U , and any one column corresponding to a vertex. The matrix-tree theorem states that det EG (U ) ∈ {0, ±1}, and is non-zero if and only if U is a spanning tree of G. Proposition 8. Suppose that I ∩ J = ∅. Then I,J G,∅ = αu det(EG (U ∪ I )) det(EG (U ∪ J )), / U ⊂G\(I ∪J ) u ∈U

where U ranges over all subgraphs of G\(I ∪ J ) which have the property that U ∪ I and U ∪ J are both spanning trees in G. I,J . The first goal of this paper is to provide combinatorial interpretations of the G,K The objects we will use to this end are spanning forests, that is, subgraphs of G which contain all vertices of G and are disjoint unions of trees.

Definition 9. Let P = P1 ∪ . . . ∪ Pk be a set partition of a subset of the vertices of G. Define αe ,

GP = F e ∈ F

where the sum runs over spanning forests F = T1 ∪. . .∪Tk , where each tree Ti of F contains the vertices in Pi and no other vertices of P, i.e., V (Ti ) ⊇ Pi and V (Ti ) ∩ P j = ∅ for j = i. Trees consisting of a single vertex are permitted. Call GP a spanning forest polynomial of G. 2 We require that the contraction of tadpoles be zero.

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Graphically we will represent GP by associating a colour to each part of P and drawing G with the vertices in P coloured accordingly. For example, let G be the wheel with three spokes, labelled as illustrated.

Let P = {1}, {2, 4}. Then

GP = a f (be + cd + ce + de). Graphically we represent this

Spanning forest polynomials are well behaved under contraction and deletion. The following propositions can be most easily understood simply by drawing each GP . Proposition 10. Let e be an edge variable of G and let P be a set partition of some vertices of G. Then

GP =

⎧ P ⎨e G\e ⎩e P

G\e

if the ends of e are in different parts of P P/e

+ G/e

otherwise

,

where P/e is the set partition made from P by identifying the two ends of e should they appear in P. Proof. If the ends of e are in different parts of P then the edge e must not appear in the spanning forest polynomial. Thus factoring out e is equivalent to cutting e. This leaves a spanning forest of G\e compatible with P. The second case follows as in the graph polynomial case.


363

Proposition 11. Let v and w be the two ends of e. Assume that v and w are not in different parts of P. Then ⎧ P\ p, p ∪{v}, p2 ∪{w} ⎪

G\e 1 v, w ∈ P ⎪ ⎪ ⎪ ⎪ p ∪ p = p p part of P 1 2 ⎨ p1 ∩ p2 =∅ P/e

G/e = , P\ p, p1 ∪{v}, p2 ∪{w} ⎪

v in part p of P ⎪ G\e ⎪ ⎪ ⎪ ⎩ p1 ∪ p2 = p∩(V (G)\{v,w}) p1 ∩ p2 =∅

where P\ p means the partition consisting of the parts of P other than p, and P, q means the partition with the parts of P and with the part q. Note that p1 ∪ p2 and p2 ∪ p1 are counted as two different terms of the sums provided p1 = p2 . Proof. Let u be the vertex v ∼ w in G/e. Suppose v and w are not in P. Then u is not in P/e, so P/e = P and u may belong to any tree of a spanning forest of G/e giving P\ p, p∪{u} P/e

G/e =

G/e . p part of P

Any spanning forest corresponding to the partition P\ p, p ∪ {u} in G/e, is also a spanning forest with one more tree in G. This extra tree splits p ∪ {u} into p1 ∪ {v} and p2 ∪ {w}. Specifically, P\ p, p∪{u} P\ p, p ∪{v}, p2 ∪{w} =

G\e 1 ,

G/e p1 ∪ p2 = p p1 ∩ p2 =∅

giving the formula for v, w ∈ P. Now suppose v is in part p of P, and so w is either not in P or is also in p. Then P/e = P\ p, p ∪ {u}, where p = p ∩ (V (G)\{v, w}). Any spanning forest in G/e corresponding to this set partition is again also a spanning forest with one more tree in G. This extra tree splits p ∪ {u} into p1 ∪ {v} and p2 ∪ {w}, where p1 ∪ p2 = p , leading, as above, to the desired sum decomposition.

2. Signs in Dodgson Polynomials I,J can be expanded in terms of spanning forest polynomials. This expansion The G,K I,J . provides a simple combinatorial explanation for the signs of the monomials of the G,K What we do here is a variant on the all minors matrix tree theorem [5] in a form which is convenient for our uses. We are grateful to Christian Bogner and Stefan Weinzierl [2] for pointing this out. I,J In view of Corollary 7 it suffices to consider G,K with K = ∅ and I ∩ J = ∅. Thus we will suppress K = ∅ from the notation.

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Proposition 12. Let I and J be sets of edges of G with |I | = |J | and I ∩ J = ∅. Then we can write f k GPk, GI,J = k

where the sum runs over partitions of V (I ∪ J ) and f k ∈ {−1, 0, 1}. Proof. Take a particular monomial m in GI,J . Let M denote the set of edges in m, and N the complementary set of edges in G\(I ∪ J ). We know that N ∪ I and N ∪ J are spanning trees in G, and so N is a forest. The coefficient in front of m is obtained by setting all Schwinger parameters in N to zero, and taking the coefficient of all remaining parameters M. In other words, the coefficient of m is exactly I,J G\M/N

which by Proposition 8 is given by det(EG\M/N (I )) det(EG\M/N (J )) ∈ {0, ±1}. This vanishes if I and J are not spanning trees in G\M/N . The only information of m which remains in G\M/N is which end points of edges of I and J belong to the same tree of N . Thus every monomial which gives the same set partition of V (I ∪ J ) has the same sign, and every monomial corresponds to some such set partition.

Example 13. One simple example is the case where I and J each consist of a single edge with a common vertex v. Let u and w be the other two end points. The only set partition with a nonzero coefficient in this case is {v}{u, w}. Graphically, if

Definition 14. Let I, J be two subsets of edges of G with |I | = |J | and let P be a partition of V (I ∪ J ). Let I P (resp. J P , (I ∪ J ) P ) denote the graph obtained from the subgraph I (resp. J , I ∪ J ) by identifying vertices which lie in the same partition. If the edges of G are oriented, or are ordered, then the graphs I P , J P , (I ∪ J ) P inherit these extra structures also. If the vertices of G are ordered then the graphs I P , J P , (I ∪ J ) P inherit this structure by using the first vertex in each part to give the order. The proof of the previous proposition shows that f k = (II,J∪J ) P . The coefficient is non-zero precisely when both I P and J P are trees, and have exactly one vertex of every colour. Definition 15. Let T be a rooted tree, with edges labelled from {1, . . . , n} and non-root vertices labelled {1, . . . , n}. Choose an orientation on its edges. We define a number by constructing a bijection φ : E(T ) → V (T )


365

and a map s : E(T ) → {±1} as follows. To each e ∈ E(T ), φ(e) associates the vertex meeting e which is furthest from the root, and s(e) is +1 if φ(e) is the endpoint of the oriented edge e, and −1 if it is the initial point. Define the sign of the (oriented, numbered) tree T to be: ε(T ) = sgn(φ) s(e). e∈E(T )

Proposition 16. Choose an ordering on the edges and vertices of G and an orientation of its edges. Let I, J be subsets of edges of G and P a partition on V (I ∪ J ). Then (II,J∪J ) P = ε(I P )ε(J P ). Proof. From the proof of Proposition 12, we have (II,J∪J ) P = det(E(I ∪J ) P (I P )) det(E(I ∪J ) P (J P )). This vanishes if I P and J P are not both trees. In the case where they are, an inspection of the matrices E(I ∪J ) P (I P ) shows that its determinant is exactly ε(I P ). The choice of vertex v to remove in (II,J∪J ) P becomes the choice of roots for I P and J P by taking as roots the vertices of I P and J P which correspond to the part of v.

As a corollary we can easily understand how the sign changes under simple transformations of P. Let I and J be sets of edges of G with |I | = |J | and I ∩ J = ∅. Let GI,J = f k GPk k

be as in Proposition 12. Suppose Pi and P j are set partitions appearing in the sum which agree on V (I ). Without loss of generality, order the vertices so that all of V (I ) comes before V (J ). Then the matrices on the I side are identical for Pi and P j and we have an order of the parts of Pi and P j on J (coming from the vertex order on I ) which gives the vertex order on J Pi and J P j . Corollary 17. Suppose Pi and P j differ on V J by a transposition of two vertices in the same tree of J . Then fi = − f j . Proof. The choice of a root r in J P determines a choice of root in each tree of J recursively. To see this, first take each vertex which reduces to r as a root in its tree. Next, for each tree which under P has a vertex v which is identified with a vertex in an already rooted tree of J , take v as the root. Continue until all trees are rooted. Since only the vertex of each edge which is furthest from r in J P contributes to the sign, we can compute the sign tree by tree and multiply, using the order of the vertices given by the order of the parts on J . Let v and w be the two transposed vertices in the statement above. If neither are a root, then the permutation φ from Definition 15, taken tree by tree, is composed with a transposition, changing the sign. If it is not possible to choose the root of J P so that this occurs, then v and w must be the only vertices in their tree of J . In this case switching the parts of v and w has the same effect as switching the orientation of the edge between them, again changing the sign.

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Corollary 18. Suppose Pi and P j differ on V J by one vertex having switched parts. Then fi = f j . Proof. Let v be the vertex which changes part. Let F be the forest of two trees given by J with the identifications of Pi or P j except that v is not identified with its part-mates. As in the previous corollary, we can compute the sign at the level of F. By taking the root of J Pi and J P j to be in the tree of F which does not contain v we get v to be the root of the other tree of F. Thus, by the definition of φ, v does not contribute to the sign.

Example 19. Let

with the three vertical edges on the left being I and the three vertical edges on the left being J . Consider the two set partitions

which agree on the ends of I and differ by a transposition on the ends of J . Orient the edges of I and J downwards. The graphs J P1 , J P2 are

so with the filled square as the removed vertex ⎡

1 E(I ∪J ) P1 (I P1 ) = ⎣0 0

−1 1 1

⎤ ⎡ 0 1 −1⎦ and E(I ∪J ) P2 (I P2 ) = ⎣0 0 0

−1 −1 0

⎤ 0 1⎦ . 1

Calculate det(E(I ∪J ) P1 (I P1 )) = −1 and det(E(I ∪J ) P2 (I P2 )) = 1 as expected from Proposition 17.


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Example 20. Let

with the four vertical edges on the left being I and the four vertical edges on the left being J . Consider the two set partitions

which agree on the ends of I and differ by a single vertex having changed colour on the ends of J . Orient the edges of I and J downwards. With the empty square as the removed vertex ⎡

1 ⎢0 E(I ∪J ) P1 (I P1 ) = ⎣ 0 0

0 −1 1 1

0 0 −1 0

⎤ ⎡ 0 1 0⎥ ⎢0 and E(I ∪J ) P2 (I P2 ) = ⎣ 0⎦ 1 −1 1

0 −1 0 0

0 0 −1 0

⎤ 0 0⎥ . 0⎦ −1

Both determinants are −1 as expected from Proposition 18.

3. Identities Spanning forest polynomials give a nice way to look at graph polynomial identities. To illustrate this we will first recast useful known identities and then prove a new identity which generalizes results of [4]. We say a graph is two vertex reducible if we can remove two vertices of the graph, and the adjoining edges, and the resulting graph is disconnected. Let G 1 and G 2 be two graphs. Let e1 be an edge of G 1 and e2 an edge of G 2 . Then define a two-vertex join of G 1 and G 2 to be the graph resulting from identifying e1 and e2 and then cutting the new edge. Given e1 and e2 there are two ways to do this identification. However, this ambiguity is of little interest to us because the period of the graph does not depend on it [7], nor does the graph polynomial.

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Proposition 21. Let G be a two-vertex join as above with v1 and v2 the join vertices. Suppose that edges i, j, k, l ∈ G are such that i, j ∈ G 1 and k, l ∈ G 2 . Then {v },{v }

{v },{v }

G = G 11\e1 2 G 2 \e2 + G 1 \e1 G 21\e2 2 , i j,kl

G

= 0,

ik, jl G

= G

il, jk

.

Proof. The first identity holds because every spanning tree of G either connects v1 and v2 on the G 1 side or on the G 2 side. In either case the remaining side has a forest of two trees, one connected to v1 , the other to v2 . Pairings of such a forest with a spanning tree on the other side give all spanning trees of G. Consider i j,kl . Any monomial appearing in this polynomial is a monomial in G/{i, j} and in G/{k,l} . Thus any spanning forest polynomial appearing in the expansion of i j,kl comes from a partition with exactly three parts and all three parts are represented among the end points of i and j as well as among the end points of k and l. This means there are three trees which connect G 1 to G 2 in G. However G 1 and G 2 join at only two vertices. This is a contradiction and so there are no such monomials. Then ik, jl = il, jk by the Plücker identity [4] i j,kl

G

ik, jl

− G

il, jk

+ G

= 0.

Proposition 22. Let u, v, and w be vertices of a graph G. Then {u,v,w}

G

{u},{v},{w}

G

{u,v},{w}

= G

{u,w},{v}

G

{u,v},{w}

+ G

{u},{v,w}

G

{u,w},{v}

+ G

{u},{v,w}

G

.

Graphically,

This identity is essentially the so-called Dodgson identity in this context. Proof. Use the Dodgson identity, det(MG (12, 12)) det(MG ) = det(MG (1, 1)) det(MG (2, 2)) − det(MG (1, 2)) det(MG (2, 1)) along with Proposition 6 and Example 13.

The graphical formulation suggests interpreting this proposition as a result about transferring an extra edge from any term in the left-hand factor of the left-hand side to the right-hand factor of the left-hand side, thus cutting a spanning tree into two in the left-hand factor and joining two of the three trees together in the right-hand factor. The proposition says that transferring an edge in this way results in two spanning forests with exactly two trees in all possible ways. Carrying this idea through to a proof is delicate as there are many ways the cutting can be done and many ways to build any particular forest of two trees with an extra edge. We have recently been made aware [2] of a classic combinatorial proof of the Dodgson identity [8] which can be straightforwardly translated into graph language to give a proof along these lines.


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Theorem 23. Let

be a graph which is three vertex reducible at vertices {u, v, w}. Let G and G be the two halves of G when separated at u, v, and w. Let {u},{v,w}

f 1 = G f2 = f3 = f =

f =

,

{v},{u,w}

G , {w},{u,v}

G , {u,v,w} , G = G {u},{v},{w} ,

G

{u},{v,w}

g1 = G g2 = g3 = g=

g=

,

{v},{u,w}

G , {w},{u,v}

G , {u,v,w} G = G , {u},{v},{w}

G .

Then f deg(g)+1 gG = f 1 f 2 + f 1 f 3 + f 2 f 3 + f 1 g2 + f 1 g3 + f 2 g1

+ f 2 g3 + f 3 g1 + f 3 g2 + g1 g2 + g1 g3 + g2 g3 α

1 ,α2 ,..., fβ1 , fβ2 ,...

,

where f i = f i g, α1 , . . . are the variables for the edges of G and β1 , . . . are the variables for the edges of G . Note that the piece in parentheses of the expression for G , f 1 f 2 + f 1 f 3 + f 2 f 3 + f 1 g2 + f 1 g3 + f 2 g1 + f 2 g3 + f 3 g1 + f 3 g2 + g1 g2 + g1 g3 + g2 g3 is itself the graph polynomial of the following graph with the indicated polynomials as edge variables

Graphically, then, we can represent the theorem as

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F. Brown, K. Yeats

Note however that this picture does not capture all the details of the theorem as it does not indicate the scalings by powers of

Proof. Any tree contributing to a term of G breaks up uniquely into a spanning forest of G and a spanning forest of G . A pair of a spanning forest of G and a spanning forest of G give a tree of G precisely when each tree of each forest contains at least one of u, v, and w and when there is exactly one path between each pair of u, v, and w using both spanning forests. Thus G =

f g + f 1 g2 + f 1 g3 + f 2 g1 + f 2 g3 + f 3 g1 + f 3 g2 + f

g. Let n = deg g. Note that deg gi = n + 1 and deg

g = n + 2. Multiplying by f n+1 and using Proposition 22 we get f n+1 G = f n ( f 1 f 2 g + f 1 f 3 g + f 2 f 3 g) + f n+1 ( f 1 g2 + f 1 g3 + f 2 g1 + f 2 g3 + f 3 g1 + f 3 g2 ) + f n+2

g. Let α1 , . . . , be the edges of G and let β1 , . . . , be the edges of G . Scale the edges of G by f giving f n+1 G = f 1 f 2 g + f 1 f 3 g + f 2 f 3 g + f 1 g2 + f 1 g3 + f 2 g1 + f 2 g3 + f 3 g1 + f 3 g2 +

g . α1 ,α2 ,..., fβ1 , fβ2 ,...

Next multiply by g and use Proposition 22 f n+1 gG = f 1 f 2 + f 1 f 3 + f 2 f 3 + f 1 g2 + f 1 g3 + f 2 g1

+ f 2 g3 + f 3 g1 + f 3 g2 + g1 g2 + g1 g3 + g2 g3 α

1 ,α2 ,..., fβ1 , fβ2 ,...

where f i = f i g, which is the desired result.

,

4. Weight Drop in Feynman Graphs 4.1. Hyperlogarithms. Let σ0 = 0, and let σ1 , . . . , σ N denote N distinct points in C∗ . Let = {σ0 , . . . , σ N }, and let Y = C\. Let A = {x0 , . . . , x N } denote an alphabet with N + 1 letters, and let A∗ denote the free non-commutative monoid on X , which consists of the set of all words w in the alphabet A and the empty word e. Let CA denote the ring of non-commutative formal power series in A, equipped with the concatenation product. For any element S ∈ CA, let Sw denote the coefficient of w in S, i.e., Sw w, Sw ∈ C. S= w∈A∗


371

Consider the trivial bundle E = Y ×CA over Y , and consider the following one-form on Y : N dz xi (z) = . z − σi i=0

Since d(z) = (z) ∧ (z) = 0 this defines a flat connection on E. There is a unique multivalued section L : Y → E which satisfies: d L(z) = (z)L(z), L(z) ∼ exp(x0 log z),

(1)

where the notation L(z) ∼ exp(x0 log z) means that there exists a function h(z) holomorphic in the neighbourhood of the origin, such that h(0) = 1 and L(z) = h(z) exp(x0 log z) for z near 0. If w = xi1 . . . xir , where ir = 0, then one can show that the coefficient L w (z) of L(z) is an iterated integral: dtr dt1 dt2 ... L w (z) = t − σ t − σ t ir 2 i 2 1 − σi 1 0≤tr ≤tr −1 ≤...≤t1 ≤z r for z ∈ R in a neighbourhood of 0. Equations (1) are equivalent to the following system of differential equations on the coefficients L w (z), and determine them uniquely: 1 ∂ L xi w (z) = L w (z) for i = 0, . . . , N , ∂z z − σi 1 L x0n (z) = logn (z), n! L w (z) ∼ 0 as z → 0 if w = x0n . Lemma 24. We deduce the following indefinite integrals (with constants of integration omitted), where the denominators are of degree at most 2 in z: L w (z) i) dz = L xi w (z), z − σi 1 L w (z) dz = L xi w (z) − L x j w (z) , ii) (z − σi )(z − σ j ) σi − σ j n iii) L xi1 ...xin (z)dz = (−1)k−1 (z − σik )L xik+1 ...xin (z), iv)

L r xi x j w (z)

k=1

1 dz = (z − σi )2 z − σi

r k+1 (−1) L xr−k x j w + k=1

i

1 L xi w (z) − L x j w (z) , σi − σ j

where i = j, r ≥ 0, and i 1 , . . . , i n are any indices in {0, . . . , n}. Proof. (i) and (ii) follow from the definition of the functions L w (z) and partial fractions. (iii) and (iv) follow by integration by parts and induction.

Definition 25. Let L denote the Q-vector space spanned by the multivalued functions L w (z), for w ∈ A∗ (which can be shown to be linearly independent). It is graded by the weight, a word w ∈ A∗ is the number of letters in w. We write where the weight |w| of L = n≥0 gr nW L, and Wk L = 0≤n≤k gr nW L.

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It turns out that L is closed under multiplication (by the shuffle product formula), and is in fact a graded Hopf algebra. The various cases of the previous lemma are summarized in the following corollary. Corollary 26. Let F(z) be an element of gr kW L of weight k, and let P(z) = az 2 + bz + c be a polynomial in z of degree at most 2 with zeros in . Let (P) = b2 − 4ac denote the discriminant of P(z). Then √ 1 gr W L if (P) = 0, (“no weight drop”) F(z) (P) k+1 dz ∈ . P(z) Wk L if (P) = 0 (“weight drop”) Note that in the case when a weight drop occurs, the primitive is not necessarily pure as there may be mixing of weights. Proof. The case (P) = 0 corresponds to (iii) and (iv) in the previous lemma, and the remaining cases (i) and (ii) correspond to (P) = 0.

4.2. Initial integrations. One can use the algebras L to integrate out the first few variables in a Feynman integral. Let G be a primitive-divergent graph, and choose an order on its edges. Consider the residue: eG 1 IG = dαi δ(αeG = 1). 2 [0,∞]eG G i=1 We can successively integrate out the variables α1 , . . . , α5 using Lemma 24. Dropping the δs from the notation, this gives [4]: eG 1 dαi , IG1 = G1,1 G,1 i=2 1,1 2,2 eG log G,2 + log G,1 − log G12,12 − log G,12 dαi , IG2 = (G1,2 )2 i=3 ⎛ 123,123 log 123,123 G,123 log G,123 IG3 = ⎝ G12,13 12,23 G 13,23 − 2,3 1,3 1,2 G G G G,1 G,2 G,3 ⎞ ij ij eG i i G, G,k log G,k jk log G, jk ⎠ + − i j,ik i j, jk i, j dαi , i j,ik i,k i, j G G G,k i=4 {i, j,k} G G, j G,k where the sum runs over permutations of {1, 2, 3} and so there are 8 terms in the last integral. Continuing in a similar way and exploiting the many algebraic relations between the polynomials KI,J [4], one verifies that: e G A B C 4 + + dαi , (2) IG = G12,34 G13,24 G14,23 G13,24 G12,34 G13,24 i=5 eG F dαi , (3) IG5 = 5 (1, 2, 3, 4, 5) G i=6


373

where A, B, C are hyperlogarithms of weight 2, and F is a hyperlogarithm of weight 3 I,J = 0}, where I ∪ J ∪ K = {1, 2, 3, 4, 5}. The 5-invariant with singularities in {G,K 5 (1, 2, 3, 4, 5) is defined as follows: Definition 27. The 5-invariant of any 5 edges i, j, k, l, m in G is: ⎞ ⎛ i j,kl i jm,klm G,m G 5 ⎠ G (i, j, k, l, m) = ± det ⎝ ik, jl ikm, jlm G,m G It is well-defined, i.e., permuting i, j, k, l, m in the above only changes the sign of the determinant.

4.3. Denominator reduction. The denominator reduction is an algorithm for computing the denominators at successive stages of integration using Corollary 26. Definition 28. Let G be a primitive-divergent graph and choose an ordering on its set of edges. Let D5 = 5 G (1, 2, 3, 4, 5). Let n ≥ 5, and suppose inductively that Dn factorizes into a product of linear factors in αn+1 , i.e., Dn = (aαn+1 + b)(cαn+1 + d). Then we define Dn+1 = (Dn ) = ±(ad − bc), where the discriminant is taken with respect to αn+1 . A graph G for which the polynomials Dn can be defined for all n is called denominator-reducible. If for some n, Dn is a perfect square in αn+1 , then Dn+1 , and all Dm for m ≥ n + 1 are identically 0. In this case, we say that G has a weight drop. Otherwise, G is non-weight drop. Remark 29. The interpretation of Dn as a denominator proves that Dn does not depend on the chosen order of variables up to that point. We will frequently use the notation n

G (e1 , . . . , en )

n ≥ 5,

to denote the denominator Dn of the graph G after reducing with respect to the edges e1 , . . . , en . We have the following rather naive definition of the transcendental weight of a period: Definition 30. Let P, Q be polynomials in Q[x1 , . . . , xn ] and consider an absolutely convergent period of the form: P(x1 , . . . , xn ) I = d x1 . . . d xn . n Q(x 1 , . . . , xn ) [0,∞] We say that such a period has weight at most n if it can be written as a sum of convergent period integrals as above with at most n integrations.3 This defines a filtration on the set of these periods. 3 This corresponds to the fact that the mixed Hodge structure of a complex of open affine varieties of dimension at most n has weights contained in [0, 2n].

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The definition of weight as given above is compatible with the weight filtration on the elements L w (z) of L. It is satisfactory for periods of mixed Tate motives, and in particular should give back the usual notion of weight for multiple zeta values, however, in the general case it is a little simplistic; as remarked earlier, a more sophisticated approach to the weight is to view I as a period of a mixed Hodge structure. It is remarked in [4] that the arguments we give in this paper do in fact prove an analogous result on the weights in the Hodge-theoretic sense. It follows from the computations of IG4 above that I can be written as a 2 + eG − 5 = eG − 3 fold integral (each term A, B, C is of weight 2 and can therefore be written as a 2-fold integral), and there remain eG − 5 Schwinger parameters to integrate out owing to the δ(αeG − 1) term. It follows that the weight of IG is at most eG − 3. Theorem 31 [4]. Suppose that G is primitively divergent as above and has a weight drop at the n th stage of its denominator reduction. If furthermore G is linearly reducible up to the n th point, then the weight of IG is at most eG − 4. The linear reducibility condition guarantees that the integrands in the integration process can indeed be written as hyperlogarithms (i.e., they are multivalued functions on a Zariski open subset of projective space and have global unipotent monodromy). The previous calculations make it clear that every graph G is linearly reducible up to the 5th stage. Sufficient conditions for linear reducibility are given in [4]. Remark 32. If G is linearly reducible and has no weight drop, then the expected transcendental weight of IG is eG − 3. The purpose of this paper is to investigate the combinatorial conditions under which a weight drop (defined by the vanishing of a denominator Dn ) occurs.

4.4. Weight-preserving operations. Let

where the circled vertices indicate where the explicitly drawn edges attach to the rest of the graph, which is left undrawn to avoid clutter. Let K be the rest of the graph, K = G\{1, 2, 3, 4, 5}. Let


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Proposition 33. 5

13,45 4,5 G (1, 2, 3, 4, 5) = ± H H,13 {A,B},{C,D}

= ± K

{A,C},{B,D}

− K

!

{A,D},{B},{C}

K

.

Proof. Since 123 forms a triangle, we have 5

14,35 G (1, 2, 3, 4, 5) = ±G123,245 G,2 .

Drawing K as a blob and using ∩ to indicate the polynomial formed terms common to each argument with signs as in Proposition 12, we have

4,5 14,35 13,45 H,13 gives the same intersection of blobs as G,2 and H gives the same inter-

section of blobs as G123,245 . The result follows.

Consider a ‘double triangle’:

Let K againbe the rest of the graph, K = G \{1, 2, 3, 4, 5, 6, 7}. Note that H = G \{3, 7}/2 |6↔3 .

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F. Brown, K. Yeats

Proposition 34. The denominator D7 after reducing G with respect to the seven edges 1 − 7 indicated above, is given by 13,45 4,5 D7 (G ) = ± H H,13 {A,B},{C,D}

= ± K

{A,C},{B,D}

− K

!

{A,D},{B},{C}

K

.

Proof. By Proposition 33 applied to edges 1, 3, 2, 4, 6 we know that 5

{A,C},{B},{E}

G (1, 2, 3, 4, 6) = ± K ∪{5,7}

{A,B},{C,E}

K ∪{5,7}

{A,E},{B,C}

− K ∪{5,7}

!

.

Notice that in the partition {A, C}, {B}, {E} the two ends of edge 7 are in different parts. Thus, by Proposition 10, {A,C},{B},{E}

K ∪{5,7}

{A,C},{B},{E}

= α7 K ∪5

.

This removes one term in the discriminant so we can easily apply a denominator reduction with respect to the edge 7. We deduce that

From the pictures we can read off the contractions and deletions of edge 5 and deduce that the reduction with respect to edge 5 is 7

G (1, 2, 3, 4, 5, 6, 7) {A,C},{B}

= ± K

{A,B},{C},{D}

K

{A,C},{B},{D}

− K

{A,B},{C}

K

! .

But this is itself a five-invariant, 5 G (1, 2, 3, 4, 5), expanded as 12,34 14,23 125,345 G145,235 G,5 − G,5 G ,

where G is as in the previous proposition. Applying Proposition 33 to rewrite this 5-invariant completes the proof.

Theorem 35. Let G and G be obtained, as above, by splitting a triangle. Suppose that G is linearly reducible with respect to a set of edges {1, . . . , 7} ∪ S (in that order), where S ⊂ G \{1, . . . , 7}. Then G is linearly reducible with respect to {1, . . . , 5} ∪ S and has a weight drop if and only if G has a weight drop. Proof. It follows from the two previous propositions that: 5

G (1, 2, 3, 4, 5) = ±7 G (1, 2, 3, 4, 5, 6, 7).


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Note that G is always linearly reducible with respect to {1, 2, 3, 4, 5, 6, 7} (the case S = ∅), because every 5-edge minor of G has either a triangle or a 3-valent vertex, and so G contains no non-trivial 5-invariants. By deleting edges 6 and 7 in G we get a special case of the double triangle where a single triangle with two three-valent vertices is contracted to two three-valent vertices connected by an edge. If we also consider the two remaining edges adjacent to B, then by similar arguments 7 and 5 integrations respectively give the same denominator. By deleting edge 4 we get a special case with a three-valent vertex in the double triangle contracting to a single triangle with a three-valent vertex. If we also consider one more edge adjacent to C then by similar arguments 7 and 5 integrations respectively again give the same denominator. 4.5. Families of weight drop graphs. The first family of weight drop graphs is already well known. Proposition 36. Let G be two vertex reducible. Then G has a weight drop. Proof. Write G as the 2-vertex join of two graphs G 1 and G 2 . Number the edges of G in any way so that edges 1, 2 lie in G 1 and edges 3, 4 lie in G 2 . Then by Proposition 21, G12,34 = 0 and G13,24 = G14,23 . At the fourth stage of integration (IG4 above) the denominator reduces to G13,24 G14,23 = (G13,24 )2 . Thus we have a weight drop.

The same argument shows that any graph with a double edge and more than 4 edges in total has weight drop. The first family of weight drop graphs which goes beyond 2-vertex reducible graphs was observed empirically by one of us (KY) and independently by Oliver Schnetz, who later also found a proof. The most beautiful proof, also observed by Oliver Schnetz, is in terms of the material of this section. Example 37. Every graph of the form

has a weight drop. To see this consider the pair of triangles marked below by heavy edges

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F. Brown, K. Yeats

By Theorem 35 this graph has weight drop iff

does. The latter is two vertex reducible at the marked vertices and hence by Proposition 36 has weight drop. Further families can be built along these lines by repeated double triangles. To make a more systematic search for weight preserving operations and weight drop families we can use the 3-vertex join result as in the next section. 4.6. Operations on 3-connected graphs. Let G be a 3-connected graph. We can write G = L ∪3 R as the join of two graphs L and R along three distinguished vertices v1 , v2 , v3 (below left). By Theorem 23 there is a universal formula for the graph polynomial of G in terms of Dodgson polynomials. Thus for any fixed left-hand side L, we can consider the graph

L obtained by joining a vertex v to v1 , v2 , v3 (right):

Suppose that L has at least 5 edges, and let x, y, z denote the Schwinger parameters of the edges {v, v1 }, {v, v2 }, {v, v3 } respectively. Suppose that

L is denominator reducible. Then we define ρ L (x, y, z) = e L

L (α1 , . . . , αe L ) ∈ Q[x, y, z], where the edges of L are numbered 1, . . . , e L . By Theorem 23, the polynomial ρ L computes the general shape of the denominator reduction of any graph G = L ∪3 R after reducing out all the edges in L. Proposition 38. Suppose that G is the 3-vertex join of L and R. Let {1},{2,3}

x R = R

{2},{1,3}

, yR = R

{1,2},{3}

, z R = R {1,2,3}

We have shown that x R y R + x R z R + y R z R = R R

.

(Proposition 22), and

3 2 1 x R + y R = R,12

, x R + z R = R,13

, y R + z R = R,23

.

If G is denominator reducible, then Theorem 23 implies that: eL

G (α1 , . . . , αe L ) = ( R )2−deg ρ L ρ L (x R , y R , z R ).

It follows in particular that if ρ L is of degree 0 or is a perfect square, then G has a weight drop.


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There are a limited number of possibilities for the polynomials ρ L . This gives rise to families of weight-preserving operations as follows. Corollary 39. Suppose that ρ L 1 = ρ L 2 , and let G = L 1 ∪3 R and G 2 = L 2 ∪3 R for any R. Then the denominator reductions of G 1 and G 2 are the same after reducing out all the edges in L 1 and L 2 : eL 1

G 1 (α1 , . . . , αe L 1 ) = e L 2 G 2 (α1 , . . . , αe L 2 ).

Thus G 1 has a weight drop after reducing with respect to the edges E(L 1 ) ∪ S, where S ⊂ E(R), if and only if G 2 has a weight drop after reducing with respect to E(L 2 ) ∪ S. We begin a classification of graphs L and compute their polynomials ρ L as follows. In the following diagrams, the white vertices are v1 , v2 , v3 from top to bottom, and the polynomials ρ L are indicated underneath. At 5 edges, there are only two possibilities for L which have neither a double edge nor a two-valent vertex (these are the simple 5-local minors in the terminology of [4]):

Note that graphs 51 and 52 have a split triangle (resp. split 3-valent vertex), so if they occur in a graph, we can, as noted after Theorem 35, reduce to a smaller graph, except in trivial cases where the extra edges are not available. At 6 edges, there are exactly six such 6-local minors:

The graphs 61 and 66 have weight drops, and we obtain the first identity: ρ63 = ρ65 . However, most of the above graphs (except for 62 and 66 ) contain a double-triangle or

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split 3-vertex and do not tell us anything new. From now on, we only consider graphs which do not contain a double triangle, rather than giving the complete list. It turns out that at 7 edges, we obtain polynomials ρ L which have already appeared above, except for the following graph:

Similarly, at 8 edges, we get a new identity for the graphs:

neither of which is amenable to a double-triangle type reduction. We conclude that ρ8a = ρ8b . We conclude with a few more examples to illustrate the general principle:

Thus ρ10a has weight drop, and we get an identity ρ10b = ρ62 . One can continue generating larger and larger graphs L (provided they are denominator-reducible) and compute their polynomials ρ L , giving rise to more and more complicated identities of the form ρL 1 = ρL 2 . 4.7. Examples of 3-connected operations. The previous discussion enables one to prove results about graphs which are inaccessible by double-triangle type arguments. Corollary 40. The following graphs have a weight drop:


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The same result holds more generally for K 3,4 ∪3 R or G ∪3 R, where R is any 3-vertex reducible graph connected to the 3 white vertices. Proof. The graph K 3,4 = 62 ∪3 62 on the left is denominator reducible and has a weight drop by direct calculation [4]. Since ρ10b = ρ62 it follows that G = 10b ∪3 62 also has a weight drop by Corollary 39. The last statement follows from a similar argument, after noting (again by direct computation) that ρ = 0.

K 3,4

Now we can consider the graphs obtained by gluing 8a and 8b together. There are three possibilities which preserve the symmetries of the ρ polynomials: 8a ∪ 8a , 8a ∪ 8b , 8b ∪ 8b

(4)

Arguing as above we conclude that all 3 have the same weight. Remark 41. It is known that two graphs G 1 , G 2 , which, when completed by adjoining a new vertex to all 3-valent vertices give rise to the same graph, necessarily have the same period. O. Schnetz has shown this nicely in [7]. In particular, they have the same weight. This should imply that G 1 has a weight drop in the denominator sense if and only if G 2 does. Can one find a combinatorial proof of this fact? Remark 42. We obtained a list of 3-vertex connected operations on graphs simply by calculating their universal polynomials ρ. Is it possible to do the same for graphs with higher degrees of vertex connectivity? In other words, for every n ≥ 3, is there a finite list of ‘right-hand sides’ R1 , . . . , R Nn such that if a property holds for all L ∪n Ri for 1 ≤ i ≤ Nn then it holds for all graphs L ∪n R? In the 3-vertex connected case we have shown that N3 = 1. It would be interesting to draw up a list of 4-vertex connected weight-preserving operations, of which the triangle splitting operation is one example. The results above are almost sufficient to explain all known weight-drops. Of the graphs up to 8 loops which have been calculated to be weight drop, the application of Theorem 35 and Proposition 36 without any further identities explains all but seven of the graphs. One of these can be explained immediately by planar duality. Two more of them are K 3,4 and G from Corollary 40. Three more are the graphs of (4) which all must have the same weight. The remaining graph is

(5)

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So the methods of this paper suffice to prove all known weight drops except for two. Fortunately, these graphs are amenable to the denominator reduction algorithm. Alternately, if we further allow ourselves Schnetz’ completion and twist operations, which preserve the period [7], but which are not understood in this language of graph polynomials, then the graphs of (5) and (4) must all have the same value as G from Corollary 40, giving weight drop for all of them. Acknowledgement. Very many thanks to S. Bloch, D. Broadhurst, D. Kreimer, and O. Schnetz for discussions and enthusiasm.

References 1. Bloch, S., Esnault, H., Kreimer, D.: On motives associated to graph polynomials. Commun. Math. Phys. 267, 181–225 (2006) 2. Bogner, C., Weinzierl, S.: Feynman graph polynomials. http://arXiv.org/abs/1002.3458v3 [hep-th], 2010 3. Broadhurst, D.J., Kreimer, D.: Knots and numbers in φ 4 theory to 7 loops and beyond. Int. J. Mod. Phys. C 6, 519–524 (1995) 4. Brown, F.: On the periods of some Feynman integrals. http://arXiv.org/abs/0910.0114v2 [math.AG], 2010 5. Chaiken, S.: A combinatorial proof of the all minors matrix tree theorem. SIAM J. Alg. Disc. Meth. 3(3), 319–329 (1982) 6. Kontsevich, M., Zagier, D.: Periods. In: Mathematics Unlimited–2001 and Beyond. Berlin-HeidelbergNew York: Springer, 2001, pp. 771–808 7. Schnetz, O.: Quantum periods: A census of φ 4 -transcendentals. Comm. Numb. Th. 4(1), 1–48 (2010) 8. Zeilberger, D.: Dodgson’s determinant-evaluation rule proved by two-timing men and women. Elec. J. Combin. 4(2), R22 (1997) Communicated by A. Connes


Communications in


Local Well-Posedness for Membranes in the Light Cone Gauge Paul T. Allen1,2 , Lars Andersson1 , Alvaro Restuccia3 1 Albert Einstein Institute, Am Mühlenberg 1, 14476 Potsdam, Germany. E-mail: [email protected] 2 Department of Mathematical Sciences, Lewis & Clark College, 0615 SW Palatine Hill Road, Portland,

OR 97219, USA. E-mail: [email protected]

3 Department of Physics, Simon Bolivar University, Caracas, Venezuela. E-mail: [email protected]

Received: 17 November 2009 / Accepted: 14 March 2010 Published online: 17 October 2010 – © Springer-Verlag 2010

Abstract: In this paper we consider the classical initial value problem for the bosonic membrane in light cone gauge. A Hamiltonian reduction gives a system with one constraint, the area preserving constraint. The Hamiltonian evolution equations corresponding to this system, however, fail to be hyperbolic. Making use of the area preserving constraint, an equivalent system of evolution equations is found, which is hyperbolic and has a well-posed initial value problem. We are thus able to solve the initial value problem for the Hamiltonian evolution equations by means of this equivalent system. We furthermore obtain a blowup criterion for the membrane evolution equations, and show, making use of the constraint, that one may achieve improved regularity estimates. 1. Introduction The initial value problem for the classical evolution of a physical system seeks to characterize critical points of the action functional associated to the problem in terms of an appropriate set of initial data. Once the well-posedness of the classical field equations has been shown, the initial data not only determines the classical motion for some time interval (0, T ), where T is the time of existence corresponding to the data, but also the Hilbert space of wave functions of the corresponding quantum-mechanical system. For physical systems without gauge symmetries described by conjugate pairs (x, p) satisfying Hamilton’s equations, the initial data is given directly by specifying the conjugate pair (x0 , p0 ) at an initial time. The wave functions, in the Schrödinger picture, are precisely the space of functions ϕ(x0 ) with ϕ ∈ L 2 . In the presence of gauge symmetries the initial data is restricted by constraints and gauge-fixing conditions. One may solve these restrictions at the classical level in terms of conjugate pairs and then determine the wave function as before, or one may consider general wave functions ϕ(x0 ), ϕ ∈ L 2 , and restrict them at the quantum level to a subspace H ⊂ L 2 ; the domain of the quantum operators should then be dense in H. In this latter case, one may then extend the phase space in order to realize the BRST symmetry

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of the quantum system. A key point in both procedures is to determine both the classical and quantum restrictions associated to the gauge symmetries. The formulation of the initial value problem for gauge theories, including Einstein gravity, electromagnetism, Yang-Mills, string, and membrane theories, as well as their supersymmetric extensions, is formally solved by the approach of Dirac [19].1 It determines in a constructive and systematic manner, the constraints associated to the gauge theory. Moreover the procedure ensures that the constraints are preserved in time by the Hamiltonian flow, with the Lagrange multipliers associated to second-class constraints determined by the conservation procedure and Lagrange multipliers associated to firstclass constraints remaining as gauge-dependent variables. The resulting Hamiltonian formulation of the gauge theory ensures that if the constraints are satisfied initially, then they are also satisfied on (0, T ), for some time of existence T . The first-class constraints close as an algebra under a Poisson bracket constructed from the symplectic structure of the Hamiltonian formulation. The generators of this algebra realize the gauge symmetry of the action functional, and consequently of the field equations themselves. (For a detailed exposition of this systematic approach in the case of the Einstein gravity, including a discussion of first- and second-class constraints, see [30].) A well-known gauge choice of the above mentioned field equations is the light cone gauge (LCG). In this gauge, one can solve the constraints, together with the gauge fixing conditions, in terms of unconstrained physical degrees of freedom. This property of the LCG becomes very useful when proving relevant properties of the corresponding quantum theories, one example being arguments concerning the unitarity of the S-matrix in superstring theory. A treatment of Einstein gravitation has been considered in [4,31,43]. In [4] the positivity of the energy and coupling to gauge fields was analyzed. For a review of non-covariant gauges in gauge field theories, including the light cone gauge, see [34]. The LCG has proven particularly fruitful in classical and quantum analyses of the D = 11 supermembrane theory [11,16,27], which is also a a relevant ingredient of M-theory. The supermembrane theory has first-class constraints associated to the generators of diffeomorphisms of the world volume and the local fermionic symmetry, known as κ-symmetry, as well as second-class fermionic constraints. The complete set of constraints is very difficult to treat in a covariant formulation. However, the LCG allows an explicit solution of all first- and second-class constraints. It is also convenient in the case of both membrane and supermembrane theory to fix all symmetries up to area-preserving diffeomorphisms, which in this case are in fact symplectomorphisms (with respect to a symplectic structure defined as part of the gauge choice).2 The resulting Hamiltonian, in LCG with these residual symmetries, may be analyzed without difficulty. The constraint associated to the area-preserving diffeomorphisms may be interpreted as a symplectic generalization of the Gauss law, with nonlinear terms à la Yang-Mills, but arising from the symplectic bracket rather than the bracket of the SU (n) Lie algebra [38]. Once the LCG is implemented in a Hamiltonian formulation of the supermembrane, and the first- and second-class constraints are solved, one obtains directly a canonical Hamiltonian reduction of the original formulation. That is, the elimination of phasespace variables occurs in canonical-conjugate pairs. The Hamiltonian in the LCG is polynomial in the remaining variables and their derivatives and, because it is formulated in terms of physical degrees of freedom (the residual gauge symmetry may be fixed 1 For the formal computations to be properly defined, the well-posedness of the Hamiltonian equations must be established. 2 In even dimensions different from two, these symplectomorphisms are volume-preserving, but not all volume-preserving diffeomorphisms are symplectomorphisms.

Membranes in Light Cone Gauge

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in a convenient manner), it is essentially the same Hamiltonian which appears in the path-integral formulation of the quantum theory. This means that properties of the same potential will determine both classical and quantum aspects of the theory. The classical and quantum stability properties of the membrane or supermembrane theories are determined by the nonlinear dependence of the potential, along the configurations for which the potential becomes zero. These properties have been analyzed by considering a SU (n) regularization of the membrane or supermembrane theory [17,18,27]. This regularization is itself an interesting physical model, and was the starting point in the introduction of the matrix model. The SU (n)-regularized supermembrane is a maximally supersymmetric Yang-Mills theory in 1+0 dimensions [15], the classical field equations being ordinary differential equations and the corresponding quantum problem finite dimensional. The regularized membrane potential has “valleys” extending to infinity; the value of the potential at the bottom of these valleys is zero. Thus the static solutions of the equations of motion with zero value of the potential are unstable. Nonetheless, the quantum-mechanical Hamiltonian of the membrane theory has a discrete spectrum, due to the structure of the valleys. In dimensions greater than one, the discreteness of the spectrum of the regularized Hamiltonian, a Schrödinger operator, is determined by the behavior of the mean value of the potential in the sense of Molˇcanov [39,41]. This mean value tends to infinity as one moves outward along the valleys in configuration space, thus ensuring that the operator has discrete spectrum [25,36,47]. In the supermembrane case, the potential becomes unbounded from below, due to Fermionic contributions, in a manner which renders the spectrum continuous [17]. However, the supermembrane with central charges generated by the wrapping of the supermembrane on a compact sector of the target space has discrete spectrum [14]. In fact the topological condition ensuring the nontrivial wrapping, which does not modify the number of local degrees of freedom, eliminates the nontrivial configurations with zero potential. Besides this interesting relation between classical and quantum stability properties of the membrane and supermembrane theories, there are other aspects of classical supermembrane theory which reproduce quantum α effects of string theory. (That is, perturbative quantum effects in string theory where the perturbative parameter is the inverse of the string tension.) The closure of the κ symmetry in supermembrane theories with a general background metric on the target space is only possible provided the background satisfies the D = 11 supergravity equations [12]. The analogous result for superstring theory arises only when α quantum effects are taken into account. Furthermore, the IIA D-brane action in D = 10 may be obtained from a duality transformation from the D = 11 supermembrane compactified on a circle. The same D-brane action arises from Dirichlet strings only when quantum effects are considered [44,49]. In light of these considerations, it is natural to analyze the classical initial-value problem of the membrane and supermembrane in the LCG. This not only provides a foundation for the study of the quantum mechanical systems corresponding to the membrane and supermembrane in LCG, by putting the classical theory on a firm basis but, in establishing a criterion for continuing the classical solution in time, is an important first step towards the identification and characterization of whatever singularities may develop. This is interesting not only from a classical perspective, but also from a quantum mechanical one, as singularities are expected to play an important role in the quantum theory [2,6]. It may also provide a framework to analyze the large n limit of the regularized theories and the related stability problems, one may hope to extrapolate consequences for the quantum stability problem.

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From a mathematical perspective, the problem is interesting as the membrane equations, which correspond to the supermembrane field equations after the spinor dependence (i.e., the fermionic sector) has been annihilated, are one case of an important class of geometric wave equations. Geometrically, membranes are timelike submanifolds with vanishing mean curvature; the equation governing this condition is the Lorentzian analogue of the minimal submanifold equations (much as wave maps are the Lorentzian analogue of harmonic maps). Attempts to approach the problem of existence of such submanifolds by applying techniques from the theory of differential equations are complicated by the inherent diffeomorphism invariance of the problem; in an arbitrary coordinate system the equations are not strictly hyperbolic (i.e., wave equations). As in the case of the mathematical study of the Einstein field equations, this difficulty can be overcome by choosing a gauge, which eliminates (or at least reduces) the diffeomorphism freedom and yields a system of equations to which PDE theory can be applied. In fact, there are actually two levels at which the mathematical problem of local existence can be posed. As the equations governing the embedding are wave-type equations, one expects to pose an initial value problem. The first is at the level of geometry: Given an initial spacelike submanifold and timelike vectorfield on the submanifold, can one extend the submanifold in the direction of the vectorfield such that the mean curvature of the extension vanishes? The second is at the level of embedding functions: Given an embedding function for an initial spacelike slice, an initial ‘velocity’ for that function, and a gauge condition, can one find an embedding function satisfying the relevant PDE (as expressed under the gauge condition)? In either case, one would also like to show Cauchy stability as well: that not only do local solutions exist, but that they are unique and depend continuously upon the given data. The problem of local well-posedness at the level of geometry has been recently addressed in a very general setting in [40]. At the level of the PDE, that work makes use of a variation of the harmonic gauge condition, which was first applied to the membrane problem in a Minkowski ambient spacetime by [7], and has been used extensively in the study of the initial value problem for the Einstein equations ([23], see also [22]). These works provide a very satisfactory resolution to the geometric local-existence problem, as well as a rather complete solution to the local existence problem for the PDE in harmonic coordinates. It leaves open, however, the problem of local existence of embedding functions satisfying other gauge conditions which may be better-suited for addressing questions of the lifespan of solutions (and/or singularity formation of solutions) or questions arising when considering aspects of the quantum problem. In fact, relatively little is known concerning the existence of solutions to the PDE when reduced by even the simplest gauge conditions. A Hamiltonian reduction under the partial gauge condition that the time coordinate of the submanifold coincide with the Minkowski time coordinate was considered in [42] (see also [26]). Under this choice of foliation, the equations for codimension-1 membranes reduce to a first-order system. A large number of examples of solutions satisfying this gauge condition have been constructed; see for example [28]. In this work we address the well-posedness of the membrane field equations in the LCG. In particular, we show that for any non-degenerate initial data satisfying the constraints there exists a time interval (0, T ), with T depending on the initial data, and a unique solution to equations of motion of the reduced (in LCG) Hamiltonian corresponding to the initial data. Furthermore, both T and the solution depend continuously on the initial data in a suitable topology. The result is obtained by application of the theory of hyperbolic partial differential equations.


387

The system of field equations obtained by taking variations of the membrane action in light cone gauge is a fully non-linear system which is not, however, hyperbolic. We therefore introduce a modified system which is quasi-linear and hyperbolic, and has the property that solutions of this modified system satisfy the constraints and field equations if these are satisfied initially. It would be interesting if this modified system has a (super-) membrane action associated to it, as it has additional constraints which are preserved by evolution and which should appear in any quantum formulation of such a theory. The existence of such preserved constraints is an indication that there may be a gauge theory, with gauge symmetries generated by both constraints, which under some appropriate gauge fixing reduces to the modified system we employ. The modified system is obtained by differentiating the LCG field equations with respect to time, and eliminating terms which vanish due to the constraints. Similar techniques have been used to extract hyperbolic systems for the Einstein and Yang-Mills equations without gauge fixing, see [1] and references therein. We are able to show the existence of solutions to the modified equation using a standard argument based on energy estimates. The structure of the modified system is such that the normal procedure for obtaining energy estimates, commuting spatial derivative operators through the equation, leads to a loss (see commutator estimate (5.11c)). We recover this loss by means of an elliptic estimate and by commuting a time derivative through the equation. The result of our method is a local existence result which requires slightly more regularity than expected for arguments based on the Sobolev embedding. We are nevertheless able, by making use of the constraint, to obtain an improved energy estimate for solutions to the original equation. With this improved energy estimate we are able to show that the time of existence depends on the “classically-expected” norm of the initial data. In this paper we work in terms of integer order Sobolev spaces only. By making use of more sophisticated techniques, the results presented here can be improved as far as the regularity requirements are concerned. The algebraic structure of the reduced field equation, makes it interesting to ask for the optimal well-posedness result from the point of view of the regularity of initial data. The matrix analog of the system gives an ODE analog of the membrane system, which has been extensively studied (see for example [5,13,29]). It is interesting to consider this from an analytical point of view as a consistent truncation of the system, and to make use of related ideas to study the local well-posedness of the system for rough initial data. The organization of the remainder of this paper is as follows. In the next section we outline our notational conventions, and list a number of functions spaces and related estimates appearing in the local existence proof. In the subsequent Sect. 3, we introduce the Lagrangian formulation of the membrane problem and perform a Dirac-style canonical analysis of the membrane problem, deriving the reduced equations of motion under the light cone gauge condition. In Sect. 4 we give a treatment of the initial value problem by means of the modified system described above. The modified system is hyperbolic and well-posedness follows along essentially standard lines, with the additional difficulty that the system is fully nonlinear. See [32,35] for treatments of related problems. For completeness, we give a self-contained proof in Sect. 5. Finally in Sect. 6 we derive the improved energy estimate, which gives us the improved estimate for the time of existence. 2. Preliminaries We consider 3-dimensional submanifolds M of D-dimensional Minkowski space R D with M ∼ = R × and some compact 2-manifold.

388


We make use of a number of index sets: Greek indices μ, ν, . . ., ranging over 0, . . . , D − 1, refer to Cartesian coordinates in Minkowski space. Middle Latin indices m, n, . . ., ranging 1, . . . , D −2, refer to Cartesian coordinates on Euclidean space R D−2 . Lower-case early Latin indices a, b, . . . , take values 1, 2 and refer to coordinates on . Upper-case Latin indices A, B, . . ., take values 0, 1, 2 and refer to coordinates on M. We now describe the various coordinates in more detail and indicate which metrics are used to raise/lower each set of indices. In all cases we sum over repeated indices. In Cartesian coordinates (x μ ) the Minkowski metric ημν is given by ημν d x μ d x ν = −(d x 0 )2 + (d x 1 )2 + · · · + (d x D−1 )2 .

(2.1)

Greek indices are raised/lowered using ημν so that xμ = ημν x ν . We also make use of null coordinates x ± = √1 (x 0 ± x D−1 ) for Minkowski space, 2

denoting by (x m ), the remaining coordinates (x 1 , . . . , x D−2 ) in Euclidean space R D−2 . With respect to these coordinates the Minkowski metric is given by η++ = η−− = 0, η+− = η−+ = −1, and ηmn = δmn is the Euclidean metric. Let τ be some global coordinate whose level sets τ foliate M. When there is no confusion, we denote τ by . On each we use (τ -independent) local coordinates (σ a ) with a = 1, 2. Together (τ, σ a ) = (ξ A ) give coordinates on M. Any (embedded) submanifold M ⊂ R D is determined by the function x = (x μ ) : M → R D which induces a metric g AB on M which in local coordinates is given by ∂x μ g AB = ημν ∂ A x μ ∂ B x ν ; here ∂ A x μ = ∂ξ A . In what follows we restrict attention to the The metric case where the induced metric g AB is Lorentzian with ∂τ a timelike √ direction. 0 ∧dξ 1 ∧dξ 2 with g induces a volume form μ , given in coordinates by μ = |g| dξ AB g g √ √ |g| = − det [g AB ] , used below to define the Lagrangian action for the membrane system. We denote by γab the metric on induced by g AB ; we require γab to be Riemannian. Note that γab = ημν ∂a x μ ∂b x ν = −(∂a x + ∂b x − + ∂a x − ∂b x + ) + ∂a x m ∂b xm .

(2.2)

Where we desire to indicate explicitly the x-dependence of γab we write γ (x)ab . The √ metric γab gives rise to a volume form μγ = γ dσ 1 ∧ dσ 2 ; here γ is the determinant of γab . By the usual formula for the inverse of a matrix, the inverse γ ab of γab can be expressed γ ab =

1 ac bd ∈ ∈ γcd . γ

(2.3)

Here ∈ab is the anti-symmetric symbol with two indices. (Explicitly, ∈12 = − ∈21 = 1, ∈11 = ∈22 = 0.) Using the anti-symmetric symbol, the determinant γ can be written as γ =

1 ab cd ∈ ∈ γac γbd . 2

(2.4)

2.1. Symplectic structure. The √the choice of 2a fixed √ light cone gauge condition requires to be (τ -independent) area form w dσ 1 ∧ dσ 2 on . We presume w dσ 1 ∧ dσ√ the area element arising from some fixed background metric wab on ; thus w = √ det wab . We may without loss of generality assume that wab is real analytic.


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The area form gives rise to a symplectic structure on . In local coordinates, the Poisson bracket associated to this symplectic structure is given by ∈ab { f, g} = √ ∂a f ∂b g, w

(2.5)

where f, g are functions on . Recall that the Poisson bracket is bilinear, skew ({ f, g} = −{g, f }), and satisfies the Jacobi identity 0 = { f, {g, h}} + {g, {h, f }} + {h, { f, g}}. (2.6) √ 1 2 In what follows we make use of√the area form w dσ ∧ dσ when integrating on , but denote the form simply by w (i.e. the expression dσ 1 ∧ dσ 2 is to be implicitly understood). Note that Stokes’ theorem implies √ √ { f, g}h w = − f {h, g} w . (2.7)

2.2. Derivatives and function spaces. Here we outline our notational conventions regarding derivatives and introduce some function spaces used in the local existence results below. These spaces are defined with respect to a fixed atlas of coordinate charts on . For function v defined on [0, T ]×, we denote by Dv any coordinate derivative ∂a v. By Dl v, with l a non-negative integer, we mean an arbitrary combination of l coordinate derivatives of v. We furthermore denote by ∂v any of ∂a v or ∂τ v. In a slight abuse of notation, in the presence of norms we implicitly sum over all coordinate derivatives, so that |∂v|2 = |∂τ v|2 + |Dv|2 ;

(2.8)

|Dl v|,

|D∂v|, etc. are defined analogously. The following is an overview of the norms and function spaces used. Sobolev spaces: Denote by H l the Sobolev space of functions on whose derivatives (in a fixed coordinate atlas) of up to order l are square-integrable. Thus √ 2 2√ 2 2 v L 2 = |v| w , v H 1 = v L 2 + |Dv|2 w , etc. (2.9)

L ∞ spaces: The following norms control the (essential) supremum of functions: v L ∞ = ess sup |v|.

(2.10)

When v ∈ C 0 () one can replace ess sup with max . Furthermore define v W 1,∞ = v L ∞ + Dv L ∞ .

(2.11)

Curves in function spaces: The set of maps w : [0, T ] → H l which are r times differentiable with respect to τ is denoted C r ([0, T ]; H l ) and given the norm w C r ([0,T ];H l ) = sup

r

[0,T ] i=0

∂τi w H l .

For functions w ∈ C r ([0, T ]; H l ), we use a subscript to denote restriction to τ = 0. It is important that this restriction is made after any differentiation, so that (for example) |∂w0 |2 = |∂τ w(0)|2 + |Dw(0)|2 .

390


Spacetime norms: We make use of two particular norms for curves of H l functions. For v ∈ C Tl :=

2

C i ([0, T ]; H l−i )

(2.12)

i=0

we make use of the norms v l2 =

2

∂τi v 2H l−i

(2.13)

|v| l,τ = sup v(·) l .

(2.14)

i=0

and [0,τ ]

Note that v l is equivalent to v H l + ∂∂τ v H l−2 . We also use the norms x l2 = x 2H l + ∂τ x 2H l

(2.15)

x l,τ = sup x l

(2.16)

and [0,τ ]

for x ∈ C 1 ([0, T ]; H l ). 2.3. Results from analysis. We make use of the following estimates, which can be proven using classical methods of calculus; see, for example, Chapter 13 §3 of [48]. Note that while versions of these estimates hold in all dimensions, as presented here the estimates are dependent on the dimension of being 2. Here, and in the application of these estimates below, C is a constant independent of the function(s) being √ estimated (and unless otherwise specified depends only on , our coordinate charts, w , and the number of derivatives being estimated). Sobolev inequality: For l > 1 and v ∈ H l we have v ∈ C 0 and v L ∞ ≤ C v H l . Product estimate: For v, w ∈ L ∞ ∩ H l we have vw H l ≤ C v L ∞ w H l + w L ∞ v H l .

(2.17)

(2.18)

Note that this estimate, together with (2.17) implies that x y l ≤ C x l y l provided l > 1. Elliptic regularity: For uniformly elliptic operator L = ∂a [a ab ∂b (·)] + ba ∂a with a ∈ W 1,∞ , ba ∈ L ∞ , there exists a constant C, depending on the norm of the coefficients and the ellipticity constant, such that (2.19) v H 2 ≤ C v L 2 + Lv L 2 .


391

Gagliardo-Nirenberg-Moser estimate: For l < k we have 1− l

l

Dl v L 2k/l ≤ C v L ∞k D k v Lk 2 .

(2.20)

An easy consequence of Hölder’s inequality and (2.20) is the following product estimate for u, v ∈ H 2 : (Du)(Dv) L 2 ≤ C u H 2 v H 2 .

(2.21)

Commutator estimate: The estimate (2.20) also implies the following commutator estimate: [D j , v]w H l ≤ C w L ∞ v H l+ j + Dv L ∞ w H l+ j−1 . (2.22) 3. Canonical Analysis and the Light Cone Gauge We now perform an ADM-style canonical analysis of the membrane system. Beginning with a Lagrangian action integral, we give a brief treatment for the membrane system in general before analyzing in detail the reduction under the light cone gauge condition. 3.1. Lagrangian formulation. We seek an embedding x = (x μ ) : M 1+2 → R D ,

(3.1)

which is critical with respect to the action given by the induced volume element μg . (3.2) S=− M

In string theory (3.2) corresponds to the Nambu-Goto action. An equivalent formulation, expressed in terms of a polynomial Lagrangian density and corresponding to the Polyakov action of string theory, is given by 1 ημν g AB ∂ A x μ ∂ B x ν − 1 μg , (3.3) SP = − 2 M where the inverse metric g AB is treated as an independent variable. Both (3.2) and (3.3) lead to the same Hamiltionian formulation. Critical points of the functional (3.2) give rise to submanifolds M ⊂ R D with vanishing mean curvature; the Euler-Lagrange equations for (3.2) are (3.4) |g| g x μ = ∂ A |g| g AB ∂ B x μ = 0, μ = 0, . . . D − 1, which can be seen by computing

When

√

1 |g| g AB δg AB = |g| g AB ∂ A x μ ∂ B [δxμ ]. δ |g| = 2 |g| = 0, the system (3.4) can be expressed (in any coordinates) as

δμν − g C D ∂C xμ ∂ D xν g AB ∂ A ∂ B x ν = 0.

(3.5)

(3.6)

392


One can interpret (3.4) as evolution equations for x(τ ) : → R D . As the induced metric g AB is Lorentzian, one expects the system to be hyperbolic (i.e., a wave-type equation) and thus to be able to pose the following initial value problem: For initial embedding x0 , and initial velocity u 0 does there exists an interval [0, T ] on which x(τ ) is defined and satisfies x(0) = x0 , ∂τ x(0) = u 0 , and (3.4)? As expressed above, the evolution equations for x μ are not strictly hyperbolic; this is due to the diffeomorphism invariance of (3.2) under re-parameterizations of M. (Note that it is also invariant under choice of coordinates in the Minkowski spacetime, but we have fixed these degrees of freedom.) Thus we turn to the issue of gauge choice by performing a canonical analysis. 3.2. Canonical (Hamiltonian) analysis. The Hamiltonian approach plays a fundamental role in the classical and quantum analysis of field theories. The formulation for gauge theories briefly described below was developed by Dirac in [19]. For a detailed description of this approach applied to classical covariant field theories see [30]; for a description of path-integral quantum analysis in the presence of general constraints see [45]. The starting point, as proposed by Dirac [19], is an action integral expressed in terms of a Lagrangian density L, defined on some time-foliated manifold R × . The density L depends on some number of independent fields φ and their spatial derivatives up to order k, Dφ, . . . , D k φ, as well as on the derivatives of the first time derivative of φ, ∂t φ, D∂t φ, . . . , Dl ∂t φ. One then introduces the Hamiltonian density H via a Legendre transformation H= (3.7) (π ∂t φ − L) ,

where the conjugate momenta π associated to φ, defined as π=

δL , δ(∂t φ)

(3.8)

appear in the functional derivative of the Lagrangian action L = R× L with respect to independent variations δφ and δ(∂t φ):

δL δL δφ + δ(∂t φ) . δL = (3.9) δ(∂t φ) R× δφ The canonical variables φ, π take values in an infinite-dimensional manifold referred to as the phase space of the theory, which is equipped with a Poisson structure given (as a density) by [u, v]P =

δv δu δu δv − . δφ ∂π δφ δπ

(3.10)

The symplectic structure determined by the Poisson bracket plays a fundamental role in both the classical analysis of the field theory, as well as in the canonical quantization of the theory. It is also the main algebraic structure in the deformation quantization approach [10,21,33]. In gauge theories, ∂t φ cannot be expressed in terms of unconstrained momenta as the Hessian of L with respect to ∂t φ becomes singular; thus there are constraints on


393

the phase space for (φ, π ). Further constraints, or restrictions on the Lagrange multipliers associated to these constraints, may arise upon imposing the requirement that the vanishing of the constraints be preserved by evolution under the Hamiltonian flow (see below). These constraints are of two types. First-class constraints are those which commute, on the constraint submanifold (defined to be the submanifold where all first and second class constraints are satisfied), with all other constraints; the Lagrange multipliers associated to these first-class constraints are gauge-dependent fields and remain undetermined during the canonical analysis. Second-class constraints do not commute (on the constraint submanifold) with all other constraints and the associated Lagrange multipliers are determined by the condition that the constraints be preserved. Notice that there is an implicit assumption in the Dirac approach concerning the structure of the constraints: they must be regular. A point x0 is a regular point of φ : X → Y if φ (x0 ) is onto. The constraint φ = 0 is regular when each point of the constraint submanifold {x : φ(x) = 0} is a regular point of φ. Irregular constraints must be treated in a separate manner as the usual theory for Lagrange multipliers assumes regular constraints. Once the Hamiltonian H = H and the constraints have been determined, one can reformulate the action integral in terms of the canonical fields S[φ, π] = π ∂t φ − H − λa C a , (3.11) R×

where C a define the constraints on the phase space and λa are the associated Lagrange multipliers. The canonical Hamiltonian density defined by Hc = Hc = H + λa C a (3.12)

determines the evolution of the canonical fields via the Hamiltonian field equations δHc , δπ δHc ∂t π = [π, Hc ]P = − , δφ ∂t φ = [φ, Hc ]P =

(3.13)

which are obtained by varying (3.11), and are equivalent to the Euler-Lagrange field equations associated to the Lagrangian L. Note that for a general quantity F = f (φ, π, t) one has ∂t F = [ f, Hc ]P + ∂t f

(3.14)

along the Hamiltonian flow. The action (3.11) is also the starting point for the Feynman path-integral formulation of quantum field theory. Under some assumptions on the dependence of H on π , the path integral defined from S[φ, π] is formally equivalent to the one defined from the Lagrangian action integral. The action integral S[φ, π] is invariant under the gauge transformations generated by the first class constraints, this can be easily seen by noting that the procedure above ensures that any first class constraint C has [C, H ]P = 0 on the constraint submanifold. (An interesting feature of diffeomorphism-invariant gauge theories is that H = 0, i.e. Hc = λa C a ). This gauge invariance leads to degeneracies for the classical field equations (3.13); thus one typically performs a (partial) gauge fixing before proceeding to analyze the equations of motion. There is a general method for introducing gauge-fixing terms,

394


and the corresponding Faddeev-Popov terms in the path-integral formulation, without solving the constraints. The resulting effective action becomes BRST-invariant; it may be obtained from the Hamiltonian formulation following [8,24], or from the Lagrangian formulation following [9]. Both approaches present difficulties in the presence of complicated second class constraints, such as those arising in supermembrane theory. In that case, it is most convenient to explicitly solve the constraints at the classical level (in the light cone gauge) and then proceed to quantize the theory. Thus we follow this latter approach in this paper. 3.3. Canonical analysis for membranes. We start from the Lagrangian action (3.3), where x μ and g AB are independent fields. Alternatively, one may start from (3.2); the resulting Hamiltonians are exactly the same. We perform the usual ADM decomposition of g with respect to the foliation τ , √ = −g 00 the lapse, and by N a = N 2 g 0a denoting by γab the metric on , by N = √|g| γ the shift vector. We raise and lower a, b with γab , γ ab ; thus Na = γab N b , etc. To be explicit, the metric g AB and its inverse g AB are given by

N −2 N b −N −2 −N 2 + N c Nc Nb AB [g ] = . (3.15) [g AB ] = Na γab N −2 N a γ ab − N −2 N a N b Treating x μ as canonical variables, we see that the conjugate momenta to x μ are given by √ γ ∂τ xμ − N a ∂a xμ . pμ = (3.16) N Instead of introducing the conjugate momenta to N , Na , and γab , it is convenient to treat them as auxiliary fields. Furthermore, γab may be eliminated as an independent field and expressed in terms of the x μ using γab = ∂a x μ ∂b xμ . We are able to solve (3.16) for ∂τ x μ in terms of pμ , N ∂τ x μ = √ p μ + N a ∂a x μ . γ The canonical Hamiltonian density is therefore given by 1 N 2 p + γ + N a pμ ∂a x μ . H= √ 2 γ

(3.17)

(3.18)

Here p 2 = pμ p μ . There are two constraints 1 2 p + γ = 0, 2 a = pμ ∂a x μ = 0, =

(3.19) (3.20)

with Lagrange multipliers N λ= √ γ

and

λa = N a .

(3.21)


395

The action can now be written S= λ + λa a . p μ ∂τ x μ − M

(3.22)

M

Note that a is given independently of the target (Minkowski) metric, while depends on the ambient metric η. The constraints , a are first-class constraints in the sense of Dirac. The quantities , a are the generators of time and spatial diffeomorphisms, respectively. One can compare the structure of this Hamiltonian to that arising in the theory of general relativity (see, for example [30]); the Hamiltonian has the same linear structure (i.e., it is linear in , a ) since both are invariant under diffeomorphisms. We also point out that has the same quadratic dependence on the momentum as the corresponding time generator in general relativity. 3.4. Light cone gauge. We consider a (partial) gauge fixing of the above action: the light cone gauge. It has the property that the gauge fixing procedure gives rise to a canonical reduction of the above action, and is also the only known gauge where the κ-symmetry constraints of the supermembrane can be explicitly solved. In order to specify the light cone gauge, we make use of the null coordinates (x + , x − , √ m x ) in Minkowski space and also the (τ -independent) volume form w on . The light cone gauge condition for the membrane Hamiltonian is determined by taking 0 0√ x + = − p− τ and p− = p− w, (3.23) 0 is some constant.3 As is made evident below, this is only a partial gauge fixing, where p− the resulting system √ being invariant under diffeomorphisms which are area-preserving with respect to w . Note that under this gauge choice, the metric γab = ∂a x m ∂b xm ; i.e., it does not depend on derivatives of x ± . We now proceed to construct the (partial) gauge-fixed Hamiltonian in light cone gauge. The constraint = 0 may be solved algebraically for p+ in terms of x m and pm ,

1 1 pm p m + γ , p+ = √ c w 2

(3.24)

while the constraint a = 0 determines x − (in terms of x m and pm ) via the relation 1 ∂a x − = − √ pm ∂a x m , c w provided the integrability condition C

(3.25)

pm √ dxm = 0 w

(3.26)

holds for all closed curves C in . The conjugate pairs x + , p+ and x − , p− are thus eliminated provided this condition holds. 3 The constant p 0 is related to the total momentum (in the x − direction) P 0 = − − 0 = P 0 /vol(). Here the volume is measured with respect to √w . p− −

p− by the relation

396


The Poisson bracket analysis of (3.26) shows that it is a first-class constraint generating area-preserving diffeomorphism and is equivalent to the local constraint pm m =0 (3.27) √ ,x w in combination with the global constraint pm √ dxm = 0 w Ci

(3.28)

on some basis {Ci } of the homology of . Together, (3.27)–(3.28) generate area-preserving diffeomorphisms homotopic to the identity. The distinction between them is the following. The left side of (3.27) generates area-preserving diffeomorphisms under the Poisson bracket, with infinitesimal parameter , a time-dependent single-valued function on , i.e. √ pn δ x m = x m , √ , xn w = {x m , }. (3.29) w P The left side of (3.28) generates area-preserving diffeomorphisms with infinitesimal parameter ˆ , where d ˆ is a harmonic 1-form on . Thus we may write pn m m n δˆ x = x , î = {x m , }, ˆ (3.30) √ dx w Ci P where î are time dependent functions such that d ˆ = î ωi for some basis ωi of harmonic 1-forms on normalized with respect to the homology basis {Ci }. The constraint (3.28) is used in the quantum theory as a matching level condition. We may now determine N and N a so that the gauge conditions are preserved under evolution in τ . Ensuring that x + = −cτ by requiring that x + + cτ, (λ + λa a ) + c = 0 (3.31)

P

we find that √ γ N=√ . w

(3.32)

In order that the second light cone gauge condition be preserved, we require that √ p− − c w , (λ + λa a ) = 0, (3.33)

P

which leads to the condition √ ∂a [ w N a ] = 0.

(3.34)

√ Let = 21 w ∈ab N a dσ b ; the condition (3.34) is equivalent to d = 0, i.e. that is closed. Consequently, on each contractible domain in , we have that is (locally)


397

exact: = d f for some locally defined function f . The one form may be globally decomposed, uniquely, into its harmonic and exact parts = h + dexact .

(3.35)

In any local coordinates, = (h a + ∂a exact ) dσ a . Thus ∈ab N a = √ (h b + ∂b exact ) . w

(3.36)

The form h may be expressed as h = dharmonic for some multi-valued function harmonic ; write = harmonic + exact . Note that while is multi-valued, d is a well-defined geometric object. It is also useful to keep in mind that d x m are exact one forms, as they are single-valued functions → R D . This need not be true if the target space R D is replaced by a manifold with non-trivial topology. The light cone action may be obtained from (3.22) by noticing that (3.19)–(3.20) hold and thus μ S= pm ∂τ x m − cp+ + ∂τ [ p− x − ] , p μ ∂τ x = (3.37) M

M

where we may drop the last term as it is a total derivative. The reduced Hamiltionian in light cone gauge is then given by H=

√

w

1 pm p m 1 √ 2 + |{x, x}|2 + pm {, x m }, 2 w 4

(3.38)

where |{x, x}|2 = {x m , x n }{xm , xn } and we have made use of the expression (2.4) for γ . Notice that the constraints (3.19)–(3.20) are implemented in the action by the reduction procedure. The last term in H is well-defined as it is expressed in terms of d. In contrast, the expression { pm , x m } is ill-defined. We now verify that the term pm {, x m } indeed corresponds to a Lagrange multiplier term multiplied by the constraints. The harmonic one form h may be expressed in terms of a basis {ωi } of harmonic 1-forms4 . The basis contains 2g elements, where g is the genus of . Thus h = λi ωi , i = 1, . . . , 2g,

(3.39)

where the coefficients λi are functions only of τ . Making use of the bilinear Riemann identities (see, for example [20]) we have

pm {, x m } = −

√ pm pm exact √ , x m w + λi √ dxm w w Ci

(3.40)

and arrive at the constraints (3.27)–(3.28). We furthermore can interpret exact and the λi as the Lagrange multipliers associated to these constraints; thus our remaining gauge freedom lies in the choice of these functions. 4 Defined eg. with respect to w . ab

398


We now turn to the equations of motion associated to the reduced Hamiltonian (3.38). Defining Dτ = ∂τ + {·, }, they are pm Dτ x m = ∂τ x m + {x m , } = √ , w

∂τ p m pm pm = √ + √ , = {{xm , xn }, x n }. Dτ √ w w w

(3.41) (3.42)

Note that the constraints (3.27)–(3.28) are preserved under this evolution. This system transforms covariantly under area-preserving diffeomorphisms provided transforms appropriately. We see that under area-preserving diffeomorphisms generated by (3.27)–(3.28), we have dδ = d∂τ ζ + d{ζ, } = d(Dτ ζ ),

(3.43)

where ζ is + ˆ as above. The harmonic part of d transforms as δλi = ∂τ î .

(3.44)

δexact = ∂τ + { + ˆ , exact + harmonic },

(3.45)

The exact part of d transforms as

where we have used that d{ + ˆ , harmonic } is an exact 1-form. 4. The Initial Value Problem The gauge freedom present in the system allows us to fix exact and the λi . Making the simple choice of setting each of these functions to zero, we see that Hamilton’s equations of motion reduce to the second-order system (see also [27]) ∂τ2 x m = {x m , x n }, xn .

(4.1)

It is interesting to note that under these choices, which fix î and up to time-independent parameters, the coordinate function τ is in fact harmonic: g τ = 0, i.e., this the co-moving gauge. The remainder of this paper is devoted to studying the initial value problem for classical membranes in light cone gauge, as formulated in (4.1): For functions (x0 , u 0 ) defined on we show the existence of a function x : [0, T ] × → R D−2 satisfying (4.1) and x(0) = x0 ,

∂τ x(0) = u 0 .

(4.2)

We require that the initial data (x0 , u 0 ) lie in an appropriate function space, as well as satisfy an appropriate version of the constraints in order that {∂τ x m , xm } = 0 be satisfied by the corresponding solution x.

(4.3)


399

4.1. The degenerate hyperbolic system (4.1). The main difficulty presented by the system (4.1) is that it is not strictly hyperbolic. First, note that for fixed x, the operator 1 1 y → {y m , x n }, xn = √ ∂a √ γ (x) γ (x)ab ∂b y m w w

(4.4)

is elliptic with symbol w1 γ (x) γ (x)ab . However, when the operator (4.4) is applied to x itself, one must consider L : x → {x m , x n }, xn .

(4.5)

The first variation of L is given by δL[x]y m = {{y m , x n }, xn } − {{y n , xn }, x m } + 2{{x m , x n }, yn },

(4.6)

where we have made use of the Jacobi identity (2.6). The first term is the elliptic operator (4.4), but the second term also contributes to the symbol and thus δL(x) need not be strictly elliptic even if the metric γ (x)ab is Riemannian. Note that by the constraint (4.3), the second term in (4.6) vanishes when we take y = ∂τ x. Thus Eq. (4.1), together with the constraint (4.3), imply a non-degenerate hyperbolic equation for u = ∂τ x, which we use below to construct the solution x. We also make use of the constraint when estimating solutions to the main equation (4.1), once their existence has been established. Returning to the operator L, we compute in local coordinates L(x)m = {{xm , x n }, xn } 1 ab ab γ γ δmn − mn ∂a ∂b x n + lower order terms, = w

(4.7)

ab =∈ac ∈bd ∂ x ∂ x . Writing where mn c m d n

L ab mn =

1 ab ab γ γ δmn − mn w

(4.8)

we have m n L ab mn Va Vb

2

1 2 ab m γ |V |γ − ∈ ∂a xm Vb = w

(4.9)

for the vector Vam ∈ R2 × R D−2 . The first term appearing in L ab mn is diagonal and positive-definite when γab is Riemannian, but the second term can cause the symbol to be degenerate. For example, consider a (local) situation with x 1 = σ 1 , x 2 = σ 2 , V21 = 1 and all other components of x and V zero. These degeneracies associated to L prevent us from constructing solutions by direct application of standard energy methods, as the energy-type quantities associated to L cannot be shown to adequately control approximate solutions. Thus we approach the initial value problem by considering a modified system, motivated by the linearization presented above, which we now describe.

400


4.2. The modified system. Differentiating (4.1) with respect to τ , which has the effect of linearizing the system, and making use of the Jacobi identity (2.6) we obtain ∂τ2 ∂τ x m = {{∂τ x m , x n }, xn } + 2{{x m , x n }, ∂τ xn } − {{∂τ x n , xn }, x m }.

(4.10)

When the constraint (4.3) is satisfied, the third term on the right vanishes; the remaining terms, when viewed as an operator acting on ∂τ x, are non-degenerate. We take advantage of this structure in the following manner. For functions x, u let D = D(x, u) = {u m , xm }, Jm = Jm (x, u) = ∂τ u m − {{x m , x n }, xn }, A[x]u m = {{u m , x n }, xn } + 2{{x m , x n }, u n }.

(4.11) (4.12) (4.13)

A computation making use of the Jacobi identity and integration by parts shows that d dt

√ D2 + |J|2 D{u m , ∂τ xm } + Jm (∂τ2 u m − A[x]∂τ x m ) w =2 √ + (D − {∂τ x m , xm }){∂τ u m , ∂τ xm } w . (4.14)

Thus if x, u satisfy

∂τ x = u, ∂τ2 u = A[x]u,

(4.15)

then the conditions D = 0, Jm = 0 are preserved if initially satisfied. In particular, the function x is a solution to the main equation (4.1) satisfying the constraint (4.3). We thus approach the initial value problem for Eq. (4.1) by considering the modified system (4.15) with initial data x(0) = x0 , u(0) = u 0 , ∂τ u(0) = u 1 .

(4.16)

In order that the solution (x, u) to the modified system (4.15) give rise to a solution of the main equation (4.1), we require5 n D0 = {u m 0 , x 0 }δmn = 0, and

J0m

=

um 1

− {{x0m , x0n }, x0l }δnl

= 0.

(4.17) (4.18)

In the local existence result stated below, we require initial data with x0 ∈ H k , u 0 ∈ H k , and u 1 ∈ H k−1 . The condition (4.18) imposes an extra regularity condition on x0 . Note, however, that due to the degeneracy discussed in §4.1 the condition (4.18) does not imply x0 ∈ H k+1 .

5 One should also impose the global constraint condition Ci u 0 · d x 0 = 0 on the data in order to have a

solution to the Hamiltonian system; however, this condition does not play a role in our method for constructing solutions to the reduced equations (4.1).


401

Provided γ (x)ab is non-degenerate, the operator A[x] is (quasi-diagonal) elliptic which in local coordinates can be written in divergence form: γ (x) m ab m A[x]u = ∂a γ (x) ∂b u w ab cd ∈ ∈ + ∂a 2 √ √ ∂c x m ∂d x n ∂b u n w w cd ab ∈ ∈ γbd ∂a u m + 2∂a x m ∂b x n ∂d u n . − √ ∂c √ (4.19) w w Note that by the anti-symmetry of ∈ab the second line does not contain second derivatives of u. In particular, the symbol of A[x] is γ (x) γ (x)ab . For notational convenience we denote the third line of (4.19) by b[x]u = b(x)a ∂a u, and the sum of the second and third lines by B[x]u = B(x)a ∂a u. Since γ (x)ab = ∂a x n ∂b xn , the operator can be schematically written in two forms, A[x]u = ∂a γ γ ab ∂b u + B[x]u ∼ D (Dx)2 Du = ∂a

+ (D 2 x)(Dx)Du + (Dx)2 Du Aab ∂b u + b[x]u ∼ D (Dx)2 Du + (Dx)2 Du

(4.20) (4.21)

(recall D represents spatial derivative); here ∈ab ∈cd ab Aab = Aab √ ∂ c x m ∂d x n . mn = δmn γ (x) γ (x) + 2 √ w w

(4.22)

4.3. Well-posedness results. We show the existence of a solution (x, u) to the modified system (4.15); by the discussion above if the constraints (4.17)–(4.18) are initially satisfied, this leads to a solution x to the main equation (4.1) satisfying the constraint (4.3). This procedure implies no restriction on the solution, in particular all solutions of the membrane initial value problem with the appropriate regularity can be obtained in this way. τ Our approach views x = x0 + 0 u as a functional of u; thus x and u are required to have the same degree of spatial regularity. Theorem 4.1. Let k ≥ 4 and let x0 , u 0 , u 1 ∈ H k × H k × H k−1 such that γ (x0 )ab is a non-degenerate Riemannian metric, and such that (4.17)–(4.18) are satisfied. Then we have the following. (1) There exists T > 0, depending continuously on the norm of the initial data and unique (x, u) ∈ C 1 ([0, T ]; H k ) × C Tk satisfying (4.15)–(4.16). In particular, x ∈ C 1 ([0, t]; H k ) is a solution to (4.1)–(4.2) which satisfies (4.3). (2) Let T∗ be the maximal time of existence for (x, u) as given above. Then either T∗ = ∞ or

sup γ −1 L ∞ + Dx W 1,∞ + Du L ∞ = ∞. (4.23) [0,T∗ )

402


Remark 4.1. An adaption of standard arguments, see [3, §2.3] and citations therein, shows that the solution map H k ×H k ×H k−1 → C 1 ([0, T ], H k ) given by (x0 , u 0 , u 1 ) → (x, u) ∈ C 1 ([0, T ], H k ), is continuous. Thus the initial value problem for the system (4.17)–(4.18) is strongly well-posed. 5. Proof of Theorem 4.1 In this section we prove the existence of a solution (x, u) as in Point 1 of Theorem 4.1 to the initial value problem for the modified system (4.15)–(4.16), and establish the continuation criterion stated in Point 2 of that theorem. First, we prove energy estimates for the linear system ∂τ2 − A[x] = F, (, ∂τ )τ =0 = (u 0 , u 1 ), (5.1) associated to (4.15). Once such energy estimates have been established, a standard sequence of arguments (see [37,46] for example) implies the theorem. Below we use expressions like ∂0 L 2 , 0 l to denote norms calculated in terms of the initial data at τ = 0. For the higher order norms, the higher order τ -derivatives are calculated formally. 5.1. The linear system. We consider first the linear system (5.1) for some fixed x ∈ C 1 ([0, T ]; H k ), k ≥ 4, be such that the metric γ (x)ab is non-degenerate on [0, T ]. In this section we generally suppress the x-dependence of A, γ , and other quantities defined below. We furthermore denote by any quantity which can be bounded by a constant times Dx W 1,∞ + D∂τ x L ∞ . Let T = sup[0,T ] . Define the energy E = E[],

√ 1 |∂τ |2 + γ γ ab ∂a m ∂b m w. (5.2) E(τ ) = 2 Define E = E [x] to be the smallest constant such that 2 2 2 −2 E ∂ L 2 ≤ E[] ≤ E ∂ L 2

(5.3)

on [0, T ] for all v. Note that E is bounded when Dx L ∞ is. Differentiating with respect to τ and integrating by parts we estimate ∂τ E ≤ ∂τ L 2 F L 2 + 2 D 2L 2 .

(5.4)

Integrating (5.4) and applying Grönwall’s lemma yields the standard basic energy estimate. Lemma 5.1. If ∈ C 1 ([0, T ]; L 2 ) ∩ C 0 ([0, T ]; H 1 ) solves (5.1) with F ∈ L 1 ([0, T ]; L 2 ) then τ τ 2 0 ∂0 L 2 + (5.5) ∂ L 2 ≤ E e F L 2 . 0


403

This estimate is used below to construct estimates for l using the identity ∂τ2 (Dl ∂τ ) − A(Dl ∂τ ) = Dl ∂τ F + Dl , A ∂τ + Dl ([∂τ , A] ) .

(5.6)

Applying the energy estimate (5.5) to the identity (5.6) yields the following: τ 2 ∂∂τ H l ≤ E e 0 ∂∂τ 0 H l τ

∂τ F H l + Dl , A ∂τ L 2 + Dl ([∂τ , A] ) L 2 . +

(5.7)

0

In order to obtain an energy estimate which closes, we need an estimate for spatial derivatives of . This is accomplished by use of the elliptic estimate (2.19) which implies

Dl L 2 ≤ Dl−2 L 2 + ADl−2 L 2 .

(5.8)

Making use of the linear equation (5.1) we have

Dl L 2 ≤ Dl−2 L 2 + F H l−2 + Dl−2 ∂τ2 L 2 + Dl−2 , A L 2 .

(5.9)

Combining this with (5.7) yields l ≤ (1 + T ) E e

τ 0

2

0 l + F(0) H l−2 + Dl−2 , A L 2 τ

∂τ F H l−2 + l−1 + 0 + Dl−2 , A ∂τ L 2 l−2 (5.10) + D ([∂τ , A] ) L 2 .

We estimate the commutator terms as follows: Lemma 5.2. Suppose ∈ H 2 , then [D, A] L 2 ≤ Dx H 3 H 2 , D ([∂τ , A]) L 2 ≤ Dx 3 H 2 .

(5.11a) (5.11b)

If furthermore ∈ H l for 3 ≤ l ≤ k, then by (2.17) D ∈ L ∞ and [Dl−2 , A] L 2 ≤ D H l−2 + Dx H l−1 D L ∞ , Dl−2 ([∂τ , A]) L 2 ≤ D H l−1 + Dx l−1 D L ∞ .

(5.11c) (5.11d)

404


Proof. Direct computation shows [D, A] L 2 ≤ (D 2 A)(D) L 2 + (D A)(D 2 ) L 2 + (Db)(D) L 2 .

(5.12)

Application of (2.21), (2.18) and (2.17) implies (5.11a). Similar considerations imply (5.11b). To see (5.11c) write [Dl−2 , A] = Dl−2 D, A D − (D A)Dl−2 D + Dl−2 (bD) − bDl−2 D. (5.13) By the product estimate (2.18) we have A H l−1 ≤ Dx H l−1 ,

b H l ≤ Dx H l−2 ,

(5.14)

bD H l−2 ≤ D H l−2 + Dx H l−2 D L ∞ .

(5.15)

and

the second of which implies

The estimate (5.11c) follows from the commutator estimate (2.22). Finally, the product estimate (2.18) and Sobolev inequality (2.17) imply (5.11d). As an immediate application of (5.11) we have [D, A] ∂τ L 2 ≤ Dx H 3 ∂τ H 2 Dl−2 , A ∂τ L 2 ≤ D∂τ H l−2 + Dx H l−1 D∂τ L ∞ .

(5.16)

Furthermore

τ [D, A] L 2 ≤ Dx H 3 0 H 2 + 3 , 0

τ Dl−2 , A L 2 ≤ 2 D0 H l−2 + D∂τ H l−2 0

τ + Dx H l−1 D0 H 2 + D∂τ H 2 , (5.17) 0

where we have used the Sobolev inequality (2.17) in the last line. Applying the commutator estimates above to (5.10), followed by application of Grönwall’s lemma, yields the following energy inequalities. Lemma 5.3. A solution to the linear system (5.1) can be estimated for τ ∈ [0, T ] by

τ E (1+τ )τ Dx 3,τ 3 ≤ E (1 + T )e (1 + Dx H 3 ) 0 3 τ (5.18) + F(0) H 1 + ∂τ F H 1 0

and, for k ≥ 4, by k ≤ E (1 + T )e

τ E (1+τ )τ Dx k−1,τ

+ F(0) H k−2 +

τ 0

(1 + Dx H k−1 ) 0 k

∂τ F H k−2 .

(5.19)


405

5.2. Application to non-linear system. In order to apply the above energy estimates to the non-linear system (4.15) we fix k ≥ 4 and (x0 , u 0 , u 1 ) ∈ H k × H k × H k−1 . For T, R > 0 let k (5.20) = v ∈ C Tk : |v| k,T ≤ R, v(0) = u 0 , ∂τ v(0) = u 1 . V R,T k , let x(v) = x + τ v. There exists K such that the restriction For any v ∈ V R,T 0 0 0 T < K 0 /R implies E [x] ≤ 2 E [x0 ]. We restrict to such values of T ; thus γ [x(v)]ab k . is uniformly elliptic for all v ∈ V R,T Define = [v] to be the solution to ∂ 2 − A[x(v)] = 0, (, ∂τ ) = (u 0 , u 1 ). (5.21) τ

τ =0

The linear system (5.21) is hyperbolic and it follows from our assumptions that the coefficients are sufficiently regular that standard existence results apply, see [46]. Estimating

(5.22) Dx H k−1 ≤ Dx 2H k−1 ≤ 4 1 + Dx0 2H k−1 + τ 2 |Dx| H k−1 ,τ the energy estimate (5.19), together with the Sobolev inequality (2.17) which provides control of , implies the following. Lemma 5.4. There exists R0 > 0 depending on x0 , u 0 , u 1 , such that for each R ≥ R0 k to itself. there exists TR > 0 such that for all T ≤ TR the map v → [v] takes V R,T Fixing some such R, we now show that for a possibly smaller T > 0, the map v → [v] is a contraction with respect to the C T3 norm. Lemma 5.5. Let R, TR be as given by Lemma 5.4. For a possibly smaller value of TR , we have for each T ≤ TR that |[v 1 ] − [v 2 ]| 3,τ < |v 1 − v 2 | 3,τ for all

v1, v2

∈

(5.23)

k . V R,T

Proof. Let i = [v i ] and x i = x(v i ) for i = 1, 2. Then ∂τ2 (1 − 2 ) − A[x 1 ](1 − 2 ) = (A[x 1 ] − A[x 2 ])2 .

(5.24)

Schematically,

∂τ (A[x 1 ] − A[x 2 ]) = D ∂τ A(x 1 ) − ∂τ A(x 2 ) D

+ D A(x 1 ) − A(x 2 ) D∂τ

+ ∂τ b(x 1 ) − ∂τ b(x 2 ) D

+ b(x 1 ) − b(x 2 ) D∂τ .

(5.25)

Using the mean value theorem, along with the product estimate (2.18) and the Sobolev inequality (2.17) the H 1 norm of each of the first two lines in (5.25) is controlled by C(R)D(x 1 −x 2 ) 2 . A direct estimate yields the same bound for the latter terms. Applying the energy estimate (5.18), one can choose T , depending on R, such that v → [v] is a contraction. A standard argument (see [37,46]) using that the solution map v → [v] is bounded in C Tk and a contraction in C T3 yields Part 1 of Theorem 4.1.

406


5.3. Continuation criterion. We now establish the second part of Theorem 4.1. Consider the solution (x, u) to (4.15)–(4.16) and suppose [0, T∗ ) is the maximal interval of existence. Let (5.26) τ = sup Dx W 1,∞ + Du L ∞ + E [x] . [0,τ )

Note that E is finite if and only if Dx L ∞ and |γ (x)|−1 L ∞ are. Using (5.11) and (5.16) applied to (5.10) we see for any τ < T∗ that τ τ 2 2 2 0 (1 + ) u 0 k + (1 + τ ) u k . u k ≤ (1 + τ ) E e

(5.27)

0

Thus Grönwall’s lemma implies that |u| k,τ ≤ (1 + 2τ )2 E eτ (1+τ )

2 2

E

u 0 k .

(5.28)

If T∗ < ∞ and T∗ < ∞, then one may extend x, u, ∂τ u to [0, T∗ ] such that their restriction to τ = T∗ satisfies the hypotheses of the local existence theorem. As this contradicts the maximality of T∗ , we obtain the second part of Theorem 4.1. 6. Improved Energy Estimate Having established the existence of a solution x ∈ C 1 ([0, T ]; H k ) to the main equation (4.1), we are able to establish an improved energy estimate by making use of the constraint (4.3). In particular, we obtain the following. Theorem 6.1. Let k ≥ 4. For a solution x ∈ C 1 ([0, T ], H k ) (4.1) with initial data (x0 , u 0 ), the maximal time of existence T∗ depends continuously on x0 H k + u 0 H k−1 . Remark 6.1. Comparing with Theorem 4.1, we see that the regularity requirement on the initial data is less. In particular, in Theorem 4.1 it is required that x0 , u 0 are in H k , while in Theorem 6.1, the regularity condition is of the type usually encountered for hyperbolic systems. Thus, from this point of view, when taking the area preserving constraint (4.3) into account, the degenerate hyperbolic system (4.1) behaves very much like a hyperbolic system. From the extension criterion, we know that x may be continued so long as τ is finite. The Sobolev estimate (2.17) implies that 2τ can be estimated by ∂x0 2H k−1 = Dx0 2H k−1 + u 0 2H k−1 . We now establish an estimate for this quantity by deriving an energy estimate for the main equation (4.1) itself. Retaining the notation above, we have (6.1) = [x] = C Dx W 1,∞ + Du L ∞ , where we let the constant C increase (independent of x) as needed and τ = sup[0,τ ) . Following the discussion in §4.1 we write (4.1) as n ∂τ2 xm − L ab mn ∂a ∂b x = Fm ,

(6.2)

where Fm = Fm (Dx). In order to estimate derivatives of x we make use of the identity ∂τ2 (Dl x) − L(Dl x) = Dl (F) − Dl , L x, (6.3) where L = L ab mn ∂a ∂b and we have dropped the m, n indices for notational convenience.


407

Define the energy associated to ∂τ2 − L by

√ 1 m n |∂τ y|2 + L ab E[y] = E[y](τ ) = w. mn (Dx)(∂a y )(∂b y ) 2

(6.4)

We are of course interested in the cases y = Dl x for l = 0, . . . , k − 1. From (4.9) the non-degeneracy of γ (x) implies there is a constant E = E[x; T ] such that on [0, T ],

√ γ 2 2 2 2 1 |∂τ y|2 + |Dy|2γ w . (6.5) E[y] ≤ E ∂ y L 2 and ∂ y L 2 ≤ E 2 w Note that since E essentially controls the ellipticity of γ (x)ab , it is effectively equivalent to E above. A priori, the energy E[y] does not necessarily control ∂ y L 2 . Rather ∂ y 2L 2 ≤ 2E (E[y] + Z[y]) , where Z[y] =

1 2

y m , xm

(6.6)

2 √ w.

(6.7)

The following lemma shows that the “error” Z can be controlled for y = Dl x, if x satisfies the constraint (4.3). Lemma 6.1. Since the solution x ∈ C 1 ([0, T ]; H k ) satisfies the constraint (4.3), then for 0 ≤ l ≤ k − 1 we have

τ l τ l 2 l 2 D ∂x L 2 Z[D x] ≤ e Z[D x](0) + 0

(6.8) ≤ eτ C ∂x0 2H l + τ 2τ |Dl ∂x| 2L 2 ,τ . Proof. Applying Dl to the constraint yields

D u , xm = − l m

l

Dl− j u m , D j xm

(6.9)

j=1

from which we see that ∂τ Z[Dl x] ≤ Z[Dl x] +

l j=1

Dl− j u m , D j xm

2 √

w.

(6.10)

When j = 1 or l we have 2 √ Dl− j u m , D j xm w ≤ D∂x 2L ∞ Dl ∂x 2L 2 .

(6.11)

When 2 ≤ j ≤ l − 1 the Hölder estimate implies 2 √ Dl− j u m , D j xm w ≤ Dl− j Du L 2 p D j−1 D 2 x L 2q

(6.12)

408


with

1 p

=1−

1 q

=

l− j l−1 .

From the interpolation estimate (2.20) we have 1

1− 1

Dl− j Du L 2 p ≤ C Dl−1 Du Lp 2 Du L ∞p , 1− 1p

(6.13)

1 p

D j−1 D 2 x L 2q ≤ C Dl−1 D 2 x L 2 D 2 x L ∞ .

(6.14)

∂τ Z[Dl x] ≤ Z[Dl x] + 2 Dl ∂x 2L 2 .

(6.15)

Thus

Integrating and applying Grönwall’s lemma yields (6.8). The estimate (6.8) for Z , together with (6.6), implies that when −2 T e T < −2 E T

then

(6.16)

|∂ Dl x| 2L 2 ,τ ≤ CE ∂x0 2H l + sup[0,τ ] E[Dl x] .

(6.17)

This facilitates the construction of the following energy estimate. Lemma 6.2. When (6.16) holds the first derivatives ∂x of solution x ∈ C 1 ([0, T ]; H k ) can be controlled in L ∞ ([0, T ]; H l ) for 0 ≤ l ≤ k − 1 by |∂x| H l ,τ ≤ CEe

τ 0

2

∂x0 H l .

(6.18)

Proof. A straightforward computation using integration by parts yields ∂τ E[y] ≤ ∂τ y L 2 ∂τ2 y − L y L 2 + 2 Dy 2L 2 .

(6.19)

In order to apply this to y = Dl x we must estimate the right side of (6.3), which we write schematically as

l ab Dl F − Dl , L x = Dl F − Dl D, L ab Dx + D (DL )(Dx) . (6.20) mn mn Applying the commutator estimate (2.22) and product estimate (2.18) we have Dl F − Dl , L x L 2 ≤ 2 Dx H l . (6.21) Thus integrating (6.19) and using (6.17) we have

2 2 |∂x| H l ,τ ≤ CE ∂x0 H l +

0

Applying Grönwall’s lemma we obtain (6.18).

τ

2

Dx 2H l

.

(6.22)

We now show that there exists T > 0, depending only on ∂x0 H 3 , such that the solution x to (4.1) is defined on [0, T ]. The key is to establish estimates for T and E, and to ensure that (6.16) is satisfied.


409

In order to estimate E note that there exists K , depending on E[x0 ], such that E[x] ≤ 2E[x0 ] whenever T ≤ K −1 T .

(6.23)

For such T , the condition (6.16) follows from T e T ≤ CE[x0 ]−2 −2 T .

(6.24)

Making use of the Sobolev inequality (2.17) the energy estimate (6.18) implies that when (6.23) is satisfied T ≤ CE[x0 ]e T T ∂x0 H k−1 . 2

(6.25)

Since (6.23)–(6.24)–(6.25) all hold with strict inequality for T = 0, we may choose T > 0, depending on ∂x0 H k−1 , such that they continue to hold on [0, T ]. Acknowledgements. LA and PA thank the Mittag-Leffler Institute for hospitality and support during the 2008 program Geometry, Analysis, and Gravitation. The work of LA was supported in part by the NSF, grant no. DMS 0707306, and work of AR was supported by a grant from the Albert Einstein Institute. Part of this work was done while PA held a post-doc position at the Albert Einstein Institute, and during a long-term research visit at the same institution.

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Commun. Math. Phys. 301, 411–441 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1155-z

Communications in


Singular Harmonic Maps and Applications to General Relativity∗ Luc Nguyen1,2,∗∗ 1 Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA 2 Present address: OxPDE, Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB,

UK. E-mail: [email protected] Received: 18 November 2009 / Accepted: 18 July 2010 Published online: 26 October 2010 – © Springer-Verlag 2010

Abstract: The study of axially symmetric stationary multi-black-hole configurations and the force between co-axially rotating black holes involves, as a first step, an analysis on the “boundary regularity” of the so-called reduced singular harmonic maps. We carry out this analysis by considering those harmonic maps as solutions to some homogeneous divergence systems of partial differential equations with singular coefficients. Our results extend previous works by Weinstein (Comm Pure Appl Math 43:903–948, 1990; Comm Pure Appl Math 45:1183–1203, 1992) and by Li and Tian (Manu Math 73(1):83–89, 1991; Commun Math Phys 149:1–30, 1992; Differential geometry: PDE on manifolds, vol 54, pp. 317–326, 1993). This paper is based on the Ph.D. thesis of the author (Singular harmonic maps into hyperbolic spaces and applications to general relativity, PhD thesis, The State University of New Jersey, Rutgers, 2009). 1. Introduction It is an open problem in general relativity that whether or not there exists a vacuum or electrovac stationary black hole spacetime that is “regular” and has “non-degenerate” disconnected event horizon. A relevant question to this problem is whether there is a strut singularity in any of the axially symmetric stationary multi-black-hole solutions constructed by Weinstein [24,25,28]. In [1], Bach and Weyl showed that any axially symmetric and static vacuum spacetime outside two bodies possesses a conical singularity along the axis connecting the bodies. Later, Bunting and Massood-ul-Alam [3] cleverly used the positive mass theorem of Schoen and Yau to show that there does not exist any regular static “multiple-body” vacuum spacetimes. For non-static spacetimes, much less is known. Regardless of regularity, Weinstein [24,25,28] used the harmonic map reduction, known as the Ernst-Geroch ∗ This article was funded in part by a grant from the Vietnam Education Foundation (VEF). The opinions, findings, and conclusions stated herein are those of the author and do not necessarily reflect those of VEF. ∗∗ Partially funded by a Rutgers University and Louis Bevier Dissertation Fellowship.

412

L. Nguyen

reduction, to construct a family of multi-black-hole axially symmetric and stationary vacuum/electrovac solutions, of which a regular one must be a member. In the vacuum case, Li and Tian [14,15] and Weinstein [24–26] independently proved some regularity results for the reduced harmonic maps and then used them to show that within the solutions constructed by Weinstein, there is a continuum of irregular solutions. In this paper we bridge the methods used by Li and Tian and by Weinstein to extend the above regularity results to the axisymmetric stationary electrovac case. Let (M, ds 2 ) be the domain of outer communications of an asymptotically flat, globally hyperbolic, axially symmetric, stationary charged spacetime and assume that M is simply connected and has non-degenerate horizons. Roughly speaking, the ErnstGeroch formulation gives a representation of each such (M, ds 2 ) as (R3 \Γ ) × R equipped with a metric of the form (see [11,28]) ds 2 = −ρ 2 e2u dt 2 + e−2u (dϕ − p dt)2 + e2μ+2u (dρ 2 + dz 2 ),

(1)

where Γ is the z-axis in R3 with a few line segments removed, ρ is the distance to the z-axis, ϕ is the cylindrical angle around the z-axis and u, p and μ are determined by an axially symmetric harmonic map Φ from R3 \Γ into the complex hyperbolic plane HC whose boundary data are singular on Γ . (See Sect. 2 for a precise statement.) For our present discussion, it should be noted that the number of connected components of the complement of Γ relative to the z-axis is the number of black holes and that the singular behavior of Φ near Γ forces Φ to have infinite energy in any open set that intersects Γ non-trivially. Using this reduction and a clever renormalization of the energy functional, Weinstein [28] proved the existence of a family of (possibly singular) spacetimes which can be interpreted as equilibrium configurations of asymptotically flat co-axially rotating electrovac black holes. Moreover, he showed that this family is uniquely parametrized by 4n − 1 parameters, where n is the number of black holes. These parameters can be interpreted as the masses, the angular momenta, the charges of the black holes, and the distances between them. This result implies in particular the uniqueness of the KerrNewman solutions (among axially symmetric stationary spacetimes), which was proved independently earlier by Mazur [17,18] and by Bunting [2]. For a historical account, we note that the uniqueness of the Kerr solutions (among axially symmetric stationary spacetimes) was done earlier by Robinson [20,21], while the analogue of Weinstein’s result in vacuum was established earlier by Weinstein himself [24,25]. We also note a recent paper by Chru´sciel and Costa [4] where the uniqueness problem in vacuum is revisited. At a first glance at the reduction, one might have the impression that given any solution to the singular harmonic map equations, one could easily cook up a solution to the Einstein vacuum equations. This is in fact not always the case, which was probably first observed by Bach and Weyl [1] in the static vacuum case. As one tries to construct a spacetime out of a singular harmonic map, one might possibly introduce a conical singularity along Γ . A necessary and sufficient condition for regularity is lim (μ + 2u + log ρ) = 0 along Γ.

ρ→0

(2)

Bach and Weyl showed in [1] that if Γ has more than two components, i.e. the spacetime has disconnected event horizon, (2) is violated in the static vacuum setting. Their method does not seem welcoming of the general setting since it relies on the explicit form of the solution. Even though there has been some progress in getting the explicit form of solutions for multiple-body spacetimes, e.g. the famous double Kerr solutions

Singular Harmonic Maps and Applications to General Relativity

413

of Kramer and Neugebauer [12], the dependence of the validity or invalidity of (2) on the parameters seems still unclear in general. We however mention here a remarkable result by Hennig and Neugebauer [10] which shows under appropriate assumptions that (2) is always violated in the case of two vacuum black holes. Concerning Eq. (2) itself, some “regularity structure” across the symmetry axis Γ of the harmonic map Φ is required in order to make sense of the limit on the left-hand side. Note that because of the singular behavior of Φ near Γ , Φ is not harmonic across Γ . In this paper, we would like to address the above regularity issue. To describe our results, it is convenient to use the following weighted spaces. Definition 1. Let Ω be a domain in Rn , Σ be a subset of Ω, and w be a positive measurable function in Ω. We denote by L p (Ω, w) (1 ≤ p ≤ ∞) the space of p-integrable functions with respect to the measure w(x) d x, equipped with its standard Banach space 1, p 1, p structure. The spaces WΣ (Ω, w) and W0,Σ (Ω, w) are respectively the completions of ¯ Cc∞ (Ω\Σ) and Cc∞ (Ω\Σ) with respect to the norm 1/ p p p gW 1, p (Ω,w) = |g| + |Dg| w d x . Σ

Ω

1 (Ω, w) for W 1,2 (Ω, w) and When p = 2, we also write HΣ1 (Ω, w) and H0,Σ Σ 1,2 W0,Σ (Ω, w).

We now state our main result. Theorem A. Let Γ be the z-axis in R3 with some bounded line segments removed and ρ be the distance function to the z-axis. Let h be the Newtonian potential created by a uniform line charge distribution of density γ > 0 along Γ . If Φ = (u, v, χ , ψ) is a singular harmonic map from R3 \Γ into the complex hyperbolic plane H controlled by h h 1 3 2 and some ideal points on ∂H in the sense of Definition 2 and if u − 2 ∈ H (R ), h 1 3 2h 1 3 h k,α v ∈ H (R , e ), and χ , ψ ∈ H (R , e ), then u − 2 , v, χ , ψ ∈ C across the interior of Γ for any k + α < 2γ . In addition, near any compact subset of the interior of any component of Γ , v, χ and ψ admit the asymptotic expansion v = C1 + C3 χ − C2 ψ + O(ρ 2k+2α ), χ = C2 + O(ρ k+α ), ψ = C3 + O(ρ k+α ), where C1 , C2 and C3 are constants (which might vary for different components of Γ ). Moreover, if (u, v, χ , ψ) is axially symmetric about Γ and γ = 1, u − h, v, χ , ψ are everywhere C ∞ except possibly at the endpoints of Γ . Remark 1. (a) The regularity of v can be “slightly improved” as follows: v − C3 χ + C2 ψ ∈ C k,α for k + α < 4γ . (b) In the statement of Theorem A, one can replace R3 by any open set that intersects Γ non-trivially. (c) The conclusions of Theorem A can be extended to singular harmonic maps with values in any real, complex or quaternionic hyperbolic spaces, i.e. symmetric spaces of rank one. See Remark 3. (d) In vacuum, i.e. χ ≡ ψ ≡ 0, we can allow k +α = 2γ in Theorem A. See Corollary 2 and Remark 5.

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Remark 2 (added in proof). After submitting this work, the author realized that, for singular harmonic maps arising from asymptotically flat spacetimes, by the uniqueness arguments in [28] and in [5], especially [5, Prop. C.4], the condition that u− h2 ∈ H 1 (R3 ), v ∈ H 1 (R3 , e2h ), and χ , ψ ∈ H 1 (R3 , eh ) in Theorem A can be dropped. For such integrability condition is then a consequence of the fact that Φ is controlled by h2 . As an immediate consequence of Theorem A, we prove: Theorem B. The metric components of the co-axially rotating, stationary, multipleblack-hole, charged spacetimes constructed by Weinstein in [28] are C ∞ outside the event horizons. Also, we have: Corollary 1. For any solution to the reduced harmonic map equations of the EinsteinMaxwell equations, the limit on the left hand side of (2) exists and is finite along Γ . In the vacuum case, Theorem A was settled independently by Weinstein [24,25] and by Li and Tian [14,15]. In fact, after settling the regularity issue, Li and Tian [13–15] and Weinstein [26] went forward to study if there is a conical singularity in an axially symmetric stationary vacuum spacetime. They proved that there are several continuous sets of parameters which give rise to spacetimes violating (2). Unfortunately, their results do not reveal whether there is any set of parameter that realizes (2) besides those corresponding to the Kerr spacetimes, i.e. one-body spacetimes. In the electrovac case, it is possible to use Theorem A to carry out an analysis similar to that in the aforementioned works to prove nonexistence of regular spacetimes corresponding to certain class of parameters. However, we will not pursue this direction in the present work. The rest of the paper is organized as follows. In Sect. 2 we describe the Ernst-Geroch reduction. In Sect. 3, we carry out some preliminary analysis on the harmonic map equations (3)–(6) which allows us to consider it as a special case of a broader class of singular quasi-linear elliptic systems. In Sect. 4 we consider the case when the model problem is a single linear equation. In Sect. 5, we return to the study of the model problem in the general setting. The proofs of Theorems A and B are carried out in Sect. 6. Finally, for completeness, we include in Appendix A a brief study the weighted Sobolev and 1, p Lebesgue spaces WΣ (Ω, w) and L p (Ω, w) defined in Definition 1. 2. The Ernst-Geroch Reduction To describe the Ernst-Geroch reduction, we first introduce the notion of singular harmonic maps into the complex hyperbolic plane (see [27]). Let H = HC ∼ = U (1, 2)/ (U (1) × U (2)) be the complex hyperbolic plane. In the disk model, the hyperbolic plane H is modeled by the disk D = {ζ = (ζ 1 , ζ 2 ) ∈ C2 : |ζ |2 < 1} with the metric element ds 2 =

|dζ |2 |ζ · dζ |2 + . 1 − |ζ |2 (1 − |ζ |2 )2

For our purpose, it is more convenient to use a different model which takes into account the structure of the horocycles of H. See [27] for a detailed discussion. Define


415

|1 + ζ1 | 1 u = log , v = Im w1 , χ = Re w2 , ψ = Im w2 , 2 1 − |ζ |2 where w1 =

1 − ζ1 , 1 + ζ1

w2 =

ζ2 . 1 + ζ1

The inverse transformation is given by ζ1 =

1 − w1 2w2 , ζ2 = , w1 = e−2u + χ 2 + ψ 2 + 2i v, w2 = χ + i ψ. 1 + w1 1 + w1

In short, H can be modeled by R4 = {(u, v, χ , ψ)} with the metric ds 2 = du 2 + e4u (dv − ψ dχ + χ dψ)2 + e2u (dχ 2 + dψ 2 ). Let Γ be a subset of the z-axis in R3 obtained by removing some bounded line segments. Let h be the Newtonian potential created by a uniform line charge distribution of density γ > 0 along Γ . Then given a geodesic ζ in H, ζ ◦ h is a harmonic map from R3 \Γ into H. Moreover, as x → Γ , ζ ◦ h(x) approaches the ideal point ζ (+∞) ∈ ∂H. Recall that a map (u, v, χ , ψ) : Ω ⊂ R3 → H is harmonic if it satisfies Δu − 2 e4u |Dv − ψ Dχ + χ Dψ|2 − e2u (|Dχ |2 + |Dψ|2 ) = 0, div e4u (Dv − ψ Dχ + χ Dψ) = 0,

(4)

div (e2u Dχ ) − 2e4u Dχ · (Dv − ψ Dχ + χ Dψ) = 0, div (e2u Dψ) + 2e4u Dψ · (Dv − ψ Dχ + χ Dψ) = 0.

(5) (6)

(3)

Definition 2. Let Γ be a subset of the z-axis in R3 obtained by removing n bounded line segments. To each component Γ j of Γ , 1 ≤ j ≤ n + 1, we associate an ideal point p j ∈ ∂ H . Pick any normalized geodesic ζ j so that ζ j (+∞) = p j . Let h be the Newtonian potential created by a line charge distribution of strictly positive density on Γ . We say that a map Φ : R3 \Γ → H is a singular harmonic map controlled by the distribution potential h and the ideal points p j if Φ is harmonic and near each component Γ j the hyperbolic distance between Φ and ζ j ◦ h is bounded. Sometimes, we simply say that Φ is a singular harmonic map controlled by h or that h controls Φ. We next state the conditions that determine the metric (1) according to the ErnstGeroch reduction. (i) u, p and μ are independent of the angle variable ϕ. (ii) u is the first component of some axially symmetric singular harmonic map Φ = (u, v, χ , ψ) from R3 \Γ into the complex hyperbolic plane H. Moreover, (u, v, χ , ψ) is singular near Γ and the singular rate is controlled by the Newtonian potential h2 created by a uniform line charge distribution of density 21 and some ideal points on ∂H.

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(iii)

L. Nguyen

p and μ satisfy (7) dp = −ρ e4u (vz − ψ χz + χ ψz )dρ + ρ e4u (vρ − ψ χρ + χ ψρ ) dz,

4u e dμ = ρ (u 2ρ − u 2z ) + (vρ − ψ χρ + χ ψρ )2 − (vz − ψ χz + χ ψz )2 4 + e2u (χρ2 − χz2 + ψρ2 − ψz2 ) dρ

e4u (vρ − ψ χρ + χ ψρ )(vz − ψ χz + χ ψz ) +2ρ u ρ u z + 4 + e2u (χρ χz + ψρ ψz ) dz. (8)

(iv) The metric (1) is asymptotically flat. (See [28, pp. 1406–1408] for a precise statement on asymptotic flatness. We suppress this point here for simplicity as it will not involve our later discussion.) We note that the harmonic map equations (3)–(6) give the integrability conditions for (7) and (8). The case where χ ≡ ψ ≡ 0 corresponds to vacuum spacetimes while the case where v ≡ 0 corresponds to static spacetimes. Also, as pointed out in the Introduction, condition (ii) forces (u, v, χ , ψ) to have infinite energy in any open set that intersects Γ non-trivially. 3. The Model PDE Problem We reformulate the reduced singular harmonic map into another form which is more convenient for our later study. We start with a study of condition (ii) stated in the previous section. Since our regularity result is local, it suffices to restrict our attention to a subset Ω of R3 such that Σ = Ω ∩ Γ has only one component whose endpoints lie on ∂Ω. In particular, we only need to pick one controlling ideal point p ∈ ∂H. Using an isometry of H, we can assume without loss of generality that p is the +∞ point sitting on the u-axis of H. We pick the geodesic going to p to be t → ζ (t) = (t, 0, 0, 0) ∈ H. As noted earlier, ζ ( h2 ) is a singular harmonic map from Ω\Σ into H. Also, as h2 and

p control (u, v, χ , ψ), we must have dH ( 21 h, 0, 0, 0), (u, v, χ , ψ) < C < ∞ in Ω. After some calculation, this turns out to be equivalent to u − h + e2h |v| + eh (|χ | + |ψ|) < C < ∞ in Ω. 2 We have shown: Lemma 1. Let Ω be a subset of R3 and Σ be a curve in Ω. Let h be the Newtonian potential created by a uniform line charge distribution of density γ > 0 along Σ. Then (u, v, χ , ψ) is a singular harmonic map from Ω\Σ into the complex hyperbolic plane H controlled by h2 and the ideal point (+∞, v0 , χ0 , ψ0 ) ∈ ∂H if and only if it satisfies the harmonic map equations (3)–(6) and the estimate u − h + e2h |v − v0 − ψ0 χ + χ0 ψ| + eh (|χ − χ0 | + |ψ − ψ0 |) < C < ∞ in Ω. 2 (9)


417

This a priori estimate shows that the harmonic map equations (3)–(6) is a singular semilinear elliptic system. As h ‘behaves like’ −2γ log ρ near Σ, the singularity types are negative powers of the distance function to a line. Next, we observe that our target manifold is H, a symmetric space. By Noether’s theorem, the harmonic map equations (3)–(6) induce several conservation laws, each for an isometry of H. In fact, there are even enough symmetries to turn (3)–(6) into divergence form: div Du − 2e4u v Dv + (2e4u v ψ − e2u χ )Dχ − (2e4u v χ + e4u ψ)Dψ = 0, div e4u (Dv − ψ Dχ + χ Dψ) = 0, div −2e4u ψ Dv + (e2u + 2e4u |ψ|2 )Dχ − 2e4u χ ψ Dψ = 0, div 2e4u χ Dv − 2e4u χ ψ Dχ + (e2u + 2e4u |χ |2 )Dψ = 0. (In the vacuum case, this divergence structure is more apparent (cf. [23, Chap. 7]).) Writing u 1 = u − h/2, u 2 = v, u 3 = χ , u 4 = ψ, and noting that h is harmonic, we can rewrite the above system in the form div (aαβ (x, u) Du β ) = 0,

1 ≤ α ≤ 4,

where aαβ is a 4 × 4 matrix of coefficients given by ⎡

1 ⎢0 ⎢ ⎣0 0

−2e4u v 2le4u −2le4u ψ 2le4u χ

2e4u v ψ − e2u χ −2le4u ψ le2u + 2le4u |ψ|2 −2le4u χ ψ

⎤ −2e4u v χ + e2u ψ ⎥ 2le4u χ ⎥ 4u ⎦ −2le χ ψ le2u + 2le4u |χ |2

and l is some constant. Then, for ξ ∈ R4 , aαβ (x, u) ξ α ξ β = |ξ 1 |2 + 2le4u |ξ 2 − ψξ 3 + χ ξ 4 |2 + le2u (|ξ 3 |2 + |ξ 4 |2 ) −[2e4u v(ξ 2 − ψξ 3 + χ ξ 4 ) − e2u ψξ 3 + e2u χ ξ 4 ]ξ 1 . In view of (9), we can always pick l sufficiently large and positive λ, Λ such that λ[|ξ 1 |2 + e4u |ξ 2 |2 + e2u (|ξ 3 |2 + |ξ 4 |2 )] ≤ aαβ (x, u) ξ α ξ β ≤ Λ[|ξ 1 |2 + e4u |ξ 2 |2 + e2u (|ξ 3 |2 + |ξ 4 |2 )] for any ξ ∈ R4 . We are thus led to study the following problem: Problem 1. Let Ω be a domain in Rn and Σ be a (n − k)-submanifold of Ω, k ≥ 2. Consider the system div (aαβ (x, u) Du β ) = 0

in Ω\Σ,

1 ≤ α ≤ m,

(10)

where aαβ : (Ω\Σ) × Rm → R are given smooth functions and u : Ω → Rm is the unknown.

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Let d = dΣ denote the distance function to Σ. Assume that there exist constants k(1), k(2), …, k(m) ≥ 0 such that for any given R > 0, there exist λ = λ(R) > 0 and Λ = Λ(R) such that λ

m

d(x)−2k(α) |ξ α |2 ≤ aαβ (x, u) ξ α ξ β ≤ Λ

α=1

m

d(x)−2k(α) |ξ α |2

(11)

α=1

for any x ∈ Ω\Σ, ξ ∈ Rm and u ∈ Rm satisfying d(x)−k(α) |u α | ≤ R, 1 ≤ α ≤ m. Assume that u solves (10) in some appropriate sense. Determine how regular u is! Remark 3. Analogous to the case of harmonic maps into the complex hyperbolic plane, regularity for harmonic maps with prescribed singularity into any real, complex, or quaternionic hyperbolic space in the sense described in [27] can always be recast as a part of Problem 1. Consequently, the result in Theorem A extends parallelly to those cases. We now describe what we mean by a solution to (10). The singularity of the coefficients aαβ and the existence result established by Weinstein [28] make it reasonable to assume that u α ∈ HΣ1 (Ω, d −2k(α) ). In addition, in a compactly supported open subset of Ω\Σ, aαβ behaves nicely and so (10) should hold in the usual weak sense. This suggests the following definition. Definition 3. A measurable function u : Ω → Rm with u α ∈ HΣ1 (Ω, d −2k(α) ) is said to be a weak solution of (10) if aαβ (x, u) Du α Dξ β d x = 0 Ω

1 (Ω, d −2k(α) ). for any ξ α ∈ H0,Σ

4. The Case of a Single Linear Equation In this section, we consider a special case of Problem 1 where the unknown is a scalar map and the coefficient is independent of the unknown. Recall that Ω is an open subset of Rn and Σ is a (n −k)-dimensional submanifold of Ω with k ≥ 2. Let d be the distance function to Σ and w be a weight that satisfies λ d(x)−γ ≤ w(x) ≤ Λ d(x)−γ for some γ > 0, 0 < λ ≤ Λ < ∞. We are interested in the regularity of weak solutions of div (w(x)Du = 0 in Ω\Σ. (12) Here weak solutions are understood in the sense of Definition 3. This problem has been studied extensively in the literature. When γ < k − 2, w belongs to the Muckenhoupt class A2 and the result of Fabes, Kenig and Serapioni [6] implies that u is Hölder continuous across Σ. Unfortunately, in our application to the problem from general relativity, w might get too singular in a way that it is not even integrable across Σ, and so their work does not apply directly. Moreover, for other purposes, we are also interested in whether w vanishes along Σ and how fast it decays there. A result in this direction was given by Li and Tian in [14] when Σ has codimension 2. We generalize their work to the case where Σ is a general submanifold of any codimension k ≥ 2 and sharpen the decay estimate to its optimal form.


419

Theorem 1. Assume that w ∈ C 2 (Ω\Σ) and λ d −γ ≤ w ≤ Λ d −γ for some γ ≥ 0 and 0 < λ ≤ Λ < ∞. Moreover, assume that |D(d γ w)| = o(d −1 ) in a neighborhood of Σ. Then any u ∈ HΣ1 (Ω, w) which weakly solves (12) in Ω\Σ is locally Hölder continuous in Ω and enjoys γ

sup d −(γ −k+2) |u| ≤ C d − 2 u L 2 (Ω) for any ω Ω. +

ω

The constant C depends only on Λ, λ, ω and the modulus of continuity of d|D(d γ w)|. Remark 4. To see that the decay estimate is optimal, consider the case where Σ is an n−k hyperplane and w = d −γ . When γ > k − 2, (12) has a special solution u = d γ −k+2 which belongs to HΣ1 (Ω, d −γ ). This solution vanishes along Σ and exhibits Hölder continuity along Σ. When γ < k − 2, the above special solution does not belong to HΣ1 (Ω, d −γ ). Moreover, constant functions are solutions which are in HΣ1 (Ω, d −γ ) and, of course, they need not vanish along Σ. Nevertheless, as mentioned above, w is in the Muckenhoupt class A2 and so u is Hölder continuous across Σ. Before proving Theorem 1, let us give an application which improves Theorem A in the vacuum case, i.e. χ ≡ ψ ≡ 0. Corollary 2. Under the hypotheses of Theorem A and an additional assumption that χ ≡ ψ ≡ 0, we have (u + h2 , v) ∈ C k,α for any integer k and any α ∈ (0, 1) satisfying k + α ≤ 4γ . Moreover, near any compact subset of the interior of any component of Γ , v admits the expansion v = C + O(ρ 4γ ) for some constant C. Proof. Focus our attention to a small neighborhood Ω of an interior point of Γ , we can assume that v ∈ H 1 (Ω, d −4γ ) and v|Σ = 0, where Σ = Γ ∩ Ω (see Lemma 1). This implies that v ∈ HΣ1 (Ω, d −4γ ). Now, by (4), div (e4u Dv) = 0. Applying Theorem 1, we get the required decay for v near Γ and then its C k,α regularity. To get the regularity for u + h2 , we use (3) and apply a simple estimate for the Poisson equation. Remark 5. To see that the C k,α regularity in Corollary 2 is optimal, consider for example the case where Γ is the whole z-axis, h = −2γ log ρ, γ > 0 and ρ 2γ 1 , u = − log 4γ 2 ρ +1

v=

1 ρ 4γ . 2 ρ 4γ + 1

This example also shows that the C k,α regularity in Theorem A is almost optimal. We will prove Theorem 1 through a sequence of lemmas. Also, to avoid technicality, we will assume that Σ is a (n − k)-hyperplane and w = d −γ . In fact, if w¯ = d γ w is not constant, the proofs below require minor changes. Also, we will frequently use the following inequality (see Appendix A) without explicitly mentioning. For any f ∈ HΣ1 (Ω, d −γ ) and ω Ω, there holds d −γ −2 | f |2 d x ≤ C [| f |2 + d −γ |D f |2 ] d x. ω

Ω

Lemma 2. Theorem 1 holds for 0 ≤ γ < 2(k − 2).

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Proof. For 0 ≤ γ ≤ k − 2, the assertion is a consequence of a result of Fabes et al. [6]. Assume that k − 2 < γ < 2(k − 2). Set v = d −γ +k−2 u and w˜ = d −2(k−2)+γ . Then v ∈ H 1 (Ω, w) ˜ and satisfies in Ω\Σ the equation div (w˜ Dv) = 0. Since γ < 2(k − 2), w˜ belongs to the Muckenhoupt class A2 and the above equation is satisfied across Ω. By [6], v is locally bounded in Ω and for any ω Ω, γ

γ

sup |v| ≤ C d 2 −k+2 v L 2 (Ω) ≤ C d − 2 v L 2 (Ω) , ω

which completes the proof.

Lemma 3. Assume that γ ≥ 2(k − 2) and u is as in Theorem 1. Then γ

γ

sup d − 2 |u| ≤ C d − 2 u L 2 (Ω) for ω Ω. ω γ

Proof. Write v = d − 2 u. Then v ∈ H 1 (Ω) ∩ L 2 (Ω, d −2 ) and satisfies in Ω\Σ the equation Δv − c v = 0, where c = γ2 ( γ2 − k + 2) d12 > 0. We therefore can apply De Giorgi or Moser techniques to show that v is locally bounded with the desired bound (see e.g. Lemma 5 below). ¯ solves the following inhomogeneous Lemma 4. Assume that u ∈ H 1 (Ω) ∩ C 0 (Ω) equation in Ω\Σ: div (w(x)Du) = f,

(13)

where f is smooth away from Σ and | f | ≤ C d −γ −2+μ in a neighborhood of Σ for some 0 < μ < λ − k + 2. Assume in addition that u vanishes along Σ. Then for any λ ≤ μ, d −λ u is locally bounded and for any ω Ω, sup d −λ |u| ≤ C sup |u|. ω

Ω

Proof. Fix some ω Ω. Fix some 0 < h < 1. Assume for the moment that d −c u is bounded. We will show that if c + h ≤ μ, then supω d −c−h |u| ≤ C supω d −c |u|. This obviously implies our result. Fix x0 ∈ Ω. Consider the function θ (x) = d c+h (x) + d c (x) |x − x0 |2 . Using h < 1, it is straightforward to show that div (d −γ Dθ ) ≤ C(c + h)(c + h − γ + k − 2) d −γ +c+h−2 ≤ −C | f |. in some neighborhood U of Σ. Take another neighborhood V of Σ such that V¯ ⊂ U . Set δ = dist (V, ∂U ) > 0. For x0 ∈ V and x ∈ ∂ Bδ (x0 ), we have |u| ≤ d −c u L ∞ (U ) d c ≤ d −c u L ∞ (U ) δ −2 θ. Also, θ vanishes on Σ. Hence, by the maximum principle, |u| ≤ d −c u L ∞ (U ) δ −2 θ in Bδ (x0 ). In particular, for x = x0 , we have |u(x0 )| ≤ d −c u L ∞ (U ) δ −2 d c+h (x0 ). Since x0 is arbitrary, this proves the lemma.


421

Lemma 5. Let u ∈ H 1 (Ω) ∩ L 2 (Ω, d −2 ) solve the following equation in Ω\Σ: Δu − p

1 Dd · Du − q 2 u = 0, d d

(14)

where p and q are two real numbers satisfying, for some μ > 0, (i) q ≥ 0, (ii) ( p − k + 2)2 + 4q − p 2 > μ, 2q 4q 2 (iii) and p − k + 2 ≤ k−2 or (k−2) 2 −

4q( p−k+2) k−2

+ p 2 − 4q < −μ.

∞ (Ω) and for any ω Ω, Then u ∈ L loc

sup |u| ≤ C(ω, μ) u L 2 (Ω) . ω

Proof. We can assume that Ω is the unit ball B1 . We will show that u is bounded in B1/2 . The idea is to adapt De Giorgi’s proof for local boundedness for solutions of a linear elliptic equation. As is well known, his proof applies to broader classes of functions which are usually referred to as De Giorgi classes. The argument provided here is mainly to show that our solution u belongs to one such class. Let A(k, ρ) = {x ∈ Bρ : u(x) > k}. Let η be a standard cut-off function supported in Bρ . Observe that we can take η2 (u − k)+ as a test function for (14). This yields ε η2 |Du|2 + Cε |u − k|2 |Dη|2 η2 |Du|2 d x ≤ A(k,ρ) A(k,ρ) Dd 1 Du (u − k)+ − q η2 2 |u − k|2 d x. (15) − p η2 d d On the other hand, by integrating by parts, we see that

Dd Dd 1 ± η2 div η2 · Du (u − k)+ d x = ∓ |u − k|2 d x d 2 A(k,ρ) d A(k,ρ) 1 k−2 + ε ) ≤ −(± η2 2 |u − k|2 d x 2 d A(k,ρ) |u − k|2 |Dη|2 d x. + Cε A(k,ρ)

Substituting the above estimate into (15), we arrive at 2 2 ε η2 |Du|2 + C |u − k|2 |Dη|2 η |Du| d x ≤ A(k,ρ)

A(k,ρ)

Dd Du (u − k)+ d 1 (k − 2)b − ε ) η2 2 |u − k|2 d x, − (q − 2 d − ( p − b) η2

2q where b is any real number. If we require that b < k−2 , this implies α η2 |Du|2 + C |u − k|2 |Dη|2 d x, η2 |Du|2 d x ≤ A(k,ρ)

A(k,ρ)

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L. Nguyen

where α=

( p − b)2 4(q −

(k−2)b 2

− ε )

+ ε.

2q Our hypotheses on p and q are exactly to allow us to find some b < k−2 so that α < 1. We have therefore shown that 2 2 η |Du| d x ≤ C |u − k|2 |Dη|2 d x. (16) A(k,ρ)

A(k,ρ)

This estimate shows that u belongs to the De Giorgi class DG +2 (Ω) as defined by [9, Def. 7.1]. An identical argument shows that u ∈ DG − 2 (Ω). De Giorgi’s proof for local boundedness (see e.g. [9, Theorem 7.2]) then implies sup u ≤ C u + L 2 (B1 ) . B1/2

The lemma follows.

Proof of Theorem 1. The case 0 ≤ γ < 2(k −2) is taken care of by Lemma 2. We hence assume that γ ≥ 2(k − 2). γ By Lemma 3, d − 2 u is locally bounded. Thus, by elliptic theory, u is continuous in Ω and vanishes along Σ. Lemma 4 hence shows that d −λ u is locally bounded for any λ < γ − k + 2. By elliptic theory, we infer that d −λ+1 |Du| is locally bounded for the same λ’s. Take a sequence λm < γ − k + 2 converging to γ − k + 2. Set vm = d −λm u. Since λm < γ − k + 2, the above estimates on the sizes of u and Du show that vm ∈ H 1 (Ω) ∩ L 2 (Ω, d −2 ). Moreover, in Ω\Σ, vm satisfies Δvm − pm

Dd 1 · Dvm − qm 2 vm = 0, d d

where pm = γ − 2λm and qm = λm (γ − k + 2 − λm ) > 0. An application of Lemma 5 shows that, for any ω Ω, sup d −λm |u| = sup |vm | ≤ Cm (ω) vm L 2 (Ω) = Cm (ω) d −λm u L 2 (Ω) . ω

ω

More importantly, as λm → γ − k + 2, the constant Cm does not blow up as 2 → (γ − k + 2)2 − (γ − 2k + 4)2 > 0, ( pm − k + 2)2 + 4qm − pm 2qm → −γ + k − 2 < 0. pm − k + 2 − k−2

As a consequence, if ω ω Ω, sup d −γ +k−2 |u| ≤ C d −γ +k−2 u L 2 (ω ) ≤ C sup d −γ +k−2+ε |u| ω

≤ C sup d

− γ2

ω

In any case, the theorem is ascertained.

|u| ≤ C d

− γ2

ω

u L 2 (Ω) .


423

5. Hölder Regularity We next switch our attention back to the system (10). Throughout the section, aαβ are assumed to satisfy the ellipticity condition (11). In general, without any additional assumption on the coefficients aαβ or the unknowns u α , one does not expect u α to be regular, or even Hölder continuous, across Σ. For example, when the coefficients aαβ satisfy the classical ellipticity, i.e. k(α) = 0, it is well-known that the regular set of u consists of points x ∈ Σ at which 1 lim inf n−2 |Du|2 d x = 0. δ→0 δ Bδ (x) ij

(See [7], Chap. 4, for example.) If aαβ do not depend on u and are smooth across Σ, ij

this set is empty and so u is regular in Ω. Nonetheless, if aαβ is allowed to depend on u, the points where regularity fails might constitute a non-vacuous subset of Σ, even if one assumes additionally that u is bounded (see [7], Chap. 2). We show that the above phenomenon also persists for the problem we are considering. For any Bδ (x) ⊂ Ω, we define E δ (x) =

1 δ n−2

m Bδ (x) α=1

d(z)−2k(α) |Du α (z)|2 dz.

(17)

Also, define bαβ (x, u) ˜ = d(x)−[k(α)+k(β)] a(x, d(x)−k(γ ) u˜ γ ),

x ∈ Ω\Σ, u˜ ∈ Rm .

˜ remain bounded as long as {u˜ α }m By (11), the functions bαβ (x, u) α=1 stays bounded. i, j

Theorem 2. Let Ω be a domain in Rn and let Σ be a (n − k)-submanifold of Ω, k ≥ 2. Assume that aαβ are smooth functions on (Ω\Σ) × Rm which satisfy the ellipticity condition (11) with k(α) being either zero or bigger than k − 2. Let τ be the smallest index so that k(τ ) > k − 2. (If all k(α) vanish, we set τ = m + 1.) Assume that bαβ (x, u˜ 1 , . . . , u˜ τ −1 , 0) = 0 unless α = β.

(18)

Let u α ∈ HΣ1 (Ω, d −2k(α) ) be a weak solution of (10). If, in addition, u α ∈ L ∞ (Ω, then there exists ε0 > 0 such that if E δ (x0 ) ≤ ε0 for some 0 < δ < dist (x0 , ∂Ω)/4, then d −k(α) ),

E σ (x) ≤ C σ 2λ ,

x ∈ Bδ/2 (x0 ), σ < δ/4,

for some positive constants C and λ that depend only on k(1), …, k(m), and u α L ∞ (Ω,d −k(α) ) . In particular, u is Hölder continuous in Bδ/4 (x0 ) and for α ≥ τ , d −k(α)−λ |u α | is bounded in Bδ/4 (x0 ). Remark 6. The requirement that k(τ ) > k − 2 is probably merely technical. It would be interesting to remove this extra hypothesis. However, in doing so, one must be careful with the assumption that d −k(α) u α is bounded. For, in case of a single linear equation, if k(α) < k − 2, there are examples that u α may not vanish along Σ as fast as d k(α) (see Remark 4).

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In the context of harmonic maps, an ε-regularity result is usually established using some appropriately devised monotonicity formula. For example, this is the case in the work of Li and Tian [14]. It seems that their proof does not apply in our context. However, as demonstrated in Sect. 3, our harmonic map problem has an equivalent homogeneous divergence form. This special structure makes it apparent to obtain a Caccioppoli inequality which we will state shortly. When restricted to the vacuum case in [14], this Caccioppoli inequality implies the monotonicity formula therein. Observe that, as long as we stay away from Σ, (10) is an elliptic system for u with bounded smooth coefficients, so classical elliptic theory tells us that a Caccioppoli inequality holds there. However, as we move towards Σ, the coefficients of these equations behave badly signifying some possible deterioration in the structure of the Caccioppoli inequality. This is indeed true as stated in the following result. Proposition 1 (Caccioppoli Inequality). Let u α ∈ HΣ1 (Ω, d −2k(α) ) ∩ L ∞ (Ω, d −k(α) ) be a weak solution of (10). Let x0 be a point in Ω and δ0 the distance from x0 to ∂Ω. There exists C = C(Λ, λ) > 0 such that for δ ≤ δ0 /2 and b ∈ Rm whose last m − τ + 1 components vanish when B2δ (x0 ) ∩ Σ = ∅, m m C −2k(α) α 2 d |Du | dz ≤ 2 d −2k(α) |u α − bα |2 dz. δ Bδ (x0 ) Bδ/2 (x0 ) α=1

α=1

Rn

→ R be a cut-off function satisfying η(z) = 0 Proof. Fix ε > 0 and δ < δ0 . Let η : if |z − x0 | ≥ δ and η(z) = 1 if |z − x0 | ≤ δ/2, and |Dη| ≤ 4δ −1 . Observe that our constraints on b imply that u α − bα ∈ HΣ1 (Ω, d −2k(α) ). Thus, by Corollary 3 in Appendix A, we can take ξ = η2 (u − b) as a test function for (10). Using this special choice of ξ and recalling (11) yields m 2 η d −2k(α) |Du α |2 dz Ω

≤C

α=1

Ω

η2 aαβ Du α Du β dz = −C

Ω

ηaαβ Du β Dη (u α − bα ) dz

m η2 d −2k(α) |Du α |2 + |Dη|2 d −2k(α) |u α − bα |2 dz. ≤C Ω α=1

This implies the inequality in question.

We next study the limiting system when doing a blow-up analysis and then prove that smallness in energy density implies regularity. We first recall a well known result (see [7], Chap. 4). ij

Lemma 6. Let aαβ(h) be a sequence of measurable functions satisfying β

λ| p|2 ≤ aαβ(h) (x, v) piα p j ≤ Λ| p|2 , ij

(x, v, p) ∈ Ω × Rm × Rmn ij

for some 0 < λ ≤ Λ < ∞ and converging in L 2 (B1 (0)) to aαβ verifying the same ellipticity bound. Let f (h) be a sequence in L 2 (B1 (0); Rm ) converging weakly in L 2 (B1 (0)) 1 (B (0); Rm ) ∩ L 2 (B (0)) such that to f . Let u (h) be a sequence in Hloc 1 1 β ∂u (h) ∂ ij α aαβ(h) = f (h) i ∂x ∂x j


425

in the weak sense in B1 (0). If u (h) converges weakly in L 2 (B1 (0); Rm ) to u then u ∈ 1 (B (0); Rm ) and Hloc 1 (i) for any ρ < 1, u (h) converges strongly to u in L 2 (Bρ (0); Rm ) and Du (h) converges weakly to Du in L 2 (Bρ (0); Rm ), (ii) and more importantly, u satisfies in the weak sense in B1 (0) the system β ∂ i j ∂u a = f. αβ ∂ xi ∂x j Lemma 7. Let u α ∈ HΣ1 (Ω, d −2k(α) ) ∩ L ∞ (Ω, d −k(α) ) be a weak solution to (10). Let d(x(h) ) are all less B(h) = Bδ(h) (x(h) ) be a sequence of balls in Ω such that the ratios δ(h) than some fixed κ. Let Σ(h) be the image of Σ under the map x → (x − x(h) )/δ(h) and d(h) the distance function to Σ(h) . Assume that Σ(h) stabilizes to some Σ∗ , i.e. d(h) converges uniformly away from Σ∗ to d∗ , the distance function to Σ∗ . Set ε(h) = E δ(h) (x(h) )1/2 . Define u α(h) (x) = where u¯ α(h)

1 k(α)

ε(h) δ(h)

=

[u α (x(h) + δ(h) x) − u¯ α(h) ],

x ∈ B1 (0),

the average o f u α over B(h) i f k(α) = 0, 0 other wise.

1 If ε(h) → 0, there exists u α∗ ∈ Hk(α) (B1/2 (0)) such that, up to extracting a subsev¯∗0 quence,

(i) u α(h) converges weakly in H 1 (B1/2 (0)) and strongly in L 2 (B1/2 (0)) to u α∗ ,

−k(α) α −k(α) u (h) converges strongly to d∗−k(α) u α∗ in L 2 (B1/2 (0)) and d(h) Du α(h) con(ii) d(h) −k(α)

Du α∗ in L 2 (B1/2 (0)). −[k(α)+k(β)] = 0 in the weak sense in Moreover, u ∗ satisfies div d∗ bαβ∗ Du β B1/2 (0)\Σ∗ , where bαβ∗ are constant and satisfy verges weakly to d∗

ν1

m α=1

−[k(α)+k(β)]

d∗−2k(α) | p α |2 ≤ d∗

bαβ∗ p α p β ≤ ν2

m α=1

d∗−2k(α) | p α |2 ,

and the ratio ν2 /ν1 does not depend on the sequence of balls Bδ(h) (x(h) ). Additionally, bαβ∗ can be written in the form bαβ∗ = bαβ (x∗ , u˜ 1∗ , . . . , u˜ τ∗−1 , 0) for some accumulation point x∗ of the sequence x(h) . Proof. Let’s assume for the moment that the above convergences have been established. −k(α) α Passing to a subsequence, we can assume that x h → x∗ , d(h) u (h) → d∗−k(α) u α∗ a.e. Also, as C −2k(α) α 2 d −2k(α) |u α |2 dz ≤ C, δ(h) |u¯ (h) | ≤ n δ(h) B(h)

426

L. Nguyen −k(α)

u¯ α(h) → u˜ α∗ . It follows that −k(α) −k(α) k(α) u˜ α (x(h) + δ(h) x) := d(h) (x) δ(h) ε(h) δ(h) u α(h) (x) + u¯ α(h)

we can assume that δ(h)

→ u˜ α∗ d∗ (x)−k(α) a.e. Therefore, if we define −[k(α)+k(β)]

aαβ(h) (x) = d(h) then

bαβ x(h) + δ(h) x, u(x ˜ (h) + δ(h) x) ,

aαβ(h) (x) → d∗ (x)−[k(α)+k(β)] bαβ x∗ , u˜ α∗ d∗ (x)−k(α) a.e.

By the Lebesgue dominated convergence theorem, this can be regarded as convergence 2 (B (0)\Σ ). An application of Lemma 6 yields the second half of the lemma. in L loc 1 ∗ It remains to check the convergences. We have m m 1 −2k(α) α 2 d(h) |Du (h) | d x = n−2 2 d −2k(α) |Du α |2 d x = 1. δ(h) ε(h) B(h) α=1 B1 (0) α=1 2 . In particular, this implies Here we have used the assumption that E δ(h) (x(h) ) = ε(h) α 2 that the L norm of Du (h) over B1 (0) is uniformly bounded for α < τ . To proceed, we consider the two cases α < τ and α ≥ τ separately. When α < τ , k(α) = 0 and so |Du α(h) |2 d x = 1. B1 (0)

On the other hand, by definition, the integral of u α(h) over B1 (0) vanishes. The standard Poincaré inequality then implies that the H 1 -norm of u α(h) over B1 (0) is uniformly bounded for α < τ . The required convergence for such α follows. Consider the remaining case where α ≥ τ , i.e. k(α) > 0. By Corollary 3 in Appendix A, the L 2 -norm and so the HΣ1 -norm of u α(h) over B3/4 (0) are uniformly bounded. One can then modify the proof of Proposition 5 in Appendix A to get the required convergences. (The conditions (C1)–(C5) therein are satisfied with ϕ(h) being the distance function to Σ(h) . The condition on the smallness of one weight with respect to the other holds thanks to Remark 7(ii). Furthermore, the bound of this smallness relation is uniform as Σ(h) stabilizes to Σ∗ .) The details work out in exactly the same manner and hence are omitted. Lemma 8. Let u α ∈ HΣ1 (Ω, d −2k(α) ) ∩ L ∞ (Ω, d −k(α) ) be a weak solution to (10). Let d(x(h) ) are all at least B(h) = Bδ(h) (x(h) ) be a sequence of balls in Ω such that the ratios δ(h) some fixed κ. Set ε(h) = E δ(h) (x(h) )1/2 and define u α(h) (x) =

1 [u α (x(h) + δ(h) x) − u¯ α(h) ], ε(h) d(x(h) )k(α)

= B where u¯ α(h) is the average of u α over B(h) κδ(h) /2 (x (h) ).

x ∈ B1 (0),


427

If ε(h) → 0, there exists u ∗ ∈ H 1 (Bκ/2 (0)) such that, up to extracting a subsequence, 1 2 u (h) converges

weakly in βH (Bκ/2 (0)) and strongly in L (Bκ/2 (0)) to u ∗ . Moreover, u ∗ satisfies div aαβ∗ (x) Du = 0 in the weak sense in B1 (0)\Σ∗ , where aαβ∗ are smooth and satisfy ν1

m

| p α |2 ≤ aαβ∗ p α p β ≤ ν2

α=1

m

| p α |2 ,

α=1

and the ratio ν2 /ν1 does not depend on the sequence of balls Bδ(h) (x(h) ) and does not exceed C κ −2 max k(α) . Proof. The proof of this lemma is similar to but easier than that of Lemma 7 and is omitted. Having a Caccioppoli inequality at hand and some understanding of the limiting system, we are ready to go forward with our proof of Theorem 2, i.e. smallness in density E δ (x) implies regularity. Recall that we require here that any limiting system decouples into m independent equations by asking (18). Lemma 9. Let u α ∈ HΣ1 (Ω, d −2k(α) ) ∩ L ∞ (Ω, d −k(α) ) be a weak solution to (10). Assume in addition that (18) holds and all the k(α) are either zero or bigger than k − 2. Let x0 be a point in Ω and δ0 = dist (x0 , ∂Ω). There exist ε0 and λ0 such that for any x ∈ Bδ0 /2 (x0 ) and δ ∈ (0, δ0 /4) satisfying E δ (x) ≤ ε02 there holds E λ0 δ (x) ≤ E δ (x)/2. Proof. Arguing indirectly, we assume that the conclusion fails. Fix some λ0 and κ for the moment. They will be determined by a set of constraints which will be formulated in the sequel. We can find sequences ε(h) → 0, x(h) ∈ Bδ0 /2 (x0 ), and δ(h) ∈ (0, δ0 /4) 2 but E 2 such that E δ(h) (x(h) ) = ε(h) λ0 δ(h) (x (h) ) > ε(h) /2. In addition, we can assume that one of the following two cases occurs: All the ratios Case 1:

d(x(h) ) δ(h)

d(x(h) ) δ(h)

are less than κ or all are not.

< κ for all h. Define d(h) , u α(h) , d∗ , Σ∗ , u α∗ and bαβ∗ as in Lemma 7. In

the ball B1/2 (0), u α∗ satisfies div (d −2k(α) Du α ) = 0. By Theorem 1, |u α∗ (x) − u α∗ (0)| ≤ C|x|, |u α∗ (x)|

≤ C d∗ (x)

α < τ, 2k(α)−k+2

,

α ≥ τ,

for some C independent of the balls Bδ(h) (x(h) ). Therefore, by the Caccioppoli inequality, for B(h) = B2λ0 δ(h) (x(h) ), E λ0 δ(h) (x(h) ) 2 ε(h)

C ≤ 2 ε(h) (λ0 δ(h) )n +

m

B(h)

α=1

|u α − u¯ α(h) − ε(h) m (h) u α∗ (0)|2 k(α)

d −2k(α) |u α |2 dz

α=τ

C ≤ n λ0

τ −1

τ −1

B2λ0 (0)

α=1

|u α(h)

− u α∗ (0)|2

+

m α=τ

−2k(α) α 2 d(h) |u (h) |

dz

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L. Nguyen

≤

C λn0 C + n λ0

≤

C λn0

m B2λ0 (0) α=1

−k(α)

|d(h)

B2λ0 (0)

|z|2 +

m B2λ0 (0) α=1

u α(h) − d∗−k(α) u α∗ |2 dz

m α=τ

d∗2k(α)−2k+4 dz

−k(α) α |d(h) u (h) − d∗−k(α) u α∗ |2 dz

+ C1 λ20 + C2 (κ + 2λ0 )2k(τ )−2k+4 . Hence if C1 λ20 + C2 (κ + 2λ0 )2k(τ )−2k+4 ≤

1 , 8

(19)

2 ε(h) ε2 2 < E λ0 δ(h) (x (h) ) ≤ 4 for h large enough, whence a contradiction. d(x(h) ) Case 2: δ(h) ≥ κ for all h. Let u α(h) , u α∗ and aαβ∗ be as in Lemma 8. In the ball

Bκ/2 (0), u ∗ satisfies div aαβ∗ (x)Du β = 0 where the aαβ∗ are smooth and satisfy m α 2 α β α 2 −2 max k(α) . Also, the ν1 m α=1 | p | ≤ aαβ∗ p p ≤ ν2 α=1 | p | with ν2 /ν1 ≤ C κ 2 L norm of u ∗ in Bκ/2 (0) is universally bounded. Therefore

we infer that

m α=1

|u α∗ (x) − u α∗ (0)| ≤ C κ −

n+A 2

|x|, x ∈ Bκ/4 (0)

for some A > 0. Like in Case 1, the constant C is independent of the balls Bδ(h) (x(h) ). We again invoke the Caccioppoli inequality and obtain E λ0 δ(h) (x(h) ) 2 ε(h)

m C d −2k(α) |u α − u¯ α(h) − ε(h) d(x(h) )k(α) u α∗ (0)|2 dz 2 (λ δ )n ε(h) B(h) α=1 0 (h) m C |u α(h) − u α∗ (0)|2 dz ≤ n λ0 B2λ0 (0) α=1 m C C |u α(h) − u α∗ |2 dz + n+A n |z|2 dz ≤ n λ0 B2λ0 (0) κ λ0 B2λ0 (0) α=1 m C |u α(h) − u α∗ |2 dz + C3 κ −n−A λ20 . ≤ n λ0 B2λ0 (0)

≤

α=1

Similar to Case 1, if we can choose λ0 and κ such that C3 κ −n−A λ20 ≤ this will lead us to an absurdity.

1 , 8

(20)


429

To complete the proof, we need to furnish (19) and (20). This is always doable by 1 1 1 ) B − 2λ0 , first picking λ0 small enough such that λ0 ≤ 4√1C and (8C3 λ20 ) n+A ≤ ( 16C 2 1 and then picking κ in between the last two figures. Here B = 2k(τ ) − 2k + 4 > 0. Proof of Theorem 2. Let u α ∈ HΣ1 (Ω, d −2k(α) ) ∩ L ∞ (Ω, d −k(α) ) be a weak solution of (10). Let ε0 be as in Lemma 9. Assume that E δ (x0 ) ≤ ε0 for some 0 < δ < dist (x0 , ∂Ω)/4. It follows from Lemma 9 and standard iteration techniques (cf. [8]) that there exist positive constants C and λ depending only on k(α) and the L ∞ (Ω, d −k(α) )-norm of u α such that m 1 E σ (x) = n−2 d −2k(α) |Du α |2 dz ≤ C σ 2λ σ Bσ (x) α=1

for any x ∈ Bδ/2 (x0 ) and σ < δ/4. It follows from Morrey’s lemma that u is Hölder continuous in Bδ/2 (x0 ). Moreover, if k(α) > 0, the above estimate implies a rate of vanishing of u α near Σ. To see this, observe that if σ < d(x)/2, then |Du α |2 dz ≤ C d(x)2k(α) σ n−2+2λ , 1 ≤ α ≤ m, Bσ (x)

which implies osc u α ≤ C d(x)k(α) σ λ .

Bσ/2 (x)

(See Theorem 7.19 in [8], for example.) Since u α vanishes along Σ when k(α) > 0, this implies that d k(α)−λ u α is bounded in Bδ/4 (x0 ). 6. Applications We now turn to the proof of Theorem A and B. We first consider a special case of Problem 1 in which the smallness assumption in Theorem 2 is fulfilled. The condition (21) below that we impose additionally on the system (10) is a common feature that harmonic maps into hyperbolic spaces share. This was first used by Li and Tian in [14] in the case of real hyperbolic spaces. We think that this phenomenon has some connection to the underlying geometry of the target manifolds, but we have very limited evidence. Proposition 2. Suppose all the assumptions of Theorem 2 hold. Assume in addition that τ = 2. If aβ0 = δβ0 and a0β = −l(α) aαβ u α ,

2 ≤ β ≤ m,

(21)

uα

are Hölder continuous across Σ. for some l(1) = 0, l(2), …, l(m) > 0, then the Moreover, for α > 1, |u α | ≤ C d k(α)+λ for some λ > 0. First observe that, due to (21), the first equation in the system (10) can be rewritten as Δu 1 =

l(α) aαβ Du α Du β .

(22)

2≤α,β≤m

Note that the right-hand side, which we will abbreviate as F(x, u, Du), is always nonnegative by ellipticity.

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L. Nguyen

Lemma 10. Let u α ∈ HΣ1 (Ω, d −2k(α) ) ∩ L ∞ (Ω, d −k(α) ) satisfy (22) in the sense of distributions. Let x0 be a point in Ω and δ0 = dist (x0 , ∂Ω). Then there exists K such that if δ < δ0 /4 and x ∈ Bδ0 /2 (x0 ), Bδ (x)

m

|x − z|−n+2

d −2k(α) |Du α |2 dz ≤ K ,

α=1

and so E δ (x) =

1 δ n−2

m Bδ (x) α=1

d −2k(α) |Du α |2 dz ≤ K .

Proof. Note that (22) is satisfied in the sense of distributions throughout Ω. For ε sufficiently small, let ηε : [0, ∞) → R be a cut-off function satisfying ηε = 0 in [0, ε] ∪ [2δ, ∞), ηε = 1 in [2ε, δ], |ηε |, |ηε | ≤ C in [δ, 2δ], and |ηε | ≤ Cε−1 , |ηε | ≤ Cε−2 in [ε, 2ε]. Let R be a positive number such that |u 1 | ≤ R. Let G x (z) = |x − z|−n+2 and set ε G x (z) = ηε (|x − z|)G x (z). We will write Br for Br (x). Inserting G εx (z)(u 1 + R + 1) as a test function into Eq. (22) and note that F is nonnegative, we get ε G x (z) F(z, u, Du) dz ≤ F(z, u, Du) G εx (z) (u 1 + R + 1) dz B2δ B2δ G εx (z)|Du 1 |2 + (u 1 + R + 1) Du 1 DG εx dz. =− B2δ

Hence Bδ \B2ε

G εx (z)

Bδ \B2ε

≤ −C =C ≤ K.

d −2k(α) |Du α |2 dz

α=1

≤C

m

B2δ

B2δ

G x (z) F(z, u, Du) + |Du 1 |2 dz

(u 1 + R + 1) Du 1 DG εx dz.

(u 1 + R + 1)2 Δ[ηε (|x − z|) G x (z)] dz

The conclusion then follows from Lebesgue’s monotone convergence theorem.

Proposition 3 (Smallness of density). Let u α ∈ HΣ1 (Ω, d −2k(α) ) satisfy (22) in the sense of distributions and assume that u α ∈ L ∞ (Ω, d −k(α) ). Let x0 be a point in M and δ0 = dist (x0 , ∂ M). Let K be the constant obtained in Lemma 10. For given ε > 0, δ < δ0 /4, and x ∈ Bδ0 /2 (x0 ), there exists σ = σ (ε, δ, x) between e−2K (n−2)/ε δ and δ such that E σ (x) ≤ ε.


431

Proof. Fix ε > 0. For δ = e−2K (n−2)/ε δ, we have

δ δ

E σ (x) dσ = σ

δ

δ

σ −n+1

m

d −2k(α) |Du α |2 dz dσ

Bσ α=1

= (−n + 2)σ

−n+2

d

−2k(α)

Bσ α=1

+ (n − 2)

m

Bδ \Bδ

|x − z|−n+2

δ |Du | dz

m

α 2

δ

d −2k(α) |Du α |2 dz,

α=1

where in the above, all balls are centered at x. This together with Lemma 10 implies δ E σ (x) dσ ≤ 2K (n − 2). σ δ The assertion follows immediately from an argument by contradiction.

Proof of Proposition 2. By Proposition 3, lim inf E σ (x) = 0 ∀ x ∈ Ω. σ →0

Theorem 2 then applies yielding the assertion.

Finally, we consider Theorem A. Recall that u, v, χ and ψ satisfy (3)–(6) We will use various equations derivable from (3)–(6). The first set gives the formulas for the Laplacians of u, χ and ψ, h ) = Δu = 2e4u |Dv − ψ Dχ + χ Dψ|2 + e2u (|Dχ |2 + |Dψ|2 ), 2 Δχ = −2Du · Dχ + 2e2u Dψ · (Dv − ψ Dχ + χ Dψ), Δψ = −2Du · Dψ − 2e2u Dχ · (Dv − ψ Dχ + χ Dψ).

Δ(u −

(23) (24) (25)

The second makes way to apply Lemma 4, div (e4u Dv) = e4u Du · (ψ Dχ − χ Dψ) + e4u (ψ Δχ − χ Δψ), div (e2u Dχ ) = 2e4u Dψ · (Dv − ψ Dχ + χ Dψ), div (e2u Dψ) = −2e4u Dχ · (Dv − ψ Dχ + χ Dψ).

(26) (27) (28)

The proof consists of two parts. In the first part, we prove the general case where γ is arbitrary. In the second part, we consider the physical case where γ = 1 and (u, v, χ , ψ) is axially symmetric around the z-axis. For the first part, we apply Theorem 2 and Proposition 2 to obtain Hölder continuity. Note that in this step, condition (21) is crucial. For higher regularity, unlike proving Corollary 2, we cannot apply Theorem 1 due to the coupling of the unknowns. We instead appeal to Lemma 4, a primitive of Theorem 1. For the second part, we follow the reversed Ernst-Geroch reduction scheme to prove smoothness under axial symmetry. This was used by Weinstein in his work on the vacuum case. His idea is the following. The system (3)-(6) was obtained by applying the ErnstGeroch formulation for axisymmetric stationary vacuum solutions, which was done in

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L. Nguyen

the order of the axial Killing vector field being applied first and the stationary Killing vector field second. This choice of order made it more convenient to establish existence and uniqueness but regularity. If one applies the reduction scheme in the reversed order, i.e. to the stationary Killing vector field first and then to the axial Killing vector field, one obtains a different harmonic map problem from a domain of R3 into the real hyperbolic plane. Similarly, if one applies the reversed Ernst-Geroch reduction scheme to the Einstein-Maxwell equations, the system acquired is, though no longer a harmonic map problem, still elliptic. To finish one needs to prove some initial regularity, e.g. energy estimate, for the acquired system. In the vacuum case, this can be done much more simply by a clever use of the De Giorgi-Nash-Moser regularity result for scalar elliptic equation (see [24]). It seems unlikely that this technique applies in the electrovac case. It is precisely at this point that we need to use the C k,α regularity obtained in the general case to go forward. Proof of Theorem A. Note that since the complex hyperbolic plane is a manifold of negative sectional curvature, (u, v, χ , ψ) is C ∞ in R3 \Γ . Thus we only need to consider the regularity question around Γ . Without loss of generality, we assume that the origin is an interior point of Γ , h = −2γ log ρ and that the controlling ideal point on ∂H is (+∞, 0, 0, 0) as in Sect. 3. Step 1. C k,α regularity in the general case. Let u¯ = u − h. Then, as shown in Sect. 3, (u, ¯ v, χ , ψ) satisfies the hypotheses of Theorem 2. (Note that the required L ∞ bounds are a consequence of Lemma 1). Furthermore, by a direct verification, it also satisfies the hypotheses of Proposition 2. (In particular, (21) is satisfied.) Therefore, (u, ¯ v, χ , ψ) is locally Hölder continuous in Ω and there is some λ > 0 so that 1 2 −4γ 2 −2γ 2 2 |D u| ¯ + ρ |Dv| + ρ (|Dχ | + |Dψ| ) ≤ C δλ. δ n−2 Bδ It follows that |D u| ¯ = O(ρ −1+λ ), |Dv| = O(ρ −1+2γ +2λ ), and |Dχ | + |Dψ| = −1+γ +λ O(ρ ). Thus, by (23), |Δu| = O(ρ −2+2λ ). The above estimates show that |Dψ| |Dv − ψ Dχ + χ Dψ| = O(ρ −2+3γ +2λ ). Thus, if λ < γ2 , an application of Lemma 4 to (27) shows that |χ | = O(ρ γ +2λ ). Similarly, |ψ| = O(ρ γ +2λ ). Applying basic gradient estimates for Poisson equations to (27) and (28), we deduce that |Dχ | + |Dψ| = O(ρ −1+γ +2λ ). Using Eqs. (24) and (25) we infer that |Δχ | + |Δψ| = O(ρ −2+γ +2λ ). Hence the right-hand side of (26) is at most O(ρ −2−2γ +4λ ). Lemma 4 applies again, but to (26), yielding |v| = O(ρ 2γ +4λ ). Gradient estimates for Poisson equations then show that |Dv| = O(ρ −1+2γ +4λ ). Repeating the argument in the previous paragraph we see that |v| = O(ρ 2γ +2λ ) and |χ | + |ψ| = O(ρ γ +λ ) for any λ < γ . The regularity result in the general case follows. Step 2. Smoothness when (u, v, χ , ψ) is symmetric about Γ and γ = 1. It suffices to show that (u, v, χ , ψ) is C ∞ in Bδ (0) for some δ > 0 small enough. Step 2(a). Construction of new functions. We first exploit the symmetry to construct new functions defined in a neighborhood of the origin. Let D2 denote the gradient operator in the half plane {(ρ, z) ∈ R2 |ρ > 0}. Set ω = (ωρ , ωz ) = D2 v − ψ D2 χ + χ D2 ψ. Using (4)–(6), we define p, χ˜ and ψ˜ by dp = −2e4u ρ ωz dρ + 2e4u ρ ωρ dz,

(29)

d χ˜ = −(e ρ ψz + p χρ )dρ + (e ρ ψρ − p χz )dz, d ψ˜ = (e2u ρ χz − p ψρ )dρ − (e2u ρ χρ + p ψz )dz.

(30)

2u

2u

(31)


433

Note that they are well-defined up to a constant. Also, by the C k,α regularity result from the general case, p, χ˜ and ψ˜ are locally bounded. Set 1 −4u −1 2 ω˜ = (ω˜ ρ , ω˜ z ) = (e ρ p + ρ) pz − 4ρ p u z , 2 − (e−4u ρ −1 p 2 + ρ) pρ + 4ρ p u ρ + 2 p 1 2 2 p ωρ + ρ pz − 4ρ p u z , 2 p 2 ωz − ρ pρ + 4ρ p u ρ + 2 p . = 2 A straightforward calculation using (3), (29), (30) and (31) shows that 2(ω˜ ρ,z − ω˜ z,ρ ) = 4(ψ˜ ρ χ˜ z − χ˜ ρ ψ˜ z ).

(32)

Thus there exists v˜ such that d v˜ = (ω˜ ρ + ψ˜ χ˜ ρ − χ˜ ψ˜ ρ )dρ + (ω˜ z + ψ˜ χ˜ z − χ˜ ψ˜ z )dz.

(33)

˜ v˜ is defined up to a constant but remains locally bounded. Like χ˜ and ψ, Finally, define u˜ by 1 1 u˜ = − log(e2u ρ 2 − e−2u p 2 ) = − log(e2u¯ − e−2u¯ ρ 2 p 2 ). 2 2

(34)

Recall that u¯ = u − h = u + γ log ρ. Since u¯ and p are locally bounded, u˜ is bounded in B2δ (0) for δ small enough. ˜ By construction, (u, ˜ is smooth in Step 2(b). Equations governing (u, ˜ v, ˜ χ˜ , ψ). ˜ v, ˜ χ˜ , ψ) Bδ (0)\Γ . We claim that it satisfies the following equations in Bδ (0)\Γ : ˜ 2 + e2u˜ (|D χ| ˜ 2 ) = 0, Δu˜ − 2 e4u˜ |D v˜ − ψ˜ D χ˜ + χ˜ D ψ| ˜ 2 + |D ψ| ˜ = 0, div e4u˜ (D v˜ − ψ˜ D χ˜ + χ˜ D ψ)

(35)

˜ = 0, div (e2u˜ D χ) ˜ + 2e4u˜ D ψ˜ · (D v˜ − ψ˜ D χ˜ + χ˜ D ψ) 2u˜ 4 u ˜ ˜ − 2e D χ˜ · (D v˜ − ψ˜ D χ˜ + χ˜ D ψ) ˜ = 0. div (e D ψ)

(37)

(36)

(38)

Define p˜ =

e−2u p e−2u p = . e2u ρ 2 − e−2u p 2 e−2u˜

(39)

Then d p˜ = −2e4u˜ ρ ω˜ z dρ + 2e4u˜ ρ ω˜ ρ dz,

(40)

˜ satisfies (36). which implies that (u, ˜ v, ˜ χ˜ , ψ) By (30), (31) and (39), dχ = −(e2u˜ ρ ψ˜ z − p˜ χ˜ ρ )dρ + (e2u˜ ρ ψ˜ ρ + p˜ χ˜ z )dz. This implies that −(e2u˜ ρ ψ˜ z + p˜ χ˜ ρ )z = (e2u˜ ρ ψ˜ ρ − p˜ χ˜ z )ρ .

(41)

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L. Nguyen

In view of (40), (38) follows. Similarly, (37) holds due to dψ = (e2u˜ ρ χ˜ z + p˜ ψ˜ ρ )dρ − (e2u˜ ρ χ˜ ρ − p˜ ψ˜ z )dz.

(42)

We next verify (35). In the interior of { p˜ = 0}, p and ω˜ vanish, and u˜ = −u, ¯ so (35) follows immediately from (3), (30) and (31). Thus, by continuity, it suffices to consider the region where p˜ = 0. We note that p=

e−2u˜ p˜ e−2u˜ p˜ = . e−2u e2u˜ ρ 2 − e−2u˜ p˜ 2

In view of (29), this implies that 1 −4u˜ −1 2 (e ω= ρ p˜ + ρ) p˜ z − 4ρ p˜ u˜ z , −(e−4u˜ ρ −1 p˜ 2 + ρ) p˜ ρ + 4ρ p˜ u˜ ρ + 2 p˜ , 2 and so, by (32) and (39), ωρ,z − ωz,ρ = 4e4u˜ ρ p˜ |ω| ˜ 2 − 2 p˜ [(ρ u˜ z )z + (ρ u˜ ρ )ρ ] + 2(e4u˜ ρ 2 + p˜ 2 )(ψ˜ ρ χ˜ z − χ˜ ρ ψ˜ z ). On the other hand, by (41) and (42), ωρ,z − ωz,ρ = 2(ψρ χz − χρ ψz ) ˜ 2 ) + 2(e4u˜ ρ 2 + p˜ 2 )(ψ˜ ρ χ˜ z − χ˜ ρ ψ˜ z ). ˜ ˜ 2 + |D2 ψ| = 2e2u˜ ρ p(|D 2 χ| The above relations imply that ˜ 2 ). 4e4u˜ ρ p˜ |ω| ˜ 2 − 2 p˜ [(ρ u˜ z )z + (ρ u˜ ρ )ρ ] = 2e2u˜ ρ p(|D ˜ ˜ 2 + |D2 ψ| 2 χ| Since p˜ is nonzero, (35) follows. ˜ We will use the following lemma, which is easy Step 2(c). Smoothness of (u, ˜ v, ˜ χ˜ , ψ). to prove. Lemma 11. Assume that f is a regular function in B1 (0)\Γ and that | f | = O(ρ −1+α ) for some α! ∈ (0, 1). If N f is the Newtonian potential of f with respect to B1 (0), i.e. N f (x) = B1 (0) Φ(x − y) f (y) dy where Φ is the fundamental solution of the Laplace equation, then N f is C 1,α in B1 (0). As a consequence, if u ∈ L 1 (B1 (0)) is a weak solution of Δu = f in B1 (0), then u is C 1,α in B1 (0). We have shown that (35)–(38) hold in Bδ (0)\Γ . Moreover, by the regularity result for the general case, u, ˜ v, ˜ χ˜ and ψ˜ belong to H 1 (Bδ (0)) ∩ C 0,α (B1 (0)) for any α ∈ (0, 1). Hence, since Γ is of codimension 2, they satisfy (35)-(38) in Bδ (0) in the weak sense. On the other hand, using the regularity result for the general case, we can show that ˜ = u, ˜ v, ˜ χ˜ and ψ˜ are C 0,α in Bδ (0) for any α ∈ (0, 1). Moreover, |D v| ˜ + |D χ| ˜ + |D ψ| O(ρ α−1 ). Applying Lemma 11 to (35), (37), (38) and then (36), we can show that u, ˜ v, ˜ χ˜ and ψ˜ are C 1,α . This allows us to bootstrap in between (35)–(38) to obtain smoothness for u, ˜ v, ˜ χ˜ and ψ˜ in Bδ (0). Step 2(d). Smoothness of (u, ¯ v, χ , ψ). The smoothness of χ and ψ follows from (41) and (42). By (40), p˜ and so e−2u p = e−2u˜ p˜ are smooth. That of e−2u and of u¯ follows from the identity ρ 2 e−2u¯ = e−2u = e2u˜ ρ 2 − e−2u˜ p˜ 2 . Next, since ρ 2 p is smooth and p ∈ C 0,α , p is smooth too. The smoothness of v follows from (29). The proof of the theorem is complete.


435

Proof of Theorem B. The theorem follows immediately from Theorem A and the results in [28]. Acknowledgments. The author is deeply grateful to his advisor, Professor Yanyan Li, who suggested to him this problem as the subject of a Ph.D. dissertation and has given much advice since. He is also thankful to Professor Piotr T. Chru´sciel and Professor Michael Kiessling who read the draft and gave various constructive comments which helped improve the presentation of the paper. He also wishes to thank Professor Penny D. Smith for drawing his attention to [22]. Special thanks go to the anonymous referees who made useful comments and suggestions. 1 ,p

A. The Space W (, w) 1, p

In this appendix, we study the space WΣ (Ω, w) defined in the Introduction. Most of the results here are probably known to readers. However we show them for completeness. Throughout this appendix, we will frequently make the following assumptions on Ω, Σ and w. (C1) Ω is a bounded domain in Rn and Σ is a (n − k)-dimensional submanifold of Rn (possibly with or without a boundary). (C2) ϕ is a positive C 2 function in a punctured neighborhood of Σ such that ϕ(x) → 0 as x → Σ. (C3) w is a positive measurable function defined on Σ such that w = w(ϕ) in the domain of ϕ. ¯ (C4) w is bounded on any compact subset of Ω\Σ. ¯ (C5) w is bounded from below by a positive number on any compact subset of Ω\Σ. It should be noted that, in this appendix, Σ needs not be a subset of Ω. Examples of interest include the case where Σ is C 2 , ϕ is the distance function to Σ and w is a (positive or negative) power of the distance function to Σ. When Σ is less smooth, results for distance-function-related weights can be obtained from the results presented in this section via the so-called “regularized distance function” (see [16]). Lemma 12. Let 1 < p < ∞. Assume (C1)–(C3). Let α and β be functions on (0, ∞) which are locally bounded measurable functions on compact subsets of (0, ∞) and satisfy α(t) ≥ sup

ϕ(x)=t

Δϕ(x) , |Dϕ(x)|2

β(t) ≥ sup |Dϕ(x)|−( p−2)/( p−1) . ϕ(x)=t

Let A be an anti-derivative of α and define t − p p μ A(t) −1/( p−1) −1/( p−1) μ A(s) w˜ μ (t) = e w (t) β(s) w (s) e ds 0

and w˜ μ (x) = w˜ μ (ϕ(x)). (i) If β w −1/( p−1) e−A/( p−1) is locally integrable near 0, then there exists δ > 0 such that d(Ω c , Σ) < δ implies 1, p p | f | w˜ −1/( p−1) d x ≤ C |D f | p w d x f ∈ W0,Σ (Ω, w). Ω

The constant C depends only on p.

Ω

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L. Nguyen

(ii) If, in addition, α is positive and the level surfaces of ϕ are regular hypersurfaces, then there exists δ > 0 such that d(Ω c , Σ) < δ implies 1, p | f | p w˜ μ d x ≤ C |D f | p w d x f ∈ W0,Σ (Ω, w), Ω

Ω

for any μ for which β w−1/( p−1) eμ A is locally integrable near 0. The constant C depends only on p and μ. Proof. It suffices to consider non-negative f ∈ Cc∞ (Ω\Σ). (i) Let Bt = {d(x, Σ) ≥ t}. For t sufficiently small, f vanishes outside of Bt . Thus, if →

F is a C 1 vector-valued function on Ω\Σ, then → → → 0= div ( f p F ) d x = [ f p div ( F ) + p f p−1 D f · F ] d x. Bt

Bt

→

Hence, if div ( F ) is positive,

→

→

f div ( F ) d x ≤ C p

Bt

|∇ f |

p

| F |p

d x.

→

(div ( F )) p−1

Bt

(43)

Define F(t) = −e

−A(t)

t

β(s) w

−1/( p−1)

(s) e

−A(s)/( p−1)

− p+1 ds

< 0,

0

so that F + α F = ( p − 1) β w −1/( p−1) |F| p/( p−1) . →

Hence, if F (x) = F(ϕ(x))Dϕ(x), then → Δϕ F ≥ |Dϕ|2 (F + α F) div ( F ) = |Dϕ|2 F + |Dϕ|2 = ( p − 1) |Dϕ|2 β w −1/( p−1) |F| p/( p−1) ≥ ( p − 1) w −1/( p−1) |F| p/( p−1) |Dϕ| p/( p−1) →

= ( p − 1) w −1/( p−1) | F | p/( p−1) = ( p − 1)w˜ −1/( p−1) , →

which gives

| F |p →

(div ( F )) p−1

≤ ( p − 1) p−1 w. Assertion (i) follows from (43).

(ii) We localize the proof of (i). Since α is positive, A is strictly increasing. Let t j be an increasing sequence of positive real numbers such that 0 ≤ A(t j+1 ) − A(t j ) ≤ 1. Let S j = {d(x, Σ) = t j } and A j = {t j+1 ≤ d(x, Σ) ≤ t j }. Similar to (i), any vector-valued →

→

function F which is differentiable in A j such that div ( F ) > 0 gives rise to − Dj +

1 2

→

→

f p div ( F ) d x ≤ C Aj

|∇ f | p Aj

| F |p →

(div ( F )) p−1

d x,

(44)


where

→

Dj =

437

→

f p F ·ν j dσ − Sj

f p F ·ν j+1 dσ. S j+1

Here ν j denotes the normal vector to S j in the direction of increasing distance. For t j ≤ t ≤ t j+1 , define F j (t) = −e

−A(t)

− p+1

t

β(s) w

−1/( p−1)

(s) e

−A(s)/( p−1)

< 0,

ds + K j

tj →

where K j is a constant to be specified later. Set F j (x) = F j (ϕ(x)) Dϕ(x). Again, F j (t) + α F j (t) = ( p − 1)β w −1/( p−1) (t) |F j (t)| p/( p−1) , →

→

→

which implies div ( F j ) ≥ C w −1/( p−1) | F j | p/( p−1) and

|Fj |p →

(div ( F j )) p−1

≤ Cw.

On the other hand, recalling the definition of F j and the sequence {t j }, we have for t ∈ [t j , t j+1 ], |F j | p/( p−1) (t) − p t −1/( p−1) (A(t)−A(s))/( p−1) A(t)/( p−1) = βw e ds + e Kj tj

≥C

t

− p βw

tj

≥ Ce

p μ A(t)

−1/( p−1) −μ(A(t)−A(s))

e

t

ds + e

A(t)/( p−1)

Kj − p

βw

−1/( p−1) μ A

e

ds + e

(μ+1/( p−1))A(t j )

Kj

.

tj

Hence, by setting K j = e−(μ+1/( p−1))A(t j )

tj

β w −1/( p−1) eμ A ds,

0

we arrive at w −1/( p−1) |F j | p/( p−1) ≥ C w˜ μ (t). The above estimates allow us to rewrite (44) as −D j + C1 f p w˜ d x ≤ C2 Aj

|D f |2 w d x, Aj

where C1 and C2 is independent of j. Summing over j and recall that f ∈ Cc∞ (Ω\Σ), we get the assertion. Remark 7.

(i) It happens in many cases that w = o(w˜ μ ).

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L. Nguyen

(ii) For ϕ(x) = d(x, Σ), we can take α(t) = k/t and β(t) = 1. Here k is the codimension of Σ. In that case, t − p w −1/( p−1) (s) s k μ ds . w˜ μ (t) = t p k μ w −1/( p−1) (t) 0

˜ = d(x, Σ)−γ − p . In particular, if w(x) = d(x, Σ)−γ , we can take w(x) 1, p (iii) The proof is actually valid if we only assume that f ∈ WΣ (Ω, w) and f = 0 at points on ∂Ω where the normal vector is not perpendicular to Dϕ. Proposition 4. Let 1 < p < ∞. Assume (C1)-(C4). Let δ and μ be as in Lemma 12. Define w˜ μ (x) if d(x, Σ) < δ, w(x) ˜ = 1 otherwise. 1, p

˜ and (i) For any f ∈ W0,Σ (Ω, w), f ∈ L p (Ω, w) | f | p w˜ d x ≤ C [| f | p + |D f | p w] d x. Ω

(ii) For any f ∈

Ω

1, p WΣ (Ω, w),

ω

f ∈

p L loc (Ω, w) ˜

| f | p w˜ d x ≤ C

Ω

and

[| f | p + |D f | p w] d x for ω Ω.

Proof. Part (i) follows immediately from Lemma 12 and the classical Poincaré inequality. For part (ii), it suffices to consider ω for which there is some r such that each connected component of ω ∩{d(x, Σ) ≤ r } is bounded by the hypersurface {d(x, Σ) = r } and two ¯ arbitrary hypersurfaces tangential to Dϕ. Also, we can assume that f ∈ Cc∞ (Ω\Σ). We will show that p | f | w˜ d x ≤ C [| f | p + |D f | p w] d x. (45) ω

ω

Let η be a standard cut-off function which vanishes in {d(x, Σ) > r } and is identically 1 in {d(x, Σ) < r/2}. Split f = f 1 + f 2 = f η + f (1 − η). Then f 1 vanishes on {x ∈ ∂ω|d(x, Σ) = r }. The remainder of ∂ω is tangential to Dϕ. Thus by the remark following Lemma 12, p | f 1 | w˜ d x ≤ C |D( f η)| p w d x ω ω | f |p w dx + C |D f | p w d x ≤C {r/2 βc := 1/(2σ ) + d/4, the initial data are too small to ignite the nonlinearity at leading order, and the leading order behavior of ψ ε as ε → 0 is the same as in the linear case λ = 0, up to Ehrenfest time. On the other hand, if β = βc , the function ψ ε is given at leading order by a wave packet whose envelope satisfies a nonlinear equation, up to a nonlinear analogue of the Ehrenfest time. We show moreover a nonlinear superposition principle: when the initial data is the sum of two wave packets of the form (1.2), then ψ ε is approximated at leading order by the sum of the approximations obtained in the case of a single initial coherent state. Up to changing ψ ε to ε−β ψ ε , we may assume that the initial data are of order O(1) in L 2 (Rd ), and we consider ⎧ ε2 ⎪ ⎪ ⎨ iε∂t ψ ε + ψ ε = V (x)ψ ε + λεα |ψ ε |2σ ψ ε , (t, x) ∈ R+ × Rd , 2 x − x0 i(x−x0 )·ξ0 /ε ⎪ ⎪ ⎩ , e ψ ε (0, x) = ε−d/4 a √ ε

(1.3)

where α = 2βσ . 1.1. The linear case. In this paragraph, we assume λ = 0: iε∂t ψ ε +

ε2 ψ ε = V (x)ψ ε ; ψ ε (0, x) = ε−d/4 a 2

x − x0 √ ε

ei(x−x0 )·ξ0 /ε .

(1.4)

The assumption we make on the external potential throughout this paper (even when λ = 0) is the following: Assumption 1.1. The external potential V is smooth, real-valued, and subquadratic: γ V ∈ C ∞ (Rd ; R) and ∂x V ∈ L ∞ Rd , ∀|γ | 2.

Nonlinear Coherent States and Ehrenfest Time

Consider the classical trajectories associated with the Hamiltonian

445 |ξ |2 2

+ V (x):

x(t) ˙ = ξ(t), ξ˙ (t) = −∇V (x(t)); x(0) = x0 , ξ(0) = ξ0 .

(1.5)

These trajectories satisfy |ξ(t)|2 |ξ0 |2 + V (x(t)) = + V (x0 ), ∀t ∈ R. 2 2 The fact that the potential is subquadratic implies that the trajectories grow at most exponentially in time. Notation. For two positive numbers a ε and bε , the notation a ε bε means that there exists C > 0 independent of ε such that for all ε ∈ ]0, 1], a ε Cbε . Lemma 1.2. Let (x0 , ξ0 ) ∈ Rd × Rd . Under Assumption 1.1, (1.5) has a unique global, smooth solution (x, ξ ) ∈ C ∞ (R; Rd )2 . It grows at most exponentially: ∃C0 > 0, |x(t)| + |ξ(t)| eC0 t , ∀t ∈ R.

(1.6)

Sketch of the proof. We explain the exponential control only. We infer from (1.5) that x solves an Hamiltonian ordinary differential equation, x(t) ¨ + ∇V (x(t)) = 0. Multiply this equation by x(t), ˙

d 2 ˙ + V (x(t)) = 0, (x) dt and notice that in view of Assumption 1.1, V (y) y 2 : x(t) ˙ x(t) , and the estimate follows. Remark 1.3. The case V (x) = −|x|2 shows that the result of Lemma 1.2 is sharp. We associate with these trajectories the classical action t 1 |ξ(s)|2 − V (x(s)) ds. S(t) = 2 0 We observe that if we change the unknown function ψ ε to u ε by x − x(t) i(S(t)+ξ(t)·(x−x(t)))/ε ε −d/4 ε e ψ (t, x) = ε u t, √ , ε

(1.7)

(1.8)

then, in terms of u ε = u ε (t, y), (1.4) is equivalent: 1 i∂t u ε + u ε = V ε (t, y)u ε ; u ε (0, y) = a(y), 2

(1.9)

where the external time-dependent potential V ε is given by V ε (t, y) =

√ √ 1 V (x(t) + εy) − V (x(t)) − ε ∇V (x(t)), y . ε

(1.10)

446

R. Carles, C. Fermanian-Kammerer

This expression reveals the first terms of the Taylor expansion of V about the point x(t). Passing formally to the limit, V ε converges to the Hessian of V at x(t) evaluated at (y, y). One does not even need to pass to the limit if V is a polynomial of degree at most two: in that case, we see that the solution ψ ε remains exactly a coherent state for all time. Let us denote by Q(t) the symmetric matrix Q(t) = Hess V (x(t)). It is well-known that if v solves 1 1 i∂t v + v = Q(t)y, y v; v(0, y) = a(y), 2 2 then the function x − x(t) i(S(t)+ξ(t)·(x−x(t)))/ε ε −d/4 e ϕlin (t, x) = ε v t, √ ε

(1.11)

(1.12)

approximates ψ ε for large time in the sense that there exists C > 0 independent of ε such that √ ε

ψ ε (t, ·) − ϕlin (t, ·) L 2 (Rd ) C εeCt . See e.g. [3,11–13,22–24] and references therein. We give a short proof of this estimate, which can be considered as the initial step toward the nonlinear analysis which is presented in the next paragraph. We first notice that since V is subquadratic, we have the following pointwise estimate: ε √ V (t, y) − 1 Q(t)y, y C ε|y|3 , (1.13) 2 ε = u ε − v satisfies for some constant C independent of t. The error rlin 1 ε 1 1 ε ε + rlin = V ε u ε − Q(t)y, y v = V ε rlin + V ε − Q(t)y, y v, i∂t rlin 2 2 2 ε along with the initial value rlin|t=0 = 0. Since V ε is real-valued, the classical energy estimate for Schrödinger equations yields, in view of (1.13), t √ ε

rlin

L ∞ ([0,t];L 2 (Rd )) ε

|y|3 v(τ, y) L 2 (Rd ) dτ. 0

Since Q is bounded (V is subquadratic), we have the control

|y|3 v(τ, y) L 2 (Rd ) CeCτ for some constant C > 0; see Proposition 2.1 below. We then have to notice that the wave packet scaling is L 2 -unitary: ε

ψ ε (t, ·) − ϕlin (t, ·) L 2 (Rd ) = u ε (t, ·) − v(t, ·) L 2 (Rd ) .

To summarize, we have: Lemma 1.4. Let d 1 and a ∈ S(Rd ). There exists C > 0 independent of ε such that √ ε (t, ·) L 2 (Rd ) C εeCt . (1.14)

ψ ε (t, ·) − ϕlin In particular, there exists c > 0 independent of ε such that sup 0t c log

1 ε

ε

ψ ε (t, ·) − ϕlin (t, ·) L 2 (Rd ) −→ 0. ε→0


447

1.2. The nonlinear case. We now consider the nonlinear situation λ = 0. Resuming the same change of unknown function (1.8), then adapting the above computation leads to 1 i∂t u ε + u ε = V ε u ε + λεα−αc |u ε |2σ u ε , 2

(1.15)

where V ε is given by (1.10) as in the linear case, and αc = 1 +

dσ . 2

(1.16)

The real number αc appears as a critical exponent. In the case α > αc , we can approximate the nonlinear solution u ε by the same function v as in the linear case, given by (1.11). The space will turn out to be quite natural for energy estimates. Introduce the operators Aε (t) =

√

ξ(t) ε∇ − i √ ; ε

B ε (t) =

x − x(t) . √ ε

Note that A and B are essentially ∇ and x, up to the wave packet scaling, in the moving frame. From this point of view, our energy space is quite different from the one associated with the Lyapounov functional considered in [15], and more related to the one considered in [26], since we pay attention to the localization of the wave packet, through B ε . For f ∈ , we set

f H = f L 2 (Rd ) + Aε (t) f L 2 (Rd ) + B ε (t) f L 2 (Rd ) , where we do not emphasize the fact that this norm depends on ε and t. Proposition 1.5. Let d 1, a ∈ S(Rd ). Suppose that α > αc . There exist C, C1 > 0 independent of ε, and ε0 > 0 such that for all ε ∈ ]0, ε0 ], 1 1 ε , α − αc . (t) H εγ eC1 t , 0 t C log , where γ = min

ψ ε (t) − ϕlin ε 2 In particular, there exists c > 0 independent of ε such that sup 0t c log

1 ε

ε

ψ ε (t) − ϕlin (t) H −→ 0. ε→0

The proof is more complicated than in the linear case (see §2). The solution of (1.3) is linearizable in the sense of [19] (see also [9]), up to an Ehrenfest time. In the critical case α = αc with λ = 0, the solution of (1.3) is no longer linearizable. Indeed, passing formally to the limit ε → 0, Eq. (1.15) becomes 1 1 i∂t u + u = Q(t)y, y u + λ|u|2σ u; u(0, y) = a(y). 2 2

(1.17)

Remark 1.6 (Complete integrability). The cubic one-dimensional case d = σ = 1 is special: if Q˙ = 0, then (1.17) is completely integrable ([1]). However, if Q˙ = 0, there exists no Lax pair when the nonlinearity is autonomous as in (1.17); see [31,39]. Note also that if Q˙ = 0, then u ε = u for all time.

448


As in the linear case, we note that if V is exactly a polynomial of degree at most two, then u is actually equal to u ε for all ε. The global well-posedness for (1.17) has been established in [7]. We first prove that u yields a good approximation for u ε on bounded time intervals: Proposition 1.7. Let d 1, σ > 0 with σ < 2/(d − 2) if d 3, and a ∈ S(Rd ). Let u ∈ C(R; ) be the solution to (1.17), and let x − x(t) i(S(t)+ξ(t)·(x−x(t)))/ε ε −d/4 ϕ (t, x) = ε u t, √ . (1.18) e ε For all T > 0 (independent of ε > 0), we have sup ψ ε (t) − ϕ ε (t) L 2 (Rd ) = O

0t T

If in addition σ > 1/2, sup ψ ε (t) − ϕ ε (t) H = O

0t T

√

ε .

√

ε .

Remark 1.8. The presence of u, which solves a nonlinear equation, clearly shows that the nonlinearity modifies the coherent state at leading order. Note however that the Wigner measure of ψ ε (see e.g. [20,33]) is not affected by the nonlinearity: w(t, x, ξ ) = u(t) 2L 2 (Rd ) δ (x − x(t)) ⊗ δ (ξ − ξ(t)) = a 2L 2 (Rd ) δ (x − x(t)) ⊗ δ (ξ − ξ(t)). The Wigner measure remains the same because the nonlinearity alters only the envelope of the coherent state, not its center in phase space. Remark 1.9 (Supercritical case). Consider the case α < αc , and assume for instance V = 0. Resuming the scaling (1.8), Eq. (1.15) becomes 1 i∂t u ε + u ε = λεα−αc |u ε |2σ u ε . 2 At time t = 0, u ε is independent of ε: u ε|t=0 = a. Setting 2 = εαc −α and changing the time variable to s = t/, the problem reads 2 u = |u |2σ u ; u (0, x) = a(x). (1.19) 2 Therefore, to understand the asymptotic behavior of u as ε → 0 (or equivalently, as → 0) for t ∈ [0, T ], we need to understand the large time (s ∈ [0, T /]) behavior in (1.19). This corresponds to a large time semi-classical limit in the (supercritical) WKB regime. Describing this behavior is extremely delicate, and still an open problem; see [6]. i∂s u +

In order to prove the validity of the approximation on large time intervals, we introduce the following notion: Definition 1.10. Let u ∈ C(R; ) be a solution to (1.17), and k ∈ N. We say that (E x p)k is satisfied if there exists C = C(k) such that ∀α, β ∈ Nd , |α| + |β| k, x α ∂xβ u(t) L 2 (Rd ) eCt .


449

Note that reasonably, to establish (E x p)k , the larger the k, the smoother the nonlinearity z → |z|2σ z has to be. For simplicity, we shall now assume σ ∈ N. Proposition 1.11 (from [7]). Let d 3, σ ∈ N with σ = 1 if d = 3, a ∈ S(Rd ) and k ∈ N. Then (E x p)k is satisfied (at least) in the following cases: • σ = d = 1 and λ ∈ R (cubic one-dimensional case). • σ 2/d, λ > 0 and Q(t) is diagonal with eigenvalues ω j (t) 0. • σ 2/d, λ > 0 and Q(t) is compactly supported. It is very likely that this result remains valid under more general assumptions (see in particular [7, §6.2] for the case σ = 2/d). Yet, we have not been able to prove it. Let us comment a bit on these three cases. The first case is the most general one concerning the potential V and the classical trajectory x(t): the only assumption carries over the nonlinearity (the important aspect is that it is L 2 -subcritical). The other two cases concern L 2 -critical or supercritical defocusing nonlinearities. In the second case, V is required to be concave (along the classical trajectory), and the last case corresponds for instance to a compactly supported HessV, when the classical trajectory is not trapped. In this last case, we have actually better than an exponential decay: Sobolev norms are bounded, and momenta grow algebraically in time. The following result could be improved in this case. Theorem 1.12. Let a ∈ S(Rd ). If (E x p)4 is satisfied, then there exist C, C2 > 0 independent of ε, and ε0 > 0 such that for all ε ∈ ]0, ε0 ], √ 1 ε exp (exp(C2 t)), 0 t C log log . ε In particular, there exists c > 0 independent of ε such that

ψ ε (t) − ϕ ε (t) H

sup 0t c log log

1 ε

ψ ε (t) − ϕ ε (t) H −→ 0. ε→0

In the one-dimensional cubic case, this result can be improved on two aspects. First, we can prove a long time asymptotics in L 2 provided (E x p)3 is satisfied. More important is the fact that we obtain an asymptotics up to an Ehrenfest time: Theorem 1.13. Assume d = σ = 1, and let a ∈ S(R). If (E x p)3 is satisfied, then there exist C, C3 > 0 independent of ε, and ε0 > 0 such that for all ε ∈ ]0, ε0 ], √

1 ε exp(C3 t), 0 t C log . ε In particular, there exists c > 0 independent of ε such that

ψ ε (t) − ϕ ε (t) L 2 (R)

sup 0t c log

1 ε

ψ ε (t) − ϕ ε (t) L 2 (R) −→ 0. ε→0

If in addition (E x p)4 is satisfied, then for the same constants as above,

ψ ε (t) − ϕ ε (t) H

√

1 ε exp(C3 t), 0 t C log , ε

and sup 0t c log

1 ε

ψ ε (t) − ϕ ε (t) H −→ 0. ε→0

450


The technical reason which explains the differences between Theorem 1.12 and Theorem 1.13 is that the one-dimensional cubic case is L 2 -subcritical. This aspect has several consequences regarding the Strichartz estimates we use in the course of the proof. These nonlinear results are to be compared with previous ones concerning the interaction between a linear dynamics (classical trajectories) and nonlinear effects. Consider the WKB regime ε2 ψ ε = V (x)ψ ε + λ|ψ ε |2σ ψ ε ; ψ ε (0, x) = εβ a(x)ei x·ξ0 /ε , (1.20) 2 with V satisfying Assumption 1.1. Like above, it is equivalent, up to a rescaling, to iε∂t ψ ε +

ε2 α ψ ε = V (x)ψ ε + λε |ψ ε |2σ ψ ε ; ψ ε (0, x) = a(x)ei x·ξ0 /ε , 2 . The critical value in this regime is with α = 2σ β αc = 1 (see [6]). In (1.20), this corresponds to initial data of order ε1/(2σ ) in L ∞ , like in the present case of wave packets. However, the critical nonlinear effects are very different in the case of (1.20). The following asymptotics holds in L 2 (Rd ) (see [6]): iε∂t ψ ε +

ψ ε (t, x) ∼ a(t, x)eig(t,x) eiφ(t,x)/ε , ε→0

as long as the phase φ, solution to the Hamilton–Jacobi equation 1 ∂t φ + |∇φ|2 + V = 0; φ(0, x) = x · ξ0 , 2 remains smooth. More general initial phases are actually allowed: we consider an initial phase linear in x for the comparison with (1.3). The amplitude a solves a linear transport equation: at leading order, nonlinear effects show up through the phase modulation g (which depends on λ and σ ). This result calls for at least two comments. First, this nonlinear effect is rather weak: for instance, it does not affect the main quadratic ε observables at leading order, |ψ ε |2 (position density) and ε Im ψ ∇ψ ε (current density). In the case of (1.3), the profile equation is, in a sense, more nonlinear, even though in both cases, Wigner measures are not affected by the critical nonlinearity. Second, the validity of WKB analysis is limited in general, even if V is a polynomial. If V = 0, φ(t, x) = x ·ξ0 −t|ξ0 |2 /2 is smooth for all time, a(t, x) = a0 (x −tξ0 ) remains bounded, and the asymptotics can be justified up to Ehrenfest time, by simply resuming the proof given in [6]. If V (x) = E · x, Avron–Herbst formula shows that this case is essentially the same as V = 0. On the other hand, if V (x) = ω2 |x|2 /2, classical trajectories in (1.5) are explicit: sin(ωt) . ω They all meet at ξ0 /ω at time t∗ = π/(2ω): the phase φ becomes singular as t → t∗ , and WKB analysis ceases to be valid, while the wave packets approach yields an exact result for all time in such a case. In [5,15,18,25,26,29,30], the authors have considered a similar problem, in a different regime though: ε2 x − x0 iξ0 ·x/ε iε∂t ψ ε + ψ ε = V (x)ψ ε − |ψ ε |2σ ψ ε ; ψ ε (0, x) = Q e , 2 ε (1.21) x(t) = x0 cos(ωt) + ξ0


451

where Q is a ground state solution to a nonlinear elliptic equation. They prove, with some precision depending on the papers: x − x(t) iξ(t)·x/ε+iθ ε (t) e ψ ε (t, x) ∼ Q , θ ε (t) ∈ R. ε→0 ε As pointed out in [25], such results may be extended to an Ehrenfest time. An important difference with our paper must be emphasized, besides the scaling: the particular initial data makes it possible to rely on rigidity properties of the solitary waves, which do not hold for general profiles. In [10], some results concerning a defocusing equation with more general initial profiles are proved (or cited), in the same scaling as in (1.21): however, it seems that unless V is a polynomial of degree at most two, only partial results are available then (that is, on relatively small time intervals). Finally, even when ∂ γ V = 0 for all |γ | 3, the time intervals on which some asymptotic results are proved must be independent of ε. 1.3. Nonlinear superposition. We still suppose α = αc . For simplicity, in this paragraph, we assume that σ is an integer: this is compatible with the fact that the nonlinearity is energy-subcritical only if d 3. We consider initial data corresponding to the superposition of two wave packets: x − x1 i(x−x1 )·ξ1 /ε x − x2 i(x−x2 )·ξ2 /ε ε −d/4 −d/4 ψ (0, x) = ε a1 +ε a2 , e e √ √ ε ε 2 with a1 , a2 ∈

S(R), (x1 , ξ1 ), (x2 , ξ2 ) ∈ R , and (x1 , ξ1 ) = (x2 , ξ2 ). For j ∈ {1, 2}, x j (t), ξ j (t) are the classical trajectories solutions to (1.5) with initial data (x j , ξ j ). We denote by S j the action associated with (x j (t), ξ j (t)) by (1.7) and by u j the solution of (1.17) for the curve x j (t) and with initial data a j . We consider ϕ εj associated by (1.18) with u j , x j , ξ j , S j , and ψ ε ∈ C(R; ) solution to (1.3) with α = αc and the above initial data. The functional setting used to describe the function ψ ε must be changed in the case of two initial wave packets: recall that H is defined through Aε and B ε , which are related to the Hamiltonian flow. The geometric meaning of Aε and B ε becomes irrelevant in the case of two wave packets. Instead, we use norms on whose geometric meaning is weaker, since essentially, they reflect the fact that we consider ε-oscillatory functions, which remain somehow localized in space (before Ehrenfest time):

f ε = f L 2 (Rd ) + ε∇ f L 2 (Rd ) + x f L 2 (Rd ) . For finite time, we have: Proposition 1.14. Let d 3, σ ∈ N (σ = 1 if d = 3), and a1 , a2 ∈ S(Rd ). For all T > 0 (independent of ε), we have, for all γ < 1/2:

sup ψ ε (t) − ϕ1ε (t) − ϕ2ε (t) ε = O εγ . 0t T

Besides, nonlinear superposition holds for large time (at least) in the one-dimensional case, if the points (x1 , ξ1 ) and (x2 , ξ2 ) have different energies.

452


Theorem 1.15. Assume that d = 1, σ is an integer, and let a1 , a2 ∈ S(R). Suppose that E 1 = E 2 , where ξ 2j

+ V xj . 2 Suppose that (E x p)k is satisfied for some k 4 (for u 1 and u 2 ). 1. There exist C, C3 > 0 independent of ε, and ε0 > 0 such that for all ε ∈ ]0, ε0 ], Ej =

1

ψ ε (t) − ϕ1ε (t) − ϕ2ε (t) ε εγ exp (exp(C3 t)), 0 t C log log , ε with γ =

k−2 2k−2 .

In particular, there exists c > 0 independent of ε such that sup 0t c log log

1 ε

ψ ε (t) − ϕ1ε (t) − ϕ2ε (t) ε −→ 0. ε→0

2. Suppose in addition that σ = 1. There exist C, C4 > 0 independent of ε, and ε0 > 0 such that for all ε ∈ ]0, ε0 ], k−2 1 .

ψ ε (t) − ϕ1ε (t) − ϕ2ε (t) ε εγ eC4 t , 0 t C log , with γ = ε 2k − 2 In particular, there exists c > 0 independent of ε such that sup 0t c log

1 ε

ψ ε (t) − ϕ1ε (t) − ϕ2ε (t) ε −→ 0. ε→0

It is interesting to see that even though the profiles are nonlinear, the superposition principle, which is a property of linear equations, still holds. There are many other such nonlinear superposition principles in the literature, and we cannot mention them all. We emphasize however that this superposition principle is another difference with the weakly nonlinear geometric optics regime mentioned in the previous paragraph: iε∂t ψ ε +

ε2 ψ ε = ε|ψ ε |2σ ψ ε ; ψ ε (0, x) = a j (x)ei x·ξ j /ε . 2 N

j=1

It is proved in [8] that if N 2, then nonlinear interaction effects are present in the leading order description of ψ ε : there cannot be a nonlinear superposition principle. This result is to be compared with those in [32] (see also references therein), for several reasons. In [32], the authors construct a solution for the three-dimensional Schrödinger–Poisson system which behaves, in H 1 (R3 ) and asymptotically for large time, like the sum of two ground state solitary waves. The two solitary waves are centered, in the phase space, at the solution of a two-body problem: unlike what happens in our case, there exists an interaction between the trajectories, due to the fact that the Poisson potential is long range. In our case, the long range aspect of the nonlinearity (when d = σ = 1; see [35]) does not have such a consequence: we will see that the key point in the proof of the above two results is the fact that in the wave packet scaling, the two functions ϕ1ε and ϕ2ε do not interact at leading order in the limit ε → 0: the nonlinear effects concentrate on the profiles, along the classical trajectories, and it turns out that these trajectories do not meet “too much”. In [9], another nonlinear superposition principle was proved, in the scaling of (1.21). However, in [9], nonlinear effects were localized in space and time, so most of the time, the nonlinear superposition was actually a linear one.


453

1.4. Outline of the paper. In Sect. 2, we first analyze the linearizable case and prove Proposition 1.5 after a short analysis of the linear case. Then, in Sect. 3, we recall basic facts about Strichartz estimates in this semi-classical framework and prove the consistency of our approximation on bounded time intervals. Theorem 1.12 is proved in Sect. 4. Finally, Sect. 5 is focused on the one-dimensional cubic case and Sect. 6 on the analysis of the nonlinear superposition. Notation. Throughout the paper, in the expression eCt , the constant C will denote a constant independent of t which may change from one line to the other. 2. The Linearizable Case In this section, we assume α > αc and we prove Proposition 1.5. We first recall estimates in the linear case λ = 0 which are more precise than in §1.1. 2.1. The linear case. We suppose here λ = 0. The first remark concerns the properties of the profile v. It is not difficult to prove the following proposition. Proposition 2.1. Let d 1 and a ∈ S(Rd ). For all k ∈ N, there exists C > 0 such that the solution v to (1.11) satisfies ∀α, β ∈ Nd , |α| + |β| k, x α ∂xβ v(t) L 2 (Rd ) eCt . A general proof of Proposition 2.1 is given for instance in [7, §6.1]. Let us now consider ε . We have w ε (0) = 0 and wε = ψ ε − ϕlin iε∂t w ε +

ε2 ε w ε = V (x)w ε − (V (x) − T2 (x, x(t))) ϕlin , 2

where T2 corresponds to a second order Taylor approximation: T2 (x, a) := V (a) + ∇V (a), x − a +

1 HessV (a)(x − a), x − a . 2

We have seen in §1.1 that the standard L 2 estimate for Schrödinger equations yields √ √

w ε (t) L 2 (Rd ) ε y 3 v(t) L 2 (Rd ) εeCt . In order to analyze the convergence in , we can write

ε2 iε∂t + − V (x) ε∇w ε = ε∇V w ε − ε∇ L ε , 2 2

ε ε2 , x w ε − x L ε = ε2 ∇w ε − x L ε , iε∂t + − V (x) xw ε = 2 2 where ε L ε (t, x) := (V (x) − T2 (x, x(t))) ϕlin (t, x).

(2.1)

454


Typically if d = 1, 1 1 3 ε L (t, x) = (x − x(t)) ϕlin (t, x) V (x(t) + θ (x − x(t))) θ 2 dθ 2 0 x − x(t) (x − x(t))3 −i(S(t)+ξ(t)·(x−x(t)))/ε = I (x, x(t)), e v t, √ 2ε1/4 ε ε

where

1

I (x, x(t)) =

V (x(t) + θ (x − x(t))) θ 2 dθ.

0

Energy estimates make it possible to show

ε∇wε (t) L 2 (Rd ) + xw ε (t) L 2 (Rd )

√

εeCt .

However, the operators Aε and B ε defined in the Introduction yield more precise results. For instance, ε∇ϕlin L 2 is of order O(1) exactly, because of the phase factor in (1.12). We note the formula

√ ξ(t) √ Aε (t) = ε∇ − i √ = εei/ε ∇ e−i/ε · ; = S(t) + ξ(t) · (x − x(t)), (2.2) ε √ so Aε (t)ϕlin L 2 is of order O(1): morally, we have gained a factor ε. Lemma 2.2. The operators Aε and B ε , defined by Aε (t) =

√ ξ(t) x − x(t) ε∇ − i √ ; B ε (t) = √ , ε ε

satisfy the commutation relations: √ ε2 ε iε∂t + − V, A (t) = ε (∇V (x) − ∇V (x(t))), 2 ε2 iε∂t + − V, B ε (t) = ε Aε (t). 2 We can then write √ ε2 iε∂t + − V (x) Aε (t)w ε = ε (∇V (x) − ∇V (x(t))) w ε − Aε (t)L ε , 2 ε2 iε∂t + − V (x) B ε (t)w ε = ε Aε (t)w ε − B ε (t)L ε . 2 In view of (2.2), we observe

Aε (t)L ε L 2 (Rd ) ε3/2 x 2 v(t) L 2 (Rd ) + ε3/2 x 3 ∇v(t) L 2 (Rd ) + ε2 x 3 v(t) L 2 (Rd ) ε3/2 eCt , thanks to Lemma 2.1. Similarly,

B ε (t)L ε L 2 (Rd ) ε3/2 eCt .


455

Since we have the pointwise estimate √ ε (∇V (x) − ∇V (x(t))) w ε ε B ε (t)w ε , energy estimates yield t

√

w (t) H

w ε (s) H + εeCs ds. ε

0

We conclude by the Gronwall Lemma:

wε (t) H

√

εeCt .

We will see in the following subsection that the arguments are somehow more complicated in the nonlinear setting. 2.2. Proof of Proposition 1.5. We now assume λ = 0, and α > αc . For the simplicity of the presentation, we give the detailed proof in the case d = 1 only. ε and we write the equation satisfied by w ε : We set again wε = ψ ε − ϕlin iε∂t w ε +

ε2 2 ε ε ε ∂ w = V (x)w ε − (V (x) − T2 (x, x(t))) ϕlin + N ε ; w|t=0 = 0, 2 x

where the nonlinear source term is given by ε ε + w ε |2σ (ϕlin + w ε ). N ε = λεα |ϕlin ε + w ε |2σ ∈ R, the L 2 energy estimate for w ε yields First, since λεα |ϕlin 1 1 ε ε

w ε (t) L 2 (R) L ε L 1 ([0,t];L 2 (R)) + εα |ϕlin + w ε |2σ ϕlin , 1 L ([0,t];L 2 (R)) ε ε

where we have kept the notation (2.1). The contribution of N ε cannot be studied directly, since we do not know yet how to estimate wε : since w ε will turn out to be small, we use a bootstrap argument. ε (t) ∞ −1/4 v(t) ∞ Since we have ϕlin L (R) = ε L (R) , Proposition 2.1 and Sobolev embedding show that there exists C0 > 0 such that ε

ϕlin (t) L ∞ (R) C0 ε−1/4 eC0 t , ∀t 0.

The bootstrap argument goes as follows. We suppose that for t ∈ [0, τ ] we have

w ε (t) L ∞ ε−1/4 eC0 t ,

(2.3)

ε = 0 and ψ ε ∈ C(R; ), there exists τ ε > 0 (a with the same constant C0 . Since w|t=0 priori depending on ε) such that (2.3) holds on [0, τ ε ]. So long as (2.3) holds, α ε ε εα−σ/2 a L 2 (R) e2σ C0 t . 2 ε |ϕlin + w ε |2σ ϕlin L (R)

We infer

w ε (t) L 2 (R)

√

εeCt + εα−αc e2σ C0 t .

456


Applying the operators Aε and B ε to the equation satisfied by w ε , we find:

√ ε2 2 iε∂t + ∂x − V (x) Aε w ε = ε V (x) − V (x(t)) w ε − Aε L ε + Aε N ε, 2 ε2 2 iε∂t + ∂x − V (x) B ε w ε = ε Aε w ε − B ε L ε + B ε N ε . 2 We observe that in view of (2.2), Aε acts on gauge invariant nonlinearities like a derivative. Therefore, so long as (2.3) holds,

ε ε ε 2σ ε ε A (t)N ε (t) 2 εα ϕlin (t) 2σ L ∞ (R) + w (t) L ∞ (R) A (t)ϕlin (t) L 2 (R) L (R)

ε ε 2σ ε ε + εα ϕlin (t) 2σ +

w (t) ∞ ∞ L (R) L (R) A (t)w (t) L 2 (R)

εα−σ/2 e2σ C0 t eCt + Aε (t)w ε (t) L 2 (R) . Similarly, we obtain ε

B (t)N ε (t) 2 εα−σ/2 e2σ C0 t eCt + B ε (t)w ε (t) L 2 (R) . L (R) We infer, thanks to the linear estimates,

√

Aε (t)w ε (t) L 2 (R) B ε w ε L 1 ([0,t];L 2 (R)) + εeCt t

+ εα−αc e2σ C0 s eCs + Aε (s)w ε (s) L 2 (R) ds, 0 √

B ε (t)w ε (t) L 2 (R) Aε w ε L 1 ([0,t];L 2 (R)) + εeCt t

+ εα−αc e2σ C0 s eCs + B ε (s)w ε (s) L 2 (R) ds. 0

The Gronwall Lemma yields, so long as (2.3) holds: t t

wε (t) H εγ eCs exp Cεα−αc e2σ C0 s ds ds 0 s

t

α−αc 2σ C0 t exp Cε e εγ eCs ds exp Cεα−αc e2σ C0 t εγ eCt , 0

where γ = min(1/2, α − αc ). First, we notice that εα−αc e2σ C0 t 1 for 0 t

α − αc 1 log . 2σ C0 ε

c Then, setting κ = α−α 2σ C0 , the Gagliardo-Nirenberg inequality yields, so long as (2.3) holds, with also t κ log 1ε , and thanks to the factorization (2.2),

w ε (t) L ∞ (R)

1 ε1/4

w ε (t) L 2 (R) Aε (t)w ε (t) L 2 (R) εγ −1/4 eCt . 1/2

1/2

This is enough to show that the bootstrap argument (2.3) works provided the time variable is restricted to Cεγ eCt 1,


457

that is, 0 t C log 1ε for some C > 0 independent of ε. Proposition 1.5 follows in the case d = 1. To prove Proposition 1.5 when d 2, one can use Strichartz estimates. This approach is more technical. Since the case α > αc does not seem the most interesting one, and since we will use Strichartz estimates in the fully nonlinear case, we choose not to present the proof of Proposition 1.5 when d 2. 3. Fully Nonlinear Case: Bounded Time Intervals In this section, we prove Proposition 1.7. This gives us the opportunity to introduce some technical tools which will be used to study large time asymptotics.

3.1. Strichartz estimates. Definition 3.1. A pair (q, r ) is admissible if 2 r 2 r < ∞ if d = 2) and 1 2 = δ(r ) := d − q 2

2d d−2

(resp. 2 r ∞ if d = 1,

1 . r

Following [21,28,42], Strichartz estimates are available for the Schrödinger equation without external potential. Thanks to the construction of the parametrix performed in [16,17], similar results are available in the presence of an external potential satisfying Assumption 1.1 (V could even depend on time). Denote by U ε (t) the semi-group 2 associated to − ε2 + V : φ ε (t, x) = U ε (t)φ0 (x) if iε∂t φ ε +

ε2 φ ε = V φ ε ; φ ε (0, x) = φ0 (x). 2

From [16], it satisfies the following properties: • • • • •

The map t → U ε (t) is strongly continuous. U ε (t)U ε (s) = U ε (t + s). U ε (t)∗ = U ε (t)−1 = U ε (−t). U ε (t) is unitary on L 2 : U ε (t)φ L 2 (Rd ) = φ L 2 (Rd ) . Dispersive properties: there exist δ, C > 0 independent of ε ∈ ]0, 1] such that for all t ∈ R with |t| δ,

U ε (t) L 1 (Rd )→L ∞ (Rd )

C . (ε|t|)d/2

We infer the following result, from [28]: Lemma 3.2 (Scaled Strichartz inequalities). Let (q, r ), (q1 , r1 ) and (q2 , r2 ) be admissible pairs. Let I be some finite time interval. 1. There exists C = C(r, |I |) independent of ε, such that for all φ ∈ L 2 (Rd ), ε1/q U ε (·)φ L q (I ;L r (Rd )) C φ L 2 (Rd ) .

(3.1)

458


2. If I contains the origin, 0 ∈ I , denote D εI (F)(t, x) =

I ∩{s t}

U ε (t − s)F(s, x)ds.

There exists C = C(r1 , r2 , |I |) independent of ε such that for all F ∈ L q2 (I ; L r2 ), ε1/q1 +1/q2 D εI (F) L q1 (I ;L r1 (Rd )) C F q2 r2 d . (3.2) L

I ;L (R )

3.2. Proof of Proposition 1.7. Denote the error term by wε = ψ ε − ϕ ε , where ϕ ε is now given by (1.18), and u ∈ C(R; ) satisfies (1.17). This remainder solves iε∂t w ε +

ε2 ε w ε = V w ε − Lε +λεαc |ψ ε |2σ ψ ε − |ϕ ε |2σ ϕ ε ; w|t=0 = 0, 2

(3.3)

where Lε (t, x) = (V (x) − T2 (x, x(t))) ϕ ε (t, x)

(3.4)

is the nonlinear analogue of L ε given by (2.1). Duhamel’s formula for w ε reads t+τ U ε (t + τ − s)Lε (s)ds w ε (t + τ ) = U ε (τ )w ε (t) + iε−1 t t+τ

U ε (t + τ − s) |ψ ε |2σ ψ ε − |ϕ ε |2σ ϕ ε (s)ds. − iλεαc −1 t

Introduce the following Lebesgue exponents: θ=

4σ + 4 2σ (2σ + 2) ; q= ; r = 2σ + 2. 2 − (d − 2)σ dσ

Then (q, r ) is admissible, and 1 2σ 1 + ; = q θ q

1 2σ 1 + . = r r r

Let t 0, τ > 0 and I = [t, t + τ ]. Lemma 3.2 yields

w ε L q (I ;L r ) ε−1/q w ε (t) L 2 + ε−1−1/q Lε L 1 (I ;L 2 ) + εαc −1−2/q |ψ ε |2σ ψ ε − |ϕ ε |2σ ϕ ε q

L (I ;L r )

.

In view of the pointwise estimate

ε 2σ ε |ψ | ψ − |ϕ ε |2σ ϕ ε |w ε |2σ + |ϕ ε |2σ |w ε |, we infer

w ε L q (I ;L r ) ε−1/q w ε (t) L 2 + ε−1−1/q Lε L 1 (I ;L 2 )

2σ 2σ + εαc −1−2/q w ε L θ (I ;L r ) + ϕ ε L θ (I ;L r ) w ε L q (I ;L r ) .

(3.5)


459

Thanks to [7], we know that the rescaled functions for ψ ε and ϕ ε , are such that u ε , u ∈ C(R; ), with estimates which are uniform in ε ∈ ]0, 1]. Typically, for all T > 0, there exists C(T ) independent of ε such that

Pu ε L ∞ ([0,T ];L 2 ) + Pu L ∞ ([0,T ];L 2 ) C(T ),

P ∈ {Id, ∇, x}.

In terms of ψ ε and ϕ ε , this yields

P ε ψ ε L ∞ ([0,T ];L 2 ) + P ε ϕ ε L ∞ ([0,T ];L 2 ) C(T ), P ε ∈ {Id, Aε , B ε }.

(3.6)

The formula (2.2) and Gagliardo–Nirenberg inequality yield, if 0 δ( p) < 1,

f L p (Rd )

C( p) 1−δ( p) Aε (t) f δ(2p) d , ∀ f ∈ H 1 (Rd ), ∀t ∈ R. (3.7)

f 2 d L (R ) L (R ) εδ( p)/2

We infer that there exists C(T ) independent of ε such that

ψ ε (t) L r (Rd ) + ϕ ε (t) L r (Rd ) C(T )ε−δ(r )/2 , ∀t ∈ [0, T ].

(3.8)

Recalling that I = [t, t + τ ], we deduce from (3.5):

wε L q (I ;L r ) ε−1/q w ε (t) L 2 + ε−1−1/q Lε L 1 (I ;L 2 ) + εαc −1−2/q τ 2σ/θ ε−σ δ(r ) w ε q r . L (I ;L )

Since (q, r ) is admissible, we compute αc − 1 −

2 dσ 2σ + 2 − σ δ(r ) = − = 0, q 2 q

(3.9)

hence

wε L q (I ;L r ) ε−1/q w ε (t) L 2 + ε−1−1/q Lε ϕ ε L 1 (I ;L 2 ) + τ 2σ/θ w ε q r .

(3.10)

L (I ;L )

Choosing τ sufficiently small, and repeating this manipulation a finite number of times to cover the time interval [0, T ], we obtain

w ε L q ([0,T ];L r ) ε−1/q w ε L 1 ([0,T ];L 2 ) + ε−1−1/q Lε L 1 ([0,T ];L 2 ) . Using Strichartz estimates again and (3.8), we have, with J = [0, t], 0 t T ,

wε L ∞ (J ;L 2 ) ε−1 Lε L 1 (J ;L 2 ) + εαc −1−1/q |ψ ε |2σ ψ ε − |ϕ ε |2σ ϕ ε q

(3.11)

L (J ;L r )

w ε L 1 (J ;L 2 ) + ε−1 Lε L 1 (J ;L 2 ) + εαc −1−1/q ε−1−1/q−σ δ(r ) Lε L 1 (J ;L 2 ) w ε L 1 (J ;L 2 ) + ε−1 Lε L 1 (J ;L 2 ) , where the last estimate stems from (3.9). We have the pointwise control

|Lε | |x − x(t)|3 |ϕ ε (t, x)| = ε3/2 ε−d/4 |y|3 |u(t, y)| x−x(t) . y=

√

ε

460


We infer ε−1 Lε L 1 ([0,T ];L 2 (Rd ))

√

ε |y|3 u(t, y) L 1 ([0,T ];L 2 (Rd )) ,

and the first part of Proposition 1.7 follows from the Gronwall Lemma. To establish a control of the H-norm, we notice that in view of (2.2), we have, for P ε ∈ {Aε , B ε },

P ε |φ ε |2σ φ ε ≈ |φ ε |2σ P ε φ ε , where the symbol “≈” is here to recall the abuse of notation when P ε = Aε (there should be two terms on the right-hand side, with coefficients). Lemma 2.2 shows that we have √ ε2 iε∂t + − V Aε w ε = ε (∇V (x) − ∇V (x(t))) w ε − Aε Lε 2

+ λεαc Aε |ψ ε |2σ ψ ε − |ϕ ε |2σ ϕ ε . The first term of the right-hand side is controlled pointwise by Cε|B ε w ε |. The L 2 -norm of the second term is estimated by

Aε (t)Lε (t) L 2 (Rd ) ε3/2 |y|2 u(t, y) L 2 (Rd ) + |y|3 ∇u(t, y) L 2 (Rd ) . Finally, we have

Aε |ψ ε |2σ ψ ε − |ϕ ε |2σ ϕ ε ≈ |ψ ε |2σ Aε ψ ε − |ϕ ε |2σ Aε ϕ ε ≈ |w ε + ϕ ε |2σ (Aε w ε + Aε ϕ ε ) − |ϕ ε |2σ Aε ϕ ε

≈ |w ε + ϕ ε |2σ Aε w ε + |w ε + ϕ ε |2σ − |ϕ ε |2σ Aε ϕ ε . (3.12) The first term of (3.12) is handled like in the first step. For the second term, we have, since σ > 1/2,

ε |w + ϕ ε |2σ − |ϕ ε |2σ |w ε |2σ −1 + |ϕ ε |2σ −1 |w ε |. Following the same lines as for the L 2 estimate, we find

Aε w ε L q (I ;L r ) ε−1/q Aε (t)w ε (t) L 2 + ε−1/q B ε w ε L 1 (I ;L 2 ) + ε−1−1/q Aε Lε L 1 (I ;L 2 ) + τ 2σ/θ Aε w ε L q (I ;L r ) + τ 2σ/θ w ε L q (I ;L r ) , by using the estimate ) )

Aε (t)2 ϕ ε (t) δ(r ,

Aε (t)ϕ ε (t) L r (Rd ) ε−δ(r )/2 Aε (t)ϕ ε (t) 1−δ(r L 2 (Rd ) L 2 (Rd )

and the remark

Aε (t)2 ϕ ε (t) L 2 (Rd ) = ∇ 2 u(t) L 2 (Rd ) .


461

Since σ > 1/2, the nonlinearity z → |z|2σ z is twice differentiable, and one can prove u ∈ C(R; H 2 (Rd )) ([7]). Using (3.11) and the same argument as in the first step, we infer

Aε w ε L q (I ;L r ) ε−1/q Aε (t)w ε (t) L 2 + ε−1/q B ε w ε L 1 (I ;L 2 ) + ε−1−1/q Aε Lε L 1 (I ;L 2 ) + ε−1−1/q Lε 1

L (I ;L 2 )

,

hence, using Strichartz estimates again,

Aε w ε L ∞ (I ;L 2 ) Aε (t)w ε (t) L 2 + B ε w ε L 1 (I ;L 2 ) + ε−1 Aε Lε L 1 (I ;L 2 ) 1/q ε w + ε1/q Aε w ε q r +ε q r ε

ε

L (I ;L ) ε

ε

L (I ;L ) −1 ε

A (t)w (t) L 2 + B w L 1 (I ;L 2 ) + ε A Lε L 1 (I ;L 2 ) + ε−1 Lε L 1 (I ;L 2 ) + ε1/q w ε L q (I ;L r ) . Since we have similar estimates for B ε w ε , we end up with

Aε w ε L ∞ (J ;L 2 ) + B ε w ε L ∞ (J ;L 2 ) Aε w ε L 1 (J ;L 2 ) + B ε w ε L 1 (J ;L 2 ) + ε−1

P ε Lε L 1 (J ;L 2 )

P ε ∈{Id,Aε ,B ε }

Aε w ε L 1 (J ;L 2 ) + B ε w ε L 1 (J ;L 2 )

+

√

ε.

Proposition 1.7 then follows from the Gronwall Lemma. 4. Fully Nonlinear Case: Proof of Theorem 1.12 To prove Theorem 1.12, the strategy consists in examining more carefully the dependence of the L θ L r -norms with respect to time in the previous proof. Also, since (E x p)4 concerns only u, not u ε , we need a bootstrap argument in order to use the same control for the error term w ε as for the approximate solution ϕ ε . This control carries on the L r (Rd )-norms, for fixed t. By (E x p)1 , the relation

Aε (t)ϕ ε (t) L 2 (Rd ) = ∇u(t) L 2 (Rd ) , and the modified Gagliardo–Nirenberg inequality (3.7), we have the following estimate, for all time:

ϕ ε (t) L r (Rd ) ε−δ(r )/2 eκt .

(4.1)

We will use the following bootstrap argument:

w ε (t) L r (Rd ) ε−δ(r )/2 eκt , t ∈ [0, T ],

(4.2)

with the same constant κ as in (4.1) to fix the ideas. By Proposition 1.7, for any T > 0 independent of ε, (4.2) is satisfied provided 0 < ε ε(T ). By this argument only, it may very well happen that ε(T ) → 0 as T → +∞. The goal of the bootstrap argument is to show that we can take T ε = C log log 1ε for some C > 0 independent of ε, provided that ε is sufficiently small.

462


The key step to analyze is the absorption argument, which made it possible to infer (3.11) from (3.10). We resume the computations of §3.2 from the estimate (3.5). Rewrite this estimate with I = [t, t + τ ], t, τ 0:

w ε L q (I ;L r ) ε−1/q w ε (t) L 2 + ε−1−1/q Lε L 1 (I ;L 2 )

2σ 2σ + εαc −1−2/q w ε L θ (I ;L r ) + ϕ ε L θ (I ;L r ) w ε L q (I ;L r ) . For simplicity, assume τ 1: (4.1) and (4.2) yield, in view of (3.9),

wε L q (I ;L r ) M ε−1/q w ε (t) L 2 + ε−1−1/q Lε L 1 (I ;L 2 )

+ τ 1/θ e2σ κt w ε L q (I ;L r ) , for some constant M independent of ε, t 0 and 0 τ 1. In order for the last term to be absorbed by the left-hand side, we have to assume Mτ 1/θ e2σ κt

1 , that is, τ Ce−Ct 2

for some C independent of ε, t 0 and 0 τ 1. Proceeding with the same argument as in §3.2, we come up with: t t ε Cs ε −1

w L ∞ ([0,t];L 2 ) e w L ∞ ([0,s];L 2 ) ds + ε eCs Lε (s) L 2 ds 0 0 t √ eCs w ε L ∞ ([0,s];L 2 ) ds + εeCt , 0

where we have used (E x p)3 . The Gronwall Lemma yields: √ √

w ε L ∞ ([0,t];L 2 ) ε exp (C exp(Ct)) ε exp (exp(2Ct)). Mimicking the computations of §3.2, we have, thanks to (E x p)4 and so long as (4.2) holds, √

Aε w ε L ∞ ([0,t];L 2 ) + B ε w ε L ∞ ([0,t];L 2 ) ε exp (exp(Ct)). To conclude, we check that (4.2) holds for t c log log 1ε , provided c is sufficiently small. The Gagliardo–Nirenberg inequality (3.7) yields ) )

Aε w ε δ(r

w ε (t) L r (Rd ) ε−δ(r )/2 w ε 1−δ(r L ∞ ([0,t];L 2 ) L ∞ ([0,t];L 2 ) √ Mε−δ(r )/2 ε exp (exp(Ct)).

Therefore, taking ε sufficiently small, (4.2) holds as long as √ M ε exp (exp(Ct)) eκt . We check that for large t and sufficiently small ε, this remains true for t c log log 1ε , with c possibly small, but independent of ε ∈ ]0, ε0 ]. This completes the proof of Theorem 1.12.


463

5. Ehrenfest Time in the One-Dimensional Cubic Case As pointed out in the Introduction, since we consider nonlinearities of the form z → |z|2σ z with σ ∈ N, the one-dimensional cubic case is special. Not because it is integrable (see Remark 1.6: (1.17) is not completely integrable, unless no approximation is needed to describe the wave packets, ψ ε ≡ ϕ ε ), but because it is the only case where the nonlinearity is L 2 -subcritical, σ < 2/d. This case is in contrast with the general case of energy-subcritical nonlinearities: without any other assumption on Q(t) than Q ∈ C ∞ (R; R) ∩ L ∞ (R), it seems that the only a priori control that we have for u, solution to (1.17), is

u(t) L 2 (Rd ) = a L 2 (Rd ) , ∀t ∈ R.

(5.1)

A remarkable case where other a priori estimates are available is when Q is constant, but in this case, ψ ε ≡ ϕ ε . Otherwise, the most general reasonable assumption seems to be (E x p)k , which has been considered in the previous section. Note also that if d = 1, the notations of §3.2 become: θ=

8 ; q = 8; r = 4. 3

So to improve the result of Theorem 1.12, we assume σ = d = 1 and start with the crucial remark: Lemma 5.1. Suppose σ = d = 1, and for a ∈ L 2 (R), consider u ∈ C(R; L 2 (R)) the solution to (1.17). Then there exists C such that

u L 8 ([t,t+1];L 4 (R)) C a L 2 (R) , ∀t ∈ R. Proof. First, recall that since σ = d = 1 and a ∈ L 2 (R), (1.17) has a unique solution,

u ∈ C(R; L 2 (R)) ∩ L 8loc R; L 4 (R) . In addition, (5.1) holds. Denoting W (t, x) =

1 V (x(t)) x 2 , 2

it has been established in [7] that since V ∈ L ∞ (R; R), uniform local Strichartz estimates are available for the linear propagator. Following [16,17], let U (t, s) be such that as u(t, x) = U (t, s)u 0 (x) is the solution to 1 i∂t u + u = W (t, x)u; u(s, x) = u 0 (x). 2 Then Lemma 3.2 remains true (with ε = 1) when U ε (t − s) is replaced with U (t, s), t, s ∈ R. Let t, τ 0, with τ 1, and denote I = [t, t + τ ]. Strichartz inequalities yield:

u L 8 (I ;L 4 ) C(τ ) u(t) L 2 + C(τ ) |u|2 u 8/7 . 4/3 L

(I ;L

)

In view of (5.1), and using the Hölder inequality after the decomposition 3 1 3 = + ; 4 4 ∞

7 3 1 = + , 8 8 2

464


we infer √

u L 8 (I ;L 4 ) C(τ ) a L 2 + C(τ ) τ u 3L 8 (I ;L 4 ) . Since τ 1, we may assume that C(τ ) does not depend on τ : √

u L 8 (I ;L 4 ) C a L 2 + C τ u 3L 8 (I ;L 4 ) . We use the following standard bootstrap argument, borrowed from [2]: Lemma 5.2 (Bootstrap argument). Let f = f (t) be a nonnegative continuous function on [0, T ] such that, for every t ∈ [0, T ], f (t) M + δ f (t)θ , where M, δ > 0 and θ > 1 are constants such that 1 1 1 ; f (0) . M < 1− θ (θ δ)1/(θ−1) (θ δ)1/(θ−1) Then, for every t ∈ [0, T ], we have f (t)

θ M. θ −1

Lemma 5.1 follows with [t, t + 1] replaced with [t, t + τ ] for 0 < τ τ0 1. We then cover any interval of the form [t, t + 1] by a finite number of intervals of length at most τ0 , and Lemma 5.1 is proved. Proof (Proof of Theorem 1.13). Like in the previous section, we resume the proof of Proposition 1.7, and pay more precise attention to the dependence of various constants upon time. We modify the bootstrap argument of §4: in view of Lemma 5.1, (4.2) is replaced by

w ε L 8 ([t,t+1];L 4 (R)) ε−1/8 a L 2 (R) , ∀t ∈ [0, T ].

(5.2)

By Proposition 1.7, for any T > 0 independent of ε, (5.2) remains true provided 0 < ε ε(T ). Keeping the notations of §3.2, we have: θ=

8 ; q = 8; r = 4. 3

With t 0, τ ∈ ]0, 1] and I = [t, t + τ ], (3.5) becomes

wε L 8 (I ;L 4 ) ε−1/8 w ε (t) L 2 + ε−1−1/8 Lε L 1 (I ;L 2 )

2 2 + ε1/4 w ε L 8/3 (I ;L 4 ) + ϕ ε L 8/3 (I ;L 4 ) w ε L 8 (I ;L 4 ) ε−1/8 w ε (t) L 2 + ε−1−1/8 Lε L 1 (I ;L 2 )

2 2 + ε1/4 τ 1/4 w ε L 8 (I ;L 4 ) + ϕ ε L 8 (I ;L 4 ) w ε L 8 (I ;L 4 ) ε−1/8 w ε (t) L 2 + ε−1−1/8 Lε L 1 (I ;L 2 ) + τ 1/4 w ε 8

L (I ;L 4 )

, (5.3)


465

where we have used Lemma 5.1 and (5.2). Choosing τ sufficiently small and independent of t, we come up with

wε L ∞ ([0,t];L 2 ) w ε L 1 ([0,t];L 2 ) + ε−1 Lε L 1 ([0,t];L 2 ) t √ w ε L 1 ([0,t];L 2 ) + ε eCs ds, 0

by (E x p)3 . The Gronwall Lemma yields

w ε L ∞ ([0,t];L 2 )

√

εeCt .

Back to (5.3), we infer, with τ 1,

w ε L 8 (I ;L 4 ) ε1/4 eCt . Therefore, there exists c > 0 such that (5.2) holds for T = c log 1ε provided ε is sufficiently small, hence the first part of Theorem 1.13. It is then quite straightforward to infer the estimates in H, by rewriting the end of the proof of Proposition 1.7, with (5.2) in mind. 6. Nonlinear Superposition 6.1. General considerations. The proof of Proposition 1.14 and Theorem 1.15 follows the same lines as the proof of Proposition 1.7 and Theorem 1.13. The main difference comes from the way one deals with the nonlinearity, since new terms appear. These terms come from the nonlinear interaction between the two profiles ϕ1ε and ϕ2ε . Denote wε = ψ ε − ϕ1ε − ϕ2ε . It solves iε∂t w ε +

ε2 ε = 0, w ε = V w ε − Lε + λN ε ; w|t=0 2

where we have now Lε (t) = (V (x) − T2 (x, x1 (t))) ϕ1ε (t, x) + (V (x) − T2 (x, x2 (t))) ϕ2ε (t, x), and

N ε = εαc |w ε + ϕ1ε + ϕ2ε |2σ (w ε + ϕ1ε + ϕ2ε ) − |ϕ1ε |2σ ϕ1ε − |ϕ2ε |2σ ϕ2ε .

Decompose N ε as the sum of a semilinear term and an interaction source term: N ε = N Sε + N Iε , where

N Sε = εαc |w ε + ϕ1ε + ϕ2ε |2σ (w ε + ϕ1ε + ϕ2ε ) − |ϕ1ε + ϕ2ε |2σ (ϕ1ε + ϕ2ε ) ,

N Iε = εαc |ϕ1ε + ϕ2ε |2σ (ϕ1ε + ϕ2ε ) − |ϕ1ε |2σ ϕ1ε − |ϕ2ε |2σ ϕ2ε . We see that the term N Sε is the exact analogue of the nonlinear term in (3.3), where we have simply replaced ϕ ε with ϕ1ε + ϕ2ε . We can thus repeat the proofs of Proposition 1.7 and Theorem 1.13, respectively, up to the control of the new source term N Iε (the linear source term Lε is treated as before). More precisely, we have to estimate 1

N Iε L 1 ([0,t];L 2 (Rd )) . ε

466


The first remark consists in noticing that if σ is an integer, N Iε can be estimated (pointwise) by a sum of terms of the form εαc |ϕ1ε |1 × |ϕ2ε |2 , 1 , 2 1, 1 + 2 = 2σ + 1. To be more precise, we have the control, for fixed time, 1

N Iε (t) L 2 (Rd ) εdσ/2 ε

ε 1 |ϕ1 | × |ϕ2ε |2

1 ,2 1, 1 +2 =2σ +1

L 2 (Rd )

.

We will see below why the right-hand side must be expected to be small, when integrated with respect to time. We need to estimate 1 x1 (t) − x2 (t) 2 u εdσ/2 ϕ1ε 1 ϕ2ε 2 2 d = t, x − (t, x) u √ 1 2 2 d , L (R ) ε L (Rx ) with 1 , 2 1, 1 + 2 = 2σ + 1. We have the following lemma: Lemma 6.1. Suppose d 3, and σ is an integer. Let T ∈ R, 0 < γ < 1/2, and I ε (T ) = {t ∈ [0, T ], |x1 (t) − x2 (t)| εγ }.

(6.1)

Then, for all k > d/2, 1 ε

T 0

N Iε (t) ε dt (Mk+2 (T ))2σ +1 T εk(1/2−γ ) + |I ε (T )| eC T,

where Mk (T ) = sup x α ∂xβ u j L ∞ ([0,T ];L 2 (Rd ) ;

j ∈ {1, 2}, |α| + |β| k .

Proof. We observe that in view of Peetre inequality (see e.g. [41]), for η ∈ Rd ,

1 1 . sup x −1 x − η −1 η |η| x∈Rd With ηε (t) = 1 ε

x1 (t)−x √ 2 (t) , ε

[0,T ]\I ε (T )

we infer (forgetting the sum over 1 , 2 ),

N Iε (t) L 2 (R) dt

−k −k

x − ηε (t) x −k x − ηε (t) u 11 t, x − ηε (t) u 22 (t, x) 2 ε L [0,T ]\I (T ) dt k 2 x k u 11 ∞ . x u 2 ∞ L ([0,T ];L 4 ) L ([0,T ];L 4 ) [0,T ]\I ε (T ) |ηε (t)|k

We have, for j ∈ {1, 2}, j −1 k j k u j ∞ x u x u j ∞ j L ([0,T ]×Rd ) L ([0,T ];L 4 ) L ∞ ([0,T ];L 4 ) u j 1∞−1 x k u j ∞ Mk+1 (T ) j , L ([0,T ];H k ) 1 L ([0,T ];H )


467

where we have used H 1 (Rd ) ⊂ L 4 (Rd ) since d 3. On the other hand, dt εk/2 dt εk(1/2−γ ) T. ε k k [0,T ]\I ε (T ) |η (t)| [0,T ]\I ε (T ) |x 1 (t) − x 2 (t)| On I ε (T ), we simply estimate 1

N Iε (t) L 2 (R) dt u 1 L1∞ ([0,T ]×R) u 2 L2∞−1([0,T ]×R) u 2 L 1 (I ε (T );L 2 (R)) ε I ε (T ) Mk (T )2σ |I ε (T )| u 2 L ∞ ([0,T ];L 2 (R)) Mk (T )2σ +1 |I ε (T )|. The L 2 estimate follows, without an exponentially growing factor. This factor appears when dealing with the ε -norm. Typically, √

ε∇ϕ εj (t) L 2 (Rd ) ε ∇u j (t) L 2 (Rd ) + |ξ j (t)| u j (t) L 2 (Rd ) , √

xϕ εj (t) L 2 (Rd ) ε xu j (t) L 2 (Rd ) + |x j (t)| u j (t) L 2 (Rd ) . The result then follows from the above computations, and Lemma 1.2. At this stage, the main difficulty is to estimate the length of I ε (t). We do this in two cases: bounded t, and large time when d = 1. 6.2. Nonlinear superposition in finite time. In the proof of Proposition 1.7, we have only used the fact that u ε ∈ C(R; ), with estimates which are independent of ε. Recall that in the case of a single wave packet, ψ ε and u ε are related through (1.8): in the case of two wave packets, there is no such natural rescaling. So in the case of two initial wave packets, we are not able to prove uniform estimates for ψ ε , like in (3.6). Even to prove Proposition 1.14, which is the analogue of Proposition 1.7, we need to use a bootstrap argument. We know that for j ∈ {1, 2},

ϕ εj (t) L r (Rd ) C(T )ε−δ(r )/2 , ∀t ∈ [0, T ]. The bootstrap argument is of the form:

w ε (t) L r (Rd ) C(T )ε−δ(r )/2 , ∀t ∈ [0, T ], with the same constant C(T ) if we wish. Repeating the computations of §3.2, we first have, for t ∈ [0, T ], and so long as the above condition holds, 1 ε 1

L L 1 ([0,T ];L 2 ) + N Iε L 1 ([0,T ];L 2 ) . ε ε As we have seen in §2.1, (ε∇wε , xw ε ) solves a system which is formally analogous to the system satisfied by (Aε w ε , B ε w ε ). Therefore, under the bootstrap condition, we come up with

w ε L ∞ ([0,t];L 2 )

w ε L ∞ ([0,t];ε )

1 ε 1

L L 1 ([0,T ];ε ) + N Iε L 1 ([0,T ];ε ) . ε ε

We easily estimate √ 1 ε

L L 1 ([0,T ];ε ) ε, ε so in view of Lemma 6.1, the point is to estimate the length of I ε (T ).

468


Lemma 6.2. For T > 0 (independent of ε), we have

|I ε (T )| = O εγ , where I ε (T ) is defined by (6.1). Proof. The key remark is that since (x1 , ξ1 ) = (x2 , ξ2 ), the trajectories x1 (t) and x2 (t) may cross only in isolated points: by uniqueness, if x1 (t) = x2 (t), then x˙1 (t) = x˙2 (t). Therefore, there is only a finite number of such points in the interval [0, T ]: (x1 (·) − x2 (·))−1 (0) ∩ [0, T ] = {t j }1 j J , where J = J (T ). If we had J = ∞, then by compactness of [0, T ], a subsequence of (t j ) j would converge to some τ ∈ [0, T ], with x1 (τ ) = x2 (τ ). By uniqueness for the Hamiltonian flow, x˙1 (τ ) = x˙2 (τ ): τ cannot be the limit of times where x1 (t j ) = x2 (t j ). By uniqueness for the Hamiltonian flow, continuity and compactness, there exists δ > 0 such that inf{|x˙1 (t) − x˙2 (t)| ; t ∈ I(δ, T )} = m > 0, where I(δ, T ) =

J

[t j − δ, t j + δ],

j=1

and there exists ε(δ, T ) > 0 such that for ε ∈ ]0, ε(δ, T )], I ε (T ) ⊂ I(δ, T ). Let t ∈ I ε (T ) ∩ [t j − δ, t j + δ]. Taylor’s formula yields x1 (t) − x2 (t) = x1 (t j ) − x2 (t j ) + (t − t j ) (x˙1 (τ ) − x˙2 (τ )), τ ∈ [t j − δ, t j + δ]. We infer εγ |x1 (t) − x2 (t)| |t − t j |m, and Lemma 6.2 follows. Back to the bootstrap argument, we infer

w ε L ∞ ([0,t];ε )

√ ε + εk(1/2−γ ) + εγ.

Fix γ ∈ ]0, 1/2[. By taking k sufficiently large in Lemma 6.1, this yields

w ε L ∞ ([0,t];ε ) εγ . The Gagliardo–Nirenberg inequality yields 1−δ(r )

w ε (t) L r ε−δ(r ) w ε (t) L 2

δ(r )

ε∇w ε (t) L 2 ε−δ(r ) w ε L ∞ ([0,t];ε ) εγ −δ(r ) .

To close the argument, we note εγ −δ(r ) ε−δ(r )/2 provided ε 1 and γ > The last condition is equivalent to γ > the nonlinearity is energy-subcritical.

dσ 4σ +4 ,

δ(r ) . 2

which is compatible with γ < 1/2 since


469

6.3. Nonlinear superposition for large time. Things become more complicated when T is large. We first need to control Mk : this is achieved assuming (E x p)k , and we have Mk (t) eCt . The main point is to estimate |Iε |. This is achieved thanks to the following proposition, whose proof relies heavily on the fact that the space variable is one-dimensional. Proposition 6.3. Under the assumptions of Theorem 1.15, there exist C, C0 > 0 independent of ε such that 1 |Iε (t)| εγ eC0 t |E 1 − E 2 |−2 , 0 t C log . ε Proof (Proof of Theorem 1.15). Before proving Proposition 6.3, we show why this is enough to infer Theorem 1.15. By Lemma 6.1, we have, if (E x p)k is satisfied,

1

N Iε L 1 ([0,t];ε ) eCt tε(k−2)(1/2−γ ) + εγ eC0 t ε(k−2)(1/2−γ ) + εγ eCt . ε Optimizing in γ , we require (k − 2)(1/2 − γ ) = γ , that is γ =

k−2 . 2k − 2

We can thus resume the bootstrap arguments as in §4 and §5, respectively. The key is to notice that this works like in the previous paragraph, since γ =

k−2 σ > (k 4). 2k − 2 4σ + 4

This yields Theorem 1.15 Proof (Proof of Proposition 6.3). We consider J ε (t) an interval of maximal length included in I ε (t) and N ε (t) the number of such intervals. The result comes from the estimate |I ε (t)| N ε (t) × max |J ε (t)|, with |J ε (t)| εγ eCt |E 1 − E 2 |−1 , ε

N (t) te

2Ct

|E 1 − E 2 |

We first prove prove (6.2). Let t1 , t2 ∈

−1

J ε (t).

e

(6.2) 3Ct

|E 1 − E 2 |

−1

.

t∗

∈ [t1 , t2 ] such that |(x1 (t1 ) − x2 (t1 )) − (x1 (t2 ) − x2 (t2 ))| = |t2 − t1 | ξ1 (t ∗ ) − ξ2 (t ∗ ) , There exists

whence |t1 − t2 | |ξ1 (t ∗ ) − ξ2 (t ∗ )|−1 × 2εγ . On the other hand,

|ξ1 (t ∗ )|2 − |ξ2 (t ∗ )|2 ∗ ∗ . |ξ1 (t ) − ξ2 (t )| |ξ1 (t )| − |ξ2 (t )| |ξ1 (t ∗ )| + |ξ2 (t ∗ )| ∗

∗

(6.3)

470


Using |ξ1 (t ∗ )| + |ξ2 (t ∗ )| eCt ,

|ξ1 (t ∗ )|2 − |ξ2 (t ∗ )|2 = 2 E 1 − E 2 − V (x1 (t ∗ )) + V (x2 (t ∗ )) , V (x1 (t ∗ )) − V (x2 (t ∗ )) εγ eCt , we get ||ξ1 (t ∗ )|2 − |ξ2 (t ∗ )|2 | |E 1 − E 2 | − εγ eCt , whence |t1 − t2 | εγ eCt |E 1 − E 2 |−1 , provided εγ eCt 1. Let us now prove (6.3). We use that as t is large, N ε (t) is comparable to the number of distinct intervals of maximal size where |x1 (t) − x2 (t)| εγ . We consider Jε = [t1 , t2 ] such an interval. We have |x1 (t1 ) − x2 (t1 )| = |x1 (t2 ) − x2 (t2 )| = εγ , and ∀t ∈ [t1 , t2 ], |x1 (t) − x2 (t)| εγ . Therefore, for t ∈ [t1 , t2 ], the quantity x1 (t) − x2 (t) has a constant sign: we suppose that x1 (t) − x2 (t) is positive. We then have ξ1 (t1 ) − ξ2 (t1 ) > 0 and ξ1 (t2 ) − ξ2 (t2 ) < 0. Using the exponential control of V (x j (t)) for j ∈ {1, 2}, we obtain

ξ1 (t1 ) − ξ2 (t1 ) − ξ1 (t2 ) − ξ2 (t2 ) eCt |t1 − t2 |. We write ξ1 (t1 ) − ξ2 (t1 )

=

|ξ1 (t1 ) − ξ2 (t1 )|

|ξ1 (t )|2 − |ξ2 (t )|2 1

1

|ξ1 (t1 )| + |ξ2 (t1 )| e−Ct |ξ1 (t1 )|2 − |ξ2 (t1 )|2 , |ξ1 (t )|2 − |ξ2 (t )|2 2 2 −ξ1 (t2 ) + ξ2 (t2 ) = |ξ1 (t2 ) − ξ2 (t2 )| |ξ1 (t2 )| + |ξ2 (t2 )| e−Ct |ξ1 (t2 )|2 − |ξ2 (t2 )|2 . Besides, in view of

1 |ξ1 (t1 )|2 − |ξ2 (t1 )|2 = E 1 − E 2 − V (x1 (t1 )) + V (x2 (t1 )) 2 = E 1 − E 2 − V (x ∗ ) x1 (t1 ) − x2 (t1 ) with x ∗ ∈ [x2 (t1 ), x1 (t1 )], we have ∗ V (x ) x1 (t ) − x2 (t ) eCt x1 (t ) − x2 (t ) εγ eCt . 1 1 1 1


471

Therefore, if εγ eCt 1, and if for example E 1 − E 2 > 0 then we have 1 1 |ξ1 (t1 )|2 − |ξ2 (t1 )|2 (E 1 − E 2 ). 2 2 The same holds for t2 , which yields

ξ1 (t1 ) − ξ2 (t1 ) − ξ1 (t2 ) − ξ2 (t2 ) e−Ct (E 1 − E 2 ), whence the existence of a constant c > 0 such that |t1 − t2 | ce−2C T (E 1 − E 2 ) and |Jε | c e−2C T (E 1 − E 2 ). The number N˜ ε (t) of intervals of the type Jε satisfies N˜ ε (t) × ce−2Ct (E 1 − E 2 ) t, whence the second point of the claim. References 1. Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge: Cambridge University Press, 1991 2. Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121(1), 131–175 (1999) 3. Bambusi, D., Graffi, S., Paul, T.: Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time. Asymptot. Anal. 21(2), 149–160 (1999) 4. Bily, J.M., Robert, D.: The semi-classical Van Vleck formula. Application to the Aharonov-Bohm effect. In: Long time behaviour of classical and quantum systems (Bologna, 1999), Ser. Concr. Appl. Math., Vol. 1, River Edge, NJ: World Sci. Publ., 2001, pp. 89–106 5. Bronski, J.C., Jerrard, R.L.: Soliton dynamics in a potential. Math. Res. Lett. 7(2–3), 329–342 (2000) 6. Carles, R.: Semi-classical analysis for nonlinear Schrödinger equations. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2008 7. Carles, R.: Nonlinear Schrödinger equation with time dependent potential. http://arXiv.org/abs/0910. 4893v2 [math.AP], 2010 8. Carles, R., Dumas, E., Sparber, C.: Multiphase weakly nonlinear geometric optics for Schrödinger equations. SIAM J. Math. Anal. 42(1), 489–518 (2010) 9. Carles, R., Fermanian, C., Gallagher, I.: On the role of quadratic oscillations in nonlinear Schrödinger equations. J. Funct. Anal. 203(2), 453–493 (2003) 10. Carles, R., Miller, L.: Semiclassical nonlinear Schrödinger equations with potential and focusing initial data. Osaka J. Math. 41(3), 693–725 (2004) 11. Combescure, M., Robert, D.: Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. Asymptot. Anal. 14(4), 377–404 (1997) 12. Combescure, M., Robert, D.: Quadratic quantum Hamiltonians revisited. Cubo 8(1), 61–86 (2006) 13. Combescure, M., Robert, D.: A phase-space study of the quantum Loschmidt echo in the semiclassical limit. Ann. Henri Poincaré 8(1), 91–108 (2007) 14. Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463–512 (1999) 15. Fröhlich, J., Gustafson, S., Jonsson, B.L.G., Sigal, I.M.: Solitary wave dynamics in an external potential. Commun. Math. Phys. 250(3), 613–642 (2004) 16. Fujiwara, D.: A construction of the fundamental solution for the Schrödinger equation. J. Analyse Math. 35, 41–96 (1979) 17. Fujiwara, D.: Remarks on the convergence of the Feynman path integrals. Duke Math. J. 47(3), 559– 600 (1980) 18. Gang, Z., Sigal, I.M.: Relaxation of solitons in nonlinear Schrödinger equations with potential. Adv. Math. 216(2), 443–490 (2007) 19. Gérard, P.: Oscillations and concentration effects in semilinear dispersive wave equations. J. Funct. Anal. 141(1), 60–98 (1996)

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Communications in


The Critical Z -Invariant Ising Model via Dimers: Locality Property Cédric Boutillier1,2, , Béatrice de Tilière1,3, 1 Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI Pierre et Marie Curie,

Case courrier 188, 4 place Jussieu, F-75252 Paris Cedex 05, France. E-mail: [email protected]; [email protected]

2 École Normale Supérieure, DMA, 45 rue d’Ulm, 75005 Paris, France 3 Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, CH-2007 Neuchâtel,

Switzerland. E-mail: [email protected] Received: 14 December 2009 / Accepted: 20 May 2010 Published online: 10 November 2010 – © Springer-Verlag 2010

Abstract: We study a large class of critical two-dimensional Ising models, namely critical Z -invariant Ising models. Fisher (J Math Phys 7:1776–1781, 1966) introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z -invariant Ising model, consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph. Our main result is an explicit local formula for the inverse Kasteleyn matrix, in the spirit of Kenyon (Invent Math 150(2):409–439, 2002), as a contour integral of the discrete exponential function of Mercat (Discrete period matrices and related topics, 2002) and Kenyon (Invent Math 150(2):409–439, 2002) multiplied by a local function. Using results of Boutillier and de Tilière (Prob Theor Rel Fields 147(3–4):379–413, 2010) and techniques of de Tilière (Prob Th Rel Fields 137(3–4):487–518, 2007) and Kenyon (Invent Math 150(2):409– 439, 2002), this yields an explicit local formula for a natural Gibbs measure, and a local formula for the free energy. As a corollary, we recover Baxter’s formula for the free energy of the critical Z -invariant Ising model (Baxter, in Exactly solved models in statistical mechanics, Academic Press, London, 1982), and thus a new proof of it. The latter is equal, up to a constant, to the logarithm of the normalized determinant of the Laplacian obtained in Kenyon (Invent Math 150(2):409–439, 2002). 1. Introduction In [Fis66], Fisher introduced a correspondence between the two-dimensional Ising model defined on a graph G, and the dimer model defined on a decorated version of this graph. Since then, dimer techniques have been a powerful tool for solving pertinent questions Supported in part by the Swiss National Foundation Grant 200020-120218/1.

Supported in part by the Swiss National Foundations grants 47102009 and 200020-120218/1.

474

C. Boutillier, B. de Tilière

about the Ising model, see for example the paper of Kasteleyn [Kas67], and the book of Mc Coy and Wu [MW73]. In this paper, we follow this approach to the Ising model. We consider a large class of critical Ising models, known as critical Z -invariant Ising models, introduced in [Bax86]. More precisely, we consider Ising models defined on graphs which have the property of having an isoradial embedding. We suppose that the Ising coupling constants naturally depend on the geometry of the embedded graph, and are such that the model is invariant under star-triangle transformations of the underlying graph, i.e. such that the Ising model is Z -invariant. We suppose moreover that the coupling constants are critical by imposing a generalized self-duality property. The standard Ising model on the square, triangular and honeycomb lattice at the critical temperature are examples of critical Z -invariant Ising models. In the mathematics literature, the critical Z -invariant Ising model has been studied by Mercat [Mer01b], where the author proves equivalence between existence of Dirac spinors and criticality. In [CS09], Chelkak and Smirnov prove a fundamental contribution by establishing universality and conformal invariance in the scaling limit. Technical aspects of this paper rely on the paper [CS10] by the same authors, which provides a comprehensive toolbox for discrete complex analysis on isoradial graphs. Note that some key ideas of the first paper [CS09] were presented in [Smi06]. In [BdT08], we give a complete description of the equivalent dimer model, in the case where the underlying graph is periodic. The critical Z -invariant Ising model has also been widely studied in the physics literature, see for example [Bax86,AYP87,AP07,Mar97,Mar98,CS06]. Let G = (V (G), E(G)) be an infinite, locally finite, isoradial graph with critical coupling constants on the edges. Then by Fisher, the Ising model on G is in correspondence with the dimer model on a decorated graph G, with a well chosen positive weight function ν on the edges. We refer to this model as the critical dimer model on the Fisher graph G of G. It is defined as follows. A dimer configuration of G is a subset of edges of G such that every vertex is incident to exactly one edge of M. Let M(G) be the set of dimer configurations of G. Dimer configurations of G are chosen according to a probability measure, known as Gibbs measure, satisfying the following properties. If one fixes a perfect matching in an annular region of G, then perfect matchings inside and outside of this annulus are independent. Moreover, the probability of occurrence of an interior matching is proportional to the product of its edge-weights given by the weight function ν. The key objects used to obtain explicit expressions for relevant quantities of the dimer model are the Kasteleyn matrix, denoted by K , and its inverse. A Kasteleyn matrix is an oriented adjacency matrix of the graph G, whose coefficients are weighted by the function ν, introduced by Kasteleyn [Kas61]. Our main result is Theorem 1, consisting of an explicit local expression for an inverse K −1 of the Kasteleyn matrix K . It can loosely be stated as follows: refer to Sect. 4.2 for detailed definitions and to Theorem 5 for a precise statement. Theorem 1. Let x, y be two vertices of G. Then the infinite matrix K −1 , whose coefficient −1 is given below, is an inverse Kasteleyn matrix, K x,y 1 −1 = f x (λ) f y (−λ) Expx,y (λ) log λdλ + C x,y , K x,y (2π )2 Cx,y where f x is a complex-valued function depending on the vertex x only; Expx,y is the discrete exponential function introduced in [Mer01a], see also [Ken02]; C x,y is a constant equal to ± 41 when x and y are close, and 0 otherwise; Cx,y is a simple closed curve oriented counterclockwise containing all poles of the integrand, and avoiding a half-line dx,y starting from zero.

Critical Z -Invariant Ising Model via Dimers: Locality Property

475

Before stating implications of Theorem 1 on the critical dimer model on the Fisher graph G, let us make a few comments. 1. Theorem 1 is in the spirit of the work of Kenyon [Ken02], where the author obtains explicit local expressions for the critical Green’s function defined on isoradial graphs, and for the inverse Kasteleyn matrix of the critical dimer model defined on bipartite isoradial graphs. Surprisingly both the expression for the Green’s function of [Ken02] and our expression for the inverse Kasteleyn matrix involve the discrete exponential function. This relation is pushed even further in Corollary 14, see the comment after the statement of Theorem 3. Note that the expression for the inverse Kasteleyn matrix obtained in [Ken02] does not hold in our setting, since the dimer model we consider is defined on the Fisher graph G of G, which is not bipartite and not isoradial. 2. Theorem 1 should also be compared with the explicit expression for the inverse Kasteleyn matrix obtained in [BdT08], in the case where the graph G is periodic. Then, by the uniqueness statement of Proposition 5 of [BdT08], the two expressions coincide. The proofs are nevertheless totally different in spirit, and we have not yet been able to understand the identity by an explicit computation, except in the case where G = Z2 . 3. The most interesting features of Theorem 1 are the following. First, there is no −1 is local, periodicity assumption on the graph G. Secondly, the expression for K x,y meaning that it only depends on the local geometry of the underlying isoradial graph G. More precisely, it only depends on an edge-path of G between vertices xˆ and yˆ of G, naturally constructed from x and y. This implies that changing the isoradial −1 . Thirdly, graph G away from this path does not change the expression for K x,y −1 explicit computations of K x,y become tractable, whereas, even in the periodic case, −1 given in [BdT08]. they remain very difficult with the explicit expression for K x,y 4. The structure of the proof of Theorem 1 is taken from [Ken02]. The idea is to find complex-valued functions that are in the kernel of the Kasteleyn matrix K , then to define K −1 as a contour integral of these functions, and to define the contours of integration in such a way that K K −1 = Id. The great difficulty lies in actually finding the functions that are in the kernel of K , since there is no general method to construct them; and in defining the contours of integration. Indeed, the Fisher graph G is obtained from an isoradial graph G, but has a more complicated structure, so that the geometric argument of [Ken02] does not work. The proof of Theorem 1 being long, it is postponed until the last section of this paper. Using Theorem 1, an argument similar to [dT07b], and the results obtained in [BdT08], yields Theorem 2 giving an explicit expression for a Gibbs measure on M(G). When the graph G is periodic, this measure coincides with the Gibbs measure obtained in [BdT08] as the weak limit of Boltzmann measures on a natural toroidal exhaustion of the graph G. A precise statement is given in Theorem 9 of Sect. 5.1. Theorem 2. There is a unique Gibbs measure P defined on M(G), such that the probability of occurrence of a subset of edges {e1 = x1 y1 , . . . , xk yk } in a dimer configuration of G, chosen with respect to the Gibbs measure P is: k T , P(e1 , . . . , ek ) = K xi ,yi Pf (K −1 ){x 1 ,y1 ,...,x k ,yk } i=1

K −1

where is given by Theorem 1, and (K −1 ){x1 ,y1 ,...,xk ,yk } is the sub-matrix of K −1 , whose lines and columns are indexed by vertices {x1 , y1 , . . . , xk , yk }.

476


Theorem 2 is a result about Gibbs measures with no periodicity assumption on the underlying graph. There are only very few examples of such instances in statistical mechanics. Moreover, the Gibbs measure P inherits the properties of the inverse Kasteleyn matrix K −1 of Theorem 1: it is local, and allows for explicit computations, examples of which are given in Appendix A. Let us now assume that the graph G is Z2 -periodic. Using Theorem 1, and the techniques of [Ken02], we obtain an explicit expression for the free energy of the critical dimer model on the graph G, see Theorem 12, depending only on the angles of the fundamental domain. Using Fisher’s correspondence between the Ising and dimer models, this yields a new proof of Baxter’s formula for the free energy of the critical Z -invariant Ising model, denoted by f I , see Sect. 5.2 for definitions. Theorem 3 ([Bax89]). f I = −|V (G 1 )|

π θe log 2 1 − log tan θe + L(θe ) + L − θe . 2 π π 2 e∈E(G 1 )

Note that the free energy f I of the critical Z -invariant Ising model is, up to a multiplicative constant − 21 and an additive constant, the logarithm of the normalized determinant of the Laplacian obtained by Kenyon [Ken02]. See also Corollary 14, where we explicitly determine the constant of proportionality relating the characteristic polynomials of the critical Laplacian and of the critical dimer model on the Fisher graph G, whose existence had been established in [BdT08]. Outline of the paper Section 2: Definition of the critical Z -invariant Ising model. Section 3: Fisher’s correspondence between the Ising and dimer models. Section 4: Definition of the Kasteleyn matrix K . Statement of Theorem 5, giving an −1 of the inverse Kasteleyn explicit local expression for the coefficient K x,y −1 , as |x − y| → ∞, using techniques matrix. Asymptotic expansion of K x,y of [Ken02]. Section 5: Implications of Theorem 5 on the critical dimer model on the Fisher graph G: Theorem 9 gives an explicit local expression for a natural Gibbs measure, and Theorem 12 gives an explicit local expression for the free energy. Corollary: Baxter’s formula for the free energy of the critical Z -invariant Ising model. Section 6: Proof of Theorem 5. 2. The Critical Z-Invariant Ising Model Consider an unoriented finite graph G = (V (G), E(G)), together with a collection of positive real numbers J = (Je )e∈E(G) indexed by the edges of G. The Ising model on G with coupling constants J is defined as follows. A spin configuration σ of G is a function of the vertices of G with values in {−1, +1}. The probability of occurrence of a spin configuration σ is given by the Ising Boltzmann measure, denoted P J : ⎛ ⎞ 1 P J (σ ) = J exp ⎝ Je σu σv ⎠, Z where Z J =

σ ∈{−1,1}V (G)

exp

e=uv∈E(G)

e=uv∈E(G)

Je σu σv , is the Ising partition function.


477

θe e

Fig. 1. Left: example of isoradial graph. Center: corresponding diamond graph. Right: rhombus half-angle associated to an edge e of the graph

We consider Ising models defined on a class of embedded graphs which have an additional property called isoradiality. A graph G is said to be isoradial [Ken02], if it has an embedding in the plane such that every face is inscribed in a circle of radius 1. We ask moreover that all circumcenters of the faces are in the closure of the faces. From now on, when we speak of the graph G, we mean the graph together with a particular isoradial embedding in the plane. Examples of isoradial graphs are the square and the honeycomb lattice, see Fig. 1 (left) for a more general example of an isoradial graph. To such a graph is naturally associated the diamond graph, denoted by G , defined as follows. Vertices of G consist in the vertices of G, and the circumcenters of the faces of G. The circumcenter of each face is then joined to all vertices which are on the boundary of this face, see Fig. 1 (center). Since G is isoradial, all faces of G are sidelength-1 rhombi. Moreover, each edge e of G is the diagonal of exactly one rhombus of G ; we let θe be the half-angle of the rhombus at the vertex it has in common with e, see Fig. 1 (right). The additional condition imposed on circumcenters ensures that we can glue rhombic faces of G along edges to get the whole plane, without having some “upside-down” rhombi. The same construction can be done for infinite and toroidal isoradial graphs, in which case the embedding is on a torus. When the isoradial graph is infinite and non-periodic, in order to ensure that the embedding is locally finite, we assume that there exists an ε > 0 such that the half-angle of every rhombus of G lies between ε and π2 − ε. This implies in particular that vertices of G have bounded degree. It is then natural to choose the coupling constants J of the Ising model defined on an isoradial graph G, to depend on the geometry of the embedded graph: let us assume that Je is a function of θe , the rhombus half-angle assigned to the edge e. We impose two more conditions on the coupling constants. First, we ask that the Ising model on G with coupling constants J as above is Z -invariant, that is, invariant under star-triangle transformations of the underlying graph. Next, we impose that the Ising model satisfies a generalized form of self-duality. These conditions completely determine the coupling constants J , known as critical coupling constants: for every edge e of G, J (θe ) =

1 + sin θe 1 log . 2 cos θe

(1)

The Z -invariant Ising model on an isoradial graph with this particular choice of coupling constants is referred to as the critical Z -invariant Ising model. This model was introduced by Baxter in [Bax86]. A more detailed definition is given in [BdT08].

478


G Fig. 2. Left: a vertex of G with its incoming edges. Right: corresponding decoration in G

3. Fisher’s Correspondence Between the Ising and Dimer Models Fisher [Fis66] exhibits a correspondence between the Ising model on any graph G drawn on a surface without boundary, and the dimer model on a “decorated” version of G. Before explaining this correspondence, let us first recall the definition of the dimer model. 3.1. Dimer model. Consider a finite graph G = (V (G), E(G)), and suppose that edges of G are assigned a positive weight function ν = (νe )e∈E(G ) . The dimer model on G with weight function ν is defined as follows. A dimer configuration M of G, also called perfect matching, is a subset of edges of G such that every vertex is incident to exactly one edge of M. Let M(G) be the set of dimer configurations of the graph G. The probability of occurrence of a dimer configuration M is given by the dimer Boltzmann measure, denoted P ν : νe ν P (M) = e∈Mν , Z where Z ν = M∈M(G ) e∈M νe is the dimer partition function. 3.2. Fisher’s correspondence. Consider an Ising model on a finite graph G embedded on a surface without boundary, with coupling constants J . We use the following slight variation of Fisher’s correspondence [Fis66]. The decorated graph, on which the dimer configurations live, is constructed from G as follows. Every vertex of degree k of G is replaced by a decoration consisting of 3k vertices: a triangle is attached to every edge incident to this vertex, and these triangles are linked by edges in a circular way, see Fig. 2. This new graph, denoted by G, is also embedded on the surface without boundary and has vertices of degree 3. It is referred to as the Fisher graph of G. Fisher’s correspondence uses the high temperature expansion1 of the Ising partition function, see for example [Bax89]: ⎛ ⎞ cosh(Je )⎠ 2|V (G)| tanh(Je ), ZJ = ⎝ e∈E(G)

C∈P e∈C

1 It is also possible to use the low temperature expansion, if the spins of the Ising model do not sit on the vertices but on the faces of G.


1

2

3

4

479

1

2

1

1

2

2

3

4

3

3

4

4

Fig. 3. Polygonal contour of Z2 , and corresponding dimer configurations of the associated Fisher graph

where P is the family of all polygonal contours drawn on G, for which every edge of G is used at most once. This expansion defines a measure on the set of polygonal contours P of G: the probability of occurrence of a polygonal contour C is proportional to the product of the weights of the edges it contains, where the weight of an edge e is tanh(Je ). Here comes the correspondence: to any contour configuration C coming from the high-temperature expansion of the Ising model on G, we associate 2|V (G)| dimer configurations on G: edges present (resp. absent) in C are absent (resp. present) in the corresponding dimer configuration of G. Once the state of these edges is fixed, there is, for every decorated vertex, exactly two ways to complete the configuration into a dimer configuration. Figure 3 gives an example in the case where G is the square lattice Z2 . Let us assign, to an edge e of G, weight νe = 1, if it belongs to a decoration; and weight νe = coth Je , if it corresponds to an edge of G. Then the correspondence is mea(G)| ) of images sure-preserving: every contour configuration C has the same number (2|V by this correspondence, and the product of the weights of the edges in C, e∈C tanh(Je ) is proportional to the weight e∈C coth(Je ) of any of its corresponding dimer configu rations for a proportionality factor, e∈E(G) tanh(Je ), which is independent of C. As a consequence of Fisher’s correspondence, we have the following relation between the Ising and dimer partition functions: ⎛ ⎞ J Z =⎝ sinh(Je )⎠ Z ν . (2) e∈E(G)

Fisher’s correspondence between Ising contour configurations and dimer configurations naturally extends to the case where G is an infinite planar graph.

3.3. Critical dimer model on Fisher graphs. Consider a critical Z -invariant Ising model on an isoradial graph G on the torus, or on the whole plane. Then, the dimer weights of the corresponding dimer model on the Fisher graph G are: νe =

1

if e belongs to a decoration, ν(θe ) = cot θ2e if e comes from an edge of G.

We refer to these weights as critical dimer weights, and to the corresponding dimer model as critical dimer model on the Fisher graph G.

480


Fig. 4. An example of Kasteleyn orientation of the Fisher graph of Z2 , in which every triangle of every decoration is oriented clockwise

4. Kasteleyn Matrix on Critical Infinite Fisher Graphs In the whole of this section, we let G be an infinite isoradial graph, and G be the corresponding Fisher graph. We suppose that edges of G are assigned the dimer critical weight function denoted by ν. Recall that G denotes the diamond graph associated to G. 4.1. Kasteleyn and inverse Kasteleyn matrix. The key object used to obtain explicit expressions for the dimer model on the Fisher graph G is the Kasteleyn matrix introduced by Kasteleyn in [Kas61]. It is a weighted, oriented adjacency matrix of the graph G defined as follows. A Kasteleyn orientation of G is an orientation of the edges of G such that all elementary cycles are clockwise odd, i.e. when traveling clockwise around the edges of any elementary cycle of G, the number of co-oriented edges is odd. When the graph is planar, such an orientation always exists [Kas67]. For later purposes, we need to keep track of the orientation of the edges of G. We thus build a specific Kasteleyn orientation of the graph G in the following way: choose a Kasteleyn orientation of the planar graph obtained from G by contracting each decoration triangle to a point, which exists by Kasteleyn’s theorem [Kas67], and extend it to G by imposing that every triangle of every decoration is oriented clockwise. Refer to Fig. 4 for an example of such an orientation in the case where G = Z2 . The Kasteleyn matrix corresponding to such an orientation is an infinite matrix, whose rows and columns are indexed by vertices of G, defined by: K x,y = εx,y νx y , where εx,y

⎧ ⎪ ⎨1 = −1 ⎪ ⎩0

if x ∼ y, and x → y if x ∼ y, and x ← y otherwise.


wk(x) x

481

vk(x) zk(x)

v (x) k+1

x

G Fig. 5. Notations for vertices of G

Note that K can be interpreted as an operator acting on CV (G ) : ∀ f ∈ CV (G ) , (K f )x = K x,y f y . y∈V (G )

An inverse of the Kasteleyn matrix K , denoted K −1 , is an infinite matrix whose rows and columns are indexed by vertices of G, and which satisfies K K −1 = Id. 4.2. Local formula for an inverse Kasteleyn matrix. In this section, we state Theorem 5 proving an explicit local expression for the coefficients of an inverse K −1 of the Kasteleyn matrix K . This inverse is the key object for the critical dimer model on the Fisher graph G. Indeed, it yields an explicit local expression for a Gibbs measure on dimer configurations of G, see Sect. 5.1. It also allows for a simple derivation of Baxter’s formula for the free energy of the critical Z -invariant Ising model, see Sect. 5.2. This section is organized as follows. The explicit expression for the coefficients of K −1 given by Theorem 5 below, see also Theorem 1, is a contour integral of an integrand involving two quantities: • a complex-valued function depending on a complex parameter, and on the vertices of the graph G only, defined in Sect. 4.2.2. • the discrete exponential function which first appeared in [Mer01a], see also [Ken02]. It is a complex valued function depending on a complex parameter, and on an edgepath between pairs of vertices of the graph G, defined in Sect. 4.2.3. Theorem 5 is then stated in Sect. 4.2.4. Since the proof is long, it is postponed until Sect. 6. In Sect. 4.2.5, using the same technique as [Ken02], we give the asymptotic −1 of the inverse Kasteleyn matrix, as |x − y| → ∞. expansion of the coefficient K x,y 4.2.1. Preliminary notations. From now on, vertices of G are written in normal symbol, and vertices of G in boldface. Let x be a vertex of G, then x belongs to the decoration corresponding to a unique vertex of G, denoted by x. Conversely, vertices of G of the decoration corresponding to a vertex x of G are labeled as follows, refer to Fig. 5 for an example. Let d(x) be the degree of the vertex x in G, then the corresponding decoration of G consists of d(x) triangles, labeled from 1 to d(x) in counterclockwise order. For the k th triangle, let vk (x) be the vertex incident to an edge of G, and let wk (x), z k (x) be the two other vertices, in counterclockwise order, starting from vk (x). Later on, when no confusion occurs, we will drop the argument x in the above labeling. Define a vertex x of G to be of type ‘v’, if x = vk (x) for some k ∈ {1, . . . , d(x)}, and similarly for ‘w’ and ‘z’.

482


eiαzk eiαwk = ei αzk+1

wk(x) z k+1(x)

x

vk(x) zk(x) x

G Fig. 6. Rhombus vectors of the diamond graph G assigned to vertices of G

The isoradial embedding of the graph G fixes an embedding of the corresponding diamond graph G . There is a natural way of assigning rhombus unit-vectors of G to vertices of G: for every vertex x of G, and every k ∈ {1, . . . , d(x)}, let us associate the rhombus unit-vector eiαwk (x) to wk (x), eiαzk (x) to z k (x), and the two rhombus-unit vectors eiαwk (x) , eiαzk (x) to vk (x), as in Fig. 6. Note that eiαwk (x) = eiαzk+1 (x) . 4.2.2. Complex-valued function on the vertices of G. Let us introduce the complex-valued function defined on vertices of G and depending on a complex parameter, involved in the integrand of the contour integral defining K −1 given by Theorem 5. Define f : V (G) × C → C, by: f (wk (x), λ) := f wk (x) (λ) =

ei

αw (x) k 2

eiαwk (x) − λ

f (z k (x), λ) := f z k (x) (λ) = −

ei

αz (x) k 2

,

, eiαzk (x) − λ f (vk (x), λ) := f vk (x) (λ) = f wk (x) (λ) + f z k (x) (λ),

(3)

for every x ∈ G, and every k ∈ {1, . . . , d(x)}. In order for the function f to be well defined, the angles αwk (x) , αz k (x) need to be well defined mod 4π , indeed half-angles need to be well defined mod 2π . Let us define them inductively as follows, see also Fig. 7. The definition strongly depends on the Kasteleyn orientation introduced in Sect. 4.1. Fix a vertex x0 of G, and set αz 1 (x0 ) = 0. Then, for vertices of G in the decoration of a vertex x ∈ G, define: αwk (x) = αz k (x) + 2θk (x), where θk (x) > 0 is the rhombus half-angle of Fig. 7, αwk (x) if the edge wk (x)z k+1 (x) is oriented from wk (x) to z k+1 (x) αz k+1 (x) = αwk (x) + 2π otherwise. (4) Here is the rule defining angles in the neighboring decoration, corresponding to a vertex y of G. Let k and be indices such that vk (x) is adjacent to v (y) in G. Then, define: αwk (x) − π if the edge vk (x)v (y) is oriented from vk (x) to v (y) (5) αw (y) = αwk (x) + π otherwise.


483

y z (y)

y

e i αw

l

e i αz l e i αwk= e i αzk+1

l

θl θk

e

i αz

k

x

w (y) vl (yl )

vk(x) wk(x) zk(x) zk+1(x) x

G Fig. 7. Notations for the definition of the angles in R/4π Z

C1

C2

x

z k+1(x) wk(x) z k (x)

vlk xk+1

wlk +1

vl

k +1

vlk +1

wl k zlk +1 xk

Fig. 8. The two types of cycles on which the definition of the angles in R/4π Z needs to be checked. Left: C1 , the inner cycle of a decoration, oriented counterclockwise. Right: C2 , a cycle coming from the boundary of a face of G, oriented clockwise

Lemma 4. For every vertex x of G, and every k ∈ {1, . . . , d(x)}, the angles αwk (x) , αz k (x) , are well defined in R/4π Z. Proof. It suffices to show that when doing the inductive procedure around a cycle of G, we obtain the same angle modulo 4π . There are two types of cycles to consider: inner cycles of decorations, and cycles of G coming from the boundary of a face of G. These cycles are represented in Fig. 8. Let C1 be a cycle of the first type, that is C1 is the inner cycle of a decoration corresponding to a vertex x of G, oriented counterclockwise. Let d = d(x) be the degree of the vertex x in G, then: C1 = (z 1 (x), w1 (x), . . . , z d (x), wd (x), z 1 (x)) = (z 1 , w1 , . . . , z d , wd , z 1 ). According to our choice of Kasteleyn orientation, all edges of this cycle are oriented counterclockwise with respect to the face surrounded by C1 , except an odd number n of edges of the form wk z k+1 . By definition of the angles (4), when jumping from z k to z k+1 , the change in angle is 2θk if wk z k+1 is oriented counterclockwise, and 2θk + 2π otherwise. The total change along the cycle is thus, d

2θk + 2π n = 2π(n + 1) ≡ 0 mod 4π.

k=1

Let C2 be a cycle of the second type, around a face touching m decorations corresponding to vertices x1 , . . . , xm of G. Suppose that C2 is oriented clockwise, C2 = w 1 (x1 ), z 1 +1 (x1 ), v 1 +1 (x1 ), . . . , . . . , v m (xm ), w m (xm ), z m +1 (xm ), v m +1 (xm ), v 1 (x1 ), w 1 (x1 ) .

484


The cycle C2 contains 4m edges, and the number n of co-oriented edges along C2 is odd by definition of a Kasteleyn orientation. By our choice of Kasteleyn orientation, the only edges that may be co-oriented along the cycle, are either of the form w k (xk )z k +1 (xk ), or of the form v k +1 (xk )v k+1 (xk+1 ). For every k ∈ {1, . . . , m}, there are two different contributions to the total change in angle from w k (xk ) to w k+1 (xk+1 ). First, from w k (xk ) to z k +1 (xk ), by (4), there is a contribution of 2π if the corresponding edge is co-oriented. Then, from z k +1 (xk ) to w k+1 (xk+1 ), by (5), the angle is changed by 2θ k +1 (xk ) − π = 2θ k+1 (xk+1 ) − π , plus an extra contribution of 2π if the edge v k +1 (x)v k+1 (xk+1 ) is co-oriented. Thus the total change in angle is: m

(2θ k (xk ) − π ) + 2π n.

k=1

But the terms π − 2θ k (xk ) are the angles of the rhombi at the center of the face surrounded by C2 , which sum to 2π . Therefore, the total change is equal to 2π(n − 1) which is congruent to 0 mod 4π . These angles at vertices are related to the notion of spin structure on surface graphs (see [CR07,CR08,Kup98]). A spin structure on a surface is equivalent to the data of a vector field with even index singularities. Kuperberg explained in [Kup98] how to construct from a Kasteleyn orientation a vector field with odd index singularities at vertices of G. One can then obtain the spin structure by merging the singularities into pairs using a reference dimer configuration. The angles αwk (x) and αz k (x) defined here are directly related to the direction of the vector field with even index singularities obtained from Kuperberg’s construction applied to our choice of Kasteleyn orientation, and from the pairing of singularities corresponding to the following reference dimer configuration: • wk (x) ↔ z k (x), • vk (x) ↔ v (y) if they are neighbors. 4.2.3. Discrete exponential functions. Let us define the discrete exponential function, denoted Exp, involved in the integrand of the contour integral defining K −1 given by Theorem 5. This function first appeared in [Mer01a], see also [Ken02]. In order to simplify notations, we use a different labeling of the rhombus vectors of G . Let x, y be two vertices of G, and let y = x1 , x2 , . . . , xn+1 = x be an edge-path of G from y to x. The complex vector x j+1 − x j is the sum of two unit complex numbers eiβ j + eiγ j representing edges of the rhombus in G associated to the edge x j x j+1 . Then, Exp : V (G) × V (G) × C → C is defined by: iγ j n iβ j e +λ e +λ . Exp(x, y, λ) := Expx,y (λ) = eiβ j − λ eiγ j − λ j=1

(6)

The function is well defined (independent of the choice of edge-path of G from y to x) since the product of the multipliers around a rhombus is 1.


485

4.2.4. Inverse Kasteleyn matrix. We now state Theorem 5 proving an explicit local formula for the coefficients of an inverse K −1 of the Kasteleyn matrix K , constructed from the functions f defined in (3) and the discrete exponentials defined in (6). The vertices x and y of G in the statement should be thought of as being one of wk (x), z k (x), vk (x) for some x ∈ G and some k ∈ {1, . . . , d(x)}, and similarly for y. The proof of Theorem 5 is postponed until Sect. 6. Theorem 5. Let x, y be any two vertices of G. Then the infinite matrix K −1 , whose −1 is given by (7) below, is an inverse Kasteleyn matrix, coefficient K x,y 1 −1 = f x (λ) f y (−λ) Expx,y (λ) log λdλ + C x,y . (7) K x,y (2π )2 Cx,y The contour of integration Cx,y is a simple closed curve oriented counterclockwise containing all poles of the integrand, and avoiding the half-line dx,y starting from zero.2 The constant C x,y is given by ⎧1 if x = y = wk (x) ⎪ ⎪ 4 ⎪ ⎨− 1 if x = y = z k (x) 4 C x,y = (−1) (8) n(x,y) ⎪ if x = y, x = y, and x and y are of type ‘w’ or ‘z’ ⎪ 4 ⎪ ⎩ 0 otherwise, where n(x, y) is the number of edges oriented clockwise in the clockwise arc from x to y of the inner cycle of the decoration corresponding to x. −1 given in (7) has the very interesting feature Remark 6. The explicit expression for K x,y of being local, i.e. it only depends on the geometry of the embedding of the isoradial graph G on a path between xˆ and yˆ , where xˆ and yˆ are vertices of G at distance at most 2 from x and y, constructed in Sect. 6.3. A nice consequence of this property is the following. For i = 1, 2, let G i be an isoradial graph with corresponding Fisher graph Gi . Let K i be the Kasteleyn matrix of the graph Gi , whose edges are assigned the dimer critical weight function. Let (K i )−1 be the inverse of K i given by Theorem 5. If G 1 and G 2 coincide on a ball B, and if the Kasteleyn orientations on G1 and G2 are chosen to be the same in the ball B, then for every couple of vertices (x, y) in B at distance at least 2 from the boundary, we have −1 (K 1 )−1 x,y = (K 2 )x,y .

4.2.5. Asymptotic expansion of the inverse Kasteleyn matrix. As a corollary to Theorem 5, and using computations analogous to those of Theorem 4.3 of [Ken02], we obtain −1 of the inverse Kasteleyn matrix, as the asymptotic expansion for the coefficient K x,y |x − y| → ∞. In order to give a concise statement, let us introduce the following simplified notations. When x is of type ‘w’ or ‘z’, let eiα denote the corresponding rhombus unit-vector of G ; and when x is of type ‘v’, let eiα1 , eiα2 be the two corresponding rhombus unit-vectors of G , defined in Sect. 4.2.1. Moreover, set: 1 if x is of type ‘w’ or ‘v’

x = −1 if x is of type ‘z’. 2 In most cases, the half-line d x,y is oriented from xˆ to yˆ , where xˆ , yˆ are vertices of G at distance at most two from x, y, constructed in Sect. 6.3. Refer to this section and Remark 19 for more details.

486


A superscript “prime” is added to these notations for the vertex y, and y is defined in a similar way. −1 of the Corollary 7. The asymptotic expansion, as |x − y| → ∞, of the coefficient K x,y inverse Kasteleyn matrix of Theorem 5 is: −1 K x,y

⎧ α+α i 2 ⎪ 1 ⎪ ⎪ e x−y + o |x−y| if x and y are of type ‘w’ or ‘z’, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ α1 α2 ⎪ iα i i ⎨

x y ⎝e 2 (e 2 −e 2 )⎠ + o 1 if x is of type ‘w’ or ‘z’, y is of type ‘v’, = x−y |x−y| 2π ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ α α ⎪ α α ⎪ i 21 i 22 i 21 i 22 ⎪ )(e −e )⎠ ⎪ 1 ⎪ ⎝ (e −e x−y if x and y are of type ‘v’. + o ⎪ |x−y| ⎩

−1 is given Proof. We follow the proof of Theorem 4.3 of [Ken02]. By Theorem 5, K x,y by the integral: 1 −1 K x,y = f x (λ) f y (−λ) Expx,y (λ) log λdλ + C x,y , (9) (2π )2 Cx,y

where Cx,y is a simple closed curve oriented counterclockwise containing all poles of the integrand, and avoiding the half-line dx,y starting from 0 in the direction from xˆ to yˆ , where xˆ , yˆ are vertices of G , at distance at most 2 from x and y, constructed in Sect. 6.3. We choose dx,y to be the origin for the angles, i.e. the positive real axis R+ . The coefficient C x,y equals 0 whenever x and y are not in the same decoration. It will be a fortiori the case when they are far away from each other. It suffices to handle the case where both x and y are of type ‘w’ or ‘z’. The other cases are then derived using the relation between the functions f : f vk (λ) = f wk (λ) + f z k (λ). Define, ξ =x−y=

n

eiβ j + eiγ j ,

j=1

where eiβ1 , eiγ1 , . . . , eiβn , eiγn are the steps of a path in G from y to x. When x and y → are far apart, since xˆ and yˆ are at distance at most 2 from x and y, the direction of − xy is − → not very different from that of xˆ yˆ , ensuring that (ξ ) < 0. The contour Cx,y can be deformed to a curve running counterclockwise around the ball of radius R (R large) around the origin from the angle 0 to 2π , then along the positive real axis, from R to r (r small), then clockwise around the ball of radius r from the angle 2π to 0, and then back along the real axis from r to R. The rational fraction f x (λ) f y (−λ) Expx,y (λ) behaves like O(1) when λ is small, and like O λ12 , when λ is large. As a consequence, the contribution to the integral around the balls of radius r


487

and R converges to 0, as we let r → 0 and R → ∞. The logarithm differs by 2iπ on the two sides of the ray R+ , the integral (9) is therefore equal to ∞ 1 −1 f x (λ) f y (−λ) Expx,y (λ)dλ. K x,y = 2iπ 0 As in [Ken02], when |x − y| is large, the main contribution to this integral comes from a neighborhood of the origin and a neighborhood of infinity. When λ is small, we have: α

α ei 2 = e−i 2 +O(λ) , eiα − λ

ei e

iα

α 2

+λ

= e−i

α 2 +O(λ)

eiβ + λ −iβ 3 = exp 2e λ + O(λ ) . eiβ − λ

,

Thus, for small values of λ: α

α

ei 2 eiβ j + λ eiγ j + λ ei 2 f x (λ) f y (−λ) Expx,y (λ) = x y iα e − λ eiα + λ eiβ j − λ eiγ j − λ j=1 α+α = x y e−i 2 exp 2ξ λ + O(λ) + O(nλ3 ) . Integrating this estimate from 0 to

√1 n

0

√1 n

n

gives

f x (λ) f y (−λ) Expx,y (λ)dλ = x y

0

√1 n

e−i

α+α 2

α+α

e−i 2 = − x y 2ξ¯

1 exp 2ξ¯ λ + O(n − 2 ) dλ

1 1+O √ . n

Similarly, when λ is large, we have: α

α

α

−1

ei 2 ei 2 +O(λ ) ei 2 ei = − , = eiα − λ λ eiα + λ iβ e +λ iβ −1 −3 = − exp 2e λ + O(λ ) . eiβ − λ

α −1 2 +O(λ )

λ

,

Thus for large values of λ, α+α

ei 2 f x (λ) f y (−λ) Expx,y (λ) = − x y 2 exp 2ξ λ−1 + O(λ−1 ) + O(nλ−3 ) . λ √ Computing the integral of this estimate for λ ∈ ( n, ∞) gives

∞ √ n

α+α

ei 2 f x (λ) f y (−λ) Expx,y (λ)dλ = x y 2ξ 1

1

1 . 1+O √ n

The rest of the integral between n − 2 and n 2 is negligible (see [Ken02]). As a consequence,

488


−1 K x,y

⎞ ⎛ α+α −i α+α 2

x y ei 2 e ⎝ = (1 + o(1))⎠ (1 + o(1)) − 2iπ 2ξ 2ξ¯ ⎛ ⎞ i α+α 2

x y e 1 ⎝ ⎠ = . +o 2π x−y |x − y|

Remark 8. Using the same method with an expansion of the integrand to a higher order −1 . would lead to a more precise asymptotic expansion of K x,y 5. Critical Dimer Model on Infinite Fisher Graphs Let G be an infinite Fisher graph obtained from an infinite isoradial graph G. Assume that edges of G are assigned the dimer critical weight function ν. In this section, we give a full description of the critical dimer model on the Fisher graph G, consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph G. More precisely, in Sect. 5.1, using the method of [dT07b], we give an explicit local formula for a Gibbs measure on dimer configurations of G, involving the inverse Kasteleyn matrix of Theorem 5. When the graph is periodic, this measure coincides with the Gibbs measure of [BdT08], obtained as the weak limit of Boltzmann measures on a natural toroidal exhaustion. Then, in Sect. 5.2, we assume that the graph G is periodic and, using the method of [Ken02], we give an explicit local formula for the free energy of the critical dimer model on G. As a corollary, we obtain Baxter’s celebrated formula for the free energy of the critical Z -invariant Ising model. 5.1. Local formula for the critical dimer Gibbs measure. A Gibbs measure on the set of dimer configurations M(G) of G, is a probability measure on M(G), which satisfies the following. If one fixes a perfect matching in an annular region of G, then perfect matchings inside and outside of this annulus are independent. Moreover, the probability of occurrence of an interior matching is proportional to the product of the edge weights. In order to state Theorem 9, we need the following definition. Let E be a finite subset of edges of G. The cylinder set AE is defined to be the set of dimer configurations of G containing the subset of edges E; it is often convenient to identify E with AE . Let F be the σ -algebra of M(G) generated by the cylinder sets (AE )E finite . Recall that K denotes the infinite Kasteleyn matrix of the graph G, and let K −1 be the matrix inverse given by Theorem 5. Theorem 9. There is a unique probability measure P on (M(G), F), such that for every finite collection of edges E = {e1 = x1 y1 , . . . , ek = xk yk } ⊂ E(G), the probability of the corresponding cylinder set is k T P(AE ) = P(e1 , . . . , ek ) = , (10) K xi ,yi Pf (K −1 ){x 1 ,y1 ,...,x k ,yk } i=1

where K −1 is given by Theorem 5, and (K −1 ){x1 ,y1 ,...,xk ,yk } is the sub-matrix of K −1 whose rows and columns are indexed by vertices {x1 , y1 , . . . , xk , yk }.


489

Moreover, P is a Gibbs measure. When G is Z2 -periodic, P is the Gibbs measure obtained as the weak limit of the Boltzmann measures Pn on the toroidal exhaustion {Gn = G/(nZ2 )}n≥1 of G. Proof. The idea is to use Kolmogorov’s extension theorem. The structure of the proof is taken from [dT07b]. Let E = {e1 = x1 y1 , . . . en = xn yn } be a finite subset of edges of G. Denote by FE the σ -algebra generated by the cylinders (AE )E ⊂E . We define an additive function PE on FE , by giving its value on every cylinder set AE , with E = {ei1 , . . . , eik } ⊂ E: ⎛ ⎞ k T . PE (AE ) = ⎝ K xi j ,yi j ⎠ Pf (K −1 ){x i ,yi ,...,xi ,yi } 1

j=1

1

k

k

It is a priori not obvious that PE defines a probability measure on FE . Let us show that this is indeed the case. Let VE be the subset of vertices of G to which the decorations containing x1 , y1 , . . . , xn , yn retract. Define Q to be a simply-connected subset of rhombi of G containing all rhombi adjacent to vertices of VE . By Proposition 1 of [dT07b], there exists a Z2 -periodic rhombus tiling of the plane containing Q, which we denote by G p . Moreover, the rhombus tiling G p is the diamond graph of a unique isoradial graph G p whose vertices contain the subset VE . Let G p be the Fisher graph of G p , then the graphs G and G p coincide on a ball B containing x1 , y1 , . . . , xn , yn . Endow edges of G p with the critical weights and a periodic Kasteleyn orientation, coinciding with the orientation of the edges of G on the common ball B. We know by Remark 6, that the coefficients of the inverses (K p )−1 and K −1 given by Theorem 5 are equal for all pairs of vertices in B. On the other hand, we know by Corollary 7 that the coefficient (K p )−1 x,y → 0, as |x − y| → ∞. By Proposition 5 of [BdT08], stating uniqueness of the inverse Kasteleyn matrix decreasing at infinity in the periodic case, we deduce that (K p )−1 is in fact the inverse computed in [BdT08] by Fourier transform. In Theorem 6 of [BdT08], we use the inverse (K p )−1 to construct the Gibbs measure P p on M(G p ) obtained as the weak limit of Boltzmann measures on the natural toroidal exhaustion of G p , which has the following explicit expression: let {e1 = x1 y1 , . . . , ek = xk yk } be a subset of edges of G p , then ⎛ ⎞ k p T K x y ⎠ Pf ((K p )−1 ){x P p (e1 , . . . ek ) = ⎝ ,...,y } . j=1

j j

1

k

In particular, the expression of the restriction of P p to events involving edges in E (seen as edges of G p ) is equal to the formula defining PE . As a consequence, PE is a probability measure on FE . The fact that Kolmogorov’s consistency relations are satisfied by the collection of probability measures (PE )E is immediate once we notice that any finite number of these probability measures can be interpreted as finite-dimensional marginals of a Gibbs measure on dimer configurations of a periodic graph with a large enough fundamental domain. The measure P of Theorem 9 is thus the one given by Kolmogorov’s extension theorem applied to the collection (PE )E . The Gibbs property follows from the Gibbs property of the probability measures P p .

490


The probability of single edges are computed in details in Appendix A, using the explicit form for K −1 given by Theorem 5. Consider a rhombus of G with vertices x, t, y, u: x and y are vertices of G; t and u are vertices of G ∗ . Let θ be the half-angle of this rhombus, measured at x. Denote by e the edge of G coming from the edge xy of G in the decoration process. Then, Proposition 10. The probability of occurrence P(e) of the edge e in a dimer configuration of G is: P(e) =

1 π − 2θ + . 2 2π cos θ

(11)

From this, we can deduce the following probability for the Ising model on the dual graph G ∗ . In Fisher’s correspondence, the presence of the edge e in the dimer configuration corresponds to the fact that xy is not covered by a piece of contour. In the low temperature expansion of the Ising model, this thus means that the two spins at t and u have the same sign. Since they can be both + or −, the probability that they are both +, as a function of φ = π2 − θ , the angle of the rhombus measured at u, is: Corollary 11. ⎛ ⎜ ⎜ P⎜ ⎝

+

⎞

φ

+ ⎟ ⎟

1 ⎟= ⎠ 2

1 π − 2θ + 2 2π cos θ

=

1 φ + . 4 2π sin φ

5.2. Free energy of the critical dimer model. In this section, we suppose that the isoradial graph G is periodic, so that the corresponding Fisher graph G is. A natural exhaustion of G by toroidal graphs is given by {Gn }n≥1 , where Gn = G/nZ2 , and similarly for G. The graphs G1 and G 1 are known as the fundamental domains of G and G respectively. In order to shorten notations in Theorem 12 below and in the proof, we write |E n | = |E(G n )|, and |Vn | = |V (G n )|. The free energy per fundamental domain of the critical dimer model on the Fisher graph G is denoted f D , and is defined by: f D = − lim

n→∞

1 log Znν , n2

where Znν is the dimer partition function of the graph Gn , whose edges are assigned the critical weight function ν. Theorem 12. The free energy per fundamental domain of the critical dimer model on the Fisher graph G is given by: log 2 f D = −(|E 1 | + |V1 |) 2

π π − 2θe 1 θe 1 L(θe ) + L , + log tan θe − log cot − − θe 2π 2 2 π 2 e∈E(G 1 )

where L is the Lobachevsky function, L(x) = −

x 0

log 2 sin t dt.


491

Proof. For the proof, we follow the argument given by Kenyon [Ken02] to compute the normalized determinant of the Laplacian and of the Dirac operator on isoradial graphs. See also [dT07a] for a detailed computation of the free energy of dimer models on isoradial bipartite graphs. In [BdT08], we proved that fD

1 =− 2

T2

dz dw log det Kˆ (z, w) , 2iπ z 2iπ w

where Kˆ (z, w) is the Fourier transform of the infinite periodic Kasteleyn matrix of the critical dimer model on G. We need the following definition of [KS05]. Recall that G is the diamond graph associated to the isoradial graph G. A train-track of G is a path of edge-adjacent rhombi of G , which does not turn: on entering a face, it exits along the opposite edge. As a consequence, each rhombus in a train-track has an edge parallel to a fixed unit vector. We assume that each train-track is extended as far as possible in both directions, so that it is a bi-infinite path. The idea of the proof is to first understand how the free energy f D is changed when the isoradial embedding of the graph G is modified by tilting a family of parallel traintracks. If we can compute the free energy in an extreme situation, that is when all the rhombi are flat, i.e with half-angles equal to 0 or π/2, then we can compute the free energy of the initial graph by integrating the variation of the free energy along the deformation from the trivial flat embedding back to the initial graph, by tilting all the families of train-tracks successively until all rhombi recover their original shape. Let G flat be the trivial flat isoradial graph, and let G flat be the corresponding Fisher graph. Consider a train-track T of G . All the rhombi of the train-track have a common parallel, with an angle equal to α. Let us compute the derivative of f D as we tilt the train-track T as well as all its copies. The function log det Kˆ (z, w) is integrable over the unit torus, is a differentiable function of α for all (z, w) ∈ T2 \{1, 1}, and the derivative is also uniformly integrable on the torus. The dimer free energy f D is thus differentiable with respect to α, and 1 d fD =− dα 2 =− =−

1 2

dz dw d log det Kˆ (z, w) 2iπ z 2iπ w T2 dα d Kˆ (z, w)u,v dz dw Kˆ −1 (z, w)v,u dα 2iπ z 2iπ w T2

u,v∈V (G1 )

e=uv∈E(G1 )

−1 K v,u

dK u,v d log νe =− . P(e) dα dα e∈E(G 1 )

In the last line, we used the uniqueness of the inverse Kasteleyn matrix whose coefficients decrease at infinity, given by Proposition 5 of [BdT08]. We also used the fact that if −1 . Note that the sum is restricted to edges e = uv, then νe = |K u,v | and P(e) = K u,v K v,u coming from E(G 1 ), because they are the only edges of G1 with a weight depending on an angle. There is in this sum a term for every rhombus in the fundamental domain G 1 , and that term only depends on the half-angle of this rhombus. The variation of the free energy of the dimer model along the deformation from G flat to G is thus, up to an additive constant,

492


the sum over all rhombi in a fundamental domain of the variation of a function f(θ ) depending only on the geometry of the rhombus: f(θe ) − f(θeflat ), f D = f D (G) − f D (G flat ) = e∈E(G 1 )

with df(θ ) d log ν(θ ) = −P(θ ) , dθ dθ where P(θ ) is the probability of occurrence in a dimer configuration of the edge of G coming from an edge of G, whose half rhombus-angle is θ ; and θeflat is the angle, equal to 0 or π2 , of the degenerate rhombus in G flat associated to the edge e. Recalling that by definition of ν, and by Eq. (11): θ 1 π − 2θ ν(θ ) = cot , P(θ ) = + , 2 2 2π cos θ and using the fact that derivative of f(θ ): df(θ ) = dθ

d dθ

log cot

θ 2

= − sin1 θ , we deduce an explicit formula for the

1 π − 2θ + 2 2π cos θ

Using integration by parts, a primitive of

1 1 π − 2θ = + . sin θ 2 sin θ π sin 2θ df(θ) dθ

is given by: π 1 θ π − 2θ 1 L(θ ) + L f(θ ) = log tan + log tan θ − −θ . 2 2 2π π 2 The problem is that f(θ ) goes to −∞ when θ approaches 0, and thus the free energy diverges when the graph G becomes flat. One can instead study s D , the entropy of the corresponding dimer model, defined by P(e) log νe , (12) sD = − f D − e∈E(G1 )

which behaves well when the embedding degenerates, since it is insensible to gauge transformations, and thus provides a better measure of the disorder in the model. The sum in (12) is over all edges in the fundamental domain G1 of the Fisher graph G. But since edges of the decoration have weight 1, only the edges coming from edges of G contribute. Thus, as for f D , the variation of s D can be written as the sum of contributions s(θe ) of all rhombi in a fundamental domain G 1 , s D = s D (G) − s D (G flat ) = s(θe ) − s(θeflat ), (13) e∈E(G 1 )

where s(θ ) = −f(θ ) − P(θ ) log ν(θ ) θ π − 2θ 1 log tan θ = − log tan − 2 2 2π π 1 π − 2θ 1 θ + L(θ ) + L −θ − + log cot π 2 2 2π cos θ 2 θ π log cot 2 π − 2θ 1 L(θ ) + L −θ . = log cot θ − + 2π cos θ π 2


Since limθ→0 log cot θ −

log cot θ2 cos θ

493

= − log 2, and L(0) = L

π 2

= 0, we deduce

that the values of s(θ flat ) are given by the following limits: 1 lim s(θ ) = − log 2, and θ→0 2

lim s(θ ) = 0.

θ→ π2

Let us now evaluate s D (G flat ). Since the sum of rhombus-angles around a vertex is 2π , and since rhombus half-angles are equal to 0 or π/2, there is in G flat , around each vertex, exactly two rhombi with half-angle θ flat equal to π2 . Let us analyze the dimer model on Gnflat : the “long” edges with weight equal to ∞ (corresponding to rhombus half-angles 0) are present in a random dimer configuration with probability 1. The configuration of the long edges is thus frozen. The “short” ones with weight equal to 1 (corresponding to rhombus half-angles π/2) are present in a random dimer configuration with probability 1 1 2 + π. As noted above, around every vertex of G flat n , there are exactly two “short” edges. This implies that “short” edges form a collection of k disjoint cycles covering all vertices of the graph G flat n , for some positive integer k. Such a cycle cannot be trivial, because surrounding a face would require an infinite number of flat rhombi, but there is only a finite number of them in G flat n . Since these cycles are disjoint, their number is bounded 2 by the number of edges crossing the “boundary” of G flat n . Therefore k = O(n) = o(n ). Since all “long” edges are taken in a random dimer configuration, this implies that, for every cycle of “short” ones, edges are either all present, or all absent. The logarithm of the number of configurations for these cycles grows slower than O(n 2 ), and therefore does not contribute to the entropy. The main contribution comes from the decorations, which have two configurations each: s D (G flat ) = |V1 | log 2. The total number of “short” edges is |Vn | (two halves per vertex of G n ), and thus the number of long ones is |E n | − |Vn |. From (13), we deduce that: log 2 s D = (|E 1 | + |V1 |) 2 π log cot θ2e π − 2θe 1 L(θe ) + L − θe + + log cot θe − , 2π cos θe π 2 e∈E(G 1 )

and using (12), we deduce Theorem 12.

The free energy per fundamental domain of the critical Z -invariant Ising model, denoted f I , is defined by: 1 log Z nJ , n→∞ n 2

f I = − lim

where Z nJ is the partition function of the critical Z -invariant Ising model on the toroidal graph G n . In [Bax86], Baxter gives an explicit expression for the free energy of Z -invariant Ising models (not only critical), by transforming the graph G with star-triangle transformations to make it look like large pieces of Z2 glued together, and making use of the celebrated computation of Onsager [Ons44] on Z2 . As a corollary to Theorem 12, we obtain an alternative proof of this formula at the critical point.

494


Theorem 13 ([Bax86]). f I = −|V1 |

π θe log 2 1 − log tan θe + L(θe ) + L − θe . 2 π π 2 e∈E(G 1 )

Proof. Using the high temperature expansion, and Fisher’s correspondence, we have from Eq. (2) that: ⎛ ⎞ ⎛ ⎞n 2 Z nJ = ⎝ sinh J (θe )⎠ Znν = ⎝ sinh J (θe )⎠ Znν . e∈E(G n )

e∈E(G 1 )

As a consequence,

fI = fD −

log sinh J (θe ).

e∈E(G 1 )

Moreover, ! sinh J (θ ) = sinh log

" 1 + sin θ = cos θ

tan θ2 tan θ , 2

so that by Theorem 12: log 2 f I = −(|E 1 | + |V1 |) 2

π π − 2θe θe 1 1 log tan θe − log cot − L(θe ) + L − θe + 2π 2 2 π 2 e∈E(G 1 )

1 θe log 2 − log tan + log tan θe +|E 1 | 2 2 2 e∈E(G 1 )

π θe log 2 1 = −|V1 | − log tan θe + L(θe ) + L − θe . 2 π π 2 e∈E(G 1 )

The critical Laplacian matrix on G is defined in [Ken02] by: ⎧ ⎪ θuv if u ∼ v ⎨tan u,v = − u ∼u tan θuu if u = v ⎪ ⎩0 otherwise. The characteristic polynomial of the critical Laplacian on G, denoted P (z, w), is # (z, w), where # (z, w) is the Fourier transform of the defined by P (z, w) = det critical Laplacian . In a similar way, the characteristic polynomial of the critical dimer model on the #(z, w), where K #(z, w) Fisher graph G, denoted by P(z, w) is defined by P(z, w) = det K is the Fourier transform of the Kasteleyn matrix K of the graph G with critical weights. In Theorem 8 of [BdT08], we prove that both characteristic polynomials are equal up to a non-zero multiplicative constant. Using Theorem 12, and [Ken02], we can now determine this constant explicitly.


495

Corollary 14. The characteristic polynomials of the critical dimer model on the Fisher graph G, and of the critical Laplacian on the graph G, are related by the following explicit multiplicative constant: θe ∀(z, w) ∈ C2 , cot 2 P(z, w) = 2|V1 | − 1 P (z, w). 2 e∈E(G 1 )

Proof. Let c = 0, be the constant of proportionality given by Theorem 8 of [BdT08]: P(z, w) = c P (z, w). Taking the logarithm on both sides, and integrating with respect to (z, w) over the unit torus yields: dz dw log P (z, w) log c = −2 f D − . (14) 2 2iπ z 2iπ w T Moreover, by [Ken02] we have:

π θe dz dw 1 =2 log tan θe + L(θe ) + L − θe . log P (z, w) 2iπ z 2iπ w π π 2 T2 e∈E(G 1 )

Plugging this and Theorem 12 into (14), we obtain: θe log cot + log cot θe log c = (|E 1 | + |V1 |) log 2 + 2 e∈E(G 1 ) ⎛ ⎞ θ e cot cot θe ⎠ = log ⎝2|E 1 |+|V1 | 2 e∈E(G 1 ) ⎛ ⎞ θ e = log ⎝2|V1 | cot 2 − 1 ⎠. 2 e∈E(G 1 )

6. Proof of Theorem 5 Let us recall the setting: G is an infinite isoradial graph, and G is the corresponding Fisher graph whose edges are assigned the dimer critical weight function; K denotes the infinite Kasteleyn matrix of the critical dimer model on the graph G, and defines an operator acting on functions of the vertices of G. We now prove Theorem 5, i.e. we show that K K −1 = Id, where K −1 is given by: 1 −1 = f x (λ) f y (−λ) Expx,y (λ) log λdλ + C x,y , ∀x, y ∈ V (G), K x,y 4π 2 Cx,y where f x and f y are defined by Eq. (3), Expx,y by Eq. (6), and C x,y by Eq. 8. This section is organized as follows. In Sect. 6.1, we prove that the function f x (λ) Expx,y (λ), seen as a function of x ∈ V (G), is in the kernel of the Kasteleyn operator K . Then, in Sect. 6.2, we give the general idea of the argument, inspired from

496


[Ken02], used to prove K K −1 = Id. This motivates the delicate part of the proof. Indeed, for this argument to run through, it is not enough to have the contour of integration Cx,y to be defined as a simple closed curve containing all poles of the integrand, and avoiding the half-line dx,y ; we need it to avoid an angular sector sx,y , which contains the halfline dx,y and avoids all poles of the integrand. These angular sectors are then defined in Sect. 6.3. The delicate part of the proof, which strongly relies on the definition of the angular sectors, is given in Sect. 6.4. 6.1. Kernel of K . In this section we prove that the function f x (λ) Expx,y (λ), seen as a function of x ∈ V (G), is in the kernel of the Kasteleyn operator K . Recall that the function f x is defined in Eq. (3), and Expx,y in Eq. (6). Proposition 15. Let x, y be two vertices of G, and let x1 , x2 , x3 be the three neighbors of x in G, then for every λ ∈ C: 3

K x,xi f xi (λ) Expxi ,y (λ) = 0.

i=1

Proof. There are three cases to consider, depending on whether the vertex x is of type ‘w’, ‘z’ or ‘v’. If x = wk (x) for some k ∈ {1, . . . , d(x)}. Then the three neighbors of x are x1 = z k (x), x2 = z k+1 (x) and x3 = vk (x). Since x, x1 , x2 , x3 all belong to the same decoration, we omit the argument x. By our choice of Kasteleyn orientation, we have: K wk ,z k = −1,

K wk ,z k+1 = εwk ,z k+1 ,

K wk ,vk = 1.

By definition of the angles in R/4π Z associated to vertices of G, see (4), we have: if εwk ,z k+1 = 1 αwk αz k+1 = αwk + 2π if εwk ,z k+1 = −1. Using the definition of the function f , we deduce that: K wk ,z k+1 f z k+1 (λ) = − f wk (λ).

(15)

As a consequence, 3

K x,xi f xi (λ) Expxi ,y (λ)

i=1

$ % = K wk ,z k f z k (λ) + K wk ,z k+1 f z k+1 (λ) + K wk ,vk f vk (λ) Expx,y (λ) $ % = − f z k (λ) − f wk (λ) + f vk (λ) Expx,y (λ).

(16)

Since by definition we have f vk (λ) = f wk (λ) + f z k (λ), we deduce that (16) is equal to 0. If x = z k (x) for some k ∈ {1, . . . , d(x)}. Then the three neighbors of x are x1 = wk−1 (x), x2 = wk (x) and x3 = vk (x). By our choice of Kasteleyn orientation, we have: K z k ,wk−1 = εz k ,wk−1 ,

K z k ,wk = 1,

K z k ,vk = −1.


497

Similarly to the case where x = wk , we have: K z k ,wk−1 f wk−1 (λ) = f z k (λ). As a consequence, 3


i=1

$ % = K z k ,wk−1 f wk−1 (λ) + K z k ,wk f wk (λ) + K z k ,vk f vk (λ) Expx,y (λ) $ % = f z k (λ) + f wk (λ) − f vk (λ) Expx,y (λ).

(17)

Again, we deduce that (17) is equal to 0. If x = vk (x) for some k ∈ {1, . . . , d(x)}. Then the three neighbors of x are x1 = z k (x), x2 = wk (x) and x3 = v (x ), where and x are such that vk (x) ∼ v (x ). Although all vertices do not belong to the same decoration, we omit the arguments x and x , knowing that the index k (resp. ) refers to the vertex x (resp. x ). By our choice of Kasteleyn orientation, we have: αwk − αz k . K vk ,z k = 1, K vk ,wk = −1, K vk ,v = εvk ,v cot 4 Using the fact that Expx ,y (λ) = Expx ,x (λ) Expx,y (λ), we deduce: 3


i=1

% $ = K vk ,z k f z k (λ) + K vk ,wk f wk (λ) + K vk ,v Expx ,x (λ) f v (λ) Expx,y (λ)

αwk − αz k = f z k (λ) − f wk (λ) + εvk ,v cot Expx ,x (λ) f v (λ) Expx,y (λ). 4

Moreover, by definition of the angles in R/4π Z associated to vertices of G, see (5), we have: αw = αwk − εvk ,v π, αz = αz k − εvk ,v π, and using the definition (3) of the function f , we deduce: αw

αz

αwk αz k ei 2 ei 2 ei 2 ei 2 f v (λ) = − = iεvk ,v

− , eiαw − λ eiαz − λ eiαwk + λ eiαzk + λ so that: K vk ,v Expx ,x (λ) f v (λ) αwk αz k ei 2 αwk − αz k (eiαwk + λ)(eiαzk + λ) ei 2 = i cot − 4 (eiαwk − λ)(eiαzk − λ) eiαwk + λ eiαzk + λ = f wk (λ) − f z k (λ). (18)

498


sx,y dx,y x,y

−1 Fig. 9. Definition of the contour of integration Cx,y of the integral term of K x,y . The poles of the integrand are the thick points

As a consequence, 3

$ % K x,xi f xi (λ) Expxi ,y (λ) = f z k (λ) − f wk (λ) + f wk (λ) − f z k (λ) Expx,y (λ) = 0.

i=1

(19) 6.2. General idea of the argument. The general argument used to prove Theorem 5, i.e. K K −1 = Id, is inspired from [Ken02], where Kenyon computes a local explicit expression for the inverse of the Kasteleyn matrix of the critical dimer model on a bipartite, isoradial graph. It cannot be applied as such to our case of the critical dimer model on the Fisher graph G of G, which is not isoradial, but it is nevertheless useful to sketch the main ideas. Let x, y be two vertices of G, and let us assume that the contour of integration Cx,y of −1 , is defined to be a simple closed curve oriented counterclockthe integral term of K x,y wise, containing all poles of the integrand and avoiding an angular sector sx,y , which contains the half-line dx,y , see Fig. 9. The argument runs as follows. Let x1 , x2 , x3 be the three neighbors of x in G. When &3 sxi ,y is non x = y, the goal is to show that the intersection of the three sectors i=1 empty. Then, the three contours Cxi ,y can be continuously deformed to a common contour C without meeting any pole, and 3 dλ K x,xi f xi (λ) f y (−λ) Expxi ,y (λ) log λ (2π )2 Cxi ,y i=1 3 dλ = K x,xi f xi (λ) Expxi ,y (λ) f y (−λ) log λ = 0, (20) (2π )2 C i=1

since by Proposition 15, the sum in brackets is zero. Note that in [Ken02], Kenyon has a result similar to Proposition 15, giving functions that are in the kernel of the Kasteleyn matrix of the dimer model on a bipartite, isoradial graph. When x = y, the goal is to show that the intersection of the three sectors is empty, and explicitly compute: 3 dλ K x,xi f xi (λ) f x (−λ) Expxi ,x (λ) log λ = 1. (21) (2π )2 Cxi ,x i=1


499

In our case things turn out to be more complicated, since we cannot define angular sectors sx,y so that (20) and (21) hold as such. We define them in such a way that when x and y do not belong to the same triangle of a decoration x, the intersection of the three sectors &3 −1 ) x,y to be equal to i=1 sxi ,y is non-empty, and (20) holds. Then, in order for (K K δx,y in all other cases, we need to have an additional constant C x,y in the expression of −1 . the inverse K x,y Remark 16. Let us mention that in [Ken02], angular sectors are not constructed explicitly neither for the inverse Kasteleyn operator nor for the Green function of the Laplacian on G, since geometric considerations suffice. For the latter, the construction has been carried out in [BMS05], by interpreting the graph G, if the train-tracks of G have only d possible directions, as the projection of a monotonic surface in Zd on the plane. The sectors are then defined as the projection of octants, that are sewed together to get a branch covering of the plane. The sectors appearing here are defined using another approach. Although the two constructions are related,3 our geometric situation is complicated by the presence of the decorations in G, and there is no natural notion of branched covering in this case. Moreover, we do not restrict ourselves on the number of possible directions for the train-tracks. Since our construction for the generic case also applies to [Ken02], it is thus slightly more general than [BMS05]. 6.3. Definition of the angular sectors sx,y . Let x, y be two vertices of G. In this section, we construct the angular sector sx,y containing the half-line dx,y and avoiding all poles −1 . In order to do this, we first recall some facts about isoradial of the integrand of K x,y −1 in an edge-path γ graphs, then we encode the poles of the integrand of K x,y x,y of the diamond graph G . Using this path γx,y , we define the angular sector sx,y . 6.3.1. Minimal paths. Let G be an infinite isoradial graph, and G be the associated diamond graph, and let x, y be two vertices of G. We now define a minimal path of G from x to y. This definition uses the notion of train-track introduced in Sect. 5.2. We say that a train-track of G separates x from y, if when deleting it, x and y are in two distinct connected components. Now, observe that each edge of G belongs to a unique train-track whose direction is given by that edge, it is called the train-track corresponding to the edge. An edge-path γ of G , from x to y, is called minimal, if all train-tracks corresponding to edges of γ separate x from y, and all those train-tracks are distinct. In general there is not uniqueness of the minimal path, but all minimal paths consist of the same steps, taken in a different order. −1 . Let x, y be two vertices of G, and let 6.3.2. Encoding the poles of the integrand of K x,y x, y be the corresponding vertices of G. In this section, we define an edge-path γx,y of the diamond graph G encoding the poles of the integrand f x (λ) f y (−λ) Expx,y (λ) log λ −1 . of K x,y By definition of the exponential function and of a minimal path, the poles of Expx,y (λ) are encoded in the steps of a minimal path of G , oriented from y to x. It would be natural −1 , as representing the steps of a path of G passing to see the poles of the integrand of K x,y 3 In [BMS05], the sectors are a priori defined, then the vertices are partitioned in classes sharing the same sector. Here we construct for each vertex in G its own sector.

500


through y and x, obtained by adding the steps corresponding to the poles of f x (λ) and f y (−λ). However the situation is a bit trickier due to possible cancellations of these poles with factors appearing in the numerator of Expx,y (λ). This is taken into account in the following way. By definition, the function f x (λ) has either 1 or 2 poles, denoted by {eiα j } j , where j j = 1 or j = {1, 2}. Let Tx = {Tx } j be the set of corresponding train-tracks. Similarly,

j

define {−eiα j } j to be the set of poles of f y (−λ), and Ty = {Ty } j to be the corresponding train-tracks. Let us start from a minimal path γx,y from y to x. For each j, we do the following j

procedure: if Ty separates y from x, then the pole −eiα j is cancelled by the exponential, and we leave γx,y unchanged. If not, this pole remains, and we extend γx,y by adding

the step −eiα j at the beginning of γx,y . The path obtained is still a path in G . Denote by yˆ the new starting point of γx,y at distance at most 2 of y. When dealing with a pole of f x (λ), one needs to be careful since, even when the corresponding train-track separates y from x, the exponential function might not cancel the pole if it has already canceled the same pole of f y (−λ). This happens when Tx and Ty have a common train-track. The procedure to extend γx,y runs as follows: for each j, j j if Tx separates y from x, and Tx ∈ Ty , then the pole eiα j is canceled by the exponential function, and we leave γx,y unchanged. If not, this pole remains, and we extend γx,y by attaching the step eiα j at the end of γx,y . The extended path γx,y obtained in this way is still a path in G , starting from yˆ . Denote by xˆ its ending point, which is at distance at most 2 from x. Note that the way to extend γx,y may not be unique, but the set of steps are always the same, only the order can differ. 6.3.3. Obtaining an angular sector sx,y from γx,y . Now that we have encoded all the poles of the integrand in a path γx,y , we want to construct the sector sx,y avoided by the −1 . contour Cx,y on which the integral is taken in the definition of K x,y Let us first suppose that γx,y is a minimal path joining yˆ and xˆ in G . Lemma 17 below describes the repartition of the steps of γx,y on the unit circle. Since the result holds for all minimal paths in G , we introduce the following more general notations. Let u, v be two vertices of G , and let γ = {u = u0 , u1 , . . . , un = v} be a minimal path from u to v. Denote by eiθ j the unit complex number representing the edge from u j−1 to u j , with θ j+1 − θ j ∈ (−π, π ). The angles θ j can be defined in R as follows. Fix some initial value for θ0 , and let θ j = θ j−1 + (θ j − θ j−1 ). Lemma 17. Let γ be a minimal path as above. Then, there exists an angular sector of size greater than π containing none of the steps eiθ j of γ . In other words, the angles θ j defined in R satisfy: ' ' ∀ j, k, 'θ j − θk ' < π. Proof. Let us begin with a preliminary remark. Take T1 and T2 two intersecting traintracks separating u from v, and orient the edges parallel to the directions of these traintracks. The two train-tracks separate the plane into four quadrants, each containing a vertex of the rhombus R at the intersection of T1 and T2 . Then the edges of R must have the following orientation: the two edges attached to the vertex in the same component as u (resp. as v) are outgoing (resp. incoming). See Fig. 10 for an illustration.


501

T2

T1

γ

v

u

Fig. 10. Orientation of a rhombus at the intersection of two train-tracks separating u from v

Tj

Dj

uj uj−1

Dk

uk−1

u uk

Tk

v Fig. 11. Two train tracks T j and Tk separating u from v, with |θ j − θk | > π

Now suppose that the conclusion of the lemma is not true. Since the path is minimal, all train-tracks are distinct, so that |θ j − θk | = π . Thus, there exists two edges in γ with angles |θ j − θk | > π . Without loss of generality, we can suppose that j < k and θ j − θk > π . The situation is represented in Fig. 11. Let D j (resp. Dk ) be the straight line defined by the vector eiθ j (resp. eiθk ). Then, since the angle θ j − θk is larger than π , these two lines must cross below γ . Consider the quadrant defined by D j and Dk , containing the part of γ from u j−1 to uk . Then, the part below γ contains both T j and Tk . These two train-tracks cannot exit above γ since each of them crosses γ exactly once, by definition of minimality. So they must exit below γ . Suppose that they do not intersect, then at least one of them, say T j , must intersect the line D j again, but this would imply having negative angles in the rhombi defining this train-track, thus yielding a contradiction. As a consequence, T j and Tk must intersect. Let R be the rhombus at the intersection. Now have a look at the orientation of the edges of R on Fig. 12. The two edges attached to the vertex in the same quadrant as y are not pointing outward, as it should be from Fig. 10. Therefore, the hypothesis we started from is false, and thus ∀ j, k |θ j − θk | < π. In the extension procedure of γx,y , we may have added steps breaking the minimality property: indeed it is possible that a single train-track corresponds to two different steps

502


Tj

u Tk

v Fig. 12. Orientation of the edges of the rhombus at the intersection between T j and Tk . Compare with the correct orientation on Fig. 10

uj γ

u

uk−1

T

e’

e

uj−1

uk

v

Fig. 13. A quasi-minimal path γ

of γx,y . This is the case when a train-track is common to both Tx and Ty . Note that since each of Tx , Ty contains at most two train-tracks, this might occur for at most two train-tracks. If this occurs for exactly one train-track T , we say that the path γx,y is quasi-minimal. Lemma 18 below is the analog, for quasi-minimal paths, of Lemma 17 describing the repartition of its steps on the unit circle. Let γ be a general quasi-minimal path from u to v, with the notations as above. Denote by T the unique train-track corresponding to two different steps e, e of γ . Then T crosses γ twice, i.e. an even number of times, implying that T does not separate u from v. Moreover, if ±eiθ denotes the direction of T , then one of e or e is ±eiθ , and the other is ∓eiθ , see Fig. 13. Lemma 18. Let γ be a quasi-minimal path from u to v, and let T denote the only traintrack corresponding to the steps e, e of γ . Then there exists an angular sector of size exactly π , delimited by the steps e, e containing none of the steps eiθ j of γ in its interior. Proof. Denote e = (u j−1 , u j ), respectively e = (uk−1 , uk ), with j < k. Suppose, as in the proof of the previous lemma, that there are two steps eiθ and eiθm such that |θ − θm | ≥ π , with < m. Then these two steps must be e and e . The other pairs of edges are excluded, by Lemma 17 applied to the pieces of γ from u to uk−1 and from u j to v, that are minimal, and to the minimal path from u to v through u j−1 and uk , using the same steps as γ , except e and e . Therefore, θ j = θk + π , and the other angles are all in one of the two sectors of angle π delimited by e and e . As a consequence of the two previous lemmas, when the path γ is minimal or quasiminimal, the unit circle deprived from the steps of γ , S 1 \{eiθ1 , . . . , eiθn }, has a connected component Sγ of size at least π . Let us now define the angular sector sx,y −1 . avoided by the contour Cx,y on which the integral is taken in the definition of K x,y


e iαwk −e i αwk

swk ,wk

e iαzk+1 −e iαwk

szk+1,wk

503

e i αwk −e i αzk+1

swk,zk+1

ei αzk+1 −e iαz k+1

szk+1, zk+1

Fig. 14. Definition of the angular sector sx,y (dashed line) in Case 1: from left to right, (x, y) = (wk , wk ), (z k+1 , wk ), (wk , z k+1 ), (z k+1 , z k+1 )

Definition 6.1 (Generic definition of sx,y ). Let x and y be two vertices of G. Let γx,y be the path of G constructed as above. When the connected component Sγx,y exists and is unique, the angular sector sx,y is defined to be the open cone {r eiθ |r > 0, θ ∈ Sγx,y }. Remark 19. 1. The definition of the sector sx,y depends on γx,y only through the set of steps, but is independent of the order of these steps. The sector is thus well defined given only the poles of the integrand. 2. The angular sector sx,y always contains the half-line dx,y starting from 0, in the direction from xˆ to yˆ . Thus, we can define the contour of integration Cx,y to be a closed contour oriented counterclockwise and avoiding the closed half-line dx,y . This definition has to be adapted in two particular cases, for different reasons: Case 1. There are two components of size π in S 1 deprived from the steps of γx,y . This occurs when the path γx,y has two steps, and crosses one train-track back and forth. This can only happen in the following four cases. Since all vertices involved belong to the same decoration, we omit the argument x in the notations. Recall that eiαwk = eiαzk+1 : ⎧ ⎪ (wk , z k+1 ) then, γx,y is the path −eiαzk+1 , eiαwk , ⎪ ⎪ ⎨(z , w ) then γ is the path −eiαwk , eiαzk+1 , k+1 k x,y (x, y) = iαwk ⎪ , w ) then γ , eiαwk , (w k k x,y is the path −e ⎪ ⎪ ⎩(z , z ) then γ is the path −eiαzk+1 , eiαzk+1 . k+1 k+1 x,y In these four cases, we set the conventions given by Fig. 14 for the angular sector sx,y . These are natural in the light of the proof of Theorem 5, see Sect. 6.4. Case 2. When Card{Tx ∩ Ty } = 2, the path γx,y does not fit in any of the two categories of paths above. This situation occurs when x and y are equal or neighbors in G, and both of type ‘v’. That is (x, y) = (vk (x), vk (x)), or (x, y) = (vk (x), v (x )), with x ∼ x in G, and k and such that vk (x) ∼ v (x ). Suppose first that (x, y) = (vk (x), v (x )). Then f vk (x) (λ) and f v (x ) (−λ) have the same poles eiαwk (x) , eiαzk (x) . The exponential function Expx,x (λ), cancels one of the pair of poles, and adds two new ones −eiαwk (x) , −eiαzk (x) . If now (x, y) = (vk (x), vk (x)). Then f vk (x) (λ) has poles eiαwk (x) , eiαzk (x) , and f vk (x) (−λ) has opposite poles. Moreover, Expx,y (λ) = 1, so that it cancels no pole. In both cases the path γx,y follows the boundary of the rhombus associated to the edge xx in G. We set the conventions given by Fig. 15 below for the angular sector sx,y . These are natural in the light of the proof of Theorem 5, see Sect. 6.4.

504


x’

e iαwk

vl

−e i α z k

vk wk

e iαzk −eiα wk

e iαwk

e iαzk

−e i α zk

−eiα wk

zk x

svk ,vk

svk ,vl

Fig. 15. Left: a piece of the graph G for the situations described in Case 2. Center: angular sector svk (x),vk (x) . Right: angular sector svk (x),v (x )

6.4. Proof of Theorem 5. In this section, we prove Theorem 5, i.e. K K −1 = Id. Let I and C be the infinite matrices whose coefficients are the integral and constant part of K −1 respectively: 1 f x (λ) f y (−λ) Expx,y (λ) log λdλ, I x,y = (2π )2 Cx,y C x,y = C x,y . Our goal is to show that (K K −1 )x,y = (K I )x,y + (K C)x,y = δx,y . Let x1 , x2 , x3 be the three neighbors of x in G. Then, (K I )x,y =

3 i=1

(K C)x,y =

3

Cxi ,y

K x,xi f xi (λ) f y (−λ) Expxi ,y (λ) log λ

dλ , (2π )2 (22)

K x,xi C xi ,y .

i=1

In Proposition 22 below, we handle the part (K I )x,y : we prove that as soon as x and &3 y do not belong to the same triangle of a decoration, i=1 sxi ,y = ∅, so that by the general argument of Sect. 6.2, (K I )x,y = 0; when x and y belong to the same triangle of a decoration, we explicitly compute (K I )x,y . Then in Lemma 23, we handle the part (K C)x,y : we show that the constants C x,y are defined so that (K K −1 )x,y = δx,y . In proving Proposition 22, we have to take into account the fact that x can be of three types, ‘w’,‘z’ or ‘v’, i.e. x = wk (x), z k (x) or vk (x) for some k ∈ {1, . . . , d(x)}. The next proposition gives relations between these three cases for K I , so that although it might seem technical at first, it is in fact very useful. Indeed, it avoids lengthy repetitions in the proof of Proposition 22. Note that whenever no confusion occurs, we omit the argument x. Proposition 20. For every vertex y of G, the quantities (K I )wk ,y , (K I )z k ,y and (K I )vk ,y satisfy the following: 1. (K I )wk ,y = −(K I )z k ,y =

−

Cz k ,y

+

Cvk ,y

f z k (λ)+ −

× f y (−λ) Expx,y (λ) log λ

dλ . (2π )2

Cwk ,y

+

Cvk ,y

f wk (λ)


2. (K I )vk ,y =

Cz k ,y

−

Cvk ,y

f z k (λ) + −


505

Cwk ,y

+

Cvk ,y

f wk (λ)

dλ . (2π )2

Proof of Proposition 20. The argument extensively uses relations between the functions Expx,y (λ) f x (λ) for neighboring vertices x. In Eqs. (16), (17), (18), (19) of the proof of Proposition 15, we explicitly computed 3

K x,xi f xi (λ) Expxi ,y (λ),

i=1

for x = wk (x), z k (x) and vk (x), respectively. Using this, Eq. (22), and the fact that f vk (λ) = f wk (λ) + f z k (λ), we obtain the following. If x = wk (x), then the three neighbors of x are x1 = z k (x), x2 = z k+1 (x), x3 = vk (x), and by (16) we have: (K I )wk ,y = − f z k (λ) − f wk (λ) + [ f wk (λ) + f z k (λ)] Cz k ,y

Cz k+1 ,y

Cvk ,y

dλ × f y (−λ) Expx,y (λ) log λ . (2π )2

(23)

If x = z k (x), then the three neighbors of x are x1 = wk−1 (x), x2 = wk (x), x3 = vk (x), and by (17) we have: (K I )z k ,y = f z k (λ) + f wk (λ) − [ f wk (λ) + f z k (λ)] Cwk−1 ,y

Cwk ,y

Cvk ,y


dλ . (2π )2

(24)

If x = vk (x), then the three neighbors of x are x1 = z k (x), x2 = wk (x), x3 = v (x ), where and x are such that vk (x) ∼ v (x ) in G. Recalling that the index refers to the decoration x , we also omit the arguments x and x . Using (18) and (19), we have: (K I )vk ,y =

Cz k ,y

f z k (λ) −

Cwk ,y

f wk (λ) +


dλ . (2π )2

Cv ,y

[ f wk (λ) − f z k (λ)] (25)

As a consequence of (23), (24), (25), Proposition 20 is proved, if we show that, for every vertex y of G, Cz k+1 ,y = Cwk ,y , and Cvk ,y = Cv ,y , which is equivalent to proving: sz k+1 ,y = swk ,y , svk ,y = sv ,y .

(26) (27)

Recall that the construction of the angular sector sx,y given in Sect. 6.3, relies on the −1 . path γx,y encoding the poles of the integrand f x (λ) f y (−λ) Expx,y (λ) log(λ) of K x,y

506


Recall also that the generic definition of the sector (Definition 6.1) has to be adapted in two specific situations (Case 1, Case 2). These cases need to be treated separately here as well. Generic case. By Remark 19, in the generic case, the definition of the sector sx,y only depends on the set of poles, hence it suffices to show that the left- and right-hand sides of (26), (27) have the same set of poles. The vertices z k+1 and wk belong to the same decoration x, so that the exponential function has the same poles in both cases. Moreover, the functions f z k+1 (λ) and f wk (λ) have the same pole eiαzk+1 = eiαwk , thus proving (26). Let us now prove (27). We have: f v (λ) f y (−λ) Expx ,y (λ) = f v (λ) Expx ,x (λ) f y (−λ) Expx,y (λ) 1 = ( f wk (λ) − f z k (λ)) f y (−λ) Expx,y (λ) (by Eq. (18)). K vk ,v

Moreover, by definition of f vk (λ): f vk (λ) f y (−λ) Expx,y (λ) = ( f wk (λ) + f z k (λ)) f y (−λ) Expx,y (λ), so that the left and right-hand sides of (27) have the same set of poles. Case 1. This case only occurs when y = wk or y = z k+1 in Eq. (26). Referring to Fig. 14, we see that by definition: sz k+1 ,z k+1 = swk ,z k+1 ,

sz k+1 ,wk = swk ,wk .

and

Case 2. This case only occurs when y = vk or y = v in Eq. (27). Referring to Fig. 15, we see that by definition: svk ,vk = sv ,vk ,

and

svk ,v = sv ,v .

A direct consequence of Proposition 20 and the general argument of Sect. 6.2 is the following corollary, which greatly reduces the number of computations for (K I )x,y . Corollary 21. 1. If swk ,y ∩ svk ,y = ∅, then (K I )wk ,y = −(K I )z k ,y = −(K I )vk ,y dλ = − + . f z k (λ) f y (−λ) Expx,y (λ) log λ (2π )2 Cz k ,y Cvk ,y 2. If sz k ,y ∩ svk ,y = ∅, then (K I )wk ,y = −(K I )z k ,y = (K I )vk ,y =

−

Cwk ,y

+

Cvk ,y

f wk (λ)

f y (−λ) Expx,y (λ) log λ

dλ . (2π )2

3. If sz k ,y ∩ swk ,y ∩ svk ,y = ∅, then (K I )wk ,y = (K I )z k ,y = (K I )vk ,y = 0. We now state Proposition 22 computing the matrix product K I of Eq. (22). Whenever no confusion occurs, we drop the argument x in wk (x), z k (x), vk (x).


507

Proposition 22. For all vertices x and y of G, we have:

(K I )x,y =

⎧ 1 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨− 1 2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩0

if (x, y) = (wk , wk ), (z k , z k ), (vk , wk ), (vk , z k ) if (x, y) = (z k , wk ), (wk , z k ) if (x, y) = (vk , vk ) otherwise.

Proof. Let x and y be two vertices of G. Then x = wk (x), z k (x) or vk (x) for some k ∈ {1, . . . , d(x)}. In order to use Corollary 21, we need to understand the intersection properties of the three angular sectors swk ,y , sz k ,y , svk ,y . The proof is divided in three cases, depending on the definition of these sectors: the generic one (Definition 6.1), and the two specific situtations (Case 1, Case 2). Generic case. Suppose that all three sectors are constructed according to the generic definition. This excludes the case where y ∈ {wk , z k+1 , z k , wk−1 , vk , v }. Since f vk (λ) = f wk (λ) + f z k (λ), the function f vk (λ) f y (−λ) Expx,y (λ) contains all poles of f wk (λ) f y (−λ) Expx,y (λ) and f z k (λ) f y (−λ) Expx,y (λ). Using Definition 6.1, and Remark 19, this implies that svk ,y ⊂ swk ,y , and svk ,y ⊂ sz k ,y . As a consequence, swk ,y ∩ sz k ,y ∩ svk ,y = svk ,y . Moreover svk ,y is an angular sector of size at least π , so that using Point 3 of Corollary 21, we deduce that (K I )x,y = 0. Case 1. This case only occurs when y = wk , z k+1 , z k , or wk−1 . In each of these cases, one of the pairs {y, z k }, {y, wk } requires the use of the convention depicted in Fig. 14 to construct the corresponding sector. For these four cases, the angular sectors sz k ,y , swk ,y , svk ,y are drawn on Fig. 16. From c) and d) we see that when y = z k+1 or y = wk−1 , the intersection sz k ,y ∩ swk ,y ∩ svk ,y = ∅. Thus, using Point 3 of Corollary 21, we deduce that (K I )x,y = 0. In the remaining two cases we do explicit computations. • Computations for y = wk . In this case, see Fig. 16 a), we have sz k ,wk ∩ svk ,wk = ∅. Hence, by Point 2 of Corollary 21, we know that: (K I )wk ,wk = −(K I )z k ,wk = (K I )vk ,wk = − + f wk (λ) Cwk ,wk

Cvk ,wk

f wk (−λ) log λ

dλ . (2π )2

Using the definition of f wk (λ), and denoting by C a generic simple closed curve oriented counterclockwise, containing all poles of the integrand, and avoiding a half-line starting from 0, yields: 1 eiαwk 1 f (λ) f (−λ) log λdλ = − log λdλ w w k k (2π )2 C (2π )2 C (λ − eiαwk )(λ + eiαwk ) ) i ( logC (eiαwk ) − logC (−eiαwk ) . (28) =− 4π

508


eiαwk eiαzk

(a)

−e iαwk

(b)

swk,wk eiαzk

eiαzk −e iαzk+1

(d)

−e iαwk−1

vk wk

swk,zk

svk,zk

eiαwk

eiαwk eiαzk

−eiαzk+1

szk,zk+1 eiαzk

eiαzk eiαwk −e iαzk

−e iαzk

szk,zk

(c)

svk,wk

eiαwk

−e iαzk

zk

y= wk

−e iαwk

−e iαwk

szk,wk

vk

eiαzk

eiαwk

−e iαzk+1

swk,zk+1

y= zk

vk zk

wk y= zk+1

svk,zk+1 eiαzk eiαwk −e iαwk−1

eiαwk −e iαwk−1

vk wk

zk y=wk−1

szk,wk−1

swk,wk−1

svk,wk−1

Fig. 16. Angular sectors sz k ,y , swk ,y , svk ,y in Case 1

As a consequence, see Fig. 16 a) (K I )wk ,wk = −(K I )z k ,wk = (K I )vk ,wk i ( − logCw ,w (eiαwk ) + logCw ,w (−eiαwk ) =− k k k k 4π ) + logCv

k ,wk

(eiαwk ) − logCv

k ,wk

(−eiαwk )

1 i [−iαwk + i(αwk + π ) + iαwk − i(αwk − π )] = . 4π 2 • Computations for y = z k . In this case, see Fig. 16 b), we have swk ,z k ∩ svk ,z k = ∅. Hence, by Point 1 of Corollary 21, we know that: =−

(K I )wk ,z k = −(K I )z k ,z k = −(K I )vk ,z k = − + f z k (λ) Cz k ,z k

Cvk ,z k

f z k (−λ) log λ

dλ . (2π )2


(a)

e iα k

e iαwk −e iαzk −eiαwk

−e iαwk −e iαzk

sz k, v k

sw ,v

k k

(b)

e iαwk −e iαzk −e iαwk

e iαzk −eiαwk −e iαzk

sw ,vl

sz k,vl

k

509

e iαwk

e iαzk

−e iαzk

−e iαwk

svk, v k e iαwk

eiαzk −e iαwk

−e iαzk

svk,vl

Fig. 17. Angular sectors swk ,y , sz k ,y , svk ,y in Case 2

Using the definition of f z k (λ), yields f z k (λ) f z k (−λ) = − using Fig. 16 b), we obtain:

eiαz k , (λ − eiαz k )(λ + eiαz k )

so that

(K I )wk ,z k = −(K I )z k ,z k = −(K I )vk ,z k i ( − logCz ,z (eiαzk ) + logCz ,z (−eiαzk ) =− k k k k 4π ) + logCv ,z (eiαzk ) − logCv ,z (−eiαzk ) k k

k k

1 i = − [−iαz k + i(αz k − π ) + iαz k − i(αz k + π )] = − . 4π 2 Case 2. This case can only occur when y = vk (x) or y = v (x ), with vk (x) ∼ v (x ). In these two cases, the angular sectors swk ,y , sz k ,y , svk ,y are drawn on Fig. 17. From b) we see that (K I )wk ,v = (K I )z k ,v = (K I )vk ,v = 0. For y = vk (x), we do explicit computations. • Computations for y = vk . We use 1 and 2 of Proposition 20. Let us first compute: dλ + . f z k (λ) f vk (−λ) log λ − (2π )2 Cz k ,vk Cvk ,vk Using the definition of f z k (λ), f vk (λ), and denoting by C a generic contour, we have: 1 f z (λ) f vk (−λ) log λdλ (2π )2 C k ⎛ ⎞ αz +αw iαz k i k2 k e 1 e ⎝ ⎠ log λ dλ =− 2 − 4π C (λ − eiαzk )(λ + eiαzk ) (λ − eiαzk )(λ + eiαwk ) ⎤ ⎡ iαz k iαwk i ⎣ (e ) − log (−e ) log C ⎦. (29) α −αC =− logC (eiαzk ) − logC (−eiαzk ) − wk zk 4π cos 2

510


As a consequence: −

Cz k ,vk

+

f z k (λ) f vk (−λ) log λ

Cvk ,vk

dλ (2π )2

⎤ 1 ⎣ π − (αwk − αz k ) ⎦ π − (αwk − αz k ) α −α − π − α −α = −π + wk zk wk zk 4π cos cos 2 2 ⎡

1 =− . 2 Let us now compute,

−

+ Cwk ,vk

Cvk ,vk

f wk (λ) f vk (−λ) log λ

dλ . (2π )2

Exchanging z k with wk in (29) yields: 1 f w (λ) f vk (−λ) log λdλ (2π )2 C k ⎤ ⎡ iαwk iαz k i ⎣ (e ) − log (−e ) log C ⎦. (30) α −αC =− logC (eiαwk ) − logC (−eiαwk ) − wk zk 4π cos 2 As a consequence: −

Cwk ,vk

+

Cvk ,vk

f wk (λ) f vk (−λ) log λ

dλ (2π )2

⎤ 1 ⎣ π − (αwk − αz k ) ⎦ π − (αwk − αz k ) α −α + π + α −α = π− wk zk wk zk 4π cos cos 2 2 ⎡

=

1 . 2

Thus, by Proposition 20, we deduce: (K I )wk ,vk = 0, (K I )z k ,vk = 0, (K I )vk ,vk = 1. By Proposition 22, proving that K K −1 = Id, amounts to proving the following lemma for K C. Lemma 23. For all vertices x and y of G, we have: ⎧ 1 ⎪ if (x, y) = (wk , wk ), (z k , z k ), (wk , z k ), (z k , wk ) ⎨2 1 (K C)x,y = − 2 if (x, y) = (vk , wk ), (vk , z k ) ⎪ ⎩0 otherwise.


511

Proof. By definition of C, we have C x,y = 0, as soon as x or y is of type ‘v’. As a consequence, using Eq. (22) and our choice of Kasteleyn orientation, we deduce: (K C)wk ,y = −C z k ,y + εwk ,z k+1 C z k+1 ,y , (K C)z k ,y = εz k ,wk−1 Cwk−1 ,y + Cwk ,y , (K C)vk ,y = C z k ,y − Cwk ,y .

(31)

Let us first prove that for every vertex y of G, we have εwk ,z k+1 C z k+1 ,y = Cwk ,y , or equivalently: C z k+1 ,y = εwk ,z k+1 Cwk ,y .

(32)

When y belongs to a different decoration than x, then both sides of (32) are equal to 0. Let us thus suppose that y is in the same decoration as x. If y = z k+1 , then by definition: C z k+1 ,y =

1 (−1)n(z k+1 ,y) = (−1)n(z k+1 ,wk ) (−1)n(wk ,y) = εwk ,z k+1 Cwk ,y , 4 4

so that (32) holds. If y = z k+1 , then by definition, C z k+1 ,z k+1 = − 41 . Moreover, 1 (−1)n(wk ,z k+1 ) εz ,w = εwk ,z k+1 k+1 k = − , 4 4 4 so that (32) also holds in this case. As a consequence, Eq. (31) becomes: εwk ,z k+1 Cwk ,z k+1 = εwk ,z k+1

(K C)wk ,y = (K C)z k ,y = −(K C)vk ,y = −C z k ,y + Cwk ,y . Let us now end the proof of Lemma 23. If y does not belong to the same decoration as x, or if y belongs to the same decoration as x and is of type ‘v’, then Cwk ,y = C z k ,y = 0, so that: (K C)wk ,y = (K C)z k ,y = (K C)vk ,y = 0. If y belongs to the same decoration as x, but not to the same triangle of the decoration, then 1 (−1)n(wk ,y) = (−1)n(wk ,z k ) (−1)n(z k ,y) = εz k ,wk C z k ,y = C z k ,y , 4 4 since by our choice of Kasteleyn orientation, εz k ,wk = 1. Thus, Cwk ,y =

(K C)wk ,y = (K C)z k ,y = (K C)vk ,y = 0. If y = wk , then by definition Cwk ,wk = 41 , and C z k ,wk = −Cwk ,z k = − 41 , thus: (K C)wk ,wk = (K C)z k ,wk = −(K C)vk ,wk =

1 . 2

Finally, if y = z k , then by definition Cwk ,z k = 41 , and C z k ,z k = − 41 , so that: (K C)wk ,z k = (K C)z k ,z k = −(K C)vk ,wk =

1 . 2

Acknowledgements. We would like to thank Richard Kenyon for asking the questions solved in this paper.

512


Fig. 18. A piece of the Fisher graph G near a rhombus with half-angle θ , adjacent to the decorations of x and y. The sides of the rhombus are represented by the unit vectors eiα and eiβ , with β − α = 2θ

A. Probability of Occurrence of Single Edges Let us compute the probability of occurrence of single edges in dimer configurations of the Fisher graph G chosen with respect to the Gibbs measure P of Theorem 9. Every edge of G is of the form wk z k , wk z k+1 , wk vk or vk v as represented on Fig. 18. The vertices z k , wk , z k+1 and vk belong to the decoration of x, and v belongs to that of y. The edge of G joining x and y is the diagonal of a rhombus with half-angle θ . In order to simplify notations, let us write αz k = α,

αwk = β = α + 2θ.

By Theorem 9, we know that the probability of an edge e = uv of G is given by P(e) = K u,v Pf

−1 K u,u −1 K u,v

−1 K v,u

−1 K v,v

−1 , = K u,v K v,u

−1 of the inverse Kasteleyn matrix is given by Theorem 5. where the coefficient K u,v

Probability of the edge wk z k . The vertices wk and z k belong to the same decoration x, so that there is no contribution from the exponential function, since Expx,x = 1. By our choice of Kasteleyn orientation, we have K z k ,wk = 1. Using Formula (7) and the definition of the function f , yields: P(wk z k ) =

K z k ,wk K w−1k ,z k

1 = f w (λ) f z k (−λ) log(λ)dλ + Cwk ,z k 4π 2 C1 k iβ iα e2 1 1 e2 log(λ)dλ + , = − iβ 2 iα 4π C1 e − λ e + λ 4


e iβ

513

e iβ

−e iα −e iβ

−1 Fig. 19. Left: the contour C1 = Cwk ,z k involved in the integral term of K w . Right: the contour C2 = k ,z k Cz k+1 ,wk involved in the integral term of K z−1 . In both cases, the arrows represent the poles of the k+1 ,wk integrand, and the shaded zone is the angular sector avoided by the contour

where C1 = Cwk ,z k is the contour defined according to Case 1 of Sect. 6.3.3, represented on Fig. 19. The integral is evaluated by Cauchy’s theorem: α+β 1 i i α+β logC1 (eiβ ) − logC1 (−eiα ) ei 2 log(λ) dλ = e 2 4π 2 C1 (λ − eiβ )(λ + eiα ) 2π eiα + eiβ =

i

4π cos( β−α 2 ) π − 2θ . = 4π cos θ Therefore, P(wk z k ) =

1 4

+

(iβ − i(α + π ))

π −2θ 4π cos θ .

Probability of the edge wk z k+1 . Since the vertices wk and z k+1 belong to the same decoration, there is no contribution from the exponential function. Moreover, by Eq. (15), we know that K wk ,z k+1 f z k+1 (λ) = − f wk (λ), and by Eq. (32), we have K wk ,z k+1 C z k+1 ,wk = Cwk ,wk = 41 . Therefore, P(wk z k+1 ) = K wk ,z k+1 K z−1 k+1 ,wk 1 = K wk ,z k+1 f z k+1 (λ) f wk (−λ) log(λ)dλ + K wk ,z k+1 C z k+1 ,wk 4π 2 C2 1 1 =− 2 f wk (λ) f wk (−λ) log(λ)dλ + . 4π C2 4 where C2 = Cz k+1 ,wk is the contour defined according to Case 2 of Sect. 6.3.3, represented on Fig. 19. Using Eq. (28) where we computed this integral for a generic contour C, we obtain: 1 i [logC2 (eiβ ) − logC2 (−eiβ )] − 2 f w (λ) f wk (−λ) log λdλ = 4π C2 k 4π 1 i (iβ − i(β + π )) = . = 4π 4 We conclude that P(wk z k+1 ) = 21 .

514


e iβ − e iα

−e iβ

e iβ

e iα

−e iα

−e iβ

−1 . Right: the contour C = C Fig. 20. Left: the contour C3 = Cwk ,vk involved in the integral term of K w v ,vk 4 k ,vk

involved in the integral term of K v−1

,vk

Probability of the edges wk vk an z k vk . Again, there is no contribution from the exponential function. By our choice of Kasteleyn orientation, we have K vk ,wk = −1, and by definition we have Cvk ,wk = 0. Using Formula (7), this yields: 1 f w (λ) f vk (−λ) log(λ)dλ, P(wk vk ) = K vk ,wk K w−1k ,vk = − 2 4π C3 k where C3 = Cwk ,vk is the contour defined according to Case 2 of Sect. 6.3.3, represented on Fig. 20. Using Eq. (30) where we computed this integral for a generic contour C, we obtain: ⎞ ⎛ logC3 (eiβ ) − logC3 (−eiα ) i ⎝ iβ iβ ⎠ P(wk vk ) = logC3 (e ) − logC3 (−e ) − 4π cos β−α 2 iβ − i(α + π ) i iβ − i(β + π ) − = 4π cos θ π − 2θ 1 . = − 4 4π cos θ By symmetry, this is also the probability of occurrence of the edge z k vk . Probability of the edge vk v . By definition, we have Cv ,vk = 0, and by Eq. (18), we know that: K vk ,v f v (λ) Expy,x (λ) = f wk (λ) − f z k (λ). Using Formula (7), this yields: P(vk v ) = K vk ,v K v−1 ,v k 1 = K v ,v f v (λ) f vk (−λ) Expy,x (λ) log λ dλ 4π 2 C4 k

$ % 1 f wk (λ) − f z k (λ) f vk (−λ) log(λ)dλ = 2 4π C4 1 1 f (λ) f (−λ) log(λ)dλ − f z (λ) f vk (−λ) log(λ)dλ, = w v k 4π 2 C4 k 4π 2 C4 k


515

where C4 = Cv ,vk is the contour defined according to Case 3 of Sect. 6.3.3, represented on Fig. 20. Using Eq. (30), the first term is equal to: 1 f w (λ) f vk (−λ) log(λ)dλ 4π 2 C4 k ⎞ ⎛ iβ ) − log (−eiα ) (e log i ⎝ C4 C4 ⎠ =− logC4 (eiβ ) − logC4 (−eiβ ) − β−α 4π cos 2 iβ − i(α + π ) i iβ − i(β − π ) − =− 4π cos θ 1 π − 2θ . = + 4 4π cos θ By Eq. (29) and Fig. 20, the second term is equal to: 1 − 2 f z (λ) f vk (−λ) log(λ)dλ = 4π C4 k ⎞ ⎛ iα ) − log (−eiβ ) (e log i ⎝ C4 C4 ⎠ = logC4 (eiα ) − logC4 (−eiα ) − β−α 4π cos 2 iα − i(β − π ) i iα − i(α + π ) − = 4π cos θ 1 π − 2θ . = + 4 4π cos θ π −2θ As a consequence P(vk v ) = 21 + 2π cos θ . Let us make a few simple comments about the values of these probabilities.

1. The value 21 for the probability of the edge wk z k+1 can be explained as follows. By Fisher’s correspondence, once the configuration of the edges coming from edges of G attached to the decoration of x is fixed, there are two possibilities for the dimer covering inside the decoration, which both have the same weight. There is always one of the two possibilities containing the edge wk z k+1 . Therefore, this edge appears in a random dimer configuration half of the time. 2. Notice that P(wk z k ) = 21 P(vk v ). This is explained by the fact that the edge wk z k appears only if vk v is not present in the dimer configuration, and it appears only in one of the two allowed configurations, once the state of the edges coming from edges of G is fixed. 3. Using the two previous points, and the fact that the probability of the edges incident to a given vertex must sum to 1, one can deduce the probability of all the edges from the probability of the edge vk v . References [AP07] [AYP87]

Au-Yang, H., Perk, J.H.H.: Q-Dependent Susceptibilities in Ferromagnetic Quasiperiodic Z -invariant Ising Models. J. Stat. Phys. 127, 265–286 (2007) Au-Yang, H., Perk, J.H.H.: Critical correlations in a Z -invariant inhomogeneous ising model. Physica A: Stat. Theor. Phys. 144(1), 44–104 (1987)

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Baxter, R.J.: Free-fermion, checkerboard and Z -invariant lattice models in statistical mechanics. Proc. Roy. Soc. London Ser. A 404(1826), 1–33 (1986) Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1989, reprint of the 1982 original Boutillier, C., de Tilière, B.: The critical Z -invariant ising model via dimers: the periodic case. Prob. Theor. & Rel. Fields. 147(3-4), 379–413 (2010) Bobenko, A.I., Mercat, C., Suris, Y.B.: Linear and nonlinear theories of discrete analytic functions. Integrable structure +and isomonodromic Green’s function. J. Reine Angew. Math. 583, 117–161 (2005) Cimasoni, D., Reshetikhin, N.: Dimers on surface graphs and spin structures. I. Commun. Math. Phys. 275(1), 187–208 (2007) Cimasoni, D., Reshetikhin, N.: Dimers on surface graphs and spin structures. II. Commun. Math. Phys. 281(2), 445–468 (2008) Costa-Santos, R.: Geometrical aspects of the Z -invariant ising model. Eur. Phys. J. B 53(1), 85–90 (2006) Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. http://arXiv.org/abs/0910.2045v1 [math-ph], (2009) Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. in Math. (to appear), 2010, available at http://arXiv.org/abs/0810.2188v1 [math.CV], 2008 de Tilière, B.: Partition function of periodic isoradial dimer models. Prob. Th. Rel. Fields 138(3-4), 451–462 (2007) de Tilière, B.: Quadri-tilings of the plane. Prob. Th. Rel. Fields 137(3-4), 487–518 (2007) Fisher, M.E.: On the Dimer Solution of Planar Iing Models. J. Math. Phys. 7, 1776–1781 (1966) Kasteleyn, P.W.: The statistics of dimers on a lattice : I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961) Kasteleyn, P.W.: Graph theory and crystal physics. In: Graph Theory and Theoretical Physics, London: Academic Press, 1967, pp. 43–110 Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002) Kenyon, R., Schlenker, J.-M.: Rhombic embeddings of planar quad-graphs. Trans. Amer. Math. Soc. 357(9), 3443–3458 (electronic) (2005) Kuperberg, G.: An exploration of the permanent-determinant method. Electron. J. Combin. 5, Research Paper 46, 34 pp. (electronic) (1998) Reyes Martìnez, J.R.: Correlation functions for the Z -invariant ising model. Phys. Lett. A 227(3-4), 203–208 (1997) Reyes Martìnez, J.R.: Multi-spin correlation functions for the Z -invariant ising model. Physica A: Stat. Theor. Phys. 256(3-4), 463–484 (1998) Mercat, C.: Discrete period matrices and related topics. http://arXiv.org/abs/math-ph/ 0111043v2, 2002 Mercat, C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218(1), 177–216 (2001) McCoy, B., Wu, F.: The two-dimensional Ising model. Cambridge, MA: Harvard Univ. Press, 1973 Onsager, L.: Crystal statistics. i. a two-dimensional model with an order-disorder transition. Phys. Rev. 65(3-4), 117–149 (1944) Smirnov, S.: Towards conformal invariance of 2D lattice models. In: Proceedings of the International Congress of Mathematicians, Madrid, Volume 2, Zürich: Eur. Math. Soc., pp. 1421–1452, 2006

Communicated by S. Smirnov


Communications in


Wall-Crossing, Free Fermions and Crystal Melting Piotr Sułkowski1,2, 1 California Institute of Technology, Pasadena, CA 91125, USA. E-mail: [email protected] 2 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Received: 15 December 2009 / Accepted: 10 June 2010 Published online: 26 October 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract: We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the noncommutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field through each two-cycle in the manifold. BPS partition functions in non-trivial chambers are given by the expectation values of these operators. Thirdly, we discuss crystal interpretation of such correlators for this whole class of manifolds. We describe evolution of these crystals upon a change of the moduli, and find crystal interpretation of the flop transition and the DT/PT transition. The crystals which we find generalize and unify various other Calabi-Yau crystal models which appeared in literature in recent years. Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Summary of the results . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Wall-crossing for local Calabi-Yau manifolds . . . . 2.2 Free fermion formalism . . . . . . . . . . . . . . . . 2.3 Crystal melting . . . . . . . . . . . . . . . . . . . . 2.4 Wall-crossing for the conifold and pyramid partitions Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Triangulations and associated operators . . . . . . . 3.2 Quantization of geometry . . . . . . . . . . . . . . .

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On leave from University of Amsterdam and Sołtan Institute for Nuclear Studies, Poland.

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518 519 520 521 523 524 525 528 528 530

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3.3 Wall-crossing operators . . . . . . . . . . . . . . . . 3.4 Crystal melting interpretation . . . . . . . . . . . . . 4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Revisiting C3 . . . . . . . . . . . . . . . . . . . . . 4.2 Orbifolds C3 /Z N +1 . . . . . . . . . . . . . . . . . . 4.3 Resolved conifold . . . . . . . . . . . . . . . . . . . 4.4 Flop transition . . . . . . . . . . . . . . . . . . . . . 4.5 Triple-P1 geometry and more general wall-operators 4.6 Closed topological vertex . . . . . . . . . . . . . . . 5. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Quantization of geometry . . . . . . . . . . . . . . . 5.2 Wall-crossing operators . . . . . . . . . . . . . . . . 6. Summary and Discussion . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P. Sułkowski

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531 535 537 537 538 539 541 542 549 554 554 555 560 561

1. Introduction Counting of BPS states is an important problem in supersymmetric theories. In the context of string compactifications one is interested in the spectrum of bound states of D-branes wrapped around cycles of the internal Calabi-Yau threefold. In recent years there has been much progress in understanding degeneracies of D0 and D2-branes bound to a single D6-brane in type IIA theory. It was found out that these degeneracies are related on one hand to the topological string theory, and on the other to statistical models of crystal melting [1–3], while the Donaldson-Thomas theory provides a mathematical framework to describe these developments [4]. In fact the number of such bound states depends on the moduli of the underlying Calabi-Yau manifold. Once these moduli are varied certain states may either get bound together or unbound and their numbers jump. The corresponding moduli space is therefore divided into distinct chambers by loci called walls of marginal stability. The term wall-crossing refers to understanding behavior of BPS counting functions upon crossing those walls of marginal stability. While the issue of stability in various supersymmetric theories has a long history, one of the sources of its new impetus in the context of D-branes in string theory was the work of Denef and Moore [5]. In a parallel development a very general mathematical theory which describes these phenomena has been formulated by Kontsevich and Soibelman [6]. Subsequently these results were applied to local Calabi-Yau manifolds from physical [7–9] and mathematical [10–16] points of view. Recently, based on various string dualities [17], these results and their relation to topological string theory were explained from the M-theory perspective for manifolds without compact four-cycles [18]. In this paper we reformulate BPS counting and wall-crossing for the entire class of local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators and crystal melting.1 There are several motivations for our work. One motivation is related to the wave-function interpretation of topological string theory [19,20]. Such interpretation was originally proposed in the context of holomorphic anomaly equations. Various explicit representations of topological string partition functions, as states in the free fermion Hilbert space, were found in [21–24]. However those 1 While this paper was being prepared for publication, the author was informed that Kentaro Nagao independently reformulated wall-crossing for local Calabi-Yau manifolds in terms of vertex operators and obtained results overlapping with ours [29]. Our papers appear simultaneously.

Wall-Crossing, Free Fermions and Crystal Melting

519

states were constructed only in one point in the moduli space, which corresponds to the large radius limit. In [21] those states were related to integrable hierarchies. In [23] by a chain of string dualities their physical origin was found in terms of open strings stretching between intersecting D4 and D6-branes. It is therefore interesting whether some analogous wave-function interpretation extends also to other chambers of the moduli space. In this paper we show that this is indeed true. Firstly, we find a new set of fermionic states which encode the quantum structure of local, toric Calabi-Yau manifolds without compact four-cycles, such that their overlaps are equal to the modulus square of the corresponding topological string partition functions. These states are different than those found in [21] for the same manifolds. In particular the states which we find here are naturally associated to the non-commutative Donaldson-Thomas chamber. Nonetheless, both of them are given by a Bogoliubov transformation of the fermionic vacuum. Secondly, we also find fermionic interpretation of wall-crossing in a large set of chambers, and realize BPS generating functions in those chambers as various fermionic correlators. While we do not provide a clear physical explanation why such fermionic quantum states should occur, we believe it should exist. Another motivation of our work is related to Calabi-Yau crystals. Similarly as in the wave-function case, originally a crystal interpretation of BPS counting was realized in one particular chamber of the moduli space and was intimately related to the vertex operators [1,2]. More recently, a generating function of pyramid partitions for the resolved conifold in another special chamber was computed using free fermion formalism [13]. It is natural to expect that such fermionic crystal representation of BPS counting should hold for all chambers and a much larger class of manifolds. In this paper we indeed find such representation for local, toric Calabi-Yau manifolds without compact four-cycles, in a large set of chambers. The main results of this paper are briefly summarized below. 1.1. Summary of the results. Let M denote a toric Calabi-Yau manifolds without compact four-cycles. Firstly, we associate to M two states |± ∈ H in the Hilbert space of free fermion H, such that Z ≡ |Ztop |2 = + |− ,

(1)

where Ztop denotes the instanton part of the topological string partition function of M, and Z is the generating function of the non-commutative Donaldson-Thomas invariants. Secondly, we find a large set of wall-crossing operators W p , W p , which inserted n times in the above correlator encode the BPS generating functions in chambers corresponding to turning on n, respectively positive and negative, quanta of the B-field through p th two-cycle of M. These generating functions, according to the prescription of [18] are given by a restriction of |Ztop |2 . In our formalism, in chambers connected by a finite number of walls of marginal stability to the non-commutative Donaldson-Thomas = 1 (i.e. corresponding to positive or negative radius region, or to the core region with Z R in the M-theory interpretation), they are given respectively by

n Zn| p = + |(W p )n |− , Zn| p = + |(W p ) |− ,

n| p = 0|(W p ) |0, Z n

n| Z p

=

0|(W p )n |0.

(2) (3)

520

P. Sułkowski

From our point of view the generating function in the core region, corresponding to a single D6-brane, is given by = 0|0 = 1. Z In particular, the change from positive to negative values of R, which corresponds for example to the so-called DT/PT transition [4,12], is represented in our formalism by the change of the ground state representing the manifold |±

←→

|0.

More precisely, the equality of the above correlators to the BPS partition functions arises upon an appropriate identification of parameters (i.e. colors of the crystal with Kähler parameters and string coupling), which we find in each case that we consider. Thirdly, we find a crystal melting interpretation of all these generating functions. They turn out to be related, respectively, to the generating functions Z,

Z n| p , Z n| p ,

Z n| p , Z n| p , Z ,

of multi-colored crystals, also under appropriate identification of crystal and stringy parameters. The shape and coloring of these crystals is encoded in the toric diagram of the Calabi-Yau manifold, and for non-trivial chambers also in the structure of the wall operators W p , W p . We also discuss evolution of crystals and find a crystal interpretation of various geometric transitions, such as the flop transition and DT/PT transition. In particular, in our framework we can easily prove the relation between certain BPS generating functions for the conifold and finite pyramid partitions conjectured in [8], and generalize it to other manifolds. Moreover, the crystals which we find provide a unifying point of view on various crystal models considered in literature in recent years. This paper is organized as follows. In Sect. 2 we review necessary background on wall-crossing, free fermion formalism and crystal melting, and set up various conventions and notation. In Sect. 3 we describe in detail the results summarized above for the class of manifolds encoded in an arbitrary triangulation of a strip. In Sect. 4 we illustrate these results in several examples and find more general wall-crossing operators. We also analyze the closed topological vertex geometry, which is a case that does not arise from a triangulation of a strip. Section 5 contains proofs of the statements given in Sect. 3. Section 6 contains a summary and discussion.

2. Preliminaries In this section we review some background material, as well as set the notation and conventions used in further parts of the paper. In Sect. 2.1 we review a physical picture of wall-crossing for manifolds without compact four-cycles, on which we will rely in derivation of some of our results. In Sect. 2.2 we review the formalism of free fermions and the construction of vertex operators. In Sect. 2.3 we recall how these can be used to solve certain models of three-dimensional melting crystals, which in particular arise in connection with enumerative invariants of Calabi-Yau threefolds. In Sect. 2.4 we review the wall-crossing for the resolved conifold, which we will generalize to a large class of manifolds in Sect. 3.


521

2.1. Wall-crossing for local Calabi-Yau manifolds. In this section we review the wallcrossing phenomena for local toric Calabi-Yau manifolds without four-cycles. One large class of such manifolds can be encoded in toric diagrams which arise from a triangulation of a strip, as we will explain in detail in Sect. 3.2. Another example of such a geometry (which does not arise from a triangulation of a strip) is the closed topological vertex presented in Sect. 4.6. Mathematically wall-crossing describes a change of generalized Donaldson-Thomas invariants upon crossing the walls of marginal stability. Generating functions of such generalized Donaldson-Thomas invariants for the local geometries mentioned above, in various chambers were derived by mathematicians in [10,13,15,16]. Physically generalized Donaldson-Thomas invariants correspond to numbers of D6-D2-D0 bound states, and for the resolved conifold they were analyzed in [7–9]. Recently a physical prescription for determining BPS generating functions in all chambers, for general manifolds without compact four-cycles, was derived in [18]. For local manifolds mentioned above this prescription agrees with mathematical results. This also makes contact with topological string theory and fits naturally in the context of our results. We therefore review the wall-crossing for local geometries from the perspective of this paper. The idea of [18] is as follows. The system of D6-D2-D0 branes in type IIA theory can be lifted to M-theory on S 1 , whereupon the D6-brane transforms into a geometric background of a Taub-NUT space with unit charge, extending in directions transverse to the D6-brane. This Taub-NUT space is a circle ST1 N fibration over R3 , with ST1 N shrinking to a point in the location of the D6-brane and attaining a radius R at infinity. The counting of bound states involving the D6-brane is then reinterpreted as the counting of BPS states of M2-branes in this Taub-NUT space. This counting does not change when the radius R grows to infinity and Taub-NUT approaches R 4 . Moreover, from the spacetime perspective, the resulting generating functions of BPS degeneracies would factorize into a product of single particle partition functions if there would be no interaction among five-dimensional particles. It is then argued that this can be achieved if the following two conditions are satisfied. Firstly, the moduli of the Calabi-Yau have to be tuned such that M2-branes wrapped in various ways have aligned central charges. This can be achieved by considering vanishing Kähler parameters of the Calabi-Yau space. At the same time, to avoid generation of massless states, non-trivial flux of the M-theory three-form field through the two-cycles of the Calabi-Yau and ST1 N have to be turned on. In type IIA this flux translates to the B-field flux B through two-cycles of Calabi-Yau. For a state arising from D2 wrapping a class β the central charge then reads Z (l, β) =

1 (l + B · β), R

(4)

where l counts the D0-brane charge, which is taken positive to preserve the same supersymmetry. The second condition requires that the only BPS states in the Taub-NUT geometry are particle-like, and therefore there are no string-like states which would arise from M5-branes wrapped on four-cycles. This is why the results of [18] hold for Calabi-Yau geometries without compact four-cycles. Under the above two assumptions, and in the limit R → ∞, the counting of D6-D2-D0 bound states translates to the counting of particle degeneracies on R4 × S 1 , arising from M2-branes wrapped on cycles β. The excitations of these particles in R4 , parametrized by two complex variables z 1, z 2 , are accounted for by the modes of the holomorphic field

522

P. Sułkowski

(z 1 , z 2 ) =

αl1 ,l2 z l11 z l22 .

l1 ,l2

Under a decomposition of the isometry group of R4 as S O(4) = SU (2) × SU (2) there are Nβm,m five-dimensional BPS states of intrinsic spin (m, m ). The degeneracies we are interested in correspond to the net number obtained by tracing over the SU (2) spins, and are expressed by Gopakumer-Vafa invariants Nβm = (−1)m Nβm,m . m

The total angular momentum of a given state contributing to the index is l = l1 + l2 + m. Now the invariant degeneracies are expressed as the trace over the corresponding Fock space, subject to the condition that all the contributing states are mutually BPS, i.e. Z (l, β) =

1 (l + B · β) > 0. R

(5)

Therefore, in a chamber specified by the moduli R and B, Z(R, B) = Tr Fock qsQ 0 Q Q 2 | Z (l,β)>0 m = (1 − qsl1 +l2 +m Q β ) Nβ | Z (l,β)>0 =

β,m l1 +l2 =l ∞

m

(1 − qsl+m Q β )l Nβ | Z (l,β)>0 ,

(6)

β,m l=1

where Q = e−T and qs = e−gs encode respectively the Kähler class T and the string coupling gs . Note that the product over β runs over both positive and negative classes, so that both M2 and anti-M2-branes contribute to the index as long as the condition (5) is satisfied. An important observation in [18] is also the fact that the BPS generating functions are simply related to the topological string partition function Z(R, B) = Ztop (Q)Ztop (Q −1 )|chamber ≡ |Ztop (Q)|2 |chamber .

(7)

Here the subscript |chamber denotes restriction to contributions from states for which Z (l, β) > 0 in a given chamber, and the topological string partition function is expressed through Gopakumar-Vafa invariants Ztop (Q) = M(q)χ /2

∞

m

(1 − Q β qsm+l )l Nβ ,

l=1 β>0,m

where M(q) = l (1−q l )−l is the MacMahon function and χ is the Euler characteristic of the Calabi-Yau manifold. There are a few interesting special cases of the relation (7). For positive R and infinite B we get contributions from states with arbitrary n and positive β, so that we get Z(R > 0, B → ∞) = M(1, q)χ /2 Ztop (Q).

(8)


523

This immediately leads to the relation between the Gromov-Witten and DonaldsonThomas invariants discussed in [2]. For positive R and B sufficiently small 0 < B 0, 0 < B 0

(−1)n−1 x n n

α±n

,

which act on fermionic states |μ corresponding to partitions μ as [13,25,26] − (x)|μ = x |λ|−|μ| |λ, + (x)|μ = x |μ|−|λ| |λ, λμ (x)|μ = −

λt μt

(11)

(12)

λ≺μ

x |λ|−|μ| |λ, + (x)|μ =

x |μ|−|λ| |λ,

(13)

λt ≺μt

where the interlacing relation between partitions is defined by λμ

⇔

λ1 ≥ μ1 ≥ λ2 ≥ μ2 ≥ λ3 ≥ · · · .

(14)

524

P. Sułkowski

The operator is in fact the inverse of with negative argument. These operators satisfy commutation relations 1 − (y) + (x), 1 − xy 1 (y) + (x), + (x) − (y) = 1 − xy − + (x) − (y) = (1 + x y) − (y) + (x), + (x) − (y) = (1 + x y) − (y) + (x).

+ (x) − (y) =

(15) (16) (17) (18)

g (a hat is to We also introduce various colors qg and the corresponding operators Q distinguish them from Kähler parameters Q i ) g |λ = qg|λ| |λ. Q

(19)

These operators commute with vertex operators up to a scaling of the argument g = Q g + (xqg ), + (x) Q g = Q g + (xqg ), + (x) Q g , Q g − g . g − (x) = − (xqg ) Q (x) = − (xqg ) Q Q

(20) (21)

2.3. Crystal melting. Since the work [1,2] it is known that various enumerative invariants of Calabi-Yau manifolds turn out to be related to statistical models of crystal melting. More precisely, generating functions of those enumerative invariants are equal to partition functions of crystal models in the grand canonical ensemble. Such crystal partition functions are computed as sums over all possible crystal configurations subject to appropriate rules, with weights given by the number (or its refinements) of (missing) elementary crystal constituents in a given configuration. Enumerative invariants of Calabi-Yau threefolds are related to three-dimensional crystal models. A simple example of such a model consists of unit cubes filling the positive octant of R3 space. A unit cube located in position (I, J, K ) can evaporate from this crystal only if all other cubes with coordinates (i ≤ I, j ≤ J, k ≤ K ) already evaporated. Therefore all missing configurations are in one-to-one correspondence with three-dimensional partitions π , also called plane partitions. Weighting each such configuration by the number of boxes it consists of |π |, the partition function of this model turns out to be the MacMahon function M(q) ≡ M(1, q), Z=

π

q |π | =

∞ l=0

p(l)q l =

∞ l=1

1 = M(q). (1 − q l )l

Mathematically the numbers p(l) encode Donaldson-Thomas invariants, while from the string theory viewpoint they count the number of bound states of l D0-branes with a single D6-brane covering C3 [1,2]. In what follows we also often use generalized MacMahon functions, M(x, q) =

∞

(1 − xq i )−i ,

q) = M(x, q)M(x −1 , q). M(x,

(22)

i=1

The above partition function Z is a prototype example which can be computed using free fermion formalism and vertex operators. Denoting the axes of R in which the crystal is embedded by x, y, z, we first slice each possible crystal configuration (or rather


525

Fig. 1. Slicing of a plane partition (left) into a sequence of interlacing two-dimensional partitions (right). A sequence of ± operators in (23) which create two-dimensional partitions is represented by arrows inserted along two axes. Directions of arrows → represent interlacing condition on partitions. We reconsider this example from a new viewpoint in Fig. 6

its missing complement) by planes given by x − y ∈ Z, as shown in Fig. 1. One can show that a configuration of boxes in each such slice corresponds to a two-dimensional partition, and two such partitions, corresponding to two neighboring slices, satisfy the interlacing condition (14). Therefore constructing all possible crystal configurations is equivalent to building them slice by slice from interlacing partitions. Precisely such an operation is performed by ± operators (12). Counting of boxes in a given plane par defined in (19). tition can also be performed slice by slice using the single operator Q Therefore the above partition function can be computed by writing infinite series of acting on two vacua representing empty two-dimensional diagrams, operators ± (1) Q and then commuting them according to (15) and (21), + (1) Q + (1) Q + (1) Q − (1) Q − (1) Q − (1) Q . . . |0 Z = 0| . . . Q = 0| . . . + (q 2 ) + (q) + (1) − (q) − (q 2 ) − (q 3 ) . . . |0 ∞ 1 = = M(q). 1 − q l1 +l2 −1

(23)

l1 ,l2 =1

This computation is represented in Fig. 1 (right), with arrows representing insertions of ± operators along two axes. and only The example presented above is very simple, as it involves just one color Q ± operators. More complicated crystal models can involve more colors, as well as both operators. A family of multicolored crystals (realized by multicolored plane ± and ± partitions) corresponding to C3 /Z N orbifolds was considered in [13]. On the other hand the crystal model encoding generalized Donaldson-Thomas invariants for the conifold involves two-colored pyramid partitions which are built by an application of interlacing sequence of and operators, as we will discuss in more detail in Sects. 2.4 below and 4.3. In Sect. 3 we find a unifying viewpoint on all these examples. 2.4. Wall-crossing for the conifold and pyramid partitions. In this section we briefly review the wall-crossing for the resolved conifold. The structure of walls and chambers has been analyzed in this case in [7,8,10,15], and it is of course consistent with the

526

P. Sułkowski

Fig. 2. Infinite pyramids with one (left) and four (right) stones in the top row. Their generating functions are pyramid pyramid given respectively by Z 0 and Z 3

results in [18] summarized in Sect. 2.1. For each chamber there is also an associated crystal model of two-colored pyramid partitions, and crossing a wall of marginal stability corresponds to the extension of the pyramid crystal. One can also translate counting of pyramid partitions into a dimer model; in this language crossing a wall corresponds to a combinatorial operation called dimer shuffling. The structure of these pyramid crystal models or dimers can also be encoded in a quiver and an associated potential. The space of stability conditions of the conifold can be divided into several infinite countable sets of chambers. As explained in Sect. 2.1, BPS generating functions in all chambers can also be related to the (square of the) topological string partition function coni f old Ztop (Q) = M(1, qs ) (1 − Qqsk )k (24) k≥1

with appropriate values of the moduli R and the B field through P1 of the conifold. Here qs = e−gs and Kähler modulus T is encoded in Q = e−T . This topological string partition function encodes the Gopakumar-Vafa invariants 0 Nβ=0 = −2,

0 Nβ=±1 = 1.

For future reference we write down BPS generating functions for two sets of chambers below. We stress that the labeling of the chambers, as well as an identification of the pyramid colors q0 , q1 with string parameters qs , Q is slightly different than in [7,15]. The conventions we use are well motivated from the point of view of the formalism developed in Sect. 3. The first set of chambers is characterized by R > 0 and positive B ∈ ]n, n + 1[ (for n ≥ 0). It extends between the non-commutative region of Szendroi (9) and the chamber with standard Donaldson-Thomas invariants (8). BPS partition functions are labeled by n and read coni f old Zn = M(1, q)2 (1 − Qqsk )k (1 − Q −1 qsk )k . (25) k≥1

k≥n+1

These partition functions are related to pyramid partitions with two colors q0 and q1 , presented in Fig. 2. The generating function of a pyramid with n + 1 yellow boxes in its top row is pyramid Zn (q0 , q1 ) = M(1, q0 q1 )2 (1 + q0k q1k+1 )k−n (1 + q0k q1k−1 )k+n (26) k≥n+1

k≥1


527

Fig. 3. Finite pyramids with m = 1, 2, 3 stones in the top row (respectively left, middle and right), whose pyramid pyramid generating functions are given by Z m+1 (note that Z1 = 1 corresponds to an empty pyramid corresponding to the pure D6-brane) coni f old

≡ Zn

under the identification of parameters

coni f old

chambers :

qs = q0 q1 ,

and it reproduces Zn Zn

pyramid

Q = −qsn q1 .

For n = 0 the non-commutative Donaldson-Thomas partition function [10,11] corresponds to a pyramid with just one stone in the top row, while n → ∞ corresponds to the pyramid which looks like a half-infinite prism. The second set of chambers is characterized by R < 0 and positive B ∈ ]n − 1, n[ (for n ≥ 1). It extends between the core region with a single D6-brane (10) and the chamber characterized by so-called Pandharipande-Thomas invariants (for the flopped geometry, or equivalently for anti-M2-branes); the BPS generating functions read n−1 j j q s coni f old n Z = . (27) 1− Q j=1

The corresponding statistical models were conjectured in [8] to correspond to finite pyramids with n − 1 stones in the top row, as shown in Fig. 3. In Sect. 4.3 we will provide a new proof of this statement,2 and show that the generating functions of such partitions are equal to pyramid (q0 , q1 ) = Zn

n−1

n− j n− j−1 j q1 ) .

(1 + q0

(28)

j=1 pyramid nconi f old ≡ The equality Z Zn arises upon an identification

nconi f old chambers : Z

qs−1 = q0 q1 ,

Q = −qsn q1 .

There are two more sets of chambers characterized by the negative value of the B-field. The corresponding generating functions are of the analogous form as above, but with Q replaced by Q −1 . We will see that in a natural way they correspond to pyramids with a vertical top row consisting of red stones (rather than the horizontal yellow top rows as in Figs. 2 and 3, even though such a correspondence would also be possible, albeit with less natural identification of parameters). 2 This statement has been proved also by mathematicians in [15].

528

P. Sułkowski

3. Results In this section we consider wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles. We reformulate it in the free fermion framework and find the corresponding crystal melting picture, as anticipated in Sect. 1.1. Unless otherwise stated, we use notation and conventions introduced in Sect. 2. There are two classes of local, toric Calabi-Yau manifolds without compact fourcycles. The first class corresponds to manifolds whose toric diagram is given by the so-called triangulation of a strip. The second class consists just of the closed topological vertex and its flop, which do not arise from a triangulation of a strip. Nonetheless the case of the closed topological vertex is closely related to one particular manifold from the first class. For this reason we focus now mainly on the first class of manifolds, and discuss the closed topological vertex later on in Sect. 4.6.

3.1. Triangulations and associated operators. To reformulate wall-crossing in the fermionic language we associate first several operators to a given local, toric manifold which arises from a triangulation of a strip. We recall first that such manifolds arise from a triangulation, into triangles of area 1/2, of a long rectangle or a strip of height 1. A toric diagram arises as a dual graph to such a triangulation. Each P1 in a Calabi-Yau geometry, represented by a finite interval in a toric diagram, corresponds to an inner line in a strip triangulation. From each vertex in a toric diagram emanates one semi-infinite vertical line, which crosses either the upper or the lower edge of the strip. Two such consecutive lines can emanate either in the same or in the opposite direction, respectively when they are the endpoints of an interval representing P1 with local O(−2) ⊕ O or O(−1) ⊕ O(−1) neighborhood. We introduce the following notation. We number independent P1 ’s from 1 to N , starting from the left end of the strip, and denote their Kähler parameters by Q i = e−Ti , i = 1, . . . , N . We also number, starting from the left, all vertices in a toric diagram, and associate to each vertex its type ti = ±1, in the following way3 : if the local neighborhood of P1 , represented by an interval between vertices i and i + 1, is O(−2) ⊕ O, then ti+1 = ti ; if this neighborhood is of O(−1) ⊕ O(−1) type, then ti+1 = −ti . The type of the first vertex could be chosen arbitrarily, but to fix attention we set t1 = +1. We also recall that the instanton part of the closed topological string partition function for such geometries reads [27] Ztop (Q i ) = M(1, q)

N +1 2

∞

−(ti t j )l 1 − q l (Q i Q i+1 · · · Q j−1 ) .

(29)

l=1 1≤i< j≤N +1

Below we introduce several operators which are the main building blocks of the fermionic and crystal construction. Their structure is encoded in the toric diagram of the manifold described above. These operators are given by a string of N +1 vertex operators ti ± (x) (introduced in (11)) which we associate to the vertices of the toric diagram, and determine their type by the type of these vertices ti such that ti =+1 ti =−1 (x) = ± (x), ± (x) = ± (x). ± 3 In the same way the type A or B was associated to vertices in a triangulation of a strip in [27].


529

ti i repreMoreover this string of ± (x) operators is interlaced with N + 1 operators Q 1 , . . . , Q N are associated to P1 in senting colors qi , for i = 0, 1, . . . , N . Operators Q 0 . We also define the toric diagram, and there is an additional Q N , 1 · · · Q = Q 0 Q q = q0 q1 · · · q N . (30) Q ti (x) and a choice of colors of the operators which we To sum up, the upper indices of ± introduce below are specified by the data of the toric manifold under consideration. The lower indices ± will be related to the wall-crossing chamber we will be interested in. Now we can introduce the operators of interest. The first two are defined as tN t N +1 t1 t2 1 ± 2 · · · ± N ± 0 . A± (x) = ± (x) Q (x) Q (x) Q (x) Q

(31)

i ’s to the left or right using (21) we also introduce related operators Commuting all Q

xq xq t N +1 t1 t2 xq t3 −1 + · · · + , A+ (x) = Q A+ (x) = + (xq) + q1 q1 q2 q1 q2 · · · q N (32) t N +1 t1 t2 t3 −1 = − A− (x) = A− (x) Q (x) − (xq1 ) − (xq1 q2 ) · · · − (xq1 q2 q N ). (33) Secondly, we define operators which we will refer to as the wall-crossing operators,

tp t1 t2 1 − 2 · · · − p W p (x) = − (x) Q (x) Q (x) Q

t p+1 · · · +t N (x) Q N +t N +1 (x) Q 0 , × +p+1 (x) Q (34)

1 +t2 (x) Q 2 · · · +t p (x) Q p W p (x) = +t1 (x) Q

t tN t N +1 p+1 · · · − N − 0 . × −p+1 (x) Q (35) (x) Q (x) Q The order of and is the same as for A± operators; the only difference is that now there are subscripts ∓ on first p operators and ± on the remaining ones. In what follows we also use the following auxiliary operators, defined for fixed p (corresponding to fixed p th P1 in the toric geometry):

xq xq tp t1 t2 xq t3 1 γ+ (x) = + (xq) + + · · · + , q1 q1 q2 q1 q2 · · · q p−1

xq xq t · · · +t N +1 , γ+2 (x) = +p+1 q1 q2 · · · q p q1 q2 · · · q N t

t1 t2 t3 γ−1 (x) = − (x) − (xq1 ) − (xq1 q2 ) · · · −p (xq1 q2 · · · q p−1 ), t

t N +1 (xq1 q2 · · · q N ). γ−2 (x) = −p+1 (xq1 q2 · · · q p ) · · · −

With these definitions we can simply write W p (x) = γ+1 (x/q)γ−2 (x) Q. W p (x) = γ−1 (x)γ+2 (x/q) Q,

(36)

Moreover A± defined in (32) and (33) can be written (for any p ∈ 1, N ) as A± (x) = γ±1 (x)γ±2 (x).

(37)

When the argument of any of these operators is x = 1, we will often use a simplified notation in which this argument is skipped, i.e. A± ≡ A± (1),

A± ≡ A± (1),

W p ≡ W p (1),

W p ≡ W p (1).

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P. Sułkowski

3.2. Quantization of geometry. In the previous section we considered a toric manifold specified by a triangulation of a strip, and its structure encoded in operators A± , which are specified by a sequence of and . We can actually do more and represent each such manifold by two states in the Hilbert space of a free fermion H, |± ∈ H. We define these states as + | = 0| . . . A+ (1)A+ (1)A+ (1) = 0| . . . A+ (q 2 )A+ (q)A+ (1), |− = A− (1)A− (1)A− (1) . . . |0 = A− (1)A− (q)A− (q 2 ) . . . |0.

(38) (39)

To define these states we used only the classical data of the toric manifold, which is encoded in operators A+ (1). Nonetheless they carry information about the full instanton part of the topological string amplitudes, to all orders in string coupling. Our first claim is that the overlap of these states Z = + |− ,

(40)

is equal to the BPS partition function Z in the chamber corresponding to the noncommutative Donaldson-Thomas invariants Z = Z ≡ |Ztop |2 ≡ Ztop (Q i )Ztop (Q i−1 ),

(41)

with Ztop (Q i ) given in (29), and under the following identification between qi parameters which enter a definition of |± and string parameters Q i = e−Ti and qs = e−gs : qi = (ti ti+1 )Q i , qs = q ≡ q0 q1 · · · q N .

(42)

The proof of the statement (41) is given in Sect. 5.1. We note that the states |± are different than the states |V which one can associate to the same geometry in the B-model picture of [21]. They have also different properties. would In particular, in the framework of [21] the expression of the form V |V = Ztop represent the topological string partition function of the manifold obtained from gluing two copies of a given manifold to each other. In our case the overlap (40) gives the square of the topological string partition function of the manifold itself. We also stress that the states |V are suitable for the large radius limit point in the moduli space, whereas |± are naturally associated with the non-commutative Donaldson-Thomas chamber. Nonetheless, similarly as for the states |V , it is tempting to think of the states |± as providing some wave-function interpretation of the underlying classical manifolds, in the spirit of [19,20]. The states |± relate to the topological string partition function and characterize an extremal chamber in which non-commutative Donaldson-Thomas invariants are defined. There is another extremal case corresponding to the chamber with a single D6-brane with no bound states, so that BPS partition function reads = 1. Z In our formalism in this chamber this partition function can be understood simply as Z = 0|0 = 1,

(43)

= and clearly Z Z . This suggests associating the vacuum state |0 to the manifold. We will see shortly that this association makes sense also in a multitude of other chambers.


531

Fig. 4. Toric Calabi-Yau manifolds represented by a triangulation of a strip. There are N independent P1 ’s with Kähler parameters Q i = e−Ti , and N + 1 vertices to which we associate and operators represented respectively by ⊕ and signs. Intervals which connect vertices with opposite signs (yellow online) represent O(−1)⊕O(−1) → P1 local neighborhoods. Intervals which connect vertices with the same signs (red online) represent O(−2) ⊕ O → P1 local neighborhoods. The first vertex on the left is chosen to be ⊕

3.3. Wall-crossing operators. In the previous section we realized BPS partition functions in two extreme chambers as fermionic correlators. Now we show that BPS generating functions can be realized as fermionic correlators also in various other chambers. As discussed in Sect. 2.1 only those bound states of D0, D2 or anti-D2-branes with D6-brane can exist for which the central charge (4) is positive Z (R, B) =

1 (n + β · B) > 0. R

The fermionic correlators we are after must therefore contain information about the moduli R and B. Our first claim is that the information about R is encoded in the ground state which represents a manifold. This ground state depends only on the sign of R and should be chosen as follows: R>0

−→

|± ,

R 0. Consider a chamber characterized by positive R and positive B-field through p th two-cycle R < 0,

B ∈ ]n − 1, n[ for 1 ≤ n ∈ Z.

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P. Sułkowski

The BPS partition function in this chamber contains only those factors which include Q p and it reads n| p = Z

p N +1 n−1

1−

i=1 s=1 r = p+1

qsi Q s Q s+1 · · · Q r −1

−tr ts i .

On the other hand we can compute the expectation value of n wall-crossing operators W p . We find that Z n| p = 0|(W p )n |0 =

p N n−1 +1

1 − (tr ts )

i=1 s=1 r = p+1

q n−i qs qs+1 · · · qr −1

−tr ts i .

(45)

The proof of this equality is given in Sect. 5.2. Therefore, under the following change of variables: Q p = (t p t p+1 )q p qsn ,

Q i = (ti ti+1 )qi for i = p, qs =

1 , q

(46)

the correlator (45) reproduces the BPS partition function n| p = Z Z n| p .

(47)

An insertion of W p has an interpretation of turning on a positive quantum of B-field, and the redefinition of Q p can be interpreted as effectively enlarging the Kähler parameter T p by one unit of gs . In fact the minimal number of insertions is n = 1. Because in W p all + operators are to the right of − , an insertion of one operator does not have any effect, so Z 1| p = 0|W p |0 = 0|0 = 1 still represents a chamber with a single D6-brane and no other branes bound to it. Chambers with R > 0, B > 0. In the second case we consider the positive value of R and the positive flux through p th P1 , R > 0,

B ∈ ]n, n + 1[ for 0 ≤ n ∈ Z.

Denote the BPS partition function in this chamber by Zn| p . We find that the expectation value of n wall-crossing operators W p in the background of | has the form (0) (1) (2) Z n| p = + |(W p )n |− = M(1, q) N +1 Z n| p Z n| p Z n| p ,

(48)

(0) ±1 which would include q , while where Z n| p p does not contain any factors (qs · · · qr −1 ) (1)

(2)

Z n| p contains all factors qs · · · qr −1 which do include q p , and Z n| p contains all factors


533

(qs · · · qr −1 )−1 which also include q p : (0) Z n| p =

∞

1 − (tr ts )

l=1 p ∈s,r / +1⊂1,N +1

ql qs qs+1 · · · qr −1

−tr ts l

−tr ts l

× 1 − (tr ts )q l qs qs+1 · · · qr −1 , (1) Z n| p

=

∞

−tr ts l 1 − (tr ts )q l+n qs qs+1 · · · qr −1 ,

l=1 p∈s,r +1⊂1,N +1 (2)

Z n| p =

∞

1 − (tr ts )

l=n+1 p∈s,r +1⊂1,N +1

q l−n qs qs+1 · · · qr −1

−tr ts l .

Clearly the change of variables Q p = (t p t p+1 )q p qsn ,

Q i = (ti ti+1 )qi for i = p, qs = q

(49)

reproduces the BPS partition function Zn| p = Z n| p .

(50)

When no wall-crossing operator is inserted the change of variables reduces to (42) and we get the non-commutative Donaldson-Thomas partition function (41), Z 0| p = Z. The proof of (48) and in consequence (50) is given in Sect. 5.2. Chambers with R < 0, B < 0. Now we consider negative R and negative B-field R < 0,

B ∈ ] − n − 1, −n[ for 0 ≤ n ∈ Z.

For such a chamber the BPS partition function reads n| Z p =

p N +1

n −tr ts i 1 − qsi Q s Q s+1 · · · Q r −1 . i=1 s=1 r = p+1

Now we find the expectation value of n wall-crossing operators W p is equal to n Z n| p = 0|(W p ) |0 =

p N +1

n −tr ts i 1 − (tr ts )q n−i qs qs+1 · · · qr −1 . (51) i=1 s=1 r = p+1

This equality is proved in Sect. 5.2. Therefore, under a change of variables, Q p = (t p t p+1 )q p qs−n ,

Q i = (ti ti+1 )qi for i = p, qs =

1 , q

(52)

this reproduces the BPS partition function n| Z p = Z n| p .

(53)

534

P. Sułkowski

Now an insertion of W p has an interpretation of turning on a negative quantum of B-field, and the redefinition of Q p can be interpreted as effectively reducing t p by one unit of gs . As we already discussed, Z 0| p = 0|0 = 1

represents a chamber with a single D6-brane and no other branes bound to it. Contrary to the case with B > 0, an insertion of a single W p has a non-trivial effect. Chambers with R > 0, B < 0. In the last case we consider positive R and negative B, R > 0, 0 > B ∈ ] − n, −n + 1[ for 1 ≤ n ∈ Z. . We denote the BPS partition function in this chamber by Zn| p

We find that the expectation value of n operators W p in the background of |± has the form

(0)

(1)

(2)

n N +1 Z n| Z n| p Z n| p Z n| p , p = + |(W p ) |− = M(1, q)

(54)

(0) ±1 which would include q , Z (1) where Z n| p p does not contain any factors (qs · · · qr −1 ) n| p (2)

contains all factors qs · · · qr −1 which do include q p , and Z n| p contains all factors (qs · · · qr −1 )−1 which also include q p : (0)

Z n| p =

∞

1 − (tr ts )

l=1 p ∈s,r / +1⊂1,N +1

ql qs qs+1 · · · qr −1

−tr ts l

−tr ts l

× 1 − (tr ts )q l qs qs+1 · · · qr −1 , (1) Z n| p

=

∞

−tr ts l 1 − (tr ts )q l−n qs qs+1 · · · qr −1 ,

l=n p∈s,r +1⊂1,N +1 (2) Z n| p =

∞

1 − (tr ts )

l=1 p∈s,r +1⊂1,N +1

q l+n qs qs+1 · · · qr −1

−tr ts l .

Clearly the change of variables Q p = (t p t p+1 )q p qs−n−1 ,

Q i = (ti ti+1 )qi for i = p, qs = q,

(55)

reproduces the BPS partition function Zn| p = Z n| p .

(56)

We note that both Z 1| p with the above change of variables, as well as Z 0| p given in (48) with a different change of variables in (49), lead to the same BPS generating function Z which corresponds to the non-commutative Donaldson-Thomas invariants. The proof of (54) and in consequence (56) is given in Sect. 5.2.


535

Fig. 5. Assignment of arrows

3.4. Crystal melting interpretation. So far we have found a representation of D6-D2-D0 generating functions as correlators in the free fermion theory. In this section we discuss crystal melting interpretation of these correlators, and explain how to associate crystal models to local, toric manifolds without compact four-cycles, based on the results described above. Our point of view generalizes crystal models found previously, such as plane partitions for C3 crystal summarized briefly in Sect. 2.3 or pyramid partitions for the conifold crystal from Sect. 2.4, and provides an interesting unifying perspective. Furthermore we discuss evolution of crystals upon changing the moduli of the theory. Moreover, for the class of manifolds which we consider in this paper, we claim that our crystals are equivalent to colored crystals introduced in [28] in terms of quiver diagrams and relation to dimers. The construction of crystals explained in this section and their evolution upon wall-crossing is illustrated in several examples in Sect. 4. Construction of crystals. Our crystal interpretation is inherently related to the form of fermionic correlators which we found in Sects. 3.2 and 3.3. All these correlators are constructed from operators A± , W p and W p , which involve only vertex operators ± i . According to the relations (12) and (13), and ± with argument 1 and color operators Q insertion of these vertex operators can be interpreted as insertion of two-dimensional partitions satisfying interlacing, or transposed interlacing conditions. This corresponds to constructing the three-dimensional crystal from two-dimensional slices. A relative position of the neighboring slices is determined by which vertex operator they are created. Additional insertions of color operators have an interpretation of coloring the i appear crystal. Because in all operators which we consider in this paper the colors Q in the same order, these colors are always repeated periodically in the full correlators, and in consequence our crystals are made of interlacing periodically colored slices. All the information about the crystal, including its shape, coloring and interlacing pattern, can be encoded in a simple graphical form. To do this we associate various arrows to the vertex operators. This assignment is shown in Fig. 5. The arrows are always drawn from left to right, or up to down (a direction of drawing is independent of the orientation of the arrow). Then we translate a sequence of vertex operators which appear in a given correlator into a sequence of corresponding arrows, and draw them such that the end of one arrow becomes the end of the next one. We also keep track of the coloring by drawing at the endpoint of each arrow a (dashed) line, rotated by 45o , i which we come across. These lines represent two-dimensional colored according to Q slices in appropriate colors. The zig-zag path which arises from the above prescription represents the shape of the crystal, as seen from the top. In particular, the corners of two-dimensional partitions arising from slicing of the crystal are located at the end-points of the arrows. The orientation of arrows represents the interlacing condition (i.e. arrows point from a larger to smaller partition), and therefore it indicates a direction in which the crystal grows. The interlacing pattern between two consecutive slices corresponds to the types of two

536

P. Sułkowski

consecutive arrows. Finally, the points from which two arrows point outwards represent those stones in the crystal, which can be removed from the initial, full crystal configurat tion. In fermionic correlators these points correspond to +ti followed by −j operators. All these statements are easy to check in the examples presented in Sect. 4. Evolution of crystals. It is also interesting to analyze the evolution of crystals upon crossing walls of marginal stability. We stress that by a fixed crystal we understand a set of all admissible configurations of its constituents (such as e.g. plane partitions) which fit into a fixed container (such as the positive octant of C3 ). The term evolution refers to the evolution of a shape of this underlying container, upon which the set of admissible crystal configurations of course changes too. Such an evolution arises from changing values of moduli, so that the values of the central charge (4), Z (R, B) =

1 (l + B · β) R

also change. We consider first increasing or decreasing B-field through a fixed p th P1 . We focus first on the noncommutative Donaldson-Thomas chamber with the BPS partition functions given by (40) Z = + |− = 0| . . . A+ A+ A+ |A− A− A− . . . |0. ti vertex operators, the crystal for this chamber is As all A± operators are built from ± built in increasing direction following the string of A+ ’s, and then in decreasing direction following the string of A− ’s. There is only one point from which two arrows point outwards, or equivalently + is followed by − : this is just the point when A+ turns into A− in the above correlator. Therefore for each manifold in this extreme chamber the crystal has only one corner which can be removed from the initial, fully filled crystal configuration. Moving to other chambers corresponds to changing the B-field and inserting walloperators, and therefore the structure of the crystal gets deformed. As explained in Sect. 3.1, the wall-crossing operators (34) and (35) consist of a string of +ti ’s followed t by a string of −j , or vice versa. Therefore insertion of each such wall-crossing operator introduces one additional corner of the crystal. One exemption from this rule arises when W p is inserted into the extreme correlator (40): then the single vertex from the extreme

configuration is replaced by a new single vertex encoded in W p . One can also start from the other extreme chamber corresponding to a single D6-brane (43). Turning on the B-field also modifies the crystal. An insertion of the first t W p operator has no effect, as it has all +ti to the right of −j . Therefore a chamber characterized by an insertion of (W p )n+1 corresponds to a crystal with n corners. Similarly, a chamber characterized by an insertion of (W p )n corresponds to a crystal with n corners. We can also consider changing the sign of the B-field. This corresponds to changing the counting of M2-branes into counting of the anti-M2-branes. At least for the conifold, this can also be interpreted as a flop transition. To change a sign one has to remove all W p operators, cross the extreme chamber, and then start adding W p operators (or vice versa). From the assignment of arrows in Fig. 5 we observe that changing a type of wall-crossing operator corresponds to a perpendicular change of the direction in which the crystal is expanding. Such a crystal interpretation of the flop transition in the conifold case will be discussed in Sect. 4.4.


537

Fig. 6. Toric diagram for C3 (upper left) consists of one triangle and has one ⊕ vertex. Therefore A± operators involve just single ± , which are represented by arrows (lower left) according to Fig. 5. The correlator (23) is translated into a sequence of arrows, with rotated dashed lines representing insertions of interlacing two-dimensional partitions. The resulting figure (right) represents MacMahon crystal for plane partitions from Fig. 1, seen from the bottom (first three layers are shown; it is assumed that each stone in layer m + 1 is blocked by only one stone located immediately above it in layer m)

Finally one can consider changing a sign of R. This corresponds to changing the ground state representing the manifold according to (44). For R > 0 the ground state is represented by |± which includes an infinite number of A± operators, and therefore the crystal extends infinitely in both associated directions. When R changes its sign to negative values the ground state gets replaced by |0, and crystal becomes finite in both associated directions, with the size specified by the number of wall-crossing operators inserted. This dramatic change of size of the crystal provides an interpretation of the so-called DT/PT transition [4,12]. Moreover, in some cases, such as the conifold discussed in Sects. 2.4 and 4.3, the whole crystal is finite for R < 0. However the crystal can also extend infinitely in the third dimension, as is the case for the orbifold of C3 discussed in Sect. 4.2. The above discussion focused on W p and W p operators. It should be possible to generalize them to account for all possible chambers, i.e. construct operators which would represent arbitrary B-fields through an arbitrary set of P1 ’s, not just one fixed P1 . Examples of such more general operators will be discussed in Sect. 4.5. In those more general cases the evolution of crystals is of course more complicated. 4. Examples In this section we illustrate the results presented in Sect. 3 in several instructive examples, which include orbifolds of C3 , resolved conifold and triple-P1 geometry. We discuss a crystal interpretation of the flop transition for the conifold. Finally we find a crystal description of the part of the chamber space of the closed topological vertex, taking advantage of its relation to the triple-P1 geometry and physical viewpoint from Sect. 2.1. 4.1. Revisiting C3 . To start with we reconsider the simplest case of C3 crystal and explain how it fits into the prescription from Sect. 3. For C3 a triangulation of a strip consists just of one triangle of area 1/2, see Fig. 6 (left). Therefore there is just one 0 ≡ Q. The operators (31) take the form vertex and only one color Q A± = ± (1) Q,

538

P. Sułkowski

Fig. 7. Toric diagram for the resolution of C3 /Z N +1 geometry has N + 1 vertices of the same type ⊕; this figure shows the case N = 1. Left: toric diagram and translation of A+ and W 1 into arrows. In the non-commutative Donaldson-Thomas chamber this leads to the same plane partition crystal as in Fig. 6, however now involving two colors (yellow and red online). Middle: for the chamber with positive R and 2 < B < 3 the crystal develops two additional corners and its partition function is given by Z 2|1 = + |(W 1 )2 |− . Right: for negative R and positive n − 1 < B < n the crystal is finite along two axes (albeit still infinite along the third axis perpendicular to the picture) and develops n − 1 corners; its generating function for the case of n = 5 shown in the picture reads Z 5|1 = 0|(W 1 )5 |0 (two external arrows, corresponding to − acting on 0| and + acting on |0, are suppressed)

and there are no wall-crossing operators. Therefore the BPS partition function (40) takes exactly the form (23) and we find that the BPS generating function is given by the MacMahon function. ± operators according to Fig. 5, To reconstruct the crystal we associate arrows to A and draw them in the order which follows the order of vertex operators in (40). The crystal which we obtain is shown in Fig. 6 (right). We indeed reproduce plane partitions from Sect. 2.3, which in our picture are seen from the other side than in the more often encountered Fig. 1. Even though there are no wall-crossing operators in this case, there is the other extreme chamber with a single D6-brane, for which the generating function is given by (43). 4.2. Orbifolds C3 /Z N +1 . We apply now the prescription from Sect. 3 to the case of (the resolution of) C3 /Z N +1 orbifold. The toric diagram looks like a big triangle of area (N + 1)/2. There are N independent P1 ’s, as well as N + 1 vertices of the same ti = +1 type, see Fig. 7 (left). Therefore operators in (31) take the form 1 ± (1) Q 2 . . . ± (1) Q N ± (1) Q 0 . A± = ± (1) Q Thus, in the non-commutative Donaldson-Thomas chamber, the corresponding crystal consists of plane partitions just as in Fig. 6, however now with periodically colored slices in N + 1 colors. The non-commutative Donaldson-Thomas partition function is given by (40). This reproduces the results for C3 /Z N +1 orbifolds from [13]. We can also analyze the wall-crossing related to turning on arbitrary B-field through a fixed P1 . Translating the wall-crossing operators (34) and (35) into arrows following Fig. 5, we obtain crystals in modified containers, as shown in the example in Fig. 7 (middle). Enlarging the B-field by one unit, which corresponds to an insertion of one wall-operator W p , adds one more yellow corner to the crystal. Applying wall-crossing operators W p would result in a crystal which develops red corners.


539

Fig. 8. Left: toric diagram for the conifold and translation of A+ and W 1 operators into arrows. Right: for chambers with negative R and positive n − 1 < B < n the crystals are given by finite pyramid partitions with n − 1 additional corners developed, represented by n − 1 stones in the top row. The corresponding partition function is given by Z n|1 = 0|(W 1 )n |0 which reproduces the result (28) reviewed in Sect. 2.4. This figure shows the case n = 4 (again two external arrows representing 0| − and + |0 are suppressed)

We also immediately find crystals corresponding to R < 0. As usual the crystal is empty in the extreme chamber with a single D6-brane (43). Adding wall-crossing operators results in a crystal which develops corners, as shown in Fig. 7 (right). This crystal is finite along two axes. Nonetheless, because W p and W p consist only of ± operators are involved), it can grow infinitely along the third axis. (and no ± 4.3. Resolved conifold. We reviewed wall-crossing for the conifold in Sect. 2.4. We show now that analyzing it from a perspective of Sect. 3 provides a new proof of some statements posed in literature and extends them to a wide class of manifolds. The toric diagram for the conifold is shown in Fig. 8 (left). There is of course just 1 and Q 0 , so that N = 1 P1 , and two colors Q 0 , q = q1 q0 . = Q 1 Q Q The operators (31) take the form 1 ± 0 , A± (x) = ± (x) Q (x) Q

while (32) and (33) read A+ (x) = + (xq) + (xq/q1 ),

A− (x) = − (x) − (xq1 ),

and satisfy A+ (x)A− (y) =

(1 + x yq/q1 )(1 + x yqq1 ) A− (y)A+ (x). (1 − x yq)2

The quantum geometry is encoded in quantum states (38) and (39) |− = A− (1)A− (q)A− (q 2 ) . . . |0,

(57)

+ | = 0| . . . A+ (q )A+ (q)A+ (1),

(58)

2

540

P. Sułkowski

Fig. 9. Conifold crystal in the chamber with positive R and 2 < B < 3 takes form of pyramid partitions with 3 stones in the top row. Its generating function is given by Z 2|1 = + |(W 1 )2 |−

and the wall-insertion operators (34) (35) are 1 + (x) Q 0 , W 1 (x) = − (x) Q 1 − (x) Q 0 . W 1 (x) = + (x) Q

(59) (60)

From our results in section (3) we immediately find that the fermionic correlators Z n|1 = + |(W 1 )n |− , Z n|1 = 0|(W 1 )n |0,

(61) (62)

result in the pyramid partition functions (26) and (28) reviewed in Sect. 2.4. With no wall-crossing operators inserted the above partition functions give respectively Szendroi’s result Z 0|1 = + |− and pure D6-brane Z = 0|0 = 1. The above correlators have crystal interpretation shown respectively in Figs. 9 and 8 (right). These are of course the same pyramid partitions that appeared in earlier lit erature, see Figs. 2 and 3. Moreover, insertion of the companion W 1 operators leads to pyramids that extend in perpendicular directions, as we will discuss in the next section. In particular, in our formalism we can automatically provide a new proof of a conjecture posed in [8],4 which states that BPS partition functions in chambers separated by finite numbers of walls from the pure D6-brane region are given by generating functions of finite pyramids. This fact is just a consequence of the existence of the fermionic representation (62): on one hand we already discussed that this fermionic representation reproduces appropriate generating functions (28). On the other hand this representation provides a crystal construction encoded in the arrow structure, as we explained in Sect. 3.4. Contrary to C3 /Z N +1 cases, now the crystal pyramids are finite in all three directions, as shown in Fig. 8 (right). This is so, because now the wall-operators operators (60) involve both and operators, which insert interlacing two-dimensional partitions that extend in opposite directions and effectively block each other beyond the length given by the number of inserted W 1 ’s. 4 This has been proved by other means by mathematicians in [15], where also the geometrical meaning of the quivers constructed in [8] was clarified. We note that our approach shows how to generalize the notion of finite pyramids to the wide class of manifolds considered in this paper.


541

Fig. 10. Flop transition of the conifold. Right: there are two equivalent representations of the non-commutative Donaldson-Thomas chamber, with yellow or red stone on top and changes of variables given respectively by (49) and (55). Upper and lower left: extension of the pyramid crystal in opposite directions for positive and negative B-field, represented respectively by insertions of W 1 and W 1 operators

4.4. Flop transition. In Sect. 3.4 we discussed evolution of crystals upon changing the moduli R and B. It is in particular interesting to focus on the case when the B-field changes the sign. According to the interpretation reviewed in Sect. 2.1 this corresponds to the counting of anti-M2-branes instead of M2-branes. However, in case of the conifold such a process can be identified with the flop transition. The crystal interpretation of the flop transition is shown in Fig. 10. The crystal representing a chamber separated by n walls of marginal stability from the non-commutative Donaldson-Thomas chamber, with Kähler parameter Q 1 = −q1 qsn

(63)

and the partition function Z n|1 = + |(W 1 )n |− , is shown in the upper left. The non-commutative Donaldson-Thomas chamber (right) is equivalently represented by Z 0|1 = + |− ≡ Z (with yellow stone on top and a change of variables Q 1 = −q1 = |W | ≡ Z (with red stone on top and a change of variables in (49)), and Z 1|1 + 1 −

Q 1 = −q1 q −1 in (55)). Further insertions of W 1 operators (lower left) extend the crystal in the perpendicular direction, with the Kähler parameter identified as q1 Q 1 = − qsn (64) q

= |(W )n | . Note that changes of variand the partition function given by Z n|1 + − 1 ables in (63) and (64) can be interpreted respectively as increasing and decreasing the Kähler parameter of the singular conifold T1 = − log Q 1 by n units of (−gs ) = log qs .

542

P. Sułkowski

Fig. 11. Toric diagram for the triple-P1 geometry (left) and the corresponding four-colored pyramid crystal in the non-commutative Donaldson-Thomas chamber (we use here subscripts b, c, a to denote various P1 ’s instead of 1, 2, 3 used earlier)

Similar picture holds for finite pyramids in chambers with R < 0, albeit in this case the singularity is represented by the empty pyramid. 4.5. Triple-P1 geometry and more general wall-operators. We discuss now the tripleP1 geometry whose toric diagram is shown in Fig. 11 (left). While this appears to be an obvious generalization of our previous examples, there are two important reasons to consider this case. Firstly, for this geometry (or any other geometry with longer but analogous toric diagram) a simple geometric intuition allows to introduce more general wall-crossing operators (related to turning on non-trivial B-field through all three P1 ’s). Secondly, this case is related to the closed topological vertex geometry, which we will discuss in the next section. We label5 the three independent P1 ’s in this geometry by B, C, A, denote by TB , TC , T A their Kähler parameters, and introduce Q A = e−T A ,

Q B = e−TB ,

Q C = e−TC .

The topological string partition function for this geometry reads tri ple

Ztop

(Q A , Q B , Q C ) = M(qs )2

M(Q A Q C , qs )M(Q B Q C , qs ) . M(Q A , qs )M(Q B , qs )M(Q C , qs )M(Q A Q B Q C , qs ) (65)

All non-zero Gopakumar-Vafa invariants for this geometry are of genus 0, and in class β they read Nβ=0 = −4,

Nβ=±A = Nβ=±B = Nβ=±C = Nβ=±(A+B+C) = 1, Nβ=±(B+C) = Nβ=±(A+C) = −1.

Let us specialize now the general structures from Sect. 3 to the present case. We introduce four colors qg , for g ∈ {b, c, a, 0} (instead of q1 , q2 , q3 , q0 used earlier). We also often use = Q a Q b Q c Q 0 , q = q0 qa qb qc , Q 5 In this section we change notation slightly: we label three P1 ’s by B, C, A instead of 1,2,3 used in Sect. 3, and denote corresponding fermionic parameters respectively by b, c, a.


543

Fig. 12. Building block of triple-P1 crystals. Operators A± and W b , W c , W a are the standard ones introduced in Sect. 3. Operators W and V are new ones, which change values of B-fields through all P1 ’s simultaneously and implement extensions of the top row of the pyramid crystal similarly as in the conifold case

and to check some results against the ones for the resolved conifold also consider specialization q 0 ≡ q c , q 1 ≡ qa ≡ q b .

(66)

Operators (32) and (33) take the form6 A+ (x) = + (xq0 qa qb qc ) + (xq0 qa qc ) + (xq0 qa ) + (xq0 ), and A− (x) = − (x) − (xqb ) − (xqb qc ) − (xqa qb qc ),

and the commutation relation (100) specialize in this case to A+ (x)A− (y) = A− (y)A+ (x) 2

×

(1 + x yqqb )(1 + x y qqb )(1 + x yqqc )(1 + x y qqc )(1+x yqqa )(1+x y qqa )(1 + x yq0 )(1 + x y qq0 ) (1 − x yq)4 (1 − x yqqb qc )(1 − x y qbqqc )(1 − x yqqa qc )(1 − x y qaqqc )

.

(67) The ground states (38) and (39) are |− = A− (1)A− (q)A− (q 2 ) . . . |0,

(68)

+ | = 0| . . . A+ (q )A+ (q)A+ (1).

(69)

2

Operators A± , as well as W b , W c , W a , are translated in Fig. 12 into the arrow form relevant to the construction of the crystals. We note that in the non-commutative Donaldson-Thomas chamber for R > 0, the crystal has the same shape as the pyramid crystal for the conifold with one stone on top, however now it has four colors, see Fig. 11 (right). Its partition function in our framework can be found as (notation (22) is used) tri ple

Z1

= + |− = M(1, q)4

a qc , q) b qc , q) M(q M(q . M(−q a , q) M(−q b , q) M(−q c , q) M(−q a qb qc , q) (70)

6 The operators A , as well as the computation of (70), appeared in [13] in the context of the closed ± topological vertex geometry. We discuss relations between both these geometries further in Sect. 4.6.

544

P. Sułkowski

Fig. 13. Four-colored pyramids: finite one with three stones in the top row, with generating function 2 tri ple = 0|W |0 (left), and infinite one with four stones in the top row and generating function Z4 tri ple

= + |W V |− (right). Identifying variables as qa = qb = q1 and qc = q0 leads to the coniZ4 fold case

This follows from general results in Sect. 3, and of course can also be found directly using commutation relations (67). Comparing with (65), we indeed find that tri ple

Z1

tri ple−P1

(qa , qb , qc ) = Ztop

tri ple−P1

(Q A , Q B , Q C )Ztop

−1 −1 (Q −1 A , Q B , QC )

under the following identification of parameters qs = q, Q A = −qa , Q B = −qb , Q C = −qc . As usual the extreme chamber with R < 0 is represented by the empty pyramid with the generating function tri ple Z1 = 0|0 = 1.

We introduce now new types of crystals which extend our prescription from Sect. 3, and associated to them new wall-crossing operators. As we already saw, in the noncommutative Donaldson-Thomas chamber the crystal with generating function (70) consists of pyramid partitions of the same type as in the conifold, however with four colors. It is reasonable to conjecture that analogous infinite or finite four-colored pyramids, albeit (similarly as in the conifold case) with a string of arbitrary number of stones in the top row, should also provide BPS generating functions in certain chambers. Below we find generating functions of such crystals and show that they indeed reproduce BPS generating functions in chambers with particular choices of B-field through all three P1 ’s upon a simple change of variables. It is clear from Fig. 12, that neither the wall-crossing operators W b , W c , W a which arise from our prescription for the triple-P1 geometry, nor their primed counterparts considered so far, could be used to construct a pyramid of arbitrary length, such as shown in Fig. 13. Therefore we have to introduce new types of operators. To preserve the periodic g operfour-colored pattern they should involve four vertex operators interlaced with Q ators, and their pattern should follow from the assignment presented in Fig. 5. Because there are four colors, the cases with even or odd stones in the top row are different. An inspection of Fig. 5 leads us to define the following operators V = − (1)Q b + (1)Q c − (1)Q a − (1)Q 0 ,

W = − (1)Q b + (1)Q c − (1)Q a + (1)Q 0 .

Their translation to the arrow structure is shown in Fig. 12 (first row on the right).

(71)


545

Insertion of these operators extends the top row respectively by one or two stones. For an arbitrary number of stones in the top row we should insert several operators tri ple W and possibly also a single V . Let Z n denote a generating function for a fourtri ple colored infinite pyramid with n stones in the top row, and Zn a generating function for four-colored finite pyramid with n − 1 stones in the top row. Therefore the generating functions in the infinite case read tri ple

Z 2n

= + |W

n−1

V |− ,

(72)

n

tri ple

Z 2n+1 = + |W |− . tri ple

In particular Z 2

tri ple

= + |V |− and Z 3 tri ple Z 2n tri ple Z 2n+1

= + |W |+ . For the finite case,

n

= 0|W |0,

(73)

n

= 0|W − (1)|0.

We compute now these partition functions explicitly, focusing first on the infinite tri ple tri ple case. To start with we determine ratios Z n+1 /Z n . In fact there are two possibilities, depending on whether n is even or odd. We note that 1 tri ple 1 − , ) − (qb qc ) + ( )| Q| Z 2n+1 = + |An−1 2 | − (1) + ( qb qa qb qc tri ple 1 − , = + |An−1 ) − (qb qc ) − (qa qb qc )| Q| Z 2n 2 | − (1) + ( qb tri ple − . = + |An−1 | − (1) − (qb ) − (qb qc ) − (qa qb qc )| Q| Z 2n−1

2

operators We see that the only difference between these quantities is in the form of ± (and their arguments) written in bold. So if commute + to the right and − to the left, the remaining correlators will be the same. In this way we find ∞ i i−1 q )(1 + q ) tri ple ∞ 0 Z 2n+1 1 − q i qa q c i=1 (1 + q qa , (74) an := tri ple = (1 − qa qc ) q i+1 qi ∞ i Z 2n i=n (1 + q0 )(1 + q qa ) i=1 1 − qa qc

as well as tri ple

bn :=

Z 2n

tri ple

Z 2n−1

∞ qi qi i ∞ 1 + qc i=1 (1 + qb )(1 + q qc ) 1 − qa qc = . 1 − qa qc ∞ (1 + q i qb )(1 + q i ) 1 − q i qa q c i=n

From these ratios and the value of tri ple mulas for all Z n . We find tri ple Z 2n+1

=

tri ple Z1

n

tri ple Z1

qc

(75)

i=1

given in (70) we can reproduce explicit for-

(ai bi )

i=1 4

b qc , q))(1 + qc )n = M(1, q) M(qa qc , q)) M(q

n+i

n+i

n+i ∞ qi qi qi 1+ 1+ 1 + qc q i )n+i 1 + × qa qb qa q b q c i=1 ∞

i−n

i−n q i i−n

i−n 1+ 1 + qa q i 1 + qb q i 1 + qa q b q c q i × , (76) qc i=n+1

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P. Sułkowski

as well as tri ple

Z 2n

tri ple

= Z1

bn

n−1

(ai bi )

i=1 i

q ∞ (1 + qc )n 1 − qa qc = M(1, q) M(qa qc , q)) M(qb qc , q)) 1 − qa q c 1 − qa q c q i i=1

n+i−1

n+i

n+i−1 ∞ qi qi qi i n+i 1+ 1+ 1+ 1 + qc q ) × qa qb qa q b q c i=1

i−n ∞ i−n+1

i−n

i−n+1

qi 1+qa q i 1 + qb q i 1 + qa q b q c q i × . (77) 1+ qc 4

i=n

We checked that upon specialization (66) both formulas above reduce to the generating functions of two-colored pyramid partitions with the same number of stones in the top row (26), as they indeed should. In the finite case we have 1 tri ple 1 ) − (qb qc ) + ( ) − (q)|0, Z 2n+1 = 0|An−1 2 | − (1) + ( qb qa q b q c tri ple 1 Z 2n = 0|An−1 ) − (qb qc )|0, 2 | − (1) + ( qb tri ple = 0|An−1 | − (1)|0. Z 2n−1

2

Commuting these expressions we find tri ple n−1 Z 2n+1 q i+1 = (1 + q i q0 )(1 + ), tri ple qb Z

(78)

tri ple n−1 Z 2n qa qi = (1 + )(1 + q i qc ). tri ple 1 + qa qa Z

(79)

a˜ n :=

2n

i=0

as well as bñ :=

2n−1

Together with Z1

tri ple

tri ple Z 2n+1 =

n

i=0

= 1 it now follows that

(a˜ i b˜i )

i=1

=

i

i n

i q n−i q n−i+1 qan 1 + 1 + 1 + qc q n−i )i 1 + q n−i q0 , (80) n (1 + qa ) qa qb i=1

as well as tri ple = bñ Z 2n

=

n−1

(a˜ i b˜i ) i=1 n

qan (1 + qa )n

1+

i=1

q n−i qa

i

i−1

i−1 q n−i+1 1+ 1 + qc q n−i )i 1 + q n−i q0 . (81) qb


547

Upon specialization (66) both these expressions reduce to the generating functions of two-colored pyramid partitions with the same number of stones in the top row (28), as they should. To sum up, we have found the generating functions of four-colored finite and infinite pyramid partitions, with arbitrary (even or odd) number of stones in the top row. The task that remains is to show that there exist such values of moduli R and B, and such identification between crystal and string parameters, so that the structure of these generating functions is consistent with the structure (6) encoded in the topological string partition function, together with the condition on central charges (5). Below we show that these conditions are indeed met, and therefore the above crystal generating functions do provide the correct BPS counting functions. We consider finite/infinite and even/odd cases separately. tri ple

Z 2n+1 : infinite pyramid, odd chambers. To start with we consider the generating function for an infinite pyramid with odd number of boxes 2n + 1 in the top row (76). Comparison of the form of products involving (1 + qa±1 q i ), (1 + qb±1 q i ) and (1 + qc±1 q i ) with the chambers in the resolved conifold suggests the following identification of the string coupling qs = e−gs and Kähler parameters: qs = q, Q A = −qa q n , Q B = −qb q n , Q C = −qc q −n ,

(82)

as well as R > 0 as was the case for infinite pyramids in the conifold ∞ case as well. In par∞ ticular, under this identification i=n+1 (1 + qa qb qc q i )i−n = i=1 (1 − Q A Q B Q C qsi )i , ∞ A Q C , qs ), M(q b qc , q) = M(Q B Q C , qs ), (1 + qc )n i=1 a qc , q) = M(Q (1 + M(q j j ∞ i n+i qc q ) = j=n (1 − Q C qs ) , etc. Now the form of products involving single Q A , Q B and Q C leads to the following identification of moduli: R > 0, n < B A < n + 1, n < B B < n + 1, −n < BC < −n + 1, for n ≥ 1. (83) This is consistent with the form of products involving Q A Q C , Q B Q C and Q A Q B Q C , however they impose respectively the following additional constraints: 0 0 implies that all factors M(1, q) should indeed be present. To sum up, under identifications (82), the crystal model for an infinite pyramid with 2n + 1 boxes in the top row leads to the generating function tri ple B Q C , qs ) A Q C , qs ) M(Q Z2n+1 = M(1, qs )4 M(Q ∞

i

i

i −1 i i 1 − Q A qsi 1 − Q B qsi 1 − QC qs ) 1 − Q A Q B Q C qsi ×

×

i=1 ∞

i

i

i−1 i i 1 − Q −1 1 − Q −1 1 − Q C qsi−1 A qs B qs

i=n+1

i

−1 −1 i × 1 − Q −1 A Q B Q C qs , which is indeed the consistent generating function of D6-D2-D0 bounds states in the triple-P1 geometry, in chambers specified by (83) and (84).

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P. Sułkowski

tri ple

Z 2n : infinite pyramid, even chambers. Next we consider the generating function for an infinite pyramid with even number of boxes 2n in the top row (77). Now the identification of the string coupling qs and Kähler parameters reads: qs = q,

Q A = −qa q n−1 ,

Q B = −qb q n ,

Q C = −qc q −n .

(85)

The form of products involving Q A , Q B and Q C leads to the following identification of moduli: R > 0, n − 1 0 implies that all factors M(1, qs ) are present. To sum up, in this case the crystal model leads to the generating function tri ple

Z2n

B Q C , qs ) A Q C , qs ) M(Q = M(1, qs )4 M(Q ∞

i

i

i −1 i i 1 − Q A qsi 1 − Q B qsi 1 − QC qs ) 1 − Q A Q B Q C qsi × ×

i=1 ∞

i 1 − Q −1 A qs

i

i+1

i i+1 i 1 − Q −1 1 − Q q q C s s B

i=n

i

−1 −1 i × 1 − Q −1 A Q B Q C qs , which is indeed the generating function of D6-D2-D0 bounds states in the triple-P1 geometry, in chambers specified by (86) and (87). tri ple Z 2n+1 : finite pyramid, odd chambers. Now we consider the generating function for a finite pyramid in odd odd chambers labeled by 2n + 1 (i.e. the pyramid with 2n stones in the top row), given by (80). The identification of the string coupling qs and Kähler parameters reads:

qs =

1 , q

Q A = −qa q −n ,

Q B = −qb q −n−1 ,

Q C = −qc q n ,

(88)

and now R < 0. The form of products involving Q A , Q B and Q C leads to the following identification of moduli: R < 0, n − 1 < B A < n, n < B B < n + 1, −n − 1 < BC < −n, for n ≥ 1, (89) together with the additional constraint: − 1 < B A + BC < 0, −1 < B B + BC < 0, n < B A + B B + BC < n + 1.

(90)

In particular R < 0 implies that factors of M(1, qs ) should indeed be absent in this case.


549

Altogether, in this case the crystal model leads to the generating function tri ple = Z 2n+1

n

i−1

i

i −1 −1 i i−1 i 1− Q −1 1− Q −1 1− Q C qsi )i 1− Q −1 A qs B qs A Q B Q C qs . i=1

This is indeed the generating function of D6-D2-D0 bounds states in the triple-P1 geometry, in chambers specified above. tri ple Z 2n : finite pyramid, even chambers. Finally we consider the generating function for a finite pyramid in even chambers labeled by 2n (i.e. with 2n − 1 stones in the top row), given by (81). The identification of the string coupling qs and Kähler parameters reads:

qs =

1 , q

Q A = −qa q −n ,

Q B = −qb q −n ,

Q C = −qc q n .

(91)

The form of products involving Q A , Q B and Q C leads to the following identification of moduli: R < 0, n − 1 < B A < n, n − 1 < B B < n, −n − 1 < BC < −n, for n ≥ 1. (92) together with additional constraint: − 1 < B A + BC < 0, −1 < B B + BC < 0,

n − 1 < B A + B B + BC < n.

(93)

In particular R < 0 implies that factors of M(1, qs ) should indeed be absent in this case. Therefore, in this case the crystal model leads to the generating function tri ple = (1 − Q C qs ) Z 2n

n−1

i 1 − Q −1 A qs

i

i i 1 − Q −1 1 − Q C qsi+1 )i+1 q B s

i=1

i

−1 −1 i × 1 − Q −1 Q Q q . A B C s This is indeed the generating function of D6-D2-D0 bounds states in the triple-P1 geometry, in chambers specified above.

4.6. Closed topological vertex. In this section we discuss wall-crossing for the closed topological vertex geometry, whose toric diagram is shown in Fig. 14. We will denote various quantities associated to it by the label C. This geometry is also the symmetric resolution of C3 /Z2 × Z2 orbifold. This is an example of the geometry which does not arise from the triangulation of the strip. Nonetheless it does not contain compact four-cycles, therefore the physical arguments reviewed in Sect. 2.1 apply in this case as well. The closed topological vertex shares important similarities with the triple-P1 geometry analyzed in the previous section. In this case let us also denote by A, B, C the three independent P1 ’s, and by T A , TB , TC the corresponding Kähler parameters, and let Q A = e−T A ,

Q B = e−TB ,

Q C = e−TC .

550

P. Sułkowski

Fig. 14. Toric diagram for the closed topological vertex geometry

In particular the topological string partition function for the closed vertex differs from the one of the triple-P1 (65) just by one factor of MacMahon function C (Q A , Q B , Q C ) = M(qs )2 Ztop

M(Q A Q B , qs )M(Q A Q C , qs )M(Q B Q C , qs ) M(Q A , qs )M(Q B , qs )M(Q C , qs )M(Q A Q B Q C , qs ) tri ple

= M(Q A Q B , qs ) · Ztop

(Q A , Q B , Q C ).

(94)

From this we find that all non-zero Gopakumar-Vafa invariants for C are of genus 0, and in class β they read Nβ=0 = −4, Nβ=±A = Nβ=±B = Nβ=±C = Nβ=±(A+B+C) = 1, Nβ=±(A+B) = Nβ=±(B+C) = Nβ=±(A+C) = −1. The close relation between partition functions of C and triple-P1 suggests the existence of similar underlying crystal models in both cases. There is a natural crystal model for the non-commutative Donaldson-Thomas chamber for C, discussed in [13], which consists of four-colored plane partitions colored according to the action of the Z2 × Z2 group. Let us denote the generating function of these partitions by Z C . In [13] it was shown that Z C indeed reproduces the non-commutative Donaldson-Thomas invariants for C. Moreover, it was also shown that it is closely related to four-colored partitions (with one stone on top) which we associated to triple-P1 in the previous section. In particular a qb , q), the generating functions of these two crystal models differ by a factor M(q a qb , q)Z tri ple , Z C = M(q 1 tri ple

(95)

where Z 1 is given by the formula (70). This relation is of course consistent with (94). Our aim now is to extend the relation between the non-commutative DonaldsonThomas invariants for these two geometries beyond the extreme chamber. We focus first on the chambers for triple-P1 associated to extended pyramid partitions from the previous section. We postulate that a generating function of each such partition can be


551

identified with a generating function of BPS invariants of C in a certain chamber, up to some overall factor of the form μn (Q A Q B , qs ).

(96)

a qb ) to reproduce (95) in the extreme chamber. This overall factor should reduce to M(q Moreover, we postulate that similar relations hold for finite pyramids, and BPS generating functions for C and triple-P1 in chambers corresponding to finite four-colored pyramids are identified up to a factor of a form μn (Q A Q B , qs ),

(97)

which reduces to 1 in the extreme chamber with a pure D6-brane. We will apply now the physical arguments of Sect. 2.1 to prove that these postulates are true. The closed vertex BPS generating functions Z C (Q A , Q B , Q C ; B, R) and C (Q A , Q B , Q C ; B, R) associated respectively to chambers corresponding to infinite Z and finite pyramids depend on the values of four moduli: the radius R and B-fields through three P1 ’s which we denote by B A , B B , BC . We prove that these postulates are true by finding the explicit form, for each such pyramid, of: • an identification between crystal q0 , qa , qb , qc and string qs , Q A , Q B , Q C parameters, • values of moduli R, B A , B B , BC , • correction factors μn (qa qb , q) and μn (qa qb , q) for infinite and finite pyramids, such that (q0 , qa , qb , qc ) μn (qa qb , q) = Z C (Q A , Q B , Q C ; B, R), Zn tri ple C (Q A , Q B , Q C ; B, R), Zn (q0 , qa , qb , qc ) μn (qa qb , q) = Z tri ple

(98) (99)

respectively for infinite and finite pyramids. The identification of crystal and string parameters, as well as values of stringy moduli, must be completely specified by n which encodes the length of a given pyramid. In fact, as a consequence of the relation between partition functions (94) we realized the above postulates to some extent already in the previous section, by matching fourcolored pyramid generating functions to BPS counting functions for triple-P1 . All we have to show now is that there exist appropriate μn (Q A Q B , qs ) and μn (Q A Q B , qs ), consistent with the values of moduli, which is a non-trivial condition. Nonetheless this is indeed the case, as we show below for each case of infinite/finite and even/odd chambers. While the identification of wall-crossing chambers completed below is satisfying from the physical point of view, there still remain several interesting questions. Firstly, one might wonder what are the crystals whose generating functions are given by the left-hand side of Eqs. (98) and (99). As reviewed above, in the extreme chamber for infinite pyramids such a crystal in given by Z2 × Z2 -four-colored plane partitions. We conjecture that for all other chambers discussed above, the corresponding crystal model is given in terms of similar four-colored plane partitions, which fill a container that develops an appropriate number of corners, similarly as in Fig. 7. Secondly, it would be interesting to find whether there exist, and if so what are the crystal models associated to other chambers of the closed topological vertex.

552

P. Sułkowski

C : infinite pyramid, odd chambers. We consider first the chambers corresponding Z2n+1 to infinite pyramids with 2n + 1 stones in the top row with generating functions (76). We postulate that the relations (82), (83) and (84) found for the triple-P1 hold also for the closed vertex:

qs = q, Q A = −qa q n , Q B = −qb q n , Q C = −qc q −n , R > 0, n < B A < n + 1, n < B B < n + 1, −n < BC < −n + 1, for n ≥ 1, 0 < B A + BC < 1, 0 < B B + BC < 1, n < B A + B B + BC < n + 1. In particular this implies that 2n < B A + B B < 2n + 1, and therefore the factor (96) can be chosen consistently as μ2n+1 (Q A Q B , qs ) =

∞ i=1

1 (1 −

∞

1

Q A Q B qsi )i i=2n+1

(1 −

−1 i i Q −1 A Q B qs )

.

Finally, the fact that R > 0 implies that all factors M(1, q) should indeed be present. To sum up, under the above identifications, the crystal model for an infinite pyramid with 2n + 1 boxes in the top row leads to the BPS generating function C B Q C , qs )μ2n+1 (Q A Q B , qs ) A Q C , qs ) M(Q Z2n+1 = M(1, qs )4 M(Q ∞ i

i

i

−1 i i 1 − Q A qsi 1 − Q B qsi × 1 − QC qs ) 1 − Q A Q B Q C qsi

×

i=1 ∞

i

i

i−1 −1 i i i−1 1 − Q −1 1 − Q 1 − Q q q q C s A s B s

i=n+1

i

−1 −1 i × 1 − Q −1 Q Q q , A B C s and this is indeed a consistent generating function of D6-D2-D0 bounds states for the closed topological vertex, in chambers specified above. C : infinite pyramid, even chambers. Next we consider the chambers related to infinite Z2n pyramids with 2n stones in the top row, with generating functions (77). Again we assume that the relations (85), (86) and (87) found for the triple-P1 hold for the closed vertex:

qs = q, Q A = −qa q n−1 , Q B = −qb q n , Q C = −qc q −n , R > 0, n − 1 < B A < n, n < B B < n + 1, −n < BC < −n + 1, for n ≥ 1, 0 < B A + BC < 1, 0 < B B + BC < 1, n − 1 < B A + B B + BC < n. These values of background fields also imply that the factor (96) can be chosen consistently as μ2n (Q A Q B , qs ) =

∞ i=1

1 (1 −

∞

Q A Q B qsi )i i=2n

1 (1 −

−1 i i Q −1 A Q B qs )

.


553

The fact that R > 0 implies that all factors M(1, qs ) are present. To sum up, in this case the crystal model leads to the BPS generating function C B Q C , qs )μ2n (Q A Q B , qs ) A Q C , qs ) M(Q Z2n = M(1, qs )4 M(Q ∞ i

i

i

−1 i i 1 − Q A qsi 1 − Q B qsi × 1 − QC qs ) 1 − Q A Q B Q C qsi i=1 ∞

i

i+1

i

i −1 −1 i i i+1 1 − Q −1 1− Q −1 1 − Q C qsi 1− Q −1 × , A qs B qs A Q B Q C qs i=n

which is a consistent generating function of D6-D2-D0 bounds states for the closed topological vertex, in chambers specified by (86) and (87). C : finite pyramid, odd chambers. We turn to consider the generating function for Z 2n+1 finite pyramids associated to odd 2n + 1 chambers, with generating functions (80). From the analysis of triple-P1 we know that (88), (89) and (90) should also hold: 1 , Q A = −qa q −n , Q B = −qb q −n−1 , Q C = −qc q n , q R < 0, n − 1 < B A < n, n < B B < n + 1, −n − 1 < BC < −n, for n ≥ 1, −1 < B A + BC < 0, −1 < B B + BC < 0, n < B A + B B + BC < n + 1. qs =

Now R < 0 implies that factors of M(1, qs ) should be absent. The above values of background fields imply that the factor (97) should be chosen as μ2n+1 (Q A Q B , qs ) =

2n i=1

1 (1 −

−1 i i Q −1 A Q B qs )

.

Altogether, in this case the crystal model leads to the BPS generating function C 2n+1 Z = μ2n+1 (Q A Q B , qs )

n

i−1

i −1 i i−1 1 − Q −1 1 − Q 1 − Q C qsi )i q q A s B s i=1

i

−1 −1 i × 1 − Q −1 Q Q q . s A B C This is the consistent generating function of D6-D2-D0 bounds states in the closed topological vertex geometry, in chambers specified above. C : finite pyramid, even chambers. Finally we consider the generating functions assoZ 2n ciated with finite pyramids and even 2n chambers, with generating functions (81). The analysis of the triple-P1 case implies (91), (92) and (93), 1 , Q A = −qa q −n , Q B = −qb q −n , Q C = −qc q n , q R < 0, n − 1 < B A < n, n − 1 < B B < n, −n − 1 < BC < −n, for n ≥ 1, −1 < B A + BC < 0, −1 < B B + BC < 0, n − 1 < B A + B B + BC < n. qs =

554

P. Sułkowski

Now R < 0 implies that factors of M(1, qs ) should indeed be absent in this case. These values of background fields also imply that the factor (97) should take the form μ2n (Q A Q B , qs ) =

2n−1 i=1

1 (1 −

−1 i i Q −1 A Q B qs )

.

Altogether, in this case the crystal model leads to the generating function C 2n Z = μ2n (Q A Q B , qs )(1− Q C qs )

× 1−

−1 −1 i Q −1 A Q B Q C qs

i

n−1

i 1 − Q −1 A qs

i

i i 1 − Q −1 1 − Q C qsi+1 )i+1 q s B

i=1

.

This is a consistent generating function of D6-D2-D0 bounds states for the closed topological vertex, in chambers specified above.

5. Proofs In this section we provide proofs of statements from Sect. 3, which relate BPS generating functions with fermionic correlators.

5.1. Quantization of geometry. In this section we prove (41), which states that Z = Z, under the identification of parameters (42). Here Z = + |− , is the overlap of the states |± which encode information about the classical geometry and Z denotes the BPS partition function in the non-commutative Donaldson-Thomas chamber. The states |± are defined in terms of A± as in (38) and (39). Using (21) one can check that A± satisfy the commutation relation A+ (x)A− (y) = C(x, y)A− (y)A+ (x), with 1 C(x, y) = 1 − ti t j x yq (qi qi+1 · · · q j−1 ) N +1 (1−x yq) 1≤i< j≤N +1

−ti t j x yq × 1 − ti t j . qi qi+1 · · · q j−1

(100)


555

The proof of (41) therefore amounts to repeated application of relations (100) in order to commute all + operators to the right of − , similarly as in the case of MacMahon function and (23). We get + |− = 0| . . . A+ (q 2 )A+ (q)A+ (1)|A− (1)A− (q)A− (q 2 ) . . . |0 =

∞

C(q r , q s )

r,s=0

= M(1, q) N +1

∞

l 1 − ti t j q l (qi qi+1 · · · q j−1 )

l=1 1≤i< j≤N +1

× 1 − ti t j

ql qi qi+1 · · · q j−1

l −ti t j .

This expression indeed reproduces Z = Ztop (Q i )Ztop (Q i−1 ), with Ztop (Q i ) given in (29) and changing variables according to (42). In particular, under the redefinition (42) each factor of the infinite product becomes

−ti t j l ±1 l 1 − (ti t j )(ti ti+1 Q i±1 )(ti+1 ti+2 Q i+1 ) · · · (t j−1 t j Q ±1 )q s j−1 = (1 − (Q i · · · Q j−1 )±1 qsl )−ti t j l . Therefore one always gets an overall minus sign inside the bracket, as expected for Ztop (Q i±1 ). On the other hand, the overall power is either +l or −l, if the i th and j th vertex are respectively of the opposite or of the same type. This means that a factor we consider appears respectively in numerator or denominator, in accordance with the “effective” local neighborhood of a chain of P1 ’s Q i · · · Q j−1 being respectively O(−1) ⊕ O(−1) or O(−2) ⊕ O. This proves (41).

5.2. Wall-crossing operators. In this section we prove statements from Sect. 3.3, considering respectively all possible signs of R and B-field. In all formulas below we represent operators W p , W p and A± as in (36) and (37), with the same fixed p. Chambers with R < 0, B > 0. Here we prove the equality (45) and in consequence (47). To compute the expectation value Z n| p = 0|(W p )n |0,

(101)

ti operators. It is convenient to approach we have to commute all +ti to the right of all − this problem recursively. We therefore assume that manipulating (W p )n to get a form ordered in such a way gives rise to an overall coefficient Cn arising from commutation ti to the right: relations between ± . Therefore, in addition moving all Q’s

(W p ) = Cn n

n−1 i=0

γ−1 (q i )

n−1 i=0

γ+2 (q i−1 )

n . Q

556

P. Sułkowski

Now an insertion of one more wall-operator can be written as (W p )

n+1

= Cn+1

n

γ−1 (q i )

i=0

= Cn

n−1

n i=0

γ−1 (q i )

n−1

i=0

= Cn

n−1

c p (q

n−i

)

i=0

n

γ+2 (q i−1 )

i=0

n

n+1 Q

γ+2 (q i−1 )

γ−1 (q i )

i=0

Q

n

γ−1 (x)γ+2 (x/q) Q n+1 , Q

γ+2 (q i−1 )

i=0

where p N +1

c p (x y) =

1 − (tr ts )

s=1 r = p+1

xy qs qs+1 · · · qr −1

−tr ts

arises from a commutation of γ+2 (x/q)γ−1 (y) = γ−1 (y)γ+2 (x/q)c p (x y). Therefore CCn+1 = n n−1 n−i ) and we get c (q i=0 p Z n| p = Cn = =

n−1

n−1 n−1 k−1 Ck+1 = c p (q k−i ) = c p (q n−i )i Ck

k=1 p +1 n−1 N

i=1 s=1 r = p+1

k=1 i=0

1 − (tr ts )

q n−i

i=1

−tr ts i

qs qs+1 · · · qr −1

,

which proves (45). A change of variables (46) leads finally to the identification of crystal and BPS partition functions (47). Chambers with R > 0, B > 0. Here we prove the equality (48). We wish to compute Z n| p = + |(W p )n |− .

(102)

This case is more involved than the previous one with R < 0, because apart from the ti ordering of ± in the product of wall-operators, we also have to commute infinite sets ti of ± ’s encoded in |± . It is again convenient to approach this problem recursively and determine Z n+1| p /Z n| p . First note that we can write Z n| p = + |(W p )n |γ−1 (1)γ−2 (1)|A− (q)A− (q 2 ) . . . |0,

Z n+1| p = + |(W p )n |γ−1 (1)γ+2 (q −1 )|A− (q)A− (q 2 ) . . . |0.

(103)

The only difference between these two expressions is that γ−2 (1) appears in the former instead of γ+2 (q −1 ) in the latter. After commuting these operators respectively to the left and to the right we will be left with the same correlator. First of all, the contribution from commuting γ−2 (1) to the left in Z n| p over γ+2 ’s (from all A+ and W p ) is the same as


557

the contribution from γ+2 (q −1 ) passing over all γ−2 in Z n+1| p . Then, commuting γ−2 (1) in Z n| p to the left over all γ+2 gives a factor p N +1

∞ −tr ts 1 − (tr ts )q l+n+1 qs qs+1 · · · qr −1 , Sn = l=0 s=1 r = p+1

while commuting γ+2 (1) to the right in Z n+1| p over all γ−1 gives Tn+1

p N ∞ +1 = 1 − (tr ts ) l=0 s=1 r = p+1

q l+1 qs qs+1 · · · qr −1

−tr ts .

Let us now assume that Z n| p has the form given in (48) (0) (1) (2) Z n| p = M(1, q) N +1 Z n| p Z n| p Z n| p ,

(104)

with the following factors: (0) Z n| p

=

∞

1 − (tr ts )

l=1 p ∈s,r / +1⊂1,N +1

ql qs qs+1 · · · qr −1

−tr ts l

−tr ts l

× 1 − (tr ts )q l qs qs+1 · · · qr −1 , (1)

Z n| p =

∞

−tr ts l 1 − (tr ts )q l+n qs qs+1 · · · qr −1 ,

l=1 p∈s,r +1⊂1,N +1 (2) Z n| p

=

∞

1 − (tr ts )

l=n+1 p∈s,r +1⊂1,N +1

q l−n qs qs+1 · · · qr −1

−tr ts l .

Then the relation between correlators in (103) implies that (0)

(1)

(2)

Z n+1| p = M(1, q) N +1 Z n| p Z n| p Z n| p

Tn+1 . Sn

Now we check that (1) Z n| p

Sn (0)

(1)

= Z n+1| p ,

(2)

(2)

Z n| p Tn+1 = Z n+1| p .

(0)

The factor Z n| p = Z n+1| p in fact does not depend on n, so we conclude that (0)

(1)

(2)

Z n+1| p = M(1, q) N +1 Z n+1| p Z n+1| p Z n+1| p . This form is consistent with the assumption of the induction proof (104). This proves (48), and further change of variables (49) leads to the identification of crystal and BPS partition functions (49).

558

P. Sułkowski

Chambers with R < 0, B < 0. Here we prove (51). We wish to compute the correlator n Z n| p = 0|(W p ) |0.

(105)

The proof is analogous as in the case R < 0 and B > 0. As we already discussed, Z 0| p = 0|0 = 1.

But contrary to the case with B > 0, an insertion of a single W p has a non-trivial effect. (W p )n

ti Again we assume that manipulating to bring all +ti to the right of − gives rise to an overall coefficient Cn : n−1 n−1 n 2 i 1 −i−1 n . (W p ) = Cn γ− (q ) γ+ (q ) Q i=0

i=0

An insertion of one more wall-operator leads to the relation n n−i Cn+1 = Cn c p (q ) , i=0

where cp (x y) =

p N +1

(1 − (tr ts )x yqs qs+1 · · · qr −1 )−tr ts

s=1 r = p+1

arises from a commutation of γ+1 (x/q)γ−2 (y) = γ−2 (y)γ+1 (x/q)cp (x y). Therefore Z n| p = C n = Z 1| p

=

n−1

k=1 p n N +1

Ck+1 = Z 1| cp (q k−i ) = cp (q n−i )i p Ck n−1 k

k=1 i=0

1 − (tr ts )q n−i qs qs+1 · · · qr −1

n

i=1

−tr ts i

.

i=1 s=1 r = p+1

This proves (51). Changing then variables according to (52) proves (53). Chambers with R > 0, B < 0. In the last case we prove (54) and compute

n Z n| p = + |(W p ) |− .

(106)

The proof is analogous to the case with B > 0. Again we have to commute all +ti to the ti right of − . First note that we can write

n 1 2 2 Z n| p = + |(W p ) |γ− (1)γ− (1)|A− (q)A− (q ) . . . |0,

n 1 −1 2 2 Z n+1| p = + |(W p ) |γ+ (q )γ− (1)|A− (q)A− (q ) . . . |0.

(107)

The only difference between these two expressions is that γ−1 (1) appears in the former instead of γ+1 (q −1 ) in the latter. After commuting these operators respectively to the


559

left and to the right we will be left again with the same correlator. The only nontrivial 2 contributions will arise from commuting γ−1 (1) in Z n| p to the left over all γ+ , Sn

=

p N ∞ +1 l=0 s=1 r = p+1

q l+n+1 1 − (tr ts ) qs qs+1 · · · qr −1

−tr ts ,

2 as well as from commuting γ+1 (q −1 ) to the right in Z n+1| p over all γ− , Tn+1

p N ∞ +1

−tr ts 1 − (tr ts )q l qs qs+1 · · · qr −1 = . l=0 s=1 r = p+1

We assume now that Z n| p has the form given in Eq. (54), (0)

(1)

(2)

N +1 Z n| p Z n| p Z n| p , Z n| p = M(1, q)

(108)

with the following factors: (0) Z n| p

=

∞

1 − (tr ts )

l=1 p ∈s,r / +1⊂1,N +1

ql qs qs+1 · · · qr −1

−tr ts l

−tr ts l

× 1 − (tr ts )q l qs qs+1 · · · qr −1 , (1)

Z n| p =

∞

l=n p∈s,r +1⊂1,N +1 (2) Z n| p

=

∞

−tr ts l 1 − (tr ts )q l−n qs qs+1 · · · qr −1 , 1 − (tr ts )

l=1 p∈s,r +1⊂1,N +1

q l+n qs qs+1 · · · qr −1

−tr ts l .

The relation between correlators in (107) implies that (0)

(1)

(2) Tn+1 . Sn

N +1 Z n+1| Z n| p Z n| p Z n| p p = M(1, q)

We now check that (2) Z n| p

Sn (0)

(2)

= Z n+1| p ,

(1)

(1)

Z n| p Tn+1 = Z n+1| p .

(0)

The factor Z n| p = Z n+1| p does not depend on n, and we conclude that (0)

(1)

(2)

N +1 Z n+1| p Z n+1| p Z n+1| p . Z n+1| p = M(1, q)

This form is consistent with the assumption of the induction proof (108), and therefore (54) is proved. Further change of variables (55) leads to the identification of crystal and BPS partition functions (56).

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P. Sułkowski

6. Summary and Discussion In this paper we developed a fermionic approach to BPS counting in local, toric Calabi-Yau manifolds without compact four-cycles. We also discussed its crystal melting interpretation, and explained the structure of crystals associated to all manifolds in this class, in a large set of chambers. There are however several issues which require further analysis. Firstly, we considered mainly chambers associated to turning on arbitrary B-field of magnitude n through one, fixed p th two-cycle in the geometry, in terms of correlators of the wall-crossing operators of the form (W p )n . In addition we analyzed a few examples of turning on B-fields of similar magnitude through all two-cycles simultaneously, and found the corresponding wall-crossing operators W and V in Sect. 4.5. It would certainly be interesting to find more general wall-crossing operators and associated crystal interpretation for all chambers, i.e. for arbitrary values of B-fields through any set of two-cycles turned on simultaneously. Secondly, and more conceptually, it would be interesting to find a physical reason for the occurrence of these free fermions. For example, such an interpretation has been found in [23] for related, but different fermions arising in the B-model topological vertex [21]. Supposedly our setup would require extension of [23] to include chamber dependence. It is also interesting to see if there is more direct connection between the framework of [21] and ours, and what would be the role of integrable hierarchies in our context. As we saw in various examples in Sect. 4, our crystals unify and generalize various other Calabi-Yau crystal models which appeared previously, such as plane partitions for C3 and pyramid partitions for the conifold. We also claim that these new crystals are equivalent to crystals introduced in [28], for the class of manifolds that we consider. They must also be related to other statistical models of crystals, dimers, and associated quivers, considered these days in [14,29,31–33]. In particular, modification of crystals upon crossing the walls of marginal stability must translate to the dimer shuffling operation of the corresponding dimer models [11,14,9]. From our results we see that this is obviously true for the conifold. In fact it is not known how to perform dimer shuffling beyond the conifold case. To define it, one could therefore try to translate the effect of the insertion of our wall-crossing operators in the corresponding dimer models. We note that there is also a different crystal model for the closed topological vertex [34], given in terms of plane partitions in a box whose sides are of finite size and translate to Kähler parameters of three P1 ’s. Furthermore, yet another model for the resolved conifold, related to the deformed boson-fermion correspondence, has been discussed in [35]. It would be interesting to relate these models to the present results. There are also further generalizations one might think of. The string coupling gs could be refined to two independent parameters 1 , 2 . In certain special cases such refinements were interpreted combinatorially, both for two-dimensional [22] and three dimensional partitions [36]. Such a refinement was also found for pyramid partitions for the conifold [9] and shown to be equivalent to considering motivic BPS invariants of Kontsevich and Soibelman [6]. It is tempting to generalize our results in this spirit, both from the perspective of fermionic states as well as crystals. Such generalization could also be translated to refined dimer shuffling operations. One could also consider open topological string amplitudes [37] and their wallcrossing from a fermionic and crystal viewpoint. In the large radius chamber open string amplitudes have been interpreted in terms of defects in crystals and analyzed in [38,39,3]. It would be nice to extend this interpretation to other chambers and crystals that we propose. The chain of dualities for open string wall-crossing proposed in [40]


561

should also be helpful in finding such an interpretation. This should also be consistent with the approach to open non-commutative Donaldson-Thomas invariants proposed in [30,31]. Finally, one could extend our results to other geometries, such as non-toric manifolds considered in [18,41,42]. Acknowledgements. I thank Jim Bryan for discussions which inspired this project. I am also grateful to Robbert Dijkgraaf, Albrecht Klemm, Hirosi Ooguri, Yan Soibelman, Balazs Szendroi, Cumrun Vafa, and Masahito Yamazaki for useful conversations. I appreciate the hospitality and inspiring atmosphere of the Focus Week on New Invariants and Wall Crossing organized at IPMU in Tokyo, International Workshop on Mirror Symmetry organized at the University of Bonn, and 7th Simons Workshop on Mathematics and Physics, as well as the High Energy Theory Group at Harvard University. This research was supported by the DOE grant DE-FG03-92ER40701FG-02, the Humboldt Fellowship, the Foundation for Polish Science, and the European Commission under the Marie-Curie International Outgoing Fellowship Programme. The contents of this publication reflect only the views of the author and not the views of the European Commission. 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.

References 1. Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and Classical Crystals. http://arxiv.org/ abs/hep-th/0309208v2, 2003 2. Iqbal, A., Nekrasov, N., Okounkov, A., Vafa, C.: Quantum foam and topological strings. JHEP 0804, 011 (2008) 3. Sułkowski, P.: Calabi-Yau crystals in topological string theory. PhD thesis, available at http://arxiv.org/ abs/0712.2173v1 [hep-th], 2007 4. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov-Witten theory and DonaldsonThomas theory, I. http://arxiv.org/abs/math/0312059v3 [math.AG], 2004 5. Denef, F., Moore, G.: Split states, entropy enigmas, holes and halos. http://arxiv.org/abs/hep-th/ 0702146v2, 2007 6. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. http://arxiv.org/abs/0811.2435v1 [math.AG], 2008 7. Jafferis, D., Moore, G.: Wall crossing in local Calabi Yau manifolds. http://arxiv.org/abs/0810.4909v1 [hep-th], 2008 8. Chunag, W., Jafferis, D.: Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions. Commun. Math. Phys. 292, 285–301 (2009) 9. Dimofte, T., Gukov, S.: Refined, Motivic, and Quantum. Lett. Math. Phys. 91, 1 (2010) 10. Szendroi, B.: Non-commutative Donaldson-Thomas theory and the conifold. Geom. Topol. 12, 1171– 1202 (2008) 11. Young, B.: Computing a pyramid partition generating function with dimer shuffling. http://arxiv.org/abs/ 0709.3079v2 [math.CO], 2008 12. Pandharipande, R., Thomas, R.: Stable pairs and BPS invariants. J. Amer. Math. Soc. 23, 267–297 (2010) 13. Bryan, J., Young, B.: Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds. http://arxiv.org/abs/0802.3948v2 [math.CO], 2008 14. Mozgovoy, S., Reineke, M.: On the noncommutative Donaldson-Thomas invariants arising from brane tilings. http://arxiv.org/abs/0809.0117v2 [math.AG], 2008 15. Nagao, K., Nakajima, H.: Counting invariant of perverse coherent sheaves and its wall-crossing. http://arxiv.org/abs/0809.2992v4 [math.AG], 2009 16. Nagao, K.: Derived categories of small toric Calabi-Yau 3-folds and counting invariants. http://arxiv.org/ abs/0809.2994 [math.AG] 17. Dijkgraaf, R., Vafa, C., Verlinde, E.: M-theory and a topological string duality. http://arxiv.org/abs/hepth/0602087v1, 2006 18. Aganagic, M., Ooguri, H., Vafa, C., Yamazaki, M.: Wall Crossing and M-Theory. http://arxiv.org/abs/ 0908.1194v1 [hep-th], 2009 19. Witten, E.: Quantum Background Independence In String Theory. http://arxiv.org/abs/hep-th/9306122v1, 1993

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20. Ooguri, H., Vafa, C., Verlinde, E.: Hartle-Hawking Wave-Function for Flux Compactifications. Lett. Math. Phys. 74, 311 (2005) 21. Aganagic, M., Dijkgraaf, R., Klemm, A., Marino, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451–516 (2006) 22. Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. http://arxiv.org/abs/hep-th/ 0306238v2, 2003 23. Dijkgraaf, R., Hollands, L., Sułkowski, P., Vafa, C.: Supersymmetric gauge theories, intersecting branes and free fermions. JHEP 0802, 106 (2008) 24. Dijkgraaf, R., Hollands, L., Sułkowski, P.: Quantum Curves and D-modules. JHEP 0911, 047 (2009) 25. Jimbo, M., Miwa, T.: Solitons and Infinite Dimensional Lie Algebras. Kyoto University, RIMS 19, 943– 1001 (1983) 26. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, Oxford: Oxford Univ. Press, 1995 27. Iqbal, A., Kashani-Poor, A.: The vertex on a strip. Adv. Theor. Math. Phys. 10, 317 (2006) 28. Ooguri, H., Yamazaki, M.: Crystal Melting and Toric Calabi-Yau Manifolds. Commun. Math. Phys. 292, 179–199 (2009) 29. Nagao, K.: Noncommutative Donaldson-Thomas theory and vertex operators. http://arxiv.org/abs/0910. 5477v3 [math.AG], 2009 30. Nagao, K.: Refined open noncommutative Donaldson-Thomas invariants for small crepant resolutions. http://arxiv.org/abs/0907.3784v2 [math.AG], 2009 31. Nagao, K., Yamazaki, M.: The Non-commutative Topological Vertex and Wall Crossing Phenomena. http:// arxiv.org/abs/0910.5479v1 [hep-th], 2009 32. Young, B.: To appear 33. Bryan, J., Cadman, C., Young, B.: The orbifold topological vertex. http://arxiv.org/abs/1008.4205v1 [math.AG], 2010 34. Sułkowski, P.: Crystal Model for the Closed Topological Vertex Geometry. JHEP 0612, 030 (2006) 35. Sułkowski, P.: Deformed boson-fermion correspondence, Q-bosons, and topological strings on the conifold. JHEP 0810, 104 (2008) 36. Iqbal, A., Kozcaz, C., Vafa, C.: The Refined Topological Vertex. JHEP 0910, 069 (2009) 37. Ooguri, H., Vafa, C.: Knot Invariants and Topological Strings. Nucl. Phys. B577, 419–438 (2000) 38. Saulina, N., Vafa, C.: D-branes as defects in the Calabi-Yau crystal. http://arxiv.org/abs/hep-th/ 0404246v2, 2004 39. Halmagyi, N., Sinkovics, A., Sułkowski, P.: Knot invariants and Calabi-Yau crystals. JHEP 0601, 040 (2006) 40. Cecotti, S., Vafa, C.: BPS Wall Crossing and Topological Strings. http://arxiv.org/abs/0910.2615v2 [hepth], 2009 41. Hollowood, T., Iqbal, A., Vafa, C.: Matrix Models, Geometric Engineering and Elliptic Genera. JHEP 0803, 069 (2008) 42. Sułkowski, P.: Matrix models for 2* theories. Phys. Rev. D 80, 086006 (2009) Communicated by A. Kapustin


Communications in


Hölder Continuity of Absolutely Continuous Spectral Measures for One-Frequency Schrödinger Operators Artur Avila1,2 , Svetlana Jitomirskaya3 1 CNRS UMR 7586, Institut de Mathématiques de Jussieu, Paris, France 2 IMPA, Estrada Dona Castorina, 22460-320 Rio de Janeiro, Brazil.


3 University of California, Irvine, California 92697, USA. E-mail: [email protected]

Received: 18 December 2009 / Accepted: 7 April 2010 Published online: 2 November 2010 – © Springer-Verlag 2010

Abstract: We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrödinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Hölder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies. 1. Introduction In this work we study absolutely continuous spectral measures of (one-frequency) quasiperiodic Schrödinger operators H = Hλv,α,θ defined on 2 (Z), (H u)n = u n+1 + u n−1 + λv(θ + nα)u n ,

(1.1)

where v is the potential, λ ∈ R is the coupling constant, α ∈ R\Q is the frequency and θ ∈ R is the phase. A central example is given by the almost Mathieu operator, when v(x) = 2 cos(2π x). Except where otherwise noted, below we assume the frequency α to be Diophantine in the usual sense (see definition in Sect. 3), and v analytic. Absolutely continuous spectrum occurs only rarely in one-dimensional Schrödinger operators [R]. Until recently it was expected that in the class of ergodic Schrödinger operators it only occurs for almost periodic potentials, a conjecture recently disproved [A2]. Quasiperiodic operators with analytic potential stand out in this respect as the family {Hλv,α,θ }λ∈R , for small couplings λ, is always in the metallic phase (has good transport properties) with zero Lyapunov exponents and absolutely continuous spectrum. This work was supported in part by NSF, grant DMS-0601081, and BSF, grant 2006483. This research was partially conducted during the period A.A. served as a Clay Research Fellow.

564

A. Avila, S. Jitomirskaya

We will be concerned with the regularity of spectral measures. More precisely, given f a function f ∈ 2 (Z) with f = 1, and letting μ f = μλv,α,θ be the associated spectral measure1 , what can be said of the modulus of continuity of the distribution of μ f ? We will assume that f is a reasonably localized function in the sense that f ∈ 1 (Z) (notice that without regularity assumptions, there are no non-trivial restrictions on μ f : any probability measure absolutely continuous with respect to some spectral measure is still a spectral measure). Our first result concerns small potentials: Theorem 1.1. For every v ∈ C ω (R/Z, R), there exists λ0 = λ0 (v) > 0 such that if f |λ| < λ0 and α is Diophantine, then μλv,α,θ (J ) ≤ C(α, λv)|J |1/2 f 21 , for all intervals J and all θ. For the almost Mathieu operator, one can take λ0 = 1. Remark 1.1. The smallness constant λ0 only depends on bounds on the analytic extension of v to some band |x| < . This is important for applications to arbitrary potentials (see below). The constant C depends on bounds on the analytic extension of λv and on the Diophantine properties of α. Recall that averaging the distributions of spectral measures with respect to the phase θ yields the integrated density of states (i.d.s.), whose regularity is therefore significantly simpler to analyze. Indeed in [AJ], it is shown that the i.d.s. is 1/2-Hölder (and no more, see below) in the setting of Theorem 1.1. The averaged 1/2-Hölder behavior is compatible with point spectrum (consider the almost Mathieu operator with λ > 1, [J,AJ]), and hence discontinuous distributions. The key point of Theorem 1.1 is that here we are able to control the behavior of each individual spectral measure, uniformly on θ . The study of small potentials is not merely interesting on its own: it gives information about the absolutely continuous spectrum of an arbitrary potential. To make this precise, one introduces the notion of almost reducibility: roughly speaking an energy is almost reducible if the associated cocycle (α, A(E−λv) ) (a dynamical system (x, w) → (x + α, A(E−λv) · w), E − λv(x) −1 , A(E−λv) = 1 0

(1.2) (1.3)

that describes the behavior of solutions of the eigenvalue equation Hλv,α,θ u = Eu), is analytically conjugate (in a uniform band) to the associated cocycle of some (α, A(E −v ) ) with v arbitrarily small. It follows from renormalization [AK1,AK2], that almost reducible energies (indeed reducible energies, for which v can be taken as 0) form an essential support of absolutely continuous spectrum. In [AJ], almost reducibility was proved for all energies in the case of small potentials (indeed the same setting of Theorem 1.1), which implies that almost reducibility is stable (in particular, the set of almost reducible energies is open). Theorem 1.2. Let v ∈ C ω (R/Z, R) and let α be Diophantine. Then for any almost ac ) there exists C, > 0 such that reducible energy E ∈ v,α , (thus for a.e. energy in v,α f

if J ⊂ (E − , E + ) is an interval then, for all θ, μv,α,θ (J ) ≤ C|J |1/2 f 1 . As far as we know Theorems 1.2,1.1 are the first results on fine properties of individual absolutely continuous spectral measures of ergodic operators. 1 That is, μ f (X ) = ( f )2 where : 2 (Z) → 2 (Z) is the spectral projection associated to the X X Borel set X ⊂ R.

Continuity of Spectral Measures

565

Let us call attention to the following conjecture that clarifies the fundamental importance of understanding almost reducibility: Spectral Dichotomy Conjecture. For typical v, α, θ , Hv,α,θ is the direct sum of operators H+ and H− with disjoint spectra such that H+ is “localized” and H− is “almost reducible”. Remark 1.2. (1) Typical should be understood in the measure-theoretical sense of prevalence. In particular frequencies may be assumed to be Diophantine. (2) Localization for H+ means both what is usually understood as Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) or dynamical localization. Almost reducibility for H− just means that the spectrum of H− is the closure of almost reducible energies for H , but as described above, it indeed provides a very fine spectral description: in particular the results of [AJ] and this paper apply to H− , e.g., it has absolutely continuous spectral measures with 1/2-Hölder distributions. (3) By Kotani theory, see e.g. [LS], if the conjecture holds, then H+ /H− must be defined by spectral projection on the parts of the spectrum where the associated cocycle has positive/zero Lyapunov exponent, 1 ln A(E−v) (x)d x, (1.4) L(E) = lim n n R/Z A(E−v) (x) = A(E−v) (x + (n − 1)α) · · · A(E−v) (x). n

(1.5)

The result of disjointness of the spectra for this decomposition was recently established (in the typical setting) [A3,A4]. (4) With H+ defined as above, the precise spectral and dynamical description, particularly dynamical localization, follows (in the typical setting) from a minoration of the Lyapunov exponent through the spectrum of H+ , using [BG] and [BJ3]. Such minoration is a consequence of disjointness of spectra [A3,A4] and continuity of the Lyapunov exponent [BJ1]. More is known in this regime [JL3,GS2,GS3]. (5) What is still incomplete in the above picture is the description of H− . Zero Lyapunov exponent does not necessarily imply almost reducibility (consider the critical almost Mathieu operator). In [A3,A4], it is shown that (in the typical setting) energies in the spectrum of of H− satisfy not only L(E) = 0 but the stronger condition (called subcriticality) ln An (x) = o(n)

(1.6)

uniformly in some band |x| < . The Spectral Dichotomy Conjecture is thus reduced to the Almost Reducibility Conjecture (the main outstanding problem in the theory): subcriticality implies almost reducibility. 1.1. Further perspective. One should distinguish between two possible regimes of small |λ| (similar considerations can be applied to the analysis of large coupling). One is perturbative, meaning that the smallness condition on |λ| depends not only on the potential v, but also on the frequency α: the key resulting limitation is that the analysis at a given coupling, however small, has to exclude a positive Lebesgue measure set of α. Such exclusions are inherent to the KAM-type methods that have been traditionally used in this context. The other, stronger regime, is called non-perturbative, meaning that the

566


smallness condition on |λ| only depends on the potential, leading to results that hold for almost every α. A thorough study of absolutely continuous spectrum of operators (1.1) in the case of small analytic potentials in the perturbative regime was done by Eliasson [E]. He proved the reducibility of the associated cocycle for almost all energies in the spectrum and fine estimates on solutions for the other energies, by developing a sophisticated KAM scheme, which avoided the limitations of earlier KAM methods (that go back to the work of Dinaburg-Sinai [DiS] and that excluded parts of the spectrum from consideration). This allowed him in particular to conclude purely absolutely continuous spectrum. A thorough study of absolutely continuous spectrum of operators (1.1) in the nonperturbative regime of [BJ2] was done in [AJ] where we used some techniques of [BJ2] to obtain localization estimates for all energies for the dual model, and developed quantitative Aubry duality theory, which allowed us, in particular, to conclude almost reducibility for all energies (including those for which neither dual localization nor reducibility hold). The smallness condition on the coupling constant in [AJ] coincides with that of [BJ2]. In particular, for the almost Mathieu operator, all the estimates and conclusions hold throughout the subcritical regime λ < 1. The analyses of [E] and [AJ] allowed to obtain sharp bounds (Hölder-1/2 continuity) for the integrated density of states, for Diophantine frequencies. This was done, in perturbative and non-perturbative regimes in correspondingly [Am] and [AJ]. Earlier, Goldstein-Schlag [GS2] had shown Hölder continuity of the integrated density of states for a full Lebesgue measure subset of Diophantine frequencies in the regime of positive Lyapunov exponents, with the result becoming almost sharp for the super-critical almost Mathieu operator: (1/2 − )-Hölder for any , and |λ| > 1. For this model their result also gives the same bound in the sub-critical regime |λ| < 1, by duality. Before that Bourgain [B1] had obtained almost 1/2-Hölder continuity for almost Mathieu type potentials in the perturbative regime, for Diophantine α and ln |λ| large (depending on α). There were no results however, neither recently nor previously, on the modulus of continuity of the individual spectral measures, even in the perturbative regime. In this paper we achieve this by applying methods developed in [AJ] combined with a dynamical reformulation of the power-law subordinacy techniques of [JL2,JL3]. As mentioned above, our all energy results hold throughout the regime of [BJ2], and in particular, for all sub-critical almost-Mathieu operators. The general absolutely continuous case is obtained through a reduction to the small potential case and almost reducibility result of [AK1]. Our estimate is optimal in several ways. First, there are square-root singularities at the boundaries of gaps (e.g., [P2]), so the modulus of continuity cannot be improved. Also, since the integrated density of states satisfies n−1 1 σk( f ) μ (0, E] n→∞ n

N (E) = lim

(1.7)

k=0

(σ : 2 (Z) → 2 (Z) denotes the shift), the spectral measures of 1 functions cannot have higher modulus of continuity than N (E). There are examples with lower regularity of N (E) that demonstrate that Diophantine condition on α as well as a condition on λ are essential here. In particular, it is known that for the almost Mathieu operator for a certain non-empty set of α which satisfy good Diophantine properties (but has zero Lebesgue measure) and λ = 1, the integrated density of states is not Hölder ([B3], Remark after


567

Corollary 8.6). Additionally, for any λ = 0 and generic α, the integrated density of states is not Hölder (this is because the Lyapunov exponent is discontinuous at rational α, which easily implies that it is not Hölder for generic α. Such discontinuity holds for the almost Mathieu operator and presumably generically). Remark 1.3. As our approach is non-perturbative and non-KAM, it is not expected to break down at the Brjuno condition and can potentially be extended much further. While, as mentioned above, the exact modulus of continuity should depend on the Diophantine properties for very well approximated α we expect the same methods to work for small rate of exponential approximation as well. We do not pursue it here though. 2. Preliminaries For a bounded analytic (perhaps matrix valued) function f defined on a strip {|z| < } and extending continuously to the boundary, we let f = sup|z|< | f (z)|. If f is a bounded continuous function on R, we let f 0 = supx∈R | f (x)|. 2.1. Cocycles. Let α ∈ R\Q, A ∈ C 0 (R/Z, SL(2, C)). We call (α, A) a (complex) cocycle. The Lyapunov exponent is given by the formula 1 L(α, A) = lim (2.1) ln An (x)d x, n→∞ n where An , n ∈ Z, is defined by (α, A)n = (nα, An ), so that for n ≥ 0, An (x) = A(x + (n − 1)α) · · · A(x).

(2.2)

We say that (α, A) is uniformly hyperbolic if there exists a continuous splitting C2 = E s (x) ⊕ E u (x), x ∈ R/Z such that for some C > 0, c > 0, and for every n ≥ 0, An (x) · w ≤ Ce−cn w, w ∈ E s (x) and A−n (x) · w ≤ Ce−cn w, w ∈ E u (x). In this case, of course L(α, A) > 0. Given two cocycles (α, A(1) ) and (α, A(2) ), a (complex) conjugacy between them is a continuous B : R/Z → SL(2, C) such that A(2) (x) = B(x + α)A(1) (x)B(x)−1 .

(2.3)

We assume now that (α, A) is a real cocycle, that is, A ∈ C 0 (R/Z, SL(2, R)). The notion of real conjugacy (between real cocycles) is the same as before, except that we ask for B ∈ C 0 (R/Z, PSL(2, R)). Real conjugacies still preserve the Lyapunov exponent. We say that a (real) cocycle (α, A) is analytically reducible if it is (real) conjugate to a constant cocycle, and the conjugacy is analytic. We say that it is almost reducible if there exists a sequence A(n) ∈ C ω (R/Z, R) converging (uniformly in some band {|z| < }) to a constant, such that (α, A(n) ) is conjugated to (α, A), and the conjugacies extend holomorphically to some fixed band: B (n) ∈ C ω (R/Z, PSL(2, R)).2 2.2. Schrödinger operators. We consider now Schrödinger operators {Hv,α,θ }θ∈R (we incorporate the coupling constant into v). The spectrum = v,α does not depend on 2 In fact this last property is automatic, but this is non-trivial (it follows from the openness of almost reducibility).

568


θ , and it is the set of E such that (α, A(E−v) ) is not uniformly hyperbolic, with A(E−v) as in the Introduction. f For f ∈ l 2 (Z) the spectral measure μ = μx is defined so that 1 dμ(E ) (2.4) (Hx − E)−1 f, f = R E −E holds for E in the resolvent set C\ . Alternatively, for a Borel set X , f

μv,α,θ (X ) = X f 2 ,

(2.5)

where X is the corresponding spectral projection. The integrated density of states is the function N : R → [0, 1] that can be defined by (1.7). It is a continuous non-decreasing surjective (for bounded potentials) function. The Thouless formula relates the Lyapunov exponent to the integrated density of states, L(E) = ln |E − E|d N (E ). (2.6) R

2.3. Almost reducibility and the support of absolutely continuous spectrum. We justify the claim made in the Introduction that almost reducible energies support the absolutely continuous part of the spectral measures. By [LS], the set 0 = {L(E) = 0} (which is closed by [BJ1]) is the essential support of ac spectrum, so that the ac spectral measures are precisely those probability measures on 0 which are equivalent to Lebesgue. Theorem 2.1 ([AFK]). If α ∈ R\Q, then for almost every E ∈ 0 , (α, A(E−v) ) is real analytically conjugated to a cocycle of rotations, i.e., taking values in SO(2, R). This result was proved in [AK1] under a full measure condition on α (which is stronger than Diophantine). For Diophantine α, it can also be obtained as a consequence of [AJ] and [AK2]. It is easy to see that for any α ∈ R\Q, analytic cocycles of rotations are almost reducible. Moreover, if α is Diophantine (more generally, if the best rational approximations to α are subexponential), then analytic cocycles of rotations are reducible. 2.4. Almost reducibility in Schrödinger form. While almost reducibility allows one to conjugate the dynamics of the cocycle close to a constant, it is rather convenient to have the conjugated cocycle in Schrödinger form, since many results (particularly the ones depending on Aubry duality, as the ones obtained in [AJ]) are obtained only in this setting. The following result takes care of this. Lemma 2.2. Let (α, A) ∈ R\Q × C ω (R/Z, SL(2, R)) be almost reducible. Then there exists 0 > 0 such that for every γ > 0, there exists v ∈ C ω0 (R/Z, R) with v 0 < γ , E ∈ R and B ∈ C ω0 (R/Z, PSL(2, R)) such that B(x + α)A(x)B(x)−1 = A(E−v) (x). Moreover, for every 0 < ≤ 0 , there exists δ > 0 such that if A˜ ∈ C ω (R/Z, SL(2, R)) is such that A˜ − A < δ then there exists v˜ ∈ C ω (R/Z, R), such that v ˜ < γ ˜ + α) A(x) ˜ B(x) ˜ −1 = and B˜ ∈ C ω (R/Z, PSL(2, R)) such that B˜ − B < γ and B(x ˜ (x). A(E−v)


569

For the proof, one basically just needs to be able to convert non-Schrödinger perturbations of Schrödinger cocycles to Schrödinger form. This problem is studied in [A4]. For completeness, we will give a much simpler (unpublished) argument of Avila-Krikorian which is enough for our purposes. Lemma 2.3. Let v ∈ C ω (R/Z, R) and α ∈ R\Q, be such that 1/v ∈ C ω (R/Z, R). If A ∈ C ω (R/Z, SL(2, R)) and A − A(v) is sufficiently small (depending on v and 1/v ), then there exists v ∈ C ω (R/Z, R) and B ∈ C ω (R/Z, SL(2, R)) such that v − v and B − id are small and B(x + α)A(x)B(x)−1 = A(v ) (x). w1 w2 ∈ C ω (R/Z, sl(2, R)) be such that w is small and Proof. Let w = w3 −w1 s s2 A = A(v) ew . Let s = 1 ∈ C ω (R/Z, sl(2, R)) be defined by s1 = 0, s2 (x) = s3 −s1 1 (x) 1 (x−α) , s3 (x) = − wv(x−α) , and let v˜ ∈ C ω (R/Z, R) be given by w2 (x) + wv(x)

v(x) ˜ = v(x) − w3 (x) + w2 (x + α) +

w1 (x + α) w1 (x − α) + v(x)w1 (x) − . v(x + α) v(x − α)

˜ ew˜ , where w Then v˜ − v ≤ Cw and es(x+α) A(x)e−s(x) is of the form A(v) ˜ ≤ Cw2 , for some constant C depending on v and 1/v . The result follows by iteration.

Remark 2.1. (1) This result with only assuming v ∈ C ω (R/Z, R) to be non-identically zero is proved in [A4]. (2) For Diophantine α the result holds with no conditions on v ∈ C ω (R/Z, R). Proof of Lemma 2.2. Since (α, A) is almost reducible, if 0 > 0 is small then there exists a sequence B (n) ∈ C ω0 (R/Z, PSL(2, R)) and A∗ ∈ SL(2, R) such that B (n) (x + α)A(x)B (n) (x)−1 − A∗ 0 → 0. Let us show that, up to changing B (n) to C (n) B (n) for an appropriate choice of C (n) ∈ C ω0 (R/Z, SL(2, R)), we may assume that A∗ is of the E −1 form with E = 0. Indeed: 1 0 ˜ (1) If |trA∗ | > 2, by converting first to the diagonal form, we can find C A∗ ∈ SL(2, R) trA −1 ∗ , so we can just take C (n) = C A∗ . such that C˜ A∗ A∗ C˜ −1 A∗ = 1 0 (2) If |trA∗ | < 2, there exists C˜ A∗ ∈ SL(2, R) such that C˜ A∗ A∗ C˜ −1 A∗ = Rθ for some θ = k/2, k ∈ Z. If 0 < sin 2π θ < 1, then let Cθ−1

1 = (sin 2π θ )1/2

0 − sin 2π θ 1 − cos 2π θ

,

2 cos 2π θ −1 , and we can take C (n) = Cθ C˜ A∗ . Otherwise, 1 0 let k ∈ Z be such that 0 < sin 2π(θ + kα) < 1, and take C (n) (x) = Cθ+kα Rkx C˜ A∗ .

so that Cθ Rθ Cθ−1 =

570


(3) If |trA∗ | = 2, there exists C˜ (n) ∈ SL(2, R) such that C˜ (n) A∗ (C˜ (n) )−1 → Rθ , where θ = 0 or θ = 1/2. Indeed, either A∗ is equal to such Rθ or we can assume it is in the Jordan form, in which case one can take n 0 (n) ˜ . C = n 1 n By choosing n appropriately, we may also assume that C˜ (n) 2 B (n) (x + α)A(x)B (n) (x)−1 − A∗ 0 → 0. Choosing again k ∈ Z such that 0 < sin 2π(θ + kα) < 1 we can take C (n) = Cθ+kα Rkx C˜ (n) . Now the first statement follows from Lemma 2.3. For the second statement, apply again Lemma 2.3. 3. Estimates on the Dynamics Here we describe the [AJ] estimates on the dynamics of almost reducible cocycles. 3.1. Rational approximations. Let qn be the denominators of the approximants of α. We recall the basic properties: qn αR/Z =

inf

1≤k≤qn+1 −1

kαR/Z ,

1 ≥ qn+1 qn αR/Z ≥ 1/2. We say that α is Diophantine if numbers.

ln qn+1 ln qn

(3.1) (3.2)

= O(1). Let DC ⊂ R be the set of Diophantine

3.2. Resonances. Let α ∈ R, θ ∈ R, 0 > 0. We say that k is an 0 -resonance if 2θ − kαR/Z ≤ e−|k| 0 and 2θ − kαR/Z = min 2θ − jαR/Z . | j|≤|k|

Remark 3.1. In particular, there always exists at least one resonance, 0. If α ∈ DC(κ, τ ), 2θ − kαR/Z ≤ e−|k| 0 implies 2θ − kαR/Z = min 2θ − jαR/Z | j|≤|k|

for k > C(κ, τ ). For fixed α and θ , we order the 0 -resonances 0 = n 0 < |n 1 | ≤ |n 2 | ≤ · · ·. We say that θ is 0 -resonant if the set of resonances is infinite. If θ is non-resonant, with the set of resonances {n 0 , . . . , n j } we formally set n j+1 = ∞. The Diophantine condition immediately implies exponential repulsion of resonances: Lemma 3.1. If α ∈ DC, then c 0 |n j | |n j+1 | ≥ c2θ − n j α−c , R/Z ≥ ce

where c = c(α, 0 ) > 0.


571

3.3. Dynamical estimates. Let us say that a cocycle (α, A) ∈ R\Q×C ω (R/Z, SL(2, R)) is (C, c, 0 )-good if there exists θ ∈ R with the following property: for any finite 0 -resonance n j associated to α and θ , denoting n = |n j | + 1 and N = |n j+1 |, there exists : R/Z → SL(2, C) analytic with cn −C ≤ Cn C such that (x + α)A(x)(x)−1 =

0 e2πiθ 0 e−2πiθ

+

q1 (x) q(x) , q3 (x) q4 (x)

(3.3)

with q1 cn −C , q3 cn −C , q4 cn −C ≤ Ce−cN

(3.4)

and qcn −C ≤ Ce−cn(ln(1+n))

−C

.

(3.5)

The following is one of the main estimates of [AJ] (combining Theorems 3.3, 3.4 and 5.1 of [AJ]): Theorem 3.2 [see Theorems 3.4 and 5.1 of [AJ]]. There exists a constant c0 > 0 with the following property. Let v ∈ C ω (R/Z, R) and E ∈ v,α , (α, A). If for some 0 < < 1, v < c0 3 then (α, A) is (C, c, 0 )-good for some constants c = c( , α) > 0, C = C( , α) > 0 and 0 = 0 ( ). A more precise result is available for the almost Mathieu operator (still a combination of Theorems 3.4 and 5.1 of [AJ]): Theorem 3.3 [see Theorems 3.4 and 5.1 of [AJ]]. For every 0 < λ0 < 1 and α Diophantine, there exists C = C(λ0 , α), c = c(λ0 , α), 0 = 0 (λ0 ) > 0 such that for v = 2λ cos 2π(x + θ ) with |λ| < λ0 , and E ∈ v,α , (α, A(E−v) ) is (C, c, 0 )-good. Coupling Theorem 3.2 and Lemma 2.2 we immediately get: Theorem 3.4. Let α be Diophantine and let A ∈ C ω (R/Z, SL(2, R)). If (α, A) is almost reducible, then there exists ¯ > 0 such that for every 0 < < ¯ there exist δ, C, c, 0 > 0 ˜ is not unisuch that if A˜ ∈ C ω (R/Z, SL(2, R)) is such that A˜ − A < δ and (α, A) ˜ is (C, c, 0 )-good. formly hyperbolic then (α, A) Remark 3.2. Using [A1], Theorem 3.8, one can consider a stronger definition of goodness, so that Theorem 3.2, and hence Theorem 3.4, and Theorem 3.3, still hold: c ≤ Cn C , q j c ≤ Ce−cN , j = 1, 3, 4, and qc ≤ Ce−cn . An immediate consequence of (C, c, 0 )-goodness is (see [AJ] for the easy argument): Lemma 3.5. If (α, A) is (C, c, 0 )-good, then for every s ≥ 0 we have As 0 ≤ C (C, c, 0 , α)(1 + s).

572


4. Regularity of the Spectral Measures at Good Energies −1 0 + μev,α,x and ei is the Dirac mass at Let v ∈ C ω (R/Z, R), E ∈ v,α . Let μx = μv,α,x i ∈ Z. Our main estimate is:

e

Theorem 4.1. If (α, A(E−v) ) is (C0 , c0 , 0 )-good, then for every 0 < < 1,μx (E − , E + ) ≤ C (C0 , c0 , 0 , α) 1/2 . Proof of Theorems 1.1 and 1.2. We first prove Theorem 1.2. By Theorem 3.4, (α, A(E −v) ) is (C0 , c0 , 0 )-good for any E near E which is in the spectrum. By Theorem 4.1, we get μx (J ) ≤ C |J |1/2

(4.1)

for any interval containing such an E , and hence (since μx is supported on the spectrum), for any interval contained in a neighborhood of E. Let σ : l 2 (Z) → l 2 (Z) be σf f the shift f (i + 1) = σ f (i). Then σ Hv,α,x σ −1 = Hv,α,x+α . Thus μx+α = μx and f ek e0 μx = μx+kα ≤ μx+kα . By (2.5), μx (E − , E + )1/2 defines a semi-norm on l 2 (Z). f Therefore, by the triangle inequality, μx (J )1/2 ≤ k∈Z | f (k)|(μx+kα (J ))1/2 , and the result follows immediately from (4.1). Theorem 1.1 is proved analogously, using Theorems 3.2 and 3.3 to establish appropriate (C, c)-goodness. It therefore remains to prove Theorem 4.1 which we do in Sect. 4.2. Through the end of this section, A = A(E−v) . We will use C and c for large and small constants that only depend on C0 , c0 , 0 , and α. 4.1. Spectral measures and m-functions. In the study of μ = μx , we will use a result of [JL2] (or its improvement in [KKL]), interpreted in terms of cocycles. In the definition of the m-functions below, we follow the notation of [JL3]. We will consider energies E + i , E ∈ R, > 0. Then there are non-zero solutions u ± of H u ± = (E + i )u ± which are l 2 at ±∞, well defined up to normalization. We define m± = ∓

u± 1 u± 0

.

(4.2)

It coincides with the Weyl-Titchmarsh m-function which is the Borel transform of the spectral measure μ± = μ± e0 of the corresponding half-line problem with Dirichlet boundary conditions: dμ± (x) ± m (z) = , x−z (e.g. [CL]). Thus m ± has positive imaginary part for every > 0. Let 1 dμ(E ). M(E + i ) = E − (E + i )

(4.3)


573

Notice that M(E + i ) ∈ H = {z, z > 0}. We have M(E + i ) ≥

1 μ(E − , E + ). 2

(4.4)

Then, as discussed in [JL3], M=

m+m− − 1 . m+ + m−

(4.5)

As in [JL3], we define m +β = R−β/2π · m + , or, more generally, z β = R−β/2π z. β

Those are Borel transforms of the half-line spectral measures μβ = μe0 of operator H on l 2 ([0, ∞)) with boundary conditions u 0 cos β + u 1 sin β = 0. Here we make use of the action of SL(2, C) on C, az + b a b ·z = . c d cz + d Let ψ(z) = supβ |z β |. We have ψ(z)−1 ≤ z ≤ |z| ≤ ψ(z),

(4.6)

where the first inequality easily follows from the invariance of φ, see below. It was shown in [DKL] that, as a corollary of the maximal modulus principle, one obtains |M| ≤ ψ(m + ).

(4.7)

This also can be shown directly by the following computation, that gives some more quantitative estimates. Let φ(z) =

1 + |z|2 . 2z

If z ∈ H then φ(z) ≥ 1. φ(z) is invariant with respect to the action of Rβ . Thus the maximum of |z β | is attained when z β is purely imaginary with z β > 1 and is easily checked to be equal to φ(z) + (φ(z)2 − 1)1/2 . Thus ψ(z) = φ(z) + (φ(z)2 − 1)1/2 . We can compute φ(M) =

φ(m + )φ(m − ) + 1 , φ(m + ) + φ(m − )

(4.8)

which implies φ(M) ≤ φ(m + ) and hence ψ(M) ≤ ψ(m + ) (whatever the value of m − ∈ H). By (4.6), this gives (4.7).

(4.9)

574


For k ≥ 1 integer, let P(k) =

k

A∗2 j−1 (x + α)A2 j−1 (x + α).

(4.10)

j=1

Then P(k) is an increasing family of positive self-adjoint linear maps. In particular, det P P(k) , P(k)(k) and det P(k) are increasing positive functions. It is not difficult to see that P(k) (and hence det P(k) ) is also unbounded (since A j ∈ SL(2, R) implies trP(k) ≥ 2k). Lemma 4.2. Let be such that det P(k) = C −1
or j = r . Thus, by (3.5) and (4.27), −C | yˆ j | + | yˆ j | ≤ Ce−cn(ln(1+n)) (1 + |r |)C . Y − id 0 = y0 ≤ j≤C(1+|r |)C

C(1+|r |)C < j≤

(4.31) Let = Y . By (C0 , c0 , 0 )-goodness, 0 ≤ Cn C , which together with (4.31) gives 0 ≤ C(n C + e−cn(ln(1+n))

−C

(1 + |r |)C ).

(4.32)

Let k ≥ 1 be maximal such that for 1 ≤ k < k , if we let T˜ (x) = (x + α)A(x)(x)−1 and X˜ =

k

T˜2∗j−1 T˜2 j−1 ,

(4.33)

j=1

then X˜ − X 0 ≤ 1,

(4.34)

where X is as in Lemma 4.3. Notice that T˜ − T = Y (x + α)H (x)Y (x)−1 , so T˜ − T 0 ≤ Y 20 H 0 . By Lemma 4.4, (4.34) implies −2 −4 Y 20 H 0 ≥ ck (1 + 2k |qˆr |)−2 ≥ ck

(4.35)

(since (3.5) implies |qˆr | ≤ C), so that (4.29) and (4.31) imply k ≥ c min{ecn

−C

, −C ecN }.

(4.36)


579

Notice that P(k) (x) ≤ 40 X˜ (x + α) and −1 −1 −1 −1 ˜ ≥ −4 P(k) 0 X (x + α) .

Since X˜ ≤ X + 1 and X˜ −1 −1 ≥ X −1 −1 − 1 for 1 ≤ k < k , Lemma 4.3 and (4.32) imply P(k) ≤ C(n C + e−cn(ln(1+n)) −1 −1 P(k) ≥ c(n C + e

−C

(1 + |r |)C )k(1 + |qˆr |2 k 2 ), C ≤ k < k ,

−cn(ln(1+n))−C

(1 + |r |)C )−1 k, C ≤ k < k .

(4.37) (4.38)

Thus P(k)

−1 −3 P(k)

≤C

(2) n

+ e−c

C (2)

+ e−c

(3) n(ln(1+n))−C (2)

(1 + |r |)C

(2)

k2 (3) n(ln(1+n))−C (2)

+ C (2) (n C

(2)

(2)

(1 + |r |)C )|qˆr |2 , for C (2) < k < k .

(4.39)

By (3.5), C (2) (n C

(2)

+ e−c

(3) n(ln(1+n))−C (2)

(2)

(1 + |r |)C )|qˆr |2 ≤ C.

(4.40)

Let − k = (n C

(2)

+ e−c

(3) n(ln(1+n))−C (2)

(2)

(1 + )C )1/2 .

(4.41)

Then (4.39) and (4.40) imply P(k)

−1 −3 P(k)

− ≤ C, k < k < k .

(4.42)

In order to conclude, we have to show that for Cn C < k < cecN there exists > n − such that k < k < k . But this is clear from (4.36) and (4.41) that one can find such with ≈ n C ln k. Corollary 4.6. For k ≥ 1, we have P(k) ≤ C(P(k) )−1 −3 . Proof. It follows from Theorem 4.5 and Lemma 3.1.

Set k = 4 det1 P(k) . −1/2

Corollary 4.7. We have ψ(m + (E + i k )) ≤ C k

Proof. We have P(k) = det P(k) (P(k) )−1 < and the statement follows from (4.11).

.

C P(k) −1/3 . k2

−3/2

Thus P(k) ≤ C k

580


Proof of Theorem 4.1. By (4.4) it is enough to show that c 1/2 ≤ M(E + i ) ≤ C −1/2 .

(4.43)

For any bounded potential and any solution u we have u L+1 ≤ Cu L , thus by (4.17), k+1 > c k . Since 1 M(E + i ) is monotonic in it therefore suffices to prove (4.43) for = k . But it follows immediately from Corollary 4.7 and (4.6), (4.9). Remark 4.3. Corollary 4.7 can be refined: for any 0 < < 1 we have ψ(m + (E + i )) ≤ C −1/2 (thus it is not necessary to restrict to a subsequence of the k ). Indeed, by the definition of ψ(z) it is enough to show that |m +β | ≤ C 1/2 for arbitrary β. Our proof can be easily adapted to show 1/2-Hölder continuity of spectral measures μ+β associated to half-line problems (with appropriate boundary conditions) whose Borel transform is m +β , so that

|m +β (E + i )| ≤

dμ+β

(x − E)2 + 2 1 = dt μ x ∈ R |x − E| < − 2 t2 0 ∞ x 1/4 −1/2 < C dx . (x + 1)3/2 0

1

(4.44)

Remark 4.4. It would be interesting to obtain estimates on the modulus of absolute continuity of the spectral measures. It does not seem unreasonable that for all X, μ(X ) ≤ C|X |1/2 . Heuristically, the densities of the spectral measures are unbounded just because of the presence of those countably many (but quickly decaying) square-root singularities located at the gap boundaries. We point out that this is extremely similar to what is expected from the densities of physical measures of typical chaotic unimodal maps. References [Am] [A1] [A2] [A3] [A4] [AFK] [AJ] [AK1] [AK2] [Be] [B1]

Amor, S.: Hölder continuity of the rotation number for quasi-periodic co-cycles in SL(2, R). Commun. Math. Phys. 287(2), 565–588 (2009) Avila, A.: Absolutely continuous spectrum for the almost Mathieu operator with subcritical coupling. Preprint available at http://arxiv.org/abs/1006.0704 [math.DS], 2010 Avila, A.: Absolute continuity without almost periodicity. In preparation. Avila, A.: Global theory of one-frequency Schrödinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity. Preprint http://arXiv.org/abs/ 0905.3902v1 [math.DS], 2009 Avila, A.: Global theory of one-frequency Schrödinger operators II: acriticality and the finiteness of phase transitions. Preprint, avilable at http://w3impa.br/~avila/global2.pdf, 2010 Avila, A., Fayad, B., Krikorian, R.: A KAM scheme for SL(2, R) cocycles over Liouvillean rotations. http://arXiv.org/abs/1001.2878v1 [math.DS], 2010 Avila, A., Jitomirskaya, S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010) Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2) 164(3), 911–940 (2006) Avila, A., Krikorian, R.: Monotonic cocycles. In preparation Berezanskii, Y.: Expansions in eigenfunctions of selfadjoint operators. Transl. Math. Monogr., Vol. 17. Providence, RI: Amer. Math. Soc. 1968 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)


[B3] [BG] [BJ1] [BJ2] [BJ3] [CL] [DKL] [DiS] [E] [GS2] [GS3] [J] [JL2] [JL3] [KKL] [LS] [P2] [R]

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Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematics Studies, 158. Princeton, NJ: Princeton University Press, 2005 Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. of Math. (2) 152(3), 835–879 (2000) Bourgain, J., Jitomirskaya, S.: Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Stat. Phys. 108(5-6), 1203–1218 (2002) Bourgain, J., Jitomirskaya, S.: Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148(3), 453–463 (2002) Bourgain, J., Jitomirskaya, S.: Anderson localization for the band model. Lecture Notes in Math. 1745, Berlin: Springer, 2000, pp. 67–79 Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Boston, MA: Birkhauser, 1990 Damanik, D., Killip, R., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals. iii. α-continuity. Commun. Math. Phys. 212(1), 191–204 (2000) Dinaburg, E., Sinai, Ya.: The one-dimensional schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9, 279–289 (1975) Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic schrödinger equation. Commun. Math. Phys. 146(3), 447–482 (1992) Goldstein, M., Schlag, W.: Fine properties of the integrated density of states and a quantitative separation property of the dirichlet eigenvalues. Geom. Funct. Anal. 18(3), 755–869 (2008) Goldstein, M., Schlag, W.: On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations. Ann. Math. (to appear) Jitomirskaya, S.Ya.: Metal-insulator transition for the almost mathieu operator. Ann. of Math. (2) 150(3), 1159–1175 (1999) Jitomirskaya, S., Last, Y.: Power-law subordinacy and singular spectra. i. half-line operators. Acta Math. 183(2), 171–189 (1999) Jitomirskaya, S., Last, Y.: Power law subordinacy and singular spectra. ii. line operators. Commun. Math. Phys. 211(3), 643–658 (2000) Killip, R., Kiselev, A., Last, Y.: Dynamical upper bounds on wavepacket spreading. Amer. J. Math. 125(5), 1165–1198 (2003) Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators. Invent. Math. 135(2), 329–367 (1999) Puig, J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19(2), 355–376 (2006) Remling, C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. (2010, to appear)

Communicated by B. Simon


Communications in


Holography and Wormholes in 2+1 Dimensions Kostas Skenderis1,2 , Balt C. van Rees1,3 1 Institute for Theoretical Physics, P. O. Box 94485, 1090 GL Amsterdam, The Netherlands.


2 Korteweg-de Vries Institute for Mathematics, P. O. Box 94248, 1090 GE Amsterdam,

The Netherlands. E-mail: [email protected]

3 YITP, State University of New York, Stony Brook, NY 11794-3840, USA

Received: 22 December 2009 / Accepted: 21 June 2010 Published online: 1 December 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract: We provide a holographic interpretation of a class of three-dimensional wormhole spacetimes. These spacetimes have multiple asymptotic regions which are separated from each other by horizons. Each such region is isometric to the BTZ black hole and there is non-trivial spacetime topology hidden behind the horizons. We show that application of the real-time gauge/gravity duality results in a complete holographic description of these spacetimes with the dual state capturing the non-trivial topology behind the horizons. We also show that these spacetimes are in correspondence with trivalent graphs and provide an explicit metric description with all physical parameters appearing in the metric. Contents 1. 2. 3. 4. 5.

Introduction and Summary of Results . . . . . . . . . . . Lorentzian Wormholes . . . . . . . . . . . . . . . . . . . 2.1 Wormholes as quotient spacetimes . . . . . . . . . . 2.2 Physical properties . . . . . . . . . . . . . . . . . . Euclidean Wormholes . . . . . . . . . . . . . . . . . . . 3.1 Construction . . . . . . . . . . . . . . . . . . . . . . Holographic Interpretation of Euclidean Wormholes . . . 4.1 Bulk interpretation and relation to Teichmüller theory 4.2 Non-handlebodies . . . . . . . . . . . . . . . . . . . Holographic Interpretation of Lorentzian Wormholes . . . 5.1 Naive computation . . . . . . . . . . . . . . . . . . 5.2 Lorentzian gauge/gravity prescription . . . . . . . . 5.3 Gauge/gravity duality for Lorentzian wormholes . . . 5.4 More fillings . . . . . . . . . . . . . . . . . . . . . . 5.5 State dual to wormholes . . . . . . . . . . . . . . . . 5.6 2-point functions . . . . . . . . . . . . . . . . . . .

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Remarks . . . . . . . . . . . . . . 6.1 Other bulk spacetimes . . . . . 6.2 Rotating wormholes . . . . . . 7. Outlook . . . . . . . . . . . . . . . A. Coordinate Systems . . . . . . . . A.1 Fenchel-Nielsen coordinates . A.2 Construction of the charts . . . A.3 Parameters . . . . . . . . . . . A.4 Fatgraph description . . . . . . A.5 Transition functions . . . . . . B. Eternal Black Holes and Filled Tori

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608 608 609 610 611 611 613 619 619 620 623

1. Introduction and Summary of Results The gauge/gravity duality [1–3] has significantly enhanced our understanding of gravity and gauge theory. This can be ascribed largely to a well developed dictionary that translates results between string and gauge theory. Although the entries in the dictionary are by now well understood for Euclidean backgrounds, a real-time dictionary along the lines of [3] was developed only recently in [4,5]. This real-time dictionary uses a construction that is a holographic version of the closed time path method of non-equilibrium QFT [6–9] and results in a prescription that incorporates in the bulk the information about the QFT initial and final states via a Hartle-Hawking type construction [10,11]. Thus this prescription although originated from QFT considerations is also in line with expectations from quantum gravity. In this paper we apply the prescription of [4,5] to a class of 2+1-dimensional ‘wormhole’ spacetimes that were found and studied in [12,13]. Our main motivation is to investigate global issues in gauge/gravity duality. Three dimensional gravity is an ideal setup to study this problem because of the absence of local degrees of freedom. In the holographic context one finds that the general solution of the bulk Einstein equations with a cosmological constant in the Fefferman-Graham gauge can be explicitly obtained for general Dirichlet boundary conditions specified by an arbitrary boundary metric [14]. In contrast to the higher dimensional case, where in general the Fefferman-Graham expansion contains an infinite number of terms, in three dimensions the series terminates (see (33)) and all coefficients can be expressed explicitly in terms of the boundary metric and boundary stress energy tensor. What is left to be done is to impose regularity in the interior and this step requires global analysis.1 The wormholes are global solutions of 2+1 dimensional gravity with a negative cosmological constant. They can be thought of as generalized eternal BTZ black holes. Whereas the spatial slices of an eternal BTZ black hole have a cylindrical topology, in the wormholes the spatial slices are general two-dimensional Riemann surfaces with boundary. We sketch an example of a wormhole in Fig. 1. These spacetimes have a number of different asymptotic regions, which we will call outer regions in this paper, one for each boundary component of the Riemann surface. The outer regions are separated by horizons and there is a non-trivial topology behind the horizons. The wormholes are locally AdS3 and should have a holographic interpretation. Each outer region, however, is isometric to the static BTZ black hole and it would seem as though holographic data, 1 Note also that the Fefferman-Graham coordinates are in general well-defined only in a neighborhood of the boundary and they may not cover the entire spacetime.

Holography and Wormholes in 2+1 Dimensions

585

Fig. 1. A wormhole spacetime with two outer regions corresponding to a Riemann surface of genus 2 with 2 boundary components

which are obtained from the behavior of the solution near the conformal boundary, do not contain enough information to completely describe the wormhole spacetime. This follows from a simple counting argument. The spacetimes are uniquely determined given a Riemann surface of genus g with m boundaries. Such a Riemann surface is determined by 6g − 6 + 3m parameters. Each of the outer regions however depends on only one parameter, the mass of the BTZ black hole, so the holographic data from the m outer regions would seem to provide only m parameters. We will shortly describe how the real-time dictionary resolves this puzzle. There are corresponding Euclidean solutions which have been discussed in [15]. These spaces are handlebodies, i.e. closed surfaces of genus g filled in with hyperbolic three-space. These are also generalizations of BTZ whose Euclidean counterpart is a solid torus, i.e. a handlebody of genus 1. For these spacetimes a fairly straightforward application of the Euclidean gauge/gravity prescription shows there is no corresponding puzzle: the holographic one-point function captures the non-trivial topology and in particular does contain enough parameters to completely describe these spaces. This indicates that it is the real-time issues that are crucial in understanding holography for the Lorentzian wormholes. We will indeed find that once we properly apply the real-time gauge/gravity prescription of [4,5] there is a direct and unambiguous holographic interpretation of the entire Lorentzian wormhole spacetimes. The real-time prescription relies on gluing to a given Lorentzian spacetime Euclidean spaces that provide the initial and final states. A class of such Euclidean spaces are the handlebodies described above, but we emphasize that there are also other choices one can make. Once the complete spacetime has been specified (with the Euclidean parts representing initial/final states included), the holographic one-point functions do carry enough information about the spacetime and in particular the geometry behind the horizons. This information is encoded in the initial and final states.

586


The way this happens is instructive and reflects a number of subtle points about the holographic dictionary. Recall that because of the holographic conformal anomaly [16,17] the theory depends on the specific boundary metric, not just its conformal class. In particular, the expectation value of the stress energy tensor changes anomalously under bulk diffeomorphisms that induce a boundary Weyl transformation [18,19]. Now as mentioned earlier, one can choose coordinates such that the metric in any of the outer regions of the wormhole is exactly that of the BTZ black hole. In these coordinates the boundary metric is flat. According to the prescription of [4,5], however, the Lorentzian solution should be matched in a smooth fashion to a corresponding Euclidean solution. Euclidean solutions that satisfy all matching conditions are provided by the handlebodies but these can never have a boundary metric that is globally flat (because the Euler number of the boundary Riemann surface is negative). One can arrange for an everywhere smooth matching by performing a bulk diffeomorphism on the Lorentzian side that induces an appropriate boundary Weyl transformation such that the Lorentzian boundary metric now matches with that of the handlebody. This has the effect that the expectation value of the stress energy tensor changes from its BTZ value to a new value, which is smooth as we cross from the Euclidean side to the Lorentzian side (as it should be [5]). In other words, the initial state via the matching conditions dictates a specific bulk diffeomorphism on the outer regions of the Lorentzian solution and as a result the holographic data extracted using the solution in this coordinate system encode the information hidden behind the horizon. Our results indicate that the dual state for a wormhole with n outer regions is an entangled state in a Hilbert space that is the direct product of n Hilbert spaces, one for each component. A reasonable guess for this state is that it is the state obtained by the Euclidean path integral over the conformal boundary of half of the Euclidean space glued at the t = 0 surface of the Lorentzian wormhole. This is a Riemann surface with n boundaries and in the case of the handlebodies discussed above, it is precisely the Riemann surface that serves as the t = 0 slice of the wormhole. If one traces out all components but one, then the reduced description is given in terms of a mixed state in the remaining copy. This paper is organized as follows. In the next section we describe the wormhole spacetimes in detail and in Sect. 3 and we discuss the handlebodies. In Sect. 4 and 5 we discuss holography for the handlebodies and the Lorentzian wormholes, respectively. We emphasize that our analysis applies only to non-rotating wormholes. The interesting possibility of extending the analysis to rotating wormholes is discussed in Sect. 6 along with several other general remarks. We conclude in Sect. 7 with an outlook. In all of the previous literature and in the main text of this paper the wormhole spacetimes are described abstractly as quotients of a domain in AdS3 . While this presents no loss of information, this description is abstract and requires mastering prerequisite mathematical background in order to understand the properties of these spacetimes. One should contrast this with the case of the BTZ black hole [20,21] where one has an explicit metric containing the physical parameters (the mass and angular momentum). The BTZ also has an abstract representation as a quotient of a domain of AdS3 but this has not been used as much as the explicit metric description. With the hope that a more concrete description of the wormholes would make them more readily accessible we derive in Appendix A an explicit metric description where all parameters that determine the spacetime appear in the metric and we summarize this result here. All information about the wormhole can be summarized in an oriented trivalent fatgraph, like the one in Fig. 2. For a wormhole that is based on a Riemann surface of genus


587

Fig. 2. A fatgraph representing the wormhole spacetime sketched in Fig. 1

g with m boundaries, this graph should have m outer edges (ends) and (3g −3 + m) inner edges. With every outer edge we associate one parameter Mk and with every inner edge two parameters Mi , χi , where k = 1, . . . , m and i = m + 1, . . . , 3g − 3 + 2m. This yields a total of (6g − 6 + 3m) parameters, which is indeed the correct number of moduli for a Riemann surface of genus g with m boundaries.2 We now associate a coordinate chart for every edge of the fatgraph and every such chart has a canonical metric on it. The precise definition of the coordinate charts as well as the meaning of the orientation is given in Appendix A. To complete the description we need to specify the transition functions in the overlap regions and these are also given in Appendix A. Thus, the spacetime is described by the graph and two different metrics, one for the outer charts and one for the inner charts. The metric in the k th outer chart takes the form: dsk2 =

ρ 2 + Mk dρ 2 2 2 + dϕ ) + . (−d τ ˜ √ ρ 2 + Mk cosh2 ( Mk τ˜ )

The corresponding (τ˜ , ρ, ϕ) coordinate system has coordinate ranges, √ cosh( Mk τ˜ )ρ β2 τ˜ ∈ R, ϕ ∼ ϕ + 2π, >− , 1 + β2 ρ 2 + Mk

(1)

(2)

where β is defined in Appendix A. These coordinates extend beyond the future and past horizons, which lie at ρ = Mk | sinh( Mk τ˜ )|. (3) If we restrict ourselves to the region outside of the horizons we may also put the metric in the static BTZ form, dsk2 = −(r 2 − Mk )dt 2 +

r2

dr 2 + r 2 dφ 2 , − Mk

(4)

with coordinate ranges, r > Mk , t ∈ R and φ ∼ φ + 2π . In these metrics Mk is the parameter of the corresponding outer edge. The metric in the i th inner chart is given by μi2 dr 2 1 2 + Mi 1 + (μi r + νi )2 dψ 2 dsi = −dt 2 + 2 2 2 (μi r + νi ) + cos (χi ) cosh (t) √ 2μi M i sin(χi ) − dψdr , (5) (μi r + νi )2 + cos2 (χi ) 2 As we review in Appendix A, the parameters {M , χ } (I = k, i) are directly related to the Fenchel-NielI i sen coordinates of the moduli space of the Riemann surface.

588


with coordinate ranges, r ∈ [−1, 1], τ ∈ R and ψ ∼ ψ + 2π . This is a time-dependent metric of constant negative curvature which (as far as we know) has not appeared before in the literature. The parameters Mi and χi are the parameters associated with the i th inner edge. The parameters μi , νi on the other hand are functions of the M parameters, see the discussion in Sect. A.3. Note that both metrics (1) and (5) have a U (1) isometry, the transition functions however do not respect this symmetry and the entire spacetime is not U (1) symmetric. 2. Lorentzian Wormholes In this section we describe the Lorentzian wormholes. We show how they can be obtained as quotients of a part of AdS3 and discuss their physical properties. The material in this section summarizes discussions in [12,13,22–24]. We will occasionally use results from Teichmüller theory; more information on this topic can be found in [24–27]. 2.1. Wormholes as quotient spacetimes. The wormholes are obtained as follows. One starts with a Riemann surface S which is a quotient of the upper half plane H with respect to some discrete subgroup of S L(2, R). The upper half plane is then embedded into AdS3 and the action of is extended to AdS3 entirely. After removing certain regions in AdS3 that would lead to pathologies, one may take the quotient of the remainder with respect to , which will give us the wormhole spacetime we are after. The topology of such a spacetime is S × R, with S the Riemann surface we started with and R the time direction. The aim of this subsection is to discuss this procedure in more detail. 2.1.1. Riemann surfaces. Consider a Riemann surface S with m > 0 circular boundaries but no punctures.3 As follows from the uniformization theorem, such a Riemann surface can be described as a quotient of the upper half plane H by some discrete subgroup of P S L(2, R): S = H/ , where the action of

a b c d

(6)

∈ P S L(2, R) ≡ S L(2, R)/{±1}

(7)

on H is given by z →

az + b . cz + d

(8)

Since these transformations act as isometries for the standard negatively curved metric on H , ds 2 =

dzd z¯ , Im(z)2

(9)

this metric descends to a metric on S. Up to a constant rescaling, this is the unique hermitian metric of constant negative curvature on S (given the complex structure of S) 3 Recall that a Riemann surface is a topological two-dimensional surface equipped with a complex structure. One can distinguish between punctures and circular boundaries precisely because of the complex structure.


589

Fig. 3. On the left we sketched two fundamental domains in H . The boundaries are pairwise glued together as indicated by the arrows. After the gluing we find the Riemann surfaces shown on the right

and is unique up to conjugation. We shall require absence of conical singularities on S, which means that the nontrivial elements of cannot have fixed points in H . A simple analysis of the fixed points of (8) tells us that we should require that for all elements γ ∈ we have |a + d| ≥ 2.

(10)

Furthermore, absence of any punctures on S translates into |a + d| > 2 for all nontrivial γ . We then say that consists of only hyperbolic elements (and the identity), and we call it a Fuchsian group of the second kind. A particularly convenient way to visualize S as a quotient of H is to define a fundamental domain in H , basically a domain in H whose boundary in H consists of various segments that are pairwise identified by generators of . For convenience we may take these segments to be geodesic segments, which are circular arcs in H . Two examples of a fundamental domain are sketched in Fig. 3. From the theory of Fuchsian groups we obtain that the fixed points of such a group form a nowhere dense subset of the conformal boundary ∂ H of H , which is the real line plus a point at infinity. We will call this set the limit set and denote it as (). Notice that () is invariant under the action of . 2.1.2. AdS3 . To find the wormhole spacetime associated to S, we first fix some coordinate systems and conventions for AdS3 . We define AdS3 as the surface − U 2 − V 2 + X 2 + Y 2 = −1, in

R2,2 ,

(11)

where the metric has the form ds 2 = −dU 2 − d V 2 + d X 2 + dY 2 ,

(12)

and we have set the AdS radius 2 = 1. By combining (U, V, X, Y ) into a matrix, V + X Y +U , (13) Y −U V − X we may identify the hyperboloid with the space of real unit determinant matrices, i.e. the group S L(2, R). The connected component of the identity of the isometry group of AdS3 , Isom0 (AdS3 ) = (S L(2, R) × S L(2, R))/Z2 ,

(14)

590


acts by left and right multiplication: if (γ1 , γ2 ) ∈ S L(2, R) × S L(2, R) then their action on AdS3 is defined by V + X Y +U V + X Y +U → γ1 γ2T . (15) Y −U V − X Y −U V − X Taking the transpose of γ2 is a convention which will turn out to be convenient below. We may describe a patch in the hyperboloid with Poincaré coordinates (t, x, y) defined by U Y , x= , V−X V−X In these coordinates, the metric takes the form t=

ds 2 =

y=

1 . V−X

−dt 2 + d x 2 + dy 2 . y2

(16)

(17)

Although the Poincaré coordinate system may not cover the entire region of interest, the coordinate horizon at y → ∞ will not be important in what follows. 2.1.3. Constructing wormholes. We can now construct a three-dimensional wormhole spacetime from the Riemann surface S = H/ . We begin by extending the action of the isometries of H to isometries on AdS3 via the homomorphism: P S L(2, R) → (S L(2, R) × S L(2, R))/Z2 ,

(18)

which is given explicitly by P S L(2, R) γ → (γ , γ ) ∈ S L(2, R) × S L(2, R). One may check that elements of the form (γ , γ ) leave the slice U = 0 invariant when they act on AdS3 according to (15). Furthermore, their action on the slice U = 0 is exactly of the form (8) when we define z = x + i y with (x, y) the Poincaré coordinates on this slice. The image of under this homomorphism is a discrete subgroup of Isom0 (AdS3 ) ˆ One may now try to take a quotient which is isomorphic to and which we denote as . ˆ like AdS3 /, which clearly contains S = H/ as the slice given by U = 0. However, away from the slice U = 0 this quotient turns out to have closed null or timelike curves. To get a spacetime free of pathologies we proceed as follows. The embedding of H in AdS3 as the slice U = 0 can be directly extended to an embedding of ∂ H in the conformal boundary of AdS3 . This extension maps the limit ˆ We set () to a subset of the conformal boundary of AdS3 , which we denote as (). then pass to the universal covering space of the hyperboloid and remove from it all points ˆ (after a standard conformal rescaling of with a timelike or lightlike separation to () the metric that brings the radial boundary to finite distance). Informally speaking, we are removing the filled forward and backward semi-lightcones emanating from every ˆ We call the remainder point in (). AdS3 which notably includes the original slice ˆ invariant and, being isometries, they map U = 0 entirely. The elements of ˆ leave () lightcones to lightcones so they also leave AdS3 invariant. Furthermore, the quotient M = AdS3 /ˆ

(19)

is a spacetime that is free of closed timelike curves and conical singularities [22,23] and contains S = H/ as a hypersurface. These spacetimes are what we call the 2 + 1dimensional wormholes.


591

2.2. Physical properties. We briefly discuss some physical properties of the wormholes. First of all, they are of course locally AdS3 but, as was mentioned above, their global topology is of the form S × R with S a surface with m > 0 circular boundaries and R representing time. We sketched an example in Fig. 1, where S has genus 2 and has 2 boundary components. The wormholes can have an arbitrary number m > 0 cylindrical boundaries, and S can have arbitrary genus g ≥ 0. There are two special cases: when m = 2 and g = 0 we obtain the eternal static BTZ black hole and the case m = 1, g = 0 is just AdS. Except for the eternal BTZ black hole described already in [21], none of the wormholes have globally defined Killing vector fields since no such isometry of AdS3 commutes with all the elements in . On the other hand, all wormholes admit a discrete Z2 isometry, which acts as time reflection U ↔ −U and therefore leaves the U = 0 slice invariant. The wormholes are not geodesically complete and begin with and end on locally Milne-type singularities. Furthermore, these singularities have associated black and white hole horizons (not drawn in Fig. 1). Perhaps surprisingly, the m segments of the spacetime between the horizons and the conformal boundaries are exactly the same as for the BTZ black hole [12]. More precisely, we find that these segments can be covered by a (t, r, φ) coordinate system with the coordinate ranges r > M, t ∈ R and φ ∼ φ + 2π , in which the metric is of the form dr 2 + r 2 dφ 2 . (20) r2 − M The mass M can be different for the m different boundaries, but it should always be strictly positive so we do not ‘pinch off’ the rest of the wormhole. We will call these m segments the outer regions of the wormhole, and what remains when we excise these segments we call the inner region. Notice that what we call the outer region is precisely the domain of outer communication [12]. What was called the ‘exterior region’ in [12] is obtained by keeping only the region outside of the future horizon, but we will never consider this region here. The fact that the nontrivial topology is hidden behind the horizons is in agreement with the general discussion of [28]. Depending on the genus of S, the geometry in the inner region is specified by a discrete number of parameters, namely the moduli of S. One may for example think of these parameters as the elements (ai , bi , ci , di ) of a set {γi } of generators of . It will be important for what follows to notice that these parameters do not show up in the metric on the outer regions if we put the metric in the form (20). On the other hand, in Appendix A we present a set of different coordinate systems that can be used to describe the wormholes as well. In these coordinate systems the coordinate ranges are natural and the metric features several parameters that are geometric (rather than abstract matrix elements). For example, some of the parameters are directly related to the lengths of certain cycles on the surface. As we explain in more detail in Appendix A, the combination of all parameters from the different charts that make up the surface can be used to completely describe the spacetime.4 It is straightforward to embed the wormholes into string theory, since the wormholes are locally just AdS3 . For example, a wormhole times S 3 × T 4 with a constant dilaton and three-form flux is an asymptotically locally AdS3 solution of type IIB supergravity. However, these solutions are not supersymmetric. ds 2 = −(r 2 − M)dt 2 +

4 The parameters (a , b , c , d ) are similar to the Fricke coordinates on the moduli space of S, whereas i i i i the metric we find in Appendix A features parameters that are similar to Fenchel-Nielsen coordinates on the moduli space of S. These coordinate systems on the moduli or rather Teichmüller space of S are described in more detail in for example [25].

592


3. Euclidean Wormholes In this section we describe ‘Euclidean wormholes’. These Euclidean spaces are handlebodies and one may think of them as closed Riemann surfaces filled in with three-dimensional hyperbolic space. They are a natural generalization of the Euclidean BTZ black hole, which is a solid torus [29]. These spaces were considered first in a holographic context in [15], where it was argued that they are natural Euclidean analogues of the Lorentzian wormholes, even though they are not obtained by analytic continuation of a globally defined time coordinate. We will see later that they are indeed suitable Euclidean counterparts of the Lorentzian wormholes, in the sense of the real-time gauge/gravity prescription of [4,5], but we will also show that they are not the only possible Euclidean counterparts. 3.1. Construction. We will again describe the handlebodies via a quotient construction. Recall that Euclidean (unit radius) AdS3 , denoted by H 3 , is defined as the hyperboloid U 2 − V 2 + X 2 + Y 2 = −1,

(21)

with V > 0 in R1,3 with the metric: ds 2 = dU 2 − d V 2 + d X 2 + dY 2 .

(22)

We may again combine (U, V, X, Y ) into a matrix: V + X Y + iU Y − iU V − X

(23)

which maps H 3 into the space of hermitian unit determinant matrices. An element γ of the connected component of the identity of the isometry group of H 3 , Isom0 (H 3 ) = P S L(2, C), acts on H 3 as

V + X Y + iU Y − iU V − X

→ γ

(24)

V + X Y + iU Y − iU V − X

γ †.

(25)

Notice that P S L(2, C) maps the upper hyperboloid to itself. We may again define Poincaré coordinates (τ, x, y) via τ=

Y U , x= , V−X V−X

y=

1 . V−X

(26)

In these coordinates, the metric takes the form ds 2 =

dτ 2 + d x 2 + dy 2 . y2

(27)

This time there are no coordinate singularities and this metric covers all of H 3 . To find the Euclidean analogue of the wormholes, we again start with the Riemann surface S = H/ . The action of on H can again be extended to an action on H 3 entirely, this time via the trivial homomorphism P S L(2, R) → P S L(2, C),

(28)


593

Fig. 4. The Schottky double of the Riemann surfaces of Fig. 3 is constructed by gluing two copies of the fundamental domain to each other and identifying the boundaries. The line τ = 0 is invariant and the Schottky ˆ is a subset of the line τ = 0 but double surface is symmetric under reflection in this line. The limit set () is not shown here. It has to be removed from the (τ, x) plane before taking a quotient

(i.e. any element of P S L(2, R) is also an element of P S L(2, C)). One may again check that real elements in P S L(2, C) leave the slice U = 0 invariant when they act on H 3 according to (25). Furthermore, their action on the slice U = 0 is again of the form (8) if we define z = x + i y with (x, y) the Poincaré coordinates on this slice. After using this homomorphism to map to ˆ in Isom0 (H 3 ), we can define the quotient ˆ Me = H 3 /,

(29)

which now never leads to pathologies; Me is a smooth and geodesically complete manifold. This quotient again contains S = H/ as the U = 0 slice, and Me also admits a Z2 isometry that leaves this surface invariant. Let us now show why we call Me a handlebody. We can extend the action of ˆ to ˆ i.e. a real the conformal boundary of H 3 which is an S 2 . Consider an element γ of , element of P S L(2, C), acting as (8) on the U = 0 slice. Its extension to H 3 entirely is found most easily by noticing that, according to (25), real elements of Isom0 (H 3 ) leave slices of constant U = τ/y invariant and act on these slices exactly as on the slice U = 0. In the limit where y → 0, we recover the action of γ on the conformal boundary, which is just the same as on the slice U = 0, γ : w →

aw + b , cw + d

(30)

but this time with w = x + iτ . From (30) we find that the great circle τ = 0 is invariant because a, b, c, d in (30) ˆ After are all real. Just as in the Lorentzian case, this circle contains the limit set (). ˆ with respect to ˆ is a removing the limit set, the quotient of the remainder S 2 \() smooth manifold. As can be seen from Fig. 4, it consists of two copies of S, one from

594


Fig. 5. The extension of the fundamental domain for ˆ from the S 2 to H 3 is bounded by a set of hemispheres that should be pairwise identified. We recover S as the surface given by τ = 0

the upper and one from the lower half plane, glued together along their m boundaries. This surface is called the Schottky double Sd of S. If S has genus g and m holes, then Sd has genus 2g + m − 1 and no holes. Since Sd is just the conformal boundary of Me , we may think of Me as a filled Sd . This shows that Me is indeed a handlebody. A fundamental domain for Me in H 3 is sketched in Fig. 5 and can be found by extending the circles on the boundary S 2 to hemispheres in H 3 . The fundamental domain for the original surface S is then embedded in this three-dimensional fundamental domain as the surface given by τ = 0. 4. Holographic Interpretation of Euclidean Wormholes We discuss in this section the holographic interpretation of the Euclidean wormholes. Our discussion, which builds on [15,24,30,31], is a fairly straightforward application of Euclidean holography. In the next section we will turn to Lorentzian wormholes, where things are more subtle. Recall that the boundary Sd of the handlebody is a closed Riemann surface with g > 1 and therefore naturally has a metric of constant negative curvature. Below, following [24,30], we holographically compute the one-point function of the stress energy tensor for this background metric. The negative curvature metric on Sd is obtained by describing Sd as a quotient of H , that is Sd = H/ d . Above we described Sd as a quotient of the conformal boundˆ removed, with respect to the group , ˆ that is ary S 2 of H 3 , with the limit set () 2 ˆ ˆ Sd = (S \())/. As sketched in Fig. 6, this is just a different description of the same Riemann surface. Therefore, there should be a locally biholomorphic map J : H → S 2 between the two descriptions. Such a map should be compatible with the actions of d and , in the sense that for every γd ∈ d there should exist a γ ∈ ˆ such that J ◦ γd = γ ◦ J . Now consider the case where γ is trivial for a nontrivial γd . Since γd corresponds to a nontrivial one-cycle on Sd , the image of this one-cycle under J must be ˆ The only way to do this is to let this curve encircle a nontrivial closed curve on S 2 \(). ˆ a nonempty subset of () on the S 2 , but such a one-cycle is contractible in the bulk manifold. For example, the dashed circle drawn within the fundamental domain of Fig. 6 can be continuously shrunk to a point by moving it inside the bulk, as can be seen from Fig. 5. Therefore, precisely those γd for which J ◦ γd = J correspond to contractible cycles in the bulk. The map J thus determines the filling of Sd : different maps J (up to composition with an element of P S L(2, R) or P S L(2, C)) precisely correspond to the different fillings of Sd . It therefore suffices to know J in order to know which cycles of Sd are filled to give a handlebody and therefore to determine the Euclidean bulk geometry.


595

ˆ . ˆ However, like any closed Riemann Fig. 6. The Riemann surface Sd was originally obtained as (S 2 \())/ surface with g > 1 it can also be described as H/ d for some d for which we have drawn a fundamental domain in the bottom figure. J is a locally biholomorphic map interpolating between the two descriptions. The dashed circle is a homotopically nontrivial closed curve on Sd that can be contracted in the bulk

Since J is by construction locally biholomorphic, we can use the locally defined J −1 to pull back the metric (9) from H to S 2 \(). In Poincaré coordinates for H 3 defined 2 = dwd w in (26), the induced metric on the boundary S 2 was the flat metric ds(0) ¯ with −1 w = x + iτ . On the other hand, when we pull back the metric from H using J , we find a metric on this S 2 which is of the form: −1 2 d J dwd w¯ 2 ds(0) = ≡ e2σ dwd w. ¯ (31) dw Im(J −1 (w))2 This metric is just a Weyl rescaling of the original metric: dwd w¯ → e2σ dwd w, ¯

(32)

where we note that σ becomes singular whenever Im(J −1 (w)) vanishes, which is ˆ on S 2 . precisely at the fixed point set () We may now investigate what happens to the one-point function of the stress energy tensor. Recall that in three dimensions the metric near the conformal boundary can always be put in the Fefferman-Graham form [14], ds 2 =

dρ 2 1 + 2 (g(0)i j + ρ 2 g(2)i j + ρ 4 g(4)i j )d x i d x j , 2 ρ ρ

g(4)i j =

1 −1 (g(2) g(0) g(2) )i j , 4 (33)

and the one-point function of the stress energy tensor in the dual state is given by [18]

Ti j = 2g(2)i j + R(0) g(0)i j , with R(0) the scalar curvature of g(0)i j and we set 16π G N = 1.

(34)

596


2 = dwd w In the case at hand, starting with the bulk metric (27), we find that ds(0) ¯ and Ti j g(0) = 0. A bulk diffeomorphism that induces the Weyl rescaling in (32) has the effect of transforming the g(2) such that [19]

Tww e2σ g = Tww g + 2∂w2 σ − 2(∂w σ )2 , in agreement with CFT expectations. Since in our case 1 1 −1 1 −1 −1 −1 ¯ σ = ln(∂w J ) + ln(∂w J ) − ln (J − J ) , 2 2 2i

(35)

(36)

we obtain directly that

Tww e2σ g

∂ 3 J −1 3 = w −1 − ∂w J 2

∂w2 J −1 ∂w J −1

= S[J −1 ](w),

with S[ f ](w) the Schwarzian derivative of f (w), 3 f 2 f . S[ f ] = − f 2 f

(37)

(38)

We therefore find that in the metric (31), the one-point function of the energy-momentum tensor is given by (37). This is already an encouraging result: we mentioned above that the bulk geometry is captured by J and here we find that the same J arises in the boundary energy-momentum tensor, which therefore provides the holographic encoding of the bulk geometry. However, the boundary metric (31) also depends on J which is not completely intuitive. This can be avoided by using J once more to pull back everything to H . If we use a complex coordinate z on H , so w = J (z), then we find that:

Tzz = −S[J ]

2 ds(0) =

dzd z¯ , Im(z)2

(39)

where we used that (S[J −1 ] ◦ J )(d J/dz)2 = −S[J ], which follows from [27] S[ f ◦ g] = (S[ f ] ◦ g)(dg/dz)2 + S[g].

(40)

This equation may be directly verified by using the chain rule for differentiation, which in our notation is written as ( f ◦ g) = ( f ◦ g)g . Equation (39) is the result we are after: if we describe the boundary Sd of the handlebody as the quotient H/ d (corresponding to the bottom picture in Fig. 6), then the one-point function of the stress energy tensor in the constant negative curvature metric is given by minus the Schwarzian derivative of the map J to S 2 . If we now recall that J dictates which cycles in d are contractible in the bulk, namely precisely those for which J ◦ γd = J , then this implies that Tzz indeed encodes the precise filling and therefore the bulk geometry. Notice also that S[J ] has the right transformation properties under composition of J with S L(2, R) from the right, under which it transforms covariantly, and with S L(2, C) from the left, under which it is invariant. These transformation properties follow from (40) and the fact that S[ f ] = 0 if f is a Möbius transformation [27]. Finally, let us mention that the renormalized on-shell bulk gravity action has been computed in [15] and shown to be equal to the on-shell Liouville action on Sd , computed earlier in the mathematics literature [32]. Note also that the map J implicitly defines a solution to the Liouville equation.


597

4.1. Bulk interpretation and relation to Teichmüller theory. We can make the holographic encoding of the spacetime a little more explicit. As mentioned above, from the bulk perspective the boundary Weyl rescaling is induced by a bulk diffeomorphism that preserves the Fefferman-Graham form of the metric but introduces a new FeffermanGraham radial coordinate ρ [19,33]. For the case at hand, the precise bulk diffeomorphism is given in [24,30] and leads to the bulk metric ds 2 =

dρ 2 (1 + 41 ρ 2 )2 |dz + μρ d z¯ |2 + , ρ2 ρ2 Im(z)2

(41)

with z again a coordinate on H and μρ (z, z¯ ) = −

1 ρ2 S[J ](z) Im(z)2 , 2 1 + 41 ρ 2

(42)

where the bar indicates complex conjugation and we dropped the primes on the new coordinates. Indeed, by expanding this metric in ρ 2 and using (34) we obtain again the result (39). It is noteworthy to mention that in the new coordinates the action of leaves slices of constant ρ invariant, so its elements γ just act as (ρ, z) → (ρ, γ (z)) with γ (z) given by (8). We expect these new coordinates to become ill-defined somewhere inside the handlebody since the contractible cycles shrink to zero length at a certain point. By inspection of (41), this only happens when |S[J ](z)| Im(z)2 > 21 . This bound on the Schwarzian derivative is familiar from Teichmüller theory as it figures prominently in the Ahlfors-Weil theorem concerning a local inverse of Bers’ embedding of Teichmüller spaces [27] in the space of holomorphic quadratic differentials. The physical relevance of the bound is the following. When this bound is nowhere satisfied the coordinate system is nonsingular all the way to ρ → ∞, where we recover another asymptotically AdS region. We then do not describe a wormhole but rather a spacetime with two disconnected boundaries which are simultaneously uniformized in the boundary S 2 , as expected from Teichmüller theory. These do not correspond to wormholes and we refer to [34] for more information as well as open questions regarding these spaces. For a handlebody there are no other asymptotic regions and we may therefore assume on physical grounds that the bound is everywhere satisfied. In that case, the coordinate system becomes degenerate at a surface given by ρ 2 = ρc2 ≡

1 |S[J ]| Im(z)2 −

1 2

.

(43)

At the surface ρ = ρc , the metric is everywhere degenerate since |μρc | = 1. We then describe a point in the boundary of Teichmüller’s compactification of the Teichmüller space [27]. It would be interesting to verify explicitly that the contractible cycles are indeed the degenerate cycles on this surface. 4.2. Non-handlebodies. The discussion so far was about Euclidean handlebodies, but these are not the only 3-manifolds that have a genus g Riemann surface as their conformal boundary. We briefly discuss an example of such non-handlebody spacetimes in this subsection.5 A simple example can be constructed from the spacetimes described 5 We thank Alex Maloney for discussions about the material in this subsection.

598


in [34]. These are obtained by starting from H 3 written in hyperbolic slicing and quotienting the boundary by a discrete subgroup of H to obtain a compact, finite volume, genus g > 1 surface, g . This yields the metric with two boundaries, 2 , ds 2 = dr 2 + cosh2 (r )ds g

(44)

where r ∈ (−∞, ∞) and 2 = ds g

dzd z¯ Im(z)2

(45)

is the constant negative curvature metric on g which has scalar curvature R = −2. To produce a manifold with a single boundary, one may try to quotient by r → −r . This procedure however introduces a singularity at r = 0. The singularity can be avoided if the surface g has a fixed point free involution I , since then we can combine r → −r together with the action of I to obtain a smooth hyperbolic 3-manifold with conformal boundary the Riemann surface g . Such involutions are discussed, for example, in [35]. In this case the singularity at r = 0 is replaced by the smooth Riemann surface g /I . The resulting 3-manifold is a quotient of H 3 which has no contractible cycles so it is not a handlebody. This 3-manifold has the same conformal boundary as the handlebody build from d but it has a different expectation value of the energy momentum tensor. Changing variable, ρ = 2e−r , the metric becomes of the form (33)) with: 2 g(0)i j = 2g(2)i j = ds . g

(46)

We may then use (34) to obtain that:

Ti j = −g(0)i j .

(47)

We see that the one-point function of the energy-momentum tensor is notably different from that of a handlebody. However, we also observe that any involution that ‘ends’ the spacetime at r = 0 (with or without fixed points, orientation-reversing or orientationpreserving) results in the same one-point function, so the holographic one-point function of Ti j does not seem able to distinguish these geometries. This is an interesting subtlety of the Euclidean dictionary due to global issues. Let us recall why we expect that locally, in the Euclidean setup, g(0)i j and Ti j uniquely fix a bulk solution (in any dimension). Intuitively, this is because the bulk equations of motion are second order differential equations and g(0)i j and Ti j provide the correct initial data. One can indeed show rigorously that given this data there exists a unique bulk solution in a thickening of the conformal boundary, see [36] and references therein. Furthermore, one can show that (g(0)i j , Ti j ) are coordinates in the covariant phase space of the theory [37] and thus each such pair specifies a solution. In the case at hand this data indeed produces a unique metric for r > 0, but the way the spacetime is capped off at r = 0 depends on the fixed point free involution used. One can presumably distinguish the different spacetimes by using higher point functions and non-local observables, such as the expectation values of Wilson loops, i.e. minimal surfaces that end at a loop in the conformal boundary of the 3-manifold. It would be interesting to verify this explicitly.


599

5. Holographic Interpretation of Lorentzian Wormholes We now move to discuss the holographic interpretation of the Lorentzian wormholes. We start by demonstrating in the next subsection that a naive adaptation of the analysis of the previous section leads to incomplete results where, in contrast with the Euclidean results, the spacetime geometry does not seem to be captured by the dual field theory on the Lorentzian side. This is then resolved using the real-time gauge/gravity prescription of [4,5]. 5.1. Naive computation. As we mentioned earlier, the metric in the outer regions can always be cast in the BTZ form (20). When using a new coordinate ρ defined via r=

M 2 4 ρ

ρ

+1

,

(48)

the metric takes the form in (33) with g(0)i j = ηi j , g(2)i j =

M M2 δi j , g(4)i j = ηi j . 2 16

(49)

The one-point function of the stress energy tensor in the dual state can be computed from (34) yielding,

Ti j = Mδi j .

(50)

On the other (m − 1) conformal boundaries, we obtain similar one-point functions (with different values of M) and all the other one-point functions vanish. This is problematic, since we obtain no information whatsoever about the inner part of the geometry and the Lorentzian one-point functions therefore seem to be insufficient to reconstruct the wormhole spacetime. The holographic encoding of the spacetime appears to fail, which would contradict standard expectations from the gauge/gravity duality. This apparent contradiction comes from the fact that we have not taken into account the holographic interpretation of Cauchy data. This can be done using the real-time gauge/gravity prescription of [4,5], which we review in the next subsection. 5.2. Lorentzian gauge/gravity prescription. In this subsection, we prepare for the discussion below by reviewing some known facts about Lorentzian quantum field theory. Afterwards, we show how one may use these facts to obtain a consistent real-time prescription for the gauge/gravity duality. We then return to the wormholes in Subsect. 5.3. 5.2.1. States in field theory. The prescription in [4,5] is based on the fact that any Lorentzian field theory path integral requires a specification of the initial and final states as well. Such a state | may be specified via path integrals on a Euclidean space Y with a boundary and possible operator insertions away from this boundary. If we want to compute, say, |O(t)| = |ei H t Oe−i H t |, we continue to path integrate along a Lorentzian segment with length t that is glued to the boundary of the Euclidean space, then insert the operator, and finally go back in time for a period t before we attach a second copy of the Euclidean space. For Euclidean spaces which are topologically R × X with X a real space and R representing Euclidean time, the overall field theory

600


Fig. 7. A contour in the complex time plane; the cross signifies the operator insertion

background manifold corresponds to a contour in the complex boundary time plane of the form sketched in Fig. 7 times a real space. Notice that extending the contour beyond the point t, say to a point T > t, amounts to an extra insertion of ei H (T −t) e−i H (T −t) = 1 which does not affect the correlation function. A similar story holds for higher-point correlation functions, but in those cases an operator ordering has to be specified. Although the contour may often be deformed to a simpler version, we emphasize that a procedure like the above is always necessary for Lorentzian quantum field theory. 5.2.2. Translation to gravity. In the Lorentzian gauge/gravity prescription of [4,5], one incorporates the Euclidean segments for the path integral into the holographic description and ‘fills’ them with a bulk solution as well. For example, to the contour of Fig. 7 may correspond a bulk manifold consisting of two Lorentzian and two Euclidean segments. These segments are then glued to each other along spacelike hypersurfaces that should end on the corners of the boundary contour. The behavior of the fields at these hypersurfaces is then determined using matching conditions. These guarantee the C 1 continuity of the fields. More precisely, for the metric one imposes continuity of the induced metric h AB and the extrinsic curvature K AB with a factor of i: h AB = Eh AB ,

L

L

K AB = −i EK AB ,

(51)

with the superscript indicating the Lorentzian or the Euclidean side and the extrinsic curvature on either side is defined using the outward pointing unit normal. There is also a corner matching condition,6 which is defined at the intersection between S and the conformal (radial) boundary. It dictates that the inner product between the unit normal to S, denoted as n μ , and the unit normal to the radial boundary, written as nˆ μ , is continuous across the boundary (up to appropriate factors of i). For a Lorentzian-Euclidean gluing, using outward pointing unit normals, it becomes: (nˆ μ n μ ) = i E(nˆ μ n μ ).

L

(52)

As discussed in [5], all the matching conditions arise naturally from a saddle-point approximation. Although they are equivalent to analytic continuation in many simple cases, they do not rely on a globally defined time coordinate and are therefore more generally applicable. This construction is an essential ingredient in the Lorentzian gauge/gravity dictionary. For example, it allows us to understand precisely how changing the initial and final states modifies the Lorentzian spacetime, gives the correct initial and final conditions for the bulk-boundary and bulk-bulk propagators, and also cancels surface terms from timelike infinity in the on-shell action, which would otherwise lead to additional infinities. Furthermore, the boundary correlators directly come in the in-in form as in quantum field theory. 6 It is likely that this condition follows from the matching for the induced metric and the extrinsic curvature in (51) but in the absence of a general proof we treat it as an additional matching condition.


601

Fig. 8. We take half of a genus two surface and attach Lorentzian cylinders to the boundary. This boundary manifold can be filled in with half an Euclidean handlebody plus a Lorentzian wormhole with spatial topology of a pair of pants. On the right, we shaded the matching surface between the Euclidean and the Lorentzian segment. It indeed has three boundaries and no handles

5.3. Gauge/gravity duality for Lorentzian wormholes. Let us now apply the construction outlined in the previous subsection to the wormholes. To this end, we have to cut off the wormhole along some spatial bulk hypersurface and find a Euclidean space that we may glue to this hypersurface such that the matching conditions are satisfied. Of course the field theory contour also has a backward-going segment and a final state. To fill this in, we have to cut off the wormhole along some final time slice as well and glue a second Lorentzian and Euclidean segment to this final surface. These second copies can be taken to be identical to the first ones, which correspond to taking the final and the initial state to be just the same. In [5], we performed this procedure for the eternal BTZ black hole. As long as we do not switch on any perturbations, we may take the second Lorentzian and Euclidean segment to be completely identical to the first one. This also means that the matching conditions are trivially satisfied along the final gluing surface, so these do not have to be investigated separately. Therefore, it will be sufficient to focus on a single Euclidean-Lorentzian gluing below. A candidate for the Euclidean space is half of the Euclidean handlebody Me that we obtained in Sect. 3. (It is not however the only candidate, as we will explicitly demonstrate in Subsect. 5.4.) Indeed, we may cut this handlebody and the wormhole spacetime in two halves along the surface S and glue them together along S. In the case S is a pair of pants (a surface of genus zero with three circular boundaries, so g = 0 and m = 3), the procedure is sketched in Fig. 8 and the filling of the full field theory contour, including the backward-going segment, is sketched in Fig. 9. Let us now verify that the matching conditions are satisfied at the shaded matching surface in Fig. 8. On both sides, the induced metric is locally just the unique negative curvature metric on S described as H/ , so it is the same metric indeed. Also, the extrinsic curvature vanishes completely on both sides because of the Z2 time-reversal symmetry. Therefore, the first and second matching conditions are satisfied indeed. Finally, the extra corner matching discussed in [5] is also satisfied: in our case S intersects the conformal boundary orthogonally (again because of the Z2 symmetry) and the inner product n μ nˆ μ thus vanishes both for M and for Me . However, there is still a subtlety with the boundary metric which we now discuss. If we use the BTZ coordinate system on the Lorentzian side, then the boundary metric on this side is flat. The boundary metric on the Euclidean side, however, can never be globally flat because Sd has negative Euler number. On the other hand, to match M and Me , we should also take the boundary metric to be smooth (in the sense specified in [5]). This can be done by Weyl rescaling the Lorentzian boundary metric to a metric of

602


Fig. 9. Analogous to Fig. 7, the full field theory contour has a forward- and a backward-going segment and two Euclidean segments to specify the initial and final state. Similarly, the full bulk spacetime consists of four segments as well. They should be glued along the matching surfaces which we shaded in this picture

constant negative curvature, as we discuss below. The boundary metric is then smooth across the corner and the discrepancy between the boundary metrics on either side is removed.7 5.3.1. Matching Euclidean and Lorentzian wormholes and Ti j . We now discuss the consequences of the continuity of the boundary metric across the matching surface. As described above we match the initial U = 0 surface of the Lorentzian wormhole to half of the Euclidean handlebody. On the boundary of the spacetime, the Lorentzian cylinders are glued to the boundary of the Euclidean handlebody along the m circles that form the boundary of the U = 0 Riemann surface. These m circles lift to segments of the great circle given by τ = 0 in the Poincaré coordinates (26) on the boundary S 2 of H 3 . Let us now focus on one of the m circles. After conjugation, we can always ensure that it lifts to the half-line l given by: l : x > 0, τ = 0

(53)

on the S 2 . Its projection down to Sd is then given via the identification w ∼ λw

(54)

with w = x + iτ and for some positive real λ = 1. The relevant part of the fundamental domain is then sketched in Fig. 10. To find the boundary metric of constant curvature on this surface we again have to ˆ to that of a quotient of H , for pass from the description of Sd as quotient of (S 2 \()) which we defined the map J in Sect. 4. Using J −1 , we now map the half-line (53) to H , where we use the coordinate z. Although J −1 is multi-valued, we will need only one of the images of l in H . We can again use conjugation freedom to make sure that the image under consideration is the half-line: l : Re(z) = 0. In H , the identification (54) becomes an isometry of S L(2, R) that leaves Such an isometry is necessarily of the form: z ∼ μz,

(55) l

invariant. (56)

7 Another possibility would be to Weyl rescale the metric on the Euclidean side such that it is flat in the vicinity of the gluing circles. Although the gluing is then smooth, the Euclidean boundary metric can then no longer be analytic.


603

Fig. 10. Part of a fundamental domain of Sd on S 2 . We will eventually replace the part with τ > 0 with a Lorentzian wormhole. The single identification is given by w ∼ λw and the line l is the entire positive x-axis

Fig. 11. Under the locally defined map J −1 the domain in Fig. 10 maps to the sketched domain in H , where we use a coordinate z. The identification is given by z ∼ μz. The line l is the positive imaginary axis. We will replace the part Re(z) < 0 with a Lorentzian wormhole

for some real μ = 1 given implicitly by J (μz) = λJ (z).

(57)

The construction in H is sketched in Fig. 11. Notice that J (z) is an analytic map from the imaginary axis to the (positive) real axis, that is J (i y) = J (i y),

y > 0.

(58)

Notice also that the Z2 symmetry w ↔ w¯ maps under J −1 to reflection in the imaginary axis, that is z ↔ −¯z . (Again, as J −1 is multi-valued, it maps the original Z2 to many other reflections in H as well, but we do not need them here.) We now ready to attach a Lorentzian cylinder to the boundary. The procedure is sketched in Fig. 12. On H , this means that we cut away the half given by Re(z) < 0 and attach the universal covering of a Lorentzian cylinder to the gluing line Re(z) = 0. In the bulk, we can use the metric (41) with the matching surface given by Re(z) = 0, at least up to the point ρ = ρc . We now need to find a Lorentzian bulk metric that satisfies the matching conditions of [5] when glued to this surface. Both in the bulk and on the boundary, it is straightforward to obtain the explicit matching Lorentzian metric by analytic continuation. We first introduce a coordinate

604


Fig. 12. On the left, the Euclidean boundary geometry in the coordinate z . In the center figure we sketched the Lorentzian boundary geometry and on the right the glued-together geometry

z = −i z. In the z plane Fig. 11 is rotated clockwise by 90 degrees which slightly simplifies the matching below. In the coordinate z the metric (41) becomes: ds 2 =

dρ 2 (1 + 14 ρ 2 )2 |dz + μρ d z¯ |2 + , ρ2 ρ2 Re(z )2

(59)

where now μρ (z , z¯ ) =

1 ρ2 ˜ S[ J ](z ) Re(z )2 , 2 1 + 41 ρ 2

J˜(z ) = J (i z ),

(60)

where we used that S[J ](i z ) = −S[ J˜](z ), which follows from (40). The gluing takes place along the half-line Im(z ) = 0, Re(z ) > 0. We then replace z → u and z¯ → v to find the Lorentzian bulk metric: ds 2 =

dρ 2 (1 + 41 ρ 2 )2 (du + μρ (v)dv)(dv + μρ (u)du) + , 1 2 ρ2 ρ2 4 (u + v)

(61)

with μρ (u) =

1 ρ2 ˜ S[ J ](u) (u + v)2 , 8 1 + 41 ρ 2

(62)

and a similar expression with u → v. Note that J˜(x) is real-analytic for x > 0 and monotonic, so S[ J˜](x) is real-analytic too. This Lorentzian metric is thus real and covers the bulk spacetime up to ρ = ρc . Since z ∼ μz , the periodicity on the Lorentzian side is (u, v) ∼ μ(u, v). The point (u, v) = (0, 0) on the boundary is a fixed point of this identification and therefore we need to exclude the forward lightcone emanating from this point from the spacetime (the backward lightcone is already replaced by the Euclidean geometry). Since we also demanded Re(z ) > 0, so u + v > 0, we need only the part of the Lorentzian boundary with u > 0 and v > 0. On the boundary we find the metric: 2 ds(0) =

dudv 1 4 (u

+ v)2

(63)

which has scalar curvature R(0) = −2. Using once more (34) we obtain for the one-point functions: −1 . (64)

Tuu = −S[ J˜](u), Tvv = −S[ J˜](v), Tuv = 8(u + v)2


605

Fig. 13. Adding a handle as indicated does not change the properties of the gluing surface or the Lorentzian spacetime

Fig. 14. As indicated on the left, one may add generators to the Schottky group ˆ without breaking the Z2 symmetry. The resulting surface has two extra handles: one for the half corresponding to the initial state and another one for the final state

c Notice that one expects that Tii = 24π R(0) and we obtained here Tii = −2. Reinstating the factors of 16π G N , we find c = 24π/(16π G N ) = 3/(2G N ) which is indeed the correct central charge. Equation (64) is the main result of this section and demonstrates that the Lorentzian one-point function of the stress energy tensor as obtained from the metric (61) does contain information about the dual geometry that is hidden behind the horizons.

5.4. More fillings. In the previous section we glued a particular handlebody to the Lorentzian wormhole. There exist a variety of handlebodies {Me } that all have a hypersurface S where the matching conditions are satisfied as we discuss now. In particular, one may attach an extra filled handle to Me somewhere away from the matching surface to obtain a manifold Me with conformal boundary with = Sd . An example of this is sketched in Fig. 13. This procedure does not change any properties like the induced metric or the extrinsic curvature of the matching surface. Geometrically, this can be seen by going to the universal covering: one may add generators to ˆ to obtain a group ˆ and as long as Me = H 3 /ˆ has S as a surface of Z2 symmetry we may slice open Me along this surface and glue the Lorentzian wormhole Ml = AdS3 /ˆ to it. For the boundary surface this procedure is sketched in Fig. 14. In the figure we represented the addition of two generators to ˆ by cutting four circles out of the fundamental domain that are pairwise identified, all done in such a way that the original Z2

606


symmetry remains intact. Although we have not sketched it here, this procedure directly extends to the entire three-dimensional space: the Z2 symmetry is also present for the new handlebody and S is again the invariant surface given by τ = 0. We conclude that we can glue to the Lorentzian wormholes also half of Me . A similar analysis as in the previous section establishes that the 1-point functions captures the fact that the initial state is different than the one corresponding to Me .

5.5. State dual to wormholes. Let us now discuss what our results imply about the QFT state dual to the wormholes. Since there are m boundaries, the Hilbert space consists of a tensor product of m Hilbert spaces, one for each boundary component. From the fact that the wormholes are manifolds that interpolate between the m segments, we expect to find nonzero correlations between the m boundaries and the initial state to be an entangled state. Indeed, this is precisely what we find. To see this, suppose the initial state is separable, namely of a product form |α1 ⊗ · · · ⊗ |αm . Then the 1-point functions would necessarily take a factorizable form. More precisely, suppose the state was separable and consider the insertion of a stress energy tensor in, say, the first boundary component,

Ti j (x1 ) = α1 | ⊗ · · · ⊗ αm |Ti j (x1 )|α1 ⊗ · · · ⊗ |αm = α1 |Ti j (x1 )|α1

2

|||αk ||2 .

(65)

k=2

Now the naive 1-point function in (50) would support the view that the state is separable. Namely, in that case we could naively say that the state |α1 in the first copy depends only on the corresponding mass parameter M1 and not on the other variables that determine the spacetime. This would lead to one- and higher-point functions of the energy momentum tensor in the first copy which up to an overall factor only depend on M1 . The one-point functions that we got, however, in (64) are not of that form, as the Schwarzian S[ J˜] does not have such a factorizable form and does contain all the variables that determine the spacetime. Another check on the non-separability of the state is provided by the computation of a two-point function. An argument analogous to the one above implies that if the state is separable then the 2-point function would have a factorizable form. We illustrate that this is not the case in the next subsection. A natural guess for the dual state is that it is the state obtained by an Euclidean path integral over a Riemann surface with m circular boundaries. According to the reasoning of [11,38], this surface can be taken to be precisely the conformal boundary of one half of the Euclidean manifold Me . This can be = S, with S the surface of time reversal symmetry of the wormhole spacetime, for the case of the handlebodies of Sect. 3, or = S if the initial state is that of the previous subsection. If we now trace over all components but one, all wormholes with m > 1 can be thought of as having been associated with a mixed state in the remaining copy and this explains the presence of horizons. The m = 1 case is special in that we only have a single copy of the CFT so there are no copies to trace out. Nevertheless results for the 1- and 2-point functions indicate that there is an entanglement between the outer region and the region behind the horizon. These spacetimes were also analyzed in [39] which suggested that the dual state is in some respects similar to a thermal state. We leave a better understanding of this case for future work.


607

5.6. 2-point functions. We discuss in this subsection the computation of the 2-point function for a scalar operator O of dimension . In the bulk it suffices to consider a free massive scalar field, as interaction terms contribute only to higher point functions. We glue an Euclidean handlebody at t = 0 and take the initial and final states to be the same. It follows from an analysis along the lines of [5] that the different real-time correlators (time-ordered, Wightman, etc.) are obtained by suitable analytic continuations of the Euclidean correlator in the handlebody geometry. The two-point function of a scalar operator on the Euclidean plane is uniquely fixed by conformal invariance and takes the form:

O(τ, x)O(τˆ , x) ˆ =

[(τ

− τˆ )2

1 , + (x − x) ˆ 2 ]

(66)

where we normalized the operators so that the coefficient in the numerator equals one. ˆ whose For the handlebody, we have to sum over the elements of the Schottky group , elements γ act as Möbius transformations on the boundary, γ : ω = x + iτ →

a(x + iτ ) + b , c(x + iτ ) + d

(67)

with real a, b, c, d and ad − bc = 1. This can also be written as 1 2 τ, (ax + b)(cx + d) + acτ . γ : (τ, x) → (γτ , γx ) ≡ (cx + d)2 + c2 τ 2

(68)

Using the complex coordinate w on the boundary S 2 of H 3 , we obtain ¯ 1 ) =

O(w, w)O(w ¯ 1, w

γ ∈ˆ

1 , |cw + d|2 |γw − w1 |2

(69)

where the boundary metric is locally dwd w. ¯ We then Weyl transform to the metric (31) which is globally well-defined to find

O(w, w)O(w ¯ ¯ 1 ) = 1, w

¯ e−σ (w1 ,w¯ 1 ) e−σ (w,w) , 2 |cw + d| |γw − w1 |2

¯ ds 2 = e2σ (w,w) dwd w, ¯

γ ∈ˆ

(70) with e

2σ (w,w) ¯

−1 2 d J 1 = . −1 dw Im(J (w))2

(71)

We can now pull back to H , using z = J −1 (w), to obtain

O(z, z¯ )O(z 1 , z¯ 1 ) =

γ ∈ˆ

|J (z)J (z 1 )| (Im(z)Im(z 1 )) dzd z¯ , ds 2 = . |c J (z) + d|2 |γ (J (z)) − J (z 1 )|2 Im2 (z) (72)

As before, we may assume the covering groups are such that J (λz) = μJ (z). We then again introduce the coordinate z = −i z and the map J˜(z ) = J (i z ) = J (z), replace

608


z → u and z¯ → v to obtain the Lorentzian metric. We recall that in this case J˜(x) is real-analytic for real positive x. More precisely, following the steps in [5], one finds that the time-ordered correlator is obtained by replacing z → u − iu and z¯ → v + iv, where the i insertions push the singularity everywhere away from the real-time contour. To avoid clutter we will however not write the i insertions explicitly below. The final answer is then

T O(u, v)O(u 1 , v1 ) 2−2 (u + v) (u 1 + v1 ) [ J˜ (u) J˜ (v) J˜ (u 1 ) J˜ (v1 )]/2 = (c J˜(u) + d) (c J˜(v) + d) (γ ( J˜(u)) − J˜(u 1 )) (γ ( J˜(v)) − J˜(v1 )) ˆ

(73)

γ ∈

in the metric ds 2 =

4dudv . (u + v)2

(74)

This 2-point function does not take a factorizable form supporting the view that the dual state is entangled, as anticipated. We would like to note however that at late times ˆ More precisely, if one defines the correlator is dominated by the BTZ elements in . 8 coordinates u = exp(x + t), v = exp(x − t), then in the limit t, t1 → ∞ with (t − t1 ) fixed, all terms in (73) go to zero, except the ones with either b = 0 or c = 0. These are precisely the elements associated with the BTZ black hole. 6. Remarks In this section we discuss some general remarks concerning the wormhole spacetimes.

6.1. Other bulk spacetimes. The question we addressed in this paper is what is the holographic interpretation of any given wormhole spacetime. One can also ask: given a geometry at infinity, how many different bulk spacetimes can one have? In general, all such saddle points contribute and should be taken into account, although typically one of the saddle points dominates at large N at any given regime. A well-known example is that associated with the Hawking-Page transition [3,40]. In that case the boundary is S 1 × S d−1 and there are two possible (Euclidean) bulk manifolds corresponding to making contractible in the interior either S 1 or S d−1 , namely the Euclidean Schwarzschild AdS solution and thermal AdS. This question is usually addressed in Euclidean signature, but it is clearly also relevant in Lorentzian signature. In this context the question is now: given the conformal boundary of the complete Euclidean and Lorentzian pieces how many different bulk manifolds can one have? Naively, one might think that for every Euclidean solution there would be a corresponding Lorentzian plus Euclidean solution, but this turns out not to be the case. This can be demonstrated with the case where the conformal boundary is a torus, S 1 × S 1 . As in the higher dimensional case, there are two solutions that correspond to either the first or the second circle being contractible in the interior (which correspond to thermal AdS and Euclidean BTZ), but there are now new possibilities obtained by considering a contractible cycle that is a linear combination of the above cycles [41]. These solutions 8 In these coordinates the Lorentzian cylinder is (t, x) ∼ (t, x + log λ).


609

Fig. 15. The analogue of thermal AdS for the pair of pants wormhole. In this case three copies of Lorentzian AdS3 are attached to the three boundaries of the pair of pants

are called the ‘S L(2, Z) family’ of black holes as they relate to the Euclidean BTZ black hole by a modular transformation. In Appendix B we however show that none of these solutions can be used in the real-time gauge/gravity prescription as the Euclidean part associated with a vertical segment of the QFT contour. The reason is that the matching conditions force the Lorentzian bulk metric to be complex and this results in a energy momentum tensor that does not satisfy the correct reality conditions. For a higher genus Riemann surface there also exists a similar family of solutions [42] as well as the aforementioned non-handlebody solutions. As for the other fillings of handlebody-type, we expect that only the analogues of the BTZ and the thermal AdS would be relevant for holography of the Lorentzian wormholes. The analogue of thermal AdS is obtained by attaching m copies of empty Lorentzian AdS to the m boundary components of the handlebody. The case of a pair of pants wormhole is sketched in Fig. 15. For the non-handlebodies the corresponding Lorentzian solution remains to be investigated.

6.2. Rotating wormholes. In the previous sections we considered non-rotating wormholes. Rotating wormholes do exist [43,44] and are obtained by taking a quotient with respect to a group generated by elements of the form (γ1 , γ2 ) ∈ S L(2, R) × S L(2, R) with γ1 = γ2 . A similar region like AdS3 exists such that the quotient AdS3 / is a good spacetime [22,23] and the metric in the outer regions is isometric to the rotating BTZ metric [23]. The corresponding ‘Euclidean spaces’ for these wormholes, however, are not so straightforward. A prescription for obtaining them has been proposed in [45] and was critically analyzed in [24]. From the holographic perspective, the reality condition of the bulk fields, especially on the Euclidean caps, should be dictated by the standard reality condition of the dual QFT. In the case of the rotating BTZ we have demonstrated in [5] (Sect. 4.5) that the matching conditions result in a complex metric on the Euclidean caps. It is likely that the same would be true here, namely the Euclidean solution that should be glued to the rotating Lorentzian wormhole would be complex. There are several issues that need to be resolved in order to understand the rotating case. Firstly, it is not straightforward to find in the rotating wormhole the analogue of a U = 0 slice of the non-rotating wormhole [23]. One approach to this problem is to consider the rotating wormholes as deformations of the non-rotating wormholes. In the Lorentzian case such deformations might be described by a Lorentzian version of the standard quasiconformal mappings [27], one for each S L(2, R) factor. One would then need to extend these deformations to the ‘Euclidean’ solutions, which, as mentioned above, are likely to be complex solutions that possess a real slice where the Lorentzian solution can be glued. It would be interesting to further develop this direction.

610


7. Outlook We have discussed the holographic interpretation of a class of 2+1-dimensional wormhole spacetimes. They are interesting toy models for the analysis of global issues in the real-time gauge/gravity correspondence. We have shown that the asymptotics of the complete solution that includes both the Lorentzian solution and the Euclidean caps completely characterize the geometry including the regions behind the horizons. This came about by a subtle interplay between global issues and the real-time gauge/gravity dictionary. In particular, the real-time gauge/gravity prescription requires gluing a smoothly Euclidean solution to the Lorentzian solution at early and late times. This in turn fixes the apparent freedom for independent Weyl rescaling at different outer components and results in holographic data that contain information about the complete geometry. We thus find that the Lorentzian CFT correlators encode in a very precise sense the parts of the geometry that lie behind the horizons. This presents a unique opportunity to study and settle classic questions and puzzles in black hole physics. The way the information is given to us, however, (i.e. in terms of CFT correlators) is very different from the way the black holes puzzles are usually formulated (e.g. using bulk local observers) and this presents the main obstacle in directly addressing these issues. In this respect, one of the most interesting cases to further understand is that of spacetimes with m = 1 and g > 0. As discussed earlier, this has only one outer region. The form of the 1-point and 2-point functions indicate entanglement between the outer region and the region behind the horizon. It is not clear however which modes are entangled in the CFT, since unlike the cases with m > 1 the dual state seems to be defined in only a single copy of the Hilbert space. We can however suggest some possibilities. Note that all wormholes can be viewed as quotients of a part of BTZ, since the group associated with them always contains a subgroup isomorphic to that of BTZ (namely Z) and so one can take the quotient first with respect to this group, resulting in BTZ, and then with respect to the rest of the group elements (modulo issues related to the regions one needs to remove to avoid closed timelike curves that need to be investigated). Thus we find a state in the tensor product of two Hilbert spaces (associated with the two boundaries of BTZ) with certain correlations between the two components because of the final quotient. It would be interesting to make this more precise and understand its relation with the apparent entanglement between the outer and inner regions. As mentioned earlier, there is a reasonable guess for the dual state: this would be the pure state obtained by performing the Euclidean path integral over the Riemann surface that is the conformal boundary of the Euclidean 3-manifold that we glue to the Lorentzian spacetime at t = 0. However, this appears at odds with the presence of a bulk horizon. It would be interesting to clarify this and also check the identification of the state by computing in the CFT the expectation value of the stress energy tensor in this state and see if the results agree with our bulk computation. One of the main reasons the black hole entropy has been so puzzling is that classically black holes appear to be unique (they have “no-hair”) so their phase space is zero dimensional. In a typical quantum system the correspondence principle relates the quantum states to the classical phase space and the entropy of the system to the volume of phase space in Planck units. Thus since the phase space for black holes appears to be zero dimensional, they should not carry any entropy. As was discussed earlier, however, the outer region of the wormholes is isometric to the BTZ black hole. Thus one can view the ‘wormhole’ spacetimes with a single outer region as ‘BTZ hair’, where the ‘hair’ is essentially the non-trivial topology hidden behind the horizon. It is thus natural to ask


611

whether this classical phase space can account for the entropy of the BTZ black hole upon quantization.9 In other words, these spacetimes would then be the semi-classical approximation of the underlying black hole microstates. This is similar in spirit to the fuzzball proposal (whose relation to holography was discussed extensively in the review [47]) although here the geometries counted contain horizons and singularities. Let us outline how one would do such a computation. We have seen that these spacetimes are uniquely specified by a Riemann surface with one boundary and the mass of the BTZ black hole is determined by one of the moduli of the Riemann surface. Thus the classical phase space is the moduli space of Riemann surfaces of arbitrary genus with a single fixed modulus, corresponding to the length of the horizon (in other words the BTZ mass parameter), which is the only parameter accessible to an observer outside of the horizon. More precisely, if one uses the Fenchel-Nielsen coordinates on the Teichmüller space (described in detail in Appendix A.1) the restriction to a fixed BTZ mass amounts to considering a codimension one hypersurface in Teichmüller space. This hypersurface is invariant under the mapping class group and therefore directly descends to the moduli space. The complete phase space is then the union of these hypersurfaces for different genera. Classically, the volume of this phase space is infinite and one should proceed by geometric quantization. One can readily compute the symplectic form on the covariant space following [48–50] and proceed to quantize. It would be interesting to carry out this computation. The explicit form of the metric derived in Appendix A should facilitate this. Acknowledgments. We would like to thank Alex Maloney, Jan Smit and Erik Verlinde for discussions. KS acknowledges support from NWO via a VICI grant. 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. Coordinate Systems The description of the wormholes in Sect. 2 as a quotient AdS3 /ˆ is precise but rather abstract. This Appendix presents a metric description of the wormholes, building on [24]. More details are presented in [51]. Concretely, this description consists of covering the spacetime with a set of charts for which the coordinates have natural ranges. We then show that on each of these charts we can put an explicit metric, which features several natural parameters that describe the local geometry (similar to the mass M for a BTZ metric). We will show that one may arrive at a complete description of the spacetime by combining the parameters from all the charts plus specifying some combinatorial data, which can be combined in a single labelled fatgraph. An example of such a fatgraph is given in Fig. 2, which completely describes a spacetime with the topology sketched in Fig. 1. The particular parameters that will appear in the metric are very similar to Fenchel-Nielsen coordinates on Teichmüller space, so we begin with a review of these coordinates. A.1. Fenchel-Nielsen coordinates. In this section we review the definition of the Fenchel-Nielsen coordinates on the Teichmüller space of Riemann surfaces of genus g with m > 0 circular boundaries (and no punctures). As we discussed in the main text, 9 This question has been independently pursued by Alex Maloney [46].

612


Fig. 16. Defining Fenchel-Nielsen coordinates on a Riemann surface. We cut the Riemann surface into pairs of pants along simple closed geodesics and assign lengths li to all the edges of every pants plus a twisting parameter t j for every gluing involving two pairs of pants

all such Riemann surfaces are quotients of the upper half plane, from which they all inherit a canonical metric of constant negative curvature. It can be shown that in this metric there is precisely one smooth periodic geodesic corresponding to every nontrivial primitive loop on the surface. After a little counting one finds that one can pick a maximum of 3g − 3 + 2m of such periodic geodesics that do not intersect each other, see Fig. 16 for an example. We then cut the Riemann surface along these geodesics, i.e. we remove these geodesics from the surface. This leaves us with 2g −2+m disconnected so-called ‘pairs of pants’, that is Riemann surfaces of genus 0 with three circular boundary components, as well as m annuli. The annuli correspond to the regions on the Riemann surface between a periodic geodesic that is retractable into a boundary component and the boundary component itself. The Fenchel-Nielsen coordinates are now based on the idea that we can reconstruct the complete Riemann surface from this collection of pairs of pants and annuli, provided we also specify how to glue these ‘building blocks’ together. Therefore, we can define coordinates on the Teichmüller space of Riemann surfaces of the given type by specifying enough data to first of all construct the pairs of pants and annuli that make up the original surface, plus some rules on how to glue them together. Let us begin with a description of the individual pairs of pants and annuli. Using some simple hyperbolic geometry, see for example [25], one finds that the pairs of pants are completely described by only three real moduli which one may take to be the strictly positive lengths of the periodic geodesics along which we made the cuts. A similar statement is true for the annuli: these are completely specified by giving the length of the periodic geodesic as well. Since we cut along 3g − 3 + 2m periodic geodesics, we find that we can reconstruct the individual pairs of pants and annuli by the specification of precisely 3g − 3 + 2m strictly positive lengths. Next, we have to specify the way in which the various components are glued together. More specifically, we have to specify the angle that the various components have to be twisted with before we perform the gluing. Notice that these angles are actually only relevant when we glue two pairs of pants together, since twisting an annulus is an isometry. The angles are defined as follows, see Fig. 17. On every pair of pants we may define


613

Fig. 17. The twist parameter t is defined by the angle between two points p and q that lie at the intersection of the dashed geodesics with a boundary circle

three distinguished geodesics, namely the shortest non-intersecting geodesics that run from one boundary circle to another. A given boundary circle of the pants intersects with two of these geodesics, say at the points p and p . (Figure 17 is drawn slightly distorted since these points actually lie diametrically opposite of each other. This follows from a reflection isometry of the pair of pants whose fixed points are precisely the three geodesics we just defined.) Following the same reasoning on the other pair of pants we find two more points, say q and q , on this boundary circle. The twist parameter describing the gluing is now precisely the angle between, say p and q, on the boundary circle.10 Since we cut along 3g − 3 + 2m geodesics, we have as many gluings to perform. For precisely m of these we glue annuli to pairs of pants, which leaves us with 3g − 3 + m gluings between pairs of pants for which we need to specify an angle. Adding these to the 3g − 3 + 2m lengths precisely gives the required number of 6g − 6 + 3m parameters. Indeed, it can be shown that these lengths and angles provide good coordinates that cover the Teichmüller space of Riemann surfaces of the given type, which is therefore isomorphic to (R+ )3g−3+2m × R3g−3+m . This is then the Fenchel-Nielsen description of the Teichmüller space. A.2. Construction of the charts. The procedure to obtain our charts is sketched in Fig. 18 and is described in words as follows. We first restrict ourselves to the U = 0 Riemann surface S = H/ . Just as in the Fenchel-Nielsen description of the surface, we begin by picking a maximal set of 3g − 3 + 2m primitive periodic geodesics. We now consider one geodesic and ‘thicken’ it, i.e. we define a small cylindrical neighborhood around the geodesic. When we try to extend this ‘collar’ further, eventually we might wrap another cycle and the cylinder will then start to overlap with itself. We then stop the thickening when the boundary circles just touch each other, as indicated in Fig. 18. In the cases where the periodic geodesic we consider is retractable into a boundary component we extend the thickening on that end all the way to this boundary. Except for the BTZ black hole, the other end of the cylinder is then never extendable to another boundary component and pinches as usual. This procedure results in two types of cylindrical domains: those where both boundary circles are pinched on S, which we call ‘inner domains’, and those where precisely one end extends to a boundary component, which we call ‘outer domains’. An inner domain covers part of two pairs of pants, whereas an outer domain covers an annulus and part of a pair of pants. 10 A shift of 2π in the angles corresponds to an element of the mapping class group and therefore to two different points in Teichmüller space. Strictly speaking, therefore, these angles take values in R in order to properly parametrize the Teichmüller space. We will be rather loose in this distinction.

614


(a)

(b)

(c)

Fig. 18. (a) Charts are defined around a closed periodic geodesic on the Riemann surface. (b) We begin by thickening this geodesic to obtain a cylinder. (c) We extend the cylinder as far as possible, until the bounding circles just touch, in this case on the black dots. We define coordinates (r, φ) as indicated, as well as a third time coordinate which is not shown

Notice that the inner and outer domains we define here are not precisely the inner and outer regions we defined in the main text. Namely, the inner and outer regions in the main text were separated by the horizons, whereas the outer domains we define here do extend beyond the horizons. The inner domains that we define here never cross the horizons and therefore lie entirely in what we called the inner region in the main text. Consider now a single pair of pants. It intersects with precisely three (inner or outer) domains, namely those that are defined around each of its boundary circles. Of course, the domains overlap with each other on the pants but more importantly it can be shown that the entire pair of pants is covered by these three domains. (This follows from direct computation using hyperbolic geometry, see [51] for details.) Since the domains also cover the annuli completely, it follows that the entire surface at U = 0 is covered by these domains. Below, we will use these domains as the U = 0 slice of analogously defined three-dimensional coordinate patches, which taken together cover the entire spacetime. We will then find a suitable coordinate system on these patches to complete our description of the wormholes. To define more precisely the inner and outer domains let us lift them to the universal cover H of S, where we will use a complex coordinate z. Consider one of the periodic geodesics around which we defined a chart. We assume that on H the homotopy class of the periodic geodesic is generated by the identification γ : z → λz,

(75)

which can always be realized using the conjugation freedom of . If z = x + i y, then the periodic geodesic lifts to the line x = 0. The corresponding lift of the cylindrical neighborhood around it is a region D given by D : −βy < x < αy,

(76)

for some positive real α and β (which are given in terms of the Fenchel-Nielsen parameters that fix the geometry of the pairs of pants, as it will become clear from the analysis below). For an inner domain α and β are finite whereas for an outer domain either α or


615

Fig. 19. A lift to the upper half plane of the cylindrical region. The dotted lines bounding the darker region should be identified under the map w ∼ λw. Under γβ , the region maps to the smaller, lighter shaded region on the left. The image γ (lβ ) of lβ just touches lβ at the indicated point. A similar thing happens on the right for lα , but we have not sketched the second image

β are equal to +∞ and the domain extends all the way to the boundary. A region D with finite α and β, i.e. corresponding to an inner domain, is sketched in Fig. 19. The lines lα and lβ , given by αy = x and −βy = x respectively, determine the bounding circles of the cylindrical neighborhood. We have deliberately chosen the shape of these bounding circles such that they lift to straight lines on H which are called hypercycles. (Recall that geodesics on H are either semicircles that are orthogonal to the real axis or straight vertical lines; hypercycles, on the other hand, are straight lines or circle segments that end on the real axis but not at a right angle. Examples are lα , lβ and γ (lβ ) in Fig. 19.) As we mentioned above, the cylindrical neighborhood is ‘maximally extended’ in the sense that its bounding circles on S touch themselves somewhere on S. Correspondingly, there must exist γα , γβ ∈ that map lα , lβ to circle segments that just touch lα , lβ on H . We have sketched this in Fig. 19. The cylinder can now be extended to a region on the full three-dimensional wormhole geometry. We first extend the action of the isometry (75) to the Poincaré patch: γˆ : (t, x, y) ∼ λ(t, x, y),

(77)

and then extend the domain D to an invariant domain Dˆ in the full three-dimensional geometry. For inner domains it is defined as 2 2 ˆ D : −β y − t < x < α y 2 − t 2 , (78) with y 2 − t 2 > 0. For outer domains either α or β are equal to +∞ and correspondingly there is no restriction on the sign of y 2 − t 2 when x > 0 or x < 0, respectively. On that end the outer domain extends all the way to the conformal boundary of the spacetime. We note that the region with −t 2 + x 2 + y 2 ≤ 0 has to be excluded because it lies within the future and past lightcone of the origin, which is a fixed point of the isometry (77). One may check that (77) indeed leads to closed timelike or lightlike curves in this region. There are other excluded regions that are bounded by lightcones with their vertex at the point at infinity but these are precisely the regions in AdS3 that are not covered by the Poincaré coordinate system anyway.

616


Let us now sketch a proof for the covering of the entire spacetime by these domains. First of all, notice that the future and past Cauchy development of the t = 0 slice (which we will call C) is covered by the part of the Poincaré coordinate system with y 2 −t 2 > 0. Then from (78) we see that the inner charts all lie within this domain. The domain C can be foliated with slices of constant U = t/y on which the quotient group ˆ acts just as on the initial U = 0 surface. The covering of C then follows straightforwardly from the fact that the U = 0 surface is covered. However, the wormholes are not globally hyperbolic and a part of the wormhole spacetime near the conformal boundary lies outside of C. To find the shape of this part of the spacetime we notice the following. Near the conformal boundary the spacetime has the form of an annulus times a time coordinate and when we move inward this annulus pinches just as in Fig. 18c. It follows from (78) that the pinching occurs either at x = −β y 2 − t 2 or at x = α y 2 − t 2 for some finite α, β. Either way this ‘pinching surface’ must lift to a region with y > |t| and therefore always lies entirely within C. It follows that the parts of the spacetime outside of C must have the shape of an annulus times time. It is then easy to verify that these regions of the spacetime outside of C can be described in Poincaré coordinates by starting with the region where y < |t| and x > 0, excluding the lightcones where −t 2 + x 2 + y 2 ≤ 0 and taking the quotient of the remainder with respect to the cyclic group generated by (77). Indeed, the domains so obtained are bounded by the lightlike surfaces y = |t| that bound C, extend all the way to the conformal boundary y = 0 and the action of the cyclic covering group guarantees that the quotient has the form of an annulus times time. These regions are by construction also completely covered by an outer domain and therefore indeed the entire spacetime is covered. One may also explicitly verify that the coordinate systems on the inner and outer domains as given in (79) and (87) below are everywhere well-defined on these domains. A.2.1. Coordinate systems on inner domains. We may now define new coordinates on ˆ For inner domains we define a coordinate system the three-dimensional domains D. (τ, r, φ) via: tanh(τ ) =

t , y

μr + ν =

x y2 − t 2

,

e2

√

Mφ

= −t 2 + x 2 + y 2 ,

(79)

with coefficients e2π

√

M

= λ,

μ + ν = α,

μ − ν = β.

(80)

From (77) and (78) we find the coordinate ranges: τ ∈ R, φ ∼ φ + 2π, r ∈ [−1, 1], and the metric takes the form: 1 μ2 dr 2 2 2 2 2 ds = −dt + + M(1 + (μr + ν) )dφ . (μr + ν)2 + 1 cosh2 (t)

(81)

(82)

This metric already features several parameters M, μ, ν which inform us about the geometry at least in this local patch. We can however introduce one more parameter which is related to the Fenchel-Nielsen twist described above.


617

Fig. 20. The phase in φ can be fixed by letting φ = 0 correspond to the intersection of the boundary of the chart with the unique shortest length geodesic (dashed line) between the two boundary circles of the corresponding pair of pants

To find this parameter, let us begin by considering one edge of a particular chart, say at r = +1. As indicated in Fig. 20, such an end lies at a pair of pants that is used in the Fenchel-Nielsen description of the surface. We described before that there are three shortest geodesics on this pair of pants that run between the three boundary components, see Fig. 17. As we sketched in Fig. 20, two of these geodesics intersect the boundary circle of the chart. We can now shift φ such that one of these intersection points corresponds to φ = 0 and from the aforementioned reflection symmetry it follows that the other one automatically lies at φ = π . (It can also be shown that the third of these geodesics precisely touches the boundary of the charts at the pinching point which is indicated by the black dot in Fig. 20.) After having implemented this shift at the side r = +1 we find that the corresponding points at the side r = −1, which lie on another pair of pants, generally lie at a value φ = φ0 and φ = φ0 + π , all modulo 2π . In fact, the angle φ0 is precisely the FenchelNielsen twist coordinate (denoted t above) that is associated to the gluing. We can make this twist explicit in the metric by introducing a new ‘twisted’ coordinate ψ given by: √ √ exp( Mψ − k) = exp( Mφ) f (μr + ν, χ ) (83) with

ρ sin(χ ) + ρ 2 + cos2 (χ ) . f (ρ, χ ) = ρ2 + 1

(84)

This coordinate transformation features two new parameters k and χ . If they are chosen such that e−k = f (μ + ν, χ ) = e

√

Mφ0

f (−μ + ν, χ ),

(85)

then the aforementioned distinguished points are given by ψ = 0 and ψ = π on both sides. The coordinate range of ψ is the same as φ, so ψ ∼ ψ + 2π . The parameter χ now shows up explicitly in the metric, which takes the form: 1 μ2 dr 2 2 2 2 −dt ds = dψ 2 + + M 1 + (μr + ν) (μr + ν)2 + cos2 (χ ) cosh2 (t) √ 2μ M sin(χ ) − dψdr . (86) (μr + ν)2 + cos2 (χ )

618


This is the final metric on the inner chart. The four different parameters M, μ, ν, χ appearing in it inform us about some ‘local’ geometrical aspects of the spacetime. Namely, the periodic geodesic√around which we defined the chart lies at the point r = −ν/μ and has length 2π M. The angle χ reflects the twisting of the pairs of pants with respect to each other and the parameters μ and ν are related to the shapes of these pairs of pants: for example, the distance between the periodic geodesic and the pinched hypercycle at r = 1 is μ + ν + (μ + ν)2 + 1 | ln |, √ ν + ν2 + 1 and the distance to the hypercycle at r = −1 has the same form with the replacement μ → −μ. A.2.2. Coordinate systems on outer domains. For the outer domains we may always conjugate such that α = ∞ and β is finite. We can then use a (τ˜ , ρ, ϕ) coordinate system defined as √ √ x √ t , ρ = M , e2 M(ϕ−h) = −t 2 + x 2 + y 2 , (87) tanh( M τ˜ ) = y y2 + x 2 where, as in (80), e2π

√

M

= λ,

(88)

which is again related to the length of the periodic geodesic. The parameter h shifts the coordinate ϕ such that the aforementioned special points on the bounding circle lie again at ϕ = 0 and ϕ = π . We will not need the explicit value of h below. The bounding circle itself is given by √ β2 cosh( M τ˜ )ρ =− , (89) 1 + β2 ρ2 + M and the coordinate ranges are given by τ˜ ∈ R, ϕ ∼ ϕ + 2π,

√ β2 cosh( M τ˜ )ρ >− . 1 + β2 ρ2 + M

(90)

The radial boundary of the spacetime lies at ρ → ∞. The metric takes the form: ds 2 =

ρ2 + M dρ 2 2 2 . (−d τ ˜ + dϕ ) + √ ρ2 + M cosh2 ( M τ˜ )

(91)

Notice that these coordinate systems extend beyond the future and past horizons, which lie at the surfaces x = |t| or √ √ ρ = M| sinh( M τ˜ )|. (92) The metric in the region outside of these horizons (which we called the outer region in the main text) can be put back in BTZ form (20) by the coordinate transformation: √ √ ρ2 + M 2 tanh( M τ˜ ), φ = ϕ. r2 = , tanh( Mt) = 1 + M/ρ (93) √ cosh2 ( M τ˜ ) Notice that the parameter M in (91) agrees with the BTZ mass M.


619

A.3. Parameters. The above charts can be combined to cover the wormhole spacetime completely. More specifically, for a wormhole of genus g and with m boundaries, we can cover the entire spacetime with 3g −3+m inner charts plus m outer charts. For every inner chart we have four parameters, M, μ, ν, χ , and for every outer chart we have a single parameter M. As we showed above, the angles χ and the parameters M are directly related to the Fenchel-Nielsen twists and length parameters associated to the periodic geodesics and should therefore completely determine the surface. The remaining μ and ν parameters are therefore expressable in terms of those. The precise relation takes the following form. Consider a pair of pants in the surface. In our description of the surface it is covered by three (inner or outer) charts, in fact it is already completely covered by only half of each of these three charts. Suppose now that chart number 3 is an inner chart (with parameters μ3 , ν3 , M3 , χ3 ) and that it is the half with r > 0 that lies on the pair of pants under consideration. Denote the M parameters in the other two charts as Mi with i ∈ {1, 2}. One then finds the relation: C12 + C22 + 2C1 C2 C3 μ3 + ν3 = , (94) √ sinh(π M 3 ) √ with Ci = cosh(π M i ). This relation follows from a straightforward computation in the upper half plane using hyperbolic geometry. A similar relation can be found at the other side of chart number 3, which has r < 0 and lies on another pair of pants. Namely, using the parameters M1 and M2 of the two other charts on that pair of pants we find: C 21 + C 22 + 2C1 C2 C3 , (95) − μ3 + ν3 = √ sinh(π M 3 ) with Ci = cosh(π Mi ). Using these formulae, we can determine all the μ, ν parameters in the inner charts if we are only given the M parameters in every chart. This reduces the number of independent parameters to two per inner chart and still one per outer chart, just as for the Fenchel-Nielsen description of the surface. A.4. Fatgraph description. To completely specify the spacetime we need to specify both the parameters and the way the charts are glued together. This combinatorial data can be nicely summarized in an oriented trivalent fatgraph as shown in Fig. 2. (In the usual Fenchel-Nielsen description of the surface this combinatorial data is implicitly specified, for example by using a reference surface. The description given below, on the other hand, explicitly fixes the required combinatorial data and it is then no longer necessary to use a reference surface.) The data in the fatgraph is translated to the coordinate systems as follows. Every edge represents a periodic geodesic and therefore a chart. Every vertex represents a pair of pants. The orientation of the edges indicates the direction of increasing r (and by convention always points outward for outer charts), and the ‘fattening’ is necessary to indicate how three charts come together on a pair of pants. If we add to this fatgraph two parameters M, χ for every interior edge of the graph and a single parameter M for every outer edge, then the wormhole spacetime is completely specified. At this point we should note that there are two discrete ambiguities in the above definitions of the coordinates ψ and ϕ on the inner and outer charts that we have not yet

620


Fig. 21. Fixing the ambiguities in the definition of ψ and ϕ

dealt with. Although these ambiguities do not affect the metric or the coordinate ranges given above, they will affect the transition functions below and therefore they should be fixed. The first ambiguity involves the direction of increasing ψ and ϕ. With the fatgraph description this can be easily fixed by fixing the handedness of the (r, ψ) or (r, ϕ) coordinate system to be the same in every chart. The second ambiguity is the fact that we have only ψ or ϕ up to an overall shift by π . To see this, recall that we decided that the point ψ = 0 or ϕ = 0 would correspond to one of the distinguished points on the boundary circle (sketched in Fig. 20) and by the reflection isometry the other point would then be at ψ = π or ϕ = π . We however did not yet specify which point we chose to be at 0 and which one at π . This ambiguity can be fixed from the fatgraph. We first demand that at an overlap between two charts the point where ψ = 0 on one chart corresponds to ψ = π on the other chart (and similarly for ϕ), as indicated in Fig. 21. Furthermore, for an inner chart we should alternately associate ψ = 0 and ψ = π to the four corners of the corresponding edge in the fatgraph, which is indicated in Fig. 21 as well. This fixes the ambiguity up to an overall shift of ψ or ϕ with π in all charts at the same time, which is however irrelevant for the description of the manifold. A.5. Transition functions. With all the ambiguities fixed, we may proceed to define transition functions on the overlap between two different charts. These follow from the coordinate transformations (79) and (87) plus the explicit form of the elements of ˆ in Poincaré coordinates (which can be deduced from (15) and (16). An important subtlety is that we find different transition functions depending on the gluings and the orientations of the charts. For example, if we consider the vertex in Fig. 2 where we may go from chart 2 to chart 3 or chart 4, we find different transition functions because we turn ‘right’ at the vertex if we go to chart 3, whereas we turn ‘left’ if we go to chart 4. As another example, the transition functions between chart 2 and chart 3 (on both vertices) are different from those between chart 5 and chart 6 because (again on both vertices) the orientation of chart 3 and chart 6 are not the same. When we define the transition functions below we will have to take into account these different possibilities.


621

Fig. 22. Possible transitions between inner charts. The transition functions are by definition always taken from unprimed to primed coordinate systems: for example, in the first line the unprimed coordinates in (96) are the coordinates in chart 1 and the primed coordinates are those of chart 2

In the transition functions we will not use the ‘twisted’ coordinate ψ defined in (83). Instead, we will use the coordinate φ which agrees with ψ at the bounding circle of the chart where we define the transition function. Of course, it is not hard to compose the transition functions with (83) and its inverse, or a similar function when the transition takes place at r = −1. A.5.1. Transitions between two inner charts. The complete set of possibilities for the transitions between two inner charts is depicted in Fig. 22. As one may expect, the transition functions are almost the same for either one of these possibilities and it is convenient to give them in a general form with certain parameters , , d and d whose value depends on these possibilities and is given in the table in Fig. 22. Using these parameters, one finds for the transition functions, t = t

√ − (μr + ν ) = cosh(A)(μr + ν) − sinh(A) (μr + ν)2 + 1 cosh( M(φ − d)) (96) √ √ 2 (μr + ν) − (μr + ν) + 1 cosh( M(φ − d) − g) e2 M (φ −d ) = √ (μr + ν) − (μr + ν)2 + 1 cosh( M(φ − d) + g) with cosh(A) =

cosh(π

√ √ M) cosh(π M ) + cosh(π M ) √ √ sinh(π M) sinh(π M )

√

(97)

and sinh(A) sinh(g) = 1. Here M and M denote mass parameters in the metric on the unprimed and the primed chart between which we define the transition functions, and M denotes the mass parameter from the metric of the third chart that joins this vertex. We therefore have to inspect the metric of all three charts at the vertex in order to obtain the transition functions between only two of these charts.

622


Fig. 23. Possible transitions involving outer charts. In this picture the charts 1, 2, 4 and 6 are outer charts and the charts 3 and 5 are inner charts. Conventions are as in Fig. 22

Notice that the transition functions are not automatically periodic in φ or φ ; they are in fact only valid for φ, φ ∈ [0, 2π ). Of course, this is by no means a restriction as this is sufficient to cover the entire chart. The other boundaries of the domain of validity of the transition functions are obtained from the coordinate ranges (81). For example, substituting r = 1 in the second equation of (96) one finds an equality involving r and φ which defines the boundary of the domain of definition of the transition functions. A.5.2. Transitions involving outer charts. If the transitions involve outer charts we need the (τ˜ , ρ, ϕ) coordinate system. Since we always pick the ρ coordinate to increase towards the boundary there is no ambiguity on the orientation of this coordinate. We are however still left with the left/right ambiguity and correspondingly need a discrete parameter f associated to every outer chart. For the transition functions between two outer charts we find,

2+M √ M ρ ρ = − , cosh(A)ρ + sinh(A) cosh( M(ϕ − f )) √ M cosh( M τ˜ ) √ √ √ √ M tanh( M τ˜ ) ρ 2 + M = M tanh( M τ˜ ) ρ 2 + M, (98) √ √ √ ρ cosh( M τ˜ ) + ρ 2 + M cosh( M(ϕ − f ) − g) , e2 M (ϕ − f ) = √ √ ρ cosh( M τ˜ ) + ρ 2 + M cosh( M(ϕ − f ) + g) with the possible values of f and f given in Fig. 23 and the same values of A and g as before. The transition function on the second line is slightly implicit but it is straightforward to plug in the solution for ρ of the first line and then solve for τ˜ .


623

Similarly, between an inner and an outer chart we find, √ √ M cosh(A)(μr + ν) − sinh(A) (μr + ν)2 + 1 cosh( M(φ − d)) , ρ = cosh(t) √ √ (99) tanh( M τ˜ ) ρ 2 + M = M tanh(t), √ √ 2 (μr + ν) − (μr + ν) + 1 cosh( M(φ − d) + g) , e2 M (ϕ − f ) = √ (μr + ν) − (μr + ν)2 + 1 cosh( M(φ − d) − g) and conversely, √ √ M tanh(t ) = tanh( M τ˜ ) ρ 2 + M,

√ (μr +ν ) 1 ρ2 + M = cosh(A)ρ +sinh(A) cosh( M(ϕ − f )) , √ cosh(t ) M cosh( M τ˜ ) √ √ √ ρ cosh( M τ˜ ) + ρ 2 + M cosh( M(ϕ − f ) + g) 2 M (φ −d ) . = e √ √ ρ cosh( M τ˜ ) + ρ 2 + M cosh( M(ϕ − f ) − g)

(100)

Again, these transition functions are not obviously periodic in φ and ϕ are are only valid in the interval [0, 2π ) and the other boundaries are again found by inserting the coordinate ranges (81) and (90) in the transition functions. One may again compose the transition functions with (83) and its inverse to obtain the transition functions for the twisted coordinate ψ on the inner charts. B. Eternal Black Holes and Filled Tori In this Appendix we discuss the genus 1 handlebodies. We show that the ‘S L(2, Z) family’ of black holes cannot be used in the real-time gauge/gravity prescription as the bulk filling of a vertical segment of the QFT contour because the matching conditions lead to a complex Lorentzian metric (and therefore Ti j does not satisfy the correct reality conditions, either). Consider a Euclidean field theory on a torus with modular parameter τ = τ1 + iτ2 . Without loss of generality we can pick the circle given by z ∼ z + 1 as the spatial circle along which we will cut open the Euclidean path integral and glue the Lorentzian solutions. More precisely, we will glue two Lorentzian cylinders to the lines y = 0 and y = τ2 /2, where z = x + i y. As we discussed in [5], τ1 is then i times the angular momentum chemical potential, but since we are not interested in rotating black holes here, we, will set τ1 to zero throughout this Appendix (it is straightforward to generalize to τ1 = 0), so τ = iτ2 is purely imaginary. The torus so defined admits multiple bulk fillings, which are given by the specification of a contractible cycle z ∼ z + aτ + b with (a, b) two relatively prime integers. For each of these fillings, one may obtain a complete Euclidean metric which is locally H 3 . After cutting the torus in half, we will glue a Lorentzian bulk solution to the bulk hypersurface ending on the lines y = 0 and y = τ2 /2. This hypersurface has the shape of an annulus, except when (a, b) = (0, 1), when it consists of two disks. In this case the matching Lorentzian solution is two segments of thermal AdS. Notice also that for (a, b) = (1, 0) we obtain the rotating BTZ black hole. To find the matching Lorentzian solutions in the general case, we will first explicitly write down the Euclidean bulk metric. We then investigate how the Lorentzian metric is determined by the matching conditions.

624


Euclidean geometries. Let us give a brief review of the possible fillings of the torus. We will again use the Poincaré coordinates (τ, x, y) defined in (26) on H 3 , as well as the complex coordinate w = x + iτ on the boundary of H 3 . (Notice that the τ here is a coordinate and not the modular parameter of the torus. We henceforth exclusively use the coordinate w so no confusion should arise.) Any torus handlebody can be obtained as a quotient of H 3 by a cyclic group of identifications generated in Poincaré coordinates by: (w, y) ∼ (e2πiβ w, |e2πiβ |y),

(101)

with β = β1 + iβ2 a complex number. Let us now compute the bulk metric when we use the complex boundary coordinate z which has the natural periodicity z ∼ z + 1 ∼ z + τ . We can do so using the map J of Sect. 4. In this case, J is a locally biholomorphic map from C rather than H , since the universal covering of the torus is C and not H . If the contractible cycle is given by (a, b), the corresponding map J : C → S 2 is given by: J : z → w = eαz ,

(102)

with α = 2πi(aτ + b)−1 . The identifications z ∼ z + 1 ∼ z + τ become w ∼ weα ∼ weατ

(103)

w ∼ eα(cτ +d) w.

(104)

which implies

Now, since one trivially has that w ∼ eα(aτ +b) w, it follows that the single identification (104) is equivalent to both identifications in (103) provided ad − bc = 1. Comparing (104) with (101), we then read off that β=

cτ + d . aτ + b

(105)

Following the same steps as in Sect. 4, we find that the bulk metric in the z coordinate becomes ds 2 =

dρ 2 1 α¯ 2 ρ 2 2 d z¯ | . + |dz + ρ2 ρ2 4

(106)

This metric is of the Fefferman-Graham form (33) and we can read off that the one-point function of the stress energy tensor is given by:

Tzz =

α2 , 2

(107)

which is again −S[J ], just as we found for the higher genus handlebodies in Sect. 4.


625

Lorentzian geometry. Let us now consider the continuation to the Lorentzian geometry. On the boundary we cut open the Euclidean geometry along the circles given by y = 0 and y = τ2 /2. In every case except thermal AdS these circles are the boundary of a single annular region in the bulk manifold. Locally the unique solution is simply given by analytic continuation. Using the boundary lightcone coordinates (u, v), we find the Lorentzian metric, ds 2 =

dρ 2 1 α¯ 2 2 α2 2 ρ ρ du). + (du + dv)(dv + ρ2 ρ2 4 4

(108)

The periodicity for the boundary coordinates is (u, v) ∼ (u + 1, v + 1), and (u, v) are real, whereas ρ has the same range as above. This metric is however complex unless α 2 is real, which only happens if either a = 0 or b = 0. This is problematic both from the bulk and the holographic perspective. In particular, the expectation value of the dual stress energy tensor can be computed using (34), 1 2 1 α ,

Tvv = α¯ 2 , 2 2 and is complex, which cannot be the case for a hermitian operator.

Tuu =

(109)

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Communications in


Asymptotic Infinitesimal Freeness with Amalgamation for Haar Quantum Unitary Random Matrices Stephen Curran1 , Roland Speicher2,3, 1 Department of Mathematics, University of California, Los Angeles, CA 90095, USA.


2 Saarland University, FR 6.1 - Mathematik, Campus E 2.4, Saarbrucken 66123, Germany.


3 Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston,

Ontario K7L3N6, Canada. E-mail: [email protected] Received: 4 January 2010 / Accepted: 27 June 2010 Published online: 21 November 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract: We consider the limiting distribution of U N A N U N∗ and B N (and more general expressions), where A N and B N are N × N matrices with entries in a unital C∗ -algebra B which have limiting B-valued distributions as N → ∞, and U N is a N × N Haar distributed quantum unitary random matrix with entries independent from B. Under a boundedness assumption, we show that U N A N U N∗ and B N are asymptotically free with amalgamation over B. Moreover, this also holds in the stronger infinitesimal sense of Belinschi-Shlyakhtenko. We provide an example which demonstrates that this result may fail for classical Haar unitary random matrices when the algebra B is infinite-dimensional. 1. Introduction One of the most important results in free probability theory is Voiculescu’s asymptotic freeness for random matrices [20]. One simple form of this result is the following. Let A N and B N be (deterministic) N × N matrices with complex entries, and suppose that A N and B N have limiting distributions as N → ∞ with respect to the normalized trace on M N (C). Let (U N ) N ∈N be a sequence of N × N unitary random matrices, distributed according to Haar measure. Then U N A N U N∗ and B N are asymptotically freely independent as N → ∞. Moreover, when computing a fixed moment in U N A N U N∗ and B N , the error is O(N −2 ) as N → ∞ (see e.g. [10]), which can be interpreted as asymptotic infinitesimal freeness in the sense of Belinschi-Shlyakhtenko [6]. On the other hand, it is becoming increasingly apparent that in free probability, the roles of the classical groups are played by certain “free” quantum groups. This can most clearly be seen in the study of quantum distributional symmetries, originating with the free de Finetti theorem of Köstler and Speicher [16] and further developed in [4,12,13], in which the classical permutation, orthogonal and unitary groups are replaced by Wang’s Research supported by a Discovery grant from NSERC.

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S. Curran, R. Speicher

universal compact quantum groups [21,22]. For a general discussion of the passage from classical groups to free quantum groups, see [5]. In this paper, we will consider the limiting distribution of U N A N U N∗ and B N , where A N and B N are as above, but U N is now a Haar distributed N × N quantum unitary random matrix, in the sense of Wang [21]. We will show that asymptotic (infinitesimal) freeness now holds even if the entries of A N and B N are allowed to take values in an arbitrary unital C∗ -algebra B: Theorem 1. Let B be a unital C∗ -algebra and let A N , B N ∈ M N (B) for N ∈ N. Assume that there is a finite constant C such that A N ≤ C, B N ≤ C for all N ∈ N. For each N ∈ N, let U N be a Haar distributed N × N quantum unitary random matrix, with entries independent from B. (1) Suppose that there are linear maps μ A , μ B : Bt → B such that for any b0 , . . . , bk ∈ B, lim (tr N ⊗ idB )[b0 A N b1 · · · A N bk ] − μ A [b0 tb1 · · · tbk ] = 0,

N →∞

lim (tr N ⊗ idB )[b0 B N b1 · · · B N bk ] − μ B [b0 tb1 · · · tbk ] = 0,

N →∞

where tr N denotes the normalized trace on M N (C). Then U N A N U N∗ and B N are asymptotically free with amalgamation over B. (2) Suppose that in addition, the limits lim N {(tr N ⊗ idB )[b0 A N b1 · · · A N bk ] − μ A [b0 tb1 · · · tbk ]}

N →∞

lim N {(tr N ⊗ idB )[b0 B N b1 · · · B N bk ] − μ B [b0 tb1 · · · tbk ]}

N →∞

converge in norm for any b0 , . . . , bk ∈ B. Then U N A N U N∗ and B N are asymptotically infinitesimally free with amalgamation over B. We will present more general asymptotic freeness results in Sect. 5, in particular Theorem 1 will be a special case of Corollary 5.9. We note that Theorem 5.1 holds equally well if U N is a Haar distributed N × N quantum orthogonal random matrix [21], indeed it follows from the results of Banica in [1] that U N A N U N∗ and B N have the same joint distribution in both cases. However, the more general results given in Sect. 5 do require that we work in the unitary case. For finite-dimensional B, we show in Proposition 5.11 that classical Haar unitary random matrices are sufficient to obtain such a result. However, classical unitaries are in general insufficient for asymptotic freeness with amalgamation, even within the class of approximately finite dimensional C∗ -algebras, and so it is indeed necessary to allow quantum unitary transformations. We will discuss this further in the second part of Sect. 5, see in particular Example 5.12 and the remarks which follow. We note that random matrix models for free products with amalgamation have also been considered by Brown, Dykema and Jung [8]. The difference between our frameworks is that we work with matrices whose entries take value in the algebra which we amalgamate over, while they consider random matrices with complex entries which approximate generating sets of certain amalgamated free products in distribution. Our paper is organized as follows: Section 2 contains notations and preliminaries. Here we collect the basic notions from free and infinitesimally free probability and introduce the quantum unitary group Au (N ). Section 3 contains some combinatorial results,

Asymptotic Infinitesimal Freeness with Amalgamation

629

related to the “fattening” operation on noncrossing partitions, which will be required in the sequel. In Sect. 4 we recall the Weingarten formula from [2] for computing integrals over Au (N ), and prove a new estimate on the entries of the corresponding Weingarten matrix. Section 5 contains our main results, and a discussion of their failure for classical Haar unitaries. 2. Preliminaries and Notations 2.1. Free probability. We begin by recalling the basic notions of noncommutative probability spaces and distributions of random variables. Definition 2.2. (1) A noncommutative probability space is a pair (A, ϕ), where A is a unital algebra over C and ϕ : A → C is a linear functional such that ϕ(1) = 1. Elements in a noncommutative probability space will be called random variables. (2) A W∗ -probability space (M, τ ) is a von Neumann algebra M together with a faithful, normal, tracial state τ . The joint distribution of a family (xi )i∈I of random variables in a noncommutative probability space (A, ϕ) is the collection of joint moments ϕ(xi1 · · · xik ) for k ∈ N and i 1 , . . . , i k ∈ I . This is nicely encoded in the linear functional ϕx : Cti |i ∈ I → C determined by ϕx ( p) = ϕ( p(x)) for p ∈ Cti |i ∈ I , where p(x) means of course to replace ti by xi for each i ∈ I . These definitions have natural “operator-valued” extensions given by replacing C by a more general algebra of scalars, which we now recall. Definition 2.3. An operator-valued probability space (A, E : A → B) consists of a unital algebra A, a subalgebra 1 ∈ B ⊂ A, and a conditional expectation E : A → B, i.e., E is a linear map such that E[1] = 1 and E[b1 ab2 ] = b1 E[a]b2 for all b1 , b2 ∈ B and a ∈ A. Example 2.4. Let B be a unital algebra over C, and let Mn (B) = Mn (C) ⊗ B be the algebra of n × n matrices over B, with the natural inclusion of B as In ⊗ B. Let tr n = n −1 Tr n denote the normalized trace on Mn (C). Then (Mn (B), tr ⊗idB ) is a B-valued probability space. Note that if B = (bi j )i,n j=1 ∈ Mn (B), (tr n ⊗ idB ) (B) =

n 1 bii . n i=1

The B-valued joint distribution of a family (xi )i∈I of random variables in an operatorvalued probability space (A, E : A → B) is the collection of B-valued joint moments E[b0 xi1 · · · xik bk ]

630


for k ∈ N, i 1 , . . . , i k ∈ I and b0 , . . . , bk ∈ B. Again this is conveniently encoded in the B-linear functional E x : Bti |i ∈ I → B determined by E x [ p] = E[ p(x)] for p ∈ Bti |i ∈ I , the algebra of noncommutative polynomials with coefficients in B. Definition 2.5. Let (A, E : A → B) be an operator-valued probability space, and let (Ai )i∈I be a collection of subalgebras B ⊂ Ai ⊂ A. The algebras are said to be free with amalgamation over B, or freely independent with respect to E, if E[a1 · · · ak ] = 0 whenever E[a j ] = 0 for 1 ≤ j ≤ k and a j ∈ Ai j with i j = i j+1 for 1 ≤ j < k. We say that subsets i ⊂ A are free with amalgamation over B if the subalgebras Ai generated by B and i are freely independent with respect to E. Remark 2.6. Voiculescu first defined freeness with amalgamation, and developed its basic theory in [19]. Freeness with amalgamation also has a rich combinatorial structure, developed in [18], which we now recall. For further information on the combinatorial theory of free probability, the reader is referred to the text [17]. Definition 2.7. (1) A partition π of a set S is a collection of disjoint, non-empty sets V1 , . . . , Vr such that V1 ∪ · · · ∪ Vr = S. V1 , . . . , Vr are called the blocks of π , and we set |π | = r . If s, t ∈ S are in the same block of π , we write s ∼π t. The collection of partitions of S will be denoted P(S), or in the case that S = {1, . . . , k} by P(k). (2) Given π, σ ∈ P(S), we say that π ≤ σ if each block of π is contained in a block of σ . There is a least element of P(S) which is larger than both π and σ , which we denote by π ∨ σ . (3) If S is ordered, we say that π ∈ P(S) is non-crossing if whenever V, W are blocks of π and s1 < t1 < s2 < t2 are such that s1 , s2 ∈ V and t1 , t2 ∈ W , then V = W . The non-crossing partitions can also be defined recursively, a partition π ∈ P(S) is non-crossing if and only if it has a block V which is an interval, such that π \V is a non-crossing partition of S\V . The set of non-crossing partitions of S is denoted by N C(S), or by N C(k) in the case that S = {1, . . . , k}. (4) Given π, σ ∈ N C(S), the join π ∨ σ taken in P(S) may not be non-crossing. However, there is a least element of N C(S) which is larger than π and σ , which we will denote by π ∨nc σ . Note that in this paper we will always use π ∨ σ to denote the join in P(S), even when π, σ are assumed noncrossing. (5) Given i 1 , . . . , i k in some index set I , we denote by ker i the element of P(k) whose blocks are the equivalence classes of the relation s ∼ t ⇔ is = it . Note that if π ∈ P(k), then π ≤ ker i is equivalent to the condition that whenever s and t are in the same block of π , i s must equal i t . (6) With 0n and 1n we will denote the smallest and largest element, respectively, in P(n); i.e., 0n has n blocks, each consisting of one element, and 1n has only one block. Of course, both 0n and 1n are in N C(n).


631

Definition 2.8. Let (A, E : A → B) be an operator-valued probability space. (1) A B-functional is a n-linear map ρ : An → B such that ρ(b0 a1 b1 , a2 b2 , . . . , an bn ) = b0 ρ(a1 , b1 a2 , . . . , bn−1 an )bn for all b0 , . . . , bn ∈ B and a1 , . . . , an ∈ A. Equivalently, ρ is a linear map from A⊗ B n to B, where the tensor product is taken with respect to the obvious B-B-bimodule structure on A. (2) For each k ∈ N, let ρ (k) : Ak → B be a B-functional. For n ∈ N and π ∈ N C(n), we define a B-functional ρ (π ) : An → B recursively as follows: If π = 1n is the partition containing only one block, we set ρ (π ) = ρ (n) . Otherwise let V = {l + 1, . . . , l + s} be an interval of π and define ρ (π ) [a1 , . . . , an ] = ρ (π \V ) [a1 , . . . , al ρ (s) (al+1 , . . . , al+s ), al+s+1 , . . . , an ] for a1 , . . . , an ∈ A. Example 2.9. Let (A, E : A → B) be an operator-valued probability space, and for k ∈ N let ρ (k) : Ak → B be a B-functional as above. If π = {{1, 8, 9, 10}, {2, 7}, {3, 4, 5}, {6}} ∈ N C(10), 1

2

3

4

5

6

7

8

9 10

then the corresponding ρ (π ) is given by ρ (π ) [a1 , . . . , a10 ] = ρ (4) (a1 · ρ (2) (a2 · ρ (3) (a3 , a4 , a5 ), ρ (1) (a6 ) · a7 ), a8 , a9 , a10 ). Remark 2.10. Note that if B is commutative, then ρ (π ) [a1 , . . . , an ] = ρ(V )[a1 , . . . , an ], V ∈π

where if V = (i 1 < · · · < i s ) is a block of π , we set ρ(V )[a1 , . . . , an ] = ρ (s) [ai1 , . . . , ais ]. Definition 2.11. Let (A, E : A → B) be an operator-valued probability space. (1) For k ∈ N, define the B-valued moment functions E (k) : Ak → B by E (k) [a1 , . . . , ak ] = E[a1 · · · ak ]. (k)

(2) The operator-valued free cumulants κ E : Ak → B are the B-functionals defined by the moment-cumulant formula: (π ) E[a1 · · · an ] = κ E [a1 , . . . , an ] π ∈N C(n)

for n ∈ N and a1 , . . . , an ∈ A.

632


Note that the right hand side of the moment-cumulant formula above is equal to κ E(n) (a1 , . . . , an ) plus products of lower order terms and hence can be solved recursively (n) for κ E . In fact the cumulant functions can be solved from the moment functions by the following formula from [18]: for each n ∈ N, π ∈ N C(n) and a1 , . . . , an ∈ A, (π ) κ E [a1 , . . . , an ] = μn (σ, π )E (σ ) [a1 , . . . , an ], σ ∈N C(n) σ ≤π

where μn is the Möbius function on the partially ordered set N C(n). The Möbius function μn (σ, π ) is defined to be 0 unless σ ≤ π , is 1 if σ = π , and for σ < π is given by −1 +

(−1)l+1 #{(ν1 , . . . , νl ) ∈ N C(n)l : σ < ν1 < · · · < νl < π }.

l≥1

The key relation between operator-valued free cumulants and freeness with amalgamation is that freeness can be characterized in terms of the “vanishing of mixed cumulants”. Theorem 2.12 ([18]). Let (A, E : A → B) be an operator-valued probability space, and let (Ai )i∈I be a collection of subalgebras B ⊂ Ai ⊂ A. Then the family (Ai )i∈I is free with amalgamation over B if and only if (π )

κ E [a1 , . . . , an ] = 0 whenever a j ∈ Ai j for 1 ≤ j ≤ n and π ∈ N C(n) is such that π ≤ ker i. 2.13. Infinitesimal free probability. We will now introduce the notions of operatorvalued infinitesimal probability spaces and infinitesimal freeness with amalgamation. This is a straightforward generalization of the framework of [6], and we refer the reader to that paper for further discussion of infinitesimal freeness and its relation to the type B free independence of Biane, Nica and Goodman [7]. See [14] for a more combinatorial treatment of infinitesimal freeness. Definition 2.14. (1) If B is a unital algebra, a B-valued infinitesimal probability space is a triple (A, E, E ) where A is a unital algebra which contains B as a unital subalgebra and E, E are B-linear maps from A to B such that E[1] = 1 and E [1] = 0. (2) Let (A, E, E ) be a B-valued infinitesimal probability space, and let (Ai )i∈I be a collection of subalgebras B ⊂ Ai ⊂ A. The algebras are said to be infinitesimally free with amalgamation over B, or infinitesimally free with respect to (E, E ), if (i) (Ai )i∈I are freely independent with respect to E. (ii) For any a1 , . . . , ak so that a j ∈ Ai j for 1 ≤ j ≤ k with i j = i j+1 , we have E [(a1 − E[a1 ]) · · · (ak − E[ak ])] =

k j=1

E (a1 − E[a1 ]) · · · E [a j ] · · · (ak − E[ak ]) .


633

We say that subsets (i )i∈I are infinitesimally free with amalgamation over B if the subalgebras Ai generated by B and i are infinitesimally free with respect to (E, E ). Remark 2.15. The motivating example is given by a family (Ai (s))i∈I of B-valued random variables for s > 0 which are free “up to o(s)” as s → 0. This is made precise in the next proposition. Note that there we make the notion “free up to o(s)” precise by comparing the family (Ai (s))i∈I with a family (ai (s))i∈I which is free for all s. Infinitesimal freeness will then occur at s = 0 (both for the Ai and the ai ). Since 0 is not necessarily in K , we define the states E and E on the free algebra A := BAi |i ∈ I generated by non-commuting indeterminates Ai =A ˆ i (0)=a ˆ i (0). Proposition 2.16. Let B be a unital C∗ -algebra and K a subset of R for which 0 is an accumulation point. Suppose that for each s ∈ K we have a B-valued probability space (A(s), E s : A(s) → B), where A(s) is a unital C∗ -algebra which contains B as a unital subalgebra and E s is contractive. Furthermore, suppose that, for each s ∈ K , there are variables (Ai (s))i∈I belonging to A(s) such that the following hold: (1) There are B-linear maps E, E : BAi |i ∈ I → B such that E[ p(A)] = lim E s [ p(A(s))] , s→0

E [ p(A)] = lim

s→0

1 {E s [ p(A(s))] − E[ p]} , s

for p ∈ Bti |i ∈ I , where the limits hold in norm. (2) For each i ∈ I , lim sup Ai (s) < ∞. s→0

Let I = j∈J I j be a partition of I . For s ∈ K , let (ai (s))i∈I be a family in some B-valued probability space (C, F : C → B) and suppose that (1) For any j ∈ J , p ∈ Bti |i ∈ I j , and s ∈ K , E s [ p(A(s))] = F[ p(a(s))]. (2) The sets ({ai (s)|s ∈ K , i ∈ I j }) j∈J are free with respect to F. (3) For any p ∈ Bti |i ∈ I we have E s [ p(A(s))] − F[ p(a(s))] = o(s)

(as s → 0).

Then the sets ({Ai |i ∈ I j }) j∈J ⊂ BAi |i ∈ I are infinitesimally free with respect to (E, E ). Proof. Since E, E only depend on the distribution of the variables Ai (s) up to first order, it clearly suffices to assume that the sets ({Ai (s) : i ∈ I j }) j∈J are freely independent with respect to E s for all s ∈ K . It is then clear that the sets ({Ai : i ∈ I j }) j∈J ⊂ BAi |i ∈ I are free with respect to E, so it suffices to show that E satisfies condition (ii) of Definition 2.14. Let j1 = · · · = jk in J and pl ∈ Bti |i ∈ I jl for 1 ≤ l ≤ k, and consider

634


E [( p1 (A) − E[ p1 (A)]) · · · ( pk (A) − E[ pk (A)])] 1 = lim {E s [( p1 (A(s)) − E[ p1 (A)]) · · · ( pk (A(s)) − E[ pk (A)])] s→0 s −E [( p1 (A) − E[ p1 (A)]) · · · ( pk (A) − E[ pk (A)])]} 1 = lim {E s [( p1 (A(s)) − E[ p1 (A)]) · · · ( pk (A(s)) − E[ pk (A)])]} , s→0 s where we have used freeness with respect to E. Rewrite this expression as 1 {E s [(( p1 (A(s)) − E s [ p1 (A(s))]) + (E s [ p1 (A(s))] − E[ p1 (A)])) s · · · (( pk (A(s)) − E s [ pk (A(s))]) + (E s [ pk (A(s))] − E[ pk (A)]))]} ,

lim

s→0

and consider the terms which appear in the expansion. First observe that E s [ pl (A(s))] − E[ pl (A)] is O(s) for 1 ≤ l ≤ k. By the boundedness assumption on the norms of Ai (s), and the contractivity of E s , it follows that those terms involving more than one expression (E s [ pl (A(s))] − E[ pl (A)]) vanish in the limit. The term involving none of these expressions is E s [( p1 (A(s)) − E s [ p1 (A(s))]) · · · ( pk (A(s)) − E s [ pk (A(s))])] which is zero by freeness. So we are left to consider only the terms involving one such expression, which gives k l=1

lim

s→0

1 {E s [( p1 (A(s)) − E s [ p1 (A(s))]) s

· · · (E s [ pl (A(s))] − E[ pl (A)]) · · · ( pk (A(s)) − E s [ pk (A(s))])]} , which again by invoking the boundedness assumptions on Ai (s) and contractivity of E s , converges to k

E ( p1 (A) − E[ p1 (A)]) · · · E [ pl (A)] · · · ( pk (A) − E[ pk ])

l=1

as desired. 2.17. Quantum unitary group. We now recall the definition of the quantum unitary group from [21], which is a compact quantum group in the sense of Woronowicz [23]. Definition 2.18. Au (n) is the universal C∗ -algebra generated by {Ui j : 1 ≤ i, j ≤ n} such that the matrix U = (Ui j ) ∈ Mn (Au (n)) is unitary. Au (n) is a C∗ -Hopf algebra with comultiplication, counit and antipode given by (Ui j ) =

n

Uik ⊗ Uk j

k=1

(Ui j ) = δi j S(Ui j ) = U ∗ji . The existence of these maps is given by the universal property of Au (n).


635

Remark 2.19. A fundamental result of Woronowicz [23] guarantees the existence of a unique Haar state ψn : Au (n) → C which is left and right invariant in the sense that (ψn ⊗ id)(a) = ψn (a)1 Au (n) = (id ⊗ ψn )(a) for a ∈ Au (n). We will discuss this further in Sect. 4. Wang also introduced the free product operation on compact quantum groups in [21]. We will use Au (n)∗∞ to denote the C∗ -algebraic free product (with amalgamation over C) of countably many copies of Au (n). Au (n)∗∞ has a natural compact quantum group structure, given in Corollary 3.7 of [21]. The reader is referred to that paper for details, the only properties which we will use are the following: (1) Au (n)∗∞ is generated (as a C∗ -algebra) by elements {U (l)i j : l ∈ N, 1 ≤ i, j ≤ n}, such that U (l) ∈ Mn (Au (n)∗∞ ) is unitary. (2) The sets ({U (l)i j : 1 ≤ i, j ≤ n})l∈N are freely independent with respect to the Haar state ψn∗∞ on Au (n)∗∞ , and for each l ∈ N, (U (l)i j ) has the same joint distribution in (Au (n)∗∞ , ψn∗∞ ) as (Ui j ) in (Au (n), ψn ). See Proposition 3.3 and Theorem 3.4 of [21]. 3. Some Combinatorial Results In this section we introduce several operations on partitions and prove some basic results which will be required throughout the remainder of the paper. Notation 3.1. (1) Given π ∈ N C(m), we define π ∈ N C2 (2m) as follows: For each block V = (i 1 , . . . , i s ) of π , we add to π the pairings (2i 1 − 1, 2i s ), (2i 1 , 2i 2 − 1), . . . , (2i s−1 , 2i s − 1). (2) Given π ∈ N C(m), we define πˆ ∈ N C(2m) by partitioning the m-pairs (1, 2), (3, 4), . . . , (2m − 1, 2m) according to π . (3) Given π, σ ∈ P(m), we define π σ ∈ P(2m) to be the partition obtained by partitioning the odd numbers {1, 3, . . . , 2m − 1} according to π and the even numbers {2, 4, . . . , 2m} according to σ . (4) Given π ∈ P(m), let ← π− denote the partition obtained by shifting k to k − 1 for 1 < k ≤ m and sending 1 to m, i.e., s ∼← π− t

⇐⇒

(s + 1) ∼π (t + 1),

→ where we count modulo m on the right hand side. Likewise we let − π denote the partition obtained by shifting k to k + 1 for 1 ≤ k < m and sending m to 1. Remark 3.2. The map π → π is easily seen to be a bijection, and corresponds to the well-known “fattening” operation. The following example shows this for π = {{1, 4, 5}, {2, 3}, {6}}. 1 π=

2

3

4

5

6

11 π=

22

33

44

55

66

636


There is a simple description of the inverse, it sends σ ∈ N C2 (2m) to the partition τ ∈ N C(m) such that σ ∨ 0ˆ m = τˆ , where 0ˆ m = {{1, 2}, . . . , {2m − 1, 2m}}. Thus we have for π ∈ N C(m), πˆ = π ∨ 0ˆ m . Note also that 0ˆ m = 0m and that 1ˆ m = 12m . Definition 3.3. Let π ∈ N C(m). The Kreweras complement K (π ) is the largest partition in N C(m) such that π K (π ) ∈ N C(2m). Example 3.4. If π = {{1, 5}, {2, 3, 4}, {6, 8}, {7}} then K (π ) = {{1, 4}, {2}, {3}, {5, 8}, {6, 7}}, which can be seen follows: 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8

The following lemma provides the relationship between the Kreweras complement on N C(m) and the map π → π. Lemma 3.5. If π ∈ N C(m), then ← − K

(π ) = π. Proof. We will prove this by induction on the number of blocks of π . If π = 1m has one block, the result is trivial from the definitions. Suppose now that V = {l + 1, . . . , l + s} is a block of π , l ≥ 1. First note that π is obtained by taking π

\V then adding the pairs (2l + 1, 2(l + s)), (2l + 2, 2l + 3), . . . , (2(l + s) − 2, 2(l + s) − 1). Observe that K (π ) is obtained by taking K (π \V ), adding singletons {l + 1}, . . . , {l + s − 1}, then placing l + s in the block containing l. It follows that K

(π ) is the partition ←−−

obtained by taking K (π \V ), which by induction is π \V , then moving the leg connected to 2l to 2(l + s) and adding the pairs (2l, 2(l + s) − 1), (2l + 1, 2l + 2), . . . , (2(l + s) − 3, 2(l + s) − 2). The result now follows. We will also need the following relationship between π → π and the Kreweras complement on N C(2m). This is a generalization of the relation K (π) ˆ = K ( 0m ∨ π ) = 0m K (π )

(π ∈ N C(m)),

which is obvious from the definition of πˆ . Lemma 3.6. If π, σ ∈ N C(m) and σ ≤ π , then σ ∨ π ∈ N C(2m) and K ( σ ∨ π ) = σ K (π ).


637

Proof. We will prove this by induction on the number of blocks of π . First suppose that π = 1m , then we have − − → −−−→ ← −−−−← − −−−−−−−→ (σ ) ∨ 0ˆ m = K (σ ) σ ∨ π= σ ∨ π = K

is noncrossing. Moreover, K ( σ ∨ π) = K

−−−−−−−→ −−−→ K (σ ) = 0m K 2 (σ ),

where for the last equality we used the equation for K (πˆ ) mentioned before Lemma 3.6 and the fact that the Kreweras complement commutes with shifting. But, by [17, Exercise 9.23], we have that K 2 (σ ) = ← σ− and thus we finally get −−−−→ σ− = σ 0m . K ( σ ∨ π ) = 0m ← Now suppose that V = {l + 1, . . . , l + s}, l ≥ 1 is an interval of π . Observe that σ ∨ π is the partition obtained by partitioning {1, . . . , 2l} ∪ {2(l + s) + 1, . . . , 2m}

|V ∨ π

according to σ \σ \V , and {2l + 1, . . . , 2(l + s)} according to σ

|V ∨ 1 V . It follows that σ ∨ π is noncrossing and that K ( σ ∨ π ) is the partition obtained by partitioning

|V ∨π

{1, . . . , 2l}∪{2(l+s)+1, . . . , 2m} according to K (σ \σ \V ) and {2l+1, . . . , 2(l+s)} according to K (σ

|V ∨ 1 ), then joining the blocks containing 2l and 2(l + s). On the V other hand, K (π ) is equal to the partition obtained by taking K (π \V ) then adding {l + 1}, . . . , {l + s − 1} and joining l + s to l, and the result now follows by induction. We will need to compare the number of blocks in the join of two partitions before and after fattening. For this purpose we will use the following linearization lemma of Kodiyalam-Sunder [15] and, independently, Chen-Przytycki [9]. Note that the notation S → S used in [15] corresponds to the inverse of the fattening procedure π → π used here. Theorem 3.7 ([15]). Let π, σ ∈ N C(m). Then | π ∨ σ | = m + 2|π ∨ σ | − |π | − |σ |. In particular, if σ ≤ π then | π ∨ σ | = m + |π | − |σ |. We now introduce some special classes of noncrossing partitions and prove some basic results. These are related to integration on the quantum unitary group via the Weingarten formula to be discussed in the next section. Notation 3.8. Let 1 , . . . , 2m ∈ {1, ∗}. (1) N C h (2m) denote the set of partitions π ∈ N C(2m) such that each block V of π has an even number of elements, and |V is alternating, i.e., |V = 1 ∗ 1 ∗ · · · 1∗ or ∗1 ∗ 1 · · · ∗ 1. (2) N C2 (2m) will denote the collection of π ∈ N C2 (2m) such that each pair in π connects a 1 with a ∗, i.e., s ∼π t ⇒ s = t . (3) N C (m) will denote the collection of π ∈ N C(m) such that π ∈ N C2 (m).

638


Lemma 3.9. Let 1 , . . . , 2m ∈ {1, ∗}. If σ, π ∈ N C (m) and σ ≤ π , then σ ∨ π is in N C h (2m). Conversely, if τ ∈ N C h (2m) then there are unique σ, π ∈ N C (m) such that σ ≤ π and τ = σ ∨ π. Proof. First suppose that τ ∈ N C h (2m). Since each block of τ has an even number of elements, we have K (τ ) = σ K (π ) for some σ, π ∈ N C(m) such that σ ≤ π . By Lemma 3.6 we have τ = σ ∨ π , and this clearly determines σ and π uniquely. If V is a block of τ , then |V is alternating and hence π |V , σ |V ∈ N C2 (V ). It follows that π, σ ∈ N C (m). Conversely, let σ, π ∈ N C (m) such that σ ≤ π . Let ˆ = ( 1 , 1 , 2 , 2 , . . . , 2m , 2m ). Observe that if τ ∈ N C(2m), then τ ∈ N C h (2m) if and only if τ ∈ N C2 ˆ (4m). So let τ = σ ∨ π , we need to show τ ∈ N C2 ˆ (4m). Now ← − τ = K

(τ ) = σ K

(π ), ← − where we have applied Lemmas 3.5 and 3.6. In other words, τ is the partition given by partitioning {1, 2, 5, 6, . . . , 4m − 3, 4m − 2} according to σ and {3, 4, 7, 8, . . . , ← −

4m − 1, 4m} according to K (π ) = π . Now since σ, π ∈ N C (m), it follows that ← − ← − ← − τ ∈ N C2 ˆ (4m). τ ∈ N C2 ˆ (4m), where ˆ = ( 1 , 2 , 2 , . . . , 2m , 2m , 1 ), and hence Lemma 3.10. N C (m) is closed under taking intervals in N C(m), i.e., if σ, π ∈ N C (m) and τ ∈ N C(m) is such that σ < τ < π , then τ ∈ N C (m). Proof. Let σ, π ∈ N C (m), and τ ∈ N C(m) such that σ < τ < π . From the inductive definition of τ , to show that τ ∈ N C (m) it suffices to consider π = 1m . Now by the previous lemma, we have σ ∨ 1 m ∈ N C h (2m). By Lemma 3.5, ←−−−− σ ∨ 1 (σ ) ∨ 0ˆ m = K (σ ). m = K

Since σ ≤ τ , we have 0m ≤ K (τ ) ≤ K (σ ). Let δ = ( 2 , . . . , 2m , 1 ), and suppose that K (τ ) ∈ / N C hδ (2m). Let V be a block of K (τ ), and note that V is of the form (2i 1 − 1, 2i 1 , . . . , 2i s − 1, 2i s ) for some i 1 < · · · < i s . Since 0ˆ m ∈ N C hδ (2m), it follows that there is a 1 ≤ l < s with δ2il = δ2il+1 −1 . Now since 0m ≤ K (τ ) ≤ K (σ ), it follows that the block W of K (σ ) which contains V must have an even number of elements between 2il and 2il+1 − 1. But then δ|W cannot be alternating, which contradicts K (σ ) ∈ N C hδ (2m). So we have shown that K (τ ) ∈ N C hδ (2m), and since −−→ −−−−−−−→ (τ ) ∨ 0ˆ m = K (τ ) = K

τ ∨ 1m , we have τ ∨ 1 m ∈ N C h (2m). But then by the previous lemma, there is a γ ∈ N C (m) with γ ∨ 1m = τ ∨ 1m , and by Lemma 3.6 this implies τ = γ is in N C (m) as claimed.


639

4. Integration on the Quantum Unitary Group We begin by recalling the Weingarten formula from [2] for computing integrals with respect to the Haar state on Au (n). Let 1 , . . . , 2m ∈ {1, ∗} and define, for n ∈ N, the Gram matrix G n (π, σ ) = n |π ∨σ |

(π, σ ∈ N C2 (2m)).

It is shown in [2] that G n is invertible for n ≥ 2, let W n denote its inverse. Theorem 4.1 [2]. The Haar state on Au (n) is given by 2m ψn (Ui 11j1 · · · Ui 2m j2m ) =

W n (π, σ ),

π,σ ∈N C2 (2m) π ≤ker i σ ≤ker j

2m+1 ψn (Ui 11j1 · · · Ui 2m+1 j2m+1 ) = 0,

for 1 ≤ i 1 , j1 , . . . , i 2m+1 , j2m+1 ≤ n and 1 , . . . , 2m+1 ∈ {1, ∗}. Remark 4.2. Note that the Weingarten formula above is effective for computing integrals of products of the entries in U and its conjugate U , the matrix with (i, j)-entry Ui∗j . We will also need to compute integrals of products of entries from U and its adjoint U ∗ , whose (i, j)-entry we denote (U ∗ )i j to distinguish from the conjugate U . To do this we will use the following proposition, which allows us to reduce to the former case. Note that such a formula clearly fails for the classical unitary group. Indeed we have

∗

∗

(U )21 (U )43 U12 U34 = Un

U 12 U 34 U12 U34 = Un

n2

1 −1

while U 21 U 34 U12 U43 = 0, Un

as can be seen by using the Weingarten formula from [10]. Proposition 4.3. Let 1 ≤ i 1 , i 2 , . . . , i 4m ≤ n and 1 , . . . , 2m ∈ {1, ∗}. Then 2m ψn (U 1 )i1 i2 (U 2 )i3 i4 · · · (U 2m )i4m−1 i4m = ψn Ui 11i2 Ui 42i3 · · · Ui 4m i 4m−1 . Proof. We will use the fact from [1] that the joint ∗-distribution of (Ui j )1≤i, j≤n with respect to ψn is the same as that of (z Oi j )1≤i, j≤n , where z and (Oi j ) are random variables in a ∗-probability space (M, τ ) such that: (1) z is ∗-freely independent from {Oi j : 1 ≤ i, j ≤ n}. (2) z has a Haar unitary distribution. (3) (Oi j ) are self-adjoint, and have the same joint distribution as the generators of the quantum orthogonal group Ao (n).

640


The joint distribution of (Oi j ) can also be computed via a Weingarten formula, see [2] for details. The only fact that we will use is that the joint distribution is invariant under transposition, i.e., the families (Oi j )1≤i, j≤n and (O ji )1≤i, j≤n have the same joint distribution. Now let 1 , . . . , 2m ∈ {1, ∗}. Let A = { j : j is even and j = ∗} ∪ { j : j is odd and j = 1}, and B = {1, . . . , 2m}\A. Let 1 ≤ i 1 , j1 , . . . , i 2m , j2m ≤ n. For 1 ≤ k ≤ 2m, define ik , k ∈ A j , k∈A ik = , jk = k . jk , k ∈ B ik , k ∈ B We claim that 2m 1 2m , ψn Ui 11j1 · · · Ui 2m j2m = ψn Ui j · · · Ui j 1 1

2m 2m

from which the formula in the statement follows immediately. As discussed above, we have 1 2m 2m ψn Ui 11j1 · · · Ui 2m . j2m = τ (z Oi 1 j1 ) · · · (z Oi 2m j2m ) Note that the expression (z Oi1 j1 ) 1 · · · (z Oi2m j2m ) 2m can be written as a product of terms of the form z Oik jk or Oik jk z ∗ , depending if k is 1 or ∗. After rewriting the expression in this form, let C be the subset of {1, . . . , 4m} consisting of those indices corresponding to z or z ∗ , and let D be its complement. Explicitly, if k = 1 then 2k − 1 is in C and 2k is in D, and if k = ∗ then 2k is in C and 2k − 1 is in D. Given partitions α, β ∈ N C(2m), let (α, β) ∈ P(4m) be given by partitioning C according to α and D according to β. By freeness, we have τ (z Oi1 j1 ) 1 · · · (z Oi2m j2m ) 2m = κα [z 1 , . . . , z 2m ]κβ [Oi1 j1 , . . . , Oi2m j2m ]. α,β∈N C(2m) (α,β)∈N C(4m)

Now since Haar unitaries are R-diagonal, we have κα [z 1 , . . . , z 2m ] = 0 unless each block of α contains an even number of elements. So assume that α has this property, we claim that if β is such that (α, β) is noncrossing, then β does not join any element of A with an element of B. Indeed, suppose that β joins k1 < k2 and that one of k1 , k2 is in A and the other is in B. If k1 , k2 have the same parity, then it follows that one of k1 , k2 is a 1 while the other is a ∗. Suppose that k1 = 1, k2 = ∗; the other case is similar. Then we have 2k1 connected to 2k2 − 1 in (α, β). Since (α, β) is noncrossing, α cannot join any element of {k1 + 1, . . . , k2 − 1} to an element outside of this set. But since this set contains an odd number of elements, we obtain a contradiction to the choice of α. If k1 , k2 have different parity, then it follows that k1 = k2 . Suppose that k1 = k2 = 1; the other case is similar. Then 2k1 is connected to 2k2 in (α, β). It follows that α cannot connect any element of {k1 + 1, . . . , k2 } to an element outside of this set, and again this set has an odd number of elements which contradicts the choice of α. So the only nonzero terms appearing in the expression above come from β ∈ N C(2m) which split into noncrossing partitions π of A and σ of B. In this case, if A = (a1 < · · · < as ) and B = (b1 < · · · < br ), we have


641

κβ [Oi1 j1 , . . . , Oi2m j2m ] = κπ [Oia1 ja1 , . . . , Oias jas ]κσ [Oib1 jb1 , . . . , Oibr jbr ] = κπ [Oia1 ja1 , . . . , Oias jas ]κσ [O jb1 ib1 , . . . , O jbr ibr ] j ], = κβ [Oi1 j1 , . . . , Oi2m 2m where we have used the invariance of the distribution of (Oi j ) under transposition. Putting this all together, we have 1 2m 2m ψn Ui 11j1 · · · Ui 2m j2m = τ (z Oi 1 j1 ) · · · (z Oi 2m j2m ) κα [z 1 , . . . , z 2m ]κβ [Oi1 j1 , . . . , Oi2m j2m ] = α,β∈N C(2m) (α,β)∈N C(4m)

=

α,β∈N C(2m) (α,β)∈N C(4m)

j ] κα [z 1 , . . . , z 2m ]κβ [Oi1 j1 , . . . , Oi2m 2m

j ) 2m = τ (z Oi1 j1 ) 1 · · · (z Oi2m 2m 2m 1 = ψn Ui j · · · Ui j 1 1

2m 2m

as desired. We can now extend this result to the free product Au (n)∗∞ . Corollary 4.4. Let l1 , . . . , l2m ∈ N, 1 , . . . , 2m ∈ {1, ∗} and 1 ≤ i 1 , j1 , . . . , i 2m , j2m ≤ n. In Au (n)∗∞ , we have ψn∗∞ (U (l1 ) 1 )i1 i2 (U (l2 ) 2 )i3 i4 · · · (U (l2m ) 2m )i4m−1 i4m 2m = ψn∗∞ U (l1 )i 11i2 U (l2 )i 42i3 · · · U (l2m )i 4m i 4m−1 . Proof. First we claim that in Au (n), we have κ (2m) [(U 1 )i1 i2 , (U 2 )i3 i4 , . . . , (U 2m )i4m−1 i4m ] 2m = κ (2m) [Ui 11i2 , Ui 42i3 , . . . , Ui 4m i 4m−1 ].

(Note that any cumulant of odd length is zero by Theorem 4.1). Indeed, we have κ (2m) [(U 1 )i1 i2 , (U 2 )i3 i4 , . . . , (U 2m )i4m−1 i4m ] = μ2m (σ, 12m ) ψn (V )[(U 1 )i1 i2 , (U 2 )i3 i4 , . . . , (U 2m )i4m−1 i4m ]. σ ∈N C(2m)

V ∈σ

Now it is clear from Theorem 4.1 that ψn (V )[(U 1 )i1 i2 , (U 2 )i3 i4 , . . . , (U 2m )i4m−1 i4m ] = 0 unless V has an even number of elements. So the nonzero terms in the expression above come from those σ ∈ N C(2m) for which every block has an even number of elements.

642


For such a σ , the noncrossing condition implies that each block V = (l1 < · · · < ls ) must be alternating in parity. By Proposition 4.3 we have ψn (V )[(U 1 )i1 i2 , (U 2 )i3 i4 , . . . , (U 2m )i4m−1 i4m ] = ψn (U l1 )i2l1 −1 i2l1 (U l2 )i2l2 −1 i2l2 · · · (U ls )i2ls −1 i2ls l l = ψn Ui2l1 −1 i2l Ui2l2 i2l −1 · · · Ui2lls i2l −1 s s 1 2 2 1 l2 ls l1 = ψn Ui2l i2l −1 Ui2l −1 i2l · · · Ui2l −1 i2l , 1

1

2

2

s

s

where the last equation follows from the invariance of the joint ∗-distribution of (Ui j ) under transposition. It follows that κ (2m) [(U 1 )i1 i2 , (U 2 )i3 i4 , . . . , (U 2m )i4m−1 i4m ] = μ2m (σ, 12m ) ψn (V )[(U 1 )i1 i2 , (U 2 )i3 i4 , . . . , (U 2m )i4m−1 i4m ] V ∈σ

σ ∈N C(2m)

= =

μ2m (σ, 12m )

2m ψn (V )[Ui 11i2 , Ui 42i3 , . . . , Ui 4m i 4m−1 ]

V ∈σ σ ∈N C(2m) 1 2 (2m) 2m [Ui1 i2 , Ui4 i3 , . . . , Ui 4m κ i 4m−1 ]

as claimed. Now by free independence, in Au (n)∗∞ we have ψn∗∞ (U (l1 ) 1 )i1 i2 (U (l2 ) 2 )i3 i4 · · · (U (l2m ) 2m )i4m−1 i4m = κ(V )[(U (l1 ) 1 )i1 i2 , (U (l2 ) 2 )i3 i4 , . . . , (U (l2m ) 2m )i4m−1 i4m ]. σ ∈N C(2m) V ∈σ σ ≤ker l

Since κ(V ) is zero unless V has an even number of elements, the only terms which contribute to the sum above come again from σ ∈ N C(2m) for which each block has an even number of elements. From the previous claim, we have κ(V )[(U (l1 ) 1 )i1 i2 , (U (l2 ) 2 )i3 i4 , . . . , (U (l2m ) 2m )i4m−1 i4m ] 2m = κ(V )[U (l1 )i 11i2 , U (l2 )i 42i3 , . . . , U (l2m )i 4m i 4m−1 ]

for each block V ∈ σ , and the result follows immediately. We will now give an estimate on the asymptotic behavior of the entries of W n as n → ∞. This improves the estimate given in [2]. Note that by taking = 1 ∗ · · · 1∗, this estimate also applies to the quantum orthogonal group, see [2]. Theorem 4.5. Let 1 , . . . , 2m ∈ {1, ∗}. Let π, σ ∈ N C (m). Then W n ( π, σ ) = O(n 2|π ∨σ |−|π |−|σ |−m ). Moreover, n m+|σ |−|π | W n ( π, σ ) = μm (σ, π ) + O(n −2 ), where μm is the Möbius function on N C(m), and we use the convention that μm (σ, π ) = 0 if σ ≤ π .


643

Proof. We use a standard method from [10,11], further developed in [2,3,12,4]. First observe that 1/2

1/2

G n = n (1 + B n ) n , where

nm , π = σ , 0, π = σ

n (π, σ ) = B n (π, σ ) =

0,

π =σ

n |π ∨σ |−m ,

π = σ

.

Note that the entries of B n are O(n −1 ), in particular for n large we have the geometric series expansion (1 + B n )−1 = 1 − B n +

l+1 (−1)l+1 B n . l≥1

Hence −m n −1/2 l+1 −1/2 (l+1) W n ( π, σ) = (−1) ( n B n n )( π, σ) + π ∨ σ |−2m −n | l≥1

π = σ, π = σ.

Now for l ≥ 1, we have −1/2

( n

−1/2

l+1 B n n

)( π, σ) =

π ∨ ν1 |+| ν1 ∨ ν2 |+···+| νl ∨ σ |−(l+2)m n | .

∈N C (m)

ν1 ,...,νl π =ν1 =··· =νl =σ

Now we claim that | π ∨ ν1 | + · · · + | νl ∨ σ | ≤ | π ∨ σ | + | ν1 | + · · · + | νl | ≤ | π ∨ σ | + l · m, from which the first equation follows from the above equation and Theorem 3.7. Indeed, the case l = 1 follows from the semi-modular condition: | π ∨ ν1 | + | ν1 ∨ σ | ≤ |( π ∨ ν1 ) ∨ ( ν1 ∨ σ )| + |( π ∨ ν1 ) ∧ ( ν1 ∨ σ )| ≤ | π ∨ σ | + | ν1 | = | π ∨ σ | + m. The general case follows easily from induction on l. For the second part, apply Theorem 3.7 to find that | π ∨ ν1 | + · · · + | νl ∨ σ | = 2(|ν1 ∨ ν2 | + · · · + |νl ∨ σ | − |ν1 | − · · · − |νl |) +2|π ∨ ν1 | − |π | − |σ | + (l + 1)m ≤ |π | − |σ | + (l + 1)m,

644


where equality holds if σ < νl < · · · < ν1 < π and otherwise the difference is at least 2. It then follows from the equation above that, up to O(n −2 ), n m+|σ |−|π | W n ( π, σ ) is equal to 0 if σ ≤ π , 1 if σ = π and otherwise is given by −1 +

∞ (−1)l+1 |{(ν1 , . . . , νl ) ∈ (N C (m))l : σ < νl < · · · < ν1 < π }|. l=1

Since

N C (m)

is closed under taking intervals in N C(m), this is equal to μm (σ, π ).

As a corollary, we can give an estimate on the free cumulants of the generators Ui j of Au (n). (Note that the cumulants of odd length are all zero since the generators have an even joint distribution). Corollary 4.6. Let 1 , . . . , 2m ∈ {1, ∗} and i 1 , j1 , . . . , i 2m , j2m ∈ N. For ω ∈ N C(2m), we have for the moment functions 2m n |π |−|σ |−m (μm (σ, π ) + O(n −2 )), ψn(ω) [Ui 11j1 , . . . , Ui 2m j2m ] = σ,π ∈N C (m) π ≤ker i∧ω σ ≤ker j∧ω

and for the cumulant functions 2m κ (ω) [Ui 11j1 , . . . , Ui 2m j2m ] =

n |π |−|σ |−m (μm (π, σ ) + O(n −2 )).

π,σ ∈N C (m) π ≤ker i σ ≤ker j σ =ω π ∨nc

(ω)

2m Proof. First note that ψn [Ui 11j1 , . . . , Ui 2m j2m ] = 0 unless ω ∈ N C h (2m), i.e., unless each block of ω has an even number of elements. So suppose this is the case, then by for some α, β ∈ N C(m) with α ≤ β. By the Weingarten Lemma 3.9 we have ω = α ∨β formula, we have 2m ψn(ω) [Ui 11j1 , . . . , Ui 2m W |V n ( π |V , σ |V ). j2m ] =

π,σ ∈N C (m) V ∈ω π ≤ker i∧ω σ ≤ker j∧ω

Let V = (l1 < · · · < ls ) be a block of ω. In order to apply Theorem 4.5 we have to write π |V and σ |V as π V and σ V , respectively, for some πV , σV ∈ N C(|V |/2). Since μ|V |/2 (σV , πV ) = μ|V | ( σV , π V ), it suffices to recover the doubled versions σ V,π V from π |V and σ |V . But this can be achieved as follows: ˆ |V |/2 = π π |V ∨ {(l1 , l2 ), . . . , (ls−1 , ls )}. V =π V ∨0 So it remains to write {(l1 , l2 ), . . . , (ls−1 , ls )} intrinsically in terms of ω. Recall from Lemma 3.6 that we have K (ω) = α K (β). It follows that for 1 ≤ r ≤ s such that lr is odd, α has a block whose least element is lr 2+1 and greatest element is lr2+1 . Therefore lr is joined to lr +1 in α . So if l1 is odd, then α |V is equal to {(l1 , l2 ), (l3 , l4 ), . . . , (ls−1 , ls )}. In this case, from Theorem 4.5 we have W |V n ( π |V , σ |V ) α |V |−| α |V |−|V |/2 π |V ∨ σ |V ∨ (μ|V | ( σ |V ∨ α |V , π |V ∨ α |V ) + O(n −2 )). = n |


645

On the other hand, if l1 is even then α |V = {(l1 , ls ), (l2 , l3 ), . . . , (ls−2 , ls−1 )}. In this case we have W |V n ( π |V , σ |V ) =n =n

−→ −→ α |V − α |V −|V |/2 σ |V ∨ π |V ∨ ←− ←−− σ |V ∨ α |V − α |V −|V |/2 π |V ∨

−→ −→ (μ|V | ( π |V ∨ α |V , σ |V ∨ α |V ) + O(n −2 )) ←− ←− (μ|V | ( π |V ∨ α |V , σ |V ∨ α |V ) + O(n −2 )),

where here the arrows act on the legs of V . Since this corresponds, by Lemma 3.5, to the Kreweras complement on N C|V |/2 , we have ←− σ |V ∨ σ |V ∨ α |V = |V |/2 + 1 − | α |V | and ←− ←− μ|V | ( π |V ∨ α |V , σ |V ∨ α |V ) = μ|V | ( σ |V ∨ α |V , π |V ∨ α |V ). So it follows that, as in previous case, we have W |V n ( π |V , σ |V ) α |V |−| α |V |−|V |/2 π |V ∨ σ |V ∨ (μ|V | ( σ |V ∨ α |V , π |V ∨ α |V ) + O(n −2 )). = n |

Therefore, 2m ψn(ω) [Ui 11j1 , . . . , Ui 2m j2m ] α |V |−| α |V |−|V |/2 π |V ∨ σ |V ∨ = n |

σ,π ∈N C (m) V ∈ω π ≤ker i∧ω σ ≤ker j∧ω

· (μ|V | ( σ |V ∨ α |V , π |V ∨ α |V ) + O(n −2 )) π ∨ α |−| σ ∨ α |−m = n | (μ2m ( σ ∨ α, π ∨ α ) + O(n −2 )), σ,π ∈N C (m) π ≤ker i∧ω σ ≤ker j∧ω

where we have used the multiplicativity of the Möbius function on N C(2m). , taking the Kreweras complement and applying Now since σ = σ ∨ σ ≤ α∨β Lemma 3.6 gives α K (β) ≤ σ K (σ ). So we have α ≤ σ ≤ β. By Theorem 3.7, we then have | σ ∨ α | = |σ | + m − |α|. Also, we have μ2m ( σ ∨ α, π ∨ α) = = = =

μ2m (K ( π ∨ α ), K ( σ ∨ α )) μ2m (α K (π ), α K (σ )) μm (K (π ), K (σ )) μm (σ, π ).

Plugging this into the equation above, we have 2m ψn(ω) [Ui 11j1 , . . . , Ui 2m n |π |−|σ |−m (μm (σ, π ) + O(n −2 )). j2m ] = σ,π ∈N C (m) π ≤ker i∧ω σ ≤ker j∧ω

646


For the cumulant function this gives 2m κ (τ ) [Ui 11j1 , . . . , Ui 2m j2m ] 2m = μ2m (ω, τ )ψn(ω) [Ui 11j1 , . . . , Ui 2m j2m ]

ω∈N C(2m) ω≤τ

=

μ2m (ω, τ )

n |π |−|σ |−m (μm (σ, π ) + O(n −2 ))

σ,π ∈N C (m) π ≤ker i∧ω σ ≤ker j∧ω

ω∈N C(2m) ω≤τ

=

n |π |−|σ |−m (μm (σ, π ) + O(n −2 ))

σ,π ∈N C (m)

μ2m (ω, τ ).

ω∈N C(2m) π ∨nc σ ≤ω≤τ

π ≤ker i σ ≤ker j

Since

μ2m (ω, τ ) =

ω∈N C(2m) π ∨nc σ ≤ω≤τ

σ =τ 1, π ∨nc , 0, otherwise

the result follows. As a corollary, we can give an estimate on the Haar state on the free product Au (n)∗∞ . Corollary 4.7. Let l1 , . . . , l2m ∈ N, 1 , . . . , 2m ∈ {1, ∗} and i 1 , j1 , . . . , i 2m , j2m ∈ N. In Au (n)∗∞ , we have 2m ψn∗∞ U (l1 )i 11 j1 · · · U (l2m )i 2m n |π |−|σ |−m (μm (σ, π ) + O(n −2 )). j2m = π,σ ∈N C (m) π ≤ker i∧ker l σ ≤ker j∧ker l

Proof. Since the families ({U (l)i j })l∈N are freely independent, we have by the vanishing of mixed cumulants 2m 2m κ (τ ) [U (l1 )i 11 j1 , . . . , U (l2m )i 2m ψn∗∞ U (l1 )i 11 j1 · · · U (l2m )i 2m j2m = j2m ]. τ ∈N C(2m) τ ≤ker l

Since the families ({U (l)i j })l∈N are identically distributed, we have 2m 1 (τ ) 2m κ (τ ) [U (l1 )i 11 j1 , . . . , U (l2m )i 2m j2m ] = κ [U (1)i 1 j1 , . . . , U (1)i 2m j2m ]

for any τ ∈ N C(2m) such that τ ≤ ker l. Applying the previous corollary, we have 2m ψn∗∞ U (l1 )i 11 j1 · · · U (l2m )i 2m j2m n |π |−|σ |−m (μm (σ, π ) + O(n −2 )) = τ ∈N C(2m) π,σ ∈N C (m) τ ≤ker l π ≤ker i σ ≤ker j π ∨nc σ =τ

=

π,σ ∈N C (m) π ≤ker i∧ker l σ ≤ker j∧ker l

n |π |−|σ |−m (μm (σ, π ) + O(n −2 )).


647

5. Asymptotic Freeness Results Throughout the first part of this section, the framework will be as follows: B will be a fixed unital C∗ -algebra, and (D N (i))i∈I will be a family of matrices in M N (B) for N ∈ N, which is a B-valued probability space with conditional expectation E N = tr N ⊗ idB . Consider the free product Au (N )∗∞ , generated by the entries in the matrices (U N (l))l∈N ∈ M N (Au (N )∗∞ ). By a family of freely independent Haar quantum unitary random matrices, independent from B, we will mean the family (U N (l) ⊗ 1B )l∈N in M N (Au (N )∗∞ ⊗ B) = M N (C) ⊗ Au (N )∗∞ ⊗ B, which we will still denote by (U N (l))l∈N . We also identify D N (i) = D N (i) ⊗ 1 Au (N )∗∞ for i ∈ I . We will consider the B-valued joint distribution of the family of sets ({U N (1), U N (1)∗ }, {U N (2), U N (2)∗ }, . . . , {D N (i)|i ∈ I }) with respect to the conditional expectation ψ N∗∞ ⊗ E N = tr N ⊗ ψ N∗∞ ⊗ idB . We can now state our main result. Theorem 5.1. Let B be a unital C∗ -algebra, and let (D N (i))i∈I be a family of matrices in M N (B) for N ∈ N. Suppose that there is a finite constant C such that D N (i) ≤ C for all i ∈ I and N ∈ N. Let (U N (l))l∈N be a family of freely independent N × N Haar quantum unitary random matrices, independent from B. Let (u(l), u(l)∗ )l∈N and (d N (i))i∈I,N ∈N be random variables in a B-valued probability space (A, E : A → B) such that (1) (u(l), u(l)∗ )l∈N is free from (d N (i))i∈I with respect to E for each N ∈ N. (2) ({u(l), u(l)∗ })l∈N is a free family with respect to E, and u(l) is a Haar unitary, independent from B for each l ∈ N. (3) (d N (i))i∈I has the same B-valued joint distribution with respect to E as (D N (i))i∈I has with respect to E N . Then for any polynomials p1 , . . . , p2m ∈ Bt (i)|i ∈ I , l1 , . . . , l2m ∈ N and 1 , . . . , 2m ∈ {1, ∗}, ∗∞ 1 2m (ψ p2m (D N )] N ⊗ E N )[U N (l1 ) p1 (D N ) · · · U N (l2m ) −E[u(l1 ) 1 p1 (d N ) · · · u(l2m ) 2m p2m (d N )] is O(N −2 ) as N → ∞. Observe that Theorem 5.1 makes no assumption on the existence of a limiting distribution for (D N (i))i∈I . If one assumes also the existence of a limiting (infinitesimal) B-valued joint distribution, then asymptotic (infinitesimal) freeness follows easily. We will state this as Theorem 5.3 below, let us first recall the relevant notions. Definition 5.2. Let B be a unital C∗ -algebra, and for each N ∈ N let (D N (i))i∈I be a family of noncommutative random variables in a B-valued probability space (A(N ), E N : A(N ) → B). (1) We say that the joint distribution of (D N (i))i∈I converges weakly in norm if there is a B-linear map E : BD(i)|i ∈ I → B such that lim E N [b0 D N (i 1 ) · · · D N (i k )bk ] − E[b0 D(i 1 ) · · · D(i k )bk ] = 0

N →∞

for any i 1 , . . . , i k ∈ I and b0 , . . . , bk ∈ B. If B is a von Neumann algebra with faithful, normal trace state τ , we say the joint distribution of (D N (i))i∈I converges weakly in L 2 if the equation above holds with respect to | |2 .

648


(2) If I = j∈J I j is a partition of I , we say that the sequence of sets of random variables ({D N (i)|i ∈ I j }) j∈J are asymptotically free with amalgamation over B if the sets ({D(i)|i ∈ I j }) j∈J are freely independent with respect to E. (3) We say that the joint distribution of (D N (i))i∈I converges infinitesimally in norm if there is a B-linear map E : BD(i)|i ∈ I → B such that E [b0 D(i 1 ) · · · D(i k )bk ] = lim N {E N [b0 D N (i 1 ) · · · D N (i k )bk ] N →∞

−E[b0 D(i 1 ) · · · D(i k )bk ]}

with convergence in norm, for any b0 , . . . , bk ∈ B and i 1 , . . . , i k ∈ I . If B is a von Neumann algebra with faithful, normal trace state τ , we say the joint distribution of (D N (i))i∈I converges infinitesimally in L 2 if the equation above holds with respect to | |2 . (4) If I = j∈J I j is a partition of I , we say that the sequence of sets of random variables ({D N (i)|i ∈ I j }) j∈J are asymptotically infinitesimally free with amalgamation over B if the sets ({D(i)|i ∈ I j }) j∈J are infinitesimally freely independent with respect to (E, E ). Theorem 5.3. Let B be a unital C∗ -algebra, and let (D N (i))i∈I be a family of matrices in M N (B) for N ∈ N. Suppose that there is a finite constant C such that D N (i) ≤ C for all i ∈ I and N ∈ N. For each N ∈ N, let (U N (l))l∈N be a family of freely independent N × N Haar quantum unitary random matrices, independent from B. (1) If the joint distribution of (D N (i))i∈I converges weakly (in norm or in L 2 with respect to a faithful trace), then the sets ({U N (1), U N (1)∗ }, {U N (2), U N (2)∗ }, . . . , {D N (i)|i ∈ I }) are asymptotically free with amalgamation over B as N → ∞. (2) If the joint distribution of (D N (i))i∈I converges infinitesimally (in norm or in L 2 with respect to a faithful trace), then the sets ({U N (1), U N (1)∗ }, {U N (2), U N (2)∗ }, . . . , {D N (i)|i ∈ I }) are asymptotically infinitesimally free with amalgamation over B as N → ∞. Theorem 5.3 follows immediately from Theorem 5.1 and Proposition 2.16. The proof of Theorem 5.1 will require some preparation, we begin by computing the limiting distribution appearing in the statement. Proposition 5.4. Let (u(l), u(l)∗ )l∈N and (d N (i))i∈I,N ∈N be random variables in a Bvalued probability space (A, E : A → B) such that (1) (u(l), u(l)∗ )l∈N is free from (d N (i))i∈I with respect to E for each N ∈ N. (2) ({u(l), u(l)∗ })l∈N is a free family with respect to E, and u(l) is a Haar unitary, independent from B for each l ∈ N. Let a(1), . . . , a(2m) be in the algebra generated by B and {d(i)|i ∈ I }, and let l1 , . . . , l2m ∈ N and 1 , . . . , 2m ∈ {1, ∗}. Then E[u(l1 ) 1 a(1) · · · u(l2m ) 2m a(2m)] μm (σ, π )E (σ K (π )) [a(1), . . . , a(2m)]. = π,σ ∈N C (m) σ ≤π π ∨ σ ≤ker l


649

Note that elements of the form appearing in the statement of the proposition span the algebra generated by (u(l), u(l)∗ )l∈N and (d(i))i∈I , and so this indeed determines the joint distribution. Proof. We have

E[u(l1 ) 1 a(1) · · · u(l2m ) 2m a(2m)] =

κ Eα [u(l1 ) 1 , a(1), . . . , a(2m)].

α∈N C(4m)

By freeness, the only non-vanishing cumulants appearing above are those of the form τ γ , where τ, γ ∈ N C(2m), τ ≤ ker l and γ ≤ K (τ ). So we have E[u(l1 ) 1 a(1) · · · u(l2m ) 2m a(2m)] (τ γ ) κ E [u(l1 ) 1 , a(1), . . . , a(2m)]. = τ ∈N C(2m) γ ∈N C(2m) τ ≤ker l γ ≤K (τ )

Since the expectation of any polynomial in (u(l), u(l)∗ )l∈N with complex coefficients is scalar-valued, it follows that E[u(l1 ) 1 a(1) · · · u(l2m ) 2m a(2m)] κ E(τ ) [u(l1 ) 1 , . . . , u(l2m ) 2m ] = τ ∈N C(2m) τ ≤ker l

=

(γ )

κ E [a(1), . . . , a(2m)]

γ ∈N C(2m) γ ≤K (τ ) (τ )

κ E [u(l1 ) 1 , . . . , u(l2m ) 2m ]E (K (τ )) [a(1), . . . , a(2m)].

τ ∈N C(2m) τ ≤ker l

Since Haar unitaries are R-diagonal ([17, Example 15.4]), we have (τ )

κ E [u(l1 ) 1 , . . . , u(l2m ) 2m ] = 0 unless τ ∈ N C h (2m). By Lemmas 3.6 and 3.9, we have E[u(l1 ) 1 a(1) · · · u(l2m ) 2m a(2m)] ( σ ∨ π) κE [u(l1 ) 1 , . . . , u(l2m ) 2m ]E (σ K (π )) [a(1), . . . , a(2m)]. = π,σ ∈N C (m) σ ≤π σ ∨ π ≤ker l

So it remains only to show that if σ, π ∈ N C (m) and σ ≤ π then ( σ ∨ π)

μm (σ, π ) = κ E

[u(l1 ) 1 , . . . , u(l2m ) 2m ].

Since the Möbius function is multiplicative on N C(m), we have μ|W | (σ |W , 1W ), μm (σ, π ) = W ∈π

and so it suffices to consider the case π = 1m . By [17, Prop. 15.1],

σ ∨1 m ) [u(l1 ) 1 , . . . , u(l2m ) 2m ] = κ E(

V ∈ σ ∨1 m

(−1)|V |/2−1 C|V |/2−1 ,

650


where Cn is the n th Catalan number. Since −−−−− → − −−−−−−−→ −−−→ ← − ← σ ∨ 1 (σ ) ∨ 0 (σ ), σ ∨ 1m = K

m = m = K we have ( σ ∨1 m)

κE

[u(l1 ) 1 , . . . , u(l2m ) 2m ] =

(−1)|W |−1 C|W |−1 .

W ∈K (σ )

On the other hand, we have μm (σ, 1m ) = μm (0m , K (σ )) = μ|W | (0W , 1W ) W ∈K (σ )

=

(−1)|W |−1 C|W |−1 ,

W ∈K (σ )

where we have used the formula for μm (0m , 1m ) from [17, Prop. 10.15]. Proposition 5.5. Let B be a unital algebra, A(1), . . . , A(2m) ∈ M N (B) and π, σ ∈ N C(m). Let E N = tr N ⊗ idB . If σ ≤ π , then

A(1) j1 j2 A(2)i1 i2 · · · A(2m)i2m−1 i2m

1≤ j1 ,..., j2m ≤N 1≤i 1 ,...,i 2m ≤N σ ≤ker j K

(π )≤ker i (σ K (π ))

= N |σ |+|K (π )| E N

[A(1), . . . , A(2m)].

Proof. First observe that the sum above can be rewritten as

A(1)i1 i2 · · · A(2m)i4m−1 i4m .

1≤i 1 ,...,i 4m ≤N

σ K (π )≤ker i

So this will follow from the formula (σ ) A(1)i1 i2 · · · A(m)i2m−1 i2m = N |σ | E N [A(1), . . . , A(m)] 1≤i 1 ,...,i 2m ≤N σ ≤ker i

for any σ ∈ N C(m). We will prove this by induction on the number of blocks of m. If σ = 1m has only one block, then we have

A(1)i1 i2 · · · A(m)i2m−1 i2m =

1≤i 1 ,...,i 2m ≤N σ ≤ker i

A(1)i1 i2 A(2)i2 i3 · · · A(m)im i1

1≤i 1 ,...,i m ≤N

= N · E N (A(1) · · · A(m)).


651

Suppose now that V = {l + 1, . . . , l + s} is an interval of σ . Then A(1)i1 i2 · · · A(m)i2m−1 i2m 1≤i 1 ,...,i 2m ≤N σ ≤ker i

=

⎛

A(1)i1 i2 · · · ⎝

1≤i 1 ,...,i 2l−2 , i 2(l+s)+1 ,...,i 2m ≤N σ

\V ≤ker i

⎞ A(l + 1) j1 j2 · · · A(l + s) js j1 ⎠

1≤ j1 ,..., js ≤N

· · · A(m)i2m−1 i2m = A(1)i1 i2 · · · (N · E N (A(l + 1) · · · A(l + s))) · · · A(m)i2m−1 i2m , 1≤i 1 ,...,i 2l−2 , i 2(l+s)+1 ,...,i 2m ≤N σ

\V ≤ker i

which by induction is equal to (σ \V )

N |σ | E N =N

|σ |

[A(1), . . . , A(l)E N (A(l + 1) · · · A(l + s)), . . . , A(m)]

(σ ) E N [A(1), . . . ,

A(m)].

Remark 5.6. We will also need to control the sum appearing in the proposition above for σ, π ∈ N C(m) with σ ≤ π . If B is commutative this poses no difficulty, as then A(1) j1 j2 A(2)i1 i2 · · · A(2m)i2m−1 i2m 1≤ j1 ,..., j2m ≤N 1≤i 1 ,...,i 2m ≤N σ ≤ker j K

(π )≤ker i

⎞

⎛

⎜ =⎜ ⎝

1≤ j1 ,..., j2m ≤N σ ≤ker j

⎟ A(1) j1 j2 · · · A(2m − 1) j2m−1 j2m ⎟ ⎠

⎛ ⎜ ·⎜ ⎝

1≤i 1 ,...,i 2m ≤N K

(π )≤ker i

⎞ ⎟ A(2)i1 i2 · · · A(2m)i2m−1 i2m ⎟ ⎠

(σ )

(K (π ))

= N |σ |+|K (π )| E N [A(1), . . . , A(2m − 1)]E N

[A(2), . . . , A(2m)].

However, when B is noncommutative it is not clear how to express this sum in terms of expectation functionals. Instead, we will use the following bound on the norm: Proposition 5.7. Let B be a unital C∗ -algebra, and A(1), . . . , A(2m) ∈ M N (B). If σ, π ∈ N C(m) then A(1) A(2) · · · A(2m) j1 j2 i1 i2 i 2m−1 i 2m 1≤ j1 ,..., j2m ≤N 1≤i1 ,...,i2m ≤N

σ ≤ker j K (π )≤ker i

≤N

|σ |+|K (π )|

A(1) · · · A(2m).

652


Proof. For this proof, we extend the definition of π to all partitions π ∈ P(m) in the obvious manner. We can rewrite the expression above as A(1)i1 i2 · · · A(2m)i4m−1 i4m , 1≤i 1 ,...,i 4m ≤N

σ K (π )≤ker i

and so the result will follow from |σ | A(1)i1 i2 · · · A(m)i2m−1 i2m ≤ N A(1) · · · A(m) 1≤i1 ,...,i2m ≤N σ ≤ker i for any partition σ ∈ P(m). The idea now is to realize this expression as the trace of a larger matrix. For each V ∈ σ , let M NV be a copy of M N (C). Consider the algebra M NV M N |σ | (C), V ∈σ

with the natural unital inclusions ιV of M NV for V ∈ σ . For 1 ≤ l ≤ m, let M NV ⊗ B M N |σ | (B), X (l) = ισ (l) ⊗ idB A(l) ∈ V ∈σ

where we have used the notation σ (l) for the block of σ which contains l. In other words, X (l) is the matrix indexed by maps i : σ → [N ] = {1, . . . , N } such that X (l)i j = A(l)i(σ (l)) j (σ (l)) δi(V ) j (V ) . V ∈σ l ∈V /

Consider now the trace (Tr N |σ | ⊗ idB )(X (1) · · · X (m)) = =

X (1)i1 i2 · · · X (m)im i1

i 1 ,...,i m il :σ →[N ]

A(1)i1 (σ (1))i2 (σ (1)) · · · A(m)im (σ (m))i1 (σ (m))

i 1 ,...,i m il :σ →[N ]

δil (V )iγ (l) (V ) ,

1≤l≤m V ∈σ l ∈V /

where γ ∈ Sm is the cyclic permutation (123 · · · m). The nonzero terms in this sum are obtained as follows: For each block V = (l1 < · · · < ls ) of σ , choose 1 ≤ il1 (V ), i γ (l1 ) (V ), . . . , ils (V ), i γ (ls ) (V ) ≤ N with the restrictions i γ (l1 ) (V ) = il2 (V ), . . . , i γ (ls−1 ) (V ) = ils (V ) and i γ (ls ) (V ) = il1 (V ). Comparing with the definition of σ , it follows that (Tr N |σ | ⊗ idB )(X (1) · · · X (m)) = A(1)i1 i2 · · · A(m)i2m−1 i2m 1≤i 1 ,...,i 2m ≤N σ ≤ker i


653

is the expression to be bounded. However, (tr N |σ | ⊗ idB ) = N −|σ | (Tr N |σ | ⊗ idB ) is a contractive conditional expectation onto B and so (Tr N |σ | ⊗ idB )(X (1) · · · X (m)) ≤ N |σ | X (1) · · · X (m). Since (ιV ⊗ idB ) is a contractive ∗-homomorphism, we have X (l) = (ισ (l) ⊗ idB )(A(l)) ≤ A(l) and the result follows. We are now prepared to prove the main theorem. Proof of Theorem 5.1. Fix p1 , . . . , p2m ∈ Bt (i)|i ∈ I , and set A N (k) = pk (D N ) for 1 ≤ k ≤ 2m. For notational simplicity, we will suppress the subscript N in our computations. Let l1 , . . . , l2m ∈ N, 1 , . . . , 2m ∈ {1, ∗} and consider (ψ N∗∞ ⊗ E N )[U (l1 ) 1 A(1)U (l2 ) 2 · · · U (l2m ) 2m A(2m)] = (ψ N∗∞ ⊗ idB )N −1 (U (l1 ) 1 )i1 i2 A(1)i2 i3 (U (l2 ) 2 )i3 i4 · · · A(2m)i4m i1 =

1≤i 1 ,...,i 4m ≤N

N −1 ψ N∗∞ (U (l1 ) 1 )i1 i2 · · · (U (l2m ) 2m )i4m−1 i4m

1≤i 1 ,...,i 4m ≤N

· A(1)i2 i3 · · · A(2m)i4mi1 . By Corollary 4.4, this is equal to 1≤i 1 ,...,i 4m ≤N

! " 2m N −1 ψ N∗∞ U (l1 )i 11i2 U (l2 )i 42i3 · · · U (l2m )i 4m i 4m−1

· A(1)i2 i3 · · · A(2m)i4mi1 . After reindexing, this becomes

! " 2m 2 1 N −1 ψ N∗∞ U (l1 )i 2m j1 U (l2 )i 1 j2 · · · U (l2m )i 2m−1 j2m

1≤i 1 ,...,i 2m ≤N 1≤ j1 ,..., j2m ≤N

· A(1) j1 j2 A(2)i1 i2 · · · A(2m)i2m−1 i2m . Applying Corollary 4.7, we have

N −|K (π )|−|σ | (μm (σ, π ) + O(N −2 ))

1≤i 1 ,...,i 2m ≤N 1≤ j1 ,..., j2m ≤N π,σ ∈N C (m) −−→ π ≤ker i∧ker l σ ≤ker j∧ker l

· A(1) j1 j2 A(2)i1 i2 · · · A(2m)i2m−1 i2m = (μm (σ, π ) + O(N −2 ))N −|K (π )|−|σ | π,σ ∈N C (m) π ≤ker l σ ≤ker l

×

1≤ j1 ,..., j2m ≤N 1≤i 1 ,...,i 2m ≤N σ ≤ker j K

(π )≤ker i

A(1) j1 j2 A(2)i1 i2 · · · A(2m)i2m−1 i2m .

654


By Propositions 5.5 and 5.7, this is equal to (σ K (π )) μm (σ, π )E N [A(1), . . . , A(2m)], π,σ ∈N C (m) σ ≤π π ∨ σ ≤ker l

up to O(N −2 ) with respect to the norm on B. Set a(k) = pk (d N ) for 1 ≤ k ≤ 2m, then by Proposition 5.4 we have E[u(l1 ) 1 a(1) · · · u(l2m ) 2m a(2m)] μm (σ, π )E (σ K (π )) [a(1), . . . , a(2m)] = π,σ ∈N C (m) σ ≤π π ∨ σ ≤ker l

=

π,σ ∈N C (m)

(σ K (π ))

μm (σ, π )E N

[A(1), . . . , A(2m)],

σ ≤π π ∨ σ ≤ker l

and the result now follows immediately. 5.8. Randomly quantum rotated matrices. It follows easily from Theorem 5.3 and the definition of asymptotic freeness that under the hypotheses of the theorem, the sets ({D N (i) : i ∈ I }, ({U N (l)D N (i)U N (l)∗ : i ∈ I })l∈N ) are asymptotically (infinitesimally) free with amalgamation over B as N → ∞. The condition on existence of a limiting joint distribution can be weakened slightly as follows: Corollary 5.9. Let B be a unital C∗ -algebra, and let (D N (i))i∈I and (D N ( j)) j∈J be two families of matrices in M N (B) for N ∈ N. Suppose that there is a finite constant C such that D N (i) ≤ C and D N ( j) ≤ C for N ∈ N, i ∈ I and j ∈ J . For each N ∈ N, let U N be a N × N Haar quantum unitary random matrix, independent from B. (1) If the joint distributions of (D N (i))i∈I and (D N ( j)) j∈J both converge weakly (in norm or in L 2 with respect to a faithful trace), then (U N D N (i)U N∗ )i∈I and (D N ( j)) j∈J are asymptotically free with amalgamation over B as N → ∞. (2) If the joint distribution of (D N (i))i∈I and (D N ( j)) j∈J both converge infinitesimally (in norm or in L 2 with respect to a faithful trace), then the families (U N D N (i)U N∗ )i∈I and (D N ( j)) j∈J are asymptotically infinitesimally free with amalgamation over B. Proof. The only condition of Theorem 5.3 which is not satisfied is that {D N (i) : i ∈ I } ∪ {D N ( j) : j ∈ J } should have a limiting (infinitesimal) joint distribution as N → ∞. We can see that this is not an issue as follows. Let p1 , . . . , pm ∈ Bt (i)|i ∈ I and q1 , . . . , qm ∈ Bt ( j)| j ∈ J and set A N (k) = pk (D N ), B N (k) = qk (D N ) for 1 ≤ k ≤ m. From the proof of Theorem 5.1, we have (ψ N ⊗ E N )[U A(1)U ∗ B(1) · · · U A(m)U ∗ B(m)] (σ K (π )) μm (σ, π )E N [A(1), B(1), . . . , A(m), B(m)], = π,σ ∈N C(m) σ ≤π

up to O(N −2 ). But the right-hand side depends only on the distributions of (D(i))i∈I and (D ( j)) j∈J , and so the result follows from Theorem 5.3.


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5.10. Classical Haar unitary random matrices. In the remainder of this section, we will discuss the failure of these results for classical Haar unitaries. First we show that if B is finite dimensional, then classical Haar unitaries are sufficient. Proposition 5.11. Let B be a finite dimensional C∗ -algebra, and let (D N (i))i∈I be a family of matrices in M N (B) for each N ∈ N. Assume that there is a finite constant C such that D N (i) ≤ C for all N ∈ N and i ∈ I . For each N ∈ N, let (U N (l))l∈N be a family of independent N × N Haar unitary random matrices, independent from B. Let (u(l), u(l)∗ )l∈N and (d N (i))i∈I,N ∈N be random variables in a B-valued probability space (A, E : A → B) such that (1) (u(l), u(l)∗ )l∈N is free from (d N (i))i∈I with respect to E for each N ∈ N. (2) ({u(l), u(l)∗ })l∈N is a free family with respect to E, and u(l) is a Haar unitary, independent from B for each l ∈ N. (3) (d N (i))i∈I has the same B-valued joint distribution with respect to E as (D N (i))i∈I has with respect to E N . Then for any polynomials p1 , . . . , p2m ∈ Bt (i) : i ∈ I , l1 , . . . , l2m ∈ N and 1 , . . . , 2m ∈ {1, ∗}, ∗∞ 1 2m (ψ p2m (D N )] N ⊗ E N )[U N (l1 ) p1 (D N ) · · · U N (l2m ) 1 2m −E[u(l1 ) p1 (d N ) · · · u(l2m ) p2m (d N )] is O(N −2 ) as N → ∞. Proof. Let e1 , . . . , eq be a basis for B with er = 1 for 1 ≤ r ≤ q. Let p1 , . . . , p2m ∈ Bt (i)|i ∈ I , let A N (k) = pk (D N ) and let A N (k, r ) ∈ M N (C) be the matrix of coefficients of the entries of A N (k) on er for 1 ≤ k ≤ 2m and 1 ≤ r ≤ q. Let a N (k, r ) and (u(l), u(l)∗ )l∈N be random variables in a noncommutative probability space (A, ϕ) such that (1) {a N (k, r ) : 1 ≤ k ≤ 2m, 1 ≤ r ≤ q} and (u(l), u(l)∗ )l∈N are free with respect to ϕ. (2) (a N (k, r ))1≤k≤2m,1≤r ≤q has the same joint distribution with respect to ϕ as (A N (k, r ))1≤k≤2m,1≤r ≤q with respect to tr N . (3) (u(l), u(l)∗ )l∈N are freely independent with respect to ϕ and u(l) has a Haar unitary distribution. # For 1 ≤ k ≤ 2m and N ∈ N, let a N (k) = a N (k, r ) ⊗ er ∈ A ⊗ B, and note that the family (an (k))1≤k≤2m has the same joint distribution with respect to E = ϕ ⊗ idB as does (A N (k))1≤k≤2m with respect to E N . Identifying u(l) = u(l) ⊗ 1 in A ⊗ B, it is also easy to see that (u(l), u(l)∗ ) and (a N (k))1≤k≤2m are freely independent with respect to E. Now let 1 , . . . , 2m ∈ {1, ∗} and consider (tr N ⊗ E ⊗ idB )[U (l1 ) 1 A(1) · · · A(2m)U (l2m ) 2m ] = (tr N ⊗ E)[U (l1 ) 1 A(1, r1 ) · · · A(2m, r2m )U (l2m ) 2m ]er1 · · · er2m . 1≤r1 ,...,r2m ≤q

656


Since er = 1, it follows that (tr N ⊗ E ⊗ idB )[U (l1 ) 1 A(1) · · · U (l2m ) 2m A(2m)] −E[u(l1 ) 1 a(1) · · · u(l2m ) 2m a(2m)] (tr N ⊗ E)[U (l1 ) 1 A(1, r1 ) · · · U (l2m ) 2m A(2m, r2m )] ≤ 1≤r1 ,...,r2m ≤q −ϕ[u(l1 ) 1 a(1, r1 ) · · · u(l2m ) 2m a(2m, r2m )] .

From standard asymptotic freeness results (see e.g. [10]), this expression is O(N −2 ) as N → ∞. We will now give an example to show that Theorem 5.1 may fail for classical Haar unitaries if the algebra B is infinite dimensional. First we recall the Weingarten formula for computing the expectation of a word in the entries of a N × N Haar unitary random matrix and its conjugate: 2m E[Ui 11j1 · · · Ui 2m W cN (π, σ ), j2m ] = π,σ ∈P2 (2m) π ≤ker i σ ≤ker j

where P2 (2m) is the set of pair partitions for which each pairing connects a 1 with a ∗ in the string 1 , . . . , 2m , and W cN is the corresponding Weingarten matrix, see [5,10]. Example 5.12. Let B be a unital C∗ -algebra, and for each N ∈ N let {E i j (N , l) : 1 ≤ i, j ≤ N , l = 1, 2} be two commuting systems of matrix units in B, i.e., (1) (2) (3) (4)

E i1 j1 (N , 1)E i2 j2 (N , 2) = E i2 j2 (N , 2)E i1 j1 (N , 1) for 1 ≤ i 1 , j1 , i 2 , j2 ≤ N . E i j (N , l)∗ = E ji (N , l) for 1 ≤ i, j ≤ N . E ik1 (N , l)E k2 j (N , l) = δk1 k2 E i j (N , l) for 1 ≤ i, j, k1 , k2 ≤ N . E ii (N , l) is a projection for 1 ≤ i ≤ N , and N

E ii (N , l) = 1.

i=1

For N ∈ N, define A N , B N ∈ M N (B) by (A N )i j = E ji (N , 1), (B N )i j = E ji (N , 2). Note that A N , B N are self-adjoint and A2N , B N2 are the identity matrix, indeed (A2N )i j =

N

E ki (N , 1)E jk (N , 1) = δi j

k=1

N

E kk (N , 1) = δi j · 1,

k=1

and likewise for B N . It follows that A N = B N = 1 for N ∈ N. For each N ∈ N, let U N be a N × N Haar unitary random matrix, independent from B. Since (tr N ⊗ idB )[A N ] =

N 1 1 ·1 E ii (N , 1) = N N i=1


657

converges to zero as N → ∞, and likewise for B N , for asymptotic freeness we should have lim (tr N ⊗ E ⊗ id)[(U N A N U N∗ B N )3 ] = 0.

N →∞

However, we will show that this limit is in fact equal to 1. Indeed, suppressing the subindex N we have (tr ⊗ E ⊗ idB )[(U AU ∗ B)3 ] 1 = E[Ui1 i2 U i4 i3 · · · U i12 i11 ]Ai2 i3 Bi4 i5 · · · Bi12 i1 N 1≤i 1 ,...,i 12 ≤N = E[Ui6 j1 U i1 j2 · · · U i5 j6 ]A j1 j2 A j3 j4 A j5 j6 Bi1 i2 Bi3 i4 Bi5 i6 . 1≤i 1 , j1 ,...,i 6 , j6 ≤N

Applying the Weingarten formula, we obtain ⎛ π,σ ∈P2 (6)

⎜ N −1 W cN (π, σ ) ⎜ ⎝

⎛ ⎜ ·⎜ ⎝

⎞

1≤ j1 ,..., j6 ≤N σ ≤ker j

⎟ A j1 j2 A j3 j4 A j5 j6 ⎟ ⎠

⎞

1≤i 1 ,...,i 6 ≤N ← π−≤ker i

⎟ Bi1 i2 Bi3 i4 Bi5 i6 ⎟ ⎠.

Note that P2 (6) has 6 elements, namely the 5 noncrossing pair partitions and τ = {(1, 4), (2, 5), (3, 6)}. The noncrossing pair partitions can be expressed as σ for some σ ∈ N C(3), in which case we have (σ ) A j1 j2 A j3 j4 A j5 j6 = N |σ | E N [A, A, A]. 1≤ j1 ,..., j6 ≤N σ ≤ker j

Using E N [A] = E N [A3 ] = N −1 and E N [A2 ] = 1, one easily sees that this expression is O(N ) for the 5 noncrossing pair partitions. For τ , we have A j1 j2 A j3 j4 A j5 j6 = A j1 j2 A j3 j1 A j2 j3 1≤ j1 ,..., j6 ≤N τ ≤ker j

1≤ j1 , j2 , j3 ≤N

=

E j2 j1 (N , 1)E j1 j3 (N , 1)E j3 j2 (N , 1)

1≤ j1 , j2 , j3 ≤N

=

E j2 j2 (N , 1)

1≤ j1 , j2 , j3 ≤N

= N 2 · 1, and likewise for B N . Also we have N 3 W cN (π, σ ) = δπ σ + O(N −1 ). Putting these statements together, we find that the only term which remains in the limit comes from π = σ = τ , which gives 1.

658


Remarks. (1) We note that M N 2 (C) = M N (C) ⊗ M N (C) has a natural pair of commuting systems of matrix units, so this example demonstrates that Theorem 5.1 fails for any unital C∗ -algebra B which contains M N 2 (C) as a unital subalgebra for k some increasing sequence of natural numbers (Nk ). (2) It is a natural question whether the matrices A N , B N in the above example have limiting B-valued distributions, which would demonstrate that Theorem 1 also fails for classical Haar unitaries. First observe that 1, k is even lim (tr N ⊗ B)[AkN ] = , 0, k is odd N →∞ which follows from the case k = 1 and the fact that A2N is the identity matrix. However, it is not clear that moments of the form b0 A N · · · A N bk will converge for arbitrary b0 , . . . , bk ∈ B. Let us point out a special case in which the limiting distribution does exist. Suppose that there is a dense ∗-subalgebra F ⊂ B such that each element of F commutes with the matrix units E i j (N , l) for N sufficiently large. Then for any b0 , . . . , bk ∈ B we have b0 b1 · · · bk , k is even lim (tr N ⊗ B)[b0 A N · · · A N bk ] = , 0, k is odd N →∞ and likewise for B N , indeed this holds for b0 , . . . , bk ∈ F by hypothesis and for general b0 , . . . , bk by density. In particular, we may take B to be the C∗ -algebraic infinite tensor product M N (C) B= N ∈N

with the obvious systems of matrix units E(N , l)i j ∈ M N 2 = M N (C) ⊗ M N (C) ⊂ B, and F ⊂ B to be the image of the purely algebraic tensor product. Note that B is uniformly hyperfinite, in particular approximately finitely dimensional in the C∗ -sense. (3) Note that if B is a von Neumann algebra with a non-zero continuous projection p, then pB p contains M N (C) as a unital subalgebra for all N ∈ N and hence (1) applies to pB p. It follows that Theorem 5.1 fails also for B. To obtain a contradiction to Theorem 1 for classical Haar unitaries in the setting of a von Neumann algebra with faithful, normal trace, we may modify the example in (2) by taking (B, τ ) to be the infinite tensor product (B, τ ) = (M N (C), tr N ) N ∈N

taken with respect to the trace states tr N on M N (C), which is the hyperfinite I I1 factor. Acknowledgements. We would like to thank T. Banica, M. Neufang, and D. Shlyakhtenko for several useful discussions. S.C. would like to thank his thesis advisor, D.-V. Voiculescu, for his continued guidance and support while completing this project. 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.


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Communications in


Nice Inducing Schemes and the Thermodynamics of Rational Maps Feliks Przytycki1, , Juan Rivera-Letelier2, 1 Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich ´ 8, 00956 Warszawa, Poland.


2 Facultad de Matemáticas, Campus San Joaquín, Pontificia Universidad Católica de Chile,

Avenida Vicuña Mackenna, 4860 Santiago, Chile. E-mail: [email protected] Received: 5 January 2010 / Accepted: 27 July 2010 Published online: 26 November 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract: We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential −t ln f , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in R, with at most two exceptions. Contents 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . Nice Sets, Pleasant Couples and Induced Maps From the Induced Map to the Original Map . . Whitney Decomposition of a Pull-back . . . . The Contribution of a Pull-back . . . . . . . . Proof of the Main Theorem . . . . . . . . . .

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662 668 672 679 684 687 691

Partially supported by Polish MNiSW Grant NN201 0222 33 and the EU FP6 Marie Curie ToK and RTN programmes SPADE2 and CODY. Partially supported by Research Network on Low Dimensional Dynamical Systems, PBCT/CONICYT, Chile, Swiss National Science Foundation Projects No. 200021-107588 and 200020-109175, and IMPAN.

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F. Przytycki, J. Rivera-Letelier

Appendix A. Puzzles and Nice Couples . . . . . . . . . . . . . . . . . . . . . . 695 Appendix B. Rigidity, Multifractal Analysis, and Level-1 Large Deviations . . . 701 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 1. Introduction The purpose of this paper is to study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere C. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential −t ln f and the (real) analytic dependence on t of the corresponding pressure function. Our particular choice of potentials is motivated by the close connection between the corresponding pressure function and various multifractal spectra. In fact, we give applications of our results to rigidity, multifractal analysis of dimension spectrum for Lyapunov exponents and for pointwise dimensions, as well as level-1 large deviations. See [BS96,BS05,Erë91] for other applications of the thermodynamic formalism of rational maps to complex analysis. For t < 0 and for an arbitrary rational map f , a complete description of the thermodynamic formalism was given by Makarov and Smirnov in [MS00]. They showed that the corresponding transfer operator is quasi-compact on a suitable Sobolev space, see also [Rue92]. For t = 0 and a general rational map f , there is a unique equilibrium state of f for the constant potential equal to 0 [Lju83,FLM83]. To the best of our knowledge it is not known if for a general rational map f the pressure function is real analytic on a neighborhood of t = 0. For t > 0 the only results on the analyticity of the pressure function that we are aware of, are for generalized polynomial-like maps without recurrent critical points in the Julia set. For such a map the analyticity properties of the pressure function were studied in [MS03,SU03], using a Markov tower extension and an inducing scheme, respectively. Under very weak hypotheses on a rational map f , we show that the pressure function is real analytic at each parameter t in R, with at most two exceptions. In other words, the pressure function can have at most two phase transitions and thus at most three phases. It turns out that the parameter t = 0 is always contained in one of the phases, which is characterized as the only phase where the measure theoretic entropy of an equilibrium state can be strictly positive. We show that for every parameter in this phase there is a unique equilibrium state that has exponential decay of correlations and that satisfies the Central Limit Theorem. Our results apply to well-known classes of non-uniformly hyperbolic rational maps. Furthermore our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and to all quadratic polynomials with real coefficients. The main ingredients in our approach are the distinct characterizations of the pressure function given in [PRLS04] and the inducing scheme introduced in [PRL07], which we develop here in a more general setting. It is worth noticing that to study a rational map with a recurrent critical point in the Julia set, it is usually not enough to consider an induced map defined with the first return time. The induced maps considered here are constructed with higher return times, which makes the estimates more delicate. As in [PRL07], our key estimates are based on controlling a discrete version of conformal mass. However, the “density” introduced in [PRL07] for this purpose does not work in the more general setting considered here. We thus introduce a different technique, based on a Whitney type decomposition.

Thermodynamics of Rational Maps

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There have been several recent results on the thermodynamic formalism of multimodal interval maps with non-flat critical points, by Bruin and Todd [BT08,BT09] and Pesin and Senti [PS08]. Besides [BT08, Theorem 6], that gives a complete description of the thermodynamic formalism for t close to 0 and for a general topologically transitive multimodal interval map with non-flat critical points, all the results that we are aware of are restricted to non-uniformly hyperbolic maps. It is possible to apply the approach given here to obtain a fairly complete description of the thermodynamic formalism of a general topologically transitive multimodal interval map with non-flat critical points. We obtain in particular that the pressure function of such a map is real analytic at each parameter in R, with at most two exceptions.1 We are in the process of writing these results. After reviewing some general properties of the pressure function in §1.1, we state our main result in §1.2. The applications to rigidity, multifractal analysis, and level-1 large deviations are given in Appendix B. Throughout the rest of this Introduction we fix a rational map f of degree at least two, we denote by Crit( f ) the set of critical points of f and by J ( f ) the Julia set of f . 1.1. The pressure function and equilibrium states. We give here the definition of the pressure function and of equilibrium states, see §2 for references and precise formulations. Let M ( f ) be the space of all probability measures supported on J ( f ) that are invariant by f . We endow M ( f ) with the weak∗ topology. For each ∈ M ( f ), denote μ by h μ ( f ) the measure theoretic entropy of μ, and by χμ ( f ) := ln f dμ the Lyapunov exponent of μ. Given a real number t we define the pressure of f | J ( f ) for the potential −t ln f by P(t) := sup h μ ( f ) − tχμ ( f ) | μ ∈ M ( f ) . (1.1) For each t ∈ R we have P(t) < +∞,2 and the function P : R → R so defined will be called the pressure function of f . It is convex, non-increasing and Lipschitz continuous. An invariant probability measure μ supported on the Julia set of f is called an equi librium state of f for the potential −t ln f , if the supremum (1.1) is attained for this measure. The numbers, χinf ( f ) := inf χμ ( f ) | μ ∈ M ( f ) , χsup ( f ) := sup χμ ( f ) | μ ∈ M ( f ) , will be important in what follows. We call t− := inf{t ∈ R | P(t) + tχsup ( f ) > 0} t+ := sup{t ∈ R | P(t) + tχinf ( f ) > 0}

(1.2) (1.3)

1 Recently Iommi and Todd [IT09] have shown similar results for transitive multimodal maps with non-flat critical points as those presented here, but only obtaining that the pressure function is continuous differentiable, and without statistical properties of the equilibrium states. 2 When t ≤ 0, the number P(t) coincides with the topological pressure of f | J ( f ) for the potential −t ln f , defined with (n, ε)-separated sets. However, these numbers do not coincide when t > 0 and there are critical points of f in J ( f ). In fact, since ln f takes the value −∞ at each critical point of f , in this case the topological pressure of f | J ( f ) for the potential −t ln f is equal to +∞.

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the condensation point and the freezing point of f , respectively. We remark that the condensation (resp. freezing) point can take the value −∞ (resp. +∞). We have the following properties (Proposition 2.1): • t − < 0 < t+ ; • for all t ∈ R\(t− , t+ ) we have P(t) = max{−tχsup ( f ), −tχinf ( f )}; • for all t ∈ (t− , t+ ) we have P(t) > max{−tχinf ( f ), −tχsup ( f )}. 1.2. Nice sets and the thermodynamics of rational maps. A neighborhood V of Crit( f )∩ J ( f ) is a nice set for f , if for every n ≥ 1 we have f n (∂ V ) ∩ V = ∅, and if each connected component of V is simply connected and contains precisely one critical point , V ) for f such that V ⊂ V of f in J ( f ). A nice couple for f is a pair of nice sets (V = ∅. We will say that a nice couple and such that for every n ≥ 1 we have f n (∂ V ) ∩ V , V ) is small, if there is a small r > 0 such that V ⊂ B(Crit( f ) ∩ J ( f ), r ). (V We say that a rational map f is expanding away from critical points, if for every neighborhood V of Crit( f ) ∩ J ( f ) the map f is uniformly expanding on the set z ∈ J ( f ) | for every n ≥ 0, f n (z) ∈ V . Main Theorem. Let f be a rational map of degree at least two that is expanding away from critical points, and that has arbitrarily small nice couples. Then the following properties hold. Analyticity of the pressure function: The pressure function of f is real analytic on (t− , t+ ), and linear with slope −χsup ( f ) (resp. −χinf ( f )) on (−∞, t− ] (resp. [t+ , +∞)). Equilibrium states: For t0 ∈ (t− , t+ ) there is a unique equilibrium state of f for each the potential −t0 ln f . Furthermore this measure is ergodic and mixing. We now list some classes of rational maps for which the Main Theorem applies. • Using [KvS09] we show that each at most finitely renormalizable polynomial without indifferent periodic orbits satisfies the hypotheses of the Main Theorem, see Theorem C in §A.1. • Quadratic polynomials with real coefficients satisfy the hypothesis of the Main Theorem, with two exceptions: Maps with an indifferent periodic point, which are considerably simpler to treat, and maps having a renormalization conjugated to the Feigenbaum polynomial, for which we show that a slightly more general version of the Main Theorem applies (Theorem B in §7). In particular our results imply that the conclusions of the Main Theorem hold for each quadratic polynomial with real coefficients, see §A.3 for details. • Topological Collet-Eckmann rational maps have arbitrarily small nice couples [PRL07, Theorem E] and are expanding away from critical points. These maps include Collet-Eckmann rational maps, as well as maps without recurrent critical points and without parabolic periodic points; see [PR98] and also [PRLS03, Main Theorem]. • Each backward contracting rational map has arbitrarily small nice couples [RL07, Prop. 6.6]. If in addition the Julia set is different from C, such a map is also expanding away from critical points [RL07, Coro. 8.3]. In [RL07, Theorem A] it is shown that a rational map f of degree at least two satisfying the summability condition with


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exponent 1: f does not have indifferent periodic points and for each critical value v in the Julia set of f we have +∞ n −1 f (v) < +∞ n=1

is backward contracting, and it thus has arbitrarily small nice couples. In [Prz98] it is shown that each rational map satisfying the summability condition with exponent 1 is expanding away from critical points. Using a stronger version of the Main Theorem (Theorem B in §7), we show that each infinitely renormalizable quadratic polynomial for which the diameters of the small Julia sets converge to 0 satisfies the conclusions of the Main Theorem, see §A.2 in Appendix A. In particular the conclusions of the Main Theorem hold for each infinitely renormalizable polynomial with a priori bounds; see [KL08,McM94] and references therein for results on a priori bounds. Remark 1.1. In the proof of the Main Theorem we construct the equilibrium states through an inducing scheme with an exponential tail estimate, that satisfies some additional technical properties; see §4.3 for precise statements. The results of [You99] imply that the equilibrium states in the Main Theorem are exponentially mixing and that the Central Limit Theorem holds for these measures. It also follows that these equilibrium states have other statistical properties, such as the “almost sure invariant principle”, see e.g. [Gou05,MN05,MN08,TK05]. We obtain as a direct consequence of the Main Theorem the following result on the integral means spectrum. Corollary 1.2. Let f be a monic polynomial with connected Julia set and degree d ≥ 2, that is expanding away from critical points and that has arbitrarily small nice couples. Let φ : {z ∈ C | |z| > 1} → C\J ( f ) be a conformal representation that is tangent to the identity at infinity. Then the integral means spectrum of φ, t 2π ln 0 φ (r exp(iθ )) dθ βφ (t) := lim sup , | ln(r − 1)| r →1+ is real analytic on (t− , t+ ) and linear with slope 1−χsup ( f )/ ln d (resp. 1−χinf ( f )/ ln d) on (−∞, t− ] (resp. [t+ , +∞)). This corollary follows directly from the fact that for each t ∈ R we have βφ (t) = P(t)/ ln d + t − 1, see for example [BMS03, Lemma 2]. We will now consider several known results related to the Main Theorem. As mentioned above, Makarov and Smirnov showed in [MS00] that the conclusions of the Main Theorem hold for every rational map on (−∞, 0). Furthermore, they characterized all those rational maps whose condensation point t− is finite; see §B.1. For a uniformly hyperbolic rational map we have t− = −∞ and t+ = +∞, and for a sub-hyperbolic polynomial with connected Julia set we have t+ = +∞ [MS96]. The freezing point t+ is finite whenever f does not satisfy the Topological Collet-Eckmann

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Condition3 (Proposition 2.1). In fact, in this case the freezing point t+ is the first zero of the pressure function. On the other hand, there is an example in [MS03, §3.4] of a generalized polynomial-like map satisfying the Topological Collet-Eckmann Condition4 and whose freezing point t+ is finite. When f is a generalized polynomial-like map without recurrent critical points, the part of the Main Theorem concerning the analyticity of the pressure function was shown in [MS00,MS03,SU03]. Note that the results of [SU03] apply to maps with parabolic periodic points. Let us also mention that, if f is an at most finitely renormalizable polynomial without indifferent periodic points and such that for every critical value v in J ( f ), lim | f n (v)| = +∞, n→+∞

and if t0 > 0 is the first zero of the pressure function, then the absolutely continuous invariant constructed in [RLS10] is an equilibrium state of f for the potential measure −t0 ln f , see also [GS09,PRL07]. In the case of a general transitive multimodal interval map with non-flat critical points, a result analogous to the Main Theorem was shown by Bruin and Todd in [BT08, Theorem 6] for t in a neighborhood of 0. Similar results for t in a neighborhood of [0, 1] were shown by Pesin and Senti in [PS08] for multimodal interval maps with non-flat critical points satisfying the Collet-Eckmann condition and some additional properties (see also [BT09, Theorem 2]) and by Bruin and Todd in [BT09, Theorem 1], for t in a one-sided neighborhood of 1, and for multimodal interval maps with non-flat critical points and with a polynomial growth of the derivatives along the critical orbits; see also [BK98]. In [Dob09, Prop. 7], Dobbs shows that there is a quadratic polynomial with real coefficients f 0 such that the pressure function, defined for the restriction of f 0 to a certain compact interval, has infinitely many phase transitions before it vanishes. This behavior of f 0 as an interval map is in sharp contrast with its behavior as a complex map: Our results imply that the pressure function of f 0 , viewed as a map acting on the (complex) Julia set of f 0 , is real analytic before it vanishes. 1.3. Notes and references. See the book [Rue04] for an introduction to the thermodynamic formalism and [PU02,Zin96] for an introduction in the case of rational maps. For results concerning other potentials, see [DU91,GW07,Prz90,Urb03] for the case of rational maps, [Bal00] for piecewise monotone maps, and [BT08,PS08] and references therein for the case of multimodal interval maps with non-flat critical points. For a rational map f satisfying the Topological Collet-Eckmann Condition and for t = HDhyp ( f ), the construction of the corresponding equilibrium state given here gives a new proof of the existence of an absolutely continuous invariant measure, with respect to a conformal measure. More precisely, it gives a new proof of [PRL07, Key Lemma].

1.4. Strategy and organization. We now describe the strategy of the proof of the Main Theorem, and simultaneously describe the organization of the paper. Our results are 3 By [PRLS03, Main Theorem] f satisfies the Topological Collet-Eckmann Condition if, and only if, χinf ( f ) > 0. 4 In fact this map has the stronger property that no critical point in its Julia set is recurrent.


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either well-known or vacuous for rational maps without critical points in the Julia set, so we will (implicitly) assume that all the rational maps we consider have at least one critical point in the Julia set. In §2 we review some general results concerning the pressure function, including some of the different characterizations of the pressure function given in [PRLS04]. We also review some results concerning the asymptotic behavior of the derivative of the iterates of a rational map. These results are mainly taken or deduced from results in [Prz99,PRLS03,PRLS04]. To prove the Main Theorem we make use of the inducing scheme introduced in [PRL07], which is developed in the more general setting considered here in §§3, 4. In §3.1 we recall the definitions of nice sets and couples, and introduce a weaker notion of nice couples that we call “pleasant couples”. Pleasant couples will allow us to handle non-primitive renormalizations, see Remark A.6. Then we recall in §3.2 the definition of the canonical induced map associated to a nice (or pleasant) couple. We also review the decomposition of its domain of definition into “first return” and “bad pull-backs” as well as the sub-exponential estimate on the number of bad pull-backs of a given order (§3.3). In §3.4 we consider a two variable pressure function associated to such an induced map, that will be very important for the rest of the paper. This pressure function is analogous to the one introduced by Stratmann and Urbanski in [SU03]. In §4 we give sufficient conditions on a nice (or pleasant) couple so that the conclusions of the Main Theorem hold for values of t in a neighborhood of an arbitrary t0 ∈ (t− , t+ ) (Theorem A). These conditions are formulated in terms of the two variable pressure function defined in §3.4. We follow the method of [PRL07] for the construction of the conformal measures and the equilibrium states, which is based on the results of Mauldin and Urbanski in [MU03]. As in [PS08], we use a result of Zweimüller in [Zwe05] to show that the invariant measure we construct is in fact an equilibrium state. The uniqueness is a direct consequence of the results of Dobbs in [Dob08], generalizing [Led84]. Finally, we use the method introduced by Stratmann and Urbanski in [SU03] to show that the pressure function is real analytic. Here we make use of the fact that the two variable pressure function is real analytic on the interior of the set where it is finite, a result shown by Mauldin and Urbanski in [MU03]. The proof of the Main Theorem is contained in §§5, 6, 7. The proof is divided into two parts. The first, and by far the most difficult one, is to show that for t0 ∈ (t− , t+ ) the two variable pressure associated to a sufficiently small nice (or pleasant) couple is finite on a neighborhood of (t, p) = (t0 , P(t0 )). To do this we use the strategy of [PRL07]: we use the decomposition of the domain of definition of the induced map associated to a nice (or pleasant) couple, into first return and bad pull-backs evoked in §3.3. Unfortunately, for values of t such that P(t) < 0, there does not seem to be a natural way to adapt the density introduced in [PRL07] to estimate the contribution of a bad pullback. Instead we use a different argument involving a Whitney type decomposition of a pull-back, which is one of the main technical tools introduced in this paper. Roughly speaking, we have replaced the “annuli argument” of [PRL07, Lemma 5.4] by an argument involving “Whitney squares”, that allow us to make a direct estimate avoiding an induction on the number visits to the critical point. The Whitney type decomposition is introduced in §5 and the estimate on the contribution of a (bad) pull-back is given in §6. The finiteness of the two variable pressure function is shown in §7.1. The second part of the proof, that for each t close to t0 the two variable pressure function vanishes at (t, p) = (t, P(t)), is given in §7.2. Here we have replaced the analogous (co-)dimension argument of [PRL07], with an argument involving the pressure function of the rational map.

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Appendix A is devoted to show that the conclusions of the Main Theorem hold for several classes of polynomials. In §A.1 we show that each at most finitely renormalizable polynomial without indifferent periodic points satisfies the hypotheses of the Main Theorem (Theorem C). Then in §A.2 we show that each infinitely renormalizable quadratic polynomial for which the diameters of small Julia sets converge to 0 satisfies the hypotheses of Theorem B. Finally, §A.3 is devoted to the case of quadratic polynomials with real coefficients. In Appendix B we give applications of our main results to rigidity, multifractal analysis, and level-1 large deviations. 2. Preliminaries The purpose of this section is to give some general properties of the pressure function (§§2.2, 2.3), and some characterizations of χinf and χsup (§2.4). These results are mainly taken or deduced from the results in [Prz99,PRLS03,PRLS04]. We also fix some notation and terminology in §2.1, that will be used in the rest of the paper. Throughout the rest of this section we fix a rational map f of degree at least two. We will denote h μ ( f ), χμ ( f ), . . . just by h μ , χμ , . . .. For simplicity we will assume that no critical point of f in the Julia set is mapped to another critical point under forward iteration. The general case can be handled by treating whole blocks of critical points as a single critical point; that is, if the critical points c0 , . . . , ck ∈ J ( f ) are such that ci is mapped to ci+1 by forward iteration, and maximal with this property, then we treat this block of critical points as a single critical point. 2.1. Notation and terminology. We will denote the extended real line by R := R ∪ {−∞, +∞}. Distances, balls, diameters and derivatives are all taken with respect to the spherical metric. For z ∈ C and r > 0, we denote by B(z, r ) ⊂ C the ball centered at z and with radius r . For a given z ∈ C we denote by deg f (z) the local degree of f at z, and for V ⊂ C and n ≥ 0, each connected component of f −n (V ) will be called a pull-back of V by f n . When V is clear from the context, for such a set W we put m W = n. When n = 0 we obtain that each connected component W of V is a pull-back of V with m W = 0. In the case where f n is univalent on W we will say that W is an univalent pull-back of V by f n . Note that the set V is not assumed to be connected. We will abbreviate “Topological Collet-Eckmann” by TCE. 2.2. General properties of the pressure function. Given an integer n ≥ 1 let n : C × R → R be the function defined by −t

n (z 0 , t) := f n (z 0 ) . w∈ f −n (z 0 )

Then for every t ∈ R and every z 0 in C outside a set of Hausdorff dimension 0, we have lim sup n1 ln n (z 0 , t) = P(t), n→+∞

see [Prz99,PRLS04].

(2.1)


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In the following proposition, HDhyp ( f ) := sup{HD(X ) | X compact and invariant subset of C, where f is uniformly expanding}. Proposition 2.1. Given a rational map f of degree at least two, the function t → P(t) + tχinf (resp. t → P(t) + tχsup ), is convex, non-increasing, and non-negative on [0, +∞) (resp. (−∞, 0]). Moreover t− < 0, and we have t+ ≥ HDhyp ( f ) with strict inequality if, and only if, f satisfies the TCE condition. In particular for all t in (t− , t+ ) we have P(t) > max{−tχinf , −tχsup }, and for all t in R\(t− , t+ ) we have P(t) = max{−tχinf , −tχsup }. Proof. For each μ ∈ M ( f ) the function t → h μ ( f ) − t (χμ − χinf ) (resp. t → h μ − t (χμ − χsup )) is affine and non-increasing on [0, +∞) (resp. (−∞, 0]). As by definition P(t) = sup{h μ − tχμ | μ ∈ M ( f )}, we conclude that the function t → P(t) + tχinf (resp. t → P(t) + tχinf ) is convex and non-increasing on [0, +∞) (resp. (−∞, 0]). It also follows from the definition that t → P(t) + tχinf (resp. t → P(t) + tχinf ) is non-negative on this set. The inequalities t− < 0 and t+ ≥ HDhyp ( f ) follow from the fact that χinf is non-negative and from the fact that the pressure function P is strictly positive on (0, HDhyp ( f )) [Prz99]. When f satisfies the TCE condition, then χinf > 0 [PRLS03, Main Theorem] and thus t+ > HDhyp ( f ). When f does not satisfy the TCE condition, then χinf = 0 [PRLS03, Main Theorem] and therefore the equality t+ = HDhyp ( f ) follows from the fact that HDhyp ( f ) is the first zero of the function P [Prz99]. 2.3. The pressure function and conformal measures. For real numbers t and p we will say that a finite Borel measure μ is (t, p)-conformal for f , if for each Borel subset U of C on which f is injective we have

t f dμ. μ( f (U )) = exp( p) U

By the locally eventually onto property of f on J ( f ) it follows that if the topological support of a (t, p)-conformal measure is contained in J ( f ), then it is in fact equal to J ( f ). Proposition 2.2. Let f be a rational map of degree at least two. Then for each t ∈ (t− , +∞) there exists a (t, P(t))-conformal measure for f supported on J ( f ), and for each real number p for which there is a (t, p)-conformal measure for f supported on J ( f ) we have p ≥ P(t). Proof. When t = 0, the assertions are well known, see for example [DU91, p. 104]. The case t > 0 is given by [PRLS04, Theorem A]. In the case t ∈ (t− , 0) the existence is given by [MS00, §3.5] (see also [PRLS04, Theorem A.7]), and in [PRLS04, Prop. A.11] it is shown that if for some real number p there is a (t, p)-conformal measure, then in fact p = P(t).

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2.4. Characterizations of χinf and χsup . The following proposition gives some characterizations of χinf and χsup , which are obtained as direct consequences of the results in [PRLS03]. For each α > 0 put Eα =

+∞ +∞

B f n (Crit( f )), max{n 0 , n}−α .

n 0 =1 n=1 −1 Observe that the Hausdorff dimension of E α is less

than or equal to α . It thus follows that the Hausdorff dimension of the set E ∞ := α>0 E α is equal to 0.

Proposition 2.3. For a rational map f of degree at least two, the following properties hold: 1. Given a repelling periodic point p of f , let m be its period and put χ ( p) := 1 m m ln |(( f ) ( p)|. Then we have inf{χ ( p) | p is a repelling periodic point of f } = χinf , sup{χ ( p) | p is a repelling periodic point of f } = χsup . 2. 1 n→+∞ n

lim

ln sup f n (z) z ∈ C = χsup .

3. For each z 0 ∈ C\E ∞ we have 1 n→+∞ n

lim

lim 1 n→+∞ n

ln min f n (w) w ∈ f −n (z 0 ) = χinf , ln max f n (w) w ∈ f −n (z 0 ) = χsup .

(2.2) (2.3)

Proof. 1. The equality involving χinf was shown in [PRLS03, Main Theorem]. To prove the equality involving χsup , first note that if p is a repelling periodic point of f , and if we denote by m its period, then the measure μ := m−1 j=0 δ f j ( p) is invariant by f and its Lyapunov exponent is equal to χ ( p). It thus follows that sup{χ ( p) | p repelling periodic point of f } ≤ χsup .

2.

The reverse inequality follows from the fact, shown using Pesin theory, that for every ergodic and invariant probability measure μ whose Lyapunov exponent is strictly positive and every ε > 0, one can find a repelling periodic point p such that |χμ − χ ( p)| < ε; see for example [PU02, Theorem 11.6.1]. For each integer n ≥ 1 put Mn := sup f n (z) z ∈ C . Note that for integers m, n ≥ 1 we have Mm+n ≤ Mm · Mn , so the limit χ := lim

1 n→+∞ n

ln Mn


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exists. The inequality χ ≥ χsup follows from part 1. To prove the reverse inequality, for each integer n ≥ 1 let z n ∈ C be such that ( f n ) (z n ) = Mn and put μn :=

1 n

n−1

δ f j (z n ) .

j=0

Let (n j ) j≥0 be a diverging sequence of integers so that μn j converges to a measure μ, which is invariant by f . Since the function ln f is bounded from above, the monotone convergence theorem implies that

lim max A, ln f dμ = ln f dμ. A→−∞

On the other hand, for each real number A we have

max A, ln f dμ = lim max A, ln f dμn j ≥ lim sup ln f dμn j . j→+∞

j→+∞

We thus conclude that

χsup ≥ ln f dμ ≥ lim sup ln f dμn j = χ . j→+∞

3.

For a point z 0 ∈ C which is not in the forward orbit of a critical point of f , the inequalities lim sup n1 ln min f n (w) w ∈ f −n (z 0 ) ≤ χinf , n→+∞ lim inf n1 ln max f n (w) w ∈ f −n (z 0 ) ≥ χsup . n→+∞

are a direct consequence of part 1 and the following property: For each repelling periodic point p there is a constant C > 0 such that for every integer n ≥ 1 there is w ∈ f −n (z 0 ) satisfying C −1 exp(nχ ( p)) ≤ f n (w) ≤ C exp(nχ ( p)). Part 2 shows that for each z 0 ∈ C we have lim sup n1 ln max f n (w) w ∈ f −n (z 0 ) ≤ χsup . n→+∞

It remains to show that for every z 0 ∈ C\E ∞ we have A(z 0 ) := lim inf n1 ln min f n (w) w ∈ f −n (z 0 ) ≥ χinf . n→+∞

We observe first that this inequality holds for some point z 0 in C\E ∞ . In fact, let K be a compact subset of J ( f ) of non-zero Hausdorff dimension on which f is uniformly expanding. Then for each z 0 ∈ K \E ∞ the above inequality follows from part 1 and the “specification property” of [PRLS03, Lemma 3.1]. The final observation is that the function A is constant on C\E ∞ , see [Prz99, §3] or also

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[PRLS03, §1]. To prove this fix z, w ∈ C\E ∞ and for a given integer join z to w with a certain number M of discs (U j ) M j=1 such that z ∈ U1 , z ∈ U M and such that for each j ∈ {1, . . . , M − 1} we have U j ∩ U j+1 = ∅, in such a way that for each j ∈ {1, . . . , M} the disc 2U j , with the same center as U j and twice the radius, is disjoint from i=1,....,n f i (Crit( f )). By Koebe Distortion Theorem it follows that there is a constant B > 0 such that the absolute value of the ratio of the derivative of f n at corresponding points of f −n (z) and f −n (w) is bounded by exp(B M). The main point, shown in [Prz99, §3], is that there is ∈ (0, 1) such that for every sufficiently large n such a chain of discs exists for some integer M satisfying M ≤ n . In particular, the ratio of these derivatives is sub-exponential with n. This implies that A(z) = A(w) and completes the proof of the proposition. 3. Nice Sets, Pleasant Couples and Induced Maps In §3.1 we recall the definition and review some properties of nice sets and couples. We also introduce a notion weaker than nice couple, that we call “pleasant couple”. Then we consider the canonical induced map associated to a pleasant couple in §3.2, as it was introduced in [PRL07, §4] for nice couples, and review some of its properties (§3.3). Finally, we introduce in §3.4 a two variable pressure function associated to a canonical induced map, that will be important in what follows. Throughout all this section we fix a rational map f of degree at least two. 3.1. Nice sets, nice couples, and pleasant couples. Recall that a neighborhood V of Crit( f ) ∩ J ( f ) is a nice set for f , if for every n ≥ 1 we have f n (∂ V ) ∩ V = ∅, and if each connected component of V is simply connected and contains precisely one critical point of f in J ( f ). Let V = c∈Crit( f )∩J ( f ) V c be a nice set for f . Then for every pull-back W of V we have either W ∩ V = ∅ or W ⊂ V. Furthermore, if W and W are distinct pull-backs of V , then we have either, W ∩ W = ∅, W ⊂ W or W ⊂ W. For a pull-back W of V we denote by c(W ) the critical point in Crit( f ) ∩ J ( f ) and by m W ≥ 0 the integer such that f m W (W ) = V c(W ) . Moreover we put, K (V ) = {z ∈ C | for every n ≥ 0 we have f n (z) ∈ V }. Note that K (V ) is a compact and forward invariant set and for each c ∈ Crit( f ) ∩ J ( f ) the set V c is a connected component of C\K (V ). Moreover, if W is a connected component of C\K (V ) different from the V c , then f (W ) is again a connected component of C\K (V ). It follows that W is a pull-back of V and that f m W is univalent on W . of V in C we will say that (V , V ) Given a nice set V for f and a neighborhood V by f m W is a pleasant couple for f if for every pull-back W of V , the pull-back of V containing W is if W is contained in V ; (i) contained in V (ii) disjoint from Crit( f ) if W is disjoint from V .


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, V ) is a pleasant couple for f , then for each c ∈ Crit( f ) ∩ J ( f ) we denote by V c If (V the connected component of V containing c and for each pull-back W of V we will the pull-back of V by f m W that contains W and put m W denote by W := m W and by c(W ) := c(W ). Thus, when W is disjoint from V , it is shielded from Crit( f ) by W property (ii). Otherwise, W ⊂ V and then the set W may or may not intersect Crit( f ). The latter distinction will be crucial in what follows. , V ) be a pleasant couple for f and let W be a connected component of Let (V C\K (V ). Then for every j = 0, . . . , m W − 1, the set f j (W ) is a connected component of C\K (V ) different from the V c . Thus f j (W ) is disjoint from V and by property (ii) j (W ) is disjoint from Crit( f ). It follows that f m W is univalent on W . the set f , V ) of nice sets for f such that V ⊂ V , and such A nice couple for f is a pair (V = ∅. If (V , V ) is a nice couple for f , then that for every n ≥ 1 we have f n (∂ V ) ∩ V for every pull-back W of V we have either ∩ V = ∅ or W ⊂ V. W It thus follows that each nice couple is pleasant. Remark 3.1. The definitions of nice sets and couples given here is slightly weaker than that of [PRL07,RL07]. For a set V = c∈Crit( f )∩J ( f ) V c to be nice, in those papers we required the stronger condition that for each integer n ≥ 1 we have f n (∂ V ) ∩ V = ∅, and that the closures of the sets V c are pairwise disjoint. Similarly, for a pair of nice sets , V ) to be a nice couple we required the stronger condition that for each n ≥ 1 we (V = ∅. The results we need from [PRL07] still hold with the weaker have f n (∂ V ) ∩ V property considered here. , V ) is a nice couple as defined here, then V is a nice set in the Observe that if (V sense of [PRL07,RL07]. The following proposition5 sheds some light on the definitions above, although it is not used later on; compare with the construction of nice couples in §A.1 (Theorem C) and in [RL07, §6]. Proposition 3.2. Suppose that for a rational map f there exists a nice set U = c c∈Crit( f )∩J ( f ) U such that for every integer n ≥ 1, f n (∂U ) ∩ U = ∅.

(3.1)

Suppose furthermore that the maximal diameter of a connected component of f −k (U ) converges to 0 as k → +∞. Then there exists a nice set V for f that is compactly contained in U such that (U, V ) is a nice couple for f . Proof. Since U is a nice set each connected component of the set A := C\ f −1 (K (U )) is a pull-back of U . Furthermore, by (3.1) each connected component W of A intersecting U is compactly contained in U , and m W is the first return time to U of points in W . If the forward trajectory of c visits U , take as V c the connected component containing c of A. Since U is a nice set, V c is a first return pull-back of U , and by (3.1) the set V c is compactly contained in U . In particular for each integer n ≥ 1 we have f n (∂ V c ) ∩ U = ∅. For each critical point c ∈ Crit( f ) ∩ J ( f ) whose forward trajectory never returns to U , take a preliminary disc D compactly contained in U c . By (3.1) each 5 We owe this proposition to Shen, from a personal communication.

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c be connected component of A intersecting D is compactly contained in U c . Let now V the union of D and all those connected components of A intersecting D. The hypothesis on diameters of pull-backs implies that V c is compactly contained in U , and that each c is either contained in ∂ D ∩ (C\A), or in the boundary of a connected point in ∂ V component of A intersecting D (which for is a first return pull-back of U ). Therefore c ∩ U = ∅. Finally let V c be the union of V c and all each integer n ≥ 1 we have f n ∂ V c contained in U c (We do this “filling holes” trick since connected components of C\V a priori it could happen that the union of D and one of the connected components of c , might not be simply-connected). We have ∂ V c ⊂ ∂ V c , so for A, and consequently V each integer n≥ 1 we have f n (∂ V c ) ∩ U = ∅. Set V = c∈Crit( f )∩J ( f ) V c . We have shown that for each integer n ≥ 1 we have n f (∂ V ) ∩ U = ∅, so (U, V ) is a nice couple. , V ) be a pleasant couple for f . We say that an 3.2. Canonical induced map. Let (V integer m ≥ 1 is a good time for a point z in C, if f m (z) ∈ V and if the pull-back of V m by f to z is univalent. Let D be the set of all those points in V having a good time and for z ∈ D denote by m(z) ≥ 1 the least good time of z. Then the map F : D → V , V ). defined by F(z) := f m(z) (z) is called the canonical induced map associated to (V We denote by J (F) the maximal invariant set of F and by D the collection of connected components of D. As V is a nice set, it follows that each connected component W of D is a pull-back of V and that for each z ∈ W we have m(z) = m W . Moreover, f m W is univalent on W ⊂ V . Similarly, for each integer and by property (i) of pleasant couples we have W n ≥ 1, each connected component W of the domain of definition of F n is a pull-back . Conversely, if W is a pull-back of V strictly contained of V and f m W is univalent on W m W , then there is c ∈ Crit( f ) ∩ J ( f ) and an integer in V such that f is univalent on W n ≥ 1 such that F n is defined on W and F n (W ) = V c . Indeed, in this case m W is a good time for each element of W and therefore W ⊂ D. Thus, either we have F(W ) = V c(W ) and then W is a connected component of D, or F(W ) is a pull-back of V strictly con). Thus, repeating this argument we tained in V such that f m F(W ) is univalent on F(W can show by induction that there is an integer n ≥ 1 such that F n is defined on W and F n (W ) = V c(W ) . The following result was shown in [PRL07] for nice couples. The proof applies without change to pleasant couples. Lemma 3.3 ([PRL07], Lemma 4.1). For every rational map f there is r > 0 such that , V ) is a pleasant couple satisfying if (V c ≤ r, (3.2) diam V max c∈Crit( f )∩J ( f )

, V ) is topologically then the canonical induced map F : D → V associated to (V mixing on J (F). Moreover there is c ∈ Crit( f ) ∩ J ( f ) such that the set (3.3) m W | W ∈ D contained in V c and such that F(W ) = V c is non-empty and its greatest common divisor is equal to 1. Remark 3.4. We will apply several results of [MU03] to the induced map F. However, most of the results we need from [MU03] are stated for the associated symbolic space. The corresponding results for the induced map F can be obtained using Lemma 3.1.3, Proposition 3.1.4 and Theorem 4.4.1 of [MU03].


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; on the right, the one associated Fig. 1. On the left, the family associated to a connected component Y of V to a higher order pull-back Y of V

3.3. Bad pull-backs. We will now introduce the concept of “bad pull-backs” of a pleasant couple. It is an adaptation of the concept with the same name introduced in [PRL07, §7.1] for a nice set. , V ) and an integer n ≥ 1, a point y ∈ f −n (V ) is a Given a pleasant couple (V bad iterated pre-image of order n if for every j ∈ {1, . . . , n} such that f j (y) ∈ V the by f j containing y. In this case every map f j is not univalent on the pull-back of V point y in the pull-back X of V by f n containing y is a bad iterated pre-image of order n. We could call X a bad pull-back of V of order n, although this terminology would not be used in the rest of the paper, see Fig. 1. Furthermore, a connected component Y ) is a bad pull-back of V of order n, if it contains a bad iterated pre-image of of f −n (V 6 order n. The following two lemmas will be used in the proof of the Main Theorem in §7. They are adaptations to pleasant couples of part 1 of Lemma 7.4 and of Lemma 7.1 of , V ) for f we denote by LV the collection of all [PRL07]. Given a pleasant couple (V the connected components of C\K (V ). On the other hand, let Y be a pull-back of V and recall that m Y ≥ 0 denotes the integer such that f m Y (Y ) is equal to a connected . Then we let DY be the collection of all the pull-backs W of V that component of V of V by f m W containing W are contained in Y , such that f m W maps the pull-back W c(W ) , such that f m Y (W ) ⊂ V and such that f m Y +1 (W ) ∈ LV . See univalently onto V Fig. 1. , V ) be a pleasant couple for f Lemma 3.5 ([PRL07], Part 1 of Lemma 7.4). Let (V and let D be the collection of the connected components of D. Then ⎞ ⎛ ⎞ ⎛ DVc ⎠ ∪ ⎝ DY ⎠ . D⊂⎝ c∈Crit( f )∩J ( f )

Y bad pull-back of V

Proof. Let W ∈ D. If f (W ) ∈ LV , then there is c ∈ Crit( f )∩ J ( f ) such that W ∈ DVc . Suppose now f (W ) ∈ LV , so there is an integer j ∈ {1, . . . , m W − 1} such that 6 When the pleasant couple ( V , V ) is nice, it is easy to see that the notion of bad pull-back as defined here coincides with that of [PRL07]. See also Remark 3.7.

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f j (W ) ⊂ V . Let n be the largest such integer, so we have f n+1 (W ) ∈ LV and, if we by f n containing W , then W ∈ DY . It remains to show that let Y be the pull-back of V Y is a bad pull-back of V of order n. Just observe that if we fix y ∈ W ⊂ Y , then m W is the least good time of y, so y is a bad iterated pre-image of f n (y) ∈ f n (W ) ⊂ V of of order n. This completes the proof of the lemma. order n and Y is a bad pull-back of V , V ) be a pleasant Lemma 3.6 ([PRL07], Lemma 7.1). Let f be a rational map, let (V couple for f and let L ≥ 1 be such that for every ∈ {1, . . . , L} the set f (Crit( f ) ∩ . Then for each z 0 ∈ V and each integer n ≥ 1, the number of J ( f )) is disjoint from V bad iterated pre-images of z 0 of order n is at most (2L deg( f )#(Crit( f ) ∩ J ( f )))n/L . of order n is at most In particular, the number of bad pull-backs of V #(Crit( f ) ∩ J ( f ))(2L deg( f )#(Crit( f ) ∩ J ( f )))n/L . Proof. 1. Given an integer n ≥ 1 and a bad iterated pre-image y of order n, let by (y, n) ∈ {0, . . . , n − 1} be the largest integer such that the pull-back of V n−(y,n) (y,n) f containing f (y) intersects Crit( f ) ∩ J ( f ). Using property (ii) of pleasant couples we obtain that f (y,n) (y) ∈ V , and in the case where (y, n) > 0, that the point y is a bad iterated pre-image of order (y, n). 2. Given an integer n ≥ 1 and a bad iterated preimage y of order n, define k ≥ 1 and a strictly decreasing sequence of non-negative integers (0 , . . . , k ) by induction as follows. We put 0 = n and suppose that for some integer j ≥ 0 the integer j is already defined in such a way that f j (y) ∈ V . If j = 0 then define k := j and stop. Otherwise we have f j (y) ∈ V by the induction hypothesis and therefore y is a bad iterated pre-image of f j (y) of order j . Then we define j+1 := (y, j ). As remarked above f j+1 (y) = f (y, j ) (y) ∈ V , so the induction hypothesis is satisfied. 3. Fix an integer n ≥ 1. To each bad iterated preimage y of order n we have associated in Part 2 an integer k ≥ 1 and a strictly decreasing sequence of non-negative integers (0 , . . . , k ). We have 0 = n, k = 0 and for each j ∈ {1, . . . , k} the pull-back of by f j−1 − j containing f j (y) contains an element c of Crit( f ) ∩ J ( f ) and at V most deg f (c) elements of f −( j−1 − j ) f j (y) . As for each c ∈ Crit( f ) ∩ J ( f ) and each integer m ≥ 1 there are at most #(Crit( f ) ∩ J ( f )) connected components c intersecting Crit( f ) ∩ J ( f ), it follows that there are at most of f −m V (deg( f )#(Crit( f ) ∩ J ( f )))k bad iterated pre-images of z 0 of order n whose associated sequence is equal to (0 , . . . , k ). By definition of L for each j ∈ {1, . . . , k} we have j−1 − j ≥ L. So k ≤ n/L and for each integer m ∈ {1, . . . , n} there is at most one integer r ∈ {0, . . . , L − 1} such that m + r is one of the j . Thus there are at most (L + 1)n/L such decreasing sequences. We conclude that the number of bad iterated pre-images of z 0 of order n is at most, (L + 1)n/L (deg( f )#(Crit( f ) ∩ J ( f )))n/L ≤ (2L deg( f )#(Crit( f ) ∩ J ( f )))n/L .


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Remark 3.7. The purpose of this remark is to show the reverse inclusion of Lemma 3.5. Although this is not used in the proof of the Main Theorem, we think the argument is useful to understand bad pull-backs of pleasant couples. As for each c ∈ Crit( f ) ∩ J ( f ) we clearly have DVc ⊂ D, we just need to show that for each integer n ≥ 1 and by f n we have DY ⊂ D. To do this, let y ∈ Y be a bad each bad pull-back Y of V iterated pre-image of order n and let k ≥ 1 and (0 , . . . , k ) be as in the proof of Lemma 3.6. An inductive argument using property (ii) of pleasant couples shows that for each j ∈ {0, . . . , k −1} the set f j (Y )∩ f −(n− j ) (V ) is contained in V . In particular, each element W of DY is contained in V . On the other hand, observe that the pull-back by f m W −n containing f n (W ) is univalent and by property (i) of pleasant couples it of V . Thus, to show W ∈ D, we just need to show that each element y of W is contained in V is a bad iterated pre-image of order n. Let i ∈ {1, . . . , n} be such that f i y ∈ V and let j ∈ {0, . . . , k − 1} be the largest integer such that j ≥ i. By property (i) of pleasant . by f j −i containing f i y is contained in V couples it follows that the pull-back of V − j j+1 j+1 Since the pull-back of V by f y intersects Crit( f ), it follows containing f by f i− j+1 containing f j+1 y intersects Crit( f ) and hence that the pull-back of V by f i containing y. This completes the proof f i is not univalent on the pull-back of V that y is a bad iterated pre-image of order n and that W ∈ D. , V ) be a pleasant cou3.4. Pressure function of the canonical induced map. Let (V , V ). ple for f and let F : D → V be the canonical induced map associated to (V Furthermore, denote by D the collection of connected components of D and for each c ∈ Crit( f ) ∩J ( f ) denote by Dc the collection of all elements of D contained in V c , so that D = c∈Crit( f )∩J ( f ) Dc . A word on the alphabet D will be called admissible if for every pair of consecutive letters W, W ∈ D we have W ∈ Dc(W ) . For a given n integer n ≥ 1 we denote by E the collection of all admissible words of length n. Given c(W ) of the inverse of F|W . For W ∈ D, denote by φW the holomorphic extension to V ∗ a finite word W = W1 . . . Wn ∈ E , put c W := c(Wn ) and m W = m W1 + · · · + m Wn . Note that the composition φW := φW1 ◦ · · · ◦ φWn c(W ) and takes images in V . is well defined and univalent on V For each t, p ∈ R and n ≥ 1 put t Z n (t, p) := exp −m W p sup φW (z) z ∈ V c(W ) . W ∈E n

It is easy to see that for a fixed t, p ∈ R the sequence (ln Z n (t, p))n≥1 is sub-additive, and hence that we have P F, −t ln F − pm := lim n1 ln Z n (t, p) = inf n1 ln Z n (t, p) | n ≥ 1 , (3.4) n→+∞

see for example Lemma 2.1.1 and Lemma 2.1.2 of [MU03]. Here m is the function defined in §3.2, that to each point z ∈ D it associates the least good time of z. The number (3.4) is called the pressure function of F for the potential − ln F − pm. It is easy to see that for every t, p ∈ R the sequence n1 ln Z n (t, p) n≥1 is uniformly bounded from below, so that (3.4) does not take the value −∞. Note however that if D

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has infinitely many connected components and we take t = 0 and p = 0, then we have P(F, 0) = +∞. When applying the results of [MU03] to the induced map F we will use the fact that the function − ln F defines a Hölder function on the associated symbolic space and hence that for each (t, p) ∈ R2 the same holds for the function −t ln F − pm. The following property will be important to use the results of [MU03]. (*) There is a constant C M > 0 such that for every κ ∈ (0, 1) and every ball B of C, the following property holds. Every collection of pairwise disjoint sets of the form DW , with W ∈ E ∗ , intersecting B and with diameter at least κ · diam(B), has cardinality at most C M κ −2 . In fact, F determines a Conformal Graph Directed Markov System (CGDMS) in the sense of [MU03], except maybe for the “cone property” (4d). But in [MU03] the cone property is only used in [MU03, Lemma 4.2.6] to prove (*). Thus, when property (*) is satisfied all the results of [MU03] apply to F. In [PRL07, Prop. A.2] we have shown , V ) is nice. that property (*) holds when the pleasant couple (V The function, P : R2 → R ∪ {+∞} (t, p) → P F, −t ln F − pm , 2 will be important in what follows. Notice thatif P is finite at (t, p) = (t0 , p0 ) ∈ R , then by Proposition 2.1.9 the function −t0 ln F − p0 m defines a summable Hölder potential on the symbolic space associated to the induced map F. Furthermore, it follows that P is finite on the set (t, p) ∈ R2 | t ≥ t0 , p ≥ p0

and, restricted to the set where it is finite, the function P is strictly decreasing on each of its variables. , V ) be a pleasant Lemma 3.8. Let f be a rational map of degree at least two and let (V couple for f satisfying property (*). Then the function P defined above satisfies the following properties. 1. The function P is real analytic on the interior of the set where it is finite. 2. The function P is strictly negative on {(t, p) ∈ R2 | p > P(t)}. Proof. 1. If P(t, p) < +∞, then by [MU03, Prop. 2.1.9] the function −t ln F − pm defines a summable Hölder potential on the symbolic space associated to F. Thus the desired result follows from [MU03, Theorem 2.6.12], see Remark 3.4. 2. Let (t0 , p0 ) ∈ R2 be such that p0 > P(t0 ). Then for each point z 0 ∈ V for which (2.1) holds, we have −t0 +∞ exp(− p0 m(y)) F k (y) k=1 y∈F −k (z)

≤

+∞ n=1

exp(− p0 n)

| f n (y)|−t0 < +∞,

y∈ f −n (z 0 )

which implies that P(t0 , p0 ) ≤ 0. This shows that the function P is non-positive on {(t, p) ∈ (0, +∞)×R | p > P(t)}. That P is strictly negative on this set follows from the fact that, on this set, P is strictly decreasing on each of its variables.


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4. From the Induced Map to the Original Map The purpose of this section is to prove the following theorem, which gives us some sufficient conditions to obtain the conclusions of the Main Theorem. We denote by Jcon ( f ) the “conical Julia set” of f , which is defined in §4.1. Recall that conformal measures were defined in §2.3. , V ) be a pleasant Theorem A. Let f be a rational map of degree at least two, let (V couple for f satisfying property (*), and let P be the corresponding pressure function defined in §3.4. Then for each t0 ∈ (t− , +∞), the following properties hold: Conformal measure: If P vanishes at (t, p) = (t0 , P(t0 )), then there is a unique (t0 , P(t0 ))-conformal probability measure for f . Moreover this measure is nonatomic, ergodic, and it is supported on Jcon ( f ). Equilibrium state: If P is finite on a neighborhood of (t, p) = (t0 , P(t0 )), and vanishes at this then there is a unique equilibrium measure of f for the poten point, tial −t0 ln f . Furthermore, this measure is ergodic, absolutely continuous with respect to the unique (t0 , P(t0 ))-conformal probability measure of f , and its density is bounded from below by a strictly positive constant almost everywhere. If further, V ) satisfies the conclusions of Lemma 3.3, then the equilibrium state is more (V exponentially mixing and it satisfies the Central Limit theorem. Analyticity of the pressure function: If P is finite on a neighborhood of (t, p) = (t0 , P(t0 )) and for each t ∈ R close to t0 we have P(t, P(t)) = 0, then the pressure function P is real analytic on a neighborhood of t = t0 . In §§5, 6, 7 we verify that, for a map as in the Main Theorem or more generally as in Theorem B in §7, and for a given t0 ∈ (t− , t+ ) the function P corresponding to a sufficiently small pleasant couple is finite on a neighborhood of (t, p) = (t0 , P(t0 )) and that for each t ∈ R close to t0 we have P(t, P(t)) = 0. After some general considerations in §4.1, the assertions about the conformal measure are shown in §4.2. The assertions concerning the equilibrium state are shown in §4.3, and the analyticity of the pressure function is shown in §4.4. , V ), F, P as in the statement of the Throughout the rest of this section we fix f , (V theorem. 4.1. The conical julia set and sub-conformal measures. The conical Julia set of f , denoted by Jcon ( f ), is by definition the set of all those points x in J ( f ) for which there exists ρ(x) > 0 and an arbitrarily large positive integer n, such that the pull-back of the ball B ( f n (x), ρ(x)) to x by f n is univalent. This set is also called radial Julia set. We will use the following general result, which is a strengthened version of [McM00, Theorem 5.1], [DMNU98, Theorem 1.2], with the same proof. Given t, p ∈ R we will say that a Borel measure μ is (t, p)-sub-conformal f , if for every Borel subset U of C\ Crit( f ) on which f is injective we have

t f dμ ≤ μ( f (U )). exp( p) (4.1) U

Proposition 4.1. Fix t ∈ (t− , +∞) and p ∈ [P(t), +∞). If μ is a (t, p)-sub-conformal measure for f supported on Jcon ( f ), then p = P(t), the measure μ is (t, P(t))-conformal, and every other (t, P(t))-conformal measure is proportional to μ. Moreover, every subset X of C such that f (X ) ⊂ X and μ(X ) > 0 has full measure with respect to μ. The proof of this proposition depends on the following lemma.

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Lemma 4.2. Let t, p ∈ R and let μ be a (t, p)-sub-conformalmeasure supported on Jcon ( f ). Suppose that for some p ≤ p there exists a non-zero t, p -conformal measure ν that is supported on J ( f ). Then p = p and μ is absolutely continuous with respect to ν. In particular ν(Jcon ( f )) > 0. Proof. For ρ > 0 put Jcon ( f, ρ) := {x ∈ Jcon ( f ) | ρ(x) ≥ ρ}, so that Jcon ( f ) = ρ>0 Jcon ( f, ρ). For each ρ0 > 0, Koebe Distortion Theorem implies that there is a constant C > 1 such that for every x ∈ Jcon ( f, ρ0 ) there are arbitrarily small r > 0, so that for some integer n ≥ 1 we have (4.2) μ(B(x, 5r )) ≤ C exp(−np)r t and ν(B(x, r )) ≥ C −1 exp −np r t . Given a subset X of Jcon ( f, ρ0 ), by Vitali’s covering lemma, for every r0 > 0 we can find a collection of pairwise disjoint balls (B(xj , r j )) j>0 and strictly positive integers (n j ) j>0 , such that x j ∈ X , r j ∈ (0, r0 ), X ⊂ j>0 B(x j , 5r j ) and such that for each j > 0 the inequalities (4.2) hold for x := x j and r := r j and n = n j . Moreover, for each integer n 0 ≥ 1 we may choose r0 sufficiently small so that for each j > 0 we have n j ≥ n 0 . Since by hypothesis p ≤ p, we obtain ν(X ) ≥ C −2 exp n 0 p − p μ(X ). Suppose by contradiction that p < p. Choose ρ0 > 0 such that μ(Jcon ( f, ρ0 )) > 0 and set X := Jcon ( f, ρ0 ). As in the inequality above, n 0 > 0 can by taken arbitrarily large, we obtain a contradiction. So p = p and it follows that μ is absolutely continuous with respect to ν. Proof of Proposition 4.1. Let ν be a (t, P(t))-conformal measure ν for f supported on J ( f ). By [PRLS04, Theorem A and Theorem A.7] there is at least one such measure, see also [Prz99]. So Lemma 4.2 implies that p = P(t), and that μ is absolutely continuous with respect to ν. In Parts 1 and 2 we show that ν is proportional to μ. It follows in particular that μ is conformal. In Part 3 we complete the proof of the proposition by showing the last statement of the proposition. 1. First note that ν := ν|C\Jcon ( f ) is a conformal measure for f of the same exponent as ν. Then Lemma 4.2 applied to ν = ν implies that, if ν is non-zero, then ν (Jcon ( f )) > 0. This contradiction shows that ν is the zero measure and that ν is supported on Jcon ( f ). 2. Denote by g the density μ with respect to ν. Since ν is conformal and μ sub-conformal, the function g satisfies g ◦ f ≥ g on a set of full ν-measure. Let δ > 0 be such that ν({g ≥ δ}) > 0. As ν is supported on Jcon ( f ), there is a density point of {g ≥ δ} for ν that belongs to Jcon ( f ). Going to large scale and using g ◦ f ≥ g, we conclude that {g ≥ δ} contains a ball of definite size, up to a set of ν-measure 0. It follows by the locally eventually onto property of f on J ( f ) that the set {g ≥ δ} has full measure with respect to ν. This implies that g is constant ν-almost everywhere and therefore that ν and μ are proportional. In particular μ is conformal. 3. Suppose that X is a Borel subset of C of positive measure with respect to μ and such that f (X ) ⊂ X . Then the restriction μ| X of μ to X is a (t, P(t))-sub-conformal measure supported on the conical Julia set. It follows that μ| X is proportional to μ, and thus that μ| X = μ and that X has full measure with respect to μ.


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4.2. Conformal measure. Given t, p ∈ R we will say that a measure μ supported on the maximal invariant set J (F) of F is (t, p)-conformal for F if for every Borel subset U of a connected component W of D we have

t F dμ. μ(F(U )) = exp( pm W ) U

In view of [MU03, Prop. 2.1.9], the hypothesis that P(F, −t0 ln F − P(t0 )m) = P(t0 , P(t0 )) = 0 implies that −t0 ln F − P(t0 )m defines a summable Hölder potential on the symbolic space associated to F. Furthermore, by Theorem 3.2.3 and Proposition 4.2.5 of [MU03] it follows that the induced map F admits a non-atomic (t0 , P(t0 ))-conformal measure supported on J (F). Therefore the assertions in Theorem A about conformal measures are a direct consequence of Proposition 4.1 and of the following proposition. Proposition 4.3. Let F be the canonical induced map associated to a pleasant cou, V ) for f that satisfies property (*). Then for every t ∈ (t− , +∞) and p ∈ ple (V [P(t), +∞), each (t, p)-conformal measure of F is in fact (t, P(t))-conformal, and it is the restriction to V of a non-atomic (t, p)-conformal measure of f supported on Jcon ( f ). Proof. The proof of this proposition is a straightforward generalization of that of [PRL07, Prop. B.2]. We will only give a sketch of the proof here. μ for f whose topological Since t > t− there is a (t, P(t))-conformal measure support is equal to the whole Julia set of f (Proposition 2.2). Let LV be the collection of connected components of C\K (V ). Notice that for each W ∈ LV we have μ(W ) ∼ exp(−m W P(t)) diam(W )t , for an implicit constant independent of W . Let μ be a (t, p)-conformal measure for F. For each W ∈ LV denote by φW : c(W ) → W the inverse of f m W |W V , and let μW be the measure supported on W , defined by

t φ dμ. μW (X ) = exp(−m W p) W

f m W (X ∩W )

Clearly the measure W ∈LV μW is supported on Jcon ( f ), non-atomic, and for each have μ(W ) ∼ W ∈ LV we have μW (C) ∼ exp(−m W p) diam(W )t . Since we also exp(−m W P(t)) diam(W )t , and p ≥ P(t), it follows that the measure W ∈LV μW is finite. In view of Proposition 4.1, to complete we just need to show that W ∈LV μW is (t, p)-sub-conformal for f . The proof of this fact is similar to what was done in [PRL07, Prop. B.2]. 4.3. Equilibrium state. The following are crucial estimates. Lemma 4.4. Suppose that the pressure function P is finite on a neighborhood of (t, p) = (t0 , P(t0 )) and that it vanishes at this point. If μ is the unique (t0 , P(t0 ))-conformal measure of F, then the following properties hold: 1. For every (t, p) ∈ R2 and γ > 0,

t ln F + pm γ dμ < +∞.

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2. There is ε0 > 0 such that for every sufficiently large integer n we have μ(W ) ≤ exp(−ε0 n). W connected component of D m W ≥n

In particular

m W μ(W ) < +∞.

W connected component of D

We have stated Part 1 for every (t, p) ∈ R2 , although we will only use it for (t, p) close to (t0 , p0 ). Proof. Since the function P is finite on a neighborhood of (t, p) = (t0 , P(t0 )), there is ε0 > 0 such that P(t0 − ε0 , P(t0 ) − ε0 ) < +∞. By [MU03, Prop. 2.1.9] this implies that, −(t −ε ) exp(−(P(t0 ) − ε0 )m W ) sup F (z) 0 0 | z ∈ W < +∞. W connected component of D

As for each connected component W of D we have −t μ(W ) ≤ C0 exp(−P(t0 )m W ) sup F (z) 0 | z ∈ W , we obtain the conclusion of Part 1 holds for each (t, p) ∈ R2 and that C1 := μ(W ) exp(ε0 m W ) < +∞. W connected component of D

So for each n ≥ 1 we have

exp(ε0 n)

μ(W ) ≤ C1 .

W connected component of D m W ≥n

This proves Part 2 of the lemma.

Existence. It follows from standard considerations that F has an invariant measure ρ that is absolutely continuous with respect to the (t0 , P(t0 ))-conformal measure μ of F, and that the density of ρ with respect to μ is bounded from below by a strictly positive constant almost everywhere. This result can be found for example in [Gou04, §1], by observing that F| J (F) is a “Gibbs-Markov map”. For a proof in a setting closer to ours, but that only applies to the case when V is connected, see [MU03, §6]. The measure ρ :=

m W −1

j

f ∗ ρ|W

W connected component of D j=0

is easily seen to be invariant by f and Part 2 of Lemma 4.4 implies that it is finite. Furthermore this measure is absolutely continuous with respect to the (t0 , P(t0 ))-conformal measure μ of f , and its density is bounded from below by a strictly positive constant


683

on a subset of V of full measure with respect to μ = μ|V . It follows from the locally eventually onto property of Julia sets that the density of ρ with respect to μ is bounded from below by a strictly positive constant almost everywhere; see for example [PRL07, §8] for details. As μ is ergodic (Proposition 4.1) it follows that ρ is also ergodic. We will show now that the probability proportional to ρ is an equilibrium measure ρ state of f for the potential −t0 ln f . We first observe that by Part 1 of Lemma 4.4 and [MU03, Theorem 2.2.9] the measure ρ is an equilibrium state of F for the potential −t0 ln F − P(t0 )m, see also Remark 3.4. That is, we have P F, −t0 ln F − P(t0 )m = h ρ (F) −

t0 ln F + P(t0 )m dρ,

which is equal to 0 by hypothesis. By the generalized Abramov’s formula [Zwe05, Theorem 5.1], we have h ρ (F) = h ρ( f ) ρ (C), and by definition of ρ we have m dρ = ρ (C). We thus obtain,

h ρ( f ) = ( ρ (C))−1 h ρ (F) = ( ρ (C))−1 t0 ln F + P(t0 )m dρ

−1 ρ + P(t0 ) = t0 ln f d ρ + P(t0 ). = ( ρ (C)) t0 ln f d This shows that ρ is an equilibrium state of f for the potential −t0 ln f . Uniqueness. In view of [Dob08, Theorem 8], we just need to show that the Lyapunov exponent of each equilibrium state of f for the potential −t0 ln f is strictly positive; see also [Led84]. Let ρ be an equilibrium state of f for the potential −t0 ln f . If f satisfies the Topological Collet-Eckmann Condition then it follows that the Lyapunov exponent of ρ is strictly positive, as in this case we have χinf > 0. Otherwise we have χinf = 0, and then P(t0 ) > 0 by Proposition 2.1. It thus follows that h ρ ( f ) > 0, and therefore that the Lyapunov exponent of ρ is strictly positive by Ruelle’s inequality. Statistical properties. When F satisfies the conclusions of Lemma 3.3, the statistical properties of ρ can be deduced from the tail estimate given by Part 2 of Lemma 4.4, using Young’s results in [You99]. In the case when there is only one critical point in the Julia set one can apply these results directly, and in the general case one needs to consider the first return map of F to the set V c , where c is the critical point given by the conclusion of Lemma 3.3, as it was done in [PRL07, §8.2]. In the general case one could also apply directly the generalization of Young’s result given in [Gou04, Théorème 2.3.6 and Remarque 2.3.7]. We omit the standard details.

4.4. Analyticity of the pressure function. By hypothesis for each t close to t0 we have P(t, P(t)) = 0. Since the function P is real analytic on a neighborhood of (t0 , P(t0 )) (Lemma 3.8), by the implicit function theorem it is enough to check that ∂∂p P|(t0 ,P(t0 )) = 0. By Part 1 of Lemma 4.4 and [MU03, Prop. 2.6.13] this last number is equal to the integral of the (strictly negative) function −m against ρ, see also Remark 3.4. It is therefore strictly negative.

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5. Whitney Decomposition of a Pull-back The purpose of this section is to introduce a Whitney type decomposition of a given pull-back of a pleasant couple. It is used to prove the key estimates in the next section. 5.1. Dyadic squares. Fix a square root i of −1 in C and identify C with R ⊕ iR. For integers j, k and , the set x + i y | x ∈ 2j , j+1 , y ∈ 2k , k+1 , 2 2 will be called dyadic square. Note that two dyadic squares are either nested or have disjoint interiors. We define a quarter of a dyadic square Q as one of the four dyadic squares contained in Q and whose side length is one half of that of Q. the open square having the same center as Q, Given a dyadic square Q, denote by Q sides parallel to that of Q, and length twice that of Q. Note in particular that for each dyadic square Q the set Q\Q is an annulus whose modulus is independent of Q; we denote this number by m 1 . 5.2. Primitive squares. Let f be a rational map of degree at least two. We fix r1 > 0 sufficiently small so that for each critical value v of f in the Julia set of f there is a univalent map ϕv : B(v, 9r1 ) → C whose distortion is bounded by 2. We say that a subset Q of C is a primitive square, if there is v ∈ CV( f ) ∩ J ( f ) such that Q is contained in the domain of ϕv , such that ϕv (Q) is a dyadic square, and such that ϕ contained in the image of ϕv . In this case we put v(Q) := v and v(Q) is −1 Q := ϕv ϕv (Q) . We say that a primitive square Q 0 is a quarter of a primitive square Q, if Q 0 ⊂ Q and if ϕv(Q) (Q 0 ) is a quarter of ϕv(Q) (Q). Note that each primitive square has precisely four quarters. Furthermore, each primitive square Q contained in B(CV, r1 ) is contained in a primitive square Q such that Q is a quarter of Q . Definition 5.1. Fix ∈ (0, r1 ). The Whitney decomposition associated to (the complement of) a subset F of C is the collection W (F) of all those primitive squares Q such ∩ F = ∅, and that are maximal with these properties. that diam(Q) < , Q By definition two distinct elements of W (F) have disjoint interiors, and each point in B(CV( f ) ∩ J ( f ), 9r1 )\F is contained in an element of W (F). Lemma 5.2. Let ∈ (0, r1 ), and let F be a finite subset of C. Then the following properties hold: 1. Let Q 0 be a primitive square contained in B(CV( f ) ∩ J ( f ), r1 ) and such that diam(Q 0 ) ≤ . Then either Q 0 is contained in an element of W (F), or it contains an element Q of W (F) such that √ diam(Q) ≥ 41 (2 + 3 #F)−1 diam(Q 0 ). 2. For each n ≥ 2 the number of those Q ∈ W (F) contained in B(CV( f ) ∩ J ( f ), r1 ) and such that diam(Q) ∈ [2−(n+1) , 2−n ] is less than 2599(#F).


685

Proof. 1. Let n ≥ 2 be the least integer such that (2n − 2)2 > 9(#F), so that 2n ≤ √ 2 2 + 3 #F . Put Q 0 := ϕv(Q) (Q 0 ) and denote by 0 the side length of Q 0 . For each element a of F in Q 0 , choose a dyadic square Q a whose side length is equal to 2−n 0 and that contains ϕv(Q) (a). As there are (2n − 2)2 squares of side length 2 n equal to 2−n 0 contained in the interior of Q 0 , and at most 9(#F) < (2 − 2) of them intersect one of the squares a∈F Q a , we conclude that there is at least one square Q of side length equal to 2−n 0 that is contained in the interior Q 0 −1 Q is disjoint from F. It follows that the primitive square and such that ϕv(Q) −1 Q := ϕv(Q) Q is contained in an element of W (F). As, √ diam(Q) ≥ 21 2−n diam(Q 0 ) ≥ 41 (2 + 3# F)−1 diam(Q 0 ), the desired assertion follows. 2. Let Q be an element of W (F) contained in B(CV( f ) ∩ J ( f ), r1 ), and Q be let a primitive square such that Q is a quarter of Q . Then either diam Q > Q, or Q intersects F. So, if diam(Q) ≤ 41 , then there is a ∈ F contained in 1 and therefore diam(Q) ≥ 4 dist(Q, a). So, if we let n ≥ 2 be an integer such that diam(Q) ∈ 2−(n+1) , 2−n , then Q ⊂ B a, 5 · 2−n . Since the area of Q is 1 −n 2 4 and the area of B a, 5 · 2−n is greater than or equal to 18 diam(Q)2 ≥ 32 less than 25π 4−n 2 , we conclude that there are at most 25 · 32π(#F) < 2599(#F) elements Q of W (F) satisfying diam(Q) ∈ 2−(n+1) , 2−n . 5.3. Univalent squares. For an integer n ≥ 0 we will say that a subset Q of C is a univalent square of order square Q such that Q is a connected n, if there is a primitive −(n+1) n+1 component of f Q , and such that f is univalent on the connected compo and note nent of f −(n+1) Q containing Q. In this case we denote this last set by Q, that Q\Q is an annulus of modulus equal to m 1 . It thus follows that there is a constant K 0 > 1 such that for every univalent square Q of order n and every j = 1, . . . , n + 1, the distortion of f j on Q is bounded by K 0 . , V ) be a pleasant couple for f such that f (V ) ⊂ B(CV( f ) ∩ J ( f ), r1 ). For Let (V , denote by (Y ) the number of those j ∈ {0, . . . , m Y } such that a pull-back Y of V . Moreover, let W (Y ) be the collection of all those univalent squares Q that f j (Y ) ⊂ V ⊂ Y , such that f m Y (Q) intersects V , and that are maximal are of order m Y , such that Q with these properties. Note that for Q ∈ W (Y ) we have v(Q) = f (c(Y )). By definition every pair of distinct elements of W (Y ) have disjoint interiors. On the other hand, every c(Y ) \ Crit f m Y +1 is contained in an element of W (Y ), and for point in f m Y |−1 V Y is disjoint from Crit f m Y +1 . each Q ∈ W (Y ) the set Q , V ) be a Proposition 5.3. Let f be a rational map of degree at least two and let (V pleasant couple for f . Then there is a constant C 0 > 0 such that for every ξ ∈ (0, 1) the number of those Q ∈ W (Y ) such that c(Y ) diam f m Y +1 (Q) ≥ ξ diam f V is less than

2600 deg( f )(Y ) C0 + 21 (Y ) log2 (Y ) + (Y ) log2 ξ −1 .

686


Proof. Put c = c(Y ) and v = f (c), and let ξ0 ∈(0, 1) be sufficiently so that for small c each z ∈ V c the connected component of f −1 B f (z), ξ0 diam f V contain c c . Put F = f m Y +1 Y ∩ Crit f m Y +1 , := ξ0 diam f V ing z is contained in V and consider the Whitney decomposition W (F), as defined in §5.2. Note that #F ≤ (Y ). c ⊂ 1. We prove first that for every Q ∈ W (Y ) the primitive square f m Y +1 (Q) ⊂ f V B(v, r1 ) contains an element Q of W (F) such that −1 ξ0 diam f m Y +1 (Q) . diam Q ≥ 80 #(Y ) Let Q 0 be a primitive square contained in f m Y +1 (Q) such that m Y +1 1 (Q) ≤ diam(Q 0 ) ≤ . 4 ξ0 diam f By Part 1 of Lemma 5.2 there is an element Q of W (F) that either contains Q 0 , or that it is contained in Q 0 and √ −1 diam(Q 0 ) diam Q ≥ 41 2 + 3 #F √ −1 1 2 + 3 #F ≥ 16 ξ0 diam f m W +1 (Q) . As #F ≤ (Y ) and (Y ) ≥ 1, we just need to show that Q is in fact contained in f m Y +1 (Q). Suppose by contradiction that this is not the case. Then it follows that Q m +1 −(m +1) Y Y Q contains f (Q) strictly. Let Q be the connected component of f c , so containing Q. By definition of ξ0 we have that f m Y Q is contained in V Q is contained in Y . On the other hand f m Y Q intersects V c , because it con-

tains f m Y (Q) and this set intersects V c . As by definition of W (F) the set Q is m +1 disjoint from F, it follows that f Y is univalent on Q . Thus, by definition of W (Y ), the univalent square Q is contained in an element of W (Y ). But Q ∈ W (Y ) is strictly contained in Q , so we get a contradiction. This shows that Q is in fact contained in f m Y +1 (Q) and completes the proof of the assertion. 2. For each Q ∈ W (Y ) choose an element Q of W (F) satisfying the property described in Part 1. Note that for each Q 0 ∈ W (F) the number of those Q ∈ W (Y ) such that Q = Q 0 is less than or equal to deg( f )(Y ) . As the area of a primitive square Q is 2 greater than or equal to 18 diam Q , it follows that for each ξ ∈ (0, 1) the number c is less than or equal to of those Q ∈ W (Y ) satisfying diam Q ≥ ξ diam f V −2 (Y ) 8π ξ deg( f ) . 1 Let ξ ∈ 0, 4 ξ0 be given and let n 0 be the least integer n ≥ 2 such that √ √ −(n 0 −1) 80 (Y ). If Q ∈ W (Y ) is such that ξ ≥ 2−n 80 (Y ), so that ξ c < 2 m +1 , then we have diam f Y (Q) ≥ ξ diam V −1 c . diam Q ≥ 80 (Y ) ξ0 diam f m Y +1 (Q) ≥ 2−n 0 ξ0 diam V So Part 2 of Lemma 5.2 implies that for each n ≥ 2 the number of those Q ∈ W (Y ) such that c −n c , 2 ξ0 diam f V diam Q ∈ 2−(n+1) ξ0 diam f V ,


687

is less than 2599(#F) deg( f )(Y ) ≤ 2599(Y f ) (Y ) . So we conclude that the m) deg( +1 Y c is less than number of those Q ∈ W (Y ) such that diam f (Q) ≥ ξ diam V −2 + (n 0 − 2)2599(Y ) deg( f )(Y ) 8π 41 ξ0 −2 ≤ deg( f )(Y ) 8π 14 ξ0 + 2599(Y ) log2 ξ −1 + log2 (80) + 21 log2 ((Y )) . This completes the proof of the lemma.

6. The Contribution of a Pull-back , V ) for f . Recall Fix a rational map f of degree at least two, and a pleasant couple (V that LV is the collection of all the connected components of C\K (V ) and that for a we denote by DY the collection of all the pull-backs W of V that are pull-back Y of V and such contained in Y , such that f m Y (W ) ⊂ V , such that f m Y +1 is univalent on W m +1 Y that f (W ) ∈ LV ; see §3.3. Furthermore, we denote by (Y ) the number of those . j ∈ {0, . . . , m Y } such that f j (Y ) ⊂ V The purpose of this section is to prove the following. Proposition 6.1 (Key estimates). Let f be a rational map of degree at least two that is expanding away from critical points. Then for each sufficiently small pleasant couple , V ) for f the following properties hold: (V 1. For every t0 ∈ R, and every (t, p) ∈ R2 sufficiently close to (t0 , P(t0 )), we have exp(− pm W ) diam(W )t < +∞. (6.1) W ∈LV

2. Let t, p ∈ R be such that (6.1) holds and such that p > max{−tχinf , −tχsup }. Then for every ε > 0 such that |t|ε 0 such that for each pull-back Y of V exp(− pm W ) diam(W )t W ∈DY

≤ C1 (deg( f ) + 1)(Y ) exp −m Y ( p − max{−tχinf , −tχsup } − |t|ε) .

To prove this proposition we start with the following lemma. Lemma 6.2. Let f be a rational map that is expanding away from critical points. Then for every compact and forward invariant subset K of the Julia set of f that is disjoint from the critical points of f and every t > 0 we have P f | K , −t ln f < P(t).

688


Proof. By hypothesis f is uniformly expanding on K . Enlarging K if necessary we may assume that the restriction of f to K admits a Markov partition, see [PU02, Theorem 3.5.2 7 and Remark 3.5.3], so that there is at least one equilibrium state μ for f | K with potential −t ln f . We enlarge K with more cylinders to obtain a compact forward invariant subset K of J ( f ), so that f restricted to K admits a Markov partition and so that the relative interior of K in K is empty. It follows that μ cannot be an equilibrium measure for f | K for the potential −t ln f , so we have

P f | K , −t ln f = h μ ( f ) − t ln f dμ 0 as follows. If χinf = 0 we put r0 = dist(∂ V, Crit( f ) ∩ J ( f )). Suppose that χinf > 0. Then by [PRLS03, Main Theorem] there exists r0 > 0 such that for every z 0 in J ( f ), every every ε > 0, sufficiently large integer n, and every connected component W of f −n B z 0 , r0 , we have diam(W ) ≤ exp(−n(χinf − ε)). Then we put r0 = min r0 , dist(∂ V, Crit( f ) ∩ J ( f )) . Given a subset Q of C we define n Q ∈ {0, 1, . . . , +∞} as follows. If there are infinitely many integers n such that diam ( f n (Q)) < r0 , then we put n Q = +∞. Otherwise we let n Q be the largest integer n ≥ 0 such that diam ( f n (Q)) < r0 . Lemma 6.3. Let f be a rational map of degree at least two. Then for every ε > 0 there is a constant C(ε) > 1 such that for each connected subset Q of C that intersects the Julia set of f we have, C(ε)−1 exp(−n Q (χsup + ε)) ≤ diam(Q) ≤ C(ε) exp(−n Q (χinf − ε)) Proof. The inequality on the right holds trivially when χinf = 0, and when χinf > 0 it is given by the definition of r0 > 0. The inequality on the left is a direct consequence of Part 2 of Proposition 2.3. Proof of Proposition 6.1. Let r1 > 0 be as in the definition of primitive squares in §5.2, , V ) be a pleasant couple for f such that f (V ) ⊂ B(CV( f ) ∩ J ( f ), r1 ). Furand let (V thermore, let A1 > 0 and K 1 > 1 be given by Koebe Distortion Theorem in such a way we have diam(W ) ≤ that for each pull-back W of V such that f m W is univalent on W A1 dist(W, ∂ W ), and such that for each j = 1, . . . , m W the distortion of f j on W is bounded by K 1 . 1. Note that it is enough to show that there are t < t0 and p < P(t0 ) for which (6.1) holds. Let V be a sufficiently small neighborhood of Crit( f ) ∩ J ( f ) contained in V , so that for each c ∈ Crit( f ) ∩ J ( f ) the set K = z ∈ J ( f ) | for every n ≥ 0, f n (z) ∈ V 7 An analogous result in the case of diffeomorphisms is shown in [Fis06].


689

intersects V c . By Lemma 6.2 we have P f | K , −t0 ln f < P(t0 ). Let t < t0 and p < P(t0) be sufficiently close to t0 and P(t0 ), respectively, so that p > P f | K , −t ln f . For each c ∈ Crit( f ) ∩ J ( f ) choose a point z(c) in K ∩ V c . Given a univalent pullback W of V let z W be the unique point in f −m W (z(c(W ))) contained in W . Note that when W ∈ LV we have z W ∈ K . On the other hand, there is a distortion constant C > 0 such that for each pull-back W of V such that f m W maps a neighborhood of c , we have diam(W ) ≤ C ( f m W ) (z W )−1 . W univalently onto a component of V Since by hypothesis the restriction of f to K is uniformly expanding, we have −t lim sup n1 ln f n (z W ) n→+∞

W ∈LV m W =n

≤ lim sup n1 ln n→+∞

n −t f (z)

c∈Crit( f )∩J ( f ) z∈K ∩ f −n (z(c))

≤ P f | K , −t ln f ,

hence C2 :=

exp(−m W p) diam(W )t

W ∈LV

≤ C |t|

−t exp(−m W p) f m W (z W ) < +∞.

W ∈LV

2.1. Put C3 := min dist z(c), ∂ V c / diam(V c ) | c ∈ Crit( f ) ∩ J ( f ) and observe that for each pull-back W of V such W is a univalent pull-back that −1 , we have dist z W , ∂ W ≥ C3 K diam W . We will show that for each of V 1 , for each Q ∈ W (Y ), and each W ∈ DY such that z W ∈ Q, we pull-back Y of V have diam f m Y +1 (W ) ≤ 8C3−1 K 1 diam f m Y +1 (Q) . Put Q = f m Y +1 (Q) and W = f m Y +1 (W ), and suppose by contradiction that diam W > 8C3−1 K 1 diam Q . Observe that Q is a primitive square contained in B( f (c(Y )), r1 ) and that W ∈ LV . So there is a primitive square Q 0 such that Q is a quarter of Q 0 . We have diam Q 0 ≤ 8 diam Q < C3 K 1−1 diam W ≤ dist z W , ∂ W . Since by hypothesis z W ∈ Q, we have z W = f m Y +1 (z W ) ∈ Q ⊂ Q 0 , so the last inequality implies that Q 0 ⊂ W . But f m Y +1 is univalent on W , so the connected component Q 0 of f −(m Y +1) Q 0 containing Q is a univalent square of order m Y 0 ⊂ Y , that contains Q strictly. This contradicts the hypothesis that satisfying Q Q ∈ W (Y ).

690


2.2. We will now show that there is a constant C4 > 0 such that for each pull-back Y and each Q ∈ W (Y ) we have of V exp (− pm W ) diam(W )t ≤ C4 exp − pn Q diam(Q)t . (6.2) W ∈DY z W ∈Q

and let Q ∈ W (Y ). Put Q = f m Y +1 (Q), and let B be a Let Y be a pull-back of V ball whose center belongs to Q and of radius equal to 8C3−1 K 1 + 1 diam Q . By Part 2.1, for each W ∈ DY such that z W ∈ Q we have f m Y +1 (W ) ⊂ B. Since the distortion of f m Y +1 is bounded by K 0 on Q, and by K 1 on each element of DY , we obtain, exp(− pm W ) diam(W )t W ∈DY z W ∈Q

!

diam(Q) ≤ exp(− p(m Y + 1))(K 0 K 1 ) diam (Q ) t exp (− pm W ) diam W . · |t|

"t ·

W ∈LV W ⊂B

If there is no W ∈ DY such that z W ∈ Q, then there is nothing to prove. So we assume that there is an element W0 of DY such that z W0 ∈ Q. Then Q , and hence B, intersects K , as it contains the point z f m Y +1 (W0 ) . Since by hypothesis the restriction of f to K is uniformly expanding, there is n 0 ≥ 0 independent of Y , such that n Q ≤ n B + n 0 and such that there is an integer n B ≥ 0 satisfying n − n B ≤ n 0 , such that f n B is univalent on B and has distortion bounded by 2 B on this set. We have n Q − n B ≤ 2n 0 , so there is a constant C5 > 0 independent

of B such that diam f n B (B) > C5 . So, if we put

|t| C6 := exp(| p|2n 0 ) 2C5−1 2 8C3−1 K 1 + 1 C2 , then we have t exp (− pm W ) diam W W ∈LV W ⊂B

⎛ ≤ exp − pn B 2|t| ⎝

⎞t diam(B) ⎠ · diam f n B (B)

t exp (− pm W ) diam W

W ∈LV W ⊂ f n B (B)

|t| ≤ exp − pn Q exp(−| p|2n 0 ) 2C5−1 diam(B)t C2 t ≤ C6 exp − pn Q diam Q . Inequality (6.2) with constant C4 := C6 (K 0 K 1 )|t| , is then a direct consequence of the last two displayed (chains of) inequalities.


691

2.3. We will now complete the proof of the proposition. For each Q ∈ W (Y ) put Q := f m Y +1 (Q). Let Q ∈ W (Y ) be such that there is W ∈ DY satisfying z W ∈ Q. As this last point is in the Julia set of f , by Lemma 6.3 we have diam(Q)t ≤ C(ε)|t| exp n Q (max{−tχsup , −tχinf } + |t|ε) . Since the elements of W (Y ) cover Y \ Crit f m Y +1 , if we put γ := exp(− p + max{−tχsup , −tχinf } + |t|ε) ∈ (0, 1), then by summing over Q ∈ W (Y ) in (6.2) we obtain exp(− pm W ) diam(W )t W ∈DY

≤ C4 C(ε)

γ n Q = C4 C(ε)γ m Y +1

Q∈W (Y ) Q∩J ( f ) =∅

γ n Q .

Q∈W (Y ) Q∩J ( f ) =∅

To estimate this last number, observe that by Lemma 6.3, for each Q ∈ W (Y ) intersecting the Julia set of f we have diam Q ≥ C(ε)−1 exp −n Q (χsup + ε) . ln 2

c(Y )

So, if we put γ = γ χsup +ε , C7 = γ − log2 C(ε)−log2 diam(V ) and for each c(Y ) ), then we have γ n Q ≤ Q ∈ W (Y ) we put ξ Q = diam Q / diam(V γ − log2 ξ ( Q ) . So Proposition 5.3 implies that C7 γ n Q ≤ C7 γ − log2 ξ ( Q ) Q∈W (Y ) Q∩J ( f ) =∅

Q∈W (Y ) Q∩J ( f ) =∅

≤ 2600C7 deg( f )

# (Y )

C0 +

1 2 (Y ) log2 (Y ) + (Y )

+∞

$ γ

n

.

n=0

This completes the proof of the proposition.

7. Proof of the Main Theorem The purpose of this section is to prove the following version of the Main Theorem for pleasant couples. Recall that each nice couple is pleasant and satisfies property (*), see §3.4. Theorem B. Let f be a rational map of degree at least two that is expanding away from critical points, and that has arbitrarily small pleasant couples having property (*). Then following properties hold: Analyticity of the pressure function: The pressure function of f is real analytic on (t− , t+ ), and linear with slope −χsup ( f ) (resp. −χinf ( f )) on (−∞, t− ] (resp. [t+ , +∞)).

692


Equilibrium states: For each t0 ∈ (t− , t+ ) there is a unique equilibrium state of f for the potential −t0 ln f . Furthermore this measure is ergodic and mixing. Throughout the rest of this section we fix a rational map f and t0 ∈ (t− , t+ ) as in the statement of the theorem. Recall that by Proposition 2.1 we have P(t0 ) > max{−t0 χinf , −t0 χsup }. Put γ0 := exp − 21 (P(t0 ) − max{−t0 χinf , −t0 χsup }) ∈ (0, 1), and choose L ≥ 0 sufficiently large so that (2L deg( f )(deg( f ) + 1)#(Crit( f ) ∩ J ( f )))1/L γ0 < 1.

(7.1)

, V ) be a pleasant couple for f that is sufficiently small so that for each ∈ Let (V (recall that our standing con{1, . . . , L} the set f (Crit( f ) ∩ J ( f )) is disjoint from V vention is that no critical point of f in its Julia set is mapped to a critical point under , V ) has property (*). By Lemma 3.6 forward iteration.) We assume furthermore that (V it follows that for each integer n ≥ 1 and each z 0 ∈ V the number of bad iterated pre-images of z 0 of order n is at most (2L deg( f )#(Crit( f ) ∩ J ( f )))n/L , of order n is at most and that the number of bad pull-backs of V #(Crit( f ) ∩ J ( f ))(2L deg( f )#(Crit( f ) ∩ J ( f )))n/L . We show in §7.1 that the pressure function P of the canonical induced map associ, V ), defined in §3.4, is finite on a neighborhood of (t, p) = (t0 , p0 ). In §7.2 ated to (V we show that for each t close to t0 the function P vanishes at (t, p) = (t, P(t)). Then Theorem B follows from Theorem A. 7.1. The function P is finite on a neighborhood of (t, p) = (t0 , P(t0 )). By the considerations in §3.4, to show that P is finite on a neighborhood of (t, p) = (t0 , p0 ) we just need to show that there are t < t0 and p < P(t0 ) such that exp(− pm W ) diam(W )t < +∞. (7.2) W ∈D

Let t < t0 and p < P(t0 ) be given by Part 1 of Proposition 6.1. Taking t and p closer to t0 and P(t0 ), respectively, we assume that there is ε > 0 sufficiently small so that p − max{−tχinf , −tχsup } − |t|ε > 21 (P(t0 ) − max{−t0 χinf , −t0 χsup }), and put γ := exp(− p + max{−tχinf , −tχsup } + |t|ε) ∈ (0, γ0 ). For each c ∈ Crit( f ) ∩ J ( f ) we have, by applying Part 2 of Proposition 6.1 to c , Y =V exp(− pm W ) diam(W )t ≤ C1 (deg( f ) + 1). (7.3) W ∈DVc


693

we have (Y ) ≤ n/L + 1, using Part 2 of Proposition 6.1 Since for each pull-back Y of V again and letting C2 = C1 (deg( f ) + 1)#(Crit( f ) ∩ J ( f )) we obtain exp(− pm W ) diam(W )t W ∈DY Y bad pull-back of V

≤ C1

(deg( f ) + 1)(Y ) · γ m Y

Y bad pull-back of V

≤ C2

+∞

(2L deg( f )(deg( f ) + 1)#(Crit( f ) ∩ J ( f )))1/L · γ

n

.

n=1

As γ ∈ (0, γ0 ), we have by (7.1) that the sum above is finite. Then (7.2) follows from (7.3) and Lemma 3.5. 7.2. For each t close to t0 we have P(t, P(t)) = 0. Recall that for a given t0 ∈ (t− , t+ ) , V ), and that we denote by P the we have fixed a sufficiently small pleasant couple (V corresponding pressure function defined in §3.4. Furthermore, in §7.1 we have shown that the function P is finite on a neighborhood of (t0 , P(t0 )). We will show now that for t close to t0 the function P vanishes at (t, P(t)), thus completing the proof of Theorem B. In view of Lemma 3.8 we just need to show that for each t close to t0 we have P(t, P(t)) ≥ 0. Suppose by contradiction that in each neighborhood of t0 we can find t such that P(t, P(t)) < 0. As P is finite on a neighborhood of (t, p) = (t0 , P(t0 )), it follows that P is continuous at this point (Lemma 3.8). Thus there are t ∈ (t− , t+ ) and p ∈ (max{−tχinf , −tχsup }, P(t)), such that P(t, p) < 0, and such that the conclusion of Part 1 of Proposition 6.1 holds for these values of t and p. However, this contradicts the following lemma. Lemma 7.1. Let t ∈ (t− , t+ ) and p > min{−tχinf , −tχsup } be such that P(t, p) < 0 and such that the conclusion of Part 1 of Proposition 6.1 holds for these values of t and p. Then p ≥ P(t). Proof of Proposition 6.1. Fix z 0 ∈ V such that all, (2.1), (2.2), and (2.3) hold. To prove the lemma we just need to show that +∞ n=1

exp(− pn)

−t n f (y) < +∞.

y∈ f −n (z 0 )

1. Given an integer n ≥ 1 an element y ∈ f −n (V ) is a univalent iterated pre-image by f n containing y is univalent. Recall that for an of order n if the pull-back of V integer n ≥ 1 an element y of f −n (V ) is a bad iterated pre-image of z 0 of order n if by f n containing y for every j ∈ {1, . . . , n} such that f j (y) ∈ V the pull-back of V is not univalent. For y ∈ f −n (z 0 ) there are three cases: y is univalent, bad, or there is m ∈ {1, . . . , n − 1} such that f m (y) ∈ V , such that f m (y) is a bad iterated pre-image of z 0 of order n − m and such that y is a univalent iterated pre-image of f m (y) of order m. In fact, if y ∈ f −n (z 0 ) is not bad, then there is m ∈ {1, . . . , n −1} such that f m (y) ∈ V

694


and such that y is a univalent iterated pre-image of f m (y). If m is the largest integer with this property, then there are two cases. Either m = n and then y is a univalent iterated pre-image of z 0 , or m < n and then f m (y) is a bad iterated pre-image of z 0 . Therefore, if for each w ∈ V we put U (w) := 1 +

+∞

−t n f (y) ,

exp(− pn)

y∈ f −n (w), univalent

n=1

then we have 1+

+∞

exp(− pn)

−t n f (y)

y∈ f −n (z 0 )

n=1

= U (z 0 ) +

+∞

w∈ f −n (z

n=1

−t n f (w) U (w).

exp(− pn)

0 ),

(7.4)

bad

As p > max{−tχinf , −tχsup }, by Lemma 3.6 and (7.1) it follows that +∞

−t n f (w) < +∞.

exp(− pn)

w∈ f −n (z 0 ), bad

n=1

So by (7.4), to prove the lemma it is enough to prove that the supremum supw∈V U ( p, w) is finite. 2. Denote by L V the first entry map to V , which is defined on the set of points y ∈ C\V having a good time, by L V (y) = f m(y) (y). Note that for each w0 ∈ V , each integer n ≥ 1 and each univalent iterated pre-image y ∈ f −n (w0 ) of w0 of order n, we have that m(y) ≤ n and that L V (y) ∈ V is a univalent iterated pre-image of w0 of order n − m(y). Moreover, note for each k ≥ 1, each element of F −k (w0 ) is a univalent iterated pre-image of w0 . Conversely, for each univalent iterated pre-image y of w0 there is an integer k ≥ 1 such that F k is defined at y and F k (y) = w0 (see §3.2). Therefore, if for z ∈ V we put −t m(y) L(z) := 1 + exp(− pm(y)) f (y) , y∈L −1 V (z 0 )

then we have, U (w0 ) = L(w0 ) +

+∞

k=1 y∈F −k (w0 )

−t exp(− pm(y)) F k (y) L(y).

Since by hypothesis P(t, p) < 0, for each w ∈ V the double sum TF (w) :=

+∞

k=1 y∈F −k (w)

is finite.

−t k exp(− pm(y)) F (y) ,

(7.5)


695

On the other hand, since the conclusion of Part 1 of Proposition 6.1 holds, for each z ∈ V the sum L(z) is finite. By bounded distortion it follows that C := sup L(z) < +∞. z∈V

Thus, by (7.5) for each w ∈ V we have U ( p, w) ≤ C TF (w) < +∞, and by bounded distortion supw∈V U ( p, w) < +∞. This completes the proof of the lemma. Acknowledgements. We are grateful to Weixiao Shen and Daniel Smania for their help with references, Weixiao Shen again and Genadi Levin for their help with the non-renormalizable case and Henri Comman for his help with the large deviations results. We also thank Neil Dobbs, Godofredo Iommi, Jan Kiwi and Mariusz Urbanski for useful conversations and comments. Finally, we are grateful to Krzysztof Baranski for making Fig. 1 and the referee for his suggestions and comments that helped to clarify some of the concepts introduced in the paper. 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. Puzzles and Nice Couples This Appendix is devoted to showing that several classes of polynomials satisfy the conclusions of the Main Theorem. In §A.1 we consider the case of at most finitely renormalizable polynomials without indifferent periodic points, in §A.2 we consider the case of some infinitely renormalizable quadratic polynomials, and finally in §A.3 we consider the case of quadratic polynomials with real coefficients. A.1. At most finitely renormalizable polynomials. The purpose of this section is to prove the following result. We thank Weixiao Shen for providing the main idea of the proof. Theorem C. Every at most finitely renormalizable complex polynomial without indifferent periodic points has arbitrarily small nice couples. Furthermore, these nice couples can be formed by nice sets that are finite unions of puzzle pieces. See [CL09, Prop. 5] for a somewhat similar result in the case of multimodal maps. The proof relies on the fundamental result that diameters of puzzles tend uniformly to 0 as their depth tends to ∞ [KvS09]; see also [QY09] for the case when the Julia set is totally disconnected. Let f be an at most finitely renormalizable polynomial, and consider the puzzle construction described in [KvS09, §2.1]. Given an integer n ≥ 0 we denote by ϒn the collection of all puzzles of depth n, which are by definition open sets. For P ∈ ϒn and p ∈ P we put Pn ( p) := P. We will assume that every critical point of f in J ( f ) is contained in a puzzle piece. It is always possible to do the puzzle construction with this property. This follows from the fact that in each periodic connected component of J ( f ) that is not reduced to a single point, there are infinitely many separating periodic points, see for example [LS09, §A.1]. We remark that the main technical results of [KvS09], including the “complex a priori bounds”, are stated for “complex box mappings”, and they are thus independent ofthe periodic points used to construct the puzzle pieces. For z ∈ C put O f (z) = n≥1 f n (z), and put δ0 := min dist c, O f (c ) | c, c ∈ Crit( f ) ∩ J ( f ), c ∈ O f (c ) .

696


Let n 0 ≥ 1 be a sufficiently large integer so that the diameter of each puzzle piece of depth n 0 is strictly smaller than δ0 /2. Furthermore, let n 1 > n 0 be a sufficiently large integer such that for each c ∈ Crit( f ) ∩ J ( f ) we have Pn 1 (c) ⊂ Pn 0 (c), and such that for distinct c, c ∈ Crit( f ) ∩ J ( f ) we have Pn 1 (c) ∩ Pn 1 (c ) = ∅. The following is a straightforward consequence of our choice of n 1 . Lemma A.1. For each c, c ∈ Crit( f ) ∩ J ( f ) such that c ∈ O f (c), and each n ≥ 1, m ≥ 1, n ≥ n 1 we have f m (∂ Pn (c)) ∩ Pn (c ) = ∅. Given a subset C of Crit( f ) ∩ J ( f ) and a function ν : C → N, we put Pν(c) (c). P(ν) := c∈C

We will say that ν is nice (resp. strictly nice) if for every c ∈ C we have ν(c) ≥ n 1 , and if for every integer m ≥ 1, we have f m (∂ P(ν)) ∩ P(ν) = ∅ (resp. f m (∂ P(ν)) ∩ P(ν) = ∅). The following lemma is Part 2 of Lemma 2.2 of [KvS09]. Lemma A.2. For every recurrent critical point c in J ( f ) there is a strictly nice function defined on {c}. Proof. Let 0 ≥ 0 be a sufficiently large integer so that P0 (c) ⊂ P0 (c), and note that for every m ≥ 0 the set f m ∂ P0 (c) is disjoint from P0 (c) by the puzzle structure, and hence it is disjoint from P0 (c). Define inductively a strictly increasing sequence of integers (k )k≥1 as follows. Suppose that for some k ≥ 0 the integer k is already defined. Then we denote by m k the least integer such that f m k (c) ∈ Pk (c), and put k+1 = k + m k . Clearly the sequence (k )k≥0 is strictly increasing, so diam(Pk (c)) → 0 as k → +∞, and therefore m k → +∞ as k → +∞. Let k ≥ 1 be sufficiently large so that m k ≥ 0 , and so that for every m ∈ {1, . . . , 0 } the sets f m (P k(c)) and Pk (c) have disjoint closures. We will show that for each m ≥ 1 the set f m ∂ Pk (c) and Pk (c) are disjoints, which shows that the function ν : {c} → N defined by ν(c) = k is strictly nice. Suppose by contradiction that for some m ≥ 1 the set f m ∂ Pk (c) intersects P k (c). This − +m m − ∂ Pk (c) = f ∂ P0 (c) intersects f k 0 Pk (c) = P0 (c). implies that f k 0 By our choice of k we have m ≥ 0 , so we get a contradiction with our choice of n. For a strictly nice function ν : C → N, denote by Dν the set of those points z ∈ C for which there is an integer m ≥ 1 such that f m (z) ∈ P(ν), and for each z ∈ Dν denote by m ν (z) the least such integer, and by cν (z) the critical point c in C such that f m ν (z) (z) ∈ Pν(c) (c). Furthermore we denote by Eν : Dν → P(ν) the map defined by Eν (z) := f m ν (z) (z). For a subset C of Crit( f ) ∩ J ( f ) we put NC := {c ∈ C such that O f (c) ∩ C = ∅}. For a strictly nice function ν defined on C let Rν : C \NC → N be the function defined by Rν(c) = ν(cν (c)) + m ν (c). By definition, PR ν(c) (c) is the pull-back of Pν(cν (c)) (cν (c)) by f m ν (c) containing c.


697

Now we shall prove the key technical lemma which provides the inductive step to construct nice couples starting from the existence of “nice couples” around a single recurrent critical point. This procedure resembles the procedure to build ‘complex box mappings’ in [KvS09] or τ -nice sets in [CL09, Proposition 5] (in the multimodal interval setting). Lemma A.3. For a subset C of Crit( f ) ∩ J ( f ) the following properties hold: 1. Let ν : C \NC → N be a strictly nice function. Then for each sufficiently large integer n the function ν˜ : C → N defined by ν˜ |C \NC = ν and ν −1 (n) = NC , is strictly nice. 2. Let ν : C → N be strictly nice. Then for each sufficiently large integer n ≥ n 1 the −1 function ν : C → N defined by ν (n) = NC , and ν |C \NC = Rν is strictly nice, we have P (ν ) ⊂ P(ν), and for each integer m ≥ 1 we have f m ∂ P ν ∩ P(ν) = ∅. 3. Let c˜ ∈ Crit( f ) ∩ J ( f ) not in C , be such that O f (c) ˜ ∩ C = ∅. If there is a strictly nice function defined on C , then there is also one defined on C ∪ {c}. ˜ Proof. 1. Put k = max{ν(c) | c ∈ C \NC } ≥ n 1 , let n > k be a sufficiently large integer so that for all c ∈ NC we have Pn (c) ⊂ Pk (c), and consider the function ν˜ defined as in the statement of the lemma for this choice of n. Then for each c ∈ C \NC and m ≥ 1 we have f m ∂ Pν˜ (c) (c) ∩ P(˜ν ) = ∅. On the other hand, since n > k ≥ n 1 , by Lemma A.1 the same property holds for each c ∈ NC . 2. Let n ≥ n 1 be a sufficiently large integer such that for all c ∈ NC we have Pn (c) ⊂ Pν(c) (c), and let ν be the function defined in the statement of the lemma for this choice of n. Since ν is strictly nice we have PR ν (C \NC ) ⊂ Pν (C ). So, by our choice of n we have Pν (C ) ⊂ Pν (C ). On the other hand, by the definition of Rν it follows that for each c ∈ C \NC and each m ≥ 1 the set f m ∂ PR ν(c) (c) is disjoint from Pν (C ), and hence from Pν (C ). Finally, by Lemma A.1, for each c ∈ NC and each m ≥ 1 the set f m (∂ Pn (c)) is disjoint from Pν (C ), and hence from Pν (C ). 3. Let ν0 : C → N be a strictly nice function. By Part 2 there is a sequence of strictly nice functions (νk )k≥1 defined on C , such that for each k ≥ 1 we have νk |C \NC = Rνk−1 , and P(νk ) ⊂ P(νk−1 ). Put L = # Crit( f ) ∩ J ( f ), and let cˆ be the critical point defined as follows.8 If Eν L (c) ˜ ∈ C , then put cˆ := c, ˜ vˆ := Eν L (c), ˜ and = 0. Otherwise we let ∈ {1, . . . , L} be the largest integer such that for all j ∈ {0, . . . , − 1} we have Eν L− j ◦ · · · ◦ Eν L (c) ˜ ∈ C, and then put

cˆ := Eν L−(−1) ◦ · · · ◦ Eν L (c) ˜ , and vˆ := Eν L− cˆ .

8 The proof of this part is simpler in the case when the forward orbit of c˜ is disjoint from Crit( f ). We advise to restrict to this case on a first reading, taking L = 0 and cˆ = c. ˜

698


By definition we have vˆ ∈ P (ν L− ), but vˆ ∈ C . Let k ≥ L − bethe largest integer such that vˆ ∈ P(νk ), and note that m νk cˆ = m ν L− cˆ , and cνk cˆ = cν L− cˆ . Put nˆ := νk (cνk vˆ ) + m νk vˆ + m νk cˆ , ˆ so that Pnˆ cˆ is the pull-back of P(νk ) by f m νk (vˆ )+m νk (cˆ) containing c. 3.1. We will show now that for every m ≥ 1 the set f m ∂ Pnˆ cˆ is disjoint from P(νk+1 ). We will use several times the fact that νk is strictly nice. By definition of nˆ component of P(νk ). So for the image of Pnˆ cˆ by fm ν k (vˆ )+m νk (cˆ) is a connected each m ≥ m νk vˆ + m νk cˆ the set f m (∂ Pnˆ cˆ ) is disjoint from P(νk ), and hence from P(νk+1 ). Since m νk vˆ is the first time of vˆ to P(ν k ), it follows that the entry same property holds for each m ∈ m νk cˆ + 1, . . . , m νk vˆ + m νk cˆ − 1 . On the other hand, by definition of nˆ the set f m νk (cˆ) Pnˆ cˆ is the pull-back of P(νk ) ˆ and by definition of k we have vˆ ∈ P(νk )\P(νk+1 ). As νk by f m νk (vˆ ) containing v, m νk (cˆ) Pnˆ cˆ and P(νk+1 ) have disjoint is strictly nice, it follows that the sets f of cˆ to P(νk ), it follows that closures. Finally, since mνk cˆ is the first entry time for all m ∈ 1, . . . , m νk cˆ − 1 the sets f m Pnˆ cˆ and P(νk+1 ) have disjoint closures. 3.2. We will show now that for each integer m ≥ 1 the set f m (∂ P(νk+1 )) is disjoint from Pnˆ cˆ . Suppose by contradiction that there is an integer m ≥ 1 and c ∈ C such that f m ∂ Pνk+1 (c) (c) intersects Pnˆ cˆ . Then the set f m νk (cˆ)+m ∂ Pνk+1 (c) (c) intersects the closure of f m νk (cˆ) Pnˆ cˆ . This last set is a first return domain of P(νk ), and it is thus compactly contained in the open set P(νk ), because νk is strictly nice. We conclude that the set f m νk (cˆ)+m ∂ Pνk+1 (c) (c) intersects the open set P(νk ). However, this contradicts the fact that Pνk+1 (c) (c) is a first return domain of P(νk ). 3.3. If cˆ = c, ˜ then the properties shown in Parts 3.1 and 3.2 imply that the function ν˜ : C ∪ {c} ˜ → N defined by ν˜ (c) ˜ = nˆ and ν˜ |C = νk+1 is strictly nice. If cˆ = c, ˜ then we let m˜ be the integer such that f m˜ (c) ˜ = Eν L−(−1) ◦ · · · ◦ Eν L (c), ˜ c ˆ by E ◦ · · · ◦ E . Then we define so that Pn+ c) ˜ is the pull-back of P ( ν L−(−1) νL ˆ m˜ nˆ ν˜ : C ∪ {c} ˜ → N by ν˜ (c) ˜ = nˆ + m, ˜ and ν˜ |C = νk+ . The properties shown in Parts 3.1 and 3.2 imply that ν˜ is strictly nice. Proof of Theorem C. In view of Part 2 of Lemma A.3 it is enough to show that there is a strictly nice function defined in all of Crit( f ) ∩ J ( f ). Let us say a critical point c ∈ Crit( f ) ∩ J ( f ) is corresponded if for each c ∈ Crit( f ) ∩ J ( f ) such that c ∈ O f (c) we have c ∈ O f (c ). Denote by C0 the set of corresponded critical points in J ( f ). Note that for each critical point c in J ( f ) that is not corresponded, the set O f (c) intersects C0 . So, using Part 3 of Lemma A.3 inductively, it follows that to show the existence of a strictly nice function defined in all of Crit( f ) ∩ J ( f ) it is enough to show the existence of a strictly nice function defined on C0 . Let ∼ be the relation on C0 defined by c ∼ c if c = c or c ∈ O f (c ). It follows from the definition of C0 that ∼ is an equivalence relation. Let C1 be a subset of C0 containing a unique element in each equivalence class of ∼. Thus for each c ∈ C0 the


699

set O f (c) intersects C1 . Using Part 3 of Lemma A.3 inductively it follows that to show that there is a strictly nice function defined on C0 , it is enough to show that there is one defined on C1 . By definition of C1 for each c ∈ C1 the set O f (c) is disjoint from C1 \{c}. Thus the set NC defined as in the statement of Lemma A.3 for C = C1 , is equal to the set of those c ∈ C1 such that c ∈ O f (c). Equivalently, NC is the set of those non-recurrent critical points in C1 . Thus, by Part 1 of this lemma we just need to show that there is a strictly nice function defined on C2 := C1 \NC1 . For each c ∈ C2 let νc be a strictly nice function defined on {c}, given by Lemma A.2. Let ν : C2 → N be defined for each c ∈ C2 by ν(c) = νc (c). As for each c ∈ C2 we have O f (c) ∩ (C2 \{c}) = ∅, by Lemma A.1 the function ν is strictly nice. This completes the proof of the theorem. A.2. Infinitely renormalizable quadratic maps. The purpose of this section is to show that each infinitely renormalizable polynomial or polynomial-like map whose small critical Julia sets converge to 0 satisfy the hypotheses of Theorem B. This includes the case of infinitely renormalizable quadratic maps with a priori bounds; see [KL08,McM94] and references therein for results on a priori bounds. The post-critical set of a rational map f is by definition P( f ) :=

+∞

f n (Crit( f )).

n=1

If f is an infinitely renormalizable quadratic-like map, then P( f ) does not contain pre-periodic pionts [McM94, Theorem 8.1]. Lemma A.4. Let f be a rational map and let V be a nice set for f such that ∂ V is of V there is disjoint from the post-critical set of f . Then for every neighborhood V ⊂V such that (V , V ) is a pleasant couple. V Proof. We will assume that P( f ) contains at least three points; otherwise f is conjugated to a power map [McM94, Theorem 3.4] and then the assertion%is vacuously true. % We will denote by disthyp the hyperbolic distance on C\P( f ) and by % f % the derivative % % of f with respect to it. Then by Schwarz lemma we have % f % ≥ 1 on C\ f −1 (P( f )) (cf., [McM94, Theorem 3.5]). Furthermore, for z ∈ C\P( f ) and r > 0 we denote by Bhyp (z, r ) the ball corresponding to the hyperbolic metric on C\P( f ). and put Let ε > 0 be sufficiently small such that Bhyp (∂ V, 2ε) ⊂ V := V ∪ Bhyp (∂ V, ε). V is a neighborhood of V in C and the set V \V is disjoint from P( f ). By construction V \W is disjoint from Crit( f ). We thus have So for each pull-back W of V the set W % % ∩Crit( f ) = ∅ when W ∩V = ∅. On the other hand, since % f % ≥ 1 on C\ f −1 (P( f )), W when W ⊂ V we have , V \P( f )) ≤ disthyp (∂ W , ∂ W ) ≤ disthyp (∂ V , ∂ V ) ≤ ε. disthyp (∂ W ⊂V . This shows that (V , V ) is a pleasant couple for f . Hence W

700


In what follows we shall use some terminology of [McM94] and [AL08, §2.4, App. A]. Proposition A.5. Let f be an infinitely renormalizable quadratic-like map for which the diameters of small critical Julia sets converge to 0. Then f is expanding away from critical points and has arbitrarily small pleasant couples having property (*). In particular the conclusions of Theorem B hold for f . Proof. We will show that there are arbitrarily small puzzles containing the critical point whose boundaries are disjoint from the post-critical set. Then Lemma A.4 implies that there are arbitrarily small pleasant couples. That each of these pleasant couples satisfies property (*) is a repetition of the proof of [MU03, Lemma 4.2.6], using the fact that each puzzle is a quasi-disk and thus that it has the “cone property” of [MU03, §4.2] with “twisted angles”. Let SR( f ) be the set of all integers n ≥ 2 such that f n is simply renormalizable and let Jn be the corresponding small critical Julia set. Then Jn is decreasing with n. For each k ≥ 1 we denote by m(k) the k th element of SR( f ). We consider the usual puzzle construction with the α-fixed point of f . Then for each ≥ 1 there is a puzzle of depth , that we denote by P , whose closure contains Jm(1) .

We have +∞ =1 P = Jm(1) . More generally, by induction it can be shown that if for a given s ≥ 1 we consider the puzzle construction with the α-fixed points of the renormalizations of f m(1) , f m(2) , …, f m(s) , then for each ≥ 1 there is a puzzle of depth that contains Jm(s) . We will denote it by Ps, . Thus Ps, is bounded by a finite number of arcs in an equipotential line and by the closure of some pre-images of external rays landing at the α-fixed points of the renormalizations of f m(1) , f m(2) , …, f m(s) . In particular the intersection of ∂ Ps, with the Julia set is a finite set of pre-periodic points and it is thus disjoint from P( f ) by [McM94, Theorem 8.1]. Furthermore we have +∞

Ps, = Jm(s) ,

=1

and hence lim

lim diam(Ps, ) = 0.

s→+∞ →+∞

This completes the proof that f has arbitrarily small pleasant couples having property (*). To show that f is expanding away from critical points we just need to show that for each s ≥ 1 and ≥ 1 the map f is uniformly expanding on K (Ps, )∩ Jm(s) % . As % this set is compactly contained in C\P( f ), it is enough to show that the derivative % f % of f with respect of the hyperbolic metric on this set is strictly larger than 1 on C\ f −1 (P( f )). Since f −1 (P( f )) contains P( f ) strictly, this is a consequence of Schwarz lemma. A.3. Quadratic polynomials with real coefficients. In this section we show that each quadratic polynomial satisfies the conclusions of the Main Theorem. If f is at most finitely renormalizable without indifferent periodic points, then by Theorem C the map f satisfies the hypotheses of the Main Theorem. If f is infinitely renormalizable, then it has a priori bounds by [McM94], so the diameters of the small


701

Julia set converge to 0 and then the assertion follows from Proposition A.5. See also Remark A.6. It remains to consider the case when f has an indifferent periodic point. Fix t0 ∈ (t− , t+ ). Since f has real coefficients it follows that f has a parabolic periodic point, and since f is quadratic it follows that f does not have critical points in the Julia set. Therefore the function ln f is bounded and Hölder continuous on J ( f ), and since the measure theoretic entropy of f is upper semi-continuous [FLM83,Lju83], there is an equilibrium state ρ of f for the potential −t0 ln f . Since f has a parabolic periodic point it follows that t+ is the first zero of P, so we have P(t0 ) > 0 and therefore the Lyapunov exponent of ρ is strictly positive. Since by [PRLS04, Theorem A and Theorem A.7] there is a (t0 , P(t0 ))-conformal measure of f (see also [Prz99]), [Dob08, Theorem 8] implies that ρ is in fact the unique equilibrium state of f for the potential −t0 ln f . The analyticity of P at t = t0 is given by [MS00] when t0 < 0 and when t0 ≥ 0 the fact that P is analytic at t = t0 can be shown in an analogous way as in [SU03], using an induced map defined with puzzles pieces. Remark A.6. We will now explain why we have introduced pleasant couples to deal with infinitely renormalizable quadratic-like maps as in Proposition A.5 and with quadratic polynomials with real coefficients in particular. Following [McM94] we call a renormalization of a quadratic-like map primitive if the corresponding small Julia sets are pairwise disjoint. If the first renormalization of a quadratic-like map f is primitive, then the usual puzzle construction produces a puzzle piece P containing the small critical Julia set, in such a way that the first return puzzle P0 to P containing the critical point is compactly contained in P. These puzzle pieces form a nice couple (P, P0 ) for f . Since the puzzle P can be made arbitrarily close to the small critical Julia set, a slightly more general argument shows that a map as in Proposition A.5 having infinitely many primitive renormalizations admits arbitrarily small nice couples. The Feigenbaum quadratic polynomial is an example of an infinitely renormalizable quadratic map having no primitive renormalization and it is possible to show that as such it does not have arbitrarily small nice couples. However, the Feigenbaum polynomial does have arbitrarily small pleasant couples by Proposition A.5. Appendix B. Rigidity, Multifractal Analysis, and Level-1 Large Deviations The purpose of this Appendix is to prove that, apart from some well-known exceptional maps, the pressure function of each of the maps considered in this paper is strictly convex on (t− , t+ ). We derive consequences for the dimension spectrum for Lyapunov exponents (§B.1) and for pointwise dimensions of the maximal entropy measure (§B.2), as well as some level-1 large deviations results (§B.3). See [Pes97,Mak98] for background in multifractal analysis, and [DZ98] for background in large deviation theory. In what follows by a power map we mean a rational map P(z) ∈ C(z) such that for some integer d we have P(z) = z d . Theorem D. Let f be a rational map satisfying the hypotheses of Theorem B. If f is not conjugated to a power, Chebyshev or Lattès map, then for every t ∈ (t− , t+ ) we have P (t) > 0. In particular ∗ ∗ χinf := inf −P (t) | t ∈ (t− , t+ ) < χsup := sup −P (t) | t ∈ (t− , t+ ) . It is well known that for a power, Chebyshev or Lattès map, t+ = +∞ and the pressure ∗ = χ ∗ . For a function P is affine on (t− , +∞); in particular in this case we have χinf sup

702


general rational map f and for t0 ∈ (t− , 0), a result analogous to Theorem D was shown by Makarov and Smirnov in [MS00, §3.8]. , V ) be a pleasant Proof. Suppose that for some t0 ∈ (t− , t+ ) we have P (t0 ) = 0. Let (V couple as in §7, so that the corresponding pressure function P is finite on a neighborhood of (t, p) = (t0 , P(t0 )), and such that for each t ∈ R close to t0 we have P(t, P(t)) = 0, see §3.4 for the definition of P. Then the implicit function theorem implies that the function, p0 (τ ) := P t0 + τ, P(t0 ) + τ P (t0 ) = P F, −t0 ln F − P(t0 )m − τ ln F + P (t0 )m , defined for τ ∈ R in a neighborhood of t = 0, satisfies p0 (0) = 0. Let ρ be the measure of F for the potential −t0 ln F − P(t0 )m and equilibrium put ψ = − ln F − P (t0 )m. Since for each t close to t0 we have P(t, P(t)) = 0, the implicit function theorem gives p0 (0) = 0. Thus, by Part 1 of Lemma 4.4 and [MU03, Prop. 2.6.13] we have

ψdρ =

− ln F − P (t0 )mdρ = p0 (0) = 0,

see also Remark 3.4. On the other hand, by Part 1 of Lemma 4.4 and [MU03, Prop. 2.6.14]

0=

p0 (0)

=

# +∞

"2 $

!

ψ ◦ F · ψdρ − k

ψdρ

,

k=0

is the asymptotic variance of ψ with respect to ρ, see also Remark 3.4. By Part 1 of Lemma 4.4 and [MU03, Lemma 4.8.8] it follows that there is a measurable function u : J (F) → R such that ψ = u ◦ F − u, see also Remark 3.4. Put J := {z ∈ C\K (V ) | f m(z) (z) ∈ J (F)} and extend u to a function defined on J, that for each z ∈ J\J (F) it is given by, m(z)−1 u(z) = u f m(z) (z) − − ln f f j (z) − P (t0 ) . j=0

An argument similar to the construction of the conformal measure given in the proof of Proposition 4.3, shows that we have ln f = −P (t0 ) + u ◦ f − u on J; see also [PRL07, Prop. B.2]. By construction this last set has full measure with respect to the equilibrium state of f for the potential −t0 ln f , cf. §4.3. Thus, an argument similar to the proof of [Zdu90, §§5–8] (see also [MS00, §3.8] or [May02, Theorem 3.1]) implies that f is a power, Chebyshev or Lattès map.


703

B.1. Dimension spectrum for Lyapunov exponents. Let f be a rational map of degree at least two. For z ∈ C we define 1 n ln f (z) , χ (z) = lim n→+∞ n whenever the limit exists; it is called the Lyapunov exponent of f at z. The dimension spectrum for Lyapunov exponents is the function L : (0, +∞) → R defined by, L(α) := HD ({z ∈ J ( f ) | χ (z) = α}) . Following [MS96] we will say that f is exceptional if there is a finite subset of C such that f −1 ()\ Crit( f ) = ,

(B.1)

see also [MS00, §1.3]. A rational map f is exceptional if and only if t− > −∞. Furthermore, in this case there is a set f containing at most four points such that (B.1) is satisfied with = f , and such that each finite set satisfying (B.1) is contained in f . Power, Chebyshev and Lattès maps are all exceptional. See [MS96] for other examples of exceptional rational maps. It has been recently shown in [GPR08, Theorem 2] that if f is not exceptional, or if f is exceptional and f ∩ J ( f ) = ∅, then for each α ∈ (0, +∞) we have 1 inf{P(t) + αt | t ∈ R}. α Equivalently, the functions α → −αL(α) and s → P(−s) form a Legendre pair. Note that a Chebyshev or a Lattès map f is exceptional and f intersects J ( f ). The following is a direct consequence of Theorem D. L(α) =

Corollary B.1. Let f be a rational map satisfying the hypotheses of Theorem B. Suppose furthermore that f is not conjugated to a power map, and that either f is not exceptional, or that f is exceptional and f is disjoint from J ( f ). Then the dimension ∗ , χ ∗ ). spectrum for Lyapunov exponents of f is real analytic on (χinf sup B.2. Dimension spectrum for pointwise dimension. Let ρ0 be the unique measure of maximal entropy of f . Then for z ∈ J ( f ) we define ln ρ0 (B(z, r )) , r →0 ln r whenever the limit exists; it is called the pointwise dimension of ρ0 at z. The dimension spectrum for pointwise dimensions is defined as the function α(z) := lim+

D(α) := HD({z ∈ J ( f ) | α(z) = α}). When f is a polynomial with connected Julia set we have P (0) = − ln deg( f ), so by [MS00, §5] it follows that for α ≤ 1 we have, ' & P(t) |t ≤0 . D(α) = inf t + α ln deg( f ) Equivalently, the function β → −β D β1 on β ≥ 1 and the function s →

(ln deg( f ))−1 P(−s) on s ≥ 0 form a Legendre pair. So the following is a direct consequence of Theorem B and Theorem D.

704


Corollary B.2. Let f be a polynomial with connected Julia set satisfying the hypotheses of Theorem B. If f is not a power or Chebyshev map, then the dimension spectrum for pointwise dimensions of the maximal entropy measure of f is real analytic on α < 1. Remark B.3. In the uniformly hyperbolic case one has D(α) = L(ln deg( f )/α).

(B.2)

This also holds when the set of those z ∈ J ( f ) for which χ (z) exists and satisfies χ (z) ≤ 0 has Hausdorff dimension equal to 0, like for rational maps satisfying the TCE condition [PRL07, §1.4]. In fact, it is easy to see that for z ∈ J ( f ) belonging to the “conical Julia set” and for which both α(z) and χ (z) exists, and χ (z) > 0, we have α(z) = ln deg( f )/χ (z). Then (B.2) follows from [GPR08, Prop. 3], that the set of those z ∈ J ( f ) that are not in the conical Julia set and χ (z) > 0 has Hausdorff dimension equal to 0.

B.3. Large deviations. The purpose of this section is to present a sample application of Theorem D to level-1 large deviations, using the characterizations of the pressure function given in [PRLS04]. See [CRL08] and references therein for some level-2 large deviation principles for rational maps. Corollary B.4. Let f be a rational map satisfying the hypotheses of Theorem B, and that is not conjugated to a power, Chebyshev, or Lattès Fix t0 ∈ (t− , t+ ) and let ρt0 map. be the equilibrium state of f for the potential −t0 ln f . Fix x0 ∈ J ( f ) such that (2.1) holds, and for each n ≥ 1 put n ( f ) (x)−t0 ωn := δx . ( f n ) (y)−t0 −n −n x∈ f

(x0 )

y∈ f

(x0 )

∗ , let t (ε) ∈ (t , t ) be determined by P (t (ε)) = P (t )−ε. Given ε ∈ 0, −P (t0 ) − χinf − 0 0 Then we have,

& ' 1 j 1 lim ln ωn x ∈ J ( f ) ln f (x) > ln f dρt0 + ε n→+∞ n n = P(t (ε)) − P(t0 ) − (t (ε) − t0 )P (t (ε)) < 0. ∗ + P (t ) let t˜(ε) ∈ (t , t ) be determined by P t˜(ε) = Similarly, given ε ∈ 0, χsup 0 0 + P (t0 ) + ε. Then we have,

& ' 1 1 ln ωn x ∈ J ( f ) ln f j (x) < ln f dρt0 − ε lim n→+∞ n n = P t˜(ε) − P(t0 ) − t˜(ε) − t0 P t˜(ε) < 0. For a rational map satisfying the TCE condition, or the weaker “Hypothesis H” of [PRLS04], a similar result can be obtained for periodic points. See [Com09] and references therein for analogous statements in the case of uniformly hyperbolic rational maps, and [KN92] for similar results in the case of Collet-Eckmann unimodal maps and t0 near 1.


705

Proof. First observe that by the choice of x0 , for each s ∈ R we have −t0 +s n

1 1 x∈ f −n (x0 ) ( f ) (x) n ln exp s ln f dωn = lim ln lim −t0 n→+∞ n n→+∞ n n y∈ f −n (x0 ) ( f ) (y) = P(t0 − s) − P(t0 ).

( We will apply the theorem in p. 343) of [PS75] to the space J := +∞ n=1 J ( f ) endowed ω . Furthermore for each n ≥ 1 we take the with the probability measure P := +∞ n n=1 n (+∞ random variable Wn : J → R as Wn x j := ln ( f ) (xn ) . So for each s ∈ R j=1

we have

exp(sWn )dP =

exp s ln f n dωn ,

and by the computation above,

1 ln exp(sWn )dP = P(t0 − s) − P(t0 ). lim n→+∞ n Using that ln ( f n ) dρt0 = −P (t0 ) and that the function s → P(t0 − s) − P(t0 ) is real analytic and strictly convex on (t0 − t+ , t0 − t− ) by Theorem D, we obtain by the theorem in p. 343 of [PS75] that & '

1 n n 1 ln ωn n ln f > ln f dρt0 + ε lim n→+∞ n = P(t (ε)) − P(t0 ) − (t (ε) − t0 )P (t (ε)). The second assertion n is obtained analogously with Wn replaced by the function (+∞ ( f ) (xn ). x → − ln j j=1

References [AL08] [Bal00] [BK98] [BMS03] [BS96] [BS05] [BT08] [BT09] [CL09] [Com09]

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[CRL08] [DMNU98] [Dob08] [Dob09] [DU91] [DZ98] [Erë91] [Fis06] [FLM83] [Gou04] [Gou05] [GPR08] [GS09] [GW07] [IT09] [KL08] [KN92] [KvS09] [Led84] [Lju83] [LS09] [Mak98] [May02] [McM94] [McM00] [MN05] [MN08] [MS96] [MS00] [MS03]


Comman, H., Rivera-Letelier, J.: Large deviation principles for non-uniformly hyperbolic rational maps. arXiv:0812.4761v1, to appear in Ergodic Theory Dynam. Systems, doi:10. 1017/s0143385709001163, 2010 Denker, M., Mauldin, R.D., Nitecki, Z., Urbański, M.: Conformal measures for rational functions revisited. Fund. Math. 157(2-3), 161–173 (1998) Dobbs, N.: Measures with positive lyapunov exponent and conformal measures in rational dynamics. http://arXiv.org/abs/0804.3753v2 [math.DS], 2010, to appear in Trans. Amer. Math. Soc. Dobbs, N.: Renormalisation-induced phase transitions for unimodal maps. Commun. Math. Phys. 286(1), 377–387 (2009) Denker, M., Urbański, M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4(1), 103–134 (1991) Dembo, A., Zeitouni, O.: Large deviations techniques and applications, Volume 38 of Applications of Mathematics (New York). New York: Springer-Verlag, second edition, 1998 Erëmenko, A.È.: Lower estimate in Littlewood’s conjecture on the mean spherical derivative of a polynomial and iteration theory. Proc. Amer. Math. Soc. 112(3), 713–715 (1991) Fisher, T.: Hyperbolic sets that are not locally maximal. Erg. Th. Dyn. Syst. 26(5), 1491– 1509 (2006) Freire, A., Lopes, A., Mañé, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14(1), 45–62 (1983) Gouezël, S.: Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes. PhD thesis, 2004 Gouëzel, S.: Berry-Esseen theorem and local limit theorem for non uniformly expanding maps. Ann. Inst. H. Poincaré Probab. Statist. 41(6), 997–1024 (2005) Gelfert, K., Przytycki, F., Rams, M.: Lyapunov spectrum for rational maps. Math. Ann. 348, 965–1004 (2010) Graczyk, J., Smirnov, S.: Non-uniform hyperbolicity in complex dynamics. Invent. Math. 175(2), 335–415 (2009) Gelfert, K., Wolf, C.: Topological pressure for one-dimensional holomorphic dynamical systems. Bull. Pol. Acad. Sci. Math. 55(1), 53–62 (2007) Iommi, G., Todd, M.: Natural equilibrium states for multimodal maps. Commun. Math. Phys. 300, 65–94 (2009) Kahn, J., Lyubich, M.: A priori bounds for some infinitely renormalizable quadratics. II. Decorations. Ann. Sci. Éc. Norm. Supér. (4) 41(1), 57–84 (2008) Keller, G., Nowicki, T.: Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Commun. Math. Phys. (4) 149(1), 31–69 (1992) Kozlovski, O., van Strien, S.: Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. Proc. Lond. Math. Soc. (3) 99(2), 275–296 (2009) Ledrappier, F.: Quelques propriétés ergodiques des applications rationnelles. C. R. Acad. Sci. Paris Sér. I Math. 299(1), 37–40 (1984) Ljubich, M.Ju.: Entropy properties of rational endomorphisms of the Riemann sphere. Erg. Th. Dyn. Systs. 3(3), 351–385 (1983) Li, H., Shen, W.: On non-uniformly hyperbolicity assumptions in one-dimensional dynamics. Science in China: Math. 53(7), 1663–1677 (2010) Makarov, N.G.: Fine structure of harmonic measure. Algebra i Analiz 10(2), 1–62 (1998) Mayer, V.: Comparing measures and invariant line fields. Erg. Th. Dyn. Syst. 22(2), 555– 570 (2002) McMullen, C.T.: Complex dynamics and renormalization, Volume 135 of Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 1994 McMullen, C.T.: Hausdorff dimension and conformal dynamics. II. geometrically finite rational maps. Comment. Math. Helv. 75(4), 535–593 (2000) Melbourne, I., Nicol, M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Commun. Math. Phys. 260(1), 131–146 (2005) Melbourne, I., Nicol, M.: Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360(12), 6661–6676 (2008) Makarov, N., Smirnov, S.: Phase transition in subhyperbolic Julia sets. Erg. Th. Dyn. Syst. 16(1), 125–157 (1996) Makarov, N., Smirnov, S.: On “thermodynamics” of rational maps. I. Negative spectrum. Commun. Math. Phys. 211(3), 705–743 (2000) Makarov, N., Smirnov, S.: On thermodynamics of rational maps. II. Non-recurrent maps. J. London Math. Soc. (2) 67(2), 417–432 (2003)


[MU03] [Pes97] [PR98] [PRL07] [PRLS03] [PRLS04] [Prz90] [Prz98] [Prz99] [PS75] [PS08] [PU02] [QY09] [RL07] [RLS10] [Rue92] [Rue04] [SU03] [TK05] [Urb03] [You99] [Zdu90] [Zin96] [Zwe05]

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Communicated by S. Smirnov


Communications in


A∞ -Algebra of an Elliptic Curve and Eisenstein Series Alexander Polishchuk Department of Mathematics, University of Oregon, Eugene, OR 97403, USA. E-mail: [email protected] Received: 1 February 2010 / Accepted: 30 July 2010 Published online: 15 November 2010 – © Springer-Verlag 2010

Abstract: We compute explicitly the A∞ -structure on the algebra Ext ∗ (OC ⊕ L , OC ⊕ L), where L is a line bundle of degree 1 on an elliptic curve C. The answer involves higher derivatives of Eisenstein series. 0. Introduction The bounded derived category D b (X ) of coherent sheaves is an important invariant of an algebraic variety X (see [1] for a survey). This category can be described in a purely algebraic way if one has a generator, i.e., an object G such that the smallest full triangulated subcategory containing G and closed under passage to direct summands is the entire D b (X ). With such an object one can associate a graded algebra E G = ⊕n∈Z Ext n (G, G). However, in order to recover D b (X ) from E G one has to take into account certain higher products which fit together into a structure of an A∞ − algebra on E G (see [3] for an introduction into A∞ -algebras). Namely, one can realize E G as the cohomology of a dg-algebra and then apply a general algebraic construction that gives an A∞ -structure on such cohomology (see [7]). This A∞ -structure is minimal in the sense that m 1 = 0, and is canonical up to A∞ -equivalence. Now the category D b (X ) can be shown to be equivalent to the derived category of perfect A∞ -modules over A (see [4, Thm. 3.1], [6, Sec. 7.6]). Thus, it is of interest to compute explicitly higher products on algebras of the form Ext ∗ (G, G) as above. In this paper we solve this problem in the case when X is a complex elliptic curve and G = O X ⊕ L, where L is a line bundle of degree 1. Namely, we compute the A∞ -structure arising from the harmonic representatives (with respect to natural metrics) in the Dolbeault complex computing Ext∗ (G, G). The resulting formulas involve higher derivatives of Eisenstein series (see Theorem 2.5.1). More precisely, we have to use the well-known non-holomorphic (but modular) modification e2∗ of the standard Eisenstein series e2 along with all the higher Eisenstein series e2k , k ≥ 2. Supported in part by NSF grant.

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It is interesting that Eisenstein series appear not in their usual form but rather as some rapidly decreasing series, similar to those considered in [9] (see Theorem 1.2.1). The A∞ -constraint gives rise to some quadratic relations involving derivatives of Eisenstein series, some of them well known (see Proposition 2.6.1). 1. Eisenstein Series 1.1. Definitions. Let us recall some basic definitions and facts (see [10, Ch. III]). We consider C ∞ -functions F(ω1 , ω2 ) on the space of all oriented bases of C = R2 . Recall that such a function is said to be of weight k ∈ Z if F(λω1 , λω2 ) = λ−k F(ω1 , ω2 ). Such F is called modular (with respect to SL(2, Z)) if it is invariant with respect to SL(2, Z) base changes of (ω1 , ω2 ) and F(1, τ ) = f (e2πiτ ), where f (q) is meromorphic at q = 0. The Eisenstein series e2k for k ≥ 2 is defined by

e2k (ω1 , ω2 ) =

ω∈L\{0}

1 , ω2k

where L = Zω1 + Zω2 . The function e2k is modular of weight 2k. One can also consider the analogous series for k = 1 using Eisenstein’s summation rule: e2 (ω1 , ω2 ) =

m n;n=0 if m=0

1 . (mω2 + nω1 )2

The function e2 is not modular but admits a simple non-holomorphic correction that makes it SL(2, Z)-invariant. Namely, let us set e2∗ (ω1 , ω2 ) = e2 (ω1 , ω2 ) −

π ω¯1 , a(L) ω1

where a(L) = Im(ω1 ω2 ) is the area of C/L. Then e2∗ (ω1 , ω2 ) is SL(2, Z)-invariant (and ∗ = e for k ≥ 2. Eisenstein series appear of weight 2). For convenience we set also e2k 2k as coefficients in the expansion of the Weierstrass zeta-function (cf. [10, Ch. III, formula (9)]): 1 ζ (z; ω1 , ω2 ) = − e2k (ω1 , ω2 )z 2k−1 . (1.1.1) z k≥2

Following [2, Sec. 1.5] we consider the normalized Weil operator π ∂ ∂ . W =− ω1 + ω2 a(L) ∂ω1 ∂ω2 This operator is SL(2, Z)-invariant and is of weight two. Slightly modifying the definition in [10, Ch. VI] for a pair of integers b > a ≥ 0 of different parity we set ∗ ga,b = gb,a = (b − a)!W a (eb−a+1 ). ∗ Note that (a(L)/π )a · ga,b differs by a rational factor from Weil’s ea,b+1 . As shown in ∗ [10, Sec. VI.5], ga,b is a polynomial in e2 , e4 , . . . , ea+b+1 with rational coefficients.

A∞ -Algebra of an Elliptic Curve and Eisenstein Series

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1.2. Presentation by rapidly decreasing series. For m, n ∈ Z and a lattice L ⊂ C we set m ω¯ m π π 2 f m,n (L) = . |ω| · exp − a(L) ωn a(L) ω∈L\{0}

Note that f m,n = 0 unless m+n is even (as one can see making the substitution ω → −ω). Theorem 1.2.1. For n = 2k ≥ 2 one has

π n Pn−1 a(L) |ω|2 1 2 π 2 en∗ = exp − f n−1,1 + f n−m,m = |ω| , (n − 1)! (n − m)! ωn a(L) ω=0

m=2

where z n−m 2z n−1 + . (n − 1)! (n − m)! n

Pn−1 (z) =

m=2

Proof. This follows easily from Theorem 1 of [9] stating that π exp − a(L) |ω + z|2 ζ (z; ω1 , ω2 ) − z 1 η1 − z 2 η2 = ω+z ω∈L Im(ωz) π exp − a(L) |ω|2 + 2πi a(L) , − ω ω∈L\{0}

where z = z 1 ω1 + z 2 ω2 with z 1 , z 2 ∈ R. Indeed, we can use the expansion (1.1.1) to check the assertion for n ≥ 4: one has to subtract 1/z from both parts of the above n−1 ∂ and then evaluate at z = 0. The case n = 2 is slightly different: identity, apply ∂z we again subtract 1/z from both parts, then apply ∂z∂ 1 (taking (z 1 , z 2 ) as independent variables) and evaluate at z = 0. The required formula follows the fact that e2 = −η1 /ω1 (see e.g., [2, Sec. 1.2]).

Remark. The fact that en∗ is holomorphic in (ω1 , ω2 ) for n > 2 is equivalent to the identity 2 f n−1,−1 = (n − 1) f n−2,0 . For n = 2 we have instead 2 f 1,−1 = f 0,0 + 1. These identities can be derived π from the Poisson summation formula and Fourier self-duality of exp − a(L) |z|2 (see [9, Sec. 1.1, Remark 1]; for n > 2 one also has to use differentiation). One can immediately check that W ( f m,n ) = f m+2,n + n f m+1,n+1 . Hence, from Theorem 1.2.1 we get the following formula for ga,b . Corollary 1.2.2. For a pair of integers a, b ≥ 0 of different parity one has a b ga,b = k! + f a+b−k,k+1 . k k k≥0

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2. Minimal A∞-Algebra of an Elliptic Curve 2.1. General construction. Let us first recall the general construction of the A∞ structure on the cohomology of a dg-algebra (A, d) equipped with a projector : A → B onto a subspace of ker(d) and a homotopy operator Q such that 1 − = d Q + Qd. Merkulov’s formula for this A∞ -structure (see [7]) was rewritten in [5] as a sum over trees: m n (b1 , . . . , bn ) = − (T )m T (b1 , . . . , bn ). T

Here T runs over all oriented planar rooted 3-valent trees with n leaves (different from the root) marked by b1 , . . . , bn left to right, and the root marked by (we draw the tree in such a way that leaves are above, and every vertex has two edges coming from above and one from below). The expression m T (b1 , . . . , bn ) is obtained by going down from leaves to the root, applying the multiplication in A at every vertex and applying the operator Q at every inner edge (see [5, Sec. 6.4] for details). The sign (T ) has form (T ) = (−1)|e1 (v)|+(|e2 (v)|−1) deg(e1 (v)) , v

where v runs through vertices of T, (e1 (v), e2 (v)) is the pair of edges above v, for an edge e we denote by |e| the total number of leaves above e and by deg(e) the sum of degrees of all leaves above e (recall that leaves are marked by bi ). Lemma 2.1.1. Assume in addition that Q = Q = Q 2 = 0. Let (b1 , . . . , bn ) be a collection of elements in B, where n ≥ 3, such that bi = 1 for some i. Then m n (b1 , . . . , bn ) = 0. Proof. It is convenient to use Merkulov’s original formula m n (b1 , . . . , bn ) = λn (b1 , . . . , bn ), where λn : A⊗n → A are defined for n ≥ 2 by the following recursion: λ2 (a1 , a2 ) = a1 a2 , λn (a1 , . . . , an ) = ± Q(λn−1 (a1 , . . . , an−1 )) · an ± a1 · Q(λn−1 (a2 , . . . , an )) + ±Q(λk (a1 , . . . , ak )) · Q(λl (ak+1 , . . . , an )). k+l=n;k,l≥2

Since, Q = 0, it is enough to prove that λn (b1 , . . . , bn ) ∈ Q(A). Let us use induction in n. In the case n = 3 we have λ3 (b1 , b2 , b3 ) = Q(b1 b2 )b3 ± b1 Q(b2 b3 ) and the assertion follows immediately from the fact that Q(B) = 0. Suppose now that n ≥ 4 and the assertion holds for all n < n. Since Q 2 = 0, the induction assumption easily implies that the first two terms in the recursive formula for λn belong to Q(A). Similarly, all the remaining terms vanish if n ≥ 5. In the case n = 4 the term Q(b1 b2 ) · Q(b3 b4 ) also vanishes because either b1 b2 ∈ B or b3 b4 ∈ B and Q(B) = 0.


713

2.2. The case of an elliptic curve. Let C = C/(Z ⊕ Zτ ) be a complex elliptic curve. We denote by L the holomorphic line bundle of degree 1 on C, such that the classical theta-function θ (z, τ ) descends to a global section of L. We consider the Dolbeault dg-algebra A = ( 0,∗ ⊗ End(OC ⊕ L), ∂). Its cohomology B is the direct sum of the following components: (i) (ii) (iii) (iv)

Hom(O, O) and Hom(L , L), both generated by identity maps; Hom(O, L), one-dimensional space; Ext 1 (L , O), one-dimensional space; Ext 1 (O, O) and Ext1 (L , L), both isomorphic to the one-dimensional space H 1 (O).

To construct the homotopy operator Q, as in [8], we use the flat metric on C and the hermitian metric on L given by y2 ( f, g) = d xd y, f (z)g(z) exp −2π Im(τ ) C ∗

where z = x + i y. Then we set Q = ∂ G, where G is the Green operator corresponding ∗ ∗ to the Laplacian ∂ ∂ + ∂∂ . Then B ⊂ A is exactly the space of harmonic forms, and

: A → B is the orthogonal projection. Let us fix the following harmonic generators in the above components: (i) idO and id L ; (ii) θ = θ (z, τ ) viewed as a holomorphic section of L; √ Im(z) dz viewed as a (0, 1)-form with values (iii) η := 2 Im(τ ) · θ (z, τ ) exp −2π Im(τ )2 in L −1 ; (iv) ξ = dz. When it is viewed as an element of Ext 1 (L , L) we write ξ L .

Note that we have a natural symmetric bilinear pairing on A = A0 ⊕ A1 given by 1 α, β = · Tr(α ◦ β) ∧ dz, 2i Im(τ ) C where α and β are homogeneous elements such that deg(α) + deg(β) = 1. The normalization is chosen in such a way that 1 ξ, 1 = · dz ∧ dz = 1. 2i Im(τ ) C By Serre duality, the induced pairing between B 0 and B 1 is nondegenerate. Also, by [8, Thm. 1.1], the A∞ -structure on B satisfies the following cyclic symmetry: m n (α1 , . . . , αn ), αn+1 = (−1)n(deg(α1 )+1) α1 , m n (α2 , . . . , αn+1 ).

(2.2.1)

The product m 2 on B is just the induced product on cohomology. The only interesting products are m 2 (θ, η) ∈ Ext 1 (O, O) and m 2 (η, θ ) ∈ Ext 1 (L , L). Both are proportional to the generator ξ . To find the coefficient of proportionality it is enough to compute 1 m 2 (θ, η), idO = m 2 (η, θ ), id L = · θ · η ∧ dz. 2i Im(τ ) C

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The above integral is well known: √ 2 Im(τ ) Im(z) dz ∧ dz = 1. · θ (z, τ )θ (z, τ ) exp −2π 2i Im(τ ) Im(τ )2 C Thus, we obtain m 2 (θ, η) = ξ, m 2 (η, θ ) = ξ L .

(2.2.2)

By Lemma 2.1.1, every higher product m n containing idO or id L vanishes. Together with the cyclic symmetry (2.2.1) this implies that the only potentially nonzero higher products are of the following types: (I) (II) (III) (IV)

m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ) ∈ Hom(O, L), m n ((ξ L )a , η, (ξ )b , θ, (ξ L )c , η, (ξ )d ) ∈ Ext 1 (L , O), m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d , η, (ξ )e ) ∈ Hom(O, O), m n ((ξ L )a , η, (ξ )b , θ, (ξ L )c , η, (ξ )d , θ, (ξ L )e ) ∈ Hom(L , L),

where we denote by (ξ )a the string (ξ, . . . , ξ ) with ξ repeated a times. By the cyclic symmetry (2.2.1), we have m n ((ξ L )a , η, (ξ )b , θ, (ξ L )c , η, (ξ )d ), θ = m n ((ξ )b , θ, (ξ L )c , η, (ξ )d , θ, (ξ L )a ), η, m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d , η, (ξ )e ), ξ = m n ((ξ )a+e+1 , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ), η, m n ((ξ L )a , η, (ξ )b , θ, (ξ L )c , η, (ξ )d , θ, (ξ L )e ), ξ L = m n ((ξ )b , θ, (ξ L )c , η, (ξ )d , θ, (ξ L )a+e+1 ), η. Hence, it is enough to compute the products of type (I), i.e., the coefficients m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ), η. 2.3. Calculation I: combinatorial part. We start by computing some signs (T ). For a pair of oriented planar rooted 3-valent trees T1 , T2 , let us denote by join(T1 , T2 ) the tree (of the same type) obtained by joining together the roots of T1 and T2 and adding a root to the obtained new vertex (we keep T1 on the left from T2 in the plane). Lemma 2.3.1. Let T = join(T1 , T2 ), where each Ti has n i + 1 leaves (i = 1, 2), exactly one of which is marked by a degree 0 element, and the rest marked by degree 1 elements. Assume also that in each Ti no two leaves of degree 1 can be attached to the same vertex. Then (T ) = (−1)(

n 1 +n 2 +2 2

)+n 2 .

Proof. By definition, (T ) = (T1 )(T2 ) · (−1)(n 1 +1)+n 1 n 2 , so it remains to compute (Ti ) for i = 1, 2. Note that under our assumptions each tree Ti has a very simple structure: it has the main stem from the root to the leaf of degree 0, to which leaves of degree 1 can attach on the left and on the right:


715

deg=1

deg=1 deg=0

Ti

Suppose we have a pair of consecutive vertices v and w on the stem (with v above w) such that there is a degree 1 leaf attaching to v on the left and a degree 1 leaf attaching to w on the right. Let a be the number of degree 1 leaves above v. Then the contribution of v into the product defining (Ti ) is equal to (−1)a+1 , while the contribution of w is equal to (−1)a+2 . Hence, the contribution of both v and w is −1. It is easy to check that if T is the tree obtained from T by reversing the order of attaching these two leaves at vertices v and w then the contribution of these vertices into (T ) will still be −1. Assume that Ti has exactly a (resp., b) leaves to the left (resp., to the right) of the degree 0 leaf, and let v1 , . . . , va (resp., w1 , . . . , wb ) be the vertices on the stem to which they attach. By the above observation, it is enough to consider the case when all the vertices v1 , . . . , va are above w1 , . . . , wb :

a

b deg=0

Ti

Then one can easily calculate that (Ti ) = (−1)(

).

a+b+1 2

Hence, n 1 +1 n 2 +1 n 1 +n 2 +2 (T ) = (−1)( 2 )+( 2 )+(n 1 +1)+n 1 n 2 = (−1)( 2 )+n 2 .

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Next, we consider the terms m T ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ). Note that if two leaves of degree 1 in T are attached to the same vertex then m T vanishes because it will involve taking products of two elements of degree 1 in A. Henceforward, we assume that no two leaves of degree 1 can be attached to the same vertex. It is convenient to introduce the operator HL : C ∞ (L) → C ∞ (L) : s → Q(s · dz) and a similar operator HO on C ∞ -functions. The next result follows immediately from the definition. Lemma 2.3.2. (i) Let T = join(T1 , T2 ), where the leaves of T1 are marked (from left to right) by (ξ )a , θ, (ξ L )b , η, (ξ )c1 , while the leaves of T2 are marked by (ξ )c2 , θ, (ξ L )d . Let a = a1 + a2 , where a1 leftmost leaves of T1 get attached to the main stem of T1 below the vertex where η attaches to the main stem, and the next a2 leaves get attached to the stem above this vertex: a1

Then

a2

b

c1

c2

d

m T (ξ )a , θ, (ξ L )b , η, (ξ )c1 +c2 , θ, (ξ L )d

a1 +c1 = HO Q HLa2 +b (θ ) · η · HLc2 +d (θ ) .

(ii) Let T = join(T1 , T2 ), where the leaves of T1 are marked by (ξ )a , θ, (ξ L )b1 , while the leaves of T2 are marked by (ξ L )b2 , η, (ξ )c , θ, (ξ L )d . Let d = d1 + d2 , where exactly d2 rightmost leaves of T2 get attached to the main stem of T2 below the vertex where η attaches to the main stem. Then m T (ξ )a , θ, (ξ L )b1 +b2 , η, (ξ )c , θ, (ξ L )d b2 +d2 = HLa+b1 (θ ) · HO Q HLc+d1 (θ ) · η .


717

It is easy to see that all the trees giving a nonzero contribution to m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ) are of the form considered in Lemma 2.3.2. Hence, combining Lemmas 2.3.1 and 2.3.2 we arrive at the following expression for this higher product. Lemma 2.3.3. One has m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ), η n a 2 + b a 1 + c 1 c2 + d +1 ) ( 2 · φ(a2 + b, a1 + c1 , c2 + d) = (−1) a1 a2 c2 a=a1 +a2 ;c=c1 +c2 n c + d1 b2 + d2 a + b1 +n+1 ) ( 2 +(−1) · φ(c + d1 , b2 + d2 , a +b1 ), c b2 a b=b1 +b2 ;d=d1 +d2

where n p Q(HLm (θ ) · η) · HL (θ ) , η . φ(m, n, p) = (−1) p HO 2.4. Calculation II: analytic part. We will use real coordinates u, v on C such that z = u + vτ . Let us also set a = Im(τ ). Lemma 2.4.1. Consider the differential operator D = − πa one has: HLk θ =

∂ ∂z

− 2iav. Then for k ≥ 0

1 (−2ia)k · Dk θ = · (n + v)k exp(πiτ n 2 + 2πinz), k! k!

(2.4.1)

n∈Z

√

2a(D k θ ) · θ exp(−2πav 2 ) =

(−1)mn (mτ − n)k

(m,n)∈Z2

π × exp − |mτ − n|2 + 2πi(mu + nv) . 2a

(2.4.2)

Proof. The case k = 0 of the identity (2.4.2) is well-known (see [8, Eq. (2.2)]). The general case follows easily by applying D k . Next, let us prove (2.4.1) by induction in k. Recall that the operator Q : 0,1 (L) → 0,0 (L) is uniquely determined by the following two properties: ∂ ◦ Q = id and the image of Q is orthogonal to θ . Thus, HLk θ is D θ equal to the unique function f such that ∂∂zf = (k−1)! and ( f, θ ) = 0. Thus, it is enough to check the identities (n + v)k exp(πiτ n 2 + 2πinz), (2.4.3) D k θ = (−2ia)k · k−1

n∈Z

∂ k D θ = k D k−1 θ, ∂z (D k θ, θ ) = 0

(2.4.4)

for k > 0. The latter follows immediately from the Fourier expansion (2.4.2), since (D k θ, θ ) is proportional to the Fourier coefficient of this expansion corresponding to (m, n) = (0, 0). To check (2.4.3) one can apply D to the similar expansion for D k−1 θ . ∂ to the right-hand side of (2.4.3). Finally, one checks (2.4.4) by applying ∂z

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Now we can calculate the expressions φ(m, n, p) (see Lemma 2.3.3). Lemma 2.4.2. One has φ(k, l, p) =

1 a l+1 · k! p! π

ω∈Z+Zτ \{0}

π ωk+ p 2 |ω| exp − ωl+1 a

1 a k+l+ p+1 = f k+ p,l+1 (Z + Zτ ). k! p! π Proof. It is easy to see that the operator Q : 0,1 → 0,0 is given by a π(mτ −n) exp(2πi(mu + nv)), (m, n) = (0, 0), Q(exp(2πi(mu + nv))dz) = 0 (m, n) = (0, 0). Hence, using Lemma 2.4.1 we obtain 1 √ Q( 2a(D k θ ) · θ exp(−2πav 2 )) k! π (mτ − n)k exp − |mτ − n|2 + 2πi(mu + nv) . (−1)mn (mτ − n) 2a 2

Q(HLk (θ ) · η) = =

a · k!π

(m,n)∈Z \{(0,0)}

Therefore, 1 a l+1 l Q HLk (θ ) · η = HO k! π π (mτ − n)k 2 |mτ − n| · (−1)mn exp − + 2πi(mu + nv) . (mτ − n)l+1 2a 2 (m,n)∈Z \{(0,0)}

(2.4.5) Next, comparing the formulas for η and for the metric on L we observe that for a C ∞ -section f of L one has ( f, θ ) = 0 if and only if f, η = 0. Hence, for f ∈ C∞ (L) one has ( f ), η = f, η (since is the orthogonal projection onto Cθ ). Therefore,

p l φ(k, l, p) = (−1) p HO Q(HLk (θ ) · η) · HL (θ ), η p l = (−1) p HO Q(HLk (θ ) · η), HL (θ ) · η . p

Now the right-hand side can be computed using the Fourier expansion for HL (θ ) · η from Lemma 2.4.1 and the Fourier expansion (2.4.5): φ(k, l, p) =

1 a l+1 · k! p! π

(m,n)∈Z2 \{(0,0)}

π (mτ − n)k+ p 2 |mτ − n| . exp − (mτ − n)l+1 a


719

2.5. Calculation III: conclusion. It remains to put everything together. Substituting the expressions for φ(k, l, p) found in Lemma 2.4.2 into the formula of Lemma 2.3.3 we get m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ), η n Im(τ ) n−2 (a1 + c1 )! f a +c +b+d,a1 +c1 +1 = (−1)(2)+1 · π a1 !a2 !c1 !c2 !b!d! 2 2 a=a1 +a2 ;c=c1 +c2 n−2 n Im(τ ) (b2 + d2 )! f b +d +a+c,b2 +d2 +1 . + (−1)(2)+n+1 · π b1 !b2 !d1 !d2 !a!c! 1 1 b=b1 +b2 ;d=d1 +d2

Since f k,l = 0 unless k + l is even, this immediately implies that m n = 0 for odd n. Now assuming that n is even we can rewrite the above equation as follows (denoting k = a1 + c1 and l = b2 + d2 ):

n−2 π · m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ), η Im(τ ) 1 1 = · · C(a, c, k) f a+c−k+b+d,k+1 + C(b, d, l) f b+d−l+a+c,l+1 , b!d! a!c!

(−1)(2)+1 n

k≥0

l≥0

where

C(a, c, k) =

a1 +c1 =k;a1 ≤a,c1 ≤c

k! a + c = a!c! k

a c k! k! = a1 c1 a1 !c1 !(a − a1 )!(c − c1 )! a!c! a1 +c1 =k

Thus, our formula for m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ), η takes the form

n−2 π · m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ), η Im(τ ) a + c b + d 1 · k! + f n−3−k,k+1 . = a!b!c!d! k k

(−1)(2)+1 n

k≥0

Taking into account Corollary 1.2.2 we obtain for even n, n m n (ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d , η = (−1)(2)+1

1 · a!b!c!d!

Im(τ ) π

n−2 · ga+c,b+d .

(2.5.1) Let us summarize our calculations. We set a+b+c+d+1 M(a, b, c, d) := (−1)( 2 )

1 · a!b!c!d!

Im(τ ) π

a+b+c+d+1 · ga+c,b+d .

(2.5.2)

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A. Polishchuk

Theorem 2.5.1. The only non-trivial higher products m n of the A∞ -structure on B = Ext ∗ (O ⊕ L , O ⊕ L) are of the form m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d ) m n ((ξ L )a , η, (ξ )b , θ, (ξ L )c , η, (ξ )d ) m n ((ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d , η, (ξ )e ) m n ((ξ L )a , η, (ξ )b , θ, (ξ L )c , η, (ξ )d , θ, (ξ L )e )

= = = =

M(a, b, c, d) · θ, M(a, b, c, d) · η, M(a + e + 1, b, c, d) · idO , M(a + e + 1, b, c, d) · id L .

All products m n with odd n vanish. Remarks. 1. Assume that τ belongs to the ring of integers of an imaginary quadratic field (so that our elliptic curve admits complex multiplication). Set = 2π |η(q)|2 , where η(q) is the Dedekind’s η-function, and q = exp(2πiτ ). Then the numbers −a−b−1 · ga,b are algebraic over Q (see [10, Sec. VI.6]). Hence, if we multiply our basis elements of degree 1 (η, ξ and ξ L ) by the factor |η(q)|−2 then the structure constants of our A∞ -structure with respect to the new basis will be algebraic over Q. 2. Another meaningful rescaling is obtained if we multiply our basis elements of degree 1 by the factor π/ Im(τ ). Then the new structure constants M (a, b, c, d) will all have limit at the cusp Im(τ ) → +∞. Namely, using the well known q-series for the Eisenstein series, one can easily check that the only nonzero limiting values will be i+ j+1 i+j M (i, 0, j, 0) = M (0, i, 0, j) → (−1)( 2 ) · · 2ζ (i + j + 1), i where i + j is odd.

2.6. A∞ -constraint. The A∞ -axiom (we follow [4, Sec. 3.1] for sign conventions) gives certain quadratic equations on the structure constants (M(a, b, c, d)) and hence leads to identities for gm,n . For example, applying this axiom to the string (ξ )a , θ, (ξ L )b , η, (ξ )c , θ, (ξ L )d , η, (ξ )e , θ, (ξ L ) f , where a, b, c, d, e, f are positive, we get

(−1)(a2 +b+c+d1 +1)(a1 +d2 +e+ f )+a1 M(a2 , b, c, d1 )M(a1 , d2 , e, f )

a=a1 +a2 ;d=d1 +d2

+

(−1)(b2 +c+d+e1 +1)(a+b1 +e2 + f +1)+a+b1 +1 M(b2 , c, d, e1 )M(a, b1 , e2 , f )

b=b1 +b2 ;e=e1 +e2

+

(−1)(c2 +d+e+ f1 +1)(a+b+c1 +1+ f2 )+a+b+c1 M(c2 , d, e, f 1 )M(a, b, c1 , f 2 ) = 0.

c=c1 +c2 ; f = f 1 + f 2

When one of a, b, c, d, e, f is zero, additional terms will arise due to the presence of double products. For example, for the string (ξ )a , θ, η, θ, η, θ, (ξ L )b ,


we get

(−1)(a2 +1)(a1 +b)+a1 M(a2 , 0, 0, 0)M(a1 , 0, 0, b)

a=a1 +a2

+

721

(−1)(b1 +1)(a+b2 +1)+a M(0, 0, 0, b1 )M(a, 0, 0, b2 )

b=b1 +b2

+ (−1)a [M(a + 1, 0, 0, b) − M(a, 1, 0, b) + M(a, 0, 1, b) − M(a, 0, 0, b + 1)] + δb,0 (−1)a M(a + 1, 0, 0, 0) − δa,0 M(b + 1, 0, 0, 0) = 0. Substituting into the above identities the expressions for M(a, b, c, d) from (2.5.2) we arrive at the following result. Proposition 2.6.1. (i) For positive integers a, b, c, d, e, f one has a d ga2 +c,b+d1 ga1 +e,d2 + f (−1)c+d1 +1 a1 d1 a=a1 +a2 ;d=d1 +d2 e b2 +1 b + (−1) gb2 +d,c+e1 ga+e2 ,b1 + f b1 e1 b=b1 +b2 ;e=e1 +e2 c f gc2 +e,d+ f1 ga+c1 ,b+ f2 = 0. + (−1)c1 c1 f1 c=c1 +c2 ; f = f 1 + f 2

(ii) For integers a, b ≥ 0 one has a a + 2 + δb,0 ga1 ,0 ga2 ,b − ga+1,b a+1 a 1 a=a1 +a2 b b + 2 + δa,0 g0,b1 ga,b2 − ga,b+1 . = b1 b+1 b=b1 +b2

Note that the identity in (ii) gives a recursive formula for ga+1,b in terms of all ga ,b ∗ , we recover the fact that all g with a ≤ a. Since g0,n = n!en+1 a,b are polynomials in ∗ (en ) with rational coefficients. For example, in the case a = 0 the obtained identity (for even n) n n+3 g0,m g0,k + gn+1,0 2g1,n = − m n+1 n=m+k

is equivalent to the formula 1 ∗ ∗ W en∗ = − em+1 ek+1 + (n + 3)en+2 n n=m+k

(see [10, VI.5]). Acknowledgements. I am grateful to Alexandr Usnich and Dmytro Shklyarov for helpful discussions. This paper was written during a stay at the Institut des Hautes Études Scientifiques. I’d like to thank that institution for hospitality and support.

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References 1. Bondal, A., Orlov, D.: Derived categories of coherent sheaves. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Beijing: Higher Ed. Press, 2002, pp. 47–56 2. Katz, N.: p-adic interpolation of real analytic Eisenstein series. Ann. Math. 104, 459–571 (1976) 3. Keller, B.: Introduction to A-infinity algebras and modules. Homology Homotopy Appl. 3(1), 1–35 (2001) 4. Keller, B.: A-infinity algebras, modules and functor categories. In: Trends in representation theory of algebras and related topics, Providence, RI: Amer. Math. Soc., 2006, pp. 67–93 5. Kontsevich, M., Soibelman Y.: Homological mirror symmetry and torus fibration. In: Symplectic geometry and mirror symmetry (Seoul, 2000), River Edge, NJ: World Sci. Publishing, 2001, pp. 203–263 6. Lefèvre-Hasegawa, K.: Sur les A∞ -catégories. Thèse de doctorat, Université Denis Diderot – Paris 7, 2003, available at http://www.math.jussieu.fr/~keller/lefevre/publ.html, 2005 7. Merkulov, S.: Strong homotopy algebras of a Kähler manifold. Internat. Math. Res. Notices 3, 153–164 (1999) 8. Polishchuk, A.: Homological mirror symmetry with higher products. In: Proceedings of the Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, Providence, RI: AMS and International Press, 2001, pp. 247–259 9. Polishchuk, A.: Rapidly converging series for the Weierstrass zeta-function and for the Kronecker function. Math. Research Letters 7, 493–502 (2000) 10. Weil, A.: Elliptic functions according to Eisenstein and Kronecker. Berlin-Heidelberg-New York: Springer-Verlag, 1976 Communicated by N.A. Nekrasov


Communications in


A Note on Dimer Models and McKay Quivers Kazushi Ueda1 , Masahito Yamazaki2 1 Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1,

Toyonaka, Osaka 560-0043, Japan. E-mail: [email protected]

2 Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku,

Tokyo 113-0033, Japan. E-mail: [email protected] Received: 5 February 2010 / Accepted: 9 March 2010 Published online: 31 July 2010 – © Springer-Verlag 2010

Abstract: We give one formulation of a procedure of Hanany and Vegh (J High Energy Phys 0710(029):35, 2007) which takes a lattice polygon as an input and produces a set of isoradial dimer models. We study the case of lattice triangles in detail and discuss the relation with coamoebas following Feng et al. (Adv Theor Math Phys 12(3):489–545, 2008). 1. Introduction Dimer models were introduced in the 1930s as a statistical mechanical model for adsorption of di-atomic molecules on the surface of a crystal [8]. A dimer model in this sense is a graph consisting of the set N of nodes and the set E of edges, together with a function E : E → R which represents the energy of adsorption. A perfect matching is a subset D of E such that for any node n ∈ N , there exists a unique edge e ∈ D adjacent to n, and one is interested in the asymptotic behavior of the partition function Z= exp −β E(e) D : a perfect matching

e∈D

as the graph becomes large. Many interesting statistical mechanical models such as domino tilings, Ising models and plane partitions can be realized as a dimer model on a particular planar graph. See e.g. [20] for a review on dimer models. In 1961, Kasteleyn [18] introduced a method to represent the partition function as the determinant of a weighted adjacency matrix, called the Kasteleyn matrix. This is used to great effect by Kenyon, Okounkov and Sheffield [21] to study dimer models on periodic bipartite graph. The determinant of the Kasteleyn matrix with respect to a particular weighting is called the characteristic polynomial, and plays an essential role in their work. It is a Laurent polynomial in two variables whose Newton polygon, called

724

K. Ueda, M. Yamazaki

the characteristic polygon, contains the information of possible asymptotic behavior of the periodic dimer model on the universal cover of the torus. Recent advances in string theory have uncovered a new connection between dimer models and geometry, culminating in the proposal that a bicolored isoradial graph on a torus produces an AdS/CFT dual pair of a toric Sasaki-Einstein 5-manifold and an N = 1 superconformal field theory in four dimensions [9,10,14]. Isoradiality is needed for the existence of R-charges in superconformal field theory, which is dual to volumes of Sasaki-Einstein manifolds [1,3,4,19,23–25]. The toric Calabi-Yau 3-fold, obtained as the metric cone over the Sasaki-Einstein 5-manifold, is determined by the characteristic polygon. The dual superconformal field theory is determined by a quiver with potential, which is obtained as the dual graph of the bicolored graph. Isoradiality implies [2,5,16,27] that the path algebra of this quiver with potential is a Calabi-Yau algebra in the sense of Ginzburg [12], and the toric Calabi-Yau 3-fold can be obtained as the moduli space of representations of the quiver [11,17]. An inverse construction of a bicolored isoradial graph on a torus from a convex lattice polygon is studied by Hanany and Vegh [15]. Following their work, we formulate the linear Hanany-Vegh procedure, which takes a convex lattice polygon as an input and produces a set of bicolored graphs on a torus. The adjective linear comes from the fact that zig-zag paths in the resulting graph behave like lines, which enables one to show the following: Theorem 1.1. Let be a convex lattice polygon and G be a bicolored graph obtained from by the linear Hanany-Vegh procedure. Then G is isoradial and coincides with the characteristic polygon of G up to translation. Although it is difficult to enumerate the output of the linear Hanany-Vegh procedure in general, one can give a complete description when the input is a triangle: Theorem 1.2. If the input of the linear Hanany-Vegh procedure is a lattice triangle, then the output consists of a unique graph which is dual to the McKay quiver. See [33] for another attempt to give a mathematical formulation of the procedure of Hanany and Vegh, and [13,16,32] for an alternative algorithm due to Gulotta. An important aspect of dimer models is their relation with mirror symmetry. The mirror of the toric Calabi-Yau 3-fold is an algebraic curve in (C× )2 , which governs the limit shape of the melting crystal model associated with the bicolored graph [29,30]. The coamoeba of this curve is defined by Passare and Tsikh as its image by the argument map (R/Z)2 ∈

∈

Arg : (C× )2 →

(1.1) 1 (arg(x), arg(y)) , 2π whose behavior is expected by Feng, He, Kennaway and Vafa [7] to be described by the bicolored graph. We show the following in this paper: (x, y) →

Theorem 1.3. Let be a lattice triangle and W be the Laurent polynomial obtained as the sum of monomials corresponding to the vertices of . Then the bicolored graph obtained in Theorem 1.2 is a deformation retract of the coamoeba. A more detailed description of the coamoeba in this case in terms of the McKay quiver of an abelian subgroup of S L 3 (C) is given in Theorem 7.1. The proof is based on

Note on Dimer Models and McKay Quivers

725

the identification of the torus (C× )2 with the dual of a maximal torus of S L 3 (C). This point of view will be used in [34] to study equivariant homological mirror symmetry for two-dimensional toric Fano stacks. The organization of this paper is as follows: In Sect. 2, we recall basic definitions on dimer models. In Sect. 3, we formulate the linear Hanany-Vegh procedure. Theorem 1.1 is proved in Sect. 4. In Sect. 5, we recall the identification of McKay quivers with hexagonal tilings of a real 2-torus and prove Theorem 1.2. We discuss the asymptotic behavior of coamoebas in Sect. 6, and study the case of the sum of three monomials in detail in Sect. 7. 2. Characteristic Polygon of a Dimer Model Let N = Z2 be a free abelian group of rank two and M = Hom(N , Z) be the dual group. Put MR = M ⊗ R and T ∨ = MR /M. We fix the standard orientation and the Euclidean inner product on MR ∼ = R2 . • A graph on T ∨ consists of – a finite subset N ⊂ T ∨ called the set of nodes, and – another finite set E called the set of edges, consisting of continuous maps e : [0, 1] → T ∨ , such that e(0) ∈ N, e(1) ∈ N, e((0, 1)) ∩ N = ∅, and e((0, 1)) ∩ e ((0, 1)) = ∅ • • • • • •

for not necessarily distinct edges e, e ∈ E. A pair (n, e) of a node and an edge of a graph is said to be adjacent if e(0) = n or e(1) = n. A graph is said to be bipartite if the set of nodes can be divided into two disjoint sets N = B W so that every edge is adjacent both to an element of B and W . A bicolored graph is a bipartite graph together with a choice of a division N = B W satisfying the condition above. The elements of B and W are said to be black and white respectively. A face of a graph on T ∨ is a connected component of the complement of the set of (the images of) edges. A bicolored graph on T ∨ is a dimer model if every face is simply-connected. A perfect matching on a dimer model G = (B, W, E) is a subset D ⊂ E such that for every node n ∈ B ∪ W , there is a unique edge e ∈ D adjacent to it.

on MR obtained from Let G be a dimer model and consider the bicolored graph G ∨ G by pulling-back by the natural projection MR → T . The set of perfect matchings Fix a of G can naturally be identified with the set of periodic perfect matchings of G. reference perfect matching D0 . Then for any perfect matching D, the union D ∪ D0 divides MR into connected components. The height function h D,D0 is a locally-constant function on MR \(D ∪ D0 ) which increases (resp. decreases) by 1 when one crosses an edge e ∈ D with the black (resp. white) vertex on the right or an edge e ∈ D0 with the white (resp. black) vertex on the right. This rule determines the height function up to an additive constant. The height function may not be periodic even if D and D0 are

726


Fig. 1. A dimer model

periodic, and the height change h(D, D0 ) = (h x (D, D0 ), h y (D, D0 )) ∈ N of D with respect to D0 is defined as the difference h x (D, D0 ) = h D,D0 ( p + (1, 0)) − h D,D0 ( p), h y (D, D0 ) = h D,D0 ( p + (0, 1)) − h D,D0 ( p) of the height function, which does not depend on the choice of p ∈ MR \(D ∪ D0 ). The dependence of the height change on the choice of the reference matching is given by h(D, D1 ) = h(D, D0 ) − h(D1 , D0 ) for any three perfect matchings D, D0 and D1 . We will often suppress the dependence of the height difference on the reference matching and just write h(D) = h(D, D0 ). • The characteristic polynomial of G is the Laurent polynomial in two variables defined by Z (x, y) = x h x (D) y h y (D) . D : a perfect matching

• The characteristic polygon is the Newton polygon of the characteristic polynomial, i.e. the convex hull of {(h x (D), h y (D)) ∈ N | D is a perfect matching}. As an example, consider the dimer model given in Fig. 1. This dimer model has six perfect matchings D1 , . . . , D6 shown in Figs. 2–7. The captions in these figures show the height changes with respect to D4 , and Fig. 8 shows an example of a height function. The characteristic polygon is shown in Fig. 9. 3. Description of the Procedure An element m ∈ M is primitive if it cannot be written as km , where m ∈ M and k is an integer greater than one. The set of primitive elements of M will be denoted by Mprim . • An oriented line on T ∨ is a pair L = (π(), m) of the image π() of a line in MR by the natural projection π : MR → T ∨ = MR /M and a primitive element m ∈ Mprim in the tangent space of .


727

Fig. 2. D1 : (1, 0)

Fig. 3. D2 : (0, 1)

Fig. 4. D3 : (−1, −1)

• The map (π(), m) → m from the set of oriented lines to Mprim is denoted by p. • The image p(L) is called the slope of L. • The set of oriented lines on T ∨ can be identified with the product Mprim × S 1 , where Mprim parametrizes the slope and S 1 parametrizes the position of π() in the direction perpendicular to the slope. • An oriented line arrangement on T ∨ is a finite set of oriented lines on T ∨ . • An oriented line arrangement on T ∨ is simple if no three lines intersect at one point. • A cell of an oriented line arrangement is a connected component of the complement of the union of lines.

728


Fig. 5. D4 : (0, 0)

Fig. 6. D5 : (0, 0)

Fig. 7. D6 : (0, 0)

• An edge of a cell is a connected component of the intersection of the boundary of the cell and a line in the arrangement. • A vertex of a line arrangement is an intersection point of two lines. The following concepts are at the heart of the procedure of Hanany and Vegh: • An edge of a cell of an oriented line arrangement has two orientations; one comes from the orientation of the line, and the other is the boundary orientation of the cell. A cell is said to be white if these two orientations agree for any of its edge, and black if they are opposite for any of its edge.


729

Fig. 8. The height function for D1 with respect to D4

Fig. 9. The characteristic polygon

Fig. 10. A triangle 2

• A cell is colored if it is either black or white. Note that an edge in an oriented line arrangement is bound by two cells, and at most one of them can be colored. • An oriented line arrangement is admissible if every edge bounds a colored cell. Let ⊂ NR be a convex lattice polygon, i.e. the convex hull of a finite subset of N . • An edge of is a connected component of ∂\(∂ ∩ N ). The set of edges of will be denoted by E. • The primitive outward normal vector to an edge e of is denoted by m(e) ∈ Mprim . • The map → {m(e)}e∈E is a bijection from the set of convex lattice polygons up to translations to the set of finite collections of elements of Mprim summing up to zero. • An oriented line arrangement A is said to be associated with if corresponds to { p(L)} L∈A by the bijection above. For example, the triangle 2 shown in Fig. 10 has four edges. There are two combinatorially distinct ways shown in Fig. 11 and Fig. 12 to arrange four lines in the directions of outward normal vectors of the edges of 2 . In Fig. 11, all edges bound colored polygons, whereas in Fig. 12, the edges drawn in dotted lines do not bound any colored polygon. Hence the polygon 2 in Fig. 10 has a unique admissible arrangement. Now we define the map D from the set of admissible arrangements to the set of dimer models:

730


Fig. 11. The admissible arrangement

Fig. 12. The non-admissible arrangement

Fig. 13. The dimer model

Definition 3.1. Let A be an admissible oriented line arrangement. Then the dimer model D(A) = (B, W, E) associated with A is defined as follows: • The set W of white nodes is the set of the centers of gravities of white cells. • The set B of black nodes is the set of the centers of gravities of black cells. • Two nodes are connected by a straight line segment if the corresponding cells share a vertex. As an example, Fig. 13 shows the dimer model associated with the admissible arrangement in Fig. 11.


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Fig. 14. The lattice polygon 4 with its primitive normal vectors

Fig. 15. The unique admissible arrangement corresponding to 4

Now we can formulate the linear Hanany-Vegh procedure: Definition 3.2. The linear Hanany-Vegh procedure is defined as follows: • Take a convex lattice polygon ⊂ NR as an input. • The set A() of oriented line arrangements {L a }a associated with is a real n-torus, where n is the number of edges of . • The subset of A() corresponding to simple arrangements will be denoted by U(). The complement A()\U() is a closed subset of real codimension one. • Take a representative V() ⊂ U() of π0 (U()) and let W() ⊂ V() be the subset consisting of admissible arrangements. • The output is the set D(W()) of dimer models associated with arrangements in W(). As an example, take the convex hull 4 of v1 = (1, 0), v2 = (0, 1), v3 = (−1, 0), and v4 = (−1, −1) shown in Fig. 14. The primitive outward normal vectors of the edges of 4 are given by m 1 = (1, 1), m 2 = (−1, 1), m 3 = (−1, 0), and m 4 = (1, −2). Lemma 3.3. There is a unique admissible oriented line arrangement associated with 4 .

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Fig. 16. The unique arrangement of three lines in the directions of m 1 , m 2 and m 4

Fig. 17. Five ways to insert the fourth line in the direction of m 3

Fig. 18. One way to perturb three lines to insert the fourth line

Proof. Figure 16 shows the unique arrangement of three oriented lines on the torus in the directions of m 1 , m 2 , and m 4 . There are five combinatorially distinct ways to insert a line into Fig. 16 in the direction of m 3 shown in Fig. 17. In addition, since two of six intersection points of the three lines in Fig. 16 are on the same horizontal level, there are two combinatorially distinct ways shown in Fig. 18 and Fig. 19 to perturb them a little and insert the fourth line. It is easy to see that out of these seven arrangements, only the one shown in Fig. 15 is admissible.


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Fig. 19. Another way to perturb three lines to insert the fourth line

Fig. 20. The dimer model

Hence the output of the linear Hanany-Vegh procedure consists of only one dimer model shown in Fig. 20 in this case. 4. Zig-zag Paths and Isoradiality The following notion is due to Duffin [6] and Mercat [26]: Definition 4.1. A dimer model is isoradial if one can choose an embedding of the graph into the torus T ∨ so that every face of the graph is a polygon inscribed in a circle of a fixed radius with respect to a flat metric on T ∨ . Here, the circumcenter of any face must be contained in the face. The following notion is introduced by Kenyon and Schlenker [22]. Definition 4.2. A zig-zag path is a path on a dimer model which makes a maximum turn to the right on a white node and maximum turn to the left on a black node. If a dimer model comes from a lattice polygon through the linear Hanany-Vegh procedure, then the set of zig-zag paths can naturally be identified with the set of oriented lines in the admissible arrangement. Figure 21 shows an example of this identification. A dimer model is isoradial if and only if zig-zag paths behave like straight lines: Theorem 4.3 (Kenyon and Schlenker [22, Theorem 5.1]). A dimer model is isoradial if and only if the following conditions are satisfied:

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Fig. 21. An admissible arrangement and zig-zag paths

1. Every zig-zag path is a simple closed curve. 2. The lift of any pair of zig-zag paths to the universal cover of the torus intersect at most once. Since zig-zag paths in a dimer model obtained through the linear Hanany-Vegh procedure clearly satisfy these conditions, Theorem 4.3 immediately implies the following: Corollary 4.4. A dimer model obtained by the linear Hanany-Vegh procedure is isoradial. The following notion is slightly weaker than isoradiality: Definition 4.5 ([16, Def. 5.2]). A dimer model is consistent if • no zig-zag path has a self-intersection on the universal cover, • no zig-zag path on the universal cover is a closed path, and • no pair of zig-zag paths intersect each other on the universal cover in the same direction more than once. One can formulate the consistent Hanany-Vegh procedure by enlarging the domain of the brute-force search from the set of oriented line arrangements to the set of arrangements of oriented curves satisfying the three conditions in Definition 4.5. The orientations are given by the homology classes of the curves instead of the tangent spaces to the lines. Note that for a convex lattice polygon , the number of combinatorial types of arrangements of oriented curves associated with satisfying the conditions in Definition 4.5 is finite. One can also formulate the isoradial Hanany-Vegh procedure by restricting to arrangements satisfying the conditions in Theorem 4.3. The following theorem shows that Hanany-Vegh procedures are inverse procedures: Theorem 4.6 (Gulotta [13, Theorem 3.3] Ishii and Ueda [16, Cor. 8.3]). If the set of zig-zag paths on a consistent dimer model G is associated with a lattice polygon , then the characteristic polygon of G coincides with up to translation. 5. Dimer Models and McKay Quivers We recall the description of the McKay quiver for an abelian subgroup of S L 3 (C) as the dual of a hexagonal tiling on a torus [28,31] and prove Theorem 1.2 in this section.


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5.1. Quivers with potentials. A quiver consists of • a set V of vertices, • a set A of arrows, and • two maps s, t : A → V from A to V . For an arrow a ∈ A, the vertices s(a) and t (a) are said to be the source and the target of a respectively. A path on a quiver is an ordered set of arrows (an , an−1 , . . . , a1 ) such that s(ai+1 ) = t (ai ) for i = 1, . . . , n −1. We also allow for a path of length zero, starting and ending at the same vertex. The path algebra CQ of a quiver Q = (V, A, s, t) is the algebra spanned by the set of paths as a vector space, and the multiplication is defined by the concatenation of paths; (bm , . . . , b1 , an , . . . , a1 ) s(b1 ) = t (an ), (bm , . . . , b1 ) · (an , . . . , a1 ) = 0 otherwise. A quiver with relations is a pair of a quiver and a two-sided ideal I of its path algebra. Let p = (an , . . . , a1 ) be an oriented cycle on the quiver, i.e. the class of a path satisfying s(a1 ) = t (an ) up to a cyclic change of the starting point. For an arrow b, the derivative of p by b is defined by ∂p = δai ,b (ai−1 , ai−2 , . . . , a1 , an , an−1 , . . . , ai+1 ), ∂b n

i=1

where δa,b

1 a = b, = 0 otherwise.

This derivation extends to formal sums of oriented cycles by linearity. A potential on a quiver is a formal sum of oriented cycles, which defines an ideal I = (∂) of relations generated by the derivatives ∂/∂a for all the arrows a of the quiver: ∂ (∂) = . ∂a a∈A The most basic example is the quiver with V = {v} and A = {x, y, z}. Since there is only one vertex, the maps s and t must be constant. Consider the potential = yx z − xyz. The corresponding ideal is given by ∂ ∂ ∂ (∂) = , , ∂ x ∂y ∂z = (yz − zy, zx − x z, xy − yx). The resulting path algebra with relations is just the polynomial ring in three variables; Cx, y, z/(∂) ∼ = C[x, y, z]. See Ginzburg [12] and references therein for more on algebras associated with quivers with potentials.

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5.2. McKay quivers. Let G be a finite subgroup of G L n (C). The McKay quiver of G is a quiver whose set of vertices is the set of irreducible representations of G, and the number aσ τ of arrows from σ to τ is given by the multiplicity of τ in the tensor product ∨ of the natural representation of G: of σ with the dual representation ρNat ∨ = τ ⊗aσ τ . σ ⊗ ρNat τ

A path of length two from σ to τ corresponds to an element of ∨ ∨ Hom G (σ ⊗ ρNat ⊗ ρNat , τ ).

The McKay quiver comes with relations generated by the kernel of the map ∨ ∨ ∨ ⊗ ρNat → σ ⊗ (Sym2 ρNat ). σ ⊗ ρNat

When G is an abelian subgroup A of S L 3 (C), the McKay quiver has the following description: Assume that A is contained in the diagonal subgroup, and let πi : A → C× ,

i = 1, 2, 3,

be the projections to the i th diagonal component, viewed as a one-dimensional representation of A. The set of vertices is the set Irrep(A) of irreducible representations of A, and for each vertex ρ ∈ Irrep(A), there exist three arrows xρ , yρ , and z ρ , ending at the vertices ρ ⊗ π1∨ , ρ ⊗ π2∨ , and ρ ⊗ π3∨ respectively. The relations come from the potential

where x =

ρ

xρ , y =

= x zy − xyz,

ρ yρ , and z = ρ zρ .

(5.1)

= N ⊕ Z be a 5.3. Toric Calabi-Yau 3-folds associated with lattice polygons. Let N free abelian group of rank three and M = M ⊕ Z be the dual group. For a lattice polygon R and be the fan in N R consisting of σ ⊂ NR , let σ be the cone over × {1} ⊂ N and its faces. Let { vi }ri=1 be the set of primitive generators of one-dimensional cones of , and : Zr → N φ . Let vi ∈ N be the map which sends the i th standard coordinate vector ei ∈ Zr to ⊗ C× = Spec C[ M] T=N be a three-dimensional torus. The group ⊗ C× : (C× )r → T) ⊂ (C× )r K = Ker(φ naturally acts on Cr , and the toric variety X associated with is the quotient of Cr by the action of K : X = Cr /K .


737

The monoid ring R = C[σ ∨ ] of the dual cone

| m, n ≥ 0, n ∈ σ ⊂ M σ∨ = m ∈ M

is the coordinate ring of X : X = Spec R. When is the convex hull of v1 = ( p, q), : then the map φ

Z3

v2 = (r, s),

v3 = (0, 0),

is given by →N (a, b, c) → ( pa + r b, qa + r b, a + b + c),

⊗ C× : Z3 → T is given by so that the map φ (α, β, γ ) → (α p β r , α q β s , αβγ ). It follows that the toric variety associated with is the quotient X = C3 /A, where

A = (α, β, γ ) ∈ (C× )3 | α p β r = α q β s = αβγ = 1 .

5.4. McKay quivers on the torus. The McKay quiver of an abelian subgroup of S L 3 (C) can naturally be drawn on a real 2-torus as follows: Let N1 = Z2 be a free abelian group of rank two and M1 = Hom(N1 , Z) be the dual group. Identify the torus T1 = N1 ⊗ C× = Spec C[M1 ] with a maximal torus {diag(α, β, γ ) ∈ S L 3 (C) | αβγ = 1} of S L 3 (C), so that the group A = diag(α, β, γ ) ∈ S L 3 (C) | α p β r = α q β s = αβγ = 1 is identified with the kernel of the map →

T ∈

∈

φ ⊗ C× : T1

(α, β) → (α p β r , α q β s ), where φ : N1 → N is the linear map given by the matrix p r . P= q s

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Fig. 22. The periodic quiver

The short exact sequence φ⊗C×

1 → A → T1 −−−→ T → 1 of abelian groups induces an exact sequence ψ

0→M− → M1 → Irrep(A) → 0 of characters, where ψ is the adjoint map of φ : N1 → N represented by the transposed matrix of P. Now consider the infinite quiver in Fig. 22 drawn on M1 ⊗ R ∼ = MR , whose set of vertices is M1 and whose set of arrows consists of xi, j , yi, j and z i, j for (i, j) ∈ M1 such that s(xi, j ) = s(yi, j ) = s(z i, j ) = (i, j) and t (xi, j ) = (i + 1, j), t (yi, j ) = (i, j + 1), t (z i, j ) = (i − 1, j − 1). The McKay quiver for A is the quotient of this quiver by the natural action of M. As an example, consider the case when is the convex hull of (2, 1), (1, 3) and (0, 0) shown in Fig. 23. The map φ is presented by the matrix 2 1 , P= 1 3


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Fig. 23. The triangle 5

Fig. 24. The McKay quiver

and the subgroup A is given by

A = (α, β, γ ) ∈ T1 | α 2 β = αβ 3 = αβγ = 1 , which is isomorphic to the image of T ∈

∈

ρNat : Z/5Z →

[1] → (ζ, ζ 3 , ζ ) √ where ζ = exp(2π −1/5). The quotient of the periodic quiver in Fig. 22 by the action of M is shown in Fig. 24, where fundamental regions of the action are shown as parallelograms.

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Fig. 25. The dual honeycomb graph

5.5. A quiver with potential from a dimer model. Note that the dual graph of the quiver in Fig. 24 is a hexagonal tiling of the torus T ∨ = MR /M shown in Fig. 25. The colors of the nodes of the dual graph is chosen so that a white node is always on the right of an arrow. Observe that the potential (5.1), which gives the relations of the McKay quiver, is the signed sum of cycles around the nodes of this graph: = pw − pb . (5.2) w∈W

b∈B

This motivates the following definition of the quiver with potential associated with a dimer model G = (B, W, E): • The set V of vertices is the set of faces of G. • The set A of arrows is the set E of edges of the graph. The directions of the arrows are determined by the colors of the vertices of the graph, so that the white vertex w ∈ W is on the right of the arrow. • A node n ∈ B W determines an oriented cycle pn of the quiver around it, and the potential is defined as the signed sum (5.2) over such cycles. A hexagonal tiling of a torus can naturally be identified with the McKay quiver of an abelian subgroup of S L 3 (C) in this way. 5.6. The linear Hanany-Vegh procedure on triangles. Now we prove Theorem 1.2. First note that given three slopes, there is only one way to form a polygon whose edges has one of these slopes and the boundary is positively oriented. This unique arrangement is the triangle shown in Fig. 26, and an example of a polygon with unoriented boundary is shown in Fig. 27.


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Fig. 26. The unique positively-oriented polygon

Fig. 27. An example of an unoriented pentagon

Fig. 28. Two of the adjacent triangles

Fig. 29. Two more triangles

For an arrangement of oriented lines to be admissible, each of the vertices of the triangle in Fig. 26 must be contained in another triangle, whose boundary is negatively oriented. Again, there is a unique such arrangement, and Fig. 28 shows two of these adjacent triangles. By repeating the same argument, one obtains two new triangles in Fig. 29, and then another triangle in Fig. 30. It follows that the hexagonal tiling of the the torus which locally looks as in Fig. 31 is the unique output of the linear Hanany-Vegh procedure on

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Fig. 30. Yet another triangle

Fig. 31. The hexagonal tiling

a triangle . It is clear that this tiling is dual to the McKay quiver associated with , and Theorem 1.2 is proved. 6. Coamoebas and Newton Polygons For a Laurent polynomial

W (x, y) =

ai j x i y j

(i, j)∈N

in two variables, its Newton polygon is defined as the convex hull of (i, j) ∈ N such that ai j = 0; = Conv{(i, j) ∈ N | ai j = 0} ⊂ NR . It gives a regular map W : T∨ → C from the torus T∨ = M ⊗ C× = Spec C[N ] dual to T = N ⊗ C× . The coamoeba of W −1 (0) ⊂ T∨ is its image under the argument map

(x, y) →

T∨ ∈

∈

Arg : T∨ →

1 (arg x, arg y). 2π

For an edge e of , let (n(e), m(e)) ∈ M be the primitive outward normal vector of e and l(e) be the integer such that the defining equation for the edge e is given by n(e)i + m(e) j = l(e).


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The leading term of W with respect to the edge e is defined by We (x, y) = ai j x i y j . n(e)i+m(e) j=l(e)

This is indeed the leading term if we put (x, y) = (r n(e) u, r m(e) v),

r ∈ R and u, v ∈ C× ,

and take the r → ∞ limit: P(r n(e) u, r m(e) v) = r l(e) We (u, v) + O(r l(e)−1 ). Now assume that for an edge e, the leading term We (x, y) is a binomial We (x, y) = a1 x i1 y j1 + a2 x i2 y j2 , where a1 , a2 ∈ C and (i 1 , j1 ), (i 2 , j2 ) ∈ N . Put αi = arg(ai ) for i = 1, 2 and (x, y) = (r n(e) |a2 |e(θ ), r m(e) |a1 |e(φ)), √ where e(x) = exp(2π −1x) for x ∈ R/Z. Then the leading behavior of W as r → ∞ is given by r l(e) We (e(θ ), e(φ)) = r l(e) |a1 a2 |{e(α1 + i 1 θ + j1 φ) + e(α2 + i 2 θ + j2 φ)}.

(6.1)

Hence the coamoeba of W −1 (0) asymptotes in this limit to the line 1 = 0 mod Z 2 on the torus T ∨ . This line will be called an asymptotic boundary of the coamoeba of W −1 (0). The asymptotic boundary has a natural orientation coming from the outward normal vector of the edge of . To understand the role of this orientation, take a pair of adjacent edges e and e of as in Fig. 32 and consider the behavior of the coamoeba of W −1 (0) near the intersection of asymptotic boundaries corresponding to e and e . Assume that the leading terms corresponding to e and e are binomials (α2 − α1 ) + (i 2 − i 1 )θ + ( j2 − j1 )φ +

We (x, y) = e(α)x i1 y j1 + e(β)x i2 y j2 , We (x, y) = e(β)x i2 y j2 + e(γ )x i3 y j3 for some α, β, γ ∈ R/Z and (i 1 , j1 ), (i 2 , j2 ), (i 3 , j3 ) ∈ N . Put Wee (x, y) = e(α)x i1 y j1 + e(β)x i2 y j2 + e(γ )x i3 y j3 . Assume further that all the coefficients of W corresponding to interior lattice points of the Newton polygon of Wee vanish. Then Wee is the sum of the leading term and the sub-leading term of W as one puts

(x, y) = (r −n(e ) u, r −m(e ) v),

r ∈ R>0 and u, v ∈ C× ,

and take the r → ∞ limit. Here, (n(e ), m(e )) ∈ M is the primitive outward normal vector of the edge e of the Newton polygon of Wee shown in Fig. 32. The coamoeba for the sum of three monomials will be analyzed in Sect. 7; the asymptotic boundaries coincide with the actual boundaries of the coamoeba, and the orientations on the asymptotic boundaries determine which side of the boundary belongs to the coamoeba. Hence the orientations of asymptotic boundaries determine the leading behavior of the coamoeba near the intersections of asymptotic boundaries as in Fig. 33.

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Fig. 32. A pair of adjacent edges of the Newton polygon

Fig. 33. The leading behavior of the coamoeba near an intersection of asymptotic boundaries

7. Coamoebas for Triangles We discuss the coamoeba of W −1 (0) ⊂ T∨ in this section, where W is the sum of three Laurent monimials. The strategy is to reduce to the simplest case W (x, y) = 1 + x + y by a finite cover of the torus. Theorem 7.1. Let be a lattice triangle and W be the Laurent polynomial obtained as the sum of monomials corresponding to the vertices of . Then the coamoeba of W −1 (0) has the following description: • The set A of asymptotic boundaries, equipped with the orientation coming from the outward normal vectors of , is an admissible arrangement of oriented lines. • The coamoeba is the union of colored cells and vertices of A. • The restriction of the argument map to the inverse image of a colored cell is a diffeomorphism. It is orientation-preserving if the cell is white, and reversing if the cell is black. • The inverse image of a vertex of A by the argument map is homeomorphic to an open interval. Proof. We first consider the simplest case W1 (x, y) = 1 + x + y ∈ C[N1 ], where N1 ∼ = Z2 is a free abelian group of rank two. It gives a regular map × T∨ 1 = M1 ⊗ C → C,


745

Fig. 34. The coamoeba for 1

where M1 = Hom(N1 , Z) is the dual group of N1 , and the coamoeba of W1−1 (0) is a subset of T1∨ = M1 ⊗ R/M1 . The zero locus of W1 is obtained by gluing two disks D1 = {(x, y) ∈ (C× )2 | Im(x) > 0, y = −1 − x} and D2 = {(x, y) ∈ (C× )2 | Im(x) < 0, y = −1 − x} along three intervals I1 = {(x, y) ∈ (R)2 | x < −1, y = −1 − x}, I2 = {(x, y) ∈ (R)2 | −1 < x < 0, y = −1 − x}, and I3 = {(x, y) ∈ (R)2 | x > 0, y = −1 − x}. Define U1 = {(θ, φ) ∈ T1∨ | 0 < θ
0, the space with LeBrun metric gi converges to the space equipped with the LeBrun metric having the fixed (n − 1) points as the set of monopole points. The degeneration from nCP2 to (n − 1)CP2 constructed in Sect. 2 is the twistor translation of this degeneration. Next consider the same sequence of n points as in the previous paragraph but this time as base points we choose the moving point pi1 , viewed as a point on nCP2 . Then since the distance between the fixed (n − 1) points and the base point goes to infinity as i → ∞, the limit as pointed space should be the space equipped with the LeBrun metric with one monopole point; namely the Burns metric. This is rather the space which disappeared in the last degeneration, and thus changing a base point can give a different limit. Also it is natural to expect that in the degeneration of the twistor spaces constructed in Sect. 2, at the limit (central fiber), we find not only a twistor space of (n − 1)CP2 but also the twistor space of the Fubini-Study metric (which is a conformal compactification of the Burns metric). However, even if we regard our degeneration as a family Z → C(λ) and try to perform the birational transformations within the 4-fold Z , I could not find the flag twistor space at the central fiber. We note that the existence of such a model is guaranteed by the framework of Donaldson-Friedman [2]. If one can find the flag twistor space at the central fiber of the family, it provides an explicit realization of the Donaldson-Friedman model for the case of the present degeneration, not relying on deformation theory of complex spaces. Next in accordance with the degeneration taken up in Sect. 3 we consider the sequence {{ pi1 , pi2 , . . . , pin } ⊂ H 3 | i = 1, 2, . . .} of n points for which all pi j -s approach to a point p ∈ H 3 as i → ∞. In this case, as was explicitly shown in coordinates by LeBrun [7, pp. 235–236], the limit is a space equipped with LeBrun’s scalar flat Kähler metric on O(−n). However we note that we have to choose a conformal gauge which is different from the above LeBrun metrics on the collinear blowup of C2 ; instead another Kähler representative found in [7, pp. 243–244] has to be chosen to get the above limit. As was discovered in [7, pp. 236–237], if we look at this degeneration more carefully, it provides a typical example of bubbling off phenomena of ALE spaces. Namely, when all the monopole points pi1 , · · · , pin of LeBrun metrics become closer to a point p ∈ H 3 in such a way that their angular positions relative to p as well as the ratio of the distances from p are preserved, around the point p we make a sequence of rescalings in a way that the distances from p become constant. Then at the limit the curvature of the hyperbolic space becomes zero and as a result a Gibbons-Hawking space with n monopole points on R3 bubble off from the point p. To understand this bubble off through our degeneration of the twistor spaces constructed in Sect. 4, for each s ∈ R with 0 < s < 1, we define a dilation ψs : R3 → R3 by ψs (b, c) = (b/s, c/s) for (b, c) ∈ R × C = R3 . Then for the ellipsoids we have ψs (B(1)) = B(s). Hence pulling back by ψs , B(s) becomes B(1) and the point p j = (b j , c j ) ∈ B(s) is pulled back to s. p j = (sb j , sc j ). Thus by letting s → 0, all the monopole points become closer to the origin, while their relative position is exactly as explained in the last paragraph. Finally it seems natural to expect that, when any sequence {{ pi1 , pi2 , . . . , pin } ⊂ H 3 | i = 1, 2, . . .} of n points is given, possible limits (as pointed spaces) of the

770

N. Honda

associated LeBrun metrics, implied by the above Tian-Viaclovsky’s convergence theorem, are LeBrun orbifolds with k (0 ≤ k ≤ n) monopole points, where the LeBrun orbifold metric means a natural generalization of the LeBrun metrics, allowing some of the monopole points to coincide (see [12, Sect. 2.2] for the precise definition). Then it would be possible to say that the degenerations considered in Sects. 2 and 3 are two extremal cases among these degenerations. Acknowledgements. First of all, I would like to express my sincere gratitude to Jeff Viaclovsky for a number of helpful discussions about the material in this paper. Certainly, without these discussions, I could not get any result in this paper. Also I would like to thank Claude LeBrun for suggesting the use of minitwistor spaces for connecting his twistor spaces with Hitchin’s one. This suggestion was crucial for obtaining the result in Sect. 4. I also would like to thank the referees for a lot of kind and insightful suggestions, which yields Sect. 5. At this point I also would like to thank Kazuo Akutagawa for teaching materials on convergence of critical metrics on 4-manifolds, and kindly giving me a lot of suggestions.

References 1. Atiyah, M., Hitchin, N., Singer, I.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London, Ser. A 362, 425–461 (1978) 2. Donaldson, S.K., Friedman, R.: Connected sums of self-dual manifolds and deformations of singular spaces. Nonlinearity 2, 197–239 (1989) 3. Gibbons, G.W., Hawking, S.W.: Gravitational multi-instantons. Phys. Lett. 78B, 430–432 (1978) 4. Hitchin, N.: Polygons and Gravitons. Math. Proc. Cambridge Philos. Soc. 85, 465–476 (1979) 5. Honda, N., Viaclovsky, J.: Conformal symmetries of self-dual hyperbolic monopole metrics. http://arxiv./ org/abs/0902.2019v1 [math.DG], 2009 6. LeBrun, C.: Counter-examples to the generalized positive action conjecture. Commun. Math. Phys. 118, 591–596 (1988) 7. LeBrun, C.: Explicit self-dual metrics on CP2 · · · CP2 . J. Diff. Geom. 34, 223–253 (1991) 8. LeBrun, C.: Twistors, Kähler manifolds, and bimeromorphic geometry I. J. Amer. Math. Soc. 5, 289–316 (1992) 9. Pontecorvo, M.: On twistor spaces of anti-self-dual Hermitian surfaces. Tran. Amer. Math. Soc. 331, 653– 661 (1992) 10. Poon, Y.S.: Compact self-dual manifolds of positive scalar curvature. J. Diff. Geom. 24, 97–132 (1986) 11. Tian, G., Viaclovsky, J.: Moduli spaces of critical Riemannian metrics in dimension four. Adv. Math. 196, 346–372 (2005) 12. Viaclovsky, J.: Yamabe invariants and limits of self-dual hyperbolic monopole metrics. Ann. Inst. Fourier (Grenoble) (accepted) 2010 Communicated by P.T. Chru´sciel

Commun. Math. Phys. 301, 771–809 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1157-x

Communications in


Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3) David E. Evans, Mathew Pugh School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG Wales, United Kingdom. E-mail: [email protected]; [email protected] Received: 12 February 2010 / Accepted: 15 June 2010 Published online: 4 December 2010 – © Springer-Verlag 2010

Abstract: We complete the computation of spectral measures for SU (3) nimrep graphs arising in subfactor theory, namely the SU (3)ADE graphs associated with SU (3) modular invariants and the McKay graphs of finite subgroups of SU (3). For the SU (2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whilst for the SU (3) graphs the spectral measures distill onto very special subsets of the discoid/torus. The theory of nimreps allows us to compute these measures precisely. We have previously determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU (2) modular invariants, all subgroups of SU (2), the torus T2 , SU (3), and some SU (3) graphs. 1. Introduction The Verlinde algebra of SU (n) at level k is represented by a non-degenerately braided μ system of endomorphisms N X N on a type III1 factor N with fusion rules λμ = ν Nλν ν σ [35]. The fusion matrices Nλ = [Nρλ ]ρ,σ are a family of commuting normal matrices, and themselves give a representation of the fusion rules of thepositive energy repreμ sentations of the loop group of SU (n) at level k, Nλ Nμ = ν Nλν Nν , the regular representation. This family {Nλ } of fusion matrices can be simultaneously diagonalised: Nλ =

Sσ,λ Sσ Sσ∗ , S σ,1 σ

(1)

where 1 is the trivial representation, and the eigenvalues Sσ,λ /Sσ,1 and eigenvectors Sσ = [Sσ,μ ]μ are described by the symmetric modular S matrix. Braided subfactors N ⊂ M (the dual canonical endomorphism is in ( N X N ), i.e. decomposes as a finite linear combination of endomorphisms in N X N ) yield modular invariants through the procedure of α-induction which allows two extensions of λ on N

772

D. E. Evans, M. Pugh

to endomorphisms αλ± of M, such that the matrix Z λ,μ = αλ+ , αμ− is a modular invariant [9,6,17]. The action of the N -N sectors N X N on the M-N sectors M X N produces a nimrep (non-negative matrix integer representation of the fusion rules) μ Nλν G ν GλGμ = ν

whose spectrum reproduces exactly the diagonal part of the modular invariant, i.e. Gλ =

Si,λ ψi ψi∗ , Si,1

(2)

i

with the spectrum of G λ = {Sμ,λ /Sμ,1 with multiplicity Z μ,μ } [10]. The labels μ of the non-zero diagonal elements are called the exponents of Z , counting multiplicity. Every SU (2) and SU (3) modular invariant can be realised by α-induction for a suitable braided subfactor [4,5,9,10,33,34,36], [4,5,7–9,19,20,33,34,36] respectively. For SU (2), the classification of Cappelli, Itzykson and Zuber [12] of SU (2) modular invariants is understood in the following way. Suppose N ⊂ M is a braided subfactor which realises the modular invariant Z G . Evaluating the nimrep G at the fundamental representation ρ, we obtain for the inclusion N ⊂ M a matrix G ρ , which is the adjacency matrix for the AD E graph G which labels the modular invariant. Since these AD E graphs can be matched to the affine Dynkin diagrams – the McKay graphs of the finite subgroups of SU (2) – di Francesco and Zuber [14] were guided to find candidates for classifying graphs for SU (3) modular invariants by first considering the McKay graphs of the finite subgroups of SU (3) to produce a candidate list of ADE graphs whose spectra described the diagonal part of the modular invariant. The classification of SU (3) modular invariants was shown to be complete by Gannon [25], and the complete list is given in [20]. Ocneanu claimed [33,34] that all SU (3) modular invariants were realised by subfactors and this was shown in [20]. The figures for the list of the ADE graphs are given in [2], or in [19,20]. However this list of nimreps has not been shown to be complete. In general, different inclusions which yield different nimreps may still realise the (12) same modular invariant, as is the case in SU (3) with the inclusions for the graphs E1 and E2(12) , which both realise the modular invariant Z E (12) [8, Sect. 8]. Thus any modular invariant may have more than one nimrep associated to it (although this is not the case in SU (2)). However, in SU (3) there is uniqueness in the reverse direction, that is, each nimrep has an unique modular invariant associated to it, due to the coincidence that at any level k each SU (3) modular invariant has a different trace. Unlike the situation for SU (2), there is a mismatch between the list of nimreps associated to each SU (3) modular invariant and the McKay graphs of the finite subgroups of SU (3) which are also the nimreps of the representation theory of the group. The latter also have a diagonalisation as in (1), with diagonalising matrix S = {Si j } usually non-symmetric, where i labels conjugacy classes and j the irreducible characters (see [18, Sect. 8.7] and [22, Sect. 4]). Both of these kinds of nimreps will play a role in this paper. In [22] we determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU (2) modular invariants and all subgroups of SU (2). Our methods gave an alternative approach to deriving the results of Banica and Bisch [1] for AD E graphs and subgroups of SU (2), and explained the connection between their results for affine AD E graphs and the Kostant polynomials. We also determined spectral measures for the torus T2 and SU (3), and some SU (3) graphs, namely

Spectral Measures for Nimrep Graphs in Subfactor Theory II: SU (3)

773

A(n) , D(3k) and A(n)∗ , for integers n ≥ 4, k ≥ 2. We now complete the computation of the spectral measures for the SU (3)ADE graphs in this present work, as well as all finite subgroups of SU (3). We note that the proof of [22, Eq. (71)] will now appear in [23]. Suppose A is a unital C ∗ -algebra with state ϕ. If b ∈ A is a normal operator then there exists a compactly supported probability measure μb on the spectrum σ (b) ⊂ C of b, uniquely determined by its moments ϕ(bm b∗n ) =

σ (b)

z m z n dμb (z),

(3)

for non-negative integers m, n. We computed in [22] such spectral measures and generating series when b is the normal operator = G ρ acting on the Hilbert space of square summable functions on the graph, for the nimreps G described above, i.e. G ρ is the adjacency matrix of the AD E and affine AD E graphs in SU (2) and certain graphs in SU (3). We computed the spectral measure for the vacuum, i.e. the distinguished vertex of the graph which has lowest Perron-Frobenius weight. However the spectral measures for the other vertices of the graph could also be computed by the same methods. In particular, for SU (2), we can understand the spectral measures for the torus T and SU (2) as follows. If w Z and w N are the self adjoint operators arising from the McKay graph of the fusion rules of the representation theory of T and SU (2) respectively, then the spectral measures in the vacuum state can be described in terms of semicircular law, on the interval [−2, 2] which is the spectrum of either as the image of the map z ∈ T → z + z −1 [22, Sects. 2 & 3.1]: dim

⊗ M2 k

T

=

ϕ(w 2k Z )

1 = π

2

x 2k √

1

dx, 4 − x2 −2 2 SU (2)

1 2k = ϕ(w 2k ) = x 4 − x 2 dx . dim ⊗k M2 N 2π −2 The fusion matrix for S O(3) is just 1 + , where is the fusion matrix for SU (2), and thus is equal to the infinite SU (3)ADE graph A(∞)∗ . Thus the spectral measure μ in the vacuum state (over [−1,

3]) for S O(3) has semicircle distribution with mean 1 and variance 1, i.e. dμ(x) = 4 − (x − 1)2 dx [22, Sect. 7.3]. The spectral weight for SU (2) arises from the Jacobian of a change of variable between the interval [−2, 2] and the circle. Then for T2 and SU (3), the 3-cusp discoid D in the complex plane is the image of the two-torus under the map (ω1 , ω2 ) → ω1 + ω2−1 + ω1−1 ω2 , which is the spectrum of the corresponding normal operators v Z and v N on the Hilbert spaces of the fusion graphs of T2 and SU (3) respectively. The corresponding spectral measures are then described by a corresponding Jacobian or discriminant as [22, Theorems 3 & 5]: T2 k = ϕ(|v Z |2k ) dim ⊗ M3 3 1 = 2 |z|2k

dz , π D 27 − 18zz + 4z 3 + 4z 3 − z 2 z 2

774


dim

SU (3) = ϕ(|v N |2k ) ⊗k M 3

1 2k = |z| 27 − 18zz + 4z 3 + 4z 3 − z 2 z 2 dz , 2π 2 D

where dz := d Rez d Imz denotes the Lebesgue measure on C. For the SU (2) and SU (3) graphs, the spectral measures distill onto very special subsets of the semicircle/circle (SU (2)) and discoid/torus (SU (3)), and the theory of nimreps allows us to compute these measures precisely. In the present work we complete the computation of the spectral measures for the SU (3)ADE graphs in Sect. 2, and compute the spectral measures for the finite subgroups of SU (3) in Sect. 3. 2. Spectral Measures of the SU(3) ADE Graphs Let G denote the adjacency matrix of a finite graph G for which G is normal, let V(G) denote the set of vertices of G and let ev = (δa,v )a∈V(G ) denote the basis vector in 2 (G) ∗ n corresponding to the vertex v of G. The inner product m G ( G ) ev , ev defines a spec ∗ n tral measure μ of (G, v) which has m, n th moment z m z n dμ(z) = m G ( G ) ev , ev . In this work we will compute the spectral measure in the vacuum state, i.e. when v is the distinguished vertex ∗ of G which has lowest Perron-Frobenius weight. However the same method will work for any vertex v of G. Let β j be the eigenvalues of G, with corresponding eigenvectors x j , j = 1, . . . , s, ∗ n where s is the number of vertices of G. Then as for SU (2) [22, Sect. 3], m G ( G ) = m ∗ n ∗ 1 2 s UG (G ) U , where G is the diagonal matrix G = diag(β , β , . . . , β ) and U = (x 1 , x 2 , . . . , x s ), so that ∗ n ∗ m ∗ n ∗ ∗ z m z n dμ(z) = Um G (G ) U e1 , e1 = G (G ) U e1 , U e1 =

s (β j )m (β j )n |y j |2 ,

(4)

j=1 j

where y j = x1 is the first entry of the eigenvector x j . Now suppose G is a finite ADE graph with distinguished vertex ∗ which is the vertex with lowest Perron-Frobenius weight. Every eigenvalue β (λ) of G is a ratio of the S-matrix given by β (λ) = Sρλ /S0λ , for a Dynkin label λ, with corresponding eigenvector (ψaλ )a∈V(G ) . Suppose G is the nimrep given by a braided subfactor which realises the modular invariant Z . Then the spectrum of the adjacency matrix G of G is σ ( G ) = {β (λ) | λ ∈ Exp}, where Exp is the set of exponents of Z counting multiplicity. The moments of G over T2 are given by (4) or [22, Eq. (48)]: (β (λ) )m (β (λ) )n |ψ∗λ |2 . (5) z m z n dμ(z) = λ∈Exp

The spectrum σ ( G ) of the adjacency matrix G of any SU (3) ADE graph G is contained in the spectrum σ ( ) = D = {ω1 + ω2−1 + ω1−1 ω2 | ω1 , ω2 ∈ T} of the adjacency matrix of A(∞) . The (3-cusp) discoid D is the surface given by the union of the deltoid and its interior, illustrated in Fig. 1.


775

Fig. 1. The (3-cusp) discoid D, the union of the deltoid and its interior

Thus the support of the probability measure μ of G is contained in the discoid D. There is a map : T2 → D from the torus onto D given by (ω1 , ω2 ) = ω1 + ω2−1 + ω1−1 ω2 ,

(6)

where ω1 , ω2 ∈ T. The Weyl group of SU (3) is the permutation group S3 . Consider the group S3 as a subgroup of G L(2, Z), generated by the matrices T2 , T3 , of orders 2, 3 respectively, given by T2 =

0 −1 , −1 0

T3 =

0 −1 . 1 −1

(7)

The action of S3 on T2 is given by T (ω1 , ω2 ) = (ω1a11 ω2a12 , ω1a21 ω2a22 ), for T = (ai j ) ∈ S3 . For (ω1 , ω2 ) = (e2πiθ1 , e2πiθ2 ), we will define the action of S3 on (θ1 , θ2 ) ∈ [0, 1] × [0, 1] by T (θ1 , θ2 ) = (a11 θ1 + a12 θ2 , a21 θ1 + a22 θ2 ) for T = (ai j ) ∈ S3 (notice that T (e2πiθ1 , e2πiθ2 ) = (e2πiθ1 , e2πiθ2 ), where (θ1 , θ2 ) = T (θ1 , θ2 )). The quotient T2 /S3 is topologically homeomorphic to the discoid D [22, Sect. 6.1]. The deltoid, which is the boundary of the discoid D, is given by the lines θ1 = 1 − θ2 , θ1 = 2θ2 and 2θ1 = θ2 . The diagonal θ1 = θ2 in T2 is mapped to the real interval [−1, 3] ⊂ D. A fundamental domain C of T2 under the action of the group S3 is illustrated in Fig. 2, where the axes are labelled by the parameters θ1 , θ2 in (e2πiθ1 , e2πiθ2 ) ∈ T2 . The boundaries of C map to the deltoid. The torus T2 contains six copies of C. Any probability measure ε on T2 produces a probability measure μ = ∗ ε on D. There is a bijection between S3 -invariant probability measures on T2 and probability measures on D. We will compute S3 -invariant spectral measures of the SU (3) graphs on T2 . For convenience we will use the notation Rm,n (ω1 , ω2 ) := (ω1 + ω2−1 + ω1−1 ω2 )m (ω1−1 + ω2 + ω1 ω2−1 )n ,

(8)

776


Fig. 2. A fundamental domain C of T2 /S3

∗ n so that T2 Rm,n (ω1 , ω2 )dε(ω1 , ω2 ) = z m z n dμ(z) = m G ( G ) e1 , e1 . It was shown in [22, Sect. 7.1] that the eigenvalues β (λ) ∈ D, λ ∈ Exp, of an SU (3)ADE graph G are given by β (λ) = (e2πiθ1 , e2πiθ2 ),

(9)

where θ1 = (λ1 + 2λ2 + 3)/3n and θ2 = (2λ1 + λ2 + 3)/3n. Under the change of variable z = e2πiθ1 + e−2πiθ2 + e2πi(θ2 −θ1 ) , we have x := Re(z) = cos(2π θ1 ) + cos(2π θ2 ) + cos(2π(θ2 − θ1 )), y := Im(z) = sin(2π θ1 ) − sin(2π θ2 ) + sin(2π(θ2 − θ1 )). Then

(ω1 + ω2−1 + ω1−1 ω2 )m (ω1−1 + ω2 + ω1 ω2−1 )n dω1 dω2 = 6 (x + i y)m (x + i y)n |J |−1 dx dy,

T2

D

(10)

where the Jacobian J = det(∂(x, y)/∂(θ1 , θ2 )) is the determinant of the Jacobian matrix. The Jacobian J = J (θ1 , θ2 ) is given by [22, Eq. (39)] J (θ1 , θ2 ) = 4π 2 (sin(2π(θ1 + θ2 )) − sin(2π(2θ1 − θ2 )) − sin(2π(2θ2 − θ1 ))). (11) The Jacobian is real and vanishes on the deltoid, the boundary of the discoid D. For the values of θ1 , θ2 such that (e2πiθ1 , e2πiθ2 ) are in the interior of the fundamental domain C illustrated in Fig. 2, the value of J is always negative. In fact, restricting to any one of the fundamental domains shown in Fig. 2, the sign of J is constant. It is negative over three


777

Fig. 3. The points (θ1 , θ2 ) such that (e2πiθ1 , e2πiθ2 ) ∈ D6

of the fundamental domains, and positive over the remaining three. When evaluating J at a point in z ∈ D, we pull back z to T2 . However, there are six possibilities for (ω1 , ω2 ) ∈ T2 such that (ω1 , ω2 ) = z, one in each of the fundamental domains of T2 in Fig. 2. Thus over D, J is only determined up to a sign. To obtain a positive measure over D we take the absolute value |J | of the Jacobian in the integral (10). Since J 2 is invariant under the action of S3 it can be written in terms of z, z, namely J (z, z)2 = 4π 4 (27 − 18zz + 4z 3 + 4z 3 − z 2 z 2 ) for z ∈ D [22, Sect. 6.1]. Since J is real, J 2 ≥ 0, so we can write

|J (z, z)| = 2π 2 27 − 18zz + 4z 3 + 4z 3 − z 2 z 2 . (12) In [22, Theorem 4] it was shown that the spectral measure (on T2 ) for the graph A(n) is the measure J 2 d(n) (up to a factor of 16π 4 ), where d(n) is the uniform measure on Dn = {(e2πiq1 /3n , e2πiq2 /3n ) ∈ T2 | q1 , q2 = 0, 1, . . . , 3n − 1; q1 + q2 ≡ 0 mod 3}. (13) The points (θ1 , θ2 ) ∈ [0, 1]2 for which (e2πiθ1 , e2πiθ2 ) ∈ D6 are illustrated in Fig. 3. Notice that the points in the interior of the fundamental domain C (those enclosed by the dashed line) correspond to the vertices of the graph A(6) . The Jacobian J of Eq. (12) will appear in Sect. 3 as the discriminant in solutions for the inverse image −1 (z) ∈ T2 of z ∈ D. This discriminant also appears in the work of Gepner [26, Eq. (2.64)] as the measure required to make the polynomials Sμ (z, z) orthogonal, where the polynomials Sν (x, y) are defined by S(0,0) (x, y) = 1, and x Sν (x, y) = T (n) μ A (ν, μ)Sμ (x, y) and y Sν (x, y) = μ A (ν, μ)Sμ (x, y) for vertices ν of A . 2 The Jacobian may be written in terms of (ω1 , ω2 ) ∈ T as [22, Eq. (40)]: i J (ω1 , ω2 )/2π 2 = ω1 ω2 − ω1−1 ω2−1 − ω12 ω2−1 + ω1−2 ω2 − ω1−1 ω22 + ω1 ω2−2 . (14) Remark. Let 1 denote the trivial representation, ρ the fundamental representation of SU (3) and ρ its conjugate representation. Kuperberg [29, Conjecture 3.4] conjectured that certain A2 -(k, n)-tangles in the sense of [21, Sect. 2.3] which do not contain elliptic faces in the sense of [30, Sect. 4] are a basis for Hom(1, ρ n ρ k ) , and observed that it is

778


sufficient to show that the number of such A2 -(k, n)-tangles is given by the coefficient of the term ω1−1 ω22 in the polynomial: (ω1 + ω2−1 + ω1−1 ω2 )k (ω1−1 + ω2 + ω1 ω2−1 )n ×(ω1−1 ω22 − ω1 ω2 + ω12 ω2−1 − ω1 ω2−2 + ω1−1 ω2−1 − ω1−2 ω2 ).

(15)

In the special case of k = n, this follows from [22, Theorem 5]. Denote by p(k, n) the polynomial in (15). The coefficient c of the term ω1−1 ω22 in p(k, k) is given by the integral T2 p(k, k)ω1 ω2−2 dω1 dω2 . Averaging over the orbit of ω1 ω2−2 under the action of the Weyl group S3 of SU (3) gives i J/2π 2 as in (14), thus c = T2 (ω1 + ω2−1 + ω1−1 ω2 )k (ω1−1 + ω2 + ω1 ω2−1 )k J 2 dω1 dω2 /24π 4 . This is the dimension of the path alge bra ( k M3 ) SU (3) [22, Corollary 1], which has basis given by the A2 -(k, k)-tangles which do not contain elliptic faces [21, Lemma 2.12]. It was shown in [22, Sects. 7.4 & 7.5] that the spectral measure for the graphs (12) E (8) and E1 cannot be written as a linear combination of measures of the form ( p) ( p) 2 2 d , J d , J d p/2 ×d p/2 and d p/2 ×d p/2 for p ∈ N, where d p is the uniform measure on the 2 p th roots of unity. However, if we introduce two new measures d((n)) , d(n,k) , we (12) can write the spectral measures for E (8) , E1 and the other SU (3)ADE nimrep graphs as linear combinations of these measures. Definition 1. Let ω = e2πi/3 , τ = e2πi/n . We define the following measures on T2 : (1) dm × dn , where dk is the uniform measure on the k th roots of unity, for k ∈ N. (2) d(n) , the uniform measure on Dn for n ∈ N. (3) d((n)) , the uniform measure on the S3 -orbit of the points (τ, τ ), (ω τ , ω) and (ω, ω τ ), for n ∈ Q, n ≥ 2. (4) d(n,k) , the uniform measure on the S3 -orbit of (τ e2πik , τ ), (τ, τ e2πik ), (ω τ , ω e2πik ), (ω e2πik , ω τ ), (ω τ e−2πik , ω e−2πik ) and (ω e−2πik , ω τ e−2πik ), for n, k ∈ Q, n > 2, 0 ≤ k ≤ 1/n. Let Supp(dμ) denote the set of all (θ1 , θ2 ) ∈ [0, 1]2 such that (e2πiθ1 , e2πiθ2 ) is in the support of the measure dμ. The sets Supp(d((n)) ), Supp(d(n,k) ) are illustrated in Figs. 4, 5 respectively. The white circles in Fig. 5 denote the points given by the measure d((n)) . The cardinality |Supp(dm × dn )| of Supp(dm × dn ) is mn, whilst |Supp(d(n) )| = |Dn | = 3n 2 was shown in [22, Sect. 7.1]. For n > 2 and 0 < k < 1/n, |Supp(d((n)) )| = 18, whilst |Supp(d(n,k) )| = 36. The cardinalities of the other sets are |Supp(d(n,0) )| = |Supp(d(n,1/n) )| = 18 for n > 2, and |Supp(d((2)) )| = 9. It is easy to check that the following relations hold: 3J 2 d(3) = J 2 d3 × d3 , 1 J 2 d(4) , d((4)) = 24π 4 d(n,0) = d((n)) , 1 d(6,1/6) = d((2)) = (4d(2) − d(1) ). 3

(16) (17) (18) (19)


779

Fig. 4. Supp(d((n)) )

Fig. 5. Supp(d(n,k) )

2.1. Graphs D(n)∗ . Let A denote the automorphism of order 3 on the exponents μ of A(n) given by A(μ1 , μ2 ) = (n − 3 − μ1 − μ2 , μ1 ). This induces the orbifold invariant Z D(n) , which is treated in [22], and hence its conjugate orbifold invariant Z D(n)∗ = Z D(n) C, where C = [δλ,λ ] is the conjugate modular invariant. The conjugate orbifold invariant is given by Z D(3k)∗ = Z D(n)∗ =

1 3

∗ (χμ + χ Aμ + χ A2 μ )(χμ∗ + χ Aμ + χ ∗ 2 ), A μ

(3k) μ∈P+ μ1 −μ2 ≡0 mod 3

(n) μ∈P+

χμ χ ∗ (n−3)(μ A

1 −μ2 ) μ

,

n ≥ 5, n ≡ 0 mod 3.

k ≥ 2,

780


The exponents of Z D(n)∗ are Exp = {Ak (λ1 , λ1 )| λ1 = 0, 1, . . . , (n − 3)/2; k = 0, 1, 2}. It was shown in [20] that this modular invariant is realised by a braided subfactor with nimrep D(n)∗ [19, Fig. 12]. Then as in (9), with θ1 = (λ1 + 2λ2 + 3)/24, θ2 = (2λ1 + λ2 + 3)/24, we have for each λ1 = 0, 1, . . . , (n − 3)/2: λ ∈ Exp (λ1 , λ1 ) (n − 2λ1 − 3, λ1 ) (λ1 , n − 2λ1 − 3)

(θ1 , θ2 ) ∈ [0,1]2 λ1 +1 λ1 +1 n , n 1 , 2 − λ1 +1 n 3 3 2 − λ1 +1 , 1 n 3 3

|ψ∗λ |2 4 2 n sin

2π(λ1 +1) n

From (5), 1 (g(λ)) m (g(λ)) n g(λ) 2 Rm,n (ω1 , ω2 )dε(ω1 , ω2 ) = (β ) (β ) |ψ∗ | , 6 T2

(20)

g∈S3 λ∈Exp

where g(λ) is uniquely given by the pair (λ 1 , λ 2 ) such that if θ1 = (λ 1 + 2λ 2 + 3)/3n and θ2 = (2λ 1 + λ 2 + 3)/3n, then (θ1 , θ2 ) = g(θ1 , θ2 ). For each j = λ1 + 1 = 1, 2, . . . , (n − 1)/2, the S3 -orbit of (e2πiθ1 , e2πiθ2 ) ∈ T2 under S3 for (θ1 , θ2 ) = ( j/n, j/n), (1/3, 2/3 − j/n), (2/3 − j/n, 1/3) give the measure d((n/j)) . Then we obtain the following result: Theorem 1. The spectral measure of D(n)∗ , n ≥ 5, (over T2 ) is 12 dε(ω1 , ω2 ) = n

(n−1)/2

sin2 (2π j/n) d((n/j)) (ω1 , ω2 ),

(21)

j=1

where d((n/j)) is defined in Definition 1. 2.2. Graph E (8) : SU (3)5 ⊂ SU (6)1 . The modular invariant realised by the inclusion SU (3)5 ⊂ SU (6)1 [4,20,36] is Z E (8) = |χ(0,0) + χ(2,2) |2 + |χ(0,2) + χ(3,2) |2 + |χ(2,0) + χ(2,3) |2 + |χ(2,1) + χ(0,5) |2 + |χ(3,0) + χ(0,3) |2 + |χ(1,2) + χ(5,0) |2 with exponents Exp = {(0, 0), (2, 0), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3), (2, 2), (5, 0), (3, 2), (2, 3), (0, 5)}.

The inclusion SU (3)9 ⊂ (E6 )1 produces the nimrep E (8) [4,20,36]. Then as in (9), with θ1 = (λ1 + 2λ2 + 3)/24, θ2 = (2λ1 + λ2 + 3)/24, for λ = (λ1 , λ2 ) ∈ Exp, we have: λ ∈ Exp (0, 0), (5, 0), (0, 5) (2, 2), (2, 1), (1, 2) (3, 0), (2, 3), (0, 2) (0, 3), (3, 2), (2, 0)

2 (θ 1 , θ2) ∈[0, 1] 1 , 1 , 1 , 13 , 13 , 1 8 8 3 24 24 3 3, 3 , 7 , 1 , 1, 7 8 8 24 3 3 24 1 , 3 , 11 , 5 , 7 , 5 4 8 24 12 24 24 3 , 1 , 5 , 11 , 5 , 7 8 4 12 24 24 24

|ψ∗λ√|2

2− 2 24 √ 2+ 2 24 1 12 1 12


781

Fig. 6. The points (θ1 , θ2 ) given by g(λ), where g ∈ S3 , λ ∈ Exp, for E (8)

Again, from (5), 1 (g(λ)) m (g(λ)) n g(λ) 2 Rm,n (ω1 , ω2 )dε(ω1 , ω2 ) = (β ) (β ) |ψ∗ | . 6 T2

(22)

g∈S3 λ∈Exp

The pairs (θ1 , θ2 ) given by g(λ) for λ ∈ Exp, g ∈ S3 , are illustrated in Fig. 6. The measure d((8)) is the uniform measure on the S3 -orbit of (e2πiθ1 , e2πiθ2 ) ∈ T2 under S3 for (θ1 , θ2 ) = (1/8, 1/8), (1/3, 13/24), (13/24, 1/3), and similarly d((8/3)) is the measure for the S3 -orbit for (θ1 , θ2 ) = (3/8, 3/8), (1/3, 7/24), (7/24, 1/3). The remaining points (θ1 , θ2 ) that appear in (22) give the measure d(24/5,1/12) . Then the spectral measure for E (8) is

√ √ 1 2 + 2 ((8/3)) 36 (24/5,1/12) 2 − 2 ((8)) d d d dε = + 18 + , 18 6 24 24 12 and we have obtained the following result: Theorem 2. The spectral measure of E (8) (over T2 ) is √ √ 2 − 2 ((8)) 2 + 2 ((8/3)) 1 (24/5,1/12) dε = d d + + d , 8 8 2

(23)

where d((n)) , d(n,k) are as in Definition 1. 2.3. Graph E1(12) : SU (3)9 ⊂ (E6 )1 . The modular invariant realised by the inclusion SU (3)9 ⊂ (E6 )1 [5,20,36] is Z E (12) = Z E (12) C = |χ(0,0) + χ(0,9) + χ(9,0) + χ(4,4) + χ(4,1) + χ(1,4) |2 + 2|χ(2,2) + χ(2,5) + χ(5,2) |2 ,

(24)

782


(12)

Fig. 7. The points (θ1 , θ2 ) ∈ {g(λ)| λ ∈ Exp, g ∈ S3 } for E1

and its exponents are Exp = {(0, 0), (4, 1), (1, 4), (4, 4), (9, 0), (0, 9), and twice (2, 2), (5, 2), (2, 5)}. (12)

The inclusion SU (3)9 ⊂ (E6 )1 produces the nimrep E1 [5,20,36]. Then as in (9), with θ1 = (λ1 + 2λ2 + 3)/24, θ2 = (2λ1 + λ2 + 3)/24, for λ = (λ1 , λ2 ) ∈ Exp, we have: λ ∈ Exp (0, 0), (9, 0), (0, 9) (4, 4), (4, 1), (1, 4) (2, 2), (5, 2), (2, 5)

2 (θ ∈ [0, 1 , θ2 ) 1] 1 , 1 , 7 , 1 , 1, 7 12 12 12 3 3 12 5 , 5 , 1, 1 , 1, 1 12 12 3 4 4 3 1 1 5 1 1 5 4 , 4 , 12 , 3 , 3 , 12

|ψ∗λ√|2

2− 3 36 √ 2+ 3 36 2 9

We illustrate the pairs (θ1 , θ2 ) given by g(λ) for λ ∈ Exp, g ∈ S3 , in Fig. 7. We (12) obtain the following spectral measure for E1 :

√ √ 1 2 + 3 ((12/5)) 2 ((4)) 2 − 3 ((12)) d d dε = + 18 + 18 d 18 6 36 36 9 √ √ 2 − 3 ((12)) 2 + 3 ((12/5)) 2 ((4)) d d = + + d . 12 12 3

(25)

Now for the pairs (θ1 , θ2 ) given by the measure d((4)) , we have J (θ1 , θ2 )2 = 64π 4 , so (12) that the spectral measure for E1 is √ √ 1 2 − 3 ((12)) 2 + 3 ((12/5)) d d + + J 2 d((4)) , dε = 12 12 96π 4 and using (17) we obtain the following result:

Spectral Measures for Nimrep Graphs in Subfactor Theory II: SU (3) (12)

Theorem 3. The spectral measure of E1

783

(over T2 ) is

√ √ 1 2 − 3 ((12)) 2 + 3 ((12/5)) + + J 2 d(4) , dε = d d 12 12 36π 4

(26)

where d(n) , d((n)) are as in Definition 1. (12)

2.4. Graph E2 : SU (3)9 ⊂ (E6 )1 Z3 . The modular invariant realised by the inclusion SU (3)9 ⊂ (E6 )1 Z3 is again the modular invariant Z E (12) given in (24). How(12) ever the nimrep obtained from the inclusion SU (3)9 ⊂ (E6 )1 Z3 is the graph E2 (12) (12) [8,20]. Hence the graphs E2 and E1 are isospectral. Then as in (9), with θ1 = (λ1 + 2λ2 + 3)/24, θ2 = (2λ1 + λ2 + 3)/24, for λ = (λ1 , λ2 ) ∈ Exp, we have: λ ∈ Exp (0, 0), (9, 0), (0, 9) (4, 4), (4, 1), (1, 4) (2, 2), (5, 2), (2, 5)

2 (θ ∈ [0, 1 , θ2 ) 1] 1 , 1 , 7 , 1 , 1, 7 12 12 12 3 3 12 5 , 5 , 1, 1 , 1, 1 12 12 3 4 4 3 1, 1 , 5 , 1 , 1, 5 4 4 12 3 3 12

λ |2 |ψ∗√

2+ 3 36√ 2− 3 36 2 9

We see that the spectral measure for E2(12) is identical to that for E1(12) , except that the √ √ weights (2 + 3)/12 and (2 − 3)/12 are interchanged, giving: (12)

Theorem 4. The spectral measure of E2

(over T2 ) is

√ √ 1 2 + 3 ((12)) 2 − 3 ((12/5)) d d + + J 2 d(4) , dε = 12 12 36π 4

(27)

where d(n) , d((n)) are as in Definition 1. (12)

2.5. Graph E5

. The Moore-Seiberg invariant

Z E (12) = |χ(0,0) + χ(0,9) + χ(9,0) |2 + |χ(2,2) + χ(2,5) + χ(5,2) |2 + 2|χ(3,3) |2 MS

+|χ(0,3) + χ(6,0) + χ(3,6) |2 + |χ(3,0) + χ(0,6) + χ(6,3) |2 ∗ +|χ(4,4) + χ(4,1) + χ(1,4) |2 + (χ(1,1) + χ(1,7) + χ(7,1) )χ(3,3) ∗ ∗ ∗ +χ(3,3) (χ(1,1) + χ(1,7) + χ(7,1) ),

(28)

has exponents Exp = {(0, 0), (3, 0), (0, 3), (2, 2), (4, 1), (1, 4), (6, 0), (0, 6), (5, 2), (2, 5), (4, 4), (9, 0), (6, 3), (3, 6), (0, 9) and twice (3, 3)}. (12)

It is realised by a braided subfactor which produces the nimrep E5 [20, Sect. 5.4]. Then as in (9), with θ1 = (λ1 + 2λ2 + 3)/24, θ2 = (2λ1 + λ2 + 3)/24, for λ = (λ1 , λ2 ) ∈ Exp, we have:

784


λ ∈ Exp (0, 0), (9, 0), (0, 9) (4, 4), (4, 1), (1, 4) (2, 2), (5, 2), (2, 5) (3, 0), (6, 3), (0, 6) (0, 3), (3, 6), (6, 0) (3, 3)

2 (θ1 , θ2 ) ∈ [0, 1] 1 , 1 , 7 , 1 , 1, 7 12 12 12 3 3 12 5 , 5 , 1, 1 , 1, 1 12 12 3 4 4 3 1, 1 , 5 , 1 , 1, 5 4 4 12 3 3 12 1, 1 , 5 , 1 , 5 , 1 6 4 12 2 12 4 1, 1 , 1, 5 , 1, 5 2 12 4 12 4 6 1, 1 3 3

|ψ∗λ |2 1 36 1 36 1 9

0

1 2

Then we obtain the following spectral measure for E5(12) : 1 18 ((12)) 18 ((12/5)) 18 ((4)) + + d d d 6 9 9 9 1 ((12)) d + d((12/5)) + d((4)) , = 3

dε =

and using (17) we obtain the following result: (12)

Theorem 5. The spectral measure of E5 dε =

(over T2 ) is

1 ((12)) 1 2 (4) + d((12/5)) + J d , d 3 24π 4

(29)

where d(n) , d((n)) are as in Definition 1. 2.6. Graph E4(12) . It has not yet been shown that the graph E4(12) is a nimrep obtained from an inclusion which realises the conjugate Moore-Seiberg modular invariant Z E (12)∗ = MS Z E (12) C, given by MS

Z E (12)∗ = |χ(0,0) + χ(0,9) + χ(9,0) |2 + |χ(2,2) + χ(2,5) + χ(5,2) |2 + 2|χ(3,3) |2 MS

∗ ∗ ∗ +(χ(0,3) + χ(6,0) + χ(3,6) )(χ(3,0) + χ(0,6) + χ(6,3) ) ∗ ∗ ∗ + χ(6,0) + χ(3,6) ) +(χ(3,0) + χ(0,6) + χ(6,3) )(χ(0,3) ∗ +|χ(4,4) + χ(4,1) + χ(1,4) |2 + (χ(1,1) + χ(1,7) + χ(7,1) )χ(3,3)

∗ ∗ ∗ +χ(3,3) (χ(1,1) + χ(1,7) + χ(7,1) ),

(30)

(12)

However, it is known that E4 is a nimrep [14], and it can be checked by hand that the eigenvalues of E4(12) are given by Sμ,ρ /Sμ,1 , where μ runs over the exponents of Z E (12)∗ : MS

Exp = {(0, 0), (2, 2), (4, 1), (1, 4), (5, 2), (2, 5), (4, 4), (9, 0), (0, 9), and twice (3, 3)}.


785

With θ1 = (λ1 + 2λ2 + 3)/24, θ2 = (2λ1 + λ2 + 3)/24, for λ = (λ1 , λ2 ) ∈ Exp, we have: λ ∈ Exp (0, 0), (9, 0), (0, 9) (4, 4), (4, 1), (1, 4) (2, 2), (5, 2), (2, 5) (3, 3)

2 (θ ∈ [0, 1 , θ2 ) 1] 1 , 1 , 7 , 1 , 1, 7 12 12 12 3 3 12 5 , 5 , 1, 1 , 1, 1 12 12 3 4 4 3 1, 1 , 5 , 1 , 1, 5 12 3 3 12 4 4 1, 1 3 3

|ψ∗λ |2 1 36 1 36 1 9 1 2

The pairs (θ1 , θ2 ) given by g(λ) for λ ∈ Exp \ {(3, 3)}, g ∈ S3 , have all appeared in the (12) (12) computations of the spectral measures for the graphs E1 , E2 , hence the measures which give these points are known. For the remaining points g((3, 3)), g ∈ S3 , we have J (g(e2πi/3 , e2πi/3 ))2 = 108π 4 , whilst J (e2πiθ1 , e2πiθ2 )2 = 0 for the other points that are given by the measure d(3) , since they map to the boundary of the discoid D. Then (12) we obtain the following spectral measure for E4 : 1 18 ((12)) 18 ((12/5)) 18 ((4)) 27 1 2 (3) dε = d d d + + + J d 6 36 36 9 2 108π 4 1 1 ((12)) 1 ((12/5)) 1 ((4)) d d + + d + J 2 d(3) , = 12 12 3 48π 4 and using (17) we obtain the following result: (12)

Theorem 6. The spectral measure of E4 dε =

(over T2 ) is

1 ((12)) 1 ((12/5)) 1 1 d d + + J 2 d(4) + J 2 d(3) , 4 12 12 72π 48π 4

(31)

where d(n) , d((n)) are as in Definition 1. 2.7. Graph E (24) : SU (3)21 ⊂ (E 7 )1 . The modular invariant realised by the inclusion SU (3)21 ⊂ (E 7 )1 [5,20,36] is Z E (24) = |χ(0,0) + χ(4,4) + χ(6,6) + χ(10,10) + χ(21,0) + χ(0,21) + χ(13,4) + χ(4,13) +χ(10,1) + χ(1,10) + χ(9,6) + χ(6,9) |2 +|χ(15,6) + χ(6,15) + χ(15,0) + χ(0,15) + χ(10,7) + χ(7,10) + χ(10,4) +χ(4,10) + χ(7,4) + χ(4,7) + χ(6,0) + χ(0,6) |2 ,

(32)

with exponents Exp = {(0, 0), (6, 0), (0, 6), (10, 1), (7, 4), (4, 7), (1, 10), (6, 6), (10, 4), (4, 10), (15, 0), (9, 6), (6, 9), (0, 15), (13, 4), (10, 7), (7, 10), (4, 13), (10, 10), (21, 0), (15, 6), (6, 15), (0, 21)}. The inclusion SU (3)21 ⊂ (E 7 )1 produces the nimrep E (24) [5,20,36]. Then as in (9), with θ1 = (λ1 + 2λ2 + 3)/24, θ2 = (2λ1 + λ2 + 3)/24, for λ = (λ1 , λ2 ) ∈ Exp, we have:

786


λ ∈ Exp (0, 0), (21, 0), (0, 21) (4, 4), (13, 4), (4, 13) (6, 6), (9, 6), (6, 9) (10, 10), (10, 1), (1, 10) (6, 0), (15, 6), (0, 15) (0, 6), (6, 15), (15, 0) (7, 4), (10, 7), (4, 10) (4, 7), (7, 10), (10, 4)

2 (θ ∈ [0, 1 , θ2 ) 1] 1 , 1 , 5, 1 , 1, 5 24 24 8 3 3 8 5 , 5 , 11 , 1 , 1 , 11 24 24 24 3 3 24 7 , 7 , 3, 1 , 1, 3 24 24 8 3 3 8 11 , 11 , 5 , 1 , 1 , 5 24 24 24 3 3 24 5 1 13 5 1 11 24 , 8 , 24 , 12 , 4 , 24 1 , 5 , 5 , 13 , 11 , 1 8 24 12 24 24 4 7 ,1 , 5 ,3 , 7 ,3 24 4 12 8 24 8 1 7 3 5 3 7 4 , 24 , 8 , 12 , 8 , 24

|ψ∗λ √ |2

√ 6−2 3− 6 144 √ √ 6+2 3− 6 144 √ √ 6+2 3+ 6 144 √ √ 6−2 3+ 6 144 √ 2− 2 48 √ 2+ 2 48

Then we obtain: Theorem 7. The spectral measure of E (24) (over T2 ) is dε =

√ √ √ √ √ √ 6 − 2 3 − 6 ((24)) 6 + 2 3 − 6 ((24/5)) 6 + 2 3 + 6 ((24/7)) + + d d d 48 48√ √ √ √ 48 6 − 2 3 + 6 ((24/11)) 2 − 2 (8,1/12) 2 + 2 (4,1/24) d d d + + + , (33) 48 8 8

where d((n)) , d(n,k) are as in Definition 1. 3. Spectral Measures for Finite Subgroups of SU(3) The classification of finite subgroups of SU (3) is due to [11,24,32,37]. Clearly any finite subgroup of SU (2) is a finite subgroup of SU (3), since we can embed SU (2) in SU (3) by sending g → g ⊕ 1 ∈ SU (3), for any g ⊂ SU (2). These subgroups of SU (3) are called type B. There are three other infinite series of finite groups, called types A, C, D. The groups of type A are the diagonal abelian groups, which correspond to an embedding of the two torus T2 in SU (3) given by ⎛

⎞ ω1 0 0 0 ⎠, (ρ|T2 )(ω1 , ω2 ) = ⎝ 0 ω2−1 0 0 ω1−1 ω2 for (ω1 , ω2 ) ∈ T2 . The groups of type C are the groups (3n 2 ), and those of type D are the groups (6n 2 ). These ternary trihedral groups are considered in [11,15,31] and generalize the binary dihedral subgroups of SU (2). There are also eight exceptional groups E-L. The complete list of finite subgroups of SU (3) is given in Table 1. Here Type denotes the type of the inclusion found in [20] which yielded the ADE graph as a nimrep. Suppose we have a braided subfactor given by the inclusion ι : N → M of type III factors, with braiding operator ε(λ, μ) intertwining λμ and μλ, for N -N sectors λ, μ ∈ N X N , and with dual canonical endomorphism θ = ιι. The inclusion N ⊂ M is called type I if and only if one of the following equivalent conditions hold, where v is an isometry which intertwines the identity and the canonical endomorphism γ = ιι [6, Prop. 3.2]:


787

Table 1. Relationship between ADE graphs and subgroups of SU (3) ADE graph

Type

Subgroup ⊂ SU (3)

||

(AD E) A(n) – D(n) (n ≡ 0 mod 3) D(n) (n ≡ 0 mod 3) –

– I – I II –

– (n − 2)2 mn 3(n − 3)2 – 3n 2

– A(n)∗ D(n)∗ (n ≥ 7) E (8) E (8)∗

– II II I II

B: finite subgroups of SU (2) ⊂ SU (3) A: Zn−2 × Zn−2 A: Zm × Zn (m = n = 3) C: (3(n − 3)2 ) = (Zn−3 × Zn−3 ) Z3 – C: (3n 2 ) = (Zn × Zn ) Z3 , (n ≡ 0 mod 3) D: (6n 2 ) = (Zn × Zn ) S3 – A: Z(n+1)/2 × Z3 E = (36 × 3) = (3.32 ) Z4 –

E1

I

F = (72 × 3)

216

E2

II

G = (216 × 3)

648

–

24

II

B ×Z3 : B D4 × Z3 L = (360 × 3) ∼ = T A6 ∼ T P S L(2, 7) K=

504

I – – –

– H = (60) ∼ = A5 I = (168) ∼ = P S L(2, 7) J∼ = T A5

– 60 168 180

(12) (12)

(12)

(E3

(12) E4 (12) E5 E (24)

– – –

)

(II)

6n 2 – 3(n + 1)/2 108 –

1080

1. Z λ,0 = θ, λ for all λ ∈ N X N . 2. Z 0,λ = θ, λ for all λ ∈ N X N . 3. Chiral locality holds: ε(θ, θ )v 2 = v 2 . Otherwise the inclusion is called type II. (In the context of nets of subfactors N (I ) ⊂ M(I ), for I a subinterval of the circle, type I means locality of the extended net M(I )). We emphasise that the type of the modular invariant is not well defined, as illustrated (12) by the case of Z E (12) in Sect. 1. The ADE graph E3 is a nimrep which is isospectral (12) (12) to the graphs E1 and E2 [14], however Ocneanu ruled it out as a candidate for the modular invariant Z E (12) by asserting that it did not support a valid cell system [34]. This graph was ruled out as a natural candidate in Sect. 5.2 of [16]. Caution should be taken regarding the type for the graph E4(12) . Although we have not yet shown that E4(12) is the nimrep obtained from an inclusion, it was shown in [20] that such an inclusion would be of type II. The fundamental representation ρ of SU (3) corresponds to the vertex (1, 0) of the graph A(∞) . The McKay graph G is the fusion graph of ρ acting on the irreducible representations of . For most of the graphs G there is a corresponding SU (3)ADE graph/quiver G, which is obtained from G by now removing more than one vertex, and all the edges that start or end at those vertices, as well as possibly some other edges, (12) as was noted in [14] (to obtain the graph E5 from the McKay graph for the subgroup ∼ K = T P S L(2, 7) an extra edge must also be added). However, unlike with SU (2), for SU (3) there is a certain mismatch between the subgroups , with their associated McKay graphs G , and the ADE graphs. The correspondence is as in Table 1, where we use the same notation as Yau and Yu [37] for the subgroups E-I. The notation x

788


Fig. 8. Zn−2 × Zn−2 for n = 6; vertices which have the same symbol are identified

Fig. 9. Z p × Z3 for p = 3

denotes the integer part of x. The subgroup T A5 (respectively T A6 , T P S L(2, 7)) is the ternary A5 group (respectively ternary A6 group, ternary P S L(2, 7) group), which is the extension of A5 (respectively A6 , P S L(2, 7)) by a cyclic group of order 3. The McKay graphs are illustrated in Figs. 8–18. We will now consider the spectral measure for the McKay graph G associated to a finite subgroup ⊂ SU (3). Any eigenvalue of can be written in the form χρ (g) = Tr(ρ(g)), where g is any element of the conjugacy class j [22]. Every element g ∈ is conjugate to an element d in the torus, i.e. ρ(h −1 gh) = ρ(d) = diag(t1 , t2 , t1 t2 ) for some (t1 , t2 ) ∈ T2 . Now Tr(ρ(g)) = Tr(ρ(d)) = t1 + t2 + t1 t2 , thus its eigenvalues are all of the form eiθ1 + eiθ2 + e−i(θ1 +θ2 ) , for 0 ≤ θ1 , θ2 < 2π , and hence the spectrum is contained in the discoid D. For T2 or the group SU (3) itself the spectrum is the whole of D [22, Sect. 6.2]. So if is SU (3) or one of its finite subgroups, the spectrum σ ( ) of is contained in D, illustrated in Fig. 1. Thus the support of μ is contained in the discoid D. Let be a finite subgroup of SU (3) and j its conjugacy classes, j = 1, . . . , s. Since the S-matrix simultaneously diagonalizes the representations of [28], then as in [22, Sect. 4] for the subgroups of

SU (2), √ √ the elements yi in (4) are then given by yi = S0, j = | j |χ0 ( j )/ || = | j |/ ||.


Fig. 10. E =

Fig. 11. F =

789

(36 × 3)

(72 × 3)

Then the m, n th moment ςm,n is given by s | j | n χρ ( j )m χρ ( j ) . ςm,n = z m z n dμ(z) = || D

(34)

j=1

Let : T2 → D be the map defined in (6). We wish to compute ‘inverse’ maps −1 : D → T2 such that ◦ −1 = id. For z ∈ D, we can write z = ω1 + ω2−1 + ω1−1 ω2

790


Fig. 12. G =

(216 × 3)

4 ⊗ σ123 Fig. 13. D


Fig. 14. L =

Fig. 15. H =

791

(360 × 3)

(60)

and z = ω1−1 + ω2 + ω1 ω2−1 . Multiplying the first equation through by ω1 , we obtain zω1 = ω12 + ω1 ω2−1 + ω2 . Then we need to find solutions ω1 to the cubic equation ω13 − zω12 + zω1 − 1 = 0.

(35)

Similarly, we need to find solutions ω2 to the cubic equation ω23 − zω22 + zω2 − 1 = 0. We see that the three solutions for ω2 are given by the complex conjugate of the three solutions for ω1 . Solving (35) we obtain solutions ω(k) , k = 0, 1, 2, given by [3, Chap. V, §6]: ω(k) = (z + 2−1/3 k P + 21/3 k (z 2 − 3z)P −1 )/3,

792


Fig. 16. I =

(168)

Fig. 17. J

where = e2πik/3 , 21/3 takes a real value, and P is the cube root P = (27−9zz +2z 3 + √ k 3 3 27 − 18zz + 4z 3 + 4z 3 − z 2 z 2 )1/3 such that P ∈ {r eiθ | 0 ≤ θ < 2π/3}. For the roots of a cubic equation, it does not matter whether the square root in P is taken to be positive or negative. We will take it to have positive value. We notice that the Jacobian J appears in the expression for P as the discriminant of the cubic equation (35).


793

Fig. 18. K 2 We can define maps −1 k,l : D → T by (k) (l) −1 k,l (z) = (ω , ω ),

k, l ∈ {0, 1, 2},

(36)

for z ∈ D. Now (−1 k,k (z)) = z, for k = 0, 1, 2, however, for the other six cases −1 (k, l ∈ {0, 1, 2} such that k = l) we do indeed have ◦ −1 k,l = id. These six k,l (z) −1 are the S3 -orbit of 0,1 (z) under the action of the group S3 . The spectral measure of (over T2 ) can then be taken as the average over these −1 k,l (z): T2

Rm,n (ω1 , ω2 )dε(ω1 , ω2 ) =

s 1 6

j=1 k,l∈{0,1,2}:

| j | (Wk,l )m (Wk,l )n , ||

(37)

k =l

where Wk,l = ω(k, j) + ω(l, j) + ω(k, j) ω(l, j) , and ω( p, j) , j = 1, . . . , s, are given by (k, j) , ω(l, j) ), for k, l ∈ {0, 1, 2}, k = l. −1 k,l (χρ ( j )) = (ω 3.1. Groups A: Z p ×Zq . We will now compute the spectral measure for the graph G corresponding to the subgroup = Z p ×Zq . When p = q = n−2, the SU (3) McKay graph of Zn−2 × Zn−2 , Fig. 8, is the “affine” version of the graph A(n) [2, Fig. 11]. The group contains || = pq elements, each of which is a separate conjugacy class k,l , where k ∈ {0, 1, 2, . . . , p}, l ∈ {0, 1, 2, . . . , q}. Now χρ (k,l ) = ω1k + ω2−l + ω1−k ω2l ∈ D, where ω1 = e2πi/ p , ω2 = e2πi/q . Let ω−l + ωl−k )m ( ω−k + ωl + ω1k−l )n . (k, l) = ( ωk +

(38)

794

D. E. Evans, M. Pugh Table 2. χρ ( j ) for group (3n 2 ), n ≡ 0 mod 3. Here ω = e2πi/3

j

1

2

3

nk,nl , (k, l) ∈ K n

j , j = 1, . . . , 6

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1, 1) (0, 0)

1 3ω (ω, ω) ( 13 , 23 )

1 3ω (ω, ω) ( 23 , 13 )

3 (e2πik , e2πil ) (e2πik , e2πil ) (k, l)

n 2 /3 0 (ω, 1) ( 31 , 0)

Then by (37), T2

Rm,n (ω1 , ω2 )dε(ω1 , ω2 ) =

q−1 p−1 1 (k, l), pq k=0 l=0

and we easily obtain: Theorem 8. For = Z p × Zq , the spectral measure of G (over T2 ) is given by the product measure dε(ω1 , ω2 ) = d p ω1 dq ω2 , where dm is the uniform measure on the m th roots of unity.

3.2. Groups C: (3n 2 ) = (Zn × Zn ) Z3 . The ternary trihedral group (3n 2 ) is the semi-direct product of Zn × Zn with Z3 , where the action of Z3 on Zn × Zn is given by left multiplication by the matrix T3 defined in (7), see [31]. This group has order || = 3n 2 . It has a presentation generated by the following matrices in SU (3): ⎛

⎛ ⎞ ⎞ 1 0 0 ω 0 0 S1 = ⎝ 0 ω 0 ⎠ , S2 = ⎝ 0 ω 0 ⎠ 0 0 ω 0 0 ω2

⎛

and

⎞ 0 1 0 T = ⎝0 0 1⎠. 1 0 0

In this presentation, the action of Z3 on Zn × Zn ∼ = S1 , S2 is given by left multiplication by the matrix T . Here g1 , g2 , . . . , gs denotes the group generated by the elements g1 , g2 , . . . , gs . We will consider the cases where n ≡ 0 mod 3, n ≡ 0 mod 3 separately. n ≡ 0 mod 3. First we consider the case where n ≡ 0 mod 3. The SU (3) McKay graph of the group (3n 2 ) (not drawn here) is the “affine” version of the graph D(n) [2, Fig. 11]. The values of the character of the fundamental representation evaluated over the conjugacy classes of (3n 2 ) are given in Table 2 (see [31]). Here K n is the region illustrated in Fig. 19 and defined by (θ1 , θ2 ) ∈ [0, 1]2 \{(0, 0), (1/3, 2/3), (2/3, 1/3)} where nθ1 , nθ2 ∈ Z and such that 2θ1 − θ2 ≥ 0, 2θ2 − θ1 ≥ 0 for 0 ≤ θ1 , θ2 ≤ 1/2, and 2θ1 − θ2 ≤ 0, 2θ2 − θ1 ≤ 0 for 1/2 < θ1 , θ2 < 1. Note that |K n | = (n 2 − 3)/3. The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )).


795

Fig. 19. Subset K n ⊂ T2

Let ω = e2πi/n , and (k, l) be as in (38). Then by (37), T2

=

Rm 1 ,m 2 (ω1 , ω2 )dε(ω1 , ω2 ) 6 1 1 3 n 2 /3 1 1 2 2 1 (0, 0)+ ( , )+ ( , )+ (k, l)+ ( 13 , 0) 3 3 3 3 3n 2 3n 2 3n 2 3n 2 3n 2 k,l∈K n

j=1

1 1 1 1 1 = 2 (0, 0)+ 2 ( 13 , 23 )+ 2 ( 23 , 13 )+ 2 (k, l)+ (g( 13 , 0)) 3n 3n 3n 3n 9 1 = 2 3n

nk,nl∈Zn

1 (k, l)+ (g( 13 , 0)), 9

k,l

g∈S3

(39)

g∈S3

where the first summation in the penultimate equality in (39) is over all (k, l) = (0, 0), (1/3, 2/3), (2/3, 1/3) such that nk, nl ∈ Zn . Now 1 (g(0, 13 )) = Rm 1 ,m 2 (ω1 , ω2 )J 2 d(3) (ω1 , ω2 ), 4π 4 T2 g∈S3

since J (ω1 , ω2 ) = 0 for (ω1 , ω2 ) ∈ T2 such that (ω1 , ω2 ) is on the deltoid, the boundary of the discoid D (c.f. Sect. 2.6). Then the spectral measure (over T2 ) for the group

(3n 2 ), n ≡ 0 mod 3, is dε(ω1 , ω2 ) =

1 1 J 2 d(3) (ω1 , ω2 ), dn ω1 dn ω2 + 3 36π 4

where dn is the uniform measure over n th roots of unity, ω = e2πi/3 , and d(n) is the uniform measure over the points in Dn . n ≡ 0 mod 3. Now consider the case where n ≡ 0 mod 3. The values of χρ ( j ) for

(3n 2 ) are given in Table 3 (see [31]). The final row in the table denotes the pair (θ1 , θ2 )

796

D. E. Evans, M. Pugh Table 3. χρ ( j ) for group (3n 2 ), n ≡ 0 mod 3

j

1

nk,nl , (k, l) ∈ K n

j , j = 1, 2

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1, 1) (0, 0)

3 (e2πik , e2πil ) (e2πik , e2πil ) (k, l)

n2 0 (ω, 1) ( 31 , 0)

given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). Then by (37), Rm 1 ,m 2 (ω1 , ω2 )dε(ω1 , ω2 ) T2

=

2 3 n2 1 (0, 0) + (k, l) + ( 13 , 0) 3n 2 3n 2 3n 2

1 = 2 3n

k,l∈K n

nk,nl∈Zn

j=1

2 1 (k, l) + ( 13 , 0) = 3 3n 2

nk,nl∈Zn

(k, l) +

1 (g( 13 , 0)), 9 g∈S3

and we obtain the same spectral measure as for the case n ≡ 0 mod 3. Summarizing, we have: Theorem 9. The spectral measure (over T2 ) for the ternary trihedral group (3n 2 ) = (Zn × Zn ) Z3 is 1 1 dn ω1 dn ω2 + J 2 d(3) (ω1 , ω2 ), (40) 3 36π 4 where dn is the uniform measure over n th roots of unity and d(3) is the uniform measure over the points in D3 . dε(ω1 , ω2 ) =

Remark. The spectral measure of the binary dihedral group B Dn ⊂ SU (2) is [1, Theorem 4.1], [22, Sect. 4.2]: 1 1 d2n−4 u + (δi + δ−i ), 2 4 where δx is the Dirac measure at the point x. The points i, −i are the two points in T which map to zero in the interval [−2, 2] under the map from T to the support of the spectral measure for (a subgroup of) SU (2). See [22] for more details. Let ω = e2πi/3 as before. Since (g( 13 , 0)) = (0, 13 ) + (0, 23 ) + ( 13 , 0) + ( 13 , 13 ) + ( 23 , 0) + ( 23 , 23 ) dε(u) =

g∈S3

it is easy to see that dε(ω1 , ω2 ) =

1 δ(1,ω) + δ(1,ω) + δ(ω,1) + δ(ω,ω) + δ(ω,1) + δ(ω,ω) dn ω1 dn ω2 + , 3 9

where now the points (1, ω), (1, ω), (ω, 1), (ω, ω), (ω, 1), (ω, ω) are the points in T2 which map to zero in the support D of the spectral measure for (a subgroup of) SU (3) under the map defined in (6).


797

Table 4. χρ ( j ) for group (6n 2 ), n ≡ 0 mod 3. Here ω = e2πi/3 j

1

2

3

(nk) , nk ∈ Zn \{0, n3 , 2n 3 }

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1,1) (0,0)

1 3ω (ω, ω) ( 13 , 23 )

1 3ω (ω, ω) ( 23 , 13 )

3 e−4πik + 2e2πik (e2πik , e−2πik ) (k, −k)

nk,nl , (k, l) ∈ K n \{( 31 , 13 )}

j , j = 1, 2, 3

(nk) , nk ∈ Zn

| j | χρ ( j )

6 e2πik + e−2πil + e2πi(l−k)

2n 2 /3 0

3n e−2πik

−1 (χρ ( j )) (θ1 , θ2 )

(e2πik , e2πil ) (k, l)

(ω, 1) ( 13 , 0)

(e2πiθ1 , e2πiθ2 ) (k) (k) (θ1 , θ2 )

j

(k)

(k)

3.3. Groups D: (6n 2 ) = (Zn × Zn ) S3 . The ternary trihedral group (6n 2 ) is the semi-direct product of Zn × Zn with S3 , where the action of S3 on Zn × Zn is given by left multiplication by the matrices T2 , T3 defined in (7) which generate S3 , see [15]. This group has order || = 6n 2 . It has a presentation generated by the generators S1 , S2 and T of (3n 2 ), and the matrix Q ∈ SU (3) given by ⎛

⎞ −1 0 0 0 −1 ⎠ . Q=⎝ 0 0 −1 0 In this presentation, the action of S3 on Zn ×Zn ∼ = S1 , S2 is given by left multiplication by the matrices T, Q. We will consider the cases where n ≡ 0 mod 3, n ≡ 0 mod 3 separately. n ≡ 0 mod 3. First we consider the case where n ≡ 0 mod 3. The values of χρ ( j ) for (6n 2 ) are given in Table 4 (see [15]). The final row in the table denotes the pair (k) (k) (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )), where θ1 , θ2 are defined by

(θ1(k) , θ1(k) ) =

⎧ ⎪ ⎪ ⎨

( 21 k + 41 , k)

if

1 6 1 2

≤ k < 21 ,

(k, 21 k + 41 ) if ≤ k < 56 , ⎪ ⎪ ⎩ ( 1 k + 1 , 1 − 1 k) otherwise. 2 4 4 2

(41)

The set K n is the subset of K n given by (θ1 , θ2 ) ∈ K n such that θ1 +θ2 < 1, 2θ1 −θ2 < 0 and 2θ2 − θ1 < 0. The points (θ1 , θ2 ) = (k, −k), for all k = 0, 1/3, 2/3 such that nk ∈ Zn , lie on the boundary of the fundamental domain C in Fig. 2 and have weight | j |/|| = 3/6n 2 , thus the S3 -orbit of (k, −k) contains the three points (k, −k), (k, 2k) and (−2k, −k), each of which has weight 1/6n 2 . The points (k, l) ∈ K n have weight | j |/|| = 6/6n 2 , thus the six points in the S3 -orbit of (k, l) ∈ K n each have weight 1/6n 2 also. Furthermore,

798


(a)

(b)

(k)

(k)

Fig. 20. The points (θ1 , θ2 ) for (a) n = 9, (b) n = 8

the S3 -orbit of (1/3, 0) contains six points. Then we see from (37) that T2

=

Rm 1 ,m 2 (ω1 , ω2 )dε(ω1 , ω2 ) 1 1 1 1 1 2 2 1 (0, 0) + ( , ) + ( , ) + (k, l) 3 3 3 3 6n 2 6n 2 6n 2 6n 2 k,l

+ =

2n 2 /3 6n 2

1 6n 2

3 j=1

nk,nl∈Zn

( 13 , 0) +

3n (k) (k) (θ1 , θ2 ) 6n 2 nk∈Zn

1 1 (k) (k) (k, l) + (g( 13 , 0)) + (g(θ1 , θ2 )), 18 12n g∈S3 nk∈Zn

g∈S3

where in the first equality the first summation is over all k, l such that nk, nl ∈ Zn and (k, l) = (0, 0), (1/3, 2/3), (2/3, 1/3). In the second equality, the second summation g∈S3 (g( 13 , 0)) is given by the measure J 2 d(3) as in Sect. 3.2. The points (k)

(k)

(e2πiθ1 , e2πiθ2 ) given by the third summation in the last equality lie on the lines θ1 + θ2 = 1/2, 2θ1 − θ2 = 1/2 and 2θ2 − θ1 = 1/2, where n points lie equidistantly along the length of each line such that there is a point at each of (1/4, 1/4), (1/4, 0) and (0, 1/4). These points are illustrated in Fig. 20(a) for n = 9. There are 6n distinct points when n ≡ 0 mod 6, but only 6n − 9 distinct points when n ≡ 0 mod 6 as the points (−1, −1), (eπi/3 , ω), (ω, eπi/3 ), and their S3 -orbits, have multiplicity two. The third summation is thus given by a different measure dεn depending on whether n is divisible by 6 or not. When n ≡ 0 mod 6, dεn

= 18d

((4))

+ 36

(n+3)/6 j=1

d(4n/(n−2 j), j/n) ,


799

whilst for n ≡ 0 mod 3, n ≡ 0 mod 6, dεn = 18d((4)) + 36

(n+3)/6

d(4n/(n−2 j), j/n) + 18d((2)) .

j=1

Thus, we have: Theorem 10. The spectral measure (over T2 ) for the ternary trihedral group (6n 2 ) = (Zn × Zn ) S3 , n ≡ 0 mod 3, is dε =

1 1 3 ((4)) 3 dn dn + d J 2 d(3) + + 4 6 72π 2n n

(n+3)/6

d(4n/(n−2 j), j/n) + d ,

(42)

j=1

where d is given by

d =

3 ((2)) 2n d

if n ≡ 0 mod 6,

0

if n ≡ 0 mod 6.

n ≡ 0 mod 3. Now consider the case where n ≡ 0 mod 3. The values of χρ ( j ) for

(6n 2 ) are given in Table 5 (see [15]). The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )) ∈ T2 , where θ1(k) , θ2(k) are defined by (41). Then from (37) we have Rm 1 ,m 2 (ω1 , ω2 )dε(ω1 , ω2 ) T2

=

1 1 2n 2 3n (k) (k) 1 (0, 0) + (k, l) + ( , 0) + (θ1 , θ2 ) 3 6n 2 6n 2 6n 2 6n 2

1 = 2 6n

nk,nl∈Zn

nk∈Zn

k,l

1 1 (k) (k) (k, l) + (g( 13 , 0)) + (g(θ1 , θ2 )), 18 12n g∈S3

g∈S3 nk∈Zn

where in the first equality the first summation is over all k, l such that nk, nl ∈ Zn and (k) (k) (k, l) = (0, 0). In the second equality, the points (e2πiθ1 , e2πiθ2 ) given by the third (k) (k) summation g∈S3 nk∈Zn (g(θ1 , θ2 )) again lie on the lines θ1 + θ2 = 1/2, 2θ1 − θ2 = 1/2 and 2θ2 − θ1 = 1/2, where n points lie equidistantly along the length of each line such that there is a point at each of (1/4, 1/4), (1/4, 0) and (0, 1/4). These points are illustrated in Fig. 20(b) for n = 8. There are 6n − 3 distinct points, the points (−1, −1), (−1, 1) and (1, −1) having multiplicity two. The measure dε given by the third summation cannot be written as a linear combination of the measures in Definition 1, since all the measures in Definition 1 are topologically invariant under a rotation of each triangular fundamental domain of T2 /S3 (see Fig. 2) by 2π/3, but dε is not. The measure dε is given by n δ(e2πi j/n ,e2πi(1+2 j)/2n ) + δ(e2πi(1+2 j)/2n ,e2πi j/n ) + δ(e2πi(1+2 j)/2n ,e2πi(1−2 j)/2n ) , j=1

where δx is the Dirac measure at the point x.

800

D. E. Evans, M. Pugh Table 5. χρ ( j ) for group (6n 2 ), n ≡ 0 mod 3. Here ω = e2πi/3

j

1

(nk) , nk ∈ Zn \{0}

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2 j

1 e2πik + e−2πil + e2πi(l−k) (e2πik , e2πil ) (k, l) nk,nl , (k, l) ∈ K n j , j = 1, 2, 3

| j | χρ ( j )

6 e−4πik + 2e2πik

2n 2 0

−1 (χρ ( j )) (θ1 , θ2 )

(e2πik , e−2πik ) (k, −k)

(ω, 1) ( 13 , 0)

3 3 (1, 1) (0, 0) (nk) , nk ∈ Zn 3n e−2πik

(k)

(k)

(e2πiθ1 , e2πiθ2 ) (k) (k) (θ1 , θ2 )

Thus, we have: Theorem 11. The spectral measure (over T2 ) for the ternary trihedral group (6n 2 ) = (Zn × Zn ) S3 , n ≡ 0 mod 3, is dε =

1 1 dn dn + J 2 d(3) 6 72π 4 n n 1 1 + δ(e2πi j/n ,e2πi(1+2 j)/2n ) + δ(e2πi(1+2 j)/2n ,e2πi j/n ) 12n 12n j=1

j=1

n 1 + δ(e2πi(1+2 j)/2n ,e2πi(1−2 j)/2n ) . 12n

(43)

j=1

3.4. Group E = (36 × 3) = (3.32 ) Z4 . The subgroup E has order 108, and its McKay graph, Fig. 10, is the “affine” version of the graph E (8) [19, Fig. 13]. It is the semidirect product of the ternary trihedral group (3.32 ) with Z4 , where the action of Z4 on (3.32 ) = S1 , S2 , T is given by left multiplication by the matrix V ∈ SU (3) given by ⎛ ⎞ 1 1 1 1 ⎝ 1 ω ω2 ⎠ . V =√ −3 1 ω2 ω Note that V 2 = Q, so that E also contains the ternary trihedral group (6.32 ) as a normal subgroup. This group is a subgroup of SU (3) but not of SU (3)/Z3 The values of χρ ( j ) for E are given in Table 6 (see [27]). The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). Then, by (37), Rm 1 ,m 2 (ω1 , ω2 )dε(ω1 , ω2 ) T2

=

1 1 (0, 0) + ( 1 , 2 ) 108 108 3 3 1 9+9 1 5 5 1 + ( 23 , 13 ) + (0, 14 ) + ( 12 , 12 ) + ( 12 , 12 ) 108 108


801

Table 6. χρ ( j ) for group E = (36 × 3). Here ω = e2πi/3 j

1

2

3

4, 5

6, 7

8, 9

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1, 1) (0, 0)

1 3ω (ω, ω) ( 13 , 23 )

1 3ω (ω, ω) ( 23 , 13 )

9 1 (1, i) (0, 41 )

9 ω (−ωi, −ωi) 1 , 5 ) ( 12 12

9 ω (−ωi, −ωi) 5 , 1 ) ( 12 12

j

10

11

12

13, 14

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

9 −1 (1,-1) (0, 21 )

9 −ω (ω, −ω) ( 26 , 16 )

9 −ω (−ω, ω) ( 16 , 26 )

12 0 (1, ω) ( 13 , 0)

12 + 12 1 9 (0, 21 ) + ( 26 , 16 ) + ( 16 , 26 ) + ( 3 , 0) 108 108 1 1 1 = (0, 0) + ( 13 , 23 ) + ( 2 , 1 ) 108 108 108 3 3 1 1 5 5 1 + (g(0, 41 )) + (g( 12 , 12 )) + (g( 12 , 12 )) 36 +

g∈S3

+

1 1 (g(0, 21 )) + (g( 26 , 16 )) + (g( 16 , 26 )) + (g( 13 , 0)). 36 27 g∈S3

g∈S3

The set of the three fixed points (0, 0), (1/3, 2/3) and (2/3, 1/3) is D1 , whilst the last summation above is given by the measure J 2 d(3) (ω1 , ω2 )/4π 4 as in Sect. 3.2. We also have ((g(0, 1/4)) + (g(1/12, 5/12)) + (g(5/12, 1/12))) g∈S3

= 18

T2

Rm 1 ,m 2 (ω1 , ω2 )d((4)) (ω1 , ω2 ),

and

((g(0, 1/2)) + (g(2/6, 1/6)) + (g(1/6, 2/6)))

g∈S3

= 12

T2

Rm 1 ,m 2 (ω1 , ω2 )d(2) (ω1 , ω2 ) − 3

T2

Rm 1 ,m 2 (ω1 , ω2 )d(1) (ω1 , ω2 ).

Then using (17) we obtain the following: Theorem 12. The spectral measure (over T2 ) for the group E = (36 × 3) is dε =

1 1 1 1 J 2 d(4) + J 2 d(3) + d(2) − d(1) , 48π 4 108π 4 3 18

where d(n) is the uniform measure over the points in Dn .

(44)

802

D. E. Evans, M. Pugh Table 7. χρ ( j ) for group F = (72 × 3). Here ω = e2πi/3

j

1

2

3

4, 5, 6

7, 8, 9

10, 11, 12

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1,1) (0,0)

1 3ω (ω, ω) ( 13 , 23 )

1 3ω (ω, ω) ( 23 , 13 )

18 1 (1, i) (0, 41 )

18 ω (−ωi, −ωi) 1 , 5 ) ( 12 12

18 ω (−ωi, −ωi) 5 , 1 ) ( 12 12

j

13

14

15

16

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

9 −1 (1,-1) (0, 21 )

9 −ω (ω, −ω) ( 26 , 16 )

9 −ω (−ω, ω) ( 16 , 26 )

24 0 (1, ω) ( 13 , 0)

3.5. Group F = (72 × 3). The subgroup F has order 216, and its McKay graph, (12) Fig. 11, is the “affine” version of the graph E1 [19, Fig. 14]. It has a presentation with generators S1 , S2 , T and V of E, and the matrix W ∈ SU (3) given by ⎛ ⎞ 2 1 1 ω 1 ⎝ W =√ 1 ω ω ⎠. −3 ω 1 ω The order of W is 4. Note that W 2 = V 2 S1 . In fact, S1 = V 2 W 2 , S2 = (W V )4 and T = V 2 W 3 V 2 W so that F = V, W . It contains the group E as a normal subgroup. This group is a subgroup of SU (3) but not of SU (3)/Z3 . The values of χρ ( j ) for F are given in Table 7 (see [27]). The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). The spectral measure for the group F = (72 × 3) is obtained in a similar way to that for the group E = (36 × 3) in Sect. 3.4, and we obtain: Theorem 13. The spectral measure (over T2 ) for the group F = (72 × 3) is dε =

1 1 1 1 J 2 d(4) + J 2 d(3) + d(2) − d(1) , 4 4 32π 216π 6 36

(45)

where d(n) is the uniform measure over the points in Dn . 3.6. Group G = (216 × 3). The subgroup G has order 648, and its McKay graph, (12) Fig. 12, is the “affine” version of the graph E2 [19, Fig. 14]. It has a presentation with generators S1 , S2 , T and V of E, and the matrix U ∈ SU (3) given by ⎞ ⎛ 2 ε 0 0 U = ⎝ 0 ε2 0 ⎠ , 0 0 ε5 where ε = e2πi/9 . The order of U is 9. Note that U 3 = S22 . It contains the group F as a normal subgroup. This group is a subgroup of SU (3) but not of SU (3)/Z3 The values of χρ ( j ) for G are given in Table 8 (see [13]). The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )).


803

Table 8. χρ ( j ) for group G = (216 × 3). Here ω = e2πi/3 j

1

2

3

4

5

6

7

8

9

| j | χρ ( j ) ∈ D (θ1 , θ2 ) ∈ [0, 1]2

1 3 (0, 0)

1 3ω ( 13 , 23 )

1 3ω ( 23 , 13 )

54 1 (0, 41 )

54 ω 1 , 5 ) ( 12 12

54 ω 5 , 1 ) ( 12 12

9 −1 (0, 21 )

9 −ω ( 26 , 16 )

9 −ω ( 16 , 26 )

j

10

11

12

13

14

15

| j | χρ ( j ) (θ1 , θ2 )

12 √ πi/18 3e ( 19 , 29 )

12 √ 13πi/18 3e ( 19 , 59 )

12 √ 25πi/18 3e ( 79 , 29 )

12 √ 11πi/18 3e ( 29 , 79 )

12 √ 23πi/18 3e ( 59 , 19 )

12 √ 35πi/18 3e ( 29 , 19 )

j

16

17

18

19

20

21

22

23, 24

| j | χρ ( j ) (θ1 , θ2 )

36 e5πi/9 2 , 7 ) ( 18 18

36 e11πi/9 8 , 1 ) ( 18 18

36 e17πi/9 5 , 1 ) ( 18 18

36 eπi/9 1 , 5 ) ( 18 18

36 e7πi/9 1 , 8 ) ( 18 18

36 e13πi/9 7 , 2 ) ( 18 18

24 0 (0, 13 )

72 0 (0, 13 )

The measures living on the points (e2πiθ1 , e2πiθ2 ) ∈ T2 for j = 1, . . . , 9 and j = 22, 23, 24 have all been computed when we considered the subgroups E and F of SU (3). j Let us denote by j12 the summation j j12

j2 1 = (g( ωθ1 + ω−θ2 + ωθ2 −θ1 ))m (g( ω−θ1 + ωθ2 + ωθ1 −θ2 ))n , 6 j= j1 g∈S3

ω = e2πi/18 , and the action of g ∈ S3 where for each j, θ1 and θ2 are given in Table 8, on ( ωθ1 + ω−θ2 + ωθ2 −θ1 ) is defined as follows: suppose g( ωθ1 , ωθ2 ) = ( ω p, ωq ), then θ −θ θ −θ p −q q− p 1 2 2 1 g( ω + ω + ω ) = ( ω + ω + ω ). For j = 10, . . . , 15, the pairs (θ1 , θ2 ) ∈ [0, 1]2 are (1/9, 2/9), (1/9, 5/9), (7/9, 2/9), (2/9, 1/9), (2/9, 7/9), (5/9, 1/9). The S3 -orbits for these pairs contain three points each. These points all lie on the boundary of the fundamental domain C, and are obtained by taking the points (θ1 , θ2 ) such that (e2πiθ1 , e2πiθ2 ) ∈ D3 and then removing the points (θ1 , θ2 ) such that θk ∈ {0, 1/3, 2/3}, k = 1, 2. Then we obtain 15 = 27 Rm 1 ,m 2 (ω1 , ω2 )d(3) (ω1 , ω2 ) − 9 Rm 1 ,m 2 (ω1 , ω2 )d3 ω1 d3 ω2 . 210 T2

T2

For j = 16, . . . , 21, the pairs (θ1 , θ2 ) ∈ are (5/18, 1/18), (2/18, 7/18), (8/18, 1/18), (1/18, 5/18), (1/18, 8/18), (7/18, 2/18). The S3 -orbits for these pairs contain six points each, which are illustrated in Fig. 21(a). At these points, J 2 = 48π 4 . We can obtain this distribution by taking the points (θ1 , θ2 ) such that (e2πiθ1 , e2πiθ2 ) ∈ D6 , illustrated in Fig. 21(b), each with the weight J 2 evaluated at that point. Since the points indicated by white circles in Fig. 21(b) map to the deltoid, the boundary of the discoid D, here J 2 = 0. We must then remove the points indicated by black circles in the interior of the triangular regions in Fig. 21(b) which are not in {g(θ1 , θ2 )|g ∈ S3 }. Then we have 21 16 = 108 Rm 1 ,m 2 (ω1 , ω2 )J 2 d(6) (ω1 , ω2 )/48π 4 T2 −36 Rm 1 ,m 2 (ω1 , ω2 )J 2 d6 ω1 d6 ω2 /12π 4 . [0, 1]2

T2

Thus we obtain the following result:

804


(a)

(b)

(c)

Fig. 21. (a) the points {g(θ1 , θ2 )|g ∈ S3 } for j = 16, . . . , 21; (b) the points (θ1 , θ2 ) such that (e2πiθ1 , e2πiθ2 ) ∈ D6 ; (c) the points (θ1 , θ2 ) such that e2πiθk is a 6th root of unity, k = 1, 2

Theorem 14. The spectral measure (over T2 ) for the group G = (216 × 3), is 7 1 1 1 1 1 (1) 2 (6) 2 (4) 2 d(3) + d(2) − + d dε = J d + J d + J 48π 4 96π 4 6 648π 4 18 108 1 1 − J 2 d6 d6 − d3 d3 , (46) 4 144π 18 where dm is the uniform measure over m th roots of unity and d(m) is the uniform measure on the points in Dm . 3.7. Group H ∼ = A5 . The subgroup H is the alternating A5 group, which has order 60. Its McKay graph, Fig. 15, is not the “affine” version of any of the SU (3)ADE graphs.


805

Table 9. χρ ( j ) for group H ∼ = A5 . Here μ± = (1 ±

√ 5)/2

j

1

2

3

4

5

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1, 1) (0, 0)

20 0 (1, ω) ( 13 , 13 )

15 −1 (1, −1) (0, 21 )

12 μ+ (1, e2πi/5 ) (0, 15 )

12 μ− (1, e4πi/5 ) (0, 25 )

Table 10. χρ ( j ) for group I = (168) ∼ = P S L(2, 7). Here ν = (−1 +

√ 7i)/2

j

1

2

3

4

5

6

| j | χρ ( j ) −1 (χρ ( j )) (θ1 , θ2 )

1 3 (1, 1) (0, 0)

21 −1 (1, −1) (0, 21 )

42 1 (1, i) (0, 41 )

56 0 (1, ω) ( 13 , 13 )

24 ν (e2πi/7 , e6πi/7 ) ( 17 , 37 )

24 ν (e6πi/7 , e2πi/7 ) ( 37 , 17 )

The values of χρ ( j ) for H are given in Table 9. The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). For j = 1, 3, the points (e2πiθ1 , e2πiθ2 ) ∈ T2 give the measure (1/5)d2 × d2 − (7/30)δ(0,0) , where δx is the Dirac measure at the point x, whilst for j = 2 we obtain the measure J 2 d(3) /72π 4 as in Sect. 3.2. Thus we have the following result: Theorem 15. The spectral measure (over T2 ) for the group H ∼ = A5 , is dε =

1 7 1 J 2 d(3) + d2 × d2 − δ(0,0) 72π 4 5 30 1 δ(e2πik/5 ,e2πil/5 ) + δ(e4πik/5 ,e4πil/5 ) , + 30

(47)

k,l

where the summation is over k, l such that (±k, ±l) ∈ {(0, 1), (1, 0), (1, 1)}. The measure d2 is the uniform measure over 1 and −1, d(3) is the uniform measure on the points in D3 and δx is the Dirac measure at the point x.

∼ P S L(2, 7). The subgroup H is the projective special lin3.8. Group I = (168) = ear group P S L(2, 7), which has order 60. The special linear group S L(2, 7) consists of all 2 × 2 matrices over F7 , the finite field with 7 elements, with unit determinant. The projective special linear group P S L(2, 7) is the quotient group S L(2, 7)/{I, −I }, obtained by identifying I and −I , where I is the identity matrix. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. Its McKay graph, Fig. 16, is not the “affine” version of any of the SU (3)ADE graphs. The values of χρ ( j ) for H are given in Table 10. The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). For j = 1, 2, the points (e2πiθ1 , e2πiθ2 ) ∈ T2 give the measure (1/6)d2 × d2 − (1/28)δ(0,0) , whilst for j = 3 we obtain the measure J 2 d4 × d4 /96π 4 . For j = 4 we obtain the measure J 2 d(3) /72π 4 as in Sect. 3.2. Thus we have the following result:

806

D. E. Evans, M. Pugh Table 11. χρ ( j ) for group J ∼ = T A5 . Here ω = e2πi/3 and μ± = (1 ±

√ 5)/2

j

1

2

3

4

5

6

7, 8, 9

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1, 1) (0, 0)

1 3ω (ω, ω) ( 13 , 23 )

1 3ω (ω, ω) ( 23 , 13 )

15 −1 (1, −1) (0, 21 )

15 −ω (ω, −ω) ( 26 , 16 )

15 −ω (−ω, ω) ( 16 , 26 )

20 0 (1, ω) ( 13 , 0)

j

10

11

12

13

14

15

| j | χρ ( j )

12 μ+

12 μ+ ω

12 μ+ ω

12 μ−

12 μ− ω

12 μ− ω

−1 (χρ ( j )) (θ1 , θ2 )

(1, e 5 ) (0, 15 )

2πi

14πi

(ω, e 15 ) 7 ) ( 23 , 15

14πi

(e 15 , ω) 7 , 2) ( 15 3

4πi

(1, e 5 ) (0, 25 )

8πi

(ω, e 15 ) 4 ) ( 23 , 15

8πi

(e 15 , ω) 4 , 2) ( 15 3

Theorem 16. The spectral measure (over T2 ) for the group I = (168) ∼ = P S L(2, 7), is dε =

1 1 1 1 J 2 d(3) + J 2 d4 × d4 + d2 × d2 − δ(0,0) 72π 4 96π 4 6 28 1 δ(e2πik/7 ,e2πil/7 ) + δ(e−2πik/7 ,e−2πil/7 ) , + 42

(48)

k,l

where the summation is over (k, l) ∈ {(1, 3), (1, 5), (2, 3), (2, 6), (4, 5), (4, 6)}. The measure dm is the uniform measure over m th roots of unity, d(3) is the uniform measure on the points in D3 and δx is the Dirac measure at the point x. ∼ T A5 . The subgroup J is the ternary alternating A5 group, which has 3.9. Group J = order 180. Its McKay graph, Fig. 17, is not the “affine” version of any of the SU (3)ADE graphs. The values of χρ ( j ) for J are given in Table 11. They are obtained from the character table of H = (60), Table 9, where each conjugacy class of the group H gives three conjugacy classes l of the group J, l = 1, 2, 3, and χρ (l ) = ωl−1 λ if χρ ( ) = λ. Here ω = e2πi/3 as usual. The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). For j = 10, 11, 12, the points (e2πiθ1 , e2πiθ2 ) ∈ T2 give the measure d((5)) , whilst for j = 13, 14, 15 they give the measure d((5/2)) , each with weight 1/5. Thus we obtain the following result: Theorem 17. The spectral measure (over T2 ) for the group J ∼ = T A5 , is dε =

1 1 1 1 1 J 2 d(3) + d(2) − d(1) + d((5)) + d((5/2)) , 4 72π 3 15 5 5

(49)

where the measures d(n) , d((n)) are as in Definition 1.

3.10. Group K ∼ = T P S L(2, 7). The subgroup K is the ternary P S L(2, 7) group, and (12) has order 504. Its McKay graph, Fig. 18, is the “affine” version of the graph E5 [19, Fig. 15]. The values of χρ ( j ) for K are given in Table 12. They are obtained from


807

Table 12. χρ ( j ) for group K ∼ = T P S L(2, 7). Here ω = e2πi/3 and ν = (−1 +

√ 7i)/2

j

1

2

3

4

5

6

| j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 3 (1, 1) (0, 0)

1 3ω (ω, ω) ( 13 , 23 )

1 3ω (ω, ω) ( 23 , 13 )

21 1 (1, i) (0, 41 )

21 ω (−ωi, −ωi) 1 , 5 ) ( 12 12

21 ω (−ωi, −ωi) 5 , 1 ) ( 12 12

j

7

8

9

10

11

12

| j | χρ ( j )

42 −1

42 −ω

42 −ω

24 ν

24 νω

24 νω

−1 (χρ ( j )) (θ1 , θ2 )

(1, −1) (0, 21 )

(ω, −ω) ( 26 , 16 )

(−ω, ω) ( 16 , 26 )

(e 7 , e ( 17 , 37 )

2πi

6πi 7

20πi

)

(e 21 , e 8 ( 10 21 , 21 )

16πi 21

)

10πi

(e 21 , e 5 , 4 ) ( 21 21

8πi 21

)

j

13

14

15

16, 17, 18

| j | χρ ( j )

24 ν

24 νω

24 νω

56 0

−1 (χρ ( j )) (θ1 , θ2 )

(e 7 , e ( 37 , 17 )

6πi

2πi 7

8πi

)

10πi 21

(e 21 , e 4 , 5 ) ( 21 21

16πi

)

(e 21 , e 8 , 10 ) ( 21 21

20πi 21

)

Table 13. χρ ( j ) for group L = (360 × 3) ∼ = T A6 . Here ω = e2πi/3 and μ± = (1 ± j | j | χρ ( j ) ∈ D −1 (χρ ( j )) ∈ T2 (θ1 , θ2 ) ∈ [0, 1]2

1 1 3 (1, 1) (0,0)

2 1 3ω (ω, ω) ( 13 , 23 )

3 1 3ω (ω, ω) ( 23 , 13 )

4 90 1 (1, i) (0, 41 )

(1, ω) ( 13 , 0) √ 5)/2

5 90 ω (−ωi, −ωi) 1 , 5 ) ( 12 12

6 90 ω (−ωi, −ωi) 5 , 1 ) ( 12 12

j

7

8

9

10

11

12

| j | χρ ( j )

45 −1

45 −ω

45 −ω

72 μ+

72 μ+ ω

72 μ+ ω

−1 (χρ ( j )) (θ1 , θ2 )

(1, −1) (0, 21 )

(ω, −ω) ( 26 , 16 )

(−ω, ω) ( 16 , 26 )

(1, e 5 ) (0, 15 )

2πi

14πi

(ω, e 15 ) 7 ) ( 23 , 15

14πi

(e 15 , ω) 7 , 2) ( 15 3

j

13

14

15

16, 17

| j | χρ ( j ) ∈ D

72 μ−

72 μ− ω

72 μ− ω

120 0

−1 (χρ ( j )) (θ1 , θ2 ) ∈ [0, 1]2

(1, e 5 ) (0, 25 )

4πi

8πi

(ω, e 15 ) 4 ) ( 23 , 15

8πi

(e 15 , ω) 4 , 2) ( 15 3

(1, ω) ( 13 , 0)

the character table of I = (168), Table 10, in the same way as the values of χρ ( j ) for J ∼ = T A5 are obtained from the character table of H = (60). The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). The measures which give the points (e2πiθ1 , e2πiθ2 ) ∈ T2 for j = 1, . . . , 9 and j = 16, 17, 18 have all been computed for previous subgroups of SU (3). For j = 10, . . . , 15 we obtain the measure d(1/21,21/4) . Thus we have: Theorem 18. The spectral measure (over T2 ) for the group K ∼ = T P S L(2, 7), is dε =

1 1 1 1 2 J 2 d(4) + J 2 d(3) + d(2) − d(1) + d(1/21,21/4) , 96π 4 72π 4 6 28 7

where the measures d(n) , d(n,k) are as in Definition 1.

(50)

808


3.11. Group L = (360 × 3) ∼ = T A6 . The subgroup L is the ternary alternating A6 group, which has order 1080. Its McKay graph, Fig. 14, is the “affine” version of the graph E4(12) [20, Fig. 7]. The values of χρ ( j ) for L are given in Table 13 (see [13]). The final row in the table denotes the pair (θ1 , θ2 ) given by (e2πiθ1 , e2πiθ2 ) = −1 (χρ ( j )). The values of χρ ( j ) for the group L = (360 × 3) ∼ = T A6 have all appeared for previous groups, and hence it is easy to compute the spectral measure: Theorem 19. The spectral measure (over T2 ) for the group L = (360 × 3) ∼ = T A6 , is dε =

1 1 1 7 (1) 1 ((5)) 1 ((5/2)) d + d J 2 d(4) + J 2 d(3) + d(2) − + d , (51) 4 4 96π 108π 6 180 5 5

where the measures d(n) , d((n)) are as in Definition 1 Acknowledgement. This work was supported by the Marie Curie Research Training Network MRTN-CT2006-031962 EU-NCG.

References 1. Banica, T., Bisch, D.: Spectral measures of small index principal graphs. Commun. Math. Phys. 269, 259– 281 (2007) 2. Behrend, R.E., Pearce, P.A., Petkova, V.B., Zuber, J.-B.: Boundary conditions in rational conformal field theories. Nucl. Phys. B 579, 707–773 (2000) 3. Birkhoff G., Mac Lane S.: A survey of modern algebra. Third edition. New York: The Macmillan Co., 1965 4. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and α-induction for nets of subfactors. II. Commun. Math. Phys. 200, 57–103 (1999) 5. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and α-induction for nets of subfactors. III. Commun. Math. Phys. 205, 183–228 (1999) 6. Böckenhauer, J., Evans, D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors. Commun. Math. Phys. 213, 267–289 (2000) 7. Böckenhauer, J., Evans, D.E.: Modular invariants and subfactors. In: Mathematical physics in mathematics and physics (Siena, 2000), Fields Inst. Commun. 30, Providence, RI: Amer. Math. Soc., 2001, pp. 11–37 8. Böckenhauer, J., Evans, D.E.: Modular invariants from subfactors. In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002, pp. 95–131 9. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: On α-induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys. 208, 429–487 (1999) 10. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733–784 (2000) 11. Bovier, A., Lüling, M., Wyler, D.: Finite subgroups of SU(3). J. Math. Phys. 22, 1543–1547 (1981) (1) 12. Cappelli, A., Itzykson, C., Zuber, J.-B.: The A-D-E classification of minimal and A1 conformal invariant theories. Commun. Math. Phys. 113, 1–26 (1987) 13. Desmier, P.E., Sharp, R.T., Patera, J.: Analytic SU(3) states in a finite subgroup basis. J. Math. Phys. 23, 1393–1398 (1982) 14. Di Francesco, P., Zuber, J.-B.: SU(N ) lattice integrable models associated with graphs. Nuclear Phys. B 338, 602–646 (1990) 15. Escobar, J.A., Luhn, C.: The flavor group (6n 2 ). J. Math. Phys. 50, 013524 (2009) 16. Evans, D.E.: Fusion rules of modular invariants. Rev. Math. Phys. 14, 709–731 (2002) 17. Evans, D.E.: Critical phenomena, modular invariants and operator algebras. In: Operator algebras and mathematical physics (Constan¸ta, 2001), Bucharest: Theta, 2003, pp. 89–113 18. Evans, D.E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford Mathematical Monographs. New York: The Clarendon Press Oxford University Press, 1998 19. Evans, D.E., Pugh, M.: Ocneanu Cells and Boltzmann Weights for the SU (3)ADE Graphs. Münster J. Math. 2, 95–142 (2009)


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20. Evans, D.E., Pugh, M.: SU (3)-Goodman-de la Harpe-Jones subfactors and the realisation of SU (3) modular invariants. Rev. Math. Phys. 21, 877–928 (2009) 21. Evans, D.E., Pugh, M.: A2 -Planar Algebras I. Quantum Topol. 1, 321–377 (2010) 22. Evans, D.E., Pugh, M.: Spectral Measures for Nimrep Graphs in Subfactor Theory. Commun. Math. Phys. 295, 363–413 (2010) 23. Evans, D.E., Pugh, M.: The Nakayama permutation of the nimreps associated to SU (3) modular invariants. Preprint: arXiv:1008.1003 (math.OA), 2010 24. Fairbairn, W.M., Fulton, T., Klink, W.H.: Finite and disconnected subgroups of SU3 and their application to the elementary-particle spectrum. J. Math. Phys. 5, 1038–1051 (1964) 25. Gannon, T.: The classification of affine SU(3) modular invariant partition functions. Commun. Math. Phys. 161, 233–263 (1994) 26. Gepner, D.: Fusion rings and geometry. Commun. Math. Phys. 141, 381–411 (1991) 27. Hanany, A., He, Y.-H.: Non-abelian finite gauge theories. J. High Energy Phys. 9902, 013 (1999), Paper 13, 31 pp. (electronic) 28. Kawai, T.: On the structure of fusion algebras. Phys. Lett. B 217, 47–251 (1989) 29. Kuperberg, G.: The quantum G 2 link invariant. Internat. J. Math. 5, 61–85 (1994) 30. Kuperberg, G.: Spiders for rank 2 Lie algebras. Commun. Math. Phys. 180, 109–151 (1996) 31. Luhn, C., Nasri, S., Ramond, P.: Flavor group (3n 2 ). J. Math. Phys. 48, 073501, (2007), 21pp. 32. Miller, G.A., Blichfeldt, H.F., Dickson, L.E.: Theory and applications of finite groups. New York: Dover Publications Inc., 1961 33. Ocneanu, A.: Higher Coxeter Systems. Talk given at MSRI, 2000. http://www.msri.org/publications/ln/ msri/2000/subfactors/ocneanu 34. Ocneanu, A.:The classification of subgroups of quantum SU(N ). In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math. 294, Providence, RI: Amer. Math. Soc., 2002, pp. 133–159 35. Wassermann, A.: Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N ) using bounded operators. Invent. Math. 133, 467–538 (1998) 36. Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 349–403 (1998) 37. Yau, S.-T., Yu, Y.: Gorenstein quotient singularities in dimension three. Mem. Amer. Math. Soc. 105 viii+88 (1993) Communicated by Y. Kawahigashi


Communications in


Effective Density of States for a Quantum Oscillator Coupled to a Photon Field Volker Betz1 , Domenico P. L. Castrigiano2 1 Department of Mathematics, University of Warwick, Coventry CV4 7AL, England.

E-mail: [email protected]; URL: http://www.maths.warwick.ac.uk/∼betz/

2 Fakultät für Mathematik, TU München, Boltzmannstraße 3, 85747 Garching, Germany.

E-mail: [email protected] Received: 23 February 2010 / Accepted: 28 July 2010 Published online: 2 December 2010 – © Springer-Verlag 2010

Abstract: We give an explicit formula for the effective partition function of a harmonically bound particle minimally coupled to a photon field in the dipole approximation. The effective partition function is shown to be the Laplace transform of a positive Borel measure, the effective measure of states. The absolutely continuous part of the latter allows for an analytic continuation, the singularities of which give rise to resonances. We give the precise location of these singularities, and show that they are well approximated by first order poles with residues equal to the multiplicities of the corresponding eigenspaces of the uncoupled quantum oscillator. Thus we obtain a complete analytic description of the natural line spectrum of the charged oscillator. 1. Introduction The standard model for a non-relativistic quantum particle interacting with the radiation field is the Pauli-Fierz model using minimal coupling. The Hamiltonian is H=

1 ( p − e A)2 + V + Hf , 2m

(1.1)

where Hf is the Hamiltonian of the free field, A is the transverse quantized field and e the electron charge, p is the particle momentum and V the particle potential. We do not introduce form factors. Over the last years, there has been much activity and significant progress in understanding the Pauli-Fierz model. Notable developments include the proof of existence for a ground state when the infrared cutoff is removed [9], a detailed spectral analysis of the Hamiltonian and the proof of the existence of resonances [1], and a proof of enhanced binding through the photon field [10]. We refrain from giving a review of the by now vast literature on the subject, and instead refer to [18] and the bibliography therein. As is apparent already from the above selection of results, the majority of works on the subject investigates the spectral structure of the full system, particle and field.

812

V. Betz, D. P. L. Castrigiano

In our work, instead we regard the photon field as an environment in which the particle is embedded. A prominent example of this point of view is Feynman’s polaron [8] by which our present investigations are inspired. The aim is to treat the coupled particle as a closed system, and to derive, for the particle alone, effective equations that capture the influence of the photon field. The particular effective quantity that we investigate (and, to our knowledge, indeed introduce) is the effective measure of states. We consider a version of (1.1) that is simplified in two ways: Firstly, we use the dipole approximation, which amounts to replacing A with A(0), and secondly we omit the self-interaction A2 of the field. (The omission of the A2 term is necessary only for a part of our calculations, and we discuss the possibility of keeping it in the final remark of Sect. 3.) The particle potential V is taken to be harmonic, so that H is completely quadratic, and we add the well-known terms of mass and energy renormalization. In this setup we compute the ratio Z of the partition functions of the full system and of the free field. We do this by first suitably discretizing both systems, and then taking the continuous-mode limit and the ultraviolet limit. This procedure yields a finite result. We find the explicit expression Z (β) =

2 3 ρ −2ρ ln(ρ) sin ϕ −iϕ (iρ e e ) 2π

(1.2)

with ρ ∼ β inversely proportional to the temperature T and sinϕ ∼ e2 determining the strength of the coupling. See Theorem 3.3 for details. From the ratio Z of the partition functions one gets an explicit expression for the difference of the respective free energies. Subtracting from this the free energy F0 of the particle leads to the excess free energy Fex = −(/β) ln(Z /Z 0 ). The latter is particularly interesting in that it is supposed to be experimentally accessible. Referring to [4] we only mention that (1.2) confirms the well-known quadratic low-temperature behavior Fex (β) ≈ −π αk 2 T 2 (3mc2 )−1

(1.3)

of the excess free energy. In the present case, however, the significance of Z goes beyond that. As we will argue in Sect. 2.3 for general coupled systems, (1.2) is the effective partition function (or partition function for the particle subsystem). This means that Z replaces Z 0 when the small system is regarded as autonomous while retaining an effective influence of the field. The central fact is that, as we prove, Z is the Laplace transform of a positive Borel measure μ. We interpret μ as the effective measure of states for the charged oscillator. It has the same significance for the effective system that the measure of states μ0 (whose Laplace transform is Z 0 , see (2.1) and (2.2)), has for the uncoupled oscillator. It determines the probabilities of energy measurements at the system in thermal equilibrium. It is worth remembering that in general the ratio of two partition functions is not the Laplace transform of some positive Borel measure. Most probably this is the case here as long as in the derivation of (1.2) the ultraviolet cutoff C is kept finite (even if the continuous-mode limit is already done). But a finite cutoff in not desirable in any case, as any result we obtain for the effective system would depend on the arbitrary parameter C. Thus it is crucial that the ultraviolet limit can be carried out, leading to an expression that is the Laplace transform of a positive measure. Formula (1.2) allows a detailed study of the effective measure of states. First of all, it turns out that μ has one atom at the ground state frequency ωϕ , is supported on the half line [ωϕ , ∞[, and is otherwise absolutely continuous with respect to Lebesgue measure.

Quantum Oscillator in a Photon Field

813

The corresponding density φ will be called effective density of states. We prove that φ has an analytic continuation with singularities and cuts along the boundary of a cone with apex ωϕ and opening angle 2ϕ. The conjugate pairs of singularities can be approximated by first order singularities in the interior of the cone, up to logarithmic corrections. The position and residue of these first order poles can be determined analytically. On the real line, they give rise to Lorentz profiles that replace the Dirac peaks present in μ0 . The mass of each Lorentz profile is equal to the mass of the corresponding Dirac peak, and its position agrees with predictions of QED time-dependent perturbation theory up to first order in the fine structure constant α. See Theorems 3.5 and 3.6 and the remarks following them for details. From the above properties, a connection of the effective density of states with the theory of resonances appears obvious. The latter occur when a small quantum system is coupled to a quantum field or reservoir, at which point eigenvalues of the small system dissolve into the continuous spectrum, and stationary states change into metastable states. In the Pauli-Fierz model, resonances have been investigated extensively by Bach, Fröhlich and Sigal in [1]. These authors employ the well-established complex dilation method. They study the spectrum of the full system, particle and field; resonances are then defined as the singularities of the analytic continuation of the resolvent to the second Riemann sheet. In contrast, we work in the context of open quantum systems, and therefore the effective small system does not even have a Hamiltonian. While there has been work on resonances and decoherence in open quantum systems, such as [2] and [14,15], all authors seem to rely on the spectral structure of the full Hamiltonian. We define a resonance of an effective system in Definition 3, in terms of the complex structure of the effective density of states. For the harmonically bound charged particle, we show that this definition, and the concept of the effective density of states in general, produce physically reasonable results. Apart from facilitating explicit formulae, we do not expect the harmonic particle potential to be a vital part of the above picture; we thus conjecture that also for other confined charged systems, the effective density of states exists, and that its complex structure yields an appropriate description of the spectral lines by Lorentz profiles. A proof of this conjecture may well turn out to be challenging, and will probably require some insight into the connections of our results with the theory of [1]. Our paper is organized as follows. In Sect. 3, we present our model in detail and state the main rigorous results of this work, along with a short discussion. Proofs are given in Sects. 4 and 5. The following Sect. 2 interprets our results in the framework of quantum statistical mechanics, and discusses their connection to the theory of resonances.

2. Partition Function of the Embedded System We regard the particle as embedded into the environment of the radiation field. An environment is characterized by the fact that it stays in thermal equilibrium if the joint system is in thermal equilibrium. Roughly speaking, the interaction has effects on the embedded system but does not change the environment. More precisely, when acting on the embedded system the environment carries out transitions between same states. A well-known example of a quantum system whose embedding in the environment determines decisively its properties is the polaron, i.e. an electron in an ionic crystal. The computations in [8, Chap. 8] confirm that the electron moving with its accompanying distortion of the lattice behaves as a free particle but with an effective mass higher than

814


that of the electron. In the same spirit we are going to treat the problem of a charged oscillator surrounded by its own radiation field. In this section we put = k = 1. 2.1. Energy distribution in thermal equilibrium. Let the positive operator H be the Hamiltonian of some quantum system. For the moment we assume that e−β H is trace class. Classically, the partition function of the system is given by Z (β) = Tr e−β H .

(2.1)

The inverse Laplace transform μ of Z , which we call the measure of states of the quantum system, is the weighted sum of Dirac measures N j δλ j , (2.2) μ= j

where λ j are the eigenvalues of H and N j are the corresponding finite multiplicities. Its physical relevance is due to the fact that the statistical uncertainty of states of the system in thermal equilibrium at the temperature T = 1/β is described by the probability 1 measure μ(β) := Z (β) e−β(·) μ. If E denotes the spectral measure of H , and a Borel subset of R, then μ(β) ( ) =

1 Tr (e−β H E( )) Z (β)

(2.3)

is the probability for a measurement of the energy to yield a value in . It is this interpretation that we will retain valid also for the effective entities.

2.2. Spectral discretization. From the basic principle of statistical mechanics, once the partition function is known, all thermodynamic properties can be found. In many interesting cases, however, including our model Hamiltonian (3.1), e−β H is not trace class. This raises the problem of how to attribute a partition function to the oscillator embedded in the photon field. The way we solve this problem uses a slight generalization of methods from the theory of random Schrödinger operators (cf. [12]). The main idea is to approximate H by operators Hn with discrete spectrum. We define Definition 1. Let H be a self-adjoint operator in a Hilbert space H, and let (Pn ) be a family of orthogonal projections on H. Put Hn = Pn H Pn . We say that Hn is a spectral discretization of H (associated to Pn ) if (i) Pn D(H ) ⊂ D(H ). (ii) limn→∞ Pn = 1 in the strong topology. (iii) The spectrum of Hn acting in Pn H consists of eigenvalues with finite multiplicity such that e−β Hn is trace class for all n.

2.3. Separation of a subsystem. Let us now consider a general model consisting of two interacting quantum subsystems with respective state spaces H1 and H2 . For convenience, the system corresponding to H1 will be called the small system, the other one


815

the large system; but note that there is actually no assumption on the dimension of H1 and H2 . The Hamiltonian of the joint system, acting on H1 ⊗ H2 is H = H1 ⊗ 1 + 1 ⊗ H2 + HI ,

(2.4)

where HI describes the interaction. Again we assume for H the existence of Tr e−β H , and also for H1 and H2 . Let the large system be in thermal equilibrium. Then the influence of it onto the small system can be described in a statistical way by averaging the states of the large system. Accordingly, the density operator attributed to the small system is W (β) := (Tr e−β H2 )−1 Tr 2 e−β H , where Tr 2 denotes the partial trace with respect to the second factor. In the case HI = 0 of non-interacting systems, W (β) equals e−β H1 . Moreover, if the joint system is in the state (Tr e−β H )−1 e−β H , any prediction on measurements which concern only the small system is given by the thus uniquely determined state (Tr W (β))−1 W (β) = (Tr e−β H )−1 Tr 2 e−β H of the small system. This is due to the fact that the formula Tr (Tr 2 e−β H A1 ) = Tr ( e−β H A1 ⊗ 1) holds for every bounded selfadjoint operator A1 on H1 and uniquely determines Tr 2 . Consequently, we regard Z (β) := Tr W (β) as the partition function attributed to the small system. It satisfies Tr e−β H , (2.5) Tr e−β H2 and we will interpret its inverse Laplace transform as the effective measure of states (cf. 2.2) for the small system when in contact with the large system. Of course, for this interpretation to make sense perfectly, Z would need to be the Laplace transform of a positive Borel measure. This is not true in general, but we will find that it does hold in our model when performing the following limiting process. Z (β) =

2.4. Infinitely large environment. The large system in (1.1) is described by H2 = Hf , which has absolutely continuous spectrum on [0, ∞[. Thus e−β H2 is not trace class, and neither is e−β H , whence (2.5) is not available. We consider spectral discretizations Hn and H2,n of H and H2 , and define Tr e−β Hn . (2.6) Tr e−β H2,n In the case we study, the limit limn→∞ Z n exists. After removing an ultraviolet cutoff we get the partition function Z , which is the Laplace transform of a positive measure μ with support in [0, ∞[. It is called the effective measure of states of the embedded system. As in Sect. 2.1 we introduce the probability measures Z n (β) =

μ(β) =

1 e−β(·) μ Z (β)

(2.7)

for β > 0, which are fundamental in that they determine the probabilities of energy measurements: if the charged oscillator is in thermal equilibrium at the temperature β1 , then μ(β) ( ) is the probability for a measurement of the energy to yield a value in . One can read off Z and μ the same amount of information about the spectrum of the embedded system as in the case of a trace class operator. We will see that the embedding into its own radiation field changes the behaviour of the oscillator qualitatively: instead of a purely discrete spectrum, we now obtain apart from the stable ground state an absolutely continuous spectrum on the positive half axis.

816


2.5. Resonances. An interesting property of the effective measure of states μ of the present system is that the analytic continuation of its density function φ has conjugate pairs of first order singularities, which for small coupling are very close to the real line, cf. Theorems 3.5, 3.6. They manifest themselves as Lorentz profiles (Breit-Wigner resonance shapes) in the effective density of states. There is a connection with the theory of resonances [11], which we will elucidate here. Generally speaking, bound states, which are perturbed, give rise to resonances. As shown in [1,11], in the case of a bounded electron coupled to a photon field, these are related to singularities p of the resolvent of the Hamiltonian H off the real axis coming from the eigenvalues of the uncoupled electron system H0 . This is in accordance with the definition of a resonance given in [16, XII.6], by which there is a dense set D of state vectors u for which the matrix elements R (u) (z) = u, (H − z)−1 u and (u) R0 (z) = u, (H0 − z)−1 u have an analytic continuation from the upper complex half-plane across the positive real axis into the lower complex half-plane. Then a point p ∈ C with Im p < 0 is called a resonance if for some u ∈ D it is a regular point for R0(u) and a pole for R (u) . However in view of the results in [1] one should consider more general singularities of R (u) than poles. There is an equivalent formulation in terms of the scalar spectral measures. Let E be the spectral measure of H . For given state vector u, denote by μ(u) the scalar spectral measure on [0, ∞[ given by μ(u) ( ) := u, E( )u . Let V be the maximal open set on which the distribution function w of μ(u) is real analytic. We consider the complex analytic continuations φ (u) of w |V on symmetric domains with respect to the real axis (u) so that φ (u) (z) = φ (u) (z) holds. Let φ0 denote the respective object for H0 . Definition 2. Let D be a dense set of state vectors such that for every u ∈ D there exist (u) complex analytic continuations φ (u) and φ0 on symmetric domains with respect to the γ real axis. Then a point p = ωˆ − i 2 ∈ C with ωˆ > 0, γ > 0 is a resonance if for some (u) u ∈ D it is a regular point of φ0 and singular for φ (u) . In the Appendix it is shown that the two definitions of a resonance are equivalent. More precisely, R (u) is analytic on C\[0, ∞[ and its analytic continuation across the positive real axis to the second Riemann sheet equals R (u) + 2π iφ (u) . The same holds true for the respective objects for H0 . The singularities of φ (u) occur in complex conjugate pairs. They manifest themselves as Lorentz profiles in φ (u) along the real axis. For an illustration assume the simplest case that the resonance p is a pole of first order and that there are no other singularities than p and its conjugate p in some open rectangle U , which is symmetric with respect to the real axis and the axis through p, p. Then there is a holomorphic function ϕ (u) on (u) (z) U such that φ (u) (z) = (z−ϕp)(z− p) for z ∈ U \{ p, p}. Hence along the real axis not far from ωˆ one has γ /2 , φ (u) (ω) constant (ω − ω) ˆ 2 + γ 2 /4 which is a Lorentz profile. A quasi energy eigenstate u, which in the energy representation shows a narrow frequency band around ω, ˆ decays exponentially. Indeed, the matrix element u, e−it H u is given by the Fourier transform μˆ (u) (t) = e−itω dμ(u) (ω) of the scalar spectral measure, which in the present case becomes ˆ μˆ (u) (t) = e−itω φ(ω)dω constant e−iωt e−tγ /2 .


817

The matrix element R (u) (z) of the resolvent equals the Stieltjes transform of the scalar spectral measure dμ(u) (ω) μ˜ (u) (z) := . ω−z The formula μ(t) ˆ = − lim→0+

1 π

∞

−∞

e−itω Im μ(ω ˜ + i)dω,

which converts the Stieltjes transform of any finite Borel measure μ on [0, ∞[ into its Fourier transform, allows to study the decay of a state u related to a resonance starting from R (u) . Cf. [1, I.32] and [11]. To summarize so far, we have seen that it is quite natural (and equivalent) to determine the resonances by the analytic continuation of the densities of scalar spectral measures of the Hamiltonian rather than by the analytic continuation of matrix elements of its resolvent. Now the question is how to apply these considerations to an effective system for which there is no Hamiltonian. Note that once the effective measure of states μ is determined, the probability distributions μ(β) of the energy are at our disposal for every temperature 1/β (see Sect. 2.4). These are the primary entities in case of an effective system. It appears natural that they have to replace the scalar spectral measures of the Hamiltonian. Hence common singular points (in the lower complex half-plane) of the analytic continuations φ (β) of the respective densities are attributed to resonances. It is immediate from (2.7) that the former are exactly the singular points of the analytic continuation φ of the density regarding the effective measure of states μ. For the uncou(β) (β) pled system, μ0 is purely atomic (by (2.2)) so that φ0 = 0. Thus we are led to the following Definition 3. Resonances of an effective system are the singularities (in the lower complex half-plane) of the analytic continuation of the effective density of states. We have seen that resonances according to Definition 3 give rise to Lorentz profiles along the real axis representing the natural lines of the system. The results in Theorems 3.5 and 3.6 then confirm that these resonances actually occur for the charged harmonic oscillator. What is more, they furnish a physically verifiable picture of the spectrum in its entirety. 3. Model and Main Results 3.1. Particle coupled to a photon field. Our starting point is the minimal coupling without a form factor (cf. [6, II.D.1 (D.1)]), in the dipole approximation of a charged particle interacting with a photon field and in a quadratic external potential. The Hamiltonian is given by H = (Hp + Hr ) ⊗ 1 + 1 ⊗ Hf + HI acting in L 2 (R3 ) ⊗ F ⊗2 . Let us explain the symbols above: Hp = −2

m 2 1 2 η q + q , 2 2m

(3.1)

818


acting in L 2 (R3 ), is the energy of the harmonically bound particle, m is the particle mass, and η is the frequency associated to the the harmonic external potential. Note that we represent the particle in momentum space, which will later turn out to be convenient. Hf = Hf,1 ⊗ 1 + 1 ⊗ Hf,2 and Hf,σ = ω(k)a∗ (k, σ )a(k, σ ) dk, σ = 1, 2 is the energy of the free field. Each Hf,σ , σ = 1, 2, acts in the symmetric Fock space (n) F= ∞ n=0 F , with F (n) = { f ∈ L 2 (R3n ) : f (k1 , . . . , kn ) = f (kπ(1) , . . . , kπ(n) ) for all permutations π, k j ∈ R3 }. ∞ (n) with (n) 2 We write an element F of the Fock space as F = ∞ n=0 F n=0 F < ∞ (n) (n) for F ∈ F . We will need the following well-known fact: if { f m : m ∈ N} is an orthonormal basis of L 2 (R3 ) or a subspace X thereof, then the system ˆ ...⊗ ˆ f m n : m 1 , . . . , m n ∈ N, n ∈ N} { fm1 ⊗ ˆ is an orthonormal basis of the Fock space over L 2 (R3 ) or X , respectively. Above, ⊗ denotes the symmetric tensor product. By definition, Hf,σ acts on F (n) as

n ω(k)a ∗ (k, σ )a(k, σ )dk F (n) (k1 , . . . , kn ) = ω(k j )F (n) (k1 , . . . , kn ), j=1

where ω(k) = c|k| is the photon dispersion relation with c the velocity of light. The interaction between the particle and the field is given by HI = HI,1 ⊗ 1 + 1 ⊗ HI,2 and

e (q · u σ (k)) a ∗ (k, σ ) + a(k, σ ) dk, HI,σ = χc (k) 2 m 4π ω(k) with an ultraviolet cutoff implemented through χc = 1{|k| 0. β β Moreover, Z (β; γ ) := lim N →∞ (Tr e− HN )/(Tr e− Hf,N ) exists for each β > 0, and ∞

ρ −3 ρ2 4 ρ −2ρ ln(1+γ ) sin ϕ 1+ 2 + sin ϕ arctan γ Z (β; γ ) = 2πρ e . l π l l l=1

(3.5) Formula (3.5) already appears in [4], where it is derived non-rigorously using path integrals. We present a conceptually simpler, and mathematically rigorous, proof of Theorem 3.2 in Sects. 5.1–5.3. Removing the ultraviolet cutoff in (3.5) results in a significantly simpler expression. Theorem 3.3. For all β > 0, Z (β) := limγ →∞ Z (β; γ ) exists, namely 2 3 ρ −2ρ ln ρ sin ϕ −iϕ e . Z (β) = (iρ e ) 2π

(3.6)

Moreover, β → Z (β) is the Laplace transform of a positive measure. The statements of Theorem 3.3 also appear in [4]. But the derivation of (3.6) is not fully rigorous there and the proof of the final statement contains a small error. Both results are improved in Sects. 5.4 and 6.1, respectively. We now turn to the effective measure of states μϕ of the charged oscillator, i.e. the Laplace inverse of Z ϕ . We display the dependence on the strength of the coupling by the −3 suffix ϕ ∈ [0, π2 ]. For ϕ = 0, we are in the case of zero coupling, Z 0 (β) = (2 sinh βη 2 ) is the partition function of the oscillator, and ∞

j +2 δ( j+3/2)η . μ0 = (3.7) 2 j=0

(β)

By (2.7), μϕ

=

temperature T =

1 −β(·) μ ϕ Z (β) e

kβ .

is the probability distribution of the energy at the

An easy first step is the following convergence result: (β)

Proposition 3.4. As ϕ → 0, convergence μϕ sense of the vague topology. (β)

(β)

→ μ0

and μϕ → μ0 holds in the (β)

Proof. One finds μˆ ϕ (t) = Z ϕ (β + it)/Z ϕ (β) for the Fourier transform of μϕ at (β) t ∈ R. Since Z ϕ (β + it)/Z ϕ (β) → Z 0 (β + it)/Z 0 (β) = μˆ 0 (t) as ϕ → 0, the vague (β) convergence of (μϕ )ϕ follows from the continuity theorem. This implies the vague convergence of (μϕ )ϕ simply noting that if ψ is a continuous function on R with compact support then so is exp(β (·))ψ. The foregoing result implies that the mass of μϕ concentrates near the lines at ( j + 3/2)η and vanishes in between as the coupling strength tends to zero. This is exactly what is expected. In the following we study the structure of μϕ more closely. We define π 3 sin ϕ + ( − ϕ) cos ϕ η, ωϕ := (3.8) π 2 and write δωϕ for the Dirac measure at ωϕ , and φλ1 for the Lebesgue measure on R with density φ.


821

Theorem 3.5. i) There exists a function φ : R → R being zero for ω < ωϕ and positive continuous for ω ≥ ωϕ with φ(ωϕ ) = πη sin ϕ such that μϕ = δωϕ + φλ1 . ii) φ is real analytic for ω > ωϕ , and extends to an analytic function on C up to singularities at p j = ωϕ + jη e−iϕ

and

p j = ωϕ + jη eiϕ ,

j ∈ Z\{0},

and cuts along z = ωϕ + s e±iϕ for s ∈ R, |s| ≥ η. So the zero-point frequency 23 η of the oscillator is shifted down to ωϕ when the oscillator couples to the radiation field, and φ is the effective density of states of the charged oscillator. Part i) above is already shown in [4], but in Sect. 6.2 an elementary proof is given. According to i) there is a stable ground state at the energy ωϕ , while at the same place the absolutely continuous part of the spectrum jumps to πη sin ϕ. Hence the ground state is no longer isolated. It also implies that there is no other stable and no singular part of the spectrum. Part ii) goes beyond these qualitative statements, and already shows the existence of resonances (cf. Definition 2) at p j , j ∈ N. Its proof will be given in Sect. 6.2. However, it still contains no information on the nature of the singularities; in order to observe unadulterated Lorentz profiles on the real line, these need to be simple poles. This, and more, is true. To formulate the corresponding result, let us define j (z) :=

1 −1 + , 2π i(z − p j ) 2π i(z − p j )

for z ∈ C. When restricted to the real line, jη sin ϕ 1 π (ω − ωϕ − jη cos ϕ)2 + ( jη sin ϕ)2 is a Lorentz profile with total mass j dω = 1. j (ω) =

Theorem 3.6. Let N ∈ N and z = ωϕ + s eiχ for 0 < s < N η and |χ | < N j+2 if |χ | < ϕ, j=1 2 j (z) + h N (z) φ(z) =

3 − j=1 (−1) j 3j j (z) + h˜ N (z) if |χ | > ϕ.

π 2.

Then

Above, h N and h˜ N are analytic up to singularities at p j , p j and cuts along z = ωϕ + s e±iϕ , s ≥ η. Moreover, there exists a constant C depending on N such that h N (and similarly h˜ N ) satisfies |h N (z)| ≤ C(1 + | ln(ϕ − |χ |)|ηN ) for |χ | < ϕ. The proof of this theorem is given in Sect. 6.3. The theorem shows that inside the cone |χ | < ϕ, the singularities of φ look like resonance poles of first order, but their overall structure is more complicated as isapparent from the behaviour for |χ | > ϕ. Nevertheless, they lead to Lorentz profiles j+2 j on the real axis. The total mass of 2

the j th Lorentz profile thus corresponds precisely to the multiplicity of the j th eigenvalue of the uncoupled harmonic oscillator, i.e. the mass of the j th delta peak of μ0 , for

822


(a)

(b)

150

1.5

100

1.0

50

0.5

j=4

j=3

j=2 j=1

2

4

6

8

j 12

j

j 12

∞

1 ∗n n=1 n! g ,

Fig. 1. a) The true effective density of states is given by an infinite sum of convolutions, = where g is given in (6.2). See Sect. 6 for details. Shown is an approximation to obtained by summing the first 15 terms, for ϕ = 1/100. This approximation is highly accurate on the displayed interval. b) shows the difference between the effective density of states and its approximation by the appropriate Lorentz-profiles ( j + 1)( j + 2) j /2, in the intervals [ j − 1/2, j + 1/2], for j = 1, 2, 3, 4. In both graphs, the units of energy on the x-axis are chosen such that the eigenvalues of the uncoupled harmonic oscillator occur at the elements of N0

all ϕ. Moreover, with Proposition 3.4 it follows that h N dω converges vaguely to zero as ϕ → 0. To summarize, the following complete picture of μϕ emerged: There is a stable ground state (Dirac peak of strength one) at ωϕ . Above, there is absolutely continuous spectrum, consisting of Lorentz profiles with widths

2α η γ j := 2 jη sin ϕ = jη , (3.9) 3 mc2 centered at

α2 ω j := ωϕ + jη cos ϕ = ωϕ + jη 1 − 18

η mc2

2

+ ... ,

(3.10)

and with total mass equal to that of the corresponding unperturbed state. Figure 1 illustrates the excellent approximation to the effective density of states obtained by the above description, even for the relatively large value ϕ = 1/100; the physical value for ϕ is about α −3 ≈ 0, 5 × 10−6 . With respect to a frequency scale starting at ωϕ the peaks at ω j for j = 1, 2, 3, . . . represent the natural lines of the charged oscillator due to transitions to the ground state (Lyman series). According to (3.10) the distance of two subsequent levels is η cos ϕ, which is smaller than η in the unperturbed case. The radiative shift (Lamb shift) η − η cos ϕ of the first excited state is of second order in α. For an observation of the natural lines the charged oscillator may be brought into contact with black body radiation at the temperature T = kβ . This causes additional negative shifts of the excited states depending on T which have to be taken into account. (β) They are determined by the positions of the peaks of μϕ . For 2T > (/k)γj one readily finds −βγ j2 /4 < ω j < −βγ j2 /8. Our results are in accordance with calculations within QED time–dependent perturbation theory up to first order in the fine-structure constant α (cf. e.g. [5, Sect. 5]). The calculations on the radiative cascade of a harmonic oscillator in [7, Exercises 15] confirm also that the shifted levels of the oscillator remain equidistant.


823

Remark. In our analysis, we have left out the self-interaction term A2 of the field. At least Theorem 3.2 can be achieved without this simplification. Actually, in the Göppert-Mayer description this term is absorbed by the coupling, compare [6, IV (B.34)] or [18, (13.124)] and [17, V.15 Eq. (15.1), (15.2)]. One is left again with an oscillator model which can be treated in the same way as we do here. The analogous formula to (3.5), without adding any extra renormalization term, is

∞

ρ −3 ρ2 4 l Z (β; γ ) = 2πρ 1 + 2 + sin ϕ γ − arctan γ . l π ρ l l=1

Again the limit γ → ∞ does not exist, but this time we do not know how to suitably renormalize this system. On the other hand, from the well-tested perturbation theory in QED one knows that for many problems concerning the interaction of bound electrons with radiation (e.g. spontaneous and stimulated emission and absorption) the linear part −ep A/m of the interaction Hamiltonian yields the main effects (see e.g. [13]). 4. Discretisation of the Photon Field Let us now give the proof of Proposition 3.1, and at the same time prepare the proof of Theorem 3.2. In effect, what we give is the reverse procedure to the one that, in the early days of QED, led from the description of the photon field as a collection of independent harmonic oscillators to the Fock space description. So in principle, what we are going to present is well known. However, we are not aware of any place where it is done carefully and in a mathematically satisfactory way, and thus present it in some detail. Let us start with some properties of the projections PN introduced in (3.3). For locally integrable f : R3 → C, and given j ∈ J N , we write 1 f¯j := 1 j , f , V for the average value of f on j . Thus, if f ∈ L 2 we have PN f (k) = f¯j 1 j (k), k ∈ R3 . j∈J N

Lemma 4.1. Assume f ∈ L 2 (R3 ), F ∈ F. Then we have: (i) As N → ∞, PN f → f in L 2 (R3 ), and P N F → F in F. (ii) If f is continuous on an open set B, then PN f (k) → f (k) for every k ∈ B. Proof. As to (ii), assume without restriction that f is real-valued. Choose k0 ∈ B. For all N large enough, we have j0 ⊂ B for the unique j0 such that k0 ∈ j0 . By continuity there are k N , k N in j0 satisfying f (k N ) = inf f | j and f (k N ) = sup f | j . Then 0 0 PN f (k0 ) = f¯j0 ∈ [ f (k N ), f (k N )], whence the assertion. Now (i) is proved in the standard fashion. Choose f to be the indicator function of a bounded measurable set B at first. Then for each ε > 0 there is a continuous function g with compact support such that g − 1 B L 2 < ε. Now PN 1 B − 1 B ≤ PN (g − 1 B ) + PN g − g + g − 1 B .

824


The middle term converges to zero by (ii) and dominated convergence, and so the left hand side is bounded by 2ε for all large N . This proves the claim for indicators of bounded measurable sets. As the latter are total in L 2 , this proves also the general result about (PN ) N . Finally, the result about ( P N ) N follows from the fact that second quantisation preserves strong convergence. It can be checked easily that P N F = F(PN L 2 (R3 )), the latter being the Fock space over the image of PN . Since {h j : j ∈ J N } is a orthonormal basis of PN L 2 (R3 ), we ˆ · · · ⊗h ˆ jn : jk ∈ J N , n ∈ N0 } is a orthonormal basis of P N F. These conclude that {h j1 ⊗ basis elements are eigenfunctions of P N Hf,σ P N , as ˆ · · · ⊗h ˆ jn = P N Hf,σ h j1 ⊗

n

ˆ · · · ⊗h ˆ jn . h j1 ⊗

ω¯ ji

i=1

This allows us to transfer the operator Hf,N unitarily to an L 2 space. For n ∈ N we write for any positive mass parameter μ, ψn (x) =

1 n 2 n!

μω¯ j π

1/4 e

−

μω¯ j 2

x2

Hn

μω¯ j x ,

for the n th Hermite function on R. For j ∈ J N and j := ( j1 , . . . , jn ) ∈ J Nn , n ∈ N let n( j, j ) denote the number of occurrences of the value j in j ; for n = 0 set j := ∅, h ∅ := 1 ∈ C and n( j, ∅) := 0 for all j ∈ J N . We put ˆ . . . ⊗h ˆ jn ∈ F (n) h j := h j1 ⊗

and

ψ j :=

ψn( j, j ) ∈ L 2 (R JN ).

j∈J N

Due to symmetrisation, h j actually only depends on the occupation numbers n( j, j ), j ∈ J N , and thus it is easy to check that the assignment θ (h j ) := ψ j defines an isomorphism from PN F onto L 2 (R JN ), and that θ P N Hf,σ θ

−1

2 ω¯ j μ 2 2 2 − ∂xσ j + ω¯ j xσ j − , = 2μ 2 2

(4.1)

j∈J N

where x1 j and x2 j are independent variables. Due to the relation 2x Hn (x) = Hn+1 (x) + 2n Hn−1 (x) we have √ √ n + 1ψn+1 (x) + nψn−1 (x) =

2μω¯ j xψn (x),

and using the definitions of a ∗ (g) and a(g), it is tedious but straightforward to check that √ Vμ e −1 θ P N HI,σ θ = ω¯ j ( y¯σ j · q)xσ j with yσ := u σ χc / ω. 2 2π m j∈J N


825

Thus we find that H N is unitarily equivalent to H˜ N = Hp + Hr,N +

j∈J N σ =1,2

ω¯ j μ 2 + − ∂x2σ j + ω¯ 2j xσ2 j − 2μ 2 2

Vμ e ω¯ j ( y¯σ j · q)xσ j , 2π 2 m (4.2)

acting in L 2 (R3+2JN ). Here we discretized also the renormalisation terms Hr,N =

V

j∈J N σ =1,2

(χc ) j e2 η2 1 e2 η2 2 ( y ¯ · q) − V σ j 2 2 2 2π m ω¯ j 4π m ω¯ j (ω¯ j + η) j∈J N

displaying the contribution to the effective oscillator mass and energy shift by each single photon. Since H˜ N is the Hamiltonian of a finite dimensional quantum oscillator, its spectrum is discrete and the eigenvalues are of finite multiplicity. Using Lemma 4.1, it is immediate to check (i) and (iii) of Definition 1 for 1 ⊗ P N ⊗ P N , and Proposition 3.1 is proved. 5. The Partition Function In this section we will prove Theorems 3.2 and 3.3. We start with a general, useful observation. 5.1. A trace formula for a quantum oscillator. Consider an operator H=

n −2 i=1

2m i

∂i2 +

n 1 Aii X i X i 2

(5.1)

i,i =1

with strictly positive real symmetric n × n-matrix A = (Aii ). H acts in L 2 (Rn ), ∂i are derivative operators with respect to xi , and X i are multiplication operators with xi . We assume that the mass parameters m i are all positive, and we define the diagonal mass matrix M := diag(m 1 , . . . , m n ). Lemma 5.1. Let λ1 , . . . , λn be the eigenvalues of M −1/2 A M −1/2 . Then −1 −1

n

β 1 β β 2 sinh( (M −1/2 A M −1/2 ) 2 λi ) = det 2 sinh Tr e− H = 2 2 i=1 −1

∞ 1 β 2 −1/2 ) M = β −n det(M −1/2 A M −1/2 )− 2 det In + ( A M −1/2 . 2πl

−1

l=1

Proof. Recall that β → 2 sinh( β2 ω) is the partition function of a one-dimensional harmonic oscillator with frequency ω. We are going to show that H is unitarily equivalent to n

−2 2 m 0 H˜ := (5.2) λi X i2 ∂i + 2m 0 2 i=1

826


with m 0 > 0 any mass parameter. Then, H˜ being the sum of n independent oscillators, the first equality holds. The second equality is obvious. The last equality holds true due to the factorization sinh z = z

∞

1+(

l=1

z 2 ) πl

valid for all z ∈ C. To show (5.2), we note that any invertible real n × n-matrix S gives 1 rise to a unitary transformation U on L 2 (Rn ) by U f := |detS| 2 f ◦ S. We choose S in the following way: Since M −1/2 AM −1/2 is real symmetric, there exists an orthogonal 1 1 1 matrix O such that O M − 2 AM − 2 O T = diag(λ1 , . . . , λn ). We put S := O( m10 M) 2 . Then S satisfies S M −1 S T =

1 In , m0

S −1T AS −1 = m 0 diag(λ1 , . . . , λn ),

and from these relations H˜ = U −1 H U follows straightforwardly.

5.2. The partition functions for Hf,N and H N . We will now use Lemma 5.1 in order to compute the partition functions for the discretized systems. Note that from (4.1) and Lemma 5.1 we immediately get −2

β β β j∈J ω¯ j N Tr e− Hf,N = e . (5.3) 2 sinh( ω¯ j ) 2 j∈J N

Now by (4.2), H˜ N has the form (5.1) up to the constant term (χ c ) j e2 η2 − . ω¯ j + V 2 4π m ω¯ 2j (ω¯ j + η) j∈J N

1 Precisely, the mass matrix M is the diagonal matrix of size 2|J N | + 3, with mη 2 as the first three diagonal elements, and μ as the remaining ones. A is a block matrix of the form ⎞ ⎛ B B1 B2 · · · B|JN | ⎜ B1T s1 I2 0 · · · 0 ⎟ ⎟ ⎜ .. ⎟ ⎜ T .. ⎟ . . 0 s I B (5.4) A=⎜ 2 2 ⎟ ⎜ 2 .. ⎟ ⎜ .. .. .. ⎝ . . . . 0 ⎠ T B|JN | 0 ··· 0 s|JN | I2

with B =

(I j denotes the j-dimensional unit matrix), s j = μω¯ 2j for 1 ≤ j ≤ |J N |, " Vμ e and B j = (v1 j , v2 j ) ∈ R2×3 , with vσ j = 2π ω¯ j y¯σ j for σ = 1, 2. It is clear that 2 m both of the matrices whose determinant appears in the second line of Lemma 5.1 are of the above form, too. We thus need a formula for determinants of matrices of the type (5.4). 1 m I3


827

Lemma 5.2. Let Q be a real symmetric (2L + 3)-dimensional matrix of the form (5.4), where B is 3-dimensional and all s j = 0. Then ⎛ ⎞ L 1 det Q = s12 . . . s L2 det ⎝ B − B j B Tj ⎠. sj j=1

Proof. We write B L = (a, b), where a and b are 3-dimensional column vectors. We assume for the moment that a and b are linearly independent. At the end this assumption can be dropped by continuity. Define a 3 × 3-matrix X by X T := (x, y, z) with x := a×b, y := b×x and z := x×a. Then X B L = |a×b|2 (e2 , e3 ) with I3 = (e1 , e2 , e3 ) and X −1 = |a × b|−2 (x, a, b) hold. The following operations do not change det Q. First pass to the equivalent matrix diag(X, I2L ) Q diag(X −1 , I2L ). Then the last block in the first block line becomes zero adding −|a × b|2 /s L times the last two lines to the second and third line of the matrix. Expanding now the determinant along the last two columns one obtains the factor s L2 times the determinant of a (2L +1)-dimensional matrix Q . One finds that diag(X −1 , I2L−2 ) Q diag(X, I2L−2 ) arises from Q by canceling the last block line and last block column and replacing B with B − s1L B L B LT . Iterating these operations the result follows. Thus by elementary computations, Lemma 5.1 and Lemma 5.2 yield the following formula for the ratio of the partition functions of the discretised systems: β

Z N (β) :=

Tr e− HN

−3

β

j∈J N

V

(χ c ) j e2 η2 4π 2 m ω¯ 2 (ω¯ j +η) j

= (βη) e β Tr e− Hf,N ⎛ ⎞−1 ∞ 2 2 η2 y¯σ j y¯σT j η e ⎠ , V × det ⎝(1 + 2 )I 3 + 2π 2 m νl ω¯ (ω¯ 2j + νl2 ) l=1 j∈J σ =1,2 j

(5.5)

N

with 1 yσ := χc √ u σ , ω

νl =

πl . β/2

5.3. The continuum limit. We will now show that in formula (5.5), the limit N → ∞ exists, yielding (3.5). We use the abbreviation Sl,N

y¯σ j y¯σT j e2 η2 V := . 2π 2 m ω¯ (ω¯ 2j + νl2 ) j∈J σ =1,2 j N

Let λ denote the Lebesgue measure on R3 and put e2 η2 u σ u σT Sl := χc dλ. 2π 2 m ω2 (ω2 + νl2 ) σ =1,2 Proposition 5.3. lim N →∞ Sl,N = Sl for each l ∈ N.

(5.6)

828


Proof. We will use dominated convergence. Put gl,N ,σ :=

y¯σ j y¯σT j

j∈J N

ω¯ j (ω¯ 2j + νl2 )

1 j .

(5.7)

e2 η2 gl,N ,σ dλ. 2π 2 m

(5.8)

By the definition of V , we have Sl,N =

σ =1,2

The functions yσ , which contain the transverse vector fields, can be chosen to be continuous outside a closed Lebesgue null subset of R3 containing the origin. Thus applying Lemma 4.1 (ii) on every factor on the right-hand side of (5.7) one deduces that gl,N converges to the integrand in (5.6) almost everywhere. Now each component of gl,N ,σ is bounded by √ 2 1/ ω j∈J N

j

√ 4 1 1 1 j ≤ 1/ ω 1 j , 2 j ν2 ω¯ j νl l j∈J N

where the inequality is obtained by applying Jensen’s inequality to the convex function √ √ φ : (0, ∞) → R, x → 1/ x, yielding 1/ ω¯ j ≤ ( 1/ ω ) j for each j. Furthermore, we claim that √ 1 1/ ω 1 j ≤ 4 √ 1 j . (5.9) j ω √ ¯ j , we replace j with the portion of the ball of radius 3/N Indeed, in the case 0 ∈ that lies in the same sector. Then j is contained in that set, and √3/N √ √ √ 1 4π 35/4 π 1/2 33/2 π N if r ≤ 3/N , c 1/ ω = r 3/2 dr = ≤ √ j V 8 0 5 5 r ¯ j , we let (k/N , l/N , m/N ) be the corner of j whence the claim in this case. If 0 ∈ closest to the origin, and find √ sup j (1/ r ) √ √ 1 (k + 1)2 /N 2 + (l + 1)2 /N 2 + (m + 1)2 /N 2 c 1/ ω ≤ √ . √ =√ j r inf j (1/ r ) r k 2 /N 2 + l 2 /N 2 + m 2 /N 2 √ The last expression is maximal for (k, l, m) = (0, 0, 1) and takes the value 6 there. Thus (5.9) holds. Hence every component of gl,N ,σ is bounded by (16/νl )2 ω−2 , which is locally integrable. This finishes the proof. A similar, but easier proof shows that lim

N →∞

j∈J N

(χ c ) j e2 η2 e2 η2 V 2 = 4π m ω¯ 2j (ω¯ j + η) 4π 2 m

χc dλ. 2 ω (ω + η)

(5.10)


829

The integrals appearing in (5.10) and (5.6) can be calculated. We have 4π χc Cc dλ = 3 ln(1 + ). ω2 (ω + η) c η Together with the prefactor, and comparing with the definitions of sin ϕ, ρ and γ , we obtain the exponent in (3.5). For (5.6), note first that σ =1,2 u σ u σT = I 3 − |k|1 2 kk T follows from the orthonormality of u 1 , u 2 and k/|k|. Thus for each component of u σ u σT χ dλ, the angular part can be calculated, and is found to be equal to σ =1,2 ω2 (ω2 +ν 2 ) c 8π 3 .

l

Then σ =1,2

8π cC u σ u σT I 3. dλ = arctan χ c 2 3 2 2 3c νl νl ω (ω + νl )

Taking the prefactor into account and computing the now trivial determinant gives the l th factor of the infinite product in (3.5). The final step is to justify the exchange of the infinite product with the limit N → ∞. To this end, we write, for each l ∈ N, Ml,N :=

η2 I 3 + Sl,N , νl2

Ml := lim Ml,N = N →∞

η2 I 3 + Sl . νl2

Lemma 5.4. We have ∞

lim

N →∞

det(I 3 + Ml,N ) =

l=1

∞

det(I 3 + Ml ).

l=1

Proof. By the continuity of the exponential function, it will suffice to prove the result for the logarithms. Then the quantity of interest is given by ∞ ∞ ln det(I 3 + Ml,N ) = ln(det(I 3 + Ml,N )). l=1

l=1

Since by continuity lim ln(det(I 3 + Ml,N )) = ln(det(I 3 + Ml ))

N →∞

for each l ∈ N, the result will follow by dominated convergence. Indeed, at the end of the proof of Proposition 5.3 we have seen that Sl,N is bounded by constant/l 2 uniformly in N . Thus so is Ml,N and hence its three eigenvalues. It follows that | det(I 3 + Ml,N )) − 1| ≤ constant/l 2 , and therefore | ln(det(I 3 + Ml,N ))| ≤ constant/l 2 uniformly in N .

We have thus proved Theorem 3.2.

830


5.4. Removing the ultraviolet cutoff. We now investigate the limit γ → ∞ in (3.5). Clearly, what we need to show is the convergence of ∞

f (γ ) := −2 sin ϕ ρ ln(1 + γ ) +

ln(1 + h l (γ )),

l=1

where h l (γ ) :=

γρ ρ2 4 ρ sin ϕ arctan . + l2 π l l

We first note that ρ2 0 < h l (γ ) < 2 l

4 1 + γ sin ϕ , π

0 < h l (γ ) h l :=

(5.11)

ρ2 ρ + 2 sin ϕ monotonically for γ ∞. 2 l l

(5.12)

By (5.11), (h l (γ )) is summable for all γ , and thus f (γ ) =

∞

h l (γ ) − 2 sin ϕ ρ ln(1 + γ ) +

l=1

∞

(ln (1 + h l (γ )) − h l (γ )) . (5.13)

l=1

We now use the formula C = lim

s→∞

∞ 2 1 s arctan( ) − ln s π l l

(5.14)

l=1

for the Euler/Mascheroni constant C. The first bracket above is then equal to

∞ ρ2 l=1

γ + 2 sin ϕ ρ ln ρ + ln 2 l 1+γ

∞ ργ 2 1 + arctan − ln(ργ ) , π l l l=1

∞ ρ 2 and by (5.14) converges to l=1 + 2 sin(ϕ)ρ(C + ln(ρ)). The last sum in (5.13) l2 ∞ converges to l=1 (ln (1 + h l ) − h l ) by (5.12) and monotone convergence, since ln(1 + x) − x is monotone decreasing. Combining these two results, we obtain f := lim f (γ ) = 2 sin ϕ ρ(C + ln(ρ)) + γ →∞

= 2 sin ϕ ρ(C + ln(ρ)) + ln

∞

∞

ln (1 + h l ) − 2

l=1

(1 + h l ) e

−2 ρl sin ϕ

ρ sin ϕ l

.

l=1

Taking into account that 1 + h l = (1 + ρl i e−iϕ )(1 + ρl i eiϕ ), and writing 2i sin ϕ = eiϕ − e−iϕ , we find


ef =

1 2ρ ln(ρ) sin ϕ e ρ2

831

∞ C ρi e−iϕ ρ −iϕ ρ i e−iφ ρ e ie 1 + el l j=1

2 .

Using the representation (z)−1 = z eC z

∞ l=1

z z (1 + ) e− l l

for Re z > 0,

for the Gamma function on the right half-plane, we arrive at (3.6). 6. The Effective Measure of States In this section, we investigate the inverse Laplace transform of Z . As a first step, we prove the second part of Theorem 3.3. 6.1. Complete monotonicity. By Bernstein’s theorem, a function f on [0, ∞[ is the Laplace transform of a positive Borel measure if and only if it is completely monotone (c.m.), i.e. (−1)n ∂xn f (x) ≥ 0 for all x ≥ 0. To show that Z is c.m., we use (3.6) in order to get 1 ln Z (β) = ln ρ − ln(2π ) − 2 sin ϕ ρ ln ρ + ln (ρ i e−iϕ ) + ln (ρ i e−iϕ ). 3 We now use Binet‘s formula ∞ −t z

1 ln(2π ) 1 e 1 1 − dt + + (z − ) ln(z) − z, ln (z) = − −t t 1− e t 2 2 2 0 valid for Re(z) > 0. Since ln(¯z ) = ln(z) for all z ∈ C, we obtain ∞ ln Z (β) = e−tρ g(t) dt − tϕ ρ

(6.1)

0

with

1 6 1 1 Re − , (6.2) − t 1 − e−τ τ 2

where τ := i e−iϕ t and tϕ := 6 sin ϕ + ( π2 − ϕ) cos ϕ . Below we will show that g(t) ≥ 0 for all t ≥ 0. This immediately implies that the first term on the right hand side of (6.1) is c.m. Since the product of two c.m.functions is c.m. ∞ also exp( f ) is c.m. if f is. Thus Z (β) is c.m. as the product of exp( 0 e−tρ g(t) dt) and the clearly c.m. function exp(−tϕ ρ). The claim g(t) ≥ 0 follows from g(t) :=

Lemma 6.1. For all w ∈ C with Re(w) > 0 we have

1 1 1 h(w) := Re − ≥ 0. − 1 − e−w w 2 Equality holds if an only if Re(w) = 0.

832


Proof. We write w = u + iv. If u = 0, then h(iv) = direct calculation yields

1−cos v 1−2 cos v+1

−

1 2

= 0. For u > 0, a

(u 2 + v 2 )(1 − e−2u ) − 2u(1 + e−2u − 2 e−u cos v) 2(1 − 2 e−u cos v + e−2u )(u 2 + v 2 ) 2 2 (u + v ) sinh(u) − 2u(cosh(u) − cos(v)) = 2(u 2 + v 2 )(cosh(u) − cos(v))

h(z) =

with nonnegative denominator. We need to show that the numerator r (u, v) is positive if u > 0. By symmetry it suffices to treat v ≥ 0. Take first v = 0. Then ∂u (r (u, 0)/u) = u cosh u − sinh u, which at u = 0 equals 0. Since ∂u2 (r (u, 0)/u) = u sinh u > 0, u → r (u, 0) > 0 for u > 0. Thus the result holds if ∂v r (u, v) = 2(v sinh u − u sin v) is nonnegative for all u > 0, v > 0. But this is clearly true, since (sinh u)/u > 1 > (sin v)/v for all u, v = 0. 6.2. Analytic continuation. Here we prove Theorem 3.5. Let us define Y (ρ) := Z (β) etϕ ρ . ηβ , the inverse Laplace transforms of Y (ρ) and Z (β) are related by translaSince ρ = 2π tion and scaling. The key observation is that, due to (6.1), Y is itself the exponential of a Laplace transform,

Y (ρ) = exp Lg(ρ) = 1 +

∞ 1 Lg ∗n (ρ). n!

(6.3)

n=1

Above, L denotes the Laplace transformation, and x h ∗ k (x) = h(t)k(x − t) dt

(6.4)

0

is the convolution of functions supported on [0, ∞[. Since g is bounded on compact intervals and nonnegative, the final sum in (6.3) converges uniformly on compact intervals. Let us define :=

∞ 1 ∗n g . n!

(6.5)

n=1

By Lemma 6.1, all terms of the above sum are positive. Thus L = Y − 1 holds by monotone convergence, and the uniqueness of Laplace transforms shows L−1 Y = δ0 +. Moreover, a direct calculation shows that g(t) → 21 sin ϕ as t → 0. Thus the same holds 2π for , and we have shown the first part of Theorem 3.5, with φ(ω) = 2π (ω − ω ) ϕ η η for ω ≥ ωϕ and zero else. We now investigate the analytic continuation of . First note that the analytic continuation of g is given by 3 1 1 2 sin ϕ g(z) = + − −1 . z 1 − exp(−i e−iϕ z) 1 − exp(i eiϕ z) z


833

Lemma 6.2. g is a meromorphic function on C with simple poles at the zeros of the denominators q j = 2π j e−iϕ The residues are Res(g; q j ) =

and q j = 2π j eiϕ , 3i 2π j ,

Res(g; q j ) =

j ∈ Z\{0}.

−3i 2π j .

The proof is routine. Note only that we have seen already above that the common zero z = 0 of the denominators actually is a regular point of g. Let us now consider the analytic continuation of the convolutions. Define P := {z : z = q j or z = q j for all j ∈ Z\{0}} and S := {z : z = s e±iϕ for real s with |s| ≥ 2π }. For brevity use A(S, P) to denote the set of functions f that are analytic on C\S such that the components of S\P are cuts for f emanating from the points of P. Analytic functions on C\P are elements of A(S, P). For any function f and any subset K of C, let f K denote the supremum of | f | on K . Lemma 6.3. Let h ∈ A(S, P) and let k be analytic on C\P. For z ∈ C\S set

z

F(z) :=

h(ζ )k(z − ζ ) dζ,

0

integrating along the straight line joining 0 and z. Then F ∈ A(S, P) and F is an analytic continuation of h ∗ k as defined in (6.4). Let K be any compact subset of C\S and let K˜ denote the union of all straight lines joining the origin with some point of K . Then F K ≤ id K h K˜ k K˜ . Proof. Plainly, F agrees with h ∗ k on the real axis and is differentiable at all z ∈ C\S. Hence F is an analytic continuation of h ∗ k on C\S. We turn to the analytic continuation F˜ of F, e.g., from above across the cut between the points q j and q j+1 with j ∈ N. Let z ∈ S1 := {z : z = s e−iϕ for s ≥ 2π } be between q j and q j+1 . Join the points 0 and z along S1 but avoiding ql , respectively z − ql , for l = 1, . . . , j, by making a small detour above, respectively below, S1 . By analogous curves γz one joins 0 to z for every z in the open disk D centered at 21 (q j + q j+1 ) with radius π . Let h˜ be an analytic continuation of h from above S1 across the cuts along S1 . Then ˜ F(z) :=

γz

˜ )k(z − ζ ) dζ h(ζ

defines an analytic function on D. It extends F, since for z ∈ D above S1 the closed curve composed by γz and the straight line from z to 0 does not contain any singularity ˜ )k(z − ζ ). — The remainder is obvious. of ζ → h(ζ n By the lemma, g ∗n ∈ A(S, P) for all n ∈ N. Moreover, g ∗n K ≤ idn−1 K g K˜ . Obviously, a similar estimate holds more generally for the analytic continuations F˜ of ˜ Hence the series in (6.5) converges uniformly on compact F on compact K ⊂ dom F. sets implying that the limiting function belongs to A(S, P). This proves the second part of Theorem 3.5 when taking into account that the translation and scaling takes q j into p j .

834


Let us comment on the cuts of F from Lemma 6.3. Even if h and k are analytic on C\P with poles at points of P, the convolution F may have cuts. More precisely, e.g., if z lies on the cut between q j and q j+1 with j ∈ N then F+ (z) − F− (z) = −2π i

j

[k(z − ql ) Res(h; ql ) + h(z − ql ) Res(k; ql )],

l=1

where F+ (z) and F− (z) denote the limit values of F at z approaching z from above and from below the cut, respectively. This is an immediate consequence of the Residue Theorem integrating the meromorphic function ζ → Mz (ζ ) := h(ζ )k(z − ζ ) along the simply closed curve, which is symmetric with respect to S1 and which joins 0 to z by γz . In case of F = g ∗ g the above formula yields the jump function F+ (z) − F− (z) =

j 6 g(z − 2π l e−iϕ ). l l=1

6.3. Analysis of the singularities. Here we prove Theorem 3.6. Again it suffices to ana 2π 2π lyse in (6.5) instead of φ as φ(ω) = η η (ω − ωϕ ) for ω ≥ ωϕ . We refer to the analytic continuations of the convolutions g ∗n and of defined in Sect. 6.2. First note the following formula for z = s eiχ with s > 0 and |χ | < ϕ,

z 0

ln(z − q j ) − ln q j + ln(z − qk ) − ln qk 1 1 −2π i dζ = + , ζ −q j z −ζ − qk z − q j − qk z − q j − qk (6.6)

1 1 1 = z−q1j −qk ζ −q + which follows from the partial fraction expansion ζ −q j z−ζ −qk j 1 −1 and z−ζ −qk evaluating the primitives ln(ζ − q j ) and − ln(z − ζ − qk ) of (ζ − q j )

(z−ζ −qk )−1 , respectively. Note that the second term in (6.6) is regular at q j +qk = q j+k . Next let us define recursively the coefficients c j1 = c˜ j1 =

3 , j

c j,n+1 =

j−1

ck1 c j−k,n , c˜ j,n+1 = −

k=1

j−1

c˜k1 c˜ j−k,n

(6.7)

k=1

for all j, n ∈ N, where the void sums for j = 1 are zero. Set s jn (z) :=

−c jn , 2π i(z − q j )

s˜ jn (z) :=

−c˜ jn . 2π i(z − q j )

In the following, we will concentrate on the fourth quadrant of C. Subsequently it will be easy to extend the result to the right half-plane. Fix N ∈ N and let z = s eiχ with 0 ≤ s ≤ 2π N and − π2 ≤ χ ≤ 0.


835

Proposition 6.4. Then g ∗n (z) =

N

s jn (z) + L n (z)

for |χ | < ϕ,

s˜ jn (z) + L˜ n (z)

for |χ | > ϕ

j=1

g ∗n (z) =

N j=1

hold with |L n (z)| and | L˜ n (z)| bounded by An (1+|ln sin |ϕ − |χ |||s ) for some constant A. Proof. We proceed by induction. The statement for n = 1 follows from Lemma 6.2; indeed, subtracting from g the first order poles at q1 , . . . , q N leaves a bounded function L 1 . Let us now assume that g ∗n has the asserted decomposition and consider the case |χ | < ϕ. Then g ∗(n+1) (z) =

N

z

s j1 (ζ )skn (z − ζ ) dζ

j,k=1 0

+

N

z

s j1 (ζ )L n (z − ζ ) + s jn (ζ )L 1 (z − ζ ) dζ.

(6.8)

j=1 0

By (6.6) we find

z

s j1 (ζ )skn (z − ζ ) dζ =

0

−c j1 ckn + L jkn (z), 2π i(z − q j+k )

(6.9)

where L jkn is regular at q j+k . The first term above contributes to s j+k,n+1 . We will show below that none of the remaining terms entering g ∗(n+1) has any first order poles, and thus by collecting all terms with j + k = m we obtain the recursive equation for cm,n+1 . The calculation for s˜ jn , i.e. for |χ | > ϕ, is very similar. The only difference is that the residue of the pole at q j+k in (6.6) is 2π i instead of −2π i, a difference which is due to two jumps of 2π i of the logarithmic terms at the cut along the negative real axis. This gives the additional minus sign in the recursion for c˜ jn . We turn to L jkn . As mentioned above, there is no singularity at q j+k . There are two logarithmic singularities at q j and qk . Other than that, L jkn is bounded. Set δ := |ϕ −|χ || and h(δ) := | ln(sin δ)|. Then |L jkn (z)| ≤ c j1 ckn K (1 + h(δ)) for some constant K . It is immediate from (6.7) that c jn = 0 for j < n. Thus there exists some constant B, independent of n, such that N

|L jkn (z)| ≤ B (1 + h(δ)).

j,k=1

Now we tackle the second line of (6.8). By the induction hypothesis, |L n (z)| ≤ An (1 + h(δ)s ). Thus z I jn (z) := s j1 (ζ )L n (z − ζ ) dζ 0 s s n ≤ A h(δ)s |s j1 (y eiχ )| h(δ)−y dy + An |s j1 (y eiχ )| dy. 0

0

836


For s ≤ π , the integrands on the right hand side above are bounded, and hence I jn (z) ≤ D An (1 + h(δ)s ) for some constant D. For s > π , we decompose the domain of integration into y ≤ π and π < y ≤ s. On the first interval, the integrands are bounded with the same result as above. On the second interval, we replace h(δ)−y by h(δ)−π . The integral over s1 j alone is clearly bounded by E(1 + h(δ)) for some constant E, and we estimate crudely N

I jn (z) ≤ 3N An (D + E)(1 + h(δ)s ).

j=1

Finally, since L 1 is bounded, we have N z ≤ F(1 + h(δ)), s (ζ )L (z − ζ ) dζ jn 1 j=1

0

where the constant F does not depend on n. Altogether, we find |L n+1 (z)| ≤ 3N An (D + E + B + F)(1 + h(δ)s ). Clearly, setting A = 3N (1 + D + E + B + F), this is bounded by An+1 (1 + h(δ)s ). In order to extend this result to the right half-plane one has to take account of the poles q j of g, too. This amounts to replacing s jn (z) by s jn (z)+s jn (z). Let Mn denote the remainder in place of L n . It satisfies the same kind of estimate with some new constant A. Using (6.5) and the fact that c jn = 0 for j < n, we now have ⎛ ⎞ j N c jn −1 1 ⎝ ⎠ (z) = + + M(z) n! 2π i(z − q j ) 2π i(z − q j ) j=1

n=1

∞

1 Mn and |M(z)| ≤ e A (1 + h(δ)s ) for −ϕ < χ < ϕ. The same forwith M := n=1 n! mula with c˜ jn and M˜ holds for |χ | > ϕ. Obviously h(δ)s can be replaced by | ln δ|2π N .

Proposition 6.5. Define c jn and c˜ jn as in (6.7). Then, for all j ∈ N, j c jn n=1

n!

=

j c˜ jn j +2 j+1 3 , and = (−1) j 2 n!

if j ≤ 3, and 0 otherwise.

n=1

j Proof. We introduce the generating functions Fn (x) = ∞ j=1 c jn x . Then, F1 (x) = ∞ 3 j n j=1 j x = −3 ln(1 − x), and Fn = F1 . The last statement follows from ⎛ ⎞ j−1 ∞ ∞ ⎝ ck1 c j−k,n ⎠ x j = c j,n+1 x j = Fn+1 (x). F1 (x)Fn (x) = j=1

k=1

Now recall that c jn = 0 for j < n. Then ∞ ∞ c jn 1 j F1 (x)n = = ∂x n! j! n! x=0 n=1 n=1 1 j −3 ln(1−x) = ∂x e = j! x=0

j=1

1 j F1 (x) − 1 ∂x e j! x=0

1 1 j j +2 . ∂x = 2 j! (1 − x)3 x=0


837

For the generating function Fñ of (c˜ jn ), a similar calculation leads to Fñ = (−1)n+1 F˜1n with F˜1 = F1 . Thus, as above, ∞ c˜ jn n=1

n!

=−

1 j 3 ln(1−x) 1 j 3 ∂x e ∂ = − (1 − x) . x j! j! x=0 x=0

This equals 3 for j = 1, −3 for j = 2, 1 for j = 3, and zero otherwise, as was claimed. The stated analyticity properties of h N (z) and h˜ N (z) from Theorem 3.6 are true by the fact that both φ (cf. Theorem 3.5) and j possess them. This concludes the proof of Theorem 3.6. Acknowledgements. We would like to thank Herbert Spohn for many useful discussions, and Marco Merkli for helpful comments. V.B. is supported by the EPSRC fellowship EP/D07181X/1.

7. Appendix Here we show that Definition 2 of a resonance is equivalent to that given e.g. in [16, XII.6]. We start with a lemma that connects the analyticity of a measure’s density with properties of its Stieltjes transform. Let μ be a finite Borel measure on [0, ∞[ and f := μ˜ its Stieltjes transform. Note that f is holomorphic on C\[0, ∞[ and satisfies f (z) = f (z). We recall the classical Stieltjes inversion formula valid for t ≥ 0 (see e.g. [19, Theorem B1]: 1 δ→0+ →0+ π

μ([0, t]) = lim lim

t+δ

Im f (s + i) ds.

(7.1)

0

Lemma 7.1. Let t0 > 0 and U ⊂ C be an open disc centered at t0 with radius smaller than t0 . Then the following two statements are equivalent: (i) There is an holomorphic function F on U which equals f on {z ∈ U : Im z > 0}. (ii) μ on U ∩ R is absolutely continuous with respect to Lebesgue measure, and its density is the restriction on U ∩ R of a holomorphic function φ on U . If (i) and (ii) hold, then obviously f (t) := lim→0+ f (t + i) converges uniformly on compact subsets of U ∩ R, and F = f + 2π iφ holds on {z ∈ U : Im z < 0}. Proof. Assume first that F exists. Then (7.1) yields immediately for t1 , t2 in U ∩R, t1 < t t2 : μ(]t1 , t2 ]) = π1 t12 Im F(s)ds. This implies that μ on U ∩ R is absolutely continuous with respect to Lebesgue measure and that the density is given by t → π1 Im F(t). Then φ(z) := 2π1 i (F(z) − F(z)) is its analytic continuation on U . For z ∈ U with Im z < 0 one has F(z) = f (z) = f (z), whence F(z) = f (z) + 2π iφ(z). Now assume the existence of φ. For z ∈ U set F(z) := f (z) if Im z > 0 and F(z) := f (z) + 2π iφ(z) if Im z < 0. Then F is holomorphic on U \R. Fix t ∈ U ∩ R. Let δ > 0 be such that {t − δ, t + δ} ⊂ U . We define three paths in C. First γ1 (s) := s for s ∈ [0, ∞[. Then γ2 differs from γ1 only in that it joins the point t − δ to t + δ not by the straight line but by the semi-circle through t + iδ. Finally the closed path γ3 joins

838


t − δ to t + δ forward by the straight line and backward by the semi-circle through t + iδ. Then for 0 < < δ one has dμ(z) dμ(z) φ(z)dz − = = 2π i φ(t + i) γ1 z − (t + i) γ2 z − (t + i) γ3 z − (t + i) by the residue theorem. Therefore lim→0+ f (t + i) =

γ2

dμ(z) + 2π i φ(t) z−t

exists. Similarly lim→0+ f (t − i) is shown to exist. Set F(t) := lim→0+ f (t + i). Thus F is defined on the whole of U . It remains to show that F stays holomorphic. By the following lemma F(t + i) − F(t − i) = f (t + i) − f (t − i) − 2π i φ(t) + 2π i (φ(t) − φ(t − i)) → 0 as → 0+ uniformly on compact subsets of U ∩ R. From this the premises on F of Morera’s Theorem easily follow, whence the result. Now the equivalence of Definition 2 with the traditional definition of resonances (u) follows by applying Lemma 7.1 to μ = μ(u) and μ = μ0 : choosing U such that the (u) (u) density φ of μ or μ0 is analytic on U , we find that the continuation F on U of the resolvent to the second Riemann sheet is given by f + 2π iφ. We close with a lemma showing that a strong version of (7.1) holds if the density of μ is continuously differentiable. Lemma 7.2. Let μ be absolutely continuous with respect to Lebesgue measure on ]A, B[ for 0 ≤ A < B and let the density φ be continuously differentiable. Then φ(t) = lim→0+ π1 Im f (t + i) holds uniformly on every compact subset of ]A, B[.

/π 1 Proof. Put χ (s; t, ) := (s−t) 2 + 2 . Then π Im f (t + i) = [0,∞[ χ (s; t, )dμ(s). Let A < A1 < B1 < B and t ∈ [A1 , B1 ]. /π i) First [B,∞[ χ (s; t, )dμ(s) ≤ (B−B 2 [0,∞[ dμ(s) → 0 uniformly as → 0+. ) 1 Similarly this holds for [0,A] χ (s; t, )dμ(s). B B−B1 t−A 1 ii) Next 1 ≥ A χ (s; t, ) ds = π1 (arctan B−t − − arctan ) ≥ π (arctan B A1 −A arctan ) → 1 as → 0+. This implies that A χ (s; t, ) ds tends uniformly to 1 as → 0+. B B iii) Because of i) it remains to show that | A φ(s)χ (s; t, )ds − φ(t)| ≤ | A (φ(s) − B φ(t))χ (s; t, ) ds|+φ(t)| A χ (s; t, )ds−1| tends uniformly to 0 as → 0+. This holds true for the second summand because of ii). As to the first summand the mean value theorem yields |φ(s)−φ(t)| ≤ c|s −t| with c := sup{φ (τ ) : τ ∈ [A1 , B1 ]}. B−t B This finishes the proof since A |s − t|χ (s; t, ) ds = π A−t |r |(r 2 + 2 )−1 dr ≤ 2 B B 2 + 2 2 2 −1 π 0 r (r + ) dr = π ln 2 → 0 as → 0+. References 1. Bach, V., Fröhlich, J., Sigal, I.M.: Quantum Electrodynamics of Confined Nonrelativistic Particles. Adv. in Math. 137, 299–395 (1998) 2. Bach, V., Fröhlich, J., Sigal, I.M.: Return to Equilibrium. J. Math. Phys. 41, 3985–4060 (2000)


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3. Betz, V., Hiroshima, F., L˝orinczi, J., Minlos, R.A., Spohn, H.: Ground state properties of the Nelson Hamiltonian - A Gibbs measure-based approach. Rev. Math. Phys. 14, 173–198 (2002) 4. Castrigiano, D.P.L., Kokiantonis, N.: Quantum oscillator in a non-self-interacting radiation field: Exact calculation of the partition function. Phys. Rev. A 35(10), 4122–4128 (1987) 5. Castrigiano, D.P.L., Kokiantonis, N., Stiersdorfer, H.: Natural Spectrum of a Charged Quantum Oscillator. Il Nuovo Cimento 108(7), 765–777 (1993) 6. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Photons and Atoms. New York: John Wiley, 1987 7. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Atom-Photon Interactions. New York: John Wiley, 1998 8. Feynman, R.P.: Statistical Mechanics. Reading, MA: Benjamin, 1972 9. Griesemer, M., Lieb, E., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3), 557–595 (2001) 10. Hiroshima, F., Spohn, H.: Enhanced binding through coupling to a quantum field. Ann. Henri Poincaré 2, 1159–1187 (2001) 11. Hunziker, W.: Resonances, Metastable States and Exponential Decay Laws in Perturbation Theory. Commun. Math. Phys. 132, 177–188 (1990) 12. Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. Proc. Symp. in Pure Math. 76(2), 649 (2007) 13. Louisell, W.H.: Quantum Statistical Properties of Radiation. New York: John Wiley, 1973, Chap. 5, and D. Marcuse: Principles of Quantum Electronics, London-New York: Academic Press, 1980, Chap. 5 14. Merkli, M., Sigal, I.M., Berman, G.P.: Decoherence and thermalization. Phys. Rev. Lett. 98, 130401 (2007) 15. Merkli, M., Sigal, I.M., Berman, G.P.: Resonance theory of decoherence and thermalization. Ann. Phys. 323, 373–412 (2009) 16. Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. New York: Academic Press, 1978 17. Simon, B.: Functional Integration and Quantum Physics. New York: Academic Press, 1979 18. Spohn, H.: Dynamics of charged particles and their radiation fields. Cambridge: Cambridge University Press, 2004 19. Weidmann, J.: Linear Operators in Hilbert Spaces. Berlin-Heidelberg-New York: Springer, 1980 Communicated by I.M. Sigal


Communications in


Rigorous Scaling Law for the Heat Current in Disordered Harmonic Chain Oskari Ajanki1, , François Huveneers2, 1 Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, 00014 Helsinki, Finland.


2 UcL, FYMA, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium.

E-mail: [email protected] Received: 4 March 2010 / Accepted: 23 June 2010 Published online: 30 November 2010 – © Springer-Verlag 2010

Abstract: We study the energy current in a model of heat conduction, first considered in detail by Casher and Lebowitz. The model consists of a one-dimensional disordered harmonic chain of n i.i.d. random masses, connected to their nearest neighbors via identical springs, and coupled at the boundaries to Langevin heat baths, with respective temperatures T1 and Tn . Let E Jn be the steady-state energy current across the chain, averaged over the masses. We prove that E Jn ∼ (T1 − Tn )n −3/2 in the limit n → ∞, as has been conjectured by various authors over the time. The proof relies on a new explicit representation for the elements of the product of associated transfer matrices. 1. Introduction In a bulk of material, Fourier’s law is said to hold if the flux of energy J is proportional to the gradient of temperature, i.e., J = −κ∇T,

(1.1)

where κ is called the conductivity of the material. This phenomenological law has been widely verified in practice. Nevertheless, the mathematical understanding of thermal conductivity starting from a microscopic model is still a challenging question [4,10] (see also [16] for a historical perspective). Since the work of Peierls [20,21], it has been understood that anharmonic interactions between atoms should play a crucial role in the derivation of Fourier’s law for perfect crystals. It has been known for a long time that the conductivity of perfect harmonic crystals is infinite. Indeed, in this case, phonons travel ballistically without any interaction. This yields a wave like transport of energy across the system, which is qualitatively different than the diffusion predicted by the Fourier law (1.1). For example, in [23], it is Partially supported by the Academy of Finland and the European Research Council. Partially supported by the Belgian IAP program P6/02 and the Academy of Finland.

842

O. Ajanki, F. Huveneers

shown that the energy current in a one-dimensional perfect harmonic crystal, connected at each end to heat baths, is proportional to the difference of temperature between these baths, and not to the temperature gradient. In addition to the non-linear interactions, also the presence of impurities causes scattering of phonons and may therefore strongly affect the thermal conductivity of the crystal. Thus, while avoiding formidable technical difficulties associated to anharmonic potentials, by studying disordered harmonic systems one can learn about the role of disorder in the heat conduction. Moreover, many problems arising with harmonic systems can be stated in terms of random matrix theory, or can be reinterpreted in the context of disordered quantum systems. Indeed, in [9] Dhar considered a one-dimensional harmonic chain of n oscillators connected to their nearest neighbors via identical springs and coupled at the boundaries to the rather general heat baths parametrized by a function μ : R → C and the temperatures T1 and Tn of the left and right baths, respectively. Dhar expressed the steady state (μ) heat current Jn as the integral over oscillation frequency w of the modes: −2 T (w)An (w) · · · A1 (w)vμ,1(w) dw. (1.2) Jn(μ) = (T1 − Tn ) vμ,n R

Here Ak (w) ∈ R2×2 is the random transfer matrix corresponding to the mass of the k th oscillator, while vμ,1 (w) and vμ,n (w) are C2 -vectors determined by the bath function μ and the masses of the left and the right most oscillators, respectively. Standard multiplicative ergodic theory [2] tells that asymptotically the norm of Q n (w) := An (w) · · · A1 (w) grows almost surely like eγ (w)n , where the non-random function γ (w) ≥ 0 is the associated Lyapunov exponent. In the context of heat conduction this corresponds to the localization of the eigenmodes of one-dimensional chains while in disordered quantum systems one speaks about the one-dimensional Anderson localization [1]. However, in the absence of an external potential (pinning), the Lyapunov exponent scales like w 2 , when w approaches zero, and this makes the scaling behavior of (1.2) non-trivial as well as highly dependent on the properties of the bath. Indeed, only those modes for which the localization length 1/γ (w) is of equal or higher order than the length of the chain, n, do have a non-exponentially vanishing contribution in (1.2). Thus the heat conductance of the chain depends crucially on how the bath vectors vμ,1 (w), vμ,n (w) weight the critical frequency range w2 n 1. In other words, explaining the scaling of the heat current in disordered harmonic chains reduces to understanding the limiting behavior of the matrix product Q n (w) when w ≤ n −1/2+ for some > 0. The evolution of n → Q n (w) reaches stationarity only when w 2 n ∼ 1 while the com2 ponents of Q n (w) oscillate in the scale wn ∼ 1 with a typical amplitude of w−1 eγ0 w n as observed numerically in [9]. Thus the challenge when working in this small frequencies regime is that the analysis fall back neither to classical asymptotic estimates for large n, nor to the estimate of the Lyapunov exponent for small w. Of course, the difficulty of this analysis depends also on the exact form of the vectors u μ,k in (1.2), i.e., on the choice of the heat baths. Besides some rather recent developments, most of the studies so far have concentrated on two particular models. In the first model, introduced by Rubin and Greer [24], the heat baths themselves are semi-infinite ordered harmonic chains distributed according to Gibbs equilibrium measures of temperatures T1 and Tn , respectively. Rubin and Greer were able to show that E JnRG n −1/2 with E[ • ] denoting the expectation over the masses. Later Verheggen [25] proved that E JnRG ∼ n −1/2 .

Rigorous Scaling Law for the Heat Current in Disordered Harmonic Chain

843

In the second model the heat baths are modeled by adding stochastic OrnsteinUhlenbeck terms to the Hamiltonian equations of the chain (see (1.4) below). This model was first rigorously analyzed by Casher and Lebowitz [6] in the context of heat conduction. In an appendix of [6] the lower bound E JnCL n −3/2 is claimed. However, the argument contains a gap, and no strict scaling bounds for E JnCL have been proven until now. According to [6], the scaling law E JnCL ∼ n −3/2 was first conjectured by Visscher.

1.1. Casher-Lebowitz model and results. The Hamiltonian of the isolated one-dimensional disordered chain is H (q1 , . . . qn , p1 , . . . pn ) =

n n pk2 1 + (qk+1 − qk )2 , 2m k 2 k=1

(1.3)

k=0

where qk ∈ R is the displacement of the k th mass m k from its equilibrium position and pk is the associated momentum. We consider fixed boundaries, i.e., q0 = qn+1 = 0. The usual Hamilton’s equations are modified at the endpoints in order to include an interaction with heat baths. In the Casher-Lebowitz model, this interaction consists of adding white noise and viscous friction terms to the Hamiltonian equations of p1 and pn : Suppose λ > 0 is the coefficient of viscosity, let T1 ≥ Tn > 0 be the respective temperatures of the reservoirs, and let W1 , Wn be two independent Brownian motions. The equations of motion for the Casher-Lebowitz chain then take the form of the stochastic differential equation ∂H dt, ∂ pk ∂H d pk = − dt + (δk,1 + δk,n )(−λpk dt + 2λTk m k dWk ), ∂qk dqk =

(1.4)

with 1 ≤ k ≤ n. If {e1 , e2 } is the canonical basis of C2 , then, as far as the scaling behavior goes, the choice (1.4) of heat baths corresponds (see [6], and (2.5) below) to setting vμ,1 (w) = |w|−1/2 e1 + i|w|1/2 e2 and vμ,n (w) = |w|−1/2 e1 − i|w|1/2 e2 in (1.2). The resulting current, denoted by JnCL (m 1 , . . . , m n ), is then by definition the average rate at which energy is carried from the left to the right heat bath over the stationary measure of (1.4) for fixed masses m k . Now, suppose that the masses are random variables Mk . Our main result is the following strict scaling relation for the mass averaged, or annealed, stationary current. Theorem 1.1. Assume that the masses (Mk : k ∈ N) are independent and identically distributed. Suppose that the common probability distribution of the masses Mk admits a density, compactly supported on ]0, ∞[, continuously differentiable inside its support, with an uniformly bounded derivative. Denote by E[ • ] the expectation over the masses. Then there exist K , K > 0 such that the heat current JnCL satisfies the relation K

T1 − Tn T1 − Tn ≤ E JnCL(M1 , . . . Mn ) ≤ K

. 3/2 n n 3/2

(1.5)

844


The proof is based on a new representation of the matrix Q n (w) in terms of a discrete time Markov chain on a circle. Based on this representation we obtain a good control of the joint behavior of the matrix elements of Q n (w) for the most important regime w ≤ n −1/2+ , where > 0 is small. Moreover, together with O’Connor’s decay estimates [19] for high frequencies we have a good control of the exponential decay of Q n (w) whenever w ≥ n −1/2+ . Therefore, the possibility of generalizing Theorem 1.1 to a quite large class of heat baths seems possible by extending our analysis. Indeed, in Subsect. 6.3 we sketch how one can derive the scaling behavior of the stationary heat current for Dhar’s modified version of the Casher-Lebowitz model as well as to prove the analogue of Theorem 1.1 for the Rubin-Greer model. Also two other ways to extend Theorem 1.1 are discussed, at the speculative level, in Subsect. 6.3. We first consider the possibility of loosening the conditions on the mass distribution, so that the theory would become applicable, for example, in the case of binary masses. Secondly, we describe briefly the problem of proving a quenched, or almost sure, version of Theorem 1.1. The organization of the paper is as follows. In Sect. 2, we present the practical expression for the current JnCL , after first introducing some conventions and notations to be used in the rest of the paper. In the end of Sect. 2 our strategy to obtain Theorem 1.1 is outlined. Sections 3 to 5 contain the three main technical results needed for the proof. The actual proof of Theorem 1.1 is then presented in Sect. 6. 2. Conventions and Outline of Paper For the rest of this manuscript we are going to assume that the conditions of Theorem 1.1 hold. In particular, this means that the zero mean random variables Bk :=

Mk − E Mk , E Mk

(2.1)

are i.i.d., have a (Lebesgue) probability density τ that satisfies supp(τ ) ⊂ [b− , b+ ], and τ ∈ C1 ([b− , b+ ]), for some constants −1 < b− < b+ < ∞. Here Ck ([a, b]) denotes j the set of continuous functions f : [a, b] → R such that d fj exist for j ≤ k, and that dx

these derivatives are bounded and continuous on ]a, b[. The transfer matrices appearing in (1.2) are related to Bk : 2 − π 2 w 2 (1 + Bk ) −1 , (2.2) Ak ≡ Ak (w) = 1 0 where the frequency variable w is related to the frequency variable ω in [6] by ω = π −1 (E Mk )1/2 w. As already pointed out in the Introduction, O’Connor has shown (see Theorem 6 and its proof in [19]) that for any reasonable heat baths the frequencies above any fixed w0 > 0 have exponentially small contribution to the total current (1.2) as n grows. Therefore, one may consider an arbitrary small but fixed interval ]0, w0 ] of frequencies w in order to prove Theorem 1.1. ¯ = R ∪ {∞} We write N = {1, 2, 3, . . .}, N0 = {0, 1, 2, . . .}, R+ = ]0, ∞[ and R ¯ with ∞ = ±∞. We denote by C the extended complex plane C ∪ {∞}, where ∞ is here the point at infinity. Additionally, following conventions are used frequently. Probability. Since all the randomness of the stationary state current JnCL originates from the random masses we define the probability space ( , F, P) as the semi-infinite countable product of spaces ([b− , b+ ], B([b− , b+ ]), τ (b)db). Here B(S) denotes


845

the Borel σ -algebra of the topological space S. The filtration generated by the sequence B ≡ (Bk : k ∈ N) is denoted by F = (Fk : k ∈ N), Fk = σ (B j : 1 ≤ j ≤ k) ⊂ F. As a convention, the names of new random variables on ( , F, P) will be generally written in capital letters. A discrete time stochastic process (Z n : n ∈ K) is denoted by Z ≡ (Z n ) when index set K is known or not relevant. Finally, we write Z n = Z n − Z n−1 . Constants and scaling. Because we are interested only in the scaling relations many expressions can be made more manageable by using the following conventions. First, we use letters C, C , C1 , C2 , . . . to denote strictly positive finite constants, whose value may vary from place to place. Except otherwise stated, these values depend only on τ, λ, T1 − Tn and w0 , but never on w or n. Secondly, suppose f, g, h are functions, we write f g, or equivalently, g f provided f ≤ C g pointwise. If f g and f g, then we write f ∼ g. Moreover, the expression f = g + O(h) means | f − g| ≤ C|h|. Periodicity. In the following we are going to deal with functions that are defined and/or take values on the unit circle T = R/Z. The following conventions are practical on such occasions. When x ∈ R, write |x|T = min(x − x, x − x), where x (x) denotes the largest (smallest) integer smaller (larger) than x. We identify 1-periodic functions on R with functions on T. Similarly, a function g : R → R of the form g(x) = x + f (x), where f is 1-periodic, is identified with a function from T to itself. 2.1. Heat current in terms of matrix elements. Let v = [v0 v−1 ]T ∈ C2 , and denote by D(v) ≡ (Dn (v) : n ∈ N) the discrete time stochastic process that solves for n ∈ N: Dn (v) = (2 − π 2 w 2 (1 + Bn )) Dn−1 (v) − Dn−2 (v) D0 (v) = v0 , D−1(v) = v−1 .

(2.3)

By definition one then has for n ∈ N Q n = An An−1 · · · A1 =

Dn (e1 )

Dn (e2 )

Dn−1 (e1 ) Dn−1 (e2 )

,

(2.4)

where Ak is the transfer matrix (2.2) and e1 = [1 0]T and e2 = [0 1]T . As a remark it is worth noting that in the derivation of the stationary heat current one actually starts with (2.3) where Dn (ek ) are certain real valued (sub-)determinants (see (3.29) below) of a semi-infinite matrix and then expresses the final formula conveniently in terms of the product (2.4). Now, in [6] it was proven that the Casher-Lebowitz model corresponds to setting the (μ) bath vectors vμ,1 and vμ,n in the general expression (1.2) of Jn equal to (α M1 |w|)−1/2 (α Mn |w|)−1/2 and vCL,n (w) = . (2.5) vCL,1 (w) = +i(α M1 |w|)1/2 −i(α Mn |w|)1/2 Here the constant α > 0 depends on the units of the frequency variable w, etc. Since the masses have a compact support, [m − , m + ] ⊂ ]0, ∞[ , and the bath vectors are symmetric in w, one has ∞ −2 T JnCL ∼ (T1 − Tn ) vCL,n (w)Q n (w)vCL,1 (w) dw ∼ jn (w)dw =: Jn , (2.6) R

0

846


where jn (w) := |vnT (w)Q n (w)v1 (w)|−2 , with v1 (w) = w −1/2 e1 +iw 1/2 e2 and vn (w) = w−1/2 e1 − iw 1/2 e2 . By using Dn (e1 )Dn−1 (e2 ) − Dn−1 (e1 )Dn (e2 ) = det(An · · · A1 ) = 1n = 1 to get rid of the mixed terms of Dn (ek ) ≡ Dn (ek ; w) one obtains: −2

. (2.7) jn (w) = 1 + w −2 Dn (e1 )2 + Dn−1 (e1 )2 + Dn (e2 )2 + w 2 Dn−1 (e2 )2 This is the form we are going to use for the proof of Theorem 1.1. 2.2. Outline of the proof. It follows from (2.6) and (2.7) that the scaling bounds of E JnCL ∼ E Jn rely on the good understanding of the processes D(v) defined in (2.3). Thus, the first natural step towards the proof of the theorem is the derivation of an easier representation for Dn (v). This is the purpose of Sect. 3 where one constructs (Proposition 3.5 and Corollary 3.6) the representations: Dn (e1 ) ∼ w −1 nϑ · sin π X nϑ ,

Dn (e2 ) ∼ w −1 n0 · sin π X n0 .

and

(2.8)

Here ϑ ≡ ϑ(w) = w + O(w3 ) is non-random, the phases (X nx : n ∈ N0 ) form a Markov process on T, x x X nx = X n−1 + w + wφ(X n−1 )Bn + O(w 2 )

with

X 0x = x,

(2.9)

and the amplitude nx ∈ ]0, ∞[ is an exponential functional of (x, B1 , . . . , Bn ): nx = ew

n

x k=1 s(X k−1 )Bk

+ w2

n

x 2 k=1 r (X k−1 )Bk

+ O (w3 n)

.

(2.10)

The smooth functions φ, s, r : T → R are explicitly known. The process X ≡ X x is specified precisely in Definition 3.3 and Lemma 3.2, and its most important qualitative properties are listed in Corollary 3.4. The main advantage of the representation (2.8) is that, unlike the recursion relations (2.3) of D(v), it allows us to treat both the scaled noise w Bn and the initial values e2 of Dn (e2 ) as small perturbations around 0 and e1 , respectively. A physical interpretation of X n and n by means of response functions of the chain is provided at the end of Subsect. 3.1. Since (X n − nw : n ∈ N0 ) is a martingale to the first order in w, standard martingale central limit theorems [15] suggest that X n ≡ X nx , considered as taking values on R, starts to resemble a Gaussian with a mean x + wn + O(w2 n) and variance O(w 2 n) as n becomes large. Similarly, one sees that as w2 n becomes of order unity and larger, the chain X , now considered again on T, should reach its stationary state, which is expected to be close to the uniform measure on T. This later point is supported by Lemma 4.4 and Proposition 5.1 below. As the first sum inside the exponent in (2.10) is a martingale with bounded increments of size w, standard results on martingales imply that typically n ∼ 1 as well as En , En−1 ∼ 1 for w 2 n 1. On the other hand, for w 2 n 1 and w 3 n 1 one sees that the second sum in (2.10) must produce (see Eq. (4.2) below) the Lyapunov exponent γ (w) ∼ w2 associated to the transfer matrices Ak in (2.2). This clearly indicates that the parameter w2 n should play a crucial role in the analysis of the problem with w2 n ∼ 1 defining the borderline between two qualitatively different regimes of behavior for both X n (w) and n (w). Based on (2.8) and our intuition about X and , let us now carry out a heuristic derivation of E Jn ∼ n −3/2 which forms the outline for the actual proof. Along these


847

lines we will point out the properties of X n and n which must be proven to make these calculations rigorous. We start with the upper bound. By using Theorem 6 of [19] to bound the integral over region [w0 , ∞[ in (2.6) and then dropping positive terms from the denominator in (2.7) yields ∞ CL E Jn ∼ E Jn = E jn (w)dw w00 √ 1 dw + O e−C n (2.11a) E ≤ −2 2 1 + w Dn (e1 ; w) 0 w0 1 E P(X ∈ dx | ) dw, (2.11b) n n −2 2 0 T 1 + (w n sin π x) where the exponential bound has been dropped from the second line as we anticipate it to be insignificant compared to the remaining integral. We would now like to drop the conditioning on n in the inner integral of (2.11b) without changing the scaling. Boldly assuming that this is possible as the next step we would like to replace X n with a Gaussian of mean x + wn and variance O(w2 n) without affecting the scaling. However, since the integrand of (2.11b) depends on n and w, this replacement can not be justified by applying typical weak convergence results such as martingale central limit theorems. Indeed, the crucial contribution to the integral over T comes from |x| w2 / n which is expected to be a very unlikely event, e.g., assuming w 2 n ∼ 1, n ∼ 1, and P(X n ∈ dx) ∼ dx yields a probability of order n −1 . Thus, to continue from (2.11b), we would like to have two estimates: (a) Independence: P(X n ∈ dx |n ) ∼ P(X n ∈ dx), x ∈ T. dx √ (b) Pointwise bound: χB(wn,Cw√n) (x) · min(1,w P(X n ∈ dx) n) x ∈ T.

dx √ , min(1,w n)

The purpose of Sect. 5 is to prove Proposition 5.1 which together with the bounds in Subsect. 6.2 implies that as far as (2.11b) goes one may think that both (a) and (b) hold literally. Besides the potential theory, the proof of (a) relies on the fact that n , roughly, speaking, stays of order 1 as long as w2 n 1 while X n 0 +n , on the other hand, becomes uncorrelated with (Bk : k ≤ n 0 ), and consequently also with n 0 , when w 2 n 1. Applying (a-b) in (2.11b) and then parametrizing T with [−1/2, 1/2] in (2.11b) one gets w0 1/2 1 dx dw E Jn E · √ −2 2 min(1, w n) −1/2 1 + (w n x) 0

w0 1 arctan(w −2 n ) dw √ E w −2 n min(1, w n) 0 n −1/2 w0 w En−1 dw. (2.12) √ En−1 dw + n 0 n −1/2 Here we have used the upper bound in (a), approximated sin z ∼ z and then performed a change of variables x → w−2 n x. To get the last line we have approximated arctan q 1 for q ∈ R+ . In Sect. 4 we bound the only remaining unknown in (2.12), the expectation En−1 , by proving that there exists a constant α > 0 such that for every x ∈ T: E{1/ nx (w)} e−αw n , when 0 < w ≤ w0 . 2

(2.13)

848


The first sum Sn inside the exponent of (2.10) is a martingale with a variance of order 2 w 2 n. Thus standard large deviation results imply Ee Sn ∼ eO(w n) , and we see that (2.13) amounts to showing that this effect does not completely cancel the decay e−γ (w)n produced by the second sum in (2.10) when w 2 n 1. Applying the bound (2.13) in (2.12), yields the upper bound for the total current:

n −1/2w

E Jn 0

√ · 1 dw + n

w0

n −1/2

w 2 e−γ w n dw ∼ n −3/2 . 2

To prove the lower bound it sufficesto show that there is ε > 0 such that for every w ∈ I := [n −1/2 , 2n −1/2 ] one has P jn (w) ≥ εn −1 1. Indeed, if this bound is verified then E Jn E jn (w)dw ≥ n −1/2 · (εn −1 ) · P jn (w) ≥ εn −1 ∼ n −3/2 . I

Just like with the upper bound, the main contribution of E jn (w) comes from the unlikely events, e.g., when |X n | w 2 . For this reason one needs again the pointwise bounds (a) and (b). However, unlike in (2.11a), the lower bound depends in a non-trivial way also on Dn (e2 ) since for sufficiently small ε and every w ∈ I one has by (2.7): P jn (w) ≥ εn −1 ≥ P jn (w) ≥ εw 2 P |Dn (e1 ; w)| ≤ w 2 , |Dn (e2 ; w)| ≤ w . (2.14) Thus, to prove the lower bound one has to be able to analyze the joint behavior of the matrix elements (Dn (e1 ), Dn (e2 )), or equivalently, (X nϑ , X n0 , nϑ , n0 ). These dependencies are first addressed in Subsect. 3.2 by deriving martingale exponent representations for both X nϑ − X n0 and n0 / nϑ . In Subsect. 6.1 these representations are used to extract (Lemma 6.2) the typical joint behavior of the processes D(ek ), k = 1, 2. Based on this typical behavior one is then able to proof that the right side of (2.14) scales like O(1). 3. Representation of Matrix Elements The purpose of this section is to derive the representation (2.8) of processes D(v), v ∈ R2 , (Proposition 3.5 and Corollary 3.6) in terms of the Markov process (X n ) on the unit circle T. The first step of this derivation is to use the Möbius transformation, associated to the average of the transfer matrix E(An ), to construct w-depended change-of-coordi¯ to nates g which maps the evolution of the quotients ξn = Dn /Dn−1 bijectively from R −1 T. It turns out that in these new coordinates x = g (ξ ) the noise, w Bn , can be considered as a small perturbation around the zero noise evolution, which in turn is reduced to ¯ where the the simple shift x → x + ϑ. This is unlike in the original coordinates ξ ∈ R, effect of noise is typically of order O(1) regardless how small w is. The Markov process (X n ) is now defined by X n := g −1 (Dn /Dn−1 ) while the representation for the matrix elements is obtained by first writing Dn = g(X n ) · · · g(X 1 ) · D0 and then using the explicit knowledge of g for expanding the resulting expression w.r.t. the small disorder (w Bn : n ∈ N). The representation (2.8) is new. Besides having the benefits already mentioned before, it also has the nice property of reducing in the zero noise case to the explicit expression D1,n ≡ Dn = sin ππϑ(n+1) , which was already discovered by Casher and Lebowitz ϑ


849

(consider the 1-periodic chain in Eq. (3.5) in [6]). The change-of-coordinates g, on the other hand, is not really new as it was already discovered in a slightly different form by Matsuda and Ishii [17]. However, since our method of deriving g is different than in [17] we have decided to include it here for the convenience of the reader. In a more general context, our representation (2.8) is similar to some standard decomposition of products on Markov chains. Indeed, since Dn = ξn · · · ξ1 D0 with ξk = Dn /Dn−1 , and since the transfer operator of the chain (ξn ) admits a spectral gap [19], a general argument [14] allows us to write the decomposition |Dn | = eγ n+Mn u(ξn ), where γ is a Lyapunov exponent, (Mn ) is a martingale, and u is a function on R. Although, one is not in general able to determine Mn and u, it turns out that, in the special case of random matrices, Raugi [22] has been able to compute them explicitly, up to the knowledge of the invariant measure of the chain (ξn ). Still, the derivation of our formula (2.8) is much more straightforward than the use of Raugi’s formula. ¯ = C ∪ {∞}. Let us asso3.1. Expansion around zero noise evolution. We recall that C ¯ ¯ ciate a Möbius transformation M A : C → C to a 2 × 2 square matrix A by setting az + b a b M A (z) := for A = . c d cz + d The association A → M A preserves the matrix multiplication M A ◦ M B = M AB ,

(A, B ∈ C2×2 )

(3.1)

)−1

= M A−1 whenever either side of the equality exists. so that (M A By writing Dn ≡ Dn (v), v = [v0 v−1 ]T ∈ C2 , and using (2.3) one sees that the ratios Dn ξn := , (3.2) Dn−1 form a Markov process ξ ≡ (ξn : n ∈ N0 ) which satisfies a simple recursion relation: ξn = M An (ξn−1 ) v0 . ξ0 = v−1

(n ∈ N),

(3.3a) (3.3b)

Here the random matrices An depend on Bn through the relation (2.2). Since M An (±∞) = 2 − π 2 w 2 (1 + Bn ) we identify ±∞ = ∞. By using (3.2) and (3.3) we get Dn = ξn ξn−1 · · · ξ1 D0 ,

(3.4)

¯ = R ∪ {∞} provided no ξk ∈ {0, ∞}. In the following we shall consider (3.3) on R ¯ instead of C. ¯ such that Lemma 3.1. There exists a coordinate transformation g : T → R (g −1 ◦ ME(Ak ) ◦ g)(x) = x + ϑ

(x ∈ T),

where Ak is the random matrix (2.2), and the constant shift is given by 1 π 2 w2 ϑ ≡ ϑ(w) = arccos 1 − = w + O(w 3 ). π 2

(3.5)

(3.6)

850


The function g and and its inverse g −1 are given by tan π x , g(x) = MG ◦ E −1 (x) = cos π ϑ tan π x + sin π ϑ (sin π ϑ) ξ 1 g −1 (ξ ) = E ◦ MG −1 (ξ ) = arctan , π (cos π ϑ) ξ − 1 where E : ∂ D := {z ∈ C : |z| = 1} → T is the bijection eiφ → G consists of eigenvectors of E(Al ).

φ 2π ,

(3.7) (3.8)

and the columns of

Proof. By diagonalizing, we get E(Al ) = GG −1 , where iπ ϑ iπ ϑ 1 1 −1 0 e e −1 , G = , G = = e−iπ ϑ −eiπ ϑ 0 e−iπ ϑ 2i sin πϑ e−iπ ϑ

−1 , −1

(3.9) ¯ = ∂ D. Since the matrix G is and ϑ is given in (3.6). From (3.9) we see that MG −1 (R) invertible, the property (3.1) implies that the associated Möbius transformation is also invertible. In particular, the restrictions MG |∂ D and M−1 ¯ = MG −1 |R ¯ are bijections G |R ¯ and R ¯ into ∂ D, respectively. Using these observations we identify mapping ∂ D into R ¯ and its inverse g −1 : R ¯ → T by regrouping the coordinate transformation g : T → R as follows: ME(Al ) = MG ◦ M ◦ MG −1 = MG ◦ E −1 ◦ E ◦ M ◦ E −1 ◦ E ◦ M−1 G = g ◦ λ ◦ g −1 ,

(3.10)

where λ equals the shift function on the right of (3.5). In order to derive (3.7) and (3.8) the easiest way is to first solve g −1 using E(z/z ∗ ) = 2E(z) = π −1 arctan [(z)/(z)]: iπ ϑ e ξ −1 ξ sin(π ϑ) −1 −1 x = g (ξ ) ≡ E −iπ ϑ = π arctan . e ξ −1 ξ cos(π ϑ) − 1 The formula for g follows now by simply inverting the above function.

¯ and ξ = ME(A ) (ξ ). An important property of the new coordinates Suppose ξ ∈ R l x is as follows: although ξ − ξ can take values from order w 2 to ∞, one has always x − x := g −1 (ξ ) − g −1 (ξ ) = ϑ ∼ w. ¯ be the w-dependent coordinate Lemma 3.2. Let w > 0 be fixed and let g : T → R tranformation (3.7). Then for any b ∈ ]0, ∞[ the function 2 − π 2 w 2 (1 + b) −1 −1 , f b := g ◦ M A ◦ g : T → T where A ≡ A(b) := 1 0 (3.11) is a bijection, that can be written as f b (x) −1 f b (y)

= x + ϑ + (x, b),

(3.12a)

= y − ϑ + (y − ϑ, −b),

(3.12b)


851

where the constant ϑ = w + O(w 3 ) is given in (3.6) and the smooth function : T× ]0, ∞[ → T is specified by 1 (π w/2) [1 − cos(2π x)] b (x, b) = arctan (3.13a) π 1 − (π w/2)2 − (π w/2) sin(2π x) b = sin2 (π x) wb + w 2 b2 (π/2) sin(2π x) + w 3 b R3 (w, x, b) . (3.13b) The remainder term R3 : [0, w0 ] × T × [b− , b+ ] is a smooth and bounded function. The lemma says that in x-coordinates the system ξn = M An(ξn−1 ), n ∈ N, and ξ0 = g −1 (x) is described by the following process on a circle. The proof which is just a mechanical calculation can be found in Appendix A.1. Definition 3.3. Let x ∈ T. Markov process X x ≡ (X nx : n ∈ N0 ) on T is defined by setting x X nx = f Bn(X n−1 )

(n ∈ N),

X 0x = x.

(3.14)

When the starting point x is known from the context or its specific value is not relevant we write simply X and X n instead of X x and X nx , respectively. The main properties of f b (x) are best seen by expanding it into the power series w.r.t. w. Indeed, by using (3.6), (3.12a) and (3.13) one gets: f b (x) = x + w + wφ(x)b + w 2 ψ(x)b2 + O(w 3 ), φ(x) = sin2 π x, ψ(x) = π sin3 π x cos π x.

(3.15a) (3.15b) (3.15c)

Let us denote Z k := Z k − Z k−1 for a stochastic process (Z k ). By using the expansion (3.15) together with E(Bk ) = 0 and Bk ≥ b− > −1, the following qualitative properties of X emerge. Corollary 3.4. The process X has the following three useful properties: 2 (i) Uniform monotonicity: 0 < (1 + b− )w + O(w 2 ) ≤ X k ≤ (1 + b+ )w + O(w ); (ii) O(w1 )-martingale property modulo constant shift: E X k − w |Fk−1 = X k−1 + O(w 2 ); (iii) Uniform diffusion outside any neighborhood of zero: There are constants α(ε), β > 0 such that E (X k − w)2 |X k−1 = x ∈ [α(ε)w 2 , βw 2 ] for |x|T ≥ ε.

Having found good coordinates x = g(ξ ), where ξn = Dn /Dn−1 evolves in w-sized steps in a relatively simple manner, our next step is to express the matrix elements of Q n in terms of these new coordinates. ¯ 2 with v0 = 0. Then there is a constant w0 > 0 Proposition 3.5. Let v = [v0 v−1 ]T ∈ R such that for w ∈ ]0, w0 ] the solution of (2.3) is Dn (v) = v0 · nx ·

sin π X nx sin π [x + (x, B1 )]

with x = g −1 (v0 /v−1 ),

(3.16)

852


almost surely. Here the random amplitude nx : → ]0, ∞[ has an exponential representation n n x x nx = exp w s(X l−1 )Bl + w 2 r (X l−1 )Bl2 + O(w 3 n) , (3.17) l=1

l=1

where the smooth functions r, s : T → R are specified by s(x) = − r (x) =

π sin 2π x, 2

(3.18a)

π2 (cos2 2π x − cos 2π x). 4

(3.18b)

Proof. Denote Dn := Dn (v), ξn = Dn /Dn−1 and set x := g −1 (ξ0 ) ≡ g −1 (v0 /v−1 ). By definition (3.3) the process (ξn ) is described in x-coordinates by the process (X nx ). Set X n := X nx and use (3.7) to write ξl = g ◦ X l = (MG ◦ E −1 )(X l ) = MG (ei2π X l ).

(3.19)

By using (3.9) to write out the Möbius transformation we obtain: MG (eiφ ) =

sin φ2 eiφ − 1 . = ei(φ−π ϑ) − eiπ ϑ sin φ2 − π ϑ

By combining this with (3.19), reorganizing the resulting product and then using (3.12a) to write f in terms of yields n n−1 sin π X n sin π X l sin π X l Dn ξn ξn−1 · · · ξ1 v0 = = = v0 v0 sin π(X l − ϑ) sin π(X 1 − ϑ) sin π(X l+1 − ϑ)

=

l=1 n−1

sin π X n sin π [x + (x, B1 )]

l=1

l=1

sin π X l . sin π [X l + (X l , Bl+1 )]

(3.20)

Here the possible extreme values ξk ∈ {0, ∞} do not cause problems because we assumed ξ0 = v0 /v−1 = 0 and (3.3) implies P ( ξk ∈ {0, ∞} for some k ∈ N | ξ0 = 0 ) = 0 . We must now show that the product of sin ratios in (3.20) equals the exponent nx . Since, the terms in the product are all similar let us consider only one such factor. From (3.13b) one sees that (x, b) = O(w). This suggests expressing the denominators on the last line of (3.20) as power series of π (x, b) around zero: sin π(x + (x, b)) = sin π x cos π (x, b) + cos π x sin π (x, b)

1 2 2 = sin π x 1− π (x, b) +π (x, b) cos π x +O 3 (x, b) . 2 (3.21) The expression (3.13b) also shows that k (x, b)/ sin π x = O(w k ) for k ≥ 1/2. Thus using (3.21) to rewrite the denominators in (3.20) and then dividing the numerator and


853

the denominator by sin π x yields the expression for the geometric sum of variable 2 q = −π (x, b) cot π x + π2 2 (x, b) + O(w 3 ) = O(w). Expanding this geometric sum gives the first line of sin π x π2 = 1−π (x, b) cot π x + 2 (x, b)+π 2 2 (x, b) cot 2 π x +O(w 3 ) sin π(x +(x, b)) 2 π2 π = 1 − w sin 2π x b+w 2 (1 − cos 2π x)2 b2 + O(w 3 ), 2 8 while the last line follows from (3.13b) and trigonometric double angle formulae. By using 1 + z = exp ◦ ln(1 + z) = exp z − 21 z 2 + O(z 3 ) , with |z| ≤ Cw0 , for the last expression we get sin π x = exp −w(π/2) sin 2π x b + w 2 (π/2)2 sin π(x + (x, b)) × cos2 2π x − cos 2π x b2 + O(w 3 ) . Identifying functions s and r on the right side and then applying this bound term by term for the product in (3.20) yields the expression on the right side of (3.17). It is worth remarking that the proposition does not apply directly for v ∈ C2 since it relies on Lemmas 3.1 and 3.2 which apply only when (ξn ) takes values on R. Of course, by the linearity of the system (2.3) one still has Dn (v R + iv I ) = Dn (v R ) + iDn (v I ) for any v R , v I ∈ R2 . The next corollary shows that the generic choice Dn (v) with v = ek , k = 1, 2, is often a convenient choice as D(e2 ) can be treated as a perturbation of D(e1 ). Corollary 3.6. There is a constant w0 > 0 such that for w ∈ ]0, w0 ]: sin π X nϑ ∼ w −1 nϑ · sin π X nϑ , sin π [ϑ + (ϑ, B1 )] sin π X n0 ∼ w −1 n0 · sin π X n0 . Dn (e2 ) = n0 · sin π [ϑ + (ϑ, B2 )] Dn (e1 ) = nϑ ·

(3.22a) (3.22b)

Proof. By (3.8) we get g −1 (ξ0 ) = g −1 (1/0) = ϑ and thus (3.22a) follows directly from Proposition 3.5. In order to prove (3.22b) one can not directly apply the proposition since the first component of e2 is zero. However, from (2.3) one sees that [D1 (e2 ) D0 (e2 )]T = [−1 0]T = −e1 and Dn (−v) = −Dn (v). Thus, by defining θ : → by θ ω = (b2 , b3 , . . . ) for ω = (b1 , b2 , . . . ) and denoting the associated pullback θ∗ on random variables Z by θ∗ Z (ω) = Z (θ ω), one can write ϑ Dn (e2 ) = −θ∗ Dn−1 = θ∗ n−1 ·

ϑ sin π θ∗ X n−1

sin π [ϑ + (ϑ, θ∗ B1 )]

,

(3.23)

where by the definition: n−1 n−1 ϑ ϑ 2 ϑ 2 3 θ∗ n−1 = exp w s(θ∗ X l−1 ) θ∗ Bl + w r (θ∗ X l−1 )(θ∗ Bl ) + O(w n) . l=1

l=1

(3.24)

854


Now, since (0, b) = 0 it follows that X 10 = f B1 (0) = ϑ + (0, B1 ) = ϑ = θ∗ X 0ϑ ϑ regardless of the value of B1 . But (X n0 : n ∈ N) and (θ∗ X n−1 : n ∈ N) also satisfy 0 , n ∈ N . Also, ϑ the same recursion relations for n ≥ 2 and therefore θ∗ X n = X n+1 0 ϑ with X 0 and by definition θ∗ Bl (ω) = bl+1 = Bl+1 (ω). Thus we may replace θ∗ X l−1 l write θ∗ Bl = Bl+1 in (3.23) and (3.24). Moreover, if we also reindex the sums in (3.24) we obtain an exponential representation for θ∗ n−1 that is up to missing first terms ws(X 00 )B1 and w 2 r (X 00 )B12 equal to n0 . However, these missing terms are both zero due to the “coincidence” s(0) = r (0) = 0, and thus we get θ∗ n−1 = n0 . This proves (3.22b). The physical interpretation of the random variables X n and n is not very obvious from what we have presented so far. One can, however, relate them to certain impulse response functions of the damped disordered harmonic chain. To see this let ( p1 , . . . , pn ; q1 . . . , qn ) : R → R2n satisfy the Hamiltonian (1.3) equations of the chain subjected to the external forces t → f k (t), k = 1, n, and dissipation at the boundaries. By letting the dissipation be strong enough and assuming f 1 and f n to be compactly supported, the Fourier transforms qˆk and pˆ k of the oscillator coordinates w.r.t. time t can be shown to exist. The equations of motion can be then written as1 qˆk+1 − (2 − Mk ω2 ) · qˆk + qˆk−1 = −(δk1 + δkn )(μk qˆk + fˆk ) for 1 ≤ k ≤ n, (3.25) with the convention qˆ0 = qˆn+1 = 0. Here the Fourier multipliers μ1 , μn : R → C determine the dissipation, e.g., in the Casher-Lebowitz model one would have μk (ω) = −iλω. The equation (3.25) can be alternatively written as qˆk qˆk qˆk+1 2 − Mk ω2 −1 = = Ak , (3.26) qˆk qˆk−1 qˆk−1 1 0 provided we redefine qˆ0 and qˆn+1 so that qˆ1 1 0 qˆ1 = and μ1 1 fˆ1 qˆ0

qˆn+1 1 = qˆn 0

μn 1

fˆn , qˆn

(3.27)

hold. Here the transfer matrix A√k appears already in (2.2) but is now expressed in terms of Mk and ω = αw, α = π −1 E Mk , instead of Bk and w. Using (2.4) to express the product matrix An · · · A2 A1 in terms of the processes (Dn (ek ) : n ∈ N0 ), k = 1, 2, and then applying (3.26) and (3.27) yields Dn (e2 ) Dn (e1 ) 1 0 qˆ1 1 −μn fˆn ˜ n qˆ1 . = =: Q Dn−1 (e1 ) Dn−1 (e2 ) μ1 1 fˆ1 0 1 qˆn fˆ1 The elements of the matrix Q˜ n are explicitly known so one may now solve qˆ1 and qˆn in terms of fˆ1 , fˆn . Rather than writing down the solution in all generality let us instead look at the special case f 1 = μ1 = 0 and μn (ω) = −iλω, and consider only the left most oscillator qˆ1 =

sin π [X 1ϑ − ϑ] fˆn = ϑ · fˆn . ϑ sin π X ϑ Dn (e1 ) + μn Dn−1 (e2 ) n sin π X nϑ − iλω · n−1 n−1

(3.28)

1 In the rest of this subsection ω denotes frequency instead of the element of the probability space .


855

This shows that X nϑ and nϑ determine the response of the oscillator at site 1 to the external ϑ (w) ∼ w ∼ ω force applied at the other end of the chain. Especially, since X nϑ (w)− X n−1 one observes that an oscillating force applied on the n th oscillator at the frequency ω ϑ(w ) close to a resonance frequency ω0 = αw0 , satisfying X nϑ (w0 ) ≡ X n 0 (w0 ) = 0, is ϑ carried across the chain most effectively. Naturally, n can be interpreted as describing the decay of the connection between two oscillators separated by a distance n in the chain. Note also, that a non-vanishing imaginary part of function μn guarantees that the divisor in (3.28) can not become zero. This makes the Fourier transforms well defined even around the resonance frequencies. The resonant frequencies are nothing but the normal modes of the free chain (μk = f k = 0, k = 1, n). This follows from (3.29) Dn (e1 ; ω/α) = det (n) − ω2 M (n) , (n)

where the force and the mass matrices (n) , M (n) ∈ Rn×n are defined by i j

=

(n) Mi j

= δi j Mi , respectively. This representation follows −δi, j+1 + 2δi j − δi, j−1 and by showing that the determinants K n (ω) on the right of (3.29) satisfy [6] the recursion relation (2.3) with initial conditions K 0 (ω) = 1 and K 1 (ω) = 2 − ω2 M1 . As these are equivalent to setting K −1 (ω) = 0 and K 0 (ω) = 1, equality in (3.29) follows. 3.2. Joint behavior. In order to prove E Jn n −3/2 we analyze the current density jn defined in (2.7). This leads us to consider the properties of the quadruple (X nϑ , X n0 , nϑ , n0 ). Since X 0ϑ − X 00 = ϑ ∼ w one can consider X n0 and n0 as perturbations around X nϑ and nϑ , respectively. Based on this simple idea one proves the following. Lemma 3.7. Let us treat X x , x ∈ R as real valued processes. Then for all n ∈ N and w ∈ ]0, w0 ]: X nϑ − X n0 = w e Mn +L n +O(w n0 / nϑ = e K n

+O (w+w2 n)

2 n)

,

,

(3.30) (3.31)

where (Mn ), (L n ), (K n ) are R-valued F-martingales such that M0 = L 0 = K 0 = 0 and n ∈ N: ϑ Mn = wφ (X n−1 )Bn ,

(3.32)

+O (w2 n)

Hn−1 Bn ,

(3.33)

2 Mn−1 +L n−1 +O (w2 n)

Un−1 Bn .

(3.34)

L n = w 2 e Mn−1 +L n−1 K n = w e

The processes (Hn ) and (Un ) are F-adapted and bounded such that: sup |Hn |, |Un |, w −1|L n |, w −1|K n | : n ∈ N ≤ C.

(3.35)

Proof. From (3.13b) and (3.15b) one sees that (x, b) = wφ(x)b + w2 R2 (x, b), where R2 is a smooth and bounded function. Using (3.12a) we get f b (x) − f b (x − z) = z + (x, b) − (x − z, b)

R2 (x, b) − R2 (x − z, b) φ(x) − φ(x − z) b + w2 z, = 1+w z z (3.36)

856


for any z ∈ R. By the mean value theorem there are functions ζ1 (x, z), ζ2 (x, z, b) ∈ [x − z, x] such that for any x ∈ R, z ≥ 0 and b ∈ [b− , b+ ], we have

1 f b (x) − f b (x − z) = 1 + wφ (x)b − wz φ

(ζ1 (x, z))b + w 2 ∂x R2 (ζ2 (x, z, b), b) z 2 1 (3.37) = exp wφ (x)b − wz φ

◦ ζ1 (x, z)b + O(w 2 ) z 2 Now, set n := (X nϑ − X n0 )/w

1 Hn := − (φ

◦ ζ1 )(X nϑ , wn ). 2

and

(3.38)

Then (3.37) and (3.14) yield 1 ϑ f B (X ϑ ) − f Bn(X n−1 − wn−1 ) w n n−1 1 ϑ ϑ )Bn − w 2 n−1 φ

◦ ζ1 (X n−1 , wn−1 ) Bn + O(w 2 ) · n−1 = exp wφ (X n−1 2 ⎡ ⎤ n n

ϑ 2 2 = exp ⎣w φ (X j−1 )B j + w j−1 H j−1 B j + O(w n)⎦ · 0 . (3.39)

n =

j=1

j=1

By using (3.32) and (3.33) we identify the two sums inside the exponent in (3.39) as Mn and L n , respectively. Together with 0 = (X 0ϑ − X 00 )/w = ϑ/w = 1 + O(w2 ) this 2 gives n = e Mn +L n +O(w n) and by the definition (3.38) this equals (3.30). Moreover, w −1 L n+1 = wn Hn Bn+1 , where using (3.37), (3.38) and the definition of ζ1 we get wn Hn =

φ(X nϑ ) − φ(X n0 ) − φ (X nϑ ) =: φ (ζ0 ) − φ (X nϑ ), X nϑ − X n0

for some ζ0 ∈ [X n0 , X nϑ ], and therefore w −1 |L n+1 | ≤ 2φ ∞ · max{−b− , b+ } =: C. In order to prove (3.31) we use again the mean value theorem to write s(X n0 ) = s(X nϑ − wn ) = s(X nϑ ) − wn · s ◦ ζ3 (X nϑ , wn ),

(3.40)

where X nϑ − wn ≤ ζ3 (X nϑ , wn ) ≤ X nϑ . Using this in (3.17) yields n 0 0 2 n = exp w s(X l−1 )Bl + O(w n) = exp w

l=1 n

ϑ s(X l−1 )Bl

l=1

=: nϑ e K n +O(w+w

−w

2

n

l−1 · s

ϑ ◦ ζ3 (X l−1 , wl−1 ) Bl

+ O(w n) 2

l=1 2 n)

.

Above, we have identified Un = −(s ◦ ζ3 )(X nϑ , wn ) in (3.34). Finally, by Eq. (3.40) w −1 K n+1 = wn Un Bn+1 = [s(X n0 ) − s(X nϑ )] Bn+1 . Since s is a bounded function (3.18a) this implies w −1 |K n | ≤ C.


857

4. Expectation of 1/ n In this section we prove the following result. Proposition 4.1. For sufficiently small w0 ∼ 1 there exists α ≡ α(w0 ) > 0 such that for n ∈ N, 2 sup E 1/ nx e−αw n , w ∈ ]0, w0 ]. (4.1) x∈T

The content of this result is best understood by using (3.17) to write 1/ n as exponent 2 1/2 3 e−Rn w n+wn Sn +O(w n) , where the normalized random variables Sn =

n −1 s(X k−1 )Bk n 1/2

and

Rn =

k=1

n 1 r (X k−1 )Bk2 , n k=1

are in average of order 1. Consider now any 1/w consecutive values Sk (w) := {X j (w) : j = k, . . . , k + 1/w}, k ∈ N0 , of the chain (X n ). During these 1/w steps the accumulated noises M j−k := X j − X k − w( j − k) form a martingale up to the leading order in w, and are √ therefore typically bounded in magnitude by a term of order w 1/w ∼ w 1/2 . Since the deterministic part of the evolution is essentially the simple shift map, x → x + w mod 1, one concludes that the random set Sk (w) should sample T increasingly evenly as w → 0. The proof of Proposition 4.1 consists of two steps which both rely on this sampling property. First, Lemma 4.4 is used to show that Rn ≡ Rn (w) can be replaced by the constant γ (w)/w2 without introducing too large errors in E(1/ n ) provided wn → ∞. Here

π 2 E(B12 ) 2 2 γ (w) = E(B1 ) · r (x)dx w 2 + O(w 3 ) = w + O(w 3 ), (4.2) 8 T is the Lyapunov exponent associated to the norm of Q n in (2.4). Secondly, the uniform monotonicity (property (i) of Corollary 3.4) of the process X is used to bound the conditional variance (see (4.3)) of the martingale n 1/2 Sn so that Freedman’s exponential 1/2 2 martingale bound, i.e., Lemma 4.2, can be applied to obtain a bound Eewn Sn ≤ eβw n , where γ (w)/w2 − β =: α ∼ 1. The following lemma provides two powerful exponential martingale bounds due to Freedman [13] and Azuma [3]. Lemma 4.2. Let (Mn ) be a (Fn )-martingale, and define a process (Vn ) by setting V0 = 0 and Vn :=

n E (Mi − Mi−1 )2 |Fi−1 ,

n∈N.

(4.3)

i=1

Suppose there exists a constant m and a sequence (vn ) ⊂ R+ such that |Mn − Mn−1 | ≤ m and Vn ≤ vn for all n ∈ N. Then for any t ∈ R and n ∈ N: e κm (t)vn , ”Freedman’s bound”; t Mn (4.4) Ee ≤ t2 2 e 2 m n, ”Azuma’s bound”;

858


where κm (t) =

emt − 1 − mt t 2 m m|t| 3 ≤ + e |t| . m2 2 6

(4.5)

For the convenience of readers the proofs of these bounds are included in Appendix A.2. The next inequality (4.6) is often referred as Azuma’s inequality. Corollary 4.3. Suppose (Mk ) satisfies the hypothesis of Lemma 4.2. Then for any n ∈ N and r > 0: P(|Mn | ≥ r ) ≤ 2 e

−

r2 2m 2 n

.

(4.6)

Proof. The proof follows by using Markov’s inequality: P(|Mn | ≥ r ) = P(Mn ≥ r ) + P(−Mn ≥ r ) ≤ e−sr E es Mn + e−sr E e−s Mn , and then use Azuma’s bound (4.4) with t = r/(m 2 n). Lemma 4.4. Suppose u is a Lipshitz-function on T, i.e., there is a constant L u > 0 such that for all x, y ∈ T: |u(x) − u(y)| ≤ L u |x − y|T . Then: ⎧ p⎫ ⎬ ⎨ 1/w p sup E w u(X xj ) − u(y)dy (4.7) ≤ C p L u w p/2 , T x∈T ⎩ ⎭ j=0

where C p does not depend on u. Proof. Fix x and set X := (X x and I j := [x + w ( j − 1), x + w j[ . Define for each j some x˜ j ∈ I j by requiring I j u(x)dx = w u(x˜ j ), and set x¯ j := E(X j ). The properties (3.15) of the chain X imply |x¯ j − x˜ j | ≤ w for all j ≤ 1/w. By writing the integral on the left side of (4.7) as a sum over u(x˜ j ) and then applying the Lipshitz-property of u one gets ⎧ p⎫ p ⎨ 1/w ⎬ p u(X j ) − u(x˜ j ) E w E |X jl − x˜ jl | . (4.8) ≤ Lu w p ⎩ ⎭ j=0 j1 ,..., j p l=1 j Now, X j = x + wj + w1/2 M j + O(w) with M j = w 1/2 i=1 φ(X i−1 )Bi uniformly for any 0 ≤ j ≤ 1/w. This means X j − x˜ j = w 1/2 (M j + O(w 1/2 )). By applying the generalized Hölder’s inequality one has, p p p/2 1/2 M jl + O(w ) |X jl − x˜ jl | = w E E l=1

l=1

) p *1/ p p p/2 1/2 ≤w E M jl + O(w ) .

(4.9)

l=1

The last expectations of (4.9) can be bounded with Azuma’s inequality (4.6). Indeed, |M j − M j−1 | ≤ w 1/2 max(−b− , b+ )φ∞ ≡ Cw 1/2 for each j. This implies


859

2 2 2

P |M j | ∈ [k, k + 1[ ≤ 2P(|M j | ≥ k) ≤ 2e−k /(2C w1/w) = e−k /C which, in turn, yields ∞

p E M j + O(w 1/2 ) ≤ (k + 1 + O(w 1/2 )) p P |M j | ∈ [k, k + 1[ k=0 ∞

≤2

k p e−k

2 /C

=: C p .

k=0

Since this bound holds uniformly for all j = 0, 1, . . . , 1/w we may apply it term by term in (4.9). Using the resulting bound again term by term in (4.8) yields the bound (4.7). Proof of Proposition 4.1. When wn ∼ 1 the sums inside the exponent of (3.17) are at most of order one since s, r : T → R are bounded. Thus 1/ nx (w) ∼ 1, and conse2 quently it only remains to show that E(1/ nx ) ≤ C e−αw n for n = 1/wm, m ∈ N. Moreover, since X n ≥ Cw we may for the same reason fix some arbitrary starting point x ∈ T and denote X nx and nx by X n and n , respectively. We begin the proof by decomposing the second sum in the exponent of (3.17) into the double sum w2

n

r (X i−1 )Bi2 = w

i=1

m

w

ik

r (X i−1 )Bi2

k=1 i=i k−1 +1

=w

m

γ (X ik−1 ) + w

k=1

m

Zk ,

(4.10)

k=1

where i k = 1/wk + 1, k = 1, 2, . . . , m is roughly the time the averaged process x¯ j := Ex (X j ) = x + wj + O(w 3 j) has passed its starting point kth time. In the rightmost expression of (4.10) we have further divided the inner sums into the conditional expectations and the fluctuation parts: Z k := w

ik

r (X i−1 )Bi2 − γ (X ik−1 )

i=i k−1 +1

⎧ ⎨

γ (y) := E w ⎩

1/w i=1

(4.11a)

⎫ ⎬

y r (X i−1 )Bi2 . ⎭

(4.11b)

The motivation behind the decomposition (4.10) is twofold. First, Lemma 4.4 tells us that the function γ is almost constant for small w, and especially ⎧ ⎫ ⎨ 1/w ⎬ y 2 2 γ (y) = E(B ) E w r (X i−1 ) ≥ E(B ) r (z)dz − β0 w 1/2 =: γ˜− , (4.12) ⎩ ⎭ T i=1

that does not depend on y. Here the first equality folwhere β0 > 0 is a finite constant lows from E r (X i−1 )Bi2 = E(B 2 ) E(r (X i−1 )), while the last expression comes from

860


Lemma 4.4 with p = 1 and L u := r ∞ . Using (4.12) to bound each term γ (X ik−1 ) in (4.10) yields the bound: n m 2 s(X i−1 )Bi − w Zk , E (1/ n ) ≤ e−γ− w n E exp −w i=1

k=1

with γ := γ˜ + O(w),

(4.13)

where the O(w 3 n)-term inside the exponent (3.17) of n has been also absorbed into the constant γ− . The second property of the decomposition (4.10) is that (Z k : k ∈ N) constitutes a sequence of bounded martingale increments in the sparse filtration F = (Fk ), Fk := Fik ≡ σ (B1 , B2 , . . . , Bik ): the boundedness of Z k is obvious as it is an average of 1/w uniformly bounded increments, while the martingale property holds, since X is Markov: ⎧ ⎫ ⎛ ⎞ ik ⎨ 1/w ⎬ X i (ω)

⎠ Bi2 ≡ γ (X ik−1 (ω)), E ⎝w r (X i−1 )Bi2 Fk−1 r X i−1k−1 (ω) = E w ⎩ ⎭ i=i k−1 +1

i=1

for a.e. ω ∈ . We want to consider both sums in the right side of (4.13) as martingales. Since this is not possible under the same expectation, we apply Hölder’s inequality to divide the expectation into the product of separate expectations E(1/ n ) ≤ e

−γ− w2 n

E exp −pw

n

1/ p s(X i−1 )Bi

i=1

E exp −p w

m

1/ p

Zk

,

k=1

(4.14) where p, p ≥ 1 and 1/ p + 1/ p = 1. We can now bound both of these expectations with the help of Lemma 4.2. Azuma’s exponential bound (4.4) is sufficient for the second factor: if |Z k | ≤ C Z , then

E exp − p w

m k=1

1/ p

Zk

1/ p

(− p w)2 2

3 C Z nw ≤ exp ≤ eβ2 p w n , (4.15) 2

for some constant β2 . In order to handle the first expectation of (4.14) we note that the martingale (M j ), defined by M j := s(X j−1 )B j , j ∈ N and M0 = 0, has bounded increments. Moreover, since E[(Mi )2 |Fi−1 ] = E(B 2 ) · s 2 (X i−1 ), we see that for sufficiently small ε > 0: n n Vn := E (Mi − Mi−1 )2 |Fi−1 = E(B 2 ) s 2 (X i−1 ) ≤ (1 − ε)E(B 2 )s2∞ n. i=1

i=1

In order to get the last bound above, one uses the property (i) of Corollary 3.4, the continuity of s and s(0) = 0, to conclude that there must exist ε > 0 such that

0 ≤ i ≤ n − 1 : |s 2 (X i )| ≤ s2∞ /2 ≥ 2εn.


861

This, by definition, implies the bound of Vn above. Applying Freedman’s bound of Lemma 4.2 with vn := E(B 2 )(1 − ε)s2∞ n and |Mi − Mi−1 | ≤ C M =: m yields

E exp (−wp)

n

1/ p s(X i−1 )Bi

1/ p ≤ exp κC M (−wp)E(B 2 )(1 − ε)s2∞ n

i=1 1

≤ e 2 pw

2 (1−ε)E(B 2 )s2 n+β p 2 w 3 n 1 ∞

,

(4.16)

where β1 > (1/6)(1 − ε)E(B 2 )C M eC M pw ∼ 1. Plugging (4.16) and (4.15) along with the estimate (4.12) for γ− into (4.14) results into the total bound

E(1/ n ) ≤ e

−E(B 2 )

(

T r (y)dy− p(1−ε)

s2∞ 2

w2 n+β0 w5/2 n+β1 p 2 w3 n+β2 p w3 n+Cw3 n

. (4.17)

Here the term inside curly brackets would disappear if p = 1, ε = 0 because ( r (y)dy = s2∞ /2 = π 2 /8. However, since ε > 0 we can take p > 1 such that it T remains positive. However, by taking w0 sufficiently small, the last three terms, regardless of the size of p or β1 , β2 , β3 , C, can be made arbitrary small compared to the first part. 5. Potential Theory This section is devoted to the statement and the proof of Proposition 5.1 below. The derivation of the inequalities (5.1a) and (5.1b) constitutes a relatively classical problem in potential theory for Markov chains. However, it does not seem possible to apply classical results (see e.g. [5] and [11], but also [7] and [8]), since the chain X is neither reversible, nor uniformly diffusive. In particular, little appears to be known on lower bounds of the type (5.1b) for non-reversible Markov chains. Results for Markov chains on a lattice [18], or for differential equations in non-divergence form [12], do not adapt straightforwardly (and maybe not at all) to our case. Instead, since we consider only the case w → 0, it has been possible to treat the left hand side of (5.1a) and (5.1b) as a perturbation of quantities that can be computed explicitly. We are then able to handle both of these bounds with a single method. Proposition 5.1. Let κ > 0, and let h ∈ C1 (T). There exist K , K , w0 > 0 such that, for every w ∈ ]0, w0 ], for every function u ∈ L1 (T; R+ ), for every x ∈ R, and for every n ∈ N, one has n x K E ew k=1 h(X k−1 )Bk u(X nx ) ≤ √ u(y)dy (wn ≥ κ, w2 n ≤ 1), (5.1a) w n T n x )B w k=1 h(X k−1 x

k u(X n ) ≥ K u(y)dy (1/2 ≤ w2 n ≤ 1). (5.1b) E e T

Before starting the proof let us make a few definitions: First, for A ⊂ T and 1 ≤ p ≤ ∞ we define the space p

L A (T) := {u ∈ L p (T) : supp(u) ⊂ A}. Secondly, let S be a continuous operator from L p (T) toLq (T), for 1 ≤ p, q ≤ ∞, and de note the associated operator norm by S p→q := sup Suq : u ∈ L p (T), u p ≤ 1 .

862


The content of Proposition 5.1 is twofold. First, it describes the approach to equilibrium of the chain X . To see this, let us consider the case h = 0, and√let us take some subset A ⊂ T. Equation (5.1a) implies that P(X nx ∈ A) max{1/(w n), 1} · Leb(A) when wn ≥ κ, whereas (5.1a) and (5.1b) imply that P(X nx ∈ A) ∼ Leb(A) when w2 n ≥ 1/2. This is obvious when w 2 n ≤ 1. If w 2 n > 1, one decomposes n = n 1 + n 2 such that n 2 w 2 = 1, and then writes Eu(X nx ) = E u(X nx ) X nx1 = y P(X nx1 ∈ dy). T

y The result now follows since, by definition E u(X nx ) X nx1 = y = Eu(X n 2 ) and (5.1) y gives Eu(X n 2 ) ∼ u1 . Secondly, Proposition 5.1 asserts that the result obtained for h = 0 is not destroyed when some specific perturbation is added (h = 0). On the other hand, if h = 0 but u = 1, results (5.1a) and (5.1b) are trivial. Indeed, by Azuma’s inequality (4.6), one finds some C > 0 such that, for every n ∈ N and for every a > 0, one has 2 n x − Ca P e−a ≤ ew k=1 h(X k−1 )Bk ≤ ea ≥ 1 − 2e w2 n . So, in general, one sees that the rare events where ew close to zero may essentially be neglected. In the sequel, one assumes that

n

x k=1 h(X k−1 )Bk

is very large or very

(A1) κ > 0 and h ∈ C1 (T) are given, (A2) w ∈ ]0, w0 ], where w0 is small enough to make all our assertions valid. All the constants introduced below may depend on κ and h. In order to prove Proposition 5.1, let us introduce a continuous operator T on L p (T), 1 ≤ p ≤ ∞, by setting T u(x) := E [(1 + wh(x)B) (u ◦ f B )(x)] =

b+

(u ◦ f b )(x) (1 + wh(x)b) τ (b)db.

b−

(5.2) ( Since E(B) = b τ (b)db = 0, one has T 1 = 1 and T ∞→∞ = 1. Moreover, for w small enough, the operator T maps positive functions to positive functions, and may thus be considered as a transfer operator of a Markov chain on the circle. On the other hand, for every b ∈ [b− , b+ ] and every x ∈ T, one has ewh(x)b = (1 + wh(x)b) · eO(w ) . 2

Therefore, for every u ∈ L1 (T; R+ ), for every n ∈ N satisfying w 2 n ≤ 1, and for almost every x ∈ T, one has n x T n u(x) ∼ E ew k=1 h(X k−1 )Bk u(X nx ) . (5.3) Let y ∈ T. The proof of Proposition 5.1 rests on the fact that, when T n acts on a function u ∈ L1B(y,w2 ) (T), it can be well approximated by an operator S y,n which can be explicitly


863

studied. In order to define S y,n , let us first introduce the convolution operators Ty on L p (T), 1 ≤ p ≤ ∞, by setting Ty u(x) := (u ◦ gb )(x, y) (1 + wh(y)b) τ (b)db, (5.4) where gb (x, y) := x + ϑ + (y, b) = x + w + wφ(y)b + w 2 ψ(y)b2 + O(w 3 ),

(5.5)

with defined as in (3.13), and φ and ψ defined as in (3.15b) and (3.15c). Then, one sets S y,0 := Id, and defines for each n ∈ N the operators S y,n := Ty−nw · · · Ty−w , R y := T − Ty .

(5.6a) (5.6b)

The core of our approximation scheme is described by Eq. (5.32) below. Let us however first sketch the idea behind it. Let z ∈ T, and let u ∈ L1B(z,w2 )(T; R+ ). The support of u ◦ f b should be centered at z − w, and so gb ( • , z − w) is likely to be the best approximation of f b , among all the maps gb ( • , y), y ∈ T. Consequently, Tz−w u is likely to be the best approximation of T u among all the choices Ty u , y ∈ T. One writes T n u = T n−1Rz−w u + T n−1 Tz−w u,

(5.7)

where Rz−w is defined by (5.6b). The first term in the right hand side of (5.7) can be bounded by using our estimates on R y , that is Lemmas 5.11 or 5.14 below. One is thus left with the second term. By the definition (5.4) the function Tz−w u will be approximately centered at z − w so it makes sense to now approximate T Tz−w u by Tz−2w Tz−w u so that one gets T n−1 Tz−w u = T n−2 Rz−2w Tz−w u + T n−2 Tz−2w Tz−w u. Again, one is left with the second term. Continuing this procedure one finally needs to handle the term T Tz−(n−1)w · · · Tz−w u, and arrives at T Tz−(n−1)w · · · Tz u = Rz−nw Tz−(n−2)w · · · Tz−w u + Tz−nw · · · Tz−w u.

(5.8)

By the definition (5.6a), one has Tz−nw · · · Tz−w u = Sz,n u. So, in this case, the second term in (5.8) can be bounded from above and below by some explicit estimates contained in Lemma 5.2 below. By means of Lemmas 5.11 and 5.14, one thus needs to show that the sum over the terms containing operators R y , y ∈ T, does not destroy the estimate on Sz,n u. The rest of the section is organized as follows. In Lemma 5.2, one obtains some bounds on the functions S y,n u for u ∈ L1B(y,w2 ) (T). These bounds show that at the level of scaling the shape of the function S y,n u, with u1 ∼ 1, resembles a Gaussian density of mean y − wn and variance O(w2 n). The proof turns out to be a straightforward computation as the operators Ty are diagonal in Fourier space. Next, Lemmas 5.11 and 5.14 give us bounds on R y . Lemma 5.14 is actually not crucial, and needs only to be used when n < 8, since then the function S y,n u may not be smooth enough for Lemma 5.11 to be applied. Some easy results about the localization of the functions T n u and S y,n u, for u ∈ L1B(y,Cw) (T), are given in Lemma 5.5. Finally, the proof of Proposition 5.1 is given.

864


Let us notice that, in Lemma 5.2, and consequently in the proof of Proposition 5.1, one has to distinguish the case where |y|T ∼ 1 does not hold. This comes from the lack of diffusivity of the chain X around 0 (see property (iii) of Corollary 3.4). Lemma 5.2. Let > 0. There exists K > 0 such that, for every n ∈ N satisfying 8 ≤ n ≤ w −2 , for every y ∈ T\B(0, ), and for every u ∈ L1B(y,w2 ) (T; R+ ), one has S y,n u ∈ C2 (T) and, for every x ∈ T,

K u1 , l = 0, 1, 2, √ (w n)(l+1) k 1 sin π(x + wn − y) · ∂ k S y,n u(x) ≤ K u √ , k = 1, 2. x w n |∂xl S y,n u(x)| ≤

(5.9a) (5.9b)

Moreover, when is small enough, there exists K () > 0, with K () → ∞ as → 0, such that, for every n ∈ N satisfying ≤ w2 n ≤ 2, for every x, y ∈ T, and for every u ∈ L1B(y,w2 ) (T; R+ ), |S y,n u(x)| ≥ K ()u1 when |x + nw − y|T ≤ 10.

(5.10)

The proof is deferred to Appendix A.3. Lemma 5.3. There exists K > 0 such that, for every u ∈ C2 (T) and every y ∈ T, one has

/ / R y u∞ ≤ K w 2 / sin π( • − y − w) · u /∞ + wu ∞ / / (5.11) +/ sin2 π( • − y − w) · u

/∞ + wu

∞ . Proof. One takes some u ∈ C2 (T), and one fixes x, y ∈ T. From the definitions (5.2) and (5.4), one has R y u(x) ≡ (T − Ty )u(x) = ((u ◦ f b )(x) − (u ◦ gb )(x, y)) (1 + wh(x)b) τ (b)db +w(h(x)−h(y)) (u ◦ gb )(x, y) b τ (b)db =: A1 + A2 . It is enough to bound |A1 | and |A2 | by the right hand side of (5.11). Let us first bound |A1 |. By the mean value theorem, and the definitions (3.15) and (5.5) of f b and gb , one has (u ◦ f b )(x) − (u ◦ gb )(x, y) = u (x + w + ξ1 ) × w [φ(x) − φ(y)] b + w 2 [ψ(x) − ψ(y)] b2 + O(w 3 ) , where ξ1 ≡ ξ1 (b) is such that |ξ1 | ≤ w|φ(x) − φ(y)| + O(w 2 ). By the mean value theorem again, one has u (x + w + ξ1 ) = u (x + w) + u

(x + w + ξ2 ) ξ1 , where ξ2 ≡ ξ2 (b) is such that |ξ2 | ≤ |ξ1 |.

(5.12)


865

Therefore, setting τ˜ (b) = (1 + wh(x)b) τ (b), one can write A1 as w [φ(x) − φ(y)] b + w 2 [ψ(x) − ψ(y)] b2 + O(w 3 ) τ˜ (b)db A1 = u (x + w) + u

(x + w + ξ2 (b)) ξ1 (b) w [φ(x) − φ(y)] b + O(w 2 ) τ˜ (b)db. One has |φ(x) − φ(y)| | sin π(x − y)|

|ψ(x) − ψ(y)| | sin π(x − y)|. ( So, taking into account the bound (5.12) and the fact that bτ (b)db = 0, one gets and

|A1 | w 2 |u (x + w)| | sin π(x − y)| + w 3 u ∞ 2 +w |u

(x + w + ξ2 (b))| sin2 π(x − y) τ˜ (b)db + w 3 u

∞ .

(5.13)

But one has sin2 π(x − y) ≤ sin2 π(x + ξ2 − y) + O(w). So, inserting this last bound in (5.13), one sees that |A1 | is bounded by the right hand side of (5.11). Let us then bound |A2 |. By the mean value theorem and the definition (5.5) of gb , one writes (u ◦ gb )(x, y) = u(x + w) + u (x + w + ξ ) O(w), ( where ξ ≡ ξ(b) = O(w). Therefore, taking into account that bτ (b)db = 0 and that |h(x) − h(y)| | sin π(x − y)|, one obtains 2 |A2 | w | sin π(x − y)| · |u (x + w + ξ(b))| · |b|τ (b)db w 2 sin π(Id − y − w) · u ∞ + wu ∞ . This finishes the proof.

Lemma 5.4. Let K , > 0. Let y ∈ T be such that |y|T ≥ . Then there exists K > 0 such that, for every u ∈ L1B(y,K w) (T), one has R y u1 ≤ K wu1 .

(5.14)

Moreover T u ∈ L∞ (T), and one has T u∞ ≤ K w −1 u1 .

(5.15)

Proof. The constants introduced in this proof may depend on K and . Let u ∈ L1B(y,K w) (T). One writes T u(x) = t (x, z)u(z)dz and Ty u(x) = t y (x, z)u(z)dz, (5.16) B(y,K w)

B(y,K w)

where the functions t and t y are obtained by performing a change of variables in the definitions (5.2) and (5.4) of T and Ty . Setting Fx (b) := f b (x) and G x (b) := gb (x, y), where f b and gb are defined in (3.15) and (5.5), one obtains t (x, z) = (1 + wh(x)Fx−1 (z)) τ (Fx−1 (z)) ∂z Fx−1 (z), −1 −1 t y (x, z) = (1 + wh(y)G −1 x (z)) τ (G x (z)) ∂z G x (z).

(5.17)

866


Let z ∈ B(y, K w) be given. Let us see that t ( • , z) and t y ( • , z) are well defined functions. The support of t ( • , z) (respectively of t y ( • , z)) is the support of (τ ◦ F(−1 • ) )(z) −1 (resp. of (τ ◦ G −1 ( • ) )(z)). The support of (τ ◦ F( • ) )(z) is made of all the x such that

b− ≤ Fx−1 (z) ≤ b+ ⇔ f b− (x) ≤ z ≤ f b+ (x) ⇔ f b−1 (z) ≤ x ≤ f b−1 (z). + − One obtains a similar relation for the support of (τ ◦ G −1 ( • ) )(z) and one gets therefore supp(t ( • , z)), supp(t y ( • , z)) ⊂ B(z, Cw) ⊂ B(y, C w). The hypothesis |y|T ≥ ensures that the maps Fx and G x are invertible when x ∈ B(y, C w), and actually that ∂b Fx (b) w and ∂b G x (b) w.

(5.18)

This shows in particular that t ( • , z) and t y ( • , z) are bounded functions. Let us now show (5.14). Taking (5.17) into account, one has, from the definition (5.6b) of R y , R y u1 ≤ |u(z)|dz |t (x, z) − t y (x, z)|dx. (5.19) B(y,K w)

B(y,C w)

It is therefore enough to show that, for every z ∈ B(y, K w), one has |t (x, z) − t y (x, z)|dx = O(w). B(y,C w)

(5.20)

Let us take some z ∈ B(y, K w) and some x ∈ B(y, C w). Since b− ≤ Fx−1 (z), G −1 x (z) ≤ b+ , since τ is bounded, and since (5.18) holds, one finds, starting from (5.17), that −1 −1 −1 |t (x, z) − t y (x, z)| |∂z Fx−1 (z) − ∂z G −1 x (z)| + w |τ (Fx (z)) − τ (G x (z))| + C. (5.21)

For every b ∈ [b− , b+ ], one has ∂b Fx (b) = wφ(x) + O(w 2 ) and ∂b G x (b) = wφ(y) + O(w 2 ). Therefore 1 1 − (z)| ≤ |∂z Fx−1 (z) − ∂z G −1 x wφ(x) + O(w 2 ) wφ(y) + O(w 2 ) w −1 |φ(y) − φ(x) + O(w)| 1,

(5.22)

since |y − x| = O(w). Inserting thus (5.22) in (5.21), and then (5.21) in (5.20), one finds |t (x, z) − t y (x, z)|dx w −1 |τ (Fx−1 (z)) − τ (G −1 x (z))|dx + O(w) B(y,Cw)

B(y,Cw)

=: w −1 I + O(w).

(5.23)

It remains thus to show that I = O(w 2 ). For this, let us define D1 := {x ∈ T : b− ≤ Fx−1 (z) ≤ b+ }, and D2 := {x ∈ T : b− ≤ G −1 x (z) ≤ b+ }.


867

One writes

I =

D1 ∩D2

(. . . ) +

(D1 ∩D2 )c

(. . . ) =: I1 + I2 .

First, when x ∈ D1 ∩ D2 , one uses the fact that τ ∈ C1 ([b− , b+ ]), that z − x − w z−x −w −1 −1 |Fx (z) − G x (z)| = − + O(w) = O(w), wφ(x) wφ(y) since |z−x −w| = O(w) and |φ(y)−φ(x)| = O(w), and that Leb(D1 ∩ D2 ) = O(w), to conclude that I1 = O(w 2 ). Next, when x ∈ (D1 ∩ D2 )c , one has t (x, z) = t y (x, z) = 0, except on D1 D2 . But, for every b ∈ [b− , b+ ], one has | f b (x) − gb (x, y)| = O(w 2 ), since |x − y| = O(w). So, one has Leb(D1 D2 ) = O(w 2 ), and thus I2 = O(w 2 ). Let us finally show (5.15). From (5.16), one has that |T u(x)| ≤ supz∈B(y,K w) |t (x, z)|. The relations (5.17) and (5.18) allow us to obtain the result. In order to prove the next lemma, we introduce the adjoint T ∗ of T with respect to the Lebesgue measure. This operator is defined on L p (T) (1 ≤ p ≤ ∞) and is such that,

for every u ∈ L p (T) and every v ∈ L p (T), with 1/ p + 1/ p = 1, one has ∗ v T u dx = u T v dx. (5.24) T

T

From the definition (5.2) of T , one concludes that T ∗ u(x) =

b+ b−

(u ◦ f b−1 )(x) 1 + w (h ◦ f b−1 )(x)b ∂x f b−1 (x) τ (b)db. (5.25)

Therefore, when u ≥ 0, one has T ∗ u(x) ≥ e−O(w)

(u ◦ f b−1 )(x) τ (b)db.

(5.26)

For z ∈ R, let us define the chain Y = (Ynz : n ∈ N0 ) by Y0z := z and z z z (Yn−1 ) = Yn−1 − w − wφ(Yn−1 )Bn + O(w 2 ). Ynz := f B−1 n

(5.27)

Lemma 5.5. Let K > 0. There exist K 2 ≥ K 1 > 0 such that, for every n ∈ N, for every y ∈ T, and for every u ∈ L1B(y,K w) (T), one has supp(T n u), supp(S y,n u) ⊂ [y − K 2 wn, y − K 1 wn] .

(5.28)

Morover, for every R > 0 large enough, there exists K > 0 such that, for every n ∈ N satisfying wn ≤ 1, for every y ∈ T, and for every u ∈ L1B(y,w) (T; R+ ), one has √ B(y−wn,R w)

T n u(z)dz ≥ K u1 .

(5.29)

868


Proof. Let us first show (5.28). Let us consider the case of T n u ; the case of S y,n u is strictly analogous. From the definition (5.2), one sees that −n supp(T n u) ⊂ f b−n (y − K w/2), f (y + K w/2) . b− + This implies the result, since, by the definition (3.15) of f b , one has, for every x ∈ T and every b ∈ [b− , b+ ], (1 + b− )w − O(w 2 ) ≤ x − f b−1 (x) ≤ (1 + b+ )w + O(w 2 ). Let us then show (5.29). Let u ∈ L1B(y,w2 ) (T; R+ ), let R > 0, and let n ∈ N be such that wn ≤ 1. From the definition (5.24) of the adjoint T ∗ , one has n T u(z)dz = T ∗nχ B(y−wn,R √w)(z) u(z)dz. √ B(y−wn,R w)

B(y,w)

It is therefore enough to show that, for every z ∈ B(y, w), one has T ∗n χ B(y−wn,R √w) (z) 1, if R is large enough. But, since wn ≤ 1, (5.26) implies that −1 T ∗n χ B(y−wn,R √w) (z) E (χ B(y−wn,R √w) ◦ f B−1 ◦ · · · ◦ f )(z) B1 n z √ (5.30) = 1 − P |Yn − (y − wn)| ≥ R w , where Y is defined in (5.27). Therefore, since |z − y| = O(w) and since w2 n = O(w), one obtains, from the definition (5.27) of Y , and from Azuma’s inequality (4.6), that ) n * z √ √ C R2 z P |Yn − (y − wn)| ≥ R w = P w φ(Yk−1 ) + O(w) ≥ R w ≤ 2e− nw . k=1

(5.31) The proof is finished by taking R large enough, and inserting (5.31) in (5.30).

Proof of proposition 5.1. Let n ≥ 9 be such that nw 2 ≤ 1. Let us make three observan x tions. First, by (5.3), it is enough to show the proposition with E(ew k=1 h(X k−1 )Bk u(X nx )) replaced by T n u(x) in (5.1a) and (5.1b). Second, it is enough to prove the proposition for functions in L1B(y,w2 )(T; R+ ) with y ∈ T arbitrary. So, throughout the proof, one assumes that y ∈ T is given, and the symbol v denotes a function in L1B(y,w2 ) (T; R+ ).

Third, it is enough to show (5.1b) for some n satisfying w 2 n ≤ 1/2. Indeed, let us now assume that (5.1b) is shown for this n , and let n be such that 1/2 ≤ w 2 n ≤ 1. From the definition (5.2), one sees that, if u 1 ≥ u 2 , one has T u 1 ≥ T u 2 . So, one

writes n = n + n

and, for every u ∈ L1 (T; R+ ), one gets T n u(x) = T n T n u(x)

u1 · T n 1(x) ∼ u1 , where the fact that T n 1 ∼ 1 directly follows from the definition (5.2) of T , Azuma’s bound (4.4), and the hypothesis w 2 n ≤ 1. The proof is now divided into three steps, but the core is entirely contained in the first one.

Step 1: Approximating T n by S y,n . One here shows the bounds (5.1a) and (5.1b) under two particular assumptions:


869

1. One supposes that |y|T ≥ 1 , for some 1 > 0. The constants introduced below may depend on 1 . 2. Only for (5.1b), one assumes that n is such that 2 ≤ n ≤ 22 and that |x +nw−y|T ≤ 102 for some 2 > 0 small enough. By the definition (5.6a) of S y,n , one can write T n v = S y,n v +

8

T n−k R y−kw S y,k−1 v +

k=1

n−1

T n−k R y−kw S y,k−1 v

k=9

=: S y,n v + Q 1 + Q 2 .

(5.32)

Let us bound Q 1 ∞ . Let k ∈ N be such that 1 ≤ k ≤ 8. By (5.28), one has supp(R y−kw S y,k−1 v) ⊂ B(y, Cw).

(5.33)

Recalling that T ∞→∞ = 1, one uses (5.14) and (5.15) to obtain that Q 1 ∞ ≤

8 8 T 9−k R y−kw S y,k−1 v∞ w −1 R y−kw S y,k−1 v1 k=1

k=1

7 S y,k−1 v1 v1 ,

(5.34)

k=0

where, for the last inequality, one has used the fact that Ty 1→1 = 1 for every y ∈ T. Let us bound Q 2 ∞ . By Lemma 5.11 and estimates (5.9b) and (5.9a) in Lemma 5.2, one has, for 8 ≤ k ≤ w −2 , T n−k R y−kw S y,k−1 v∞ ≤ R y−kw S y,k−1 v∞

1 1 w w 2 · v1 . + √ + w √ + w k w 2 k w k w 3 k 3/2 Therefore, since w2 n ≤ 1 by hypothesis, one gets √ Q 2 ∞ (w n + w log n + C)v1 v1 .

(5.35)

So, from (5.32), (5.34) and (5.35), one has T n v − S y,n v∞ ≤ C v1 , where the constant C is independent of 2 . Therefore, in the particular case considered, (5.1a) follows from (5.9a) with l = 0, and (5.1b) follows from (5.10), if 2 has been chosen small enough. Step 2: Proof of (5.1a). By Step 1, (5.1a) is known to hold when |y|T ≥ 1 , and one may now assume that |y|T < 1 . Moreover, one has still the freedom to take 1 as small as we want. One now uses the hypothesis wn ≥ κ. Let m ∈ N be such that mw = , for some ∈ ]0, c/2]. If 1 is small enough, it follows from (5.28) that one can choose such that supp(T m u) ∩ B(0, 1 ) = ∅. But the particular case considered in Step 1 1 implies that (5.1a) is valid for any function in LT \B(0,1 ) (T), and thus one has T n v(x) = T n−m T m v(x)

T m v1 v1 √ , √ w n w n−m

870


where the last inequality follows from the fact that T 1→1 ≤ eO(w) , as can be seen from the definition (5.2). Step 3: Proof of (5.1b). One first will establish (5.1b) for n such that n = 2 w −2 , and for x such that |x + wn − y|T ≤ 102 . By Step 1, it is now enough to consider the case |y|T < 1 . Let now m = 21 w −1 , and let R > 0. If R is taken large enough, it follows from (5.29), and from the particular case of (5.1b) already established in Step 1, that m T n v(x) ≥ T n−m χ B(y−wm,R √w) T m v (x) √ T v(z)dz v1 . B(y−wm,R w)

One finally needs to get rid of the assumption |x + wn − y|T ≤ 102 . One use a classical technique [8]. One shows (5.1b) for n = kq, with k ≥ 1/182 , and q such that q = 2 w −2 . One already knows that T q v χ B(y−wq,102 ) v1 .

(5.36)

But one now will show that, for every z ∈ T, and for every s ∈ [2 , 1], one has T q χ B(z,s) 2 χ B(z−qw,s+92 ) .

(5.37)

This will imply the result : T n v = T kq v T (k−1)q χ B(y−wq,102 ) v1 · · · 2k−1 χ B(y−wkq,(10+9(k−1))2 ) v1 2k−1 v1 . show (5.37). Let z ∈ T and s ∈ [2 , 1]. Let us write T q u(x) = ( Let us thus

tq (x, z )u(z )dz for any u ∈ L1 (T). Relation (5.36) implies in fact that tq (x, • ) χ B(x+wq,102 ) ( • ) (which may be formally checked by taking u(x) = δ(y − x)). Therefore T q χ B(z,s) (x) χ B(x+wq,102 ) (z ) · χ B(z,s) (z )dz

2 χ B(z,s+92 ) (x + wq) = 2 χ B(z−wq,s+92 ) (x). This finishes the proof.

6. Putting Everything Together In [6] p. 1710, Casher and Lebowitz derive the lower bound E Jn (T1 − Tn )n −3/2 . However, their argument contains a gap, and consequently this lower bound remains still to be proven. Indeed, their proof is based on the following estimate of Dn (e1 ) (K 1,n in their notation): 2 E Dn (e1 )2 ∼ eCnw as w 0. (6.1) This bound is obtained by computing the eigenvalues of a 4 × 4 matrix F, defined in [6] p. 1710. But this estimate cannot hold. Indeed, we know for example, from Corollary 3.6 and Proposition 5.1, that E[Dn (e1 )2 ] ∼ w −2 when w 2 n ∼ 1. Although the computation of the eigenvalues of F is correct, the authors do not take into account the fact that a w-dependent change of variables is needed to obtain a correct estimate on E[Dn (e1 )2 ].


871

6.1. Proof of the lower bound. We begin by a lemma. Let (L n ) and (K n ) be the processes defined in Lemma 3.7. Lemma 6.1. For every α > 0, there exists C(α) > 0, such that, for every a > 0, and every n ∈ N satisfying w 2 n ≤ 1, one has P(|K n | ≥ a), P(|L n | ≥ a) ≤ C(α)w α .

(6.2)

Proof. Let (An : n ∈ N0 ) be a F-adapted process such that An := e Mn +L n +O(w

2 n)

(6.3)

for every n ∈ N0 , with Mn as defined in Lemma 3.7. From the expressions (3.33) and (3.34), both K n and L n are of the form Rn := w 2

n

A j−1 S j−1 B j ,

j=1

where (S j ) is F-adapted, and satisfies |S j | 1 for j ∈ N0 . Let a > 0. One writes ⎛ ⎞ n P(Rn ≥ a) = P ⎝w 3/2 w 1/2 A j−1 S j−1 B j ≥ a, max w 1/2A j−1 ≤ 1⎠ ⎛

1≤ j≤n

j=1

+ P ⎝w 3/2

n

⎞ w 1/2 A j−1 S j−1 B j ≥ a, max w 1/2A j−1 > 1⎠ . 1≤ j≤n

j=1

Let us now define a process ( A˜ n : n ∈ N0 ) by setting A˜ n := An · χ[0,1](w 1/2 An ). One has ⎛ ⎞ n n P(Rn ≥ a) ≤ P ⎝w 3/2 w 1/2 A˜ j−1 S j−1 B j ≥ a ⎠ + P(w 1/2 A j−1 > 1). j=1

j=1

(6.4) ≤ 1, one has First, by Azuma’s inequality (4.6), and since ⎛ ⎞ n 2 3 2 −1 3/2 ⎝ P w w 1/2 A˜ j−1 S j−1 B j ≥ a ⎠ ≤ 2 e−Ca /w n ≤ e−Ca w . w2 n

(6.5)

j=1

Next, it follows from (3.32), (3.33) and (3.35) that An defined in (6.3) if also of the form n 2 An = ew j=1 G j−1 B j +O(w n) , where (G j ) is F-adapted, and |G j | 1 for j ∈ N0 . So, applying again Azuma’s inequality, one gets ⎛ ⎞ j−1 1 1 P(w 1/2 A j−1 > 1) = P ⎝w G k−1 Bk + O(w 2 n) > log ⎠ 2 w k

n

e

2 − C log (1/w) ( j−1)w2

≤ e−C

Therefore j=1 P(w 1/2 A j−1 > 1) w −2 e−C ing this last bound and (6.5) in (6.4).

log2 (1/w)

log2 (w−1 )

.

. The proof is finished by insert-

872


With the help of this lemma we can now prove the lower bound E JnCL n −3/2 of Theorem 1.1. Indeed, from (2.6), it follows that E JnCL

2n −1/2

n −1/2

E jn (w)dw,

with jn (w) defined in (2.7). Thus it suffices to show that 1 ≤ w 2 n ≤ 2 implies E jn (w) w 2 ∼ n −1 . So assume 1 ≤ w 2 n ≤ 2, and use Corollary 3.6 in (2.7) to write −1 ϑ sin X ϑ )2 (nϑ sin X nϑ )2 (n−1 (n0 sin X n0 )2 n−1 0 0 2 jn (w) 1+ + + +(n−1 sin X n−1 ) . w4 w2 w2 (6.6) Let us take some R, c > 1. The constants introduced below may depend on R and c. Let us observe that, by point (i) of Corollary 3.4, one has |X n−1 |T w provided |X n |T w 2 , and that, from the definition (3.17), one has n−1 ∈ [0, 2R] when n ∈ [0, R]. It follows therefore from (6.6) that E jn (w) P |X nϑ |T ≤ w 2 , nϑ ≤ R, |X n0 |T ≤ cw, n0 ≤ c R . (6.7) We now use Lemma 3.7. First, by (3.30), one has χ B(cw) (X n0 ) ≥ χ[0,R] (e Mn ) · χ B(0,1) (L n ) · χ B(0,w2 ) (X nϑ ),

(6.8)

provided c is large enough. Secondly, by (3.31), one has χ[0,c R] (n0 ) ≥ χ B(0,1) (K n ) · χ[0,R] (nϑ ),

(6.9)

again, provided c is large enough. Using then (6.8) and (6.9) in (6.7), one obtains E jn (w) P |X nϑ |T ≤ w 2 , nϑ ≤ R, e Mn ≤ R, |L n | ≤ 1, |K n | ≤ 1 ≥ P |X nϑ |T ≤ w 2 , |L n | ≤ 1, |K n | ≤ 1 − P |X nϑ |T ≤ w 2 , nϑ > R − P |X nϑ |T ≤ w 2 , e Mn > R ≥ P |X nϑ |T ≤ w 2 − P (|L n | > 1) − P (|K n | > 1) − P |X nϑ |T ≤ w 2 , nϑ > R − P |X nϑ |T ≤ w 2 , e Mn > R . Applying then Markov’s inequality to the two last terms, one gets E jn (w) P |X nϑ |T ≤ w 2 − P (|L n | > 1) − P (|K n | > 1) 1 1 − E χ B(0,w2 ) (X nϑ ) · nϑ − E χ B(0,w2 ) (X nϑ ) · e Mn . R R Proposition 5.1 and Lemma 6.2 allow then to conclude that E( jn (w)) w 2 if R is chosen large enough. This finishes the proof.


873

6.2. Proof of the upper bound. Let n ∈ N. Let c > 0 to be fixed later. Starting from (2.6), one writes c/n w0 ∞ E JnCL ∼ E jn (w)dw + E jn (w)dw + E jn (w)dw =: J1 + J2 + J3 , c/n

0

w0

(6.10) 2 (e ), D 2 (e ), D 2 (e ) ≥ 0 in with jn defined in (2.7). Using the crude bounds Dn−1 1 n 2 n−1 2 the definition of jn , and applying then Corollary 3.6, one obtains

jn (w)

1 1 + w −2 Dn2 (e1 )

h(nϑ sin π X nϑ ) with h(r ) =

1 . (6.11) 1 + w −4 r 2

Let us first bound J1 . Let w ∈ [0, c/n[. First, nϑ 1, as can be checked from its definition (3.17). Next, if c is small enough, one has, by point (i) of Corollary 3.4, that wn X n ≤

1 1 wn ≤ . 2 2

Therefore one has sin2 π X nϑ w 2 n 2 , and thus c/n dw n −3 . J1 1 + w −2 n 2 0

(6.12)

Let us next bound J2 . Let w ∈ [c/n, w0 [ , define m := min{n, w−2 }, and write ϑ ϑ ϑ ϑ E jn (w) = E jn (w) X n−m = x, n−m = a P(X n−m ∈ dx, n−m ∈ da). (6.13) R T

ϑ ϑ To simplify notations, set E( • |x, a) := E • X n−m = x, n−m = a . If x ∈ T and a ∈ R are given, it follows from (6.11) that E( jn (w)|x, a) E h(a · mx sin π X mx ), (6.14) 0n ϑ ϑ ϑ since, by the definition (3.17), one may write n = l=1 g(X l−1 , Bl ) = n−m 0n ϑ 4 −2 l=n−m+1 g(X l−1 , Bl ), for some function g. Because h(r ) ≤ 1 and h(r ) ≤ w r for every r ∈ R, one has, for every event A, the bound h(a mx sin π X mx ) ≤ 1 A + 1 Ac · w 4 · (a mx sin π X mx )−2 .

(6.15)

So, taking 1 A = χ[0,1] (w −4 a 2 sin2 π X mx ), and using (6.15) in (6.14) one obtains E( jn (w)|x, a)

E χ[0,1] (w −4 a 2 sin2 π X mx )+χ]1,∞[ (w −4 a 2 sin2 π X mx )·w 4 ·(amx sin π X mx )−2 . Therefore, Proposition 5.1 implies E( jn (w)|x, a)

1 √ χ[0,1] (w −4 a sin2 π y) + χ]1,∞[ (w −4 a sin2 π y) w 4 a −2 sin−2 π y dy w m T 1/2 1 dy dy 1 √ √ w m T 1 + w −4 a 2 sin2 π y w m −1/2 1 + (w −2 a y)2 w 2 a −1 +∞ dz w ≤ √ √ a −1 , 2 w m −∞ 1 + z m

874


where one has used the change of variables z = w−2 ay to get the third line. One now inserts this last bound in (6.13). Applying Proposition 4.1 yields ϑ w ϑ ∈ dx, n−m ∈ da E jn (w) √ a −1 P X n−m R T m

w w w 2 2 ϑ √ e−αw (n−m) max √ , w 2 e−αw n . = √ E 1/ n−m m m n Therefore 1 J2 √ n

n −1/2

we−αw n dw +

0

2

∞

n −1/2

w 2 e−αw n dw n −3/2 . 2

(6.16)

It has already been shown by O’Connor [19] that J3 e−Cn . The proof is completed by inserting this last estimate, together with (6.12) and (6.16) in (6.11). 1/2

6.3. Generalizations. We start by analyzing possible extensions of Theorem 1.1 to some classes of heat baths. Associate a heat bath to a function μ : R → C as described by Dhar [9]. One may then obtain, at least formally, a new heat bath by replacing μ with a function μ˜ : R → C defined by scaling μ(w) ˜ ∼ μ(sgn(w)|w|s ), s > 0. In [9] Dhar argued based on numerics and a non-rigorous approximation that Casher-Lebowitz and Rubin-Greer bath functions μCL (w) ∼ −iw and μRG (w) ∼ e−iπ ϑ(w) , with ϑ(w) given 1 1 in (3.6), yield E JnCL ∼ n −(1+s/2) and E JnRG ∼ n −(1+|s−1|)/2 , respectively. The first of these statements can be proven rigorously by directly adapting the proof of Theorem 1.1. The second case, however, does not follow directly from the proof of E J RG ∼ n −1/2 , even though we believe it should not be too difficult to prove by using our results. To see where the difficulties within this second case lie, as well as to further demonstrate our approach, let us sketch how E J RG ∼ n −1/2 , first proven by Verheggen [25], can be obtained by using our representation of Dn (v). Indeed, the choices e˜1 := 2−1/2 (e1 + e2 ) and e˜2 := 2−1/2 (e1 − e2 ) yield (Proposition 3.5) Dn (e˜1 ) ∼ nx1 sin π X nx1 and Dn (e˜2 ) ∼ w −1 nx2 sin π X nx2 with x1 = 1/2 + O(w) and x2 = w/2 + O(w 2 ), respectively. If one substitutes these in the expression for the current density jnRG (w) of the Rubin-Greer model (the equation between 3.1 and 3.2 in [25]) one ends up with an estimate

−1 −1

1 + (nx1 )2 + (nx2 )2 jnRG 1 + (nx2 )2 on w ∈ ]0, w0 ], (6.17) after making use of the basic properties of X -processes (Corollary 3.4). This reveals that the Rubin-Greer model is special in the sense that the random phases X nxk in the expressions Dn (e˜k ) ∼ nxk sin π X nxk do not have any direct role in the scaling behav1 ior of the current. The reason why proving E JnRG ∼ n −(1+|s−1|)/2 , s = 1, is again more difficult is that the bounds analogous to (6.17) become again explicitly dependent on X xk . Now continuing with the RG-model, based on (6.17) one can prove E jnRG (w) ∼ ( 2 e−Cw n which then implies the scaling: EJnRG = R E jnRG (w)dw ∼ n −1/2 . Indeed, for 2 the lower bound E jnRG (w) e−Cw n one considers the typical behavior, which is easier to analyze than in the Casher-Lebowitz model since X -processes are not present. The respective upper bound follows from Proposition 4.1.


875

There are also two other directions which one could naturally consider generalizing Theorem 1.1 into. Let us first consider dropping the assumption about the continuity of the mass density τ , so that the theorem would become applicable, for example, in the case binary distribution Mk ∈ {1, 2}. One finds out that the existence of continuous derivative τ is only needed in the proof of the pointwise density bounds on X n (Proposition 5.1) while all the other steps used to prove Theorem 1.1 rely only on the minimal assumption E Bk2 > 0. Actually, it is obvious that Proposition 5.1 can not even be expected to hold for a mass distribution which assigns non-zero weights on atoms. Luckily however, Proposition 5.1 is only needed to bound the probabilities of X n visiting some small intervals of size O(w2 ).2 Thus one could attempt to carry out the generalization of the theorem for the non-smooth mass distributions by trying to replace the pointwise density bounds on X n with slightly weaker version of Proposition 5.1 which considers the densities up to the critical resolution O(w2 ) only. It would be also desirable to prove a quenched version of Theorem 1.1, i.e., to prove that there exist constants 0 < C1 ≤ C2 < ∞ such that, for every n ∈ N, and for almost every realization of the masses, one has C1 ≤ n 3/2 Jn ≤ C2 .

(6.18)

One is not yet in the state to establish this. Indeed, in order to prove the annealed version, i.e., Theorem 1.1, it was possible to analyze the expectation E jn (w) for each frequency w separately, and then in the end, just integrate over the frequencies to obtain E Jn . On the other hand, to show (6.18), one has to, at least in principle, face a harder ( ∞problem: for a given realization of the masses, one needs to study directly the integral 0 jn (w)dw of the random function jn . It seems therefore that the techniques of our paper alone do not allow to establish (6.18). It is however worth to notice that some aspects may become, on the other hand, easier in the quenched setting. For example, Proposition 4.1 is non-trivial only because one has to take into account some bad but very unlikely events. For the almost sure convergence, such events could probably just be ignored. Acknowledgements. A. Kupiainen deserves a special acknowledgement for introducing this problem to us, and always being available for useful comments and enlightening ideas. We are grateful to J. Bricmont for helpful discussions and valuable feedback. We benefited from various illuminating discussions with M. Jara, J. Lukkarinen, M. Pakkanen, W. de Roeck, L. Saloff-Coste, A. Raugi and C. Liverani. We both thank the Academy of Finland for Financial support. Additionally, O. Ajanki thanks the European Research Council and F. Huveneers thanks the Belgian Interuniversity Attraction Poles Program for additional financial support.

A. Appendix A.1. Proof of Lemma 3.2. By using (3.1) one gets f b ≡ g −1 ◦ M A ◦ g = E ◦ MG −1AG ◦ E −1 , where G −1 AG =

(1 + iδ)eiπ ϑ iδe−iπ ϑ

−iδeiπ ϑ , (1 − iδ)e−iπ ϑ

(A.1)

2 This is for the Casher-Lebowitz model. For other heat baths intervals of size O(w α ), α > 0 have to be considered.

876


and δ=

(π w/2)b π 2 w2 b = = (π w/2) b + O(w 3 b). 2 sin π ϑ 1 − (π w/2)2

(A.2)

Here the second equality follows from (3.6). The map MG −1 AG describes the evolution ξ → M A (ξ ) on the complex unit circle ∂ D: MG −1 AG (eiφ ) = ei(φ+2π ϑ)

1 + iδ (1 − e−iφ ) ˜ =: exp i(φ + 2π ϑ + 2 (φ, δ)) . 1 − iδ (1 − eiφ )

Here the effect of noise δ comes through 1 − cos φ ˜ (φ, δ) = arg 1 + iδ(1 − e−iφ ) = arctan δ 1 − δ sin φ = (1 − cos φ) δ + (1 − cos φ) sin φ δ 2 + O (1 − cos φ)δ 3 .

(A.3)

By substituting φ = 2π x and using the middle expression of (A.2) in place of δ we obtain (3.13a). Let h(w, x, b) be a function so that wb h(w, x, b) sin2 π x equals the argument of arctan in (3.13a). It is easy to see that h is a smooth bounded function on [0, w0 ] × T × [b− , b+ ]. We may then write (x, b) ≡ (w; x, b) as (w, x, b) =

3 1 1 wb h(w, x, b) sin2 π x + arctan

(s) wb h(w, x, b) sin2 π x , π 6π (A.4)

where the third derivative arctan

(s) of arctan is bounded on 0 ≤ s ≤ wb sin2 (π x) h(w, x, b) = O(w). By expanding h(w, x, b) = π + O(w) similarly, and then substituting the result back into (A.4) one obtains (3.13b). −1 i2π y ) = To prove the formula (3.12b) for f b−1 we note that ei2π fb (y) = M−1 2 (e i2π y ), where 2 is the matrix in (A.1). After replacing 2 by its inverse, the proof M 2−1 (e proceeds just like before. The identity involving (x, −b) follows by expressing f b and f b−1 in terms of in x = f b−1 ( f b (x)). A.2. Proof of Lemma 4.2. Both proofs are rather directly adapted from Freedman’s paper [13]. We start with Freedman’s bound. To this end define a function g : R → R: g(0) = 1/2, g(t) := (et − 1 − t)/t 2 for t = 0. Let t, y ∈ R so that |y| ≤ 1. By definition we have then et y = 1 + t y + (t y)2 g(t y). It is not too difficult to see that g is an increasing function. Therefore, g(t y) ≤ g(t) above, and et y ≤ 1 + t y + y 2 t 2 g(t) = 1 + t y + y 2 (et − 1 − t) ≡ 1 + t y + y 2 κ1 (t).

(A.5)


877

Suppose Y is a random variable such that |Y | ≤ 1 and E(Y ) = 0. Setting y = Y in (A.5) and taking expectation yields 2 (A.6) E etY ≤ E 1 + tY + κ1 (t)Y 2 = 1 + κ1 (t)E(Y 2 ) ≤ eκ1 (t)E(Y ) . Now, set Yi := (Mi − Mi−1 )/m, so that |Yi | ≤ 1 and E(Yi |Fi−1 ) = 0. By using κm (t) = m −2 k1 (tm) to write κm (t)(Mi − Mi−1 )2 = κ1 (mt)Yi2 , the estimate (A.6) implies that for any t ∈ R: 2 2 E et (Mi −Mi−1 )−κm (t)E[(Mi −Mi−1 ) |Fi−1 ] |Fi−1 ) = E etmYi −κ1 (tm)E[Yi |Fi−1 ] Fi−1 ≤ 1. (A.7) Recall the definition (4.3) of Vn and the pointwise bound Vn ≤ vn . Apply these to get the first two lines below. Then use (A.7) iteratively to get Freedman’s bound: E et Mn ≤ eκm (t)vn E et Mn −κm (t)Vn

2 = eκm (t)vn E et Mn−1 −κm (t)Vn−1 E et (Mn −Mn−1 )−κm (t)E[(Mn −Mn−1 ) |Fn−1 ] |Fn−1 ) ≤ eκm (t)vn E et Mn−1 −κm (t)Vn−1 ≤ · · · ≤ eκm (t)vn .

(s)t 3 = (1/2)t 2 + The bound (4.5) comes from the power expansion km (t) = (1/2)t 2 +km ms 3 (m/6)e t , with s ∈ [0, t], by taking s = |t|. The proof of Azuma’s bound proceeds in a very similar way: First, one uses the convexity of the exponent function to get a bound

et y = e

1+y 1−y 2 t+ 2 (−t)

≤

1 + y t 1 − y −t 2 e + e = cosh t + y sinh t ≤ et /2 + y sinh t, 2 2

for every t, y ∈ R with |y| ≤ 1. Using this instead of (A.5) in the first inequality 1 2 of (A.6) yields the bound EetY ≤ e 2 t , and consequently E et (Mi −Mi−1 ) Fi−1 = 2 E e(tm)Yi Fi−1 ≤ e(tm) /2 . Iterating this finishes the proof:

2 2 2 2 Eet Mn = E et Mn−1 E(et (Mi −Mi−1 ) |Fn−1 ) = et m /2 E(et Mn−1 ) ≤ · · · ≤ et m n/2 . A.3. Proof of Lemma 5.2. Let us start with some conventions and definitions: For k ∈ N0 , y ∈ T we define: yk := y − wk,

αk := φ(yk ),

γk := h(yk ).

For > 0 and n 0 ∈ N, one defines H (, n 0 ) := (y, n) ∈ T × N : |y|T ≥ , n ≥ n 0 , w 2 n ≥ . For u ∈ L1 (T) and ξ ∈ Z, one defines u(ξ ˆ )= e−i2π ξ x u(x)dx. T

The operators Tyk , k ∈ N, are diagonal in Fourier space: for every ξ ∈ Z, one has i2π wξ (T λk (wξ ) · u(ξ ˆ ), yku)(ξ ) = e

(A.8)

878


where λk is a function on R defined by −1 λk (z) := ei2π zw (ϑ−w+(yk ,b)) (1 + wγk b)τ (b)db = ei2π z(αk b+O(w)) (1 + wγk b)τ (b)db.

(A.9)

Let y ∈ T, let u ∈ L1B(y,w2 ) (T; R+ ), and let v ∈ L1B(0,w2 ) (T; R+ ) be such that v(x) = u(x + y).

(A.10)

One writes S y,n u(x) = Tyn · · · Ty1 v(x − y) =

ei2π ξ(x+nw−y) n (ξ )v(ξ ˆ ),

(A.11)

ξ ∈Z

where n is a function on R defined by n (ξ ) :=

n

λ j (ξ w) (n ≥ 1).

(A.12)

j=1

But, if (y, n) ∈ H (, 8) for some > 0, the right hand side of (A.11) represents actually a C2 -function. This follows directly from (A.17) with l = 0 in Lemma A.1 below, and the fact that |v(ξ ˆ )| ≤ v1 for every ξ ∈ Z. Lemma A.1. Let > 0. There exist K , K , > 0 such that, for every (y, n) ∈ H (, 1), and for every ξ ∈ R satisfying |ξ w| ≤ , one has e−K nw

2ξ 2

| n (ξ )| |

n (ξ )|

≤ |n (ξ )| ≤ e−K

nw 2 ξ 2

≤ K nw (1 + |ξ |)e 2

,

−K nw2 ξ 2

(A.13) ,

≤ K nw (1 + nw + nw ξ )e 2

2

2 2

(A.14) −K nw2 ξ 2

| arg(n (ξ ))| ≤ K nw (|ξ | + w|ξ | ). 2

3

,

(A.15) (A.16)

For every > 0, there exist K , K > 0 such that, for every (y, n) ∈ H (, 1), and for every ξ ∈ Z satisfying |ξ w| > , one has |∂ξl n (ξ )| ≤

K (wn)l , l = 0, 1, 2. (1 + K |ξ w|)n/2

(A.17)

Proof. The constants introduced in this proof may depend on . For the whole proof, one sets z := wξ . Before starting, let us make two observations. First, one has |αk | 1 and |γk | 1 for every k ∈ N0 . Secondly, for every (y, n) ∈ H (, 1), there exists an integer m ≥ n/2 independent of y, and a subsequence {k j } ≡ {k j : 1 ≤ j ≤ m} ⊂ {1, 2, . . . , n} such that |αk j | 1.

(A.18)


879

Let us first prove the formulas (A.13) up to (A.16). ( One takes (y, n) (∈ H (, 1). A Taylor expansion in (A.1), taking into account that τ (b)db = 1 and bτ (b)db = E(B) = 0, gives λk (z) = 1 + iO(w|z|) −

(2π )2 2 2 z αk E(B 2 ) + O(wz 2 ) + iO(|z|3 ) + O(z 4 ), 2

as z → 0. Therefore, one has (2π )2 2 2

|λk (z)| = e− 2 z αk E(B )+O(z | arg(λk (z))| = O(|z|w + |z|3 ), 2

2 w+|z|3 )

,

(A.19) (A.20)

as z → 0. Similarly, a Taylor expansion in (A.1) gives |∂z λk (z)| = O(w + |z|),

(A.21)

|∂z2 λk (z)|

(A.22)

= O(1),

as z → 0. First, by (A.19) with z = ξ w, and by the definition (A.12) of n , one obtains n 1 2 2 2 2 |n (ξ )| = exp − (ξ w) E(B ) αk + O n(ξ w) (|ξ w| + w) as ξ w → 0. 2 k=1

This shows (A.13), taking into account the two observations at the beginning of this proof. Next, with z = ξ w, one has ∂ξ n (ξ ) = w

n

∂z λ j (z)

j=1

∂ξ2 n (ξ ) = w 2

j=1

λk (z),

(A.23)

1≤k≤n

⎛ n

k= j

⎞

⎜ 2 ⎟ ⎜∂ λ j (z) ⎟ . (A.24) λ (z) + ∂ λ (z) ∂ λ (z) λ (z) k z j z k l z ⎝ ⎠ 1≤k≤n

k= j

1≤k≤n

k= j

1≤l≤n

l= j,k

One then obtains (A.14) and (A.15), by using these last formulas together with (A.21), (A.22), and the fact that |λk (z)| ≤ 1 for every k ∈ N and every z ∈ R, which follows from the definition (A.1). Finally, (A.16) directly follows from (A.20). Let us now show (A.17). Let > 0, and let (y, n) ∈ H (, 1). The constants introduced below may depend on . One proceeds in two steps. First, one shows (A.17) for |z| = |ξ w| ∈ [ , 1/ [ . It is actually enough to show that |λk (z)| ≤ 1 − 1

(A.25)

for some 1 > 0 and for every k ∈ {k j }, with {k j } as defined in (A.18). Indeed, from the definition (A.1), one has |λk (z)| ≤ 1 and |∂zl λk (z)| 1 for l = 1, 2, for every k ∈ N and every z ∈ R. So, inserting (A.25) in (A.12), (A.23) or (A.24), respectively for l = 0, l = 1 or l = 2, will imply n

|∂ξl n (ξ )| (wn)l (1 − 1 ) 2 −2 , which is equivalent to (A.17) when ≤ |ξ w| < 1/ .

880


So let us show (A.25). By continuity of τ , one finds an interval J on which τ ≥ 2 for some 2 > 0. One has iz(αk b+O (w)) e (1 + wγk b)τ (b)db = eizαk b τ (b)db + O(w), J

J

and, for some 3 > 0,

eizαk b τ (b)db ≤ (1 − 3 ) τ (b)db. J

Therefore

J

|λk (z)| ≤ (1 − 3 )

τ (b)db + J

Jc

τ (b)db + O(w) = 1 − 3

τ (b)db + O(w) J

≤ 1 − 2 3 Leb(J ) + O(w).

One thus may take 1 = 21 2 3 Leb(J ). Next, one shows (A.17) for |z| = |ξ w| ≥ 1/ . Here, it is enough to show that, for some C > 0, one has |∂zl λk (z)| ≤ C/|z|,

(A.26)

for l = 0, 1, 2, and for every k ∈ {k j }. Indeed, if has been taken small enough, one finds some C > 0 such that C/|z| ≤ 1/(1 + C |z|), when |z| ≥ 1/ . So, inserting now (A.26) in (A.12), (A.23) or (A.24), respectively for l = 0, l = 1 or l = 2, one will obtain (A.17) for |ξ w| ≥ 1/ . So let us show (A.26). It follows from (A.1) that ∂zl λk (z) can be written under the ( form ∂zl λk (z) = eizμ(b) ρl (b)db. An integration by parts gives ρ(b) 1 eizμ(b)ρ(b) b+ 1 b+ izμ(b) l ∂z λk (z) = db. − e ∂b z i∂b μ(b) b− z b− i∂b μ(b) j

Here, one has ρl ∈ C1 ([b− , b+ ]), since τ ∈ C1 ([b− , b+ ]), and |∂b ρ(b)| 1 for j = 0, 1. Moreover, one checks from the definition (3.13) of that |∂b μ(b)| 1 and |∂b2 μ(b)| 1. This finishes the proof. One now lets (y, n) ∈ H (, 8). The constants introduced below depend on y only through . Proof of (5.9a). By (A.11), (A.13) and (A.17) (with l = 0), there exists > 0 such that, |∂xl S y,n u(x)| ≤ (2π )2 |ξ |l |n (ξ )| u1 ξ ∈Z

u1

ξ :|ξ w|≤

|ξ |l e−Cn(ξ w) + u1 2

|ξ |l

ξ :|ξ w|>

(1 + C|ξ w|) 2

n

y l dy u1 ∞ l −Cny 2 u1 ∞ l+1 ye dy + l+1 n w w 0 (1 + C y) 2 u1 =: l+1 (I1 + I2 ). w


881

But one has I1 n −(l+1)/2 , and ∞ dy 1 1

I2 ≤ l ≤ e−C( )n . n n n −l −l−1 l+1

C (1 + C y) 2 C ( 2 − l − 1)(1 + C ) 2

(A.27)

This finishes the proof.

Proof of (5.9b). We will only consider the case k = 2 ; the case k = 1 can be handled similarly, and turns out to be easier. To simplify the notations, one writes A := sin2 π(x + nw − y) · ∂x2 S y,n u(x). We recall that the function v defined in (A.10) satisfies v(x) = u(x + y). One has sin2 z = 41 (2 − ei2z − e−i2z ), and thus, by (A.11), one has

A = −π 2 2 − ei2π(x+nw−y) − e−i2π(x+nw−y) ξ 2 ei2π ξ(x+wn−y) n (ξ )v(ξ ˆ ) = −π

2

ξ ∈Z

e

i2π ξ (x+wn−y)

2ξ 2 n (ξ )v(ξ ˆ ) − (ξ − 1)2 n (ξ − 1)v(ξ ˆ − 1)

ξ ∈Z

−(ξ + 1)2 n (ξ + 1)v(ξ ˆ + 1) .

Since

(A.28)

|v(ξ ˆ ) − v(ξ ˆ − 1)| ≤

B(0,w2 )

|v(x)| |1 − ei2π x |dx w 2 u1 ,

for every ξ ∈ Z, one has, for every > 0, |A| u1 2ξ 2 n (ξ )−(ξ − 1)2 n (ξ − 1)−(ξ + 1)2 n (ξ + 1)+(ξ w)2 |n (ξ )| ξ ∈Z

u1

ξ 2 |2n (ξ ) − n (ξ − 1) − n (ξ + 1)|

ξ ∈Z

u1

+|ξ | · |n (ξ − 1) − n (ξ + 1)| + (1 + (ξ w)2 )|n (ξ )|

ξ 2 |

n (ξ1 (ξ ))| + |ξ | · | n (ξ2 (ξ ))| + (1 + (ξ w)2 )|n (ξ )|

ξ ∈Z

= u1 ·

⎧ ⎨ ⎩

ξ :|ξ w|≤

(···) +

ξ :|ξ w|>

(···)

⎫ ⎬ ⎭

=: u1 · (I1 + I2 ).

(A.29)

The numbers ξ1 (ξ ) and ξ2 (ξ ) in (A.29) are obtained by a Taylor expansion and satisfy |ξ1 (ξ ) − ξ | ≤ 2 and |ξ2 (ξ ) − ξ | ≤ 2. If is taken small enough, then, by (A.13), (A.14) and (A.15), and because nw 2 ≤ 1 by hypothesis, one has ∞

√ √ √ 1 2 1 + (ξ w n)2 + (ξ w n)4 e−C( nwξ ) dξ √ . I1 (A.30) w n 0

882


By (A.17), one gets as for (A.27), I2

(ξ wn)2 + ξ wn + 1 |ξ |w≥

(1 + Cξ w)

n 2

1 w

∞

(yn)2 + yn + 1 dy (1 + C y)

Inserting (A.30) and (A.31) in (A.29) gives the result.

n 2

1 −C n e . w

(A.31)

Proof of (5.10). Let > 0 be as small as we want. One takes x, y ∈ T such that |x + nw − y|T ≤ 10. The constants introduced below do not depend on . We recall that the function v defined in (A.10) satisfies v(x) = u(x + y). Starting from (A.11), one obtains S y,n u(x) ≥ e2iπ ξ(x+nw−y) n (ξ )v(ξ ˆ )− |n (ξ )| u1 . (A.32) ξ :|ξ |≤ −2/3

ξ :|ξ |> −2/3

On the one hand, mimicking the proof of (5.9a) with l = 0, and taking the hypothesis nw 2 ≥ into account, one finds, for some > 0,

dy 1 1 ∞ 2 |n (ξ )| e−Cny dy + n w −2/3 w w (1 + C y) 2 ξ :|ξ |> −2/3 ∞ 1 1 2

−1/3 √ e−C z dz + √ e−C ( )n e−C , (A.33) −1/6 where, to get rid of the term

√1 e−C ( )n ,

one has used the hypothesis nw 2 ≥ , which

implies n ≥ −1/3 when w is small enough. On the other hand, n (−ξ ) = ∗n (ξ ) by (A.12), v(−ξ ˆ ) = vˆ ∗ (ξ ) since u is real, and v(0) ˆ = u1 since u ≥ 0. Therefore e2iπ ξ(x+nw−y) n (ξ )v(ξ ˆ ) ξ :|ξ |≤ −2/3

= u1 + 2

|n (ξ )| |v(ξ ˆ )| cos arg e2iπ ξ(x+nw−y) n (ξ )v(ξ ˆ ) . (A.34)

1≤ξ ≤ −2/3

ˆ )) |ξ |w 2 for every ξ ∈ Z. So, by (A.16) and Since v ∈ L1B(0,w2 ) (T), one has arg(v(ξ the hypothesis |x + nw − y|T ≤ 10, one obtains ˆ ) ξ 1/3 , (A.35) arg e2iπ ξ(x+nw−y) n (ξ )v(ξ when 1 ≤ ξ ≤ −2/3 . But, if 1 ≤ ξ ≤ −2/3 , one has 1 |v(ξ ˆ )| ≥ v(x)dx − |v(x)| |e−i2π ξ x − 1|dx ≥ (1 − Cξ w 2 )u1 ≥ u1 , 2 B(0,w2 ) (A.36) and, by (A.13), one has |n (ξ )| ≥ e−Cnw ξ ≥ e−Cξ , since nw 2 ≥ . Therefore, using this last estimate, (A.35) and (A.36) in (A.34) gives

2 e2iπ ξ(x+nw−y) n (ξ )v(ξ ˆ ) e−C ξ v1 , (A.37) 2 2

ξ :|ξ |≤ −2/3

if is small enough.

2

|ξ |≤ −2/3


883

Therefore, inserting (A.33) and (A.37) in (A.32), one gets ⎛ ⎞

2

−1/3 S y,n u(x) ≥ u1 ⎝C1 e−C ξ − C2 e−C ⎠ , |ξ |≤ −2/3

and this tends to ∞ as → 0.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Anderson, P.W.: Absence of Diffusion in Certain Random Lattices. Phys. Rev. 109(5), 1492–1505 (1958) Arnold, L.: Random Dynamical Systems. Berlin: Springer, 1998 Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. 19(3), 357–367 (1967) Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Mathematical physics 2000. London: Imp. Coll. Press, 2000, pp. 128–150 Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré, section B 23(2 suppl.), 245–287 (1987) Casher, A., Lebowitz, J.L.: Heat Flow in Regular and Disordered Harmonic Chains. J. Math. Phys. 12(8), 1701–1711 (1971) Coulhon, T., Saloff-Coste, L.: Puissances d’un opérateur régularisant. Ann. Inst. H. Poincaré, section B, 26(3), 419–436 (1990) Coulhon, T., Saloff-Coste, L.: Minoration pour les chaînes de Markov unidimensionnelles. Prob. Th. and Rel. Fields 97, 423–431 (1993) Dhar, A.: Heat Conduction in the Disordered Harmonic Chain Revisited. Phys. Rev. Lett. 86(26), 5882– 5885 (2001) Dhar, A.: Heat transport in low-dimensional systems. Adv. in Phys. 57(5), 457–537 (2008) Diaconis, P., Saloff-Coste, L.: Nash Inequalities for Finite Markov chains. J. Theor. Prob. 9(2), 459–510 (1996) Escaurazia, L.: Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form. Comm. Part. Diff. Eq. 25(5-6), 821–845 (2000) Freedman, D.A.: On tail probabilities for martingales. Ann. of Prob. 3, 100–118 (1975) Guivarc’h, Y.: Limit Theorems for Random Walks and Products of Random Matrices. In: CIMPA-TIFR School on Probability Measures on Groups: Recent Directions and Trends (Mumbai) 2002, New Delhi Natosa pub., 2006 Hall, P., Heyde, C.C.: Martingale limit theory and its application. London-New York: Academic Press, 1980 Lepri, S., Livi, R., Politi, A.: Anomalous heat conduction. In: Anomalous Transport: Foundations and Applications, Klages, R., Radons, G., Sokolov, I.M. (eds.), Weinheim: Wiley-VCH Verlag, 2008, Ch. 10 Matsuda, H., Ishii, K.: Localization of Normal Modes and Energy Transport in the Disordered Harmonic Chain. Suppl. Prog. Theor. Phys. 45, 56–86 (1970) Mustapha, S.: Gaussian estimates for spatially inhomogeneous random walks on Z d . Ann. Prob 34(1), 264–283 (2006) O’Connor, A.J.: A Central Limit Theorem for the Disordered Harmonic Chain. Commun. Math. Phys. 45(1), 63–77 (1975) Peierls, R.E.: Zur kinetischen Theorie der Wärmeleitung in Kristallen. Ann. der Phys. 395(8), 1055–1101 (1929) Peierls, R.E.: Quantum Theory of Solids. London: Oxford University Press, 1955 Raugi, A.: Théorème ergodique multiplicatif. Produits de matrices aléatoires indépendantes (Rennes, 1997), Rennes: Publ. Inst. Rech. Math. Rennes, 1997, pp. 1–43 Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of Harmonic Crystal in a Stationary Nonequilibrium State. J. Math. Phys. 8(5), 1073–1078 (1967) Rubin, R.J., Greer, W.L.: Abnormal Lattice Thermal Conductivity of a One-Dimensional, Harmonic Isotopically Disordered Crystal. J. Math Phys. 12, 1686–1701 (1971) Verheggen, T.: Transmission Coefficient and Heat Conduction of a Harmonic Chain with Random Masses: Asymptotic Estimates on Products of Random Matrices. Commun. Math. Phys. 68(3), 69–82 (1979)

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Communications in Mathematical Physics - Volume 235

Communications in Mathematical Physics - Volume 223

Communications In Mathematical Physics - Volume 283

Communications In Mathematical Physics - Volume 270

Communications in Mathematical Physics - Volume 208

Communications in Mathematical Physics - Volume 186

Communications In Mathematical Physics - Volume 294

Communications in Mathematical Physics - Volume 217

Communications In Mathematical Physics - Volume 274

Communications in Mathematical Physics - Volume 239

Communications in Mathematical Physics - Volume 306

Communications in Mathematical Physics - Volume 264

Communications in Mathematical Physics - Volume 227

Communications in Mathematical Physics - Volume 184

Communications in Mathematical Physics - Volume 261

Communications in Mathematical Physics - Volume 225

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Communications in Mathematical Physics - Volume 211

Communications In Mathematical Physics - Volume 293

Communications in Mathematical Physics - Volume 246

Communications In Mathematical Physics - Volume 298

Communications in Mathematical Physics - Volume 234

Communications In Mathematical Physics - Volume 288

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Communications In Mathematical Physics - Volume 292

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Communications in Mathematical Physics - Volume 253

Communications in Mathematical Physics - Volume 222

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