Communications In Mathematical Physics - Volume 272

Commun. Math. Phys. 272, 1–23 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0204-8 Communications in Mathe...

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Commun. Math. Phys. 272, 1–23 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0204-8

Communications in

Mathematical Physics

Perturbative Solutions of the Extended Constraint Equations in General Relativity Adrian Butscher Department of Mathematics, University of Toronto, Toronto, ON M1C 1A4, Canada. E-mail: [email protected] Received: 26 June 2003 / Accepted: 4 October 2006 Published online: 7 March 2007 – © Springer-Verlag 2007

Abstract: The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface Z in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on Z, and are equivalent to the usual constraint equations that Z satisfies as a space-like hypersurface in a space-time satisfying Einstein’s vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the ‘classical’ method of Lichnerowicz and York that is used to solve the usual constraint equations.

1. Introduction The notion of an asymptotically simple space-time was first proposed by Penrose in the 1960s as a means for studying the asymptotic properties of isolated solutions of Einstein’s equations in General Relativity [15]. The central idea is to define a class of space-times which are conformally diffeomorphic to the interior of a Lorentz manifold with boundary, called the unphysical space-time, where the boundary is identified in a certain way with points at infinity in the original space-time. Einstein’s vacuum equations can be rephrased in terms of this construction and can be used to describe the metric and conformal boundary of the conformally rescaled space-time. However, this description is rather awkward in some respects because of the following phenomenon. The metric g of the unphysical space-time is related to the metric g˜ of the original spacetime by g˜ = −2 g, where the conformal factor vanishes at the boundary (thereby encoding the fact that the boundary is at infinite distance). Consequently, Einstein’s

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A. Butscher

equation in the unphysical space-time is Ric(−2 g) = 0, which degenerates at the boundary. Helmut Friedrich has developed a new approach for the mathematical description of the unphysical space-time, known as the conformal Einstein equations, which avoids this difficulty. The reader is asked to consult [9–11] for a review of these ideas. Essentially, Friedrich has discovered equations which are equivalent to Einstein’s equations in the unphysical space-time but which do not degenerate at the boundary. Using these equations, Friedrich has made significant steps in the development of the Penrose model of asymptotic simplicity; in particular, he has been able to prove the first semi-global existence and stability results for the Cauchy problem of evolving such space-times from initial data. Semi-global in this context means that, starting with so-called asymptotically hyperbolic initial data, it is possible to generate the entire future development of the data under Cauchy evolution, all the way up to time-like infinity. Recently, thanks to work of Corvino [8] and Chru´sciel and Delay [7], this result has been extended to a global existence result, in which a large class of asymptotically simple space-times can be constructed by Cauchy evolution from asymptotically flat initial data. As is well known, the Cauchy problem near a space-like hypersurface Z consists of splitting Einstein’s equations into constraint equations satisfied by initial data on Z and evolution equations that describe how the initial data evolve in time to produce a space-time neighbourhood around Z. The conformal Einstein equations also admit such a Cauchy problem. Indeed, given a space-like hypersurface Z in the unphysical spacetime, one can deduce a system of constraint equations and evolution equations for initial data on Z. The constraint equations on Z that arise in this way are called the conformal constraint equations. The conformal constraint equations are far more complex than the ‘usual’ constraint equations satisfied by the hypersurface Z when it is seen as an infinitely extended, spacelike hypersurface of the original space-time. The reason is that the conformal constraint equations combine two sub-structures — namely ‘geometric’ constraints (i.e. resulting from the Gauss-Codazzi equations) and a boundary value problem at the boundary of Z — in a highly coupled and non-linear way. Although general techniques do not yet exist for analyzing these equations, there is one approach which may be fruitful: to tackle these two sub-structures individually and in isolation from one another by considering special cases. The hope is that the knowledge gained in this way can be re-assembled in order to develop an understanding of the equations in more general cases. Just such an approach has been commenced by the author in [3] and is extended in this article. The Extended Constraint Equations. In the author’s previous paper [3], the Ansatz ≡ 1 is used to eliminate the boundary value problem for and bring the ‘geometric’ constraint problem to the forefront. This happens because the Ansatz ≡ 1 essentially assumes that the conformal rescaling is trivial so that Z is a boundaryless space-like hypersurface in the original space-time. The system of equations which results from this Ansatz reduces the conformal constraint equations considerably. The only remaining unknowns are the induced metric g and the second fundamental form χ of Z, as well as the so-called electric part S and the magnetic part S¯ of the Weyl tensor W of the space-time relative to the 3 + 1 splitting induced by Z. (That is, Sab = Wa00b is the electric part of W and S¯ab = εast Wb0st is the magnetic part of W , where εabc is the fully antisymmetric permutation symbol associated to the metric g.) Note that both S and S¯ are trace-free and symmetric. Explicitly, the conformal constraint equations reduce to the following system under the ≡ 1 Ansatz, called the extended constraint equations:

Perturbative Solutions of the Extended Constraint Equations

3

e ¯ ∇c χab − ∇b χac = εbc Sae , a¯ bc a ∇ Sae = εe χ Rac , b

Rab = Sab − χcc χab + χac χcb , ∇ a Sab = −χ ac εe S¯ae ,

(1)

bc

where ∇ is the covariant derivative of the metric g and Rab (g) is its Ricci curvature. (Actually, (1) is a slightly different yet equivalent formulation of the extended constraint equations found in [3].) As equations involving the metric g and the second fundamental form χ of the space-like hypersurface Z in the original space-time, one expects the extended constraint equations to be related in some way to the ‘usual’ constraint equations satisfied by g and χ . In fact, the following lemma shows that the extended constraint equations are fully equivalent to the usual constraint equations. Lemma 1. Let Z be a space-like hypersurface in the original space-time having induced metric g and second fundamental form χ . Then g and χ satisfy the usual constraint equations on Z if and only if g and χ , along with two tensors S and S¯ defined via Sab ≡ Rab (g) + χcc χab − χac χcb and S¯ae ≡ εebc ∇c χab satisfy the extended constraint equations. ¯ satisfy (1). Then by taking traces in the first and Proof. Suppose first that (g, χ , S, S) third equations, one obtains the usual constraint equations, namely that R = χbc χcb − (χcc )2 and 0 = ∇ c χcb − ∇b χcc . Next suppose that g and χ satisfy the usual constraint equations. Let S and S¯ be defined as in the hypotheses of the lemma, in which case the desired symmetries of S and S¯ follow from the constraint equations and the first and third equations of (1) follow automatically. Some simple calculations are now needed to show that the second and fourth equations also hold. First, the Bianchi identity for the Ricci curvature states ∇ a Rab − 21 ∇b R = 0. Substitute for Rab and R using the definition of S to derive 1 0 = ∇ a Sab − χ cc χ ab + χ ac χ cb − ∇b − (χ cc )2 + χ ac χ ac 2 = ∇ a Sab − χ cc δba − χ ab h uv S¯uav + χ ac S¯abc ,

(2)

where S¯abc is the dual of the tensor S¯ae defined by S¯abc ≡ ∇c χab − ∇b χac . By symmetry, the middle term in (2) vanishes, leaving 0 = ∇ a Sab + χ ac S¯abc , which is exactly the fourth equation of (1) when written in terms of this dual. ¯ That is, Next, consider the commutator of the second covariant derivatives of S. εebc ∇e S¯abc = 2εebc ∇e ∇c χ ab = εebc ∇e ∇c χ ab − ∇c ∇e χ ab s χ = εebc Reca sb

(3)

using the symmetries of Rabcd . Substitute in (3) the well-known decomposition s Reca = gea Rcs − δes Rca + δcs Rea − gca Res −

1 R gea δcs − δes gca , 2

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A. Butscher

of the curvature tensor in three dimensions to obtain εebc ∇e S¯abc = 2εabc χ sb Rcs .

(4)

The second equation of (1) is now a consequence of (4) after invoking the symmetries of S¯ once again. The proof of Lemma 1 reveals an important feature of the extended constraint equations that will be exploited in a crucial way in the proof of the Main Theorem of this paper. Indeed, the calculations above show that the extended constraint equations break up into two pairs of equations, each consisting of a primary ‘geometric’ equation along with its integrability condition, i.e. an equation derived from the primary one by commuting covariant derivatives. This is explicitly true of the second equation of (1) but it is also true of the fourth equation since the Bianchi identity is the integrability condition for the Riemann curvature tensor. Remark. Another way of thinking about this system of equations, given that S and S¯ are the electric and magnetic parts of the Weyl tensor, is that (1) is equivalent to the vanishing of the full four-dimensional Ricci tensor along Z. The Statement of the Main Theorem. The extended constraint equations given in (1) are still too complicated to solve for general initial data on general non-compact, boundaryless space-like hypersurfaces in an infinitely extended space-time. An appropriate simplification is to consider perturbative solutions near a given known solution. An obvious known solution of the equations is the trivial solution in which Z = R3 and g = δ (the Euclidean metric) along with χ = S = S¯ = 0. This paper will therefore investigate the nature of a certain class of perturbative solutions of the extended constraint equations decaying asymptotically to this trivial solution and ‘near’ it in some appropriate sense. In [3], an approach for producing such solutions was developed under the further simplifying assumption of vanishing second fundamental form (the so-called time-symmetric case). The present paper will extend the approach used there to the full, non-time-symmetric case. (Note: perturbations near general asymptotically flat metrics on more general asymptotically flat manifolds will not be considered here.) The Main Theorem to be proved in this paper is the following. Main Theorem. There exists an infinite-dimensional Banach space B of so-called free data consisting of triples (T, T¯ , φ), where T , T¯ are symmetric, δ-trace-free and δ-divergence-free two-tensors on R3 and φ is a function on R3 , which parametrizes solutions of the extended constraint equations near the trivial solution as follows. 1. For every triple (T, T¯ , φ) whose norm is sufficiently small, there is a unique solu¯ of the extended constraint equations that is an asymptotically flat tion (g, χ , S, S) perturbation of the trivial solution g = δ and χ = S = S¯ = 0 (in a sense to be defined later in this paper). 2. The point (0, 0, 0) ∈ B corresponds to the trivial solution. 3. The association of triples (T, T¯ , φ) to solutions maps onto a neighbourhood of the trivial solution.


5

Furthermore, the map taking (T, T¯ , φ) to the solution of the extended constraint equations is smooth in the sense of Banach spaces. Sketch of Proof. The Main Theorem will be proved by applying the Implicit Function Theorem to the extended constraint equations in a suitable way. But because these equations are not fully-determined elliptic equations, it is not possible to apply this theorem in a simple-minded way. Rather, a two-step approach must be used. The first step, called Theorem A in this paper, consists of deriving and solving a related system of equations which is fully-determined elliptic. This system will be called the secondary system, and it will be shown that it determines the quantities g, χ , S and S¯ uniquely in terms of certain free data in a Banach space B provided the norm of the free data is sufficiently small. These solutions are of course not a priori solutions of the extended constraint equations and thus it is still necessary to prove that a solution found in Theorem A is also a solution of the extended constraint equations. This second step of the proof is called Theorem B in this paper; it asserts that solutions of Theorem A are solutions of the extended constraint equations. A surprising technical difficulty arises in the proof of Theorem A. In order to prove the existence of solutions via the Implicit Function Theorem, one must study the linearization of the secondary system at the trivial solution and show that it is bijective. However, the fact of the matter is that this operator is injective but not surjective. Consequently, it is possible only to solve the secondary system up to an error term belonging to the co-kernel of the linearization (which is finite-dimensional by Fredholm theory). This error term may be non-zero, and its presence complicates the proof of Theorem B. However, the lack of surjectivity is really an artifact of the two-step method of the proof of the Main Theorem, since the integrability conditions are exploited in a subtle way in Theorem B to prove that the potential error terms produced in Theorem A must in fact be zero at the same time that the solution found in Theorem A does indeed satisfy the extended constraint equations. The nature of the free data of the Main Theorem can be characterized as follows. Recall first that the extended constraints are equivalent to the usual Einstein constraint equations, so that the method outlined above for solving the extended constraints can be interpreted as a completely new method, different from the ‘classical’ LichnerowiczYork method, for solving the usual constraint equations. In the classical method (under the assumption of constant mean curvature), one freely prescribes a metric g0 (without loss of generality whose components have unit determinant) and a symmetric tensor χ0 , divergence free with respect to g0 , on R3 . The number of ‘freely prescribable’ functions for data defined in this way is thus seven, or four if one first makes a specific choice of coordinates. Then, one considers the conformally rescaled metric g = u 4 g0 , where u : R3 → R is an unknown function, and reduces the constraint equations to a semi-linear elliptic equation for u. Once this equation is solved, a further rescaling of χ0 yields a tensor χ which is trace-free and divergence-free with respect to g. The Lichnerowicz-York method thus produces a solution (g, χ ) in terms of the four degrees of freedom represented by g0 and χ0 . In contrast, the present method treats the metric g and certain components of χ , S and S¯ as the unknowns and leads to a quasi-linear elliptic system for these quantities that will be solved in terms of totally different free data. In the derivation of the secondary system in Sect. 2.4, it will be shown that the free data consists of certain components of the curvature of g, amounting to four degrees of freedom, along with a single degree of freedom corresponding to the trace of the second fundamental form (which is absent in the Lichnerowicz-York solution since the mean curvature is kept constant there).

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2. Preliminaries 2.1. Spaces of tensors and tensor operators. For the sake of convenience, it is worthwhile to assign names to the various tensors spaces and tensor operators that appear in (1) and elsewhere in this paper. Suppose M is any Riemannian manifold with metric g. Let S(M) denote the space of symmetric two-tensors over M and let Sg (M) denote the symmetric two-tensors of M that are in addition trace-free with respect to g. Additionally, let 1 (M) denote the oneforms of M. Finally, let J (M) denote those three-tensors which possess the symmetries Jabc + Jcab + Jbca = 0 and Jabc + Jacb = 0. Such tensors will make an appearance at one point in this paper. Next, define the following operators: • divg : S 2 (M) → 1 (M), given in local coordinates by [divg (S)]b = ∇ a Sab ; • Ric : S 2 (M) → S 2 (M), given in local coordinates by [Ric(g)]ab = Rab (g). This is naturally only defined on the non-degenerate symmetric tensors. These operators appear in the system (1). The following additional operators will play an important role in the sequel. First, there are the operators • Dg : S 2 (M) → J (M), given in local coordinates by [Dg (χ )]abc = ∇c χab −∇b χac ; • Qg : J (M) → 1 (M) defined in local coordinates by [Qg (J )]a = εebc ∇e Jabc , which will be used in exploiting the integrability conditions built into the extended constraint equations. Second, there are the formal adjoints (up to numerical factors) of the operators Dg and divg , given by: • Dg∗ : J (M) → Sg (M), given by [Dg∗ (J )]ac = ∇ b Jabc + ∇ b Jcba − 23 ∇ b Jubv g uv gac . Note that the adjoint of the restriction of the operator Dg to the space Sg (M) is being given here. • div∗g : 1 (M) → Sg (M), given by [div∗g (X )]ac = ∇a X c + ∇c X a − 23 ∇ b X b gac . Note that this operator is also known as the conformal Killing operator. It is a simple matter of calculating these adjoints by means of the integration by parts formula for covariant derivatives and so this will not be carried out here. 2.2. Weighted Sobolev spaces. In order to proceed with the solution of the extended constraint equations, it is first necessary to specify in what Banach spaces the various unknown quantities lie. The notion of asymptotic flatness in R3 should be encoded rigorously into these spaces by requiring that the relevant objects belong to a space of tensors with built-in control at infinity. Furthermore, the spaces should be chosen to exploit the Fredholm properties of the operators above. Both these ends will be served by weighted Sobolev spaces, whose definition and some of whose properties will be given in this section as a reminder to the reader (details can be found in such works as [2, 4–6, 12, 13]). The actual choice of Banach spaces in which the solutions of the Main Theorem will be found will then be given in the next section. Let T be any tensor on R3 . (This tensor may be of any order — the norm · appearing in the following definition is then simply the norm on such tensors that is induced from the Euclidean metric of R3 . Note that the following definitions can also be made for general metrics on R3 but this will not be necessary in this paper.) The H k,β Sobolev norm of T is the quantity k 1/2 l 2 −2(β−l)−3 T H k,β = D T σ , l=0

R3


7

where σ (x) = (1 + r 2 )1/2 is the weight function and r 2 = (x 1 )2 + (x 2 )2 + (x 3 )2 is the squared distance to the origin. Note that Bartnik’s convention [2] for the power of σ in the definition of the weighted spaces is being used (the reason for this is psychological: if f ∈ H k,β and f is smooth enough to invoke the Sobolev Embedding Theorem (again, see [2]), then f (x) = o(r β ) as r → ∞, which is easy to remember). The space of H k,β functions of R3 will be denoted by H k,β (R3 ) and the space of k,β H sections of a tensor bundle B over R3 will be denoted by H k,β (B). As an abbreviation, or where the context makes the bundle clear, such a space may be indicated simply by H k,β . Note also that the following convention for integration will be used in the rest of this paper. An integral of the form R3 f , as in the definition above, denotes an integral of f with respect to the standard Euclidean volume form. Integrals of quantities with respect to the volume form of a different metric will be indicated explicitly, as, for example, R3 f dVolg . Constant-coefficient elliptic partial differential operators acting on spaces of H k,β tensors satisfy several important analytic properties, and two of these will be used in a crucial way in the sequel. The first property is a characterization of the kernels and co-kernels of such operators that will be used in Theorem A. The second property concerns the stability of the co-kernels of these operators that will be used in Theorem B. Proposition 2 (Kernel/Co-Kernel). Let B be a tensor bundle over R3 and let Q : H k,β (B) → H k−2,β−2 (B) any linear, second order, homogeneous, elliptic partial differential operator with constant coefficients, where β ∈ Z, and k ≥ 2. Denote by Ker(Q : β) the kernel of Q acting on H k,β . Then the following is true: 1. Q is Fredholm; 2. Q is injective if β < 0 and surjective if β > −1; 3. If β > 0, then Ker(Q : β) consists of polynomials of degree equal to the integer part of β and has non-zero, finite dimension. 4. If β < −1, then the image of Q is the space Im(Q) = y ∈ H k−2,β−2 (B) : y, z = 0 ∀ z ∈ K er (Q ∗ : −β − 1) , R3

where Q ∗ is the formal adjoint of Q. The image thus has non-zero finite codimension. Proof. The fact that the operator Q is Fredholm and has finite dimensional kernel and co-kernel is a classical result that can be found in [12, 14]. Standard Schauder theory then asserts that any solution of the equation Q(u) = 0 is smooth, and if it has polynomial growth or decay at infinity, then it is a polynomial or is zero, respectively. The characterization of the image of Q as the orthogonal complement of the kernel of the adjoint Q ∗ is elementary functional analysis. An excellent source for understanding the motivation behind this theorem can be found in [13] in which the behaviour of the Laplace operator on Rn is explained. Proposition 3 (Stability). Let B be any tensor bundle over R3 and let Q ε : H k,β (B) → H k−2,β−2 (B), ε ∈ [0, 1], be a continuous family of linear, elliptic operators. Furthermore, suppose that Q ε is uniformly injective; i. e. there is a constant C independent of ε so that Q ε (y) H k−2,β−2 ≥ Cy H k,β . Finally, suppose C is a finite-dimensional linear subspace of H k−2,β−2 (B). If C ∩ Im(Q 0 ) = {0}, then there exists ε0 > 0 so that C ∩ Im(Q ε ) = {0} for all ε < ε0 .

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A. Butscher

Proof. Suppose that this proposition is false; that is, for every ε > 0, let z ε be a non-zero element of C ∩ Im(Q ε ) and without loss of generality, it is possible to take z ε H k−2,β−2 = 1. Since z ε ∈ C for every ε, the finite-dimensionality of C implies that there is a subsequence ε j → 0 and z ε j ≡ z j that converges in the H k−2,β−2 norm to a non-zero element z ∈ C, with z H k−2,β−2 = 1. Furthermore, since z j ∈ Im(Q j ), there is an element y j ∈ H k,β (B) so that Q j (y j ) = z j . Claim. There is y ∈ H k,β (B) so that y j → y and Q 0 (y) = z. First, by the uniform injectivity of Q j , y j H k,β ≤ Cz j H k−2,β−2 = C,

(5)

so that the sequence y j is uniformly bounded in H k,β . Next, by the injectivity of the operator Q 0 , yi − y j H k,β ≤ CQ 0 (yi − y j ) H k−2,β−2 ≤ C (Q 0 − Q i )(yi ) H k−2,β−2 + Q i (yi ) − Q j (y j ) H k−2,β−2 + (Q 0 − Q j )(y j ) H k−2,β−2 ≤ C Q 0 − Q i op + z i − z j H k−2,β−2 + Q j − Q 0 op

(6)

by (5), where · op denotes the operator norm in the space of linear operators on H k−2,β−2 (B). Now the first and last terms in (6) go to zero with sufficiently large i and j because the family Q ε is continuous, while the middle term goes to zero by construction. Hence the sequence y j is Cauchy in H k,β and thus converges to an element y ∈ H k,β (B). By similar estimates as above, it is straightforward to show that z − Q 0 (y) H k−2,β−2 is zero. Thus z = Q 0 (y). But z is a non-zero element in C ∩ Im(Q 0 ). This contradicts the hypotheses of the theorem. 2.3. Choosing the Banach spaces. Solutions of the extended constraint equations will be found in the following Banach spaces. Pick any β ∈ (−1, 0) and any k ≥ 4. Then choose: • metrics δ + h so that h ∈ H k,β S 2 (R3 ) ; • tensors χ in H k−1,β−1 S 2 (R3 ) ; • tensors S and S¯ in H k−2,β−2 Sg2 (R3 ) . The preceding choice of Banach spaces is necessitated by the following two considerations. First, in order to ensure that the metric δ + h is asymptotically flat, h must decay as r → ∞, and this holds by the Sobolev Embedding Theorem when β < 0. Next, a non-trivial, asymptotically flat metric satisfying the constraint equations must satisfy the Positive Mass Theorem [16] and consequently must have non-zero ADM mass. Thus the r −1 term in the asymptotic expansion of h must be allowed to be non-zero, which imposes the further requirement that β > −1. Then, as a consequence of the choice made for h, the χ , S and S¯ quantities must be chosen as above because of the differing numbers of derivatives taken on these quantities in the equations (1): since the equations are meant to define maps between weighted Sobolev spaces, the weightings on χ and S and S¯ must match together properly and match the weighting on the metric δ + h. (For example, the Ricci curvature operator is of degree two and sends a metric δ + h with


9

h ∈ H k,β to a tensor in H k−2,β−2 . Thus S must lie in H k−2,β−2 to match Ric(δ + h).) Finally, k ≥ 4 is required in order to apply the Sobolev Embedding Theorem at one stage of the proof of the Main Theorem. Remark. One issue has been glossed over in the previous paragraph, and this is the effect of the non-linear terms. Because of this, for example, it is not immediately obvious that Ric(δ + h) is in H k−2,β−2 when h ∈ H k,β because this expression involves products of the metric and its first and second derivatives. However, the Multiplication Theorem for weighted Sobolev spaces [5] implies that the choice for β and k made above ensures that the various Banach spaces are in fact Banach algebras, and so this is not a problem.

2.4. The secondary system to be used in Theorem A. The first step of the proof of the Main Theorem is to derive and solve a secondary system of equations that is related to the extended constraint equations in a clever way. The key modification is that the secondary system is fully-determined and elliptic. The secondary system will be given immediately to streamline the presentation. The objects that will appear in the definition of the secondary system are these. 1. Let ST T (R3 ) denote the space of symmetric, δ-trace-free and δ-divergence-free two-tensors and define the map σ : H k−1,β−1 1 (R3 ) × H k−2,β−2 ST2 T (R3 ) → H k−2,β−2 Sg2 (R3 )

(7)

by σ (V, W ) = div∗g (V ) + W − 13 Tr g (W )g. Then put S(X, T ) = σ (X, T ) and ¯ X¯ , T¯ ) = σ ( X¯ , T¯ ). S( 2. Define the map χ : H k−1,β−1 Sδ2 (R3 ) × H k−1,β−1 (R3 ) → H k−1,β−1 S 2 (R3 ) via χ (K , φ) = K + 13 φδ. 3. Let Ric H (g) denote the formal expression of the Ricci curvature of a metric g in harmonic coordinates. In local coordinates, this is given by 1 H Rab = Rab + ( a;b + b;a ), 2

(8)

a and a are the Christoffel symbols of g. where a = g st st st

The secondary system can now be defined as follows. ¯ X¯ , T¯ ) and Definition 4. Let g = δ + h be the metric of R3 and let S = S(X, T ), S¯ = S( χ = χ (K , φ) be as above. Then the secondary system is:

Dg∗ Dg (χ ) − S¯ = 0, ab

st u ¯ − εa χs Rut (g) = 0, divg ( S) (9)

a Ric H (δ + h) − Sab + χss χab − χas χbs = 0, ab

u ¯ divg (S) a + χ st εat Ssu = 0, e . where S¯ denotes the tensor given in local coordinates by S¯ae εbc

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A. Butscher

Motivation for the secondary system. Each equation of (1) has a corresponding equation in (9) and the motivation behind each of these equations will be given in turn. Begin by considering the first equation of (1). In [3], it was found that the symbol of this equation has a one dimensional kernel consisting of symmetric, pure-trace tensors. The symbol is thus injective when restricted to the space of symmetric, trace-free tensors and thus the equation is over-determined elliptic when so restricted. A related fully-determined elliptic equation can thus be derived as follows. First, one decomposes χ into its trace and trace-free parts (with respect to the metric δ) as χ = K + 13 φg and substitutes this into the first equation of (1). One then composes both sides of this equation with the adjoint of the operator Dg , so that the resulting equation contains the operator Dg∗ ◦ Dg , which is elliptic on K (the injectivity of the symbol of Dg implies the bijectivity of the symbol of Dg∗ ◦ Dg by straightforward algebra). Therefore K should be viewed as the unknown quantity in this equation and φ should be viewed as a free datum. Note that the correspondence between pairs (K , φ) and tensors χ is bijective. Now consider the second and fourth equations of (1). Again, in [3], both these equations contain the operator divg which has surjective symbol, making it under-determined elliptic. One should thus imagine that each of these equations determines only ¯ say in some direct sum decomposition, while part of the unknown tensors S and S, leaving the other part free. This idea is implemented by using a modified version of the well-known York decomposition of trace-free, symmetric tensors in the form of the mapping σ . This is needed to guarantee the uniqueness of the solutions of the extended constraint equations. (The modification is that traces and divergences are being taken with respect to the fixed metric δ rather than the solution metric g whenever the tensors in question correspond to free data.) One can easily check that σ is a bijection. Substituting this decomposition into the second and fourth equations of (1) yields the second and fourth equations of the secondary system, and the operators divg ◦ div∗g (X ) and divg ◦ div∗g ( X¯ ) appearing there are elliptic on X and X¯ (again, the surjectivity of the symbol of divg implies the bijectivity of the symbol of divg ◦ div∗g ). Therefore X and X¯ should once again be viewed as the unknowns in these equations and T and ¯ and (X, X¯ , T, T¯ ) is T¯ can be treated as free data. The correspondence between (S, S) bijective. Finally, consider the third and remaining equation of (1). As pointed out in [3], it is a well-known fact that the Ricci operator is not elliptic on h because of its gauge invariance. Again as in [3], the standard trick of using harmonic coordinates will be employed to break the gauge invariance. Recall that such coordinates x a are defined by the requirement that g x a = 0 for each a, making the x a harmonic functions. Under this coordinate condition, the Ricci operator can be written as in (8) with a = 0 and that

1 H Rab = − gr s gab,r s + q(Dg), 2

(10)

where q(Dg) denotes a term that is quadratic in the first derivatives of the components of g. The operator Ric H is clearly elliptic when acting on g. The third and remaining equation of the secondary system, then, is obtained simply by replacing the Ricci operator in (1) by its formal expression in harmonic coordinates.


11

3. Statement and Proof of Theorem A 3.1. The statement. The Implicit Function Theorem will be used to solve the secondary system near the trivial solution and so it is restated here for ease of reference. For an excellent discussion and proof of this theorem, see [1]. Implicit Function Theorem. Let : X × B → Y be a smooth map between Banach spaces and suppose that (0; 0) = 0. If the restricted linearized operator D (0; 0)X ×{0} : X → Y is an isomorphism, then there exists an open set U ⊂ B contain ing 0 and a smooth function ψ : U → X with ψ(0) = 0 so that ψ(b), b = 0. The Implicit Function Theorem allows solutions of the equation (x, b) = 0, with (x, b) sufficiently close to (0; 0) in the Banach space norm of X × B to be parametrized over the Banach space B of free data. To apply this theorem to the secondary system, identify the three Banach spaces X , B and Y with the space of unknown quantities (h, K , X, X¯ ) appearing in the secondary system (9), the space of free data (T, T¯ , φ) appearing there, and the space corresponding to the range of the secondary system, respectively. That is, define X = H k,β S 2 (R3 ) × H k−1,β−1 Sδ2 (R3 ) ×H k−1,β−1 1 (R3 ) × H k−1,β−1 1 (R3 ) , B = H k−2,β−2 ST2 T (R3 ) × H k−2,β−2 ST2 T (R3 ) × H k−1,β−1 (R3 ), Y = H k−2,β−2 Sg2 (R3 ) × H k−3,β−3 1 (R3 ) × H k−2,β−2 S 2 (R3 ) ×H k−3,β−3 1 (R3 ) . Now use the secondary system to define a map : X × B → Y by

⎛ ⎞ Dg∗ Dg (χ ) − S¯ ab

⎜ ⎟ ¯ ⎜ ⎟ divg ( S) − εast χsu Rut (g) (h, K , X, X¯ ; T, T¯ , φ) = ⎜ a ⎟, ⎝ Ric H (δ + h) − Sab + χss χab − χas χbs ⎠ ab

u S¯ . divg (S) a + χ st εat su

(11)

¯ X¯ , T¯ ) and χ (K , φ) are as in Definition 4. where g = δ + h and S(X, T ), S( The map is a well-defined map of Banach spaces by the considerations of Sect. 2.3 (and the remark made there about non-linear terms) and is clearly smooth. Since the flat solution g = δ and χ = S = S¯ = 0 satisfies the extended constraint equations, (0; 0) = 0, and any other asymptotically flat solution of the secondary system satisfies (h, K , X, X¯ ; T, T¯ , φ) = 0. The first theorem to be proved in this paper is the following. Theorem A. Let : X × B → Y be the map of Banach manifolds corresponding to the secondary system given above. Then there is a neighbourhood U of zero in B and a smooth map of Banach manifolds ψ : U → X with ψ(0) = 0 so that for each b ∈ U, the point (ψ(b), b) solves the secondary system up to a finite dimensional error. In other words, there is a finite-dimensional linear subspace C ⊆ Y such that (ψ(b), b) ∈ C. Furthermore, the map ψ is injective. 3.2. The proof. Analysis of the linearization The first step of the proof is to show that the linearization of in the X directions at the origin is Fredholm and to find its kernel and co-kernel. This begins with a lemma.

12

A. Butscher

Lemma 5. The operator divδ ◦ div∗δ : H k−1,β−1 1 (R3 ) → H k−3,β−3 1 (R3 ) is Fredholm and injective with image equal to the codimension-3 space 1 3 a k−3,β−3 I1 = X = X a dx ∈ H Xa = 0 ∀ a . (R ) : R3

2 3 Sδ (R ) → H k−3,β−3 Sδ2 (R3 ) is ◦ Dδ : Furthermore, the operator Fredholm and injective with image equal to the codimension-5 space I2 = T = Ti j dx i ⊕ dx j ∈ H k−3,β−3 Sδ2 (R3 ) : Ti j = 0 ∀ i, j . Dδ∗

H k−1,β−1

R3

Here, x i are the standard coordinate functions on R3 . Proof. This lemma follows from Proposition 2 on the mapping properties of the linear, homogeneous elliptic operators with constant coefficients, since divδ ◦ div∗δ and Dδ∗ ◦ Dδ are such operators. It is necessary only to identify the kernels of these operators and the kernels of their adjoints. Note that this task is simplified somewhat, since these operators are formally self-adjoint. Let P be any constant-coefficient elliptic operator on R3 . By Proposition 2 the components of a solution of Pu = 0 with growth or decay at infinity are polynomials of degrees less than or equal to the growth rate, or zero, respectively. The operators considered by the lemma are thus injective since β ∈ (−1, 0). Furthermore, β ∈ (−1, 0) implies that −β − 1 ∈ (0, 1) and so the tensors in the kernels of the adjoints of these operators have components that are polynomials of degree zero. That is, these components must all be constants. Consequently, the images of the operators considered in the lemma are tensors orthogonal to the constant 1-forms λi dx i (λi ∈ R for all i) and the constant trace-free symmetric tensors µi j dx i ⊕ dx j (µi j ∈ R for each i, j and δ i j µi j = 0), respectively. The dimension of each of these spaces is obviously 3 and 5, respectively. Proposition 6. The linearized operator P ≡ D (0; 0)X ×{0} in the X directions is given by ⎛ ∗ ⎞ Dδ Dδ (K ) − div∗δ ( X¯ ) ⎟ ⎜ divδ ◦ div∗δ ( X¯ ) ⎟ (12) P(h, K , X, X¯ ) = ⎜ ⎝ − 1 δ h − div∗ (X ) ⎠ , δ 2 divδ ◦ div∗δ (X ) where δ is the Laplacian corresponding to the Euclidean metric. The principal symbol of P is bijective, making P an elliptic operator of Banach spaces. Furthermore, P is Fredholm and injective with image equal to the codimension-11 space Im(P) = I2 × I1 × H k−2,β−2 S 2 (R3 ) × I1 , where I1 and I2 are the spaces defined in Lemma 5. Proof. One can break up the calculation of the linearization into two pieces: P(h, K , X, X¯ ) = D (0; 0)(h, 0, 0, 0; 0) + D (0; 0)(0, K , X, X¯ ; 0). First,

⎞ Dδ∗ Dδ (K ) − div∗δ ( X¯ ) ⎟ ⎜ divδ ◦ div∗δ ( X¯ ) ⎟ D (0; 0)(0, K , X, X¯ ; 0) = ⎜ ∗ ⎠ ⎝ −divδ (X ) divδ ◦ div∗δ (X ) ⎛

(13)


13

since (0, K , X, X¯ ; 0) consists of a sum of differential operators that are linear in K , X and X¯ with terms which are quadratic in K , X and X¯ . Second, ⎞ ⎞ ⎛ 0 0 d ⎜ 0 ⎟ ⎜ 0 ⎟ D (0; 0)(h, 0, 0, 0; 0) = =⎝ 1 ⎝ Ric H (δ + sh) ⎠ − 2 δ h ⎠ ds 0 0 s=0 ⎛

(14)

by definition of the reduced Ricci operator and using the fact that the terms in appearing there are quadratic. Since the operator P is upper-triangular and the operators appearing on the diagonal are all elliptic, P is itself elliptic. Furthermore, Proposition 2 shows that P is Fredholm; and by the choice of β ∈ (−1, 0) made in Sect. 2.3, both β and β − 1 are less than 0, so that each of the operators appearing on the diagonal are injective on their respective domains. The operator P is thus itself injective. To find the image of P, one must attempt to solve the equations ⎛ ∗ ⎞ ⎛ ⎞ Dδ Dδ (K ) − div∗δ ( X¯ ) f1 ∗( X ⎜ ⎟ ⎜ f2 ⎟ ¯) ◦ div div δ ⎜ ⎟ δ (15) ⎝ − 1 δ h − div∗ (X ) ⎠ = ⎝ f 3 ⎠ δ 2 f4 divδ ◦ div∗δ (X ) with ( f 1 , f 2 , f 3 , f 4 ) ∈ Y. According to Lemma 5, the second and fourth equations can be solved if and only if f 2 and f 4 are in I1 . The third equation can now be solved since δ is an isomorphism according to Proposition 2. The remaining equation of (15) can be solved if and only if [ f 1 ]i j + [Dδ∗ div∗δ ( X¯ )]i j = 0 R3

for every i and j. But this is true if and only if f 1 ∈ I2 by the definition of the adjoint and the fact are in the kernel of Dδ . Thus ( f 1 , f 2 , f 3 , f 4 ) ∈ I2 × I1 × that constants H k−2,β−2 S 2 (R3 ) × I1 , which is of codimension 11 in Y. Existence of solutions in Theorem A. The conclusion to be drawn from Proposition 6 is that solutions of equation (x; b) = 0 can not be found using the Implicit Function Theorem for x = (h, K , X, X¯ ) near 0 in terms of b = (T, T¯ , φ). The non-trivial co-kernel of the linearization P is the essential obstruction. However, the existence of solutions up to an error term transverse to the image of P can be proved using the following technique. Since P is Fredholm with 11-dimensional co-kernel, the image of P is closed and one can find (in many different ways) an 11-dimensional subspace C so that Y = Im(P) ⊕ C. If η : R3 → R denotes a smooth, positive function on R3 with compact support satisfying R3 η = 1, then one such choice is given by C = ηµi j dx i ⊗ dx j : µi j ∈ R × ηλ¯ i dx i : λ¯ i ∈ R × {0} × ηλi dx i : λi ∈ R .

14

A. Butscher

If π : Y → Im(P) denotes the projection operator corresponding to this decomposition, namely the operator given by ⎞ [ f 1 ]i j dxi ⊗ dx j f 1 − η R3 ⎟ ⎜ f 2 − η R3 [ f 2 ]i dx i ⎟, π( f 1 , f 2 , f 3 , f 4 ) = ⎜ ⎠ ⎝ f i 3 f 4 − η R3 [ f 4 ]i dx ⎛

then the operator π ◦ : X × B → Im(P) has linearization in the X direction equal to π ◦ P, which is injective (proved as in Proposition 6) and is surjective since the composition with π forces it to map onto its image. Consequently, the Implicit Function Theorem can be applied to the equation π ◦ (x, b) = 0. The result is as follows. There is a neighbourhood U of 0 in B and a smooth function ψ : B → X such that ¯ (b) , then ψ(b), b ∈ X × B if b = (T, T¯ , φ) ∈ U and ψ(b) ≡ h(b), K (b), X (b), X satisfies the equation π ◦ ψ(b), b = 0. This is equivalent to the statement that the tensors g = δ + h, χ = K + 13 φg, S = div∗g X + T − 13 T r g (T )g, S¯ = div∗g X¯ + T¯ − 13 T r g (T¯ )g satisfy the equations

⎞ ⎛ ⎞ Dg∗ Dg (χ ) − S¯ µab ab

⎜ ⎟ st u ¯ ⎜ ⎟ ⎜ λ¯ ⎟ ut (g) ⎜ H divg ( S) a − εa χs R ⎟ = η ⎝ a ⎠ ∈ C, 0 ⎝ Ric (δ + h) − Sab + χss χab − χas χbs ⎠ ab λ st u a divg (S) + χ εat S¯su ⎛

(16)

a

where µab = λ¯ a = λa =

R

3

R

3

R3

Dg∗ Dg (χ ) − S¯ , ab

¯ − εast χsu Rut (g) , divg ( S) a

u ¯ divg (S) a − χ st εat Ssu ,

in the standard coordinates of R3 , provided that the free data (T, T¯ , φ) is chosen sufficiently small in the norm of the Banach space B. Uniqueness of solutions in Theorem A. It remains to show that the mapping from (T, T¯ , φ) to the solution (h, K , X, X¯ ) is injective. This will follow from the Implicit Function Theorem if it can be shown that the linearization of in the direction of the


15

free data, namely D(π ◦ (0; 0)){0}×B , is injective. But a calculation similar to that in Proposition 6 gives D π ◦ (0; 0) {0}×B (T, T¯ , φ) d = π ◦ (0, 0, 0, 0; sT, s T¯ , sφ) ds s=0 ⎞ ⎛ ∗ 1 Dδ Dδ 3 φδ − T¯ − 13 T rδ (T¯ )δ ab ⎟ ⎜ d divδ T¯ − 13 T rδ (T¯ )δ a ⎟ ⎜

s = 1 ⎠ ⎝ −T + T r (T )δ ds s=0 δ 3 ab 1 divδ T − T rδ (T )δ a ⎞ ⎛ ∗ 1 3 ¯ 1 δ (T¯ )δ η D δ Dδ 3 φδ − T − 3 T r R3 ab ⎟ ⎜ d η R3 divδ T¯ − 13 T rδ (T¯ )δ a ⎟ ⎜ − s⎝ ⎠ 0 ds s=0 1 η R3 divδ T − 3 T rδ (T )δ a ⎛ ⎞ eb eb ¯ ¯ ∇a ∇b φ − εc ∇bebTae − εc ∇bebTce ⎜ −η R3 ∇a ∇b φ − εc ∇b Tae − εc ∇b Tce ⎟ ⎜ ⎟ =⎜ (17) ⎟, 0 ⎝ ⎠ −Tab 0 using the fact that T and T¯ were chosen to be transverse-traceless with respect to δ. Suppose now that D(π ◦ ){0}×B (0; 0)(T, T¯ , φ) = (0, 0, 0, 0). Then clearly T = 0. Taking the trace of what remains of the first equation in (17) yields δ φ − η δ φ = δ φ = 0 R3

by the divergence theorem (valid because of the decay property of φ). But now, one can invoke the injectivity of on the space H k−1,β−1 (R3 ) to conclude that φ = 0. Finally, e T¯ + ε e T¯ + ε e T¯ = 0 and the fact that T¯ is divergence free at use the identity εbc ae ca be ab ce the Euclidean metric to conclude that the remaining equation for T¯ reads eb ¯ εa ∇b Tce − η εaeb ∇b T¯ce = 0. R3

The integrand above is an exact vector-valued differential. Thus by using Stokes’ Theorem (valid because of the decay property of T¯ ), one can eliminate the integral above and end up with εaeb ∇b T¯ce = 0. This equation shows that the vector-valued one-form T¯ab dx b is closed and hence exact, so that T¯ab = ∇b T¯a for some vector-valued zero form T¯a . The divergence-free condition on T¯ab then shows that δ T¯b = 0 and so T¯b = 0 by the injectivity of δ on the space of H k−2,β−2 tensors over R3 . These calculations show that the operator D(π ◦ ){0}×B (0; 0) is injective, thereby proving the uniqueness claim contained in Theorem A and concluding the proof of Theorem A.

16

A. Butscher

4. Statement and Proof of Theorem B 4.1. The statement. In order to complete the proof of the Main Theorem, it remains to show that the solution (16) of the secondary system constructed in the previous section actually satisfies the extended constraint equations. The second theorem to be proved in this paper establishes this fact. Theorem B. If (ψ(b), b) is a solution of the equation π ◦ (ψ(b), b) = 0 as in Theorem A, with b = (T, T¯ , φ) sufficiently small in the norm of the Banach space B and ψ(b) = (h(b), K (b), X (b), X¯ (b)), then the quantities g = δ + h(b), χ = K (b) + 13 φδ, S = div∗g X (b) + T − 13 T r g (T )g, S¯ = div∗g X¯ (b) + T¯ − 13 T r g (T¯ )g, satisfy the extended constraint equations (1). Strategy of proof. Theorem A has produced g ∈ H k,β (S 2 (R3 )), χ ∈ H k−1,β−1 (S 2 (R3 )), S ∈ H k−2,β−2 (Sg2 (R3 )), S¯ ∈ H k−2,β−2 (J (R3 )), and error terms λa , λ¯ a and µab in R that satisfy Eq. (16). Let J ∈ H k−2,β−2 (J (R3 )) be ¯ Then (g, χ , S, S) ¯ satisfies the extended constraint equations the tensor J ≡ Dg (χ )− S. (1) if and only if Jabc = 0, a = 0, (18) error terms = 0, a where is the quantity arising from the harmonic gauge choice as explained in Sect. 2.2. Here is a schematic of how it will be shown that (18) holds. Let z ε = (J, ) represent a solution of the secondary system, where the norm of the free data is represented by the small parameter ε, and let ε represent an error term lying in a fixed, finitedimensional subspace C. Then the integrability conditions contained within the extended constraint equations will be used to show that z ε satisfies a system of equations of the form Pε (z ε ) = ε , where Pε is a family of linear, elliptic operators with coefficients depending smoothly on ε. These equations will be called the auxiliary equations. Then, it will be shown that the operator Pε is (1) injective and uniformly elliptic, and thus uniformly injective, all for sufficiently small ε, and (2) that the finite-dimensional subspace C is transverse to the image of P0 . Consequently, the stability of co-kernels of linear elliptic operators proved in Proposition 3 shows that C remains transverse to the image of Pε for sufficiently small ε. Since ε ∈ C ∩ Im(Pε ), therefore ε = 0. But now the uniform injectivity can be invoked yet again to show that z ε = 0.


17

4.2. The proof. The auxiliary equations The first step in the proof of Theorem B is to derive the auxiliary equations satisfied by J and which are a consequence of the integrability conditions in the conformal constraint equations. Proposition 7. The quantities J , and the real numbers µab , λa , λ¯ a satisfy the equations 1 g b − Rba a = −ηλb − χ st Jsbt + Aa χab − χss gab , 2 where Ab = g ac Jabc , and

∗ Dg (J ) ab = ηµab ,

Qg (J ) a = 2ηλ¯ a ,

(19)

(20)

where Qg is the operator defined at the beginning of Sect. 2.1. Proof. The equation satisfied by will be deduced from the Bianchi identity ∇ a Rab − 1 2 ∇b R = 0. In fact, substitute the reduced Ricci operator into this identity to obtain 1 1 H 0 = ∇ a Rab − ∇b R H + g b + ∇ a ∇b a − ∇b ∇ a a 2 2 1 1 a a s = ∇ Sab − ∇ (χs χab ) + ∇ a (χas χsb ) + ∇b (χss )2 − χ st χst + g b + Rba a 2 2 1 st e ¯ a = ηλb − χ εbt Sse + g b + Rb a 2 − ∇ s χsa − ∇ a χss gab χss − χab − ∇b χst − ∇s χb χ st . (21) Now let Ab = g ac Jabc . According to the definition of J made above, Ab = ∇ a χab − ∇b χaa . Substituting this into (21) leads to 1 g b + Rba a = −ηλb − χ st Jsbt + Aa χab − χss gab , 2 which is the desired equation for a . Next, the first equation of (20) holds for J by definition. For the second equation, consider

Qg (J ) a = εebc ∇e Jabc s ¯ = εebc ∇e (∇c χab − ∇b χac ) − εbc Sas . At this stage, one must commute the second derivatives, bringing in curvature terms. Thus,

s ∇e S¯as Qg (J ) a = Recas χbs + Recbs χas εebc − εebc εbc = 2Recas χbs εecb − 2∇ s S¯as

(22)

by the algebraic Bianchi identity satisfied by the curvature tensor. Substitute the second equation of (16) into (22) to obtain

Qg (J ) a = 2Recas χbs εecb − 2εast χsu Rut + 2ηλ¯ a .

18

A. Butscher

The fact that only curvature terms remain is a manifestation of the integrability conditions. Finally, use the decomposition Recas = Rea gcs − Res gca + Rcs gea − Rca ges − 1 s ecb = ε st χ u R and thus that a s ut 2 R (gea gcs − ges gca ) to find that Recas χb ε

Qg (J ) a = 2ηλ¯ a , which is the second equation of (20).

Notation. Denote the pair of operators appearing in (20) by Pε (J ) = Dg∗ (J ), Qg (J ) . Thus Pε is an operator from H k−2,β−2 J (R3 ) to H k−3,β−3 Sg2 (R3 ) × 1 (R3 ) . Uniform injectivity and ellipticity. The operators appearing on the left-hand sides of Eqs. (19) and (20) should be viewed as linear, elliptic operators for J and whose coefficients depend on the solution g, χ , S and S¯ of Theorem A and thus on the free data (T, T¯ , φ). The uniform injectivity and ellipticity of these operators will be established one at a time using the following lemma. Lemma 8. Let g = δ + h be an asymptotically flat metric on R3 with h ∈ H k,β S 2 (R3 ) and let R(x) be a quantity that is quadratic in the argument x with coefficients proportional to the curvature of g. For any l ≥ 1 and γ < −1, there exists a number ε > 0 so that if h H k,β ≤ ε, then the only function u ∈ H l,γ (R3 ) satisfying the inequality ∇u2 dVolg + C R(u)dVolg (23) 0≤− R3

R3

for some constant C, is the zero function. Remark. This lemma is the generalization of a result from [3]. Proof. Because h ∈ H k,β with k ≥ 4 > 23 and β < 0, the Sobolev Embedding Theorem [2] guarantees that |h|C 2 ≤ Ch H k,β . Consequently, the coefficients of R are bounded in C 0 by h H k,β and it follows that R(u)dVolg ≤ R |u|2 dVolg R3 R3 ≤ Ch H k,β |u|2 σ −2 dVolg R3 ≤ Ch H k,β ∇u2 dVolg (24) R3

by the Poincaré inequality for weighted Sobolev norms proved in [2]. (Actually, [2] proves this only for the metric g = δ, but the result is also valid for the metric g = δ + h with sufficiently small h H k,β because the norms with the metric g are uniformly equivalent to the norms with the metric h and no Christoffel symbols appear.) Using (24) in inequality (23) leads to 0 ≤ Ch H k,β − 1 ∇u2 , R3

so if h H k,β is sufficiently small, the right hand side above becomes negative. Avoiding this contradiction requires ∇u = 0. But since u decays at infinity when γ < −1, it must be true that u = 0.


19

It is now straightforward to deduce the uniform injectivity and ellipticity for the operator g − Ric appearing in (19). In fact, all that is needed is to show the injectivity of this operator, since uniform ellipticity is clear. To see this, contract the equation g a − Rab b = 0 with a to obtain 0 = 2 − ∇ − 2Ric( , ). Then integrate over R3 and apply the divergence theorem as well as the Cauchy-Schwarz inequality to produce an expression of the form considered in Lemma 8. The injectivity of g − Ric follows. The analogous result for the operator Pε appearing in (20) is not quite so straightforward. Another lemma is needed; then a lengthy calculation will transform the injectivity equation for Pε into a form suitable for the application of Lemma 8. Lemma 9. Let M be any Riemannian manifold. Then J (M) is isomorphic to Sg2 (M) × 1 (M). In local coordinates, this isomorphism is given by Jabc = e 1 2 εbc Fae + Ab gac − Ac gab , where εabc is the fully antisymmetric permutation symbol associated to the metric of M. Proof. Given any tensor J ∈ J (M), one can define a 2-tensor T by the prescription J (X, Y, Z ) = T X, (Y ∧ Z ) , for any three vectors X , Y , and Z , which is well-defined and unique because of the antisymmetry in the last two slots of J . Here, and are the raising and lowering operators associated to the metric of M and is its Hodge star operator. Moreover, one can check that the trace of T vanishes by virtue of the symmetries of J . The tensor T can now be decomposed into its symmetric part F and its antisymmetric part written in terms of a 1-form A by means of Hodge duality. It is easy to see that this association is bijective and yields the local coordinate expression of the lemma. Proposition 10. The family of operators Pε is injective provided that ε ≤ ε0 is sufficiently small. e F + A g − A g ) into the expression for P to obtain Proof. Substitute Jabc = 21 (εbc ae b ac c ab ε after some straightforward calculation ⎞ ⎛ 2 e e εcb ∇ c Fae + εca ∇ c Fbe − ∇a Ab − ∇b Aa + ∇ c Ac gab ⎠ 3 Pε (F, A) = ⎝ (25) b bc ∇ Fab + εa ∇b Ac for F and A. Now suppose that (F, A) ∈ H k−2,β−2 Sg2 (R3 ) × 1 (R3 ) and that Pε (F, A) = (0, 0). The key to showing (F, A) = (0, 0) is to find an inequality satisfied by F and A of the type needed to invoke Lemma 8. Begin with the identity e F + εe F + εe F εbc ae ca be = 0 that F satisfies by virtue of being trace-free. Then ab ce take its divergence with respect to ∇ c and substitute from the second equation of Pε (F, A) = (0, 0) to find e e e st εca ∇ c Fbe = εcb ∇ c Fae + εab εe ∇s At e c = εcb ∇ Fae + ∇a Ab − ∇b Aa .

(26)

20

A. Butscher

b Next, substitute (26) into the first equation of Pε (F, A) = (0, 0) and contract with εuv to get

1 b e b 0 = εuv εcb ∇ c Fae − εuv ∇b Aa + ∇ s As εauv 3 1 b = ∇v Fau − ∇u Fav − εuv ∇b Aa + ∇ s As εauv . 3

(27)

Now contract (27) with ∇ v : 1 b ∇ v ∇b Aa + ∇ v ∇ s As εauv 0 = g Fau − ∇ v ∇u Fav − εuv 3 1 b vs 1 vs = g Fau − ∇u ∇ v Fav − εuv Rba As + ∇ v ∇ s As εauv − Rus Fas − Rua Fsv 2 3 1 1 b vs pq vs = g Fau + εa ∇u ∇ p Aq + ∇ v ∇ s As εauv − εuv Rba As − Rus Fas − Rua Fsv (28) 3 2 after using the second equation of Pε (F, A) = (0, 0) once again. The advantage of this expression is that the only derivatives of F appearing here are in the Fau term. A similar expression can be obtained for A by computing the antisymmetric part of (28) and contracting with εtau . That is, 1 1 pq b vs vs Rba As − εtau Rus Fas − εtau Rua Fsv 0 = εtau εa ∇u ∇ p Aq + ∇ v ∇ s As εauv εtau − εtau εuv 3 2 2 = g At − ∇ u ∇t Au + ∇t ∇ b Ab − Rus Fas εtau − Rts As 3 1 s = g At − ∇t ∇ As − Rus Fas εtau . (29) 3 Since F ∈ H k−2,β−2 , the integral of F au g Fau over R3 is well-defined and can be integrated by parts. In what follows, R denotes a curvature quantity as in Lemma 8. Now, 0=

R3

F au g Fau dVolg +

R3

pq

εa ∇u ∇ p Aq F au dVolg +

1 g F2 dVolg − ∇ F2 dVolg 3 2 R3 R pq au + εa ∇u ∇ p Aq F dVolg + R(A, F)dVolg 3 R3 R pq =− ∇ F2 dVolg + εa ∇u ∇ p Aq F au dVolg +

R3

R(A, F)dVolg

=

R3

R3

R3

R(A, F)dVolg , (30)


21

since R3 g udVolg = 0 for any u ∈ H k−2,β−2 (R3 ). But now, pq pq εa ∇u ∇ p Aq F au dVolg = − εa ∇ p Aq ∇u F au dVolg R3 R3 pq = εa ∇ p Aq εr sa ∇r As dVolg 3 R = ∇ A2 dVolg − ∇ s Ar ∇r As 3 3 R R 2 = ∇ A dVolg + (∇ s As )2 dVolg + R3

R3

R3

R(A)dVolg , (31)

using ∇u F au = −εr sa ∇r As and repeated use of integration by parts. Substituting (31) into (30) yields 2 2 s 2 0=− ∇ F dVolg + ∇ A dVolg − (∇ As ) dVolg + R(A, F)dVolg . R3

R3

R3

R3

Similar calculations involving Eq. (29) for g At yield the identity 1 2 s 2 0=− ∇ A dVolg + (∇ As ) dVolg + R(A)dVolg . 3 R3 R3 R3

(32)

(33)

Adding one third of Eq. (32) to Eq. (33) allows the divergence term to cancel and leads to the identity 1 2 0=− ∇ F2 + ∇ A2 dVolg + R(A, F)dVolg . (34) 3 R3 3 R3 Now R(A, F) ≤ CR(F2 + A2 ), where R denotes the supremum norm of curvature coefficients and C is a general numerical constant. Consequently, one obtains the estimate ∇ F2 + ∇ A2 dVolg + C 0=− R A2 + F2 dVolg . (35) R3

R3

Lemma 8 along with the simple estimate ∇T 2 ≤ C∇T 2 (by Cauchy-Schwarz) can now be applied to conclude that (F, A) = (0, 0) when h H k,β is sufficiently small. But since h depends smoothly on (T, T¯ , φ), and vanishes when these quantities are zero, the operator Pε is injective when ε is sufficiently small. Proposition 11. For every 0 ≤ ε ≤ ε0 , the operator Pε is elliptic. Proof. The symbol of Pε , rewritten in terms of the isomorphism of Lemma 9 is given by ⎛ ⎞ 2 c e c e c ε ξ Fae + εca ξ Fbe − ξa Ab − ξb Aa + ξ Ac gab ⎠ 3 (F, A) → ⎝ cb b bc ξ Fab + εa ξb Ac

22

A. Butscher

for any non-zero ξ ∈ R3 . If one follows the same algebraic steps as were used in the previous proposition, one ends up with the two equations 1 b v 0 = ξ 2 Fau − ξ v ξu Fav − εuv ξ ξb Aa + ξ v ξ s As εauv , 3 1 s 2 0 = ξ − ξt ξ As . 3 (These are the analogs of Eqs. (28) and (29), where curvature terms do not appear since commuting derivatives reduces to commuting multiplication by ξ at the level of the symbol.) Contracting the second equation against ξ t yields 23 ξ 2 ξ t At = 0 or ξ t At = 0. If this is substituted back into the first equation, one has ξ 2 At = 0, or simply At = 0. In a similar manner, one then concludes that Fau = 0. The symbol of Pε is thus injective. But since it is a map from one 8-dimensional space of tensors to another, it must be surjective as well. Consequently, Pε is elliptic. Corollary 12. The family of operators Pε is uniformly injective, provided ε is sufficiently small. Proof. The injectivity of Pε together with the elliptic estimate yields an estimate of the form u ≤ CPε u in the appropriate norms. Since the coefficients of Pε depend smoothly on ε and P0 is not degenerate, the constant C can be made independent of ε. But this is the uniform injectivity of the operators Pε . Transversality and completion of the proof. The equation for J in (20) is schematically of the form Pε (J ) = ε , where ε depends on ε is the error term. However, for each ε ≤ ε0 , ε belongs to the fixed, finite dimensional subspace C = ηµi j dx i ⊗ dx j : µi j ∈ R × ηλ¯ i dx i : λ¯ i ∈ R , of H k−3,β−3 Sg2 (R3 ) × 1 (R3 ) . But the following transversality result makes this impossible unless ε = 0. Proposition 13. The family of operators Pε satisfies C ∩ Im(Pε ) = {(0, 0)} provided ε is sufficiently small. Proof. Apply the divergence theorem to the equation P0 (J ) = η(µ, λ¯ ). Integrating over R3 gives zero on the left-hand side, while by construction, integrating the right-hand side yields (µab , λ¯ a ). Thus the conclusion of the theorem holds for P0 . Consequently, by Proposition 3 and the uniform injectivity of Pε , the image of Pε remains transverse to C for sufficiently small ε. Equation (20) states that η(µ, λ¯ ) ∈ C ∩ Im(Pε ). Thus (λ, µ) must vanish, and J must satisfy Pε (J ) = (0, 0). Thus one can invoke the uniform injectivity of Pε once again to conclude that J itself must of vanishing J , λ¯ and µ vanish. The consequence is that the equation for reads 21 g b + Rba a = −ηλb . The vanishing of both and λ can now be shown in the same way as above. That is, since −ηλ belongs to the fixed, finite-dimensional subspace C ≡ {ηλi dx i : λi ∈ R} and g + Ric is uniformly injective and elliptic, all that is needed is the transversality of δ to C by Proposition 3. But this is again a simple consequence of the divergence theorem. Consequently, λ = 0 and so = 0 by uniform injectivity.


23

This completes the proof of Theorem B. In combination with Theorem A, this yields a construction of perturbative solutions of the extended constraint equations and concludes the proof of the Main Theorem. Acknowledgements. The research and most of the writing for this paper was carried out while I was a postdoctoral fellow at the Max Planck Institute for Gravitational Physics in Golm, Germany. I would like to thank Helmut Friedrich of this institute for suggesting the topic of the extended constraint equations as well as for suggesting the two-step approach for its solution. I would also like to thank him for providing guidance and support during and after the course of the research. Furthermore, I would like to thank Piotr Chru´sciel, Justin Corvino, Jörg Frauendiener and Gerhard Huisken for their helpful suggestions and comments. Erratum. Note that [3] contains an error in the section pertaining to that paper’s analogue of Theorem B. The Stability Lemma in Sect. 10.3.6 of [3] is mis-stated and its proof is incorrect. It is then applied to the associated equations in the way the correctly-stated version should have been applied. The result in [3] is thus still correct. In any case, the results of this paper subsume and generalize those of [3].

References 1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Second edition New York: Springer-Verlag, 1988 2. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39, 661–693 (1986) 3. Butscher, A.: Exploring the conformal constraint equations. In: Proc. of the Int’l Workshop on the Conformal Structure of Spacetime (Tubinger, Germany, April 2001), in Lect. Notes in Phys. 604, Berlin Heidelberg-New York: Springer, 2002, pp. 195–222 4. Cantor, M.: Elliptic operators and the decomposition of tensor fields. Bull. Amer. Math. Soc. 5(3), 235– 262 (1981) 5. Choquet-Brouhat, Y., Christodoulou, D.: Elliptic systems in Hs,δ spaces on manifolds which are Euclidean at infinity. Acta Math. 146, 129–150 (1981) 6. Christodoulou, D., O’Murchadha, N.: The boost problem in general relativity. Commun. Math. Phys. 80, 271–300 (1981) 7. Chru´sciel, P., Delay, E.: Existence of non-trivial, vacuum, asymptotically simple space-times. Class. Quant. Grav. 19 (2002) 8. Corvino, J.: Scalar curvature deformation and a gluing construction for the einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000) 9. Frauendiener, J.: Conformal infinity. Living Rev. Relativity 3 (2000), no. 4, Online Article: http://relativity.livingreviews.org/Articles/Irr-2004-/, 2004 10. Friedrich, H.: Asymptotic structure of space time. In: Recent Advances in General Relativity, A. Janis, J. Porter, eds., Einstein Studies, Vol. 4, Basel-Boston: Birkäuser, 1992 11. Friedrich, H.: Einstein’s equations and conformal structure. In: The Geometric Universe. Science, Geometry and the work of Roger Penrose S. Huggett, L. Mason, K.P. Tod, S. Tsou, N.M.J. Woodhouse, eds., Oxford: Oxford University Press, 1998 12. Lockhart, R.: Fredholm properties of a class of elliptic operators on non-compact manifolds. Duke Math. J. 48, 289–312 (1981) 13. McOwen, R.C.: The behavior of the Laplacian on weighted Sobolev spaces. Comm. Pure Appl. Math. 32, 783–795 (1979) 14. McOwen, R.C.: On elliptic operators in Rn . Comm. Part. Differ. Eqs. 5, 913–933 (1980) 15. Penrose, R.: Asymptotic properties of fields and spacetimes. Phys. Rev. Lett. 10, 66–68 (1963) 16. Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979) Communicated by G.W. Gibbons

Commun. Math. Phys. 272, 25–52 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0190-x

Communications in


Convergence of the Wick Star Product Svea Beiser, Hartmann Römer, Stefan Waldmann Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann Herder Straße 3, D 79104 Freiburg, Germany. E-mail: [email protected]; [email protected]; [email protected] Received: 14 July 2005 / Accepted: 27 September 2006 Published online: 7 March 2007 – © Springer-Verlag 2007

Abstract: We construct a Fréchet space as a subspace of C ω (Cn ) where the Wick star product converges and is continuous. The resulting Fréchet algebra A is studied in detail including a ∗ -representation of A in the Bargmann-Fock space and a discussion of star exponentials and coherent states. 1. Introduction Deformation quantization usually comes in two flavours: formal and strict deformations: In formal deformation quantization as introduced by [2], see also [20, 25] for recent reviews, one considers the Poisson algebra of smooth complex-valued functions C ∞ (M) on a Poisson manifold as observable algebra in classical mechanics. A formal star product is a C[[λ]]-bilinear associative product for C ∞ (M)[[λ]] such that in zeroth order f g is the pointwise product and in first order of λ the -commutator gives i times the Poisson bracket. Usually one requires the higher orders to be given by bidifferential operators. The algebra C ∞ (M)[[λ]] then serves as a model for the quantum mechanical observables corresponding to the classical system described by M. In particular, the formal parameter λ corresponds to Planck’s constant and should be replaced by whenever one can establish convergence of the formal series. On the other hand, in strict deformation quantization as introduced in [44], see also [35], one works on the framework of C ∗ -algebras where the dependence of the deformed product on the deformation parameter is now required to be continuous. This is made precise using the notion of continuous fields of C ∗ -algebras. While in the first approach one has very strong existence [18,22–24,34,41] and classification results [3, 17, 26, 34, 38, 39, 47], the formal character of Planck’s constant is, of course, physically not acceptable. Here the second approach is much more appealing as it directly uses the analytical framework suitable for quantum mechanics. On the other hand, however, a general construction and reasonable classification of strict quantizations seems still to be missing.

26

S. Beiser, H. Römer, S. Waldmann

Many examples like the global symbol calculus on cotangent bundles [6–8], Berezin-Toeplitz quantization on Kähler manifolds [5,13–16,33] as well as [4] suggest that the formal star products should be seen as asymptotic expansions for → 0 of their convergent counterparts in strict quantization. On the other hand, many formal star products allow for large subalgebras, where the formal series actually converge, whence in some sense the asymptotics can be used again to recover the strict result, a heuristic statement for which a general theorem unfortunately is still missing. The above examples also suggest that there is a framework in between formal and C ∗ -algebraic, namely one can try to construct deformations of C ∞ (M) (or suitable subalgebras of it) in the framework of Fréchet or more generally locally convex algebras. Early results in this direction have been obtained in [27,30,36], see also [40]. Moreover, a general set-up of smooth deformations has been established and exemplified in [21]; in [43] holomorphic deformations were studied. The example we are going to discuss will provide an entire holomorphic deformation of a Fréchet subalgebra of C ω (M). More specifically, we consider the most simple phase space M = Cn with its canonical Poisson structure {z k , z } = 2i δ k and the formal Wick star product ∞ (2λ)r ∂r f ∂r g , (1.1) f Wick g = r! ∂z i1 · · · ∂z ir ∂z i1 · · · ∂z ir r =0

i 1 ,...,ir

where z 1 , . . . , z n are the canonical, global, holomorphic coordinates on Cn . Our convergence scheme to construct the ‘convergent’ subalgebra A is then based on the crucial observation that the Wick star product enjoys a very strong positivity property [10–12]: every δ-functional is a positive C[[λ]]-linear functional in the sense of formal power series. After choosing a point p ∈ Cn and > 0 this will allow us to construct recursively a system of seminorms for which the Wick star product is continuous, thereby defining our algebra A p,. Moreover, we shall construct, via the GNS construction corresponding to the positive functional δ p , a faithful ∗ -representation of A p, on a dense subspace of the Bargmann-Fock space giving an interpretation of the δ-functionals as coherent states with respect to the Heisenberg group Hn acting on Cn . We treat this example in quite some detail as we believe that it may serve as a good starting point for geometric generalizations to Wick star products on Kähler manifolds [9,31] suitable for a bottom-up approach to [5, 13–16, 33]. Moreover, in a future project we shall discuss the field-theoretic generalization for infinitely many degrees of freedom. The paper is organized as follows: In Sect. 2 we briefly recall some basic properties of Wick , the formal GNS construction for δ-functionals and the Bargmann-Fock space. Section 3 is devoted to the construction of the seminorms, depending on p ∈ Cn and > 0. This gives the space A p, which is shown to be a subspace of C ω (Cn ) with a Fréchet topology. In Sect. 4 we show that A p, is a subalgebra of C ω (Cn ) such that the pointwise product as well as the Poisson bracket are continuous, i.e. A p, becomes a Fréchet-Poisson algebra. Moreover, we show that the formula (1.1) for Wick actually converges on A p, in the Fréchet topology resulting in a continuous product. This way, A p, becomes a holomorphic deformation. In Sect. 5 we discuss the dependence on the a priori chosen point p and on the value > 0 of Planck’s constant. It turns out that the translation group acts on A p, by inner ∗ -automorphisms whence A p, = A does not depend on the choice of p. As a side remark we show that the star exponential, see [2], of linear functions converges in the topology of A. Moreover, for all values > 0 the algebras A are isomorphic in a canonical way. Finally, in Sect. 6 we show how the GNS construction yields a ∗ -representation of A in the sense of [45] in the Bargmann-Fock

Convergence of the Wick Star Product

27

space. The action of the Heisenberg group by inner ∗ -automorphisms gives easily the coherent states. 2. Preliminary Results In this section we shall collect some well-known results on the Wick star product which we shall use in the sequel, see e.g. [9, 10]. On the classical phase space R2n ∼ = Cn with standard symplectic form ω = 2i d z k ∧ k d z one defines the formal Wick star product by f Wick g =

∞ (2λ)r r =0

r!

i 1 ,...,ir

∂z i1

∂r f ∂r g , i i · · · ∂z r ∂z 1 · · · ∂z ir

(2.1)

where f, g ∈ C ∞ (Cn )[[λ]], the formal parameter λ corresponds to Planck’s constant without any further prefactors and z, z denote the usual global holomorphic/antiholomorphic coordinates on Cn . Then Wick is known to be an associative star product quantizing the canonical Poisson bracket corresponding to ω. It has the separation of variable property in the sense of Karabegov [31, 32] and is in fact the name-giving example of a star product of Wick type in the sense of [9]. Moreover, Wick is Hermitian f Wick g = g Wick f ,

(2.2)

where according to our interpretation of λ the formal parameter is defined to be real λ = λ. We shall also frequently make use of multiindex notation: Let R = (r1 , . . . , rn ) ∈ Nn be a multiindex, then one defines |R| = r1 + · · · + rn , R! = r1 ! · · · rn ! as well as z R = (z1 )r1 · · · (z n )rn etc. Moreover, we define R ≤ L if ri ≤ i for all i = 1, . . . , n R! and set RL = L!(R−L)! for R ≥ L. The Wick star product can equivalently be written as f Wick g =

∞ (2λ)|R| ∂ |R| f ∂ |R| g . R! ∂z R ∂z R R=0

(2.3)

The Wick star product enjoys a very strong positivity property which e.g. the WeylMoyal star product does not share: If δ p : C ∞ (Cn )[[λ]] −→ C[[λ]] denotes the evaluation functional at p ∈ Cn then we have δ p ( f Wick f ) =

∞ (2λ)|R| R!

R=0

|R| 2 ∂ f ≥ 0, ( p) ∂z R

(2.4)

where the positivity is understood in the sense of the canonical ring ordering of R[[λ]], see [10,46] for a detailed discussion on the physical relevance of this notion of positivity. This very strong positivity property is not true for general Hermitian star products: instead one has to add ‘quantum corrections’ to a given classically positive functional (here δ p ) in order to obtain a positive functional with respect to the star product. Since positive functionals play the role of states, the above simple observation implies that for the Wick star product any classical state defines a quantum state without any quantum correction, see [46] for a detailed discussion on the general situation. In fact, the Wick star product is used in an essential way for proving that for an arbitrary Hermitian star product one can always construct quantum corrections for a classical state, see [11,12].

28


Since δ p is a positive functional for Wick we have a corresponding GNS representation of the Wick star product algebra C ∞ (Cn )[[λ]]. In fact, this example was one of the first examples of GNS constructions in deformation quantization. Since we need the construction in the following, we briefly review the results from [10]. The Gel’fand ideal J p of δ p is given by ∂ |R| f ( p) = 0 for all R . (2.5) J p = f δ p ( f Wick f ) = 0 = f ∂z R The GNS pre-Hilbert space H p = C ∞ (Cn )[[λ]] J p can canonically be identified with HBF = C[[y 1 , . . . , y n ]][[λ]]

(2.6)

with the C[[λ]]-valued positive definite inner product φ, ψ =

∞ (2λ)|R| ∂ |R| φ ∂ |R| ψ (0) (0), R! ∂yR ∂yR R=0

(2.7)

where the identification of a class ψ f ∈ H p is given by its formal anti-holomorphic Taylor expansion, i.e. H p ψ f →

∞ 1 ∂ |R| f ( p) y R , R! ∂z R R=0

(2.8)

where ψ f denotes the equivalence class of the function f in H p . Then the GNS representation on H p defined by π p ( f )ψg = ψ f Wick g is translated into

p( f ) =

∞ |R| (2λ)|R| ∂ |R+S| f S ∂ ( p) y , R!S! ∂z R ∂z S ∂yR R,S=0

(2.9)

via the unitary map (2.8). In particular, for p = 0 we see that this gives the formal analog of the usual Bargmann-Fock space and the Bargmann-Fock representation: indeed, recall that the Bargmann-Fock space is the Hilbert space

zz 1 n 2 − 2 HBF = f ∈ O(C ) e d z d z < ∞ (2.10) | f (z)| (2π )n of anti-holomorphic functions which are square-integrable with respect to the Gaussian measure, see [1]. Then it is well-known that HBF is actually a closed subspace of the L 2 -space for this measure and hence a Hilbert space itself. Moreover, the L 2 -inner product can be evaluated by the same formula (2.7) if one replaces λ by , where the series now converges absolutely. A Hilbert basis for HBF is given by the monomials 1 zR. e R (z) = |R| (2) R!

(2.11)

From (2.7) we see that the Hilbert space HBF can be interpreted as the space of those antiholomorphic functions whose Taylor coefficients at 0 form a sequence in a (weighted) 2 -space.


29

3. Construction of the Fréchet Space A p, The motivation for our convergence scheme is rather simple: we fix > 0 and we fix a point p ∈ Cn . Then we are looking for a subalgebra of C ∞ (Cn )[[λ]] such that δ p ( f ) converges for λ = . Though this looks rather innocent at the beginning, we obtain a hierarchy of conditions: For f, g in our algebra we want f Wick g to be in the algebra as well whence δ p ( f Wick g) has to converge for λ = as well. The idea is now to esti mate the convergence of δ p ( f Wick g) λ= by δ p ( f Wick f )λ= and δ p (g Wick g)λ= using the Cauchy-Schwarz inequality for the positive δ-functional. Indeed, if one of the functions f or g is a polynomial, then f Wick g is only polynomial in λ whence we can easily set λ = . We use the notation Wick for the Wick star product where λ has been replaced by . Under this hypothesis we have the Cauchy-Schwartz inequality finite 2 2 (2)|N | ∂ |N | f ∂ |N | g ( p) ( p) f Wick g ( p) = N N! ∂z N ∂z N =0 ⎛ ⎞ |N | 2 ∞ ∞ 2 |N | (2)|N | ∂ |N | f ⎠ (2) ∂ g ⎝ ≤ ( p) ( p) N N N! ∂z N! ∂z N =0 N =0 p,

p,

= h 0,0,0,0 ( f ) h 0,0,0,0 (g) . p,

(3.1)

p,

Here the quantities h 0,0,0,0 ( f ) and h 0,0,0,0 (g) are defined by the two factors on the right-hand side. An iteration of this procedure will lead to countably many unary conditions for a function f to belong to our algebra, which we shall interpret as seminorms determining the algebra. To motivate the quite involved combinatorics we first note that on the right-hand side of (3.1) we have to estimate derivatives of f and g as well. Thus we start directly with a formulation using all possible derivatives. The next iteration p, will lead to the following estimate for h 0,0,0,0 f Wick g , where again we assume that at least one function is a polynomial only, to make f Wick g well-defined as a smooth function. Then we have |R+S| ∂ |R+S| p, h 0,0,0,0 f Wick g (2) ∂z R ∂z S ∞ 2 (2)|N +R+S| ∂ |N +R+S| = ∂z N +S ∂z R f Wick g ( p) N! N =0

∞ (2)|N +R+S| = N! N =0

∞ (2)|N +R+S| N!

≤

N =0 ∞ (3.1)

≤

N =0

⎛ ×⎝

2 R N +S |I +J | f ∂ |N +S−J +R−I | g R N +S ∂ Wick ( p) I J ∂z I ∂z J ∂z R−I ∂z N +S−J I =0 J =0 2 R N +S R N +S ∂ |I +J | f ∂ |N +S−J +R−I | g I J Wick R−I N +S−J ( p) I J ∂z ∂z ∂z ∂z I =0 J =0

(2)|N +R+S| N!

R N +S R N +S I I =0 J =0

J

p, h 0,0,0,0

∂ |I +J | f ∂z I ∂z J

p, h 0,0,0,0

⎞ 2 ∂ |N +S−J +R−I | g ⎠ ∂z R−I ∂z N +S−J

30


R N +S |I +J | ∞ ∂ 1 R N +S f |I +J | p, ≤ h 0,0,0,0 I J (2) I J N! ∂z ∂z N =0 I =0 J =0 R N +S |I +J | R N +S ∂ g |I +J | p, h 0,0,0,0 × I J (2) J I ∂z ∂z I =0 J =0 R N +S |I +J | 2 R N +S ∞ 1 ∂ f p, (2)|I +J | h 0,0,0,0 ≤ I J I N! ∂z ∂z J N =0 I =0 J =0 R N +S |I +J | 2 R N +S ∞ 1 ∂ g p, |I +J | (2) × h 0,0,0,0 I J J N! ∂z ∂z I N =0 I =0 J =0 p, p, = h 1,1,R,S ( f ) h 1,0,R,S (g), p,

(3.2)

p,

where the quantities h 1,1,R,S ( f ) and h 1,0,R,S (g) are defined by the corresponding factors. In the first step we have used that the partial derivatives are still derivations with respect to the Wick star product. Moreover, we have added the power (2)|R+S| to make things more homogeneous. Note that the functions f and g enter in an asymmetric way: the multiindices I and J correspond to z and z-variables for f and vice versa for g. We use this first step of the iteration as motivation for the following definition of the seminorms. Definition 3.1. Let R, S ∈ Nn be multiindices, m ∈ N and = 0, . . . , 2m − 1. Then we define recursively for f ∈ C ∞ (Cn ), |R+S| f ∂ |R+S| f p, |R+S| ∂ h 0,0,R,S ( f ) = (2) ( p) Wick ∂z S ∂z R ∂z S ∂z R λ= |R+S+N | 2 ∞ |R+S+N | (2) f ∂ = (3.3) ∂z R ∂z N +S ( p) N! N =0 and

⎧ 2 R N +S ∞ ⎪ ⎪ 1 R N +S p, ⎪ ⎪ even ⎪ I J h m−1,/2,I,J ( f ) ⎨ N! p, N =0 I =0 J =0 (3.4) h m,,R,S ( f ) = 2 R N +S ∞ ⎪ R N +S p, ⎪ 1 ⎪ ⎪ odd. ⎪ ⎩ I J h m−1,(−1)/2,J,I ( f ) N! N =0

I =0 J =0

p,

Thanks to the positivity (2.4) of the δ-functional it is clear that h 0,0,R,S ( f ) either converges absolutely or diverges absolutely to +∞, as it is a series consisting of non-negative p, terms only. By induction, the same is true for all other h m,,R,S ( f ). Hence we have p,

h m,,R,S ( f ) ∈ [0, +∞],

(3.5)

where ‘convergence’ is always absolute and does not depend on the order of summation.


31

Definition 3.2. For f ∈ C ∞ (Cn ) we define 2m+1 p, p,

f m,,R,S = h m,,R,S ( f )

∈ [0, +∞].

Moreover, we define p, = f ∈ C ∞ (Cn ) f p, A m,,R,S < ∞ for all m, , R, S .

(3.6)

(3.7)

p, is a vector space and that the · p, The first step is now to show that A m,,R,S are seminorms on A p,. This will be a consequence of the following proposition: p,

Proposition 3.3. The maps · m,,R,S : C ∞ (Cn ) −→ [0, +∞] enjoy the following properties: p,

p,

1. α f m,,R,S = |α| f m,,R,S for α ∈ C. p,

p,

p,

2. f + g m,,R,S ≤ f m,,R,S + g m,,R,S . p,

p,

p,

p,

3. f m−1,,R,S ≤ f m,2,R,S and f m−1,,R,S ≤ f m,2+1,R,S . √ m+2 p, p, S! f m+1,2,0,0 . 4. f m,,0,S ≤ 2 √ m+2 p, p, R! f m+1,2+1,0,0 . 5. f m,,R,0 ≤ 2 √ √ m+3 m+2 p, p, R! 2 S! f m+2,4+1,0,0 . 6. f m,,R,S ≤ 2 Proof. The first part is clear by a simple induction. For the second part the case m = 0 follows directly from Minkowski’s inequality. Then m > 0 is shown inductively by using again Minkowski’s inequality, for both cases of odd and even . The remaining p, inequalities between the f m,,R,S for different values of the parameters are simply obtained by omitting all but one term for a specific N in the defining summations of (3.4). For example, in the third part one considers N = 0 and I = R, J = S only, while for the fourth and fifth one uses N = S = J and N = R = I . p, p,, give indeed seminorms. Moreover, the Thus the maps · m,,R,S , restricted to A labels R, S play only a minor role thanks to the estimates in the last part of the proposition. This motivates the following definitions. For f ∈ C ∞ (Cn ) we define p,

p,

f m, = f m,,0,0 , p, p,

f m, .

f m = maxm 0≤≤2 −1

(3.8) (3.9)

Then we have the following simple corollary: p, ⊆ C ∞ (Cn ) is a subvector space and the · p, Corollary 3.4. The set A m,,R,S are p, p, seminorms on A p,. Moreover, the seminorms · m, as well as the seminorms · m p,. determine the same locally convex topology on A p, always with this locally convex topology In the following, we shall equip A p, induced by the seminorms · m,,R,S . However, this topology has one unpleasant feature: it is non-Hausdorff as a function f whose Taylor expansion at p vanishes identically

32

S. Beiser, H. Römer, S. Waldmann p,

has clearly f m,,R,S = 0 for all parameters m, , R, S. On the other hand, as one sees p,

already from the seminorm · 1 the functions with vanishing ∞-jet j∞ p f at p are the only functions with this property. Thus we identify them to be zero in order to have a Hausdorff topology: Definition 3.5. We define p, A p, = A

f ∈ C ∞ (Cn ) j∞ f = 0 , p

(3.10)

and equip A p, with the induced locally convex topology determined by the seminorms p, p,

· m,,R,S (or equivalently, by the seminorms · m ). Clearly, A p, is now a Hausdorff locally convex topological vector space. The following theorem shows that the abstract quotient can be viewed as a certain subspace of the real-analytic functions on Cn : Theorem 3.6. Let > 0 and p ∈ Cn . 1. A p, is a Hausdorff locally convex topological vector space. 2. Every class [ f ] ∈ A p, has a unique real-analytic representative f ∈ C ω (Cn ). Therefore, we identify A p, with the corresponding subspace of C ω (Cn ) from now on. 3. Every function f ∈ A p, has a unique extension to a function fˆ ∈ O×O(Cn × Cn ), i.e. holomorphic in the first and anti-holomorphic in the second argument, such that f = ∗ fˆ,

(3.11)

where : Cn z → (z, z) ∈ Cn × Cn is the diagonal. 4. Any f ∈ C ω (Cn ) such that there exist constants a, b, c > 0 with |R+S| ∂ f |R| |S| ∂z R ∂z S ( p) ≤ ca b

(3.12)

belongs to A p,. In particular C[z, z] ⊆ A p,. Proof. The first part is clear. For the second, we consider p,

f 1,1,0,0 ≥

1 (2)2|R+S| p, ( f 0,0,R,0 )2 ≥ R! R!(S!)2

|R+S| 4 ∂ f ( p) ∂z R ∂z S

for all R, S. Thus we obtain √ √ |R+S| 4 ∂ f R! S! p, . ∂z R ∂z S ( p) ≤ f 1,1,0,0 (2)|R+S| But this implies that the series fˆ(z, w) =

∞ R,S=0

1 ∂ |R+S| f ( p)(z − p) R (w − p) S R!S! ∂z R ∂z S

(3.13)


33

converges for all z, w ∈ Cn . Thus fˆ ∈ O×O(Cn × Cn ) and clearly ∗ fˆ is in the same equivalence class as f . This shows the second and third part. Now assume f ∈ C ω (Cn ) satisfies (3.12). Then p,

h 0,0,R,S ( f ) ≤ c2 e2b n (2a 2 )|R| (2b2 )|S| , 2

whence there are constants c0,0 = c2 e2b n , a0,0 = 2a 2 and b0,0 = 2b such that 2

p,

|R|

|S|

h 0,0,R,S ( f ) ≤ c0,0 a0,0 b0,0 . We claim that for all m, there are constants am, , bm, and cm, such that p,

|R|

|S|

h m,,R,S ( f ) ≤ cm, am, bm, . Indeed, a recursive argument shows that 2 cm, = cm−1,/2 en(1+bm−1,/2 ) , am, = (1 + am−1,/2 )2 , bm, = (1 + bm−1,/2 )2 2

for even and 2 cm, = cm−1,(−1)/2 en(1+am−1,(−1)/2 ) , am, = (1 + bm−1,(−1)/2 )2 , 2

bm, = (1 + am−1,(−1)/2 )2 for odd will do the job. But then all seminorms of f are finite. Since the polynomials are in A p, we shall make intense use of them. The next proposition gives a first hint on the continuity of the Wick product. Proposition 3.7. Let f ∈ C ∞ (Cn ) and let g ∈ C[z, z] be a polynomial. 1. 2.

f Wick g is a finite sum and thus a well-defined smooth function. f Wick g

p, m,,R,S

p,

p,

≤ f m+1,2m +,R,S g m+1,,R,S .

Proof. The first part is clear from the explicit form of Wick . For the second part we note that the computation in (3.2) is precisely the case m = 0. Note the necessity that one function (we have chosen g) is polynomial since otherwise the ‘function’ f Wick g is a priori not defined as a smooth function. The general case is now obtained by a straightforward induction on m only using the Cauchy-Schwarz inequality. Corollary 3.8. The pointwise complex conjugation A p, −→ A p, is a continuous map. We have p, p, p, f m,,R,S ≤ 1 m+1,,R,S f m+1,2m +,R,S . (3.14) Note however that the seminorms themselves are not invariant under complex conjugation f ↔ f , though the complex conjugation is continuous. We also note that the second part of the proposition already shows some nice continuity properties of the Wick star product, at least if one function is a polynomial. Note however, that the above argument will not extend to arbitrary f, g whence we shall need another route.

34


Theorem 3.9. The polynomials C[z, z] ⊆ A p, are dense. More specifically, the Taylor expansion ∞ 1 ∂ |I +J | f f (z, z) = ( p)(z − p) I (z − p) J (3.15) I !J ! ∂z I ∂z J I,J =0

of f ∈ A p, ⊆ C ω (Cn ) converges unconditionally to f with respect to the topology of A p,. In particular, the truncated Taylor polynomials f (N ,M) (z, z) =

N M 1 ∂ |I +J | f ( p)(z − p) I (z − p) J I !J ! ∂z I ∂z J

(3.16)

I =0 J =0

converge unconditionally to f in the topology of A p,. Proof. First recall that in infinite dimensional locally convex vector spaces, unconditional convergence of a series, i.e. independence of the order, is in general a strictly p, weaker statement than absolute convergence. First we rewrite the seminorms · m,,R,S in the following ‘measure-theoretic’ way ∞

p,

h m,,R,S ( f ) =

I1 ,J1 ,...,Is ,Js =0

|I +J | 2 |I +J | 2 ∂ 1 1 f s s f · · · ∂ , µm,,R,S, ( p) ( p) I1 ,···Is ,J1 ,···Js J J I I s 1 ∂z 1 ∂z ∂z s ∂z

m where µm,,R,S, I1 ,···Is ,J1 ,··· ,Js ≥ 0 are numerical constants not depending on p, and s = 2 . This

can be seen by induction easily. The concrete form of the coefficients µm,,R,S, I1 ,··· ,Is ,J1 ,··· ,Js ∞ is not important for the following argument. Now we define for f ∈ C (Cn ) a non-negative function φ p ( f ) : N2sn −→ [0, ∞) by

|I +J | 2 |Is +Js | 2 ∂ 1 1 f ∂ f ( p) · · · ( p) . φ p ( f )(I1 , . . . , Is , J1 , . . . , Js ) = J1 Js I I s 1 ∂z ∂z ∂z ∂z p,

Then we can interpret h m,,R,S ( f ) as the ‘integral’ of φ p ( f ) over N2sn with respect to the

weighted counting measure d µm,,R,S, determined by the coefficients µm,,R,S, I1 ,··· ,Is ,J1 ,··· ,Js , i.e.

p, φ p ( f ) d µm,,R,S,. h m,,R,S ( f ) = N2sn

Now let K, L ⊆ Nn be finite subsets and define the polynomial f (K,L) (z, z) =

1 ∂ |I +J | f ( p)(z − p) I (z − p) J . I !J ! ∂z I ∂z J I ∈K J ∈J

Then we clearly have φp

f − f

(K,L)

!

(I1 ,...,Is ,J1 ,...,Js )

=

0 if I1 , . . . , Is ∈ K, J1 , . . . , Js ∈ L φ p ( f ) otherwise. (3.17)


35

Thus when K, L exhaust Nn , the function φ p ( f − f (K,L) ) converges pointwise and monotonically to zero, i.e. for K ⊆ K and L ⊆ L we have φ p f − f (K,L) ≥ φ p f − f (K ,L ) , and for all I1 , . . . , Is , J1 , . . . , Js , lim φ p f − f (K,L) (I1 , . . . , Is , J1 , . . . , Js ) = 0 K,L→Nn

in the sense of net convergence for the net of finite subsets of Nn . Now an order of summation in (3.15) corresponds to a strictly increasing sequence Ki × Li ⊆ Nn × Nn which exhausts Nn × Nn . Then

p, lim h m,,R,S f − f (Ki ,Li ) = lim φ p f − f (Ki ,Li ) d µm,,R,S, = 0 i→∞

i→∞

by dominated convergence. But this is equivalent to the unconditional convergence of (3.15), see also [29, Sect. 14.6, Thm. 1]. We now come to the main result of this section: Theorem 3.10. The locally convex topology of A p, is complete, i.e. A p, is a Fréchet space. Proof. Since the topology is determined by countably many seminorms we only have to consider Cauchy sequences and not Cauchy nets. Thus let f i ∈ A p, be a Cauchy p, sequence, i.e. for all seminorms · m,,R,S and all > 0 we find a K (m, , R, S, ) such that for i, j ≥ K we have fi − f j

p, m,,R,S

< .

We first evaluate this for m = 1, = 1, R, S = 0. Let f i (z, z) =

∞ I,J =0

1 (i) a (z − p) I (z − p) J I !J ! I J (i)

be the Taylor expansion of f i then from (3.13) we see that the Taylor coefficients a I J form a Cauchy sequence for each I, J . Denote their limit by a I J = lim a (i) IJ i→∞

(∗)

and define f (z, z) =

∞ I,J =0

1 a I J (z − p) I (z − p) J . I !J !

(∗∗)

Then we want to show f ∈ A p, and f i → f . To this end we first choose a smooth function f˜ ∈ C ∞ (Cn ) with Taylor coefficients at p given by (∗), which is possible p, thanks to the Borel Lemma. Since f i is a Cauchy sequence with respect to · m,,R,S

36

S. Beiser, H. Römer, S. Waldmann p,

the sequence of seminorms f i m,,R,S stays bounded as i → 0. Thus we can again use the measure-theoretic point of view and write with the notation from the previous proof

p, h m,,R,S f˜ = φ( f˜) d µm,,R,S, 2sn N

= lim φ( f i ) d µm,,R,S, N2sn i→∞

= lim inf φ( f i ) d µm,,R,S, N2sn i ≤ lim inf φ( f i ) d µm,,R,S, N2sn m+1 p, sup( f i m,,R,S )2 i i

≤

< ∞,

p, and hence f as in (∗∗) is the unique real-analytic by Fatou’s Lemma. Thus f˜ ∈ A representative in A p,. Next we compute using (3.16), p,

f − f i m,,R,S ≤ ( f − f i ) − ( f − f i )(N ,M) +

(N ,M) fj

−

m,,R,S

(N ,M) p,

+ f (N ,M) − f j

m,,R,S

(N ,M) p, fi m,,R,S

≤ ( f − f i ) − ( f − f i )(N ,M) + fj −

p,

p, f i m,,R,S

p, m,,R,S

(N ,M) p,

+ f (N ,M) − f j

m,,R,S

.

Now we fix > 0 and K such that the last term is smaller than /3 for i, j > K . For such an i we fix N , M such that the first term is smaller than /3 thanks to Theorem 3.9. Finally, for this choice of N , M we can find j large enough that the second term is smaller than /3 since we have two polynomials of fixed degree (N , M) whose p, coefficients converge. This finally proves f i → f with respect to · m,,R,S . 4. The Continuity of Wick From Proposition 3.7 we know that on the subspace C[z, z] ⊆ A p, the Wick star product Wick is well-defined and continuous with respect to the topology of A p,. Since on the other hand C[z, z] is a dense subspace by Theorem 3.9 the Wick star product extends uniquely to a continuous product on A p, which thereby becomes a Fréchet algebra. However, from this abstract extension we cannot yet conclude whether the formula (2.3) with λ being replaced by is still true. Thus we need an additional argument. Let ∈ {0, . . . , 2m − 1} be written as = m−1 2m−1 + · · · + 1 2 + 0 with m−1 , . . . , 0 ∈ {0, 1}. Then we define = (−1)m−1 +···+0 . With this notation we can prove the following continuity property of the partial derivatives: Proposition 4.1. Let f ∈ C ∞ (Cn ) then we have ! p, p, ∂ |I +J | f

f m,,R+I,S+J for = +1 |I +J | (2) ≤ p,

f m,,R+J,S+I for = −1. ∂z I ∂z J m,,R,S

(4.1)


37

Proof. Clearly, for m = 0 (and hence = 0) we even have the equality ∂ |I +J | f (2)|I +J | ∂z I ∂z J

p, 0,0,R,S

p,

= f 0,0,R+I,S+J

p, by the very definition of · 0,0,R,S . Thus we prove the claim by induction on m. Let first be even and = +1. Then /2 = +1 as well and we have by induction

∂ |I +J | f |I +J | (2) ∂z I ∂z J 2 ∞ R N +S 1 R N +S p, ≤ K L h m−1,/2,K +I,L+J ( f ) N! N =0 K =0 L=0 2 R+I N +S+J ∞ 1 R N +S p, = K −I L−J h m−1,/2,K ,L ( f ) N! N =0 K =I L=J 2 R+I N +S+J ∞ 1 R+I N +S+J p, h m−1,/2,K ,L ( f ) ≤ K L N!

p, h m,,R,S

≤

N =0 K =I L=J p, h m,,R+I,S+J ( f ) . p,

For = /2 = −1 we get h m,,R+J,S+I ( f ) instead and the two cases with odd are analogous. From Theorem 3.9 we see that the polynomials ζ pI J (z, z) = (z − p) I (z − p) J

(4.2)

form a countable unconditional topological basis for A p,. The proposition now implies that we even have a Schauder basis: Corollary 4.2. The polynomials {ζ pI J } I,J ∈Nn form an unconditional Schauder basis for A p,. Proof. By Theorem 3.9 we have the unconditional convergence f =

∞ I,J =0

1 IJ δ ( f )ζ pI J , I !J ! p

and the (I, J )th derivative δ pI J of the δ-functionals are continuous linear functionals on A p, by Proposition 4.1. This implies the result, see e.g. [29, Sect. 14.2] for a definition of a Schauder basis. The next proposition shows that the pointwise product is continuous in the topology of A p,: Proposition 4.3. Let f, g ∈ C ∞ (Cn ). Then we have p,

p,

p,

f g m,,R,S ≤ f m+1,,R,S g m+1,,R,S , whence A p, is a Fréchet ∗ -algebra with respect to the pointwise product.

(4.3)

38


Proof. First recall that from the explicit form of the Wick star product we obtain |I +J | 2 f p, |I +J | ∂ (2) ∂z I ∂z J ( p) ≤ h 0,0,I,J ( f ) .

(∗)

Now we first consider the case m = 0. Here we have by the Leibniz rule p,

h 0,0,R,S ( f g)

2 R N +S |I +J | f ∂ |R−I +N +S−J | g R N +S ∂ ( p) I J I ∂z J ∂z R−I ∂z N +S−J ∂z N =0 I =0 J =0 R N +S |R−I +N +S−J | 2 ∞ ∂ (2)|N +R+S| R N +S ∂ |I +J | f g ≤ ∂z I ∂z J ( p) ∂z R−I ∂z N +S−J ( p) I J N! N =0 I =0 J =0 2 R N +S ∞ R N +S p, (∗) 1 p, ≤ h 0,0,I,J ( f ) h 0,0,R−I,N +S−J (g) I J N! N =0 I =0 J =0 R N +S R N +S ∞ R N +S p, 1 R N +S p, ≤ I J h 0,0,I,J ( f ) I J h 0,0,I,J (g) N! N =0 I =0 J =0 I =0 J =0 2 R N +S R N +S p, ∞ 1 ≤ I J h 0,0,I,J ( f ) N! N =0 I =0 J =0 2 R N +S R N +S p, ∞ 1 × I J h 0,0,I,J (g) N! N =0 I =0 J =0 p, p, = h 1,0,R,S ( f ) h 1,0,R,S (g), ∞ (2)|N +R+S| = N!

using twice the Cauchy Schwarz inequality in the last steps. For m ≥ 1 we proceed by a straightforward induction for the two cases of even and odd separately. Corollary 4.4. A p, is a Fréchet-Poisson ∗ -algebra. Proof. Clearly, the canonical Poisson bracket is continuous as { f, g} is the sum of pointwise products of partial derivatives of f and g. Combining the last two propositions we can finally show the continuity of the Wick star product: we even show that the series (2.3) converges in the topology of A p, if we replace λ by any complex number α: Theorem 4.5. The Wick star product f αWick g =

∞ (2α)|N | ∂ |N | f ∂ |N | g N! ∂z N ∂z N N =0

(4.4)

converges absolutely in the topology of A p, for all α ∈ C and gives a continuous associative product. If α = > 0 then A p, becomes a Fréchet ∗ -algebra with respect ∗ to Wick as the product and the complex conjugation as the -involution.


39

Proof. Let f, g ∈ A p, then we have for even and = +1, ∞ (2α)|N | ∂ |N | f ∂ |N | g N! ∂z N ∂z N N =0

m,,0,0

∞ |2α||N | ∂ |N | f ∂ |N | g N! ∂z N ∂z N N =0

≤ Prop. 4.3

≤

p,

∞ |2α||N | ∂ |N | f N! ∂z N

N =0 ∞ Prop. 4.1 | α ||N |

≤

N =0

N!

p, m,,0,0

p,

∂ |N | g

m+1,,0,0

p,

∂z N

p, m+1,,0,0

p,

f m+1,,N ,0 g m+1,,0,N

∞ α |N | ∞ α |N | 2 2 | | | | p, p,

f m+1,,N ,0

g m+1,,0,N ≤ N! N! N =0 N =0 ∞ 2 √ Prop. 3.3 | α ||N | 2m+3 p, ≤ N ! f m+2,2+1,0,0 N! N =0 ∞ α |N | 2 √ | | 2m+3 p, × N ! g m+2,2,0,0 N! N =0 ∞ | α ||N | 2m+2 √ p, p, = N ! f m+2,2+1,0,0 g m+2,2,0,0 . N! N =0 " #$ % =cm ( α )

Since cm ( α ) converges for all α ∈ C we have shown the convergence of (4.4) with p,

respect to · m,,0,0 for even and = +1. The other three cases are shown analop,

gously. As the topology of A p, is already determined by the seminorms · m,,0,0 the absolute convergence in the topology of A p, follows. From the above estimate (and the analogous ones for odd , etc.) one also obtains the continuity of αWick . If α = is real, then the complex conjugation is a ∗ -involution showing the last statement. Corollary 4.6. The Wick star product αWick is a holomorphic deformation of the pointwise product in the sense of [43]. Corollary 4.7. For f, g ∈ A p, we have f Wick g

p, m,,R,S

p,

p,

≤ f m+1,2m +,R,S g m+1,,R,S .

(4.5)

Proof. This follows from Proposition 3.7, the density of C[z, z] in A p, and the continuity of Wick .

40


Though A p, becomes a Fréchet ∗ -algebra, the topology is not locally m-convex in the sense of [37], see also [43, App. A]. Recall that a locally convex algebra is called locally m-convex if there exists a set of seminorms · i defining the topology such that ab i ≤ a i b i . Such locally m-convex algebras always have a holomorphic calculus. Recall that a entire holomorphic calculus means that for any algebra element a and any entire holomorphic function f (a) is well-defined as convergent series. This property fails for A p,: Example 4.8. We consider the entire function f ∈ O(C) defined by f (z) =

∞ zr . √ 4 r! r =0

(4.6)

Then it is easy to see that f Wick f evaluated at z = 0 converges only for = 0. Since clearly the δ-functional δ p : A p, −→ C is continuous we conclude that f ∈ A0,. This shows that A0, does not allow a holomorphic functional calculus, as z ∈ A0, and the Wick -Taylor expansion of f would again coincide with f since the Wick -power of z coincides with the corresponding pointwise powers. Analogous arguments apply also for p = 0 and higher dimensions n ≥ 1. Corollary 4.9. The topology of A p, is not locally m-convex with respect to the Wick star product αWick for all α and there is no general holomorphic calculus for A p,. In the following we shall equip A p, always with the Wick star product Wick . Remark 4.10. At this point it would be interesting to compare our algebra to the construction obtained in [40]: Here the authors consider the Weyl-Moyal star product, which on the formal level is known to be equivalent to the Wick star product, and establish a convergence scheme to obtain a certain Fréchet algebra as the completion of the polynomials. However, their construction is rather different from ours whence it seems difficult to investigate whether the usual formal equivalence transformation survives the convergence conditions. 5. Translations and Rescalings We shall now discuss the dependence of A p, on the point p ∈ Cn and on the value > 0. We start with the dependence on the point p. Let α ∈ Cn then for f ∈ A p, we define

and

(τα f )(z, z) = fˆ(z + α, z)

(5.1)

(τ α f )(z, z) = fˆ(z, z + α),

(5.2)

which is well-defined according to Theorem 3.6, Part iii). Moreover, we consider the functions eα,β (z, z) = eαβ eαz+βz , (5.3) which are elements in A p, according to Theorem 3.6, Part iv). The following lemma is a simple computation, the results of which are well-known in the case of the formal Wick star product:


41

Lemma 5.1. Let α, β, γ , δ ∈ Cn . Then we have for all f ∈ A p,: (αδ−βγ ) e 1. eα,β α+γ ,β+δ . Wick eγ ,δ = e 2. eα,β Wick f = eα,β τ 2α f . 3. f Wick eα,β = eα,β τ2β f . 4. The maps τα and τ α are continuous linear bijections

τα , τ α : A p, −→ A p,.

(5.4)

Proof. The only non-trivial point here is that for f ∈ A p, we have τα f =

∞ α N ∂ |N | f , N ! ∂z N

N =0

and analogously for τ α since f has a extension to fˆ ∈ O×O(Cn × Cn ). Then the computations for the first three parts are folklore. The last part follows from α τα f = e0,− 2

α f Wick e0, 2

(∗)

and the continuity of the pointwise multiplication as well as of the continuity of Wick . The same argument applies for τ α . From (∗) one can easily work out explicit estimates p, p, for τα f m,,R,S and τ α f m,,R,S using Proposition 4.3 and Corollary 4.7. Corollary 5.2. Let α, β ∈ Cn . 1. eα,β ∈ A p, is invertible with respect to Wick with inverse given by e−α,−β . 2. eα,β is unitary iff α = −β since in general eα,β = eβ,α . We introduce now the following notation. For w ∈ Cn we denote the translation by w by Tw (z) = z + w whence we have the corresponding pull-back on functions (Tw∗ f )(z) = f (z + w). Moreover, we set uw = e

1 1 2 w,− 2 w

∈ A p,,

(5.5)

which is a unitary element of A p, according to Corollary 5.2. Proposition 5.3. The translation group Cn acts via pull-backs by continuous inner ∗ automorphisms Tw∗ = AdWick (u w ) (5.6) on A p,. Proof. This is a simple consequence of Lemma 5.1 and the fact that u w is unitary. Remark 5.4. We remark that on a heuristic level (or on polynomial functions only) the statement of this proposition is folklore. Note that it is clear that for the formal Wick star product the statement is wrong: the translations are only outer automorphisms as the elements u w are not well-defined as formal series in λ. In fact, it was one of our main motivations to find a reasonably large algebra where the statement of the proposition is still true, extending the polynomials.

42


The fact that the translations act by inner automorphisms will immediately imply the following result: Theorem 5.5. Let p, p ∈ Cn . Then A p, = A p ,

(5.7)

as Fréchet ∗ -algebras. Proof. Let f ∈ A p, be given and w = p − p. Then we have on one hand p ,

f m,,R,S = Tw∗ f

p, m,,R,S

0 and the continuity follows p, from the theorem as |δ p ( f )| ≤ f 0,0,0,0 . Corollary 5.7. The Fréchet topology of A is finer than the topology of pointwise convergence.


43

In the next step we want to analyze the continuity properties of the group representation w → Tw∗ further. To this end we consider the dependence of the elements u w on w. Since i − 2 Im(wv) u w u w+v , (5.9) Wick u v = e i

the map w → u w is not a group morphism. Since e− 2 Im(wv) is even a non-trivial group cocycle we need to pass to the central extension of the translation group Cn by this cocycle, i.e. to the Heisenberg group Hn . Here we use the convention that Hn = Cn × R with multiplication law (w, c) · (w , c ) = (w + w , c + c + Im(ww )).

(5.10)

Then it follows that i

Hn (w, c) → u (w,c) = e− 2 c u w ∈ U(A)

(5.11) i

is a group morphism from Hn into the group of unitaries U(A) in A. Since e− 2 c is central, (5.11) factors to the group morphism w → Tw∗ , i.e. we have AdWick (u (w,c) ) = AdWick (u w ) = Tw∗

(5.12)

for all (w, c) ∈ Hn . The Lie algebra hn of Hn can be identified with Hn via the exponential map. Then the Lie bracket is given by [(w, c), (w , c )] = (0, 2 Im(ww )). The next theorem shows that the group morphism Hn −→ U(A) is analytic and induces a Lie algebra morphism hn −→ A: Theorem 5.8. 1. The map hn ∼ = Hn (w, c) → u (w,c) is analytic with respect to the topology of A. 2. The generator J(w,c) of the one-parameter group t → u (tw,tc) is given by 1 d i (wz − wz), J(w,c) (z, z) = u (tw,tc) = − c + (5.13) d t t=0 2 2 and hn (w, c) → J(w,c) ∈ A is a Lie algebra morphism where A is equipped with the Wick -commutator as Lie bracket. 3. We have for all k, dk u (tw,tc) = J(w,c) · · · J(w,c) , (5.14) "Wick #$ Wick% d t k t=0 k times

and explicitly for c = 0, k/2 dk k! ww wz − wz k−2 − u (tw,0) = . d t k t=0 !(k − 2)! 4 2

(5.15)

=0

i

ww

1

i

ww

Proof. We have u (w,c) (z, z) = e− 2 c− 4 + 2 (wz−wz) . The first factors e− 2 c and e− 2 are clearly analytic as they are analytic functions times a fixed element (the identity) in A. For the remaining factor we see that the (z, z)-Taylor expansion, which converges unconditionally in the topology of A coincides up to numerical factors with the (w, w)Taylor expansion of this function. Thus the (w, w)-Taylor expansion converges also in A unconditionally which shows the first part. The second part is a trivial computation.

44


For the third part, the contribution of c is not essential whence we discuss the case c = 0 only. A straightforward computation of the Taylor expansion in t gives immediately (5.15). On the other hand, the kth power of the linear function Jw = J(w,0) satisfies the recursion formula Jwk = Jw Jw(k−1) + (k − 1)

ww (k−2) J , 2 w

which follows easily from the fact that partial derivatives are derivations of Wick and ∂ ww w ∂z Jw = 2 is central. In a last step one shows that also the right-hand side of (5.15) satisfies this recursion with the same initial conditions for k = 0, 1. Corollary 5.9. Let w ∈ Cn then the Wick -exponential function Exp (t Jw ) =

∞ k t k=0

k!

Jw · · · Jw = u tw "Wick #$ Wick%

(5.16)

k times

converges unconditionally in the topology of A for all t ∈ R. Remark 5.10. Again, the importance of this corollary is not the explicit computation of the star exponential which is folklore. Instead, we have found a well-defined analytic framework where the formula actually converges inside an algebra of functions. Note also that in general we cannot expect such a convergence as A does not allow a holomorphic functional calculus in general. Let us now discuss the dependence on the parameter . From a simple dimensional analysis we see that with our convention for the Poisson bracket {z k , z } = 2i δ k the 1

coordinates have to have the physical dimension [action] 2 . Thus a rescaling of , which physically is of course absurd, has to be reinterpreted as a rescaling of the coordinates (z, z): we are not changing the value but the unit system. On the other hand, from a purely mathematical point of view we cannot distinguish these two interpretations. As we do not have any additional absolute scale in our approach, the corresponding algebras A and A should be isomorphic in order to be physically reasonable. The following theorem will show that this is indeed the case. We define for α > 0 the diffeomorphism Rα : Cn −→ Cn , Rα (z) =

√ αz,

(5.17)

whose inverse is R 1 . α

Theorem 5.11. The pull-back Rα∗ induces an isomorphism of Fréchet ∗ -algebras Rα∗ : Aα −→ A.

(5.18)

In particular, we have for all f ∈ Aα , Rα∗ f

0, m,,R,S

= f 0,α m,,R,S .

(5.19)


45

Proof. Let f ∈ Aα be given. Then we have ∂ |I +J | (Rα∗ f ) ∂z I ∂z J

=

√

α

|I +J |

Rα∗

∂ |I +J | f ∂z I ∂z J

,

and thus Rα∗ f

0, 0,0,R,S

∞ (2)|N +R+S| N!

=

N =0 ∞

=

N =0

|N +R+S| ∗ 2 ∂ (Rα f ) (0) ∂z R ∂z N +S

(2α)|N +R+S| N!

|N +R+S| 2 ∂ f (0) ∂z R ∂z N +S

= f 0,α 0,0,R,S .

Since the higher seminorms are constructed in a purely combinatorial way out of ∗

· 0, 0,0,R,S , we can conclude (5.19). This shows that Rα : Aα −→ A is an isomorphism of Fréchet spaces as we can exchange the role of and α by passing from α to α1 . Then it is easy to see that Rα∗ is an algebra morphism: we use the convergence of (4.4) and the continuity of Rα∗ to obtain ∞ (2α)|N | ∂ |N | f ∂ |N | g Rα∗ ( f αWick g) = Rα∗ N! ∂z N ∂z N N =0

∞ (2)|N | ∂ |N | (Rα∗ f ) ∂ |N | (Rα∗ g) = N! ∂z N ∂z N

=

N =0 Rα∗ f

∗ Wick Rα g.

The compatibility with the complex conjugation Rα∗ f = Rα∗ f is obvious. Remark 5.12. Of course we can also directly compare the seminorms for different values of . Clearly, one has 0,

f 0, (5.20) 0,0,R,S ≤ f 0,0,R,S for ≤ whence by induction

0,

f 0, m,,R,S ≤ f m,,R,S

(5.21)

as well. For ≤ this gives the inclusion A ⊆ A.

(5.22)

6. The GNS Construction and Coherent States We shall now discuss the GNS construction corresponding to the positive δ-functionals δ p : A −→ C, now in the convergent situation.

(6.1)

46


Proposition 6.1. Let p ∈ Cn . Then the Gel’fand ideal of δ p is given by ∂ |I | f ( p) = 0 , (6.2) J p = f ∈ A ∀I : ∂z I and the GNS pre-Hilbert space D p = A J p is a Fréchet space in the natural way, where the topology of D p is determined by the seminorms p, p,

[ f ] m,,R,S = inf f + g m,,R,S g ∈ J p . (6.3) Proof. The statement (6.2) is obvious. Since δ p is continuous, the Gel’fand ideal is a closed subspace of A. Thus the quotient D p is again a Fréchet space by general arguments, see e.g. [29, Sect. 4.4, Prop. 1]. Since the translation group acts by inner ∗ -automorphisms we can safely specialize to the case p = 0 in a first step. Then we can describe the quotient D0 more explicitly: Theorem 6.2. Let f, g ∈ A. 1. The z-Taylor expansion : A −→ A defined by ∞ 1 ∂ |I | f I (0)z : f → z → f (z) = I ! ∂z I

(6.4)

I =0

is a continuous projection with ker = J0 . In fact, f

0, m,,R,S

(6.5)

≤ f 0, m,,R,S ,

(6.6)

where equality holds if and only if f is anti-holomorphic. 2. The quotient D0 is canonically isomorphic as a Fréchet space to the image D = im of via D0 [ f ] → f ∈ D. (6.7) In particular,

[ f ] 0, m,,R,S = f

0, m,,R,S

.

(6.8)

3. The space D is a dense subspace of the Bargmann-Fock Hilbert space HBF . The map (6.7) is an isometry of pre-Hilbert spaces and the Fréchet topology of D is finer than the topology induced from the Hilbert space HBF . 4. The GNS representation on D0 induces via (6.7) the Bargmann-Fock representation of A on D, explicitly given by ∞ ∞ |I | (2)|I | 1 ∂ |I +J | f J ∂ g (0)z , (6.9) π( f )g = J I! J ! ∂z I ∂z ∂z I I =0

J =0

where both series converge in the topology of D. 5. The bilinear map A × D ( f, g ) → π( f )g ∈ D is continuous with respect to the Fréchet topologies of A and D, respectively.


47

Proof. For the first part we have f (z) = fˆ(0, z) whence f is indeed a well-defined anti-holomorphic function. Moreover, (6.6) is clear from our consideration in the proof of Theorem 3.9 since in the seminorm of f simply less Taylor coefficients contribute compared to the corresponding seminorm of f . Thus f ∈ A and is continuous. From the explicit form of the equality (6.5) and 2 = are obvious. For the second part we first notice that (6.7) is well-defined and bijective since f − f ∈ ker = J0 , using the fact that is a projection. Moreover, (6.8) follows directly from (6.6) since 0, obviously [ f ] 0, m,,R,S ≤ f m,,R,S . Since the inverse of (6.7) is simply given by the well-defined and continuous map f → [ f ], we see that (6.7) is indeed an isomorphism of Fréchet spaces. For the third part we compute explicitly the GNS inner product on D0 , [ f ], [g] D0 = δ0 ( f Wick g) =

∞ (2)|N | ∂ |N | f ∂ |N | g (0) (0) N N! ∂z N ∂z N =0

∞ ∂ |N | g (2)|N | ∂ |N | f (0) (0) N N! ∂z ∂z N N =0 & ' = f , g BF ,

=

whence (6.7) is isometric. In particular D ⊆ HBF follows, as &

f,f

'

0, 2 = ( f Wick f )(0) = ( f 0,0,0,0 ) < ∞.

BF

(∗)

Since C[z] ⊆ D ⊆ HBF , the subspace D is dense in HBF . Moreover, since f 0, 0,0,0,0 = 0,

f 0,0,0,0 the estimate (∗) implies that the Fréchet topology of D is finer than the topology induced from HBF . This shows the third part. For the fourth part recall that the GNS representation is defined by ( f )[g] = [ f Wick g] which translates via (6.7) into π( f )g = f

Wick g

= (∞

N =0

(a)

=

(2)|N | ∂ |N | f ∂ |N | g N! ∂z N ∂z N

∞ (2)|N | ∂ |N | f N! ∂z N

N =0 ∞

∂ |N | g ∂z N

(2)|N | ∂ |N | f ∂ |N | g N! ∂z N ∂z N N =0 ∞ ∞ |N | (2)|N | 1 ∂ |N +M| f g M ∂ = (0) z , N N! M! ∂z N ∂z M ∂z (b)

=

N =0

M=0

where (a) holds since is continuous and (b) holds since obviously is a homomorphism of the pointwise product. Finally, in the last step we have used that commutes with derivatives in the z-direction. This shows the fourth part and the last part follows immediately from π( f )g = f g = f g and the continuity of and Wick . Wick Wick

48


Corollary 6.3. The Bargmann-Fock representation of A is injective. The following corollary is remarkable in so far as closed subspaces of Fréchet spaces usually do not have complementary closed subspaces: Corollary 6.4. The algebra A decomposes into two complementary closed subspaces A = D ⊕ J0 . Remark 6.5. Since the Bargmann-Fock representation is injective and since π(z i ) = 2

∂ ∂z i

= ai ,

π(z i ) = z i = ai†

(6.10) (6.11)

are the annihilation and creation operators we find another interpretation of the algebra A: it is a (rather large) completion of the polynomials in the creation and annihilation operators in a certain Fréchet topology. In particular, this completion contains the usual unitary generators of the Weyl algebra, i.e. the exponential functions of ai and ai† . They are given by π(eα,β ) for suitable α, β ∈ Cn . Remark 6.6. Since the Bargmann-Fock representation is a ∗ -representation of the ∗ -algebra A by (in general) unbounded operators with common domain D, one can investigate the resulting O ∗ -algebra of unbounded operators by techniques as developed in e.g. [45]. In particular, it would be interesting to find more concrete characterizations of the Fréchet topologies of D and A. We now discuss the action of the translation group Cn and its central extension Hn . The following statement is obvious, the representation itself being well-known: Lemma 6.7. The map Hn (w, c) → U(w,c) = π(u (w,c) ) ∈ U(HBF )

(6.12)

is a strongly continuous unitary representation of the Heisenberg group. Explicitly, i

ww

wz

(U(w,c) ψ)(z) = e− 2 c− 2 e− 2 ψ(z + w)

(6.13)

for ψ ∈ HBF . It factors to a projective representation Uw = U(w,0) of the translation group Cn . Proof. Since π is a ∗ -representation it follows that π(u (w,c) ) is a unitary operator defined on the dense domain D. Thus it extends to a unitary operator on HBF . The group representation property is obvious from (5.11). The explicit formula is a simple consequence of (6.9). The fact that (6.12) is strongly continuous is well-known& but can also' be shown within our approach directly: let φ, ψ ∈ D, then g(w, c) = ψ, U(w,c) φ BF is realanalytic since ·, · BF and π are continuous with respect to the Fréchet topology and (w, c) → u (w,c) is real-analytic according to Theorem 5.8. Since g(0, 0) = ψ, φ BF we see that on the dense domain D the representation (6.12) is weakly continuous at the identity. But this implies that it is strongly continuous on the whole group Hn and on the whole Hilbert space HBF . Since the contribution of c is only an overall phase, the representation clearly factors to a projective representation of Cn .


49

Since we have a group action of Hn we can formulate now the following covariance property of the Bargmann-Fock representation which is an obvious consequence of the fact that the ∗ -automorphisms are inner. Theorem 6.8. The Bargmann-Fock representation is Hn -covariant with respect to the action by ∗ -automorphisms on A and the action by unitaries on HBF , i.e. we have ∗ = Uw π( f )Uw∗ π(AdWick (u (w,c) ) f ) = U(w,c) π( f )U(w,c)

(6.14)

for all (w, c) ∈ Hn and f ∈ A. Since the GNS representation is cyclic with cyclic vector 1 = 1 we obtain coherent states with respect to the representation of Hn . We define the coherent state vector ψ(w,c) ∈ HBF by −1 ψ(w,c) = U(w,c) ψ1 (6.15) explicitly given by i

ww

wz

ψ(w,c) (z) = e 2 c+ 4 e 2 .

(6.16)

From the covariance property (6.14) we immediately have the following characterization of the δ-functionals at arbitrary points in Cn : Corollary 6.9. The coherent state vectors give the δ p -functionals as expectation value functionals, i.e. we have & ' (6.17) δw ( f ) = ψ(w,c) , π( f )ψ(w,c) BF for all f ∈ A and (w, c) ∈ Hn . In particular, the group action of Hn on the coherent state vectors factors through to a group action of the translation group Cn on the coherent states δw : A −→ C. Remark 6.10. This corollary gives finally the justification to view the δ-functionals of the Wick star product algebra as coherent states with respect to the translation group. Though the explicit formula (6.16) is folklore (it is just the Bergmann kernel) we would like to emphasize that in our approach the coherent states emerge out of properties of the observable algebra instead of more conventional approaches based on group actions on the state vectors in some Hilbert space, see e.g. [42, 48] for more references. In this sense our approach supports the idea that the observable algebra is the more fundamental object in both quantum and classical mechanics. Remark 6.11. We also note that the statement of Theorem 6.8 as well as the Corollary 6.9 are not possible for the formal Bargmann-Fock representation. Again, this was one of our main motivations to consider a suitable convergence scheme for the Wick star product. The result of Theorem 6.8 and Corollary 6.9 suggest the following general definition of coherent states with respect to some symmetry based on the observable algebra: Definition 6.12. Let A be a ∗ -algebra with unit 1 and let G be a group acting on A by g : A −→ A. Let ω be a state of A such that the GNS representation is G-covariant, i.e. there exists a unitary (or more general: projectively unitary) representation U of G on the GNS pre-Hilbert space Hω . Then the states ωg with ' & (6.18) ωg (a) = (ω ◦ g )(a) = ψg , π(a)ψg ,

∗ -automorphisms

where ψg = Ug∗ ψ1 ∈ Hω are called coherent with respect to G.

50


Clearly, in case of a projective representation only the coherent states ωg are welldefined, while for a unitary representation also the coherent state vectors ψg are welldefined. With the Heisenberg group acting on A and the δ-functional we are in this situation: if the action is realized by inner ∗ -automorphisms then the representation is always covariant for a (in general only projective) representation on the GNS pre-Hilbert space. Note also, that for an invariant state ω the GNS representation is trivially covariant. In this case ψg = ψ1 coincides with the vacuum vector for all g ∈ G. Thus the interesting coherent states arise from non-invariant vacua such that the GNS representation is nevertheless covariant. For a survey on covariant ∗ -representation theory for ∗ -algebras over ordered rings we refer to [28]. Let us now come to a further property of the subspace D ⊆ HBF . We have already seen that the action of Hn leaves D invariant. Theorem 6.13. The vectors in D are analytic with respect to the unitary representation U of Hn . Proof. Let ψ ∈ D be fixed, then we have to show that Hn (w, c) → U(w,c) ψ is analytic with respect to the topology of HBF . But this is simple since U(w,c) ψ = π(u (w,c) )ψ and (w, c) → u (w,c) is analytic in the topology of A. Moreover, the bilinear map ( f, ψ) → π( f )ψ is continuous in the topologies of A and D. Thus the map (w, c) → U(w,c) ψ is analytic with respect to the topology of D. Since by Theorem 6.2 this topology is finer than the one of HBF , the proof is complete. Thus it would be interesting to know whether D coincides with the space of all analytic vectors. A positive answer would help to understand the (still rather complicated) Fréchet topology of D and hence the one of A. Using the convergence of the star exponential (5.16) we even can specify the analyticity of the vectors in D further. Since π is continuous with respect to the topologies of A and D, we have ∞ r 1 π J(w,c) ψ U(w,c) ψ = π(u (w,c) )ψ = π Exp(J(w,c) ) ψ = r!

(6.19)

r =0

with respect to the topology of D. Again, since the topology of D is finer than the one of HBF , the series converges unconditionally also in the Hilbert space sense. As usual, the contribution of c is not essential. Corollary 6.14. Let ψ ∈ D and w ∈ Cn . Then the series r n ∞ 1 1 r i i † w ai − w ai ψ Uw ψ = r ! 2 r =0

(6.20)

i=1

converges unconditionally in the Hilbert space topology. Of course we can also rewrite this in terms of the position and momentum operators 1 1 (ak + ak† ) = π(z k + z k ) 2 2

(6.21)

1 1 (ak − ak† ) = π(z k − z k ), 2i 2i

(6.22)

Qk = and Pk =


51

defined as unbounded symmetric operators on D. Then (6.19) shows that for ψ ∈ D we have the unconditionally convergent series ∞ i p· Q 1 ip · Q r e ψ= ψ (6.23) r! r =0

and e

iq· P

∞ 1 iq · P r ψ= ψ r!

(6.24)

r =0

in the Hilbert space topology, for all q, p ∈ Rn , substituting w suitably. Of course, this also follows by ‘Hilbert space techniques’ from the strong continuity of the representation U and Theorem 6.13. Note however, that the above argument using the convergence of the star exponential is independent. Acknowledgement. We would like to thank Pierre Bieliavsky, Martin Bordemann, Nico Giulini, Simone Gutt, Nikolai Neumaier, Konrad Schmüdgen, Martin Schlichenmaier and Rainer Verch for valuable discussions on this example. Moreover, S. W. thanks the ULB for hospitality while part of this work was being done.

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Communications in


Continuity of Information Transport in Surjective Cellular Automata Torbjørn Helvik1 , Kristian Lindgren2 , Mats G. Nordahl3 1 Department of Mathematical Sciences, Norwegian University of Science and Technology,

NO-7491 Trondheim, Norway. E-mail: [email protected]

2 Department of Physical Resource Theory, Chalmers University of Technology and Göteborg University,

SE-41296 Göteborg, Sweden

3 Department of Applied Information Technology, Chalmers University of Technology and Göteborg

University, SE-41756 Göteborg, Sweden Received: 21 July 2005 / Accepted: 4 September 2006 Published online: 2 March 2007 – © Springer-Verlag 2007

Abstract: We introduce a local version of the Shannon entropy in order to describe information transport in spatially extended dynamical systems, and to explore to what extent information can be viewed as a local quantity. Using an appropriately defined information current, this quantity is shown to obey a local conservation law in the case of one-dimensional reversible cellular automata with arbitrary initial measures. The result is also shown to apply to one-dimensional surjective cellular automata in the case of shift-invariant measures. Bounds on the information flow are also shown. 1. Introduction A number of authors have suggested that information should be viewed as a fundamental physical quantity, starting with the vision of “It from Bit” of Wheeler [27] and the fundamental work on the thermodynamics of computation by Landauer [14] and Bennett [1]. Information theory also has a close relation to the foundations of statistical mechanics, e.g., through the information theoretic formulation introduced by Jaynes [11], where entropy is viewed as a measure of the ignorance of the actual microstate of the system. Information theory and computation theory can also be used to define an entropy for individual microstates in spatially extended systems [17, 28]. In a microscopic view, information or entropy quantified in terms of the Gibbs H-function is a globally conserved quantity due to Liouville’s theorem. A natural question to consider is to what extent this statement has a local analogue in spatially extended dynamical systems. This article explores this question for one-dimensional reversible or surjective cellular automata. Precise statements of the notion of conservation of information, and possible extensions of this formalism to other systems, could provide a more solid foundation for the use of information based concepts in different physical systems. We first introduce a local version of the Shannon entropy. In a one-dimensional system, the local information is defined in terms of the conditional probability of a local

54

T. Helvik, K. Lindgren, M. G. Nordahl

state given its left or right infinite context. Information can only be completely localized in a system without correlations. Thus, the measure we introduce is localized to the extent that correlations allow, and reduces to a completely local quantity when correlations vanish. However, even with correlations present, this quantity does obey a local continuity equation with an appropriately defined information current. For cellular automata, local conservation of information was first proposed by Toffoli [23], who derived a continuity equation for information transport in the case of small perturbations around the uncorrelated equlibrium states of particle conserving reversible cellular automata, such as lattice gases. Here we investigate how these concepts can be applied to a wider class of dynamical systems and to arbitrary measures, and how they can be given a rigorous formulation. We only consider one-dimensional systems; generalizations to systems in higher dimensions will be addressed in future work. We first consider reversible cellular automata, where the cellular automaton mapping has an inverse. Reversible cellular automata have been used to simulate physical systems, e.g., for microcanonical simulations of spin systems (e.g., [25]), and simulations of fluid dynamics [4, 7], and chemical reactions [2]. They have also been used as illustrative examples of fundamental issues in statistical mechanics [21, 22]. For one-dimensional reversible cellular automata, we show local conservation of information for any initial measure, including measures without shift-invariance. We also consider the more complicated case of surjective cellular automata, where the global mapping is finite-to-one [8]. In this case, local conservation of information is shown for all shift-invariant measures. For the simple case of permutative rules, we are able to describe the information flow in more detail. The aim of the article is to explore exactly to what extent information can be viewed as a local quantity in spatially extended systems. The main results show that important aspects of locality remain also in systems with correlations. We also give examples which illustrate the limits of locality in the formalism. The rest of this article is organized as follows. Section 2 contains background material on shift spaces and cellular automata. In Sect. 3 we introduce a local measure of information and show that it is well-defined. Section 4 contains the main results of the paper. We first define the information current, and prove that information is locally conserved for one-dimensional reversible cellular automata. We then extend this result to surjective cellular automata. In Sect. 5, we give an information theoretic interpretation of the current, and provide bounds on the information flow. We also characterize the information flow in permutative cellular automata, and study some examples illustrating the limits of locality. Section 6 contains conclusions and a discussion. 2. Preliminaries 2.1. The shift space. We study dynamical systems on the space AZ of all bi-infinite symbol sequences over a finite set A. For x ∈ AZ we write x = (xi )i∈Z . The length j j − i + 1 block (xi , xi+1 , . . . , x j ) of symbols from A will be written as xi . Likewise, i x−∞ = (. . . , xi−1 , xi ). The shift map σ is defined on AZ by σ (x)i = xi+1 . A probability measure µ on (AZ , B), where B is the Borel σ -algebra, is defined by assigning a probability µ(Cyl(aii+n )) to each cylinder set Cyl(aii+n ) = {x ∈ AZ : xii+n = aii+n } in a consistent way, see [26, §0.2]. We will usually write this probability µ(aii+n ), thus letting aii+n represent both the symbol block of length n +1 and the cylinder set. It is often convenient to consider the measure µ as defining a discrete, stochastic

Continuity of Information Transport in Surjective Cellular Automata

55

process (X n )∞ n=−∞ , X n ∈ A, with joint distributions given by Prob(X i = ai ) = µ(ai ). A measure is said to be Bernoulli if the coordinate random variables X i are all indepenj

−1 )= dent and identically distributed. The conditional probability µ(a0 |a−n −1 X −n

−1 a−n .

j

0 ) µ(a−n −1 µ(a−n )

j

is the

= probability that X 0 = a0 given that The measure µ is shift-invariant if it satisfies µ(σ −1 (B)) = µ(B) for all cylinder sets B. When µ is shift-invariant, the expectation E[ f ] of any measurable function f on AZ satisfies E[ f ] = E[ f ◦ σ ]. The Shannon entropy h(µ) of a shift-invariant measure µ can be written as h(µ) = − lim

n→∞

−1 0 µ(a−n ) log µ(a0 |a−n ).

(1)

0 ∈An+1 a−n

2.2. Cellular automata. One-dimensional cellular automata (CA) are discrete dynamical systems on AZ that commute with the shift σ . Definition 1. A cellular automaton F : AZ → AZ is a dynamical system that can be defined by non-negative integers l, r and a map f : Al+r +1 → A, such that (F x)i = f (xi−l , xi−l+1 , . . . , xi+r ) ∀ i ∈ Z.

(2)

The left and right radii of F are the smallest such integers l and r for which there is a block map f (CA rule) that generates F. Example 1. Let A = {0, 1}, and denote by F1 the simple CA on AZ defined by the radii l = 0 and r = 1 and the block map f : A2 → A given by f (x0 , x1 ) = x0 + x1 (mod 2). The global map F1 can be written as F1 (x) = x + σ (x), where addition is coordinate-wise and modulo 2. For any n ≥ 1, the block map f can be extended in a natural way to a map f n : Al+r +n+1 → An+1 by putting r +n r r +1 r +n f n (x−l ) = ( f (x−l ), f (x−l+1 ), . . . , f (xn−l )).

(3)

We will omit the subscript n and write f for the block map applied to a block of any length. For reversible CA, F has an inverse map, so that each bi-infinite sequence y ∈ AZ has exactly one preimage under F. The inverse map of a reversible CA is always itself a CA [20], but the inverse CA does not necessarily have the same radii as F (see, e.g., [24]). Example 2. Denote by F2 the reversible CA on {0, 1, 2}Z having radii l = 0, r = 1 and block map given by f (10) = f (11) = f (12) = 0, f (01) = f (20) = f (22) = 1 and f (00) = f (02) = f (21) = 2. The preimage x of a given y ∈ AZ is found by the following procedure. If yi = 0 we must have xi = 1. If yi = 1 then xi = 2 unless yi+1 = 0, in which case xi = 0. Finally, if yi = 2 then xi = 0 unless yi+1 = 0, in which case xi = 2. The inverse CA F˜2 also has l = 0 and r = 1, but a different block map f˜.

56


A more general class of CA are the surjective ones, where all y ∈ AZ have at least one preimage. The class of surjective CA includes all linear CA [10] and other permutative CA [8]. It is well known that a one-dimensional CA F is surjective if and only if all finite blocks have the same number of pre-images under f [8]. That is, if for all n ≥ 1 and n+r ∈ Al+r +n that satisfy f (z n+r ) = y n . y1n ∈ An there are exactly |A|l+r blocks z 1−l 1 1−l Furthermore, there is a constant M(F) ≤ |A|l+r such that each bi-transitive x ∈ AZ has exactly M(F) preimages. For surjective CA one can define Welch coefficients. Let x1n ∈ An with n ≥ l + r . A compatible right extension of x1n of length m is a collection B ⊂ Am such that for each z 1m ∈ B, the (n+m−l−r )-block f (x1n z 1m ) is the same. Define the integer R(F) as the maximal number of elements in any compatible right extension of any length m and of any block x1n . Define compatible left extensions and L(F) in the same way. The coefficients L(F) and R(F) are finite, and satisfy the relation L(F) · M(F) · R(F) = |A|l+r [8, Th. 14.9]. 3. Local Information The intent of introducing a local information quantity is to measure how much information that is located at each position of an infinite symbol sequence generated by some stochastic process. However, the correlations in such symbol sequences can in general be arbitrarily long, and it is impossible for information to be completely localized. The natural approach is therefore to define the local information as a limit which converges to a local analogue of the Shannon entropy as more and more distant neighbours are taken into account. While the Shannon entropy is limited to shift-invariant measures, we can define left local information for any measure. Definition 2. Let µ be a measure on AZ . The left local information at coordinate i of x ∈ AZ with respect to µ is given by i−1 SL (x; i; µ) = − lim log µ(xi |xi−n ). n→∞

(4)

i−1 The quantity − log µ(xi |xi−n ) is the information gained from the symbol at position i when only knowledge of the n left symbols is assumed. If, and only if, µ is Markov there i−1 is a fixed n such that SL (x; i; µ) = − log µ(xi |xi−n ). We will often use the intuitive i−1 notation − log µ(xi |x−∞ ) for SL (x; i; µ). The following theorem ensures that the left local information with respect to µ is a well-defined function on the probability space (AZ , B, µ). i−1 ) converges µ-almost everywhere and in Theorem 1. For each i ∈ Z, − log µ(xi |xi−n L 1 (µ). Consequently, for each fixed measure µ, SL (x; i; µ) ∈ L 1 (µ).

The validity of the theorem follows from the martingale convergence theorem, see, e.g., [12, Th. 3.1.10]. Note that from L 1 -convergence and (1) it follows that E[SL (x; i; µ)] = h(µ) for all i in the shift-invariant case. The local information SL (x; i; µ) depends on the measure µ. However, for a shiftinvariant measure µ the left local information at position i of x can be recovered with i probability one from x−∞ only by considering the empirical measure νx obtained from i , the frequencies of finite blocks in x−∞


νx (a1n ) = lim

N →∞

N −1 1 1Cyl(a1n ) (σ i−n−k x). N

57

(5)

k=0

If µ is ergodic, then νx = µ a.e. However, even when µ is only shift-invariant it suffices to look at the local information with respect to νx . Theorem 2. Let µ be a shift-invariant measure on AZ and νx the empirical measure i generated by x−∞ . Then SL (x; i; µ) = SL (x; i; νx ) µ-a.e. Proof. The result follows since the infinite history determines with probability one which ergodic component of µ x is generated by, see Lemma 8.6.2. in [6].

We can also define the right local information at coordinate i of x with respect to µ as i+n SR (x; i; µ) = − lim log µ(xi |xi+1 ). n→∞

(6)

All results we show for the left information will have corresponding results for the right information. However, although the left and right information have the same expectation for all shift-invariant measures, they are not equal nor do they in general have the same probability distribution. This is exemplified by the Markov measure on {0, 1, 2}Z defined by the following non-zero transition probabilities: p(0|0) = p(1|0) = 21 ; p(0|1) = p(2|1) = 21 ; p(0|2) = p(1|2) = 21 . 4. Information Transport In this section we investigate the transport of local information in the time-evolution of a one-dimensional surjective cellular automaton. We show that the left local information satisfies a continuity equation involving an information current JL , and supply an expression for this current. Let µ0 be a measure on AZ . The measure F(µ0 ) = µ0 ◦ F −1 gives the joint disi+r ) when (X ) tributions of the stochastic process (Yi )i∈Z with Yi = f (X i−l i i∈Z has joint 0 0 −1 1 distributions given by µ . Denote µ ◦ F by µ and, more generally, set µt = µ0 ◦ F −t . The block probabilities of µ1 can be calculated from n+r µ0 (z −l ). (7) µ1 (y0n ) = n+r ∈ f −1 (y n ) z −l 0

It is well known that h(µ1 ) = h(µ0 ) whenever F is surjective and µ0 is shiftinvariant (if F is non-surjective, this relation is replaced by h(µ1 ) ≤ h(µ0 )), see, e.g., [16]. Our goal is to prove the much stronger result that the local information obeys a local continuity equation under the time evolution of the CA. This is an equation of the form t SL + i JL = 0,

(8)

where JL (x; i; µ) is the information current. The operator is the forward difference operator, so that t SL (x; i; µt ) = SL (F(x); i; µt+1 ) − SL (x; i; µt ), i JL (x; i; µt ) = JL (x; i + 1; µt ) − JL (x; i; µt ).

(9)

58


x

τ

i-1

i

Z

y i+r −1 i+r −1 Fig. 1. An illustration of Z = Z (x−∞ ) and τ = τ (x−∞ ). In this case r = 2, |Z | = 4 and τ = i − 4. The circles represent symbols in A. These are connected by lines to semi-infinite sequences which all map to the i−1 same y−∞ and coincide with x at all j ≤ τ .

With these definitions, JL (x; i; µt ) can be interpreted as the information flow from position i − 1 to position i generated by applying the CA. Note that the local information of F(x) is taken with respect to a different measure than x, unless µt happens to be invariant for the CA. i i For a semi-infinite sequence x−∞ , define Z (x−∞ ) as the set of all semi-infinite i (see Fig. 1): sequences that have the same image and the same tail as x−∞ i Definition 3. For x ∈ AZ and a surjective CA F, define the sets Z (x−∞ ) as j

j

i i i i Z (x−∞ ) = {z −∞ : f (z −∞ ) = f (x−∞ ) and ∃ j ≤ i such that z −∞ = x−∞ }. i Note that |Z (x−∞ )| ≤ R(F) for all x by the definition of the Welch coefficient R(F). i i ) Define τ (x−∞ ) as the largest index less than i − r for which all sequences in Z (x−∞ coincide (recall that r is the right radius of F): i Definition 4. For x ∈ AZ , define τ (x−∞ ) ∈ Z as j

j

i i i τ (x−∞ ) = max{ j : j < i − r, and z −∞ = x−∞ ∀ z −∞ ∈ Z (x−∞ )}. j

We are now ready to define the information current. Definition 5. Let F be a surjective one-dimensional CA with right radius r , and µ a i+r −1 i+r −1 ) and τ = τ (x−∞ ). Define the left information measure on AZ . Put Z = Z (x−∞ current at coordinate i of x with respect to µ and F as −1 τ τ JL (x; i; µ) = − log µ(xτi−1 µ(z τi+r (10) +1 |x −∞ ) + log +1 |x −∞ ). Z i−1 τ τ The quantities µ(z τi−1 +1 |x −∞ ) are defined as lim n→∞ µ(z τ +1 |x τ −n ). Since JL is constructed entirely from conditional probabilities of this type, an analogue to Theorem 2 yields

JL (x; i; µ) = JL (x; i; νx ) µ-a.e. i ) τ (x−∞

(11)

< i − r included in Def. 4. In the Note that τ < i − 1, by the requirement that case of a reversible CA, the existence of an inverse CA ensures that τ is bounded below. Let r˜ be the right radius of the inverse CA. Then τ ≥ i − 1 − r˜ unless r˜ = 0, in which case τ = i − 2. For non-reversible CA, τ is in general unbounded but always finite. It remains to show that JL (x; i; µ) is well defined as a function on (X, B, µ). This is ascertained by the following lemma, whose proof follows from the martingale convergence theorem.


59

Lemma 1. For any measure µ on AZ , −1 −k−1 −1 |x−n ) exists for all k ≥ 0 and all a−k ∈ Ak }) = 1. µ({x : lim µ(a−k n→∞

It is furthermore the case that JL (x; i; µ) ∈ L 1 (µ), see Theorem 5 in Sect. 5.3. We now proceed to present Theorems 3 and 4, which are the main results of the paper. The first theorem states that for reversible CA the continuity equation is valid for all initial measures. Theorem 3. Let F be a reversible one-dimensional CA, and µ a measure on AZ . Then t SL (x; i; µ) + i JL (x; i; µ) = 0 for all i ∈ Z µ-a.e. For a general surjective CA, the requirement of µ being shift-invariant is necessary to ensure the validity of the continuity equation. Theorem 4. Let F be a surjective one-dimensional CA, and µ a shift-invariant measure on AZ . Then t SL (x; i; µ) + i JL (x; i; µ) = 0 for all i ∈ Z µ-a.e. Example 3 in Sect. 5 shows that the continuity equation as defined above can fail to be valid if µ is not shift-invariant and F is surjective without being reversible. Note that if one of the theorems is valid for a CA F together with an initial measure µ0 , then the continuity equation will be satisfied at all time steps of the iteration by F. Proof (of Theorem 3). We first show that it is sufficient to consider the case r = 0. Here, and in the rest of the proof, we look at the initial measure µ0 and its image µ1 . Assume that Theorem 3 is valid for CA with r = 0, and let F have right radius r . There exist a CA G with r = 0 such that F = σ r ◦ G. We have SL (F x; i; µ1 ) = i+r −1 i+r ), ), τ2 = τ (x−∞ SL (Gx; i + r ; µ1 ), since F(µ0 ) = G(µ0 ) = µ1 . Write τ1 = τ (x−∞ i+r −1 i+r Z 1 = Z (x−∞ ) and Z 2 = Z (x−∞ ). Using the formula for JL , we obtain i+r −1 SL (Gx; i + r ; µ1 ) = − log µ0 (xi+r |x−∞ ) −1 τ1 − log µ0 (xτi+r |x−∞ ) + log 1 +1

−1 τ1 µ0 (z τi+r |x−∞ ) 1 +1

Z1

+ log µ

0

(xτi+r |x τ2 ) − log 2 +1 −∞

µ0 (z τi+r |x τ2 ). 2 +1 −∞

(12)

Z2

Since τ1 ≤ i − 2 and τ2 ≤ i − 1 by definition, we can write τ2 i+r i |x τ2 ) = log µ0 (xτi 2 +1 |x−∞ ) + log µ0 (xi+1 |x−∞ ), log µ0 (xτi+r 2 +1 −∞

and

(13)

60

T. Helvik, K. Lindgren, M. G. Nordahl i+r −1 −1 τ1 − log µ0 (xi+r |x−∞ ) − log µ0 (xτi+r |x−∞ ) 1 +1

i−1 i+r i = − log µ0 (xτi−1 |x τ1 ) − log µ0 (xi |x−∞ ) − log µ0 (xi+1 |x−∞ ). 1 +1 −∞

(14)

Substituting (13) and (14) into (12) gives the correct continuity equation for F. For the rest of the proof we assume that F has right radius r = 0, and left radius l ≥ 0. We look only at coordinate i = 0. This leads to no loss of generality. Call the ˜ and let F˜ have left radius l˜ and right radius r˜ . Sequences at time t = 0 inverse CA F, are denoted by x or z and sequences at time t = 1 by y. We first define the joint measure ν of two consecutive time steps. Let ν be the measure on (A × A)Z defined by the block probabilities j

j

j

j

j

ν(xi , yi ) = µ0 ({z i−l ∈ A j−i+l+1 |z i = xi

and

j

j

f (z i−l ) = yi }).

(15)

It is easy to show that ν actually is a measure and that ν is shift-invariant if µ0 is shiftj j invariant. By summing over all possible yi or xi we obtain from the definition that j j j j ν(xi ) = µ0 (xi ) and ν(yi ) = µ1 (yi ). We will need the following lemma, which follows from the martingale convergence theorem. Z Lemma 2. Let ν be a measure on AZ 1 × A2 , where each Ai is a finite set. Let (x, y) ∈ Z Z 1 A1 × A2 . Then there is a g ∈ L (ν) such that for any k ∈ Z, −1 −1 , y−n ) = g ν-a.e. lim ν(x0 , y0 |x−n−k

(16)

n→∞

Let y = F(x). From the definition of local information we have −1 SL (x; 0; µ0 ) = − lim log µ0 (x0 |x−n ) n→∞ −1 −1 , y−n+l ) ν(x0 , y0 |x−n −1 −1 = lim log − log ν(x0 , y0 |x−n , y−n+l ) −1 n→∞ µ0 (x0 |x−n )

= lim log n→∞

0 0 ) ν(y−n+l |x−n

−1 −1 ν(y−n+l |x−n )

(17)

−1 −1 − lim log ν(x0 , y0 |x−n , y−n+l ) n→∞

−1 −1 , y−∞ ) = − log ν(x0 , y0 |x−∞ 0 0 ) = ν(y −1 |x −1 ) = 1 for all n > l, |x−n by virtue of Lemma 2 and the fact that ν(y−n+l −n+l −n 0 0 through the local map f and since y−n+l in this case is uniquely determined by x−n −1 −1 likewise for y−n+l and x−n . A similar treatment of SL (y; 0; µ1 ) yields

SL (y; 0; µ1 ) = lim log n→∞

|y 0 ) −n+l˜ −n −1 ν(x −1 ˜|y−n ) −n+l ν(x 0

−1 −1 − log ν(x0 , y0 |x−∞ , y−∞ ).

(18)

When taking the difference t SL , the last term is canceled out, so −1 0 |y 0 ) − lim log ν(x −1 ˜|y−n ). t SL (x; 0; µ0 ) = lim log ν(x−n+ l˜ −n n→∞

n→∞

−n+l

To conclude the proof, we show that −1 ) = JL (x; 0; µ0 ) − lim log ν(x −1 ˜|y−n n→∞

−n+l

(19)


61

through a sequence of transformations. Firstly, −1 −1 −1 −˜r −1 −1 log ν(x −1 ˜|y−n ) = log ν(x −˜r −1˜ |y−n ) + log ν(x−˜ r |x ˜ , y−n ). −n+l

−n+l

−n+l

(20)

−1 The first term on the right hand side is zero, since x −˜r −1˜ is uniquely determined by y−n −n+l through the local map f˜ of the inverse CA. For the second term, a generalization of Lemma 2 gives −1 −˜r −1 −1 −1 −˜r −1 −1 lim log ν(x−˜ ˜ , y−n ) = lim log ν(x −˜r |x −n−l , y−n ). r |x

n→∞

−n+l

n→∞

(21)

Furthermore, for any events A, B and C in a probability space it is true that ν(A|BC) = ν(C|AB)ν(A|B) −1 −˜r −1 −1 . Let A = x−˜ ν(C|B) r , B = x −n−l and C = y−n . Then ν(C|AB) = 1. Thus, −1 −˜r −1 −1 −1 −˜r −1 0 −1 −˜r −1 log ν(x−˜ r |x −n−l , y−n ) = log µ (x −˜r |x −n−l ) − log ν(y−n |x −n−l ).

By the definition of ν, the last term can be written as −1 −˜r −1 −1 −˜r −1 |x−n−l ) = − log µ0 (z −˜ − log ν(y−n r |x −n−l ).

(22)

(23)

−1 Z (x−∞ )

Substituting (23) into (22) and taking the limit n → ∞ we arrive at the equation for JL (x; 0; µ0 ) presented in Def. 5.

For general surjective CA, there is no inverse CA and in general several possible preimages. As a consequence, the proof of Theorem 4 requires a different approach. Proof (of Theorem 4). By the same argument as in the proof of Theorem 3 it suffices to consider right radius r = 0. As before, we only look at coordinate i = 0 and consider the initial measure µ0 and its image µ1 . Let y = F(x), and define q(x) = SL (y; 0; µ1 ) − SL (x; 0; µ0 ) + JL (x; 1; µ0 ) − JL (x; 0; µ0 ).

(24)

Our goal is to prove that q(x) = 0 µ0 -a.e, or equivalently that E[|q|] = 0. To prove this we will introduce a sequence qk of approximations to q which are measurable with respect to finite parts of the history. First, define the following equivalence relation on Al+n+1 for n ≥ 0: n n n n x−l ∼ z −l iff f (x−l ) = f (z −l )

and

−1 −1 x−l = z −l .

(25)

That is, two blocks in Al+n+1 are equivalent if they have the same image under f and n by [z n ]. agree on the first l coordinates. Denote the equivalence class containing z −l −l −1 Z For an x ∈ A we will in particular look at the equivalence classes [x−k−l ] for k ≥ 1. For each k ≥ 1 we have the inclusion −1 −l−k−1 −1 −1 −1 Z (x−∞ ) ⊇ {x−∞ z −l−k : z −l−k ∈ [x−l−k ]}.

(26)

−1 −1 ) is largest index such that all sequences in Z (x−∞ ) agree on and to Recall that τ (x−∞ −1 −1 the left of τ (x−∞ ). Therefore, for all k ≥ −τ (x−∞ ) − 1 Eq. (26) is an equality.

62


−1 Define τ k (x) for k ≥ 1 as the analogue of τ obtained when considering [x−k−l ] −1 rather than Z (x−∞ ), −1 −1 ∈ [x−k−l ] ⇒ z −k−l = x−k−l }. τ k (x) = max{ j ≤ −2, and z −k−l j

j

j

(27)

−1 −1 ) iff k ≥ −τ (x−∞ ) − 1. Note that τ k (x) = τ (x−∞ We now define finite versions of the current JL (x; 0; µ0 ) and of q(x). Write τ k for τ k (x) and put for k ≥ 1, τk τk JLk (x) = − log µ0 (xτ−1 µ0 (z τ−1 (28) k +1 |x −k−l ) + log k +1 |x −k−l ), −1 −1 z −k−l ∈[x−k−l ]

and −1 −1 ) + log µ0 (x0 |x−k−l ) + JLk+1 (σ x) − JLk (x). qk (x) = − log µ1 (y0 |y−k

(29)

−1 ] It is straightforward to check that qk (x) → q(x) a.e. by using the properties of [x−k−l and τ k discussed above. Proceeding, we can write |q|dµ0 ≤ |qk |dµ0 + |q − qk |dµ0 . (30)

The theorem will follow if we can prove that both integrals on the right-hand side converge to zero. In order to do this we investigate the stochastic process (gn )n≥0 on AZ defined by gn (x) = n ]) means Here, µ0 ([x−l relationship

µ1 (y0n ) n ]) , µ0 ([x−l

0 n ). n ∈[x n ] µ (z z −l −l −l

n where y0n = f (x−l ).

(31)

The interest in this process is due to the

log gn−1 (x) − log gn (x) = qn (σ n x).

(32)

We will prove that the process (gn )n≥0 is a supermartingale with respect to a filtration that we now will describe. Let P n be the partition of AZ defined by the equivalence relation (25) on Al+n+1 . n ∈ [u]} for all equivalence classes That is, the elements of P n are the sets P[u] = {x : x−l [u] of Al+n+1 . Let Fn = σ (P n ). Then, Fn is the σ -algebra generated by gn . We have to show that Fn ⊆ Fn+1 for all n. This follows if we can show that the partition P n+1 is a n ] of P n . We claim that refinement of P n . Consider a general element P[w−l n ] = n a] P[w−l P[z −l . (33) n ∈[w n ] z −l −l

a∈A

n ] , then P n+1 clearly is a member of the double union. Conversely, if x is If x ∈ P[w−l [x ] −l

−1 −1 −1 n n n n a] in the union, then x in some P[z −l −l = z −l = w−l and f (x −l ) = f (z −l ) = f (w−l ). n ] . The claim follows, and (Fn : n ≥ 0) is a filtration. Thus, x ∈ P[w−l


63

To prove that gn is a supermartingale with respect to this filtration we show that E[g |F ] ≤ g . Since each sub-σ -algebra F is finite it suffices to show gn+1 dµ0 ≤ n+1 n n n 0 n gn dµ over any P[u] ∈ P . We find that gn dµ0 = P[u]

n µ0 (x−l )

n ∈[u] x−l

µ1 (y0n ) 1 n n ]) = µ (y0 ). µ0 ([x−l

(34)

For gn+1 , we split the integral into cylinder sets where gn+1 is constant: gn+1 dµ0 = gn+1 (x)dµ0 P[u]

n ∈[u] a∈A x−l

=

n ∈[u] a∈A x−l

=

n a) Cyl(x−l

n µ0 (x−l a)

n a)) µ1 (y0n f (xn−l+1 n a]) µ0 ([x−l

(35)

ψb · µ1 (y0n b),

b∈A

where ψb =

n a) µ0 (x−l n ∈[u] a∈A x−l

n a]) µ0 ([x−l

n · 1{ f (xn−l+1 a)=b} (x).

(36)

We claim that for each b, the quantity ψb is equal either to 0 or to 1. Fix a b. First n a in the double sum in (36) that satisfy f (x n note that all blocks x−l n−l+1 a) = b must n a] = [v]. Conversely, each block z n+1 ∈ [v] generate the same equivalence class [x−l −l n ∈ [x n ] = [u]. Consequently, is an element of the double sum, since it will satisfy z −l −l ψb = 1 by summation of the fractions and cancelation. The exception is the case where n ∈ [u] can be extended with one symbol to the right such that the new block maps no x−l n to y0 b under f . For such b, ψb = 0. We can conclude that gn+1 dµ ≤ 0

P[u]

b∈A

µ

1

(y0n b)

=µ

1

(y0n )

=

gn dµ0 .

(37)

P[u]

Finally, it is easy to prove that E[|g0 |] ≤ |A|l+1 < ∞. This finalizes the proof that gn is a supermartingale, so by the martingale convergence theorem gn converges a.e. to a g ∈ L 1 (µ0 ). We now proceed to show L 1 -convergence of log gn . A family ( f n )n≥0 of measurable functions is uniformly integrable if ([5, Sect. 1.14]) lim sup (| f n | − M)+ dµ = 0. (38) M→∞ n

We claim that (log gn )n≥0 is an uniformly integrable family. Note first that n ]) < 2−t · µ1 (y0n ). log gn (x) > t ⇔ µ0 ([x−l

(39)

64


n : µ0 ([x n ]) < 2−t · µ1 ( f (x n ))}. We obtain Define An,t ⊂ Al+n+1 as An,t = {x−l −l −l

µ0 ({log gn > t}) =

n µ0 (x−l )≤

An,t

n 2−t µ1 ( f (x−l ))

An,t

≤

n 2−t µ1 ( f (x−l )) = 2−t · |A|l ,

(40)

Al+n+1

for all n. A simple application of Fubini’s theorem yields ∞ µ0 ({log gn > t}) dt sup (| log gn | − M)+ dµ0 = sup n

n

M

∞

≤ |A|l

2−t dt = 2−M ·

M

|A|l . ln 2

(41)

Thus, uniform integrability is satisfied and limn→∞ E[| log g − log gn |] = 0. From (32) and shift-invariance, lim E[|qn |] = lim E[|qn | ◦ σ n ] = lim E[| log gn−1 − log gn |] = 0.

n→∞

n→∞

n→∞

(42)

Hence, the first integral on the right-hand side in (30) converges to zero. Regarding the second integral on the right-hand side, we know that limn→∞ qn = q a.e. Thus, if we can prove that (qn )n≥0 is itself a uniformly integrable family then we are done. From (32) and the fact that log gn ≥ 0 for all n it follows that {|qn | ◦ σ n > M} ⊆ {log gn−1 > M}

{log gn > M}.

We can conclude that |A|l + 0 , sup (|qn | − M) dµ = sup (|qn ◦ σ n | − M)+ dµ0 ≤ 2−M+1 · ln 2 n n and the result follows.

(43)

(44)

A corresponding continuity equation can also be written for right local information. The right variants of the set Z and variable τ are defined by ∞ Z (xi∞ ) = {z i∞ : f (z i∞ ) = f (xi∞ ) and ∃ j ≥ i such that z ∞ j = x j }, ∞ ∞ ∞ τ (xi∞ ) = min{ j : j > i − l, and z ∞ j = x j ∀ z i ∈ Z (x i )}. j

∞ ) and τ = τ (x ∞ ), and define the right information current at coordinate Put Z = Z (xi−l i−l i of x with respect to µ and F by

JR (x; i; µ) = log µ(xiτ −1 |xτ∞ ) − log

τ −1 ∞ µ(z i−l |xτ ).

(45)

Z

Then, JR (x; i; µ) satisfies the continuity equation t SR (x; i; µ) + i JR (x; i; µ) = 0 at all i ∈ Zµ-a.e., under the same conditions as in Theorem 3 or Theorem 4.


65

5. Further Aspects of Information Transport 5.1. Information Theoretic Interpretation. We now describe a way of decomposing JL (x; i) into JL (x; i) = JL+ (x; i) − JL− (x; i),

(46)

with JL+ , JL− ≥ 0, such that JL+ has a natural interpretation in terms of information flowing to the right between coordinates i − 1 and i, and JL− in terms of information flowing to the left. Here, and in the rest of the section, we omit µ from the notation in JL and SL when considering some fixed measure µ. i First recall the definition of Z (x−∞ ) and define i−r i−r i i i . (47) Z 0 (x−∞ ) = z −∞ ∈ Z (x−∞ ) : z −∞ = x−∞ i+r −1 We will consider the set Z 0 (x−∞ ), which consists of the semi-infinite sequences which i+r −1 i+r −1 up to index i − 1. Define JL+ have the same image as x−∞ and coincide with x−∞ − r −1 r −1 r −1 and JL at coordinate i = 0, with τ = τ (x−∞ ), Z = Z (x−∞ ) and Z 0 = Z 0 (x−∞ ), as −1 JL− (x; 0) = − log µ(z r0−1 |x−∞ ), (48) Z0

Z0

JL+ (x; 0) = − log

Z

τ µ(z rτ −1 +1 |x −∞ )

τ µ(z rτ −1 +1 |x −∞ )

.

(49)

It is straightforward to confirm that JL− (x; 0) and JL+ (x; 0) are non-negative and satisfy (46). We first examine JL− . Using the joint measure ν defined in (15) we can write JL− (x; 0) = − lim log

−1 −1 , y−r ) ν(x−n

n→∞

−1 ν(x−n )

−1 −1 = − log ν(y−r |x−∞ ).

(50)

−1 when The equation states that JL− (x; 0) is the information gained by observing y−r −1 −1 having knowledge of x−∞ . This is what one should expect. Indeed, since x−∞ is known, −r −1 is uniquely determined by the CA map. Hence, all the semi-infinite sequence y−∞ −1 −1 , and this uncertainty comes from lack of uncertainty about y−∞ is with respect to y−r −1 −1 −1 ∞ . The quantity − log ν(y−r |x−∞ ) is thus knowledge about the continuation x0 of x−∞ −1 −1 ∞ . This the further information about the continuation x0 found in y−∞ but not in x−∞ ∞ information has been transported from x0 , and adds a negative contribution JL− to the information current. Considering JL+ , we can write

JL+ (x; 0)

Z0

= − lim log n→∞

= − lim log n→∞

−1 µ(z r−n )

r −1 Z µ(z −n ) −1 −1 ν(x−n , y−n+l ) τ , y −1 ) ν(x−n −n+l

−1 τ = − log ν(xτ−1 +1 |x −∞ , y−∞ ).

(51)

66


−1 τ Thus, JL+ (x; 0) is the information gained from observing xτ−1 +1 when x −∞ as well as y−∞ −1 r −1 r −1 is determined up to the set Z (x is known. Since y−∞ is known, the preimage x∞ −∞ ). r −1 This is illustrated by Fig. 1, where xτ +1 must be one of the “branches” to the right, but −1 r −1 r −1 which one. However, which member of Z (x−∞ ) that x−∞ it is not decidable from y−∞ −1 actually is will, with probability one, be determined by the continuation y0∞ of y−∞ . −1 τ −1 Therefore, the information − log ν(xτ +1 |x−∞ , y−∞ ) flows to the right and is found to the right of coordinate −1 in y.

5.2. Permutative Cellular Automata. For the class of permutative CA the information dynamics has a particularly simple form. A CA F is called right permutative if R(F) = 1 i or, equivalently, if |Z (x−∞ )| = 1 for all pairs x and i. Thus, for right permutative CA − i−1 ) and (49) gives JL+ ≡ 0. This gives the (48) gives JL (x; i; µ) = − log µ(xii+r −1 |x−∞ following corollary to Theorem 4. Corollary 1. Let µ be any shift-invariant measure on AZ and F : AZ → AZ a right permutative CA with right radius r . Then, µ-almost everywhere, SL (F x; i; µ ◦ F −1 ) = SL (x; i + r ; µ).

(52)

In particular, if r = 0, then SL (F t x; i; µt ) = SL (x; i; µ0 ) for all t ≥ 0 so that the local information is locally constant. As a result of Corollary 1, the distribution of local information will also remain unchanged. Corollary 2. Let µ0 be any ergodic measure on AZ and F : AZ → AZ a right permutative CA. Then, for all measures ν being a weighted sum of the measures µt , t ≥ 0, the random variables SL (x; i; ν) have the same distribution. Note that even though the behaviour of the local information is very simple in the case of permutative CA, the sequence µt of measures generated by a linear CA under iteration is quite complicated. Block probabilities and the structure of correlations in the system varies widely with t [16]. On the other hand, for many bipermutative CA large classes of initial measures weak∗ converge in Cesàro mean to the uniform Bernoulli measure. That is, n−1 1 t k 1 lim µ (a1 ) = n→∞ n |A|k

(53)

t=0

for all k ≥ 1 and finite blocks a1k ∈Ak . This was first proved for the linear CA F = σ +σ −1 on {0, 1}Z with µ0 a Bernoulli measure by Lind [15]. It has later been extended to a larger subclass of the permutative CA and classes of measures [3, 9, 18, 19]. For the uniform Bernoulli measure µ¯ the local information has a uniform distribution, i.e., SL (x; i; µ) ¯ = log |A| for all x and i. We can use the result on Cesàro convergence to demonstrate that convergence of a sequence (µn )n≥0 of measures to a limit measure µ in the weak∗ -topology does not, in any sense, mean that SL (x; i; µn ) converges to SL (x; i; µ). Indeed, let F and µ0 be any combination of a CA and an ergodic measure t such that (53) is valid, and put µn = n1 n−1 t=0 µ . Then µn converges to the uniform Bernoulli measure, but by Corollary 2 all SL (x; 0; µn ) have the same non-uniform probability distribution on R. The reason that the distribution of SL (x; i; µn ) can remain unchanged even though µn → µ, ¯ with µ¯ uniform Bernoulli, is that local information


67

takes all correlations in the system into account while the weak∗ topology only considers finite blocks. In this sense local information yields a different, more microscopic, view of the system than the weak∗ topology does. We now use Corollary 1 to demonstrate the necessity of the shift invariance condition on the measure in Theorem 4. Example 3. Let F be the CA σ −1 ◦ F1 , with F1 from Example 1. Let µ0 be the uniform Bernoulli measure on {0, 1}Z , except that x0 always is zero. Then, for all x ∈ AZ we have SL (x; i, µ0 ) = 1 for i = 0 and SL (x; 0, µ0 ) = 0. However, we claim that SL (y; i, µ1 ) = 1 for all i. Each sequence y ∈ AZ has two preimages under f , call them z and w. These have the property that z i and wi always are different, z i = 1 − wi . Assume that z 0 = 0. We obtain

2 j−k 0 if j ≤ 0 ≤ k if j ≤ 0 ≤ k 0 k 0 k , µ (w j ) = . (54) µ (z j ) = j−k−1 j−k−1 2 otherwise otherwise 2 The local information SL (y; i, µ1 ) is the limit n → ∞ of i−1 ) = − log − log µ1 (yi |yi−n

i i µ0 (z i−n−1 ) + µ0 (wi−n−1 ) i−1 i−1 µ0 (z i−n−1 ) + µ0 (wi−n−1 )

.

(55)

In all three cases: i < 0, i = 0 and i > 0, inserting the probabilities from (54) gives i−1 − log µ1 (yi |yi−n ) = 1 for all n. Now assume that t SL + i JL = 0 at all iµ0 -a.e. Since F is right permutative and has r = 0, Corollary 1 states that JL = 0 µ0 -a.e. However, this makes SL (y; 0, µ1 ) = 0, which is not satisfied for the image of any x ∈ AZ . An alternative way to appreciate that SL (y; 0, µ1 ) = 1 for the system in the example −1 does not give any information about x−1 . Therefore, even though is to realize that y−∞ −1 ) = 21 . Note that once y0 is observed, we will have x0 = 0 with certainty, µ1 (y0 |y−∞ −1 perfect knowledge of x−∞ . Thus, information about the preimage that is not contained in the tail of y is made available at some position in the sequence. In similar constructs the information need not appear at a single position as it did in Example 3. A case illustrating this would be to let µ0 (X i = 1) = 41 for all i ≥ 0 and µ0 (X i = 1) = 21 for i ≤ 0. Then the correct preimage will be learnt gradually from observing y0 , y1 , y2 , . . . , since the fraction of 1’s in the preimage block x0n will converge either to 41 or 43 . In this case, the continuity equation will in general not be satisfied at any i ≥ 0, but will be an increasingly better approximation as i increases. Finally in this section, we use a left and right permutative CA to illustrate the difference between SL and SR , and the effect of the choice of a frame of reference on the distribution of information in the system. We consider the CA on {0, 1}Z defined by the radii l = r = 1 and local rule f (x−1 , x0 , x1 ) = x−1 + x0 + x1

(mod 2).

(56)

Let the initial measure µ0 be Bernoulli with a very small probability for a 1, say µ(1) = 2−10 . Assume that i = 0 is the only coordinate in the interval −100 ≤ i ≤ 100 initially having xi = 1. Figure 2 shows the configurations of the interval −50 ≤ i ≤ 50 for all iteration up to time t = 50. Coordinate i = 0 initially has 10 bits of left and right

68

T. Helvik, K. Lindgren, M. G. Nordahl 0 White: 0 Black: 1

Time t

10 20 30 40 50 0

50

Coordinate i

Fig. 2. The evolution of the symbol sequence with a single 1 located at i = 0 under the CA rule defined in (56). The left local information from the initial 1 is located at the left boundary of the expanding pattern while the right local information is located at the right boundary.

local information, SL (x; 0; µ0 ) = SR (x; 0; µ0 ) = 10. Let y = F t (x). An observer which knows the left history will by observing y−t = 1 learn that x0 = 1 and gain 10 bits of information. However, from each of the subsequent symbols the observer will gain only − log(1−2−10 ) bits of information. This is in agreement with Corollary 1. On the other hand, an observer knowing the right history will gain the 10 bits of information by observing that yt = 1. Thus, the question about where in the pattern the information generated by the unlikely event {x0 = 1} is located at time t cannot be answered without also taking into account which frame of reference an observer has.

5.3. Bounds on the Current. We first consider bounds for the local information flow. i+r −1 Let τ = τ (x−∞ ). Then, from (46), (48), and monotonicity of log x, −

i+r −1 k=i

SL (x; k) ≤ JL (x; i) ≤

i−1

SL (x; k).

(57)

k=τ +1

Thus, the amount of information that flows from coordinate i − 1 to i is limited by the amount of information available in the intervals [τ + 1, i − 1] and [i, i + r − 1]. The appearance of τ + 1 rather than i − l in the first interval warrants a closer examination, because a perturbation of one symbol in the initial configuration only can propagate a distance l per time step. For reversible CA the existence of an inverse CA ensures that τ + 1 ≥ i − r˜ , where r˜ is the right radius of the inverse CA. Therefore, the distance over which information can flow in a single iteration is uniformly bounded for a given reversible CA. For surjective non-reversible CA, τ is in general unbounded. In the following discussion, assume that r = 0, since this case gives the maximal flow of left local information to the right. We look at coordinate i = 0. The appearance of τ + 1 in (57) is related to the −1 interpretation of JL+ as the information gained by observing xτ−1 +1 when the image y−∞ as τ well as the history x−∞ is known. The second inequality in (57) is an equality if and only −1 leads to no reduction in information gain compared if the additional knowledge of y−∞ τ τ to knowledge of only x−∞ . From (49) this is equivalent to having Z µ(z τ−1 +1 |x −∞ ) = 1. However, since |Z | ≤ R(F) the sum is rarely close to this magnitude, particularly when


(a) x

0 1 2 2

1 1

1

0 0 2 2

69

(b) x

0 0 2

0 2 2 0 1 2 n

n

y

2 2 0

0 0 n-1

y

0 1 2

2 2 0 n-1

−1 1 ) for the CA in Example 4 when x −1 = 22 . . . 2 Fig. 3. An illustration of the sets (a) Z (x−∞ ) and (b) Z (x−∞ −n and x−n−2 = x−n−1 = x0 = x1 = 0.

|τ | is large. A further argument that large information flows are improbable is the obser −1 −1 τ τ −s vation that JL (x; i; µ) > s requires µ(xτ−1 Z µ(z τ +1 |x −∞ ) < 2 , so x τ +1 +1 |x −∞ )/ τ must be a very unlikely continuation of x−∞ to generate a large current s. We illustrate these considerations with the following example. Example 4. Let the surjective CA F on {0, 1, 2}Z be defined by the radii l = 1, r = 0 and local function f given by f (10) = f (11) = f (22) = 0, f (12) = f (20) = f (21) = 1 and f (00) = f (01) = f (02) = 2. Unlikely events in this system can generate information flows over large distances in a single iteration. Let µ be Bernoulli with a low probability p = µ(2) for the symbol 2 occurring, and q = µ(0) = µ(1) = 1−2 p . Although p is small, long blocks of successive 2’s will occur −1 −n−2 at some points. Assume that x−n = 22 . . . 2 while x−n−1 = 00 and x03 = 0000. The set −1 Z (x−∞ ) is illustrated to the left in Fig. 3 using the representation introduced in Fig. 1. −1 τ τ For p small, the quantity Z µ(z τ−1 +1 |x −∞ ) is much larger than µ(x τ +1 |x −∞ ), since the −1 ) consist entirely of 0’s and 1’s. It follows from (49) that two other elements in Z (x−∞ + JL is large and an increasing function of the number n of 2’s. Equation (10) yields for 1 ≤ k ≤ n, k q q . (58) JL (x; −n + k; µ) = log 1 + 2 ≈ 1 + k log p p Only approximately − log q bits of information remain at each coordinate −n ≤ i ≤ −1, while the surplus information is transported to the right of i = −1, see Fig. 4. Most of the information is accumulated at position i = 1. The reason is that observing the value 0 y1 = 2 while knowing y−∞ establishes that the actual preimage was the one containing the large block of 2’s, see the right part of Fig. 3. This preimage was highly improbable, and a high local information results. Finally, note that since F is left permutative, the transport of right local information is simply given by JR (x; i; µ) = SL (x; i − 1; µ). i−1 i−l−1 |x−∞ ) can be better appreciated by The possibility that JL (x; i) > − log µ(xi−l recalling that local information SL is defined with respect to an infinite frame of reference. Therefore, a permutation arbitrary far to the left of i can alter the conditional i−1 probability µ(xi |x−∞ ) and hence SL (x; i). Contrary to this, the propagation of a perturbation in the initial configuration consists of the symbols at an increasing number of coordinates deviating from some reference symbols. Clearly, no frame of reference is needed to detect these deviations. We can compare the results above to a situation that involves communication between 0 two parts of the lattice. Consider an observer A who knows the initial configuration x−∞

70

T. Helvik, K. Lindgren, M. G. Nordahl 20 Local Information, t=0 Local Information, t=1 Information Current

[bit]

15 10 5 0 −9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

Position i

Fig. 4. The distribution of local information SL and the corresponding information currents JL (x; i; µ) for the situation in Example 4 with n = 7 and p = 0.1. The information is accumulated at coordinates i = 0 and i = 1 at time t = 1.

of the negative part of the lattice. How much information about the continuation x1∞ can A gain by observing the configurations (F k x)0−∞ for 0 < k ≤ t? This question can be answered by using the concept of relative entropy, or Kullback Liebler distance [13]. The relative entropy of a posterior measure µ with respect to a prior measure µ0 satisfying µ µ0 is defined as D(µ||µ0 ) =

log X

dµ dµ, dµ0

(59)

dµ where dµ is the Radon-Nikodym derivative. The quantity D(µ||µ0 ) is interpreted as 0 the Shannon information gained by going to the posterior. The posterior is in our case expressed in terms of the joint measure ν t of all times 0 ≤ k ≤ t obtained as a straightforward generalization of ν defined in (15). For x ∈ AZ , define the measure µtx on σ (X i : i > 0) as 0 µtx (z 1n ) = ν t (z 1n |x−∞ , (F x)0−∞ , . . . , (F t x)0−∞ ), n ≥ 1.

(60)

The information the observer A gains by time t is given by the relative entropy D(µtx ||µ0x ). The following relations are valid Proposition 1. The measures defined in Eq. (60) satisfy

D(µtx ||µ0x ) ≤ D(µ1x ||µ0x ) =

rt

SL (x; i; µ), i=1 JL− (x; 1; µ).

(61) (62)

This means that A during the first t iterations of F cannot gain more information about 0 than the left local information initially located within the interval [1, r t]. Similarly, x−∞ if B is an observer knowing x0∞ and observing the symbols at i ≥ 0 for times 1 ≤ k ≤ t, −1 −1 his information gain about x−∞ would be bounded by i=−lt SR (x; i; µ0 ).


71

0 z kr ) = f (x kr ) for 1 ≤ k ≤ t}. Both Proof. Define Bx ⊆ Ar t as Bx = {z r1t : f k (x−kr 1 −kr results follow from

D(µtx ||µ0x ) = lim

n→∞

= lim

n→∞

µtx (z 1n ) log

z 1n

µtx (z 1n ) µ0x (z 1n )

0 µ(z 1n |x−∞ , z r1t ∈ Bx ) log

0 , zr t ∈ B ) µ(z 1n |x−∞ x 1

z 1n

0 ) µ(z 1n |x−∞

0 0 = − log µ({z r1t ∈ Bx }|x−∞ ) ≤ − log µ(x1r t |x−∞ ).

We now move on to determine bounds on the average information flow generated by a surjective one-dimensional CA. Theorem 5. Let JL (x; i; µ) be the information current with respect to a surjective CA F and a measure µ. Then, for each i ∈ Z, JL (x; i; µ) ∈ L 1 (µ). Furthermore, if µ is shift-invariant, then E[JL ] satisfies the relationship −r h(µ) ≤E[JL ] ≤ log R(F) − r h(µ).

(63)

The term log R(F) on the right hand-side in (63) is related to the interpretation of JL+ as r −1 r −1 r −1 ) that x−∞ is. Since |Z (x−∞ )| ≤ R(F), the information about which member of Z (x−∞ the average of this information cannot exceed log R(F). The term −r h(µ) is related to JL− . Proof. We look at coordinate i = 0. The current can be written as µ(x r −1 |x τ ) −1 JL (x; 0; µ) = − log τ +1r −1−∞τ + log µ(x0r −1 |x−∞ ), µ(x |x ) Z τ +1 −∞

(64)

where the first term is non-negative and the second term is non-positive. To bound the r −1 integral of the first term we divide AZ into the sets Tk = {x : τ (x−∞ ) = k} for k ≤ −2. Furthermore, we wish to subdivide each Tk through an equivalence relation similar to that defined in (25). Define the following relation on A|k|+2r +l−1 : r −1 −1 −1 −1 k k ∼ z rk+1−l−r iff f (rk+1−l−r ) = f (rk+1−l−r ) and xk+1−l−r = z k+1−l−r . xk+1−l−r

Then, transfer the equivalence relation to Tk through r −1 −1 ∼ z rk+1−l−r . x ∼ z iff xk+1−l−r

We denote the equivalence classes of Tk by Pk, j with j in some finite index set. Furthermore, for each Pk, j denote the corresponding equivalence class of A|k|+2r +l−1 by P¯k, j . Each P¯k, j has at most R(F) members. k−l−r For each j there is a set Pk,−j ∈ σ (X i : i ≤ k − l − r ) of histories x−∞ such that Pk, j = Pk,−j

P¯k, j

k Cyl(z k+1−l−r ).

(65)

72


If | P¯k, j | = R(F) then Pk,−j = AZ , but otherwise Pk,−j can be a subset of AZ . For instance,

−1 let F be the CA from Example 4 and x have xi = 1 for all i. Then τ (x−∞ ) = −2, − | P¯−2, j | = 2, and P−2, j = {z : z i = 0 for i ≤ −3}. Using the subdivision, we can write µ(x r −1 |x τ ) − dµ log τ +1r −1−∞τ AZ Z µ(x τ +1 |x −∞ ) ⎛ ⎞ −2 k−l−r ⎜ −1 k−l−r ⎟ k−l−r = k, j (x−∞ )⎝ µ(z rk+1−l−r |x−∞ )⎠ dµ(x−∞ ), (66) k=−∞ Pk, j ∈Tk

− Pk, j

P¯k, j

where k−l−r k, j (x−∞ )=−

P¯k, j

−1 k µ(z rk+1 |x−∞ ) P¯k, j

−1 k µ(z rk+1 |x−∞ )

log

−1 k µ(z rk+1 |x−∞ ) P¯k, j

−1 k µ(z rk+1 |x−∞ )

.

(67)

k−l−r k−l−r The function k, j (x−∞ ) is for each x−∞ the entropy of a discrete random variτ −l−r ) ≤ log R(F). By using this able with at most R(F) outcomes. Therefore, k, j (x−∞ inequality, we obtain from (66) that

−

−2 µ(x r −1 |x τ ) dµ ≤ log R(F) log τ +1r −1−∞τ µ(Pk, j ) = log R(F). (68) AZ Z µ(x τ +1 |x −∞ ) k=−∞ T k

Considering the second term in (64), −1 log µ(x0r −1 |x−∞ )dµ ≥ −r log |A|. AZ

Therefore, E[|JL (x; 0; µ)|] ≤ r log |A| + log R(F) < ∞, so JL (x; 0; µ) ∈ L 1 (µ). The second statement follows since for µ shift-invariant, −1 log µ(x0r −1 |x−∞ )dµ = −r h(µ). AZ

From Theorem 5 it follows that for any surjective CA and any measure µ, the following uniform bound is valid: |E[JL (x; i; µ)]| ≤ 1. (l + r ) log |A|

(69)

When µ is the uniform Bernoulli measure µ, ¯ the inequality is sharp for all CA with r = 0 and maximal R. We close this section by looking briefly at information transport for µ, ¯ which is invariant for all surjective CA. With µ¯ there are no correlations in the system, so SL (x; i; µ) ¯ ≡ SR (x; i; µ) ¯ ≡ log |A|. The information currents JL and JR are consequently also constant, but these


73

depend on the radii and the Welch coefficients L(F), M(F) and R(F). Using (10) and (45) we obtain JL (x; i; µ) ¯ ≡ log R − r log |A|, JR (x; i; µ) ¯ ≡ − log L + l log |A|.

(70) (71)

By using the relation L · M · R = |A|l+r , we obtain JR − JL ≡ log M.

(72)

Thus, the sum of the velocity of left information to the left and right information to the right only depends on M(F), the number of preimages that almost all bi-infinite sequences possess. The higher M is, the higher the potential for information transport. The choice of radii decides how the potential is allocated to transport of information to the right and to the left. 6. Conclusions and Discussion The main concern of the paper has been to investigate transport of local information in the time evolution of a cellular automaton F. In particular, we have introduced an information current JL (x; i; µ) such that the continuity equation t SL +i JL = 0 holds under very general conditions. This is expressed in our main results, Theorems 3 and 4 in Sect. 4. We have also given an information theoretic interpretation of the current, and shown bounds for the information flow. The fact that the local information is a locally conserved quantity for all measures under iteration of any reversible CA is a clear indication that the function SL is an appropriate local information measure in a spatially extended system. However, we still need to consider the fact that information is not a strictly local quantity when correlations are present, and that it depends on the choice of context. In one dimension, this is illustrated by the fact that both the left and right local information are locally conserved, and in general different (as seen, e.g., for bipermutative CA). We have also given other examples which illustrate the limits of locality when correlations are present. We are currently investigating how the continuity equation can be extended to other classes of cellular automata. In particular this includes non-surjective and probabilistic CA in dimension one as well as CA in dimension two and higher. For non-surjective or non-deterministic systems a continuity equation must take loss and production of information into account. One can also look at local information and transport of local information for other types of spatially extended dynamical systems, in particular coupled map lattices. Extensions of the formalism to other systems will also bring us closer to addressing fundamental issues relating to information transport and conservation in physical systems. The continuity equation is a fundamental property of information transport in reversible systems. But we expect local information in cellular automata to have further interesting properties. In particular, a continuity equation is a constraint, rather than an equation that determines the dynamics of the system. One may expect information flow to have different dynamic characteristics in different cellular automata. In particular it should be investigated whether some systems allow a description of the dynamics of information separate from the underlying dynamical system, which would provide an additional argument for viewing information as a fundamental physical quantity.

74


Acknowledgements. Helvik acknowledges support from by the Research Council of Norway. Lindgren acknowledges support from PACE (Programmable Artificial Cell Evolution), a European Integrated Project in the EU FP6-IST-FET Complex Systems Initiative, and from EMBIO (Emergent Organisation in Complex Biomolecular Systems) under EU FP6.

References 1. Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Develop. 17(6), 525–532 (1973) 2. Dab, D., Lawniczak, A., Boon, J.P., Kapral, R.: Cellular-automaton model for reactive systems. Phys. Rev. Lett. 64, 2462–2465 (1990) 3. Ferrari, P., Maass, A., Martínez, S., Ney, P.: Cesàro mean distribution of group automata starting from measures with summable decay. Ergodic Theory Dynam. Systems. 20(6), 1657–1670 (2000) 4. Frisch, U., Hasslacher, B., Pomeau, Y.: Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett. 56, 1505–1508 (1986) 5. Gänssler, P., Stute, W.: Wahrscheinlichkeitstheorie. Berlin Heidelberg New York: Springer Verlag 1977 6. Gray, R.M.: Probability, random processes, and ergodic properties. New York: Springer-Verlag 1988 7. Hardy, J., Pomeau, Y., de Pazzis, O.: Time evolution of two-dimensional model system. I. invariant states and time correlation functions. J. Math. Phys. 14, 1746–1759 (1973) 8. Hedlund, G.A.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory. 3, 320–375 (1969) 9. Host, B., Maass, A., Martínez, S.: Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6), 1423–1446 (2003) 10. Ito, M., Osato, N., Nasu, M.: Linear cellular automata over Z m . J. Comput. System Sci. 27(2), 125– 140 (1983) 11. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957) 12. Keller, G.: Equilibrium states in ergodic theory, Volume 42 of London Mathematical Society Student Texts. Cambridge: Cambridge University Press 1998 13. Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951) 14. Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Develop. 5(3), 183–191 (1961) 15. Lind, D.A.: Applications of ergodic theory and sofic systems to cellular automata. Physica D. 10, 36–44 (1984) 16. Lindgren, K.: Correlations and random information in cellular automata. Complex Systems. 1, 529–543 (1987) 17. Lindgren, K.: Microscopic and macroscopic entropy. Phys. Rev. A. 38, 4794–4798 (1988) 18. Pivato, M., Yassawi, R.: Limit measures for affine cellular automata. Ergodic Theory Dynam. Systems. 22(4), 1269–1287 (2002) 19. Pivato, M., Yassawi, R.: Limit measures for affine cellular automata II. Ergodic Theory Dynam. Systems. 24(6), 1961–1980 (2004) 20. Richardson, D.: Tesselations with local transformations. J. Comput. System Sci. 5, 373–388 (1972) 21. Takesue, S.: Reversible cellular automata and statistical mechanics. Phys. Rev. Lett. 59, 2499–2502 (1987) 22. Takesue, S.: Fourier’s law and the Green-Kubo formula in a cellular-automaton model. Phys. Rev. Lett. 64, 252–255 (1990) 23. Toffoli, T.: Information transport obeying the continuity equation. IBM J. Res. Develop. 32(1), 29–36 (1988) 24. Toffoli, T., Margolus, N.H.: Invertible cellular automata: a review. Physica D. 45(1–3), 229–253 (1990) 25. Vichniac, G.: Simulating physics with cellular automata. Physica D. 10, 96–115 (1984) 26. Walters, P.: An Introduction to Ergodic Theory. Number 79 in Graduate Texts in Mathematics. Berlin Heidelberg New York: Springer 1982 27. Wheeler, J.A.: Information, physics, quantum: The search for links. In: Zurek, WH (ed.): Complexity, Entropy and the Physics of Information. Redwood City, CA: Addison-Wesley 1989 28. Zurek, W.H.: Algorithmic randomness and physical entropy. Phys. Rev. A. 40, 4731–4751 (1989) Communicated by G. Gallavotti


Communications in


A Liouville-type Theorem for Schrödinger Operators Yehuda Pinchover Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel. E-mail: [email protected] Received: 19 December 2005 / Accepted: 12 June 2006 Published online: 2 March 2007 – © Springer-Verlag 2007

Abstract: In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P1 , such that a nonzero subsolution of a symmetric nonnegative operator P0 is a ground state. Particularly, if P j := − + V j , for j = 0, 1, are two nonnegative Schrödinger operators defined on ⊆ Rd such that P1 is critical in with a ground state ϕ, the function ψ 0 is a subsolution of the equation P0 u = 0 in and satisfies ψ+ ≤ Cϕ in , then P0 is critical in and ψ is its ground state. In particular, ψ is (up to a multiplicative constant) the unique positive supersolution of the equation P0 u = 0 in . Similar results hold for general symmetric operators, and also on Riemannian manifolds. 1. Introduction 2

Let ⊂ Rd be a domain. We assume that A : → Rd is a measurable symmetric matrix valued function such that for every compact set K ⊂ there exists µ K > 1 such that ∀x ∈ K , (1.1) µ−1 K Id ≤ A(x) ≤ µ K Id where Id is the d-dimensional identity matrix, and the matrix inequality A ≤ B means p that B − A is a nonnegative matrix on Rd . Let V ∈ L loc (; R), where p > d/2. We consider the quadratic form a A,V [u] := A∇u · ∇u + V |u|2 dx (1.2)

on

C0∞ ()

associated with the Schrödinger equation Pu := (−∇ · (A∇) + V )u = 0

in .

We say that a A,V is nonnegative on C0∞ (), if a A,V [u] ≥ 0 for all u ∈ C0∞ ().

(1.3)

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1 () is a (weak)sol ut i on of (1.3) if for every Definition 1.1. We say that v ∈ Hloc ∞ ϕ ∈ C0 (), (A∇v · ∇ϕ + V vϕ) dx = 0. (1.4)

1 () is a subsol ut i on of (1.3) if for every nonnegative ϕ ∈ C ∞ (), We say that v ∈ Hloc 0 (A∇v · ∇ϕ + V vϕ) dx ≤ 0. (1.5)

v∈

1 () Hloc

is a super sol ut i on of (1.3) if −v is a subsolution of (1.3).

Let C P () be the cone of all positive solutions of the equation Pu = 0 in , and let λ0 (P, ) := sup{λ ∈ R | C P−λ () = ∅}

(1.6)

be the generalized principal eigenvalue of the operator P in . By the AllegrettoPiepenbrink theory (see for example, [1, 21]), the form a A,V is nonnegative on C0∞ () if and only if λ0 (P, ) ≥ 0. Let K (i.e. K is relatively compact in ). Recall [1, 21] that u ∈ C P ( \ K ) is said to be a positive solution of the operator P of minimal growth in a neighborhood of infinity in , if for any K K 1 , with a smooth boundary, and any v ∈ C P ( \ K 1 ) ∩ C(( \ K 1 ) ∪ ∂ K 1 ), the inequality u ≤ v on ∂ K 1 implies that u ≤ v in \ K 1 . A positive solution u ∈ C P () which has minimal growth in a neighborhood of infinity in is called a ground state of P in . The operator P is said to be critical in , if P admits a ground state in . The operator P is called subcritical in , if C P () = ∅, but P is not critical in . If C P () = ∅, then P is supercritical in . It is known that the operator P is critical in if and only if the equation Pu = 0 in admits (up to a multiplicative constant) a unique positive supersolution. In particular, in the critical case we have dim C P () = 1 (see for example [18, 21] and the references therein). On the other hand, P is subcritical in , if and only if P admits a positive minimal Green function G P (x, y) in . For each y ∈ , the function G P (·, y) is a positive solution of the equation Pu = 0 in \ {y} that has minimal growth in a neighborhood of infinity in and has a (suitably normalized) nonremovable singularity at y (see for example [18, 21] and the references therein). The following basic example will be used several times throughout the paper. Example 1.2. Let P = − and = Rd . It is well known that λ0 (−, Rd ) = 0. In addition, the positive Liouville theorem asserts that C− (Rd ) = {c1 | c > 0}, where 1 is the constant function taking at any point the value 1. Moreover, − is critical in Rd if and only if d ≤ 2. Recently, Berestycki, Hamel, and Roques [6] have introduced the following definition which arises naturally in the study of some semilinear equations. Definition 1.3. λ0 (P, ) := inf{λ ∈ R | ∃φ ∈ C 2 () ∩ W 2,∞ (), φ > 0, (P − λ)φ ≤ 0 in , φ = 0 on ∂, if ∂ = ∅}.

A Liouville-type Theorem for Schrödinger Operators

77

Before formulating our results, we present four basic problems (Problems 1–4) concerning the particular case = Rd , which have been solved in the past few years. It turns out that (almost) all these previous special results can be recovered directly from our general result (Theorem 1.7). Problem 1 ([23, 24]). Let V ∈ L 2loc (Rd ). Does the existence of a positive bounded solution to the equation HV u := (− + V )u = 0

on Rd

(1.7)

imply that HV is critical in Rd ? Problem 2 ([5]). Suppose that V is smooth and bounded. Does the existence of a signchanging bounded solution to Eq. (1.7) imply that λ0 (HV , Rd ) < 0? Problem 3 ([5, 14]). Let σ be a strictly positive C 2 -function on Rd , and consider the divergence form operator L = −∇ · (σ 2 ∇) on Rd . Suppose that the equation Lψ = 0 in Rd admits a nonzero solution ψ such that ψσ is bounded. Is ψ necessarily the constant function? Problem 4 ([7, Conjecture 4.6]). Suppose that P = −∇ · (A∇) + V is a uniformly elliptic operator with smooth bounded coefficients on Rd . Does the inequality λ0 (P, Rd ) ≤ λ0 (P, Rd ) hold true in any dimension d? The answers to the above four problems for the free Laplacian in Rd are well known. Nevertheless, the above problems are not of perturbational nature since there is no assumption on the asymptotic behavior of the coefficients of the given operator near infinity. Problem 1 was posed by B. Simon in [23, 24]. Clearly, the answer to Problem 1 is ‘no’ for d ≥ 3. Partial results concerning Problem 1 for d ≤ 2 were obtained under the assumption that V is a short-range potential (see for example, [12, 13, 16, 18]). On the other hand, Gesztesy, and Zhao showed in [12, Example 4.6] that there is a critical Schrödinger operator on R with ‘almost’ short-range potential such that its ground state behaves logarithmically at infinity. In a recent article Damanik, Killip, and Simon proved a result which reveals a complete answer to Problem 1. Theorem 1.4 (cf. [10, Theorem 5]). The answer to Problem 1 is “yes” if and only if d = 1, 2. Indeed, for d = 1, 2, it is shown in [10] that if the equation HV u = 0 admits a positive bounded solution, then any W ∈ L 2loc (Rd ) satisfying HV ±W ≥ 0 is necessarily the zero potential. But this property holds if and only if HV is critical (see [18]). Let us turn to Problem 2 which was raised by Berestycki, Caffarelli, and Nirenberg [5]. This problem is closely related to De Giorgi’s conjecture [11] (see [3–5, 14]). In [14], Ghoussoub and Gui showed a connection between Problem 2 and Problem 3 which concerns the Liouville property for operators in divergence form (see also the proof of Theorem 1.7 in [5]). In fact, the following result is proved in [5, 14, 3]. Theorem 1.5. The answers to Problems 2 and 3 are “yes” if and only if d = 1, 2.

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Note that Ghoussoub and Gui [14] used this Liouville-type theorem for d = 2 [5], to prove De Giorgi’s Conjecture in R2 . Problem 3 was posed by Berestycki and Rossi [7] who also solved it for d ≤ 3: Theorem 1.6 ([7, Theorem 4.1]). Suppose that P = −∇ · (A∇) + V is a uniformly elliptic operator with smooth bounded coefficients on Rd . If d ≤ 3, then λ0 (P, Rd ) ≤ λ0 (P, Rd ). The purpose of the present article is to (partially) generalize Theorems 1.4, 1.5, and 1.6 to general symmetric operators which are defined on an arbitrary domain ⊆ Rd , or on a noncompact Riemannian manifold. Our main result is as follows. Theorem 1.7. Let be a domain in Rd , d ≥ 1. Consider two Schrödinger operators defined on of the form P j := −∇ · (A j ∇) + V j

j = 0, 1,

(1.8)

p

such that V j ∈ L loc (; R) for some p > d/2, and A j satisfy (1.1). Assume that the following assumptions hold true. (i) The operator P1 is critical in . Denote by ϕ ∈ C P1 () its ground state. 1 () such that ψ = 0, (ii) λ0 (P0 , ) ≥ 0, and there exists a real function ψ ∈ Hloc + and P0 ψ ≤ 0 in , where u + (x) := max{0, u(x)}. (iii) The following matrix inequality holds: (ψ+ )2 (x)A0 (x) ≤ Cϕ 2 (x)A1 (x)

a. e. in ,

(1.9)

where C > 0 is a positive constant. Then the operator P0 is critical in , and ψ is its ground state. In particular, dim C P0 () = 1 and λ0 (P0 , ) = 0. Corollary 1.8. Suppose that all the assumptions of Theorem 1.7 are satisfied except possibly the assumption that λ0 (P0 , ) ≥ 0. Assume further that either ψ changes its sign in , or ψ is not a solution of the equation P0 u = 0 in . Then λ0 (P0 , ) < 0. Theorem 1.7 and Corollary 1.8 imply in particular the sufficiency parts of Theorem 1.4 and Theorem 1.5, and also Theorem 1.6 for d = 1, 2. We point out that Theorem 1.6 holds also for d = 3, and this case is not covered by our result. Note that in contrast to the assumptions of Theorem 1.5 and 1.6, in Theorem 1.7 we assume neither that the functions V j are bounded and smooth, nor that the matrix valued functions A j are smooth. The outline of the paper is as follows. In Sect. 2, we present some results from [19] that will be used in the proof of Theorem 1.7. Sect. 3 is devoted to the proof of Theorem 1.7 and its consequences. We conclude the paper in Sect. 4, where we pose two open problems suggested by the results of the present paper.


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2. Preliminary Results Definition 2.1. We say that a sequence {u k } ⊂ C0∞ () is a null sequence for the quadratic form a A,V if there exists an open set B such that nonnegative 2 B |u k | dx = 1, and lim a A,V [u k ] = 0. (2.1) k→∞

We say that a positive function ϕ is a null st at e for the nonnegative quadratic form a A,V , if there exists a null sequence {u k } for the form a A,V such that u k → ϕ in L 2loc (). Remark 2.2. The requirement that u k ⊂ C0∞ (), can clearly be weakened by assuming only that {u k } ⊂ H01 (). Also, the requirement that B |u k |2 dx = 1 can be replaced by 2 B |u k | dx 1, where f k gk means that there exists a positive constant C such that −1 C gk ≤ f k ≤ Cgk for all k ∈ N. The following auxiliary lemma is well known (see, e.g. [9, 17, 19]). 1 () be a nonnegative subsolution of the equation Pψ = 0 in Lemma 2.3. Let ψ ∈ Hloc . Then for any nonnegative v ∈ C0∞ () we have a A,V [ψv] ≤ (ψ)2 A∇v · ∇v dx. (2.2)

Moreover, if ψ is a (real valued) solution of the equation Pψ = 0 in , then for any v ∈ C0∞ () we have a A,V [ψv] = (ψ)2 A∇v · ∇v dx. (2.3)

Proof. Follows from the definition of a weak (sub)solution and elementary calculation. The following theorem was recently proved by K. Tintarev and the author [19] (see also [20]). Theorem 2.4. Suppose that a A,V ≥ 0 on C0∞ (). Then a A,V has a null sequence if and only if the corresponding operator P = −∇ · (A∇) + V is critical in . In this case, any null sequence converges in L 2loc () to cϕ, where ϕ is a ground state of the operator P and c is a nonzero constant. Moreover, there exists a null sequence {u k } of nonnegative functions that converges to ϕ locally uniformly in \ {x0 }, where x0 is some point in . 3. Proof of Theorem 1.7 In this section we prove Theorem 1.7 and some consequences. Proof of Theorem 1.7. Since ψ satisfy P0 ψ ≤ 0 in , it follows that P0 ψ+ ≤ 0 in (see for example [1, Lemma 2.7]). By Theorem 2.4 and our assumptions, there exists a null sequence {u k } for the quadratic form a A1 ,V1 of nonnegative functions which converges locally uniformly in \{x0 } and in L 2loc () to the ground state ϕ of the operator P1 , and satisfies B (u k )2 dx = 1 for some open set B \ {x0 } and all k ∈ N.

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Denote wk := u k /ϕ. Since wk → constant locally uniformly in \{x0 } and ψ+ = 0, it follows that B1 (ψ+ wk )2 dx 1 for some open set B1 and every k ≥ k0 . Moreover, by Lemma 2.3 and our assumptions, we have a A0 ,V0 [ψ+ wk ] ≤ (ψ+ )2 A0 ∇wk · ∇wk dx ≤

C

ϕ 2 A1 ∇wk · ∇wk dx = Ca A1 ,V1 [ϕwk ] = Ca A1 ,V1 [u k ] → 0. (3.1)

Therefore, {ψ+ wk } is a null sequence for P0 . By Theorem 2.4, P0 is critical in and ψ+ is its ground state. In particular, ψ+ is strictly positive, and hence ψ− = 0, and ψ = ψ+ is the ground state of P0 . Remark 3.1. Suppose that all the assumptions of Theorem 1.7 are satisfied except possibly the assumption that λ0 (P0 , ) ≥ 0. One can show directly that λ0 (P0 , ) ≤ 0. Indeed, using the notations of the proof of Theorem 1.7, we have that for some C1 > 0, (ψ+ wk )2 dx ≥ (ψ+ wk )2 dx ≥ C1 = C1 (u k )2 dx ∀k ≥ k0 .

B1

B

Moreover, by Lemma 2.3 and our assumptions, we have 2 a A0 ,V0 [ψ+ wk ] + ) A0 ∇wk · ∇wk dx (ψ ≤ ≤ 2 2 (ψ+ wk ) dx B1 (ψ+ wk ) dx C˜

ϕ

1 ∇wk · ∇wk 2 B (u k ) dx

2A

dx

˜ A1 ,V1 [ϕwk ] = Ca ˜ A1 ,V1 [u k ] → 0. = Ca (3.2)

In light of the Rayleigh-Ritz variational formula and the Allegretto-Piepen-brink theorem, estimate (3.2) implies that λ0 (P0 , ) ≤ 0. It follows that λ0 (P0 , ) ≤ inf { λ ∈ R | ∃ ψ 0, (P0 − λ)ψ ≤ 0 in s.t. ψ+2 (x)A0 (x) ≤ Cϕ 2 (x)A1 (x) in for some critical operator P1 with a ground state ϕ } . In particular, if P = −∇ · (A∇) + V is an elliptic operator on Rd , d ≤ 2, with a bounded matrix A, then λ0 (P, Rd ) ≤ λ0 (P, Rd ) (cf. Theorem 1.6). Recall that if P := −∇ · (A∇) + V is Zd -periodic on Rd , then P − λ0 admits a (unique) periodic positive solution (see for example [15, 21]). On the other hand, − is critical in Rd if and only if d ≤ 2 (see Example 1.2). Therefore, Theorem 1.7 implies the following result of R. Pinsky (who proved it for general second-order elliptic Zd -periodic operators). Corollary 3.2 ([22]). Assume that the coefficients of the elliptic operator P := −∇ · (A∇) + V are Zd -periodic on Rd . Then the operator P − λ0 is critical in Rd if and only if d ≤ 2.


81

Remark 3.3. Suppose that P j are two nonnegative symmetric operators which are defined on a noncompact Riemannian manifold M of dimension d, where j = 0, 1. Since Lemma 2.3 holds true also in this case (see [17]), it follows that Theorem 2.4 is valid on Riemannian manifolds, which in turn implies that Theorem 1.7 holds true also in this case. Recall that a Riemannian manifold M is called recurrent if the Laplace-Beltrami operator on M is critical (see [21]). Therefore, we have in particular, the following generalization of Theorem 1.4 and Theorem 1.5. Theorem 3.4. Let M be a recurrent Riemannian noncompact manifold of dimension d. Let V ∈ L 2loc (M). Suppose that HV := − + V ≥ 0 on C0∞ (M), and that the equation HV u = 0 in M admits a nonzero bounded subsolution ψ such that ψ+ = 0. Then HV is critical in M and ψ is a ground state of HV in M. In particular, λ0 (HV ) = 0, the space of all bounded solutions of the equation HV u = 0 in M is one-dimensional, and dim C HV (M) = 1 . In addition, one can use the results in [15] and [8, Theorem 5.2.11] to extend Corollary 3.2 to the case of equivariant Schrödinger operators on cocompact nilpotent coverings. Corollary 3.5. Let M be a noncompact nilpotent covering of a compact Riemannian manifold of dimension d. Suppose that P := − + V is an equivariant operator on M with respect to the (nilpotent) deck group G. Then P − λ0 is critical in M if and only if G has a normal subgroup of finite index isomorphic to Zn for n ≤ 2. Theorem 1.7 might be called a Liouville comparison theorem and can be considered as a sufficient condition for the removability of singularity at infinity in or as a Phragmén-Lindelöf type principle. A positive solution of (1.3) in \ K , where K , is called singular at infinity if it does not have minimal growth in a neighborhood of infinity in . Accordingly, Theorem 1.7 implies that the behavior of a positive solution of minimal growth in a neighborhood of infinity in of an equation of the form (1.8), dictates some ‘growth’ on all positive singular at infinity solutions of any equation of the form (1.8). More precisely, we have the following result. Corollary 3.6. Suppose that for j = 0, 1 the operators P j are of the form (1.8), and A j satisfy (1.1). Let u 1 be a positive solution of the equation P1 u = 0 of minimal growth in a neighborhood of infinity in , and let u 0 be a positive solution of the equation P0 u = 0 in \ K , where K . If (u 0 )2 A0 ≤ C(u 1 )2 A1 in \ K , then u 0 is nonsingular at infinity, i.e., u 0 is a positive solution of the equation P0 u = 0 of minimal growth in a neighborhood of infinity in . 1 () be positive functions which are defined Proof of Corollary 3.6. Let u0 , u1 ∈ Hloc in such that uj |\K 1 = u j , and uj | K 1 are sufficiently smooth, where K 1 , and j = 0, 1. Then for j = 0, 1, uj ∈ C P j (), where the operators P j are of the form (1.8) and satisfy P j |\K 2 = P j for some K 2 . Since u 1 (and hence also u1 ) is a positive solu1 u = 0 of minimal growth in a neighborhood of infinity in , it tion of the equation P 1 in . Therefore, Theorem 1.7 follows that u1 is a ground state of the critical operator P 0 in . Hence, u 0 is a positive implies that u0 is a ground state of the critical operator P solution of the equation P0 u = 0 of minimal growth in a neighborhood of infinity in .

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Example 3.7. Let d ≥ 2, and V ∈ L loc (Rd ), where p > d/2. Suppose that HV := − + V ≥ 0 on C0∞ (Rd ), and the equation HV u = 0 on Rd has a subsolution ψ 0 satisfying ψ+ (x) = O(|x| Since ϕ(x) := |x|

2−d 2

2−d 2

)

as |x| → ∞.

(3.3)

is a positive solution of the Hardy-type equation d −2 2 u −u − =0 2 |x|2

of minimal growth in a neighborhood of infinity in Rd , it follows from Corollary 3.6 that HV is critical in Rd and ψ is its ground state (cf. Theorem 1.7 in [5]). p

Example 3.8. Let d = 1, and V ∈ L loc (R), where p > 1. Suppose that HV := −d2 /dx 2 + V ≥ 0 on C0∞ (R), and the equation HV u = 0 on R has a subsolution ψ 0 satisfying ψ+ (x) = O(log |x|) as |x| → ∞. (3.4) It follows from [12, Example 4.6] and Corollary 3.6 that HV is critical in R and ψ is its ground state. 4. Open Problems We conclude the paper with two open problems suggested by the above results which are left for future investigation. Problem 5. Generalize Theorem 1.7 to the class of nonsymmetric second-order linear elliptic operators with real coefficients which have the same (or even comparable) principal parts, or at least to the subclass of operators which differ only by their zero-order terms. Remark 4.1. 1. Clearly, the condition (1.9) which involves the squares of the corresponding solutions of the symmetric operators P j , for j = 0, 1, should be replaced in the nonsymmetric case by a condition which involves products of the form u j u ∗j , where u j (resp. u ∗j ) are the corresponding solutions of the operators P j (resp. of the formal adjoint operators PJ∗ ) for j = 0, 1. 2. Let u be a positive solution of an equation of the form (1.3) of minimal growth in a neighborhood of infinity in , then Corollary 3.6 implies that any positive solution v of another equation of the form (1.3) (with a comparable principal part) in a neighborhood of infinity in which is singular at infinity satisfies lim inf x→∞

u(x) =0 v(x)

in the one-point compactification of (∞ denotes the point at infinity in ). Ancona proved [2] that a subcritical symmetric (or even quasi-symmetric) operator P in always admits v ∈ C P (), such that lim

x→∞

G P (x, x 0 ) = 0. v(x)


83

Moreover, it is shown in [2] that such a positive solution does not always exist for general nonsymmetric operators. This result indicates that the answer to Problem 5 in the nonsymmetric case might be more involved. The second problem that we pose deals with Liouville-comparison theorems for the p-Laplacian with a potential term. Let be a domain in Rd , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (). Recently the criticality theory for linear equations was extended in [20] to quasilinear equations of the form −∇ · (|∇u| p−2 ∇u) + V |u| p−2 u = 0

in .

(4.1)

In particular, Theorem 2.4 was proved also for such equations. Therefore, it is natural to pose the following problem. Problem 6. Assume that 1 < p ≤ d. Generalize Theorem 1.7 to positive solutions of quasilinear equations of the form (4.1). Remark 4.2. An answer to Problem 6 (for 1 < p < ∞) was obtained after the present paper was submitted, and will appear elsewhere. Acknowledgements. The author wishes to thank H. Brezis, F. Gesztesy, and M. Marcus for valuable discussions and N. Schwartz for the musical inspiration. The author is also grateful to the anonymous referee for his careful reading and useful comments. This work was partially supported by the RTN network “Nonlinear Partial Differential Equations Describing Front Propagation and Other Singular Phenomena”, HPRN-CT2002-00274, the Israel Science Foundation (grant No. 1136/04) founded by the Israeli Academy of Sciences and Humanities, and the Fund for the Promotion of Research at the Technion.

References 1. Agmon, S.: On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. In: “Methods of Functional Analysis and Theory of Elliptic Equations” (Naples, 1982), Naples: Liguori, 1983, pp. 19–52 2. Ancona, A.: Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators. Nagoya Math. J. 165, 123–158 (2002) 3. Barlow, M.T.: On the Liouville property for divergence form operators. Canad. J. Math. 50, 487–496 (1998) 4. Barlow, M.T., Bass, R.F., Gui, C.: The Liouville property and a conjecture of De Giorgi. Comm. Pure Appl. Math. 53, 1007–1038 (2000) 5. Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25, 69–94 (1997) 6. Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: I - Influence of periodic heterogeneous environment on species persistence. J. Math. Biology 51, 75– 113 (2005) 7. Berestycki, H., Rossi, L.: On the principal eigenvalue of elliptic operators in R N and applications. J. Europ. Math. Soc. (2006), in press 8. Davies E.B.: “Heat Kernels and Spectral Theory”. Cambridge Tracts in Mathematics, 92, Cambridge: Cambridge University Press, 1989 9. Davies, E.B., Simon, B.: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984) 10. Damanik, D., Killip, R., Simon, B.: Schrödinger operators with few bound states. Commun. Math. Phys. 258, 741–750 (2005) 11. De Giorgi, E.: Convergence problems for functionals and operators. In: “Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis” (Rome, 1978), Bologna: Pitagora Ed., 1979, pp. 131–188 12. Gesztesy, F., Zhao, Z.: On critical and subcritical Sturm-Liouville operators. J. Funct. Anal. 98, 311– 345 (1991) 13. Gesztesy, F., Zhao, Z.: On positive solutions of critical Schrödinger operators in two dimensions. J. Funct. Anal. 127, 235–256 (1995)

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14. Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998) 15. Lin, V., Pinchover, Y.: “Manifolds with Group Actions and Elliptic Operators”. Memoirs AMS, no. 540, 1994 16. Murata, M.: Positive solutions and large time behaviors of Schrödinger semigroups, Simon’s problem. J. Funct. Anal. 56, 300–310 (1984) 17. Murata, M.: Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations. J. Funct. Anal. 194, 53–141 (2002) 18. Pinchover Y.: Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations. In: “Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday”, Proceedings of Symposia in Pure Mathematics, Providence, RI: Amer. Math. Soc. to appear 19. Pinchover, Y., Tintarev, K.: Ground state alternative for singular Schrödinger operators. J. Funct. Anal. 230, 65–77 (2006) 20. Pinchover, Y., Tintarev, K.: Ground state alternative for p-Laplacian with potential term, to appear in Calc. Var. Partial Differential Equations, http://arxiv.org/PS_cache/math/pdf/0511/0511039.pdf 21. Pinsky, R.G.: “Positive Harmonic Functions and Diffusion”. Cambridge Studies in Advanced Mathematics 45, Cambridge: Cambridge University Press, 1995 22. Pinsky, R.G.: Second order elliptic operators with periodic coefficients: criticality theory, perturbations, and positive harmonic functions. J. Funct. Anal. 129, 80–107 (1995) 23. Simon, B.: Large time behavior of the L p norm of Schrödinger semigroups. J. Funct. Anal. 40, 66– 83 (1981) 24. Simon, B.: Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7, 447–526 (1982) Communicated by B. Simon


Communications in


Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures Maciej Dunajski, Simon West Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK. E-mail: [email protected]; [email protected] Received: 27 January 2006 / Accepted: 4 October 2006 Published online: 2 March 2007 – © Springer-Verlag 2007

Dedicated to the memory of Jerzy Plebański Abstract: Using twistor methods, we explicitly construct all local forms of four–dimensional real analytic neutral signature anti–self–dual conformal structures (M, [g]) with a null conformal Killing vector. We show that M is foliated by antiself-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of (M, [g]) by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface orthogonal in (M, [g]). We give examples of conformal classes which contain Ricci–flat metrics on compact complex surfaces and discuss other conformal classes with no Ricci–flat metrics.

1. Introduction The anti–self–duality (ASD) condition in four dimensions seems to underlie the concept of integrability of ordinary and partial differential equations [29]. Many lower dimensional integrable models (KdV, NlS, Sine–Gordon, ...) arise as symmetry reductions of the ASD Yang–Mills equations on a flat background, and various solution generation techniques are reductions of the twistor correspondence [19]. Other integrable models (dispersionless Kadomtsev–Petviashvili, SU (∞) Toda, ...) are reductions of the ASD conformal equations which say that the self–dual Weyl tensor of a conformal class of metrics vanishes [30, 7]. Generalisations to ASD Yang–Mills on ASD conformal background are also possible [27, 4]. In all cases the main interest is in conformal structures of signature + + −− which are called neutral, as the reductions can lead to interesting hyperbolic and parabolic equations. There are no non–trivial ASD structures in the Lorentzian signature + + +−, and the reductions from Riemannian manifolds can only yield elliptic equations thus ruling out interesting soliton dynamics.

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The main gap in the programme to classify the reductions of ASD neutral conformal structures was understanding the reductions by a null conformal Killing vector. We embarked on this project hoping to incorporate more integrable systems into the framework of anti–self–duality, but we have found (Theorem 2) that the resulting geometry is a completely solvable system. Let (M, [g]) be a four dimensional real analytic neutral ASD conformal structure. We say that K is a null conformal Killing vector if it satisfies L K g = ηg,

g(K , K ) = 0,

(1.1)

for some g ∈ [g], where η is a function on M, and L is the Lie-derivative. When studying conformal structures with non-null conformal Killing vectors, it is natural to look at the space of Killing vector trajectories, since this will inherit a nondegenerate conformal structure. In the case of a null conformal Killing vector, the situation is different. The space of trajectories inherits a degenerate conformal structure. We find that it is necessary to go down one dimension more, and consider a two dimensional space U of anti-self-dual totally null surfaces in M, called β–surfaces, containing K , which exist as a consequence of the conformal Killing equation. It turns out that there is a naturally defined projective structure [] on U . Moreover, we show that the twistor spaces of (M, [g]) and (U, []) are related by dimensional reduction. Specifically, the twistor space Z of (U, []) is the space of trajectories of a vector field on the twistor space PT of (M, [g]) corresponding to K . Projective structures are just equivalence classes of torsion-free connections, which do not need to satisfy any equations; this underlies the complete solvability of null reductions, and contrasts with the non-null case where one obtains Einstein-Weyl structures [13], and associated integrable systems [30, 7, 6, 4]. In Sect. 2 we derive some elementary properties of null conformal Killing vectors. Section 3 is an introduction to projective structures. In Sect. 4 we prove the following: Theorem 1. Let (M, [g]) be a four dimensional real analytic neutral ASD conformal structure with a null conformal Killing vector K . Let U be the two dimensional space of β-surfaces containing K . Then there is a naturally defined projective structure on U , induced on PT by whose twistor space is the space of trajectories of a distribution K the action of K on M. In Sect. 5 we investigate the local form of ASD conformal structures with null Killing vectors. This is expressed in the following theorem: Theorem 2. Let (M, [g], K ) be a smooth neutral signature ASD conformal structure with null conformal Killing vector. Then there exist local coordinates (t, x, y, z) and g ∈ [g] such that K = ∂t and g has one of the following two forms, according to whether the twist K ∧ dK vanishes or not (K := g(K , .)): 1. K ∧ dK = 0. g = (dt + (z A3 − Q)dy)(dy − βd x) − (dz − (z(−β y + A1 + β A2 + β 2 A3 ))d x −(z(A2 + 2β A3 ) + P)dy)d x, (1.2) where A1 , A2 , A3 , β, Q, P are arbitrary functions of (x, y). 2. K ∧ dK = 0. g = (dt + A3 ∂z Gdy + (A2 ∂z G + 2 A3 (z∂z G − G) − ∂z ∂ y G)d x)(dy − zd x) −∂z2 Gd x(dz − (A0 + z A1 + z 2 A2 + z 3 A3 )d x),

(1.3)

ASD Conformal Structures with Null Killing Vectors from Projective Structures

87

where A0 , A1 , A2 , A3 are arbitrary functions of (x, y), and G is a function of (x, y, z) satisfying the following PDE: (∂x + z∂ y + (A0 + z A1 + z 2 A2 + z 3 A3 )∂z )∂z2 G = 0.

(1.4)

The functions Aα (x, y) in the metrics (1.2) and (1.3) determine projective structures on the two dimensional space U in the following way. A general projective structure corresponds to a second-order ODE, dy 3 dy 2 dy d2 y + A0 (x, y). = A (x, y) + A (x, y) + A (x, y) 3 2 1 dx2 dx dx dx

(1.5)

In (1.3) all the Aα , α = 0, 1, 2, 3 functions occur explicitly in the metric. In (1.2) the function A0 does not explicitly occur. It is determined by the following equation: A0 = βx + ββ y − β A1 − β 2 A2 − β 3 A3 ,

(1.6)

as is shown in the proof of the theorem. If one interprets z as a fibre coordinate on the projective tangent bundle of the (x, y) space, then (1.4) says that ∂z2 G is constant along the projective structure spray (compare formula 3.4). Note that in both cases the Killing vector is ∂t and is pure Killing (this comes from choosing a suitable g ∈ [g]). The non-twisting case (1.2) is a natural conformal generalisation of Ricci–flat pp waves. The twisting case (1.3) is a neutral analog of the Fefferman conformal class [10]. As special cases of (1.3) we recover some examples of [20], where neutral metrics were related to second order ODEs. The aim of Sect. 6 is to put our results into a broader context. We examine some examples found by different means in the light of our results. We find necessary and sufficient conditions on the underlying projective structure in order for there to exist (pseudo) hyper–complex metrics with triholomorphic K within a conformal class. A special case of the metric (1.2) yields a compact example of a Ricci–flat metric on a Kodaira surface of a special type. We consider how to construct conformal structure twistor spaces from projective structure twistor spaces in Sect. 7. The more involved spinor calculations are moved to the Appendix. 2. Null Conformal Killing Vectors 2.1. Spinors in neutral signature. We will denote by (M, [g]) a local patch of R4 endowed with a neutral signature conformal structure [g]. That is, [g] is an equivalence class of neutral signature metrics with the equivalence relation g ∼ ec g for some function c on M. Any neutral metric g on M can be put in the following form:

g = 2(θ 00 θ 11 − θ 01 θ 10 ) = AB A B θ A A ⊗ θ B B ,

(2.1)

where AB , A B are antisymmetric matrices with 01 = 0 1 = 1. The four (real) basis one-forms θ A A for A = 0, 1, A = 0, 1 are called a tetrad. The algebraic dual vector basis is denoted e A A , and is defined by θ A A (e B B ) = δ BA δ BA . Any vector V at a point can be written V A A e A A , and this exhibits an isomorpism TM ∼ = S ⊗ S,

(2.2)

88


where S, S are two-dimensional real vector bundles known as the unprimed spin bundle and the primed spin bundle respectively. For a general manifold M there is a topological obstruction to (2.2) but we are working locally so it always holds. Using a particular choice of tetrad, a section µ of S is denoted µ A , A = 0, 1. Simi larly ν A is a section of S ∗ , κ A a section of S and τ A a section of S ∗ , where ∗ denotes the dual of a bundle. The natural pairing S × S ∗ → R is given by µ A ν A , using the summation convention, and similarly for primed spinors. We sometimes use the notation µ A ν A = µ.ν. This product is not commutative; we have µ.ν = −ν.µ. It follows from (2.1) that g(V, V ) = det V A A . If V is null, then this gives V A A = µ A κ A . Abstractly, if V is null then V = µ ⊗ κ under the isomorphism (2.2), where µ and κ are sections of S and S respectively. The relation (2.1) can be written abstractly as g = ⊗ under the isomorphism (2.2). and are symplectic structures on S and S . These give isomorphisms S ∼ = S ∗ and S ∼ = S ∗ by µ → (µ, .), for µ a section of S, and similarly for S . Given a choice of tetrad, the spinors and are written AB and A B , where we drop the prime on the latter because no confusion can arise due to the indices. Note these are anti-symmetric in AB and A B . Then the isomorphism S ∼ = S ∗ is given in the A B trivialization by µ → µ B A := µ A and similarly for primed spinors. There are useful isomorphisms 2+ ∼ = Sym(S ∗ ⊗ S ∗ ), 2− ∼ = Sym(S ∗ ⊗ S ∗ ),

(2.3)

where + , − are the bundles of self-dual and anti-self-dual two-forms, using an appropriate choice of volume form for the Hodge-∗ operator. In the local trivialization, the isomorphisms (2.3) are expressed by the following formula for a two-form F in spinors: Fab = FA A B B = φ A B AB + ψ AB A B , where φ A B , ψ AB are symmetric. The φ A B term is the self-dual component of F and the ψ AB is the anti-self-dual component. The vector bundles S, S and their duals inherit connections from the Levi-Civita connection of T M (see Appendix A). These are the unique torsion-free connections defined so that the sections and are covariantly constant. Then covariant differentiation on either side of (2.2) is consistent. A primed spinor κ A at a point corresponds to a totally null self-dual two-plane spanned by κ A e A A , A = 1, 2, whilst an unprimed spinor corresponds to an anti-selfdual two-plane in a similar way. In twistor theory, these two-planes are called α-planes and β-planes respectively. 2.2. Null conformal Killing vectors in neutral signature. Suppose g is a neutral metric with a conformal Killing vector K . Then L K (ec g) = (K (ec ) + ec η)g, so K is a conformal Killing vector for the conformally rescaled metric, and we can refer to K as a conformal Killing vector for the conformal structure [g]. Now suppose g has a null conformal Killing vector K . We shall show (Lemma 1) that M is foliated in two different ways, by self-dual and anti-self-dual surfaces, whose leaves intersect tangent to K . This is a property of the conformal structure [g], since the Hodge-∗ acting on 2-forms is conformally invariant.


89

The spinor form of the conformal Killing equation is: 1 ∇a K b = φ A B AB + ψ AB A B + η AB A B , 2

(2.4)

where φ A B , ψ AB are the self-dual and anti-self dual parts of the 2-form ∇[a K b] , and η is a function on M. Since K is null, we have K = ι ⊗ o, where ι is a section of S and o a section of S . Choosing a null tetrad, and a trivialization of S and S , we have K A A = ι A o A . These spinors are defined up to multiplication by a non-zero function α, since K A A = ι A o A = (αι A )(o A /α).

Lemma 1. Let K = ι A o A e A A be a null conformal Killing vector. Then 1. The following algebraic identities hold: ι A ι B ψ AB = 0,

(2.5)

o o φ A B = 0.

(2.6)

A B

2. ι A and o A satisfy A

ι A ι B ∇ B B ι A = 0, B

o o ∇

B B

o

A

= 0.

(2.7) (2.8)

Remark. Equations (2.7), (2.8) are equivalent to the statement that the distributions spanned by ι A e A A and o A e A A are Frobenius integrable (see the Appendix). Equations of this type are often called ‘geodesic shear free’ equations, since in the Lorentzian case they result in shear-free congruences of null geodesics. Proof. Using K A A = ι A o A , the Killing equation (2.4) becomes 1 o A ∇ B B ι A + ι A ∇ B B o A = φ A B AB + ψ AB A B + η AB A B . 2

(2.9)

Contracting both sides with ι A o A gives 1 0 = o A ι B φ A B + ι A o B ψ AB + ηι B o B . 2

Multiplying by ι B and o B respectively leads to (2.5) and (2.6). To get (2.7) and (2.8),

multiply (2.9) by ι A ι B and o A o B . We have found that M is foliated in two different ways by totally null surfaces. Those determined by o A are self-dual and are called α-surfaces, and those determined by ι A are anti-self-dual and are called β-surfaces. It is clear that the α-surfaces and β-surfaces of Lemma 1 intersect on integral curves of K . Denote the β-surface distribution by Dβ ; this will be used later. It is appropriate here to recall the Petrov-Penrose classification [22] of the algebraic type of a Weyl tensor. In split signature this applies separately to C ABC D and C A B C D . In our case C A B C D = 0 and we are concerned with the algebraic type of C ABC D . When we refer to the algebraic type we will be referring to the algebraic type of C ABC D . One can form a real polynomial of fourth order P(x) by defining µ A = (1, x) and setting

90


P(x) = µ A µ B µC µ D C ABC D . The Petrov-Penrose classification refers to the position of roots of this polynomial, for example if there are four repeated roots then we say C ABC D is type N. If there is a repeated root the metric is called algebraically special. There are additional complications in the split signature case [16] arising from the fact that real polynomials may not have real roots. The split signature version of the Goldberg-Sachs theorem together with (2.7) implies that any Ricci-flat ASD space with null conformal Killing vector is algebraically special. In fact the vacuum condition can be removed if K is non-twisting; we will discuss this further in Sect. 6.5. It also follows from the Killing equations and the fact that K is null that K b ∇b K a =

1 ηK a . 2

Thus K is automatically geodesic, and if it is pure then its trajectories are parameterized by an affine parameter. 3. Projective Structures Let (U, []) be a local two dimensional real projective structure. That is, U is a local patch of R2 , and [] is an equivalence class of torsion-free connections whose unparameterized geodesics are the same. Then in a local trivialization, equivalent torsion-free connections are related in the following way: ˜ ijk − ijk = a j δki + ak δ ij ,

(3.1)

for functions ai on U , and i, j, k = 1, 2. Note that this is a tensor equation since the difference between two connections is a tensor. The ai on the RHS are the components of a one-form. The geodesics satisfy the following ODE: j k d 2si ds i i ds ds = v , + jk dt 2 dt dt dt

where s i are local coordinates of U , and t is a parameter, which is called affine if v = 0. One can associate a second-order ODE to a projective structure by picking a connection in the equivalence class, choosing local coordinates s i = (x, y) say, and eliminating the parameter from the geodesic equations. The resulting equation determines the geodesics in terms of the local coordinates, without the parameter. The equation is as follows: dy 3 dy 2 d2 y y y dy y x x − x x . = + (2 − ) + (xx x − 2x y ) yy yy xy 2 dx dx dx dx

(3.2)

A general projective structure is therefore defined by a second-order ODE (1.5). In fact, two of the four functions A0 , A1 , A2 , A3 can be eliminated by a coordinate transformation (x, y) → (x(x, ˆ y), yˆ (x, y)) which introduces two arbitrary functions. On T U , the horizontal lifts of ∂/∂s i are defined by Si =

∂ ∂ j − ik v k j , ∂s i ∂v


91

where v i , i = 1, 2 are the fibre coordinates of T U . The geodesics on U lift to integral curves of the following spray on T U : = v i Si = v i

∂ ∂ − ijk v j v k i . i ∂s ∂v

(3.3)

Now is homogeneous of degree 1 in the v i , so it projects to a section of a one dimensional distribution on P T U . P T U is the quotient of T U − {0-section} by the vector field v i ∂v∂ i . If λ is a standard coordinate on one patch of the RP1 factor,1 then the spray has the form = ∂x + λ∂ y + (A0 (x, y) + λA1 (x, y) + λ2 A2 (x, y) + λ3 A3 (x, y))∂λ .

(3.4)

There is a unique curve in any direction through a point in U , which is why the curves can be lifted to a foliation of the projective tangent bundle U × RP1 . ˜ is the spray corresponding to a different To obtain (3.1) we argue as follows. If ˜ ˜ ˜ push down to connection , then and are in the same projective class if and the same spray on P T U . This gives ∂ , ∂v i from which (3.1) follows, using the fact that the connections are torsion-free (i.e. symmetric in their lower indices). ˜ ∝ vi −

3.1. The twistor space of a projective structure. Now suppose we have a holomorphic projective structure on a local patch of C2 , which we still denote U . All of the above is still valid, with real coordinates replaced by complex ones. The functions ijk are now required to be holomorphic functions of the coordinates. Given a real-analytic projective structure, one can complexify by analytic continuation to obtain a holomorphic projective structure that will come equipped with a reality structure (see below). ∂ The space P T U is obtained from T U on quotienting by µi ∂µ i , which defines a tautological line bundle O(−1) over P T U . As the Si are weight zero in the µi coordinates, they push down to vector fields on P T U , giving a two-dimensional distribution S. Since is weight one in the µi , one must divide by a homogeneous polynomial of degree one in the µi to get something that pushes down to a vector field on P T U . The resulting vector field will have a singularity at a single point on each fibre, where the degree one polynomial vanishes. Different choices of polynomial will result in different vector fields on P T U , but they will always be in the same direction. In other words, defines a one dimensional distribution which we shall call D . Restricting to a CP1 fibre, D defines a line bundle over CP1 . A section of this line bundle corresponds to a vector field in D , i.e. a choice of polynomial as described above, and has a pole at a single point. Therefore by the classification of holomorphic line bundles over CP1 , it must be O(−1).2 1 By standard coordinates λ, λ ˜ on RP1 or CP1 , we mean the usual coordinates v 1 /v 0 and v 0 /v 1 , where v 0 , v 1 are homogeneous coordinates. 2 Coordinatize CP1 using two patches, U with coordinate λ ∈ C, and U with coordinate η ∈ C, and 0 1 transition function λ = 1/η. The holomorphic line bundle O(n) over CP1 is defined by the transition function a = λn b, where a ∈ C is the fibre coordinate over U0 and b(η) ∈ C is the fibre coordinate over U1 . The Birkhoff-Grothendieck theorem states that any holomorphic line bundle over CP1 is O(n) for some n. A global section of O(n) has |n| zeroes or poles, for n positive or negative respectively.

92


Restricting to a CP1 fibre, one obtains the following exact sequence of vector bundles over CP1 : 0 → O(−1) → O ⊕ O → S/D → 0,

(3.5)

where the first bundle is D , the second is S, and the last is the quotient. In fact, the quotient is O(1), for the following reason. Consider for instance the push down of S0 to P T U . This defines a subbundle of S that is different to everywhere except at a single point, the image of µ1 = 0. Hence it determines a section of S/ which vanishes at a single point. Therefore, again using the classification of holomorphic line bundles over CP1 , we have S/D ∼ = O(1). The twistor space Z is the two dimensional quotient of P T U by D . A point u ∈ U corresponds to a twistor line uˆ ⊂ Z corresponding to all the geodesics through u. The normal bundle of an embedded uˆ = CP1 ⊂ Z is given by the quotient bundle in the above sequence, i.e. O(1). This is summarized by a double fibration picture: U

U × CP1

Z

The left arrow denotes projection to U , and the right arrow denotes the quotient by D . The converse is also valid: Theorem 3 [11, 17]. There is a 1-1 correspondence between local two dimensional holomorphic projective structures and complex surfaces containing an embedded CP1 with normal bundle O(1). A vector V ∈ Tu U corresponds to a global section of the normal bundle O(1) of u. ˆ Such a section vanishes at a single point p ∈ Z . The geodesic of the projective structure through this direction is given by points in U corresponding to twistor lines in Z that intersect uˆ at p. That there is a one-parameter family of such lines can be shown by blowing up Z at the vanishing point and using Kodaira theory, see [11].

3.2. Flatness of projective structures. A projective structure is said to be flat if the corresponding second order ODE (1.5) can be transformed to the trivial ODE d2 y =0 dx2

(3.6)

by coordinate transformation (x, y) → (x(x, ˆ y), yˆ (x, y)). The terminology comes from the fact that given any second order ODE one can construct a Cartan connection on a certain G-structure [2], and when this connection is flat the equation can be transformed to the trivial ODE (3.6). It turns out that a second order ODE must be of the form (1.5) to be flat, and in addition the functions A0 , A1 , A2 , A3 must satisfy some PDEs. Defining F(x, y, λ) = A0 (x, y) + λA1 (x, y) + λ2 A2 (x, y) + λ3 A3 (x, y), the following must hold [2]: d d2 d F11 − 4 F01 − F1 F11 + 4F1 F01 − 3F0 F11 + 6F00 = 0, dx 2 dx dx

(3.7)


93

where F0 =

∂F , ∂y

F1 =

∂F , ∂λ

d ∂ ∂ ∂ = +λ +F . dx ∂x ∂y ∂λ

This is a set of PDEs for the functions A0 , A1 , A2 , A3 .

3.3. Reality conditions for projective structures. A reality structure for Z is an antiholomorphic involution that leaves invariant a two real parameter family of twistor lines, and fixes an equator of each line. Given a line in this real family, all the sections pointing to nearby lines in the real family have a zero at some point, and the union of these points gives an equator of the line; this equator must be fixed by the reality structure. The real family of twistor lines then corresponds to a real manifold U with a projective structure. In this paper all holomorphic projective structures have reality structures since they occur as complexifications of real projective structures.

4. Null Killing Vectors and Twistor Space 4.1. The twistor space of an ASD conformal structure. In the following and for the rest of the paper, e˜ A A denote the horizontal lifts of e A A to S , or their push-down to P S . Abstractly, the integral curves of these horizontal lifts define parallelly transported primed spinors using the connection on S (see Appendix A). We can abstractly define the two-dimensional twistor distribution on S as follows. A point s ∈ S is determined by a primed spinor π at a point x ∈ M. The null vectors π ⊗ µ for all unprimed spinors µ span an α-plane at x. Define the twistor distribution at s to be the subspace of horizontal vectors at s whose push-down to the base lies in this α-plane. Concretely, the twistor distribution is spanned by vectors L A (A = 0, 1) on S , defined with a choice of tetrad by ∂ L A = π A e˜ A A = π A e A A − A A BC π B , ∂π C

(4.1)

where π A are the local coordinates on the fibres of S . In the Appendix it is shown that the twistor distribution is integrable for ASD conformal structures, which is a seminal result of Penrose [21]. In other words, given a neutral ASD conformal structure [g], each self-dual two plane at a point is tangent to a unique α-surface through that point, which is the push down of a leaf of the twistor distribution. In the holomorphic case, the space of leaves of the twistor distribution (locally, over a suitably convex region of the base), is a three dimensional complex manifold PT called the twistor space [21, 11]. The double fibration picture is very similar to the projective structure case discussed in Sect. 3.1. The projective primed spin bundle P S is the quotient of S by the vector field π A ∂ A . P S is fundamental in the fibration picture, as each α-surface in M has a ∂π unique lift, in the same way that each geodesic of a projective structure has a unique lift to the projective tangent bundle. The horizontal vectors e˜ A A are weight zero in the π A coordinates, so push down to vector fields on P S , giving a four-dimensional distribution on P S . The L A vectors (4.1) are weight one, so together define a two dimensional

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subdistribution L of , which restricts to O(−1) ⊕ O(−1) on a CP1 fibre; we also refer to this as the twistor distribution. Over a CP1 fibre, there is an exact sequence 0 → O(−1) ⊕ O(−1) → O ⊗ C4 → /L → 0.

(4.2)

The first term is L, the second is . As in the projective structure case, one can show that /L is O(1) ⊕ O(1). The twistor space PT is the quotient of P S by L. The image of a CP1 fibre over x ∈ M is an embedded CP1 ∈ PT , and has normal bundle O(1) ⊕ O(1), the quotient bundle in (4.2). It corresponds to all the α-surfaces through x. The twistor correspondence is summarized by the double fibration: M

P S

PT

Here the left arrow denotes projection to M, and the right arrow denotes the quotient by L. Again, there is a converse: Theorem (Penrose [21]). There is a 1-1 correspondence between local four dimensional holomorphic ASD conformal structures (M, [g]) and three dimensional complex manifolds PT with an embedded CP1 ⊂ PT , with normal bundle O(1) ⊕ O(1). The essential fact is that an embedded CP1 with the above normal bundle belongs to a family of embedded CP1 s parameterized by a complex 4-manifold M. Vectors at x ∈ M correspond to sections of the normal bundle of x, ˆ and null vectors are given by sections with a zero. This defines a conformal structure, because a global section of O(1) ⊕ O(1) is given by (aλ + b, cλ + d) for affine coordinate λ ∈ C, (a, b, c, d) ∈ C4 , and this can only be (0, 0) when ad − bc = 0, which is a quadratic condition. In this case there is a zero at a single point. The conformal structure is anti-self-dual, with α-surfaces defined by families of twistor lines through a fixed point in PT . In this picture, the α-surfaces are obtained as follows. Let xˆ ⊂ PT be the twistor line corresponding to a point x ∈ M. Let V ∈ Tx M be a null vector. We want to show that V lies in a unique α-surface through x. The corresponding section of the normal bundle of xˆ has a zero at some point p ∈ PT because V is null. The α-surface corresponds to all the twistor lines that intersect xˆ at p. There is a two-parameter family of such lines. It is easy to see that there is a two-parameter family of sections that vanish at p. To show that these are tangent to a two-parameter family of lines one must blow-up PT at p and use Kodaira theory; see [11] for details. 4.1.1. Reality conditions for split signature. In order to obtain a real split signature metric from a twistor space, we must be able to distinguish a four real parameter family of twistor lines, whose parameter space will be the four real dimensional manifold. In addition we require that given a line in this real family, the sections of the normal bundle that point to others in the family inherit a split signature conformal structure. As described above, a section of O(1) ⊕ O(1) is defined by four complex numbers (a, b, c, d), with a quadratic form defined by ad − bc. If we restrict (a, b, c, d) to be real we obtain a real split signature conformal structure. The sections tangent to the real family are of this type. The zero of such sections occurs for real λ, that is, on an equator of CP1 . The conformal structure is thus invariant under an anti-holomorphic involution of the CP1 that


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has the equator fixed. A split signature real structure on PT is an anti-holomorphic involution that leaves invariant a four real parameter family of twistor lines, and when restricted to one of these fixes an equator. Not all holomorphic metrics have real structures, but all the holomorphic metrics in this paper have obvious real ‘slices’ because they are complexifications of real metrics, obtained by letting the real coordinates be complex. 4.2. Lift of K to P S . Now given a null conformal Killing vector for an ASD conformal structure, the fact that M is foliated by α-surfaces (Lemma 1) is not very illuminating, since they must already exist by anti-self-duality. The foliation by β-surfaces is more interesting, since these do not exist generically. In this section we will prove that in the analytic case, the space of β-surfaces inherits a natural projective structure. We then explain how this arises geometrically, due to the presence of α-surfaces ensured by anti-self-duality. Let K be a null conformal Killing vector for (M, [g]). We assume K is without fixed points, which can always be arranged by restricting M to a suitable open set. Since K preserves the conformal structure, the corresponding diffeomorphism maps α-surfaces to α-surfaces, and hence it induces a vector field K on PT . We now translate this fact into a statement on the projective primed spin bundle P S . Each α-surface has a unique lift and these lifts foliate P S . The following proposition shows how to lift K to P S , giving a vector field that is Lie-derived along the lifts of the α-surfaces.

Proposition 1. Let K = K A A e A A be a conformal Killing vector for an ASD metric g. Define a vector field K˜ on S by ∂ 1 A ∂ ηπ K˜ := K A A e˜ A A + π A φ A B + . 2 ∂π B ∂π A

(4.3)

∂ 3 [ K˜ , L A ] = (K B B B B AD − ψ A D )L D + (e AB η)π B π C . 4 ∂π C

(4.4)

Then this satisfies

Proof. See the Appendix.

Remark. Since K˜ is weight zero in the π A coordinates, it defines a vector field on P S , which we will also refer to as K˜ by abuse of notation. The last term on the right-hand side of (4.4) is proportional to the Euler vector field, so does not contribute to K˜ on P S . Hence (4.4) shows that K˜ commutes with the twistor distribution L on P S . The vector field K on PT is the push-forward of K˜ to PT , which is well defined because K˜ is Lie-derived along L. 4.3. Projective structure from a quotient. In this section we assume that [g] is analytic, so we can complexify by analytic continuation. Thus we are now working on a local patch of C4 , with a holomorphic conformal structure. We assume that we have restricted to a suitable open set on the base so that all the spaces of leaves involved are non-singular complex manifolds. As in Sect. 2, write K = ι A o A e A A , where now e A A is a holomorphic tetrad and ι A , o A are complex spinor fields that vary holomorphically.

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When K is null, it is easy to see that K, the induced vector field on twistor space PT , will vanish on a hypersurface H in PT , because K fixes a two-parameter family of α-surfaces, which are those to which it is tangent. These are the ‘special’ α-surfaces of Lemma 1. We now explain how this is seen from the lift K˜ to P S . On S , K˜ from Proposition 1 is given by: ∂ 1 A ∂ K˜ = ι A o A e˜ A A + π A φ A B ηπ + . B 2 ∂π ∂π A

Now when π A ∝ o A , one has π A φ A B ∝ o B from (2.5), so the second term on the RHS is proportional to the Euler vector field ϒ = π A ∂ A . The last term is everywhere ∂π proportional to the Euler vector field. To go from S to P S one quotients S −{0-section} by the integral curves of ϒ. So we have shown that on the section [π A ] = [o A ] of P S , K˜ is the push down of ι A o A e˜ A A only. But this is in L, so K˜ pushes down to the zero vector under the quotient of P S by L. So there is a (complex) hypersurface in P S , defined by the section [π A ] = [o A ], on which K˜ lies in the twistor distribution. One can also define this hypersurface as the image in P S of the hypersurface π.o = 0 in S , under the quotient by ϒ. We will refer to this hypersurface as H . It is easy to see by pushing down to the base that K˜ is linearly independent of the twistor distribution everywhere else on P S . Define a vector field

V = ι A L A = ι A π A e˜ A A

on S . This is weight one in the π A coordinates, so gives a one dimensional distribution on P S which restricts to O(−1) on fibres. Together with span{ K˜ }, we get a two dimensional distribution on P S − H . On H , the distribution drops its rank from two to one. The two dimensional distribution defined by {V, K˜ } on P S − H pushes down to the β-plane distribution Dβ on the base. Lemma 2. The two dimensional distribution on P S − H determined by {V, K˜ } is integrable. Proof. We work on S for convenience, and push down to P S at the end. The distri bution span{ K˜ , V } on S is two dimensional on S when π A o A = 0. Multiples of the Euler field ϒ are therefore irrelevant: [V, K˜ ] = [ K˜ , ιC L C ] = ιC [ K˜ , L C ] + K˜ (ι B )L B

∂ 3 = ιC ((K B B B B CD − ψC D )L D + (eC B η)π B π C ) 4 ∂π C +K B B e B B (ιC )L C

= (K B B ∇ B B ιC − ι D ψ DC )L C + #ϒ

= (ι B o B ∇ B B ιC − ι D ψ DC )L C + #ϒ.

From (2.5) we have ι D ψ DC ∝ ιC , and from (2.7) we have ι B o B ∇ B B ιC ∝ ιC . Hence the RHS is proportional to V , ignoring the irrelevant Euler vector field part.


97

Next we will show that it is possible to continue this distribution over the hypersurface H so it is rank two on the whole of P S , and that the resulting distribution commutes on the hypersurface. It will then be possible to quotient P S by the leaves of this distribution. Lemma 3. There is a two-dimensional integrable distribution D over P S , which on P S − H is determined by { K˜ , V }. Let be the projection P S → M. Then for every p ∈ P S , we have ∗ (D | p ) = Dβ . Remark. Intuitively one can think of D as a lift of the β-surfaces to P S , where each β-surface has a CP1 of lifts.

Proof. Choose a spinor ι A satisfying o A ι A = 1. Define the following (singular) vector field on S : W =

1 (V − (π D ι D ) K˜ ). π C oC

(4.5)

This is weight zero in the π A , so defines a vector field on P S by push-forward, which we shall also call W . We will now show that W is well defined even over H ⊂ P S , despite the 1/(π C oC ) factor in (4.5). Without loss of generality, choose a tetrad such that

K = ι A o A e A A = e00 .

That is, ι A = (1, 0), o A = (1, 0). Define λ = π 1 /π 0 to be the coordinate on the π 0 = 0 patch of CP1 , and extend this to a patch of P S ; we call the patch U. Then H lies entirely within U at λ = 0. We have the following expression for K˜ , obtained by ‘projectivizing’ (4.3): ∂ K˜ = e˜ 00 + (φ01 + λ(φ11 − φ00 ) + λ2 φ10 ) ∂λ ∂ 1 0 2 0 = e˜ 00 + (λ(φ1 − φ0 ) + λ φ1 ) , ∂λ

where φ01 = 0 due to (2.6). In the above conventions, we have V = π A e˜ 0 A . On U ⊂ P S , the push forward of 1 V is C π

oC

1 e˜ 00 + e˜ 01 , λ

which is singular at H , corresponding to λ = 0. Choosing ι A = (0, −1), we then obtain the following expression for W on U: W =

1 1 ∂ e˜ 00 + e˜ 01 − K˜ = e˜ 01 − ((φ11 − φ00 ) + λφ10 )) . λ λ ∂λ

This is a non-singular vector field on U. By construction, away from H this lies in span{ K˜ , V˜ }. Define D on U to be span{ K˜ , W }. This is clearly non-degenerate everywhere on U. Note that W is also well defined over the other patch (i.e. at λ = ∞) so we can define D as span{ K˜ , W } over the whole of P S .

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We now want to show that D is integrable over H . We know (Lemma 2) that D is integrable away from H . Therefore on U we have [ K˜ , W ] = f K˜ + gW + Y, where f, g are holomorphic functions on U and Y is a holomorphic vector field vanishing on U − H . But such a vector field must vanish, otherwise it is not even continuous, so is certainly not holomorphic. The last part of the lemma is obvious, just from inspecting the coordinate expressions of K˜ , W .

We now have a three dimensional integrable distribution L+D. It is three dimensional because at each point L and D have a direction in common, which is the one-dimensional distribution defined on P S by the push-forward of V on S . From Lemma 3, D is an integrable subdistribution. Note that D consists of a CP1 of lifts of each β-surface in the base. If we pick a suitably convex set on the base so that the space of β-surfaces U intersecting it is a Hausdorff complex manifold, then the quotient P S /D will also be a Hausdorff complex manifold. A point in this quotient is a point in U together with a choice of lift. In fact we can canonically identify P S /D with P T U , the projective tangent bundle of U , as follows. Using the conventions of Lemma 3, the tangent planes to the β-surfaces in the base are spanned at each point by e00 , e01 . Now L 1 has the form e10 + λe11 + (. . .)∂λ , so at each point in the fibre above a point x ∈ M, L 1 pushes down to a different null direction transverse to the β-plane at x. Now suppose we take a lift of a β-surface , i.e. a leaf of D that projects down to . Push down L 1 at each point over this lift. This will give a vector field = e10 + λe11 over , where λ is now a function on the M. We want to show that this determines a projective vector at the point s ∈ U corresponding to S. This means we require [e00 , ] ∝ mod{e00 , e01 }, [e01 , ] ∝ mod{e00 , e01 }. But it is easy to show that this is satisfied, using the fact that the distribution spanned by K˜ , W, L 1 commutes. Hence to determine the projective vector corresponding to a leaf of D, just choose a point on the leaf and push down L 1 . Because of the above considerations, this direction will be independent of the choice of point on the leaf. Proof of Theorem 1. Define Z as the quotient of P S by L ∪ D. Equivalently, this is the quotient of PT by a one-dimensional distribution which on PT − H is span{K}. The image of a CP1 fibre of P S under the quotient is a twistor line in Z . On a CP1 fibre, the horizontal part of D defines a subbundle O ⊗ C2 of the horizontal distribution = O ⊗ C4 , corresponding to the horizontal parts of the vectors K˜ and W . Choosing a spinor o A such that ι A o A = 1, we can form the vector field o A L A on S , which pushes down to a horizontal distribution on P S that is always linearly independent of D. Since the L A are weight one, this is O(−1) when restricted to a CP1 fibre. Because L ∪ D is integrable (Lemma 3), this distribution determines a one dimensional distribution D on P T U = P S /D. The spray of a projective structure is a section of D ⊗ O(1) where here O(1) is dual to the tautological line bundle over P T U . The situation is described by the following commuting diagram: 0 → O(−1) ⊕ O(−1) → O ⊗ C4 → O(1) ⊕ O(1) → 0 ↓V ↓D ↓ 0→ O(−1) → O ⊗ C2 → O(1) →0


99

PS’ L D

M Dβ

PT PTU

DΘ

U

K

Z Fig. 1. Relationship between foliation spaces

where these are bundles over a CP1 fibre of P S . The vector field o A L A on S constructed above corresponds to the O(−1) in the bottom row after quotienting by V , and gives the projective structure spray. The bottom row is the sequence (3.5) on P T U = P S /D, where U is the space of β-surfaces in M. Thus there is a projective structure on U .

Remark. The real space of β-surfaces has a system of curves that comes from the quotienting operations described above but with real spaces instead of complex. These real curves are described by the holomorphic projective structure with a reality structure. Figure 1 illustrates the situation. Here p and q are the obvious projections. Dβ repre labelling the map from PT to Z requires sents the β-surface distribution on M. The K ˜ some explanation. The vector field K over P S commutes with the twistor distribution (Lemma 1), so determines a vector field K on PT . This vector field vanishes on a hypersurface H ⊂ PT , corresponding to the α-surfaces to which K is tangent; these are the α-surfaces appearing in Lemma 1. Now K on PT only depends on K˜ modulo L. Lemma 3 shows that we can multiply K˜ modulo L by a meromorphic function (1/λ) and obtain a vector field W commuting with the twistor distribution. This means that over the whole of PT that never degenerates, there is a one-dimensional distribution K and which agrees with span {K} on PT − H. The quotient of PT by this distribution gives Z , as illustrated in the diagram. One can rephrase this in terms of divisor line bundles. That is, there is a holomorphic line bundle E over PT defined by the property that it has a meromorphic section ζ with a pole of order one on H. Then ζ ⊗ K defines a non-vanishing section of E ⊗ T PT . over PT described above. To This is equivalent to the one dimensional distribution K obtain the distribution one simply finds trivializations of E and T PT over a patch, and expresses ζ in this trivialization. Its direction will be independent of the trivialization of E, and defines the distribution over the patch. 4.4. Relationship of the twistor spaces. Here we discuss the relationship between the twistor spaces without the foliation space picture. Incidence relation between various objects in M and PT is represented by Fig. 2. First one must understand what a β-surface corresponds to in PT . The answer is a two-parameter family of twistor lines, each of which intersects any other at a single point. This is because all points on a β-surface are null separated. However, unlike the case of an α-surface, there is not just a single point of intersection of the whole family. To construct the family, pick a point on the β-surface, say x. Then xˆ is a twistor line in PT . Now K determines a section of the normal bundle with a zero. Twistor lines intersecting

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PT M

p1 β p3 p2 K

β

p2

α

p3 p1

α

Fig. 2. The α and β surfaces in M intersect along a trajectory of K which is a null geodesic. This corresponds to a point α lying on a surface β in PT . Points p1 , p2 , p3 in M correspond to projective lines in PT

xˆ at this zero are on the β-surface, and correspond to those along the trajectory of K through x. In fact this is a null geodesic, since null Killing vector fields have geodesic integral curves. Now pick another section of the normal bundle with a zero at a different point, such that all linear combinations of this with the section determined by K also have a zero. The resulting two dimensional distribution in M at x is a β-plane. Doing this for each x ∈ M gives a β-plane distribution which is integrable. The diagram (Fig. 3) illustrates the situation. In M, a one parameter family of β-surface is shown, each of which intersects a one parameter family of α-surfaces, also shown. The β-surfaces correspond to a projective structure geodesic in U , shown at the bottom left. The β-surfaces in M correspond to surfaces in PT , as discussed above. These surfaces intersect at the dotted line, which corresponds to the one parameter family of

Fig. 3. Relationship between M, U , PT and Z


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α-surfaces in M. When we quotient PT by K to get Z , the surfaces become twistor lines in Z , and the dotted line becomes a point at which the twistor lines intersect; this is shown on the bottom right. This family of twistor lines intersecting at a point corresponds to the geodesic of the projective structure. 5. Local Classification The second theorem stated in the Introduction gives a local expression for any smooth neutral signature ASD conformal structure. We now prove this theorem. In the proof we continue to work in the holomorphic category for continuity, but all the arguments work when the variables are real rather than complex, CP1 is replaced by RP1 , and functions are smooth rather than holomorphic. The smooth generalization of Theorem 1 could perhaps be established using techniques introduced in [18]. In the proof we will often use the following shorthand for coordinate transformations: t → F(t, x, y, z) means define a new coordinate t˜ = F(t, x, y, z) and then relabel it t again. This avoids having to introduce new symbols for new coordinates. We will denote partial derivatives by subscripts, for example Fz := ∂z F. Proof of Theorem 2. In what follows, we will use coordinates (x, y) for the twodimensional space of β-surfaces U . We will always work on a single patch of P S , with λ a standard coordinate on one patch of the CP1 fibre. The projectivization of (4.1) is L 0 = e00 + λe01 + ( f 0 + λ f 1 + λ2 f 2 + λ3 f 3 )∂λ , L 1 = e01 + λe11 + (A0 + λA1 + λ2 A2 + λ3 A3 )∂λ ,

(5.1) (5.2)

where the f α and Aα are functions on M derived from primed connection coefficients. We can trivialize P T U by first choosing a two dimensional surface in M, transverse to the β-surfaces, and trivializing P S over this, using the standard two patch coordinates for CP1 . Then define a trivialization over the rest of P S by requiring constant coordinate on each leaf of D (this will be a base dependent Möbius transformation of any other trivialization of P S using a standard two patch trivialization of CP1 , since any two standard trivializations of CP1 are related by a Möbius transformation). This gives a trivialization P T U ∼ = U × CP1 . The special feature of this particular trivialization is ˜ that K and W will have no vertical terms, because it was defined by saying that the fibre coordinate is constant along them. We will use the conventions of Lemma 3, that is we choose a tetrad with K = e00 , and the tangent planes to the β-surfaces are spanned by K and e01 . Now choose a coordinate system (t, x, y, z) such that K = ∂t , and a conformal factor so that K is pure Killing. Any tetrad can then be written in these coordinates without any t dependence. Then [e00 , e01 ] = 0 and we can in addition choose the z coordinate such that e01 = ∂z . Then we have K˜ = ∂t , L 0 = ∂t + λ∂z + f (x, y, z, λ)∂λ . Note that f does not depend on t because it is composed from connection coefficients, which do not depend on t since it does not occur in the metric. Also note that K˜ = ∂t because L 0 , L 1 do not contain functions of t so it commutes with both. As vector fields on the base, ∂x and ∂ y are transverse to the β-surfaces, and so are coordinates on U .

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Now we will alter the λ coordinate, using a trivialization as described above, so that L 0 has no ∂λ terms. This is achieved by a Möbius transformation, λ → (β + δλ)/(α + γ λ), where α, β, γ , δ are functions on M. Now the new λ coordinate satisfies K˜ (λ) = L 0 (λ) = 0. Therefore α, . . . , δ do not depend on t, from the first of these. This gives the following general form: K˜ = ∂t , L 0 = α ∂t + β ∂z + λ(γ ∂t + δ ∂z ).

(5.3) (5.4)

Now from Theorem 1, we know that L 1 must define a projective structure on U , the space of β-surfaces. In fact this can be seen directly using our coordinate choices. Clearly U has coordinates (x, y), since the β-surfaces are spanned by (∂t , ∂z ). Also, λ is a fibre coordinate on P T U , since D is defined by constant λ. Since {L 0 , L 1 } is an integrable distribution, one can find a non-zero function f on P S such that [L 0 , f L 1 ] ∝ L 0 . We may therefore assume we have chosen an L 1 such that [L 0 , L 1 ] ∝ L 0 . It follows from (5.4) that the coefficients in front of the ∂x , ∂ y , ∂λ terms in L 1 do not depend on z. Therefore L 1 must have the following form: L 1 := J0 (x, y)∂x + J1 (x, y)∂ y + λ(J2 (x, y)∂x + J3 (x, y)∂ y ) +(A0 (x, y) + λA1 (x, y) + λ2 A2 (x, y) + λ3 A3 (x, y))∂λ +(C(x, y, z) + λD(x, y, z))∂t + (E(x, y, z) + λF(x, y, z))∂z ,

(5.5)

where J0 J3 − J1 J2 = 0. One now observes that the ∂x , ∂ y , ∂λ terms precisely correspond to a projective structure spray on P T U . Since D is spanned by ∂t , ∂z , the quotient of L 1 by D gives a projective structure. To put the projective structure spray occurring in (5.5) into the more standard form (3.4) (i.e. J0 = J3 = 1, J1 = J2 = 0) it is necessary to perform a Möbius transformation of λ depending on (x, y). Since this does not depend on t or z, the general form (5.4) of L 0 is unchanged by this, and we can assume that the projective structure spray in L 1 is of the form (3.4), which we shall do from now on. We have found a general form that any { K˜ , L A } can be put into. For it to give an ASD conformal structure, the L A must commute modulo L A . Imposing this gives equations for the unknown functions, which will lead us to the metrics appearing in Theorem 2. First, it is convenient to change coordinates yet again, because together with conformal rescaling we can elimate three of the four functions in L 0 . We may assume δ = 0 (if δ = 0 then β = 0, in which case perform the coordinate change λ → 1/λ). Now change coordinates by (t, x, y, z) → (t + j (x, y, z), x, y, k(x, y, z)), where k z = 0. A suitable choice of j and k, and conformal rescaling, simplifies L 0 so that finally K˜ = ∂t , L 0 = ∂t − β(x, y, z)∂z + λ∂z ,

(5.6) (5.7)

L 1 = ∂x + λ∂ y + (A0 (x, y) + λA1 (x, y) + λ2 A2 (x, y) + λ3 A3 (x, y))∂λ +(C(x, y, z) + λD(x, y, z))∂t + (E(x, y, z) + λF(x, y, z))∂z .

(5.8)

One can read off a metric g ∈ [g] corresponding to the twistor distribution given by (5.7) and (5.8) by comparing with (5.1) and (5.2) and reading off a null tetrad. One finds that K ∧ dK = βz d x ∧ dy ∧ dz, where K = g(∂t , .). Thus the twist of the Killing vector ∂t vanishes iff β does not depend on z. Since existence of twist is a conformally


103

invariant property, the cases βz = 0 and βz = 0 are genuinely distinct, not an artefact of our coordinate choices. We now analyse each in turn. Twist-free case. βz = 0. Calculating the commutator of L 0 and L 1 we obtain [L 0 , L 1 ] = (−β + λ)(C z + λDz )∂t + (βx + λβ y − β E z − λβ Fz + λE z + λ2 Fz −(A0 + λA1 + λ2 A2 + λ3 A3 ))∂z .

(5.9)

Since we require {L 0 , L 1 } to be integrable, this must be a multiple of L 0 . We deduce that 3 [L 0 , L 1 ] = (−β + λ)(C z + λDz )L 0 .

(5.10)

Now comparing the ∂z coefficients of (5.9) and (5.10) we get four equations, one for each power of λ. We can solve three of them, and use L 1 → L 1 − C L 0 which does not change the conformal structure. This yields L 1 = ∂x + λ∂ y + (A0 + λA1 + λ2 A2 + λ3 A3 )∂λ +λ(−z A3 + Q)∂t + (z(−β y + A1 + β A2 + β 2 A3 ) + λ(z(A2 + 2β A3 ) + P))∂z , (5.11) where P and Q are arbitrary functions of (x, y) and we have eliminated one arbitrary function by translating the z coordinate. There is one remaining equation to solve, corresponding to the λ0 coefficient of ∂z . This equation is as follows: βx + ββ y − A0 − β A1 − β 2 A2 − β 3 A3 = 0.

(5.12)

The metric (1.2) in Theorem 2 corresponds to the twistor distribution given by L 0 , with βz = 0, and (5.11). If β(x, y) is regarded as defining a section of P T U , then (5.12) says that this section is tangent to lifted geodesics of the projective structure. In terms of the base, a solution is given by a congruence of geodesics. Twisting case. βz = 0. We may perform a coordinate transformation z → β(x, y, z). This does not affect the general form (5.8) of L 1 . Performing the coordinate change and dividing by βz gives the following form for L 0 : L 0 = H (x, y, z)∂t − z∂z + λ∂z ,

(5.13)

where H is a non-zero arbitrary function. Calculating the commutator gives [L 0 , L 1 ] = ((−z + λ)(C z + λDz ) − (E + λF)Hz )∂t +((−z + λ)(E z + λFz ) − (E + λF) − (A0 + λA1 + λ2 A2 + λ3 A3 ))∂z . We require [L 0 , L 1 ] = αL 0 for some function α(x, y, z, λ), which is at most quadratic in λ, since otherwise αL 0 will contain powers of λ greater than three, and such terms do not occur in the commutator above. We make a replacement L 1 → L 1 − F L 0 , and analyze equations obtained from comparing coefficients of ∂z , ∂t . This puts L 1 in the form L 1 = ∂x + λ∂ y + (A0 + λA1 + λ2 A2 + λ3 A3 )∂λ +(C + λD)∂t + (A0 + z A1 + z 2 A2 + z 3 A3 )∂z , 3 In [3] the resulting equations are interpreted as a special case of a gauge theory defined on a projective surface. A solution is called a projective Higgs pair. This also applies to the twisting case.

104


where C(x, y, z), D(x, y, z), H (x, y, z) satisfy C z − 2z Dz = −H A2 + Hy , Dz = −H A3 ,

(5.14) (5.15)

(∂x + z∂ y + (A0 + z A1 + z 2 A2 + z 3 A3 )∂z )H = 0.

(5.16)

and

The only things remaining now are to find expressions for C and D and construct the metric. In order to integrate Eqs. (5.14) it is convenient to express H (x, y, z) as the second derivative of another function G(x, y, z), i.e. we set H (x, y, z) =

∂2G (x, y, z). ∂z 2

Then Eqs. (5.14) integrate to give C = −G z A2 − 2 A3 (zG z − G) + G zy + ρ(x, y), D = −G z A3 + σ (x, y), where ρ and σ are arbitrary functions. Notice that G has a ‘gauge freedom’ G → G + zγ (x, y) + δ(x, y), since (1.4) will still be satisfied. Using this and a coordinate change t → t +ξ(x, y), one can set the functions ρ and σ to zero. The twistor distribution {L 0 , L 1 } is now fully determined: L 0 = G zz ∂t − z∂z + λ∂z , L 1 = ∂x + λ∂ y + (A0 + λA1 + λ2 A2 + λ3 A3 )∂λ +(−G z A2 − 2 A3 (zG z − G) + G zy ) − λ(G z A3 ))∂t +(A0 + z A1 + z 2 A2 + z 3 A3 )∂z . The distribution is integrable iff G satisfies (1.4). Calculating the corresponding null tetrad gives the conformal structure (1.3) in Theorem 2.

6. Examples 6.1. Neutral Fefferman conformal metrics. If G zz is simply a constant, then (1.4) is satisfied. So given any projective structure and setting G zz = 1 we obtain a family of conformal structures with twist which reduce to the given projective structure. Solving for G gives G=

z2 + zγ (x, y) + δ(x, y). 2

The corresponding metric takes the form z2 − δ) − γ y + ρ)d x)(dy − zd x) 2 (6.17) −(dz − (A0 + z A1 + z 2 A2 + z 3 A3 )d x)d x,

(dt + ((z + γ )A3 + σ )dy + ((z + γ )A2 + 2 A3 (


105

where we have chosen not to eliminate σ and ρ. By direct calculation one can show that the ASD Weyl tensor has Petrov-Penrose type III or N, and it is type N precisely when the following hold: 1 A2 , 3 2 γ A2 − 2 A3 δ − γ y + ρ = A1 . 3 One can always choose ρ, σ, γ , δ so that these are satisfied. In this case, the metric is the same as (31) in [20], with their Q cubic in p. These are neutral signature analogues of Fefferman metrics. γ A3 + σ =

6.2. ASD pp-waves. Notice that the metric (1.2) does not explicitly contain the function A0 (x, y) of the projective structure. The metric is always ASD for any choice of β, A1 , A2 , A3 ; one can regard (5.12) as giving A0 (x, y) in terms of these functions. On the other hand, if one wants to specify A0 , then one must choose a solution of (5.12) for β. In the special case A0 = 0, we have the solution β = 0. One then obtains the following metric: g = (dt + (P + z A2 )d x + (Q + z A3 dy))dy − (dz + z A1 d x)d x.

(6.18)

Different choices of function β(x, y) in (1.2) can give rise to different metrics. Suppose we choose the flat projective structure. Then β(x, y) must satisfy Eq. (1.6) with Aα = 0. By direct calculation one can show that the metric (1.2) is type III iff β yy = 0, otherwise it is type N. So the conformal structures with β yy = 0 and β yy = 0 are genuinely distinct. 6.3. Pseudo-hyper-Kähler metrics. We will find some examples of neutral ASD metrics with null conformal Killing vectors by independent means, and interpret them using our results. We will use Plebański’s method [23] adapted to neutral signature, which converts the problem of finding Ricci-flat ASD neutral metrics, or pseudo-hyper-Kähler, to the problem of solving a non-linear second order PDE. He showed that such metrics are locally of the form g = dY (dT − X X dY − T X d Z ) − d Z (d X + T T d Z + T X dY ),

(6.19)

where (T, X, Y, Z ) satisfies the ‘second Heavenly Equation’: Y T − Z X + T T X X − 2X T = 0.

(6.20)

The primed connection coefficients vanish when using the tetrad indicated in (6.19), so there is a basis of covariantly constant primed spinors o A = (1, 0), ι A = (0, −1). There is therefore also a basis A B of covariantly constant null self-dual two forms, written in spinors as follows:

1 ι A ι B AB θ A A ∧ θ B B , 2 1 = o(A ι B ) AB θ A A ∧ θ B B , 2 1 = o A o B AB θ A A ∧ θ B B . 2

0 0 =

0 1 = 1 0

1 1

(6.21) (6.22) (6.23)

106


Using the identification between two-forms and endomorphisms given by g, we can write

R = 0 0 − 1 1 , I = 0 0 + 1 1, S = 0 1 . As endomorphisms, these satisfy −I 2 = R 2 = S 2 = Id, I RS = Id,

(6.24)

which is easy to check using their spinor forms. There is a hyperboloid’s worth of almost complex structures, a I + b R + cS, where a 2 − b2 − c2 = 1, which are parallel and hence integrable. This is a pseudo-hyper-Kähler structure. Now writing (A.2) using spinors by means of (2.4) and (A.1) gives 1 ι A o A C ABC D A B C D = ∇ B B (φC D C D + ψC D C D + ηC D C D ), 2 where we have used Ricci flatness and anti-self-duality. For a pure Killing vector or a homothety (η constant), it follows that ∇ A A φ B C = 0.

(6.25)

Therefore φ B C is actually constant in the basis shown in (6.19). Now let us suppose we have a null Killing vector which preserves the α–plane distribution spanned by o A e A A . Then K = ι A o A e A A , for some ι A and using (2.6) and (6.25) we get φ B C = a1 o B oC + a2 o(B ιC ) , for constant a1 , a2 . Consider the three distinct cases: φ B C vanishing (a1 = a2 = 0), non-vanishing but degenerate (a1 = 0, a2 = 0), and non-degenerate (a1 = 0, a2 = 0). For K = ∂T we get the first case, K = Y ∂ X + Z ∂T the second, and T ∂T + X ∂ X the third, and with some effort it can be shown that these choices are canonical (the first two cases were analysed in [8]). In order for any of these to be Killing, an equation for coming from the Killing equation must be satisfied. In fact we were only able to fully solve for the first two cases. • K = ∂T Since ∂T has no twist we expect this to be of the form (1.2). It is a neutral signature version of a tri-holomorphic Killing vector; i.e. it Lie-derives I, R, S. Solving the Killing equations in conjunction with (6.20) results in the following metric: g = dY dT − d Z d X − Q(X, Y )dY 2 ,

(6.26)

where Q is an arbitrary function. This is simply the split-signature pp-wave metric, and is a special case of (6.18). Here K is a self-dual Killing vector in the sense of Gibbons et al. [1]. The local expression (6.26) in this example corresponds to a class of global neutral metrics on compact four–manifolds. To see this we compactify the flat projective space R2 to the two–dimensional torus U = T 2 with the projective structure coming from the flat metric. Both T and Z in (6.26) are taken to be periodic, thus leading


107

to πˆ : M −→ U , the holomorphic toric fibration over a torus. Assume the suitable periodicity on the function Q : U −→ R. This leads to a commutative diagram M T 2 ↓ πˆ ∗ Q Q

U −→ R. This example can be put into the framework of [15] and [9], where M is regarded as a primary Kodaira surface C2 /G and G is the fundamental group of M represented injectively in the group of complex affine transformations of C2 . In this framework the Kähler structure on M is given by f lat + i∂∂(πˆ ∗ Q), where (∂, f lat ) is the flat Kähler structure on the Kodaira surface induced from C2 . • K = Y ∂ X + Z ∂T Again, this is twist-free and we expect the metric to be of the form (1.2). Solving the Killing equations in conjunction with (6.20) results in the following metric: g = dY dT − d Z d X −

Y Z H ( Y T −Z X , Y T −Z X )

(Y T − Z X )3

(Y d Z − Z dY )2 ,

(6.27)

where H is an arbitrary analytic function of two variables. This is a generalization of the Sparling-Tod metric [24]. It is easy to show that the arguments of H are in fact constant on the special β-surfaces, so serve as coordinates on U . Using the following coordinate transformation: t=−

1 X T , + 2 Y Z 1

z = (Y Z )− 2 , YT − XZ , x= 1 (Y Z ) 2 Z , y = log Y the metric (6.27) takes the following form: g=

1 (dydt − dzd x + z A3 (x, y)dy 2 ), z2

where now the Killing vector is ∂t . Multiplying by the conformal factor z 2 , we get a special case of (6.18). The projective structure is non-trivial, unlike for the pp-wave above. The projective structure is special in that it depends on only one arbitrary function. • T ∂T + X ∂ X In this case we were not able to fully solve the Killing equations in conjunction with (6.20). This Killing vector is twisting, so the answer must be of the form (1.3). 6.4. Pseudo-hyper-hermitian conformal structures. This is a generalization of the pseudo-hyper-Kähler case discussed in the last section. We will refer to a neutral metric g as pseudo-hyper-hermitian (also called hyper-para—hermitian [12]) when there exist endomorphisms I, R, S satisfying the algebra (6.24), such that any complex structure

108


J(a,b,c) = a I + b R + cS is integrable for a 2 − b2 − c2 = 1, and g is hermitian with respect to any of these complex structures. For g to be hermitian with respect to a complex structure J means g(J X, J Y ) = g(X, Y ). Note that for pseudo-hyper-Kähler, the endomorphisms I, R, S must also be covariantly constant with respect to the Levi-Civita connection of g. In [5], it is shown that one can always find a tetrad for a pseudo-hyper-hermitian metric such that the twistor distribution has no ∂λ terms. Equivalently, the twistor space fibres over CP1 . Now let us suppose that we have a null conformal Killing that is tri-holomorphic, i.e. it preserves I , R and S and so it preserves the holomorphic fibration PT → CP1 . All such cases are classified by the following Proposition 2. All pseudo-hyper-hermitian ASD metrics with triholomorphic null conformal Killing vectors are of the form (1.2) or (1.3) up to a conformal factor, where the corresponding ODE (1.5) is point equivalent to a derivative of a first order ODE. Proof. Let g be a pseudo-hyper-hermitian ASD metric, and K be a triholomorphic conformal Killing vector. Since g is ASD, it follows from Theorem 2 that there are coordinates such that, up to a conformal factor, g is of the form (1.2) or (1.3). From [5], it is possible to find a tetrad such that the twistor distribution has no ∂λ terms. Now a change in tetrad corresponds to a Möbius transformation of λ. Since K is triholomorphic, its lift will have no ∂λ terms in the tetrad where the twistor distribution has no ∂λ terms. Therefore the Möbius transform does not depend on t, otherwise ∂t will no longer Lie-derive the twistor distribution (one would have to add ∂λ terms). Furthermore, the Möbius transformation does not depend on z, otherwise ∂λ terms will be introduced into L 0 . Hence there is a Möbius transformation of λ, depending only on (x, y), such that the ∂λ terms in L 1 are eliminated. After this change in λ, the projective structure spray in L 1 will be of the following form: = a∂x + b∂ y + λ(c∂x + e∂ y ), where a, b, c, e are functions of (x, y) with ae − bc = 0. Coordinate freedom (x, y) → (x(x, ˆ y), yˆ (x, y)) and scaling freedom (the projective structure is unchanged if is multiplied by a non-zero function) allows us to set a = 1, c = 0, e = 1, giving = ∂x + (b + λ)∂ y . Now perform another Möbius transformation λ → b + λ, which gives the following spray: ∂x + λ∂ y + (bx + λb y )∂λ .

(6.28)

This corresponds to the second-order ODE dy d2 y + A0 (x, y), = A (x, y) 1 dx2 dx where A1 = ∂b ∂ y , A0 = first-order ODE

∂b ∂x

(6.29)

for a function b(x, y). This is the derivative of the general dy = b(x, y). dx

(6.30)

Hence the original projective structure is point-equivalent to the one corresponding to (6.29).


109

Note that if a (holomorphic) projective structure spray contains no ∂λ terms, its twistor space fibres over CP1 , since each integral curve can be labelled by the λ coordinate. So a by-product of the proof of the above proposition and Theorem 3 is the following Proposition 3. There is a one to one correspondence between holomorphic 2D projective structures s.t. the corresponding second order ODE is point equivalent to the derivative of a first order ODE, and complex surfaces which contain a holomorphic curve with normal bundle O(1) and fiber holomorphically over CP1 . This is of interest purely as a statement about projective structures. Note that although all first order ODEs can be transformed to the trivial first order ODE dy/d x = 0 by coordinate transformation, this does not mean that the derivative of any such equation is flat, in the sense of Sect. 3.2. This can be shown by calculating the invariant (3.7) for (6.29) and showing that it does not necessarily vanish. 6.5. Conformal structures containing no Ricci-flat metrics. In this section we show that there are conformal structures of the form (1.2) which do not contain Ricci-flat metrics. Before doing so we discuss the Petrov-Penrose classification for the conformal structures (1.2) and (1.3).

Proposition 4. Let K A A = ι A o A be a null conformal Killing vector for ASD conformal structure. Then ι A is a principal direction, that is ι A ι B ιC ι D C ABC D = 0.

(6.31)

Moreover if the twist of K vanishes the conformal structure is of type I I I or N , that is ι A ι B C ABC D = 0.

(6.32)

Proof. From (2.5) we have ∇ A A (ιC ι D ψC D ) = 0. Expanding this out we obtain ι B ιC ∇ A A ψ BC = −2ψ BC ιC ∇ A A ι B = ι A µ A ,

(6.33)

for some spinor µ A . The last equality follows from (2.5) and (2.7). Now pick a conformal frame in which K is a pure Killing vector. The well known identity ∇a ∇b K c = Rbcad K d implies

B ∇ AA ψ BC = −2C DABC K DA − 2K (A ABC)B +

1 4 D D A R A(B K AC) − A(B C) K D D . 6 3

On contracting both sides by ι A ι B ιC and using (6.33), all terms vanish except the term involving C DABC , giving (6.31). Now let us assume that K is non–twisting, i. e. K ∧ dK = 0, where K := g(K , ). The Frobenius theorem implies the existence of functions P and Q such that K = Pd Q. We can now choose a conformal factor such that dK = 0. Then K is covariantly constant (∇a K b = 0), and we deduce ∇ A A ι B = A A A ι B , ∇ A A o B = −A A A o B ,

(6.34) (6.35)

110


for some one-form A A A . Consider the spinor Ricci identity [22] A B oC = (C A B C D −

1 R D (A B )C )o D , 12

where A B = ∇ A(A ∇ BA ) . Substituting (6.35) into this and using C A B C D = 0 gives oC ∇ A(A A BA ) = −

1 Ro(A B )C . 12

By contracting with oC we find R = 0. Now consider the Ricci identity AB ιC = (C ABC D −

1 R D(A B)C )ι D . 12

Substituting R = 0 and (6.34) into this gives

A ιC ∇ A (A A B) = C ABC D ι D .

Contracting this with ιC gives (6.32), from which it follows that the curvature is type III or N.

In the twisting case the algebraic type of the Weyl spinor can be general. This can be shown by using the following two scalar invariants [22]: I = C ABC D C ABC D ,

CD J = C AB CC EDF C E AB F .

The condition for type III is I = J = 0, and for type II that I 3 = 6J 2 . Now consider the metric (1.3), with the flat projective structure Ai = 0, i = 0, . . . , 3. The function G zz satisfies (∂x + z∂ y )G zz = 0, which is solved in general when G zz is an arbitrary function of (zx − y). Suppose G is given by: G(x, y, z) =

ezx−y + z B(x, y), x2

where B(x, y) is arbitrary, so G zz = ezx−y . Then the two scalar invariants are as follows: 3 I = − x B yy e−3(zx−y) , 2 3 J = x(x B yyx + 3B yy + x z B yyy )e−4(zx−y) . 8

(6.36) (6.37)

Therefore, from the conditions above, the metric is neither type II nor type III. To find metrics that are not conformally Ricci-flat we use results of Szekeres [25]. Although these were derived for Lorentzian signature, they can also be applied to our ASD neutral signature case, essentially because the Weyl curvature is still made up of a single spinor Cabcd = C ABC D A B C D as in the Lorentzian case (of course in the Lorentzian case it is complex hermitian, not real). Consider the metric (6.18) with A1 = 0. By direct calculation, one finds that C ABC D is type N iff (A2 )x = 0, otherwise it is type III. Now suppose (1.2) is type III, i.e.


111

(A2 )x = 0. The reason for this is that we can apply a result of Szekeres to obtain an obstruction to Ricci-flatness. It is shown in [25] that for types I, II, D or III, a necessary condition for existence of a Ricci-flat metric in the conformal class is the following tensor equation: 1 fh df df − C pq f h Cr s Cabcd ;d + (C pq Cr s fh ;h + Cr s C pqhf 2

;h )

= 0.

This is just the tensor version of the spinor identity (3.1), p. 209 [25]. Calculating this one finds that (A2 )x x is an obstruction to its vanishing (we used MAPLE for the calculation). Therefore we have a class of non-conformally vacuum type III neutral ASD conformal structures with non-twisting null conformal Killing vectors. 7. Twistor Reconstruction We have shown that when a conformal structure [g] has a null conformal Killing vector, the twistor space PT fibres over the twistor space of a projective structure, and we have classified the possible local forms for such conformal structures. The twistor lines in a projective structure twistor space Z have normal bundle O(1). The twistor lines in a conformal structure twistor space have normal bundle O(1)⊕O(1). Let B be a holomorphic fibre bundle over Z with one dimensional fibres. Let uˆ be a twistor line in Z . Then if we want B to be a conformal structure twistor space, the normal bundle of uˆ in B|uˆ must be O(1). Given a projective structure twistor space, one way of forming a fibre bundle with the correct property is to take a power of the canonical bundle κ, which reduces to O(−3) on twistor lines. The bundle κ −1/3 reduces to O(1) on twistor lines, and exists provided we take Z to be a suitably small neighbourhood of a twistor line. So the total space of κ −1/3 is a conformal structure twistor space. Consider the simplest possible case, where Z is the total space of O(1), corresponding to a flat projective structure. In this case κ −1/3 is the total space of O(1) ⊕ O(1), the twistor space of the flat conformal structure. To go further, note that given a line bundle A over Z which reduces to O(1) on twistor lines, any affine bundle modelled on A will also have the correct property on twistor lines. In the simplest case described above, taking affine bundles modelled on κ −1/3 results in the the twistor space of the pp-wave metric (6.26). In fact, this is precisely the first case discussed by Ward in [28], although he does not phrase it in this way. We will now show how this works.

7.1. Example 1. PP-waves. First we will give a twistorial demonstration of a fact shown in Sect. 6.3, namely that for a pseudo-hyper-Kähler metric with triholomorphic null Killing vector K = ι A o A e A A with o A covariantly constant, the resulting projective structure is flat. The twistor space of an analytic pseudo-hyper-Kähler metric fibres over CP1 , σ : PT → CP1 [21, 11]. There is a section of 2 PT × σ ∗ O(2). This is a symplectic form of ‘degree 2’ on the fibres. In the spin bundle picture, is the push forward to PT of the symplectic form = A B π A π B on S , where A B are defined as in Sect. 6.3. This form is Lie-derived over the twistor distribution as a consequence of the A B being covariantly constant, and is homogeneous in the π A , so the push-forward is well defined. As explained in Sect. 4, K vanishes on a hypersurface H in PT , where H is the projection to PT of the hypersurface π.o = 0 in S . For o A covariantly constant, the

112


function 1/(π.o) on S gives a section ζ of σ ∗ O(−1) on PT , which blows up on H. Then ζ ⊗ K is a non-vanishing ‘σ ∗ O(1)-valued’ vector field. Now in a local trivialization, ζ ⊗ K Lie derives the symplectic form , so it is Hamiltonian, ζ ⊗K =

∂h ∂ , ∂ω A ∂ω A

where = ω0 ∧ ω1 . Now the ω A should be regarded as coordinates of ‘degree 1’, that is they are coordinate functions multiplied by a section of σ ∗ O(1). Therefore for the weights to agree, h must be a section of κ ∗ O(1), rather than a bona fide function. This gives a projection PT → Z = O(1), with fibres the trajectories of ζ ⊗ K, so the projective structure twistor space is the total space of O(1), which corresponds to the flat projective structure. Now suppose we start with the total space of O(1) as the minitwistor space Z . The twistor lines are global holomorphic sections of O(1) → CP1 . We will use a homogeneous coordinate description of Z = O(1). Let π A be homogeneous coordinates for the base CP1 of Z = O(1), and let ω be a homogeneous coordinate for the fibre of Z = O(1). That is, O(1) = {[π0 , π1 , ω] : [cπ0 , cπ1 , cω], c ∈ C∗ , [π0 , π1 ] = [0, 0]}. Now cover the base CP1 in PT with two open sets (U0 , U1 ), and lift this covering to PT . Use homogeneous coordinates (π A , ω, ζi ) on Ui . The flat twistor space O(1) ⊕ O(1) can be formed as follows. Consider the projection τ : O(1) → CP1 . Then O(1)⊕O(1) is the pull-back bundle τ ∗ O(1) over the total space of O(1). It is easy to check that this is the same as taking κ −1/3 , where κ is the canonical bundle of Z = O(1). To obtain curved twistor spaces, we can take affine bundles over O(1) modelled on τ ∗ O(1). To form these we use the following transition functions: ζ0 = ζ1 + f (π A , ω), where f ∈ [ f ] ∈ H 1 (Z , τ ∗ O(1)), where Z is O(1). The cohomology elements f classify affine bundles over Z modelled on τ ∗ O(1). Global holomorphic sections of Z → CP1 are defined by ω = P(π A ) = π A x A , A with x = (X, Y ) say. The sections of PT → CP1 are constructed by putting ζi = π A t A + f i , where t A = −(T, Z ) say, and f = f 0 − f 1 . The reason f can be split in this way is that when restricted to a twistor line in Z , f becomes an element of H 1 (CP1 , O(1)), and this group vanishes. To realise a splitting of f we divide it by (π A o A )2 for some constant o A , to get an element of H 1 (Z , τ ∗ O(−1)). Then we can use the fact that H 0 (CP1 , O(−1)) = H 1 (CP1 , O(−1)) = 0, so any element can be written as a difference of coboundaries, and the splitting is unique. These sections are the CP1 twistor lines in PT ; we will refer to these as x, ˆ where x is the point in M with coordinates (t A , x A ). Let ρ A be homogeneous coordinates on CP1 . The splitting is given by the Sparling formula: f (ρ, P) f (ρ, P) f (π, P) = ρ.dρ − ρdρ, 2 2 2 (π.o) 0 (ρ.o) π.ρ 1 (ρ.o) π.ρ where we are using Cauchy’s integral formula, and i ⊂ xˆ ∼ = CP1 are contours that bound a region containing the point ρ A = π A . The measure ρ.dρ means A B ρ A dρ B .


113

Therefore fi =

(π.o)2 f (ρ, P) ρ.dρ. (ρ.o)2 π.ρ

i

The symplectic form discussed above is given by by = dω ∧ dζi on Ui . Restricting to a section and taking exterior derivatives keeping π A constant, we obtain a formula for , the pull-back of to S :

= d(π A x A ) ∧ d(π B t B + f 0 )

= π A π B d x A ∧ dt B + π A d x A ∧ d f 0 , where we are working over U0 . Now (π.o)2 f (ρ, ρ A x A ) ∂ d f0 = d x ⊗ B ρ.dρ 2 π.ρ ∂x 0 (ρ.o) ρ B (π.o)2 ∂ f ρ.dρ, = dx B 2 0 (ρ.o) (π.ρ) ∂ P B

where we have used

∂ ∂x A

= π A π B d x

→ ρ A ∂∂P . Using this we get

A

∧ dt

B

= π A π B d x A ∧ dt B

π A ρ B (o.π )2 ∂ f ρ.dρ dx A ∧ dx B 2 (π.ρ) ∂ P (o.π ) 0 (o.π )2 ∂ f 1 ρ.dρ dY ∧ d X + 2 0 (o.ρ)2 ∂ P +

= π A π B d x A ∧ dt B + (o.π )2 Q(X, Y )dY ∧ d X, where Q(X, Y ) =

1 2

0

1 ∂f ρ.dρ. (o.ρ)2 ∂ P

Putting o A = (1, 0), we get the following formula for pulled back to M × C2 : =π02 (dT ∧ d X +Q(X, Y )dY ∧ d X )+π0 π1 (dT ∧ dY − d X ∧ d Z )+π12 d Z ∧ dY. (7.1) Calculating in the Plebanski formalism from (6.21), (6.22) and (6.23) gives = π02 (dT − X X dY − T X d Z ) ∧ (d X + T T d Z + T X dY ) + π0 π1 (dT ∧ dY − d X ∧ d Z ) + π12 d Z ∧ dY.

Comparing gives the forms A B and hence the metric (6.26). The arbitrary function Q corresponds to some arbitrary cohomology element f .

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7.2. Example 2. Flat conformal structure. Here we show that given a conformal Killing vector for the flat conformal structure, the underlying projective structure is also flat. By the results of [26], we need only consider the conformal Killing vectors ∂T (non-twisting) and T ∂T + X ∂ X (twisting), where the flat metric is g = dT dY − d X d Z . The non-twisting case is covered by the example of the last section, with Q(X, Y ) = 0, so we know the projective structure is flat and Z = O(1). The twisting case is slightly more complicated. One can use the spray picture, but instead we will analyse the twistor space PT and show that the space of trajectories of K is the flat projective structure twistor space CP2 . We work on the non–projective twistor space T = C4 with coordinates (ω A , π A ). The projective twistor space PT is a quotient of T , and the Euler homogeneity vector field ϒ = ω A /∂ω A + π A /∂π A . The flat conformal class on the complexified R2,2 and the conformal twisting Killing vector are represented by g = ε AB dp B dq A ,

K = pA

∂ , ∂pA

where x A A := p A o A + q A ι A are coordinates on M. The point ( p A , q A ) corresponds to a two–plane in T given by solutions to the twistor equation ω A = x A A π A . The lift of K to S is ∂ , K˜ = K + π1 ∂π1 and the orbits of the induced group action on the non–projective twistor space are ω A → cω A ,

π1 −→ cπ1 ,

π0 −→ π0 .

The holomorphic vector field on T K = ωA

∂ ∂ + π1 , A ∂ω ∂π1

vanishes on the projective twistor space when it is proportional to the Euler vector field. This happens on a set B = {{ω A = 0, π1 = 0} ∪ {π0 = 0}} ⊂ T which is a union of the line and a hyperplane C3 ⊂ T . The set B descends to a union of a hypersurface and a point in the projective twistor space (Fig. 4). The minitwistor space Z corresponding to the projective structure U is the factor space of PT /B by the trajectories of K. Each trajectory in T is parametrised by its intersection with the singular surface C3 given by π · o = 0 in T so the space of trajectories in PT is Z = CP2 . Two CP1 s in CP2 intersect in a point so the normal bundle of each CP1 is O(1) and we have a projective structure. To obtain the explicit parametrisation of these CP1 s eliminate π0 from the twistor equation to get π1 = ω A u A , where, u A := p A /( p B q B ) parametrise the twistor lines in Z and are coordinates on U . The flat metric in M is conformal to (1.3) with Aα , G = z 2 /2 and conformal factor et .


4

T=C

ωA

115

PT

K π 0í π 1’ 0

ι

1

CP

Fig. 4. Quotient of the non–projective twistor space by the Euler vector field showing the singular set of K

8. Outlook We have locally classified neutral signature ASD conformal structures with null Killing vectors. Some of these are defined on compact manifolds. It would be interesting to investigate the global properties of other conformal structures we have found. It would also be interesting to understand in more detail which conformal structures admit special types of metric, for example Ricci-flat or Einstein (in this case the pure Killing vectors must be twist–free [14]). So far the only results we have in this direction are given in Sect. 6.5. The existence of these special metrics should be related to invariants of the corresponding projective structure. The recent work of Calderbank [3] extended many of the results obtained in this paper. In particular Calderbank gave a twistor characterisation of ASD conformal structures which admit a geodesic shear free congruence ι A . Not all such congruences give rise to null conformal Killing vectors K such that ι A K A A = 0, and Calderbank characterised those which do. Acknowledgements. We wish to thank Helga Baum, David Calderbank, Claude LeBrun, Lionel Mason, George Sparling and Paul Tod for helpful discussions. S.W. thanks the EPSRC for financial support.

A. Appendix Here we summarise the required spinor notation and present the calculations leading to a proof of (4.3). We use similar conventions to Penrose and Rindler [22] adapted to neutral signature, but our indices are concrete. Spin connection and curvature decomposition. As usual, we denote the Levi-Civita connection of the metric by ∇. The ‘spin connection coefficients’ are defined by

∇(eCC ) = θ D D ⊗ ( D D CE e EC + D D CE eC E ), together with the symmetry requirement D D C E = D D EC , D D C E = D D E C . These conventions result in the following expressions for differentiation of spinor components, where ι A is a two-component spinor field over the manifold: ∇ B B ι A = e B B (ι A ) + B B CA ιC , ∇ B B ι A = e B B (ι A ) − B B AC ιC ,

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and similarly for a primed spinor field. These are the concrete expressions for the covariant differentiation of spinors using the connections on S and S inherited from the Levi-Civita connection, mentioned in Sect. 2.1. One can extend the above expressions to multi-component objects in the obvious way, allowing covariant differentiation of tensors, which agrees with covariant differentiation using the Levi-Civita connection. The Riemann tensor has the following spinor decomposition ([22], p. 236): Rabcd = C ABC D A B C D + C˜ A B C D AB C D + ABC D A B C D + A B C D AB C D +2 ( AC BC A C B D − AD BC A D B C ).

(A.1)

The Weyl spinors C ABC D , C˜ A B C D are completely symmetric, and the traceless Ricci tensor ABC D is symmetric on each pair of indices. The C, C˜ spinors make up the self-dual and anti-self dual parts of the Weyl tensor. In the language of representation theory, this is the decomposition of Rabcd into irrreducible representations under the action of S L(2, R) × S L(2, R) (with R replaced by C for the holomorphic case). Note that in + + −−, spinor components are real. For analytic metrics, we can analytically continue, which amounts to allowing the spinors to be complex. The remaining calculations in this appendix are valid in both cases. Integrability of α and β surfaces. We now show that (2.7) and (2.8) are equivalent to the fact that the two-plane distributions defined by o A and ι A are integrable. The leaves are called α-surfaces and β-surfaces respectively. The argument is well-known in twistor theory. We will do the calculation for the o A case; the ι A case is identitical. Let X = α A o A e A A , Y = β A o A e A A be vector fields, which by definition are in the α-planes determined by o A . Then if they commute we have: [X, Y ] A A = ( f α A + gβ A )o A ,

for some functions f, g. Multiplying by o A gives

o A [X, Y ] A A = o A (X B B ∇ B B Y A A − Y B B ∇ B B X A A ) = 0.

Substituting the spinor expressions for X A A and Y A A results in

o A o B ∇ B B o A = 0, which is (2.8), and it is easy to show this is sufficient as well as necessary. Twistor distribution and ASD. Locally, the primed spin bundle S is isomorphic to M×C2 . We choose the coordinates on the C2 to be π A for A = 0, 1. This vector bundle has a connection inherited from the Levi-Civita connection of the metric, and therefore we can find the horizonal lifts e˜ A A of the e A A , defined by covariantly constant sections. These lifts are as follows: ∂ e˜ A A = e A A − A A BC π B . ∂π C Using the following formula ([22], p. 247) relating curvature quantities to the deriv atives of A A CD : and the spinor decomposition of the curvature (A.1) we find

[π A e˜ A A , π B e˜ B B ] = ( A A BD − B A AD )π A π B e˜ D B

+π A π B AB F

Q

∂ C˜ A B E Q π E . ∂π F


117

One can see from this that if C˜ A B C D = 0 then π A e˜ A A , A = 0, 1, forms an integrable distribution. The projection of a leaf of this distribution to M gives an α-surface. We have demonstrated that if the metric is anti-self-dual, then given any point p ∈ M and an α-plane at p, there is a unique α-surface through p tangent to this α-plane . This was first shown by Penrose [21], although without using the primed spin bundle. For our purposes the above formulation will be most useful. Proof of Proposition 1. We have the following identity: 1 K a Rabcd = ∇b ∇c K d − (η,b gcd − η,c gbd + η,d gbc ), 2

(A.2)

where η is the conformal factor appearing in (2.4). Using the curvature decomposition (A.1) to convert this into spinor form, one can calculate

[K A A e˜ A A , π B e˜ B B ] = (K A A A A BD − ψ B D )L D 1 −π B (φ B A B A + η B A B A ) e˜ A A 2

+π B π E e B B φ E F − B B EG φG F + B B GF φ E G ∂ 1 − (e B E η) FB (A.3) . 4 ∂π F

We wish to add a vertical term to K A A e˜ A A which will cancel all the non-L A terms on the RHS of (A.3). We don’t mind multiples of the Euler vector field since this gets quotiented out on projectivizing. A simple calculation shows that K˜ as defined in (4.4) does the trick.

References 1. Barrett, J., Gibbons, G.W., Perry, M.J., Pope, C.N., Ruback, P.J.: Kleinian geometry and the N = 2 superstring Int. J. Mod. Phys. A. 9, 1457–1494 (1994) 2. Bryant, R., Griffiths, P., Hsu, L.: Toward a Geometry of Differential Equations. In: Geometry, Topology, and Physics, Conf. Proc. Lecture Notes Geom. Topology, edited by S.-T. Yau, Vol. IV, Cambridge, MA: Internat. Press, 1995, pp. 1–76 3. Calderbank, D.: Selfdual 4-manifolds, projective structures, and the Dunajski-West construction. http://arxiv.org/list/math.DG/0606754, 2006 4. Calderbank, D.: Integrable Background Geometries. Preprint, 2002 5. Dunajski, M.: The twisted photon associated to hyper-Hermitian four-manifolds. J. Geom. Phys. 30, 266– 281 (1999) 6. Dunajski, M.: Anti-self dual four-manifolds with a parallel real spinor. Proc. R. Soc. Lond. A 458, 1205– 1222 (2002) 7. Dunajski, M., Mason, L.J., Tod, K.P.: Einstein–Weyl geometry, the dKP equation and twistor theory. J. Geom. Phys. 37, 63–92 (2001) 8. Finley, III, J.D., Plebański, J.F.: The classification of all H spaces admitting a Killing vector. J. Math. Phys. 20, 1938–1945 (1979) 9. Fino, A., Pedersen, H., Poon, Y-S., Sorensen, M.W.: Neutral Calabi-Yau structures on Kodaira manifolds. Commun. Math. Phys. 248, 255–268 (2004) 10. Graham, C.R.: On Sparling’s characterization of Fefferman metrics. Am. J. Math. 109, 853–74 (1987) 11. Hitchin, N.J.: Complex manifolds and Einstein’s equations. In: Twistor Geometry and Non-Linear Systems Lecture Notes in Mathematics, Vol. 970, Berlin-Heidelberg-New York: Springer Verlag, 1982 12. Ivanov, S., Zamkovoy, S.: Parahermitian and paraquaternionic manifolds. Differ. Geom. Appl. 23, 205– 234 (2005) 13. Jones, P., Tod, K.P.: Mini twistor spaces and Einstein-Weyl spaces. Class. Quant. Grav. 2, 565–577 (1985)

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14. Julia, B., Nicolai, H.: Null-Killing vector reduction and Galilean geometrodynamics. Nucl. Phys. B 439, 291–323 (1995) 15. Kamada, H.: Neutral hyper-Kahler structures on primary Kodaira surfaces. Tsukuba J. Math. 23, 321– 332 (1999) 16. Law, P.: Classification of the Weyl curvature spinors of neutral metrics in four dimensions. To be published in J. Geom. Phys., 2006 17. LeBrun, C.: Spaces of Complex Geodesics and Related Structures. D. Phil thesis, Oxford University, 1980 18. LeBrun, C., Mason, L.J.: Nonlinear Gravitons, Null Geodesics and Holomorphic Discs. http://arxiv.org/list/math.DG/0504582, 2005 19. Mason, L.J., Woodhouse, N.M.J.: Integrability, self-duality and twistor theory. LMS Monographs New Series, Vol. 15. Oxford: Oxford University Press, 1996 20. Nurowski, P., Sparling, G.A.J.: Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations. Class. Quant. Grav. 20, 4995–5016 (2003) 21. Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Relat. Grav. 7, 31–52 (1976) 22. Penrose, R., Rindler, W.: Spinors and space-time, Vols. 1 & 2, Cambridge: Cambridge University Press, 1986 23. Plebanski, J.F.: Some solutions of complex Einstein equations. J. Math. Phys. 12, 2395–2402 (1975) 24. Sparling, G.A.J., Tod, K.P.: An example of an H-Space. J. Math. Phys. 22, 331–332 (1981) 25. Szekeres, P.: Spaces conformal to a class of spaces in general relativity. Proc. R. Soc. London A 274, 206– 212 (1963) 26. Tafel, J., Wójcik, D.: Null Killing vectors and reductions of the self-duality equations. Nonlinearity 11, 835–844 (1988) 27. Tafel, J.: Two-dimensional reductions of the self-dual Yang-Mills equations in self-dual spaces. J. Math. Phys. 34, 1892–1907 (1993) 28. Ward, R.S.: A class of self-dual solutions of Einstein’s equations. Proc. R. Soc. London A 363, 289– 295 (1978) 29. Ward, R.S.: Integrable and solvable systems and relations among them. Phil. Trans. R. Soc. A 315, 451– 457 (1985) 30. Ward, R.S.: Einstein–Weyl spaces and SU (∞) Toda fields. Class. Quantum Grav. 7, L95–L98 (1990) Communicated by G.W. Gibbons


Communications in


Generalized Inverse Mean Curvature Flows in Spacetime Hubert Bray1 , Sean Hayward2,3 , Marc Mars4 , Walter Simon4 1 Mathematics Department, Duke University, Box 90320, Durham, NC 27708, USA.

E-mail: [email protected]

2 Center for Astrophysics, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China.


3 Center for Mathematical Physics, East China University of Science and Technology, 130 Meilong Road,

Shanghai 200237, China

4 Facultad de Ciencias, Universidad de Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain.

E-mail: [email protected]; [email protected] Received: 6 February 2006 / Accepted: 1 September 2006 Published online: 3 March 2007 – © Springer-Verlag 2007

Abstract: Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen’s proof of the Riemannian Penrose inequality. 1. Introduction Penrose has conjectured [1] that in an asymptotically flat spacetime satisfying the dominant energy condition, the total ADM mass M and the area |S| of an outermost apparent horizon S must satisfy |S| M≥ (1) 16π in units where Newton’s gravitational constant is unity. A similar inequality should hold for the Bondi mass as well. In the presence of a negative cosmological constant , |S| χ M≥ − |S| (2) 16π 2 12π is conjectured [2] for suitably defined masses at null infinity, where χ is the Euler characteristic of S. Penrose gave a heuristic argument for (1) based on the standard view of

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H. Bray, S. Hayward, M. Mars, W. Simon

gravitational collapse, involving cosmic censorship, a final stationary configuration and the classification of stationary black holes, among others. For inequality (1) to hold, it is known that S has to be area outer minimizing in some sense [3, 4]. For counterexamples when this condition is not fulfilled see [4, 5]. However, a rigorous proof of this Penrose inequality has been achieved only in two particular cases: in spherical symmetry [6–8] and in the time-symmetric case. The latter can be formulated as a purely Riemannian problem, termed the Riemannian Penrose inequality, which involves a Riemannian, asymptotically Euclidean 3-manifold (, γ ), with non-negative Ricci scalar, which translates the dominant energy condition, and with an outermost minimal surface S, which substitutes the apparent horizon. The inequality (1) for this case was proven by Huisken and Ilmanen [9] for connected S and by Bray [10], with a totally different method, for arbitrary S. Bray’s method uses a flow of Riemannian metrics interpolating between the starting metric γ and the metric of a t = const. slice of Schwarzschild outside the horizon. This flow of metrics keeps the area of the outermost minimal surface constant and does not increase the ADM mass. The Penrose inequality follows from the fact that equality holds for Schwarzschild. The approach of Huisken and Ilmanen makes use of the Geroch-Hawking mass [11] which is associated to any closed surface S in a hypersurface : |S| χ (S) 1 4 MG (S) = p2 + η S , (3) − 16π 2 16π S 3 where η S is the surface element, |S| is the total area of S, and p is the mean curvature of S in . The basic observations due to Geroch [11] and to Jang and Wald [12] refer to the case = 0 and are the following: (3) is monotonic for a special flow of two-surfaces called the Inverse Mean Curvature Flow (IMCF), approaches the ADM mass for suitable surfaces at infinity, such as metric 2-spheres, and equals the right-hand side of (1) at the minimal surface. Thus, if the IMCF exists globally and approaches infinitely large round spheres, the Penrose inequality follows. However, the IMCF does in general develop singularities, and Huisken and Ilmanen [9] succeeded in defining a suitable weak version of the flow for which global existence could be proven. This weak version of the IMCF allows for jumps and one needs to question monotonicity of the Geroch-Hawking mass at the jumps. The condition of connectedness of the starting minimal surface was required at this point. The Geroch-Hawking mass is a translation to 3-dimensional Riemannian manifolds of the Hawking mass [13] |S| χ (S) 1 4 M H (S) = H 2 + ηS , (4) − 16π 2 16π S 3 defined for 2 surfaces S embedded in spacetime (V, g) with mean curvature vector H , where H 2 = ( H · H ), the dot denoting the inner product on (V, g). If S is embedded in a 3-surface such that H points along the unique unit normal ν of S, one can write H = pν . However, a 2-surface S will in general have different Hawking and GerochHawking masses, the former depending only on S and the latter also on . We believe that the Hawking mass is a good candidate for proving the general Penrose inequality along the lines sketched above for MG in the Riemannian case. In spherical symmetry, there is a quite simple and otherwise general proof based on monotonicity of M H [14]. To summarize existing work, to describe the extensions achieved in the present paper

Generalized Inverse Mean Curvature Flows in Spacetime

121

and to formulate the open issues, we found it useful to distinguish the following five steps: 1. 2. 3. 4. 5.

Variation of the Hawking mass Monotonicity properties for special flows Local existence of good flows Global existence of good flows Limits at the horizon and at infinity

As to the variation of M H , in Sect. 2 we find an expression which is valid for arbitrary variations and which, when the flow is everywhere spacelike, takes the form |Sλ | d M H (Sλ ) = matter density + gravitational energy density + rest η Sλ , (5) dλ 16π Sλ where λ denotes the flow parameter and Sλ the corresponding two-surface. The names of the first two terms in the bracket mean that they vanish in vacuum and in spherical symmetry, respectively, and they turn out to be non-negative under reasonable positiveenergy conditions on the matter, and under causality conditions on H and on the flow vector ξ . A “gravitational energy density” is not well-defined in general, but the point here is to perform some intelligent splitting which allows the “rest” (or its integral) to be removed by some simple and not over-restrictive choices of the flow. Such monotonicity conditions are specified in Theorem 1 in Sect. 3, which is the main result of this paper and which we sketch here. To have the energy terms non-negative, we impose the dominant energy condition, and we require that H is spacelike or null with (ξ · H ) ≥ 0. In order to get rid of the “rest” there are, in general, many possibilities. Since we wish to determine a flow of codimension 2, it is natural to impose two conditions. While we give two alternative options for each of these conditions in Theorem 1, we focus in this paper on a particular choice for either case. As the first one we adopt the inverse mean curvature condition, namely that (ξ · H ) = a(λ) with a(λ) constant on each Sλ . This condition guarantees, in particular, an IMCF on the three-surface spanned by the evolving two-surfaces themselves, so it is a natural extension of the Geroch condition mentioned above. The second one is the dual inverse mean curvature condition, reading that (ξ · H ) should be a constant c(λ) on each Sλ , where H denotes the dual of H . To get monotonicity of M H , we take the inverse mean curvature condition and the dual inverse mean curvature condition together, for which we adopt the name uniformly expanding flow. In this case, monotonicity still holds if we allow ξ to be null somewhere, or everywhere. On regions where either ξ or H are null, the inverse mean curvature condition in fact implies the dual condition, and vice versa. If in particular ξ is null, this condition has been analyzed by Hawking himself [13], by Eardley [15] and, more extensively, by Hayward [6], while in the spacelike case, the uniformly expanding flows were investigated by Malec, Mars and Simon [16]. In the special case c(λ) = 0, the flow is tangent to the inverse mean curvature vector and was analyzed by Frauendiener [17]. While a(λ) can be set equal to 1 by a reparametrization, if it has no zeros, the presence of c(λ) provides a one-parameter freedom for the flows in the spacelike case. The alternatives to these conditions included in Theorem 1 are the following: instead of the inverse mean curvature condition we can impose a Poisson equation on S for the logarithm of the lapse function of the flow, with a source term depending on the scalar curvature of S and on H 2 . This equation can always be solved uniquely on any Sλ .

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Here we restrict ourselves to flows which are spacelike everywhere. This condition has a counterpart in codimension one, with H 2 replaced by p 2 , where it can be used as an alternative to IMCF in order to obtain monotonicity of the Geroch mass. Instead of the dual inverse mean curvature condition, we can require that a certain 1-form defined on S has vanishing divergence. This option was also mentioned in [16]. A flow of the same type, namely a Poisson equation for the logarithm of the lapse with curvature terms as sources, has been used before in the proof of global stability of Minkowski space [18–20]. In this case, the key issue is to construct a double null foliation of the given spacetime, based on a spacelike foliation which is sufficiently smooth for weak data. This is guaranteed by the above-said equation for the lapse, but could not be shown for the IMCF. While the Hawking mass is not necessarily monotonic under the flows considered in [18–20], it was shown that its derivative falls off rapidly at infinity. The conditions on the flow vector ξ as mentioned above guarantee monotonicity and they are consistent and solvable on every two-surface Sλ . However, this does not mean that we have shown existence of a flow in spacetime, i.e. a sequence of Sλ , not even locally. For the IMCF in 3-dimensions, the embedding functions satisfy a parabolic system for which local existence is guaranteed. In spacetime, the uniformly expanding condition gives a local existence result in the case when ξ is null, since only the null geodesics of the flow have to be determined in this case. On the other hand, in the spacelike case we are left with a forward-backward parabolic system which was observed by Huisken and Ilmanen in the case c(λ) = 0 and which we show in general in Sect. 4. For such systems, not even local existence results are available. A key element of the Huisken-Ilmanen proof in codimension 1 is, however, the weak or distributional formulation of the flow in terms of its level sets, which satisfy degenerate elliptic equations, if sufficiently differentiable. We have obtained a generalization of this variational principle to the spacetime case, which we also describe in Sect. 4. We finally comment on point 5 of the list above: M H obviously gives the right-hand side of (2) on any marginally trapped surface. At infinity, provided the Sλ approach surfaces of constant curvature, the Hawking mass approaches expressions which have been studied before: ADM and Bondi for = 0; for < 0, see [2]. Except for the asymptotically flat case at spatial infinity, it is an open issue if the flow necessarily leads to such surfaces. 2. Variation of the Hawking Mass We consider orientable Riemannian manifolds (V, g) and orientable closed (i.e. compact without boundary) submanifolds (S, h), where h is the induced metric on S. For our main results, we have to restrict V to be 4-dimensional with g of signature (−, +, +, +), and S to be 2-dimensional, with a positive definite metric h. We will also consider the case that S is embedded in a 3-dimensional manifold with positive definite metric, which follows from the above one by choosing a suitable embedding ⊂ V . Some auxiliary results, in particular Lemma 1 and Lemma 2, hold for arbitrary dimensions and arbitrary metric signatures. Up to and including these lemmas, we will consider this general setup, and impose suitable restrictions afterwards. Our aim in this section is to calculate the variation of the Hawking mass (4) along an arbitrary flow. We take an arbitrary C 1 embedding

: Sˆ × I −→ V, (x, λ) −→ (x, λ) ≡ λ (x),


123

where Sˆ is a copy of S viewed as an abstract manifold detached from V and I 0 is an ˆ are required to be C 2 , so that the variation open interval of R. The leaves Sλ ≡ λ ( S) of H can be defined, and we write S for S0 . Choosing an orientation on (S, h) defines uniquely a canonical two-form η S from the volume form η on (V, g). The tangent space T p of V at p ∈ S can be decomposed as T p = T p ⊕ T p⊥ , where

T p and T p⊥ are the tangent and normal spaces to S, respectively, and we shall write v = v + v⊥ for any vector v ∈ T p . The flow of submanifolds Sλ has a flow vector v = d (∂λ ) which may have an arbitrary causal character. Accordingly, the embedded submanifold N ≡ ( Sˆ × I ) need not have constant causal character at different points; in the spacetime case, the hypersurface N could be timelike, spacelike or null at different points. Let us define ξ as its normal component with respect to Sλ , ξ = v⊥ . We shall call this vector the normal flow vector and also simply flow vector when no confusion arises. Evaluating the Hawking mass for any leaf Sλ , we write M H (λ) ≡ M H (Sλ ). We will not add a subscript λ to H or to other geometric objects on Sλ , unless the meaning is not clear from the context. The derivative of (4) with respect to λ involves £ξ η Sλ = (ξ · H )η Sλ . The integral of this expression over S yields the first variation of the area. In terms of the mean value Sλ ξ · H η Sλ a(λ) ≡ |Sλ | of (ξ · H ) over each leaf of the foliation, we can write d|Sλ |/dλ = a(λ)|Sλ |. Using standard expressions for derivatives of geometric objects within an integral we now find d M H (Sλ ) 1 = dλ 16π

|Sλ | 4π χ (Sλ )a(λ) − 2 £ξ H · H + (£ξ g)( H , H )+ 16π Sλ

1 4 η Sλ . (6) + H · ξ + a(λ) H 2 + 2 3

Note that only the normal part ξ of v = d (∂λ ) appears in this expression because the tangential component of v gives rise to a divergence of a vector on T which integrates to zero, Sλ being closed. In order to reformulate (6), we recall some standard embedding formulas, e.g. [21], and collect some computations in Lemma 1 and Lemma 2 below. Let X , Y be tangent vector fields to S, ξ be a normal vector to S, and ∇ the covariant derivative on V . We can decompose the covariant derivatives along S into components tangent and normal to S as follows: ∇ X Y = D X Y − K X , Y , (7) ∇ X ξ = ∇ X⊥ ξ + Aξ ( X ).

(8)

Here D is the Levi-Civita connection on (S, h), K is the second fundamental tensor of S, ∇ ⊥ is the connection on the normal bundle of S and Aξ : T p S → T p S is the Weingarten map, both defined by the decomposition in (8).

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Lemma 1. Let S be a C 2 embedded submanifold of arbitrary codimension in a Riemannian manifold (V, g) of arbitrary signature and assume that S has a non-degenerate induced metric. Let X , Y be vector fields tangent to S and ξ a C 2 vector field normal to S defining a flow of surface Sλ as above. Then

⊥ ⊥ (£ξ K )( X , Y ) = −∇ X⊥ ∇Y⊥ ξ + ∇ D Y ξ + K Aξ (Y ), X − X ⊥ ⊥ − ∇ K ( X ,Y ) ξ − R(ξ , X )Y . Proof. By construction d λ ( X ) is a vector field tangent to Sλ . It follows directly that the commutation of the velocity vector v = d (∂λ ) and X gives a vector field tangent to S and the same happens with [ξ , X ], i.e. [ξ , X ]⊥ = 0. Using (7) and (8) and the γ = ∇α ∇ γ − ∇ ∇α γ − ∇ γ for the Riemann tensor we get definition R( α , β) β β [ α ,β] ⊥ ⊥ = ∇ξ ∇ X Y − ∇ X ∇ξ Y + K Y , [ξ , X ] = R(ξ , X )Y ⊥ = ∇ξ D X Y −∇ξ K ( X , Y )−∇ X ∇Y ξ −∇ X [ξ , Y ] + K Y , [ξ , X ] = ⊥ ⊥ − ∇ K ( X , Y ) − ∇ ⊥ ∇ ⊥ ξ + K A (Y ), X + = ∇D ξ ξ ξ X Y X Y + K X , [ξ , Y ] + K Y , [ξ , X ] . Using now ∇ξ ( K ( X , Y )) = ∇ K ( X ,Y ) ξ + £ξ ( K ( X , Y )) and the fact that £ξ ( K ( X , Y )) = £ ( K )( X , Y ) + K ([ξ , X ], Y ) + K ( X , [ξ , Y ]), the lemma follows directly.

ξ

We now treat the non-trivial term £ξ H in (6) as follows. As H is normal to S, only (£ξ H )⊥ needs to be calculated. We choose a basis {e A } of the tangent space T p . Without loss of generality, we can assume that this basis is Lie propagated along the variation vector ξ , i.e. [ξ , eA ] = 0. Indices A, B, · · · therefore refer to tensor objects within S expressed on this basis. For instance K AB = K (e A , eB ), and ∇ A = ∇eA . Such indices are lowered and raised with the induced metric h AB and its inverse h AB . Directly from the definition of H = tr Sλ K = h AB K AB

(9)

and ∂λ h λAB = −2(K λAB · ξ ), we get, after making use of Lemma 1, ⊥ ⊥ AB ⊥ £ξ Hλ = −2 K AB K AB · ξ − h AB ∇ ⊥ ∇ D A eB ξ + A ∇B ξ + h ⊥

⊥ . (10) + h AB K Aξ (e A ), eB − h AB R(ξ , eA )e B − ∇ H ξ The first and fourth termsare the same apart from coefficient, while the second and third ⊥ ⊥ terms combine as −tr Sλ ∇ ∇ ξ . Also, inserting (10) in (6) we observe that the last term, in ∇ ξ , cancels with (£ g)( H , H ). Notice that this term is not intrinsic for the H

ξ

variation because it depends on how ξ changes along normal directions. It is clear that such dependence cannot occur in a first derivative. Summarizing, we have the following result.


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Lemma 2. With the same assumptions as in Lemma 1, suppose also that S is closed and orientable. Then, the variation of the Hawking mass along ξ reads |Sλ | 1 d M H (Sλ ) = 4π χ (Sλ )a(λ)+ dλ 16π 16π

1 4 2( K AB · H )( K AB · ξ ) − ξ · H + a(λ) H 2 + + + 2 3 Sλ

+ 2h AB R(ξ , eA )e B · H + 2tr Sλ H · ∇ ⊥ ∇ ⊥ ξ η Sλ . (11) We now restrict ourselves to spacelike two-dimensional surfaces in a four dimensional spacetime of Lorentzian signature. In particular, the normal spaces Tλ⊥ are now two-dimensional and Lorentzian, and inherit an orientation from the orientation of S. µ We can introduce projection operators P µ ν and P⊥ ν which act as the identities on T p ⊥ and T p , respectively, and annihilate vectors in the other orthogonal spaces. In particular, µ µ we have δν = P⊥ ν + P µ ν. , This setup also allows us to define the Hodge dual operation on normal fields W W α =

1 α γ η W β e A eδB η SAB . λ 2 βγ δ

) = W and (W · U ) = −(W · U ). Therefore any normal vector is We have (W orthogonal to its dual and both have opposite norms. In particular, a normal null vector or anti-self dual, l = −l. We note the following completeness is either self-dual, l = l, property, valid for any pair of vectors u, v ∈ T p⊥ : αβ

v α u β + u α v β − v α u β − u α v β = 2( v · u)P⊥ .

(12)

This expression can be proven directly by noting that both terms are symmetric, orthogonal to Sλ on each index and give an identity when contracted with any tensor product of u α , vα , u α , vα . In particular, when k and l are null and linearly independent, with = 0, we have φ ≡ −(l · k) αβ

k α l β + l α k β = −φ P⊥ .

(13)

= We next introduce the Ricci scalar R(h λ ) of (Sλ , h λ ), and the trace-free part 1 T K − 2 H h of the extrinsic curvature vector K . Furthermore, we define µν , the “transverse part of the gravitational energy” by 1 AB · AB · H ξ · AB − AB ξ · H 8π T ( H , ξ ) = 2 1 AB · AB − AB ξ · H (14) AB · H ξ · = 2 for any pair of vectors H , ξ . This terminology will be justified later, and the equivalence of the two alternative expressions in (14) follows from (12) with v → H and u → ξ . It also follows from the fact that T belongs to the class of so-called super-energy tensors, see [22], and from general properties thereof. We also define =

1 1 1 R(h λ ) − H 2 − , 2 4 3

(15)

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which may be called the “Hawking energy density” since, using the Gauss-Bonnet formula, R(h λ )η Sλ = 4π χ (Sλ ) (16) Sλ

we find from (4) that

M H (S) =

|S| 16π

1 4π

η S .

(17)

S

Lemma 3. Let (V, g) be an oriented 4-dimensional spacetime of signature (−, +, +, +) and S a 2-dimensional oriented closed and spacelike C 2 surface in V . Let denote a flow of two-surfaces as defined above. Then, the derivative of the Hawking mass along the flow reads d M H (Sλ ) 1 |Sλ | = G( H , ξ ) + ( H · ξ ) + 8π T ( H , ξ )+ dλ 8π 16π Sλ

(18) + tr Sλ H · ∇ ⊥ ∇ ⊥ ξ − ξ · H − a(λ) η Sλ , where G is the Einstein tensor of (V, g). Proof. We first note that

αβ Rαβ H α ξ β = Rαβ H α ξ β + ξ · H P⊥ Rαβ ,

(19)

which follows from (12) with v → H and u → ξ by contracting with Rαβ . We next recall the Gauss identity αβ γ δ (20) P P Rαγβδ = R(h λ ) − H 2 + K AB · K AB . We can now show that 1 ξ · H H 2 − R(h λ ) − K AB · K AB h AB R(ξ , eA )e B · H = G( H , ξ ) + 2 (21) µν l, using (13) for P µν and working out by decomposing ξ , H and P⊥ in a null basis k, ⊥ the contractions, using also (19) and (20). Finally, we obtain (18) by inserting (21) into (11) and making use of the Gauss-Bonnet formula (16).

We now reformulate (18) in the case where ξ is not null, and we write (ξ · ξ ) = ξ 2 = for = ±1 and some function ψ. We decompose ξ in a null basis as follows:

e2ψ

ξ = ξl + ξk

ξk = B k for some functions A, B. where ξl = Al,

(22)

Then e2ψ = −2 ABφ = 0. Furthermore, we introduce the following one-form on Sλ 1 l · ∇ A ξk k · ∇ A ξl UA = − , 2φ B A and the “longitudinal part of the gravitational energy density” L (ξ )µν , 8π L (ξ )( H , ξ ) = (|U |2 + |Dψ|2 )(ξ · H ) − (2U · Dψ)(ξ · H ).

(23)


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Lemma 4. We require the assumptions of Lemma 3 and in addition that ξ is nowhere null. We obtain 1 |Sλ | d M H (Sλ ) = G( H , ξ ) + ( H · ξ ) + 8π T ( H , ξ )+ dλ 8π 16π Sλ

+8π L (ξ )( H , ξ )−D.U (ξ · H )+ ξ · H − a(λ) (ψ−) η Sλ , (24) where D.U denotes the divergence of U . Proof. In terms of the basis introduced above, we have ∇ ⊥ ξ = (Dψ + U )ξl + (Dψ − U )ξk . It follows that tr Sλ ∇ ⊥ ∇ ⊥ ξ = ψ + D · U + |Dψ + U |2 ξl + ψ − D · U + |Dψ − U |2 ξk , and hence tr Sλ H · ∇ ⊥ ∇ ⊥ ξ = ψ + |Dψ|2 + |U |2 (ξ · H ) − (D · U + 2U · Dψ) (ξ · H ). (25) Inserting this in (18) and using (23) gives (24).

3. Monotonicity Properties of the Hawking Mass Lemmas 3 and 4 are the basis for a systematic discussion of monotonicity of the Hawking mass. We shall discuss the terms appearing on the r.h. sides of (18) and (24) separately and show that each of them is non-negative under suitable conditions on the flow. Finally we put these conditions together to obtain the monotonicity result for M H , Theorem 1. As to the integral G( H , ξ ) + H · ξ η Sλ (26) Sλ

we require that

(G αβ + gαβ )u α v β ≥ 0

(27)

for any pair of future (or past) causal vectors. If the Einstein field equations hold, the bracket is equal to 8π times the energy momentum tensor. In any case, we call (27) the dominant energy condition. Accordingly, (26) is non-negative if H and ξ are future (past) causal. In order to consider the following terms in (18), it is convenient to define four closed subsets of the normal space T p⊥ . Being a Lorentzian vector space, we can speak of causal (i.e. timelike or null) vectors and of achronal (i.e. spacelike or null) vectors. Let ± us denote by C ± p the subset of future (past) causal vectors and A p the set of vectors such that its Hodge dual is future (past) causal. Notice that all the vectors in A± p are achronal. These four sets clearly cover the normal space T p⊥ and they are not disjoint. The 0 vector

128


Fig. 1. Decomposition of the normal space into four closed sets

belongs to all of them. Furthermore, the intersection C + ∩ A+ , dropping the subindex p from now on for simplicity, is the set of self-dual future null vectors, and similarly for C − ∩ A− . As for C + ∩ A− , this is the set of anti-self dual future null vectors, and past for C − ∩ A+ . This decomposition is visualized in Fig. 1. A positivity lemma for T ( H , ξ ) follows directly from the positivity result for the super-energy tensors, Theorem 4.1 in [22]. Since our case only requires a simpler version of the result, we add a proof for completeness. Lemma 5. The inequality AB · ξ − AB · AB ξ · H ≥ 0 AB · H 2

(28)

AB if and only if ξ , H ∈ C + , ξ , H ∈ C − , ξ , H ∈ A+ or holds for any tensor − ξ , H ∈ A . Proof. The “only if” part of the lemma is easy to prove by finding suitable counterexamρ ples. For the direct part, let p be any point in S. The object 2 AB µ νAB − AB ρAB γµν | p defines a map on T p N × T p N which is obviously continuous. Thus, it suffices to prove the inequality almost everywhere on the subspace (C + ×C + )∪(C − ×C − )∪(A+ × A+ )∪ (A− × A− ). We can assume without loss of generality that H and ξ are both non-null and linearly independent. Since, by assumption, both vectors belong to the same subset C + , C − , A+ or A− , we have H 2 = a 2 , ξ 2 = b2 , where = ±1 and a, b are strictly positive. Moreover we have (ξ · H ) > 0. On any two-dimensional Lorentzian space, the inequality ( u · v)2 ≥ v 2 u 2 holds for any pair of vectors. Hence (ξ · H ) ≥ ab. Since AB = AB H + AB ξ . Direct H and ξ are linearly independent, we can decompose substitution into the left-hand side of (28) gives ξ · H AB AB a 2 + AB AB b2 + 2a 2 b2 AB AB ≥ ab a 2 AB AB + +b2 AB AB + 2ab AB AB = ab (a AB + b AB ) a AB + b AB ≥ 0 and the lemma is proven.

Putting AB = AB , it follows that T ( H , ξ ) is monotone iff the velocity vector of the flow points into the same quadrant as the mean curvature vector, at each point on the surface, or equivalently if ξ , H ∈ C + , ξ , H ∈ C − , ξ , H ∈ A+ or ξ , H ∈ A− .


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We turn now to the positivity of L (ξ )( H , ξ ). This object is defined only when ξ is non-null. However, Eq. (29) will be used later in the limit when ξ tends to a null vector. Lemma 6. Let ξ be non-null everywhere on Sλ and assume that either (i) ξ ∈ C + , H ∈ C − , (ii) ξ ∈ C − , H ∈ C + , (iii) ξ , H ∈ A+ , or (iv) ξ , H ∈ A− holds everywhere on Sλ . Then L (ξ )( H , ξ ) ≥ 0. Proof. Conditions (i), (ii), (iii) or (iv) together with the fact that ξ is not null implies (ξ · H ) = 0 everywhere. We rewrite (23) as 2 · H ) ( ξ H 2ξ 2 L Dψ + |Dψ|2 8π (ξ )( H , ξ ) = ξ · H U − (29) (ξ · H ) (ξ · H ) after using H 2 ξ 2 = (ξ · H )2 −(ξ · H )2 , which follows from (12). Each of the hypotheses

(i) to (iv) implies H 2 ξ 2 ≥ 0 and ( H · ξ ) > 0, and the lemma follows. Turning now to the remaining terms in the derivative of the Hawking mass we first assume that ξ is non-null everywhere, in which case we are left with the expression

ξ · H − a(λ) (ψ − ) η Sλ , −D.U (ξ · H )η Sλ + (30) Sλ

Sλ

in Lemma 4. We want to remove it by imposing suitable conditions on the flow. For this purpose, we first note that the geometric flow of 2-surfaces is obviously independent of the parametrization used, and a reparametrization changes the velocity field ξ by a nowhere-zero constant on each leaf. In particular, such a reparametrization rescales a(λ) by an arbitrary nowhere-zero function of λ. Thus, if a(λ) = 0 everywhere, a = 1 can be assumed without loss of generality. If, on the other hand, a has zeros, be it at isolated points or on intervals, those values are meaningful and independent of reparametrization. In Theorem 1 below, we will restrict ourselves to conditions on the flow which make each integral of the sum (30) vanish separately. In either expression, a sufficient condition for vanishing is that either the first factor in the integrand is identically zero, or that the second factor is constant on S. More general conditions which remove (30) but not each term separately, will be discussed in Remark 1 after Theorem 1. As to removing the second term in (30), the condition (ξ · H ) = a(λ) is the generalization to codimension two of the IMCF on surfaces in a three-dimensional space. Indeed, in codimension one, the mean curvature vector and the normal velocity are obviously parallel, and setting a = 1, assuming a(λ) = 0, this condition just states ξ = µ/ p, where µ is the unit normal vector and p the mean curvature, as before. This is precisely the IMCF condition. Notice, however, that IMCF in codimension one fixes uniquely the velocity of the flow while in codimension two, it still leaves room for imposing one extra scalar condition on the velocity. We shall use the name IMCF in both cases. The requirement which could be used alternatively to remove the second integral in (30) reads ψ − = α(λ) for some constant α(λ). By Gauss’ law and by a trivial Fredholm argument, this equation has a unique solution for ψ iff α(λ) = λ η Sλ . Curiously, this integral is related to the Hawking mass itself, cf. (17). As for the IMCF condition, this one has a counterpart in codimension one as well, obtained by substituting the mean curvature p 2 for H 2 in , cf. (15). This condition could be used in place of the IMCF to

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guarantee monotonicity of the Geroch mass. A closely related flow has been employed before in the proof of global stability of Minkowski space, cf. Remark 1 after Theorem 1. As to conditions which remove the first term in (30), one such reads (ξ · H ) = const, which we call the dual inverse mean curvature condition. As remarked before, such a condition is unnecessary in the time-symmetric case, where the Geroch-Hawking mass depends only on the intrinsic geometry of the hypersurface generated by the flow. The alternative to the dual inverse mean curvature condition is to require that D.U vanishes on S. This option is also contained in (3b) of the theorem in [16]. In the case where the ξ is allowed to be null, the splitting (24) is not possible and we must return to (18). The last term in this expression can be made zero with sufficient generality only by imposing the IMCF condition (ξ · H ) = a(λ). It only remains to consider the integral I ≡ tr Sλ H · ∇ ⊥ ∇ ⊥ ξ η Sλ . (31) Sλ

Lemma 7. Under the same hypotheses of Lemma 3, (a) If H | S(λ) = B ξ | S(λ) for some function B on the surface, then I = 0 provided ξ is null all over Sλ or if B is constant on Sλ , (b) Assume (ξ · H ) = a(λ) and (ξ · H ) = c(λ) for some constant c(λ) on Sλ . Then I ≥ 0 provided (i) ξ ∈ C + , H ∈ C − , (ii) ξ ∈ C − , H ∈ C + , (iii) ξ , H ∈ A+ , or (iv) ξ , H ∈ A− holds everywhere on Sλ . Proof. Regarding part (a), if ξ is null on an open set U ⊂ Sλ , we in fact have H = ±B ξ on U and hence ∇e⊥A ξ = b A ξ on U , for some b A . Thus, ∇e⊥B ∇e⊥A ξ is parallel to ξ and hence orthogonal to H . Consequently ( H · ∇ ⊥ ∇ ⊥ ξ )|U = 0 holds irrespective of the values of B. If ξ is non-null everywhere then from the fact that (ξ · H ) =0 and (ξ · H ) = Be2ψ , (25) implies that the integrand in I can be written as −B D A ξ 2 U A which integrates to zero provided B is constant. If ξ is non-null almost everywhere the same conclusion holds by continuity. Moreover, if ξ is null on an open set, the integral on that set is zero for any B, as shown before. Thus if B is constant on Sλ the integral vanishes irrespective of the causal character of ξ . Turning to (b), we first notice that if ξ is non-null everywhere, then (25) together with the fact that (ξ · H ) and (ξ · H ) are constants on Sλ implies I = Sλ L (ξ )( H , ξ )η Sλ ≥0, where Lemma 6 has been used for the last inequality. To include the case when ξ may be null we use a continuity argument. The integral I defines a map from C 0 (Sλ )×C 2 (Sλ ) → R, where the first factor refers to H and the second factor to ξ . This map is obviously continuous with respect to the supremum norm on these spaces. Thus, it is sufficient to observe that the inequality holds for H , ξ ∈ int[C0+ (Sλ ) × C2− (Sλ )] ∪ int[C0− (Sλ ) × − + C2+ (Sλ )] ∪ int[A+0 (Sλ ) × A+2 (Sλ )] ∪ int[A− 0 (Sλ ) × A2 (Sλ )], where C 0 (Sλ ) denotes the + 0 + set of vectors h ∈ C (Sλ ) such that, for all p ∈ Sλ , h( p) ∈ C p . C2 (Sλ ) is defined analogously by replacing C 0 (Sλ ) by C 2 (Sλ ) and the other sets are defined similarly. Notice that this open set is dense in [C + (Sλ ) × C − (Sλ )] ∪ [C − (Sλ ) × C + (Sλ )] ∪ [A+ (Sλ ) × A+ (Sλ )] ∪ [A− (Sλ ) × A− (Sλ )], where we want to prove the lemma.

Remark. Condition (a) holds for instance when ( H · ξ ) = 0 everywhere on Sλ and ξ does not vanish on open sets of Sλ .


131

Among the conditions (i) to (iv) that ensure monotonicity in part (b) of Lemma 7, only (iii) and (iv), i.e. velocity vector and mean curvature vector both achronal and to the same quadrant, give conditions which are compatible with those ensuring non-negativity of (26) and T ( H , ξ ). Regarding the case a(λ) = 0 on Sλ , the orthogonality of H and ξ implies that they cannot belong to the same quadrant, thus violating the non-negativity condition in Lemma 5, unless ξ is null and H = B ξ , with a non-negative arbitrary function on Sλ . Thus, when the surface is marginally trapped and the flow is null, monotonicity is very easily achieved initially, the only condition being that the velocity is pointwise parallel to the mean curvature of the surface. Of course, the property of H being null will not be maintained by the flow and the flow vector will have to adjust, if at all possible, to keep a monotone flow. Summarizing all the results above yields the following theorem. Theorem 1. Let S be an oriented, spacelike, closed, C 2 embedded 2-surface in a spacetime (V, g). Let ξ be a C 2 normal vector field on S, H the mean curvature vector of the surface and M H (S) its Hawking mass. Let Sλ be any flow of spacelike two-surfaces starting at S with ξ as normal component of the initial velocity. Then, d M H (Sλ )/dλ|λ=0 ≥ 0 holds whenever the following four conditions hold: (1) The spacetime satisfies the dominant energy condition (27). (2) The mean curvature vector H is spacelike or null on S, and ξ points into the same causal quadrant as H . (3) Either (a) the IMCF condition holds, i.e. (ξ · H ) = a0 , where a0 is a non-negative constant, or (b) ξ 2 = 0 everywhere on S and ψ =

1 1 1 R(h λ ) − H 2 + − α, 2 4 3

(32)

where α is constant. (4) Either (a) the dual IMCF condition holds, i.e. (ξ · H ) = c0 , where c0 is a constant, or (b) ξ 2 = 0 everywhere on S and D.U = 0. Remark 1. As mentioned earlier, there are more general conditions to remove (30) than those given in Theorem 1. We restrict ourselves to flows which do not get null and for which H does not get null anywhere on S. They can be written in the form ξ = H −2 ( f H − g H ∗ )

(33)

for some functions f and g. Let h be another function and let b(λ), c(λ) be constants satisfying g = c(λ) + h( f − a(λ)), f 2 − g 2 = H 2 e2ψ, ψ − = b(λ) + h(D.U )

(34)

on S. Here the second condition serves just to recall the definition of ψ, viz. ξ 2 = exp(2ψ). It is now easy to see that the expression (30) vanishes for flows (33) satisfying (34). Thus, conditions (3) and (4) of the theorem can be replaced by (34). Note also that conditions (3b) and (4a) correspond to the particular case h ≡ 0 of (34), in which H must be non-null.

132


While conditions (34) leave a large freedom for the flow, their coupling nevertheless makes them rather difficult to analyze in general. One of the reasons for displaying (34) lies in its relation to the proof of the non-linear stability of Minkowski space by Christodoulou and Klainerman [18] described in more detail in [19, 20]. This work contains a construction of a double null foliation of the spacetime based on a spacelike foliation such that the logarithm of its lapse function satisfies a Poisson equation similar to (32), cf the “canonical foliation” in Definition 3.3.2 of [19]. Due to the elliptic character of this flow, it could be shown to remain smooth for data near to the Minkowski ones. As the Hawking mass is not monotonic in general under this flow, it is not contained in the class (34). Nevertheless, for weak data, the same construction as in Christodoulou and Klainerman should be applicable to (34). This might be useful for proving stability in more general cases, in particular for Schwarzschild. Remark 2. We now assume that ξ 2 = 0. Under any of the conditions that make (30) vanish, e.g. hypotheses (3)–(4) of the theorem or (34) in the previous remark, Eq. (24) can be reformulated so that the energy interpretation becomes transparent. We can write the change of M H as follows: 1 d M H (Sλ ) T L = (G µν + gµν ) + µν + (ξ )µν χ µ ξ ν . η Sλ (35) dλ 8π Sλ √ Here we have introduced χ = H |Sλ |/16π ; the normalization is chosen such that χ coincides with the asymptotically unit timelike Killing vector when S is a 2-sphere on a time-symmetric slice in Schwarzschild. In more general situations with timelike Killing vectors, we are not aware if they necessarily coincide with χ. We note that on any marginally trapped surface, H and therefore χ is a null vector. A spin-coefficient version of the energy conservation law (35) is given in [23]. While T is a tensorial object (14) depending only on the geometry, L depends on the velocity vector ξ as l) with well. In any case, the components of these objects with respect to a null basis (k, = −φ read (k · l) T T T AB )2 , AB )2 , 8π kk = (k · kl = lk = 0, 8π llT = (l · L L L 2 L (ξ )ll = (ξ )kk = 0, 8π (ξ )lk = φ(Dψ − U ) , 8π (ξ )kl = φ(Dψ +U )2. (36)

These tensors may be interpreted respectively as encoding transverse and longitudinal modes of the gravitational field, with ≡ T + L taking the same form as in a corresponding energy conservation law along black-hole horizons [24]. In particular, the T can be interpreted as energy densities of ingoing and outgoing components llT and kk gravitational radiation, taking the same form as standard definitions at null infinity in an asymptotically flat spacetime. In that case, the conservation law corresponding to (35) is the Bondi mass equation [25]. We proceed by discussing the requirements of Theorem 1 in some detail, restricting ourselves to the IMCF conditions, i.e. (3.a) and (4.a). First a remark on terminology: recalling that £ξ η Sλ = (ξ · H )η Sλ , we see that the first of these conditions implies that the area element is preserved under the flow ξ , while the second one implies the same along the dual ξ . This motivates the terminology uniformly expanding flow (UEF) [6] for both (ξ · H ) = a(λ) and (ξ · H ) = c(λ), while we reserve


133

the term inverse mean curvature flow (IMCF) for the first one alone. The UEFs have some interesting special cases which we discuss in turn. We first assume that the flow velocity is null everywhere, while the mean curvature vector of the surface is achronal everywhere. To obtain monotonicity, we impose the IMCF condition with a(λ) = 1. This implies the dual IMCF condition since ξ = ±ξ , and therefore ξ · H = − ξ · H = ∓ ξ · H = ∓1. All the conditions of the theorem hold provided ξ and H point into the same quadrant. Notice that the IMCF condition actually prevents ξ or H from vanishing anywhere on the surface. This null monotone flow was discovered by Hayward [6]. Since ξ is null and hypersurface orthogonal, the flow is geodesic, i.e. ∇ξ ξ = Y ξ for some function Y . In terms of an affinely parametrized geodesic with tangent l, we have ξ = θ −1l and the l

= 1. We emphasize that flow can be determined by solving the ODE ξ(λ) = θ−1l(λ) l this is the only case for which we obtain local existence. We next assume that H is spacelike everywhere, i.e. H 2 > 0, which allows us to write the velocity vector in the form ξ = H −2 H − c(λ) H , (37) choosing a(λ) = 1. A special case is that c(λ) vanishes identically, so that the velocity vector of the flow is ξ = H −2 H . This flow was mentioned by Huisken and Ilmanen in [9] and analyzed in more detail by Frauendiener [17], who showed monotonicity in an explicit way. In the general case covered by (37), we can set a(λ) = 1 so that the velocity flow contains one arbitrary function of one variable c(λ) satisfying |c(λ)| ≤ 1. In the particular case |c(λ)| < 1, this is the spacetime version of the flows found by Malec, Mars and Simon [16] in terms of initial data sets. Indeed, in this case the vector ξ in (37) is spacelike by construction, so that the hypersurface generated by the flow of 2-surfaces will be spacelike, if it exists. Thus, when the flow is viewed as a flow of 2-surfaces in this 3-dimensional Riemannian manifold, the IMCF condition (ξ · H ) = 1 translates into the condition that S flows within as an inverse mean curvature flow. The other condition (ξ · H ) = c(λ) becomes a condition involving p, the mean curvature of S ⊂ , and q, the trace on S of the extrinsic curvature of as a hypersurface in spacetime, which explicitly reads q/ p = c(λ). This is precisely the condition obtained in [16] (see also [26]) in order to ensure monotonicity of the Hawking mass in an initial data setting. These flows significantly generalize flows along the inverse mean curvature vector, since the one-parameter freedom after fixing a(λ) = 1 allows flows from a given initial surface to cover a spacetime region. In particular, while H /|H |2 can only approach spacelike infinity in an asymptotically flat spacetime, ξ of the form (37) can also approach null infinity. This suggests the possibility to prove the Penrose inequality (1) not only for the ADM mass M but also for the Bondi mass M B ≤ M. The latter inequality is significantly stronger than the former, but it also follows from the heuristic argument involving cosmic censorship [1]. The final special case of the UEF guaranteeing monotonicity of the Hawking mass are surfaces with null H and velocity vector which is either spacelike or null, towards the same quadrant as H . Since H = ±H , the dual IMCF condition implies the IMCF

134


condition, as in the case where ξ is null. The surface S is restricted to being a future (past) marginally trapped surface, which is precisely the starting surface for flows aiming at proving the Penrose inequality. Given H , the velocity vector ξ at S has a freedom of one arbitrary function. Thus, we are allowed to flow out of a marginally trapped surface along a large class of flows for which M H is monotone. 4. On Existence of Uniformly Expanding Flows In this section we discuss the prospects of obtaining existence of the UEF defined above. Recall that the method of Huisken and Ilmanen in codimension 1 rests mainly on two properties of the IMCF: firstly, the embedding functions determining the flow satisfy a parabolic system. Secondly, the level sets of the flow satisfy a degenerate elliptic system. As to codimension 2, Huisken and Ilmanen noticed that the embedding functions describing a flow along the inverse mean curvature vector, i.e. (37) with c ≡ 0, satisfy a backward-forward parabolic system. There is no general theory which would guarantee local existence of solutions to such systems, and counter-examples indicate that such systems have to be treated with care. Unfortunately, as we will see below, the system keeps this vicious behaviour for any choice of c. Nevertheless, the more important component on the work of Huisken and Ilmanen is the weak (variational) formulation of the level set formulation for the flow. We have obtained a generalization of this formulation which we will describe in turn. We first give the description in terms of embedding functions α (λ, x A ) for a 2-surface S embedded in spacetime, following the notation introduced in Sect. 2. In local coordinates we have ξ µ = ∂ µ /∂λ and the tangent vectors to S can be written as µ e A = ∂ µ /∂ x A . For the mean curvature vector H µ we obtain ∂ ν ∂ ρ µ , H µ = h AB K AB = − µ − h AB µνρ x= α ∂ξ A ∂ξ B

(38)

µ

where νρ are the Christoffel symbols of the ambient space, and is the Laplacian on S. In this way, (37) becomes a system of PDEs of the form ∂ α ∂ β ∂ 2 β . = J α λ, x A , β , , ∂λ ∂ x A ∂ x A∂ x B

(39)

To determine the type of this system we need to calculate the eigenvalues of the symmetrized part Q + Q † of the matrix Q αβ ≡

∂

∂ Jα ∂ 2 β ∂x A∂x B

z AzB,

(40)

where z A is a unit vector in T p , and † denotes the transpose with respect to some positive definite metric. Restricting ourselves now to codimension two, we obtain for the linearization of the operators H and H acting on ,

∂

∂ Hα ∂ 2 β ∂x A∂x B

z A z B = −P⊥αβ

∂

∂ H α 2 β z A z B = −η α S β, ∂

∂x A∂x B

(41)


135

where we have employed the dualized 2-form ηS : η α Sβ = It follows that Q αβ =

1 H2

1 α γ η βγ δ e A eδB η SAB . 2

2H α Hβ c(λ) α α α − H . − P H + H H β β β H2 H2

(42)

To do the symmetrization, we can choose the metric L αβ = H −2 (Hα Hβ + Hα Hβ ) so that the symmetrized part of Q αβ is independent of c(λ) and reads

Q + Q†

α β

=

2 α α H H + H H β β . H4

(43)

2 2 This matrix has H α and H α as eigenvectors, α with eigenvalues 2/H and 2/H = −2/H 2 , respectively. Therefore Q + Q † β is indefinite and (37) is a so-called forward-backward parabolic system. The previous reasoning applies in codimension one if all H are removed, and yields parabolicity, as is well known. It can be easily checked that the reasoning above is independent of the metric L µν used for symmetrization. Turning now to the level set formulation, we start with recalling the case of codimension 1. One can envision the family of surfaces S(λ) as level sets of a level set function u(x) defined on a spacelike hypersurface of the spacetime. If u is twice differentiable, the inverse mean curvature flow condition can be written in terms of u as a degenerate elliptic equation, namely Du D. = |Du|, (44) |Du|

where D is the covariant derivative on , and D.W denotes the divergence of some vector W . The variational formulation reduces the requirements on differentiability. While (44) are not Euler-Lagrange equations of a functional of u as the only variable, they can nevertheless be obtained by minimizing the functional E uK (v) = (45) (|Dv| + v|Du|) ∩K

with respect to variations of v on any compact set K . One can imagine this variational principle as a two-step procedure in which the term |Du| on the right-hand side of (44) is first fixed or “frozen” while E u is varied with respect to v, and in the second step |Du| is put equal to the |Dv| with the function v obtained from the minimization, with derivatives understood in a weak sense. As to codimension 2, it would be desirable to have some generalization of (44) with an extension of u to the extra dimension. While from the equation itself it is not clear how this could be accomplished, we turn directly to the variational formulation, which does, in fact, involve a natural extension of u. We start from some initial two-surface S embedded in some three-surface . The latter is arbitrary at this stage, but will be fixed automatically by the variation principle. We consider the level sets S(λ) ⊂ given by the level set function u(λ) of the IMCF near S, obtained via the Huisken-Ilmanen procedure. We denote by ν(λ) and µ(λ) = ν (λ)

136


the unit normals to S(λ) in the future timelike direction normal to and in the spacelike direction tangent to , respectively, assuming their existence. The variational principle should now move the flow in the direction of ξ given by (37), i.e. should have ξ as tangent vector whenever the latter is defined. Equivalently, the mean curvature vector to Sλ should be orthogonal to ν + c(u)µ, and this latter property is used in the weak formulation. We extend the function u off in some neighbourhood of in such a way that the directional derivative of u in the direction ν + c(u)µ, whose existence we may assume, is zero everywhere on Sλ . We denote by ϒ any other spacelike hypersurface in this neighbourhood which is equal to outside some compact region , and we take v to be a real-valued function on ϒ which agrees with u outside of as well. We then define the functional E ,u,c (ϒ, v) = (46) (|Dv| + v|Du|). ϒ∩

In analogy with (45), the “frozen” structure now consists of the hypersurface together with the level sets u, extended as above to a neighbourhood of . The objects which are varied are ϒ and v, and in the second step, and u are adjusted according to the results of the variation. This justifies the following. Definition 1. Given c(λ) ∈ [−1, 1], then (, u) is a weak solution to the corresponding uniformly expanding flow in an open region if it is a critical point of the functional E ,u,c (·, ·) for all compact ⊂ . We will also say that the 2-surfaces u are weak solutions to Eq. (37). Since u(x) may have constant regions, this allows the family of surfaces defined by the level sets of u(x) to jump, just as in the Huisken-Ilmanen case. Given this weak definition, we may check that the above weak solutions agree with the usual smooth solutions when they exist. We first use the fact that (, u) is a critical K point of E ,u,c (·, ·) with respect to variations of . Consider a variation ϒ of which is a compactly supported bump function on times ν + c(u)µ. Note that u does not change to first order in these directions. Hence, to first order, it is still true that E= (|Du| + u|Du|) , ϒ∩

where we may extend u to be constant in this variation since we are at a critical point. But by the co-area formula, if we choose to be the region where a ≤ u(x) ≤ b, b E= (1 + λ)|S(λ)|dλ, a

where |S(λ)| denotes that area of S(λ). Since we are at a critical point, the first variation of E must be zero, which, in the case that S(λ) is a smooth family of surfaces, follows if and only if our variation direction is orthogonal to the mean curvature vector of S(λ). Equivalently, it follows that ξc from Eq. (37) must be tangent to . What remains to be done is to check that the actual flow vector, which is also tangent to and orthogonal to each S(λ), has the same length as ξc . Again we use the fact that (, u) is a critical point of E ,u,c (·, ·) with respect to variations of u(x). This implies that u(x) is a weak solution to inverse mean curvature flow in in the sense defined by Huisken and Ilmanen. Thus, the level sets S(λ) of u(x) have the uniformly expanding


137

property that the rate of change of the area form of the surfaces equals the area form itself. This proves that the actual flow vector of the surfaces is not only in the same direction as ξc but also has the same length, and therefore they are equal. Thus, the smooth flow case agrees with this weak definition. To conclude this exposition we wish to emphasize that having a definition of a weak solution to Eq. (37) is still very far from an existence theorem. However, the correct notion of a weak solution is a prerequisite to an existence theorem along the lines of Huisken and Ilmanen. Acknowledgement. M.M. and W.S. are grateful to Tom Ilmanen for helpful discussions. M.M. also wishes to thank José Senovilla for interesting comments, and W.S. thanks J. Sauter for pointing out the relation of the present work to the one by Christodoulou and Klainerman. S.A.H. is supported by Shanghai Municipal Education Commission under Grant 06DZ111, by the National Natural Science Foundation of China, under Grants 10375081 and 10473007, and by Shanghai Normal University under Grant PL609. H.B. thanks the National Science Foundation for partial support during this work from grant number DMS-0533551. M.M. and W.S. were supported by the projects BFME2003-02121 of the Spanish Ministerio de Educación y Tecnología and SA010CO of the Junta de Castilla y León. W.S. was supported in part by FWF (Austria), grant P14621-N05.

References 1. Penrose, R.: Naked singularities. Ann. N. Y. Acad. Sci. 224, 125 (1973) 2. Gibbons, G.W.: Some comments on gravitational entropy and the inverse mean curvature flow. Class. Quantum Grav. 16, 1677 (1999); Chru´sciel, P.T., Simon, W. Towards the classification of static vacuum spacetimes with negative cosmological constant. J. Math. Phys. 42, 1779 (2001) 3. Horowitz, G.: The positive energy theorem and its extensions. In: Asymptotic behavior of mass and spacetime geometry, edited by F. Flaherty, Springer Lecture Notes in Physics 202, New York: Springer, 1984 4. Karkowski, J., Malec, E.: The general Penrose inquality: lessons from numerical evidence. Acta Physica Polonica B36, 59 (2005) 5. Ben-Dov, I.: The Penrose inequality and apparent horizons. Phys. Rev. D 70, 124031 (2004) 6. Hayward, S.A.: Quasi-localization of Bondi-Sachs energy-loss. Class. Quantum Grav. 11, 3037 (1994) 7. Malec, E., ÓMurchadha, N.: Trapped surfaces and the Penrose inequality in spherically symmetric geometries. Phys. Rev. D 49, 6931 (1994) 8. Hayward, S.A.: Gravitational energy in spherical symmetry. Phys. Rev. D 53, 1938 (1996) 9. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Diff. Geom. 59, 353 (2001) 10. Bray, H.: Proof of the Riemannian Penrose inequality using the Positive Mass Theorem. J. Diff. Geom. 59, 177 (2001) 11. Geroch, R.: Energy Extraction. Ann. N. Y. Acad. Sci. 224, 108 (1973) 12. Jang, P.S., Wald, R.: The positive energy conjecture and the cosmic censorship. J. Math. Phys. 18, 41 (1977) 13. Hawking, S.W.: Gravitational radiation in an expanding universe. J. Math. Phys. 9, 598 (1968) 14. Hayward, S.A.: Inequalities relating area, energy, surface gravity and charge of black holes. Phys. Rev. Lett. 81, 4557 (1998) 15. Eardley D.M.: Global problems in numerical relativity. In: Sources of Gravitational Radiation, Proceedings of the Battelle Seattle Workshop, Smarr, L.L., ed., Cambridge: Cambridge University Press, 1979, 127 16. Malec, E., Mars, M., Simon, W.: On the Penrose inequality for general horizons. Phys. Rev. Lett. 88, 121102 (2002) 17. Frauendiener, J.: On the Penrose inequality. Phys. Rev. Lett. 87, 101101–1 (2001) 18. Christodoulou, D., Klainerman, S.: The global nonlinear stability of Minkowski space. Princeton: Princeton University Press, NJ: 1993 19. Klainerman, S., Nicoló, F.: The evolution problem in General Relativity. Boston: Birkhäuser, 2003 20. Christodoulou, D.: Mathematical problems of General Relativity Theory. Zürich lectures in advanced mathematics, EMS publishing house (to appear) 21. Jost, J.: Riemannian Geometry and geometric analysis. 3rd ed., Berlin: Springer, 2001 22. Senovilla, J.M.M.: Super-energy tensors. Class. Quantum Grav. 17, 2799 (2000)

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23. Hayward, S.A.: Gravitational radiation from dynamical black holes. Class. Quantum Grav. 23, L15 (2006) 24. Hayward, S.A.: Energy conservation for dynamical black holes. Phys. Rev. Lett. 93, 251101 (2004) 25. Hayward, S.A.: Spatial and null infinity via advanced and retarded conformal factors. Phys. Rev. D 68, 104015 (2003) 26. Malec, E., Mars, M., Simon, W.: On the Penrose inequality. In: Lobo A. et. al. (eds). Proceedings of the Spanish Relativity Meeting. Universitat de Barcelona, (2002) p. 253 Communicated by G.W. Gibbons


Communications in


Integrable discrete time chains for the Frobenius-Stickelberger-Thiele polynomials V. P. Spiridonov1 , S. Tsujimoto2 , A. S. Zhedanov3 1 Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia.


2 Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University,

Kyoto 606-8501, Japan

3 Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received: 10 February 2006 / Accepted: 10 October 2006 Published online: 7 March 2007 – © Springer-Verlag 2007

Abstract: The notion of Frobenius-Stickelberger-Thiele (FST) polynomials is introduced. Spectral transformations for these polynomials analogous to the Christoffel and Geronimus transformations for orthogonal polynomials are constructed. They yield an integrable discrete time chain (the FST chain) related to the generalized ε-algorithm. Relations of the FST polynomials to the Padé interpolation problem and to general and symmetric biorthogonal rational functions are considered in detail. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. FST Polynomials and Their Properties . . . . . . . . . . . . . . . . . . 3. Spectral Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Christoffel type transformations . . . . . . . . . . . . . . . . . 3.2 The Geronimus type transformations . . . . . . . . . . . . . . . . . 4. The FST Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Connections to the Discrete Time Lotka-Volterra Chain and ε-Algorithm 6. Relations Between the R I I and FST Chains . . . . . . . . . . . . . . . . 7. An Affine Weyl Group Symmetry . . . . . . . . . . . . . . . . . . . . . 8. The Bilinear Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

139 140 145 145 147 150 151 154 157 158 164

1. Introduction Completely integrable systems defined by nonlinear partial differential equations or their finite-difference analogues have found many applications in physics [9]. Integrable This work is supported in part by the Russian Foundation for Basic Research (RFBR) grant no. 06-0100191 and the Grant-in-Aid for Scientific Research no. 15540119 from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

140

V. P. Spiridonov, S. Tsujimoto, A. S. Zhedanov

chains of the Lotka-Volterra and Toda type appeared in geometric problems a long time ago [7]. Multiparticle systems defined by them provide interesting examples of exactly solvable quantum mechanical systems. Numerical convergence acceleration algorithms, like qd, g, ε, etc. algorithms [2, 3, 5, 6, 31], are known to define some discrete integrable systems. Hirota’s equation [14] (known also as the discrete KP equation [21]) provides probably the best known example of such (2+1)-dimensional nonlinear partial difference integrable equations. Three term recurrence relations of various forms play a central role in the description of the simplest discrete integrable systems via the discrete Lax pair formalism. In particular, the three term recurrence relation for orthogonal polynomials and the associated Christoffel-Geronimus spectral transformations generate the discrete time Toda chain [12, 13, 18, 25] or the shifted qd-algorithm [8]. Similar considerations for symmetric orthogonal polynomials yield the discrete time Lotka-Volterra chain which is related to Bauer’s g-algorithm [2, 25, 26]. Other interesting examples of such relations can be found, e.g., in [22] and references cited therein. In the present paper, we deal with (1 + 1)-dimensional discrete time chains associated with the three term recurrence relation which appeared first in the Frobenius and Stickelberger paper devoted to elliptic functions [10] and later in the work of Thiele on the rational interpolation problems [30]. The corresponding continued fractions are referred to in the modern literature as the Thiele continued fractions [1, 4]. However, the same objects appeared earlier in the pioneering study by Frobenius and Stickelberger [10] and we have chosen to associate related objects with the names of all these three mathematicians. Within this framework, we introduce in the next section the notion of FrobeniusStickelberger-Thiele (FST) polynomials and describe their key properties. In Sects. 3 and 4, we introduce the Christoffel and Geronimus type transformations for the FST polynomials and derive the discrete time chain (called the FST chain) generated by them. In the following section, we describe relations of this chain to the discrete time chain introduced in [24] and to known ε-type algorithms [22]. In Sect. 6, we derive a relation between the FST chain and the R I I chain introduced in [27]. In Sect. 7, we describe an affine Weyl group symmetry for the FST chain. In the final section we introduce the corresponding τ function and describe the bilinear formalism for this chain starting from integrable discrete versions of two-dimensional Toda and KP equations [15, 21].

2. FST Polynomials and Their Properties Let an and bn , n ∈ N ≡ {0, 1, . . .}, be two sequences of complex numbers. We assume that all bn = 0 and n k=0

b2k = 0,

n

b2k+1 = 0, n ∈ N.

(2.1)

k=0

It is assumed also that the numbers an are distinct, an = am for n = m. We define the ordinary Frobenius-Stickelberger-Thiele (FST) polynomials Tn (x) as solutions of the following three term recurrence relation [10]: Tn+1 (x) = bn+1 Tn (x) + (x − an )Tn−1 (x)

(2.2)

Discrete Time Chains

141

with the initial conditions T0 (x) = 1, The associated FST polynomials

Tn(1) (x)

T1 (x) = b1 .

(2.3)

are defined by the same recurrence relation

(1)

(1)

Tn+1 (x) = bn+1 Tn(1) (x) + (x − an )Tn−1 (x)

(2.4)

with different initial conditions T0(1) (x) = b0 ,

T1(1) (x) = x − a0 + b0 b1 ,

or, equivalently, (1) T−1 (x) = 1,

T0(1) (x) = b0 .

(2.5)

The following elementary properties of the FST polynomials can be easily established from the general theory of continued fractions and three term recurrence relations (see, in particular, [10]): (1)

(1)

(i) deg(T2n (x)) = deg(T2n+1 (x)) = n, deg(T2n (x)) = deg(T2n−1 (x)) = n. (1)

(ii) Polynomials T2n = x n + O(x n−1 ) and T2n−1 = x n + O(x n−1 ) are monic and T2n+1 (x) = µn x n + O(x n−1 ),

(1)

T2n = νn x n + O(x n−1 ),

where the leading coefficients have the form µn = b1 + b3 + · · · + b2n+1 , νn = b0 + b2 + · · · + b2n . Due to condition (2.1), we have µn = 0, νn = 0 for all n ∈ N. (iii) The Casoratian identity (1)

Wn (x) ≡ Tn (x)Tn−1 (x) − Tn(1) (x)Tn−1 (x) = (−1)n (x − a0 )(x − a1 ) · · · (x − an−1 ), W0 = 1.

(2.6)

(1)

(iv) The ratio Tn (x)/Tn (x) is the n th convergent of the FST continued fraction (1)

Tn (x) = b0 + Tn (x)

x − a0 . x − a1 b1 + b2 + . . . x − an−1 + bn

(2.7)

(1)

We call FST polynomials regular if no root of the polynomials Tn (x) and Tn (x) (1) coincides with the points as , i.e. Tn (as ) = 0 and Tn (as ) = 0 for n, s ∈ N. (1)

Lemma 1. For regular FST polynomials Tn (x), Tn , the polynomials in any pair (i) Tn (x) and Tn+1 (x), (1) (x), (ii) Tn(1) (x) and Tn+1 (1) (iii) Tn (x) and Tn (x) do not have coinciding roots for all n ∈ N.

142


The proof of this lemma follows easily from the Casoratian identity (2.6). As a simple consequence, we have Lemma 2. Let Pn (x) denote regular FST polynomials (ordinary or associated ones) and suppose that the following equality holds: Pn+1 (x)A(x; n) + Pn (x)B(x; n) = 0,

(2.8)

for all n ∈ N, with A(x; n) and B(x; n) being polynomials in x of a fixed degree (i.e. only coefficients of these polynomials may depend on n, but not their degrees). Then, A(x; n) = 0 and B(x; n) = 0 identically for all x and n. Proof. We assume that A(x; n) and B(x; n) do not vanish identically. Therefore, Pn+1 (x)/Pn (x) = −B(x; n)/A(x; n). In the left-hand side we have a rational function with n distinct poles (due to Lemma 1). But in the right-hand side we have a rational function with a fixed number of poles. For n large enough we come to a contradiction which means that, necessarily, A(x; n) = B(x; n) = 0. In what follows we will assume also that Tn (0) = 0.

(2.9)

Continued fractions of the form (2.7) are related to the Thiele interpolation algorithm [1, 30]. Let F(z) be a function of a complex variable z. Given the interpolation grid a0 , a1 , . . . , we construct a family of functions Fn (z) recursively by the algorithm z − a0 , F(z) − F(a0 ) z − an , n = 1, 2, . . . . Fn+1 (z) = Fn (z) − Fn (an )

F1 (z) =

(2.10)

From these relations we see that the function F(z) is formally represented as the infinite FST continued fraction z − a0 F(z) = b0 + , (2.11) z − a1 b1 + b2 + . . . where bn = Fn (an ), and that ratios of the FST polynomials constitute its n th convergents (2.7). From this algorithm it is clear that F(as ) =

(1)

Tn (as ) , s = 0, 1, . . . , n, Tn (as )

(2.12)

i.e. ratios of the FST polynomials provide rational interpolations of the function F(z). Now we denote Pn (x; n) = T2n (x), Q n (x; n) =

(1) T2n (x),

Pn (x; n + 1) = T2n+1 (x), (1)

Q n+1 (x; n) = T2n+1 (x).

Clearly, we have deg(Pn (x; n)) = deg(Pn (x; n + 1)) = deg(Q n (x; n)) = n and the degree of Q n+1 (x; n) equals n + 1. From equality (2.12) we obtain the relations F(as ) =

Q n (as ; n) , s = 0, 1, . . . , 2n Pn (as ; n)


143

and F(as ) =

Q n+1 (as ; n) , s = 0, 1, . . . , 2n + 1. Pn (as ; n + 1)

These formulas can be put into the frames of the general Padé interpolation problem [1]: find polynomials Q m (z; n) of the degree m and polynomials Pn (z; m) of the degree n such that the rational interpolation F(as ) =

Q m (as ; n) , s = 0, 1, . . . , n + m Pn (as ; m)

(2.13)

takes place. Solution for this problem exists only under some nondegeneracy restrictions, namely, the roots of polynomials Q m (z; n) and Pn (z; m) should not coincide with the interpolation points as [1]. Sometimes such an interpolation for generic m and n is called the Cauchy-Jacobi interpolation [19]. From recurrence relation (2.2) we find Tn+2 (x) = ((1 + wn )x + vn )Tn (x) − wn (x − an−1 )(x − an )Tn−2 (x),

(2.14)

where vn = bn+1 bn+2 − an+1 − wn an , wn = bn+2 /bn (note that wn = 0, ∞ due to the conditions bn = 0). This equality allows us to obtain the following three-term recurrence relations for polynomials Pn (z; n) and Pn (x; n + 1): Pn+1 (x; n + 1) = ((1 + w2n )x + v2n )Pn (x; n) −w2n (x − a2n−1 )(x − a2n )Pn−1 (x; n − 1),

(2.15)

Pn+1 (x; n + 2) = ((1 + w2n+1 )x + v2n+1 )Pn (x; n + 1) −w2n+1 (x − a2n )(x − a2n+1 )Pn−1 (x; n).

(2.16)

and

The continued fractions associated with the recurrence relations of type (2.15) or (2.16) were named as R I I -fractions in [17]. They are also well known in the theory of the Cauchy-Jacobi interpolation problem, see, e.g. [31], where they enter under the name of osculatory continued fractions. The osculatory three term recurrence relation Pn+1 (x) = (Cn x + E n )Pn (x) + Dn (x − αn )(x − βn )Pn−1 (x)

(2.17)

with the initial conditions P−1 = 0, P0 = 1 generates some polynomials Pn (x) of degree ≤ n. We call these polynomials regular, if Dn = 0, n > 0, and Pn (x) do not vanish for x = αk or x = βk (for all k). As shown in [17], regular polynomials Pn (x) possess the orthogonality property: Pn (x)x j σ n = 0, j = 0, 1, . . . , n − 1, (2.18) k=1 (x − αk )(x − βk ) where σ is a linear functional defined on the space of rational functions with the prescribed poles at αk and βk . Similar to the FST polynomials, it is easy to see that neighbouring regular polynomials Pn (x) and Pn+1 (x) cannot have coinciding roots. Moreover, we have an important statement similar to Lemma 2.

144


Lemma 3. Suppose we have a set of regular polynomials Pn (x) generated by (2.17). Then the equality Pn+1 (x)A(x; n) + Pn (x)B(x; n) = 0, n = 0, 1, . . . ,

(2.19)

where A(x; n) and B(x; n) are polynomials of x of a fixed degree, can be satisfied only if A(x; n) = 0 and B(x; n) = 0 for all x, n. It is possible to show [28] that in the general case, for a fixed integer j and a given sequence of numbers Ys , s ∈ N, one can construct some denominator Pn (x; n + j) and numerator Q n+ j (x; n) polynomials such that the Padé interpolation condition Ys =

Q n+ j (as ; n) , Pn (as ; n + j)

(2.20)

is satisfied, where the interpolation points as are distributed among the numbers αs and βs . For example, if j = 0, we have two options: either a2n = αn , a2n−1 = βn or a2n−1 = αn , a2n = βn , as it is easy to see after comparing recurrence relations (2.15) and (2.17). The interpolated sequence Ys is related to the orthogonality functional σ in (2.18). Note that the requirement (2.20) is slightly more powerful than interpolation (2.13) because existence of a (meromorphic on a Riemannian surface) function F(z) is not assumed in general. In [34], it was shown that relations (2.18) can be extended to the biorthogonality condition σ Un (x)V j (x) = 0, j = 0, 1, . . . , n − 1, (2.21) for two rational functions Pn (x) , k=1 (x − αk )

Hn (x) k=1 (x − βk )

Un (x) = n

Vn (x) = n

with some additional set of polynomials Hn (x). For general properties of these biorthogonal rational functions, see [27, 28, 34]. Proposition 1. If the FST polynomials Tn (x), Tn(1) (x) are regular then there exist a unique sequence of nonzero complex numbers Ys , s ∈ N, such that for any n > 0, Ys =

(1)

Tn (as ) , s = 0, 1, . . . , n. Tn (as )

(2.22)

Proof. For a given set of FST polynomials, we can construct the sequence Yn =

(1)

Tn (an ) , n = 0, 1, . . . . Tn (an )

Due to regularity of the FST polynomials, we have Yn = 0. Now from the Casoratian identity and the regularity condition (2.6) we obtain by induction in j (1)

Tn+ j (an ) Tn+ j (an )

= Yn , n, j ∈ N,

and this condition is equivalent to (2.22).

(2.23)


145

Proposition 1 means that for regular FST polynomials there is a one-to-one correspondence between sequences Yn , an and polynomials Tn (x). The sequence Ys is called an interpolated sequence and an – an interpolation grid. As usual, we assume that there exists a meromorphic function F(z) such that F(as ) = Ys . From this proposition and the orthogonality relation for Padé interpolants we arrive to the orthogonality relation ζ j F(ζ )Tn (ζ )dζ = 0, j = 0, 1, . . . , [n/2], (2.24) (ζ − a0 )(ζ − a1 ) · · · (ζ − an ) where [x] means the integer part of the number x and the contour is chosen to encircle only poles at ζ = a0 , . . . , an . We also recall the definition of divided differences [1]. For any function f (z) and any sequence of different numbers an we have by definition [a0 ] f (z) = f (a0 ),

[a0 , a1 ] f (z) =

f (a1 ) − f (a0 ) , a1 − a0

and then by induction [a0 , a1 , . . . , an+1 ] f (z) =

[a0 , a1 , . . . , an−1 , an+1 ] f (z) − [a0 , a1 , . . . , an−1 , an ] f (z) . an+1 − an

There is a useful Hermite formula [1] [a0 , a1 , . . . , an ] f (z) =

f (ζ )dζ , (ζ − a0 )(ζ − a1 ) · · · (ζ − an )

(2.25)

where the contour encircles all the poles a0 , . . . , an and excludes possible singularities of the function f (z). Note that the divided difference [a0 , a1 , . . . , an−1 , z] f (z) serves as a divided difference operator of the n th order acting on a space of all complex-valued functions f (z). For some other details on the relations between the osculatory recurrence relation (2.17) and Padé approximation, see [28, 29, 35]. 3. Spectral Transformations 3.1. The Christoffel type transformations. For a given set of FST polynomials Tn (x), we introduce the transformation ξn Tn+2 (x) + (1 − ξn )(x − an+1 )Tn (x) , Tñ (x) = x −µ

(3.1)

where ξn =

(an+1 − µ)Tn (µ) bn+2 Tn+1 (µ)

(3.2)

and the parameter µ is an arbitrary complex variable differing from the numbers ak , k ∈ N, and such that Tn (µ) = 0, n = 2, 3, . . . . Equivalently, one can present Tñ (x) in the form Tn+2 (x) − Cn Tn+1 (x) , Tñ (x) = x −µ where Cn = Tn+2 (µ)/Tn (µ).

(3.3)

146


Polynomials Tñ (x) have the degrees deg(T˜2n (x)) = deg(T˜2n+1 (x))) = n. They satisfy the recurrence relation Tñ+1 (x) = bñ+1 Tñ (x) + (x − an+1 )Tñ−1 (x)

(3.4)

with the initial conditions a1 − a2 − b1 b2 . T˜0 = 1, T˜1 (x) = b1 a1 − b1 b2 − µ

(3.5)

The coefficients bñ have the form bñ = ξn bn+2 + (1 − ξn−2 )bn Tn2 (µ) Tn (µ) + (an+1 − an ) . = bn+1 Tn+1 (µ)Tn−1 (µ) Tn+1 (µ)

(3.6)

The coefficient b˜0 remains arbitrary (it is not determined by this transformation yet). In order to match with definition (3.9) given below, we define b˜0 as a1 − µ . b˜0 = b0 + b1

(3.7)

We call the transformation Tn (x) → Tñ (x) the Christoffel type transformation (CTT) for the FST polynomials, because of its similarity with Christoffel’s theory of kernel polynomials, see, e.g., [25]. Polynomials Tñ (x) define a new set of the FST polynomials with the shifted interpolation points an → an+1 and transformed recurrence coefficients bn → bñ . With the help of the original relation (2.2), we can represent CTT in a slightly different form γn Tn+1 (x) + (x − an+1 )Tn (x) , Tñ (x) = x −µ

(3.8)

where γn = bn+2 ξn . For the associated FST polynomials we have CTT of the form Tñ(1) (x) =

(1)

(1)

γn Tn+1 (x) + (x − an+1 )Tn (x) x − a0 γn Tn+1 (x) + (x − an+1 )Tn (x) . + b0 (µ − a0 ) (x − a0 )(x − µ)

(3.9)

(1) Due to the choice (3.7) both Tñ and Tñ are seen to satisfy the FST recurrence relation (3.4) starting from n = 0. The transformed interpolation problem has the form (1) Tñ (as ) , s = 1, 2, . . . , n + 1. Y˜s = Tñ (as )

This new interpolated sequence is easily found to have the form (as − µ)Ys − (a0 − µ)Y0 Y˜s = . as − a0

(3.10)


147

In terms of the interpolated function F(z) we have (z − µ)F(z) − (a0 − µ)F(a0 ) ˜ ≡ [a0 , z]{(z − µ)F(z)} F(z) = z − a0

(3.11)

instead of the relation (3.10). Thus CTT has a simple interpretation in terms of the inter˜ polated function F(z). It corresponds to a new interpolated function F(z) obtained by application of the first order divided difference operator to the function (z − µ)F(z). The transformed FST polynomials satisfy the Casoratian identity (1) W˜ n (x) ≡ Tñ (x)Tñ−1 (x) − Tñ(1) (x)Tñ−1 (x)

= (−1)n (x − a1 )(x − a2 ) · · · (x − an ), W˜ 0 = 1.

(3.12)

Finally we note that there is a special simple case of the CTT when µ = a0 . Indeed, ˜ as seen from (3.11) in this case the interpolated function F(z) remains the same F(z) = F(z), whereas the interpolated grid is shifted as → as+1 . 3.2. The Geronimus type transformations. In this section we introduce a transformation of the Thiele polynomials which is in some sense reciprocal to the Christoffel type transformation. We first take the CTT (3.8), multiply it by (x − µ) and shift the index n → n − 1, which yields: (x − µ)Tñ−1 (x) = γn−1 Tn (x) + (x − an )Tn−1 (x).

(3.13)

In this relation we replace the term Tn−1 (x) by a combination of Tn (x), Tn+1 (x) through the recurrence relation (2.2): (x − µ)Tñ−1 (x) = (γn−1 − bn+1 )Tn (x) + Tn+1 (x).

(3.14)

Considering now (3.8) and (3.14) as a system of equations, we can express Tn (x) in terms of Tñ (x) and Tñ−1 (x): Tn (x) =

(x − µ)(Tñ (x) − γn Tñ−1 (x)) . x − an+1 − γn (γn−1 − bn+1 )

(3.15)

We have (by CTT) γn = (an+1 − µ)Tn (µ)/Tn+1 (µ). Thus it is easily seen that an+1 + γn (γn−1 − bn+1 ) = µ and hence we arrive at the formula Tn (x) = Tñ (x) − γn Tñ−1 (x).

(3.16)

Tn (x) = n Tñ (x) + (1 − n )(x − an−1 )Tñ−2 (x)

(3.17)

Equivalently,

with some coefficients n . Formulas (3.16) and (3.17) will be starting points in the search of general transformations of such type. Namely, for a given set of FST polynomials Tn (x) we consider the transformation Tñ (x) = Tn (x) − γn Tn−1 (x),

(3.18)

148


where γn are some indeterminate coefficients and tildas have different meaning than in (3.16). We demand that polynomials Tñ (x) satisfy again recurrence relation (3.15) but with some transformed coefficients bñ , a˜ n . Substituting (3.18) into the recurrence relation Tñ+1 (x) = bñ+1 Tñ (x) + (x − a˜ n )Tñ−1 (x),

(3.19)

we conclude that for n ≥ 1 necessarily γn =

ϕn , ϕn−1

(3.20)

and ϕn is an arbitrary solution of the recurrence relation ϕn+1 = bn+1 ϕn + (µ − an )ϕn−1 .

(3.21)

We call (3.18) the Geronimus type transformations (GTT) due to its resemblance with the Geronimus transformation in the theory of orthogonal polynomials [11]. The parameter µ should satisfy the same restrictions as in the case of CTT. Equality (3.21) coincides with the recurrence relation for FST polynomials (2.2) but now ϕn is an arbitrary solution with unrestricted initial conditions. From the Casoratian identity (2.6) (1) we see that for fixed µ = ak , k = 1, 2, . . . , the polynomials Tn (µ) and Tn (µ) provide two linear independent solutions of the recurrence relation (3.21). We can thus write ϕn = Tn (µ) + βTn(1) (µ),

(3.22)

where β is an arbitrary parameter, β = 0. The initial conditions for ϕn can be represented in the form ϕ−1 = β, ϕ0 = 1 + βb0 . We see that Tñ (x) are again FST polynomials satisfying (3.19) with the recurrence coefficients 1 β(a0 − µ) a˜ n = an−1 , b˜0 = − , b˜1 = , β 1 + βb0 ϕn−2 ϕn − , n ≥ 2. bñ = bn + ϕn−3 ϕn−1

(3.23)

Initial conditions for the transformed polynomials Tñ (x) are T˜0 = 1,

β(a0 − µ) T˜1 (x) = . 1 + βb0

(3.24)

It should be stressed that parameters β and a−1 remain arbitrary. Thus, construction of the Geronimus type transformed polynomials Tñ (x) by (3.18) requires an extended interpolation grid as , s = −1, 0, 1, . . . , with a−1 chosen in an ad hoc manner. Using recurrence relation (2.2), we can represent the GTT in the equivalent form Tñ (x) = ηn Tñ (x) + (1 − ηn )(x − an−1 )Tñ−2 (x), where ηn =

(an−1 − µ)φn−2 . bn φn−1

(3.25)


149

For the associated polynomials we obtain the following GTT formula: µ − a−1 x − a−1 (1) (1) Tn (x) − γn Tn−1 (x) . Tñ(1) (x) = (Tn (x) − γn Tn−1 (x)) + β(x − µ) x −µ (3.26) (1)

One can verify directly that the polynomials Tñ (x) satisfy recurrence relation (3.19) together with the “correct” initial conditions: (1) T˜0 (x) = b˜0 ,

(1) T˜1 (x) = x − a−1 + b˜0 b˜1 .

Polynomials Tñ (x) and Tñ(1) (x) satisfy an interpolation problem: (1) Tñ (as−1 ) , s = 0, 1, . . . , n, Y˜s = Tñ (as−1 )

(3.27)

where the interpolated sequence has the form (as−1 − a−1 )Ys−1 + ν Y˜s = , as−1 − µ

(3.28)

with ν = (µ − a−1 )/β and Ys being the interpolated sequence corresponding to the FST polynomials Tn (x), Tn(1) (x) (see (2.22)). Note that Y˜0 = ν/(a−1 − µ) = −1/β = b˜0 as (1) prescribed by the initial condition T0 (x)/T0 (x) = b0 . For the transformed interpolated function we have (z − a−1 )F(z) + ν ˜ , F(z) = z−µ

(3.29)

˜ s−1 ), s = 0, 1, . . . . where, as usual, Y˜s = F(a It is directly verified that CTT with the parameter µ is reciprocal to the GTT with the same parameter µ: if we replace in (3.1) polynomials Tn (x) by the polynomials Tñ (x) (3.18), we obtain the identity map. However, the inverse is not true—sequential application of CTT and then GTT does not necessarily yield the identical transformation. The reason for this phenomenon is the same as in the case of the ordinary orthogonal polynomials [26, 33]: GTT contain two additional parameters β and a−1 . So, starting from given FST-polynomials Tn (x) one can obtain an infinite family of GTT polynomials Tñ (x; β). On the other hand, the CTT yields unique polynomials Tñ (x) (for a fixed parameter µ). We can illustrate this by comparing the results of two succeeding transformations upon the interpolated function F(z). If one first performs GTT and then CTT we return to the same function F(z). Assume, however, that one first performs CTT and then GTT and that the initial interpolation point a0 returns to the same value. Then we obtain the new interpolated function F(z) → F(z) + ν1 /(z − µ) with some (arbitrary!) constant ν1 . Thus in this case we obtain the function F(z) with an additional simple pole at the point z = µ with an arbitrary residue. The situation is very similar to the case of the ordinary orthogonal polynomials [33] where combination of CTT and then GTT yields the so-called Uvarov transformation (i.e. addition of a Dirac delta function to the initial weight function). However, in contrast to orthogonal polynomials, in our case there is a possibility to get a new starting interpolating point a˜ 0 = a0 (of course all other points a1 , a2 , . . . will be kept unchanged). This leads to a more complicated expression for

150


˜ the transformed interpolated function F(z). Note also that the interpolated sequence Y˜s defined by (3.28) can be obtained from the corresponding Christoffel type interpolated sequence (3.10) by a simple inversion procedure.

4. The FST Chain Let us consider now a sequence of the FST polynomials Tn (x; j) depending on a discrete time variable j in such a way that the shift j → j +1 corresponds to CTT (3.1) with some parameter µ j+1 and the shift j → j − 1 corresponds to GTT (3.18) with the parameter ( j) µ j . In these defining relations an depends on j and n in some particular way. Indeed, ( j+1) ( j) ( j−1) ( j) as we saw already, CTT lead to an = an+1 and GTT lead to an = an−1 . There( j)

fore, we have an = an+ j , a special parametrization of linear sequence of interpolation points. As mentioned already, sequential GTT and CTT do not yield in general identical transformation. In order to guarantee that j → j ± 1 shifts do not change polynomials we should impose also appropriate constraints on the parameters β j in (3.22). Such restrictions are absent for one sided time shifts, say, j ≤ 0, starting from a GTT. We can rewrite our system of transformations in the form ( j+1)

Tn (x; j + 1) =

An

Tn+1 (x; j) + (x − an+ j+1 )Tn (x; j) , x − µ j+1 ( j)

Tn (x; j − 1) = Tn (x; j) + Bn Tn−1 (x; j).

(4.1) (4.2)

We would like to derive compatibility conditions for these discrete equations for n, j ∈ Z. Shifting the time parameter j → j + 1 in (4.2) and then applying (4.1), we obtain the recurrence relation ( j+1)

An

( j+1)

Tn+1 (x; j) + (Bn ( j+1)

+ Bn

( j+1)

An−1 + µ j+1 − an+ j+1 )Tn (x; j)

(x − an+ j )Tn−1 (x; j) = 0.

(4.3)

Analogously, shifting j → j − 1 in (4.1) and then applying (4.2), we obtain ( j)

( j)

( j)

An Tn+1 (x; j) + (Bn+1 An + µ j − an+ j )Tn (x; j) ( j)

+ Bn (x − an+ j )Tn−1 (x; j) = 0. ( j+1)

We divide Eq. (4.3) by Bn This yields:

( j+1)

An

( j+1)

Bn

+

( j)

−

( j+1) An−1

An

( j)

Bn +

(4.4)

( j)

and Eq. (4.4) by Bn and then subtract one from another.

Tn+1 (x; j)

µ j+1 − an+ j+1 ( j+1)

Bn

(4.5) ( j)

−

( j)

Bn+1 An + µ j − an+ j ( j)

Bn

Tn (x; j) = 0.


151

Applying Lemma 2 to this equation, we see that the prefactors in front of Tn+1 (x; j) and Tn (x; j) vanish, i.e. ( j+1)

An

( j)

( j+1)

Bn = Bn

( j+1) ( j+1) An−1 Bn

( j)

An ,

(4.6)

+ µ j+1 − an+ j+1

( j+1) Bn

=

( j) ( j) Bn+1 An

+ µ j − an+ j ( j)

Bn ( j)

.

(4.7)

( j)

Now the first equation is easily integrated yielding Bn = gn An , where coefficients gn do not depend on j. Comparing the resulting three term recurrences with the canonical normalization, we see that it corresponds to the choice gn = −1, which can be reached by ( j) ( j) a renormalization of Tn (x) and bn . Substituting Bn = −An into the second equation, we obtain the final compatibility condition in the form ( j+1)

µ j+1 − an+ j+1 − An

( j+1)

An−1

( j+1)

An

( j)

=

( j)

µ j − an+ j − An An+1 ( j)

An

.

(4.8)

We call this discrete dynamical system the FST chain. From relations (4.3) or (4.4) we find ( j)

( j)

bn+1 = An+1 +

an+ j − µ j ( j)

An

.

(4.9)

Thus, if we know a solution of the FST chain, we can reconstruct corresponding recur( j) rence coefficients bn and, through recurrence relation (2.2), the corresponding FST polynomials themselves. In general one can abandon polynomiality constraints for solutions of the discrete Lax pair equations (4.1) and (4.2). Then the FST chain describes sequences of spectral transformations for the unrestricted FST three term recurrence relation in the same way as the unrestricted discrete time Toda chain describes spectral transformations for the finite difference Schrödinger equation. 5. Connections to the Discrete Time Lotka-Volterra Chain and ε-Algorithm Let us define polynomials Sn (x) by the formula Sn (x) = x n Tn (x −2 ).

(5.1)

Due to restriction (2.9), Sn (x) are polynomials in x of the exact degree n. Moreover, they are symmetric Sn (−x) = (−1)n Sn (x).

(5.2)

This means, in particular, that S2n (x) = pn(1) (x 2 ) and S2n+1 (x) = x pn(2) (x 2 ), where (1) (2) pn (x) and pn (x) are two polynomials of the exact degree n. From the definition (2.2) we find the recurrence relation Sn+1 (x) = bn+1 x Sn (x) + (1 − an x 2 )Sn−1 (x).

(5.3)

152


We denote as κn the leading term coefficient of the polynomial Sn (x): Sn (x) = κn x n + O(x n−1 ). From (2.2) and (5.1), the sequence κn , n ∈ N, is seen to satisfy the recurrence relation κn+1 − bn+1 κn + an κn−1 = 0.

(5.4)

For monic polynomials Sˆn (x) = Sn (x)/κn = x n + O(x n−1 ), we have the recurrence relation Sˆn+1 (x) = ζn x Sˆn (x) + (1 − ζn )(x 2 − an−1 ) Sˆn−1 (x),

(5.5)

where ζn = bn+1 κn /κn+1 . In [24], the first author introduced a class of polynomials generated by (2.17) and satisfying the condition Pn (−x) = (−1)n Pn (x) for all n ∈ N. For such symmetric polynomials we have Lemma 4. Regular symmetric polynomials generated by recurrence relation (2.17) are completely characterized by the conditions (i) E n = 0, n ∈ N, and (ii) βn = −αn , n = 1, 2, . . . . Proof. Due to the property Pn (−x) = (−1)n Pn (x), polynomials Pn (x) satisfy also the recurrence relation −Pn+1 (x) = (−Cn x + E n )Pn (x) − Dn (x + αn )(x + βn )Pn−1 (x).

(5.6)

Adding (2.17) and (5.6), we obtain 2E n Pn (x) − Dn x(αn + βn )Pn−1 (x) = 0.

(5.7)

Applying Lemma 3 to (5.7) and taking into account that Dn = 0 (the regularity condition), we arrive at the conditions (i) and (ii) of the lemma. Thus, regular symmetric osculatory polynomials are defined by the recurrence relation Pn+1 (x) = Cn x Pn (x) + Dn (x 2 − αn2 )Pn−1 (x)

(5.8)

with some coefficients Cn , Dn and αn . For monic polynomials we have the obvious condition Cn + Dn = 1. Comparing relations (5.8) and (5.5), we see that the polynomials Sˆn (x), obtained from the FST polynomials, coincide with general monic symmetric osculatory polynomials with Cn = ζn and αn2 = an−1 . Such polynomials lead to symmetric biorthogonal rational functions [24] satisfying relations similar to (2.21). The following discrete time Lotka-Volterra chain was introduced in [25, 26] (see also [16]) from spectral transformations for symmetric orthogonal polynomials j+1

j+1

j

j

Dn (κ j+1 − Dn−1 ) = Dn (κ j − Dn+1 ).

(5.9)

It is related also to Bauer’s g-algorithm [2]. In [24], the following generalization of (5.9) was proposed from spectral transformations for (5.8): j

j

2 γ2 γn+ j n+ j+2 G n (κ j − (κ j + γn+ j+1 )G n+1 ) j+1

j+1

G n (κ j+1 − (κ j+1 + γn+ j )G n−1 ) j

j

j

=

j

ξn ξn+1 , (κ j − γn+ j+1 )(κ j+1 − γn+ j+1 )

2 2 2 ξn ≡ (γn+ j − γn+ j+1 )κ j G n + γn+ j+1 (κ j − γn+ j ).

(5.10)


153

It appears that this equation is directly related to the FST chain. Indeed, let us introduce a new variable ( j)

Vn

( j)

( j)

= An An+1 .

(5.11)

Then from the FST chain (4.8) we obtain ( j+1)

µ j+1 − an+ j+1 − An ( j+1)

( j+1)

j+1) ( j+1) An−1 µ j+1 − an+ j+2 − A(n+1 An ( j+1)

An

An+1 ( j)

=

( j)

( j)

( j)

µ j − an+ j − An An+1 µ j − an+ j+1 − An+1 An+2 ( j)

( j)

An

An+1

or

µ j+1 − an+ j+1 −

( j+1) Vn−1

µ −a j+1 n+ j+2 ( j+1)

Vn

,

−1

µ −a j n+ j ( j) = µ j − an+ j+1 − Vn+1 −1 . ( j) Vn

(5.12)

Now, substituting ( j) Vn

=

−2 κ −2 j − γn+ j+1

1−

, µj =

κj

j

(κ j +γn+ j )G n

1 1 , aj = 2 2 κj γj

(5.13)

into Eq. (5.12), we obtain equality (5.10). In the limit γn → ∞, Eq. (5.10) is reduced to j j (5.9) after denoting Dn ≡ γn+ j G n and a similar limit is valid for Eq. (5.12). ( j) As a next step, we introduce new variables εn by the relation ( j)

( j+1)

An = εn

( j)

− εn .

(5.14)

Then from the FST chain (4.8) we obtain µ j − an+ j ( j+1) εn

( j) − εn

( j)

( j+1)

+ εn+1 − εn−1 = Hn ,

(5.15)

where Hn is a sequence of complex numbers independent on the discrete time j. Shifting ( j) ( j) εn → εn + h n , where h n+1 − h n−1 = Hn , we can remove completely Hn numbers from this equation. Therefore it is equivalent to the following mapping: ( j+1)

( j)

εn−1 = εn+1 +

µ j − an+ j ( j+1)

εn

( j)

− εn

.

(5.16)

This is a generalized version of Wynn’s ε-algorithm in the approximation theory [1]. Initially the ε-algorithm appeared in connection to the acceleration of series convergence. Then, it was recognized that it is related to the ordinary Padé approximation. Finally, Claessens [6] was able to construct a generalized version of this algorithm which is connected to the Padé interpolation table. It contained two arbitrary sequences similar to our Hn and an+ j (its particular subcases were considered also in [3] and [5]). In [22], the general form of the ε-algorithm (5.16) was recovered by demanding validity of Wynn’s singular rules [32] known nowadays as the singularity confinement criterion.

154


6. Relations Between the R I I and FST Chains The R I I chain was introduced in [27] as a compatibility condition of the following spectral transformations for polynomials Pn (x) satisfying the osculatory recurrence relation (2.17): ( j+1)

Pn (x; j + 1) =

Cn

( j+1)

( j+1)

Pn+1 (x; j) + Dn (x − αn x − µ j+1

( j)

( j)

)Pn (x; j)

,

( j)

Pn (x; j − 1) = E n Pn (x; j) + Fn (x − βn )Pn−1 (x; j), ( j)

( j)

( j)

(6.1) (6.2)

( j)

where coefficients Cn , Dn , E n , Fn are some sequences of numbers (they should not be confused with the coefficients in (2.17)). The compatibility condition of (6.1) and (6.2) yields the equations ( j)

( j)

( j)

( j+1)

( j)

Cn Fn+1 + Dn E n − 1

=

Dn

( j+1)

En

( j+1)

( j+1)

+ Cn−1 Fn

−1

( j) ( j) ( j+1) ( j+1) Cn E n+1 Cn En ( j) ( j) ( j) ( j) ( j) ( j) Cn Fn+1 βn+1 + Dn E n αn − µ j ( j) ( j) Cn E n+1 ( j+1) ( j+1) ( j+1) ( j+1) ( j+1) ( j+1) Dn En αn + Cn−1 Fn βn − µ j+1 = , ( j+1) ( j+1) Cn En ( j+1) ( j+1) ( j) ( j) Dn−1 Fn Dn Fn = ( j+1) ( j+1) , ( j) ( j) Cn E n+1 Cn En ( j) ( j) ( j+1) ( j+1) ( j) ( j) ( j+1) ( j+1) , αn + βn = αn−1 + βn , αn βn = αn−1 βn

,

(6.3)

(6.4) (6.5) (6.6)

called the R I I chain. The constraints (6.6) give us two possibilities ( j+1)

( j)

( j+1)

( j+1)

= βn

(1) αn−1 = αn and βn (2) αn−1 ( j) αn

=

( j)

( j)

( j+1)

and βn

( j) a2n+ j+1 , βn

( j)

( j)

= βn , then αn = αn+ j and βn = βn . ( j)

( j)

= αn , then αn

( j)

= a2n+ j , βn

= a2n+ j−1 or

= a2n+ j .

From the first case, we obtain the type A R I I chain: ( j)

( j)

( j)

( j+1)

( j)


=

Dn

( j+1)

En

( j+1)

( j+1)

+ Cn−1 Fn

( j) ( j) ( j+1) ( j+1) Cn E n+1 Cn En ( j) ( j) ( j) ( j) Cn Fn+1 βn+1 + Dn E n αn+ j − µ j ( j) ( j) Cn E n+1 ( j+1) ( j+1) ( j+1) ( j+1) Dn En αn+ j+1 + Cn−1 Fn βn = ( j+1) ( j+1) Cn En ( j+1) ( j+1) ( j) ( j) D Fn Dn Fn = (n−1 . ( j) ( j) j+1) ( j+1) Cn E n+1 Cn En

−1

− µ j+1

,

,

(6.7)


155

( j)

( j)

From αn = a2n+ j , βn = a2n+ j−1 of the second case, we obtain the type B R I I chain: ( j)

( j)

( j)

( j+1)

( j)


=

Dn

( j+1)

En

( j+1)

( j+1)

+ Cn−1 Fn

−1

( j) ( j) ( j+1) ( j+1) Cn E n+1 Cn En ( j) ( j) ( j) ( j) Cn Fn+1 a2n+ j+1 + Dn E n a2n+ j − µ j ( j) ( j) Cn E n+1 ( j+1) ( j+1) ( j+1) ( j+1) Dn En a2n+ j+1 + Cn−1 Fn a2n+ j − µ j+1 = , ( j+1) ( j+1) Cn En ( j+1) ( j+1) ( j) ( j) Dn−1 Fn Dn Fn = . ( j) ( j) ( j+1) ( j+1) Cn E n+1 Cn En ( j)

( j)

( j)

(6.8)

( j)

For monic polynomials Pn (x; j), the variables Cn , Dn , E n and Fn the constraints ( j)

( j)

( j)

Cn + Dn = 1,

( j)

E n + Fn

,

= 1,

should satisfy (6.9)

and the type B R I I chain takes a slightly simpler form ( j)

(a2n+ j+1 −a2n+ j )Cn +a2n+ j −µ j ( j) ( j) Cn E n+1 ( j)

( j+1)

=

(a2n+ j+1 −a2n+ j )E n

( j)

(1 − Cn )(1 − E n ) ( j)

( j)

Cn E n+1

=

+a2n+ j −µ j+1 , ( j+1) ( j+1) Cn En ( j+1) ( j+1) (1 − Cn−1 )(1 − E n ) . (6.10) ( j+1) ( j+1) Cn En

Hereafter we will focus on the relations between the type B R I I and FST chains. We start from the three term relation for FST polynomials ( j)

Tn+1 (x; j) + bn+1 Tn (x; j) − (x − an+ j )Tn−1 (x; j) = 0 and its shifted versions ( j)

Tn+2 (x; j) + bn+2 Tn+1 (x; j) − (x − an+ j+1 )Tn (x; j) = 0, ( j)

Tn (x; j) + bn Tn−1 (x; j) − (x − an+ j−1 )Tn−2 (x; j) = 0. Excluding Tn±1 polynomials, we obtain the three term relation connecting Tn+2 , Tn , and Tn−2 (cf. [24]) ( j)

( j)

Tn+2 (x; j) = ((1 + wn )x + vn )Tn (x; j) ( j)

− wn (x − an+ j )(x − an+ j−1 )Tn−2 (x; j), with the coefficients ( j)

( j)

( j)

( j)

( j)

vn = bn+1 bn+2 − an+ j+1 − wn an+ j , wn =

( j)

bn+2 ( j)

bn

,

(6.11)

156


and the initial conditions T−2 = 0, T0 = 1. The polynomials T2n (x; j) = Pn (x; j) are monic and satisfy the recurrence relation ( j)

( j)

( j)

( j)

( j)

Pn+1 (x; j) = (Un x + Vn )Pn (x; j) − Wn (x − αn )(x − βn )Pn−1 (x; j) (6.12) ( j)

( j)

with the parameters αn = a2n+ j , βn = a2n+ j−1 and the coefficients ( j)

Un

( j)

Vn

( j)

=

( j)

( j)

( j)

( j)

( j)

( j)

( j)

= 1 + Wn , Wn

Cn E n+1 ( j)

=

( j)

1 − Cn Fn+1 − Dn E n ( j)

( j)

=

( j)

( j)

( j)

Dn Fn

( j)

Cn E n+1

b2n+2

=

( j)

b2n

,

( j)

Cn Fn+1 a2n+ j+1 + Dn E n a2n+ j − µ j ( j)

( j)

Cn E n+1 ( j)

( j)

( j)

= b2n+1 b2n+2 − a2n+ j+1 − Wn a2n+ j . Then we obtain the direct Toda-Volterra type correspondence between the type B R I I and FST chains: ( j)

Cn

( j)

( j)

=

En =

( j)

A2n+1 A2n

( j) ( j) A2n+1 A2n

+ a2n+ j+1 − µ j

a2n+ j−1 − µ j ( j) ( j) A2n−1 A2n + a2n+ j−1

( j)

,

− µj

Dn = ( j)

,

Fn

=

a2n+ j+1 − µ j , ( j) ( j) A2n+1 A2n + a2n+ j+1 − µ j ( j) ( j) A2n−1 A2n ( j) ( j) A2n−1 A2n + a2n+ j−1 − µ j

.

(6.13)

These relations generate solutions of the type B R I I chain from given solutions of the FST chain. In a similar way, the polynomials T2n+1 (x; j) = Pn (x; j) satisfy recurrence relations ( j) ( j) (6.12) with the parameters αn = a2n+ j+1 , βn = a2n+ j and the coefficients ( j)

Un

( j)

Vn

( j)

= 1 + Wn , ( j)

( j)

Wn

( j)

=

b2n+3 ( j)

b2n+1

( j)

,

( j)

= b2n+2 b2n+3 − a2n+ j+2 − Wn a2n+ j+1 .

The corresponding type B R I I chain: ( j)

( j)

( j)

( j+1)

( j)


=

Dn

( j+1)

En

( j+1)

( j+1)

+ Cn−1 Fn

−1

( j) ( j) ( j+1) ( j+1) Cn E n+1 Cn En ( j) ( j) ( j) ( j) Cn Fn+1 a2n+ j+2 + Dn E n a2n+ j+1 − µ j ( j) ( j) Cn E n+1 ( j+1) ( j+1) ( j+1) ( j+1) Dn En a2n+ j+2 + Cn−1 Fn a2n+ j+1 − µ j+1 = , ( j+1) ( j+1) Cn En ( j+1) ( j+1) ( j) ( j) Dn−1 Fn Dn Fn = , ( j) ( j) ( j+1) ( j+1) Cn E n+1 Cn En

,

(6.14)


157

and the FST chain are related to each other by the following mapping: ( j)

( j)

Cn = ( j)

En =

( j)

A2n+2 A2n+1

( j) ( j) A2n+2 A2n+1

+ a2n+ j+2 − µ j

a2n+ j − µ j ( j) ( j) A2n A2n+1 + a2n+ j

− µj

,

( j)

,

a2n+ j+2 − µ j ( j) ( j) A2n+2 A2n+1 + a2n+ j+2 ( j) ( j) A2n A2n+1 . ( j) ( j) A2n A2n+1 + a2n+ j − µ j

Dn = ( j)

Fn

=

− µj

,

(6.15)

7. An Affine Weyl Group Symmetry Theorem 1. Let us take the FST chain and define operators Bk , k ∈ Z, acting on its entries as follows: a˜ n = Bk (an ) = an ,

for all n, ( j) ( j) ( j) µ˜ j = Bk (µ j ) = µ j , A˜ n = Bk (An ) = An , for j = k, k + 1, µ˜ k = Bk (µk ) = µk+1 , µ˜ = Bk (µk+1 ) = µk ,

k+1 − µ µ k+1 k (k) (k) (k) A˜ n = Bk (An ) = An 1 + (k) (k+1) µk − an+k + An An−1 = A(k) n + A˜ (k+1) n

=

(µk+1 − µk )A(k+1) n µk+1 − an+k+1 + A(k) A(k+1) n

n+1

Bk (A(k+1) ) n

= A(k+1) − n

=

A(k+1) n

1−

µk+1 − µk (k)

(k+1)

µk+1 − an+k+1 + An+1 An (k)

(µk+1 − µk )An (k)

,

(k+1)

µk − an+k + An An−1

.

(7.1)

Then, the FST chain is invariant with respect to the action of operators Bk , i.e. it is ( j) satisfied for the tilded variables a˜ n , µ˜ j , and A˜ n . Proof. The forms of the transformed variables are deduced from the requirement that the double CTT or GTT with discrete time values k and k + 1 give the same answer independently on whether we perform them with untilded or tilded variables. From the explicit forms of these transformations (4.1), (4.2) we immediately come to the restrictions a˜ n+k = an+k , a˜ n+k+1 = an+k+1 , µ˜ k = µk+1 , µ˜ k+1 = µk , (k+1) (k) (k+1) ˜ (k+1) = A(k) ˜ (k+1) A˜ (k) , A˜ (k) n + An n + An n An−1 = An An−1 , with all other variables being unchanged. Substituting these restrictions into the FST (k) (k+1) chain we deduce the explicit expressions for A˜ n and A˜ n . This symmetry generalizes the substitution rules derived in [23] for the discrete time Toda chain and the discrete time Volterra chain (5.9). Theorem 2. Operators Bk satisfy the affine Weyl group relations: Bk2 = 1,

Bk Bl = Bl Bk , l = k ± 1,

(Bk Bk±1 )3 = 1.

(7.2)

158


Proof. The commutativity Bk Bl = Bl Bk , l = k ± 1, and equalities Bk2 (µk ) = (k) (k) (k+1) µk , Bk2 (µk+1 ) = µk+1 are evident. The relations Bk2 (An ) = An , Bk2 (An ) = (k+1) (k) (k+1) (k) (k+1) follow from the condition Bk (An An ) = An An . An In order to verify the remaining part of identities it is sufficient to check validity of the braid type relation Bk Bk+1 Bk = Bk+1 Bk Bk+1 . Indeed, both sides of this equality map µk to µk+2 , µk+1 to µk+1 , and µk+2 to µk . Essentially more complicated computations are involved into the remaining identities. For instance, we have (k+1)

An

(k) Bk+1 Bk Bk+1 (A(k) n ) = An +

(k+2)

(µk+2 − µk )(µk+2 − an+k+1 + A(k+1) An−1 ), n

(k) dn

(7.3) where (k+1) dn(k) = (µk+2 − an+k+1 )(µk+1 − an+k+1 + A(k) ) n+1 An (k+2)

(k)

An−1 (µk+2 − an+k+1 + A(k+1) An+1 ). + A(k+1) n n The original expression for Bk Bk+1 Bk (A(k) n ) is rather cumbersome, but it can be simplified to the one given above. Similarly, we have (k+1) (k+1) Bk Bk+1 Bk (A(k) ) = A(k) n + An n + An (k+2)

+

An

(k)

cn

(k)

(k+1)

(µk+2 − µk )(µk+1 − an+k+2 + An+2 An+1 ), (7.4)

where (k)

(k+1)

cn(k) = (µk+2 − an+k+2 )(µk+1 − an+k+2 + An+2 An+1 ) (k) (k+1) + A(k+2) A(k+1) n n+1 (µk − an+k+2 + An+2 An+1 ). (k)

(k+1)

The same expression is obtained for Bk Bk+1 Bk (An + An ogous to the previous case. Finally, the obvious relations

) after computations anal-

(k+1) (k+1) Bk Bk+1 Bk (A(k) + A(k+2) ) = A(k) + A(k+2) n + An n n + An n (k+1) = Bk+1 Bk Bk+1 (A(k) + A(k+2) ) n + An n

complete the proof of the theorem. ( j+N )

For the periodic reduction An standard Aˆ N −1 affine Weyl group.

(7.5)

( j)

= An , an+N = an , µ j+N = µ j we obtain the

8. The Bilinear Formalism The bilinear form of the type A R I I chain was derived recently in [20]. In this section we introduce the τ function and corresponding bilinear formalism for the FST and type B R I I chains. We start from the equations (k ,k2 )

1 Tm+1

(k +1,k2 +1)

1 Tm−1

= Tm(k1 ,k2 ) Tm(k1 +1,k2 +1) − Tm(k1 +1,k2 ) Tm(k1 ,k2 +1) ,

(8.1)


159

known as a bilinear form of the discrete two-dimensional Toda equation [15], and (k ,k2 +1)

1 Tm(k1 +1,k2 ) Tm+1

(k +1,k2 )

1 − Tm(k1 ,k2 +1) Tm+1

(k )

(k )

(k ,k2 )

1 = (δ1 1 − δ2 2 )Tm+1

Tm(k1 +1,k2 +1) , (8.2)

representing a non-autonomous bilinear form of the discrete KP equation [21]. The arbitrary sequences δi(ki ) can be removed from (8.2) by the gauge transformation (k ,k2 )

Tm 1 Tm(k1 ,k2 ) → k −1 k −1 1 2 j=0

l=0

( j)

(l)

(δ1 − δ2 )

.

(8.3)

However, we shall be considering simultaneous solutions of (8.1) and (8.2) and transformation (8.3) introduces δi(ki ) into Eq. (8.1). More precisely, we stick to this system of equations with the parameters (k )

(k )

δ1 1 = a−k1 , δ2 2 = µk2 +1 and the independent variable transformations k1 = −l − j, k2 = j, m = n.

(8.4)

(l, j) τn

Theorem 3. Let be a function of independent discrete variables j, l, n ∈ Z. We define the τ -functions in terms of the Hankel type determinant (l, j) τn = fl, j (ν1 + ν2 − 2) 1≤ν ,ν ≤n (n = 1, 2, 3, · · · ), (8.5) 1

2

(l, j)

τ−1 = τ−2 = · · · = 0, τ0

= 1,

(8.6)

where the elements fl, j (s) satisfy the linear relations fl, j (s + 1) = fl−1, j+1 (s) + µ j+1 fl, j (s) = fl−1, j (s) + al+ j fl, j (s).

(8.7)

τ -functions (8.5) and (8.6) give particular solutions of the described above two bilinear equations: (l+2, j) (l, j+1) (l+2, j) (l, j+1) (l+1, j) (l+1, j+1) τn−1 = τn τn − τn τn , (l, j) (l, j+1) (l, j) (l, j+1) (l+1, j) (l−1, j+1) τn+1 τn − τn τn+1 = (µ j+1 − al+ j+1 )τn+1 τn .

τn+1

(8.8) (8.9)

Proof. prove Eq. (8.8). Let D = |M| be the determinant of some matrix M, and

First, we i1 i2 . . . in be the determinant of the matrix M obtained from M by removing D j1 j2 . . . jn the i 1 , . . . , i nth rows and the j1 , . . . , jnth columns. Then the following Jacobi identity is true

i1 i2 i i i1 i2 =D D −D 1 D 2 . (8.10) D·D j1 j2 j1 j2 j2 j1 The bilinear equation (8.8) follows from identity (8.10) with i 1 = j1 = 1, i 2 = j2 = n, where fl+1, j+1 (0) · · · fl+1, j+1 (n − 1) fl+2, j (0) fl, j+1 (0) · · · fl, j+1 (n − 1) fl+1, j (0) . (8.11) D = .. .. .. . . . f (n − 1) f (n − 1) · · · f (2n − 2) l+1, j

l, j+1

l, j+1

160


Using linear equations (8.7), we can easily see that fl+2, j (0) fl+1, j+1 (0) · · · fl+1, j+1 (n − 1) (l+2, j) .. .. .. D= = τn+1 , . . . fl+2, j (n) fl+1, j+1 (n) · · · fl+1, j+1 (2n − 1)

1 1n (l, j+1) (l, j+1) D = τn , D = τn−1 , 1 1n fl+1, j+1 (0) · · · fl+1, j+1 (n−2)

fl+2, j (0) n (l+2, j) .. .. .. D = , = τn . . . n fl+2, j (n−1) fl+1, j+1 (n−1) · · · fl+1, j+1 (2n−3)

1 n (l+1, j) (l+1, j+1) D = τn , D = τn . n 1

(8.12)

(8.13)

(8.14)

(8.15)

From formulas (8.12)–(8.15), the Jacobi identity (8.10) is reduced to (8.8), i.e. our τ -functions (8.5) satisfy the needed bilinear equation. Next, we prove the second equation (8.9). Consider the identity ϕ1 · · · ϕn ψ1 ∅ ψ2 ψ3 = 0, (8.16) ∅ ψ1 ϕ1 · · · ϕn−1 ψ2 ψ3 where ϕi , ψi are arbitrary (n + 1)-dimensional column vectors. Applying the Laplace expansion to the left-hand side of identity (8.16), we obtain |ϕ1 · · · ϕn−1 ϕn ψ1 | · | ϕ1 · · · ϕn−1 ψ2 ψ3 | − |ϕ1 · · · ϕn−1 ϕn ψ2 | · | ϕ1 · · · ϕn−1 ψ1 ψ3 | + |ϕ1 · · · ϕn−1 ϕn ψ3 | · | ϕ1 · · · ϕn−1 ψ1 ψ2 | = 0,

(8.17)

which is one of the Plücker relations. The bilinear equation (8.9) follows from the Plücker relation (8.17) with ϕi = fl−1, j+1 (i − 1) · · · fl−1, j+1 (i + n − 1) , ψ1 = fl, j (0) · · · fl, j (n) , ψ2 = fl, j+1 (0) · · · fl, j+1 (n) , ψ3 = 0 · · · 0 1 . Indeed, we can see that (l, j)

|ϕ1 · · · ϕn−1 ϕn ψ1 | = (−1)n τn+1 ,

|ϕ1 · · · ϕn−1 ϕn ψ3 | =

(l, j+1) (−1)n−1 τn , n (l, j+1) (−1) τn+1 , (l, j) (−1)n−1 τn , (l−1, j+1) τn ,

|ϕ1 · · · ϕn−1 ψ1 ψ2 | =

(l+1, j) (µ j+1 − al+ j+1 )τn+1 .

|ϕ1 · · · ϕn−1 ψ2 ψ3 | = |ϕ1 · · · ϕn−1 ϕn ψ2 | = |ϕ1 · · · ϕn−1 ψ1 ψ3 | =

(8.18) (8.19) (8.20) (8.21) (8.22) (8.23)

Substituting formulas (8.18)–(8.23) into (8.17), we obtain (8.9) which completes the proof.


161

Now we construct nonlinear dynamical systems from Eqs. (8.8) and (8.9). Through the dependent variable transformations (l, j)

ξn

(l, j−1)

τn+1

=

(l, j−1)

τn

,

(8.24)

these bilinear equations are transformed to µ j − al+ j

(l, j) (l, j+1) = ξ n − ξn

=

1 (l, j+1) ξn

1

1

− (l−1, j+1) (l+1, j) ξn ξn−1 1 (l−1, j+1) (l+1, j) ξn . − (l, j) − ξn+1 ξn

(8.25)

Let us split the index l to even and odd values 2n and 2n + 1 and define the dependent variables ( j)

(2n, j)

ε2n = ξn

( j)

(2n+1, j)

, ε2n+1 = 1/ξn

.

(8.26)

From relations (8.25) we find

( j) ( j+1) ( j) ( j+1) µ j − a2n+ j = ε2n − ε2n ε2n+1 − ε2n−1 , ( j+1) ( j) ( j+1) ( j) µ j − a2n+ j+1 = ε2n+1 − ε2n+1 ε2n − ε2n+2

for n ∈ Z. This discrete system coincides with the generalized ε-algorithm (5.16). The functions fl, j (s) are chosen as f (ζ )ζ s (ζ − µ1 ) · · · (ζ − µ j ) 1 fl, j (s) = dζ, (8.27) 2πi (ζ − a0 ) · · · (ζ − al+ j ) where a positively oriented contour circumscribes poles at a0 , . . . , al+ j , and they are seen to satisfy linear equations (8.7). From given solutions of the generalized ε-algorithm (5.16), relations (5.14), (5.11) and (6.13) define particular solutions of the FST chain (4.8), the generalized discrete time Lotka-Volterra (5.12) and the type B R I I chains (6.8), respectively. Using bilinear ( j) ( j) ( j) equation (8.9), we can show that dependent variables εn , An , and Vn can be fac( j) ( j+1) ( j) − εn and torized into products of τ -functions (8.5). From the relation An = εn bilinear equation (8.9), we obtain ( j)

A2n = ( j)

A2n+1 =

(2n, j)

τn+1

(2n, j)

τn

(2n, j−1)

−

τn+1

(2n, j−1)

τn

(2n+1, j)

τn

(2n+1, j)

τn+1

(2n+1, j−1)

−

τn

(2n+1, j−1)

τn+1

(2n+1, j−1) (2n−1, j) τn , (2n, j−1) (2n, j) τn τn (2n+2, j−1) (2n, j) τn+1 τn −(a2n+ j+1 − µ j ) (2n+1, j−1) (2n+1, j) . τn+1 τn+1

= (a2n+ j − µ j ) =

τn+1

These expressions provide also factorized forms of the variables (5.11). Furthermore, for the readers’ convenience, we present various bilinear relations and determinantal expressions for other dependent variables which have appeared in this paper. In a subsequent work we plan to discuss in detail these formulas together with

162


their analogues for the type A R I I chain [20] as well as some explicit solutions of the FST/R I I chains. The τ -function (8.5) and the related functions defined by (l, j) (l, j) (l, j) (l−1, j+1) σn = τn , ηn = τn , al+ j →al+ j+1

µ j+1 →x

satisfy the bilinear equations (2n, j+1) (2n, j) τn

(2n−1, j+1) (2n+1, j) ηn (2n, j) (2n, j+1) + (x − a2n+ j+1 )ηn τn , (2n+1, j+1) (2n+1, j) (2n, j+1) (2n+2, j) (x − µ j+1 )ηn τn+1 = (a2n+ j+2 − µ j+1 )τn ηn+1 (2n+1, j) (2n+1, j+1) + (x − a2n+ j+2 )ηn τn+1 , (2n, j−1) (2n, j) (2n, j) (2n, j−1) (2n+1, j−1) (2n−1, j) ηn τn = ηn τn + (a2n+ j − µ j )τn+1 ηn−1 , (2n+1, j−1) (2n+1, j) (2n+1, j) (2n+1, j−1) ηn τn+1 = ηn τn+1 (2n+2, j−1) (2n, j) + (a2n+ j+1 − µ j )τn+1 ηn .

(x − µ j+1 )ηn

= (a2n+ j+1 − µ j+1 )τn

( j)

(8.28)

(8.29) (8.30)

(8.31)

( j)

These relations and the determinantal expressions for A2n , A2n+1 induce the CTT and GTT (4.1), (4.2) for the FST polynomials, where f 2n, j (n) f 2n, j (0) · · · .. .. . . f 2n, j (n − 1) · · · f 2n, j (2n − 1) (2n, j) n 1 ··· x ηn , (8.32) T2n (x; j) = (−1)n (2n, j) = f 2n, j (0) · · · f 2n, j (n − 1) τn .. .. . . f 2n, j (n − 1) · · · f 2n, j (2n − 2) f 2n+1, j (n) f 2n+1, j (0) · · · .. .. . . f 2n+1, j (n − 1) · · · f 2n+1, j (2n − 1) (2n+1, j) n 1 ··· x ηn T2n+1 (x; j) = (−1)n (2n+1, j) = . (8.33) f 2n+1, j (0) · · · f 2n+1, j (n) τn+1 .. .. . . f 2n+1, j (n) · · · f 2n+1, j (2n) We can also show that bilinear equations for the type B R I I chain have the form (l+2, j−1) (l−1, j) (l, j−1) (l+1, j) (l, j) (l+1, j−1) τn − τn τn+1 + τn σn+1 = 0, (l+1, j−1) (l−2, j) (l−1, j−1) (l, j) (l, j−1) (l−1, j) (al+ j − µ j )τn+1 τn−1 − τn τn + τn σn = 0, (l+2, j−1) (l−1, j) (l, j−1) (l+1, j) (l, j) (l+1, j−1) (al+ j+1 − µ j )τn+1 τn − τn τn+1 + σn τn+1 = 0, (l+1, j−1) (l−2, j) (l−1, j−1) (l, j) (l, j−1) (l−1, j) (al+ j−1 − µ j )τn+1 τn−1 − τn τn + σn τn = 0,

(al+ j − µ j )τn+1

(8.34) (8.35) (8.36) (8.37)


163

and (l, j+1) (l+1, j) σn+1

(x − µ j+1 )ηn

(l−1, j+1) (l+2, j) ηn+1

= (al+ j+1 − µ j+1 )τn

(l+1, j+1) (l, j) ηn ,

+ (x − al+ j+1 )τn+1

(8.38)

(l+2, j−1) (l+1, j) σn

ηn

(l+1, j−1) (l+2, j) ηn

(l+3, j−1) (l, j) ηn−1 .

= τn

+ (al+ j+2 − µ j )(x − al+ j+1 )τn+1

(8.39)

By using (8.34) and (8.35), we obtain related determinantal expressions: ( j)

( j)

A2n A2n+1 + a2n+ j+1 − µ j

= (a2n+ j+1 − µ j ) 1 − (a2n+ j − µ j ) = (a2n+ j+1 − µ j ) ( j)

(2n+2, j−1) (2n−1, j) τn (2n, j−1) (2n+1, j) τn τn+1

τn+1

(2n, j) (2n+1, j−1) σn+1 , (2n, j−1) (2n+1, j) τn τn+1

τn

(8.40)

( j)

A2n−1 A2n + a2n+ j−1 − µ j

= (a2n+ j−1 − µ j ) 1 − (a2n+ j − µ j ) = (a2n+ j−1 − µ j )

(2n+1, j−1) (2n−2, j) τn−1 (2n−1, j−1) (2n, j) τn τn

τn+1

(2n, j−1) (2n−1, j) σn . (2n−1, j−1) (2n, j) τn τn

τn

(8.41)

( j)

In the T2n (x; j) = Pn (x; j) case, we have in (6.1), (6.2) parameters αn = a2n+ j−1 and

( j) βn

( j) Cn

=

( j)

Dn = ( j)

En = ( j)

Fn

=

(2n+2, j−1) (2n−1, j) τn , (2n, j) (2n+1, j−1) τn σn+1

(µ j − a2n+ j )τn+1

(2n, j−1) (2n+1, j) τn+1 , (2n, j) (2n+1, j−1) τn σn+1

τn

(2n−1, j−1) (2n, j) τn , (2n, j−1) (2n−1, j) τn σn

τn

(2n+1, j−1) (2n−2, j) τn−1 , (2n, j−1) (2n−1, j) τn σn

(µ j − a2n+ j )τn+1

and T2n being given by (8.32).

= a2n+ j ,

164

V. P. Spiridonov, S. Tsujimoto, A. S. Zhedanov ( j)

In the T2n+1 (x; j) = Pn (x; j) case, we have in (6.1), (6.2) parameters αn ( j) a2n+ j+1 , βn = a2n+ j , and ( j)

Cn

( j)

( j)

En = ( j)

Fn

(2n+3, j−1) (2n, j) τn , (2n+2, j−1) (2n+1, j) σn+1 τn+1 (2n+1, j−1) (2n+2, j) τn+1 τn+1 , (2n+2, j−1) (2n+1, j) σn+1 τn+1 (2n, j−1) (2n+1, j) τn τn+1 , (2n, j) (2n+1, j−1) σn τn+1 (2n+2, j−1) (2n−1, j) τ τn (µ j − a2n+ j+1 ) n+1 , (2n, j) (2n+1, j−1) σn τn+1

= (µ j − a2n+ j+1 )

Dn =

=

=

τn+2

and T2n+1 being given by (8.33). Finally we obtain the CTT and GTT (6.1), (6.2) for the polynomials associated with the type B R I I chain from the bilinear equations (8.38) and (8.39). For a fixed discrete time, the derived expressions for polynomials Pn (x; j) should be compared with the determinantal representations for them in terms of the generalized moments [28]. Acknowledgements. V.S. is grateful to the Max-Planck-Institut für Mathematik in Bonn for hospitality in the spring of 2005 when some results of the present paper have been obtained. A.Z. is indebted to Kyoto University for hospitality during visit in December of 2005.

References 1. Baker, G.A., Graves-Morris, P.: Padé approximants. Parts I and II. Encyclopedia of Mathematics and its Applications, 13, 14, Reading, MA: Addison-Wesley Publishing Co., 1981 2. Bauer, F.L.: The g-algorithm. J. Soc. Ind. Appl. Math. 8, 1–17 (1960) 3. Brezinski, C.: A general extrapolation algorithm. Numer. Math. 35, 175–187 (1980) 4. Brezinski, C.: History of continued fractions and Padé approximants. Springer Series in Comput. Math. 12, Berlin: Springer-Verlag, 1991 5. Carstensen, C.: On a general ρ-algorithm. J. Comp. Appl. Math. 33, 61–71 (1990) 6. Claessens, G.: A new algorithm for osculatory rational interpolation. Numer. Math. 27(1), 77–83 (1976/77) 7. Darboux, G.: Lecons sur la théorie génerale des surfaces et les applications géométrique du calcul infinitésimal. Paris: Gauthier Villars et Fils, 1889 8. Faddeev, D.K., Faddeeva, V.N.: Computational methods of linear algebra. San Francisco: W. H. Freeman, 1963 9. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian methods in the theory of solitons. Berlin Heidelberg NewYork: Springer-Verlag, 1987 10. Frobenius, G., Stickelberger, L.: Über die Addition und Multiplication der elliptischen Functionen. J. für die reine und angewandte Mathematik 88, 146–184 (1880) 11. Geronimus, J.: On polynomials orthogonal with respect to a given sequence of numbers and a theorem by W. Hahn. Izvestia Acad. Sc. USSR 4, 215–228 (1940) 12. Gibbons, J., Kupershmidt, B.A.: Time discretizations of lattice integrable systems. Phys. Lett. A 165, 105–110 (1992) 13. Hirota, R.: Nonlinear partial difference equations. II. Discrete time Toda equation. J. Phys. Soc. Japan 43, 2074–2078 (1977) 14. Hirota, R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan 50, 3785–3791 (1981) 15. Hirota, R.: Discrete two-dimensional Toda molecule equation. J. Phys. Soc. Japan 56, 4285–4288 (1987) 16. Hirota, R., Tsujimoto, S.: Conserved quantities of a class of nonlinear difference-difference equations. J. Phys. Soc. Japan 64, 3125–3127 (1995)


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17. Ismail, M.E.H., Masson, D.: Generalized orthogonality and continued fractions. J. Approx. Theory 83, 1– 40 (1995) 18. Matveev, V.B., Salle, M.A.: Differential-difference evolution equations. II (Darboux transformation for the Toda lattice). Lett. Math. Phys. 3, 425–429 (1979) 19. Meinguet, J.: On the solubility of the Cauchy interpolation problem. In: Approximation Theory (Proc. Sympos., Lancaster, 1969), London: Academic Press, 1970, pp. 137–163 20. Mukaihira, A., Tsujimoto, S.: Determinant structure of non-autonomous Toda-type discrete integrable systems. J. Phys A: Math. Gen. 39, 779–788 (2006) 21. Ohta, Y., Hirota, R., Tsujimoto, S., Imai, T.: Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy. J Phys. Soc. Japan 62, 1872–1886 (1993) 22. Papageorgiou, V., Grammaticos, B., Ramani, A.: Integrable lattices and convergence acceleration algorithms. Phys. Lett. A 179, 111–115 (1993) 23. Spiridonov, V.P.: Symmetries of factorization chains for the discrete Schrödinger equation. J. Phys. A: Math. Gen. 30, L15–L21 (1997) 24. Spiridonov, V.P.: Solitons and Coulomb plasmas, similarity reductions and special functions. In: Special Functions (Hong Kong Workshop, 1999), River Edge, NJ: World Sci. Publishing, 2000, pp. 324–338 25. Spiridonov, V., Zhedanov, A.: Discrete Darboux transformations, the discrete time Toda lattice, and the Askey-Wilson polynomials. Methods and Applications of Analysis 2(4), 369–398 (1995) 26. Spiridonov, V., Zhedanov, A.: Discrete-time Volterra chain and classical orthogonal polynomials. J. Phys. A: Math. Gen. 30, 8727–8737 (1997) 27. Spiridonov, V., Zhedanov, A.: Spectral transformation chains and some new biorthogonal rational functions. Commun. Math. Phys. 210, 49–83 (2000) 28. Spiridonov, V.P., Zhedanov, A.S.: To the theory of biorthogonal rational functions. RIMS Kokyuroku 1302, 172–192 (2003) 29. Spiridonov, V.P., Zhedanov, A.S.: Elliptic grids, rational functions, and the Padé interpolation. Ramanujan J. 13, 285–310 (2007) 30. Thiele, T.N.: Interpolationsrechnung. Leipzig, 1909 31. Wuytack, L.: An algorithm for rational interpolation similar to the qd-algorithm. Numer. Math. 20, 418– 424 (1972/73) 32. Wynn, P.: Singular rules for certain non-linear algorithms. BIT 3, 175–195 (1963) 33. Zhedanov, A.: Rational spectral transformations and orthogonal polynomials. J. Comput. and Appl. Math. 85, 67–86 (1997) 34. Zhedanov, A.: Biorthogonal rational functions and the generalized eigenvalue problem. J. Approx. Theory 101, 303–329 (1999) 35. Zhedanov, A.: Padé interpolation table and biorthogonal rational functions. Proceedings of RIMS Workshop on Elliptic Integrable Systems (Kyoto, 2004) Communicated by L. Takhtajan


Communications in


Gaussian Limits for Multidimensional Random Sequential Packing at Saturation T. Schreiber1, , Mathew D. Penrose2 , J. E. Yukich3, 1 Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Toruń, Poland.


2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom.

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

3 Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA.

E-mail: [email protected] Received: 28 March 2006 / Accepted: 18 October 2006 Published online: 7 March 2007 – © Springer-Verlag 2007

Abstract: Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume λ is asymptotically normal as λ → ∞. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization. 1. Main Results Given d∈ N and λ ≥ 1, let U1,λ , U2,λ , . . . be a sequence of independent random d-vectors uniformly distributed on the cube Q λ := [0, λ1/d )d . Let S be a fixed bounded closed convex set in Rd with non-empty interior (i.e., a ‘solid’) with centroid at the origin 0 of Rd (for example, the unit ball), and for i ∈ N, let Si,λ be the translate of S with centroid at Ui,λ . So Sλ := (Si,λ )i≥1 is an infinite sequence of solids arriving at uniform random positions in Q λ (the centroids lie in Q λ but the solids themselves need not lie wholly inside Q λ ). Let the first solid S1,λ be packed, and recursively for i = 2, 3, . . . , let the i th solid Si,λ be packed if it does not overlap any solid in {S1,λ , . . . , Si−1,λ } which has already been packed. If not packed, the i th solid is discarded; we sometimes use accepted as a synonym for ‘packed’. This process, known as random sequential adsorption (RSA) with infinite input, is irreversible and terminates when it is not possible to accept additional solids. At termination, we say that the sequence of solids Sλ jams Q λ or saturates Q λ . The jamming number Nλ := Nλ (Sλ ) denotes the number of solids accepted in Q λ at termination. We use the words ‘jamming’ and ‘saturation’ interchangeably in this paper. Research partially supported by the Polish Minister of Scientific Research and Information Technology grant 1 P03A 018 28 (2005-2007) Research supported in part by NSF grant DMS-0203720

168

T. Schreiber, M. D. Penrose, J. E. Yukich

Jamming numbers Nλ arise naturally in the physical, chemical, and biological sciences. They are considered in the description of the irreversible deposition of colloidal particles on a substrate (see the survey [1] and the special volume [20]), hard sphere interactions in point processes (see [26] and [7], Sect. 4.8), adsorption modelling (see [4] and the surveys [9, 25]) and also in the modelling of communication and reservation protocols (see [5, 6]). The extensive body of experimental results related to the large scale behavior of packing numbers stands in sharp contrast with the limited collection of rigorous mathematical results, especially in d ≥ 2. The main obstacle to a rigorous mathematical treatment of the packing process is that the short range interactions of arriving particles create long range spatial dependence, thus turning Nλ into a sum of spatially correlated random variables. Equilibrium systems (where particles are allowed to depart as well as arrive) present a different set of mathematical challenges and are not considered here. In the case where d = 1 and S = [−1/2, 1/2], a famous result of Rényi [21] shows that jamming limit, defined as limλ→∞ λ−1 E Nλ , exists as an integral which evaluates to roughly 0.748; also in this case, Mackenzie [11] shows that limλ→∞ λ−1 Var Nλ exists as an integral which evaluates to roughly 0.03815. Dvoretzky and Robbins [8] show that the jamming numbers Nλ are asymptotically normal as λ → ∞, but their techniques do not address the case d > 1. The above results were established in the 1960s and progress in extending them rigorously to higher dimensions has been slow until recently. Penrose [12] establishes the existence of a jamming limit for any d ≥ 1 and any choice of S, and also [13] obtains a CLT for a related model (monolayer ballistic deposition with a rolling mechanism) but comments in [13] that ‘Except in the case d = 1 ... a CLT for infinite-input continuum RSA remains elusive.’ In the present work we show for any d and any S that λ−1 Var Nλ converges to a positive limit and that Nλ satisfies a central limit theorem, i.e., the fluctuations of the random variable Nλ are indeed Gaussian in the large λ limit. This puts the recent experimental results and Monte Carlo simulations of Quintanilla and Torquato [22] and Torquato (Ch. 11.4 of [26]) on rigorous footing. We also provide a bound on the rate of convergence to the normal, and on the rate of convergence of λ−1 E Nλ to the jamming limit. Throughout N (0, 1) denotes a mean zero normal random variable with variance one. Theorem 1.1. Let Sλ be as above and put Nλ := Nλ (Sλ ). There are constants µ := µ(S, d) ∈ (0, ∞) and σ 2 := σ 2 (S, d) ∈ (0, ∞) such that as λ → ∞ we have |λ−1 E Nλ − µ| = O(λ−1/d ) and λ−1 Var Nλ → σ 2 with Nλ − E Nλ = O((log λ)3d λ−1/2 ). ≤ t − P[N (0, 1) ≤ t] sup P 1/2 (VarN ) t∈R

λ

(1.1)

(1.2)

The process of accepted solids in Q λ induces a natural random point measure νλ on [0, 1]d given by ∞ νλ := δλ−1/d Ui,λ 1{Si,λ is accepted} , (1.3) i=1

where δx stands for the unit point mass at x. It also induces a natural random volume measure νλ on Rd , normalized to have the same total measure as νλ , defined for all Borel A ⊆ Rd by

Gaussian Limits for Multidimensional Random Sequential Packing at Saturation

νλ (A) :=

λ [λ−1/d Si,λ : i ≥ 1, Si,λ is accepted] , A ∩ |S|

169

(1.4)

where | · | denotes Lebesgue measure and λ−1/d A := {λ−1/d x : x ∈ A}. The measure νλ is not necessarily supported by Q 1 due to boundary effects, but for sufficiently large λ it is supported by Q +1 , where we set Q +1 := [−1, 2)d (a fattened version of Q 1 ). Let ν¯ λ := νλ − E[νλ ] and ν¯ λ := νλ − E[νλ ]. Let R(Q +1 ) denote the class of bounded, almost everywhere continuous functions on Q +1 . For f ∈ R(Q +1 ) and µ a signed measure on Rd with finite total mass, let f, µ := R f dµ. The following theorem provides the limit theory (law of large numbers and central limit theorems) for the integrals of test functions f ∈ R(Q +1 ) against the random point measure νλ and the random volume measure νλ induced by the packing process. In particular, it shows that the finite dimensional distributions of the centered packing point measures (¯νλ )λ converge to those of a certain mean zero generalized Gaussian field, namely white noise on Q 1 with variance σ 2 per unit volume, and likewise for the centered packing volume measures (¯νλ )λ . Theorem 1.2. Let µ and σ 2 be as in Theorem 1.1. Then for any f, g in R(Q +1 ), lim λ−1 E [ f, νλ ] = µ f (x)d x λ→∞

[0,1]d

and lim λ

λ→∞

−1

Cov( f, νλ , g, νλ ) = σ

2 [0,1]d

f (x)g(x)d x.

Also, the finite-dimensional distributions of the random field (λ−1/2 f, ν¯ λ , f ∈ R(Q +1 )) converge as λ → ∞ to those of a mean zero generalized Gaussian field with covariance kernel ( f, g) → σ 2 f (x)g(x)d x, f, g ∈ R(Q +1 ). [0,1]d

Moreover, the same conclusions hold with νλ and ν¯ λ replaced by νλ and ν¯ λ respectively. The remainder of this paper is organized as follows. Section 2 provides the general limit theory (weak law of large numbers and a central limit theorem) for spatial measures which satisfy a weak dependency condition termed stabilization and which are defined in terms of point sets in Rd × R+ . Section 3 shows that the correlations of packing status of solids decay exponentially with the distance between them, thus showing that the packing measures νλ satisfy the stabilization criteria of the general results in Sect. 2. Finally, Sect. 4 shows that the convexity hypothesis implies a non-zero limiting variance σ 2 . Remarks. 1. Finite input. Let τ ∈ (0, ∞) and let x denote the smallest integer greater than or equal to x. Inputting only the first λτ solids of the sequence Sλ yields RSA packing of the cube Q λ with finite input. The finite-input packing number, i.e., the total number of solids accepted from S1,λ , S2,λ , . . . , S τ λ ,λ , is asymptotically normal as λ → ∞ with τ fixed. This is proved in [17], and extended in [3] to the case where the spatial coordinates come from a non-homogeneous point process. Packing measures induced by RSA packing with finite input have finite dimensional distributions converging to

170


those of a mean zero generalized Gaussian field with a covariance structure depending upon the underlying density of points [2, 3]. 2. Stabilization. One might expect that the restriction of the packing measure νλ or νλ to a localized region of space depends only on incoming particles with ‘nearby’ spatial locations, in some well-defined sense. This local dependency property is denoted stabilization; when the region of spatial dependency has a diameter with an exponentially decaying tail, it is called exponential stabilization. These notions are spelt out in general terms in Sect. 2. Theorem 2.1 provides a general spatial limit theory for exponentially stabilizing measures; this is an infinite-input analog to known results [3, 14–16] for the finite-input setting, and is of independent interest. A form of stabilization for infinite input RSA was proved in [12], but without any tail bounds. Exponential stabilization in the infinite input setting is perhaps not surprising, but it has been challenging to rigorously establish this key localization feature. In Sect. 3, we show that infinite-input packing measures stabilize exponentially, so that the general results of Sect. 2 are applicable to these measures. 3. Related models in the literature (see e.g. [17]) include cooperative sequential adsorption, RSA with solids of random size or shape, ballistic deposition with a rolling mechanism, and spatial birth-growth models. For all of these, limit theorems in the finite-input setting are discussed in [13]. It seems likely that these can be extended to the infiniteinput setting using the methods of this paper, although we do not discuss any of them in detail. Nor do we consider non-homogeneous point processes as input. 4. Rates of convergence. Even in d = 1, the rate given by Theorem 1.1 is new. Quintanilla and Torquato [22] use Monte Carlo simulations to predict convergence of the distribution function for Nλ to that of a normal, but they do not obtain rates. Penrose and Yukich [19] obtain rates of approximation to the normal for RSA packing with finite (Poisson) input. 5. Numerical values. We do not provide any new analytical methods for computing numerical values of µ and σ 2 when d ≥ 1. For 1 ≤ d ≤ 6, Torquato et al. [27] employ numerical and theoretical methods to estimate µ and other structural characteristics for infinite input RSA packing. 6. Jamming variability. A significant amount of work is needed (see Sect. 4) to show that the limiting variance σ 2 in Theorems 1.1 and 1.2 is non-zero, and we prove this using the following notions. Given L > 0, we shall say that a point set η ⊂ Rd \ [0, L]d is admissible if the translates of S centered at the points of η are non-overlapping. Given such an η, let N [[0, L]d |η] denote the (random) number of solids from the sequence S L d which are packed in [0, L]d given the pre-packed configuration η. In other words, N [[0, L]d |η] arises as the number of solids packed in [0, L]d in the course of the usual infinite input packing process subject to the additional rule that an incoming solid is discarded should it overlap any solid centered at a point of η. Say that the convex body S has jamming variability if there exists a L > 0 such that inf η Var N [[0, L]d |η] > 0 with the infimum taken over admissible point sets η ⊂ Rd \[0, L]d . In Proposition 4.1 we shall show that each bounded convex body S ⊂ Rd with non-empty interior has jamming variability; here we use the assumed convexity of S. 2. Terminology, Auxiliary Results Let R+ := [0, ∞). Given a point (x1 , . . . , xd , t) = (x, t) ∈ Rd × R+ , the first d coordinates of the point will be interpreted as spatial components with the (d + 1)st regarded as


171

a time mark. We shall need to consider point sets which are finite in the spatial directions and locally finite in the time direction, formally defined as follows. Definition 2.1. A point set X ⊂ Rd × R+ is temporally locally finite (or TLF for short) if X ∩ (Rd × [0, t]) is finite for all t > 0. In this section we adapt the general results and terminology from [3, 15, 16, 19] on limit theory for stabilizing spatial measures defined in terms of finite point sets in Rd , to the setting of spatial measures defined in terms of TLF point sets in Rd × R+ (typically obtained as Poisson processes). In subsequent sections, we show that these general results can be applied to obtain the limit theorems for RSA described in Sect. 1. For x ∈ Rd and r > 0, let Br (x) denote the Euclidean ball centered at x of radius r . We abbreviate Br (0) by Br . Given X ⊂ Rd × R+ , a > 0 and y ∈ Rd , we let y + aX := {(y + ax, t) : (x, t) ∈ X }; in other words, scalar multiplication and translation on Rd × R+ act only on the spatial components. For A ⊂ Rd we write y + a A for {y + ax : x ∈ A}; also, we write ∂ A for the boundary of A, and write A+ for A × R+ . Let | · | denote the Euclidean norm, and for nonempty subsets A, A of Rd , set D2 (A, A ) := inf{|x − y| : x ∈ A, y ∈ A }. Let ξ(X , A) be an R+ -valued function defined for all pairs (X , A), where X is a TLF subset of Rd × R+ and A is a Borel subset of Rd . Assume throughout this section that ξ satisfies the following criteria: 1. ξ(·, A) is measurable for each Borel A, 2. ξ(X , ·) is a finite measure on Rd for each TLF X ⊂ Rd × R+ , 3. ξ is translation invariant, that is ξ(i + X , i + A) = ξ(X , A) for all i ∈ Zd , all TLF X ⊂ Rd × R+ , and all Borel A ⊆ Rd , 4. ξ is uniformly locally bounded (or just bounded for short) in the sense that there is a finite constant ||ξ ||∞ such that for all TLF X ⊂ Rd × R+ we have ξ(X , [0, 1]d ) ≤ ||ξ ||∞ .

(2.1)

5. ξ is locally supported, i.e. there exists a constant ρ such that ξ(X , A) = 0 whenever D2 (X , A) > ρ. Note that if ξ(X , ·) is a point measure supported by the points of X , then ξ is locally supported (in fact, in this case we can set ρ = 0). For all λ > 0, let Pλ denote a homogeneous Poisson point process in Rd × R+ with intensity measure λd x × ds, with d x denoting Lebesgue measure on Rd and ds Lebesgue measure on R+ . We put P := P1 . Here and henceforth we shall assume that the point processes, random fields, and random variables considered in this paper are all defined on a common underlying probability space (, F, P). Thermodynamic limits and central limit theorems for functionals in geometric probability are often proved by showing that the functionals satisfy a type of local spatial dependence known as stabilization [3, 14–18, 24] and that will be our goal here as well. First, we adapt the definitions in [3, 14, 15] to the context of measures defined in terms of TLF point sets in Rd . Recall that Q λ denotes the cube [0, λ1/d )d . Definition 2.2. We say ξ is homogeneously stabilizing if there exists an almost surely finite random variable R (a radius of homogeneous stabilization for ξ ) such that for all TLF X ⊂ (Rd \ B R )+ we have ξ((P ∩ (B R )+ ) ∪ X , Q 1 ) = ξ(P ∩ (B R )+ , Q 1 ).

(2.2)

172


We say ξ is exponentially stabilizing if (i) it is homogeneously stabilizing and R can be chosen so that lim sup L→∞ L −1 log P[R > L] < 0, and (ii) for all λ ≥ 1 and all i ∈ Zd , there exists a random variable R := R ξ (i, λ) (a radius of stabilization for ξ at i with respect to P in (Q λ )+ ) such that for all TLF X ⊂ [Q λ\ B R (i)]+ , and all Borel A ⊆ Q 1 , we have ξ ((P ∩ [B R (i) ∩ Q λ ]+ ) ∪ X , i + A) = ξ (P ∩ [B R (i) ∩ Q λ ]+ , i + A)

(2.3)

and moreover the tail probability τ (L) defined for L > 0 by τ (L) :=

sup λ≥1, i∈Zd

P[R ξ (i, λ) > L]

(2.4)

satisfies lim sup L→∞ L −1 log τ (L) < 0. Loosely speaking, R := R ξ (i, λ) is a radius of stabilization if the ξ -measure on i + Q 1 is unaffected by changes to the Poisson points outside B R (i) (but inside Q λ ). When ξ is homogeneously stabilizing, the limit ξ(P, i + Q 1 ) := lim ξ (P ∩ (Br (i))+ , i + Q 1 ) r →∞

exists almost surely for all i ∈ Zd . The random variables (ξ(P, i + Q 1 ), i ∈ Zd ) form a stationary random field. Given ξ , for all λ > 0, all TLF X ⊂ Rd × R+ , and all Borel A ⊂ Rd we let ξ ξλ (X , A) := ξ(λ1/d X , λ1/d A). Define the random measure µλ on Rd by ξ

µλ ( · ) := ξλ (Pλ ∩ Q 1 , ·) ξ

ξ

(2.5)

ξ

and the centered version µλ := µλ − E [µλ ]. By the assumed locally supported property of ξ , µλ is supported by the fattened cube Q +1 := [−1, 2)d for large enough λ. If ξ is stabilizing, define µ(ξ ) := E [ξ(P, Q 1 )] and if ξ is exponentially stabilizing, define σ 2 (ξ ) := Cov [ξ(P, Q 1 ), ξ(P, i + Q 1 )] , i∈Zd

where the sum can be shown to converge absolutely by exponential stabilization and (2.1). The following general theorem provides laws of large numbers and normal approxξ imation results for f, µλ , suitably scaled and centered, for f ∈ R(Q +1 ). This set of results for measures determined by TLF point sets is similar to previously known results for measures determined by finite point sets (Theorem 2.1 of [18], Theorems 2.1 and 2.3 of [3], and Corollary 2.4 of [19]). Theorem 2.1. Suppose that ξ is exponentially stabilizing. Then as λ → ∞, for f and g in R(Q +1 ) we have ξ −1 lim λ E [ f, µλ ] = µ(ξ ) f (x)d x (2.6) λ→∞

[0,1]d

and lim λ

λ→∞

−1

ξ ξ Cov[ f, µλ , g, µλ ]

= σ (ξ ) 2

[0,1]d

f (x)g(x)d x.

(2.7)


Also,

173

ξ

|λ−1 E [µλ (Q +1 )] − µ(ξ )| = O(λ−1/d ).

(2.8)

Moreover, if σ 2 (ξ ) > 0 then

ξ ξ µλ (Q +1 ) − E [µλ (Q +1 )] sup P ≤ t − P[N (0, 1) ≤ t] = O((log λ)3d λ−1/2 ) ξ + 1/2 (Var[µ (Q )]) t∈R λ

1

(2.9) ξ

and the finite-dimensional distributions of the random field (λ−1/2 f, µ¯ λ , f ∈ R(Q +1 )) converge as λ → ∞ to those of a mean zero generalized Gaussian field with covariance kernel 2 ( f, g) → σ (ξ ) f (x)g(x)d x, f, g ∈ R(Q +1 ). [0,1]d

We shall use Theorem 2.1 to prove the results on RSA described in Sect. 1. It seems likely that Theorem 2.1 can also be applied to obtain similar results for the related models listed in Remark 3 of Sect. 1. For some of these, certain generalizations of Theorem 2.1 may be needed; for example, in some cases one may need to allow for the Poisson points to carry independent identically distributed random marks, and in others the boundedness condition (2.1) may need to be relaxed to a moments condition. By appropriate discretization, the proof of the weak law of large numbers (2.6, 2.8) follows from a modification of methods in [15] whereas the proof of (2.7, 2.9) follows a discretized version of the methods in [3, 16, 19]. We refer to the extended version of this paper [23] for complete details. 3. Proof of Stabilization for Packing In this section, we show that the random packing measures νλ and νλ described in Sect. 1 can each be expressed in terms of a suitably defined measure-valued functional ξ of TLF point sets in Rd × R+ , of the general type considered in Sect. 2, applied to a Poisson point process in space-time. Then we show that in both cases the appropriate choice of ξ satisfies the exponential stabilization condition described in Definition 2.2, so that Theorem 2.1 is applicable to this choice of ξ . We defer to the next section the proof that in both cases the appropriate choice of ξ satisfies σ 2 (ξ ) > 0. Throughout we let d S stand for the diameter of S. In our proofs, we shall assume that 2d S < 1. This assumption entails no loss of generality, since once Theorems 1.1 and 1.2 hold under this assumption, the results follow for general S by obvious scaling arguments. Let us say that two points (x, t) and (y, u) in Rd × R+ are adjacent if (x + S) ∩ (y + S) = ∅. Given TLF X ⊂ Rd × R+ , let us first list the points of X in order of increasing time-marks using the lexicographic ordering on Rd as a tie-breaker in the case of any pairs of points of X with equal time-marks. Then consider the points of X in the order of the list; let the first point in the list be accepted, and let each subsequent point be accepted if it is not adjacent to any previously accepted point of X ; otherwise let it be rejected. We call this the usual rule for packing points of X , since it corresponds to the packing rule of Sect. 1 with the input ordering determined by time-marks. Let A(X ) denote the subset of X consisting of all accepted points when the points of X are packed according to the usual rule.

174


We consider two specific measure-valued functionals ξ ∗ and ξ on TLF point sets in × R+ , of the general type considered in Sect. 2, which are defined as follows. For any bounded Borel A ⊂ Rd , recall that A+ := A × R+ . For any TLF point set X ⊂ Rd × R+ , let ξ ∗ (X , A) be the number of points of A(X ) which lie in A+ , and with | · | denoting Lebesgue measure, let ⎛ ⎞ ξ (X , A) := |S|−1 A ∩ ⎝ (x + S)⎠ . (x,t)∈A(X ) Rd

Then ξ ∗ and ξ are clearly translation invariant, and are bounded (i.e., satisfy (2.1)), since only a bounded number of solids can be packed in any fixed cube. Recall that Pλ denotes a homogeneous Poisson point process of intensity λ on Rd × R+ , and P = P1 . Assume Pλ is obtained from P by Pλ := λ−1/d P. For all λ > 0, recall the definition of ξλ in Sect. 2, and define the random measures ξ∗

ξ

µλ ( · ) := ξλ∗ (Pλ ∩ (Q 1 )+ , ·) and µλ ( · ) := ξλ (Pλ ∩ (Q 1 )+ , ·). ξ∗

ξ∗

Let Nλ denote the total mass of µλ , i.e. ξ∗

ξ∗

Nλ := µλ (Pλ ∩ [0, 1]d+ , [0, 1]d ). ξ∗

ξ

Then µλ and µλ are the random packing point measure and the random packing volume measure, respectively, corresponding to the random sequential adsorption process obtained by taking the spatial locations of the points of P ∩ (Q λ )+ , in order of increasing time-mark, as the input sequence. Since these spatial locations are independent and uniformly distributed on Q λ , we have the distributional equalities ξ∗ D

ξ D

ξ∗ D

µλ = νλ , µλ = νλ , and Nλ = Nλ ,

(3.1)

where the measures νλ and νλ are given in (1.3) and (1.4) and the jamming number Nλ is also given in Sect. 1. We show in Lemmas 3.5 and 3.7 below that both ξ ∗ and ξ are exponentially stabilizing, and therefore we can apply Theorem 2.1 to either of these choices of ξ . To proceed with the proof of exponential stabilization, consider a partition of Rd into translates of the unit cube C := Q 1 = [0, 1)d . It is convenient to index these translates + d as Ci , i := (i 1 , . . . , i d ) ∈ Z , with+ Ci := (i 1 , . . . , i d ) + C. We shall write Ci := j∈Zd , ||i− j||∞ ≤1 C j , that is to say Ci is the union of Ci and its neighboring cubes. We also consider the moat Ci := Ci+ \Ci . We need further terminology. Given TLF X ⊂ Rd × R+ , and given A ⊂ Rd , we say that X fully packs the region A if every point in A+ is adjacent to at least one point of A(X ). For t > 0, we say X fully packs A by time t if X ∩ (Rd × [0, t]) fully packs A. Given B ⊆ Rd , we say that a finite point configuration X ⊂ (B ∩ Ci+ )+ is maximal or strongly saturates the cube Ci in B if for each TLF external configuration Y ⊂ (B \Ci+ )+ , X ∪ Y fully packs the region B ∩ Ci (the existence of maximal configurations is guaranteed by Lemmas 3.1 and 3.3 below). We shall be interested in strong saturation of Ci in B when B = Rd or when B = Q λ . The reason for our interest is this: If we knew that there was a constant τ < ∞ such that P ∩ (C0+ × [0, τ ]) strongly saturated C0 in Rd almost surely, then points in P with time marks exceeding τ would have no bearing on the packing status of points in P ∩ (C0 )+ .


175

Thus, to check stabilization of ξ at 0 it would be enough to replace P by the Poisson point process P ∩ (Rd × [0, τ ]), and follow the stabilization arguments for packing with finite Poisson input (Sect. Four of [17]). While clearly no such constant τ exists, we shall show in Lemma 3.3 that a finite random τ exists. We say that X locally strongly saturates Ci if for each η ⊆ X ∩ ( Ci )+ , the point set (X ∩ (Ci )+ ) ∪ η fully packs Ci . The following lemma shows that local strong saturation implies strong saturation. Lemma 3.1. Suppose X ⊂ (Ci+ )+ is TLF and locally strongly saturates Ci . Then for any B ⊆ Rd with Ci ⊆ B, X ∩ B strongly saturates Ci in B. Proof. Let Y ⊂ (B \ Ci+ )+ be TLF. Let η := A((X ∩ B+ ) ∪ Y) ∩ ( Ci )+ . We claim that A((X ∩ B+ ) ∪ Y) ∩ (Ci+ )+ = A((X ∩ (Ci )+ ) ∪ η).

(3.2)

Indeed, considering each point of (X ∩(Ci )+ )∪η in the usual temporal order, we see that the decision on whether to accept is the same for these points whether we are applying the usual packing rule to (X ∩ B+ ) ∪ Y or to (X ∩ (Ci )+ ) ∪ η. Since we assume X locally strongly saturates Ci , (X ∩ (Ci )+ ) ∪ η fully packs Ci , and so by (3.2), (X ∩ B+ ) ∪ Y fully packs Ci . We will use one more auxiliary lemma. Lemma 3.2. With probability 1, P has the property that for any η ⊆ P ∩ ( C0 )+ , there exists T < ∞ such that the point set (P ∩ (C0 )+ ) ∪ η fully packs C0 by time T . Proof. Suppose that for each rational hypercube Q contained in C0 , P ∩ Q + = ∅; this event has probability 1. Take η ⊂ P ∩ ( C0 )+ . Let A := A((P ∩ (C0 )+ ) ∪ η). Clearly A is finite. Let V be the set of x ∈ C0 such that (x, 0) does not lie adjacent to any point of A. Then V is open in C0 (because we assume S is closed) and if it is non-empty, it contains a rational cube contained in C0 so that V+ contains a point of P ∩ (C0 )+ . But then this point should have been accepted so there is a contradiction. Hence V is empty and since A is finite this shows that C0 is fully packed within a finite time. For i ∈ Zd , let Ti := Ti (P) denote the time till local strong saturation, defined to be the smallest t ∈ [0, ∞] such that Ci is locally strongly saturated by the point set (P ∩ (Ci+ )+ ) ∩ (Rd × [0, t]) (and set Ti = ∞ if no such t exists). Clearly, Ti , i ∈ Zd , are identically distributed random variables depending only on P ∩ (Ci+ )+ . In particular, (Ti , i ∈ Zd ) forms a 2-dependent random field, meaning that Ti is independent of (T j , j − i∞ > 2) for each i ∈ Zd . We can now prove the key result that T0 is almost surely finite. Lemma 3.3. It is the case that P[T0 = ∞] = 0. Proof. Suppose that T0 = ∞. Then for each positive integer τ there exists ητ ⊆ P ∩ ( C0 )+ such that (P ∩ (C0 )+ ) ∪ ητ does not fully pack C0 by time τ . Assume P ∩ ( C0 )+ is locally finite (this happens almost surely). Then P ∩ ( C0 × (1) [0, 1]) is finite so that we can take a strictly increasing subsequence (τn )n≥1 of the integers τ , along which ητ (1) ∩( C0 ×[0, 1]) is the same for all n. Then we can take a further n

(2)

(1)

strictly increasing subsequence (τn )n≥1 of (τn )n≥1 along which ητ (2) ∩( C0 ×[0, 2]) n

176


is the same for all n. Repeating this procedure and using Cantor’s diagonal argument, i.e. taking τn := τn(n) for all n, we obtain a subsequence τn tending to infinity, and a limit set η ⊂ ( C0 × R+ ), such that for all k, it is the case that ητn ∩ ( C0 × [0, k]) = η ∩ ( C0 × [0, k])

(3.3)

for all but finitely many n. Let k > 0, and choose n to be large enough so that τn ≥ k and such that (3.3) holds. Then the point set (P ∩ (C0 )+ ) ∪ ητn does not yet fully pack C0 by time τn , and therefore (P ∩ (C0 )+ ) ∪ η does not yet fully pack C0 by time k. Since (P ∩ (C0 )+ ) ∪ η does not yet fully pack C0 by time k for any k, we are in the complement of the event described in Lemma 3.2. Thus by that result, the event {T0 = ∞} is contained in an event of probability zero, which completes the proof of Lemma 3.3. Using Lemma 3.3, we can now prove that ξ ∗ and ξ , defined at the start of this section, satisfy the first part of exponential stabilization (exponential decay of the tail of R ). To this end, consider the following {0, 1}-valued random field (πi , i ∈ Zd ) on (, F, P): 1, if Ti ≤ T ∗ , πi := 0, otherwise, where T ∗ is a constant to be specified below. Clearly the field (πi )i∈Zd inherits the 2-dependence property of (Ti )i∈Zd . We shall use the following auxiliary lemma showing that if T ∗ is chosen so that P[πi = 0] is small enough then the probabilities of observing long paths of zeros in πi decay exponentially in the sense made precise below. Given L > 0, let E 1 (L) be the event that there is a path of zeros from some site i ∈ {−1, 0, 1}d to the complement of B L/2−√d in the random field (πi , i ∈ Zd ). More formally, E 1 (L) is the event that there exists a sequence i 0 , i 1 , i 2 , . . . , i n , such that (a) i 0 ∈ {−1, 0, 1}d , and (b) i n ∈ / B L/2−2√d , and (c) for j = 1, . . . , n, i j ∈ Zd and i j − i j−1 ∞ = 1 and πi j = 0. Lemma 3.4. There exists δ ∗ > 0 such that if T ∗ is chosen large enough so that P[πi = 0] ≤ δ ∗ , the probability of event E 1 (L) decays exponentially in L, i.e. limsup L→∞ L −1 log P[E 1 (L)] < 0. Proof. This is a direct consequence of the product measure domination result of Liggett, Schonmann and Stacey ((7.65) of [10]), combined with the exponential decay of the cluster radius in the subcritical regime of Bernoulli percolation, see e.g. Sect. 5.2 of [10]. Alternatively, the lemma can be proved directly by a standard path-counting argument. Lemma 3.5. There exists a positive constant K 1 such that for either ξ = ξ ∗ or ξ = ξ , there is a stabilization radius R as described in Definition 2.2, satisfying P R > L ≤ K 1 exp(−L/K 1 ), ∀L > 0. Proof. Choose δ ∗ > 0 as given by Lemma 3.4. Using Lemma 3.3, take T ∗ > 0 such that P[T0 ≥ T ∗ ] ≤ δ ∗ . Let us say that the cube Ci is T ∗ -saturated iff Ti ≤ T ∗ , that is to say iff πi = 1. By Lemma 3.1, if Ci is T ∗ -saturated then for any B ⊆ Rd with Ci ⊆ B, P ∩ ([Ci+ ∩ B] × [0, T ∗ ]) strongly saturates Ci in B.


177

We declare a point (x, t) ∈ P ∩ (Ci )+ to be causally relevant if either • •

πi = 0, or πi = 1 and t ≤ T ∗ .

Otherwise the point x ∈ P ∩ (Ci )+ is declared causally irrelevant. We now argue as follows, directly adapting the oriented percolation technique introduced in Sect. Four of [17]. We convert the collection of points P (in Rd × R+ ) into a directed graph by providing a directed connection from (y, s) to (x, t) whenever |y − x| ≤ 2d S and s < t and, moreover, both (x, t) and (y, s) are causally relevant. By the causal cluster Cl[(x, t); P] of (x, t) ∈ P we understand the set of all causally relevant points (y, s) of P such that there is a directed path from (y, s) to (x, t) (referred to as a causal chain for (x, t) in the sequel). Necessarily the points in the causal cluster for (x, t) have time mark at most t. For each (x, t) ∈ P we define the causal cube cluster of (x, t) in Rd by ¯ Cl[(x, t); P] := [C +j : (C j )+ ∩ Cl[(x, t); P] = ∅] and for each i ∈ Zd we define its causal cube cluster as the union of clusters given by ¯ ¯ Cl[i; P] := Cl[(x, t); P]. (3.4) (x,t)∈P ∩(Ci+ )+

The significance of causal cube clusters is as follows. First, we assert that the pack¯ ing status of a given point (x, t) is unaffected by changes to P outside (Cl[(x, t); P])+ . Indeed, viewing the directed connections as potential direct interactions between overlapping solids in the course of the sequential packing process, we can repeat the corresponding argument from Lemma 4.1 in [17], adding the extra observation that causally irrelevant points will not be accepted regardless of the outside packing configuration and hence do not have to be taken into account. Similarly, the packing status of the totality of points falling within distance d S of the cube Ci can only be affected by the status of ¯ points with spatial locations falling in the causal cube cluster Cl[i; P]. Consequently, we see that for either ξ = ξ ∗ or ξ = ξ , we can define a radius of stabilization by ¯ R := diam(Cl[0; P]).

(3.5)

We need to show that R is almost surely finite and has an exponentially decaying tail. For L > 0, let E 1 (L) be as in Lemma 3.4. For i ∈ Zd , let E 2 (L , i) be the event that there exists (x, t) ∈ P ∩ (Ci )+ , such that t ≤ T ∗ and there exists a causal chain for (x, t) which starts at some point of P \(B L−2√d )+ . Define the event E 2 (L) :=

E 2 (L , i) : i ∈ Zd , Ci ∩ B L/2 = ∅ .

Then we assert that the event {R > L} is contained in E 1 (L) ∪ E 2 (L). Indeed, if E 2 (L) does not occur, then for any causal chain for any (x, t) ∈ P ∩ (C0+ )+ starting outside (B L−2√d )+ , all points in the causal chain of (x, t) lying inside (B L/2 )+ must have timecoordinate greater than T ∗ ; if also E 1 (L) does not occur, at least one of these points must lie in a cube which is T ∗ -saturated, and therefore be causally irrelevant, so in fact there is no causal chain for any (x, t) ∈ P ∩ (C0+ )+ starting outside (B L−2√d )+ . Hence, ¯ Cl[0, P] ⊆ B L , so that R ≤ L.

178


By the choice of δ ∗ and T ∗ as in Lemma 3.4 we have exponential decay of P[E 1 (L)]. Since T ∗ is fixed, we can use the methods of [17] for finite (Poisson) input packing, in particular the argument leading to Lemma 4.2 in [17], to see that there is a constant K 3 such that P[E 2 (L , i)] ≤ K 3 exp(−L/K 3 ) for all i ∈ Zd ∩ B L/2 . Since the number of such i is only O(L d ), we see that P[E 2 (L)] also decays exponentially in L, and hence so does P[E 1 (L)] + P[E 2 (L)]. Since the event {R > L} is contained in E 1 (L) ∪ E 2 (L), Lemma 3.5 is proved. To finish checking that ξ ∗ and ξ satisfy the conditions for Theorem 2.1, we consider strong saturation, not only of unit cubes but of cubes of slightly less than unit size. Let Q +ζ denote the cube [−ζ 1/d , 2ζ 1/d )d , i.e. the cube of side 3ζ 1/d concentric with Q ζ . Let us say that Q ζ is locally strongly saturated by a finite point set X ⊂ (Q +ζ )+ if for every η ⊆ X ∩ (Q +ζ \ Q ζ )+ , the point set (X ∩ (Q ζ )+ ) ∪ η fully packs Q ζ . Lemma 3.6. Given δ > 0, there exist constants ε > 0 and t0 < ∞ such that for all ζ ∈ [1 − ε, 1], P[P ∩ (Q +ζ × [0, t0 ]) locally strongly saturates Q ζ ] > 1 − δ.

(3.6)

Proof. By Lemma 3.3, we can choose t0 such that P ∩ (Q +1 × [0, t0 ]) locally strongly saturates Q 1 , with probability at least 1 − δ/2. Having chosen t0 in this way, we can then choose ε, with 2d S < (1 − ε)1/d , so that for any ζ ∈ [1 − ε, 1], P[P ∩ ((Q 1 \ Q ζ ) × [0, t0 ]) = ∅] < δ/2. For ζ < 1 with 2d S < ζ 1/d , if P ∩ (Q +1 × [0, t0 ]) strongly saturates Q 1 , and P ∩ ((Q 1 \ Q ζ ) × [0, t0 ]) is empty, then P ∩ (Q +ζ × [0, t0 ]) strongly saturates Q ζ . Hence, the preceding probability estimates complete the proof. Lemma 3.7. There exists a positive constant K 4 such that for either ξ = ξ ∗ or ξ = ξ , there is a family of stabilization radii R(i, λ) := R ξ (i, λ), defined for λ ≥ 1 and i ∈ Zd as described in Definition 2.2, which satisfy sup λ≥1,i∈Zd

P [R(i, λ) > L] ≤ K 4 exp(−L/K 4 ).

(3.7)

Proof. First let us restrict attention to λ with λ1/d ∈ N. Adapting notation from the proof of Lemma 3.5, for (x, t) ∈ P ∩ (Q λ )+ we let Cl[(x, t); P ∩ (Q λ )+ ] denote the set of all causally relevant points (y, s) of P ∩ (Q λ )+ such that there is a directed path from (y, s) to (x, t), with all points in the path lying inside (Q λ )+ . Then define the causal cube cluster in Q λ for (x, t) by ¯ Cl[(x, t); P ∩ (Q λ )+ ] := [C +j ∩ Q λ : (C j )+ ∩ Cl[(x, t); P ∩ (Q λ )+ ] = ∅] and for i ∈ Zd by ¯ Cl[i; P ∩ (Q λ )+ ] :=

¯ Cl[(x, t); P ∩ (Q λ )+ ].

(x,t)∈P ∩(Q λ ∩Ci+ )+

Define ¯ R(i, λ) := diam(Cl[i; P ∩ (Q λ )+ ]), λ1/d ∈ N.

(3.8)


179

Then for i ∈ Zd , the packing statuses of points of P ∩ (Ci+ ∩ Q λ )+ are unaffected by changes to P ∩ (Q λ )+ in the region (Q λ\B R(i,λ) (i))+ , by the same argument as in proof of Lemma 3.5. Here we are using the fact that λ1/d ∈ Z, and that if Ci ⊂ Q λ is T ∗ -saturated then Ci is strongly saturated in Q λ by P ∩ (Q λ × [0, T ∗ ]) (Lemma 3.1). Thus, R(i, λ) serves as a radius of stabilization in the sense of Definition 2.2 (for either ¯ ¯ ξ ∗ or ξ ). Moreover, Cl[i; P ∩ (Q λ )+ ] ⊆ Cl[i; P], and so with K 1 in as in the proof of Lemma 3.5 we have P[R(i, λ) > L] ≤ K 1 exp(−L/K 1 ), uniformly over i, λ with λ1/d ∈ N. Now suppose λ1/d ∈ / N. In this case, instead of dividing Q λ into cubes of side 1, some of which would not fit exactly, we divide Q λ into cubes of side slightly less than 1, which do fit exactly, and repeat the above argument. More precisely, we modify the proof of Lemma 3.5. With δ2 as in that proof, we use Lemma 3.6 to choose constants ε > 0 and T ∗ < ∞ (with max(2d S , 1/2) < (1 − ε)1/d ) in such a way that for any ζ ∈ [1 − ε, 1] we have P[P ∩ (Q +ζ × [0, T ∗ ]) locally strongly saturates Q ζ ] > 1 − δ2 . With ε thus fixed, for all large enough λ we can choose ζ = ζ (λ) ∈ [1 − ε, 1] in such a way that λ1/d /ζ 1/d is an integer. Partitioning Rd into cubes Ci of volume ζ , we can then follow the argument already given for the case λ1/d ∈ N, using the fact that each of the unit cubes i + Q 1 , for which we need to check conditions in Theorem 2.1, is contained in the union of at most 2d cubes in the partition {C j }. 4. Proof of Volume Order Variance Growth At the end of this section, we complete the proofs of Theorems 1.1 and 1.2. First, we need to show that the limiting variance σ 2 (S, d) is non-zero for all d and all S. This is achieved by Proposition 4.1 and Lemma 4.1 below. The first of these results establishes that any convex S ⊆ Rd with nonempty interior satisfies jamming variability (as defined in Remark 6, Sect. 1), and the second establishes that this is sufficient to guarantee that ξ∗ σ 2 (S, d) > 0. Recall from (3.1) that we can work just as well with Nλ as with Nλ . Proposition 4.1. The convex body S has jamming variability. Proof. Given S, for all x ∈ Rd define x := sup{a ≥ 0 : (x + aS) ∩ aS = ∅}. It is straightforward to verify that · is a norm on Rd , using the convexity of S to verify the triangle inequality. For nonempty A ⊂ Rd , and x ∈ Rd , write D(x, A) for inf{x − y : y ∈ A}. By our earlier assumption that 2d S < 1 we have x > x∞ for all x ∈ Rd . For L ⊂ Rd , we shall say L is packed if x − y ≥ 1 for all x ∈ L, y ∈ L, and that L is maximally packed if it is packed and D(w, L) < 1, ∀w ∈ Rd .

(4.1)

We shall say L is a periodic set if for all x ∈ L and z ∈ Zd we have x + z ∈ L. Let L be a maximally packed periodic subset of Rd (it is not hard to see that such an L exists). Then the function x → D(x, L) is a continuous function on Rd that is periodic (i.e., D(x, L) = D(x + z, L) for all x ∈ Rd , z ∈ Zd ). Hence the range of this

180


function is the continuous image of the compact torus Rd /Zd , and so is compact. Hence by (4.1) we have β := sup{D(w, L) : w ∈ Rd } < 1. Then for x ∈ Rd and α > 0, by scaling D(x, αL) = α D(α −1 x, L) ≤ αβ.

(4.2)

Choose δ > 0 such that β(1 + 6δ) < 1 − 2δ. For i = 1, 2, let Li := (1 + 3iδ)L. By (4.2) and the choice of δ we have for all x ∈ Rd and i = 1, 2 that D(x, Li ) < 1 − 2δ.

(4.3)

Let c1 denote the number of points of L in [0, 1)d . Denote by Box(L) the hypercube [−L/2, L/2]d . For i = 1, 2, let n i (L) denote the number of points of Li in Box(L − 4). Then as L → ∞, for i = 1, 2 we have n i (L) ∼ c1 (1 + 3δi)−d L d .

(4.4)

Let n 3 (L) denote the maximum integer m such that there exists a packed subset of Box(L)\Box(L − 6) with m elements. Then there is a finite constant c2 such that for all L ≥ 6 we have n 3 (L) ≤ c2 L d−1 .

(4.5)

By (4.4) and (4.5), we can choose L 0 such that for L ≥ L 0 we have n 3 (L) < n 1 (L) − n 2 (L).

(4.6)

For x ∈ Rd and r > 0, set B˜ r (x) := {y ∈ Rd : y − x ≤ r } (a ball of radius r using the norm · ). For bounded A ⊂ Rd , let T (A) denote the time of the first Poisson arrival in A, i.e. set T (A) := inf{t : P ∩ (A × {t}) = ∅}, with the convention that the infimum of the empty set is ∞. Fix L ≥ L 0 , and for i = 1, 2 define the event E i by E i := max{T ( B˜ δ (x)) : x ∈ Li ∩ Box(L − 4)} < T Box(L)\∪x∈Li ∩Box(L−4) B˜ δ (x) . Let i = 1 or i = 2. If y, y are distinct points of Li then y − y ≥ 1 + 3δ. Hence, if also w ∈ B˜ δ (y) and w ∈ B˜ δ (y ), then w − w ≥ 1 + δ by the triangle inequality. Moreover, for x ∈ Rd , by (4.3) and the triangle inequality we can find y = y(x) ∈ Li such that x −w ≤ 1−δ for all w ∈ B˜ δ (y). Hence, if E i occurs then the set of accepted points (i.e., centroids of accepted shapes) of the infinite input packing process on Box(L) induced by P with arbitrary external pre-packed configuration η in Rd \Box(L), includes one point from each B˜ δ (x), x ∈ Li ∩ Box(L − 4), and also contains no other points from Box(L − 6). Thus for any pre-packed configuration η in Rd \Box(L), if E 1 occurs the number of accepted points in Box(L) is at least n 1 (L), and if E 2 occurs the number of accepted


181

points is at most n 2 (L) + n 3 (L). Also, the probabilities P[E 1 ] and P[E 2 ] are strictly positive and do not depend on η. By (4.6), it follows that there is a constant ε > 0, ξ∗ independent of η, such that Var[N L d (Box(L))|η] ≥ ε. Thus we have established the required jamming variability. ξ∗

Lemma 4.1. It is the case that lim inf λ→∞ λ−1 Var[Nλ ] > 0. Proof. By Proposition 4.1, there exists L > 0 such that inf η Var N [[0, L]d |η] > 0, where the infimum is over all admissible η ⊂ Rd \[0, L]d . We consider λ with λ1/d /(L + 4) ∈ N. We subdivide the cube Q λ into n(λ) := λ/(L + 4)d equal-sized sub-cubes C˜ 1,λ , C˜ 2,λ , . . . , C˜ n(λ),λ arising as translates of Box(L + 4) centered at x1,λ , . . . , xn(λ),λ − respectively. For 1 ≤ i ≤ n(λ), let C˜ i,λ be the translate of Box(L) centered at xi,λ , and − ). let Mi,λ be the translate of Box(L + 2)\Box(L) centered at xi,λ (a ‘moat’ around C˜ i,λ Using terminology from Sect. 3, let Fi,λ be the event that the point set P ∩ (Mi,λ )+ fully packs Mi,λ by time 1, and let G i,λ be the event that P ∩ ((C˜ i,λ\ Mi,λ ) × [0, 1]) is empty. Let E i,λ := Fi,λ ∩ G i,λ . Then p := P[E i,λ ] satisfies p > 0, and does not depend on i or λ. Observing that the events E i,λ , 1 ≤ i ≤ n(λ), are independent (the cubes C˜ i are disjoint), denote the (random) set of indices for which E i,λ occurs by I (λ) := {i 1 , . . . , i K (λ) }. Then E [K (λ)] = pn(λ). Conditional on the event E i,λ , the packing − process inside C˜ i,λ has a particularly simple form - before time 1 there are no points in − − ˜ undergo the packing Ci,λ , and after that time the newly arriving solids centered in C˜ i,λ process according to the usual rules with the additional restriction that a solid overlapping another one packed in Mi,λ before time 1 is rejected. Note that for i ∈ I (λ), no new solids are accepted in Mi,λ after time 1 and, moreover, the acceptance times of solids accepted in Mi,λ before time 1 have no influence on the behavior of the packing process − in C˜ i,λ after time 1; only their spatial locations matter. For a configuration η of accepted points (only spatial locations taken into account) in Mi,λ , the process described above − will be referred to as packing in C˜ i,λ in the presence of the pre-packed configuration η. Let Mλ be the sigma-algebra generated by the points of P ∩ (Q λ × [0, 1]), i.e. the Poisson arrivals up to time 1. Event E i,λ is Mλ -measurable, for each i. By the conditional variance formula we have ∗ ∗ ∗ ξ ξ ξ Var Nλ = E Var Nλ |Mλ + Var E Nλ |Mλ ∗ ξ ≥ E Var Nλ Mλ ⎡ ⎛ ⎞ ⎤ ξ∗ ∗ ∗ ξ ξ − − ⎠ = E Var ⎣ Nλ [C˜ i,λ ] + ⎝ Nλ − Nλ [C˜ i,λ ] Mλ ⎦ , i∈I (λ) i∈I (λ) ∗

ξ − − − where we set Nλ [C˜ k,λ ] := ξ ∗ (P ∩ Q λ , C˜ k,λ ), the number of solids packed in C˜ k,λ . Conditionally on Mλ , the packing processes after time 1 over different sub-cubes − , i ∈ I (λ), are independent of each other and of the packing process after time 1 in C˜ i,λ − . Hence, Q λ\∪i∈I (λ) C˜ i,λ ∗ ξ ξ∗ − Var Nλ ≥ E Var[Nλ [C˜ i,λ ] | Mλ ] ≥ E [K ] inf VarN [[0, L]d |η], i∈I (λ)

η

182


where the infimum is taken over all admissible configurations η outside [0, L]d , and where N [[0, L]d |η] stands for the number of solids packed in [0, L]d in the presence of the pre-packed configuration η. By Proposition 4.1, this infimum is strictly positive, and Lemma 4.1 follows. Proof of Theorems 1.1 and 1.2. Let ξ be ξ ∗ as defined in Sect. 3. Then Lemmas 3.5 and 3.7 show that ξ = ξ ∗ satisfies the exponential stabilization conditions in Theorem 2.1, so it satisfies the conclusions (2.6), (2.7) and (2.8) of that result. The conclusion (2.8) gives us (1.1) of Theorem 1.1. Also, by putting f ≡ g ≡ 1 on Q +1 and using (2.7), we obtain the variance convergence λ−1 Var Nλ → σ 2 asserted in Theorem 1.1. By Lemma 4.1, we may therefore deduce that σ 2 > 0. Hence we may apply the last part of Theorem 2.1 to obtain the rest of the conclusions in Theorem 1.2 as they pertain to νλ ; also the conclusion (2.9) of Theorem 2.1 gives us (1.2). To get the same results for ν , we argue similarly with ξ = ξ . We need to check that the limiting means and variances are the same, i.e. µ(ξ ) = µ(ξ ∗ ) and σ 2 (ξ ) = σ 2 (ξ ∗ ). ξ ξ∗ To see this, note that if f ≡ 1 on Q +1 , then f, µλ = f, µλ so application of (2.6) to this choice of f yields ξ

ξ∗

µ(ξ ) = lim λ−1 E [ f, µλ ] = lim λ−1 E [ f, µλ ] = µ(ξ ∗ ), λ→∞

λ→∞

and a similar argument using (2.7) shows that σ 2 (ξ ) = σ 2 (ξ ∗ ).

References 1. Adamczyk, Z., Siwek, B., Zembala, M., Belouschek, P.: Kinetics of localized adsorption of colloid particles. Adv. in Colloid and Interface Sci. 48, 151–280 (1994) 2. Baryshnikov, Yu., Yukich, J.E.: Gaussian fields and random packing. J. Stat. Phys. 111, 443–463 (2003) 3. Baryshnikov, Yu., Yukich, J.E.: Gaussian limits for random measures in geometric probability. Annals Appl. Prob. 15, 213–253 (2005) 4. Bartelt, M.C., Privman, V.: Kinetics of irreversible monolayer and multilayer sequential adsorption. Internat. J. Mod. Phys. B 5, 2883–2907 (1991) 5. Coffman, E.G., Flatto, L., Jelenković, P.: Interval packing: the vacant interval distribution. Annals of Appl. Prob. 10, 240–257 (2000) 6. Coffman, E.G., Flatto, L., Jelenković, P., Poonen, B.: Packing random intervals on-line. Algorithmica 22, 448–476 (1998) 7. Diggle, P.J.: Statistical Analysis of Spatial Point Patterns. London: Academic Press 1983 8. Dvoretzky, A., Robbins, H.: On the “parking” problem. MTA Mat Kut. Int. K¨zl., (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences), 9, 209–225 (1964) 9. Evans, J.W.: Random and cooperative adsorption. Rev. Mod. Phys. 65, 1281–1329 (1993) 10. Grimmett, G.: Percolation, Second Edition, Berlin: Springer 1999 11. Mackenzie, J.K.: Sequential filling of a line by intervals placed at random and its application to linear adsorption. J. Chem. Phys. 37(4), 723–728 (1962) 12. Penrose, M.D.: Random parking, sequential adsorption, and the jamming limit. Commun. Math. Phys. 218, 153–176 (2001) 13. Penrose, M.D.: Limit theorems for monolayer ballistic deposition in the continuum. J. Stat. Phys. 105, 561–583 (2001) 14. Penrose, M.D.: Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Prob. 33, 1945–1991 (2005) 15. Penrose, M.D.: Laws of large numbers for random measures in geometric probability. Preprint, 2005 16. Penrose, M.D.: Gaussian limits for random geometric measures. Preprint, 2005 17. Penrose, M.D., Yukich, J.E.: Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12, 272–301 (2002) 18. Penrose, M.D., Yukich, J.E.: Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13, 277–303 (2003)


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19. Penrose, M.D., Yukich, J.E.: Normal approximation in geometric probability. In: Stein’s Method and Applications, Lecture Note Series, Institute for Mathematical Sciences, National University of Singapore, 5, A.D. Barbour, Louis H.Y. Chen, eds., 2005, pp. 37–58. Also available electronically from http://arxiv.org/list/math.PR/0409088, 2004 20. Privman V.: Adhesion of Submicron Particles on Solid Surfaces. In: A Special Issue of Colloids and Surfaces A 165, edited by V. Privman, 2000 21. Rényi, A.: On a one-dimensional random space-filling problem, MTA Mat Kut. Int. K¨zl., (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences) 3, 109–127 (1958) 22. Quintanilla, J., Torquato, S.: Local volume fluctuations in random media. J. Chem. Phys. 106, 2741– 2751 (1997) 23. Schreiber, T., Penrose, M.D., Yukich J.E.: Gaussian limits for multidimensional random sequential packing at saturation (extended version). http://arxiv.org/list/math.PR/0610680, 2006 24. Schreiber, T., Yukich, J.E.: Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs. Stochastic Processes and Their Applications 115, 1332–1356 (2005) 25. Talbot, J., Tarjus, G., Van Tassel, P.R., Viot, P.: From car parking to protein adsorption: an overview of sequential adsorption processes. Colloids and Surfaces A 165, 287–324 (2000) 26. Torquato, S.: Random Heterogeneous Materials, Springer Interdisciplinary Applied Mathematics, New York: Springer-Verlag 2002 27. Torquato, S., Uche, O.U., Stillinger, F.H.: Random sequential addition of hard spheres in high Euclidean dimensions. Phys. Rev. E 74, 061308 (2006) Communicated by J.L. Lebowitz


Communications in


The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices Delphine Féral1 , Sandrine Péché2 1 Institut de Mathématiques, Laboratoire de Statistique et Probabilités, Université Paul Sabatier,

31062 Toulouse Cedex 9, France. E-mail: [email protected]

2 Institut Fourier BP 74, 100 Rue des maths, 38402 Saint Martin d’Heres, France.

E-mail: [email protected] Received: 4 April 2006 / Accepted: 10 October 2006 Published online: 13 March 2007 – © Springer-Verlag 2007

Abstract: The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue for some non-necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov (cf. [11]) in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration. 1. Introduction: Model and Results The scope of this paper is to study the spectral properties of some well chosen rank one perturbation of classical complex or real large Wigner matrices. We consider a sequence (M N ) N of some complex or real Deformed Wigner matrices given by 1 MN = √ WN + A N , N where A N = (Ai, j )1≤i, j≤N is the N × N deterministic real matrix defined by Ai, j = Nθ , with θ > 0 given, independent of N . In the complex case, W N = (Wi, j )1≤i, j≤N is a N × N Wigner Hermitian matrix with non-necessarily Gaussian entries such that (i) on the diagonal, the entries are real and the {Wi,i , 1 ≤ i ≤ N } ∪ {eWi, j , mWi, j : 1 ≤ i < j ≤ N } are real independent random variables, (ii) all these real variables have symmetric laws (as a consequence, E[Wi,2k+1 j ] = 0 for all k ∈ N∗ ), 2 (iii) ∀ i < j, E[(eWi, j )2 ] = E[(mWi, j )2 ] = σ2 . The second moments of the diagonal elements Wi,i are assumed to be uniformly bounded, (iv) all their other moments are assumed to be sub-Gaussian, i.e. there exists a constant β > 0 such that, uniformly in i, j and k, E[|Wi, j |2k ] ≤ (β k)k .

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D. Féral, S. Péché

In the real setting, W N = (Wi, j )1≤i, j≤N is a N × N (non-necessarily Gaussian) Wigner symmetric matrix which satisfies the following conditions: (i’) the {Wi, j , 1 ≤ i ≤ j ≤ N } are independent random variables, (ii’) the laws of the Wi, j are symmetric (in particular, E[Wi,2k+1 j ] = 0), 2 2 (iii’) for all i < j, E[Wi, j ] = σ . The second moments of the Wi,i are assumed to be uniformly bounded, (iv’) all the other moments of the Wi, j grow not faster than the Gaussian ones. This means that there is a constant β > 0 such that, uniformly in i, j and k, E[Wi,2kj ] ≤ (β k)k . When the entries of W N are further assumed to be Gaussian (with, on the diagonal, Wi,i ∼ N (0, σ 2 )), that is, in the complex (resp. real) setting when W N is of the socalled GUE (resp. GOE), we will denote by M NG the corresponding Deformed model. Let λ1 ≥ · · · ≥ λ N be the ordered eigenvalues of M N . At this point, M N may be real or complex. If θ = 0, one recovers the classical Wigner Ensembles whose spectrum is quite well-known. Our aim is to study the influence of the parameter θ on this spectrum and mainly on the largest eigenvalues. Some answers have already been obtained. First, on a global scale, the classical Wigner Theorem is still satisfied, whatever the parameter θ ≥ 0 is. This is for example a consequence of Lemma 2.2 of [1]. Thus, the limiting N behavior of the empirical spectral measure µ N = N1 i=1 δλi of any ensemble of type (i) − (iv) (or (i ) − (iv )) is the semicircle law µσ , whose density is given by 1 2 dµσ (x) = 4σ − x 2 1[−2σ,2σ ] (x). dx 2π σ 2

(1)

On the other hand, the parameter θ may change the limiting behavior of the largest eigenvalues. Let us recall the results obtained for classical Wigner Ensembles. We denote by λ˜ 1 ≥ λ˜ 2 ≥ · · · ≥ λ˜ N the eigenvalues of ( √1 W N ) N . It is a fundamental result due N to [6] that the largest eigenvalue λ˜ 1 converges almost surely to the right endpoint 2σ of the semicircle support. Considering the special case of the GUE (resp. GOE), it was established in [12] that, for all real t, N 2/3 lim P (λ˜ 1 − 2σ ) ≤ t N →∞ σ TW = F2 (t) (resp.F1T W (t)), where F2T W (t) (resp. F1T W (t)) is the well-known GUE (resp. GOE) Tracy-Widom distribution (see [12, 13] for precise definitions). A. Soshnikov later extended in [11] these results to arbitrary complex (resp. real) non-Gaussian Wigner matrices √1 W N of type (i) − (iv) (resp. (i ) − (iv )). N Recently, the behavior of the largest eigenvalues of complex Deformed GUE was investigated in detail in [8]. It is proved therein that the fluctuations of the largest eigenvalue λ1G of (M NG ) N exhibit a phase transition according to the value of θ . Define σ2 ρθ = θ + and σθ = σ θ

θ2 − σ2 . θ2

(2)

Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices

187

Theorem 1.1. [8] For any real t, • • •

t {− u 2 } 1 2 if θ > σ, then lim P N 1/2 λ1G − ρθ ≤ t = √ e 2σθ du, N →∞ 2π σ −∞ θ 2/3

N G TW λ1 − 2σ ≤ t = F2 (t), if θ < σ, then lim P N →∞ σ 2/3

N λ1G − 2σ ≤ t = F3T W (t), where F3T W is some if θ = σ, then lim P N →∞ σ generalized Tracy-Widom distribution (see [8], pp. 2–4 and [2] Subsect. 3.3 for precise definitions).

Remark 1.2. Actually the first study of fluctuations of the largest eigenvalue for noncentered Wigner random matrices goes back to Z. Füredi and J. Komlós [5]. It is proved √ therein that, if θ is of order N , the suitably rescaled largest eigenvalue exhibits Gaussian fluctuations. The extension to the real case has not been obtained yet. Nevertheless, it can be inferred from the results of [7] and personal communications with J. Baik. In particular, denoting by F1T W the GOE Tracy-Widom distribution, one should obtain the following result. Conjecture 1.3. Let λ1G be the largest eigenvalue of the Deformed GOE. For all real t, t {− u 2 } 1 2 G (i) if θ > σ , then lim P N λ1 − ρθ ≤ t = √ e 4σθ du, N →∞ 4π σθ −∞ 2/3

N (ii) if θ < σ , then lim P λ1G − 2σ ≤ t = F1T W (t). N →∞ σ

1/2

Some generalizations of Theorem 1.1 have been investigated. In [4], Chap. 4, the almost sure limit of the largest eigenvalues of any complex or real Deformed Wigner model (M N ) N is considered, using an approach due to [3]. The main arguments are given therein to prove that, for any Deformed Wigner matrix (M N ) N , the largest eigenvalue λ1 should a.s. jump outside the support [−2σ, 2σ ] of the semicircle law to the value ρθ as soon as θ > σ . If 0 ≤ θ ≤ σ , λ1 should tend to the right edge 2σ . Our paper is mainly devoted to the study of fluctuations of the largest eigenvalue for non-necessarily Gaussian complex Deformed Wigner Ensembles of type (i) − (iv) and of parameter θ . Our main result is that universality holds for any θ > 0. Our investigation also concerns non-necessarily Gaussian real Deformed Wigner Ensembles of type (i ) − (iv ) and yields the proof of the second point of Conjecture 1.3. We first prove the following universality result. Theorem 1.4. Theorem 1.1 is true for the largest eigenvalue λ1 of any complex Deformed Wigner Ensemble of type (i) − (iv). When θ < σ , we can state a stronger result, namely that the parameter θ does not have any impact on the asymptotic distribution of the k first largest eigenvalues of any complex Deformed Wigner Ensemble of type (i) − (iv), for any fixed integer k ≥ 1. Hence, all the asymptotic results established in [11] for general non-Gaussian Hermitian Wigner Ensembles (θ = 0) extend to the case where θ < σ .

188


Theorem 1.5. Assume that θ < σ . Let k ≥ 1 be fixed. Let λi denote the i th largest eigenvalue of any complex Deformed Wigner Ensemble of type (i) − (iv). Then, for all (t1 , . . . , tk ) ∈ Rk , 2/3

N N 2/3 TW (t1 , . . . , tk ), where lim P (λ1 − 2σ ) ≤ t1 , . . . , (λk − 2σ ) ≤ tk = F2,k N →∞ σ σ T W = F T W and F T W is the Tracy-Widom limiting joint distribution of the k first F2,1 2 2,k eigenvalues of the GUE (given e.g. in [11]).

All our partial results being also true in the real setting (with minor modifications), we also consider throughout this paper the real model. Actually, once the whole real version of Theorem 1.1 is proved, our main Theorem 1.4 can readily be extended to the real framework (see Sect. 2 for a justification). Yet we prove an analog of Theorem 1.5 in the real framework which in particular gives the last point of Theorem 1.1 in the real case. Theorem 1.6. Assume that θ < σ . Let k ≥ 1. Let λi denote the i th largest eigenvalue of any real Deformed Wigner Ensemble of type (i ) − (iv ). Then, for all (t1 , . . . , tk ) ∈ Rk ,

2/3 N N 2/3 TW (t1 , . . . , tk ), where lim P (λ1 − 2σ ) ≤ t1 , . . . , (λk − 2σ ) ≤ tk = F1,k N →∞ σ σ T W = F T W and F T W is the Tracy-Widom limiting joint distribution of the k first F1,1 1 1,k eigenvalues of the GOE (given e.g. in [11]).

Remark 1.7. At this point, we would like to point out the fact that results of Theorem 1.1 have been proved for more complex Deformed GUE models. On the one hand, because of the rotational invariance of the GUE distribution, Theorem 1.1 holds for an arbitrary deterministic matrix A N of rank one and of eigenvalue θ. On the other hand, the results of [8] are stated for any deterministic deformations A N of fixed rank k ≥ 1. The natural problem of the universality of the fluctuations arises for such deformations, but is beyond the scope of this paper. In a forthcoming paper, we will prove that universality does not hold for instance if one chooses the diagonal matrix A N = diag(θ, 0 . . . , 0). We will also investigate deformations of fixed rank k ≥ 1. The derivation of our results uses ideas and combinatorial techniques similar to those used by Y. Sinai and A. Soshnikov in [9–11]. Following especially the approach developed in [11], we compute the limiting behavior of the expectation of traces of high powers of M N defined by E[Mi0 ,i1 · · · Mi2s−1(+1) ,i0 ], (3) E[Tr M N2s(+1) ] = 1≤i 0 ,i 1 ,...,i 2s−1(+1) ≤N

for some powers s = s N such that lim N →∞ s N = ∞. In particular, we study in detail the contribution to (3) from the closed paths P = {i 0 , i 1 . . . i 2s−1(+1) , i 0 } of length 2s(+1) on the set of vertices {1, · · · , N }. The strategy is to show that the leading term in the asymptotic expansion of (3), for some specific exponent s N , comes from the paths whose expectation only depends on θ and σ . This implies that, up to a negligible error, (3) has the same limiting behavior as in the case where the matrices W N are of the GUE. In 2s(+1) other words, we show that E[Tr M N ] = E[Tr (M NG )2s(+1) ](1 + o(1)). This strategy can be deepened to derive similar results for all higher moments (see Sect. 6).


189

Our paper is organized as follows. In Sect. 2, we present the main elements of the proof of our theorems and explain why they follow from the universal limiting behavior of moments of traces of high powers of M N . We then recall in Sect. 3 the needed specific terminology introduced in [11]. Sections 4, 5 and 6 are mainly devoted to the study of the case where θ > σ . The case where θ = σ is considered in Sect. 7. At last, we justify the case where 0 < θ < σ in Remark 7.13. Throughout this paper, the notations C, Ci , 1 ≤ i ≤ 8, C and C˜ will be used for different positive constants. 2. Core of the Proof Here, we first mainly concentrate on the case where θ > σ . We show how universality of the fluctuations of the largest eigenvalue for complex Deformed Wigner Ensembles of type (i) − (iv) can be derived from the computation of limiting moments of traces of high powers of M N . This is inspired from the approach of [11]. At the end of this section, we point out the main modifications needed in both the cases where θ = σ and θ < σ and also quickly discuss the real setting. √ In the case where θ > σ , we shall handle it with powers s N of order N . It is indeed expected, from Theorem 1.1, that the largest eigenvalue λ1 exhibits Gaussian fluctuations around ρθ in the scale √1 . In particular, in Sect. 5, we prove the crucial N √ fact that, for s N = [t N ] with t > 0,

M 2s N M N 2s N +1 N lim E Tr + Tr N →∞ ρθ ρθ 2s N 2s N +1 ⎞ M NG M NG ⎠ = 0. (4) + Tr −E Tr ρθ ρθ Basically, one intends to prove that only the largest eigenvalue λ1 (resp. λ1G ) contributes to the first (resp. second) expectation in the l.h.s of (4). Let λ1G ≥ · · · ≥ λG N be the ordered eigenvalues of (M NG ) N , that is of the Deformed GUE. We decompose the eigenvalues of M N and M NG as follows:

ξj τj , if λ j > 0 and λ j = −2σ + 2/3 , if λ j < 0, λ j = ρθ 1 + √ N 2 GN τ jG ξ j λGj = ρθ 1 + √ (5) , if λGj > 0 and λGj = −2σ + 2/3 , if λGj < 0. N 2 N The strategy to derive universality for the fluctuations of the largest eigenvalue from (4) can be summarized in three steps: The first step shows that, for both M N and M NG and for all t > 0, the random variable

N ,t

1 = Tr 2

MN ρθ

2[t √ N ]

+ Tr

MN ρθ

2[t √ N ]+1

−

etξ j

(6)

|ξ j |≤N 1/6

converges to 0 a.e. as in the L k -norm, for all fixed k ≥ 1. Formula (6) will be proved below. In the case where θ > σ (only), the a.e. and L 1 -norm convergence are actually

190


enough to derive the announced fluctuations for the largest eigenvalue (see also the end of the section for the other cases). The second step follows from results of [8] which ensure that there exists δ > 0 such that for |t| ≤ δ, lim E

N →∞

G

etξ j

= L θ (t),

1/6 |ξ G j |≤N

where L θ is the Laplace transform of the law N (0, σθ 2 ).

(7)

Let then P N (λ1 , . . . , λ N ) be the symmetrized joint eigenvalue distribution on R N induced by any Deformed Ensemble M N and, for all 1 ≤ m ≤ N , denote by PmN one of its m-dimensional marginals. Define then the associated m-point correlation function Rm of P N by Rm =

N! Pm . (N − m)! N

(8)

Note that Rm is a distribution in general. Using (7), the above results combined with the machinery developed in [11] yield that the rescaled one point correlation function ∞ R˜ 1 (x) = √1 R1 (ρθ + √x ) of M N satisfies, for all t ∈ R, lim N →∞ t R˜ 1 (x) 1x≤N 1/6 N ∞ N u2 d x = t 1 2 exp {− 2σ 2 } du. Actually, the following stronger result holds: 2π σθ

θ

lim

N →∞ t

∞

R˜ 1 (x) d x =

∞

t

1 2π σθ2

exp {−

u2 } du. 2σθ2

(9)

Indeed, in Lemma 2.2 given below, we establish that there exist two constants C3 , C4 > 0 such that P {i : λi > ρθ (1 + N 11/3 )} > 0 ≤ C3 exp {−C4 N 1/6 }. This yields (9) since t

∞

R˜ 1 (x) 1x>N 1/6 d x ∼

]ρθ (1+

1 ),∞[ N 1/3

R1 (y) dy

≤ N C3 exp {−C4 N 1/6 } → 0 as N → ∞.

(10)

At last, the third step is based on Lemma 2.3 proved below, which roughly states that only the largest eigenvalue of M N separates from the bulk and that it is close to ρθ . To be more precise, this lemma implies that there exist constants C1 , C2 > 0 such that, given R < 0, for all t ≥ R,

t P ∃ at least two eigenvalues of M N in ]ρθ + √ , ∞[ ≤ C1 exp {−C2 s N }, (11) N for N large enough. In this way, we claim that the largest eigenvalue λ1 of any complex Deformed Ensemble of type (i) − (iv) has the same limiting behavior as that of the


191

Deformed GUE. To see it, let D N denote the random number of eigenvalues of M N in the interval I =]ρθ + √t , ∞[. Then, with Rm (x1 , . . . , xm ) given as in (8), one has that N

√ P[ N (λ1 − ρθ ) ≤ t] = P(D N = 0) N m (−1)m = 1−E 1 I (λi ) + Rm (x1 , . . . , xm ) d xi . m! Im i=1

m≥2

i=1

(12) Then, using (11), N (−1)m Rm (x1 , . . . , xm )d x1 · · · d xm = P(D N = 0) − 1 + E 1 I (λi ) m m! I m≥2 i=1 N N = E 1 I (λi ) − P (D N ≥ 1) = E 1 I (λi )1 D N ≥2 − P (D N ≥ 2) i=1

i=1

≤ (N + 1) P(D N ≥ 2) ≤ (N + 1)C1 exp {−C2 s N } → 0 as N → ∞. ∞ N ˜ Noticing that E i=1 1 I (λi ) = I R1 (y)dy = t R1 (x) d x, we derive Theorem 1.4 for θ > σ . Let us now return to formula (6) and prove the announced convergence. We first show that the negative eigenvalues do not contribute to (6). Given a real c > 0, we define for i = 1, 2, ri =

2s N

2s N +1 i λj λj 1/2 , + 1/2 ρθ ρθ

(13)

where 1 (resp. 2 ) corresponds to the summation over { j : −2σ −c/N 2/3 < λ j < 0} (resp. { j : λ j < −2σ − c/N 2/3 }). As ρθ > 2σ , it is an easy fact that there exists a constant C > 0 such that |r1 | ≤ exp {−Cs N }, for N large enough. Considering r2 , we have that 1 + ρθ /2σ |r2 | ≤ 2

λ j 0, |ξ j | ≤ N 1/6 }, 4 over { j : λ j > 0, ξ j ≤ −N 1/6 } and 5 over { j : λ j > 0, ξ j ≥ N 1/6 }. First, it is an easy fact that r3 = |ξ j |≤N 1/6 etξ j (1 + O(N −1/6 )). We then show that the other terms lead to a negligible contribution. First, one readily has that |r4 | ≤ N exp {−C N 1/6 }. The analysis of the term r5 leans on the following lemma. Lemma 2.1. If θ > σ then for all k in N∗ , for any ti , 1 ≤ i ≤ k, in a compact subset K of R+∗ , ∃ C = C(K ) > 0, E

k i=1

Tr

MN ρθ

2[ti √ N ](+1)

2

≤ Ck .

The proof of Lemma 2.1 is obtained in Sects. 5 and 6. Thanks to this result, we can estimate the contribution of r5 . Note that

|r5 | ≤

⎛ ⎞2s N −1 λ j 2s N +1 λ j 2s N +1 λ j 2s N −1 1 ⎝ ⎠ ≤ 1/6 ρ ρ ρ θ θ θ 1 + N√ 1/6 1/6

ξ j ≥N

≤ Tr

MN ρθ

4s N

ξ j ≥N

2 N

exp {−C N 1/6 }.

From Lemma 2.1, we trivially deduce that all the moments of r5 tend to zero as N → ∞. This finishes the proof of formula (6), yielding the first step. Lemma 2.1 also ensures that the positive eigenvalues are not too large. This is stated in the next lemma which completes the proof of (10). Lemma 2.2. There exist two positive constants C3 and C4 such that, for N large enough, P {i : λi > ρθ (1 +

1 N 1/3

)} > 0 ≤ C3 exp(−C4 N 1/6 ).

Proof of Lemma 2.2. From the Chebyshev inequality, we readily have that P {i : λi > ρθ (1 +

ρθ −2[δ N 1/2 ] 2[δ N 1/2 ] , )} > 0 ≤ ρ + E TrM θ N N 1/3 N 1/3 1

where δ is a real positive number. For N large enough, we derive from Lemma 2.1 that there exist C3 > 0, C˜ 4 > 0 such that P {i : λi > ρθ (1 + N 11/3 )} > 0 ≤ ˜

C3 e−C4 δ N

1/6

.


193

At this stage, it remains to prove the following fundamental lemma which readily gives (11). Lemma 2.3. There exists a positive constant C2 such that P {i ≥ 2 : λi > 2σ + (ρθ − 2σ )/2} > 0 ≤ exp {−C2 s N }. Proof of Lemma 2.3. By the interlacing property of eigenvalues, it is clear that only one eigenvalue of M N is close to ρθ since P {i ≥ 2 : λi > 2σ + (ρθ − 2σ )/2} > 0 ≤ P {i ≥ 2 : λ˜ i > 2σ + (ρθ − 2σ )/2} > 0 √ E[Tr(W N / N )2s N ] ≤ 2s N ≤ exp {−C2 s N } 2σ + (ρθ − 2σ )/2 for some positive constant C2 . The last inequality follows from Theorem 4.2. Thus we get the statement of Theorem 1.4 in the case θ > σ . In the real setting, one can expect the same proof with N (0, 2σθ2 ) instead of the law N (0, σθ2 ) (recall Conjecture 1.3). In both other cases where θ = σ and θ < σ , the fluctuations of the largest eigenvalue are expected to occur in the scale N −2/3 around the edge 2σ . This is exactly as for classical Wigner Ensembles (θ = 0), except for the limiting distribution. The scheme to state the complete universality follows √ the same steps as in the case where θ > σ (with 2σ instead of ρθ , N 2/3 instead of N in (5) and replacing the law N (0, σθ2 ) with F3T W if θ = σ and with F2T W if θ < σ ). The asymptotics of correlation functions of the Deformed GUE required to establish the second step are straightforward from Propositions 2.1 and 2.2 in [8] and Subsect. 3.3 in [2]. The proof then mainly boils down to the universality of the limiting expectation of traces of exponents of type o(N 2/3 ) and O(N 2/3 ). Nevertheless the derivation of the result requires more complex considerations than the previous analysis. Indeed, here, the largest eigenvalue does not separate from the “bulk” and the whole spectrum lies in [−2σ − √1 , 2σ + √1 ]. In fact, the reasoning 2 N 2 N is very close to that made by A. Soshnikov for general Wigner Ensembles and we refer to Sects. 1,2 and 5 of [11] for details. In particular, universality of all higher moments of the traces is required. Note√that in the case where θ < σ , we actually prove (4) and (6) if one replaces ρθ (resp. N ) with 2σ (resp. N 2/3 ). And we also show that the same formulae hold with M NG replaced with √1 W NG . Moreover, convergence N

of (6) in the L k -norm for any fixed k ≥ 1 ensures universality of the limiting joint distribution of the k first largest eigenvalues of any Deformed Wigner Ensemble and that this limit is the same as for the GUE. A detailed proof of this fact is presented in [11]. In the real setting (and again if θ < σ ), the same reasoning shows that the fluctuations of the largest eigenvalues of M N are to be compared with those of the GOE instead of M NG . The rest of our paper is mainly devoted to the analysis of (3) for some powers L N = 2s N (+1). This is based on the combinatorial machinery developed in [9–11]. Before we proceed, we recall the main definitions needed in this paper and introduced in [9–11].

194


3. Terminology: Classification of Instants and Vertices To each term in the expectation (3), we associate a path P = {i 0 , i 1 , . . . , i L−1 , i L = i o } of length L ≥ 1 (L = 2s(+1)) and where i j ∈ N∗ (in this paper, we restrict to vertices in {1, . . . , N }). Note that loops are allowed, i.e. it may happen that i j+1 = i j . To explain our counting strategy, we need to recall some definitions given in [9–11]. Definition 3.1. The instant j = 1, . . . , L is said to be marked for the closed path P if the non-oriented edge (i j−1 , i j ) occurs an odd number of times up to time j (included). The other instants are said to be unmarked. Throughout this paper, we denote by Pm,l the set of paths P of length L = l + 2m having l +m marked instants and m unmarked instants. In particular, Pm,0 corresponds to the classical even closed paths used in the framework of the classical Wigner Ensembles. We associate to each path P a trajectory x = {x(t), 0 ≤ t ≤ L} of a simple random walk on the positive half-lattice such that x(0) = 0, x(L) = l; x(t) ≥ 0, ∀t ∈ [0, L], x(t) − x(t − 1) = 1 (resp. − 1) if t ∈ N∗ is marked (resp. unmarked). Thus, the associated trajectory x of a path P of Pm,l is such that l + m = #{t, x(t) − x(t − 1) = 1} (up steps) and m = #{t, x(t) − x(t − 1) = −1} (down steps). We define by Tm,l the set of such trajectories x and we set Tm,l = #Tm,l . The elements of Tm,0 are often called Dyck paths. Proposition 3.2. For L = l + 2m, one has Tm,l = C Ll+m − C Lm−1 =

L! (l+m+1)!m! (l

+ 1).

Remark 3.3. Proposition 3.2 is a straightforward consequence of the reflection principle used in the historical proof of the Wigner Theorem (cf. [1] for example). One can also notice that, amongst the paths of Tm,l , exactly Tm,l−1 (resp. Tm−1,l+1 ) have a last step up (resp. down). We also need to refine our classification of vertices of a path P of Pm,l . Definition 3.4. A marked instant j is called an instant of self-intersection of P if there exists a marked instant j < j such that i j = i j . Definition 3.5. A vertex i is said to be a vertex of simple (resp. k-fold) intersection of P if there exist exactly two (resp. k) marked instants such that i j = i. A path without self-intersection will be called a simple path. We can now split the vertices of P ∈ Pm,l into l + m + 1 disjoint subsets such that {1, . . . , N } =

l+m

Nk ,

(15)

k=0

where Nk is the subset of vertices of k−fold self-intersection. Setting Nk = #Nk , such a path P is said to be of type (No , N1 , . . . , Nl+m ). In particular, the simple paths of Pm,l are of type (N − (l + m), l + m, 0, . . . , 0). It may happen that the origin i o is unmarked and then belongs to No . Otherwise i o is marked and No only contains vertices not belonging to P (see Fig. 1 below).

Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices 2 sN (+1) 4. Asymptotics of E[TrM N ] for 1 0, l even Tm,l θ σ θ σ2 )(1 + o(1)). θ2

s N l 2m = o((2σ )2s N ) if θ < σ and Remark 4.4. One has that m=0 Tm,l θ σ s N l 2m 2s N = (2σ ) /2 if θ = σ . This fact combined with the following anam=0 Tm,l θ σ lysis then justifies Remark 4.3. Remark 4.5. If P is a simple path of length 2s N with marked origin, it is not hard to see that the instant To of the marked occurrence of the origin is uniquely determined. If the last step of P is up, then To = 2s N . Otherwise To = inf{t > 0 : x(t) = l and ∀t ≥ t, x(t ) ≥ l} (see Fig. 1 below). The following theorem shows that typical paths with marked origin are simple if √ s N = o( N ). √ Theorem 4.6. Assume that θ > σ . For any sequence 1 0 (and any M2 ≥ 0) is not greater than Tm,l−1 θ l σ 2m × exp{

(C(l + m))2 } × exp (C N −2/9 ) − 1 . N

(27)

As exp (C N −2/9 ) − 1 = O(N −2/9 ) = o(1), this ensures that the contribution of paths for which there exists a vertex of k−fold self-intersection for some k ≥ 11 is negligible. Finally, we shall consider paths with some multiple self-intersection of type smaller than

200


11, which amounts to consider the summation of (24) over all the (N0 , . . . , Nl+m ) such that M2 > 0. It is not greater than

(C(l + m))2 l 2m Tm,l−1 θ σ × exp{ }−1 . (28) N } − 1 = o(1) uniformly in l. As a conseAs s N = o(N 1/2 ), one has that exp{ (C(l+m)) N l 2m quence (and noticing that l≥2 Tm,l−1 θ σ = (θ −σ 2 /θ )ρθ2s N −1 (1+o(1))), we deduce that the summation of (28) over all l is negligible with respect to (18). Theorem 4.6 for paths with last step up is established. 2

4.2. Paths with a last step down. In this section, we still consider paths where at least one edge is passed an odd number of times (i.e. l > 0). Indeed, even paths (l = 0) are considered in Theorem 4.2 and give a negligible contribution (as θ > σ ). We first investigate paths whose origin is unmarked and establish the following result. √ Theorem 4.9. Assume that θ > σ . For any sequence 1 x(t1 ), ∀t > t1 }. t2 = inf{t ≥ t2 : x(t2 ) = x(t2 ) and x(t ) < x(t2 ), ∀t2 < t ≤ t3 }. The instants t1 and t2 are such that on [t j , t j ] ( j = 1, 2), the path P describes a sub-simple Dyck path Pv with origin v. Note that it may happen that t1 = t1 (and thus Pv1 is empty) or t2 = t2 . j

Let us denote by w the vertex occurring at the instant t1 + 1: (vw) is the first odd (simple) edge of P. On [t1 + 1, t2 ], the subpath Pi begins in w, ends at v by an up step and remains above the level x(t1 + 1). Consider now the rest of the path. Just after t2 , one closes the


203

13 3

12 7

6

4

4

4 10

6

5

12

l–p+1=3

2

4

1

t

o

t1

4

6

6

1

8

9

5

1

4 4 10

1

2

5 1

8 9

2

12

1

9 2

14

13

14

11 1

5

3

7

12

4

4

4

11

4

t

t’1

2

t’2 t3

9

9

Fig. 3. Left: P is in P11,3 . Its sole self-intersection is the vertex 4 which is simple. v = 4 and w = 9. Right: This is the simple path P corresponding to P. It belongs to P11,3 and p = 1.

edge e p in the reverse sense. Next we successively return in the reverse direction the other edges e p−1 , . . . , e1 and thus reach i o closing e1 at time t3 (t2 < t3 ≤ 2s). These returns can be interspersed with sub-simple Dyck paths. The edge (vw) is the first odd (simple) edge of P: this is the distinguished nonreturned edge defining the origin w of the new path P obtained from our correspondence. It is easy to see that P is simple with a marked origin w, which is well determined since the last step is up. The vertex i o is also marked and well defined. Moreover, the vertex v is now of type 1 in P : its marked occurrence at time t2 in P is its sole marked occurrence in P , whereas its marked occurrence at time t1 in P is changed to an unmarked occurrence in P . Thus, according to the 1st Step of this proof, the numbering of such simple paths P is of m

m−1 m Tm− p,l+2 p = C2s = C2s − Tm,l . N N

(33)

p=1

Then, one easily finds the r.h.s. of (29). This finishes the proof of Theorem 4.9. We illustrate our correspondence on a path whose origin is unmarked and which has only one simple self-intersection on Fig. 3 above. We now complete the proof of Theorem 4.6 (and Theorem 4.1). We shall then consider non-simple paths with marked origin and a last step down. To this aim, it is enough to notice that our correspondence still works for any path with marked origin and ending with a down step. The sole difference from the case where the origin is unmarked is that the level p (introduced in the 1st Step of the previous proof) of the first odd edge can now be equal to 0. In this way, one can note that a path P which is not simple is associated to a non-simple path P . It is then easy to see (referring to the previous 2nd Step) that the non-simple paths with last step down and marked origin give a negligible contribution to the expectation. We do not explain more. √ 2 sN (+1) 5. Computations of E[TrM N ] for sN = O( N) and θ > σ √ Here, we shall prove that, in the scale s N = O( N ) and as N → ∞, the behavior of the expectation of the trace is the same for any Deformed Wigner Ensemble of type (i)−(iv)

204


(resp. (i ) − (iv )). We use as before M NG to denote the corresponding Deformed GUE (resp. GOE) model. Theorem 5.1. Let (L N ) N be a sequence such that ∃c > 0, lim N →∞ √L N = c. Let M N N be a Deformed Wigner matrix of type (i)–(iv) (resp. (i )–(iv )). Then ∃C > 0 such that, for N large enough, L N

LN LN LN G E TrM N ≤ C ρθ (1 + o(1)). and E TrM N = E Tr M N L2 To be more precise, in the complex setting, one has E TrM NL N = ρθL N exp { 2NN ( ρσθθ )2 } (1 + o(1)). This can trivially be deduced from the result of Theorem 1.1 combined with some considerations of Sect. 2. Note that similar exact estimates, with σθ replaced by √ 2σθ , can be expected for the real model (see Conjecture 1.3). We only consider even powers L N = 2s N since the proof is similar for odd powers. The main part of this section is dedicated to paths with a last step up. We show that the typical paths with last step up have at most simple self-intersections, no loops and edges passed at most twice. The last fact ensures in particular that the expectation of the trace is the same for any Deformed Wigner Ensemble. In Subsect. 5.2, thanks to the fundamental correspondence built in Subsect. 4.2, we translate this analysis to paths with a last step down and show that universality holds too. 5.1. Paths with last step up. Throughout this section, we only consider paths with last step up. Their contribution is at least of order ρθ2s N , since it can easily be seen from the pre ceding section that the contribution of such simple paths is of order l≥2 Tm,l−1 θ l σ 2m (l+m)2

2s 2N

e{− 2N } ≥ (1 − ( σθ )2 )e{− N } ρθ2s N (1/3 + o(1)). The following proposition shows that there are at most simple self-intersections in the typical paths, i.e. those contributing to the trace in a non-negligible way. Proposition 5.2. Typical paths with last step up have no self-intersection of multiplicity k ≥ 3. Proof of Proposition 5.2. Let Z Pm,l of type o (m) denote the contribution of paths of (No , N1 , . . . , Nl+m ) such that k≥3 Nk ≥ 1. For such paths, set M1 = k>10 Nk and M2 = 10 k=3 Nk . Then, from (24), one has that

N C(l + m)2 2 l 2m 1 Z o (m) ≤ Tm,l−1 θ σ N2 ! N N2 ,M1 ,M2 ,M1 +M2 >0

1 × M2 !

(C (m + l))3 N2

M M2 C N −2 1 . M1 !

(34)

The summation of (34) over all the M1 , N2 , M2 such that M2 > 0 is not greater than

C (l + m)3 (C(l + m))2 } exp { Tm,l−1 θ l σ 2m exp { } − 1 N N2 = Tm,l−1 θ l σ 2m × O(

(l + m)3 ), N2

(35)


205

√ 3 with O( (l+m) ) = o(1) as s = O( N ). Similarly, the summation over all the N 2 N M1 , N2 , M2 such that M1 > 0 is at most of order Tm,l−1 θ l σ 2m × O(N −2 ). Thus s N −1 2s N 1 √ ), which is negligible w.r.t. the contribution of Z o := m=0 Z o (m) = O(ρθ N simple paths. Given a path P of Pm,l (with l +2m = 2s N ) with last step up and of type (No , N1 , . . . , Nl+m ), we define M := l+m k=2 (k − 1)Nk to be the number of its self-intersections. This quantity will be important in the following and is the object of the next proposition. Proposition 5.3. The number of self-intersections of typical paths satisfies M ≤ s N , for any 0 < < 1. Proof of Proposition 5.3. By Proposition 5.2 and (34), it is clear that adding the contribution of paths where k≥2 k Nk ≥ s N (for any > 0) gives a final contribution which is o(ρθ2s N ). In the following, we investigate in detail paths with a last step up and that have only simple self-intersections. Note that for such paths, each edge is passed at most four times. We first discuss paths having edges read at most twice. Then, we show that those admitting at least one edge passed three or four times and those with at least one loop can be neglected. 5.1.1. Paths with only simple self-intersections, edges read at most twice and last step up. Our goal is here to prove that, for a path with only simple self-intersections, there exist different ways of closing the path once the vertices at marked instants are chosen. In the denomination of [11], this means that there are “non-closed vertices” in typical paths. The definition will be recalled later. This explains that the expectation of the trace differs in the real and the complex setting (see the comments just before Definition 5.5 below). Define Z 1 (M, m) to be the contribution of paths of type (No , N1 , M, 0, . . . , 0) with marked origin, last step up and edges passed at most twice. Denote also by Z 1 (m) = Z (M, m) the total contribution of such paths. We want to establish that there exists 1 M a constant D, independent of N , such that, for N large enough, Z 1 :=

m≤s N −1

Z 1 (m) ≤ exp {D

s N2 } ρ 2s N . N θ

(36)

Consider a path P contributing to Z 1 (M, m). By Propositions 5.2 and 5.3, one can assume that M = N2 ≤ s N for some arbitrary 0 < < 1. Let then t j1 < t j2 < · · · < t j M be the instants of self-intersection of P. We now choose the vertices occurring at the (N − j) ∼ marked instants, and thus fix the origin of the path. First, there are l+m−M−1 j=0 (l+m)2

N l+m−M e{− 2N } different ways to choose the distinct vertices occurring in the path in the order of their appearance. If a vertex of self-intersection occurs at some instant t ji , there are ji −i possible choices for such a vertex. It is indeed chosen amongst the marked vertices which have already occurred in the path but have not yet been repeated. Note that if i o is a vertex of self-intersection, then j M = l + m, t j M = l + 2m and there are at most l + m − M choices for the vertex i o .

206


Remark 5.4. If there is no choice for closing edges at unmarked instants, then the number of such paths is at most N l+m since

l+m−M−1

M≥0 ji ≤l+m

j=0

=N

(N − j)

M

( jk − k)

k=1

l+m {−(l+m)2 /2N }

e

1 (l + m)2 M (1 + o(1)). M! 2N

M≥0

Here the o is uniform due to the fact that M ≤ s N for some 0 < < 1/32. In the general case, there are many choices for closing edges from a vertex of self-intersection and the 2 number of paths of type (No , N1 , N2 , 0, . . . , 0) can then be of order N l+m eCs N /N . We now count the number of ways to close the path at unmarked instants. One can close an edge starting from a vertex belonging to N1 . In this case, there is no choice for closing it. We can also close an edge starting from a vertex in N2 . Then we can close it in at most 3 ways: along the edge used to arrive at this vertex for the first or second time, or along the edge used to leave it for the first time. Such consideration leads to the notion of non-closed vertex. Definition 5.5. A vertex of self-intersection is said to be non-closed if there are several possibilities of return from this vertex at an unmarked instant. For example, in the left path of Fig. 1, the vertex 3 is closed whereas 4 and 6 are non-closed. Here we show that paths of type (No , N1 , N2 , 0, . . . , 0) with non-closed vertices contribute in a non-negligible way to the expectation of the trace, if l > 0. The fact that typical paths admit non-closed vertices explains that the expectation of the trace (and thus the limiting distribution of λ1 ) differs between the real and complex case. Indeed, assuming edges appear at most twice, an oriented edge repeated with the same 2 2 orientation has the weight Nθ 2 0 independent of N . Here, an extra N has been added to include the case where the last step is down. By Lemma 7.4, it is easy to see that there exists √ a constant Ccrit > 0 such that, in the large N limit, the subsum in (48) over l ≥ Ccrit N gives a contribution which is negligible with respect to (47) (i.e. to that of even paths). 3/2

Remark 7.5. For s N = o(N 2/3 ), it is enough to consider the paths with l ≤ Ccrit s N N −1/2 = o(N 1/2 ) for some constant Ccrit .

The arguments used to prove Theorem 7.1 are quite similar to those yielding Lemmas 5,6 and 7 (Sect. 4) in [11]. In particular, we show that typical paths may contain some non-closed vertices. But, using Remark 5.4, we need to have some control on the number of possibilities of choosing a non-closed vertex. In Sect. 5, given a path P in some Pm,l (with l + 2m = 2s N ) and denoting by x ∈ Tm,l the associated trajectory, the number of possible choices was estimated from above by maxt x(t) ≤ l + m. Here, in the scale


217

O(N 2/3 ), a better estimate is required. We now show that the number of choices for a non-closed vertex can be estimated through a quantity as maxt y(t), which involves the particular Dyck path y ∈ Tm,0 introduced in Subsect. 6.1. Let then z(t) denote the maximal number of ways to choose a non-closed vertex at the instant t. We now prove that l+m

z(t) ≤

t=1

l2 + m × max y(t) + lm. t 2

(49)

Recall that z(t) can be bounded from above by the number of marked vertices opened but not closed before t. If at the instant t, one crosses the subpath yi , then there are at most z(t) ≤ happen that the instant t j≤i li + yi (t) such vertices. It may also corresponds, for example, to the rise “li ” and then z(t) ≤ j≤i li (note also that in any case max z(t) ≤ l + max y(t)). Therefore, by a straightforward computation, one has p m i p l+m l2 t=1 z(t) ≤ 2 + j≥i m j . This yields (49). The estimate t=1 yi (t) + i=1 i=1 li needed on the quantity maxt≤2m y(t), with y as before, is given by Lemma 7.10 stated below. Assuming that this lemma holds, we are in position to establish Theorem 7.1. We essentially mimic the ideas in Sect. 4 of [11], distinguishing paths with a last step up or down. √ Proof of Theorem 7.1. From now on, one considers paths with l ≤ Ccrit N , where Ccrit is the constant given by Proposition 7.3. 1r st Case. Paths with last step up. It can be easily inferred from the computations of the preceding sections and of [11] (p. 41) that, up to a negligible error, for each l, one can assume that s2

(A1 ) the number of self-intersections is smaller than B NN , for some fixed B > 0 (large enough). (A2 ) the maximal type of a vertex of P, denoted ν N (P), satisfies ν N (P) ≤ C N 1/3 / ln N . (A3 ) i o is of type 1. Assumption (A1 ) is straightforward from Sect. 5 and [11]. Assumption (A2 ) can be proved using formula (28). Assumption (A3 ) follows from the fact that the contribution of paths where i o is of type at least 2 is of order M/s N = O(N −1/3 ) that of paths for which i o is of type 1. Remark 7.6. Actually, we can prove that ν N (P) ≤ 4 for paths giving a non-negligible contribution to the expectation of the trace. Yet this estimate, needed to consider edges passed at least three times, requires some technical tools we only develop in the sequel. We can now proceed to the estimation of the contribution of paths with last step up, under assumptions (A1 ) to (A3 ). Consider paths of type (No , N1 , . . . , Nl+m ). For any such path, let r denote the number of non-closed vertices of type 2 and q be the number of vertices of N2 for which the second marked occurrence is in an edge already read at least twice. Then, given (No , N1 , . . . , Nl+m ), r ≤ N2 and q ≤ N2 , it is easy to see that Proposition 4.7 reads for such paths as m Emax ≤

sN 10 θ l 2m−q q 1 r k Nk 4k Nk /3 3 σ C . √ (Ck) (Ck) N sN N k=3

k=11

218


Denote by u 1 < u 2 < · · · < u r the ranks of the instants of self-intersections of type 2 where the non-closed vertices are chosen. Let also v1 < v2 < · · · < vq be the marked instants of N2 associated to an edge read at least three times. We also set ν N := ν N (P) + ν N (P). The contribution of paths of type (No , N1 , . . . , Nl+m ) with last step up is then at most e{−

(l+m)2 2N }

e

Bs 3N N2

σ 2s N × N N2

N2

( ji − i)

j1 < j2 0 such that 3ml N 1/3 Tm,l−1 exp { } ≤ C N Ts N ,0 , N √ Ao

for some constant

C .

√

Ccrit N

N 1/3 ≤l≤C

N

crit

Indeed, as m ≤ C2 N 2/3 , one has that

N 1/3 Tm,l−1 exp {

l=Ao N 1/3

3ml } ≤ C2 N Ts N ,0 N ≤

l≥Ao N 1/3

C3 N Ts N ,0 N 1/3

≤ N Ts N ,0

C4 N 1/3

l N 2/3

l≥Ao N 1/3

exp {l(

exp {−l

l≥Ao N 1/3 2 12s N 4s N 4/3 , C N 2/3 }. oN o

O(N 1/3 ),

= straightforward that

N 1/3

Co l 3m − )} N sN

l2 Co l 2 exp {− } 4s N 2s N

where in (54), we have chosen Ao ≥ max{ C 1/3 oN {−l Co A4s }

(53)

(54)

Co Ao N 1/3 }, 4s N As

(55)

l≥Ao N 1/3

exp

we obtain that (55) ≤ C5 N Ts,0 . This yields (53). It is also

l≤Ao

N 1/3

Tm,l−1 exp {

3ml } ≤ C6 22s N , N

(56)

s N +1 since 2≤l≤Ao N 1/3 Tm,l−1 ≤ C2s ∼ s N Ts N ,0 . Then, (53) and (56) yield (52) and N Lemma 7.7. 1/2−

We then denote by Z 4 (l) the subsum of (50) over paths for which ν N (P) ≤ s N , for some (small) > 0. We then show that for such paths and any l, typical paths have edges passed at most twice. Note that in this case ν N (P) ≤ 2C N 1/3 / ln N . Thus, it 1/2−

, is nothard to see that the summation of (50) over paths for which ν N (P) ≤ s N and i≥4 Ni ≥ 1 or N3 ≥ ln ln N, whatever q is, is o(Z 3 (l)) in the large N limit. Finally, assuming that there are no self-intersections of type strictly greater than 3, it is also easy to see that the contribution of paths with q ≥ 1 gives a contribution of order Z 3 (l)s N ν N /N = o(Z 3 (l)). Assuming then that q + k≥4 k Nk = 0, we can then proceed as above to show that no vertex of type 3 is in an edge read at least three times and that there are no loops in typical paths. 1/2−

Let finally Z 5 (l) denote the contribution in (50) of paths for which ν N ≥ s N , where < 1/32.

220


Lemma 7.8. One has Z 5 :=

Ccrit √ N l=2

Z 5 (l) = o

(2σ )2s N N 1/3

.

Proof of Lemma 7.8. We first need to introduce a few notations. Given a path of type (No , N1 , . . . , Nl+m ) with r non-closed vertices, we set K := r + i≥3 i Ni + 1, K o := 200 r + i=3 Ni and K := 200K o + i≥200 i Ni . Note that K ≤ K + 1. The choice of the constant 200 is not optimal here but is enough for our next computations. Let also j be the event j := {∃ 1 ≤ t1 < t2 < · · · < t j ≤ 2s N : x(ti ) = x(t1 ), ∀i ≤ j and x(t) ≥ x(t1 ), ∀t ∈ [t1 , t j ]}. Then j+1 ⊂ j , ∀ j ≥ 2. As explained in Remark 5.7, given K ≥ 1, if a vertex is the left endpoint of ν N up edges in a path, then the associated trajectory x necessarily belongs to ν N . Note also that these returns are necessarily K +1

made inside a sub-Dyck path yi , i ≤ p of the trajectory. It is then an easy fact that there exists C1 > 0 such that, for any j, l ≥ 0 and for any class T ∈ T(m), (57) PY,T j ≤ 4s N2 exp{−C1 j}. Thus, using (50) and arguments of [11] (p. 41), there exists C > 0 such that 1 (Cs N )4i/3 Ni Cs 3 Cs N ν N } Z 5 (l) ≤ σ 2s N Tm,l−1 exp { 2N } exp { N Ni ! N i−1 N 1/2−

Ni ,i≥200

× max EY,T T ∈T(m)

1 Ko! Ko

ν N ≥s N

3/2 K o

3m max y(t) + 3ml + Cs N N

1

νN K +1

Let Z 5 (l) be the subsum over K o and Ni , i ≥ 200, such that K ≤ s N

1/2−2

.

(58)

. Then

Cs 3 Z 5 (l) ≤ σ 2s N Tm,l−1 exp { 2N } N 1/2 3/2 3m max y(t) + 3ml + Cs N } × max EY,T exp {2. N T ∈T(m) 1/2 Cs N ν N 1/2−2

}s N × exp { . max PY,T ν N 1/2−2

N T ∈T(m) s +1 1/2−

N

ν N ≥s N

3 −s N /4 )×(51). Thus Z := One 5 deduces from (57) that Z 5 (l) = O(s N e (2σ )2s N o N 1/3 , using Lemma 7.10 and (52).

Ccrit √ N l=2

Z 5 (l) =

Let then Z 5 (l) be the subsum over K ≥ s N . From Lemma 7.10 proved 1/2 below, the proportion of paths for which maxt≤2m y(t) ≥ AN 1/6 s N decreases as C exp {−co A2 N 1/3 }. Choosing A large enough then ensures that the contribution of the 1/2 sole paths for which maxt≤2m y(t) ≤ AN 1/6 s N has to be taken into account. We now restrict to such paths in Z 5 (l). This implies in particular that there exists C2 > 0 such that 1/2−2

3/2

3m max y(t)+3ml+Cs N N

1/2−

sN

≤ν N

≤ C2 N 1/6 . Using (57), we can find K 1 such that 1/2 Cs N ν N }K 1 N 1/3 max PY,T ν N /(1+K 1 N 1/3 ) exp { ≤ C3 s N3 . N T ∈T(m)


221

(l) the subsum over s Denote then Z 5,1 ≤ K ≤ K 1 N 1/3 . Then, either K o ≥ N 1/2−2

1/2−2

/400 or i≥200 i Ni ≥ s N /2. In the latter case, i≥200 Ni ≥ 1. Thus, there sN exists C > 0 such that 1/2−2

Cs 3 Z 5,1 (l) ≤ C σ 2s N Tm,l−1 exp { 2N }C3 s N3 N ⎛ Ko ⎜ 1 1/6 C ×⎜ N + 2 ⎝ Ko! 1/2−2

Ko ≥

sN

⎞

i 1 (Cs N )4i/3 N⎟ ⎟ ⎠ Ni ! N i−1

Ni ,i≥200, Ni ≥1 i≥200

400

≤ C σ 2s N Tm,l−1 exp {

Cs N3 }C3 s N3 × N −20 . N2

(59)

In the last line, we have used Stirling’s and the multinomial identity. This yields formula Ccrit √ N (2σ )2s N Z 5,1 (l) = o N 1/3 . that Z 5,1 := l=2

(l) the subsum over K ≥ K N 1/3 . Then there exists C > 0 Denote finally by Z 5,2 1 such that

Cs 3 Z 5,2 (l) ≤ σ 2s N Tm,l−1 exp { 2N } exp {C N 1/3 } N ⎛ K ⎜ C2 N 1/6 o ⎜ + ×⎝ Ko! 1/3 Ko ≥

K1 N 400

Ni ,i≥200,

i Ni ≥

K 1 N 1/3 2

i≥200

1 Ni !

⎞ N (Cs N )4i/3 i ⎟ ⎟. ⎠ N i−1 (60)

To consider the above sum, we introduce Nmax which is the maximal type of a vertex in Mo the path (in particular Nmax ≤ ν N ) and Mo := i≥200 i Ni . As i≥200 Ni ≤ 200 and N −1 max Ni ,i≤Nmax , i Ni =Mo ≤ C Mo +Nmax −1 , one has

1 i≥200 K N3 Ni , i Ni ≥ 1 2

≤

Mo ≥

≤

C

Mo

1 Ni !

K N 1/3 Mo ≥ 1 2

Mo

N − 18

Nmax ≤Mo

K 1 N 1/3 2

(Cs N )4i/3 N i−1

C

Mo

N

o −M 18

Ni

Nmax −1 CM o +Nmax −1

4

Mo

≤2

4C N 1/18

K 1 N2 1/3

.

(61)

Ccrit √ N (2σ )2s N := Inserting this in (60) then yields that Z 5,2 . This Z (l) = 1/3 l=2 5,2 N (2σ )2s N finishes the proof that Z 5 0, independent of m and T ∈ T(m), such that

max y(t) ˜ EY,T exp{C √ } ≤ C. m Remark 7.11. A similar estimate for general Dyck path y of length 2m (i.e. without assuming a particular decomposition in sub-Dyck paths) was used in [11] (Lemma 6), but not proved. Proof of Lemma 7.10. We give here a proof based on some geometrical considerations. 1r st Case. No specified number of sub-Dyck paths. We first show that there exist constants 0 , Co , C˜ o independent of m, such that, under the uniform distribution on T2m √ C˜ o Co k 2 P(max y(t) = k) ≤ √ exp {− }, if k ≥ 4 mCo . 2m m

(67)

k denote the number of paths with 2n Consider first a Dyck path y of length 2m. Let T˜2n steps ending at level k without going below 0 or above k. Then, one has that ⎛ ⎞ k T˜ k ˜ k T˜ k + T T˜2n 1 ⎝ 2m−2n 2n+1 2m−2n−1 0 0 T2m−2n⎠. P(max y(t) = k) ≤ 0 T2n 0 T0 T2m k/2≤n≤m−k/2 T2n 2m−2n

(68) k This follows from the fact that such a path is the concatenation of a path T˜2n(+1) and one k ˜ of T . Actually, in the previous sum, 2n(+1) should be seen as the first instant 2s−2n(−1)

one reaches the level k. Now, we show that there exists constants C1 , C1 , Co > 0 such √ that for k ≥ 4Co m, C k2 C k2 C k2 k2 k2 k 0 o o o k 0 0 T2n ≤ 2C1 exp − T2n T˜2n ≤ C1 exp − + √ T2n exp − , n n n n n n Co k 2 0 k2 k exp − T T˜2m−2n ≤ 2C1 . (69) m−n (m − n) 2(m−n) k . Recall first that T k denotes the number of “positive We only prove the inequality for T˜2n 2n paths”, of length 2n, beginning at 0 and ending at k. On the other hand, by a simple n+k/2+1 application of the reflection principle, it is easy to see that C2n counts the number of paths that go from 0 to k touching -1. So, one has that n+k/2+1 k k = T2n − C2n + Tˆk−1,k+1 , T˜2n


225

where Tˆk−1,k+1 is the set of paths of length 2n that go from 0 to k touching both −1 and k + 1. To estimate Tˆk−1,k+1 , one can note that either such a path goes to k + 1 after the first time it goes to −1, or it goes first to k + 1 then to −1 and joins k without reaching k + 1 afterwards. First, by a simple reflection principle, it is easy to see that paths going to k + 1 after the first time they go to −1 are in bijection with paths of length 2n beginning at 0 and n+k/2+2 ending at −(k + 4). There are exactly C2n such paths. Now, as k ≤ 2n ≤ 2m − k, by Lemma 7.4, one has n+k/2+1

k −C T2n 2n

n+k/2+2

+ C2n

=

0 T2n

k − T k+2 T2n 2n 0 T2n

≤ C2

C k2 k2 o exp − , n n

(70)

where C2 and Co are two constants independent of n. Note that Co can be chosen equal to 96−1 . The numbering of the paths which reach k + 1 before −1 and joins k without reaching k + 1 afterwards is quite more subtle and we only obtain an upper bound. The trajectory of such a path can be described as follows. Call 2t + 1 the last instant where the trajectory is at level k + 1. Call then 2t + 1 the last instant where the trajectory is at level −1. Assume first that 2t + 2 ≤ 2n(2/3). In between 0 and 2t + 1 the trajectory goes from 0 to k + 1 without touching −1. Then the trajectory in between 2t + 2 and 2n goes from k to k without touching k + 1 (and reaches −1, but we will forget this constraint). / . k ≤ k exp − Co k 2 T 0 (cf. Lemma 7.4), the Then, using that, for any n and k ≤ 2n, T2n 2n n number of such paths is at most

k+1 0 T2t+1 T2n−2t−2 ≤ 2(k + 3) exp {−

2t≤2n(2/3)

×

2≤2t≤2n(2/3)

Co k 2 } n

exp {−Co k 2

1 1 0 − }T2t0 T2n−2t−2 t n

Co k 2 n 1+3/2 0 } 3 T2n ≤ C3 (k + 3) exp {− n n 2/3 1 Co k 2 (1/u − 1)} du × exp {− u 3/2 (1 − u)3/2 n 0 k 0 Co k 2 ≤ C4 √ T2n }. exp {− n n

(71) (72)

To derive the last line, we have used the fact that Co k 2 /2n ≥ 2Co3 to bound the integral in (71). If now 2t ≥ 2n(2/3), then 2t > 2n(2/3). And the path can be described as follows. Between 0 and 2t , the path goes from 0 to 0. Then, the path goes from −1 to k without touching k + 1 in 2n − (2t + 1) steps with 2n − (2t + 1) ≤ 2n(2/3). Thus an upper bound for the number of such paths can be obtained as above. Formulas (70) and (72) finally imply formula (69). Set now n = um. There exists C5 independent of n and m, such that k k T˜2n(+1) T˜2m−2n(−1) 0 T0 T2n 2m−2n

1 0 1 Co k 2 1 + . ≤ C5 exp − 2m u 1−u

(73)

226


Thus, by (73), one has that exp −

P(max y(t) = k) ≤ C5 ×

Co k 2 2m (1/u

+ 1/(1 − u) 2

n=um 0 T0 T2n 2m−2n 0 T2m

exp {− 2Cmo k } exp {−

2Co k 2 }. m

(74)

The sum in (74) will be divided into three subsums, according to the cases where m/10 ≤ n ≤ 9m/10, n ≤ m/10 or n ≥ 9m/10. For m/10 ≤ n ≤ 9m/10, we can use Stirling’s formula to obtain that this subsum can be bounded from above by a term similar to the announced bound (67), since m/10≤n≤9m/10

C 0 0 0 T2n T2m−2n ≤ √ T2m . m

When n ≤ m/10, we use the fact that exp −

Co k 2 1 2 (n

+

1 m−n )

k2

n≤m/10

exp {− 2Cmo }

2

1 exp − Co2k ( n1 + m−n )

2 exp {− 2Cmo k }

0 0 T2m−2n ≤ T2n

n≤m/10

(75)

/ . k2 . Thus ≤ exp − Co 4n C k2 C o 6 exp − T0 4n n 3/2 2m

C7 0 ≤ √ T2m , m

(76)

for some constants C6 , C7 independent of m, by a straightforward √ comparison with an integral. Here the constant C7 does not depend on m, as k ≥ 4Co m. We can obtain by m symmetry a similar bound for the sum over n ≥ 9m 10 (as m − n ≤ 10 ). Combining then (76), (75) and (74) leads to (67). 2nd Case. A specified number of sub-Dyck paths. We can now finish the proof of Lemma 7.10. Here it is enough to prove that there exist C˜ o , Co , independent of m and T ∈ T(m), 0 and class T = ( p + such that, under the uniform distribution on trajectories of T2m 1, m o , . . . , m p ), one has √ C˜ k2 P(max y(t) = k) ≤ √ o exp {−Co }, if k ≥ 4Co m. 2m m

(77)

p Here Co is the same constant as in (67). Set αi = m i /m so that i=0 αi = 1. Obviously, √ √ if k ≥ Co m then k ≥ Co m i . Then, by the above computations, one has that P(max y(t) = k) = P(∃i ≤ p , max yi (t) = k) 2 2 p o k ( 1 −1) ok − C2m C˜ o − C2m 1 2 αi ≤ √ e √ e αi m i=0

p

C k2 C k2 C2 o o ≤ √ exp − exp − 2m 4m m i=0

1 , (78) − 1 αi2


227

2 √ 1 ok where in the last line we have used that 1/ αi ≤ C8 exp C4m − 1 as k ≥ αi2 √ 4Co m and αi ≤ 1. Note that the constant C2 does not depend on p . Then, (77) holds since one has that 2 3 p Co k 2 1 exp − −1 ≤ (q + 1) exp {4Co3 (1 − q 2 )} ≤ A, 2 4m α i q≥1 i=1 by using the fact that the number of the αi ’s in any interval [1/(q + 1), 1/q], with q ≥ 1, p is not greater than q + 1 (since i=0 αi = 1). Thus A is a constant independent of p . Note that this estimate holds for any value of the αi ’s also. This finishes the proof of Lemma 7.10. Remark 7.12. The investigation of higher moments is a mimicking of the arguments of Sect. 6 and [11] (p. 42). This is not detailed further. Remark 7.13. In the case where 0 < θ < σ , it suffices to observe that ρθ < 2σ . Thus, all the results of Sects. 4 to 7 show that contribution of paths having at least one nonreturned edge (l > 0) is negligible in the expectation and higher moments, for any scale 1 0 is some fixed constant. This problem has received a lot of attention in the last decade, as our historical review below will show. In particular it has been recognized that the energy regime given by

230

F. Bethuel, G. Orlandi, D. Smets

(H0 ) allows for the formation of topological defects called vortices. It has also been recognized that the dynamics of these vortices is non-trivial accelerating time by a factor |log ε|, that is considering the functions uε (z, s) = u ε (z, s|log ε|),

z = x + i y ≡ (x, y) ∈ R2 .

The next theorem, proved in [3, 4], gives a precise meaning to the notion of vortices in this setting, as well as a first description of their dynamics. Theorem 1 ([3, 4]). Assume (u ε )ε>0 verifies (PGL)ε and (H0 ). Then, for a subsequence εn → 0 we have l(s) z − ai (s) di (s) uεn (z, s) → u∗ (z, s) = exp[i( c(s), z + b(s))], |z − ai (s)|

(1)

i=1

where, for i = 1, . . . , (s), ai (s) ∈ R2 , di (s) ∈ Z, b(s) ∈ [0, 2π ) and c : R+ → R2 is a lipschitz function. The convergence in (1) is uniform on every compact subset of R2 × R+ \ v, where l(s)

v = ∪s>0 ∪i=1 {(ai (s), s)}. Moreover, the trajectory set v is a closed, 1-dimensional rectifiable subset of R2 × R+. We proved moreover that the numbers l(s) and di (s) are uniformly bounded by a constant depending only on M0 , and that, except for a finite number of times,1 di (s) = 0.

(2)

It is worthwile to notice that the limiting map u∗ (·, s) has modulus 1, hence with values in the circle S 1 , but is singular at the points ai (s) when di (s) = 0. In this case, it also has diverging local Dirichlet energy. The points ai (s) are called the vortices at time s, and the integers di (s) their degrees: they correspond to the winding numbers of the limiting map u∗ (., s) around the vortices ai (s). The number b(s) ∈ R corresponds to a constant phase shift, and the vector c(s) ∈ R2 is reminiscent of a wavenumber.2 The set v describes the evolution in time of the set of vortices, and therefore we refer to it as the trajectory set. The main results of this paper provide a complete description of the trajectory set v. We first have Theorem 2. There exists a finite number of times 0 = τ0 < τ1 < · · · < τq < τq+1 = +∞ such that i) The number of vortices (s) ≡ k is constant on each interval (τk , τk+1 ), for k = 0, . . . , q. ii) The restriction of v to R2 × (τk , τk+1 ) is a disjoint union of k smooth one dimensional graphs. More precisely, relabelling possibly the points a1 (s), . . . , ak (s), 1 Which are among the times τ , . . . , τ of Theorem 2. q 1 2 Notice that functions c and b depend only on the time variable s, but not on the space variable z. The

function c can be directly deduced from the initial value, for instance by Fourier transform. It accounts for persistence of low frequency oscillations in the phase over the diverging time period considered, namely t = s|log ε|. The possible presence of low frequencies is of course related to the fact that the domain R2 is unbounded.

Dynamics of Multiple Degree Ginzburg-Landau Vortices

231

R+

τ4

+1

τ3 τ2

+1 +1

τ1

−1

−1 −1 +3 R2

Fig. 1. An example of trajectory set

their degrees di (s) = di are constant in (τk , τk+1 ), and their trajectories are given by the system of ordinary differential equations di2

dai (s) = −∇ai W (a1 , . . . , ak ) + di c(s)⊥ , ds

i = 1, . . . , k ,

(3)

where W is the Kirchhoff-Onsager function defined as W (a1 , . . . , ak ) = −2

k

di d j log |ai − a j | .

(4)

i= j=1

The times τ1 , . . . , τq were already identified in [4] as the only times of dissipation in an appropriate asymptotic sense. More precisely, the following was proved there (Theorem 4 and Corollary 3.1). Theorem 3 ([4]). For s ∈ / {τ0 , . . . , τq }, (s)

vsεn (x) ≡

eεn (uεn (x, s)) d x vs∗ = π di2 (s)δai (s) |log εn |

(5)

i=1

in the sense of measures on R2 , and |∂t u εn | d xds ω∗ = π 2

q (τ k)

βik δ(ai (τk ),τk ) ,

(6)

k=0 i=1

in the sense of measures on R2 × R+ , where βik ∈ N. Since the total energy π di2 (s) is quantized and non-increasing (see [4]), it is also piecewise constant. The times τ1 , . . . , τq correspond therefore to the times of energy loss, where dissipation concentrates. The points (ai (τk ), τk ) for which βik = 0 are called the dissipation points. At this stage, we have completely described the trajectories inside the intervals (τk , τk+1 ). The next step is to understand the behavior of the trajectories accross the dissipation times. Since we already know by Theorem 1 that v is closed, the points

232


(ai (τk ), τk ) are the only possible endpoints of the trajectories in (τk−1 , τk ) and (τk , τk+1 ). For a given point (ai (τk ), τk ), let C1− , . . . , Cl−− and C1+ , . . . , Cl++ denote the vortices trai

i

jectories respectively in R2 × (τk−1 , τk ) and R2 × (τk , τk+1 ) for which (ai (τk ), τk ) is an endpoint. Accordingly, let d1− , . . . , dl−− and d1+ , . . . , dl++ be the degrees of the correi

i

sponding vortices. It follows from (5), (6) and the formula for the evolution of the energy in localized form (see e.g. [4], Prop. 3.3), that βik

−

+

j=1

j=1

li li − 2 = (d j ) − (d +j )2 .

(7)

We say that a point (ai (τk ), τk )) is a regular point of the trajectory set if v is a lipschitz graph over s in the neighborhood of (ai (τk ), τk ), or equivalently if li− = li+ = 1. If not, we say that (ai (τk ), τk ) is a branching point. Theorem 4. A point (ai (τk ), τk ) is a branching point if and only if it is a dissipation point. In this case, we have −

li

+

d− j

= di (τk ) =

j=1

d +j

(Conservation of the degree),

(8)

j=1

−

li

li

+

2 (d − j )

≥

di2 (τk )

≥

j=1

li

(d +j )2 (Energy decrease),

(9)

j=1

where the first (resp. second) inequality in (9) is strict whenever li− ≥ 2 (resp. li+ ≥ 2). In particular, −

li j=1

+

li − 2 (d j ) > (d +j )2 . j=1

Part of the statements in the previous theorems have already been known in the case |di | = 1 (see the historical review below). An important novelty here is that there are no restriction on the di ’s. One may wonder whether multiple degrees may be really observed as limits of solutions to (PGL)ε . This question is positively answered in Sect. 5.3. For a given M0 , the number of integer solutions to (8)–(9) is quite large. We may classify them into four different classes of increasing complexity. First, there are collisions with annihilation, corresponding to li+ = 0 (an example of such a collision is provided by Fig. 2a). There are also collisions without annihilation nor splitting: here li+ = 1, and li− ≥ 2 (see Fig. 2b). Next, there are splittings of single multiple degree vortices, for which li− = 1 and li+ ≥ 2 (see Fig. 2c). The remaining solutions to (8)–(9) correspond to simultaneous collisions and splittings, for which li+ ≥ 2 and li− ≥ 2 (see Fig. 2d). In Sect. 5.3, we will show examples in the classes a, b and c which may be realized through limits of solutions to (PGL)ε . In a related direction, a natural question is to know if the positions and degrees of the vortices at some time s0 completely determine their positions and degrees at future times. Whereas collisions are determined by singularities in the ordinary differential equation (3), splittings are not, and clearly Theorem 2 does not settle the question of the


a

b

233

c

d

Fig. 2. Four classes of dissipation points

occurrence of such splittings. More precisely, the dissipation times τ1 , . . . , τq as well as the number of vortices and their degrees involved after splittings are not inferred in a constructive way. Notice in particular that the algebraic relations (8) and (9) do not have in general a unique solution.3 As a matter of fact, there is no hope to determine the complete future trajectories by knowing the positions and the degrees at some fixed time: an important part of the relevant information is lost in the limiting procedure. This issue will be discussed in more details in Sect. 5.3, when we construct an example of splitting. In Sect. 6, we study the asymptotic behavior of the trajectories near a branching point. We show that, after a suitable parabolic rescaling centered at the collision point, vortices converge to the set of critical points of W restricted to the manifold M= d 2j z j z¯j = ∓4 i± , that is those for which ∇ j W = − 21 d 2j z j . Such critical points have already attracted attention in the context of fluid dynamics, in particular for rotating stationary configurations of vortices (see e.g. Palmore [16]). A brief historical review. The first works on the dynamics of Ginzburg-Landau vortices, in particular by Neu [13, 14], Peres & Rubinstein [17], Pismen & Rubinstein [18], and E [7], were based on formal matched asympotics. In these works, the vortices are described as the nodal set of the complex field u ε . Most of the first rigorous results deal with (PGL)ε on a simply connected bounded domain with fixed Dirichlet or Neumann boundary conditions: in that case the interaction energy W has to be modified appropriately in order to take into account the boundary datum. In [19], Rubinstein & Sternberg rigorously studied the dynamics of a single vortex, under the a priori assumption on the full solution that the nodal set consists of exactly one point for all positive time: they proved that the vortex speed is of order |log ε|−1 in the original time scale. Lin [11] extended their result to the case of distinct vortices of equal degree +1, removing the technical a priori assumption of [19] by a set of more natural assumptions on the initial data, among which the energy bound eε (u 0ε ) ≤ π |log ε| + O(1) as ε → 0. (10)

Concerning the dynamical law (3), the first mathematical proofs are due to Jerrard & Soner [9] and Lin [11] independently. In [12], Lin established (3) under the assumptions of [11]: in this case c = 0, di = +1, and it is therefore easily seen that the solutions of (3) exist for all time. In [9], Jerrard & Soner were able to handle the 3 Except for collisions involving clusters of ±1-vortices, with total degree equal to 0 or ±1.

234


case of vortices of degrees +1 and −1, under a set of assumptions on the initial data involving as in [11, 12] the energy bound (10). In this case, c still vanishes, but (3) has a finite life span (corresponding to our first dissipation time τ1 ) if at least two vortices have different signs. The convergence and the dynamical law was there established up to τ1 . In these papers, vortices are identified as concentration points for the energy density. Upper and lower energy estimates are crucial in their approach, an important point being that the energy is bounded off the vorticity set. In [20, 21], Sandier & Serfaty and Serfaty proposed a somehow different proof of the result of [9], introducing an abstract theory for −convergence of gradient flows. In [21], it is also shown that if E ε (u 0ε ) ≤ π |log ε| + |log ε|(log |log ε|)−β for some β > 1, then (10) is met after translating time by some Tε ≤ C log |log ε|. Concerning collisions and annihilations of vortices, the first rigorous result was obtained by Baumann, Chen, Phillips & Sternberg [1], who showed that for fixed ε and under some natural energy bounds, the solution to (PGL)ε on R2 converges to a unitary constant as time goes to infinity: in particular, if vortices exist they annihilate in finite time. A localized quantitative version of annihilation was obtained in [3] for confined clusters of vortices of total degree zero, and independently in [21] for confined ±1 dipoles. Concerning technical ingredients, in this context the important algebraic relation (24), as well as the energy quantization of energy which may be derived from it, was first introduced in [6] in the elliptic case. It was then extended independently in [4] and [21] to the parabolic setting. Although it might not be obvious at first sight, inequality (10) is a rather strong well-preparedness assumption, since the minimal energy necessary for the creation of a +1 or a −1 vortex (which plays the role of the topological ground state) is exactly π |log ε| + O(1). A first restriction related to inequality (10) is that it does not allow for initial data with multiple degree vortices, nor diverging phase energy. A second important restriction was already pointed out in [9] Remark 2.2: inequality (10) relates the energy to the degrees, and it is not clear at first that this relation remains after collisions. The assumption (H0 ), which was our framework in [3, 4] and here, was motivated by the possibility to encompass the previous restrictions. We also decided to study (PGL)ε on the whole R2 mainly for two reasons. First, this avoids some technicalities related to the boundary, and allows to use more freely the scale invariance of the equation. Second, it permits the phase and the vortices to interact, as shown by the presence of the c term in (3). In [3], we proved Theorem 1, which gives a precise meaning to the notion of vortices in our framework. In contrast to the results in [11, 9, 20], our statements require the use of subsequences. This intrinsic restriction is related to the wider class of possible initial data that (H0 ) allows for, as well as the fact that the occurrence of splittings cannot be inferred once the limit in ε has been taken. [3] also contains the previously mentioned result on annihilation. In [4], we proved Theorem 3 and established the results in Theorem 2 on (τk , τk+1 ) under the additional assumption that di (s) = ±1 for all i = 1, . . . , (s) and some s ∈ (τk , τk+1 ). As emphasized in [4], the main obstacle on the way to Theorem 2 is the possible splitting and recombination of multiple degree vortices without energy loss, i.e. outside the dissipation times. The main point in our proof of Theorem 2 is to show that they do not occur. Strategy for the proofs. The arguments involved in the proofs of Theorem 2 and 4 do not rely on additional results about (PGL)ε , but instead on properties of v established in [4] as well as new algebraic properties of W.


235

For the proof of Theorem 2 i), we need to show that for s0 ∈ / {τ1 , . . . , τk } and (ai (s0 ), s0 ) ∈ v, for s close to s0 and for some neighborhood Bi of ai (s0 ), Bi contains only a single vortex. In order to analyze the size and the spreading of the possible cluster of vortices emanating from (ai (s0 ), s0 ), we consider the variance 2 ˆ i (s)|2 a j (s)∈Bi d j (s)|a j (s) − a f i (s) = , 2 a j (s)∈Bi d j (s) where aˆ i (s) denotes the barycenter of the cluster of vortices in Bi , with weights given by the energy densities d 2j (s), namely 2 a j (s)∈Bi d j (s)a j (s) aˆ i (s) = . 2 a j (s)∈Bi d j (s) Our goal is to prove that f i (s) vanishes identically in a neighborhood of s0 , by means of a Gronwall type inequality. In order to define more precisely Bi , we recall that in [4] (Theorem 5 and identity (9)), we have shown that given s0 > 0 and i ∈ {1, . . . , (s0 )}, there exists some s0 > 0 and ri (s0 ) > 0 such that vs ∩ B(ai (s0 ), ri (s0 )) \ B(ai (s0 ), ri (s0 )/2) = ∅

(11)

for all s in [s0 − s0 , s0 + s0 ], where vs = {a1 (s), . . . , a(s) (s)}. If s0 ∈ / {τ1 , . . . , τq }, we may assume, decreasing possibly s0 , that [s0 − s0 , s0 + s0 ] contains none of the dissipation times τk . In this case, the degrees di (s) of vortices emanating from or colliding at ai (s0 ) have been shown (see Theorem 5) to be constrained by the algebraic equilibrium relation

2 d 2j (s) = d j (s) = di2 (s0 ) (12) a j (s)∈Bi

a j (s)∈Bi

for all s in [s0 − s0 , s0 + s0 ], where Bi = B(ai (s0 ), ri (s0 )). If s ∈ / {τ1 , . . . , τq }, the computation of f i (s) follows from the identities

a j (s) d +2π |x|2 dvs∗ = 4π dk (s)d j (s)Re d j (s)a j (s), c(s)⊥ ds Bi a (s) − a (s) k j a (s)∈B / a j (s)∈Bi

k i a j (s)∈Bi

(13) and

d 1 aˆ i (s) = [ c(s)⊥ + 2dk (s)∇a j (log |a j (s) − ak (s)|)] ds di (s0 ) a (s)∈B /

(14)

k i a j (s)∈Bi

for s ∈ [s0 − s0 , s0 + s0 ], which we proved in [4]. In view of the expression for vs∗ in (5), we are therefore led to

a j (s) − aˆ i (s)

d π d 2j (s)|a j (s) − aˆ i (s)|2 = 4π dk (s)d j (s)Re ds ak (s) − a j (s) a (s)∈B / a j (s)∈Bi

k i a j (s)∈Bi

+2π di (s0 )aˇ i (s) − aˆ i (s), c(s)⊥ , (15)

236


where aˇ i (s) denotes a second type of barycenter, namely a j (s)∈Bi d j (s)a j (s) · aˇ i (s) = a j (s)∈Bi d j (s)

(16)

We refer to aˆ i (s) as the center of mass of the cluster and to aˇ i (s) as the topological center of mass of the cluster. The identity (15) leads to Proposition 1. The function f i (s) is lipschitzian and verifies

| f i (s)| ≤ C(M0 , s0 ) |aˆ i (s) − aˇ i (s)| + f i (s) on [s0 − s0 , s0 + s0 ],

(17)

where C(M0 , s0 ) depends only on s0 and M0 . In order to integrate the differential inequality (17), we need some control on the term |aˇ i (s) − aˆ i (s)|. For arbitrary configurations of points and degrees, there is no reason that aˆ and aˇ should be close. However, for simple examples of critical points of W we noticed that they are equal. This observation led us to the following identity. Lemma 1. Consider points z 1 , . . . , z ∈ C, and real numbers d1 , . . . , dl whose sum is non zero. Then the following identity holds: 2 dj z j ∇z j W (z 1 , . . . , z )z 2j djz j = + , (18) dj ( d j )2 2( d j )2 where the sums are meant for j ranging from 1 to . Specifying formula (18) with z j = a j (s) − aˆ i (s), and in view of (12), we obtain Proposition 2. It holds |aˇ i (s) − aˆ i (s)| ≤ C(M0 )|∇W ({a j (s)} j∈I (s) )| f i (s),

(19)

where C(M0 ) depends only on M0 and where I (s) = { j ∈ {1, . . . , (s)}, a j (s) ∈ Bi }. Combining (17) and (19), we finally derive | f i (s)| ≤ C(M0 , s0 )(1 + |∇W ({a j (s)} j∈I (s) )|) f i (s) .

(20)

Since f i (s0 ) = 0, Gronwall’s lemma would then allow to conclude that f i (s) ≡ 0 on a neighborhood I of s0 provided that |∇W ({a j (s)} j∈I (s) )| ds < +∞. (21) I

As a matter of fact, we will even prove that |∇W ({a j (s)} j∈I (s) )|2 ds < +∞,

(22)

I

using the gradient-flow type properties of the ode (3). This last statement may seem rather odd at first reading, since the ode (3) is precisely what we wish to show. Our actual argument is by induction on di (s0 ). Indeed, when |di (s0 )| = 2, the splitting may only create ±1 vortices, for which we already established (3) in [4]. Similarly, if |di (s0 )| = k, the splitting may only involve vortices of degree at most k − 1 in absolute value, which are handled by the inductive argument. To establish (22) in view of the gradient-flow properties, we invoke


237

Proposition 3. We have, for any s > 0,

|W ({a j (s)} j∈I (s) )| ≤ C(M0 ) | log dist(s, {τ0 , . . . , τq })| + 1 .

(23)

The proof of Proposition 3 relies only on the special form of W and on the κ-confinement result of [4] which we recall next. Theorem 5 ([4] Theorem 5). Let s > 0, a ∈ R2 , r > 0 and 0 < κ ≤

1 2

be such that

∅ = vs ∩ B(a, r ) ⊂ B(a, κr ). There exist constants 0 ≤ κ1 ≤ 0 < κ ≤ κ1 and

1 4

and γ1 > 0 ,depending only on M0 , such that if

dist(s, {τ0 , . . . , τq }) ≥ γ1 κ 2 r 2 , then

i∈I (s)

di2 (s) =

2 di (s) ,

(24)

i∈I (s)

where we have set I (s) = {i ∈ 1, . . . , (s) | ai (s) ∈ B(a, r )}. Once it is proved that f i (s) ≡ 0, statement i) of Theorem 2 follows straightforwardly. Statement ii) of Theorem 2 is a direct consequence of Eq. (14) for the center of mass aˆ i (s) and the fact that there is no splitting. Remark. One may consider more generally the class of ode’s d m i ai (s) = −∇ai W (a1 , . . . , ak ) + di c(s)⊥ , ds

i = 1, . . . , k ,

(25)

where the coefficients m i > 0 may be thought of as masses. In our case, the masses and the degrees are constrained by the relation m i = di2 . Defining for the general case the center of mass a(s) ˆ as m j a j (s) a(s) ˆ = , mj then (17) still holds, with a(s) ˇ being as before the topological center of mass. However, inequality (19) in Proposition 2 does not hold in general.In particular, we do not know if a cluster verifying the algebraic equilibrium relation ( di2 ) = di2 may expand or not for the ode (25). Our arguments therefore heavily rely on the quatization of energy m i = di2 . Concerning the proof of Theorem 4, it relies essentially on refinements of the arguments involved in the proof of Theorem 2. The outline of this paper is as follows. In Sect. 2, we derive some important properties concerning the Kirchhoff-Onsager functional. In Sect. 3, we analyze the growth of cluster of vortices, and in particular give the proofs of Proposition 1, 2 and 3. Section 4 contains the proof of Theorem 2 and 4. In Sect. 5, we provide examples of branching points which are limits of solutions to (PGL)ε . Finally in Sect. 6 we describe the behavior of trajectories near branching points.

238


2. Some Properties of the Interaction Energy We consider, for distinct points z 1 , . . . , z ∈ C and real numbers d1 , . . . , d = 0, the interaction energy4 W (z 1 , . . . , z ) = − di d j log |z i − z j |2 = − log(z i − z j )(z i − z j ). i= j

i= j

This functional appears in several topics of mathematical physics. In this section, we will derive various properties of W , most of which will be used later in this paper. 2.1. Properties of the gradient of W . For fixed d1 , . . . , d = 0, the function W is clearly well defined and smooth on the open subset of C consisting of -tuples of distinct points. It possesses some elementary symmetry properties, namely Lemma 2.1. (Invariance by rotations, translations and dilations). We have ∇W (α(z 1 − β), . . . , α(z − β)) =

1 ∇W (z 1 , . . . , z ) α¯

for any α ∈ C∗ and β ∈ C. The gradient ∇W has a simple expression in complex notation Lemma 2.2. For any k ∈ {1, . . . , } we have dj 1 ∂W ∇z k W = = −dk . 2 ∂ z¯ k z j − zk

(2.1)

j=k

As a consequence,

∇k W, z k = Re ∇k W z¯k = dk d j . k=1

(2.2)

j=k

k=1

Proof. We observe that for a smooth function h on R+ , we have ∇z h(|z|2 ) = 2h (|z|2 )z = 2h (z z¯ ) and the conclusion (2.1) follows.

∂ ∂ (z z¯ ) = 2 h(z z¯ ) , ∂ z¯ ∂ z¯

(2.3)

The next formula is the starting point for the proof of Lemma 1. Lemma 2.3. We have

k=1

dk z − zk

2 =

k=1

∇z W dk2 k + . 2 (z − z k ) z − zk

(2.4)

k=1

4 Clearly W depends also on the d ’s, although this is not reflected in our notation W (z , . . . , z ). In most 1 i places the di ’s are implicit from the context, i.e. the degrees of the vortices. However, in case of possible ambiguity, we will write W ({(z i , di )}1≤i≤ ).


239

Proof. Expanding the l.h.s. of (2.4), we obtain 2

dk dk2 dk d j . = + z − zk (z − z k )2 (z − z k )(z − z j ) k=1

(2.5)

j=k

k=1

On the other hand, we have the identity 1 1 1 1 ( − ). = (z − z k )(z − z j ) zk − z j z − zk z − zj Inserting (2.6) into (2.5) we therefore derive ⎛ ⎞ 2

2 dk dk dj ⎝ ⎠ dk , = +2 z − zk (z − z k )2 zk − z j z − zk k=1

k=1

k=1

(2.6)

(2.7)

j=k

and (2.4) follows.

2.2. Proof of Lemma 1. We expand each of the terms involved in equality (2.4) as power series of 1z . 2 dk 1 − (z k /z) 1 1 1 2 dk + dk z k + O( 2 ) = 2 z z z 1 2 2 1 dk + 3 ( = 2 dk )( dk z k ) + O( 4 ), z z z dk2 dk2 1 = 2 (z − z k )2 z (1 − (z k /z))2 1 2 2 1 = 2( dk2 ) + 3 ( dk z k ) + O( 4 ), z z z

∇z W 1 ∇z k W k = z − zk z 1 − (z k /z) 1 1 1 1 = ( ∇z k W ) + 2 ( ∇z k W z k )+ 3 ( ∇z k W z k2 )+O( 4 ), z z z z as |z| → ∞. Identifying the coefficients of the expansion, in view of (2.4), we are led to

dk z − zk

2

1 = 2 z

∇z k W = 0 ,

(2.8)

k=1

(

dk )2 =

k=1

k=1

dk2 +

∇z k W z k ,

(2.9)

k=1

2( dk )( dk z k ) = 2 dk2 z k + ∇z k W z k2 . k=1

k=1

k=1

k=1

(2.10)

240


The first equality is a result of the symmetry properties of W , whereas the last one immediately yields (18). 2.3. Critical points of W . In this subsection we study critical points of W , i.e. configurations {(z i , di )}1≤i≤ verifying ∇z k W (z 1 , . . . , z ) = 0 .

(2.11)

Alternatively, it follows from Lemma 2.3 that the configuration is critical if and only if j=k=1

dj =0 zk − z j

∀k = 1, . . . , .

(2.12)

We refer to (2.12) as the geometric equilibrium condition. Although they do not enter directly in the proofs Theorem 2 and 4, critical configurations are useful for the understanding of the splitting phenomenon (see Sect. 5. They are also interesting by themselves, since they are stationary solutions for the dynamical law (3). They play a role in different topics. For instance, in fluid dynamics they describe stationary vortex solutions for the 2D Euler equation on the whole plane, and related examples may be found in electrostatics. As a consequence of identities (2.9) and (2.10), we have Proposition 2.1. Assume that W possesses a critical point (z 1 , . . . , z ), then necessarily 2 dk2 = dk (2.13) and

2 d zk dk z k = k2 · dk dk

(2.14)

Identity (2.13), which we refer to as the algebraic equilibrium relation, was already found by Kirchhoff [10] in the context of vortex solutions for the Euler equation. In the framework of Ginzburg-Landau theory, the same relation was derived by Comte and Mironescu [6] for critical points of the Ginzburg-Landau energy E ε , and exploited in [21, 4] in the asymptotics for (PGL)ε . It expresses also the conformal invariance of W , that is W (z 1 , . . . , z ) = W (λz 1 , . . . , λz )

∀λ ∈ C∗

(2.15)

if and only if (2.13) is verified. Identity (2.14) can be interpreted as the equality of the center of mass and the topological center of mass, as defined in the introduction, in other words we have zˆ = zˇ

(2.16)

for critical points (z 1 , . . . , z ). Solutions to the algebraic equilibrium relation. Set D = i=1 di . Notice first that (2.13) implies ≤ D 2 , so that there is no solution to (2.13) for D = 0, and only the trivial


241

single point solution for D = 1. More generally, let n(D) be the number of different solutions to (2.13). We observe that n(D + 1) > n(D), i.e. n(D) is strictly increasing. Indeed, there is of course the trivial single degree D solution, and for any solution with total degree D one can construct a solution with total degree D + 1 by adding D + 1 degree +1 vortices, and D degree -1 vortices. In particular, for D ≥ 2, (2.13) has at least one solution, namely (D 2 + D)/2 vortices of degree +1 and (D 2 − D)/2 vortices of degree −1. In order to illustrate the previous discussion, notice for instance that for D = 2, the only solutions to (2.13) are the trivial solution (+2), and, up to permutations, (−1, +1, +1, +1). For D = 3, besides the already mentioned solution (−1, −1, −1, +1, +1, +1, +1, +1, +1) and trivial solution (+3), there is also the solution (−1, +2, +2) and, splitting one of the +2 degrees, we obtain additionally (−1, −1, +1, +1, +1, +2). For general D, the number of solutions to (2.13) may be quite large. Indeed, the previous example D = 3 shows that the set of solutions has a sort of tree structure. Once solutions to (2.13) are found, it remains to determine the corresponding geometric equilibrium configurations, i.e. the solutions to (2.12). Solutions to the geometric equilibrium equation. For D = 2, the following solution to (2.12) is known, where the largest triangle is equilateral. +1

1

+1

+1

This construction can be generalized to arbitrary D ≥ 2 as follows. +1

+1 +1 d

+1 +1

+1

+1

where the origin is a vortex of degree −d and where 2d + 1 vortices of degree +1 are located at the vertices of a regular polygon, so that D = d + 1. For D = 3, besides the above solution, there is also the colinear configuration. +2

1

+2

242


Notice also that given a solution {(z i , di )}1≤i≤ to (2.12), one obtains further solutions of the form {(z i , kdi )}1≤i≤ for any k ∈ Z∗ . The number of solutions for large D is presumably large, however, for D = 2 we have Proposition 2.2. Assume D = 2, = 1, and that z 1 , . . . , z is a critical point of W = − di d j log |z i − z j |2 . Then necessarily = 4, and up to a rigid motion and a dilation (and to a permutation of indices), d1 = −1, d2 = d3 = d4 = +1, and (z 1 , z 2 , z 3 , z 4 ) = (0, 1, exp(2iπ/3), exp(4iπ/3)) . Proof. In view of the discussion in the previous section, we may assume that (d1 , d2 , d3 , d4 ) = (−1, +1, +1, +1) and, up to translations, rotations and dilations, z 1 = 0 and z 2 = 1. The system (2.12) is then reduced to 1 1 + = −1 z3 z4

&

1 1 + = +1. 1 − z3 1 − z4

In particular, z 32 + z 3 + 1 = 0, so that either z 3 = exp(2iπ/3) and z 4 = exp(4iπ/3) or viceversa. The proof is therefore complete. 2.4. Clusterization and computation of W . In this subsection we present two elementary lemma of somewhat combinatorial nature, which, combined with Theorem 5, will yield the proof of Proposition 3. They already appeared in [4]. We provide their proofs for completeness. Lemma 2.4. Let X be a metric space, and consider distinct points a1 , . . . , a in X . Let δ0 > 0 and 0 < κ ≤ 21 be given. Then there exists δ > 0 such that κ δ0 ≤ δ ≤ ( )− δ0 2

(2.17)

and a subset {a j } j∈J of {ai }1≤i≤ such that ∪i=1 B(ai , δ0 ) ⊂ ∪ j∈J B(ai , δ)

(2.18)

and dist(ai , a j ) ≥ κ −1 δ

∀i = j in J.

(2.19)

Proof. The proof is by iteration, in at most steps. First, consider the collection {ai }1≤i≤ . If (2.18), (2.19) is verified with δ = δ0 then there is nothing else to do. Otherwise, take two points, say a1 , a2 such that dist(a1 , a2 ) ≤ κ −1 δ0 , consider the collection a2 , a3 , . . . , al , and set δ = 2κ −1 δ0 . If (2.18) is verified, we stop. Otherwise we go on in the same way. If the process does not stop in − 1 steps, at the th step we are left with one single ball of radius δ = ( κ2 )− δ0 , and (2.18) is void. Lemma 2.5. Let X be a metric space, let 0 < κ < 1/2 and 0 < r < Rmax be given. Consider m distinct points b1 , . . . , bm in X such that dist(bi , b j ) ≥ κ −1r, Then one of the following two situations holds:

for i = j.


243

i) inf i= j dist(bi , b j ) ≥ Rmax . n J with n < m, for each i ∈ {1, . . . , n} ii) There exists a partition {1, . . . , m} = ∪i=1 i κ −m/2 some b˜i ∈ ∪ j∈Ji {b j } and r < R ≤ ( 2 ) Rmax such that m n ∪i=1 B(bi , r ) ⊂ ∪i=1 B(b˜i , R)

dist(b˜i , b˜ j ) ≥ κ −1 R,

for any i = j ∈ {1, . . . , n},

(2.20) (2.21)

and, for every d1 , . . . , dm ∈ Rm , m n di d j log dist(bi , b j ) − Di D j log dist(b˜i , b˜ j ) i= j=1 n

−

i= j=1

di d j log R ≤ C| log κ|,

(2.22)

p=1 i= j∈J p

where Di =

j∈Ji

d j , and where the constant C depends only on m and supi |di |2 .

Proof. If i) holds there is nothing left to prove. Otherwise set δ0 = inf dist(bi , b j ). i= j

Applying Lemma 2.4 we obtain a subset {b˜1 , . . . , bñ } of {b1 , . . . , bm } and δ0 ≤ δ ≤ (κ/2)−m δ0 such that m n ∪i=1 B(bi , δ0 ) ⊂ ∪i=1 B(b˜i , δ)

(2.23)

dist(b˜i , b˜ j ) ≥ κ −1 δ

(2.24)

and ∀i = j.

We choose R = δ. It follows from the definition of δ0 and (2.24) that n < m, whereas (2.20) and (2.21) follow directly from (2.23) and (2.24) respectively. We set, for i = 1, . . . , n, Ji = { j : b j ∈ B(b˜i , R)} and turn finally to (2.22). For i = j in {1, . . . , m} we distinguish two cases: −i, j belong to the same J p , for some p ∈ {1, . . . , n}: then | log dist(bi , b j ) − log R| ≤ C| log κ| which follows from the fact that δ0 ≤ dist(bi , b j ) ≤ 2R ≤ 2(κ/2)−m δ0 . −i ∈ J p and j ∈ Jq , p = q. Then | log dist(bi , b j ) − log dist(b˜ p , b˜q ) | ≤ Cκ. The proof of (2.22) follows then by summation.

244


3. Properties for Cluster of Vortices The purpose of this section is to give the proofs of Proposition 1, Proposition 2 and Proposition 3, which mainly involve properties for clusters of vortices of (PGL)ε . We also propose a local version of Proposition 3 which will be used for the proof of Theorem 4. We begin with Proof of identity 15. We write, |a j (s) − aˆ i (s)|2 = |a j (s)|2 − 2a j (s), aˆ i (s) + |aˆ i (s)|2 .

(3.1)

We have d d d π d 2j (s)a j (s), aˆ i (s) = d 2j (s)a j (s), aˆ i (s) d 2j (s)a j (s), aˆ i (s) ds ds ds a j (s)∈Bi

a j (s)∈Bi

a j (s)∈Bi

d aˆ (s) d aˆ i (s) i = di (s0 ) , aˆ i (s) − aˆ i (s), , ds ds = 0,

(3.2)

whereas, in view of (12) and (14), d aˆ i (s) d 2 d j (s)|aˆ i (s)|2 = 2di (s0 )aˆ i (s), ds ds a j (s)∈Bi

= 2di (s0 )aˆ i (s), c(s)⊥ +2 dk (s)∇a j log |ak (s) − a j (s)| .

(3.3)

ak (s)∈B / i a j (s)∈Bi

Combining (13), (3.1), (3.2) and (3.3) we deduce

d π d 2j (s)|a j (s) − aˆ i (s)|2 ds a j (s)∈Bi a j (s) − aˆ i (s) +2π dk (s)d j (s)Re d j (s)a j (s)− aˆ i (s), c(s)⊥ , = 4π a (s) − a (s) k j a (s)∈B / k i a j (s)∈Bi

a j (s)∈Bi

and the conclusion follows from the fact that di (s0 )aˇ i (s) =

(3.4) d j (s)a j (s).

/ Bi Proof of Proposition 1. By (11), we have for s ∈ [s0 − s0 , s0 + s0 ], for ak (s) ∈ and for a j (s) ∈ Bi , a (s) − aˆ (s) a j (s) − aˆ i (s) j i − Re ≤ C|a j (s) − aˆ i (s)|2 . Re ak (s) − a j (s) ak (s) − aˆ i (s) Therefore, we obtain, for the first term in the right-hand side of (15), a j (s) − aˆ i (s) = di (s0 )aˇ i (s) − aˆ i (s), γ (s) + R(s) , dk (s)d j (s)Re ak (s) − a j (s) a (s)∈B / k i a j (s)∈Bi


where γ (s) =

245

dk (s)∇ak log |ak (s) − aˆ i (s)|

ak (s)∈B / i

and |R(s)| ≤ C(M0 , s0 ) sup |a j (s) − aˆ i (s)|2 ≤ a j (s)∈Bi

Hence, d ds

C(M0 , s0 ) f i (s) . π

π d 2j (s)|a j (s) − aˆ i (s)|2 = 2π d(s0 )aˇ i (s)−aˆ i (s), c(s)⊥ +2γ (s) +4π R(s),

a j ∈Bi

and the conclusion then follows. Proof of Proposition 2. In view of Lemma 1 and the translation invariance of W we have, omitting to write the dependence in s, 2 d j (a j − aˆ i ) ∇a j W ({a j } j∈I (s) )(a j − aˆ i )2 d j (a j − aˆ i ) 2 − = , (3.5) dj ( dj) 2( d j )2 where the sums are taken over points a j (s) ∈ Bi . On the other hand, by (12) we obtain 2 d j (s)(a j (s) − aˆ i (s)) d j (s)(a j (s) − aˆ i (s)) = aˇ i (s) − aˆ i (s) and = 0, d j (s) ( d j (s))2 so that (3.5) implies |aˇ i (s) − aˆ i (s)| ≤ C|∇W ({a j (s)} j∈I (s) )|

sup |a j (s) − aˆ i (s)|2

a j (s)∈Bi

and the conclusion follows from the definition of f i (s).

In order to prove Proposition 3, we are led to consider the following situation. Let κ1 and γ1 be given by Theorem 5. Let b ∈ R2 , 0 < κ ≤ κ1 , 0 < R1 ≤ R2 and let b1 , . . . , bm be a collection of m points in R2 such that vs ∩ B(b, κ −1 R2 ) ⊂ ∪mj=1 B(b j , R1 ) ⊂ B(b, R2 ),

(3.6)

|bi − b j | ≥ κ −1 R1 ,

(3.7)

dist(s, {τ1 , . . . , τq }) ≥ (3.8) We set, for j = 1, . . . , m, D j = ak (s)∈B(b j ,R1 ) dk (s). In view of (3.6) and the uniform bound on the number of vortices, we may always assume that m ≤ C(M0 ). We first have γ1 R22 .

Lemma 3.1. If (3.6), (3.7) and (3.8) are satisfied, then m i= j=1

Di D j = 0.

(3.9)

246


Proof. We apply Theorem 5 in two situations: first with a = b and r = R2 , then with a = b j and r = R1 . The first yields

di2 (s) =

ai (s)∈B(b,R2 )

di (s)

2

=

m

ai (s)∈B(b,R2 )

2 Dj ,

j=1

whereas the second gives, for j = 1, . . . , m

di2 (s) =

ai (s)∈B(b j ,R1 )

di (s)

2

= D 2j .

ai (s)∈B(b j ,R1 )

Since

di2 (s) =

ai (s)∈B(b,R2 )

m

di2 (s)

j=1 ai (s)∈B(b j ,R1 )

we deduce m

D 2j =

m

j=1

which is equivalent to the conclusion (3.9).

2 Dj ,

j=1

Lemma 3.2. There exist constants C(M0 ) and 0 < κ(M0 ) ≤ κ1 such that if (3.6), (3.7) and (3.8) hold and if κ ≤ κ(M0 ) then m Di D j log |bi − b j | ≤ C(M0 ).

(3.10)

i= j=1

Proof. We proceed by induction on m, the number of interior balls. When m = 1, there is nothing to prove, whereas when m = 2, the r.h.s. of (3.10) is zero by (3.9), so that (3.10) follows. By induction, we assume that there exist some constants 0 < κ(m − 1) ≤ κ1 and C(m − 1) such that, for any collection of at most m − 1 balls satisfying (3.6), (3.7) and (3.8) with κ ≤ κ(m − 1), (3.10) holds with r.h.s equal to C(m − 1). For a collection of m interior balls satisfying (3.6), (3.7) and (3.8), we apply Lemma 2.5 with b1 , . . . , bm , κ, r = R1 and Rmax = R2 . This yields a new collection of n ≤ m − 1 disjoint interior balls B(b˜ j , R) which satisfy κ R1 ≤ R ≤ ( )−m/2 R2 , 2 ˜ ˜ |bi − b j | ≥ κ −1 R,

(3.11) (3.12)

and moreover m n n ˜ ˜ di d j log |bi − b j | − Di D j log |bi − b j | − di d j log R ≤ C| log κ|. i= j=1

i= j=1

p=1 i= j∈J p


Since κ ≤ κ1 ,

i= j∈J p

247

di d j = 0 by Lemma 3.1, so that

n m ˜ ˜ d d log |b − b | − D D log | b − b | i j i j i j i j ≤ C| log κ|. i= j=1

(3.13)

i= j=1

Since the balls B(b˜ j , R) are disjoints and since b˜ j ∈ B(b, R2 ), it follows that R ≤ R2 . Set b˜ = b, R˜ 1 = R, R˜ 2 = 2R2 and κ˜ = 2κ. One verifies in view of (3.11) and (3.12) ˜ the collection b˜1 , . . . , bñ , r˜1 , r˜2 that (3.6), (3.7) and (3.8) are satisfied for the point b, and κ. ˜ If κ˜ ≤ κ(m − 1), we may apply the inductive hypothesis, since n ≤ m − 1, so that n ˜ ˜ (3.14) Di D j log |bi − b j | ≤ C(m − 1). i= j=1

We choose κ(m) = 2−m κ1 and C(m) = Cm| log κ(m)|. Combining (3.13) and (3.14) we obtain, if κ ≤ κ(m), m d d log |b − b | i j i j ≤ C(m − 1) + C| log κ(m)| ≤ C(m), i= j=1

and the proof is complete.

Proof of Proposition 3. We wish to divide {a j (s) ∈ Bi } into subclusters of maximal size for which we may apply Lemma 3.2 with κ = κ(M0 ). For that purpose, we apply Lemma 2.4 to the points {a1 , . . . , a } ≡ {a j (s) ∈ Bi } , κ = κ(M0 ) and δ0 such that dist(s, {τ1 , . . . , τq }) κ(M0 ) − δ0 = . (3.15) 2 γ1 For j ∈ J we set I j = {k ∈ {1, . . . , }, ak ∈ B(a j , δ)}, and one verifies that (3.6), (3.7) and (3.8) are satisfied with κ = κ(M0 ), b = a j , {bk } = {ak }k∈I j , R2 = δ and R1 sufficiently close to zero. We may therefore apply Lemma 3.2 to obtain, for any j ∈ J , d p (s)dq (s) log |a p (s) − aq (s)| ≤ C(M0 ). p=q∈I j

For p ∈ I j and q ∈ I j with j = j , we have |a p (s) − aq (s)| ≥ (κ(M0 )−1 − 2)δ ≥ δ so that log |a p (s) − aq (s)| ≥ log δ0 ≥

1 log |dist(s, {τ1 , . . . , τq })| − C(M0 ). 2

On the other hand, we clearly have log |a p (s) − aq (s)| ≤ log(2r (s0 )) ≤ 1.

248


Since

=

p=q=1

j∈J p=q∈I j

inequality (23) follows by summation.

+

,

j= j ∈J p∈I j , q∈I j

For the proof of Theorem 4, we need also to consider the case of vortices at the dissipation times. We have the following variant of Proposition 3. Proposition 3.1. Let (ai (τk ), τk ) ∈ v and set i± =

d 2j (s) − di2 (τk )

for s → τk± .

(3.16)

j∈I (s)

Then we have W ({a j (s)} j∈I (s) ) = i± log |s − τk | + O(1)

as s → τk± .

(3.17)

Proof. We use exactly the same construction as in the proof of Proposition 3 above. For j ∈ J, we obtain, by the same argument, d p (s)dq (s) log |a p (s) − aq (s)| ≤ C(M0 ).

(3.18)

p=q∈I j

For the remaining terms, we expand further the computation in the proof of Proposition 3. For p ∈ I j and q ∈ I j with j = j , we have, by the parabolic cone property proved in [3], the definition (3.15) of δ0 and (2.17), |a p (s) − aq (s)| ≤ C(M0 )δ, provided s is sufficiently close to τk . Since on the other hand |a p (s) − aq (s)| ≥ (κ(M0 )−1 − 2)δ ≥ δ we obtain | log |a p (s) − aq (s)| − log δ| ≤ C(M0 ). Since

p= j∈I j

(3.19)

d p (s)dq (s) = 0 for all i, we are led to

j= j p∈I j , q∈I j

d p (s)dq (s) = (di2 (τk ) −

d 2j (s)) = − i± .

(3.20)

j∈I (s)

In view of (3.15) and (2.17), we have log δ = 21 log |s − τs | + O(1). The conclusion then follows from (3.18), (3.19),(3.20) and the definition of W.


249

4. Non-occurrence of Splittings without Dissipation The main purpose of this section is to provide the proofs for Theorems 2 and 4. As mentioned in the introduction, the main point is Lemma 4.1. For any (ai (s0 ), s0 ) ∈ v, with s0 ∈ / {τ1 , . . . , τq } and s0 and ri (s0 ) as in (11), there exists a neighborhood I of s0 such that 1 f i (s) ≡ 2 d 2j (s)|a j (s) − aˆ i (s)|2 = 0 (4.1) di (s0 ) a (s)∈B j

for all s ∈ I, or equivalently

i

vs ∩ Bi = 1

(4.2)

for all s ∈ I. Proof. The proof is by induction. More precisely, for k ∈ N∗ let (Pk ) be the statement that Lemma 4.1 holds for |di (s0 )| ≤ k. First notice that (P1 ) holds. Indeed, for a vortex ai (s0 ) with |di (s0 )| = 1, (4.2) has already been established in [4] Lemma 5.3, and is actually an immediate consequence of (12). For k ≥ 2, we next prove that (Pk ) holds, assuming (Pk−1 ). Let (ai (s0 ), s0 ) ∈ vs0 with s0 ∈ / {τ1 , . . . , τq } and |di (s0 )| = k, and assume by contradiction that for any interval I s0 , (4.2) does not hold on the whole I. Therefore, there exists s1 ∈ [s0 − s0 , s0 + s0 ] such that

(4.3) vs1 ∩ Bi ≥ 2. Assume first that s1 > s0 . Step 1. There exists a time s0 ∈ [s0 , s1 ) such that {vs ∩ Bi } = {vs1 ∩ Bi },

∀s ∈ (s0 , s1 ]

and s

{v0 ∩ Bi } < {vs1 ∩ B1 }. Indeed, first notice that, by identities (12) and (4.3), all vortices in vs1 ∩ Bi have degrees strictly less than k in absolute value, so that the induction hypothesis (Pk−1 ) can be used for these points. In view of (4.2) and conservation of energy (12), the number {vs ∩ Bi } is locally constant, whereas 1 = {vs0 ∩ Bi } < {vs1 ∩ Bi }. Therefore, s0 is the end point of the largest open interval containing s1 and on which the value of {vs ∩ Bi } equals {vs1 ∩ Bi }. In view of Step 1, I (s) may be chosen independently of s, and the vortices a j (s) with j ∈ I (s) may be unambiguously labelled on (s0 , s1 ]. Therefore, without loss of generality we may write {a j (s)} j∈I (s) = {a1 (s), . . . , am (s)}, ∀s ∈ (s0 , s1 ] for some m ∈ N∗ . Since d j (s) is constant, we also write d j ≡ d j (s) for j ∈ {1, . . . , m} and s ∈ (s0 , s1 ]. Step 2. We have s1 |∇W (a1 (s), . . . , am (s))|2 ds < +∞. (4.4) s0

250


In view of Lemma 5.2 of [4], we have, for any s ∈ (s0 , s1 ) and any j ∈ {1, . . . , m}, (s) d 1 c(s)⊥ + 2 dk (s)∇ak (log |ak (s) − a j (s)|)]. a j (s) = [ ds dj

(4.5)

k= j=1

In particular, for j ∈ {1, . . . , m}, d 1 a j (s) = − 2 ∇W (a1 (s), . . . , am (s)) + R j (s), ds dj

(4.6)

where R j is a continuous function bounded by a constant depending only on s0 . The system of m ordinary differential equations (4.6) would be a pseudo-gradient flow for W if R j were equal to zero. Here, we have m d d W (a1 (s), . . . , am (s)) = ∇a j W (a1 (s), . . . , am (s)) a j (s) ds ds j=1 m

=−

j=1

1 |∇a j W (a1 (s), . . . , am (s))|2 d 2j

+ ∇a j W (a1 (s), . . . , am (s)) ≤ −C(|∇W (a1 (s), . . . , am (s))|2 − 1), where C > 0. Integrating on (s0 , s1 ) we are led to s1 s1 d 2 W (a1 (s), . . . , am (s)) ds| + 1) |∇W (a1 (s), . . . , am (s))| ds ≤ C(| ds s0 s0 ≤ C( lim + |W (a1 (s1 ), . . . , am (s1 )) s→s0

−W (a1 (s), . . . , am (s))| + 1) ≤ C(2 sup |W (a1 (s), . . . , am (s))| + 1). s∈(s0 ,s1 ]

The conclusion (4.4) then follows from Proposition 3. s

Step 3. The set v0 ∩ Bi is reduced to a single point. In particular f i (s0 ) = 0. s

Indeed, assume by contradiction that v0 ∩ Bi contains at least two points. In view of the conservation of energy (12), their degrees would be, in absolute value, strictly less than k, so that using the induction hypothesis (vs ∩ Bi ) would be constant equal to s

(v0 ∩ Bi ) in some neighborhood of s0 . This would contradict Step 1. Step 4. We claim that (vs1 ∩ Bi ) = 1, which yields the desired contradiction to (4.3) when s1 > s0 . In view of (20) we have | f i (s)| ≤ C(1 + |∇W (a1 (s), . . . , am (s))|) f i (s), ∀s ∈ (s0 , s1 ). By Gronwall’s lemma, we obtain s1 f i (s) ≤ f i (s0 ) exp(C (1 + |∇W (a1 (s), . . . , am (s))|) ds), ∀s ∈ (s0 , s1 ]. s0


251

Hence f i (s) = 0 on (s0 , s1 ] in view of Step 2 and Step 3. In particular f i (s1 ) = 0 and the conclusion follows. The case s1 < s0 is treated with very similar arguments. Proof of Theorem 2. Lemma 4.1 implies that the number of vortices is locally constant out of the dissipation times. By continuation, it is therefore constant on each interval (τk , τk+1 ). Hence statement i) is proved. Once the total number of vortices is known to be constant, statement ii) follows from the evolution law for the center of mass (14), which we proved in Lemma 5.2 of [4]. For the proof of Theorem 4, we extend the result of Lemma 4.1 to vortices (ai (τk ), τk ) as follows. Lemma 4.2. Let (ai (τk ), τk ) ∈ v and assume that i

−

+

(d +j )2 = di2 (τk )

(resp.

j=1

i

2 2 (d − j ) = di (τk )).

j=1

Then i+ = 1

(resp. i− = 1).

Proof. Assume by contradiction that i+ ≥ 2. In view of Theorem 2, relabelling possibly the vortices, we have for j ∈ {1, . . . , i+ }, d 1 a j (s) = − 2 ∇W (a1 (s), . . . , ai+ (s)) + R j (s), ds dj for s ∈ (τk , τk + τk ), where R j (s) is a continuous and bounded function. Arguing as in Lemma 4.1, we obtain τk +τk |∇W (a1 (s), . . . , ai+ (s))|2 ds ≤ C( sup |W (a1 (s), . . . , ai+ (s))| + 1). τk

s∈(τk ,τk +τk )

In view of Proposition 3.1, the right-hand side of the last inequality is finite, so that τk +τk |∇W (a1 (s), . . . , ai+ (s))| ds < +∞, τk

and we may then finish the proof as in Lemma 4.1 using Gronwall’s lemma.

Proof of Theorem 4. We first notice that (8), which was already obtained in [3] is, as mentioned, a consequence of the homotopy invariance of the degree, the convergence stated in Theorem 1, and the already established regularity properties of v. Concerning (9), we turn to Proposition 3.1, and claim that i+ ≤ 0 ,

i− ≥ 0 .

(4.7)

This is a rather direct consequence of expansion (3.17) and the gradient-flow type property of (3). Inequalities (4.7) then follow from (9). We now turn to the first statement in Theorem 4, namely the identification of dissipation points and branching points. It is straightforward to show that a regular point is

252


not a dissipation point. Indeed, in this case i+ = i− = 1, di+ = di− , so that βik = 0 in view of (7) and (8). We next prove that a point which is not a dissipation point is a regular point. By Lemma 4.2, if i+ = 0 (resp. i− = 0) then i+ = 1 (resp. i− = 1). In particular, if (ai (τk ), τk ) is not a dissipation point, then i+ − i− = βik = 0. In view of (4.7) it follows in this case that i+ = i− = 0, so that i+ = i− = 1 and therefore (ai (τk ), τk ) is a regular point. 5. Examples of Branching Points in the Trajectory Set In the introduction we classified branching points by their complexity as illustrated in Fig. 2. In this section we will show that cases a), b) and c) may be observed as limits of solutions to (PGL)ε . Case d) would require more efforts. 5.1. Collisions with annihilation. We consider a well-prepared initial datum, having two vortices of degree +1 and −1 of the form u 0ε (z) = f (

|z + 1| (z − 1)(¯z + 1) |z − 1| )f( ) , ε ε |(z − 1)(¯z + 1)|

where f is a smooth nonnegative function on R+ such that f (0) = 0, f ≡ 1 outside a compact set. In particular u 0ε → u 0∗ =

(z − 1)(¯z + 1) |(z − 1)(¯z + 1)|

as ε → 0 ,

i.e. u 0∗ has a vortex of degree +1 located at a+ (0) = 1, and a vortex of degree −1 located at a− (0) = −1. The solution to the ordinary differential equation (3) with initial datum as above is given explicitly as √ a± (s) = ± 1 − s , for 0 ≤ s < 1 . Let u ε be the solution of (PGL)ε with inital datum u 0ε , and ai (s) be the points given by Theorem 1. It follows from [9] that for 0 ≤ s < 1, (s) = 2 and, after a possible relabelling, a1 (s) = a+ (s), a2 (s) = a− (s). It follows from Theorem 4 that for s > 1, (s) = 0, so that vortices have disappeared after s = 1. This provides an example for Fig. 2a. 5.2. Collisions without annihilation. We consider here a well-prepared initial datum of the form u 0ε (z) = vε

3

f(

k=1

|z − ak (0)| z − ak (0) di )( ) , ε |z − ak (0)|

where f is as in Sect. 5.1, vε is defined as vε (z) = f (

|z − ε−1 | z¯ − ε−1 ) , ε |¯z − ε−1 |


253

and ak (0) = k − 2, dk = (−1)k+1 , for k = 1, 2, 3. Since the total degree near the origin is different from zero, we added a “vortex at infinity” (by superposing vε ) in order to have total degree zero at infinity. In particular, (H0 ) is verified, with say M0 = 6π . The solution of the ordinary differential equation (3) with initial datum ak (0) = k − 2 and dk = (−1)k+1 , for k = 1, 2, 3, is given by √ ak (s) = (2 − k) 1 − 2s ,

for 0 ≤ s
1/2, a2 and a3 have disappeared and that a1 (s) = 0

for s ≥

1 . 2

In particular, the branching point (0, 21 ) is as in Fig. 2b. 5.3. On the persistency of multiple degree vortices. In view of (5), a multiple degree vortex, say of degree d ≥ 2, is energetically less favourable than d vortices of degree one. One may therefore ask whether multiple degree vortices may arise and survive as limits of solutions to the gradient flow (PGL)ε . The next construction, which in this respect complements Theorem 2 and Theorem 4 by an example, provides a positive answer to that question for d = 2. Theorem 5.1. Let s0 > 0 be given. There exists M0 and a sequence of solutions (u ε )ε>0 of (PGL)ε such that vs is reduced to a single vortex located at the origin and of degree d = +2, for s < s0 , and which splits at time s = s0 into two distinct vortices of degree +1. The idea of the proof is to approximate the trivial multiple degree solution of (3) given by (s) = 1, a1 (s) ≡ 0 for s ∈ (0, s0 ) by solutions of (3) involving only vortices of degree ±1, for which two vortices collapse at time s0 . Since ±1-vortices do not split, these solutions may be well approximated by (PGL)ε , in view of [4], Prop. 1. In order to construct these solutions, the idea is to consider as initial value the stationary solution of (3) presented in Proposition 2.2 to perturb it slightly so that the solution to the corresponding ordinary differential equation eventually breaks up, and finally to scale the whole construction down so that the configuration asymptotically appears as a single vortex of degree +2. More precisely, for δ ∈ (0, 1), we consider as initial values for Eq. (3) the configuration a1δ (0) = δ,

aiδ (0) = z i , for i = 1, 2, 3,

(5.1)

with d1 = −1, d2 = d3 = d4 = +1, and denote aiδ (s), for i = 1, 2, 3, 4, the corresponding solution to (3). We have Lemma 5.1. The maximal time of existence T ∗ (δ) of (3) with initial data (5.1) is finite. Moreover, the function δ → T ∗ (δ) is nonincreasing, continuous and satisfies T ∗ (δ) → +∞ as δ → 0, T ∗ (δ) → 0 as δ → 1.

(5.2) (5.3)

254


Proof. By symmetry, Im(aiδ (s)) = 0, a3δ (s)

=

∀s ∈ [0, T ∗ (δ) ),

a4δ (s)

i = 1, 2,

(5.4)

∗

∀s ∈ [0, T (δ) ) .

(5.5)

It is an elementary (although not straightforward) exercise to check, in view of (3), that ∀s ∈ [0, T ∗ (δ) ), d d (5.6) Re(a1δ (s)) ≥ c(δ) > 0, Re(a2δ (s)) ≤ −c(δ) < 0, ds ds d d d d Re(a3δ (s)) = Re(a4δ (s)) ≤ 0, Im(a3δ (s)) = − Im(a4δ (s)) ≥ 0. (5.7) ds ds ds ds In view of (5.7) the points a1δ and a2δ are moving towards each other at a speed bounded below by a positive constant so that they collide in finite time, in particular T ∗ (δ) is finite. On the other hand, one checks that, for any s ∈ [0, T ∗ (δ) ), d δ d δ (a (s)) > 0 , (a (s)) ≤ 0 , (5.8) dδ 1 dδ 2 which shows that T ∗ (δ) is nonincreasing. Property (5.3) is straightforward, whereas property (5.2) follows from the continuous dependence with respect to initial data and the fact that (z 1 , z 2 , z 3 , z 4 ) is a stationary solution to (3). Lemma 5.2. For every δ ∈ (0, 1) and every s ∈ [0, T ∗ (δ) ), sup |aiδ (s)| ≤ 2.

(5.9)

i=1,..,4

Proof. We have 4 d δ |ai (s)|2 = 0 . ds

(5.10)

i=1

Indeed, by (2.2), 4 4 1 d δ |ai (s)|2 = aiδ (s)∇ai W (a1δ (s), a2δ (s), a3δ (s), a4δ (s)) 2 ds i=1 i=1 = di d j = 0.

(5.11)

The conclusion follows from (5.10).

At time T ∗ (δ) the point a1δ and a2δ collide and hence we remove them from the collection, whereas the functions a3δ and a4δ can be uniquely continued beyond time T ∗ (δ) according to (3), so that ∀s ≥ T ∗ (δ), for i = 3, 4, Re(aiδ (s)) = Re(aiδ (T ∗ (δ))),

Im(aiδ (s))

= (−1)

i+1

|Im(aiδ (s))| s − T (δ) − 4 ∗

We set δ = ∪s>0 {aiδ (s)} .

1/2 .


255

Proposition 5.1. Let 0 < δ < 1 be given. There exists a family (u δε )0 τk , s˜ = − log(s − τk ),

a˜ j (˜s ) =

a j (s) − ai (τk ) . √ s − τk

The equation for a˜ j becomes d 1 1 1 a˜ j = 2 ∇ j W (a˜ 1 , . . . , a˜ k ) − exp(−˜s )c⊥ + a˜ j , d s˜ d 2 dj j

(6.1)

for which we have to consider the limit s˜ → +∞, which corresponds to s → τk+ . In view of the change of variables, the vortices a˜ j for j ∈ / {1, . . . , i+ } are sent at infinity, whereas in view of the parabolic cone property (see [3] Theorem 2) the points a˜ j for j ∈ {1, . . . , i+ } remain in a bounded set. Equation (6.1) therefore reads, for j ∈ {1, . . . , i+ }, d 1 1 s˜ a˜ j = 2 ∇ j W (a˜ 1 , . . . , a˜ i+ ) + a˜ j + O(exp(− )) d s˜ 2 2 dj as s˜ → +∞. Lemma 6.1. We have i+ + s˜ 2 2 4 + d a ˜ k k ≤ C exp(− ). i 4 k=1

Proof. In view of (6.2) and (2.2), we have i i d s˜ 2 2 + dk a˜ k = 4 i + dk2 a˜ k2 + O(exp(− )). d s˜ 2

Set E(˜s ) = (4 i+ +

+

+

k=1

k=1

i+

2 ˜ 2 )2 . k=1 dk a k

It follows from (6.1) that

d s˜ E(˜s ) = E(˜s ) + O(exp(− )). d s˜ 2 Integrating from

s˜ 2

to s˜ we obtain

s˜ s˜ E( ) exp( ) ≤ E(˜s ) + C 2 2

s˜ s˜ 2

s˜ t exp(˜s − t) exp(− ) dt ≤ C exp( ), 2 4

so that E(˜s ) ≤ C exp(− 2s˜ ) and the conclusion follows. 5 One argues similarly for s < τ . k

(6.2)


257

Arguing as in Lemme 6.1, one may prove6 similarly that + i 2 ≤ C exp(− s˜ ). d a ˜ (˜ s ) k k 4 k=1

Lemma 6.2. We have

lim sup W (a˜ 1 (˜s ), . . . , a˜ i+ (˜s )) < +∞. s˜ →+∞

Proof. By (3.17), we have, as s → τk+ , W (a1 (s), . . . , ai+ (s)) = i+ log |s − τk | + O(1) = − i+ s˜ + O(1). On the other hand, by definition of a˜ j and W ,

a j (s)

} = W ({a j (s)})− i+ log(s−τk ) = W ({a j (s)}) − i+ s˜ . W {a˜ j (˜s )} = W { √ s − τk Combining the two identities we obtain the conclusion.

Up to the exponentially decreasing error term, (6.2) represents a gradient flow of W i+ 2 2 under the constraint j=1 d j a j = −4 i+ . Lemma 6.3. For s˜ > 0, set L(˜s ) = W (a˜ 1 (˜s ), . . . , a˜ i+ (˜s )) + have

1 4

i+

2 ˜ 2 (˜ j=1 d j a j s ).

Then we

+

i 1 1 2 d 1 d 2j 2 ∇ j W + a˜ j − C exp(−˜s ). L(˜s ) ≥ d s˜ 2 2 dj

j=1

Proof. We have, by definition of L and (6.2), i i 1 d 1 2 d 1

L(˜s ) ≥ a˜ j = ∇ j W + d j a˜ j d 2j 2 ∇ j W + a˜ j d s˜ 2 d s˜ 2 dj +

+

j=1

j=1

1 1 s˜

× 2 ∇ j W + a˜ j + O(exp(− )) , 2 2 dj and the conclusion follows by Young’s inequality.

It follows from Lemmas 6.1 and 6.2 that L is uniformly bounded, and therefore we deduce from Lemma 6.3 Corollary 6.1. We have 0

2 d 2j ∇ j W ({a˜ j (˜s )}) + a˜ j (˜s ) d s˜ < +∞. 2

+∞

6 We will not make use of this fact in our subsequent arguments.

(6.3)

258


Inequality (6.3) is typical of gradient flows, where it is used to prove convergence towards critical points. In this context, we introduce the configuration set {(a j , d j )} j=1,..., , a j ∈ C, a j = ak for j = k, C= , d j = di , d 2j − ( d j )2 = i+ d j ∈ Z∗ , the constrained subset of C, M = {(a j , d j )} ∈ C, d 2j |a j |2 = −4 i+ , and the set of critical points of W restricted to M, d2 K = {(a j , d j )} ∈ M, ∇k W ({(a j , d j )}) = − k ak . 2 Notice that the configuration sets C and M have a stratified structure, each leaf corresponding to fixed values of the total number of vortices. The notion of criticality for an element in K is meant here for the restriction of W to the leaf to which it belongs. We define a distance on C by dist {(a j , d j )})1≤ j≤ , {(a j , d j )})1≤ j≤ = sup d j ξ(a j ) − d j ξ(a j ) . |∇ξ |≤1 j=1 j=1 Notice that this distance represents the flat norm (see [8]) of the current j=1 d j δa j − j=1 d j δa j . It is known (see [5]) to be equivalent to the minimal connection between the points {(a j , d j )} and {(a j , d j )}. Before we proceed to the main result of this section, we gather some properties of K which will be used later. Lemma 6.4. The set K is compact. Proof. We consider a sequence ({(a nj , d nj )}1≤ j≤n )n∈N in K, and show that a subsequence converges to an element of K. Without loss of generality, we may assume that n ≡ is constant, and that d nj = d j is independent of n. Since M is bounded, so is K, and we may therefore assume passing possibly to a subsequence that a nj → a j as n → +∞, for j = 1, . . . , . If all the points a j are distinct, then we are done by continuity of ∇W. To complete the proof, it remains to consider the situation where several points converge to the same limit. In this case, denote by {b j } j∈J the set of limit points, J j = {k ∈ 1, . . . , , akn → b j }, and set, for j ∈ J, D j = k∈J j d j . Our aim is to prove that the configuration {(b j , D j )} j∈J belongs to K. The fact that ∇k W ({a nj }) = − reads, in view of (2.1), dk dm = dk2 akn , k = 1, . . . , 4 n − an am k m=k

that is, for k ∈ J j , 4

m =k m∈J j

dk dm dk dm = −4 + dk2 akn , n n − an − ak a m k m =k

n am

m ∈J / j

dk2 n 2 ak


259

and therefore 4

k∈J j

m =k m∈J j

dk dm dk dm = −4 + dk2 akn . n − an n − an am a m k k m =k k∈J j

k∈J j

m ∈J / j

The left-hand side of the previous equality is zero, by antisymmetry. Passing to the limit as n → +∞ in the right-hand side, we are led to ⎛ ⎞ Dm D j 4 =⎝ dk2 ⎠ b j . (6.4) bm − b j m = j k∈J j

m∈J

To conclude, we claim that for all j ∈ J, dk2 = D 2j .

(6.5)

k∈J j

Indeed, by (2.9) we have dk2 = dk2 + ∇z k W ({(akn − b j , dk )}k∈J j ) (akn − b j ). k∈J j

k∈J j

k∈J j

Since ∇z k W ({(akn − b j , dk )}k∈J j ) − ∇z k W ({(akn − b j , dk )}k=1,..., ) is bounded independently of n, the conclusion (6.5) follows letting n → +∞. From (6.5) we then deduce that D 2j − ( D j )2 = i+ , j∈J

j∈J

and from (6.4) and (6.5) that ∇ j W ({(b j , D j )} j∈J ) = − The proof is complete.

D 2j 2

bj.

In the spirit of the Palais-Smale condition, we have the following variant of Lemma 6.4. Lemma 6.5. Let (a nj , d nj )1≤ j≤n be a sequence such that a nj ∈ C, a nj = akn for j = k, d nj ∈ Z∗ ,

d nj = di

2 (d nj )2 − d nj = i+ .

and

Assume moreover that for 1 ≤ j ≤ n , ∇ j W ({(akn , dkn )})

=−

d nj 2 2

a nj + o(1),

as n → +∞.

Then, up to a subsequence {(a nj , d nj )} converges towards an element in K as n → +∞. The argument is exactly the same as in the proof of Lemma 6.4: it suffices in many identities to replace zero by o(1). Therefore we omit the proof.

260


Corollary 6.2. Let δ > 0. There exists ε > 0 such that, for every configuration (a j , d j )1≤ j≤ with a j ∈ C, a j = ak for j = k, d j ∈ Z∗ , and (d j )2 − ( d j )2 = i+ , d j = di if moreover dist({(a j , d j )}, K) ≥ δ

then

|∇ j W ({(ak , dk )}) +

j

d 2j 2

a j | ≥ ε.

The proof follows from Lemma 6.5 arguing by contradiction. We come back now to the aymptotics of a˜ j (˜s ). The main result of this section is Theorem 6.1. We have dist({(a˜ j (˜s ), d j )}, K) → 0 as s˜ tends to plus infinity. Proof. This is again a standard argument for gradient flow type equations, once a Palais-Smale property has been established. Indeed, it follows from (6.3) and Lemma 6.5 that there exists a sequence (˜sn )n∈N such that sñ → +∞ and dist({(a˜ j (˜sn ), d j )}, K) → 0

as n → +∞.

It remains to show that convergence holds not only for a sequence but for s˜ → +∞. To that aim, for a given δ > 0, assume that sã < s˜b are such that dist({(a˜ j (˜sa ), d j )}, K) = δ,

dist({(a˜ j (˜sb ), d j )}, K) = 2δ,

(6.6)

and for every s˜ ∈ (˜sa , s˜b ) it holds dist({(a˜ j (˜s ), d j )}, K) ≥ δ. Then, by Corollary 6.2 we have, for some ε > 0 depending on δ,

|∇ j W ({a˜ k (˜s )}) +

j

Hence, s˜b sã

|∇ j W ({a˜ k (˜s )}) +

j

d 2j 2

d 2j 2

a˜ j (˜s )| ≥ ε

a˜ j (˜s )| d s˜ ≥ ε

s˜b

2

≥ε

sã

j

∀s ∈ (˜sa , s˜b ).

|∇ j W ({a˜ k (˜s )}) +

j s˜b sã

|

d 2j 2

a˜ j (˜s )| d s˜

d a˜ j (˜s )| d s˜ − C exp(−˜sa ) d s˜

≥ C(εδ − exp(−˜sa )). In particular, +∞ sã

j

|∇ j W ({a˜ k (˜s )}) +

d 2j 2

a˜ j (˜s )|2 d s˜ ≥ C(εδ − exp(−˜sa )).

(6.7)

In view of (6.3), the integral on the left-hand side of (6.7) tends to zero as sã goes to +∞, so that (6.6) may only happen for sã bounded by a constant depending only on δ and the conclusion follows.


261

As a consequence of Theorem 6.1, there exists a sequence sñ → +∞ an a critical point (b j , D j ) j∈J such that (a˜ j (˜sn ), d j ) → (b j , D j ) j∈J with 2 ≤ J ≤ i+ . Asymptotic self-similarity of the trajectory set near the branching point (ai (τk ), τk ) would mean that the whole family (a˜ j (˜s ), d j ) converges to the same limit (b j , D j ) j∈J . However, we do not know if this holds. References 1. Baumann, P., Chen, C-N., Phillips, D., Sternberg, P.: Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems. Eur. J. Appl. Math. 6, 115–126 (1995) 2. Bethuel, F., Orlandi, G., Smets, D.: Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature. Ann. Math. 163, 37–163 (2006) 3. Bethuel, F., Orlandi, G., Smets, D.: Collisions and phase-vortex interaction in dissipative Ginzburg-Landau dynamics. Duke Math. J. 130, 523–614 (2005) 4. Bethuel, F., Orlandi, G., Smets, D.: Quantization and motion law for Ginzburg-Landau vortices. Arch. Rat. Mech. Anal. 183, 315–370 (2007) 5. Brezis, H., Coron, J.-M., Lieb, E.H.: Harmonic maps with defects. Commun. Math. Phys. 107, 649– 705 (1986) 6. Comte, M., Mironescu, P.: Remarks on nonminimizing solutions of a Ginzburg-Landau type equation. Asymptotic Anal. 13, 199–215 (1996) 7. E, W.: Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. Phys. D 77, 383–404 (1994) 8. Federer, H.: Geometric Measure Theory. Berlin: Springer Verlag, 1969 9. Jerrard, R.L., Soner, H.M.: Dynamics of Ginzburg-Landau vortices. Arch. Rat. Mech. Anal. 142, 99– 125 (1998) 10. Kirchhoff, G.R.: Vorlesungen über Mathematische Physik (Vol. 1: Mechanik). Leipzig: Teubner, 1874 11. Lin, F.H.: Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math. 49, 323– 359 (1996) 12. Lin, F.H.: A remark on the previous paper: “Some dynamical properties of Ginzburg-Landau vortices”. Comm. Pure Appl. Math. 49, 361–364 (1996) 13. Neu, J.C.: Evolution and creation of vortices and domain walls. Department of Mathematics, University of California, Berkeley, Lecture notes (unpublished) 14. Neu, J.C.: Vortices in complex scalar fields. Phys. D 43, 385–406 (1990) 15. Ovchinnikov, Y.N., Sigal, I.M.: Symmetry-breaking solutions of the Ginzburg-Landau equation. J. Exp. Theor. Phys. 99, 1090–1107 (2004) 16. Palmore, J.: Relative equilibria of vortices in two dimensions. Proc. Nat. Acad. Sci. U.S.A. 79, 716– 718 (1982) 17. Peres, L., Rubinstein, J.: Vortex dynamics in U(1) Ginzburg-Landau models. Phys. D 64, 299–309 (1993) 18. Pismen, L.M., Rubinstein, J.: Motion of vortex lines in the Ginzburg-Landau model. Phys. D 47, 353– 360 (1991) 19. Rubinstein, J., Sternberg, P.: On the slow motion of vortices in the Ginzburg-Landau heat-flow. SIAM J. Appl. Math. 26, 1452–1466 (1995) 20. Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to GinzburgLandau. Comm. Pure App. Math. 57, 1627–1672 (2004) 21. Serfaty, S.: Vortex Collision and Energy Dissipation Rates in the Ginzburg-Landau Heat Flow. Preprint 2005, to appear in J. Eur. Math. Soc. Communicated by P. Constantin


Communications in


Hidden Grassmann Structure in the XXZ Model H. Boos1, , M. Jimbo2 , T. Miwa3 , F. Smirnov4, , Y. Takeyama5 1 Physics Department, University of Wuppertal, D-42097 Wuppertal, Germany.


2 Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan.


3 Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan.


4 Laboratoire de Physique Théorique et Hautes Energies, Université Pierre et Marie Curie, Tour 16 1er étage,

4 Place Jussieu, 75252 Paris Cedex 05, France. E-mail: [email protected]

5 Institute of Mathematics, Graduate School of Pure and Applied Sciences, Tsukuba University, Tsukuba,

Ibaraki 305-8571, Japan. E-mail: [email protected] Received: 11 July 2006 / Accepted: 3 August 2006 Published online: 8 February 2007 – © Springer-Verlag 2007

Abstract: For the critical XXZ model, we consider the space W[α] of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators b(ζ ), c(ζ ) which act on W[α] . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in W[α] can be expressed in terms of an exponential of a quadratic form of b(ζ ), c(ζ ). 1. Introduction This paper continues our study of correlation functions in lattice integrable models [1–5]. Consider the infinite XXZ spin chain with the Hamiltonian HXXZ =

1 2

∞ 1 + σ 2 σ 2 + σ 3 σ 3 σk1 σk+1 k k+1 k k+1 ,

(1.1)

k=−∞

where σ a (a = 1, 2, 3) are the Pauli matrices and = cos π ν is a real parameter. We use the usual notation q = eπiν . In our previous work [5], we obtained an algebraic representation for general correlation functions of the XXZ model. Here we generalize this result to the situation when a disorder operator is present. In the course we find a new interesting structure behind the model. We consider only the massless regime || < 1, 0 < ν < 1, since it is more On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992, Moscow, Russia

Membre du CNRS

264

H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Y. Takeyama

important for physics because of its relation to conformal field theory (CFT) discussed below. Explanation about the massive regime > 1 will be given elsewhere. Let us introduce S(k) =

1 2

k j=−∞

σ j3 .

Denote by |vac the ground state of the Hamiltonian, and let α be a parameter. We consider the normalized vacuum expectation values: vac|q 2αS(0) O|vac , vac|q 2αS(0) |vac

(1.2)

where O is a local operator. Locality of O implies that the operator q 2αS(0) O stabilizes: there exist integers k, l such that for all j > l (resp. j < k) this operator acts on the j th lattice site as 1 (resp. 3 q ασ ). If k (resp. l) is the maximal (resp. minimal) integer with this property, l − k + 1 will be called the length of the operator q 2αS(0) O. The very formulation of the problem implies that we are interested only in local operators O of total spin 0 (otherwise the correlation function vanishes). Nevertheless, for the sake of convenience we introduce the spaces Wα,s of operators q 2αS(0) O of spin s: S, q 2αS(0) O = s q 2αS(0) O, S = S(∞). Also we set Wα =

∞

Wα,s .

s=−∞

The leading long distance asymptotics of the XXZ spin chain is described √ by CFT with c = 1: that of free bosons φ, φ¯ with compactification radius β = 1 − ν. For an extensive discussion about the XXZ model as an irrelevant perturbation of CFT, we refer the reader to [6]. The space Wα,s corresponds to the space of descendants of the operator i

e2

¯ ¯ α(β −1 −β)(φ+φ)+sβ(φ− φ)

.

Similarly to the conformal case [10–12], introduction of the disorder parameter α regularizes the problem, and allows to write much nicer formulae than in the case α = 01 . Another similarity is that it is very convenient to consider, as an intermediate object which does not enter the final formulae, the following space: W[α] =

∞

Wα+k .

k=−∞

In this paper we shall introduce two anti-commuting families of operators b(ζ ) and c(ζ ) acting on W[α] : [b(ζ1 ), b(ζ2 )]+ = [b(ζ1 ), c(ζ2 )]+ = [c(ζ1 ), c(ζ2 )]+ = 0. 1 The formulae are written initially for |q α | < 1 and continued analytically in α, but α = 0 is one of singular points where l’Hôpital’s rule should be applied.

Grassmann Structure in the XXZ Model

265

The operators b(ζ ) and c(ζ ) have the following block structure: b(ζ ) : Wα+k,s → Wα+k+1,s−1 , c(ζ ) : Wα+k,s → Wα+k−1,s+1 . Hence the operator b(ζ1 )c(ζ2 ) acts from Wα,0 to itself. The operators b(ζ ), c(ζ ) are formal series in (ζ − 1)−1 . When applied to an operator 2αS(0) q O of length L, the singularity is a pole of order L, in other words, the series terminates at (ζ − 1)−L . The action of b(ζ ), c(ζ ) produces operators of the same or smaller length. The coefficients of b(ζ ), c(ζ ) give rise to an action of the Grassmann algebra with 2L generators. In particular b(ζ1 ) · · · b(ζ L+1 ) q 2αS(0) O = 0, c(ζ1 ) · · · c(ζ L+1 ) q 2αS(0) O = 0. We introduce also the linear functional on End C2 : trα (x) =

1 α 2

q +q

− α2

1 3 tr q − 2 ασ x

(1.3)

with the obvious properties: trα (1) = trα (q ασ ) = 1. 3

This gives rise to a linear functional on Wα trα (X ) = · · · tr α1 tr α2 tr α3 · · · (X ). Our main result is: vac|q 2αS(0) O|vac α 2αS(0) e q = tr O , vac|q 2αS(0) |vac

(1.4)

where2 the operator acts on W[α] :

dζ1 dζ2 , = −resζ1 =1 resζ2 =1 ω (ζ1 /ζ2 ) b(ζ1 )c(ζ2 ) ζ1 ζ2

and ω(ζ ) is a scalar operator on each Wα , ω(ζ )|Wα = ω(ζ, α)1Wα ,

(1.5)

the scalar being ω(ζ, α) =

4(qζ )α (1 + q α )2

q −α qα − 1 − q −2 ζ 2 1 − q 2ζ 2

i∞−0

+ −i∞−0

ζ u+α

sin π2 (u − ν(u + α)) du. sin π2 u cos π2ν (u + α)

⊗L For any local operator of length L, the trace is effectively taken over C2 . Comments are in order about the meaning of (1.4). In [7, 8], in the setting of inhomogeneous chains, it was conjectured that the thermodynamic limit of the ground state 2 In [5] the operator was denoted by ∗ .

266


averages in the finite XXZ chain are certain specific solutions of the reduced qKZ (rqKZ) equation given by multiple integrals. Subsequently these integral formulas were also derived from the viewpoint of algebraic Bethe Ansatz [9]. We take these formulas as the definition of the left hand side of (1.4). Following our previous works [2, 5], we present here another formula for solutions of rqKZ equations. The right-hand side of (1.4) is its specialization to the homogeneous case. We have no doubt that these two solutions coincide3 . Since a mathematical proof is lacking at the moment, we propose (1.4) as conjecture. The function ω(ζ, α) and trα develop singularities at α = ±1/ν. In view of this, we presume that the formula holds true throughout the range |Re α| < 1/ν. It will be shown that the operators b(ζ ), c(ζ ) commute with the adjoint action of the shift operator U and of local integrals of motion I p on W[α] . Since q −αS commutes with U, I p , one immediately concludes that

the vacuum expectation values of U q 2αS(0) O U −1 − q 2αS(0) O and I p , q 2αS(0) O given by (1.4) vanish, as it should be. In our opinion the appearance of anti-commuting operators b(ζ ) and c(ζ ) is quite remarkable. In the next section we explain how these operators are constructed using the q-oscillators. We explain their relation to the Jordan-Wigner fermions in the XX case in Sect. 3. In Appendix we briefly discuss the generalization of our previous formulae [5] to the case when the disorder operator is present. For the sake of simplicity we consider the homogeneous chain only. We give brief explanations about the inhomogeneous case when needed. We do not give complete proofs, but just sketch the derivation of the main statements. We tried to make this paper as brief as possible, leaving the details to a separate publication. 2. Operators b(ζ ) and c(ζ ) First we prepare our notation for the L-operators. Consider the quantum affine algebra Uq ( sl2 ). The universal R-matrix of this algebra belongs to the tensor product b+ ⊗ b− of its two Borel subalgebras. By an L-operator we mean its image under an algebra map b+ ⊗ b− → N1 ⊗ N2 , where N1 , N2 are some algebras. In this paper we always take N2 to be the algebra M = Mat (2, C) of 2 × 2 matrices. As for N1 we make several choices: Uq (sl2 ), M, the q-oscillator algebra Osc (see below) or Osc ⊗ M ± , where M ± ⊂ M are the subalgebras of upper and lower triangular matrices. For economy of symbols, we use the same letter L to designate these various L-operators. We always put indices, indicating to which tensor product of algebras they belong. We use j, k, · · · as labels for the lattice sites, and a, b, · · · as those for the ‘auxiliary’ two-dimensional space. Accordingly we write the matrix algebra as M j or Ma . Capital letters A, B, · · · will indicate the q-oscillator algebra Osc. Finally, for Osc ⊗ M ± we use pairs of indices such as {A, a}. The first case of L-operators is when N1 = Uq (sl2 ):

H +1

H +1

H −1

ζ q 2 − ζ −1 q − 2 (q − q −1 )Fq 2 L j (ζ ) = H −1 H −1 − H 2−1 −1 (q − q )q E ζ q − 2 − ζ −1 q 2

∈ Uq (sl2 ) ⊗ M j .

(2.1)

j

Here E, F, q ±H/2 are the standard generators of Uq (sl2 ). The suffix j in the right-hand side means that it is considered as a 2 × 2 matrix in M j . This is an exceptional case when 3 It is known to be the case in the massive regime, see [2]. We also confirm the coincidence at the free fermion point, see Sect. 3.


267

we do not put any index for the first (‘auxiliary’) tensor factor; we shall never use several copies of Uq (sl2 ). Mapping further Uq (sl2 ) to Ma , we obtain the second L-operator L a, j (ζ ) ∈ Ma ⊗ M j , which actually coincides with the standard 4 × 4 R-matrix. The next case is due originally to Bazhanov, Lukyanov and Zamolodchikov [12]. Let us consider the q-oscillators a, a ∗ satisfying aa ∗ − q 2 a ∗ a = 1 − q 2 . It is convenient to introduce one more element q D such that q D a ∗ = a ∗ q D+1 , q D a = aq D−1 , ∗ 2D a a = 1 − q , aa ∗ = 1 − q 2D+2 . Denote by Osc the algebra generated by a, a ∗ , q ±D with the above relations. We consider the following two representations of Osc, ∞

W+ = W− =

k=0 −1

C|k, a ∗ |k − 1 = |k,

D|k = k|k, a|0 = 0;

C|k, a|k + 1 = |k,

D|k = k|k, a ∗ | − 1 = 0.

k=−∞

In the root of unity case, if r is the smallest positive integer such that q 2r = 1, we consider the r -dimensional quotient of W ± generated by |0 or | − 1. The L-operator associated with Osc is given by 1 −ζ a ∗A q DA 0 − 21 − 14 + L A, j (ζ ) = iζ q ∈ Osc A ⊗ M j . −ζ a A 1 − ζ 2 q 2D A +2 j 0 q −D A j This L-operator satisfies the crossing symmetry relation: L +A, j (ζ )−1 =

1 + L (ζ ), ζ − ζ −1 A, j

where we have set +

L A, j (ζ ) = σ j2 L +A, j (ζ q −1 )t j σ j2 , and t j stands for the transposition in M j . We use also another L-operator 1 + 1 L− A, j (ζ ) = σ j L A, j (ζ )σ j .

Consider the product L +A, j (ζ )L a, j (ζ ). It is well known that this product can be brought to a triangular form, giving rise in particular to Baxter’s ‘T Q-equation’ for transfer matrices. Namely, introducing −D

∗

q A 0 1 aA + 1 + 1 G A,a = , G− D A,a = σa G A,a σa , A 0 1 a 0 q a

268


one easily finds that −1 + L A, j (ζ )L a, j (ζ )G +A,a L +{A,a}, j (ζ ) = G +A,a ⎛ 3 −1 −1 + −1 − ⎜(ζ q −ζ q )L A, j (ζ q )q

=⎝

(q −q −1 )L +A, j (ζ q)σ j+ q

σj

0

2

−2D A + 12

⎞

(ζ − ζ −1 )L +A, j (ζ q)q

σ 3j

⎟ − ⎠ ∈ Osc A ⊗ Ma ⊗ M j .

2

a

(2.2) For the inverse matrix one has: 1 L +{A,a}, j (ζ )−1 = (ζ − ζ −1 )(ζ q − ζ −1 q −1 )(ζ q −1 − ζ −1 q) ⎛ 3 ⎜ ×⎝

(ζ −(q

− ζ −1 )q

σj 2

+ L A, j (ζ q −1 )

1 + − q −1 )σ j+ L A, j (ζ q)q −2D A + 2

⎞ 0

(ζ q −1 − ζ −1 q)q −

σ 3j 2

+ L A, j (ζ q)

⎟ ⎠ . a

(2.3) Again we shall use another L-operator: 1 1 + 1 1 + L− {A,a}, j (ζ ) = σa σ j L {A,a}, j (ζ )σa σ j ∈ Osc A ⊗ Ma ⊗ M j . Some information will be needed about R-matrices which intertwine these L-operators. First, consider the Yang-Baxter equation: ± ± ± R A,B (ζ1 /ζ2 )L ± A, j (ζ1 )L B, j (ζ2 ) = L B, j (ζ2 )L A, j (ζ1 )R A,B (ζ1 /ζ2 ).

(2.4)

The R-matrix appearing in (2.4) is given by R A,B (ζ ) = PA,B h(ζ, u A,B )ζ D A +D B , where PA,B is the permutation, u A,B = a ∗A q −2D A a B , and the function h(ζ, u) is given by h(ζ, u) =

∞ n=0

−uq −1

n n q j−1 ζ −1 − q − j+1 ζ . q j − q− j j=1

When q is not a root of unity, the series for R A,B (ζ ) is well defined because the action of u A,B on W ± ⊗ W ± is locally nilpotent. Otherwise we replace the right-hand side by −1 the sum rn=0 , if r is the smallest positive integer such that q 2r = 1. Second, consider the Yang-Baxter equation for the L-operators L +{A,a}, j : + R{A,a},{B,b} (ζ1 /ζ2 )L +{A,a}, j (ζ1 )L +{B,b}, j (ζ2 ) + = L +{B,b}, j (ζ2 )L +{A,a}, j (ζ1 )R{A,a},{B,b} (ζ1 /ζ2 ).

(2.5)


The corresponding R-matrix has the form ⎛ ⎞ R1,1 (ζ ) 0 0 0 0 0 ⎟ ⎜R (ζ ) R2,2 (ζ ) + R{A,a},{B,b} (ζ ) = ⎝ 2,1 . R3,1 (ζ ) 0 R3,3 (ζ ) 0 ⎠ R4,1 (ζ ) R4,2 (ζ ) R4,3 (ζ ) R4,4 (ζ ) a,b

269

(2.6)

The entries Ri, j (ζ ) can be found by a direct calculation. In this paper we shall need only two of them: R1,1 (ζ ) = q −D A R A,B (ζ )q D B , R4,4 (ζ ) = −ζ 2 q D A R A,B (ζ )q −D B . Up to scalar coefficients depending on ζ , these operators can be guessed immediately, but the coefficient, especially the sign, in R4,4 (ζ ) is important for us. As usual we define: − + (ζ ) = σa1 σb1 R{A,a}{B,b} (ζ )σa1 σb1 . R{A,a},{B,b}

Now we have everything necessary for the definition of the operators b(ζ ) and c(ζ ). For two integers k ≤ l we set M[k,l] = Mk ⊗ · · · ⊗ Ml . This is the algebra of linear operators on the ‘quantum space’ on the interval [k, l]. Our main object is the monodromy matrix ± ± ∓ T{A,a},[k,l] (ζ ) = L ± {A,a},l (ζ ) · · · L {A,a},k (ζ ) ∈ Osc A ⊗ Ma ⊗ M[k,l] .

(2.7)

−1 ∈ Osc ⊗ M ∓ ⊗ End(M Define further an element T± A [k,l] ) by setting a {A,a},[k,l] (ζ ) ± ± −1 −1 · (1 T± A,a ⊗ X [k,l] ) · T{A,a},[k,l] (ζ ), (2.8) {A,a},[k,l] (ζ ) (X [k,l] ) = T{A,a},[k,l] (ζ )

where 1 A,a = 1 A ⊗ 1a and X [k,l] ∈ M[k,l] . To illustrate the definition, we have, for x{A,a} ∈ Osc A ⊗ Ma∓ and X [k,l] ∈ M[k,l] , an equality in Osc A ⊗ Ma∓ ⊗ M[k,l] , −1 T± · x{A,a} ⊗ id X [k,l] {A,a},[k,l] (ζ ) ± ± = T{A,a},[k,l] (ζ )−1 · 1{A,a} ⊗ X [k,l] · T{A,a},[k,l] (ζ ) · (x{A,a} ⊗ 1[k,l] ), where id is the identity operator in End(M[k,l] ). −1 ∈ Osc ⊗End(M −1 ∈ M ⊗End(M We define T± A [k,l] ) and Ta,[k,l] (ζ ) a [k,l] ) A,[k,l] (ζ ) in a similar manner. We understand a certain inconvenience in using the inverse operators, but it has for us an historical reason: once we define the transfer-matrix as in [7], the order of multipliers is fixed everywhere. We have the Yang-Baxter equation −1 ± −1 ± T± {A,a},[k,l] (ζ1 ) T{B,b},[k,l] (ζ2 ) R{A,a},{B,b} (ζ1 /ζ2 ) ± −1 ± −1 (ζ1 /ζ2 )T± = R{A,a},{B,b} {B,b},[k,l] (ζ2 ) T{A,a},[k,l] (ζ1 ) ,

where the identity is in Osc A ⊗ Ma∓ ⊗ Osc B ⊗ Mb∓ ⊗ End(M[k,l] ).

(2.9)

270


Of particular importance are the Q-operators acting on local operators. They are defined as Q+[k,l] (ζ, α) = tr +A q 2α D A T+A,[k,l] (ζ )−1 (1 − q 2(α−S) ), − −2α(D A +1) − Q− T A,[k,l] (ζ )−1 q 2S (1 − q 2(α−S) ), [k,l] (ζ, α) = tr A q (2.10) where S stands for the adjoint action of the total spin operator

S(X [k,l] ) = S(l) − S(k − 1) , X [k,l] , X [k,l] ∈ M[k,l] . −2α(D A +1) Y The trace functionals tr +A q 2α D A Y A and tr − A for Y A ∈ Osc A are defined A q as analytic continuations with respect to α of traces over W + and W − from the region |q α | < 1. The Q-operators (2.10) are mutually commuting families of operators. They are so normalized that Q± [k,l] (0, α) = 1. ± Regarding T{A,a},[k,l] (ζ )−1 as a matrix in Ma∓ , let us write its entries as

+ 0 A A,[k,l] (ζ ) + −1 T{A,a},[k,l] (ζ ) = , C+A,[k,l] (ζ ) D+A,[k,l] (ζ ) a

− − −1 = A A,[k,l] (ζ ) B A,[k,l] (ζ ) T− (ζ ) , {A,a},[k,l] 0 D− A,[k,l] (ζ ) a where A+A,[k,l] (ζ ), etc., are elements of Osc A ⊗ End(M[k,l] ). It follows from the defini−1 have poles of order l −k +1 at the points ζ 2 = 1, q ±2 . However, tion that T± {A,a},[k,l] (ζ ) looking at the formulae (2.2)– (2.3), one realizes that at the pole ζ 2 = 1 only C+A,[k,l] (ζ ) and B− A,[k,l] (ζ ) are singular. This motivates, at least partly, the following definition: c[k,l] (ζ, α) = q α−S 1 − q 2(α−S) sing ζ =1 ζ α−S tr +A q 2α D A C+A,[k,l] (ζ ) , (2.11) −2α(D A +1) − q b[k,l] (ζ, α) = q 2S sing ζ =1 ζ −α+S tr − B (ζ ) . (2.12) A A,[k,l] Here and after, singζ =1 [ f (ζ )] signifies the singular part of f (ζ ) at ζ = 1: 1 f (ξ ) dξ, singζ =1 [ f (ζ )] = 2πi ζ −ξ

(2.13)

where the integral is taken over a simple closed curve containing ξ = 1 inside, while ξ = ζ and other singular points of f (ξ ) are outside. We note that [S, c[k,l] (ζ, α)] = c[k,l] (ζ, α), [S, b[k,l] (ζ, α)] = −b[k,l] (ζ, α). There are several important properties of operators c[k,l] (ζ, α) and b[k,l] (ζ, α) which we formulate as lemmas. Lemma 2.1. The operators c[k,l] (ζ, α) and b[k,l] (ζ, α) satisfy the following anticommutation relations: c[k,l] (ζ1 , α − 1)c[k,l] (ζ2 , α) = −c[k,l] (ζ2 , α − 1)c[k,l] (ζ1 , α), b[k,l] (ζ1 , α + 1)b[k,l] (ζ2 , α) = −b[k,l] (ζ2 , α + 1)b[k,l] (ζ1 , α).

(2.14) (2.15)


271

Proof. Consider the Yang-Baxter equations (2.9) for +. Using the R-matrix (2.6) one finds: ζ1−2 C+A,[k,l] (ζ1 )C+B,[k,l] (ζ2 ) + ζ2 −2 q D A R A,B (ζ1 /ζ2 )q −D B · C+B,[k,l] (ζ2 )C+A,[k,l] (ζ1 ) · q −D B R A,B (ζ1 /ζ2 )−1 q D A = · · · ,

(2.16)

where · · · stands for a sum of terms which contain at least one A+[k,l] (ζi ) or D+[k,l] (ζi ), and hence have vanishing singular parts at ζi = 1. Multiplying (2.16) by q 2(α−1)D A +2α D B , taking the trace and the singular part, one immediately gets (2.14). Similarly one proves (2.15) using (2.9) for −.

Lemma 2.2. We have the following reduction relations: c[k,l] (ζ, α) X [k,l−1] · 1l = c[k,l−1] (ζ, α) X [k,l−1] · 1l , b[k,l] (ζ, α) X [k,l−1] · 1l = b[k,l−1] (ζ, α) X [k,l−1] · 1l , 3 3 c[k,l] (ζ, α) q ασk · X [k+1,l] = q (α−1)σk · c[k+1,l] (ζ, α) X [k+1,l] , 3 3 b[k,l] (ζ, α) q ασk · X [k+1,l] = q (α+1)σk · b[k+1,l] (ζ, α) X [k+1,l] .

(2.17) (2.18) (2.19) (2.20)

Proof. Equations (2.17), (2.18) are trivial consequences of the definition. In contrast, Eqs. (2.19), (2.20) are far from being obvious. Consider the first of them. By definition we have: 1 3 c[k,l] (ζ, α) q ασk · X [k+1,l] q α−S 1 − q 2(α−S) 3 = singζ =1 tr +A q 2α D A C+A,[k,l] (ζ ) q ασk · X [k+1,l] ζ α−s−1 , where s is the spin of X [k+1,l] . Let us simplify the trace. We will use the crossing symmetry P−j, j¯ L +A, j (ζ q −1 )L +A, j¯ (ζ ) = (ζ − ζ −1 )P−j, j¯ ,

(2.21)

P−j, j¯ L +{A,a}, j (ζ q −1 )L +{A,a}, j¯ (ζ ) = (ζ q − ζ −1 q −1 )(ζ − ζ −1 )(ζ q −1 − ζ −1 q)P−j, j¯ , (2.22) where P−j, j¯ is the anti-symmetrizer. Introducing consecutively some additional twodimensional spaces, we have 3 tr +A q 2α D A C+A,[k,l] (ζ ) q ασk · X [k+1,l] 3 = tra tr +A σa+ L +{A,a},k (ζ )−1 · T+{A,a},[k+1,l] (ζ )−1 X [k+1,l] · q ασk L +{A,a},k (ζ )q 2α D A =

1

− ζ −1 q) + −1 × tr k¯ tra tr +A σa+ L +{A,a},k¯ (ζ q −1 ) · 2P− X · T (ζ ) [k+1,l] {A,a},[k+1,l] ¯ k,k ασk3 + 2α D A . ·q · L {A,a},k (ζ )q (ζ

− ζ −1 )(ζ q

− ζ −1 q −1 )(ζ q −1

(2.23)

272


Now use q ασk L +{A,a},k (ζ )q 2α D A −ασa = q 2α D A −ασa L +{A,a},k (ζ )q ασk 3

3

3

3

and the cyclicity of the trace to simplify (2.23) further: 3 tr +A q 2α D A C+A,[k,l] (ζ ) q ασk · X [k+1,l] 1 (ζ − ζ −1 )(ζ q − ζ −1 q −1 )(ζ q −1 − ζ −1 q) 3 3 ×tr k¯ tra tr +A T+{A,a},[k+1,l] (ζ )−1 X [k+1,l] q 2α D A −ασa +α L(ζ )q ασk .

=

(2.24)

It is easy to see that L(ζ ) = 2P− L+ (ζ )σa+ L +{A,a},k¯ (ζ q −1 ) k,k¯ {A,a},k

ζ q − ζ −1 q −1 0 2P− L + (ζ q −1 )L +A,k¯ (ζ ) = 1 0 (q − q −1 )q −2D A − 2 a k,k¯ A,k ⎞ ⎛ σ ¯3 −σk3 σ3

k − 2k + σk¯ q 2 ⎟ (q − q −1 )q −2D A + 12 ⎜q 0 , (2.25) ×⎝ ⎠ 3 σ¯ 0 ζ q −1 − ζ −1 q a k + + + 2 σk σk¯ σk q a

where we used 1

1

L +A, j (ζ q)σ j+ q −2D A + 2 = q −2D A − 2 L +A, j (ζ q −1 )σ j+ . In view of (2.21), L(ζ ) is divisible by ζ − ζ −1 , and in (2.24) we can drop the diagonal elements of T+{A,a},[k+1,l] (ζ )−1 , arriving immediately at (2.19). The proof of (2.20) is similar.

Remark. The above construction carries over to inhomogeneous chains where an independent spectral parameter ξ j is attached to each site j. The operators c[k,l] (ζ ; ξk , · · · , ξl ), b[k,l] (ζ ; ξk , · · · , ξl ) are defined via the above construction with two modifications: ± (i) In the definition (2.7), each L ± {A,a}, j (ζ ) is replaced by L {A,a}, j ζ /ξ j . (ii) The singular part is understood as an integral (2.13) around the points ξk , · · · , ξl . Lemma 2.1 and Lemma 2.2 remain valid.

Lemma 2.2 allows us to define universal operators b(ζ, α), c(ζ, α): Definition 2.3. For any operator q 2αS(0) O ∈ Wα , let q 2αS(0) O [k,l] be its restriction to the finite interval [k, l] of the lattice. We define b(ζ, α) : Wα,s → Wα+1,s−1 , c(ζ, α) : Wα,s → Wα−1,s+1 , by setting

q 2αS(0) O [k,l] , k→−∞,l→∞ 2αS(0) c(ζ, α) q O = lim c[k,l] (ζ, α) q 2αS(0) O [k,l] .

b(ζ, α) q 2αS(0) O =

lim

k→−∞,l→∞

b[k,l] (ζ, α)

(2.26) (2.27)

(2.28) (2.29)


273

It follows from Lemma 2.2 that for any particular operator q 2αS(0) O the expressions under the limit in (2.28), (2.29) stabilize for a sufficiently large interval [k, l]. Hence the limit is well-defined. In particular we have, for any k, b(ζ, α)(q 2αS(k) ) = 0, c(ζ, α)(q 2αS(k) ) = 0.

(2.30)

Denoting by b(ζ ) and c(ζ ) the operators acting on the direct sum W[α] we have the anti-commutativity [b(ζ1 ), b(ζ2 )]+ = [c(ζ1 ), c(ζ2 )]+ = 0. In the Appendix, we give a brief summary of the algebraic formula for the correlation functions in the presence of disorder. The result is expressed in terms of the operator 1 dζ2 = −resζ1 =1 resζ2 =1 X(ζ1 , ζ2 )ω(ζ2 /ζ1 ) dζ (2.31) ζ1 ζ2 , where X(ζ1 , ζ2 )|Wα = X(ζ1 , ζ2 , α), the operator X(ζ1 , ζ2 , α) is given in either of the two formulas (A.3), (A.4), ω(ζ ) is given by (1.5). The following result allows us to express in terms of b(ζ ), c(ζ ). At the same time, the existence of two equivalent representations guarantees the anti-commutativity between the latter. Lemma 2.4. The operator X(ζ1 , ζ2 ) can be evaluated as follows: X(ζ1 , ζ2 )|Wα = b(ζ2 , α − 1)c(ζ1 , α) = −c(ζ1 , α + 1)b(ζ2 , α).

(2.32)

Proof. Consider the formula (A.3). We have: 0 tra,b Bb,a (ζ2 /ζ1 )Ta (ζ1 )−1 Tb (ζ2 )−1 Q+ (ζ1 , α + 1)Q− (ζ2 , α + 1) 0 −1 −1 + −1 − −1 = tra,b tr +A tr − B Bb,a (ζ2 /ζ1 )Ta (ζ1 ) Tb (ζ2 ) T A (ζ1 ) T B (ζ2 ) ×q 2(α+1)(D A −D B −1) (1 − q 2(α+1−S) )2 q 2S . We move Tb (ζ2 )−1 through T+A (ζ1 )−1 using the Yang-Baxter equation L +A,b (ζ1 /ζ2 ) Tb (ζ2 )−1 T+A (ζ1 )−1 = T+A (ζ1 )−1 Tb (ζ2 )−1 L +A,b (ζ1 /ζ2 ) . −1 come together. Conjugating by Now Ta (ζ1 )−1 T+A (ζ1 )−1 and Tb (ζ2 )−1 T− B (ζ2 ) − + G A,a , G B,b , we can combine them into the monodromy matrices T+{A,a} (ζ1 )−1 , −1 T− {B,b} (ζ2 ) . In these monodromy matrices we drop diagonal elements because they have no singularities at ζi = 1. Then by a straightforward calculation we come to 0 tra,b Bb,a (ζ2 /ζ1 )Ta (ζ1 )−1 Tb (ζ2 )−1 Q+ (ζ1 , α + 1)Q− (ζ2 , α + 1) − + 2(α+1)D A −2α(D B +1)−2 C (1 − q 2(α−S+1) )2 q 2S , −tr +A tr − (ζ )B (ζ )q 1 2 A B B

(2.33) where means that the singular parts are identical. Similarly we have: 1 (ζ1 /ζ2 )Tb (ζ2 )−1 Ta (ζ1 )−1 Q− (ζ2 , α − 1)Q+ (ζ1 , α − 1) (2.34) tra,b Ba,b − + 2α D A −2(α−1)(D B +1) (1 − q 2(α−S−1) )2 q 2S . −tr +A tr − B B B (ζ2 )C A (ζ1 )q Equation (2.32) follows from (2.33), (2.34) and the definition of b(ζ, α), c(ζ, α).

274


The main formula (1.4) follows from (2.31), (2.32) and (A.1). Let U be the shift operator by one lattice unit, which acts on local operators by the adjoint: a U σ ja U −1 = σ j+1 .

There is also an infinite set of local integrals of motion which commute with U and among themselves. The last important property of b(ζ ), c(ζ ) is their invariance: Lemma 2.5. The operators b(ζ ), c(ζ ) commute with the adjoint action of the shift operator U and of the local integrals of motion. Proof. For U the statement of this lemma follows immediately from the definition, essentially it is a consequence of Lemma 2.2. The local integrals of motion are of the form Ip =

∞

d j, p ,

(2.35)

j=−∞

where d j, p is an operator acting non-trivially on the sites j, · · · , j + p. We shall call operators of the type (2.35) p-local operators. Let us write the 4 × 4 R-matrix as Rˇ j,k (ξ ) = P j,k L j,k (ξ ). We set U[k,l] (ξ ) = (q − q −1 )k−l Rˇ l,l−1 (ξ ) · · · Rˇ k+1,k (ξ ). Following the remark after Lemma 2.2, consider c[k,l] with one inhomogeneity: c[k,l] (ζ ; ξ, 1, · · · , 1) and c[k,l] (ζ ; 1, · · · , 1, ξ ). It is clear from the definition that

U[k,l] (ξ ) · c[k,l] (ζ ; ξ, 1, · · · , 1) q 2αS(0) O · U[k,l] (ξ )−1 [k,l]

= c[k,l] (ζ ; 1, · · · , 1, ξ ) U[k,l] (ξ ) · q 2αS(0) O · U[k,l] (ξ )−1 . (2.36) [k,l]

Let ξ = 1 + . Then ⎛ U[k,l] (ξ ) = exp ⎝

∞

⎞

p I[k,l], p ⎠ .

p=1

Due to the Campbell-Hausdorff formula, the operators I[k,l], p are p-local. For finite k, l these operators do not commute because of some boundary terms, but in the limit k → −∞, l → ∞ they coincide with the local integrals of motion I p which are combined into the generating function: U (ξ ) = exp

∞ p=1

p Ip .


275

In the right-hand side of (2.36) we have the expression ( p) U[k,l] (ξ ) · q 2αS(0) O · U[k,l] (ξ )−1 =

p q 2αS(0) O . [k,l]

[k,l]

Here the p-local operators I[k,l], p act by multiple adjoint. It is clear that for every given degree p we can find a large enough interval [k, l] in order that ( p) ( p) q 2αS(0) O = q 2αS(0) O , [k,l]

[k,l]

where U (ξ ) · q 2αS(0) O · U (ξ )−1 = Obviously length

( p)

p q 2αS(0) O .

( p) q 2αS(0) O ≤ length q 2αS(0) O + 2 p.

Now considering (2.36) order by order in , choosing for every order sufficiently large interval [k, l] and using the inhomogeneous version of Lemma 2.2 and the definition of c(ζ ), we get: U (ξ ) · c(ζ ) q 2αS(0) O · U (ξ )−1 = c(ζ ) U (ξ ) · q 2αS(0) O · U (ξ )−1 , (2.37) which is understood as an equality of power series in .

3. Free Fermion Point Consider the point ν = 1/2, q = i. For this coupling constant the Hamiltonian turns into ∞ − + σ j+ σ j+1 , + σ j− σ j+1 HX X = j=−∞

and can be diagonalized by the Jordan-Wigner transformation: ψk± = σk± e∓πi S(k−1) . The space W[α] can be replaced by a direct sum of two components Wα ⊕ Wα+1 . We set y=e

πiα 2

,

so that the space Wα consists of operators of the form y 2S(0) O. There are two fermion operators acting in the space of states, so, there are four of them acting on the space of operators by left and right multiplication. It is convenient to introduce the following four operators: k± (X ) = ψk± X − (−1) F(X ) X ψk± , 1 ± ± ∓2 F(X ) ± ψ (X ) = X − y (−1) X ψ α,k k k , 1 − y ∓2 where F(X ) is the fermionic number of the operator X .

(3.1)

276


± ± We have ± α+2,k = α,k . These operators are natural for us because k annihilate 1 2S (recall that at plus or minus infinity y 2S(0) O stabilizes to 1 or while ± α,k annihilate y ± 2S y ). The operators k , ± α,k satisfy the canonical anti-commutation relations:

[k , l ]+ = [ α,k , α,l ]+ = 0, [k , α,l ]+ = δ + ,0 δk,l .

(3.2)

It is clear, however, that the operators b(ζ, α), c(ζ, α) cannot be constructed as linear combinations of k± , ± α,k . Indeed the operators b(ζ, α), c(ζ, α) are translationally invariant, in particular, they annihilate y 2S(k) for any k, see (2.30). Clearly this is impossible for any linear combination of k± , ± α,k . Our plan in this section is as follows. First, we find a compact expression for b(ζ, α) and c(ζ, α) in terms of k± , ± α,k . Then we show that our formula gives the same result for the correlators as the one obtained by a straightforward calculation based on normal ordering. The calculation of b(ζ, α), c(ζ, α) at the free fermon point is summarized by Lemma 3.1. At the free fermion point, the operators b(ζ, α) and c(ζ, α) are given by 2i −s+1 ζ −α+s−1 − − ζ , b(ζ, α) = sing (ζ )E (ζ, α − s) ζ =1 Wα,s 1 + (−1)s y 2 1 + ζ2 ζ , (3.3) c(ζ, α) = 2y singζ =1 ζ α−s−1 + (ζ )E + (ζ, α − s) Wα,s 1 + ζ2 where ± (ζ ) =

∞ j=−∞

and

± j

1 + ζ2 1 − ζ2

j (3.4)

2 ∓ ∓ 2 ± E ± (ζ, α) = exp N ± . (3.5) log I − ζ M − log I + ζ M α α

± In the last formula we consider ± α, j (resp. j ) as components of a row (resp. column) vector, ± u j = ± M = (1 + u)(1 − u)−1 , j+1 ,

and log 1 ± ζ 2 M are understood as Taylor series in u. N[·] stands for the normal ordering which applies only to operators acting at the same site. For them we set α, j j ( j > 0),

N[α, j j ] = (3.6) − j α, j ( j ≤ 0). Since the q-oscillators become fermions at q = i, the lemma can be shown by manipulations with exponentials of quadratic forms in fermions. Details will be given in another publication. We remark that the exponent of (3.5) is well defined as an operator on Wα . Indeed by ± ∓ definition it consists of N α,k l with l ≥ k. On a particular operator in Wα only a finite number of these operators do not vanish.


277

It has been said that, unlike b(ζ ), c(ζ ), formulae containing fermions necessarily break the translational invariance. We choose the point k = 1 as the origin and consider only operators of the form y 2S(0) O> ,

(3.7)

where O> acts only on the interval [1, ∞). Any operator in Wα can be brought to the form (3.7) by a shift, so we do not really lose generality. In the sequel we need the operators on a half line: b> (ζ, α) = b[1,∞) (ζ, α),

c> (ζ, α) = c[1,∞) (ζ, α).

± ± They are defined as in (3.3), replacing E ± (ζ, α), ± (ζ ) and ± α (ζ ) by E > (ζ, α), > (ζ ) and ± (ζ ), respectively. The latter are given by the same formulae (3.4), (3.5) with α,> non-positive components of fermions removed. In the free fermion case the function ω(ζ, α) can be calculated explicitly. Putting it together with (3.3), we rewrite our main formula in the free fermion case as follows:

vac| y 2S(0) O> |vac = trα> e> (O> ) , vac| y 2S(0) |vac i resζ1 =1 resζ2 =1 > = sin π2α ζ1α ζ2−α − 1 − dζ12 dζ22 + − + · E > (ζ2 , α)E > (ζ1 , α)> (ζ2 )> (ζ1 ) , ζ12 + ζ22 1 + ζ12 1 + ζ22

(3.8)

where trα> means that the trace is calculated over the positive half of the chain only. Now notice that ± ± > α,> (ζ ) (O> ) = 0, (ζ )(I ) = 0, tr2(α+1) >

y ∓2 ± ± (O> ). O = − ψ± j > α, j 1 − y ∓2 j

(3.9)

± , the operators ∓ So, by changing trα> to tr> α,> and > can be considered as creation-annihilation operators in the space of operators. For efficient application of them we need the following: 2(α+1)

Lemma 3.2. The following identity holds: trα> e> (O> ) = tr2(α+1) e> (O> ) , >

(3.10)

where > = i resζ1 =1 resζ2 =1 sin π2α

ζ1α ζ2−α

dζ12 − + (ζ ) (ζ ) 2 1 > > ζ12 + ζ22 1 + ζ12

dζ22 1 + ζ22

.

278


The formulae (3.9) and (3.10) allow an explicit calculation of correlators. One easily obtains: vac| y 2S(0) ψk+1 · · · ψk+p ψl−p · · · ψl−1 |vac + − ψ , (3.11) = det ψ ki l j i, j=1,··· , p vac|y 2S(0) |vac where

⎛

ζ α ζ −α i ⎝− y δk,l + resζ =1 resζ =1 1 2 ψk+ ψl− = π α 1 2 sin 2 2 ζ12 + ζ22

1 + ζ12 1 − ζ12

k

1 + ζ22 1 − ζ22

l

dζ12

dζ22

1 + ζ12 1 + ζ22

⎞ ⎠. (3.12)

On the other hand, one can calculate the correlators (3.11) directly by normal ordering y 2S(0) . The result is the same: (3.12) is the two-point function while (3.11) is obtained by the Wick theorem. This calculation is unsatisfactory because we had to pass through the fermions k± , ± α,k . It would be much better to find a basis in the space of local operators, on which the original operators b(ζ, α), c(ζ, α) act nicely. Such a construction would have a chance to generalize to an arbitrary coupling constant. For the moment we cannot do that. 4. Conclusion The main result of this paper can be formulated as follows. We consider the space W[α] of local operators in the presence of a disorder field. We have shown that the vacuum expectation values of operators in W[α] can be expressed in terms of two anti-commutative families of operators b(ζ ) and c(ζ ) acting on W[α] . At present, we do not know how to organize the space W[α] in order to describe efficiently the action of b(ζ ) and c(ζ ). The operators b(ζ ) and c(ζ ) should be considered as annihilation operators, as both of them kill the ‘vacua’, i.e., operators q 2αS(k) , for all k. What is missing is a construction of creation operators. Even in the free fermion case, we were able rather to make a detour than to actually solve the problem. In fact, the problem of constructing creation operators cannot be solved literally, because b(ζ ) and c(ζ ) have a large common kernel. Consider the restricted operators b[k,l] (ζ, α) and c[k,l] (ζ, α) acting on the space of dimension 4l−k+1 . In the free fermion case, it can be shown that the dimension of the kernel is 2l−k+1 . Numerical experiments indicate that the dimension stays the same generically. Because of this kernel, we cannot expect operators satisfying the canonical anti-commutation relations with b(ζ ) and c(ζ ). So the first problem is to understand the meaning of the kernel. Obviously, the difference of any two operators in the kernel has vanishing expectation value. The origin of these operators with zero vacuum expectation values is a mystery to us. The only operators for which this property can be easily explained are the descendants generated by adjoint action of local integrals of motion, but for them the vacuum expectation values vanish for a different reason: b(ζ ) and c(ζ ) commute with the adjoint action of local integrals of motion as is explained by Lemma 2.5. Understanding the origin of the kernel of b(ζ ) and c(ζ ), and the construction of creation operators, are the problems which we wish to solve. Appendix A In this appendix, we sketch how the results of [5] can be modified to the situation when disorder is present.


279

Consider an operator q 2αS(0) O which is stable on (−∞, k − 1] and [l + 1, ∞). Consider also an inhomogeneous chain, where spectral parameters ξk , · · · , ξl are attached to all the sites of the lattice where q 2αS(0) O acts non-trivially. Then the ground state becomes dependent on ξk , · · · , ξl , and we have vac | q 2αS(0) O |vac 2αS(0) O . = h (ξ , · · · , ξ , α) q [k,l] k l [k,l] vac | q 2αS(0) |vac ⊗(l−k+1) Here h[k,l] (ξk , · · · , ξl , α) is a linear functional on C2 subject to several requirements [7]: h[k,l] (ξk , · · · , ξ j+1 , ξ j , · · · , ξl , α)(X [k,l] ) ˇ j, j+1 (ξ j /ξ j+1 )−1 (X [k,l] ) , = h[k,l] (ξk , · · · , ξ j , ξ j+1 , · · · , ξl , α) R h[k,l] (ξk q −1 , · · · , ξl , α)(X [k,l] ) = h[k,l] (ξk , · · · , ξl , α) A[k,l] (ξk , · · · , ξl , α)(X [k,l] ) , h[k,l] (ξk , · · · , ξl , α) X [k,l−1] · 1l = h[k,l−1] (ξk , · · · , ξl−1 , α) X [k,l−1] , 3 h[k,l] (ξk , · · · , ξl , α) q ασk · X [k+1,l] = h[k+1,l] (ξk , · · · , ξl , α) X [k+1,l] , where

ˇ j, j+1 (ξ j /ξ j+1 )(X ) = Rˇ j, j+1 (ξ j /ξ j+1 )X Rˇ j, j+1 (ξ j /ξ j+1 ) −1 , R tk 3 A[k,l] (ξk , · · · , ξl , α)(X ) = (T −1 )tk · σk2 · Xq −ασk · σk2 · T, T = Rk,l (ξk /ξl ) · · · Rk,k+1 (ξk /ξk+1 ).

The way for solving these equations is absolutely parallel to the one described in [5]. The answer takes the form (A.1) h[k,l] (ξk , · · · , ξl , α) X [k,l] = trα e[k,l] (ξk ,··· ,ξl ,α) X [k,l] . In order to describe [k,l] , let T[k,l] (ζ ) ∈ Uq (sl2 ) ⊗ End(M[k,l] ) denote the monodromy matrix (2.8) constructed via the L-operator (2.1). (Normally we do not write the dependence on ξk , · · · , ξl explicitly.) i Introduce operators X[k,l] (ζ1 , ζ2 , α) (i = 0, 1) depending rationally on ζ1 , ζ2 by setting

−1 0 tra,b Ba,b (ζ1 /ζ2 )Tb,[k,l] (ζ2 )−1 Ta,[k,l] (ζ1 )−1 Tr d(ζ1 /ζ2 ) T[k,l] ζ1 ζ2 q −(α+1)H α α ζ2 ζ1 0 0 X[k,l] (ζ1 , ζ2 , α) + X[k,l] (ζ2 , ζ1 , α), = ζ2 ζ1

−1 1 (ζ1 /ζ2 )Tb,[k,l] (ζ2 )−1 Ta,[k,l] (ζ1 )−1 Tr d(ζ1 /ζ2 ) T[k,l] ζ1 ζ2 q −(α−1)H tra,b Ba,b α α ζ2 ζ1 1 1 X[k,l] (ζ1 , ζ2 , α) + X[k,l] (ζ2 , ζ1 , α). = ζ2 ζ1 Here Tr d stands for the trace functional on Uq (sl2 ) (analytic continuation of the trace with respect to dimension d) used in [2], d(ζ ) =

log ζ , log q

280


and B 0 (ζ ), B 1 (ζ ) are 4 × 4 matrices given by

⎛

0 (ζ ⎜0 B 0 (ζ ) = ⎝ −1 −1 −1 (ζ q − ζ q )(ζ q − ζ q) 0 0 − ζ −1 )

and

0 0 q −ζ −1 −ζ q −1 0 0

⎞ 0 0⎟ ⎠ 0 0

B 1 (ζ ) = σ 1 ⊗ σ 1 · B 0 (ζ ) · σ 1 ⊗ σ 1 .

The notation being as above, the formula for [k,l] is given as follows: 1 dζ1 dζ2 [k,l] (ξk , · · · , ξl , α) = − , X[k,l] (ζ1 , ζ2 , α)ω(ζ2 /ζ1 , α) 2πi ζ1 ζ2

(A.2)

where integrals are taken around ξk , · · · , ξl . The operator X[k,l] (ζ1 , ζ2 , α) is presented in either of the following two equivalent forms: α−S ! ζ1 0 X[k,l] (ζ1 , ζ2 , α) = singζ1 ,ζ2 =ξk ,··· ,ξl X[k,l] (ζ1 , ζ2 , α) ζ2 α−S ! ζ1 1 (ζ1 , ζ2 , α) = singζ1 ,ζ2 =ξk ,··· ,ξl X[k,l] . ζ2 In the formulas (A.1), (A.2) only operators of spin 0 are considered. Here we have introduced S for later convenience. Now we give the only part of the construction which has no analogues in [5]. Let us "" 1 consider the homogeneous case. In particular, we replace (2πi) by resζ1 =1 resζ2 =1 2 and drop the index [k, l]. From [12] we learn:

−1 −α H = Tr d(ζ1 /ζ2 ) T ζ1 ζ2 q α α

ζ1 ζ2 1 + − − + = Q (ζ1 , α)Q (ζ2 , α) − Q (ζ1 , α)Q (ζ2 , α) . α−S ζ2 ζ1 q − q −α+S This implies X(ζ1 , ζ2 , α)

= singζ1 =1 singζ2 =1

ζ1 ζ2

α+1−S

× Q (ζ1 , α + 1)Q (ζ2 , α + 1)

0 tra,b Bb,a (ζ2 /ζ1 )Ta (ζ1 )−1 Tb (ζ2 )−1

−1 q α+1−S − q −α−1+S α−1−S ζ1 1 tra,b Ba,b (ζ1 /ζ2 )Tb (ζ2 )−1 Ta (ζ1 )−1 = singζ1 =1 singζ2 =1 ζ2 1 × Q− (ζ2 , α − 1)Q+ (ζ1 , α − 1) α−1−S . q − q −α+1+S +

−

(A.3)

(A.4)

These two formulae will be used in Sect. 2 to derive a new expression for (see Lemma 2.4).


281

Acknowledgements. Research of HB is supported by the RFFI grant #04-01-00352. Research of MJ is supported by the Grant-in-Aid for Scientific Research B–18340035. Research of TM is supported by the Grantin-Aid for Scientific Research B–17340038. Research of FS is supported by INTAS grant #03-51-3350, by EC networks “EUCLID”, contract number HPRN-CT-2002-00325 and “ENIGMA”, contract number MRTNCT-2004-5652 and GIMP program (ANR), contract number ANR-05-BLAN-0029-01. Research of YT is supported by the Grant-in-Aid for Young Scientists B–17740089. This work was also supported by the grant of 21st Century COE Program at RIMS, Kyoto University. The authors would like to thank Boris Feigin and Leon Takhtajan for interest and discussions. HB is also grateful to Rodney Baxter, Frank Göhmann, Andreas Klümper, Ingo Peschel and Junji Suzuki for discussions. HB would like to thank the Department of Mathematics, Graduate School of Science, Kyoto University for warm hospitality. MJ is grateful to Christian Korff and the staff members of the City University of London for their kind invitation and hospitality, where part of this work has been carried out. FS is grateful to RIMS, Kyoto University always for its warm and nostalgic atmosphere.

References 1. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: A recursion formula for the correlation functions of an inhomogeneous XXX model. Algebra and Analysis 17, 115–159 (2005) 2. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Reduced qKZ equation and correlation functions of the XXZ model. Commun. Math. Phys. 261, 245–276 (2006) 3. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Traces on the Sklyanin algebra and correlation functions of the eight-vertex model. J. Phys. A: Math. Gen. 38, 7629–7659 (2005) 4. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Density matrix of a finite sub-chain of the Heisenberg anti-ferromagnet. Lett. Math. Phys. 75, 201–208 (2006) 5. Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Algebraic representation of correlation functions in integrable spin chains. http://arxiv.org/list/ hep-th/0601132 6. Lukyanov, S.: Low energy effective Hamiltonian for the XXZ spin chain. Nucl. Phys. B. 522, 533– 549 (1998) 7. Jimbo, M., Miwa, T.: Algebraic analysis of solvable lattice models, Reg. Conf. Ser. in Math. 85, Providence, RI: Amer. Math. Soc., (1995) 8. Jimbo, M., Miwa, T.: Quantum Knizhnik-Zamolodchikov equation at |q| = 1 and correlation functions of the XXZ model in the gapless regime. J. Phys. A. 29, 2923–2958 (1996) 9. Kitanine, N., Maillet, J.M., Slavnov, N.A., Terras, V.: Spin-spin correlation functions of the XXZ-1/2 Heisenberg chain in a magnetic field. Nucl. Phys. B. 641, 487–518 (2002) 10. Bazhanov, V., Lukyanov, S., Zamolodchikov, A.: Integrable structure of conformal field theory, Quantum KdV theory and thermodynamic Bethe Ansatz. Commun. Math. Phys. 177, 381–398 (1996) 11. Bazhanov, V., Lukyanov, S., Zamolodchikov, A.: Integrable structure of conformal field theory II. Q-Operator and DDV equation. Commun. Math. Phys. 190, 247–278 (1997) 12. Bazhanov, V., Lukyanov, S., Zamolodchikov, A.: Integrable structure of conformal field theory III. The Yang-Baxter Relation. Commun. Math. Phys. 200, 297–324 (1999) Communicated by L. Takhtajan


Communications in


Lace Expansion for the Ising Model Akira Sakai Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. E-mail: [email protected] Received: 26 October 2005 / Accepted: 14 November 2006 Published online: 21 March 2007 – © Springer-Verlag 2007

Abstract: The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical twopoint function for the nearest-neighbor model with d 4 and for the spread-out model with d > 4 and L 1, without assuming reflection positivity.

Contents 1. Introduction and Results . . . . . . . . . . . . . 1.1 Model and the motivation . . . . . . . . . . 1.2 Main results . . . . . . . . . . . . . . . . . 1.3 Organization . . . . . . . . . . . . . . . . . 2. Lace Expansion for the Ising Model . . . . . . . 2.1 Random-current representation . . . . . . . 2.2 Derivation of the lace expansion . . . . . . 2.3 Comparison to percolation . . . . . . . . . 3. Bounds on (j) (x) for the Ferromagnetic Models 3.1 Strategy for the spread-out model . . . . . . 3.2 Strategy for the nearest-neighbor model . . ( j) 4. Diagrammatic Bounds on π (x) . . . . . . . . 4.1 Construction of diagrams . . . . . . . . . . (0) 4.2 Bound on π (x) . . . . . . . . . . . . . . ( j) 4.3 Bounds on π (x) for j ≥ 1 . . . . . . . .

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284 284 286 288 288 288 290 298 299 299 300 302 303 305 310

284

A. Sakai

( j) 5. Bounds on π (x) Assuming the Decay of G(x) . . . . . . . . . . . . . . . 329 5.1 Bounds for the spread-out model . . . . . . . . . . . . . . . . . . . . . 329 5.2 Bounds for finite-range models . . . . . . . . . . . . . . . . . . . . . . 335

1. Introduction and Results 1.1. Model and the motivation. The Ising model is a statistical-mechanical model that was first introduced in [22] as a model for magnets. Consider the d-dimensional integer lattice Zd , and let be a finite subset of Zd containing the origin o ∈ Zd . For example, is a d-dimensional hypercube centered at the origin. At each site x ∈ , there is a spin variable ϕx that takes values either +1 or −1. The Hamiltonian represents the energy of the system, and is defined by Hh (ϕ) = − Jx,y ϕx ϕ y − h ϕx , (1.1) {x,y}⊂

x∈

where ϕ ≡ {ϕx }x∈ is a spin configuration, {Jx,y }x,y∈Zd is a collection of spin-spin couplings, and h ∈ R represents the strength of an external magnetic field uniformly imposed on . We say that the model is ferromagnetic if Jx,y ≥ 0 for all pairs {x, y}; in this case, the Hamiltonian becomes lower as more spins align. The partition function Z p,h; at the inverse temperature p ≥ 0 is the expectation of the Boltzmann factor h e− p H (ϕ) with respect to the product measure x∈ ( 21 1{ϕx =+1} + 21 1{ϕx =−1}): h e− p H (ϕ) . (1.2) Z p,h; = 2−|| ϕ∈{±1}

Then, we denote the thermal average of a function f = f (ϕ) by f p,h; =

2−|| Z p,h;

f (ϕ) e− p H (ϕ) . h

(1.3)

ϕ∈{±1}

Suppose that the spin-spin coupling is translation-invariant, Zd -symmetric and finiterange (i.e., there exists an L < ∞ such that Jo,x = 0 if x ∞ > L) and that Jo,x ≥ 0 for any x ∈ Zd and h ≥ 0. Then, there exist monotone infinite-volume limits of ϕx p,h; and ϕx ϕ y p,h; . Let M p,h = lim ϕo p,h; , G p (x) = lim ϕo ϕx p,h=0; , χ p = G p (x). ↑Zd

↑Zd

x∈Zd

(1.4) When d ≥ 2, there exists a unique critical inverse temperature pc ∈ (0, ∞) such that the spontaneous magnetization M +p ≡ lim h↓0 M p,h equals zero, G p (x) decays exponentially as |x| ↑ ∞ (we refer, e.g., to [9] for a sharper Ornstein-Zernike result) and thus the magnetic susceptibility χ p is finite if p < pc , while M +p > 0 and χ p = ∞ if p > pc (see [2] and references therein). We should also refer to [7] for recent results on the phase transition for the Ising model. We are interested in the behavior of these observables around p = pc . The susceptibility χ p is known to diverge as p ↑ pc [1, 4]. It is generally expected that lim p↓ pc M +p = limh↓0 M pc ,h = 0. We believe that there are so-called critical exponents γ = γ (d),

Lace Expansion for the Ising Model

285

β = β(d) and δ = δ(d), which are insensitive to the precise definition of Jo,x ≥ 0 (universality), such that (we use below the limit notation “≈” in some appropriate sense) p↓ pc

M +p ≈ ( p − pc )β ,

p↑ pc

χ p ≈ ( pc − p)−γ ,

h↓0

M pc ,h ≈ h 1/δ .

(1.5)

These exponents (if they exist) are known to obey the mean-field bounds: β ≤ 1/2, γ ≥ 1 and δ ≥ 3. For example, β = 1/8, γ = 7/4 and δ = 15 for the nearest-neighbor model on Z2 [26]. Our ultimate goal is to identify the values of the critical exponents in other dimensions and to understand the universality for the Ising model. There is a sufficient condition, the so-called bubble condition, for the above critical take on their respective values. Namely, the finiteness of exponents to mean-field 2 (or the finiteness of 2 uniformly in p < p ) implies that G (x) G (x) d d p p c c x∈Z x∈Z β = 1/2, γ = 1 and δ = 3 [1–4]. It is therefore crucial to know how fast G pc (x) (or G p (x) near p = pc ) decays as |x| ↑ ∞. We note that the bubble condition holds for d > 4 if the anomalous dimension η takes on its mean-field value η = 0, where the anomalous dimension is another critical exponent formally defined as |x|↑∞

G pc (x) ≈ |x|−(d−2+η) . (1.6) ik·x and G ik·x for p < p . For a ˆ p (k) = Let Jˆk = c x∈Zd Jo,x e x∈Zd G p (x) e class of models that satisfy the so-called reflection positivity [12], the following infrared bound1 holds: const. 0 ≤ Gˆ p (k) ≤ uniformly in p < pc , (1.7) ˆ J0 − Jˆk where d is supposed to be large enough to ensure integrability of the upper bound. For finite-range models, d has to be bigger than 2, since Jˆ0 − Jˆk |k|2 , where “ f g” means that f /g is bounded away from zero and infinity. By Parseval’s identity, the infrared bound (1.7) implies the bubble condition for finite-range reflection-positive models above four dimensions, and therefore p↓ pc

M +p ( p − pc )1/2 ,

p↑ pc

χ p ( pc − p)−1 ,

h↓0

M pc ,h h 1/3 .

(1.8)

The class of reflection-positive models includes the nearest-neighbor model, a variant of the next-nearest-neighbor model, Yukawa potentials, power-law decaying interactions, and their combinations [6]. For the nearest-neighbor model, we further obtain the following x-space Gaussian bound [32]: for x = o, G p (x) ≤

const. |x|d−2

uniformly in p < pc .

(1.9)

The problem in this approach to investigate critical behavior is that, since general finite-range models do not always satisfy reflection positivity, their mean-field behavior cannot necessarily be established, even in high dimensions. If we believe in universality, we expect that finite-range models exhibit the same mean-field behavior as soon as d > 4. Therefore, it has been desirable to have approaches that do not assume reflection positivity. 1 In (1.7) and (1.9), we also use the fact that, for p < p , our G (i.e., the infinite-volume limit of the c p two-point function under the free-boundary condition) is equal to the infinite-volume limit of the two-point function under the periodic-boundary condition.

286

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The lace expansion has been used successfully to investigate mean-field behavior for self-avoiding walk, percolation, lattice trees/animals and the contact process, above the upper-critical dimension: 4, 6 (4 for oriented percolation), 8 and 4, respectively (see, e.g., [31]). One of the advantages in the application of the lace expansion is that we do not have to require reflection positivity to prove a Gaussian infrared bound and mean-field behavior. Another advantage is the possibility to show an asymptotic result for the decay of correlation. Our goal in this paper is to prove the lace-expansion results for the Ising model. 1.2. Main results. From now on, we fix h = 0 and abbreviate, e.g., ϕo ϕx p,h=0; to ϕo ϕx p; . In this paper, we prove the following lace expansion for the two-point function, in which we use the notation τx,y = tanh( p Jx,y ).

(1.10)

( j) j+1) (x) and R (p; (x) for Proposition 1.1. For any p ≥ 0 and any ⊂ Zd , there exist π p; x ∈ and j ≥ 0 such that ( j) j) j+1) (x) + p; (u) τu,v ϕv ϕx p; + (−1) j+1 R (p; (x), (1.11) ϕo ϕx p; = (p;

u,v

where j (i) p; (x) = (−1)i π p; (x). ( j)

(1.12)

i=0

For the ferromagnetic case, we have the bounds ( j) π p; (x) ≥ δ j,0 δo,x ,

j+1) 0 ≤ R (p; (x) ≤

( j) π p; (u) τu,v ϕv ϕx p; .

(1.13)

u,v j+1) (i) We defer the display of precise expressions of π p; (x) and R (p; (x) to Sect. 2.2.3, since we need a certain representation to describe these functions. We introduce this representation in Sect. 2.1 and complete the proof of Proposition 1.1 in Sect. 2.2. It is worth emphasizing that the above proposition holds independently of the properties of the spin-spin coupling: Ju,v does not have to be translation-invariant or Zd -symmetric. In particular, the identity (1.11) holds independently of the sign of the spin-spin coupling. A spin glass, whose spin-spin coupling is randomly negative, is an extreme example for which (1.11) holds. Whether or not the lace expansion (1.11) is useful depends on the possibility of good control on the expansion coefficients and the remainder. As explained below, it is indeed possible to have optimal bounds on the expansion coefficients for the nearest-neighbor interaction (i.e., Jo,x = 1{ x 1 =1}) and for the following spread-out interaction:

Jo,x = L −d µ(L −1 x)

(1 ≤ L < ∞),

(1.14)

where µ : [−1, 1]d \ {o} → [0, ∞) is a bounded probability distribution, which is symmetric under rotations by π/2 and reflections in coordinate hyperplanes, and


287

piecewise continuous so that the Riemann sum L −d x∈Zd µ(L −1 x) approximates d Rd d x µ(x) ≡ 1. One of the simplest examples would be Jo,x =

1{0< x ∞ ≤L} = O(L −d ) 1{0< L −1 x ∞ ≤1}. z∈Zd 1{0< z ∞ ≤L}

(1.15)

Proposition 1.2. Let ρ = 2(d − 4) > 0. For the nearest-neighbor model with d 1 and for the spread-out model with L 1, there are finite constants θ and λ such that j) |(p; (x)−δo,x | ≤ θ δo,x +

λ(1−δo,x ) ( j ≥ 0), |x|d+2+ρ

j) |R (p; (x)| → 0 ( j ↑ ∞),

(1.16) for any p ≤ pc , any ⊂ Zd and any x ∈ . The proof of Proposition 1.2 depends on certain bounds on the expansion coefficients in terms of two-point functions. These diagrammatic bounds arise from counting the number of “disjoint connections”, corresponding to applications of the BK inequality in percolation (e.g., [5]). We prove these bounds in Sect. 4, and in anticipation of this, in Sect. 3 we explain how we use their implication to prove Proposition 1.2, with θ = O(d −1 ) and λ = O(1) for the nearest-neighbor model, and θ = O(L −2+ ) and λ = O(θ 2 ) with a small > 0 for the spread-out model. Let τo,x τo,x , D(x) = |x|2 D(x). (1.17) τ ≡ τ ( p) = , σ2 = τ x x j) Due to (1.16) uniformly in ⊂ Zd , there is a limit p (x) ≡ lim↑Zd lim j↑∞ (p; (x) such that

G p (x) = p (x) + ( p ∗ τ D ∗ G p )(x), (1.18a) λ(1 − δo,x ) | p (x) − δo,x | ≤ θ δo,x + , (1.18b) |x|d+2+ρ for any p ≤ pc and any x ∈ Zd , where ( f ∗ g)(x) = y∈Zd f (y) g(x − y). We note that the identity in (1.18) is similar to the recursion equation for the random-walk Green’s function: Sr (x) ≡

∞

r i D ∗i (x) = δo,x + (r D ∗ Sr )(x)

(|r | < 1),

(1.19)

i=0

where f ∗i (x) = ( f ∗(i−1) ∗ f )(x), with f ∗0 (x) = δo,x by convention. The leading asymptotics of S1 (x) for d > 2 is known as σad2 |x|−(d−2) , where ad = d2 π −d/2 ( d2 − 1) (e.g., [14, 15]). Following the model-independent analysis of the lace expansion in [14, 15], we obtain the following asymptotics of the critical two-point function: Theorem 1.3. Let ρ = 2(d − 4) > 0 and fix any small > 0. For the nearest-neighbor model with d 1 and for the spread-out model with L 1, we have that, for x = o, (ρ−)∧2 ad A 1 + O(|x|− d ) (NN model), G pc (x) = (1.20) × τ ( pc ) σ 2 |x|d−2 1 + O(|x|−ρ∧2+ ) (SO model),

288

where constants in the error terms may vary depending on , and −1 −1 τ ( pc ) 2 τ ( pc ) = pc (x) , A = 1+ |x| pc (x) . σ2 x x

A. Sakai

(1.21)

Consequently, (1.8) holds and η = 0. In this paper, we restrict ourselves to the nearest-neighbor model for d 4 and to the spread-out model for d > 4 with L 1. However, it is strongly expected that our method can show the same asymptotics of the critical two-point function for any translation-invariant, Zd -symmetric finite-range model above four dimensions, by taking the coordination number sufficiently large. 1.3. Organization. In the rest of this paper, we focus our attention on the model-dependent ingredients: the lace expansion for the Ising model (Proposition 1.1) and the bounds on (the alternating sum of) the expansion coefficients for the ferromagnetic models (Proposition 1.2). In Sect. 2, we prove Proposition 1.1. In Sect. 3, we reduce Proposition 1.2 to a few other propositions, which are then results of the aforementioned diagrammatic bounds on the expansion coefficients. We prove these diagrammatic bounds in Sect. 4. As soon as the composition of the diagrams in terms of two-point functions is understood, it is not so hard to establish key elements of the above reduced propositions. We will prove these elements in Sect. 5.1 for the spread-out model and in Sect. 5.2 for the nearest-neighbor model. 2. Lace Expansion for the Ising Model The lace expansion was initiated by Brydges and Spencer [8] to investigate the weakly self-avoiding walk for d > 4. Later, it was developed for various stochastic-geometrical models, such as a strictly self-avoiding walk for d > 4 (e.g., [18]), lattice trees/animals for d > 8 (e.g., [16]), unoriented percolation for d > 6 (e.g., [17]), oriented percolation for d > 4 (e.g., [25]) and the contact process for d > 4 (e.g., [27]). See [31] for an extensive list of references. This is the first lace-expansion paper for the Ising model. In this section, we prove the lace expansion (1.11) for the Ising model. From now on, (i) (i) we fix p ≥ 0 and abbreviate, e.g., π p; (x) to π (x). There may be several ways to derive the lace expansion for ϕo ϕx , using, e.g., the high-temperature expansion, the random-walk representation (e.g., [10]) or the FK random-cluster representation (e.g., [11]). In this paper, we use the random-current representation (Sect. 2.1), which applies to models in the Griffiths-Simon class (e.g., [1, 4]). This representation is similar in philosophy to the high-temperature expansion, but it turned out to be more efficient in investigating the critical phenomena [1–4]. The main advantage in this representation is the source-switching lemma (Lemma 2.3 below in Sect. 2.2.2) by which we have an identity for ϕo ϕx − ϕo ϕx A with “A ⊂ ” (the meaning will be explained in Sect. 2.1). We will repeatedly apply this identity to complete the lace expansion for ϕo ϕx in Sect. 2.2.3. 2.1. Random-current representation. In this subsection, we describe the random-current representation and introduce some notation that will be essential in the derivation of the lace expansion.


289

First we introduce some notions and notation. We call a pair of sites b = {u, v} with Jb = 0 a bond. So far we have used the notation ⊂ Zd for a site set. However, we will often abuse this notation to describe a graph that consists of sites of and are equipped with a certain bond set, which we denote by B . Note that “{u, v} ∈ B ” always implies “u, v ∈ ”, but the latter does not necessarily imply the former. If we regard A and as graphs, then “A ⊂ ” means that A is a subset of as a site set, and that BA ⊂ B . Now we consider the partition function Z A on A ⊂ . By expanding the Boltzmann factor in (1.2), we obtain

( p Ju,v )n u,v n u,v n u,v ϕu ϕv Z A = 2−|A| n u,v ! A {u,v}∈ B n ∈ Z ϕ∈{±1} u,v + A ( p Jb )n b 1 n b , (2.1) ϕv bv = n ! 2 b B n∈Z+ A

ϕv =±1

v∈A

b∈BA

where we call n = {n b }b∈ BA a current configuration. Note that the single-spin average in the last line equals 1 if bv n b is an even integer, and 0 otherwise. Denoting by ∂n the set of sources v ∈ at which bv n b is an odd integer, and defining wA (n) =

( p Jb )n b nb !

A (n ∈ ZB + ),

(2.2)

b∈BA

we obtain

ZA =

wA (n)

B n∈Z+ A

v∈A

1{bv n b even} =

wA (n).

(2.3)

∂n=∅

The partition function Z A equals the partition function on with Jb = 0 for all b ∈ B\BA . We can also think of Z A as the sum of w (n) over n ∈ ZB + satisfying n|B \BA ≡ 0, where n|B is a projection of n over the bonds in a bond set B, i.e., n|B = {n b : b ∈ B}. By this observation, we can rewrite (2.3) as ZA = w (n). (2.4) ∂n=∅ n|B \BA ≡0

Following the same calculation, we can rewrite Z A ϕx ϕ y A for x, y ∈ A as ( p Jb )n b 1 1{v∈xy}+bv n b ϕv Z A ϕx ϕ y A = nb ! 2 B n∈Z+ A

=

∂n=x y

b∈BA

wA (n) =

v∈A

ϕv =±1

w (n),

(2.5)

∂n=x y n|B \BA ≡0

where x y is an abbreviation for the symmetric difference {x} {y}: ∅ if x = y, x y ≡ {x} {y} = {x, y} otherwise.

(2.6)

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A. Sakai

y

x

Fig. 1. A current configuration with sources at x and y. The thick-solid segments represent bonds with odd currents, while the thin-solid segments represent bonds with positive even currents, which cannot be seen in the high-temperature expansion

If x or y is in Ac ≡ \A, then we define both sides of (2.5) to be zero. This is consistent c with the above representation when x = y, since, for example, if x ∈ A , then the leftmost expression of (2.5) is a multiple of 21 ϕx =±1 ϕx = 0, while the last expression in (2.5) is also zero because there is no way of connecting x and y on a current configuration n with n|B \BA ≡ 0. The key observation in the representation (2.5) is that the right-hand side is nonzero only when x and y are connected by a chain of bonds with odd currents (see Fig. 1). We will exploit this peculiar underlying percolation picture to derive the lace expansion for the two-point function. 2.2. Derivation of the lace expansion. In this subsection, we derive the lace expansion for ϕo ϕx using the random-current representation. In Sect. 2.2.1, we introduce some definitions and perform the first stage of the expansion, namely (1.11) for j = 0, simply using inclusion-exclusion. In Sect. 2.2.2, we perform the second stage of the expansion, where the source-switching lemma (Lemma 2.3) plays a significant role to carry on the expansion indefinitely. Finally, in Sect. 2.2.3, we complete the proof of Proposition 1.1. 2.2.1. The first stage of the expansion As mentioned in Sect. 2.1, the underlying picture in the random-current representation is quite similar to percolation. We exploit this similarity to obtain the lace expansion. First, we introduce some notions and notation. Definition 2.1. (i) Given n ∈ ZB + and A ⊂ , we say that x is n-connected to y in (the graph) A, and simply write x ←→ y in A, if either x = y ∈ A or there is n

a self-avoiding path (or we simply call it a path) from x to y consisting of bonds A y. We b ∈ BA with n b > 0. If n ∈ ZB + , we omit “in A” and simply write x ←→ n also define A

{x ←→ y} = {x ←→ y} \ {x ←→ y in Ac }, n n n

(2.7)

and say that x is n-connected to y through A. (ii) Given an event E (i.e., a set of current configurations) and a bond b, we define {E off b} to be the set of current configurations n ∈ E such that changing n b results in a configuration that is also in E. Let Cnb (x) = {y : x ←→ y off b}. n


o

=

x

291

o

x

+

o

x b

b

Fig. 2. A schematic representation of (2.11). The thick lines are connections consisting of bonds with odd currents, while the thin arcs are connections made of bonds with positive (not necessarily odd) currents. The shaded region represents Cnb (o)

(iii) For a directed bond b = (u, v), we write b = u and b = v. We say that a directed bond b is pivotal for x ←→ y from x, if {x ←→ b off b} ∩ {b ←→ y in Cnb (x)c } n n n y} occurs with no pivotal bonds, we say that x is n-doubly occurs. If {x ←→ n connected to y, and write x ⇐⇒ y. n We begin with the first stage of the lace expansion. First, by using the above percolation language, the two-point function can be written as w (n) w (n) ≡ 1{o←→x}. n Z Z

ϕo ϕx =

∂n=ox

(2.8)

∂n=ox

We decompose the indicator on the right-hand side into two parts depending on whether or not there is a pivotal bond for o ←→ x from o; if there is, we take the first bond n among them. Then, we have 1{o←→x} = 1{o⇐⇒x} + n

n

b∈B

1{o⇐⇒b off b} 1{n b >0} 1{b←→x in Cnb (o)c }. n

n

(2.9)

Let (0) (x) = π

w (n) 1{o⇐⇒x}. n Z

(2.10)

∂n=ox

Substituting (2.9) into (2.8), we obtain (see Fig. 2) (0) (x)+ ϕo ϕx = π

w (n) x 1{o⇐⇒b off b} 1{n b >0} 1{b←→ in Cnb (o)c }. n n Z

(2.11)

b∈B ∂n=ox

Next, we consider the sum over n in (2.11). Since b is pivotal for o ←→ x from n o (= x, due to the last indicator) and ∂n = o x, in fact n b is an odd integer. We alternate the parity of n b by changing the source constraint into o b x ≡ {o} {b, b} {x} and multiplying by ( p Jb )n /n! n odd = tanh( p Jb ) ≡ τb . (2.12) n n even ( p Jb ) /n! Then, the sum over n in (2.11) equals ∂n=obx

w (n) 1{o⇐⇒b off b} τb 1{n b even} 1{b←→x in Cnb (o)c }. n n Z

(2.13)

292

A. Sakai

Note that, except for b, there are no positive currents on the boundary bonds of Cnb (o). Now, we condition on Cnb (o) = A and decouple events occurring on BAc from events occurring on B \BAc , by using the following notation:

w˜ ,A (k) =

b∈B \BAc

( p Jb )kb kb !

B \BAc

(k ∈ Z+

).

(2.14)

Conditioning on Cnb (o) = A, multiplying Z Ac /Z Ac ≡ 1 (and using the notation k = n|B \BAc and m = n|BAc ) and then summing over A ⊂ , we have (2.13) =

A⊂

w˜ ,A (k) Z Ac wAc (m) 1{o⇐⇒b off b} ∩ {Ckb (o)=A} τb Z Z Ac k ∂k=ob

∂m=bx

×1{kb even} 1{b←→x in Ac } m

=

w (n) 1{o⇐⇒b off b} ∩ {Cnb (o)=A} τb n Z

A⊂ ∂n=ob

×1{n b even}

wAc (m) 1{b←→x (inAc )} m Z Ac ∂m=bx = ϕb ϕx Ac

w (n) = 1{o⇐⇒b off b} τb 1{n b even} ϕb ϕx C b (o)c . n n Z

(2.15)

∂n=ob

Furthermore, “off b” and 1{n b even} in the last line can be omitted, since {o ⇐⇒ b} \ n {o ⇐⇒ b off b} and {∂n = o b} ∩ {n b odd} are subsets of {b ∈ Cnb (o)}, on which n ϕb ϕx C b (o)c = 0. As a result, n

(2.15) =

w (n) 1{o⇐⇒b} τb ϕb ϕx C b (o)c . n n Z

(2.16)

∂n=ob

By (2.11) and (2.16), we arrive at (0) (x) + ϕo ϕx = π

b∈B

(0) (1) π (b) τb ϕb ϕx − R (x),

(2.17)

where (1) R (x) =

b∈B

w (n) 1{o⇐⇒b} τb ϕb ϕx − ϕb ϕx C b (o)c . n n Z

(2.18)

∂n=ob

(0) (1) This completes the proof of (1.11) for j = 0, with π (x) and R (x) being defined in (2.10) and (2.18), respectively.


293

2.2.2. The second stage of the expansion In the next stage of the lace expansion, we (1) further expand R (x) in (2.17). To do so, we investigate the difference ϕb ϕx − ϕb ϕx C b (o)c in (2.18). First, we prove the following key proposition2 : n

Proposition 2.2. For v, x ∈ and A ⊂ , we have ϕv ϕx − ϕv ϕx Ac =

∂m=∅ ∂n=v x

wAc (m) w (n) A 1{v ←→ x}. m+n Z Ac Z

(2.19)

Therefore, ϕv ϕx Ac ≤ ϕv ϕx for the ferromagnetic case. Proof. Since both sides of (2.19) are equal to 1{x∈A} when v = x (see below (2.6)), it suffices to prove (2.19) for v = x. First, by using (2.3)–(2.5), we obtain Z Z Ac ϕv ϕx − ϕv ϕx Ac = Z Ac w (n) − wAc (m) Z ∂n={v,x}

∂m={v,x}

=

w (m) w (n)

∂m=∅, ∂n={v,x} m|B \BAc ≡0

−

w (m) w (n).

(2.20)

∂m={v,x}, ∂n=∅ m|B \BAc ≡0

Note that the second term is equivalent to the first term if the source constraints for m and n are exchanged. Next, we consider the second term of (2.20), whose exact expression is

( p Jb )n b ( p Jb )m b +n b nb ! m b ! nb ! ∂m={v,x}, ∂n=∅ m|B \BAc ≡0

=

∂N={v,x}

b∈B \BAc

w (N)

b∈BAc

Nb . mb

(2.21)

∂m={v,x} b∈BAc m|B \BAc ≡0

The following is a variant of the source-switching lemma [1, 13] and allows us to change the source constraints in (2.21). Lemma 2.3 (Source-Switching Lemma).

Nb = 1{v←→x in Ac } mb N ∂m={v,x} b∈BAc m|B \BAc ≡0

∂m=∅

m|B \BAc ≡0

Nb . mb

(2.22)

b∈BAc

2 The mean-field results in [1–4] are based on a couple of differential inequalities for M p,h and χ p (under the periodic-boundary condition) using a certain random-walk representation. We can simplify the proof of the same differential inequalities (under the free-boundary condition as well) using Proposition 2.2.

294

A. Sakai

The idea of the proof of (2.22) can easily be extended to more general cases, in which the source constraint in the left-hand side of (2.22) is replaced by ∂m = V for some V ⊂ and that in the right-hand side is replaced by ∂m = V {v, x} (e.g., [1]). We will explain the proof of (2.22) after completing the proof of Proposition 2.2. We continue with the proof of Proposition 2.2. Substituting (2.22) into (2.21), we obtain

Nb (2.21) = w (N) 1{v←→x in Ac } mb N ∂m=∅ b∈BAc m|B \BAc ≡0

∂N={v,x}

=

∂m=∅, ∂n={v,x} m|B \BAc ≡0

w (m) w (n) 1{v←→x in Ac }.

(2.23)

m+n

Note that the source constraints for m and n in the last line are identical to those in the first term of (2.20), under which 1{v←→ x} is always 1. By (2.7), we can rewrite (2.20) as m+n ϕv ϕx − ϕv ϕx Ac =

∂m=∅, ∂n={v,x} m|B \BAc ≡0

w (m) w (n) Z Ac Z

A 1{v ←→ x}.

(2.24)

m+n

Using (2.3)–(2.4) to omit “m|B \BAc ≡ 0” and replace w (m) by wAc (m), we arrive at (2.19). This completes the proof of Proposition 2.2. Sketch proof of Lemma 2.3. We explain the meaning of the identity (2.22) and the idea of its proof. Given N = {Nb }b∈B , we denote by GN the graph consisting of Nb labeled edges between b and b for every b ∈ B (see Fig. 3). For a subgraph S ⊂ GN , we denote by ∂S the set of vertices at which the number of incident edges in S is odd, and let SA = S ∩ GN|B \B c . Then, the left-hand side of (2.22) equals the cardinality |S| of A

S = {S ⊂ GN : ∂S = {v, x}, SA = ∅},

(2.25)

and the sum in the right-hand side of (2.22) equals the cardinality |S | of S = {S ⊂ GN : ∂S = ∅, SA = ∅}.

(2.26)

We note that |S| is zero when there are no paths on GN between v and x consisting of edges whose endvertices are both in Ac , while |S | may not be zero. The identity (2.22) reads that |S| equals |S | if we compensate for this discrepancy. Suppose that there is a path ω from v to x consisting of edges in GN whose endvertices are both in Ac . Then, the map S ∈ S → S ω ∈ S ,

(2.27)

is a bijection [1, 13], and therefore |S| = |S |. Here and in the rest of the paper, the symmetric difference between graphs is only in terms of edges. For example, S ω is the result of adding or deleting edges (not vertices) contained in ω. This completes the proof of (2.22).


N :

0

GN :

0

295

N1 =3

N2 =3

11 12

21 22

13

23

N3 =1

S :

0 13

N5 =1

5

41 42 31

43

51

5

51

5

44 45 42

21 22

N4 =5

31

23

44 45 41 42

11

S

ω :

22

0 13

5 44

23

45

Fig. 3. N = {Nb }5b=1 = (3, 3, 1, 5, 1) is an example of a current configuration on [0, 5] ∩ Z+ satisfying ∂N = {0, 5}, and GN is the corresponding labeled graph consisting of edges e = bb , where b ∈ {1, . . . , Nb }. The third and fourth pictures show the relation between a subgraph S with ∂S = {0, 5} and its image S ω of the map defined in (2.27), where ω is a path of edges (11, 21, 31, 41, 51)

We now start with the second stage of the expansion by using Proposition 2.2 and applying inclusion-exclusion as in the first stage of the expansion in Sect. 2.2.1. First, we decompose the indicator in (2.19) into two parts depending on whether or not there A

is a pivotal bond b for v ←→ x from v such that v ←→ b. Let m+n m+n A

A

E N (v, x; A) = {v ←→ x} ∩ { pivotal bond b for v ←→ x from v such that v ←→ b}. N

N

N

(2.28)

A

On the event {v ←→ x} \ E m+n (v, x; A), we take the first pivotal bond b for v ←→ x m+n m+n A

from v satisfying v ←→ b. Then, we have (cf., (2.9)) m+n A 1{v ←→ x} = 1 E m+n (v,x;A) + m+n

b∈B

b (v)c }. 1{E m+n (v,b;A) off b} 1{m b +n b >0} 1{b←→x in Cm+n m+n

(2.29) Let v,x;A [X ] =

∂m=∅ ∂n=v x

wAc (m) w (n) 1 E m+n (v,x;A) X (m+n), Z Ac Z

v,x;A = v,x;A [1]. (2.30)

296

A. Sakai

v

x

=

v

x

+

v

x

b

b

Fig. 4. A schematic representation of (2.31). The dashed lines represent A, the thick-solid lines represent connections consisting of bonds b1 such that m b1 + n b1 is odd, and the thin-solid lines are connections made b (v) of bonds b2 such that m b2 + n b2 is positive (not necessarily odd). The shaded region represents Cm+n

Substituting (2.29) into (2.19), we obtain (see Fig. 4) ϕv ϕx − ϕv ϕx Ac = v,x;A + b∈B ∂m=∅ ∂n=v x

(2.31) wAc (m) w (n) 1{E m+n (v,b;A) off b} Z Ac Z b (v)c }, ×1{m b even, n b odd} 1{b←→x in Cm+n m+n

where we have replaced “m b + n b > 0” in (2.29) by “m b even, n b odd” that is the only possible combination consistent with the source constraints and the conditions in the indicators. As in (2.13), we alternate the parity of n b by changing the source constraint from ∂n = v x to ∂n = v b x and multiplying by τb . Then, the sum over m and n in (2.31) equals ∂m=∅ ∂n=v bx

wAc (m) w (n) b (v)c }. 1{E m+n (v,b;A) off b} τb 1{m b ,n b even}1{b←→x in Cm+n m+n Z Ac Z (2.32)

b (v) = B and decouple events occurring on Then, as in (2.15), we condition on Cm+n BBc from events occurring on B \ BBc . Let m = m|BAc \BAc ∩Bc , m = m|BAc ∩Bc , n = n|B \BBc and n = n|BBc . Note that ∂m = ∂m = ∅, ∂n = v b and ∂n = b x. Multiplying (2.32) by (Z Ac ∩Bc /Z Ac ∩Bc )(Z Bc /Z Bc ) ≡ 1 and using the notation (2.14), we obtain

(2.32) =

B⊂ ∂m =∅ ∂n =v b

w˜ Ac ,B (m ) Z Ac ∩Bc w˜ ,B (n ) Z Bc 1{E m +n (v,b;A) off b} ∩ {C b (v)=B} m +n Z Ac Z ×τb 1{m b ,n b even}

wAc ∩Bc (m ) wBc (n ) 1{b ←→ x in Bc } Z A c ∩B c Z Bc m +n ∂m =∅

∂n =bx

=

wAc (m) w (n) b (v)=B }τb 1{m ,n even} ϕ ϕ x c 1{E m+n (v,b;A) off b} ∩ {Cm+n b b b B Z Ac Z

B⊂ ∂m=∅ ∂n=v b

=

wAc (m) w (n) 1{E m+n (v,b;A) off b} τb 1{m b ,n b even} ϕb ϕx C b (v)c , m+n Z Ac Z

∂m=∅ ∂n=v b

(2.33)


297

where we have been able to perform the sum over m and n independently, due to the B c fact that 1{b ←→ x in Bc } ≡ 1 for any n ∈ Z+ B with ∂n = b x. As in the derivation m +n

of (2.16) from (2.15), we can omit “off b” and 1{m b ,n b even} in (2.33) using the source b (v). Therefore, constraints and the fact that ϕb ϕx C b (v)c = 0 whenever b ∈ Cm+n m+n

wAc (m) w (n) 1 E m+n (v,b;A) τb ϕb ϕx C b (v)c . (2.33) = m+n Z Ac Z

(2.34)

∂m=∅ ∂n=v b

By (2.30)–(2.34), we arrive at ϕv ϕx − ϕv ϕx Ac = v,x;A + −

b∈B

b∈B

v,b;A τb ϕb ϕx

v,b;A τb ϕb ϕx − ϕb ϕx C b (v)c , (2.35)

b (v) is a variable for the operation where C b (v) ≡ Cm+n v,b;A . This completes the second stage of the expansion.

2.2.3. Completion of the lace expansion. For notational convenience, we define w∅(m)/Z ∅ = 1{m≡0}. Since E n (o, x; ) = {o ⇐⇒ x} (cf., (2.28)), we can write n (0) (x) = o,x; . π

(2.36)

(1) (x) in (2.18) as Also, we can write R (1) (x) = o,b; τb ϕb ϕx − ϕb ϕx C b (o)c . R

(2.37)

b

Using (2.35), we obtain (1) o,b; τb b,x;C b (o) + R (x) = o,b; τb b,b ;C b (o) τb ϕb ϕx b

−

b

b

, (2.38) o,b; τb b,b ;C b (o) τb ϕb ϕx − ϕb ϕx C b (b)c

b where C b (o) ≡ Cnb (o) is a variable for the outer operation o,b; , and C b (b) ≡ Cm +n (b) is a variable for the inner operation b,b ;C b (o) . For j ≥ 1, we define ( j) (1) π (x) = (0) o,b ; τb1 ˜ · · · τb j−1 b1 ,...,b j

1

b1 ,b2 ;C0

( j) τ · · · , ×( j−1) b j b j−1 ,b j ;C˜ j−2 b j ,x;C˜ j−1 ( j) (1) (x) = (0) R o,b ; τb1 ˜ · · · τb j−1 b1 ,...,b j

1

×( j−1)

(2.39)

b1 ,b2 ;C0

b j−1 ,b j ;C˜ j−2

τb j ϕb j ϕx − ϕb j ϕx ˜c ··· ,

C j−1

(2.40)

298

A. Sakai

bi+1 where the operation (i) determines the variable C˜i = Cm (bi ) (provided that b0 = o). i +ni Then, we can rewrite (2.38) as (1) (1) R (x) = π (x) +

b

(1) (2) π (b ) τb ϕb ϕx − R (x).

(2.41)

As a result, (0) (0) (1) (1) (2) π (b) − π ϕo ϕx = π (x) − π (x) + (b) τb ϕb ϕx + R (x). b

(2.42) ( j) By repeated applications of (2.35) to the remainder R (x), we obtain (1.11)–(1.12) in Proposition 1.1. A For the ferromagnetic case, τb and wA (n) for any A ⊂ and n ∈ ZB + are nonnegative. This proves the first inequality in (1.13) and, with the help of Proposition 2.2, the ( j+1) ( j+1) nonnegativity of R (x) . To prove the upper bound on R (x), we simply ignore ϕb j ϕx ˜c in (2.40) and replace j by j + 1, where b j+1 = {u, v}. This completes the

C j−1

proof of Proposition 1.1.

2.3. Comparison to percolation. Since we have exploited the underlying percolation picture to derive the lace expansion (1.11) for the Ising model, it is not so surprising that the expansion coefficients (2.36) and (2.39) (also recall (2.30)) are quite similar to the lace-expansion coefficients for unoriented bond-percolation (cf., [17]): ⎧ (0) ( j = 0), E p 1{o⇐⇒x} ≡ P p (o ⇐⇒ x) ⎪ ⎪ ⎪ n0 ⎨ ( j) π p (x) = ( j) (1) ⎪ E(0) ( j ≥ 1), ⎪ p 1{o⇐⇒b1 } pb1 E p 1 E n1 (b1 ,b2 ;C˜0 ) · · · pb j E p 1 E n j (b j ,x;C˜ j−1 ) · · · ⎪ ⎩ n0 b1 ,...,b j

(2.43)

(i) where p ≡ x po,x is the bond-occupation parameter, and each E p denotes the expectation with respect to the product measure b ( pb 1{ni |b =1}+(1− pb )1{ni |b =0}). In particular, the events involved in (2.36) and (2.39) are identical to those in (2.43). However, there are significant differences between these two models. The major differences are the following: (a) Each current configuration must satisfy not only the conditions in the indicators, but also its source constraint that is absent in percolation. (b) An operation is not an expectation, since the source constraints in the numerator and denominator of in (2.30) are different. (c) In each (i) for i ≥ 1, the sum mi + ni of two current configurations is coupled with mi−1 + ni−1 via the cluster C˜i−1 determined by mi−1 + ni−1 . By contrast, in each E(i) p in (2.43), a single percolation configuration ni is coupled with ni−1 via i ˜ Ci−1 = Cnbi−1 (bi−1 ). In addition, mi is nonzero only on bonds in BC˜c , while the i−1 current configuration ni has no such restriction.


299

These elements are responsible for the difference in the method of bounding diagrams for the expansion coefficients. Take the 0th -expansion coefficient for example. For percolation, the BK inequality simply tells us that π p(0) (x) ≤ P p (o ←→ x)2 .

(2.44)

For the ferromagnetic Ising model, on the other hand, we first recall (2.10), i.e., (0) (x) = π

w (n) 1{o⇐⇒x}, n Z

(2.45)

∂n=ox

where w (n)/Z ≥ 0. Due to the indicator, every current configuration n ∈ ZB + that gives nonzero contribution has at least two bond-disjoint paths ζ1 , ζ2 from o to x such ˙ ζ2 . Also, due to the source constraint, there should be at that n b > 0 for all b ∈ ζ1 ∪ least one path ζ from o to x such that n b is odd for all b ∈ ζ . Suppose, for example, that ζ = ζ1 and that n b for b ∈ ζ2 are all positive-even. Since a positive-even integer can split into two odd integers, on the labeled graph Gn with ∂Gn = o x (recall the notation introduced above (2.25)) there are at least three edge-disjoint paths from o to x. This observation naturally leads us to expect that (0) π (x) ≤ ϕo ϕx 3

(2.46)

holds for the ferromagnetic Ising model. This naive argument to justify (2.46) will be made rigorous in Sect. 4 by taking account of partition functions. The higher-order expansion coefficients are more involved, due to the above item (c). This will also be explained in detail in Sect. 4. 3. Bounds on (j ) (x) for the Ferromagnetic Models From now on, we restrict ourselves to the ferromagnetic models. In this section, we explain how to prove Proposition 1.2 assuming a few other propositions (Propositions 3.1–3.3 below). These propositions are results of diagrammatic bounds on the expansion coefficients in terms of two-point functions. We will show these diagrammatic bounds in Sect. 4. The strategy to prove Proposition 1.2 is model-independent, and we follow the strategy in [14] for the nearest-neighbor model and that in [15] for the spread-out model. Since the latter is simpler, we first explain the strategy for the spread-out model. In the rest of this paper, we will frequently use the notation |||x||| = |x| ∨ 1.

(3.1)

We also emphasize that constants in the O-notation used below (e.g., O(θ0 ) in (3.3)) are independent of ⊂ Zd . 3.1. Strategy for the spread-out model. Using the diagrammatic bounds below in Sect. 4, we will prove in detail in Sect. 5.1 that the following proposition holds for the spread-out model:

300

A. Sakai

Proposition 3.1. Let Jo,x be the spread-out interaction. Suppose that G(x) ≤ δo,x + θ0 |||x|||−q

τ ≤ 2,

(3.2)

hold for some θ0 ∈ (0, ∞) and q ∈ ( d2 , d). Then, for sufficiently small θ0 (with θ0 L d−q being bounded away from zero) and any ⊂ Zd , we have O(θ0 )i δo,x + O(θ03 )|||x|||−3q (i = 0, 1), (i) π (x) ≤ (3.3) O(θ0 )i |||x|||−3q (i ≥ 2). The exact value of the assumed upper bound on τ in (3.2) is unimportant and can be any finite number, as long as it is independent of θ0 and bigger than the mean-field critical point 1. We note that the exponent 3q in (3.3) is due to (2.46) (and diagrammatic bounds on the higher-expansion coefficients), and is replaced by 2q with q ∈ ( 2d 3 , d) for percolation, due to, e.g., (2.44). Sketch of proof of Proposition 1.2 for the spread-out model. We will show below that, at p = pc , τ ≤ 2, G(x) ≤ δo,x + O(L −2+ )|||x|||−(d−2) ,

(3.4)

for some small > 0. Since τ and G(x) are nondecreasing and continuous in p ≤ pc for the ferromagnetic models, these bounds imply (3.2) for all p ≤ pc , with θ0 = cL −2+ > 0 and q = d − 2, where q ∈ ( d2 , d) if d > 4 and θ0 L d−q = cL > 0. Then, by Proposition 3.1, the bound (3.3) with θ0 = O(L −2+ ) and q = d − 2 holds for d > 4 and θ0 # 1 (thus L 1). Therefore, by (1.13) with ϕv ϕx ≤ 1, ( j) ( j+1) (x) ≤ τ π (u) = O(θ0 ) j → 0 ( j ↑ ∞), (3.5) 0 ≤ R u

and by (1.12) for j ≥ 0, |(j) (x) − δo,x | ≤ O(θ0 )δo,x +

O(θ02 ) O(θ02 )(1 − δo,x ) = O(θ )δ + , (3.6) 0 o,x |x|d+2+ρ |||x|||3(d−2)

where ρ = 2(d − 4). This completes the proof of Proposition 1.2 for the spread-out model, assuming (3.4) at p = pc . It thus remains to show the bounds in (3.4) at p = pc . These bounds are proved by adapting the model-independent bootstrapping argument in [15] (see the proof of [15, Prop. 2.2] for self-avoiding walk and percolation), together with the fact that G(x) decays exponentially as |x| ↑ ∞ for every p < pc [23, 30] so that supx G(x) is continuous in p < pc [28]. We complete the proof. 3.2. Strategy for the nearest-neighbor model. Since σ 2 = O(1) for short-range models, we cannot expect that θ0 in (3.2) is small, or that Proposition 3.1 is applicable to bound the expansion coefficients in this setting. Under this circumstance, we follow the strategy in [14]. The following is the key proposition, whose proof will be explained in Sect. 5.2:


301

Proposition 3.2. Let Jo,x be the nearest-neighbor or spread-out interaction, and suppose that

sup(D ∗ G ∗2 )(x) ≤ θ0 ,

τ − 1 ≤ θ0 ,

sup

x≡(x1 ,...,xd )=o l=1,...,d

x

xl2 ∨ 1 G(x) ≤ θ0 (3.7) σ2

hold for some θ0 ∈ (0, ∞). Then, for sufficiently small θ0 and any ⊂ Zd , we have (i) 1 + O(θ02 ) (i = 0), (i) π (x) ≤ |x|2 π (x) ≤ dσ 2 (i + 1)2 O(θ0 )i∨2 . i O(θ ) (i ≥ 1), 0 x x (3.8) Furthermore, in addition to (3.7) with θ0 # 1, if G(x) ≤ λ0 |||x|||−q

(3.9)

holds for some λ0 ∈ [1, ∞) and q ∈ (0, d), then we have for i ≥ 0, (i) π (x) ≤ O(θ0 )i δo,x +

λ30 (i + 1)3q+2 O(θ0 )(i−2)∨0 (1 − δo,x ). |x|3q

(3.10)

Sketch of proof of Proposition 1.2 (primarily) for the nearest-neighbor model. We first claim that the assumed bounds in (3.7) indeed hold for any p ≤ pc if d > 4 and θ0 # 1, where θ0 = O(d −1 ) for the nearest-neighbor model and θ0 = O(L −d ) for the spread-out model. The proof is based on the orthodox model-independent bootstrapping argument in, e.g., [24] (see also [21] for improved random-walk estimates; bootstrapping assumptions that are different from, but philosophically similar to, (3.7) are used in [20]). Therefore, (3.8) holds for p ≤ pc and hence ensures the existence of an infinite-volume limit (x) = lim↑Zd lim j↑∞ (j) (x) that satisfies

|(x)| = 1 + O(θ0 ),

x

|x|2 |(x)| = dσ 2 O(θ02 ).

(3.11)

x

As a byproduct, we obtain the identity in (1.21) for τ ( pc ) for both models. Suppose that G(x) ≤ λ0 |||x|||−(d−2)

(3.12)

holds at p = pc . Then, by Proposition 3.2, we obtain (3.10) with q = d − 2. Using this in (3.5)–(3.6), we can prove Proposition 1.2. To complete the proof, it thus remains to show (3.12) at p = pc . To show this, we use the following proposition: Proposition 3.3. Let G¯ (s) = sup |x|s G(x), x

W¯ (t) = sup x

|y|t G(y) G(x − y),

y

and suppose that the bounds in (3.7) hold with θ0 # 1 .

(3.13)

302

(i) If

A. Sakai

x

(x) = τ −1 and |(x)| ≤ O(|||x|||−(d+2) ), then we have ad x (x) G(x) ∼ as |x| ↑ ∞. τ x |x|2 (D ∗ )(x) |x|d−2

(3.14)

(ii) If x |x|r |(x)| < ∞ for some r > 0, then, for s, t > 0 which are not odd integers, we have G¯ (s) < ∞ if s ≤ r and s < d − 2, (3.15) W¯ (t) < ∞ if t ≤ %r & and t < d − 4. (iii) If W¯ (t) < ∞ for some t ≥ 0, then x |x|t+2 |(x)| < ∞. The above proposition is a summary of key elements in [14, Prop. 1.3 and Lemmas 1.5–1.6] that are sufficient to prove (3.12) in the current setting. The proofs of Propositions 3.3(i) and 3.3(ii) are model-independent and can be found in [14, Sects. 2 and 4], respectively. The proof of Proposition 3.3(iii) is similar to that of the first statement of Proposition 3.2: (3.7) implies (3.8). We will explain this in Sect. 5.2. Now we continue with the proof of (3.12). Fix p = pc . Since the asymptotic behavior (3.14) is good enough for the bound (3.12), it suffices to check the assumptions of Proposition 3.3(i). The first assumption on the sum of (x) is satisfied at p = pc , as d+2 mentioned below (3.11). The second assumption is also satisfied if G¯ ( 3 ) < ∞, because of the second statement of Proposition 3.2: (3.9) implies (3.10). By Proposition 3.3(ii), d+2 it thus suffices to show that x |x| 3 |(x)| is finite if d > 4. To show this, we let ri+1 = (d − 2) ∧ %ri & + 2 − , (3.16) r0 = 2, where 0 < ≤ 23 (d − 4). Note that, by this definition, ri for i ≥ 1 equals ((d − 2) ∧ (i + 3))− and increases until it reaches d − 2 − . We prove below by induction that x |x|ri |(x)| is finite for all i ≥ 0. This is sufficient for the finiteness of d+2 3 |(x)|, since x |x| lim ri = d − 2 − ≥ d − 2 − 23 (d − 4) =

i↑∞

d+2 3 .

(3.17)

Note that, by (3.11), x |x|r0 |(x)| < ∞. Suppose x |x|ri |(x)| < ∞ for some i ≥ 0. Then, by Proposition 3.3(ii), W¯ (t) is finite for t ∈ (0, %ri &] ∩ (0, d − 4). Since %r0 & = 2 and %ri & = (d − 3) ∧ (i + 2) 1, W¯ (T ) with T = (i + 2) ∧ (d − 4 − ) is for i ≥ T +2 finite. Then, by Proposition 3.3(iii), x |x| |(x)| is finite. Since T + 2 = (i + 4) ∧ (d − 2 − ) ≥ (d − 2) ∧ (i + 4) − = ri+1 , (3.18) we obtain that x |x|ri+1 |(x)| < ∞. This completes the induction and the proof of (3.12). The proof of Proposition 1.2 is now completed. ( j) 4. Diagrammatic Bounds on π (x)

In this section, we prove diagrammatic bounds on the expansion coefficients. In Sect. 4.1, we construct diagrams in terms of two-point functions and state the bounds. In Sect. 4.2, we prove a key lemma for the diagrammatic bounds and show how to apply this lemma ( j) (0) to prove the bound on π (x). In Sect. 4.3, we prove the bounds on π (x) for j ≥ 1.


303 v’2

(v2 )

PΛ(1) (v1 , v1 ) =

v1

PΛ(2) (v1 , v2 ) =

ví1

v1

(v2)

PΛ(3) (v1 , v3 ) =

(v’1 )

(v’2 )

v1

v’3

(v’1 ) (v3)

v

v

u (1)

PΛ;u (v1 , v1 ) =

(1)

v1

PΛ;u, v(v1 , v1 ) =

v’1

+

(v’) v1

(v’)

v1

v’1

u

v’1

u v

(0) PΛ;u (y, x) = y

(0) PΛ;u , v(y, x) =

x u

(v’) y

x u

(1) (1) (0) Fig. 5. Schematic representations of P (v1 , v j ) for j = 1, 2, 3, P;u (v1 , v1 ), P;u,v (v1 , v1 ), P;u (y, x) ( j)

(0) (y, x). The labels in the parentheses represent vertices that are summed over, each sequence of and P;u,v bubbles from vi and vi represents ψ (vi , vi ) − δv ,v , and the sequence of bubbles from v to v represents i

ψ (v , v).

i

4.1. Construction of diagrams. To state bounds on the expansion coefficients (as in Proposition 4.1 below), we first define diagrammatic functions consisting of two-point functions. Let ϕ y ϕb τb , (4.1) G˜ (y, x) =

b:b=x

which satisfies3 ϕ y ϕx ≤ δ y,x + = δ y,x +

b:b=x ∂n=y x n b odd

b:b=x

w (n) Z

w (n) ≤ δ y,x + G˜ (y, x). Z

τb

(4.2)

∂n=y b n b even

Let ψ (y, x) =

∞

G˜ 2

∗ j

(y, x) ≡ δ y,x +

j=0

∞ j=1

j

u 0 ,...,u j l=1 u 0 =y, u j =x

G˜ (u l−1 , u l )2 ,

and define (see the first line in Fig. 5) P(1) (v1 , v1 ) = 2 ψ (v1 , v1 ) − δv1 ,v1 ϕv1 ϕv1 ,

j ψ (vi , vi ) − δvi ,vi ϕv1 ϕv2 ϕv2 ϕv1 P( j) (v1 , v j ) = v2 ,...,v j v1 ,...,v j−1

×

j−1 i=2

(4.3)

(4.4)

i=1

ϕv ϕvi−1 i+1 ϕvi+1 ϕvi

ϕv j−1 ϕv j

( j ≥ 2),

(4.5)

where the empty product for j = 2 is regarded as 1. 3 Repeated applications of (4.2) to the translation-invariant models result in the random-walk bound: ϕo ϕx ≤ Sτ (x) for ⊂ Zd and τ ≤ 1.

304

A. Sakai

( j) Next, we define P;u (v1 , v j ) by replacing one of the 2 j − 1 two-point functions on the right-hand side of (4.4)–(4.5) by the product of two two-point functions, such as replacing ϕz ϕz by ϕz ϕu ϕu ϕz , and then summing over all 2 j − 1 choices of this replacement. For example, we define (see the second line in Fig. 5) (1) P;u (v1 , v1 ) = 2 ψ (v1 , v1 ) − δv1 ,v1 ϕv1 ϕu ϕu ϕv1 , (4.6)

and (2) (v1 , v2 ) = P;u

2

v2 ,v1

ψ (vi , vi ) − δvi ,vi

ϕv1 ϕu ϕu ϕv2 ϕv2 ϕv1

i=1

×ϕv1 ϕv2 + ϕv1 ϕv2 ϕv2 ϕu ϕu ϕv1 ϕv1 ϕv2 (4.7) +ϕv1 ϕv2 ϕv2 ϕv1 ϕv1 ϕu ϕu ϕv2 .

( j) We define P;u,v (v1 , v j ) similarly as follows. First we take two two-point functions in P( j) (v1 , v j ), one of which (say, ϕz 1 ϕz 1 for some z 1 , z 1 ) is among the aforemen tioned 2 j − 1 two-point functions, and the other (say, G˜ (z 2 , z 2 ) for some z 2 , z 2 ) is among those of which ψ (vi , vi ) − δvi ,vi for i = 1, . . . , j are composed. The product ϕz 1 ϕz G˜ (z 2 , z ) is then replaced by 1

2

v

ϕz 1 ϕv ϕv ϕz 1 ψ (v , v) ϕz 2 ϕu G˜ (u, z 2 ) + G˜ (z 2 , z 2 ) δu,z 2

ϕz 2 ϕv G˜ (v , z 2 )+ G˜ (z 2 , z 2 )δv ,z 2 ψ (v , v). +ϕz 1 ϕu ϕu ϕz 1

v

(4.8) ( j) Finally, we define P;u,v (v1 , v j ) by taking account of all possible combinations of (1) ϕz 1 ϕz 1 and G˜ (z 2 , z 2 ). For example, we define P;u,v (v1 , v1 ) as (see Fig. 5)

(1) P;u,v (v1 , v1 )

=

u ,u ,v

2ψ (v1 , u ) G˜ (u , u ) ϕu ϕu G˜ (u, u )

+G˜ (u , u ) δu,u ψ (u , v1 )ϕv1 ϕv ϕv ϕv1 ψ (v , v) +(permutation of u and v ) , (4.9)

(1) where the permutation term corresponds to the second term for P;u,v (v1 , v1 ) in Fig. 5. In addition to the above quantities, we define (see the third line in Fig. 5) (0) P;u (y, x) = ϕ y ϕx 2 ϕ y ϕu ϕu ϕx , (0) P;u,v (y, x) = ϕ y ϕx ϕ y ϕu ϕu ϕx ϕ y ϕv ϕv ϕx ψ (v , v), v

(4.10) (4.11)


305

(b1)

(b1)

(1)

πΛ (x)

o

πΛ (x)

x

(b1)

x

(2)

o

o (b2)

(1)

x

+ (b2)

(2)

Fig. 6. The leading diagrams for π (x) and π (x). The segments that terminate with bi for i = 1, 2 represent δ + G˜ (cf., (4.13)–(4.14)). The labels in the parentheses represent bonds that are summed over. There are artificial gaps in the figures to distinguish different building blocks

and let (y, x) = P;u

( j) P;u (y, x),

P;u,v (y, x) =

j≥0

( j) P;u,v (y, x),

(4.12)

j≥0

(0) (0) (y, x) and (y, x) and P;u,v (y, x) are the leading contributions to P;u where P;u P;u,v (y, x), respectively. Finally, we define Q (y, x) = (4.13) δ y,z + G˜ (y, z) P (z, x), ;u

;u

z

Q ;u,v (y, x) =

(z, x) δ y,z + G˜ (y, z) P;u,v z

+

v ,z

(z, x) ψ (v , v). δ y,v + G˜ (y, v ) G˜ (v , z) P;u (4.14)

The following are the diagrammatic bounds on the expansion coefficients (see Fig. 6): Proposition 4.1 (Diagrammatic Bounds). For the ferromagnetic Ising model, we have ⎧ (0) P;o (o, x) ≡ ϕo ϕx 3 ( j = 0), ⎪ ⎪ ⎪ ⎨

j−1 ( j) (0) π (x) ≤ P;v (o, b1 ) τbi Q ;vi ,vi+1 (bi , bi+1 ) τb j Q ;v j (b j , x) ( j ≥ 1), ⎪ 1 ⎪ ⎪ i=1 ⎩b1 ,...,b j v1 ,...,v j

(4.15) where, as well as in the rest of the paper, the empty product is regarded as 1 by convention.

(0) 4.2. Bound on π (x). The key ingredient of the proof of Proposition 4.1 is Lemma 4.2 below, which is an extension of the GHS idea used in the proof of Lemma 2.3. In this (0) subsection, we demonstrate how this extension works to prove the bound on π (x) and the inequality

w (n) (0) 1{o⇐⇒x} ∩ {o←→ y} ≤ P;y (o, x), n n Z

∂n=ox

( j) (x) for j ≥ 1. which will be used in Sect. 4.3 to obtain the bounds on π

(4.16)

306

A. Sakai

Proof of (4.15) for j = 0. Since the inequality is trivial if x = o, we restrict our attention to the case of x = o. First we note that, for each current configuration n with ∂n = {o, x} and 1{o⇐⇒ x} = 1, n there are at least three edge-disjoint paths on Gn between o and x. See, for example, the first term on the right-hand side in Fig. 2. Suppose that the thick line in that picture, ˙ 12 ∪ζ ˙ 13 from o to x, consists of bonds b with n b = 1, referred to as ζ1 and split into ζ11 ∪ζ and that the thin lines, referred to as ζ2 and ζ3 that terminate at o and x respectively, consist of bonds b with n b = 2. Let ζi , for i = 2, 3, be the duplication of ζi . Then, the ˙ ζ13 , ζ2 ∪ ˙ ζ12 ∪ ˙ ζ3 and ζ11 ∪ ˙ ζ3 are edge-disjoint. three paths ζ2 ∪ (0) Then, by multiplying π (x) by two dummies (Z /Z )2 (≡ 1), we obtain (0) π (x) =

=

∂n={o,x} ∂m =∂m =∅

w (n) w (m ) w (m ) 1{o⇐⇒x} n Z Z Z

w (N) 3 Z ∂N={o,x}

1{o⇐⇒x}

∂n={o,x} ∂m =∂m =∅ N≡n+m +m

n

b

Nb ! , n b ! m b ! m b !

(4.17)

where the sum over n, m , m in the second line equals the cardinality of the following set of partitions: ˙ S0 = (S0 , S1 , S2 ) : GN = Si , ∂S0 = {o, x}, ∂S1 = ∂S2 = ∅, o ⇐⇒ x in S0 , i=0,1,2

(4.18) where “o ⇐⇒ x in S0 ” means that there are at least two bond-disjoint paths in S0 . We will show |S0 | ≤ |S0 |, where ˙ Si , ∂S0 = ∂S1 = ∂S2 = {o, x} . S0 = (S0 , S1 , S2 ) : GN =

(4.19)

i=0,1,2

This implies (4.15) for j = 0, because

|S0 | =

∂n=∂m =∂m ={o,x} b N≡n+m +m

Nb ! , n b ! m b ! m b !

(4.20)

and

w (N) 3 Z ∂N={o,x}

∂n=∂m =∂m ={o,x} b N≡n+m +m

Nb ! = n b ! m b ! m b !

∂n={o,x}

w (n) Z

3 .

(4.21)

It remains to show |S0 | ≤ |S0 |. To do so, we use the following lemma, in which we denote by N z→z the set of paths on GN from z to z and write ω ∩ ω = ∅ to mean that ω and ω are edge-disjoint (not necessarily bond-disjoint).


307

Lemma 4.2. Given a current configuration N ∈ ZB + , k ≥ 1, V ⊂ and z i = z i ∈ for i = 1, . . . , k, we let ⎧ ⎫ k ⎪ GN = ˙ i=0 Si , ∂S0 = V, ∂Si = ∅ (i = 1, . . . , k), ⎪ ⎨ ⎬ ˙ , S = (S0 , S1 , . . . , Sk ) : ∃ ωi ∈ N (i = 1, . . . , k) such that ω ⊂ S ∪ S i 0 i z i →z i ⎪ ⎪ ⎩ ⎭ and ωi ∩ ω j = ∅ (i = j) (4.22) and define S to be the right-hand side of (4.22) with “∂S0 = V, ∂Si = ∅” being replaced by “∂S0 = V {z 1 , z 1 } · · · {z k , z k }, ∂Si = {z i , z i }”. Then, |S| = |S |.

We will prove this lemma at the end of this subsection. Now we use Lemma 4.2 with k = 2 and V = {z 1 , z 1 } = {z 2 , z 2 } = {o, x}. Note that S0 in (4.18) is a subset of S, since S includes partitions (S0 , S1 , S2 ) in which there does not exist two bond-disjoint paths on S0 . In addition, S is trivially a subset of S0 in (4.19). Therefore, we have |S0 | ≤ |S0 |. This completes the proof of (4.15) for j = 0. (0) (x) and which Here, we summarize the basic steps that we have followed to bound π ( j) we generalize to prove (4.16) below and the bounds on π (x) for j ≥ 1 in Sect. 4.3.2.

(i) Count the (minimum) number, say, k + 1, of edge-disjoint paths on Gn that satisfy the source constraint (as well as other additional conditions, if there are) of the (0) considered function f (x). For example, k = 2 for π (x) ≡ Z1 ∂n={o,x} w (n) 1{o⇐⇒ x}. n k (ii) Multiply f (x) by ( ZZ )k = i=1 ( Z1 ∂mi =∅ w (mi )) (≡ 1) and then overlap the k dummies m1 , . . . , mk on the original current configuration n. Choose k paths ω1 , . . . , ωk among k + 1 edge-disjoint paths on Gn+k mi . i=1 (iii) Use Lemma 4.2 to exchange the occupation status of edges on ωi between Gn and Gmi for every i = 1, . . . , k. The current configurations after the mapping, ˜ m ˜ 1, . . . , m ˜ k , satisfy ∂ n˜ = ∂n ∂ω1 · · · ∂ωk and ∂ m ˜ i = ∂ωi denoted by n, for i = 1, . . . , k. (0) (x). Proof of (4.16). If y = o or x, then (4.16) is reduced to the inequality for π Also, if y = o = x, then the left-hand side of (4.16) multiplied by Z /Z = w (m)/Z ≡ 1 equals ∂m=∅ w (n) w (m) w (n) w (m) 1{o←→ y} ≤ 1{o←→ y} n n+m Z Z Z Z

∂n=∂m=∅

∂n=∂m=∅

=

w (n) w (m) = ϕo ϕ y 2 , Z Z

(4.23)

∂n=∂m={o,y}

where the first equality is due to Lemma 2.3. Therefore, we can assume o = x = y = o. We follow the three steps described above. x} ∩ {o←→ y} = 1, it is not hard to see that there (i) Since y ∈ / ∂n = {o, x} and 1{o⇐⇒ n n

is an edge-disjoint cycle (closed path) o → y → x → o. Since a cycle does not have a source, there must be another edge-disjoint connection from o to x, due to the source constraint ∂n = {o, x}. Therefore, there are at least 4 (= k + 1) edge-disjoint paths on Gn : one is between o and y, another is between y and x, and the other two are between o and x.

308

A. Sakai

(ii) Multiplying both sides of (4.16) by (Z /Z )3 is equivalent to

w (N) 4 Z ∂N={o,x}

≤

1{o⇐⇒x} ∩ {o←→ y} n

∂n={o,x} ∂mi =∅ ∀ i=1,2,3 3 mi N=n+ i=1

w (N) 4 Z ∂N={o,x}

n

∂n=∂m3 ={o,x} b ∂m1 ={o,y}, ∂m2 ={y,x} 3 N=n+ i=1 mi

Nb ! (3) n b ! m b ! m (2) b ! mb ! (1)

b

Nb ! , (3) n b ! m b ! m (2) b ! mb !

(4.24)

(1)

where we have used the notation m (i) b = mi |b . Note that the second sum on the left-hand side equals the cardinality of 3 Si , ∂S0 = {o, x}, ∂S1 = ∂S2 = ∂S3 = ∅ GN = ˙ i=0 (S0 , S1 , S2 , S3 ) : , o ⇐⇒ x in S0 , o ←→ y in S0 (4.25) and the second sum on the right-hand side of (4.24) equals the cardinality of 3 (S0 , S1 , S2 , S3 ) : GN = ˙ i=0 Si , ∂S0 = ∂S3 = {o, x}, ! ∂S1 = {o, y}, ∂S2 = {y, x} .

(4.26)

Therefore, to prove (4.24), it is sufficient to show that the cardinality of (4.25) is not bigger than that of (4.26). (iii) Now we use Lemma 4.2 with k = 3 and V = {z 3 , z 3 } = {o, x}, {z 1 , z 1 } = {o, y} and {z 2 , z 2 } = {y, x}. Since (4.25) is a subset of S in the current setting, while S is a subset of (4.26), we obtain (4.24). This completes the proof of (4.16). () Proof of Lemma 4.2. We prove Lemma 4.2 by decomposing S() into ˙ ω) k Sω) k (described in detail below) and then constructing a bijection from Sω) k to Sω) k for every ω )k . To do so, we first introduce some notation. 1. For every i = 1, . . . , k, we introduce an arbitrarily fixed order among elements in N . For ω, ω ∈ N , we write ω ≺ ω if ω is earlier than ω in this order. z →z z →z i

i

i

i

˜ N be the set of paths ζ ∈ N such that there are k − 1 edge-disjoint Let z 1 →z 1 z 1 →z 1 paths on GN \ ζ (= the resulting graph by removing the edges in ζ ) each of which connects z i and z i for every i = 2, . . . , k. ˜ N , we define N;ω1 to be the set of paths ζ ∈ N on 2. Then, for ω1 ∈ z 1 →z 1 z 2 →z 2 z 2 →z 2 ˜ N earlier than ω1 . Then, we define GN \ ω1 such that ζ ⊃ ξ for any ξ ∈ z 1 →z 1

˜ N;ω1 to be the set of paths ζ ∈ N;ω1 such that there are k − 2 edge-disjoint z 2 →z 2 z 2 →z 2 ˙ ζ ) each of which is from z i to z i for i = 3, . . . , k. paths on GN \ (ω1 ∪ ˜ N , ω2 ∈ 3. More generally, for l < k and ω ) l = (ω1 , . . . , ωl ) with ω1 ∈ z →z 1

1

ω )l ˜ N;ω1 , . . . , ωl ∈ ˜ N;ω) l−1 , we define N; to be the set of paths ζ ∈ N z →z z →z z →z z 2

2

l

l

l+1

l+1

l+1 →zl+1


309

˜ N;ω) i−1 on GN \ ˙ li=1 ωi such that ζ ⊃ ξ for any ξ ∈ earlier than ωi , for every z →z i

˜ N;ω) l i = 1, . . . , l. Then, we define z →z

i

ω )l to be the set of paths ζ ∈ N; such zl+1 →zl+1 l+1 l+1 l ˙ ζ ) each of which that there are k − (l + 1) edge-disjoint paths on GN \ ( ˙ i=1 ωi ∪ is from z i to z i for i = l + 2, . . . , k. N;ω ) k−1 ˜ N;ω) k−1 4. If l = k − 1, then we simply define = z →z . We will also abuse the z →z

˜ N by ˜ N;ω) 0 . notation to denote z →z z →z 1

1

1

k

k

k

k

1

Using the above notation, we can decompose S() disjointly as follows. For a col() )k ≡ ˜ N;ω) i−1 for i = 1, . . . , k, we denote by Sω) k the set of partitions S lection ωi ∈ z →z i

i

˜ N;ω) i−1 (S0 , S1 , . . . , Sk ) ∈ S() such that, for every i = 1, . . . , k, the earliest element of z →z

˙ Si is ωi . Then, S() is decomposed as contained in S0 ∪ ˙ ˙ ˙ S() = ··· Ñ ω1 ∈

z 1 →z 1

˜ N;ω) k−1 ωk ∈

˜ N;ω1 ω2 ∈

z 2 →z 2

i

Sω() )k .

i

(4.27)

z k →z k

To complete the proof of Lemma 4.2, it suffices to construct a bijection from Sω) k to ) k ∈ Sω) , we define Sω) k for every ω ) k . For S k (0) ) k ) ≡ F (S0 ), . . . , F (k) (Sk ) = S0 ˙ k ωi , S1 ω1 , . . . ,Sk ωk , (4.28) F)ω) k (S i=1 ω )k ω )k where ∂ Fω)(0)k (S0 ) = V {z 1 , z 1 } · · · {z k , z k } and ∂ Fω)(i)k (Si ) = {z i , z i } for i = ) k )) = 1, . . . , k. Note that, by definition using symmetric difference, we have F)ω) k ( F)ω) k (S ) ˙ Sk . Also, by simple combinatorics using ωi ∩ ω j = Si ∩ S j = ∅ and ω j ⊂ S0 ∪ S j for 1 ≤ j ≤ k and i = j, we have ˙ F ( j) (S j ) = S0 ˙ i= j ωi ∪ ˙ S j . (4.29) Fω)(i)k (Si ) ∩ Fω)( j)k (S j ) = ∅, Fω)(0)k (S0 ) ∪ ω )k ˙ S j and ω j ∩ ˙ i= j ωi = ∅, we have ω j ⊂ F (0) (S0 ) ∪˙ F ( j) (S j ). Since ω j ⊂ S0 ∪ ω )k ω )k

˜ N;ω) j−1 contained in It remains to show that ω j is the earliest element of z →z j

j

˜ N;ω) j−1 ˙ F (S j ). To see this, we first recall that is a set of paths on Fω) k (S0 ) ∪ ω )k z j →z j GN \ ˙ i< j ωi , so that its earliest element contained in (S0 ˙ i< j ωi ) ∪˙ S j is still ˜ N;ω) i−1 ω j . Furthermore, since each for i > j is a set of paths that do not fully contain ( j)

(0)

z i →z i N;ω ) j−1 j →z j

˜ ω j or any earlier element of z

S0

˙ i< j

as a subset, ω j is still the earliest element of

˙ F ( j) (S j ). ˙ S j ˙ ωi = S0 ˙ ωi ∪ ˙ S j ≡ F (0) (S0 ) ∪ ωi ∪ ω )k ω )k i> j

i= j

(4.30) Therefore, F)ω) k is a bijection from Sω) k to Sω) k . This completes the proof of Lemma 4.2.

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( j) 4.3. Bounds on π (x) for j ≥ 1. First we prove (4.15) for j ≥ 1 assuming the following two lemmas, in which we recall (2.30) and use

A

E N (z, x; A) = {z ←→ x} ∩ {z ⇐⇒ x}, N

N

E N (z, x, v; A) = E N (z, x; A) ∩ {z ←→ v},

(4.31)

N

z,x;A = z,x,v;A =

wAc (m) w (n) 1 E m+n (z,x;A), Z Ac Z

∂m=∅ ∂n=z x

wAc (m) w (n) (z,x,v;A). 1 E m+n Z Ac Z

(4.32)

∂m=∅ ∂n=z x

Lemma 4.3. For the ferromagnetic Ising model, we have y,x;A ≤

δ y,z + G˜ (y, z) z,x;A ,

(4.33)

z

δ y,z + G˜ (y, z) z,x,v;A y,x;A 1{y←→v} ≤ z

+

δ y,v + G˜ (y, v ) G˜ (v , z) z,x;A ψ (v , v). (4.34) v ,z

Lemma 4.4. For the ferromagnetic Ising model, we have y,x;A ≤

u∈A

P;u (y, x),

y,x,v;A ≤

u∈A

P;u,v (y, x).

(4.35)

We prove Lemma 4.3 in Sect. 4.3.1, and Lemma 4.4 in Sect. 4.3.2. Proof of (4.15) for j ≥ 1 assuming Lemmas 4.3–4.4. Recall (2.39). By (4.33), (4.35) and (4.13), we obtain ( j−1)

b j−1 ,b j ;C˜ j−2

τb j ( j)

b j ,x;C˜ j−1

"

≤ ( j−1)

b j−1 ,b j ;C˜ j−2

τb j δb j ,z + G˜ (b j , z)

z

× ≤

vj

( j−1)

b j−1 ,b j ;C˜ j−2

v j ∈C˜ j−1

P;v (z, x) j

#

1{b j−1 ←→v j }

×τb j Q ;v j (b j , x).

(4.36)


311

For j = 1, we use (4.16) and (4.36) to obtain (1) π (x) ≡

(0) o,b

b1

=

≤

b1 ,v1

τb1 (1)

b1 ,x;C˜0

∂n=ob1

b1 ,v1

1 ;

≤

(0) o,b

b1 ,v1

1 ;

1{o←→v1 } τb1 Q ;v1 (b1 , x)

w (n) 1{o⇐⇒b1 } ∩ {o←→v1 } τb1 Q ;v1 (b1 , x) n n Z

(0) P;v (o, b1 ) τb1 Q ;v1 (b1 , x). 1

(4.37)

For j ≥ 2, we use (4.34)–(4.35) and then (4.13)–(4.14) to obtain ( j−1)

b j−1 ,b j ;C˜ j−2

≤

vj

+

v j−1 ,v j

b j ,x;C˜ j−1

τb j Q ;v j (b j , x)

v ,z

≤

τb j ( j)

≤

vj

( j−1)

b j−1 ,b j ;C˜ j−2

1{b j−1 ←→v j } τb j Q ;v j (b j , x)

δb j−1 ,z + G˜ (b j−1 , z)

z

δb j−1 ,v

+ G˜ (b j−1 , v ) G˜ (v , z)

v j−1 ∈C˜ j−2

v j−1 ∈C˜ j−2

P;v (z, b j ) ψ (v , v j ) j−1

1{v j−1 ∈ C˜ j−2 } Q ;v j−1 ,v j (b j−1 , b j ) τb j Q ;v j (b j , x).

We repeatedly use (4.34)–(4.35) to bound (i)

bi ,bi+1 ;C˜i−1

P;v (z, b j ) j−1 ,v j

(4.38)

[1{bi ←→vi+1 }] for i = j −2, . . . , 1

as in (4.38), and then at the end we apply (4.16) as in (4.37) to obtain (4.15). This completes the proof. 4.3.1. Proof of Lemma 4.3. Proof of (4.33). Recall (2.30) and (4.32). Then, to prove (4.33), it suffices to bound the ˜ contribution from 1 E m+n (y,x;A)\E m+n (y,x;A) by z G (y, z) z,x;A . First we recall (2.28) and (4.31). Then, we have $ % E m+n (y, x; A) \ E m+n (y, x; A) = E m+n (y, x; A) ∩ {y ←→ x} \ {y ⇐⇒ x} . m+n m+n (4.39) x} \ {y ⇐⇒ x}, there is at least one pivotal bond for y ←→ x from y. Let On {y ←→ m+n m+n m+n b be the last pivotal bond among them. Then, we have b ⇐⇒ x off b, m b + n b > 0, and m+n b (x)c . Moreover, on the event E y ←→ b in Cm+n b m+n (y, x; A), we have that y ←→ m+n m+n

A

A

in Ac and b ←→ x. Since {b ⇐⇒ x off b} ∩ {b ←→ x} = {E m+n (b, x; A) off b} on m+n m+n m+n

312

A. Sakai

the event that b is pivotal for y ←→ x from y, we have m+n (y, x; A) E m+n (y, x; A) \ E m+n %! $ ˙ c b c {E m+n . = (b, x; A) off b} ∩ {m b + n b > 0} ∩ y ←→ b in A ∩ C (x) m+n m+n b

(4.40) Therefore, we obtain y,x;A − y,x;A wAc (m) w (n) b (x)c }. = 1{E m+n (b,x;A) off b} 1{m b +n b >0} 1{y←→b in Ac ∩Cm+n m+n Z Ac Z ∂m=∅ ∂n=y x

b

(4.41) It remains to bound the right-hand side of (4.41), which is nonzero only if m b is even and n b is odd, due to the source constraints and the conditions in the indicators. First, as in (2.31), we alternate the parity of n b by changing the source constraint into b (x) as in (2.33) ∂n = y b x and multiplying by τb . Then, by conditioning on Cm+n b (i.e., conditioning on Cm+n (x) = B, letting m = m|BAc \BAc ∩Bc , m = m|BAc ∩Bc , n = n|B \BBc and n = n|BBc , and then summing over B ⊂ ), we obtain

B⊂ ∂m =∅ ∂n =bx

w˜ Ac ,B (m ) Z Ac ∩Bc w˜ ,B (n ) Z Bc 1{E (b,x;A) off b} ∩ {C b (x)=B} m +n m +n Z Ac Z ×τb 1{m b ,n b even}

wAc ∩Bc (m ) wBc (n ) 1{y ←→ b Z A c ∩B c Z Bc m +n ∂m =∅ ∂n =y b ∵(2.23)

=

in Ac ∩Bc }

ϕ y ϕb Ac ∩ Bc

wAc (m) w (n) = 1{E m+n b (x)c . (b,x;A) off b} τb 1{m b ,n b even} ϕ y ϕb Ac ∩ Cm+n Z Ac Z ∂m=∅ ∂n=bx

(4.42) Since ϕ y ϕb Ac ∩ C b b (x)} Cm+n

m+n (x)

c

= 0 on

E m+n (b, x; A)

\

{E m+n (b, x; A)

off b} ⊂ {b ∈

and on the event that m b or n b is odd (see below (2.15) or above (2.34)), we can omit “off b” and 1{m b ,n b even} in (4.42). Since ϕ y ϕb Ac ∩ C b (x)c ≤ ϕ y ϕb due m+n to Proposition 2.2, we have wAc (m) w (n) 1 E m+n (4.42) ≤ ϕ y ϕb τb (b,x;A) = ϕ y ϕb τb b,x;A . Z Ac Z ∂m=∅ ∂n=bx

Therefore, (4.41) is bounded by completes the proof of (4.33).

b

ϕ y ϕb τb

b,x;A

≡

(4.43)

z

G˜ (y, z) z,x;A . This


313

Proof of (4.34). Recall (2.30) and (4.32). To prove (4.34), we investigate $ % L ≡ E m+n (y, x; A) ∩ {y ←→ v} \ E m+n (y, x, v; A) m+n (y, x; A)} ∩ {y ←→ v}, = {E m+n (y, x; A) \ E m+n m+n

(4.44)

where y,x;A [1 L ] = y,x;A [1{y←→v}] − y,x,v;A .

x from y, and First we recall (4.40), in which b is the last pivotal bond for y ←→ m+n

define

% $ b (b, x, v; A) off b} ∩ {m b + n b > 0} ∩ y ←→ b in Ac ∩ Cm+n (x)c , R1 (b) = {E m+n m+n (4.45)

R2 (b) =

{E m+n (b, x; A)

off b} ∩ {m b + n b > 0} $ % b ∩ y ←→ b in Ac ∩ Cm+n (x)c , y ←→ v , m+n m+n

b (x) on R (b), while v ∈ C b (y) on R (b). Since where v ∈ Cm+n 1 2 m+n ˙ ˙ R2 (b)}, {R1 (b) ∪ L=

(4.46)

(4.47)

b

we have y,x;A [1{y←→v}] − y,x,v;A =

y,x;A 1 R1 (b) + y,x;A 1 R2 (b) .

(4.48)

b

Following the same argument as in (4.42)–(4.43), we easily obtain y,x;A 1 R1 (b) wAc (m) w (n) (b,x,v;A) off b} τb 1{m ,n even} ϕ y ϕb c = 1{E m+n b (x)c b b A ∩ Cm+n Z Ac Z ∂m=∅ ∂n=bx

≤ ϕ y ϕb τb

wAc (m) w (n) (b,x,v;A) = ϕ y ϕb τb 1 E m+n . b,x,v;A Z Ac Z

(4.49)

∂m=∅ ∂n=bx

Similarly, we have y,x;A 1 R2 (b) wAc (m) w (n) b (x)=B } τb 1{m ,n even} = 1{E m+n (b,x;A) off b} ∩ {Cm+n b b Z Ac Z B⊂ ∂m=∅ ∂n=bx

×

wAc ∩Bc (h) wBc (k) 1{y←→b in Ac ∩Bc , y←→v (in Bc )} Z Ac ∩Bc Z Bc h+k h+k ∂h=∅

∂k=y b

wAc (m) w (n) ≤ 1{E m+n b (x) , (4.50) (b,x;A) off b} τb 1{m b ,n b even} y,b,v;A,Cm+n Z Ac Z ∂m=∅ ∂n=bx

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A. Sakai

where y,z,v;A,B =

wAc ∩ Bc (h) wBc (k) 1{y←→v}. Z Ac ∩ B c Z Bc h+k ∂h=∅

(4.51)

∂k=y z

We note that, by ignoring the indicator in (4.51), we have 0 ≤ y,z,v;A,B ≤ ϕ y ϕz Bc , which is zero whenever z ∈ B. Therefore, we can omit “off b” and 1{m b ,n b even} in (4.50) to obtain wAc (m) w (n) y,x;A 1 R2 (b) ≤ 1 E m+n b (x) . (4.52) (b,x;A) τb y,b,v;A,Cm+n Z Ac Z ∂m=∅ ∂n=bx

Substituting (4.49) and (4.52) to (4.48), we arrive at y,x;A 1{y←→v} ≤ δ y,z + G˜ (y, z) z,x,v;A z

+

wAc (m) w (n) 1 E m+n b (x) . (b,x;A) τb y,b,v;A,Cm+n Z Ac Z b

∂m=∅ ∂n=bx

(4.53) The proof of (4.34) is completed by using y,z,v;A,B ≤ ϕ y ϕv ϕv ϕz ψ (v , v),

(4.54)

v

and replacing ϕ y ϕv in (4.54) by δ y,v + G˜ (y, v ), due to (4.2). To complete the proof of (4.34), it thus remains to show (4.54). First we note that, if A ⊂ B, then by Lemma 2.3 we have y,z,v;A,B =

wBc (h) wBc (k) 1{y←→v} Z Bc Z Bc h+k ∂h=∅

∂k=y z

= ϕ y ϕv Bc ϕv ϕz Bc ≤ ϕ y ϕv ϕv ϕz .

(4.55)

However, to prove (4.54) for a general A that does not necessarily satisfy A ⊂ B, we use $ % ˙ {y ←→ v} \ {y ←→ v} , {y ←→ v} = {y ←→ v} ∪ (4.56) h+k

k

h+k

k

and consider the two events on the right-hand side separately. The contribution to y,z,v;A,B from {y ←→ v} is easily bounded, similarly to (4.23), as k

wBc (k) 1{y←→v} ≤ Z Bc k

∂k=y z

wBc (k) wBc (k ) 1{y←→v} Z Bc Z Bc k+k ∂k=y z ∂k =∅

= ϕ y ϕv Bc ϕv ϕz Bc ≤ ϕ y ϕv ϕv ϕz .

(4.57)


315

Next we consider the contribution to y,z,v;A,B from {y ←→ v} \ {y ←→ v} in h+k

k

(4.56). We denote by Ck (y) the set of sites k-connected from y. Since v ∈ Ch+k (y)\Ck (y), there is a nonzero alternating chain of mutually-disjoint h-connected clusters and mutually-disjoint k-connected clusters, from some u 0 ∈ Ck (y) to v. Therefore, we have

∞ 1{y←→v}\{y←→v} ≤ 1{y←→u 0 } 1{u 2l ←→u 2l+1 } 1{u 2l−1 ←→u 2l } h+k

k

j=1

×

k

u 0 ,...,u j u l =u l ∀ l=l u j =v

l,l ≥0 l=l

h

l≥0

k

l≥1

1{Ch (u 2l ) ∩ Ch (u 2l )=∅} 1{Ck (u 2l ) ∩ Ck (u 2l )=∅} ,

(4.58)

where we regard an empty product as 1. Using this bound, we can perform the sums over h and k in (4.51) independently. For j = 1 and given u 0 = u 1 = v, the summand of (4.58) equals 1{y←→u 0 }1{u 0 ←→v}, k

h

which is simply equal to 1{y←→v} if u 0 = y. Then, by (4.57) and (4.2), the contribution h

from this to y,z,v;A,B is wBc (k) wAc ∩ Bc (h) 1{y←→u 0 } 1{u 0 ←→v} Z Bc Z Ac ∩ B c k h

∂k=y z

∂h=∅

≤ ϕ y ϕu 0 ϕu 0 ϕz G˜ (u 0 , v)2 .

(4.59)

Fix j ≥ 2 and a sequence of distinct sites u 0 , . . . , u j (= v), and first consider the contribution to the sum over k in (4.51) from the relevant indicators in the right-hand side of (4.58), which is

wBc (k) 1{y←→u 0 } 1{u 2l−1 ←→u 2l } 1{Ck (u 2l ) ∩ Ck (u 2l )=∅} Z Bc k k ∂k=y z

l,l ≥0 l=l

l≥1

wBc (k)

= 1{u 2l−1 ←→u 2l } 1{Ck (u 2l ) ∩ Ck (u 2l )=∅} Z Bc k ∂k=y z

l,l ≥1 l=l

l≥1

× 1{y←→u 0 } ∩ {Ck (u 0 ) ∩ Uk;1 =∅},

(4.60)

k

where Uk;1 = ˙ l≥1 Ck (u 2l ). Conditioning on Uk;1 , we obtain that

wBc (k)

(4.60) = 1{u 2l−1 ←→u 2l } 1{Ck (u 2l ) ∩ Ck (u 2l )=∅} Z Bc k ∂k=∅

c (k ) wBc ∩ Uk;1

∂k =y z

c Z Bc ∩ Uk;1

×

l,l ≥1 l=l

l≥1

∵(4.57)

≤

1{y←→u 0 } .

ϕ y ϕu 0 ϕu 0 ϕz

k

(4.61)

316

A. Sakai

Then, by conditioning on Uk;2 ≡ ˙ l≥2 Ck (u 2l ), following the same computation as above and using (4.2), we further obtain that wBc (k)

(4.60) ≤ ϕ y ϕu 0 ϕu 0 ϕz 1{u 2l−1 ←→u 2l } Z Bc k l≥2 ∂k=∅ w c c (k )

B ∩ Uk;2 × 1{Ck (u 2l ) ∩ Ck (u 2l )=∅} 1{u 1 ←→u 2 } . (4.62) c Z Bc ∩ Uk;2 k ∂k =∅ l,l ≥2 l=l ≤ G˜ (u 1 ,u 2 )2

We repeat this computation until all indicators for k are used up. We also apply the same argument to the sum over h in (4.51). Summarizing these bounds with (4.57) and (4.59), and replacing u 0 in (4.58)–(4.61) by v , we obtain (4.54). This completes the proof of (4.34). x} in y,x;A and 4.3.2. Proof of Lemma 4.4. We note that the common factor 1{y⇐⇒ m+n

y,x,v;A can be decomposed as 1{y⇐⇒x} = 1{y⇐⇒x} + 1{y⇐⇒x}\{y⇐⇒x}. n

m+n

(4.63)

n

m+n

x} to y,x;A and y,x,v;A in the followWe estimate the contributions from 1{y⇐⇒ n

ing paragraphs (a) and (b), respectively. Then, in the paragraphs (c) and (d) below, we x}\{y⇐⇒x} in (4.63) to y,x;A and y,x,v;A , will estimate the contributions from 1{y⇐⇒ n m+n respectively. (a) First we investigate the contribution to y,x;A from 1{y⇐⇒ x}: n ∂m=∅ ∂n=y x

wAc (m) w (n) A 1{y ←→ x} ∩ {y⇐⇒x}. n m+n Z Ac Z

(4.64)

For a set of events E 1 , . . . , E N , we define E 1 ◦ · · · ◦ E N to be the event that E 1 , . . . , E N occur bond-disjointly. Then, we have A A 1{y ←→ 1{y←→u} ◦ {u←→x} ◦ {y←→x}, (4.65) x} ∩ {y⇐⇒x} ≤ 1{y ←→x} ∩ {y⇐⇒x} ≤ n

m+n

n

n

u∈A

n

n

n

where the right-hand side does not depend on m. Therefore, the contribution to y,x;A is bounded by (4.64) ≤

u∈A ∂n=y x

(0) w (n) 1{y←→u} ◦ {u←→x} ◦ {y←→x} ≤ P;u (y, x), n n n Z

(4.66)

u∈A

where we have applied the same argument as in the proof of (4.16), which is around (4.23)–(4.26).


317

(b) Next we investigate the contribution to y,x,v;A from 1{y⇐⇒ x} in (4.63): n ∂m=∅ ∂n=y x

wAc (m) w (n) A 1{y ←→ x} ∩ {y⇐⇒x} ∩ {y←→v}. n m+n m+n Z Ac Z

(4.67)

A A Note that, by using (4.56) and 1{y ←→ x} ≤ 1{y ←→x}, we have n m+n

A A 1 ≤ 1 +1 1{y ←→ x} ∩ {y⇐⇒x} ∩ {y←→v} {y ←→x} ∩ {y⇐⇒x} {y←→v} {y←→v}\{y←→v} . m+n

n

n

m+n

n

n

n

m+n

(4.68) We investigate the contributions from the two indicators in the parentheses separately. v}, which is independent of m. Since We begin with the contribution from 1{y←→ n A

A

{y ←→ x}∩{y ⇐⇒ x}∩{y ←→ v} ⊂ {y ←→ x}◦{y ←→ x, y ←→ v}, (4.69) n n n n n n A

{y ←→ x} ⊂ n

u∈A

{y ←→ u} ◦ {u ←→ x}, n n

(4.70)

the contribution to (4.67) from 1{y←→ v} in (4.68) is bounded by n

u∈A ∂n=y x

w (n) 1{y←→u} ◦ {u←→x} ◦ {y←→x, y←→v}. n n n n Z

(4.71)

We follow Steps (i)–(iii) described above (4.23) in Sect. 4.2. Without loss of generality, we can assume that y, u, x and v are all different; otherwise, the following argument can be simplified. (i) Since y and x are sources, but u and v are not, there is an edge-disjoint cycle y → u → x → v → y, with an extra edge-disjoint path from y to x. Therefore, we have in total at least 5 (= 4 + 1) edge-disjoint paths. (ii) Multiplying by (Z /Z )4 , we have (4.71) =

w (N) 5 Z u∈A ∂N=y x

1{y←→u} ◦ {u←→x} ◦ {y←→x, y←→v} n

∂n=y x ∂mi =∅ ∀ i=1,...,4 4 mi N=n+ i=1

×

n

b

nb !

Nb ! 4

n

(i)

i=1 m b

!

,

n

(4.72)

where we have used the notation m (i) b = mi |b . (iii) The sum over n, m1 , . . . , m4 in (4.72) is bounded by the cardinality of S in Lemma 4.2 with k = 4, V = {y, x},

318

A. Sakai

{z 1 , z 1 } = {y, u}, {z 2 , z 2 } = {u, x}, {z 3 , z 3 } = {y, v} and {z 4 , z 4 } = {v, x}. Bounding the cardinality of S in Lemma 4.2 for this setting, we obtain w (N)

Nb ! (4.72) ≤ 4 (i) 5 Z ∂n=y x b n b ! i=1 m b ! u∈A ∂N=y x

≤

u∈A

∂m1 =y u, ∂m2 =u x ∂m3 =y v, ∂m4 =v x 4 N=n+ i=1 mi

ϕ y ϕx ϕ y ϕu ϕu ϕx ϕ y ϕv ϕv ϕx .

(4.73)

v}\{y←→v} in (4.68). On Next we investigate the contribution to (4.67) from 1{y←→ n m+n

the event {y ⇐⇒ x} ∩ {{y ←→ v} \ {y ←→ v}}, there exists a v0 = v such that n n m+n x} ◦ {y ←→ x, y ←→ v0 } occurs and that v0 and v are connected via a non{y ←→ n n n zero alternating chain of mutually-disjoint m-connected clusters and mutually-disjoint n-connected clusters. Therefore, by (4.58) and (4.70) (see also (4.71)), we obtain A 1{y ←→ x} ∩ {y⇐⇒x} ∩ {{y←→v}\{y←→v}} n

u∈A j≥1

v0 ,...,v j vl =vl ∀ l=l v j =v

n

m+n

≤

×

n

1{y←→u} ◦ {u←→x} ◦ {y←→x, y←→v0 } n

n

1{v2l−1 ←→v2l } n

l≥1

l,l ≥0 l=l

n

n

l≥0

1{v2l ←→v2l+1 } m

1{Cm (v2l ) ∩ Cm (v2l )=∅} 1{Cn (v2l ) ∩ Cn (v2l )=∅} .

(4.74)

For the three products of indicators, we repeat the same argument as in (4.59)–(4.62) to derive the factor ψ (v0 , v) − δv0 ,v . As a result, we have wAc (m) w (n) A 1{y ←→ x} ∩ {y⇐⇒x} ∩ {{y←→v}\{y←→v}} n n n m+n Z Ac Z ∂m=∅ ∂n=y x

≤

ψ (v0 , v) − δv0 ,v v0

u∈A ∂n=y x

w (n) 1{y←→u} ◦ {u←→x} ◦ {y←→x, y←→v0 }. n n n n Z (4.75)

Following the same argument as in (4.71)–(4.73), we obtain ψ (v0 , v) − δv0 ,v ϕ y ϕx ϕ y ϕu ϕu ϕx ϕ y ϕv0 ϕv0 ϕx (4.75) ≤ u∈A, v0

≤

u∈A

(0) P;u,v (y, x) − ϕ y ϕx ϕ y ϕu ϕu ϕx ϕ y ϕv ϕv ϕx . (4.76)

Summarizing (4.68), (4.73) and (4.76), we arrive at (0) (4.67) ≤ P;u,v (y, x). u∈A

(4.77)


319

This completes the bound on the contribution to y,x,v;A from 1{y⇐⇒ x} in (4.63). n

x}\{y⇐⇒x} in (4.63) equals (c) The contribution to y,x;A from 1{y⇐⇒ n m+n

∂m=∅ ∂n=y x

wAc (m) w (n) A 1{y ←→ x} ∩ {{y⇐⇒x}\{y⇐⇒x}}. n m+n m+n Z Ac Z

(4.78)

Note that, if 1{∂n=y x}\{y⇐⇒ x} = 1, then y is n-connected, but not n-doubly connected, n to x, and therefore there exists at least one pivotal bond for y ←→ x. Given an ordered n set of bonds b)T = (b1 , . . . , bT ), we define b1 } ∩ Hn;b)T (y, x) = {y ⇐⇒ n

T & i=1

{bi ⇐⇒ bi+1 } n %! $ x , (4.79) ∩ n bi > 0, bi is pivotal for y ←→ n

A A where, by convention, b T +1 = x. Then, by 1{y ←→ x} ≤ 1{y ←→x}, we obtain m+n

(4.78) =

T ≥1 b)T

≤

∂m=∅ ∂n=y x

T ≥1 b)T

∂m=∅ ∂n=y x

n

wAc (m) w (n) A 1{y ←→ x} ∩ Hn;b) (y,x) ∩ {y⇐⇒x} m+n m+n T Z Ac Z wAc (m) w (n) A 1{y ←→ x} ∩ Hn;b) (y,x) ∩ {y⇐⇒x}. n m+n T Z Ac Z

(4.80)

On the event Hn;b)T (y, x), we denote the n-double connections between the pivotal bonds b1 , . . . , bT by ⎧ b 1 ⎪ (i = 0), ⎨Cn (y) bi+1 Dn;i = Cn (y) \ Cnbi (y) (i = 1, . . . , T − 1), (4.81) ⎪ ⎩ bT Cn (y) \ Cn (y) (i = T ). As in Fig. 7, we can think of Cn (y) as the interval [0, T ], where each integer i ∈ [0, T ] corresponds to Dn;i and the unit interval (i − 1, i) ⊂ [0, T ] corresponds to the pivotal bond bi . Since y ⇐⇒ x, we see that, for every bi , there must be an (m + n)-bypath m+n (i.e., an (m + n)-connection that does not go through bi ) from some z ∈ Dn;s with s < i to some z ∈ Dn;t with t ≥ i. We abbreviate {s, t} to st if there is no confusion. Let (2) L(1) [0,T ] = {{0T }}, L[0,T ] = {{0t1 , s2 T } : 0 < s2 ≤ t1 < T } and generally for j ≤ T (see Fig. 7), $ % j j) L([0,T ] = {si ti }i=1 : 0 = s1 < s2 ≤ t1 < s3 ≤ · · · ≤ t j−2 < s j ≤t j−1 < t j = T . (4.82) j) For every j ∈ {1, . . . , T }, we have st∈ [s, t] = [0, T ] for any ∈ L([0,T ] , which T implies double connection. Conditioning on Cn (y) ≡ i=0 Dn;i = B (and denoting

320

A. Sakai

0

8

(4)

Fig. 7. An element in L[0,8] , which consists of s1 t1 = {0, 3}, s2 t2 = {2, 4}, s3 t3 = {4, 6} and s4 t4 = {5, 8}

k = n|BBc , h = n|B \BBc and Dn;i ≡ Dh;i = Bi ) and multiplying by Z Bc /Z Bc , we obtain wAc (m) w˜ ,B (h) Z Bc wBc (k) (4.80) = Z Ac Z Z Bc B⊂ T ≥1 b)T ∂m=∂k=∅ ∂h=y x

A × 1{y ←→ x} ∩ Hh;b)

T

h

×

T

(y,x) ∩ {Ch (y)=B} j

j=1 {s t } j ∈L( j) z 1 ,...,z j i i i=1 [0,T ] z ,...,z 1 j

1{zi ∈Bsi ,

i=1

×

z i ∈Bti } ∩ {z i ←→z i } m+k

1{Cm+k (zi ) ∩ Cm+k (zl )=∅}. (4.83)

i=l

Reorganizing this expression and then summing over B ⊂ A, we obtain (4.83) =

T ≥1 b)T ∂n=y x

×

T

w (n) A 1{y ←→ x} ∩ Hn;b) (y,x) n T Z j


×

∂m=∂k=∅

1{zi ∈Dn;s

i

,

z i ∈Dn;ti }

i=1

j

wAc (m) wD˜ c (k)

1{zi ←→zi } 1{Cm+k (zi ) ∩ Cm+k (zl )=∅}, Z Ac Z D˜ c m+k i=1

i=l

(4.84) ˜ In the rightmost expression, the first line determines where we have denoted Cn (y) by D. ˜ D that contains vertices z i , z i for all i = 1, . . . , j in a specific manner, while the second line determines the bypaths Cm+k (z i ) joining z i and z i for every i = 1, . . . , j. We first derive n-independent bounds on these bypaths in the following paragraph (c-1). Then, in (c-2) below, we will bound the first two lines of the rightmost expression in (4.84). (c-1) For j = 1, the last line of the rightmost expression in (4.84) simply equals ∂m=∂k=∅

wAc (m) wD˜ c (k) 1{z 1 ←→z 1 }. Z Ac Z D˜ c m+k

(4.85)

Since z 1 , z 1 ∈ D˜ and z 1 = z 1 , these two vertices are connected via a nonzero alternating chain of mutually-disjoint m-connected clusters and mutually-disjoint k-connected B ˜c clusters. Moreover, since z 1 , z ∈ D˜ and k ∈ Z+ D , this chain of bubbles starts and 1


321

ends with m-connected clusters (possibly with a single m-connected cluster), not with k-connected clusters. Therefore, by following the argument around (4.58)–(4.62), we can easily show ∗(2l−1) (4.85) ≤ G˜ 2 (z 1 , z 1 ). (4.86) l≥1

For j ≥ 2, since Cm+k (z i ) for i = 1, . . . , j are mutually-disjoint due to the last product of the indicators in (4.84), we can treat each bypath separately by the condition ing-on-clusters argument. By conditioning on Vm+k ≡ ˙ i≥2 Cm+k (z i ), the last line in the rightmost expression of (4.84) equals ∂m=∂k=∅

j

wAc (m) wD˜ c (k)

1{zi ←→zi } 1{Cm+k (zi ) ∩ Cm+k (zl )=∅} Z Ac Z D˜ c m+k i=2

×

∂m =∂k =∅

i,l≥2 i=l

(k ) c c wAc ∩ Vm+k (m ) wD˜ c ∩ Vm+k 1{z 1 ←→ z 1 }. c Z Ac ∩ Vm+k Z D˜ c ∩ V c m +k

(4.87)

m+k

c c , respecBy using (4.86) (and replacing Ac and D˜ c in (4.85) by Ac ∩Vm+k and D˜ c ∩Vm+k 2 ∗(2l−1) (z 1 , z 1 ). Repeating the tively), the second line of (4.87) is bounded by l≥1 (G˜ ) same argument until the remaining products of the indicators are used up, we obtain

(4.87) ≤

j

G˜ 2

∗(2l−1)

(z i , z i ).

(4.88)

i=1 l≥1

We have proved that (4.84) ≤

j

j≥1 z 1 ,...,z j z 1 ,...,z j

×

G˜ 2

∗(2l−1)

(z i , z i )

∂n=y x

i=1 l≥1

T ≥ j b)T {s t } j ∈L( j) i i i=1 [0,T ]

1 Hn;b)

T

j

(y,x)

1{zi ∈Dn;s

i

w (n) A 1{y ←→ x} n Z

, z i ∈Dn;ti }.

(4.89)

i=1

(c-2) Since (4.89) depends only on a single current configuration, we may use Lemma 4.2 to obtain an upper bound. To do so, we first simplify the second line of (4.89), which is, by definition, equal to the indicator of the disjoint union ˙ ˙

˙

T ≥ j b)T {s t } j ∈L( j) i i i=1 [0,T ]

=

˙ ˙ ˙

e1 ,...,e j

T ≥ j b)T

Hn;b)T (y, x) ∩

j & $

z i ∈ Dn;si , z i ∈ Dn;ti

%

i=1

˙

j ( j) {si ti }i=1 ∈L[0,T ] ∀ bti +1 =ei+1 i=0,..., j−1

Hn;b)T(y, x) ∩

j & $

z i ∈ Dn;si ,

z i

% ∈ Dn;ti ,

i=1

(4.90)

322

A. Sakai

z’

z

I1 (y, z, x) =

y

I2 (y, z ,x ) =

x

z

I3 (y, z, z ,x ) =

z’

y

y

x

z’

x

∪

z

y

x

Fig. 8. Schematic representations of I1 (y, z, x), I2 (y, z , x) and I3 (y, z, z , x).

where t0 = 0 by convention. On the left-hand side of (4.90), the first two unions identify the number and location of the pivotal bonds for y ←→ x, and the third union identifies n the indices of double connections associated with the bypaths between z i and z i , for every i = 1, . . . , j. The union over e1 , . . . , e j on the right-hand side identifies some of the pivotal bonds b1 , . . . , bT that are essential to decompose the chain of double connections Hn;b)T (y, x) into the following building blocks (see Fig. 8): I1 (y, z, x) = {y ⇐⇒ x, y ←→ z}, n n I2 (y, z , x) =

$

% {y ←→ u} ◦ I1 (u, z , x) , n

(4.91)

u

I3 (y, z, z , x) =

u

{I2 (y, z, u) ◦ I2 (u, z , x)} $ %! ∪ {y ←→ u} ◦ {I1 (u, z, x) ∩ I1 (u, z , x)} . n

(4.92)

For example, since L(1) [0,T ] = {{0T }}, we have ((4.90) for j = 1) = ⊂

˙ $

˙ ˙

˙

e1 T ≥1 b) :b =e 1 T 1

%! $ Hn;b)T (y, x) ∩ z 1 ∈ Dn;0 , z 1 ∈ Dn;T

% $ %! I1 (y, z 1 , e1 ) ◦ I2 (e1 , z 1 , x) ∩ n e1 > 0, e1 is pivotal for y ←→ x . n

e1

(4.93) It is not hard to see in general that ((4.90) for j ≥ 2) ˙ I1 (y, z 1 , e1 ) ◦ I3 (e1 , z 2 , z 1 , e2 ) ◦ · · · ⊂ e1 ,...,e j

· · · ◦ I3 (e j−1 , z j , z j−1 , e j ) ◦ I2 (e j , z j , x) ∩

!

j & $ % n ei > 0, ei is pivotal for y ←→ x . n i=1

(4.94)


323

To bound (4.89) using Lemma 4.2, we further consider an event that includes (4.93)– (4.94) as subsets. Without losing generality, we can assume that y = e1 , ei−1 = ei for i = 2, . . . , j, and e j = x; otherwise, the following argument can be simplified. We consider each event Ii in (4.93)–(4.94) individually, and to do so, we assume that y and e1 are the only sources for I1 (y, z 1 , e1 ), that ei−1 and ei are the only sources for , e ) for every i = 2, . . . , j, and that e and x are the only sources for I3 (ei−1 , z i , z i−1 j i I2 (e j , z j , x). This is because y and x are the only sources for the entire event (4.94), x. and every ei is pivotal for y ←→ n On I1 (y, z, x) with y, x being the only sources, according to the observation in Step (i) described below (4.23), we have two edge-disjoint connections from y to z, one of which may go through x, and another edge-disjoint connection from y to x (cf., I1 (y, z, x) in Fig. 8). Therefore, $ % I1 (y, z, x) ⊂ ∃ ω1 , ω2 ∈ ny→z ∃ ω3 ∈ ny→x such that ωi ∩ ωl = ∅ (i = l) . (4.95) Similarly, for I2 (y, z , x) with y, x being the only sources (cf., I2 (y, z , x) in Fig. 8), $ % I2 (y, z , x) ⊂ ∃ ω1 , ω2 ∈ nx→z ∃ ω3 ∈ ny→x such that ωi ∩ ωl = ∅ (i = l) . (4.96) On I3 (y, z, z , x) with y, x being the only sources, there are at least three edge-disjoint paths, one from y to z,another one from z to z , and another one from z to x. It is not hard to see this from u {I2 (y, z, u) ◦ I2 (u, z , x)} in (4.92), which corresponds to the first event depicted in Fig. 8. It is also possible to extract such three edge-disjoint paths from the remaining event in (4.92). See the second event depicted in Fig. 8 for one of the worst topological situations. Since there are at least three edge-disjoint paths between u and x, say, ζ1 , ζ2 and ζ3 , we can go from y to z via ζ1 and a part of ζ2 , and go from z to z via the middle part of ζ2 , and then go from z to x via the remaining part of ζ2 and ζ3 . The other cases can be dealt with similarly. As a result, we have $ % I3 (y, z, z , x) ⊂ ∃ ω1 ∈ ny→z ∃ ω2 ∈ nz→z ∃ ω3 ∈ nz →x such that ωi ∩ ωl = ∅ (i = l) . (4.97) Since $

! % {∃ ω ∈ nz→e } ◦ {∃ ω ∈ ne→z } ∩ {n e > 0} ⊂ {∃ ω ∈ nz→z },

(4.98)

e

we see that (4.93) is a subset of ' ∃ ω ∈ n ∃ ω , ω ∈ n ∃ ω , ω ∈ n 1 2 3 4 5 (1) →y z y→x 1 x→z 1 , I˜z ,z (y, x) = 1 1 such that ωi ∩ ωl = ∅ (i = l)

(4.99)

and that (4.94) is a subset of (see Fig. 9) ⎧∃ ⎫ n n n n ∃ ∃ ∃ ⎪ ⎨ ω1 , ω2 ∈ z 1 →y ω3 ∈ y→z 2 ω4 ∈ z 2 →z 1 ω5 ∈ z 1 →z 3 · · ·⎪ ⎬ I˜)z( j),)z (y, x) = · · · ∃ ω2 j ∈ nz →z ∃ ω2 j+1 ∈ nz →x ∃ ω2 j+2 , ω2 j+3 ∈ nx→z , j j j ⎪ j−1 j−1 j ⎪ ⎩ ⎭ such that ωi ∩ ωl = ∅ (i = l) (4.100)

324

A. Sakai

z1

z’1

y

z2 ˜( j)

Fig. 9. A schematic representation of I

()

()

)z j ,)z j

z3

z’j−1

x

z’2

zj

z’j

(y, x) for j ≥ 2 consisting of 2 j + 3 edge-disjoint paths on Gn .

()

where )z j = (z 1 , . . . , z j ). Therefore,

(4.89) ≤

j

j≥1 z 1 ,...,z j z 1 ,...,z j

G˜ 2

∗(2l−1)

(z i , z i )

∂n=y x

i=1 l≥1

w (n) A ( j) 1{y ←→ x} 1 I˜ (y,x). )z j ,)z j n Z (4.101)

Now we apply Lemma 4.2 to bound (4.101). To clearly understand how it is applied, A for now we ignore 1{y ←→ x} in (4.101) and only consider the contribution from 1 I˜( j) (y,x). n )z j ,)z j

Without losing generality, we assume that y, x, z i , z i for i = 1, . . . , j are all different. Since there are 2 j + 3 edge-disjoint paths on Gn as in (4.99)–(4.100) (see also Fig. 9), we multiply (4.101) by (Z /Z )2 j+2 , following Step (ii) of the strategy described in Sect. 4.2. Overlapping the 2 j + 3 current configurations and using Lemma 4.2 with V = {y, x} and k = 2 j + 2, we obtain ∂n=y x

×

w (n) 1 I˜( j) (y,x) ≤ ϕz 1 ϕ y 2 ϕx ϕz j 2 )z j ,)z j Z

⎧ ⎪ ⎨ϕ y ϕx ⎪ ⎩ϕ y ϕz 2 ϕz 2 ϕz 1

j−1

ϕzi−1 ϕzi+1 ϕzi+1 ϕzi ϕz j−1 ϕx

i=2

( j = 1), ( j ≥ 2). (4.102)

Note that, by (4.2), we have ⎫

z,z

˜ 2 ∗(2l−1) (y, x) ⎪ l≥1 (G ) ⎪ ⎬ 2 2 )∗(2l−1) (z, x) ˜ ϕ ϕ ( G z y z l≥1 ⎪ ⎪ 2 2 )∗(2l−1) (y, z ) ⎭ ˜ ϕ ϕ ( G x z z l≥1 ϕz ϕ y 2 ϕx ϕz 2

l≥1

G˜ 2

∗(2l−1)

≤ ψ (y, x) − δ y,x ,

(4.103)

(z, z ) ≤ 2 ψ (y, x) − δ y,x . (4.104)


325

A Therefore, (4.101) without 1{y ←→ x} is bounded by n

ϕ y ϕx

+

z 1 ,z 1

ϕz 1 ϕ y 2 ϕx ϕz 1 2

j−1

j≥2 z 2 ,...,z j z 1 ,...,z j−1

G˜ 2

∗(2l−1)

l≥1

∗(2l−1) ψ (z i , z i ) − δzi ,zi G˜ 2 ϕ y ϕz 1 2 (z 1 , z 1 ) z1

i=2

×

z j

×

(z 1 , z 1 )

j−1

ϕx ϕz j 2

G˜ 2

l≥1

∗(2l−1)

(z j , z j ) ϕ y ϕz 2 ϕz 2 ϕz 1

l≥1

ϕzi−1 ϕzi+1 ϕzi+1 ϕzi

i=2

ϕz j−1 ϕx ≤

P( j) (y, x).

j≥1

(4.105) A x} is present in the above argument, then at least one of the paths ωi for If 1{y ←→ n

i = 3, . . . , 2 j + 1 has to go through A. For example, if ω3 (∈ ny→z 2 ) goes through A, then we can split it into two edge-disjoint paths at some u ∈ A, such as ω3 ∈ ny→u and ω3 ∈ nu→z 2 . The contribution from this case is bounded, by following the same argument as above, by (4.102) with ϕ y ϕz 2 being replaced by u∈A ϕ y ϕu ϕu ϕz 2 . Bounding the other 2 j − 2 cases similarly and summing these bounds over j ≥ 1, we obtain ( j) (4.101) ≤ P;u (y, x). (4.106) u∈A j≥1

This together with (4.66) in the above paragraph (a) complete the proof of the bound on y,x;A in (4.35). x}\{y⇐⇒x} in (d) Finally, we investigate the contribution to y,x,v;A from 1{y⇐⇒ n m+n (4.63): ∂m=∅ ∂n=y x

wAc (m) w (n) A 1{y ←→ x} ∩ {{y⇐⇒x}\{y⇐⇒x}} ∩ {y←→v}. n m+n m+n m+n Z Ac Z

(4.107)

Using Hn;b)T (y, x) defined in (4.79), we can write (4.107) as (cf., (4.80)) (4.107) =

T ≥1 b)T

∂m=∅ ∂n=y x

wAc (m) w (n) A 1{y ←→ x} ∩ Hn;b) (y,x) ∩ {y⇐⇒x} ∩ {y←→v}. m+n m+n m+n T Z Ac Z (4.108)

To bound this, we will also use a similar expression to (4.84), in which k = n|BD˜ c with D˜ = Cnb (y). We investigate (4.108) separately (in the following paragraphs (d-1) and (d-2)) depending on whether or not there is a bypath Cm+k (z i ) for some i ∈ {1, . . . , j} containing v.

326

A. Sakai A A (d-1) If there is such a bypath, then we use 1{y ←→ x} ≤ 1{y ←→x} as in (4.80) to bound n m+n

the contribution from this case to (4.108) by T ≥1 b)T ∂n=y x

×

T

w (n) A 1{y ←→ x} ∩ Hn;b) (y,x) n T Z j


1{zi ∈Dn;s

i

,

z i ∈Dn;ti }

i=1

j wAc (m) w ˜ c (k)

D × 1{zi ←→zi } Z Ac Z D˜ c m+k ∂m=∅ ∂k=∅

i=1

j 1{Cm+k (zi ) ∩ Cm+k (zl )=∅} 1{v∈Cm+k (zi )}. × i=l

i=1

(4.109)

Note that the last sum of the indicators is the only difference from (4.84). When j = 1, the sum over m, k in (4.109) equals ∂m=∂k=∅

wAc (m) wD˜ c (k) 1{z 1 ←→z 1 } 1{z 1 ←→v}. Z Ac Z D˜ c m+k m+k

(4.110)

As described in (4.85)–(4.86), we can bound (4.110) without 1{z 1 ←→v} by a chain of bubm+k 2 ∗(2l−1) ˜ (z 1 , z 1 ). If 1{z 1 ←→v} = 1, then, by the argument around (4.58)– bles l≥1 (G ) m+k

(4.62), one of the bubbles has an extra vertex v that is further connected to v with another chain of bubbles ψ (v , v). That is, the effect of 1{z 1 ←→v} is to replace one m+k of the G˜ ’s in the chain of bubbles, say, G˜ (a, a ), by v (ϕa ϕv G˜ (v , a ) + G˜ (a, a )δv ,a ) ψ (v , v). Let g;y (z, z ) =

2l−1 l≥1 i=1 a,a

G˜ 2

∗(i−1)

∗(2l−1−i) (z, a) G˜ (a, a ) G˜ 2 (a , z )

× ϕa ϕ y G˜ (y, a ) + G˜ (a, a ) δ y,a .

(4.111)

Then, we have (4.110) ≤

g;v (z 1 , z 1 ) ψ (v , v).

(4.112)

v

Let j ≥ 2 and consider the contribution to (4.109) from 1{v∈Cm+k (z 1 )}; the contribution from 1{v∈Cm+k (zi )} with i = 1 can be estimated in the same way. By conditioning


327

on Vm+k ≡ ˙ i≥2 Cm+k (z i ) as in (4.87), the contribution to the sum over m, k in (4.109) from 1{v∈Cm+k (z 1 )} ≡ 1{z 1 ←→v} equals m+k

∂m=∂k=∅

j

wAc (m) wD˜ c (k)

1{zi ←→zi } 1{Cm+k (zi ) ∩ Cm+k (zi )=∅} Z Ac Z D˜ c m+k

×

i,i ≥2 i=i

i=2

∂m =∂k =∅

(k ) c c wAc ∩ Vm+k (m ) wD˜ c ∩ Vm+k 1{z 1 ←→ z 1 } 1{z 1 ←→ v}, c Z Ac ∩ Vm+k Z D˜ c ∩ V c m +k m +k

(4.113)

m+k

where the second line is bounded by (4.112) for j = 1, and then the first line is bounded j by i=2 l≥1 (G˜ 2 )∗(2l−1) (z i , z i ), due to (4.87)–(4.88). Summarizing the above bounds, we have (cf., (4.101)) (4.109) ≤

j j≥1 z 1 ,...,z j z 1 ,...,z j

g;v (z h , z h ) ψ (v , v)

h=1 v

i=h l≥1

×

∂n=y x

G˜ 2

∗(2l−1)

(z i , z i )

w (n) A ( j) 1{y ←→ x} 1 I˜ (y,x), )z j ,)z j n Z

(4.114)

to which we can apply the bound discussed between (4.80) and (4.106). (d-2) If v ∈ / Cm+k (z i ) for any i = 1, . . . , j, then there exists a v ∈ Dn;l for some l ∈ {0, . . . , T } such that v ←→ v and Cm+k (v ) ∩ Cm+k (z i ) = ∅ for any i. In addition, m+k

j since all connections from y to x on the graph D˜ ∪ ˙ i=1 Cm+k (z i ) have to go through A

A, there is an h ∈ {1, . . . , j} such that z h ←→ z h . Therefore, the contribution from this m+k

case to (4.108) is bounded by w (n) 1 Hn;b) (y,x) T Z

T ≥1 b)T ∂n=y x

×

T

j

j=1 {s t } j ∈L( j) v ,z 1 ,...,z j i i i=1 [0,T ] z 1 ,...,z j

1{zi ∈Dn;s

i

T

, z i ∈Dn;ti }

i=1

1{v ∈Dn;l }

l=0

j j

wAc (m) w ˜ c (k) D A × 1{z h ←→z h } 1{zi ←→zi } 1{v ←→v} Z Ac Z D˜ c m+k m+k m+k ∂m=∅ ∂k=∅

h=1

×

i=i

i=1

j 1{Cm+k (zi ) ∩ Cm+k (zi )=∅} 1{Cm+k (v ) ∩ Cm+k (zi )=∅}, i=1

(4.115)

328

A. Sakai

j where, by conditioning on Sm+k ≡ ˙ i=1 Cm+k (z i ), the last two lines are (see below (4.87))

j

j

wAc (m) wD˜ c (k) A 1{z h ←→ 1 1 z } {z i ←→z i } {Cm+k (z i ) ∩ Cm+k (z i )=∅} Z Ac Z D˜ c m+k h m+k

∂m=∂k=∅

h=1

×

c wAc ∩ Sm+k

∂m =∂k =∅

i=i

i=1

(m )

c Z Ac ∩ Sm+k

wD˜ c ∩ S c

m+k

Z D˜ c ∩ S c

≤ ψ

(k )

1{v ←→ v} . m +k

m+k

(v ,v)

(4.116)

When j = 1, we have

((4.116) for j = 1) ≤ ψ (v , v)

∂m=∂k=∅

wAc (m) wD˜ c (k) A . 1{z 1 ←→ z } Z Ac Z D˜ c m+k 1

(4.117)

If we ignore the “through A”-condition in the last indicator, then the sum is bounded, as in (4.86), by a chain of bubbles l≥1 (G˜ 2 )∗(2l−1) (z 1 , z 1 ). However, because of this con dition, one of the G˜ ’s in the bound, say, G˜ (a, a ), is replaced by u∈A (ϕa ϕu G˜ (u, a ) + G˜ (a, a )δu,a ). Using (4.111), we have (4.117) ≤ ψ (v , v)

g;y (z 1 , z 1 ).

(4.118)

y∈A A Let j ≥ 2 and consider the contribution to (4.116) from 1{z 1 ←→ z 1 }; the contrim+k

A

with h = 1 can be estimated similarly. By conditioning on butions from 1 A z 1 } equals Vm+k ≡ ˙ i≥2 Cm+k (z i ), the contribution to (4.116) from 1{z 1 ←→ {z h ←→z h } m+k

m+k

∂m=∂k=∅

j

wAc (m) wD˜ c (k)

1{zi ←→zi } 1{Cm+k (zi ) ∩ Cm+k (zi )=∅} Z Ac Z D˜ c m+k i,i ≥2 i=i

i=2

×ψ (v , v)

∂m =∂k =∅

(k ) c c wAc ∩ Vm+k (m ) wD˜ c ∩ Vm+k c Z Ac ∩ Vm+k

Z D˜ c ∩ V c

m+k

A , 1{z 1 ←→ z1 } m +k

(4.119) where the second line is bounded by (4.118) for j = 1, and then the first line is bounded j by i=2 l≥1 (G˜ 2 )∗(2l−1) (z i , z i ), as described below (4.113).


329

As a result, (4.115) is bounded by j

∗(2l−1) G˜ 2 ψ (v , v) g;y (z h , z h ) (z i , z i ) j≥1 v ,z 1 ,...,z j z 1 ,...,z j

×

∂n=y x

h=1 y∈A

w (n) Z

i=h l≥1

T ≥ j b)T {s t } j ∈L( j) i i i=1 [0,T ]

×

T

1 Hn;b)

T

j (y,x)

1{zi ∈Dn;s

i

, z i ∈Dn;ti }

i=1

1{v ∈Dn;l }.

(4.120)

l=0

The sum over n can be bounded by following the argument between (4.89) and (4.105); T note that the sum of the indicators in (4.120), except for the last factor l=0 1{v ∈Dn;l }, is identical to that in (4.89). First, we rewrite the sum of the indicators in (4.120) as a single indicator of an event E similar to (4.90). Then, we construct event similar to another T 1{v ∈Dn;l } in (4.120), I˜)z( j),)z (y, x) in (4.99)–(4.100), of which E is a subset. Due to l=0 j

j

n one of the paths in the definition of I˜)z( j),)z (y, x), say, ωi ∈ a→a for some a, a (dependj

j

n n ing on i) is split into two edge-disjoint paths ωi ∈ a→v and ωi ∈ v →a , followed by the summation over i = 3, . . . , 2 j + 1 (cf., Fig. 9). Finally, we apply Lemma 4.2 to obtain the desired bound on the sum over n of (4.120). Summarizing the above (d-1) and (d-2), we obtain ( j) (4.108) ≤ P;u,v (y, x). (4.121) j≥1 u∈A

This together with (4.77) in the above paragraph (b) complete the proof of the bound on y,x,v;A in (4.35). ( j) (x) Assuming the Decay of G(x) 5. Bounds on π

Using the diagrammatic bounds proved in the previous section, we prove Proposition 3.1 in Sect. 5.1, and Propositions 3.2 and 3.3(iii) in Sect. 5.2. 5.1. Bounds for the spread-out model. We prove Proposition 3.1 for the spread-out model using the following convolution bounds: Proposition 5.1. that

(i) Let a ≥ b > 0 and a + b > d. There is a C = C(a, b, d) such y

1 1 C ≤ . a b |||y − v||| |||x − y||| |||x − v|||(a∧d+b)−d

(5.1)

(ii) Let q ∈ ( d2 , d). There is a C = C (d, q) such that z

1 1 1 1 C ≤ . (5.2) |||x − z|||q |||x − z|||q |||z − y|||q |||z − y |||q |||x − y|||q |||x − y |||q

330

A. Sakai

x

(a) z

(b) uj , vj

z

x’

y

x

y

y’

x’

y’

vj

u j-1 v j-1

vj

x

uj

vj

u j-1 v j-1

u j-1 x

x v j-1

Fig. 10. (a) A schematic representation of Proposition 5.1(i), where each segment, say, from x to y represent |||x − y|||−q . (b) A schematic representation of (5.19), which is a result of successive applications of Proposition 5.1(ii) with x = x or y = y

Proof. The inequality (5.1) is identical to [15, Prop. 1.7(i)]. We use this to prove (5.2). By the triangle inequality, we have 21 |||x − y||| ≤ |||x − z||| ∨ |||z − y||| and 21 |||x − y ||| ≤ |||x − z||| ∨ |||z − y |||. Suppose that |||x − z||| ≤ |||z − y||| and |||x − z||| ≤ |||z − y |||. Then, by (5.1) with a = b = q, the contribution from this case is bounded by 22q 1 1 22q c|||x − x |||d−2q ≤ , |||x − y|||q |||x − y |||q z |||x − z|||q |||x − z|||q |||x − y|||q |||x − y |||q

(5.3)

for some c < ∞, where we note that |||x − x |||d−2q ≤ 1 because of 21 d < q. The other three possible cases can be estimated similarly (see Fig. 10(a)). This completes the proof of Proposition 5.1. Before going into the proof of Proposition 3.1, we summarize prerequisites. Recall that (4.13)–(4.14) involve G˜ , and note that, by (4.2), ϕo ϕx 3 ≤ δo,x + G˜ (o, x)3 .

(5.4)

We first show that O(θ0 ) , G˜ (o, x) ≤ |||x|||q

b:b=o

O(θ0 ) τb δb,x + G˜ (b, x) ≤ |||x|||q

(5.5)

hold assuming the bounds in (3.2). Proof. By the assumed bound τ ≤ 2 in (3.2), we have G˜ (o, x) = τ D(x) + τ D(y) ϕ y ϕx ≤ 2D(x) + 2D(y) G(x − y), y=x

(5.6)

y=x

where, and from now on without stating explicitly, we use the translation invariance of G(x) and the fact that G(x − y) is an increasing limit of ϕ y ϕx as ↑ Zd . By (1.14) and the assumption in Proposition 3.1 that θ0 L d−q , with q < d, is bounded away from zero, we obtain D(x) ≤ O(L −d )1{0< x ∞ ≤L} ≤

O(L −d+q ) O(θ0 ) ≤ . q |||x||| |||x|||q

(5.7)


331

√ √ For the last term in (5.6), we consider the cases for |x| ≤ 2 d L and |x| ≥ 2 d L separately. √ When |x| ≤ 2 d L, we use (5.7), (3.2) and (5.1) with 21 d < q < d to obtain

D(y) G(x − y) ≤

y=x

O(L −d+q ) θ0 O(θ0 L −d+q ) O(θ0 ) ≤ ≤ . q q 2q−d |||y||| |||x − y||| |||x||| |||x|||q y

(5.8)

√ When |x| ≥ 2 d L, we use the triangle inequality |x − y|√ ≥ |x|−|y| and that √ the fact 1 D(y) is nonzero only when 0 < y ∞ ≤ L (so that |y| ≤ d y ∞ ≤ d L ≤ 2 |x|). Then, we obtain

D(y) G(x − y) ≤

y=x

D(y)

y

2q θ0 2q θ0 = . |||x|||q |||x|||q

(5.9)

This completes the proof of the first inequality in (5.5). The second inequality can be proved similarly. By repeated use of (5.5) and Proposition 5.1(i) with a = b = 2q (or Proposition 5.1(ii) with x = x and y = y ), we obtain ψ (v , v) ≤ δv ,v +

O(θ02 ) . |||v − v |||2q

(5.10)

Together with the naive bound G(x) ≤ O(1)|||x|||−q (cf., (3.2)) as well as Proposition 5.1(ii) (with x = x or y = y ), we also obtain G(v − y) G(z − v ) ψ (v , v) ≤ G(v − y) G(z − v) v

+

v

≤

|||v

−

y|||q |||z

O(θ02 ) − v |||q |||v − v |||2q

O(1) . |||v − y|||q |||z − v|||q

(5.11)

The O(1) term in the right-hand side is replaced by O(θ0 ) or O(θ02 ) depending on the number of G’s on the left being replaced by G˜ ’s. (0) (x), it Proof of Proposition 3.1. Since (5.4)–(5.5) immediately imply the bound on π (i) suffices to prove the bounds on π (x) for i ≥ 1. To do so, we first estimate the build ing blocks of the diagrammatic bound (4.15): b:b=y τb Q ;u (b, x) and b:b=y τb Q ;u,v (b, x). Recall (4.10)–(4.14). First, by using G(x) ≤ O(1)|||x|||−q and (5.11), we obtain

O(1) , |||x − y|||2q |||u − y|||q |||x − u|||q O(1) (0) P;u,v (y, x) ≤ . q q |||x − y||| |||u − y||| |||x − u|||q |||v − y|||q |||x − v|||q (0) (y, x) ≤ P;u

(5.12) (5.13)

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We will show at the end of this subsection that, for j ≥ 1, O( j) O(θ02 ) j , |||x − y|||2q |||u − y|||q |||x − u|||q O( j 2 ) O(θ02 ) j ( j) (y, x) ≤ . P;u,v |||x − y|||q |||u − y|||q |||x − u|||q |||v − y|||q |||x − v|||q ( j) (y, x) ≤ P;u

(5.14) (5.15)

(0) (0) (y, x) (resp., As a result, P;u (y, x) (resp., P;u,v (y, x)) is the leading term of P;u P;u,v (y, x)), which thus obeys the same bound as in (5.12) (resp., (5.13)), with a different constant in O(1). Combining these bounds with (5.5) and (5.11) (with both G in the left-hand side being replaced by G˜ ) and then using Proposition 5.1(ii), we obtain O(θ0 ) 1 0) τb Q ;u (b, x) ≤ z |||z−y||| ≤ |||x−y|||O(θ (5.16) q q |||x−u|||2q , |||x−z|||2q |||u−z|||q |||x−u|||q b:b=y

and

τb Q ;u,v (b, x) ≤

b:b=y

O(θ0 ) 1 q q q |||z − y||| |||x − z||| |||u − z||| |||x − u|||q |||v − z|||q |||x − v|||q z +

z

≤

|||v −

O(θ0 ) O(θ0 ) 1 |||v − y|||q |||z − v|||q |||x − z|||2q |||u − z|||q |||x − u|||q O(θ0 ) . − v|||q |||x − u|||2q

(5.17)

y|||q |||x

This completes bounding the building blocks. ( j) ( j) Now we prove the bounds on π (x) for j ≥ 1. For the bounds on π (x) for j ≥ 2, we simply apply (5.12) and (5.16)–(5.17) to the diagrammatic bound (4.15). Then, we obtain j−1 O(1) ( j) O(θ0 ) π (x) ≤ 2q q q |||v −u |||q |||u i+1 −vi+1 |||q |||u i+1 −vi |||2q i=1 i+1 i u 1 ,...,u j |||u 1 ||| |||v1 ||| |||u 1 − v1 ||| v1 ,...,v j

× |||x−u

O(θ0 ) q 2q j ||| |||x−v j |||

( j ≥ 2).

(5.18)

First, we consider the sum over u j and v j . By successive applications of Proposition 5.1(ii) (with x = x or y = y ), we obtain (see Fig. 10(b)) O(θ0 ) O(θ0 ) (5.19) q |||u − v |||q |||u − v 2q |||x − u |||q |||x − v |||2q |||v − u ||| ||| j j−1 j j j j−1 j j v u j

≤

j

vj

|||v j −u j−1

|||q |||v

O(θ0 )2 O(θ0 )2 ≤ , q q 2q |||x −u j−1 |||q |||x −v j−1 |||2q j−1 − v j ||| |||x −v j−1 ||| |||x −v j |||

and thus

j−2 O(1) O(θ0 ) π (x) ≤ 2q q q |||vi+1 −u i |||q |||u i+1 −vi+1 |||q |||u i+1 −vi |||2q u 1 ,...,u j−1|||u 1 ||| |||v1 ||| |||u 1 −v1 ||| ( j)

v1 ,...,v j−1

×

i=1

O(θ0 )2 . |||x − u j−1 |||q |||x − v j−1 |||2q

(5.20)


333

Repeating the application of Proposition 5.1(ii) as in (5.19), we end up with

( j) π (x) ≤

O(1)

|||u 1 |||2q |||v1 |||q |||u 1 u 1 ,v1

− v1

|||q

O(θ0 ) j O(θ0 ) j ≤ . q 2q |||x − u 1 ||| |||x − v1 ||| |||x|||3q

(5.21)

(1) (x), we use the following bound, instead of (5.12): For the bound on π

(0) (0) (o, u) = δo,u δo,v + (1 − δo,u δo,v ) P;v (o, u) ≤ δo,u δo,v + P;v

O(θ02 ) . |||u|||2q |||v|||q |||u − v|||q (5.22)

In addition, instead of using (5.16), we use

τb Q ;v (b, x) ≤

( j) O(θ0 ) (0) δz,v δz,x + (1 − δz,x δz,v ) P;v (z, x) + P;v (z, x) |||z − u|||q z j≥1

b:b=u

≤

O(θ03 ) O(θ0 ) δ + v,x q 2q q q |||x − u|||q z |||z − u||| |||x − z||| |||v − z||| |||x − v|||

≤

O(θ03 ) O(θ0 ) δ + , v,x |||x − u|||q |||x − u|||q |||x − v|||2q

(5.23)

due to (5.5), (5.14) and (5.22). Applying (5.22)–(5.23) to (4.15) for j = 1 and then using Proposition 5.1(ii), we end up with O(θ02 ) O(θ03 ) O(θ03 ) O(θ0 ) δv,x π (x) ≤ O(θ0 ) δo,x + + + |||x|||3q u,v |||u|||2q |||v|||q |||u −v|||q |||x −u|||q |||x −u|||q |||x −v|||2q (1)

≤ O(θ0 ) δo,x +

O(θ03 ) . |||x|||3q

(5.24)

To complete the proof of Proposition 3.1, it thus remains to show (5.14)–(5.15). The (1) inequality (5.14) for j = 1 immediately follows from the definition (4.6) of P;u (see also Fig. 5) and the bound (5.10) on ψ − δ. To prove (5.15) forj = 1, we first recall (1) the definition (4.9) of P;u,v (and Fig. 5). Note that, by (5.11), v G(v − y) G(z − v ) ψ (v , v) obeys the same bound on v G(v − y) G(z − v ) (with a different O(1) term). That is, the effect of an additional ψ is not significant. Therefore, the bound (1) on P;u,v is identical, with a possible modification of the O(1) multiple, to the bound (1) (1) (or P;v ) with v (resp., u) “being embedded” in one of the bubbles consisting on P;u of ψ − δ. By (5.10), ψ (y, x) − δ y,x with v being embedded in one of its bubbles is

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bounded as ∞ k k=1 l=1 y ,x

G˜ 2

∗(l−1)

(y, y ) G˜ (y , x ) ϕ y ϕv G˜ (v, x ) + G˜ (y , x ) δv,x

∗(k−l) (x , x) × G˜ 2 = ψ (y, y ) G˜ (y , x ) ϕ y ϕv G˜ (v, x ) + G˜ (y , x ) δv,x ψ (x , x) y ,x

≤

y ,x

≤

O(1) O(θ0 ) O(θ0 ) O(1) 2q q q q − y||| |||x − y ||| |||v − y ||| |||x − v||| |||x − x |||2q

|||y

O(θ02 ) . |||x − y|||q |||v − y|||q |||x − v|||q

(5.25)

By this observation and using (3.2) to bound the remaining two two-point functions (1) consisting of P;u,v (recall (4.9)), we obtain (5.15) for j = 1. For (5.14)–(5.15) with j ≥ 2, we first note that, by applying (3.2) and (5.10) to the definition (4.5) of P( j) (y, x), we have P( j) (y, x) ≤

v2 ,...,v j v1 ,...,v j−1

|||v1

×

−

O(θ02 ) − y|||q |||v1 − v2 |||q

y|||2q |||v2

j−1

|||vi i=2 ×

O(θ02 ) |||q |||v − v |||q − vi−1 i+1 i

− vi |||2q |||vi+1

O(θ02 ) . |||x − v j |||2q |||x − v j−1 |||q

(5.26)

( j) By definition, the bound on P;u (y, x) is obtained by “embedding u” in one of the 2 j − 1 factors of ||| · · · |||q (not ||| · · · |||2q ) and then summing over all these 2 j − 1 choices. For example, the contribution from the case in which |||v2 − y|||q is replaced by |||u − y|||q |||v2 − u|||q is bounded, similarly to (5.21), by

v2 ,v1

≤

O(θ02 ) O(θ02 ) j−1 |||v1 − y|||2q |||u − y|||q |||v2 − u|||q |||v1 − v2 |||q |||x − v1 |||q |||x − v2 |||2q

v1

O(θ02 ) j O(θ02 ) j ≤ . |||v1 − y|||2q |||u − y|||q |||x − u|||q |||x − v1 |||2q |||x − y|||2q |||u − y|||q |||x − u|||q (5.27)

The other 2 j − 2 contributions can be estimated in a similar way, with the same form of the bound. This completes the proof of (5.14). ( j) By (5.25), the bound on P;u,v (y, x) is also obtained by “embedding u and v” in one of the 2 j − 1 factors of ||| · · · |||q and one of the j factors of ||| · · · |||2q in (5.26), and then summing over all these combinations. For example, the contribution from the case in


335

which |||v2 −y|||q and |||v1 −y|||2q in (5.26) are replaced, respectively, by |||u−y|||q |||v2 −u|||q and |||v1 − y|||q |||v − y|||q |||v1 − v|||q , is bounded by v2 ,v1

O(θ02 ) |||v1 − y|||q |||v − y|||q |||v1 − v|||q |||u − y|||q |||v2 − u|||q |||v1 − v2 |||q O(θ02 ) j−1 |||x − v1 |||q |||x − v2 |||2q

× ≤

v1

≤

O(θ02 ) j |||v1 − y|||q |||v − y|||q |||v1 − v|||q |||u − y|||q |||x − u|||q |||x − v1 |||2q

O(θ02 ) j . |||x − y|||q |||u − y|||q |||x − u|||q |||v − y|||q |||x − v|||q

(5.28)

The other (2 j − 1) j − 1 contributions can be estimated similarly, with the same form of the bound. This completes the proof of (5.15) and thus Proposition 3.1. 5.2. Bounds for finite-range models. First, we prove (3.8) and Proposition 3.3(iii) assuming (3.7). Then, we prove (3.10) assuming (3.7) and (3.9) to complete the proof of Propositions 3.2. Proof of (3.8) assuming (3.7). By applying (4.2) to the diagrammatic bound (4.15) on (0) π (x), it is easy to show that, for r = 0, 2, (0) |x|r π (x) ≤ δr,0 + |x|r ϕo ϕx 3 ≤ δr,0 + sup |x|r G(x) (τ D ∗ G)(x) G(x) x=o

x=o

x

x=o

2 δr,2

≤ δr,0 +(dσ )

O(θ0 ) . 2

(5.29)

For i ≥ 1, by using the diagrammatic bound (4.15) and translation invariance, we have i−1 (i) (0) π (x) ≤ P;v (o, x) sup τ y,z Q ;o,v (z, x) y z,v,x

v,x

x

× sup y

τ y,z Q ;o (z, x)

.

(5.30)

z,x

(i) The proof of the bound on x π (x) for i ≥ 1 is completed by showing that (0) P;v (o, x) − 1 ∨ sup τ y,z Q ;o,v (z, x) v,x

y z,v,x

∨ sup y

τ y,z Q ;o (z, x) = O(θ0 ).

(5.31)

z,x

The key idea to obtain this estimate is that the bounding diagrams for the Ising model are similar to those for self-avoiding walk (cf., Fig. 6). The diagrams for the self-avoiding walk are known to be bounded by products of bubble diagrams (see, e.g., [24]), and we

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can apply the same method to bound the diagrams for the Ising model by products of bubbles. Consider, for example, ˜ τ y,z Q ;o (z, x) = τ y,z δz,z + G (z, z ) P;o (z , x). (5.32) z ,x

z,x

z

The factor of θ0 is due to the nonzero line segment

z τ y,z (δz,z

+ G˜ (z, z )), because

τo,z δz,x + G˜ (z, x) = τ D(x) + τ D(z) G˜ (z, x) ≤ O(θ0 ) + τ sup G˜ (o, x),

z

x

z

G˜ (o, x) ≤ τ D(x) + τ

(5.33) G(y) D(x − y) ≤ O(θ0 ) + τ sup G(y) = O(θ0 ), (5.34) y=o

y=o

where we have used translation invariance, (3.7) and supx D(x) = O(θ0 ). By (4.12), (0) ( j) P;o (z , x)+ (5.32) ≤ O(θ0 ) P;o (z , x) = O(θ0 ) P;o (z , x) . (5.35) z ,x

z ,x

j≥1

(0) Similarly to (5.29) for r = 0, the sum of P;v (z , x) is easily estimated as 1 + O(θ0 ). ( j) ( j) We claim that the sum of P;o (z , x) for j ≥ 1 is (2 j − 1) O(θ0 ) j , since P;o (z , x) is a sum of 2 j − 1 terms, each of which contains j chains of nonzero bubbles; each chain is ψ (v, v ) − δv,v for some v, v and satisfies

2 l τ D ∗ (D ∗ G ∗2 ) (o) = ψ (v, v )−δv,v ≤ O(θ0 )l = O(θ0 ). (5.36) v

l≥1

l≥1

For example, z ,x

(z’ )

(4) P;o (z , x) =

o

+ 6 other possibilities,

(5.37)

(x )

which can be estimated, by translation invariance, as y

(y )

(z’ )

≤

o (x )

z

sup y,z

o

o

≤

(z )

ψ (o, y) − δo,y

4

o

W¯ (0)

4

y

= O(θ0 )4 , where W¯ (t) is given by (3.13).

(5.38)


337

a0

a1

a2

(i)

a3

(ii) |a3 | 2

|a0 | 2

&

(iii)

|a2 | 2

|a1 | 2

(3) Fig. 11. One of the leading diagrams for x |x|2 π (x) and its decompositions depending on whether the 2 2 2 assigned weight is (i) |a3 | , (ii) |a0 | and (iii) |an | for n = 1, 2, respectively

The sum of τ y,z Q ;o,v (z, x) in (5.31) is estimated similarly [29]. We complete the ( j) proof of the bound on x π (x) for j ≥ 1. ( j) To estimate x |x|2 π (x) for j ≥ 1, we recall that, in each bounding diagram, there are at least three distinct paths between o and x: the uppermost path (i.e., o → b1 → v2 → b3 → · · · → x in (4.15); see also Fig. 6), the lowermost path (i.e., o → v1 → b2 → v3 → · · · → x) and a middle zigzag path. We use the lowermost path to bound |x|2 as

|x|2 =

j

|an |2 + 2

n=0

am · an ≤ ( j + 1)

j

|an |2 ,

(5.39)

n=0

0≤m |z 2 | > |z 1 − z 2 | > 0. Proof. The convergence property says, in particular, that the series Y(u 1 ; z 1 , z¯ 1 )Y(u 2 ; z 2 , z¯ 2 )u 3 = Y(u 1 ; z 1 , z¯ 1 )Pn Y(u 2 ; z 2 , z¯ 2 )u 3 ,

(1.9)

n∈R

(a product of full vertex operators) converges absolutely in F for u 1 , u 2 , u 3 ∈ F when |z 1 | > |z 2 | > 0. The convergence property also says, in particular, that the series Y(Y(u 1 ; z 1 − z 2 , z¯ 1 − z¯ 2 )u 2 ; z 2 , z¯ 2 )u 3 = Y(Pn Y(u 1 ; z 1 − z 2 , z¯ 1 − z¯ 2 )u 2 ; z 2 , z¯ 2 )u 3

(1.10)

n∈R

(an iterate of full vertex operators) converges absolutely for u 1 , u 2 , u 3 ∈ F when |z 2 | > |z 1 − z 2 | > 0. Moreover, the convergence property also says that both (1.9) and (1.10) converge absolutely to m 3 (u 1 , u 2 , u 3 ; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0). This proves the associativity.

Proposition 1.5 (Commutativity). For u 1 , u 2 , u 3 ∈ F, Y(u 1 ; z 1 , z¯ 1 )Y(u 2 ; z 2 , z¯ 2 )u 3 , Y(u 2 ; z 2 , z¯ 2 )Y(u 1 ; z 1 , z¯ 1 )u 3 , are the expansions of m 3 (u 1 , u 2 , u 3 ; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0) in the domains given by |z 1 | > |z 2 | > 0 and |z 2 | > |z 1 | > 0, respectively.

(1.11) (1.12)

Full Field Algebras

351

Proof. By the convergence property, we know that (1.11) and (1.12) converge absolutely to m 3 (u 1 , u 2 , u 3 ; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0) and m 3 (u 2 , u 1 , u 3 ; z 2 , z¯ 2 , z 1 , z¯ 1 , 0, 0), respectively, when |z 1 | > |z 2 | > 0 and |z 2 | > |z 1 | > 0, respectively. By the permutation property, m 3 (u 1 , u 2 , u 3 ; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0) = m 3 (u 2 , u 1 , u 3 ; z 2 , z¯ 2 , z 1 , z¯ 1 , 0, 0). Thus (1.11) and (1.12) converge absolutely to m 3 (u 1 , u 2 , u 3 ; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0) when |z 1 | > |z 2 | > 0 and |z 2 | > |z 1 | > 0, respectively. So they are the expansions of m 3 (u 1 , u 2 , u 3 ; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0) in the sets given by |z 1 | > |z 2 | > 0 and |z 2 | > |z 1 | > 0, respectively.

Before proving more properties, we would like to discuss first the problem of constructing full field algebras. It is clear that vertex operator algebras have structures of full field algebras. Let (V L , Y L , 1 L , ω L ) and (V R , Y R , 1 R , ω R ) be two vertex operator algebras. Consider the graded vector space V L ⊗ V R equipped with the correlation function maps, the vacuum and the operator d given as follows: For n ∈ Z+ , u 1L , . . . , u nL ∈ V L and u 1R , . . . , u nR ∈ V R , m n (u 1L ⊗ u 1R , . . . , u nL ⊗ u nR ; z 1 , z¯ 1 , . . . , z n , z¯ n ) are given by the analytic extensions of (Y L (u 1L , z 1 ) ⊗ Y R (u 1R , z¯ 1 )) · · · (Y L (u nL , z n ) ⊗ Y R (u nR , z¯ n ))1. Then we take the vacuum 1 = 1 L ⊗1 R and the operators d = L L (0)⊗ I V R + I V L ⊗L R (0). In particular, the full vertex operators are given by Y(u L ⊗ u R ; z, z¯ )v L ⊗ v R = Y L (u L , z)v L ⊗ Y R (u R , z¯ )v R for u L , v L ∈ V L , u R , v R ∈ V R and z ∈ C× . We have: Proposition 1.6. The vector space V L ⊗ V R equipped with the correlation function maps and the vacuum 1 given above is a full field algebra. Proof. The proof is a straightforward and easy verification.

Note that there is also a vertex operator algebra structure on V L ⊗ V R . For simplicity, we shall use the notation V L ⊗ V R to denote both the vertex operator algebra and the full field algebra structure. It should be easy to see which structure we will be using in the remaining part of this paper. The full field algebra V L ⊗ V R in general does not give a genus-one theory, that is, suitable q-traces, even in the case that they are convergent, of the full vertex operators in general are not modular invariant. For chiral theories, we know from [H8] that if we

352

Y.-Z. Huang, L. Kong

consider the intertwining operator algebras constructed from irreducible modules for suitable vertex operator algebras, we do have modular invariance. So it is then natural to look for full field algebras from suitable extensions of V L ⊗ V R by V L ⊗ V R -modules. Note that V L ⊗ V R has an Z × Z-grading with grading operators being L L (0) ⊗ I V R and I V L ⊗ L R (0). If a full field algebra is an extension of V L ⊗V R by V L ⊗V R -modules, it has an R × R-grading. For any R × R-graded vector space F = (m,n)∈R×R F(m,n) , we have a left grading operator d L and a right grading operator d R defined by d L u = mu, d R u = nu for u ∈ F(m,n) , where m (n) is called the left (right) weight of u and is denoted by wt L u (wt R u). For m, n ∈ R, let Pm,n be the projection from F → F(m,n) . We still use F and F to denote the graded dual and the algebraic completion of F, but note that they are with respect to the R × R-grading, not any R-grading induced from the R × R-grading. Definition 1.7. An R × R-graded full field algebra is a full field algebra (F, m, 1) equipped with an R × R-grading on F (graded by left conformal weight or left weight and right conformal weight or right weight and thus equipped with left and right grading operators d L and d R ) and operators D L and D R satisfying the following conditions: 1. The grading compatibility: d = d L + d R . L R 2. The single-valuedness property: e2πi(d −d ) = I F . (1) (1) (k) 3. The convergence property: For k, l1 , . . . , lk ∈ Z+ and u 1 , . . . , u l1 , . . . , u 1 , . . . , (k)

u lk ∈ F, the series (1) (1) (1) (1) (1) (1) m k (Pp1 ,q1 m l1 (u 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , zl1 , z¯l1 ), . . . , p1 ,q1 ,..., pk ,qk

(k) (k) (k) (k) (k) (0) (0) (0) (0) Ppk ,qk m lk (u (k) 1 , . . . , u lk ; z 1 , z¯ 1 , . . . , z lk , z¯ lk ); z 1 , z¯ 1 , . . . , z k , z¯ k ) (1.13) ( j)

(0) (0) converges absolutely to (1.2) when |z (i) p | + |z q | < |z i − z j | for i, j = 1, . . . , k, i = j and for p = 1, . . . , li and q = 1, . . . , l j . 4. The d L - and d R -bracket properties: For u ∈ F, ∂ d L , Y(u; z, z¯ ) = z Y(u; z, z¯ ) + Y(d L u; z, z¯ ), (1.14) ∂z ∂ d R , Y(u; z, z¯ ) = z¯ Y(u; z, z¯ ) + Y(d R u; z, z¯ ). (1.15) ∂ z¯

5. The D L - and D R -derivative property: For u ∈ F, ∂ Y(u; z, z¯ ), D L , Y(u; z, z¯ ) = Y(D L u; z, z¯ ) = ∂z ∂ D R , Y(u; z, z¯ ) = Y(D R u; z, z¯ ) = Y(u; z, z¯ ). ∂ z¯

(1.16) (1.17)

We denote the R × R-graded full field algebra defined above by (F, m, 1, D L , D R ) or simply by F. But note that there is a refined grading on F now. Remark 1.8. Note that for R × R-graded full field algebra, there is also a weaker convergence property similar to the one in Remark 1.2.

Full Field Algebras

353

Remark 1.9. The single-valuedness property actually says that F is graded by a subgroup {(m, n) ∈ R × R | m − n ∈ Z} of R × R. This single-valuedness indeed corresponds to a certain single-valuedness condition in the geometric axioms for full conformal field theories. We have the following immediate consequences of the definition above. Proposition 1.10. 1. The pair (left weight, right weight) for 1 is (0, 0), that is, d L 1 = d R 1 = 0. 2. The pairs (left weight, right weight) for D L and D R are (1, 0) and (0, 1), respectively, that is, [d L , D L ] [d R , D L ] [d L , D R ] [d R , D R ]

= DL, = 0, = 0, = DR.

3. D L 1 = D R 1 = 0. Proof. From the identity property, z Then by (1.14), we obtain

∂ Y(1; z, z¯ ) = 0. ∂z

d L , Y(1; z, z¯ ) = Y(d L 1; z, z¯ ).

(1.18)

Since Y(1; z, z¯ ) = I F , we obtain Y(d L 1; z, z¯ ) = 0

(1.19)

from (1.18). Applying (1.19) to 1, taking the limit z → 0 on both sides of (1.19) and using the creation property, we obtain d L 1 = 0. So the left weight of 1 is 0. Similarly, we can prove that the right weight of 1 is 0. Applying both sides of (1.14) to 1, taking the limit z → 0 and then using the creation property and the fact d L 1 = 0 we have just proved, we obtain lim z

z→0

∂ Y(u; z, z¯ )1 = 0 ∂z

(1.20)

∂ for u ∈ F. Applying ∂z to both sides of (1.14) and using the D L -derivative property, we obtain d L , Y(D L u; z, z¯ )

∂ ∂ z Y(u; z, z¯ ) + Y(D L d L u; z, z¯ ) ∂z ∂z ∂ ∂ ∂ Y(u; z, z¯ ) + Y(u; z, z¯ ) + Y(D L d L u; z, z¯ ) =z ∂z ∂z ∂z ∂ L = z Y(D u; z, z¯ ) + Y(D L u; z, z¯ ) + Y(D L d L u; z, z¯ ). ∂z

=

(1.21)

354


Applying (1.21) to 1, taking the limit z → 0 on both sides of (1.21) and using the creation property, (1.20) and d L 1 = 0, we obtain d L D L u = D L u + D L d L u, proving that the left weight of D L is 1. Similarly, we can prove that the right weight of D L is 0, the left weight of D R is 0 and the right weight of D R is 1. Using the D L - and D R -derivative properties and the creation property, we see immediately that D L 1 = D R 1 = 0.

For an R × R-graded full field algebra (F, m, 1, D L , D R ), we now introduce a for× mal vertex √ operator map. We shall use the convention that for any z L∈ C , log zR = log |z| + −1 arg z, where 0 ≤ arg z < 2π . For u ∈ F, we use wt u and wt u to denote the left and right weights, respectively, of u. Let u, v ∈ F and w ∈ F be homogeneous elements. We have w , [d L , Y(u; z, z¯ )]v = (d L ) w , Y(u; z, z¯ )v − w , Y(u; z, z¯ )d L v = (wt L w − wt L v)w , Y(u; z, z¯ )v, (1.22) where (d L ) is the adjoint of d L . On the other hand, ∂ w , Y(d L u; z, z¯ )v + z Y(u; z, z¯ )v ∂z ∂ L w , Y(u; z, z¯ )v. = wt u + z ∂z

(1.23)

Let f (z, z¯ ) = w , Y(u; z, z¯ )v. Then by (1.14), (1.22) and (1.23), we have z

∂ f (z, z¯ ) = (wt L w − wt L u − wt L v) f (z, z¯ ). ∂z

(1.24)

Similarly, using (1.15), we have z¯

∂ f (z, z¯ ) = (wt R w − wt R u − wt R v) f (z, z¯ ). ∂ z¯

(1.25)

The general solution of the system (1.24) and (1.25) is C z wt

L

w −wt L u−wt L v wt R w −wt R u−wt R v

z¯

,

(1.26)

where C ∈ C. Note that f (z, z¯ ) is a single-valued function and that by the singlevaluedness of the full field algebra F, (wt L w − wt L u − wt L v) − (wt R w − wt R u − wt R v) ∈ Z.

This means that if we choose any branches of z wt w −wt u−wt v and z¯ wt w −wt u−wt v , then there must be a unique constant C such that f (z, z¯ ) is equal to (1.26). We choose L L L R R R L L L the branches ofR z wt wR −wt u−wt v and z¯ wt w −wt u−wt v to be e(wt w −wt u−wt v) log z R and e(wt w −wt u−wt v) log z , respectively. So there is a unique C ∈ C such that L

f (z, z¯ ) = Ce(wt

L

L

L

R

w −wt L u−wt L v) log z (wt R w −wt R u−wt R v) log z

e

.

R

R

(1.27)

Full Field Algebras

355

Hence Pp,q Y(u; z, z¯ )Pm,n , m, n, p, q ∈ R can be written as u m,n e( p−wt p,q

L

u−m) log z (q−wt R u−n) log z

e

,

p,q

where u m,n are linear maps from F(m,n) → F( p,q) for m, n, p, q ∈ R. Thus we have the following expansion: Yl,r (u)e(−l−1) log z e(−r −1) log z , (1.28) Y(u; z, z¯ ) = r,s∈R

where Yl,r (u) ∈ End F with wt L Yl,r (u) = wt L u −l −1 and wt R Yl,r (u) = wt R u −r −1. Moreover, the expansion above is unique. Let x and x¯ be independent and commuting formal variables. We define the formal full vertex operator Y f associated to u ∈ F by Y f (u; x, x) ¯ = Yl,r (u)x −l−1 x¯ −r −1 . (1.29) l,r ∈R

These formal full vertex operators give a formal full vertex operator map Y f : F ⊗ F → F{x, x}. ¯ For nonzero complex numbers z and ζ , we can substitute er log z and es log ζ for x r and x¯ s , respectively, in Y f (u; x, x) ¯ to obtain a map Yan (u; z, ζ ) : F → F called the analytic full vertex operator map. The following propositions are clear: Proposition 1.11. For u ∈ F and z, ζ ∈ C× , we have Yan (u; z, ζ ) = z d ζ d Y(z −d ζ −d u; 1, 1)z −d ζ −d . L

R

L

R

L

R

(1.30)

For formal full vertex operators, we have Y f (u; x, x) ¯ = x d x¯ d Y(x −d x¯ −d u; 1, 1)x −d x¯ −d . L

R

L

R

L

R

(1.31)

Proposition 1.12. For u ∈ F, lim

x→0,x→0 ¯

Y f (1; x, x)u ¯ = u, Y f (u; x, x)1 ¯ = u,

(1.32) (1.33)

where lim x→0,x→0 means taking the constant term of a power series in x and x. ¯ In ¯ particular, Yl,r (u)1 = 0 for all l, r ∈ R and Y−1,−1 (u)1 = u. Proposition 1.13. For u ∈ F, we have ∂ D L , Y f (w; x, x) Y f (w; x, x), ¯ = Y f (D L w; x, x) ¯ = ¯ ∂x ∂ Y f (w, x, x). D R , Y f (w; x, x) ¯ = Y f (D R w; x, x) ¯ = ¯ ∂ x¯ In particular, we have D L 1 = D R 1 = 0 and for l, r ∈ R, D L , Yl,r (u) = Yl,r (D L u) = −lYl−1,r (u), D R , Yl,r (u) = Yl,r (D R u) = −r Yl,r −1 (u).

(1.34) (1.35)

(1.36) (1.37)

356


We need the following strong version of the creation property: Lemma 1.14. For u ∈ F, ¯ = ex D Y f (u; x, x)1

L + x¯ D R

u.

(1.38)

Proof. Using (1.34) and (1.35), we have Y f (e x0 D

L + x¯

0D

R

u; x, x) ¯ = Y f (u; x + x0 , x¯ + x¯0 ).

(1.39)

Now let both sides of (1.39) act on the vacuum 1. Since Y f (u; x + x0 , x¯ + x¯0 )1 involves only nonnegative integer powers of x +x0 and x¯ + x¯0 , we can take the limit x → 0, x¯ → 0. Then replacing x0 and x¯0 by x and x, ¯ we obtain (1.38).

Proposition 1.15 (Skew symmetry). For any u, v ∈ F and z ∈ C× , we have Y(u; z, z¯ )v = e z D and

¯ = ex D Y f (u; x, x)v

L +¯z D R

L + x¯ D R

Y(v; −z, −z)u

Y f (v; eπi x, e−πi x)u. ¯

(1.40) (1.41)

Proof. From the convergence property, it is clear that, for any u, v ∈ F, Y(Y(u; z 1 − z 2 , z¯ 1 − z¯ 2 )v; z 2 , z¯ 2 )1

(1.42)

converges absolutely to m 3 (u, v, 1; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0) when |z 2 | > |z 1 − z 2 | > 0, and Y(Y(v; z 2 − z 1 , z¯ 2 − z¯ 1 )u; z 1 , z¯ 1 )1

(1.43)

converges absolutely when |z 1 | > |z 1 − z 2 | > 0 to m 3 (v, u, 1; z 2 , z¯ 2 , z 1 , z¯ 1 , 0, 0) which is equal to m 3 (u, v, 1; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0) by the permutation property. Hence, using (1.38), we obtain ez2 D

L +¯z

2D

R

Y(u; z 1 − z 2 , z¯ 1 − z¯ 2 )v = e z 1 D

L +¯z

1D

R

Y(v; z 2 − z 1 , z¯ 2 − z¯ 1 )u

(1.44)

when |z 2 | > |z 1 − z 2 | > 0 and |z 1 | > |z 1 − z 2 | > 0. We change the variables from z 1 , z 2 to z = z 1 − z 2 and z 2 . Then (1.44) gives ez2 D

L +¯z

2D

R

Y(u; z, z¯ )v = e(z 2 +z)D

L +(¯z

2 +¯z )D

R

Y(v; −z, −z)u

(1.45)

when |z 2 | > |z| > 0 and |z 2 + z| > |z| > 0. Notice that for fixed z = 0 and w ∈ F , w , e z 2 D

L +¯z

2D

R

Y(u; z, z¯ )v

(1.46)

involves only positive integral powers of z 2 , z¯ 2 and thus is a power series in z 2 and z¯ 2 absolutely convergent when |z 2 | > |z| > 0. From complex analysis, we know that a power series in two variables z 2 and ζ2 convergent at z 2 = z 20 and ζ2 = ζ20 must be convergent absolutely when |z 2 | < |z 20 | and |ζ2 | < |ζ20 |. In particular, when ζ20 = z¯ 20 , such a power series must be absolutely convergent when |z 2 | < |z 20 | and ζ2 = z¯ 2 . In our case, since for any fixed z, (1.46) is absolutely convergent when |z 2 | > |z| > 0, we conclude that (1.46) converges absolutely for all z 2 . Since w is arbitrary, we see that the left-hand side of (1.45) is absolutely convergent in F for all z 2 . Since z is also arbitrary,

Full Field Algebras

357

by the convergence property again, we see that the left-hand side of (1.45) is absolutely convergent in F for all z and z 2 such that z = 0. Similarly, the right-hand side of (1.45) also converges absolutely in F for all z and z 2 such that z = 0. L R If e−z 2 D −¯z 2 D gives a linear operator on F, then we can just multiply both sides of L R (1.45) by e−z 2 D −¯z 2 D to obtain (1.40). In the case that the total weights of F is lowerL R truncated, e−z 2 D −¯z 2 D is indeed a linear operator on F. In the most general case, this might not be true. But we can still obtain (1.40) as follows: Consider the formal series e x1 D

L + x¯

1D

R

= e(x1 +x2

e x2 D

L + x¯

)D L +(x¯

2D

1 + x¯2

R

Y f (u; x, x)v ¯

)D R

Y f (u; x, x)v, ¯

(1.47)

where x, x, ¯ x1 , x¯1 , x2 and x¯2 are commuting formal variables. Since Y(u; z, z¯ )v is absolutely convergent in F when z = 0, we can substitute z, z¯ , −z 2 , −¯z 2 , z 2 and z¯ 2 for x, x, ¯ x1 , x¯1 , x2 and x¯2 on the right-hand side of (1.47), respectively, and the resulting series is absolutely convergent in F. So we can do the same substitution on the left-hand side of (1.47) and the resulting series is absolutely convergent in F. Similarly, consider the formal series e x1 D

L + x¯

1D

R

e(x2 +x)D

= e(x1 +x2 +x)D L

L +( x¯

L +( x¯

¯ 2 + x)D

¯ 1 + x¯2 + x)D

R

R

Y f (v; eπi x, e−πi x)u ¯


(1.48)

R

Since e z D +¯z D Y(v; −z, −z)u is absolutely convergent in F when z = 0, we can substitute z, z¯ , −z 2 , −¯z 2 , z 2 and z¯ 2 for x, x, ¯ x1 , x¯1 , x2 and x¯2 on the right-hand side and thus also on the left-hand side of (1.48) and the resulting series is absolutely convergent in F. The convergence of these series and (1.45) with suitably chosen z 2 gives (1.40) Now (1.41) follows immediately: On the one hand, by (1.31), we have x d x¯ d Y(x −d x¯ −d u; 1, 1)x −d x¯ −d v = Y f (u; x, x)v. ¯ L

L

L

L

L

L

L +D R

Y(x −d x¯ −d v; −1, −1)x −d x¯ −d u

(1.49)

On the other hand, we have L

L

x d x¯ d e D = ex D

L + x¯ D R

L

L

L


Using skew symmetry (1.40), (1.49) and (1.50), we obtain (1.41).

L

(1.50)

Definition 1.16. An R×R-graded full field algebra (F, m, 1, D L , D R ) is called grading restricted if it satisfies the following grading-restriction conditions: 1. There exists M ∈ R such that F(m,n) = 0 if n < M or m < M. 2. dim F(m,n) < ∞ for m, n ∈ R. We say that F is lower truncated if F satisfies the first grading restriction condition. In this case, for u ∈ F and k ∈ R, we have l+r =k Yl,r (u) ∈ End F with total weight wt u − k − 2. We denote l+r =k Yl,r (u) by Yk−1 (u). Then we have the expansion Y f (u; x, x) = Yk (u)x −k−1 , (1.51) k∈R

358


where wt Yk (u) = wt u −k −1. For given u, v ∈ F, we have Yk (u)w = 0 for sufficiently large k. Let (V L , Y L , 1 L , ω L ) and (V R , Y R , 1 R , ω R ) be vertex operator algebras. Let ρ be an injective homomorphism from the full field algebra V L ⊗ V R to F. Then we have 1 = ρ(1 L ⊗ 1 R ), d L ◦ ρ = ρ ◦ (L L (0) ⊗ I V R ), d R ◦ ρ = ρ ◦ (I V L ⊗ L R (0)), D L ◦ ρ = ρ ◦ (L L (−1) ⊗ I V R ) and D R ◦ ρ = ρ ◦ (I V L ⊗ L R (−1)). Moreover, F has a left conformal element ρ(ω L ⊗ 1 R ) and an right conformal element ρ(1 L ⊗ ω R ). We have the following operators on F: ¯ L L (0) = Resx Resx¯ x¯ −1 Y f (ρ(ω L ⊗ 1 R ); x, x), ¯ L R (0) = Resx Resx¯ x −1 Y f (ρ(1 L ⊗ ω L ); x, x), ¯ L L (−1) = Resx Resx¯ x x¯ −1 Y f (ρ(ω L ⊗ 1 R ); x, x), ¯ f (ρ(1 L ⊗ ω L ); x, x). ¯ L R (−1) = Resx Resx¯ x −1 xY Since these operators are operators on F, it should be easy to distinguish them from those operators with the same notation but acting on V L or V R . Definition 1.17. Let (V L , Y L , 1 L , ω L ) and (V R , Y R , 1 R , ω R ) be vertex operator algebras. A full field algebra over V L ⊗ V R is a grading-restricted R × R-graded full field algebra (F, m, 1, D L , D R ) equipped with an injective homomorphism ρ from the full field algebra V L ⊗ V R to F such that d L = L L (0), d R = L R (0), D L = L L (−1) and D R = L R (−1). We shall denote the full field algebra over V L ⊗ V R defined above by (F, m, ρ) or simply by F. The following result allows us to construct full field algebras using the representation theory of vertex operator algebras: Theorem 1.18. Let (V L , Y L , 1 L , ω L ) and (V R , Y R , 1 R , ω R ) be vertex operator algebras. Let (F, m, ρ) be a full field algebra over V L ⊗V R . Then F is a module for V L ⊗V R viewed as a vertex operator algebra. Moreover, Y f (·, x, x), is an intertwining operator

of type FFF . Proof. Let Y L ,R be the vertex operator map for the full field algebra ρ(V L ⊗ V R ). Then we have Y L ,R (ρ(u L ⊗ u R ); z, z¯ )ρ(v L ⊗ v R ) = ρ(Y L (u L , z)v L ⊗ Y R (u R , z¯ )v R )

(1.52)

for u L , v L ∈ V L , u R , v R ∈ V R and z ∈ C× . Now we show that a splitting formula similar to (1.52) holds for vertex operators of the form Y(ρ(u L ⊗ u R ); z, z¯ ) : F → F. By the associativity of Y, we have w , Y(ρ(u L ⊗ u R ); z 1 , z¯ 1 )Y(ρ(v L ⊗ v R ); z 2 , z¯ 2 )w = w , Y(Y L ,R (ρ(u L ⊗ u R ); z 1 − z 2 , z¯ 1 − z¯ 2 )ρ(v L ⊗ v R ); z 2 , z¯ 2 )w

(1.53)

when |z 1 | > |z 2 | > |z 1 − z 2 | > 0 for u L , v L ∈ V L , u R , v R ∈ V R , w ∈ F and w ∈ F . Take v L = 1 L and u R , v R = 1 R . Then we have w , Y(ρ(u L ⊗ 1 R ); z 1 , z¯ 1 )w = w , Y(ρ(u L ⊗ 1 R ); z 1 , z¯ 1 )Y(1; z 2 , z¯ 2 )w = w , Y(Y L ,R (ρ(u L ⊗ 1 R ); z 1 − z 2 , z¯ 1 − z¯ 2 )ρ(1 L ⊗ 1 R ); z 2 , z¯ 2 )w = w , Y(ρ((Y L (u L , z 1 − z 2 )1 L ) ⊗ 1 R ); z 2 , z¯ 2 )w. (1.54)

Full Field Algebras

359

Since the right-hand side of (1.54) is independent of z¯ 1 , so is the left-hand side. Thus we see that Y(ρ(u L ⊗ 1 R ), z, z¯ ) depends only on z for all u L ∈ V L and we shall also denote it by Y L (u L , z). (Since it acts on F, there should be no confusion with the vertex operator Y L (u L , z) acting on V L .) So Y L (u L , z) is a series in powers of z. But Y L (u L , z) is also single valued. So by (1.28), there exists u nL ∈ End F for n ∈ Z such that wt L u nL = wt u L − n − 1, wt R u nL = 0 and Y L (u L , z) = u nL z −n−1 . n∈Z

Similarly, Y(ρ(1 L ⊗u R ); z, z¯ ) depends only on z¯ and will also be denoted by Y R (u R , z¯ ). (There should also be no confusion with the vertex operator Y R (u R , z) acting on V R .) For u R ∈ V R , there exists u nR ∈ End F for n ∈ Z such that wt R u nR = wt u R − n − 1, wt L u nR = 0 and Y R (u R , z) = u nR z −n−1 . n∈Z

We also have the formal vertex operator maps, denoted using the same notations Y L and Y R , associated to Y L and Y R given by Y L (u L , x) = u nL x −n−1 , n∈Z

¯ = Y (u , x) R

R

u nR x¯ −n−1

n∈Z

for u L ∈ V L and u R ∈ V R . For u L ∈ V L , u R ∈ V R and w ∈ F, w ∈ F , w , Y L (u L , z 1 )Y R (u R , z¯ 2 )w = w , Y(ρ(u L ⊗ 1 R ); z 1 , z¯ 1 )Y(ρ(1 L ⊗ u R ); z 2 , z¯ 2 )w (1.55) is absolutely convergent when |z 1 | > |z 2 | > 0, and w , Y R (u R , z¯ 2 )Y L (u L , z 1 )w = w , Y(ρ(1 L ⊗ u R ); z 2 , z¯ 2 )Y(ρ(u L ⊗ 1 R ); z 1 , z¯ 1 )w (1.56) is absolutely convergent when |z 2 | > |z 1 | > 0. They are both analytic in z 1 and z¯ 2 . By the convergence property for full field algebras, both sides of (1.55) and (1.56) can be extended to the same smooth function on {(z 1 , z¯ 2 ) ∈ (C× )2 |z 1 = z 2 }. Since the complement of the union of the sets of convergence of (1.55) and (1.56) in {(z 1 , z¯ 2 ) ∈ (C× )2 |z 1 = z 2 } is of lower dimension, by the properties of analytic functions, it is clear that the extended smooth function is actually analytic on {(z 1 , z¯ 2 ) ∈ (C× )2 |z 1 = z 2 }. By associativity, we have w , Y L (u L , z 1 )Y R (u R , z¯ 2 )w = w , Y(ρ((Y L (u L , z 1 − z 2 )1 L ) ⊗ u R ); z 2 , z¯ 2 )w (1.57) when |z 1 | > |z 2 | > |z 1 − z 2 | > 0. The right-hand side of (1.57) has a well-defined limit as z 1 goes to z 2 . Therefore (1.55) and (1.56) can be further extended to a single analytic function on {(z 1 , z¯ 2 ) ∈ (C× )2 }. This absence of singularity further implies that the left-hand sides of (1.55) and (1.56) are absolutely convergent and are equal for all

360


z 1 , z¯ 2 ∈ C× . Let z 1 = z 2 = z in (1.55), (1.56) and (1.57). Use the discussion above and the creation property for the vertex operator map Y L ; we obtain Y(ρ(u L ⊗ u R ); z, z¯ ) = Y L (u L , z)Y R (u R , z¯ ) = Y R (u R , z¯ )Y L (u L , z),

(1.58)

or equivalently, in terms of formal vertex operator, Y f (ρ(u L ⊗ u R ); x, x) ¯ = Y L (u L , x)Y R (u R , x) ¯ = Y R (u R , x)Y ¯ L (u L , x)

(1.59)

L , u R ] = 0 for all u L ∈ V L and for all u L ∈ V L and u R ∈ V R . In particular, we have [u m n R R u ∈V . Since F is lower truncated, we have

Y f (ρ(u L ⊗ u R ); x, x)v ∈ (End F)((x))

(1.60)

for u L ∈ V L , u R ∈ V R and v ∈ F. The associativity (1.53) together with (1.59) and (1.60) implies the associativity for the vertex operator map Y f (ρ(·); x, x)·. Together with the identity property this associativity implies that F is a module for the vertex operator algebra V L ⊗ V R . Next we show that Y f (·; x, x) is an intertwining operator of type FFF . Since, for given u, v ∈ F, we have Yk (u)w = 0 for sufficient large k, the lower-truncation property of Y f (·, x, x) holds. For u ∈ F, we also have Y f ((D L + D R )u; x, x) = Y f ((D L + D R )u; x, x)| ¯ x=x ¯ ∂ ∂ + Y f (u; x, x) = ¯ ∂ x ∂ x¯ x=x ¯ d Y f (u; x, x), = dx proving the D-derivative property of Y f (·; x, x). Now, we prove the Jacobi identity for Y f (·; x, x). For any fixed r ∈ R, using the associativity for the full vertex operator map Y twice, we obtain w , Y L (u L , z 1 )Y R (u R , z¯ 2 )Y(u; r, r )w = w , Y(ρ(u L ⊗ 1 L ); z 1 , z¯ 1 )Y(ρ(1 L ⊗ u R ); z 2 , z¯ 2 )Y(u; r, r )w = w , Y(Y(ρ(u L ⊗ 1 L ); z 1 − r, z¯ 1 − r )Y(ρ(1 L ⊗ u R ); z 2 − r, z¯ 2 − r )u; r, r )w = w , Y(Y L (u L , z 1 − r )Y R (u R , z¯ 2 − r )u; r, r )w (1.61) when |z 1 |, |z 2 | > r > |z 1 − r |, |z 2 − r | > 0 for all u L ∈ V L , u R ∈ V R , u, w ∈ F and w ∈ F . By the commutativity for the full vertex operator map Y,

and

w , Y L (u L , z 1 )Y R (u R , z¯ 2 )Y(u; r, r )w

(1.62)

w , Y(u; r, r )Y L (u L , z 1 )Y R (u R , z¯ 2 )w

(1.63)

are absolutely convergent in the regions |z 1 |, |z 2 | > r > 0 and r > |z 1 |, |z 2 | > 0, respectively, to the correlation function w , m 4 (u L ⊗ 1 L , 1 R ⊗ u R , u, w; z 1 , z¯ 1 , z 2 , z¯ 2 , r, r, 0, 0).

(1.64)

Full Field Algebras

361

By our discussion above, we know that the right-hand side of (1.61), (1.62) and (1.63) are all analytic in z 1 and z¯ 2 and that we can take z 1 = z¯ 2 in the right-hand side of (1.61), (1.62) and (1.63). Thus after taking z 1 = z¯ 2 , the right-hand side of (1.61), (1.62) and (1.63) are analytic in z = z 1 = z¯ 2 . Since the right-hand side of (1.61), (1.62) and (1.63) are the expansions of (1.64) in the regions r > |z 1 − r |, |z 2 − r | > 0, |z 1 |, |z 2 | > r > 0 and r > |z 1 |, |z 2 | > 0, respectively, we see that we can also let z 1 = z¯ 2 in (1.64) and the result is also analytic in z = z 1 = z¯ 2 . Thus we have proved that w , Y(Y f (ρ(u L ⊗ u R ); z − r, z − r )u; r, r )w, w , Y f (ρ(u L ⊗ u R ); z, z)Y(u; r, r )w, w , Y(u; r, r )Y f (ρ(u L ⊗ u R ); z, z)w are absolutely convergent to w , m 4 (ρ(u L ⊗ 1 L ), ρ(1 R ⊗ u R ), u, w; z, z¯ , z¯ , z, r, r, 0, 0) which is in fact analytic in z. Using the Cauchy formula for contour integrals, we obtain the Cauchy-Jacobi identity Resz=∞ f (z)w , Y f (ρ(u L ⊗ u R ); z, z)Y(u; r, r )w − Resz=0 f (z)w , Y(u; r, r )Y f (ρ(u L ⊗ u R ); z, z)w = Resz=r f (z)w , Y(Y f (ρ(u L ⊗ u R ); z − r, z − r )u; r, r )w,

(1.65)

where f (z) is a rational function of z with the only possible poles at z = 0, r, ∞. Since w and w are arbitrary, this Cauchy-Jacobi identity gives us identities for the components of the vertex operator Y f (u; x, x). These identities are the component form of the Jacobi identity for Y f (u; x, x).

Definition 1.19. Let c L , c R ∈ C. A conformal full field algebra of central charges (c L , c R ) is a grading-restricted R × R-graded full field algebra (F, m, 1, D L , D R ) equipped with elements ω L and ω R called left conformal element and right conformal element, respectively, satisfying the following conditions: 1. The formal full vertex operators Y f (ω L ; x, x) ¯ and Y f (ω R ; x, x) ¯ are Laurent series in x and x, ¯ respectively, that is, Y f (ω L ; x, x) ¯ = L L (n)x −n−2 , n∈Z

¯ = Y f (ω ; x, x) R

L R (n)x¯ −n−2 .

n∈Z

2. The Virasoro relations: For m, n ∈ Z, cL 3 (m − m)δm+n,0 , 12 cR 3 [L R (m), L R (n)] = (m − n)L R (m + n) + (m − m)δm+n,0 , 12 [L L (m), L R (n)] = 0. [L L (m), L L (n)] = (m − n)L L (m + n) +

362


3. d L = L L (0), d R = L R (0), D L = L L (−1) and D R = L R (−1). We shall denote the conformal full field algebra by (F, m, 1, ω L , ω R ) or simply by F. We have: Proposition 1.20. Let (F, m, 1, ω L , ω R ) be a conformal full field algebra. Then the following commutator formula for Virasoro operators and formal full vertex operators hold: For u ∈ F, [Y f (ω L ; x1 , x¯1 ), Y f (u; x2 , x¯2 )] x1 − x0 = Resx0 x2−1 δ Y f (Y f (ω L ; x0 , x¯0 )u; x2 , x¯2 ), x2 [Y f (ω R ; x1 , x¯1 ), Y f (u; x2 , x¯2 )] x¯1 − x¯0 = Resx¯0 x¯2−1 δ Y f (Y f (ω R ; x0 , x¯0 )u; x2 , x¯2 ). x¯2

(1.66)

(1.67)

Proof. For any v ∈ F , u, v ∈ F, we consider v , m 3 (ω L , u, v; z 1 , z¯ 1 , z 2 , z¯ 2 , 0, 0).

(1.68)

Using the convergence property and the permutation property for conformal full field algebras, we know that it is equal to v , Y(ω L ; z 1 , z¯ 1 )Y(u, z 2 , z¯ 2 )v, v , Y(u; z 2 , z¯ 2 )Y(ω L , z 1 , z¯ 1 )v,

(1.69) (1.70)

in the regions |z 1 | > |z 2 | > 0, |z 2 | > |z 1 | > 0, respectively. By the definition of conformal full field algebra, we know that for any fixed z 2 = 0, (1.69) and (1.70) are analytic as functions of z 1 in the regions |z 1 | > |z 2 | > 0 and |z 2 | > |z 1 | > 0, respectively. So (1.68) is analytic as a function of z 1 in the regions |z 1 | > |z 2 | > 0 and |z 2 | > |z 1 | > 0. But we know that (1.68) is smooth as a function of z 1 in C \ {z 2 , 0}. Thus (1.68) must be analytic in C \ {z 2 , 0}. We know that (1.68) is equal to (1.69), (1.70) and v , Y(Y(ω L ; z 1 − z 2 , z¯ 1 − z¯ 2 )u; z 2 , z¯ 2 )v in the regions |z 1 | > |z 2 | > 0, |z 2 | > |z 1 | > 0 and z 2 | > |z 1 − z 2 | > 0, respectively. Since F is lower truncated, using the Virasoro relation, we see that Y f (ω L ; x, x)u ¯ and ¯ have only finitely many terms in negative powers of x. Also using the Y f (ω L ; x, x)v lower-truncation property of F and the Virasoro relation, we see that for any w ∈ F, v , Y f (ω L ; x, x)w ¯ has only finitely many terms in positive powers of x. Using these facts, we see that the singularities z 1 = z 2 , 0, ∞ of (1.68) are all poles. Using the Cauchy formula, we obtain the component form (1.66). Similarly, we can prove (1.67).

The following is clear from the definition and Theorem 1.18: Proposition 1.21. Let (V L , Y L , 1 L , ω L ) and (V L , Y L , 1 L , ω L ) be vertex operator algebras of central charges c L and c R , respectively. A full field algebra (F, m, ρ) over

Full Field Algebras

363

V L ⊗V R equipped with the left and right conformal elements ρ(ω L ⊗1 R ) and ρ(1 L ⊗ω R ) is a conformal full field algebra. In view of this proposition, we shall call the conformal full field algebra in the proposition above, that is, a full field algebra (F, m, ρ) over V L ⊗ V R equipped with the left and right conformal elements ρ(ω L ⊗ 1 R ) and ρ(1 L ⊗ ω R ), a conformal full field algebra over V L ⊗ V R and denote it by (F, m, ρ) or simply by F.

2. Intertwining Operator Algebras and Full Field Algebras Let V L and V R be vertex operator algebras. In the preceding section, we have shown that a conformal full field algebra (F, m, ρ) over V L ⊗V R is a module for the vertex

operator algebra V L ⊗ V R and Y f (·; x, x) is an intertwining operator of type FFF . This result suggests a method to construct conformal full field algebras from intertwining operator algebras, which are algebras of intertwining operators for vertex operator algebras and were introduced and studied in [H1, H2, H3, H4, H5] and [H7] by the first author. Let V be a vertex operator algebra and for a V -module W , let C1 (W ) be the subspace of V spanned by u −1 w for u ∈ V+ = n∈Z+ V(n) and w ∈ W . We consider the following conditions for a vertex operator algebra V : 1. Every C-graded generalized V -module is a direct sum of C-graded irreducible V -modules. 2. There are only finitely many inequivalent C-graded irreducible V -modules and they are all R-graded. 3. Every R-graded irreducible V -module W satisfies the C1 -cofiniteness condition, that is, dim W/C1 (W ) < ∞. In this section, we fix vertex operator algebras (V L , Y L , 1 L , ω L ) and (V R , Y R , satisfying these conditions. Let A L and A R be the sets of equivalent classes of irreducible modules for V L and for V R , respectively. Let {W L;a | a ∈ A L } be a complete set of representatives of the equivalence classes in A L and {W R;b | b ∈ A R } a complete set of representatives of the equivalence classes in A R . 1R , ω R )

Proposition 2.1. The vertex operator algebra V L ⊗ V R also satisfies the conditions above. Proof. Let W be a generalized V L ⊗ V R -module. Then W is a generalized V L -module. So there exist M a for a ∈ A L such that W is equivalent to the generalized vector spaces L L;a V -module a∈A L (W ⊗M a ). Since W is also a generalized V R -module, M a must be V R -modules. So they can be written as direct sums of irreducible V R -modules W R;b , b ∈ A R . So W is equivalent to a∈A L ,b∈A R Nab (W L;b ⊗ W R;a ), where Nab ∈ N for a ∈ A L , b ∈ A R . By Proposition 4.7.2 of [FHL], W L;a ⊗ W R;b are irreducible V L ⊗ V R -modules. So V L ⊗ V R satisfies Condition 1. The second condition follows from Theorem 4.7.4 of [FHL]. The C1 -cofiniteness follows immediately from the fact that C1 (W L;a ) ⊗ W R;b ⊕ W L;a ⊗ C1 (W R;b ) ⊂ C1 (W L;a ⊗ W R;b ).

364


This result immediately gives: Corollary 2.2. Let F be a module for the vertex operator algebra V L ⊗ V R . Then as a module for the vertex operator algebra V L ⊗ V R , F is isomorphic to a∈A L

h ab

(W L;a )(m ab ) ⊗ (W R;b )(m ab ) .

(2.1)

m ab =1

b∈A R

Let F be a module for the vertex operator algebra V L ⊗ V R and let γ be an isomorphism from (2.1) to F. Then there exist operators L L (0) and L R (0) on F given by L L (0)ρ(w L ⊗ w R ) = ρ((L L (0)w L ) ⊗ w R ), L R (0)ρ(w L ⊗ w R ) = ρ(w L ⊗ (L R (0)w R )) for w L ∈ (W L;a )(m ab ) and w R ∈ (W R;b )(m ab ) . Clearly L L (0) and L R (0) commute with each other.

Let Y be an intertwining operator of type FFF and γ an isomorphism from (2.1) to F. Let YY : (F ⊗ F) × C× → F (u ⊗ v, z) → YY (u; z, z¯ )v and YYf : F ⊗ F → F{x, x} ¯ ¯ u ⊗ v → YYf (u; x, x)v be linear maps given by YY (u; z, z¯ )v = z L

L (0)

z¯ L

R (0))

YYf (u; x, x)v ¯ = xL

L (0)

x¯ L

R (0)

Y(u, 1)z −L

L (0)

z¯ −L

R (0)

x¯ −L

R (0)

and Y(u, 1)x −L

L (0)

,

respectively, for u ∈ F. We call YY and YYf the splitting and formal splitting of Y, respectively.

Proposition 2.3. Let Y be an intertwining operator of type FFF , YY and YYf , the splitting and formal splitting of Y, respectively, and γ an isomorphism from (2.1) to F. Then for any a1 , a2 ∈ A L , b1 , b2 ∈ A R , 1 ≤ m a1 b1 ≤ h a1 b1 and 1 ≤ m a2 b2 ≤ h a2 b2 , there exist L;m a

b

;a3

R;m a

b

;b3

intertwining operators Ya1 a2 3 3 and Yb1 b2 3 3 for a3 ∈ A L , b3 ∈ A R and m a3 b3 = (m ) (m )

(W L;a3 ) a3 b3 (W R;b3 ) a3 b3 1, . . . , h a3 b3 of types and L;a1 (m a1 b1 ) L;a2 (m a2 b2 ) R;b1 (m a1 b1 ) R;b2 (m a2 b2 ) , (W

)

(W

)

(W

)

(W

)

Full Field Algebras

365

respectively, such that for u L ⊗ u R ∈ (W L;a1 )(m a1 b1 ) ⊗ (W R;b1 )(m a1 b1 ) and v L ⊗ v R ∈ (W L;a2 )(m a2 b2 ) ⊗ (W R;b2 )(m a2 b2 ) , we have YY (γ (u L ⊗ u R ); z, z¯ )γ (v L ⊗ v R ) =

h a3 b3

L;m a3 b3 ;a3

γ (Ya1 a2

a3 ∈A L b3 ∈A R m a3 b3 =1

R;m a3 b3 ;b3

(u L , z)v L ⊗ Yb1 b2

(u R , z¯ )v R ). (2.2)

Similarly, for the formal full vertex operator, we have ¯ (v L ⊗ v R ) YYf (γ (u L ⊗ u R ); x, x)γ =

a3

h a3 b3

∈A L

b3

∈A R

L;m a3 b3 ;a3

γ (Ya1 a2

m a3 b3 =1

R;m a3 b3 ;b3

(u L , x)v L ⊗ Yb1 b2

(u R , x)v ¯ R ). (2.3)

Proof. Since YY restricted to γ ((W L;a1 )(m a1 b1 ) ⊗ (W R;b1 )(m a1 b1 ) ) ⊗ γ ((W L;a2 )(m a2 b2 ) ⊗ (W R;b2 )(m a2 b2 ) ) is an intertwining operator of type

F , γ ((W L;a1 )(m a1 b1 ) ⊗ (W R;b1 )(m a1 b1 ) ) γ ((W L;a2 )(m a2 b2 ) ⊗ (W R;b2 )(m a2 b2 ) )

it was proved in [DMZ] that (2.2) is true when z = z¯ = r > 0. Then we have YY (γ (u L ⊗ u R ); z, z¯ )γ (v L ⊗ v R ) = zL =

L (0)

z¯ L

R (0)

Y(γ (u L ⊗ u R ), 1)z −L h a3 b3

γ ((z L

L (0)

L (0)

z¯ −L

L;m a3 b3 ;a3

Ya1 a2

R (0)

γ (v L ⊗ v R )

(u L , 1)¯z −L

L (0)

vL )

a3 ∈A L b3 ∈A R m a3 b3 =1

⊗(z L =

R (0)

R;m a3 b3 ;b3

Y b1 b2

(u R , 1)¯z −L

h a3 b3

R (0)

L;m a3 b3 ;a3

γ (Ya1 a2

a3 ∈A L b3 ∈A R m a3 b3 =1

v R )) R;m a3 b3 ;b3

(u L , z)v L ⊗ Yb1 b2

(u R , z¯ )v R ).

The proof of (2.3) is completely the same.

Corollary 2.4. Let (F, m, ρ) be a conformal full field algebra over V L ⊗ V R . Then as a module for the vertex operator algebra V L ⊗ V R , F is isomorphic to (2.1). Moreover, if γ is an isomorphism from (2.1) to F, then for any a1 , a2 ∈ A L , b1 , b2 ∈ A R , 1 ≤ m a1 b1 ≤ h a1 b1 and 1 ≤ m a2 b2 ≤ h a2 b2 , there exist intertwining operaL;m a3 b3 ;a3

tors Ya1 a2 of types

R;m a3 b3 ;b3

and Yb1 b2

for a3 ∈ A L , b3 ∈ A R and m a3 b3 = 1, . . . , h a3 b3 (m )

(W R;b3 ) a3 b3 and R;b1 (m a1 b1 ) R;b2 (m a2 b2 ) , respectively, such

(m ) (W L;a3 ) a3 b3 (m a b ) (m ) L;a L;a (W 1 ) 1 1 (W 2 ) a2 b2

(W

)

(W

)

366


that for u L ⊗ u R ∈ (W L;a1 )(m a1 b1 ) ⊗ (W R;b1 )(m a1 b1 ) and v L ⊗ v R ∈ (W L;a2 )(m a2 b2 ) ⊗ (W R;b2 )(m a2 b2 ) , the formulas (2.2) and (2.3) hold when YY and YYf are replaced by Y and Y f , respectively. Proof. The first conclusion follows immediately from Corollary 2.2. Now if we consider the intertwining operator Y f (·; x, x), then the second conclusion follows immediately from Proposition 2.3.

For either the map YYf in Proposition 2.3 or the formal full vertex operator map Y f for a conformal full field algebra over V L ⊗ V R , we can substitute z and ζ for the formal ¯ or Y f (·; x, x) ¯ (that is, substitute er log z and es log ζ for variables x and x¯ in YYf (·; x, x) x r and x¯ s , respectively, for r, s ∈ R) to obtain YY an (·; z, ζ ) (called analytic splitting of Y) or Yan (·; z, ζ ). Then by (2.3), we have: Corollary 2.5. For the analytic splitting YY an of Y in Proposition 2.3, we have L R L R YY an (γ (u ⊗ u ); z, ζ )γ (v ⊗ v )

=

h a3 b3

a3 ∈A L b3 ∈A R m a3 b3 =1

L;m a3 b3 ;a3

γ (Ya1 a2

R;m a3 b3 ;b3

(u L , z)v L ⊗ Yb1 b2

(u R , ζ )v R )

(2.4)

for u L ⊗ u R ∈ (W L;a1 )(m a1 b1 ) ⊗ (W R;b1 )(m a1 b1 ) and v L ⊗ v R ∈ (W L;a2 )(m a2 b2 ) ⊗ (W R;b2 )(m a2 b2 ) . The same is also true for the analytic full vertex operator map Yan for a conformal full field algebra over V L ⊗ V R . This corollary allows us to treat the left and right variables z and z¯ in YY (·; z, z¯ ) or Y(·; z, z¯ ) independently. In particular, we have the following strong versions of associativity and commutativity for conformal full field algebra over V L ⊗ V R : Proposition 2.6 (Associativity). Let (F, m, ρ) be a conformal full field algebra over V L ⊗ V R . Then for u, v, w ∈ F and w ∈ F , w , Yan (u; z 1 , ζ1 )Yan (v; z 2 , ζ2 )w = w , Yan (Yan (u; z 1 − z 2 , ζ1 − ζ2 )v; z 2 , ζ2 )w

(2.5)

when |z 1 | > |z 2 | > |z 1 − z 2 | > 0 and |ζ1 | > |ζ2 | > |ζ1 − ζ2 | > 0. Proof. Using (2.4) and the convergence result proved by the first author in [H7] for vertex operator algebras satisfying the conditions assumed for V L and V R in the beginning of this section, the left-hand side of (2.5) converges absolutely when |z 1 | > |z 2 | > 0 and |ζ1 | > |ζ1 | > 0, and the right-hand side of (2.5) converges absolutely when |z 2 | > |z 1 − z 2 | > 0 and |ζ2 | > |ζ1 − ζ2 | > 0. By the associativity (1.8), (2.5) is true when ζ1 = z¯ 1 and ζ2 = z¯ 2 for all u, v, w ∈ F and w ∈ F . In particular, replacing u by (L L (−1))k (L R (−1))l u, v by (L L (−1))m L R (−1))n v, for k, l, m, n ∈ N and using the L L (−1)- and L R (−1)-derivative properties, we obtain ∂k ∂l ∂m ∂n w , Yan (u; z 1 , ζ1 )Yan (v; z 2 , ζ2 )w m n k l ∂z 1 ∂ζ1 ∂z 2 ∂ζ2 ζ1 =¯z 1 ,ζ2 =¯z 2 k l m n ∂ ∂ ∂ ∂ = k l m n w , Yan (Yan (u; z 1 − z 2 , ζ1 − ζ2 )v; z 2 , ζ2 )w ∂z 1 ∂ζ1 ∂z 2 ∂ζ2 ζ1 =¯z 1 ,ζ2 =¯z 2 (2.6)

Full Field Algebras

367

for all k, l, m, n ∈ N, when |z 1 | > |z 2 | > 0 and |z 2 | > |z 1 − z 2 | > 0. We know that both sides of (2.5) give branches of some multivalued analytic functions in the region given by |z 1 | > |z 2 | > 0, |ζ1 | > |ζ1 | > 0, |z 2 | > |z 1 − z 2 | > 0 and |ζ2 | > |ζ1 − ζ2 | > 0. From (2.6), we know that the power series expansions of these branches are equal in the neighborhood of those points satisfying ζ1 = z¯ 1 , ζ2 = z¯ 2 . Thus (2.6) holds in the region |z 1 | > |z 2 | > 0, |ζ1 | > |ζ1 | > 0, |z 2 | > |z 1 − z 2 | > 0 and |ζ2 | > |ζ1 − ζ2 | > 0.

Proposition 2.7 (Commutativity). Let (F, m, ρ) be a conformal full field algebra over V L ⊗ V R . Then for u, v, w ∈ F and w ∈ F ,

and

w , Yan (u; z 1 , ζ1 )Yan (v; z 2 , ζ2 )w

(2.7)

w , Yan (v; z 2 , ζ2 )Yan (u; z 1 , ζ1 )w

(2.8)

are absolutely convergent when |z 1 | > |z 2 | > 0, |ζ1 | > |ζ2 | > 0 and when |z 2 | > |z 1 | > 0, |ζ2 | > |ζ1 | > 0, respectively, and can both be analytically extended to a same multivalued analytic function of (z 1 , z 2 ; ζ1 , ζ2 ) for (z 1 , z 2 ; ζ1 , ζ2 ) ∈ M 2 × M 2 , where M 2 = {(z 1 , z 2 ) ∈ (C× )2 | z 1 = z 2 }. Proof. The convergence and the existence of analytic extensions follow immediately from Corollary 2.5 and the convergence and the existence of analytic extensions of products of intertwining operators for the vertex operator algebras V L and V R . By Proposition 1.5, we know that these two multivalued analytic functions obtained by analytically extending (2.7) and (2.8) have equal values at points of the form (z 1 , z¯ 1 , z 2 , z¯ 2 ) for (z 1 , z 2 ) ∈ F2 (C). Using the L L (−1)- and L R (−1)-conjugation properties for full vertex operators, we see that these two analytic functions are actually the same, that is, they are analytic extensions of each other.

For (z 1 , . . . , z n ), (ζ1 , . . . , ζn ) ∈ Fn (C), we denote the corresponding elements of Fn (C) × Fn (C) by (z 1 , ζ1 , . . . , z n , ζn ) instead of (z 1 , . . . , z n , ζ1 , . . . , ζn ). We have the following analyticity of the correlation functions: Proposition 2.8. Let (F, m, ρ) be a conformal full field algebra over V L ⊗ V R . For any n ∈ Z+ and u 1 , . . . , u n , there exists a multivalued analytic function of (z 1 , ζ1 , . . . , z n , ζn ) ∈ Fn (C) × Fn (C) such that for (z 1 , . . . , z n ) ∈ Fn (C), the values m n (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n ) of the correlation function is a value of this multivalued analytic function above at the point (z 1 , z¯ 1 , . . . , z n , z¯ n ). Moreover, these multivalued analytic functions are determined uniquely by the products of analytic full vertex operators in their regions of convergence. Proof. The proof of this result is basically the same as the proof of the generalized rationality for intertwining operator algebras in [H5]. We have proved the above strong versions of associativity and commutativity for analytic full vertex operators. Using these strong versions of associativity and commutativity, we see that the multivalued analytic functions in various regions obtained from all kinds of products and iterates of analytic full vertex operators are analytic extensions of each other. Thus we have such a global multivalued analytic function. Clearly these multivalued analytic functions are determined uniquely by the products of analytic full vertex operators in their regions of convergence.

368


By the results above, we see that for a conformal full field algebra over V L ⊗ V R , the correlation function maps are determined uniquely by the products of analytic full vertex operators in their regions of convergence, and thus are determined uniquely by the full vertex operator map. In view of this fact, we shall use also (F, Y, ρ) to denote a conformal full field algebra over V L ⊗ V R . We shall use E(m)n (u 1 , . . . , u n ; z 1 , ζ1 , . . . , z n , ζn ) (2.9) to denote the analytic extension obtained in the proposition above together with the preferred values m n (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n ) at the special points of the form (z 1 , z¯ 1 , . . . , z n , z¯ n ). For u 1 , . . . , u n ∈ F and a path γ : [0, 1] → Fn (C) × Fn (C) t → (z 1 (t), ζ1 (t), . . . , z n (t), ζn (t)) starting from a point of the form (z 1 , z¯ 1 , . . . , z n , z¯ n ), we shall use E(m)n (u 1 , . . . , u n ; z 1 (t), ζ1 (t), . . . , z n (t), ζn (t)) to denote the value of (2.9) at the point γ (t) obtained by analytically extending the preferred value of (2.9) at the starting point γ (0) of γ along the path γ to the point γ (t). Corollary 2.9. Let (F, Y, ρ) be a conformal full field algebra over V L ⊗ V R . Let γ : [0, 1] → Fn (C) × Fn (C) t → (z 1 (t), ζ1 (t), . . . , z n (t), ζn (t)) be a path starting from a point of the form (z 1 , z¯ 1 , . . . , z n , z¯ n ). Then we have the following permutation property: For u 1 , . . . , u n ∈ F and σ ∈ Sn , E(m)n (u 1 , . . . , u n ; z 1 (t), ζ1 (t), . . . , z n (t), ζn (t)) = E(m)n (u σ (1) , . . . , u σ (n) ; z σ (1) (t), ζσ (1) (t), . . . , z σ (n) (t), ζσ (n) (t)).

(2.10)

Proof. This follows immediately from the permutation property for full field algebras and the uniqueness of analytic extensions.

Corollary 2.10. Let (F, Y, ρ) be a conformal full field algebra over V L ⊗ V R . Let r1 , r2 ∈ R satisfying r2 > r1 > 0. Then for u, v, w ∈ F and w ∈ F , w , Yan (v; r2 , r2 )Yan (u; r1 , r1 )w,

(2.11)

can be obtained by analytically extending the analytic function (which is a branch of a multivalued function) (2.12) w , Yan (u; z 1 , ζ1 )Yan (v; z 2 , ζ2 )w, defined near the point z 1 = ζ1 = r2 , z 2 = ζ2 = r1 , in the region |z 1 | > |z 2 | > 0 and |ζ1 | > |ζ2 | > 0, along the path given by [0, 1] → M 2 × M 2 t → ((z 1 (t), z 2 (t)), (ζ1 (t), ζ2 (t))),

Full Field Algebras

369

where r1 + r2 2 r1 + r2 z 2 (t) = 2 r1 + r2 ζ1 (t) = 2 r1 + r2 ζ2 (t) = 2 z 1 (t) =

r2 − r1 , 2 r2 − r1 − eiπ t , 2 r2 − r1 + e−iπ t , 2 r2 − r1 − e−iπ t , 2 + eiπ t

to the region |z 2 | > |z 1 | > 0 and |ζ2 | > |ζ1 | > 0 and then evaluated at z 1 = ζ1 = r1 and z 2 = ζ2 = r2 . Proof. By Proposition 2.7, we know that (2.11) can indeed be obtained from (2.12) by analytic extension. What we need to show now is that the analytic extension along the path given above gives precisely (2.11). Since Y(·; z, z¯ ) = Yan (·; z, ζ )|ζ =¯z and ζ1 (t) = z 1 (t) and ζ2 (t) = z 2 (t), we see that (2.11) is equal to w , m 3 (v, u, w; r2 , r2 , r1 , r1 , 0, 0) and that w , Yan (u; z 1 (t), ζ1 (t))Yan (v; z 2 (t), ζ2 (t))w

(2.13)

when |z 2 (t)| > |z 1 (t)| > 0 and w , Yan (v; z 2 (t), ζ2 (t))Yan (u; z 1 (t), ζ1 (t))w

(2.14)

when |z 1 (t)| > |z 2 (t)| > 0 are equal to w , E(m)3 (u, v, w; z 1 (t), ζ1 (t), z 2 (t), ζ2 (t), 0, 0) and w , E(m)3 (v, u, w; z 2 (t), ζ2 (t), z 1 (t), ζ1 (t), 0, 0), respectively. By the permutation property of full field algebras and Corollary 2.9, we see that (2.11), (2.13) and (2.14) are equal to w , m 3 (u, v, w; r1 , r1 , r2 , r2 , 0, 0), w , E(m)3 (u, v, w; z 1 (t), ζ1 (t), z 2 (t), ζ2 (t), 0, 0) and w , E(m)3 (u, v, w; z 1 (t), ζ1 (t), z 2 (t), ζ2 (t), 0, 0), respectively. From this fact, we see that indeed the analytic extension of (2.12) near the point z 1 = ζ1 = r2 , z 2 = ζ2 = r1 , along the path given above gives (2.11). This result can also be proved directly using the associativity (Proposition 2.6) and the skew-symmetry (1.41) (see [K] for details).

Theorem 2.11. A conformal full field algebra over V L ⊗ V R is equivalent to a module F for the vertex operator algebra V L ⊗ V R equipped with an intertwining operator Y

370


of type FFF and an injective linear map ρ : V L ⊗ V R → F, satisfying the following conditions: 1. The identity property: Y(ρ(1 L ⊗ 1 R ), x) = I F . 2. The creation property: For u ∈ F, lim x→0 Y(u, x)ρ(1 L ⊗ 1 R ) = u. 3. The associativity: The equality (2.5) holds when |z 1 | > |z 2 | > |z 1 − z 2 | > 0 and |ζ1 | > |ζ2 | > |ζ1 − ζ2 | > 0. 4. The single-valuedness property: e2πi(L

L (0)−L R (0))

= IF .

(2.15)

YY (v; eπi , e−πi )u.

(2.16)

5. The skew symmetry: YY (u; 1, 1)v = e L

L (−1)+L R (−1)

Proof. If (F, Y, ρ) is a conformal full field algebra over V L ⊗ V R , then the results in Sect. 1 shows that F is a V L ⊗ V R -module, Y f (·; x, x) is an intertwining operator of

type FFF and the five conditions are all satisfied. We now prove the converse.

Let F be a module for V L ⊗ V R , Y an intertwining operator of type FFF and ρ : V L ⊗ V R → F an injective linear map, satisfying the five conditions above. We take the splitting YY of Y to be the full vertex operator map. For simplicity, we shall denote YY simply by Y. We now want to construct the maps m n for n ∈ N and to verify the convergence property. Using (2.4) and the convergence property of the intertwining operators for the vertex operator algebras V L and V R , we know that for u 1 , . . . , u n ∈ F and w ∈ F , w , Y(u 1 ; z 1 , ζ1 ) · · · Y(u n ; z n , ζn )1

(2.17)

is absolutely convergent when |z 1 | > · · · > |z n | > 0, |ζ1 | > · · · > |ζn | > 0, and can be analytically extended to a (possibly multivalued) analytic function of z 1 , . . . , z n , ζ1 , . . . , ζn in the region given by z i = z j , z i = 0, ζi = ζ j , ζi = 0. We use E(m)n (w , u 1 , . . . , u n ; z 1 , ζ1 , . . . , z n , ζn )

(2.18)

to denote this function. This is a function of z 1 , ζ1 , . . . , z n , ζn , where (z 1 , . . . , z n ), (ζ1 , . . . , ζn ) ∈ Fn (C). So we can view this function as a function on Fn (C) × Fn (C). In general, this function is multivalued. Using analytic extension, a value of this function at a point P1 ∈ Fn (C) × Fn (C) and a path γ in Fn (C) × Fn (C) from P1 to P2 ∈ Fn (C) × Fn (C), determines uniquely a value of the function at the point P2 . Moreover, this value depends only on the homotopy class of the path γ . We shall call the value of the function (2.18) at P2 obtained this way the value of (2.18) at P2 obtained by analytically extending the value of (2.18) at P1 along γ . We choose the correlation function w , m n (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n )

(2.19)

as follows: For z 1 = n, . . . , z n = 1, we define (2.19) to be (2.17) with z 1 = ζ1 = n, . . . , z n = ζn = 1. For general (z 1 , . . . , z n ) ∈ Fn (C), we choose a path γ from (n, . . . , 1) to (z 1 , . . . , z n ). Then we have a path γ × γ from ((n, . . . , 1), (n, . . . , 1)) ∈ Fn (C) × Fn (C)

Full Field Algebras

371

to ((z 1 , . . . , z n ), (¯z 1 , . . . , z¯ n )) ∈ Fn (C) × Fn (C). We define (2.19) to be the value of (2.18) obtained by analytically extending the value of (2.18) at ((n, . . . , 1), (n, . . . , 1)) along γ × γ . The first thing we have to prove is that the correlation function we just defined is indeed independent of the path γ . To prove this fact, we need only prove that if γ is a loop in Fn (C) based at (n, . . . , 1), then the value of (2.18) at ((n, . . . , 1), (n, . . . , 1)) obtained by analytically extending the value (2.19) of (2.18) at ((n, . . . , 1), (n, . . . , 1)) along the loop γ × γ is equal to the original value (2.19) of (2.18) at ((n, . . . , 1), (n, . . . , 1)). In other words, we need only prove that the monodromy along the path γ ×γ is trivial. Note that the group of the homotopy classes of based loops in Fn (C), that is, the fundamental group of Fn (C), is the pure braid group of n strands (see [Bi]). This group is generated by the homotopy classes of the loops given by fixing z 1 , . . . , z j−1 , z j+1 , . . . , z n to be n, . . . , n−( j −2), n− j, . . . , 1, respectively, and moving z j starting from z j = n−( j −1) around z i = n −(i −1) once (but not around other points above) in the counterclockwise direction, for i = j, i, j = 1, . . . , n. Hence we need only prove that the monodromy along the path γ × γ is trivial for (the homotopy class of) such a loop γ . We now prove that the monodromy along the path γ × γ¯ is trivial for (the homotopy classes of) such a loop γ . Let r be a positive real number satisfying n−(i −1) > r > n−i. Note that r satisfies r > n − (i − 1) − r > 0. We know that w , Y(u 1 ; n, n) · · · Y(u i ; n − (i − 1), n − (i − 1))Y(u j ; r, r )· ·Y(u i+1 ; n − i, n − i) · · · Y(u i ; n − ( j − 2), n − ( j − 2)) · ·Y(u i+1 ; n − j, n − j) · · · Y(u n ; 1, 1)1

(2.20)

can be obtained by analytically extending the value w , Y(u 1 ; n, n) · · · Y(u n ; 1, 1)1 along a path from ((n, . . . , 1), (n, . . . , 1)) to ((n, . . . , n − ( j − 2), r, n − j, . . . , 1), (n, . . . , n − ( j − 2), r, n − j, . . . , 1)). (2.21) Such a path can always be taken to be of the form γ0 × γ¯0 , where γ0 is a path in Fn (C) from (n, . . . , 1) to (n, . . . , n − ( j − 2), r, n − j, . . . , 1). This path γ0 induces an isomorphism from the fundamental group of Fn (C) based at (n, . . . , 1) to that based at (n, . . . , n − ( j − 2), r, n − j, . . . , 1). It is clear that the monodromy along a loop based at (n, . . . , 1) is trivial if and only if the monodromy along the corresponding loop based at (n, . . . , n − ( j − 2), r, n − j, . . . , 1) is trivial. So we need only prove that the monodromy along a loop of the form γ × γ¯ is trivial, where γ is a loop based at (n, . . . , n − ( j − 2), r, n − j, . . . , 1) given by fixing z 1 , . . . , z j−1 , z j+1 , . . . , z n to be n, . . . , n − ( j − 2), n − j, . . . , 1, respectively, and moving z j starting from z j = r around z i = n − (i − 1) once (but not around other points above) in the counterclockwise direction. By the definition of γ0 , the value of (2.18) at the point (2.21) obtained by analytically extending the value (2.19) of (2.18) at the point ((n, . . . , 1), (n, . . . , 1)) along γ0 is (2.20). Since we also have r > n − (i − 1) − r > 0, by associativity, (2.20)

372


is equal to w , Y(u 1 ; n, n) · · · Y(u i−1 ; n − (i − 2), n − (i − 2)) · ·Y(Y(u i ; n − (i − 1) − r, n − (i − 1) − r )u j ; r, r ) · ·Y(u i+1 ; n − i, n − i) · · · Y(u i ; n − ( j − 2), n − ( j − 2)) · ·Y(u i+1 ; n − j, n − j) · · · Y(u n ; rn , rn )1. Now let γ : [0, 1] → Fn (C) be the loop given by t → (n, . . . , n−(i −2), r +e2πit (n−(i −1)−r ), n−i, . . . , n−( j −2), r, n − j, . . . , 1). Then the value of (2.18) at (2.21) obtained by analytically extending the original value (2.20) of (2.18) at the point (2.21) along γ is w , Y(u 1 ; n, n) · · · Y(u i−1 ; n − (i − 2), n − (i − 2)) · ·Y(Y(u i ; e2πi (n − (i − 1) − r ), e−2πi (n − (i − 1) − r ))u j ; r, r ) · ·Y(u i+1 ; n − i, n − i) · · · Y(u i ; n − ( j − 2), n − ( j − 2)) · ·Y(u i+1 ; n − j, n − j) · · · Y(u n ; rn , rn )1.

(2.22)

But by the L L (0)- and L R (0)-conjugation properties and the siungle-valuedness property, we have Y(u i ; e2πi (n − (i − 1) − r ), e−2πi (n − (i − 1) − r )) = e2πi(L

L (0)−L R (0))

Y(e−2πi(L

L (0)−L R (0))

u i ; n − (i − 1) − r, n − (i − 1) − r ) ·

−2πi(L L (0)−L R (0))

·e = Y(u i ; n − (i − 1) − r, n − (i − 1) − r ).

(2.23)

Using (2.23) and the associativity again, we see that (2.22) is equal to (2.20). Thus the analytic extension along this loop indeed gives trivial monodromy. Now the correlation functions and thus the maps m n for n ∈ N are defined. The only remaining thing to be shown is the convergence property. We need to show that for any (i) (i) k ∈ Z+ , l1 , . . . , lk ∈ Z+ , (z 1 , . . . , z k ) ∈ Fn (C), (z 1 , . . . , zli ) ∈ Fli (C), i = 1, . . . , k, ( j)

(i)

(0)

(0)

the series (1.13) converges absolutely to (1.2) when |z p | + |z q | < |z i − z j | for i = j, i, j = 1, . . . , k, p = 1, . . . , li and q = 1, . . . , l j . (0) We use induction on k. We first prove the special case in which k = 2 and z 2 = (0) z¯ 2 = 0. The case k = 1 is in fact a special case. By the definition of Y, (1.13) becomes (1) (1) (1) (1) (1) (1) (0) (0) Y(Pp1 ,q1 m l1 (u 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , zl1 , z¯l1 ); z 1 , z¯ 1 ) · p1 ,q1 , p2 ,q2

(2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 )).

(2.24)

We use induction on l1 . When l1 = 1, (2.24) becomes (1) (1) (1) (0) (0) Y(Pp1 ,q1 Y(u 1 ; z 1 , z¯ 1 )1; z 1 , z¯ 1 ) · p1 ,q1 , p2 ,q2

(2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 )).

(2.25)

Full Field Algebras

373

Using the construction of Y in terms of intertwining operators, the properties of intertwin(0) (1) (0) ing operators and noticing that our condition |z 1(1) | + |z (2) j | < |z 1 | implies |z 1 | < |z 1 | (1)

(0)

(2)

and |z 1 + z 1 | > |z j |, we know that (1) (1) (1) (0) (0) Y(Pp1 ,q1 Y(u 1 ; z 1 , z¯ 1 )1; z 1 , z¯ 1 ) · p1 ,q1

(2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 )) is absolutely convergent to (1)

(1)

(0)

(1)

(0)

(0)

(0)

Y(u 1 ; z 1 + z 1 , z¯ 1 + z¯ 1 )Y(1; z 1 , z¯ 1 ) · (2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 )) (1) (0) (1) (0) = Y(u (1) 1 ; z 1 + z 1 , z¯ 1 + z¯ 1 ) · (2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 )). Then by the construction of the correlation function maps and, in particular, by the fact that the correlation functions are values of multivalued analytic functions at certain particular points, we know that the right-hand side of (2.25) is absolutely convergent to (1)

(2)

(2)

(1)

(0)

(1)

(0)

(2)

(2)

(2)

(2)

m 1+l2 (u 1 , u 1 , . . . , u l2 ; z 1 + z 1 , z¯ 1 + z¯ 1 , z 1 , z¯ 1 , . . . , zl2 , z¯l2 ). Here we have used the fact that if a certain iterated sum of a series in powers of these complex variables is convergent to an analytic function in the region above, then the multisum must also be absolutely convergent. Now we assume that for l1 < l, the conclusion holds. We want to prove the conclu(1) (1) sion for the case l1 = l. We first assume that |z 1 |, . . . , |zl | are all different from each

other. Then in particular there exists t such that |z t(1) | > |z 1(1) |, . . . , |z t(1) |, . . . , |zl(1) |, where and also below we use ˆ to denote that the item under ˆ is missing. Then (2.24) in this case is equal to (1) (1) (1) Y(Pp1 ,q1 Y(u t ; z t , z¯ t ) · p1 ,q1 , p2 ,q2 r,s

(1) (1) (1) (1) (1) (1) (1) (1) ·Pr,s m l−1 (u (1) 1 , . . . , u t , . . . , u l ; z 1 , z¯ 1 , . . . , z t , z¯ t , . . . , z l , z¯ l ); z 1(0) , z¯ 1(0) ) · (2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 )).

(2.26)

Using the construction of Y in terms of intertwining operators and the properties of (1) (2) (0) intertwining operators, and noticing that our condition |z t | + |z j | < |z 1 | implies (1)

(0)

(1)

(0)

(2)

|z t | < |z 1 | and |z t + z 1 | > |z j |, we know that (1) (1) (1) Y(Pp1 ,q1 Y(u t ; z t , z¯ t )· p1 ,q1

(1) (1) (1) (1) (1) (1) (1) (1) (1) ·Pr,s m l−1 (u 1 , . . . , u t , . . . , u l ; z 1 , z¯ 1 , . . . , z t , z¯ t , . . . , zl , z¯l ); z 1(0) , z¯ 1(0) ) · (2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ))

(2.27)

374

Y.-Z. Huang, L. Kong (1)

is convergent absolutely and, when |z t (1)

(1)

Y(u t ; z t

(0)

(1)

+ z 1 , z¯ t

(0)

(0)

+ z 1 | > |z 1 |, it is absolutely convergent to

(0)

+ z¯ 1 ) ·

(1) (1) (1) (1) (1) (1) (1) (1) (1) ·Y(Pr,s m l−1 (u 1 , . . . , u t , . . . , u l ; z 1 , z¯ 1 , . . . , z t , z¯ t , . . . , zl , z¯l ); (0)

(0)

z 1 , z¯ 1 ) · (2) (2) (2) (2) (2) ·Pp2 ,q2 m l2 (u (2) 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , z l2 , z¯ l2 )).

(2.28)

By the induction assumption, (1) (1) (0) (1) (0) Y(u t ; z t + z 1 , z¯ t + z¯ 1 ) · r,s, p2 ,q2

(1) (1) (1) (1) (1) (1) (1) (1) ·Y(Pr,s m l−1 (u (1) 1 , . . . , u t , . . . , u l ; z 1 , z¯ 1 , . . . , z t , z¯ t , . . . , z l , z¯ l ); z 1(0) , z¯ 1(0) ) · (2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ))

(2.29)

is absolutely convergent to (1)

(1)

Y(u t ; z t

(0)

(1)

+ z 1 , z¯ t

(0)

+ z¯ 1 ) ·

(1) (1) (1) (2) (2) (1) (0) (1) (0) ·m l+l2 −1 (u 1 , . . . , u t , . . . , u l , u 1 , . . . , u l2 ; z 1 + z 1 , z¯ 1 + z¯ 1 , . . . ,

(1) (0) (1) (0) (1) (0) (1) (0) (2) (2) (2) (2) + z 1 , z¯ t + z¯ 1 , . . . , zl + z 1 , z¯l + z¯ 1 z 1 , z¯ 1 , . . . , zl2 , z¯l2 ) (1) (1) (2) (2) (1) (0) (1) (0) = m l+l2 (u 1 , . . . , u l , u 1 , . . . , u l2 ; z 1 + z 1 , z¯ 1 + z¯ 1 , . . . , (1) (0) (1) (0) (2) (2) (2) (2) zl + z 1 , z¯l + z¯ 1 z 1 , z¯ 1 , . . . , zl2 , z¯l2 ).

zt

(2.30)

We know that the right-hand side of (2.29) is a value of the multivalued analytic function (1) (2) (2) (1) (0) (1) (0) E(m)l+l2 (u (1) 1 , . . . , u l , u 1 , . . . , u l2 ; z 1 + z 1 , ζ1 + ζ1 , . . . , (1)

zl

(0)

(1)

+ z 1 , ζl

(0)

(0)

(0) (2)

(2)

(2)

(2)

+ ζ1 z 1 , ζ1 , . . . , zl2 , ζl2 ) (i)

(i)

at the points satisfying ζ1 = z¯ 1 , ζ p = z¯ p for p = 1, . . . , li , i = 1, 2. Since both the sum of (2.29) and the right-hand side of (2.30) are values of multivalued analytic functions in the same region and we have proved that their values are equal when (1) (0) (0) |z t + z 1 | > |z 1 |, (2.29) must be convergent absolutely to the right-hand side of (2.30) (1) even when |z t + z 1(0) | > |z 1(0) | is not satisfied. By the properties of analytic functions, we know that (2.26) as a sum in a different order is also convergent absolutely to the right-hand side of (2.30). (1) (1) (1) (1) Now we discuss the case that some of |z 1 |, . . . , |zl | are equal. Let N (z 1 , . . . , zl ) (1) (1) be the subset of {z 1 , . . . , zl } consisting of those elements whose absolute values are (1) (1) equal to the absolute values of some other elements of {z 1 , . . . , zl }. We use induction (1) (1) on the number of elements of N (z 1 , . . . , zl ). When the number is 0, this is the case discussed above. Now assume that when the number is equal to n, the conclusion holds. When this number is equal to n + 1, let be a complex number such that the number of (1) (1) (1) (2) (0) elements of N (z 1 + , . . . , zl + ) is n and |z p + |+|z q + | < |z 1 | for p = 1, . . . , l

Full Field Algebras

375

and q = 1, . . . , l2 . Note that we can always find such an and we can take such an with | | to be arbitrarily small. By induction assumption, p1 ,q1 , p2 ,q2

(1) (1) (1) Y(Pp1 ,q1 m l (u (1) ¯ . . . , zl(1) + , z¯l(1) + ¯ ); 1 , . . . , u l ; z 1 + , z¯ 1 + , (0)

(0)

z 1 , z¯ 1 ) · (2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 + , z¯ 1 + ¯ , . . . , zl2 + , z¯l2 + ¯ )

(2.31)

is absolutely convergent to (1)

(1)

(2)

(2)

(1)

(0)

(1)

(0)

m l+l2 (u 1 , . . . , u l , u 1 , . . . , u l2 ; z 1 + z 1 + , z¯ 1 + z¯ 1 + ¯ , . . . , + , z¯l(2) + ¯ ). zl(1) + z 1(0) + , z¯l(1) + z¯ 1(0) + ¯ z 1(2) + , z¯ 1(2) + ¯ , . . . , zl(2) 2 2 We have

e− L

L (1)−¯ L R (1)

u , Y(Pr1 ,s1 e L

L (−1)+¯ L R (−1)

(2.32)

·

r1 ,s1 ,r2 ,s2 p1 ,q1 , p2 ,q2 (1) (1) (1) (1) (1) (0) (0) ·Pp1 ,q1 m l (u (1) 1 , . . . , u l ; z 1 , z¯ 1 , . . . , z l , z¯ l ); z 1 , z¯ 1 ) · (2) (2) (2) (2) (2) ·Pr1 ,s1 e L (−1)+¯ L (−1) Pp2 ,q2 m l2 (u (2) 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , z l2 , z¯ l2 ) L R = e− L (1)−¯ L (1) u , L

R

r1 ,s1 ,r2 ,s2 (1) (1) (1) (1) (1) Y(Pr1 ,s1 m l (u (1) 1 , . . . , u l ; z 1 + , z¯ 1 + ¯ , . . . , z l + , z¯ l + ¯ );

z 1(0) , z¯ 1(0) ) · (2)

(2)

(2)

(2)

(2)

(2)

·Pr1 ,s1 m l2 (u 1 , . . . , u l2 ; z 1 + , z¯ 1 + ¯ , . . . , zl2 + , z¯l2 + ¯ ). (2.33) So the right-hand side and thus also the left-hand side of (2.33) is absolutely convergent to e− L

L (1)−¯ L R (1)

(1)

(1)

(2)

(2)

(1)

(0)

u , m l+l2 (u 1 , . . . , u l , u 1 , . . . , u l2 ; z 1 + z 1 + , (1)

(0)

(1)

z¯ 1 + z¯ 1 + ¯ , . . . , zl (2)

(0)

(1)

+ z 1 + , z¯l

(2)

(0)

+ z¯ 1 + ¯ ,

(2)

(2)

¯ . . . , zl2 + , z¯l2 + ¯ ) z 1 + , z¯ 1 + , = u , e− L

L (−1)−¯ L R (−1)

(1)

(1)

(1)

(2)

(2)

(1)

(0)

m l+l2 (u 1 , . . . , u l , u 1 , . . . , u l2 ; z 1 + z 1 + ,

(0)

(1)

z¯ 1 + z¯ 1 + ¯ , . . . , zl (2)

(0)

(1)

+ z 1 + , z¯l

(2)

(0)

+ z¯ 1 + ¯ ,

(2)

(2)

¯ . . . , zl2 + , z¯l2 + ¯ ) z 1 + , z¯ 1 + , (1) (2) (2) (1) (0) (1) (0) = u , m l+l2 (u (1) 1 , . . . , u l , u 1 , . . . , u l2 ; z 1 + z 1 , z¯ 1 + z¯ 1 , . . . , (1)

zl

(0)

(1)

+ z 1 , z¯l

(0) (2)

(2)

(2)

(2)

+ z¯ 1 z 1 , z¯ 1 , . . . , zl2 , z¯l2 ).

(2.34)

376


Since the left-hand side of (2.34) is a value of a multivalued analytic function, any of its expansion must be absolutely convergent. In particular, the left-hand side of (2.33) as an expansion of the left-hand side of (2.34) is absolutely convergent. Thus we can exchange the order of the two summation signs such that the resulting series is still absolutely convergent to the left-hand side of (2.34) and thus to the right-hand side of (2.34). But for u ∈ F ,

(1)

p1 ,q1 , p2 ,q2

(1)

(1)

(1)

(1)

(1)

u , Y(Pp1 ,q1 m l (u 1 , . . . , u l ; z 1 , z¯ 1 , . . . , zl , z¯l ); (0)

(0)

z 1 , z¯ 1 ) · (2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ) L R L R = e− L (1)−¯ L (1) u , e L (−1)+¯ L (−1) · p1 ,q1 , p2 ,q2 (1)

(1)

(1)

(1)

(1)

(1)

(0)

(0)

·Y(Pp1 ,q1 m l (u 1 , . . . , u l ; z 1 , z¯ 1 , . . . , zl , z¯l ); z 1 , z¯ 1 ) · (2)

(2)

(2)

(2)

(2)

(2)

·Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ) L R L R = e− L (1)−¯ L (1) u , Y(e L (−1)+¯ L (−1) · p1 ,q1 , p2 ,q2 (1)

(1)

(1)

(1)

(1)

(1)

(0)

(0)

·Pp1 ,q1 m l (u 1 , . . . , u l ; z 1 , z¯ 1 , . . . , zl , z¯l ); z 1 , z¯ 1 ) · (2)

(2)

(2)

(2)

(2)

(2)

·e L (−1)+¯ L (−1) Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ) L R L R = e− L (1)−¯ L (1) u , Y(Pr1 ,s1 e L (−1)+¯ L (−1) · L

R

p1 ,q1 , p2 ,q2 r1 ,s1 ,r2 ,s2 (1)

(1)

(1)

(1)

(1)

(1)

(0)

(0)

·Pp1 ,q1 m l (u 1 , . . . , u l ; z 1 , z¯ 1 , . . . , zl , z¯l ); z 1 , z¯ 1 ) · ·Pr1 ,s1 e L

L (−1)+¯ L R (−1)

(2)

(2)

(2)

(2)

(2)

(2)

Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ). (2.35)

We have shown that the right-hand side of (2.35) is absolutely convergent to the righthand side of (2.34). Thus the left-hand side of (2.35) is also absolutely convergent to the right-hand side of (2.34). So we have proved the convergence property when the number (1) (1) of elements of N (z 1 , . . . , zl ) is n + 1. Thus the convergence property is proved when (1) (1) some of |z 1 |, . . . , |zl | are equal. By the induction principle, we have proved the convergence property in this special case. We now assume that when k < K , (1.13) converges absolutely to (1.2) when (0) (0) (i) (i) z p = z q for p, q = 1, . . . , K , z p = z q for p, q = 1, . . . , li and i = 1, . . . , K 1 ≤ ( j) (i) (0) (0) p, q ≤ li , and |z p | + |z q | < |z i − z j | for p = 1, . . . , li , q = 1, . . . , l j , i, j = 1, . . . , K , i = j. Now we consider the case k = K . We first consider the case that (i) (0) (0) z p ∈ R+ ∪ {0} for p = 1, . . . , li and i = 0, . . . , K and z 1 > · · · > z K . By the

Full Field Algebras

377

definition of the correlation function maps, we know that (1.13) in this case is equal to (1) (1) (1) (1) (1) (0) (0) Y(Pp1 ,q1 m l1 (u (1) 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , z l1 , z¯ l1 ); z 1 , z¯ 1 ) · p1 ,q1 ,..., p K ,q K r,s

(2)

(2)

(2)

(2)

(2)

(2)

·Pr,s m K −1 (Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ), . . . , ) (K ) (K ) ) (K ) ; z 1 , z¯ 1 , . . . , zl(K , z¯l K ); Pp K ,q K m l K (u 1(K ) , . . . , u l(K K K (0) z 2(0) , z¯ 2(0) , . . . , z (0) K , z¯ K ).

(2.36)

Using the induction assumption, we have (1) (1) (1) (1) (1) (1) (0) (0) Y(Pp1 ,q1 m l1 (u 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , zl1 , z¯l1 ); z 1 , z¯ 1 ) · r,s, p1 ,q1 p2 ,q2 ,..., p K ,q K

(2)

(2)

(2)

(2)

(2)

(2)

·Pr,s m K −1 (Pp2 ,q2 m l2 (u 1 , . . . , u l2 ; z 1 , z¯ 1 , . . . , zl2 , z¯l2 ), . . . , (K )

(K )

(K )

(K )

(K )

(K )

Pp K ,q K m l K (u 1 , . . . , u l K ; z 1 , z¯ 1 , . . . , zl K , z¯l K ); =

r,s, p1 ,q1

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(1)

z 2 , z¯ 2 , . . . , z K , z¯ K ) (1)

(1)

(0)

(0)

Y(Pp1 ,q1 m l1 (u 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , zl1 , z¯l1 ); z 1 , z¯ 1 ) · (2)

(2)

(K )

(K )

(2)

(0)

·Pr,s m l2 +···+l K (u 1 , . . . , u l2 , . . . , u 1 , . . . , u l K ; z 1 + z 2 , (2)

(0)

(2)

(0)

(2)

(0)

z¯ 1 + z¯ 2 , . . . , zl2 + z 2 , z¯l2 + z¯ 2 , . . . , (K ) (0) (K ) (0) (K ) (0) z 1(K ) + z (0) K , z¯ 1 + z¯ K , . . . , z l K + z K , z¯ l K + z¯ K ). (1)

(i)

(0)

(2.37)

(0)

Since z p + z q < z 1 − z i for p = 1, . . . , l1 , q = 1, . . . , li and i = 2, . . . , K , (1) (i) (0) (0) we have z p + (z q + z i ) < z 1 − 0. Thus by the special case we proved above, the right-hand side of (2.37) is absolutely convergent to (1)

(1)

(K )

(K )

(1)

(0)

m l1 +···+l K (u 1 , . . . , u l1 , . . . , u 1 , . . . , u l K ; z 1 + z 1 , (1)

(0)

(1)

(0)

(1)

(0)

(K )

z¯ 1 + z¯ 1 , . . . , zl1 + z 1 , z¯l1 + z¯ 1 , . . . , z 1

(0)

+ zK ,

(K ) (0) (K ) (0) z¯ 1(K ) + z¯ (0) K , . . . , z l K + z K , z¯ l K + z¯ K ).

(2.38)

Note that (2.38) is a value of the F-valued multivalued analytic function (1) (K ) (K ) (1) (0) E(m)l1 +···+l K (u (1) 1 , . . . , u l1 , . . . , u 1 , . . . , u l K ; z 1 + z 1 , (1)

(0)

(1)

(0)

(1)

(0)

(0)

(K )

(K )

ζ1 + ζ1 , . . . , zl1 + z 1 , ζl1 + ζ1 , . . . , z 1 (K )

ζ1 ( j)

( j)

( j)

(0)

(K )

(0)

+ ζ K , . . . , zl K + z K , ζl K + ζ K ) ( j)

(0)

+ zK , (2.39)

at the point z i = z i , ζi = z¯ i . Thus its expansions, no matter in which ways, must be convergent absolutely. In particular, (2.36) as one expansion of (2.38) must be convergent absolutely to (2.38), proving the convergence in this special case of the case k = K. We know that for a series in powers of several variables, if it is absolutely convergent when these variables are equal to some real numbers, then it is also convergent when the

378


variables are equal to complex numbers whose absolute values are equal to these real numbers. Using this property, we see that (1) (1) (1) (1) (1) (1) (0) (0) Y(Pp1 ,q1 E(m)l1 (u 1 , . . . , u l1 ; z 1 , ζ1 , . . . , zl1 , ζl1 ); z 1 , ζ1 ) · p1 ,q1 ,..., p K ,q K r,s

(2) (2) (2) (2) (2) ·Pr,s E(m) K −1 (Pp2 ,q2 E(m)l2 (u (2) 1 , . . . , u l2 ; z 1 , ζ1 , . . . , z l2 , ζl2 ), . . . , (K )

(K )

(K )

(K )

Pp K ,q K E(m)l K (u 1 , . . . , u l K ; z 1 , ζ1

(K )

(K )

, . . . , zl K , ζl K );

(0)

(0)

(0)

(0)

z 2 , ζ2 , . . . , z K , ζ K ) (0)

(0)

(2.40) (0)

(0)

is convergent absolutely to a branch of (2.39) when z p = z q , ζ p = ζq for p, q = ( j) (i) (i) (i) (i) (i) 1, . . . , K , z p = z q , ζ p = ζq for p, q = 1, . . . , li , i = 1, . . . , K |z p | + |z q | < ( j) (0) (0) (i) (0) (0) ||z i | − |z j ||, |ζ p | + |ζq | < ||ζi | − |ζ j ||, for p = 1, . . . , li , q = 1, . . . , l j ,

(0) (0) i, j = 1, . . . , K , i = j, and |z 1(0) | > · · · > |z (0) K |, |ζ1 | > · · · > |ζ K |. Using the permutation property for the correlation functions, we obtain that (2.40) is convergent (0) (0) (0) (0) absolutely to a branch of (2.39) when |z σ (1) | > · · · > |z σ (K ) | and |ζσ (1) | > · · · > |ζσ (K ) | for some σ ∈ S K . (0) (0) (0) (0) (0) (0) Now for fixed z 1 , . . . , z K , ζ1 , . . . , ζ K satisfying |z σ (1) | > · · · > |z σ (K ) | and (0)

(0)

|ζσ (1) | > · · · > |ζσ (K ) | for some σ ∈ S K , any branch of (2.39) can be expanded as a (i)

(i)

series in powers of z p and ζ p , p = 1, . . . , li , i = 1, . . . , K in the region (1)

(K )

(1)

(K )

(i) (i) (i) {(z 1 , . . . , zl K , ζ1 , . . . , ζl K ) | z (i) p = z q , ζ p = ζq for p, q = 1, . . . , li , i = 1, . . . , K , ( j)

(0)

|z (i) p | + |z q | < |z i

(0)

( j)

(0)

− z j |, |ζ p(i) | + |ζq | < |ζi

(0)

− ζ j |,

for p = 1, . . . , li , q = 1, . . . , l j , i, j = 1, . . . , K , i = j}.

(2.41)

But in the region (1)

(K )

(1)

(K )

(i) (i) (i) {(z 1 , . . . , zl K , ζ1 , . . . , ζl K ) | z (i) p = z q , ζ p = ζq for p, q = 1, . . . , li , i = 1, . . . , K , ( j)

( j)

(0) (0) (0) (0) (i) |z (i) p | + |z q | < ||z i | − |z j ||, |ζ p | + |ζq | < ||ζi | − |ζ j ||,

for p = 1, . . . , li , q = 1, . . . , l j , i, j = 1, . . . , K , i = j}, (2.42) we have proved that one branch of (2.39) can be expanded as the series (2.40), which can (i) (i) be further expanded as a series in powers of z p and ζ p , p = 1, . . . , li , i = 1, . . . , K in this region. Since the region (2.42) is contained in the region (2.41) and the coefficients of the expansion can be determined completely using the values of the branch in the region (2.42), we see that the restriction to the region (2.42) of the expansion in the region (2.41) is the same as the expansion in the region (2.41). Thus, the series (2.40) is convergent absolutely to a branch of (2.39) in the region (2.41). (0) (0) (i) (i) In the region (2.41), when ζ p = z p ∈ R for p = 1, . . . , K , ζ p = z p ∈ R for p = 1, . . . , li , i = 1, . . . , K , we have proved that (1.13) in this case is convergent absolutely to the right-hand side of (2.37). Thus in the region (2.41), (1.13) with k = K is convergent absolutely to the right-hand side of (2.37), the value of a branch of (2.39).

Full Field Algebras

379 (0)

(0)

Finally we consider the case that some of |z 1 |, . . . , |z K | are equal. Recall the subset (0) (0) (0) (0) N (z 1 , . . . , z K ) of {z 1 , . . . , z K } consisting of those elements whose absolute values are equal to the absolute values of some other elements of {z 1(0) , . . . , z (0) K }. We use induc(0) (0) tion on the number of elements of N (z 1 , . . . , z K ). When the number is 0, this is the case discussed above. Now assume that when the number is equal to n, the conclusion holds. When this number is equal to n + 1, let be a complex number such that the (0) (0) number of elements of N (z 1 + , . . . , z K + ) is n and that the other conditions are still satisfied. Note that we can always find such an and we can take such an with | | to be arbitrary small. By induction assumption, p1 ,q1 ,..., p K ,q K

(1) (1) (1) (1) (1) m K (Pp1 ,q1 m l1 (u (1) 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , z l1 , z¯ l1 ), . . . , (K )

(K )

(K )

(K )

(K )

(K )

Pp K ,q K m l K (u 1 , . . . , u l K ; z 1 , z¯ 1 , . . . , zl K , z¯l K ); (0)

(0)

(0)

(0)

z 1 + , z¯ 1 + ¯ , . . . , z K + , z¯ K + ¯ ) is absolutely convergent to (1)

(1)

(K )

(K )

(1)

(0)

m l1 +···+l K (u 1 , . . . , u l1 , . . . , u 1 , . . . , u l K ; z 1 + z 1 + , + z 1(0) + , z¯l(1) + z¯ 1(0) + ¯ , . . . , z 1(K ) + z (0) z¯ 1(1) + z¯ 1(0) + ¯ , . . . , zl(1) K + , 1 1 (K )

z¯ 1

(0)

(K )

(0)

(K )

(0)

+ z¯ K + ¯ , . . . , zl K + z K + , z¯l K + z¯ K + ¯ ).

Thus for u ∈ F, p1 ,q1 ,..., p K ,q K

(1) (1) (1) (1) (1) u , m K (Pp1 ,q1 m l1 (u (1) 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , z l1 , z¯ l1 ), . . . , (K )

(K )

(K )

(K )

(K )

(K )


=

(0)

(0)

(0)

z 1 , z¯ 1 , . . . , z K , z¯ K )

e L

L (1)+¯ L R (1)

u , e− L

L (−1)−¯ L R (−1)

·

p1 ,q1 ,..., p K ,q K (1) (1) (1) (1) (1) ·m K (Pp1 ,q1 m l1 (u (1) 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , z l1 , z¯ l1 ), . . . , (K )

(K )

(K )

(K )

(K )

(K )


=

(0)

(0)

(0)

z 1 , z¯ 1 , . . . , z K , z¯ K )

e L

L (1)+¯ L R (1)

u,

p1 ,q1 ,..., p K ,q K (1)

(1)

(1)

(1)

(1)

(1)

m K (Pp1 ,q1 m l1 (u 1 , . . . , u l1 ; z 1 , z¯ 1 , . . . , zl1 , z¯l1 ), . . . , (K )

(K )

(K )

(K )

(K )

(0)

(0)

(K )


(0)

¯ , . . . , z K + , , z¯ K + ¯ ) z 1 + , z¯ 1 + ,

380


is absolutely convergent to e L

L (1)+¯ L R (1)

(1)

(1)

(1)

(K )

(K )

(1)

(0)

u , m l1 +···+l K (u 1 , . . . , u l1 , . . . , u 1 , . . . , u l K ; z 1 + z 1 + ,

(0)

(1)

(0)

(1)

(0)

(K )

z¯ 1 + z¯ 1 + ¯ , . . . , zl1 + z 1 + , z¯l1 + z¯ 1 + ¯ , . . . , z 1

(0)

+ z K + ,

(K ) (0) (K ) (0) z¯ 1(K ) + z¯ (0) K + ¯ , . . . , z l K + z K + , z¯ l K + z¯ K + ¯ ) (1)

(1)

(K )

(K )

(1)

(0)

= u , m l1 +···+l K (u 1 , . . . , u l1 , . . . , u 1 , . . . , u l K ; z 1 + z 1 , (1)

(0)

(1)

(0)

(1)

(0)

(K )

z¯ 1 + z¯ 1 , . . . , zl1 + z 1 , z¯l1 + z¯ 1 , . . . , z 1 (K )

z¯ 1

(0)

(K )

(0)

(K )

(0)

+ zK , (0)

+ z¯ K , . . . , zl K + z K , z¯l K + z¯ K ).

Since u is arbitrary, we have proved that (1.13) with k = K is convergent absolutely to (0) (0) the right-hand side of (2.37) in the case that the number of elements of N (z 1 , . . . , z K ) is n + 1. Thus we have proved this conclusion in the case that some of |z 1(0) |, . . . , |z (0) K | are equal. By the principle of induction, the convergence property is proved.

Remark 2.12. Although the definition of full field algebra in Definition 1.1 is very general, it is not easy to verify all the axioms directly. Theorem 2.11 gives an equivalent definition of conformal full field algebra over V L ⊗ V R and the axioms in this definition are much easier to verify than those in Definitions 1.1, 1.7 and 1.16. In our construction of full field algebras in the next section, we shall use this definition to verify the structure we construct is indeed a full field algebra.

3. A Construction of Full Field Algebras with Nondegenerate Invariant Bilinear Forms Let V be a simple vertex operator algebra and C2 (V ) the subspace of V spanned by u −2 v for u, v ∈ V . In this section, we assume that V satisfies the following conditions: 1. V(n) = 0 for n < 0, V(0) = C1 and W(0) = 0 for any irreducible V -module W which is not equivalent to V . 2. Every N-gradable weak V -module is completely reducible. 3. V is C2 -cofinite, that is, dim V /C2 (V ) < ∞. (Note that by results of Li [L] and Abe, Buhl and Dong [ABD], Conditions 2 and 3 can be replaced by a single condition that every weak V -module is completely reducible.) Since V satisfies the conditions above, all the results in [H9] can be used. We shall use all the notations, conventions and choices used in that paper. In particular, we use the following notations and choices: A is the (finite) set of equivalence classes of irreducible V -modules; e is the equivalence class containing V ; : A → A is the map induced from the functor given by taking contragredient modules; for a ∈ A, W a is a representative of a; (·, ·) is the nondegenerate bilinear form on V normalized by (1, 1) = 1; for a3

a1 , a2 , a3 ∈ A, Vaa13a2 are the spaces of intertwining operators of type W W a1 W a2 ; σ12 and

Full Field Algebras

381

σ23 are actions of (12) and (23) on V and they generate an action of S3 on V= Vaa13a2 ; a1 ,a2 ,a3 ∈A

a ;( p)

for any bases Ya13a2 ;i , i = 1, . . . , Naa13a2 , p = 1, 2, 3, 4, 5, 6, . . . and a1 , a2 , a3 ∈ A, of Vaa13a2 , a ;(1)

a ;(2)

a ;(3)

a ;(4)

F(Ya14a5 ;i ⊗ Ya25a3 ; j ; Ya64a3 ;l ⊗ Ya16a2 ;k ) ∈ C a , Ya e are matrix elements of the fusing isomorphism; for a ∈ A, Yea;1 ae;1 and Yaa ;1 are e a a bases of Vea , Vae and Vaa chosen in [H9]; for a ∈ A, there exists h a ∈ Q such that W a = n∈h a +N W(n) . a

For a1 , a2 , a3 ∈ A, we now want to introduce a pairing between Vaa13a2 and Va 3a . 1 2

For a ∈ A, wa ∈ W a and wa ∈ (W a ) , we shall use w˜ a and w˜ a to denote e−L(1) wa and e−L(1) wa , respectively. Then we have e L(1) w˜ a , e L(1) w˜ a = wa , wa . We have: Lemma 3.1. For a ∈ A, wa ∈ W a and wa ∈ (W a ) , Resz=0 z −1 Yae a;1 (z L(0) eπi(L(0)−h a ) w˜ a , z)z L(0) w˜ a = wa , wa 1.

Proof. Since V(0) = C1 and Yae a;1 = σ23 (Yaa e;1 ), Resz=0 z −1 Yae a;1 (z L(0) eπi(L(0)−h a ) w˜ a , z)z L(0) w˜ a = (1, Yae a;1 (1 L(0) eπi(L(0)−h a ) w˜ a , 1)w˜ a )1

= (1, σ23 (Yaa e;1 )(eπi(L(0)−h a ) w˜ a i , 1)w˜ a )1

= eπi h a Yaa e;1 (e L(1) e−πi L(0) eπi(L(0)−h a ) w˜ a , 1)1, w˜ a 1

= Yaa e;1 (e L(1) w˜ a , 1)1, w˜ a 1 = e L(−1) e L(1) w˜ a , w˜ a 1 = e L(1) w˜ a , e L(1) w˜ a 1 = wa , wa 1.

For a single-valued branch f 1 (z 1 , z 2 ) of a multivalued analytic function in a region A, we use E( f 1 (z 1 , z 2 )) to denote the multivalued analytic extension together with the preferred branch f 1 (z 1 , z 2 ). Let w1 = w1 (z 1 , z 2 ) and w2 = w2 (z 1 , z 2 ) be a change of variables and f 2 (z 1 , z 2 ) a branch of E( f 1 (z 1 , z 2 )) in a region B containing w1 (z 1 , z 2 ) = 0 and w2 (z 1 , z 2 ) = 0 such that A∩ B = ∅ and f 1 (z 1 , z 2 ) = f 2 (z 1 , z 2 ) for (z 1 , z 2 ) ∈ A∩ B. Then we use Resw1 =0 | w2 E( f 1 (z 1 , z 2 ))

382


to denote the coefficient of w1−1 in the expansion of f 2 (z 1 , z 2 ) as a series in powers of w1 whose coefficients are analytic functions of w2 . By definition, we have Resw1 =0 | C1 w2 +C2 E( f 1 (z 1 , z 2 )) = Resw1 =0 | w2 E( f 1 (z 1 , z 2 ))

(3.1)

for any C1 ∈ C× , C2 ∈ C independent of z 1 and z 2 . We have: Proposition 3.2. For a1 , a2 , a3 ∈ A, wa1 ∈ W a1 , wa2 ∈ W a2 , wa1 ∈ (W a1 ) , wa 2 ∈ a

(W a1 ) , Y1 ∈ Vaa13a2 and Y2 ∈ Va 3a , there exists a constant Y1 , Y2 Vaa3a ∈ C such that 1 2

1 2

Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )

−1

E(e

L(1)

Y2 ((1 − z 1 − z 2 )

L(0)

w˜ a 1 , z 1 )w˜ a 2 ,

e L(1) Y1 ((1 − z 1 − z 2 ) L(0) w˜ a1 , z 2 )w˜ a2 ) = wa 1 , wa1 wa 2 , wa2 Y1 , Y2 Vaa3a .

(3.2)

1 2

a ;(2)

a ;(1)

a

Explicitly, for any bases {Ya13a2 ;i | i = 1, . . . , Naa13a2 } and {Ya 3a ;i | i = 1, . . . , Na 3a } of 1 2

a

1 2

Vaa13a2 and Va 3a , respectively, and for m, n, k, l ∈ Z+ , i = 1, . . . , Naa13a2 and 1 2

a

j = 1, . . . , Na 3a , we have 1 2

a ;(1)

a ;(2)

a ;(2)

a ;(1)

a2 ⊗ Yae a1 ;1 ) Ya13a2 ; j , Ya 3a ;i Vaa3a = F(σ23 (Ya 3a ;i ) ⊗ Ya13a2 ; j ; Yea 2 ;1 1 2

1 2

1

1 2

=

a3 ;(2) a2 F(σ23 (Yaa13a;(1) ) ⊗ Y ; Y a1 a2 ;i ea2 ;1 2; j

⊗ Yae a ;1 ). 1 1

(3.3)

a ;(2)

a ;(1)

a

Proof. We prove (3.2) in the case Y1 = Ya13a2 ; j and Y2 = Ya 3a ;i for i = 1, . . . , Na 3a 1 2

1 2

and j = 1, . . . , Naa13a2 , respectively, or equivalently, we prove (3.3). The general case follows immediately from the bilinearity in Y1 and Y2 of the right-hand side of (3.2). For a1 , a2 ∈ A, a1 , a2 = e, let {Yaa12a2 ;i | i = 1, . . . , Naa12a2 } and {Yaa2a ;i | i = 1 1

1, . . . , Naa2a } be an arbitrary basis of Vaa12a2 and Vaa2a , respectively. 1 1

1 1

For wa1 ∈ W a1 , wa2 ∈ W a2 , wa1 ∈ (W a1 ) , wa 2 ∈ (W a1 ) , we have a ;(2)

Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 E(e L(1) Ya 3a ;i ((1 − z 1 − z 2 ) L(0) w˜ a 1 , z 1 )w˜ a 2 , 1 2

e

Yaa13a;(1) ((1 − z 1 2; j −1 z1 − z2 ) ·

L(1)

= Res1−z 1 −z 2 =0 | z 2 (1 −

− z 2 ) L(0) w˜ a1 , z 2 )w˜ a2 )

a ;(2)

·E(Ya 3a ;i (e(1−z 1 )L(1) (1 − z 1 )−2L(0) · 1 2

·(1 − z 1 − z 2 ) L(0) w˜ a 1 , (1 − z 1 )−1 )e L(−1) e L(1) w˜ a 2 , a ;(1)

Ya13a2 ; j ((1 − z 1 − z 2 ) L(0) w˜ a1 , z 2 )w˜ a2 ) = Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 a ;(2)

E(e−L(1) e L(1) w˜ a 2 , σ23 (Ya 3a ;i )((1 − z 1 − z 2 ) L(0) e 1 2

a ;(1)

·Ya13a2 ; j ((1 − z 1 − z 2 ) L(0) w˜ a1 , z 2 )w˜ a2 )

πi(L(0)−h a ) 1

w˜ a 1 , 1 − z 1 ) ·

Full Field Algebras

N

=

383

a2

N

a4 a a

a4 a2 1 1

a4 ∈A p=1 q=1

a ;(2)

a ;(1)

F(σ23 (Ya 3a ;i ) ⊗ Ya13a2 ; j ; Yaa42a2 ; p ⊗ Yaa4a

1 1 ;q

1 2

)·

·Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 · ·E(e L(−1) e L(1) w˜ a 2 , Yaa42a2 ; p (Yaa4a

1 1 ;q

·e

πi(L(0)−h a ) 1

((1 − z 1 − z 2 ) L(0) ·

w˜ a 1 , 1 − z 1 − z 2 ) ·

·(1 − z 1 − z 2 ) L(0) w˜ a1 , z 2 )w˜ a2 ) a ;(2)

a2 ; Yea ⊗ Yae a = F(σ23 (Ya 3a ;i ) ⊗ Yaa13a;(1) 2; j 2; p

1 1 ;1

1 2

)·

·Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 · a2 (Yae a ·E(e L(−1) e L(1) w˜ a 2 , Yea 2 ;1

1 1 ;1

·e

πi(L(0)−h a ) 1

((1 − z 1 − z 2 ) L(0) ·

w˜ a 1 , 1 − z 1 − z 2 ) ·

·(1 − z 1 − z 2 ) L(0) w˜ a1 , z 2 )w˜ a2 ) a ;(2)

a ;(1)

a2 ⊗ Yae a1 ;1 ) · = F(σ23 (Ya 3a ;i ) ⊗ Ya13a2 ; j ; Yea 2; p 1

1 2

·e

L(−1) L(1)

e

a2 w˜ a 2 , Yea (wa 1 , wa1 1, z 2 )w˜ a2 2 ;1 a ;(2)

a ;(1)

a2 = wa 1 , wa1 F(σ23 (Ya 3a ;i ) ⊗ Ya13a2 ; j ; Yea ⊗ Yae a1 ;1 )e L(1) w˜ a 2 , e L(1) w˜ a2 2; p 1

1 2

=

a ;(2) a2 ; Yea wa 1 , wa1 wa 2 , wa2 F(σ23 (Ya 3a ;i ) ⊗ Yaa13a;(1) 2; j 2; p 1 2

⊗ Yae a

1 1 ;1

),

a4 = 0 for a4 = e. This proves (3.2) and also the first where we have used the fact that W(0) equality in (3.3). The second equality in (3.3) can be proved similarly or can be simply obtained using the first equality in (3.3) and symmetry.

Clearly, Y1 , Y2 Vaa3a is bilinear in Y1 and Y2 . Thus we have a pairing ·, ·Vaa3a : a

1 2

1 2

Vaa13a2 ⊗ Va 3a → C. 1 2 We need the following lemma: a

Lemma 3.3. For a1 , a2 , a3 , a4 , a5 ∈ A, Y1 ∈ Va 3a , Y2 ∈ Vaa43a5 , wa 1 ∈ (W a1 ) , wa 2 ∈ 1 2

(W a2 ) , wa4 ∈ W a1 , wa4 ∈ W a5 , if a1 = a4 or a2 = a5 , then

Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 E(e L(1) Y1 ((1 − z 1 − z 2 ) L(0) w˜ a 1 , z 1 )w˜ a 2 , e L(1) Y2 ((1 − z 1 − z 2 ) L(0) w˜ a4 , z 2 )w˜ a5 ) = 0. Proof. Using the L(1)- and L(−1)-conjugation formulas for intertwining operators, the definition of σ23 and the associativity of intertwining operators, we know that there exist W

a2

a V -module W and intertwining operators Y3 and Y4 of types WWW a5 and W a1 W a4 , respectively, such that Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 E(e L(1) Y1 ((1 − z 1 − z 2 ) L(0) w˜ a 1 , z 1 )w˜ a 2 , e L(1) Y2 ((1 − z 1 − z 2 ) L(0) w˜ a4 , z 2 )w˜ a5 )

384


= Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 · ·E(e L(−1) e L(1) w˜ a 2 , σ23 (Y1 )((1 − z 1 − z 2 ) L(0) w˜ a 1 , 1 − z 1 ) · ·Y2 ((1 − z 1 − z 2 ) L(0) w˜ a4 , z 2 )w˜ a5 ) = Res1−z 1 −z 2 =0 | z 2 (1 − z 1 − z 2 )−1 · ·E(e L(−1) e L(1) w˜ a 2 , Y3 (Y4 ((1 − z 1 − z 2 ) L(0) w˜ a 1 , 1 − z 1 − z 2 ) · ·(1 − z 1 − z 2 ) L(0) w˜ a4 , z 2 )w˜ a5 ).

(3.4)

If a1 = a4 , W a4 is not equivalent to W a1 . Thus Vae a = 0. So it is possible to find such 1 4 a V -module W which does not contain a summand equivalent to V . By the assumption on V , we have W(0) = 0. So the right-hand side of (3.4) is 0, proving the lemma in this case. If a1 = a4 , Vae a is one-dimensional. We can choose W to contain one and 1 1 a2

(that is, type only one copy of V . If a2 = a5 , any intertwining operator of type ea 5 W a2

V W a5 ) must be 0. So Y3 (1, z 2 ) = 0. Since W(0) = C1, there exists λ ∈ C such that the right-hand side of (3.4) is equal to λe L(−1) e L(1) w˜ a 2 , Y3 (1, z 2 )w˜ a2 = 0, proving the lemma in the case a2 = a5 .

As in [H9], we now choose a canonical basis of Vaa13a2 for a1 , a2 , a3 ∈ A when one a of a1 , a2 , a3 is e: For a ∈ A, we choose Yea;1 to be the vertex operator YW a defining a a the module structure on W and we choose Yae;1 to be the intertwining operator defined using the action of σ12 , or equivalently the skew-symmetry in this case, a a (wa , x)u = σ12 (Yea;1 )(wa , x)u Yae;1 a (u, −x)wa = e x L(−1) Yea;1

= e x L(−1) YW a (u, −x)wa for u ∈ V and wa ∈ W a . Since V as a V -module is isomorphic to V , we have e = e. From [FHL], we know that there is a nondegenerate invariant bilinear form (·, ·) on V e e such that (1, 1) = 1. We choose Yaa ;1 = Yaa ;1 to be the intertwining operator defined using the action of σ23 by

e a Yaa ;1 = σ23 (Yae;1 ),

that is, e πi h a a Yae;1 (e x L(1) (e−πi x −2 ) L(0) wa , x −1 )u, wa (u, Yaa ;1 (wa , x)wa ) = e

for u ∈ V , wa ∈ W a and wa ∈ W a . Since the actions of σ12 and σ23 generate the action of S3 on V, we have e Yae a;1 = σ12 (Yaa ;1 )

for any a ∈ A.

Full Field Algebras

385 a

Theorem 3.4. The pairing ·, ·Vaa3a : Vaa13a2 ⊗ Va 3a → C is nondegenerate. In particular,

a Na 3a 1 2

1 2

=

1 2

Naa13a2 .

Proof. For a1 , a2 , a3 ∈ A such that one of a1 , a2 , a3 is e, we have a canonical basis Yaa13a2 ;1 given above. For a1 , a2 , a3 = e, let Yaa13a2 ;i , i = 1, . . . , Naa13a2 , be an arbitrary basis of Vaa13a2 . For a1 , a2 , a3 ∈ A, let a

a ;(1)

Ya13a2 ; j = σ123 (Ya 1a ; j ), 2 3

a ;(2) Ya 3a ;i 1 2

=

σ23 (Yaa2a ;i ). 1 3

Then the first equality of (3.3) gives a

a ;(2)

a ;(1)

σ23 (Yaa2a ;i ), σ123 (Ya 1a ; j )Vaa3a = Ya 3a ;i , Ya13a2 ; j Vaa3a 1 3

2 3

1 2

1 2 a1

1 2

=

F(Yaa2a ;i 1 3

⊗ σ123 (Ya

2 a3 ; j

a2 ); Yea ⊗ Yae a1 ;1 ). 2 ;1 1

(3.5) In [H9], the first author proved the following formula ((4.9) in [H9]): N

a2 a a

1 3

F(Yaa22e;1 ⊗ Yae a3 ;1 ; Yaa2a 3

k=1

·F(Yaa2a

⊗ σ123 (Ya

1 3 ;k a2 a2 e;1

= δi j F(Y

1

a1

2 a3 ;i

a

3 ;k

⊗ Ya 1a ; j ) · 2 3

a2 ); Yea ⊗ Yae a1 ;1 ) 2 ;1 1

a2 ⊗ Yae a2 ;1 ; Yea ⊗ Yae2 a ;1 ). 2 ;1 2

(3.6)

2

In the same paper [H9], the first author also proved that F(Yaa22e;1 ⊗ Yae a

2 2 ;1

a2 ; Yea ⊗ Yae 2 ;1

2 a2 ;1

) = 0.

Thus from (3.5) and (3.6), we see that the matrix a

(αi j ) = (σ23 (Yaa2a ;i ), σ123 (Ya 1a ; j )Vaa3a ) 1 3

2 3

a

is left invertible. Note that when a1 , a2 , a3 = e, Yaa2a Vaa2a 1 3

and Ya 1a ; j in (3.7) are arbitrary

1 3 ;i

a Va 1a , 2 3

(3.7)

1 2

2 3

and respectively. bases of We now show that (3.7) is also right invertible. By definition, the bilinear form ·, ·Vaa3a is symmetric in the sense that 1 2

Y1 , Y2 Vaa3a = Y2 , Y1 1 2

a3 a1 a2

V

386


for a1 , a2 , a3 ∈ A. So a

σ23 (Yaa2a ;i ), σ123 (Ya 1a ; j )Vaa3a 1 3

=

2 3

1 2

a σ123 (Ya 1a ; j ), σ23 (Yaa2a ;i ) a3 2 3 1 3 V

a1 a2

a1

= σ23 (σ13 (Ya

2 a3 ; j

)), σ123 (σ13 (Yaa2a ;i ))

a3 a1 a2

V

1 3

.

(3.8)

a

Note that for a1 , a2 , a3 ∈ A, σ13 (Ya 1a ;i ) is a basis of Vaa13a2 such that when one of the 3 2 elements a1 , a2 , a3 ∈ A is e, these basis elements are equal to the special ones we chose above. Thus by the result we obtained above, the matrix a

(βi j ) = (σ23 (σ13 (Ya 1a ;i )), σ123 (σ13 (Yaa2a

1 3; j

2 3

))

a3 a1 a2

V

)

must be left invertible. So the transpose of (βi j ), that is, the matrix a

(γkl ) = (σ23 (σ13 (Ya 1a ;l )), σ123 (σ13 (Yaa2a

1 3 ;k

2 3

))

a3 a1 a2

V

),

is right invertible. By (3.8), we see that (3.7) is also right invertible. Now we have shown that the matrix (3.7) is in fact invertible. This is equivalent to a

the nondegeneracy of the bilinear form. It also implies Na 3a = Naa13a2 .

1 2

For a ∈ A, let a a e ⊗ Yae a;1 ; Yea;1 ⊗ Yaa Fa = F(Yae;1 ;1 ) = 0.

Then by (3.12) in [H9], Fa = Fa for a ∈ A. a , Ya , Ye e a a Lemma 3.5. If for a ∈ A, Yea;1 ae;1 aa ;1 are the canonical bases of Vea , Vae , Vaa ,

a , V a , V e with respectively, chosen in [H9] and above, then their dual bases in Vea a e aa a a a , ·, ·V a , ·, ·V e , respectively, are equal to Y , Y respect to the pairing ·, ·Vea ea ;1 a e;1 , ae aa

Yae a;1 Fa .

Proof. This result follows immediately from the definition of the canonical bases in [H9].

We have: Proposition 3.6. For a1 , a2 , a3 ∈ A, let {Yaa13a2 ;i | i = 1, . . . , Naa13a2 } be bases of Vaa13a2 ;a

a

and let {Ya a3 ;i | i = 1, . . . , Na 3a } be the dual bases of {Yaa13a2 ;i | i = 1, . . . , Naa13a2 } 1 2 1 2 a , Ya , Ye with respect to the pairing ·, ·Vaa3a . Assume that for a ∈ A, Yea;1 ae;1 aa ;1 are 1 2

Full Field Algebras

387

a , V a , V e , respectively, we have chosen. Then for a , a , the canonical bases of Vea 1 2 ae aa a3 , a4 ∈ A, N

a4

N

a5

a1 a5 a2 a3

a5 ∈A p=1 q=1

F(Yaa14a5 ; p ⊗ Yaa25a3 ;q ; Yaa64a3 ;m ⊗ Yaa16a2 ;k ) ·

;a

;a

;a

4 7 5 ·F(Ya a4 ; p ⊗ Ya;a a ;q ; Ya a ;n ⊗ Ya a ;l ) 1 5

2 3

= δa6 a7 δmn δkl .

7 3

1 2

Proof. For a1 , a2 , a3 , a4 ∈ A, wai ∈ W ai and wa i ∈ (W ai ) satisfying wa i , wai = 1 for i = 1, 2, 3, using (3.2) and Lemma 3.3, we have Res1−z 1 −z 3 =0 | z 3 Res1−z 2 −z 4 =0 | z 4 (1 − z 1 − z 3 )−1 (1 − z 2 − z 4 )−1 ;a

E(e L(1) Ya a4 ;k ((1 − z 1 − z 3 ) L(0) w˜ a 1 , z 1 ) · 1 5

;a ·Ya a5 ;l ((1 − z 2 2 3

− z 4 ) L(0) w˜ a 2 ;q , z 2 )w˜ a 3 ,

a

e L(1) Ya14a6 ;m ((1 − z 1 − z 3 ) L(0) w˜ a1 , z 3 ) · ·Yaa26a3 ;n ((1 − z 2 − z 4 ) L(0) w˜ a2 ;q , z 4 )w˜ a3 ) = δa5 a6 δkm Res1−z 2 −z 4 =0 | z 4 (1 − z 2 − z 4 )−1 ;a

E(Ya a5 ;l ((1 − z 2 − z 4 ) L(0) w˜ a 2 , z 2 )w˜ a 3 , 2 3

Yaa25a3 ;n ((1 − z 2 − z 4 ) L(0) w˜ a2 , z 4 )w˜ a3 ) = δa5 a6 δkm δln .

(3.9)

On the other hand, by the associativity of intertwining operators and Lemma 3.3, we have Res1−z 1 −z 3 =0 | z 3 Res1−z 2 −z 4 =0 | z 4 (1 − z 1 − z 3 )−1 (1 − z 2 − z 4 )−1 ;a

E(e L(1) Ya a4 ;k ((1 − z 1 − z 3 ) L(0) w˜ a 1 , z 1 ) · 1 5

;a ·Ya a5 ;l ((1 − z 2 2 3

− z 4 ) L(0) w˜ a 2 , z 2 )w˜ a 3 ,

a

e L(1) Ya14a6 ;m ((1 − z 1 − z 3 ) L(0) w˜ a1 , z 3 ) · ·Yaa26a3 ;n ((1 − z 2 − z 4 ) L(0) w˜ a2 , z 4 )w˜ a3 ) ;a ;a ;a ;a = F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) · 1 5

2 3

a7 ,a8 ∈A i, j,s,t ·F(Yaa14a6 ;m ⊗ Yaa26a3 ;n ; Yaa84a3 ;s

7 3

1 2

⊗ Yaa18a2 ;t ) ·

·Res1−z 1 −z 3 =0 | z 3 Res1−z 2 −z 4 =0 | z 4 (1 − z 1 − z 3 )−1 (1 − z 2 − z 4 )−1 ;a

;a

E(e L(1) Ya a4 ;i (Ya a7 ; j ((1 − z 1 − z 3 ) L(0) w˜ a 1 , z 1 − z 2 ) · 7 3

1 2

·(1 − z 2 − z 4 ) L(0) w˜ a 2 , z 2 )w˜ a 3 ,

388


e L(1) Yaa84a3 ;s (Yaa18a2 ;t ((1 − z 1 − z 3 ) L(0) w˜ a1 , z 3 − z 4 ) · ·(1 − z 2 − z 4 ) L(0) w˜ a2 , z 4 )w˜ a3 ) ;a ;a ;a ;a = F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) · a7 ,a8 ∈A i, j,s,t ·F(Yaa14a6 ;m

1 5

2 3

7 3

1 2

⊗ Yaa26a3 ;n ; Yaa84a3 ;s ⊗ Yaa18a2 ;t ) ·

·Res1−z 2 −z 4 =0 | z 4 Res1−z 1 −z 3 =0 | z 3 (1 − z 1 − z 3 )−1 (1 − z 2 − z 4 )−1 L(0) 1 − z1 − z3 L(1) ;a4 L(0) ;a7 E e Ya a ;i (1 − z 2 − z 4 ) Ya a ; j · 7 3 1 2 1 − z2 − z4 − z z 1 2 ·w˜ a 1 , w˜ a 2 , z 2 w˜ a 3 , 1 − z2 − z4 L(0) 1 − z − z 1 3 · e L(1) Yaa84a3 ;s (1 − z 2 − z 4 ) L(0) Yaa18a2 ;t 1 − z2 − z4 z3 − z4 ·w˜ a1 , w˜ a2 , z 4 w˜ a3 . (3.10) 1 − z2 − z4 We now change the variables z 1 and z 3 to z5 =

z1 − z2 1 − z2 − z4

z6 =

z3 − z4 . 1 − z2 − z4

and

Then 1 − z1 − z3 , 1 − z2 − z4 z 3 = (1 − z 2 − z 4 )z 6 + z 4 .

1 − z5 − z6 =

For any branch f (z 1 , z 2 , z 3 , z 4 ) of a multivalued analytic function of z 1 , z 2 , z 3 and z 4 on a suitable region A such that it is equal to the restriction to A ∩ B of a branch of the same analytic function on a region B containing the point 1 − z 1 − z 3 = 0, by definition, we have Res1−z 1 −z 3 =0 | z 3 E( f (z 1 , z 2 , z 3 , z 4 )) = Res1−z 5 −z 6 =0 | (1−z 2 −z 4 )z 6 +z 4 E( f (z 1 , z 2 , z 3 , z 4 ))

1 − z1 − z3 . 1 − z5 − z6

(3.11)

By (3.1), we have 1 − z1 − z3 1 − z5 − z6 1 − z1 − z3 = Res1−z 5 −z 6 =0 | z 6 E( f (z 1 , z 2 , z 3 , z 4 )) . 1 − z5 − z6

Res1−z 5 −z 6 =0 | (1−z 2 −z 4 )z 6 +z 4 E( f (z 1 , z 2 , z 3 , z 4 ))

(3.12)

Full Field Algebras

389

From (3.11) and (3.12), we obtain Res1−z 1 −z 3 =0 | z 3 E( f (z 1 , z 2 , z 3 , z 4 )) = Res1−z 5 −z 6 =0 | z 6 E( f (z 1 , z 2 , z 3 , z 4 ))

1 − z1 − z3 . 1 − z5 − z6

(3.13)

Using (3.13), the definition of the pairings ·, ·Vaa3a , Lemma 3.3, and the fact that for a1 , a2 , a3 ∈ A,

;a {Ya a3 ;i 1 2

| i = 1, . . . ,

a Na 3a } 1 2

1 2

are the dual bases of {Yaa13a2 ;i | i =

1, . . . , Naa13a2 } with respect to the pairing ·, ·Vaa3a , we see that the right-hand side of 1 2 (3.10) is equal to

;a

;a

;a

;a

F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) ·

a7 ,a8 ∈A i, j,s,t ·F(Yaa14a6 ;m

1 5

2 3

7 3

1 2

⊗ Yaa26a3 ;n ; Yaa84a3 ;s ⊗ Yaa18a2 ;t ) ·

·Res1−z 2 −z 4 =0 | z 4 Res1−z 5 −z 6 =0 | z 6 (1 − z 5 − z 6 )−1 (1 − z 2 − z 4 )−1 ;a

E(e L(1) Ya a4 ;i ((1 − z 2 − z 4 ) L(0) · 7 3

;a ·Ya a7 ; j ((1 − z 5 − z 6 ) L(0) w˜ a 1 , z 5 )w˜ a 2 , z 2 )w˜ a 3 , 1 2 e L(1) Yaa84a3 ;s ((1 − z 2 − z 4 ) L(0) · ·Yaa18a2 ;t ((1 − z 5 − z 6 ) L(0) w˜ a1 , z 6 )w˜ a2 , z 4 )w˜ a3 ) ;a ;a ;a ;a = F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) · 1 5 2 3 7 3 1 2 a7 ,a8 ∈A i, j,s,t a4 a6 a4 a7 ·F(Ya1 a6 ;m ⊗ Ya2 a3 ;n ; Ya7 a3 ;s ⊗ Ya1 a2 ;t ) · ·Res1−z 5 −z 6 =0 | z 6 Res1−z 2 −z 4 =0 | z 4 (1 − z 5 − z 6 )−1 (1 − z 2 ;a

E(e L(1) Ya a4 ;i ((1 − z 2 − z 4 ) L(0) · 7 3

;a ·Ya a7 ; j ((1 − z 5 − z 6 ) L(0) w˜ a 1 , z 5 )w˜ a 2 , z 2 )w˜ a 3 , 1 2 L(1) a4 e Ya8 a3 ;s ((1 − z 2 − z 4 ) L(0) · ·Yaa17a2 ;t ((1 − z 5 − z 6 ) L(0) w˜ a1 , z 6 )w˜ a2 , z 4 )w˜ a3 ) ;a ;a ;a ;a = F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) · 1 5 2 3 7 3 1 2 a7 ∈A i, j,s,t ·F(Yaa14a6 ;m ⊗ Yaa26a3 ;n ; Yaa74a3 ;s ⊗ Yaa17a2 ;t ) · ·Res1−z 5 −z 6 =0 | z 6 (1 − z 5 − z 6 )−1 δis ;a

E(e L(1) Ya a7 ; j ((1 − z 5 − z 6 ) L(0) w˜ a 1 , z 5 )w˜ a 2 , 1 2

e L(1) Yaa18a2 ;t ((1 − z 5 − z 6 ) L(0) w˜ a1 , z 6 )w˜ a2 )

− z 4 )−1

390


=

;a

a7 ∈A i, j,s,t ·F(Yaa14a6 ;m

=

;a

;a

;a

F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) · 1 5

2 3

7 3

1 2

⊗ Yaa26a3 ;n ; Yaa84a3 ;s ⊗ Yaa18a2 ;t )δis δ jt ;a

;a

;a

;a

F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) ·

a7 ∈A i, j ·F(Yaa14a6 ;m

1 5

2 3

7 3

1 2

⊗ Yaa26a3 ;n ; Yaa74a3 ;i ⊗ Yaa17a2 ; j ).

(3.14)

From (3.9)–(3.14), we see that the right inverse of the matrix with entries ;a

;a

;a

;a

F(Ya a4 ;k ⊗ Ya a5 ;l ; Ya a4 ;i ⊗ Ya a7 ; j ) 1 5

2 3

7 3

1 2

is the transpose of the matrix with entries F(Yaa14a6 ;m ⊗ Yaa26a3 ;n ; Yaa74a3 ;i ⊗ Yaa17a2 ; j ). Since for square matrices, right inverses are also left inverses, the proposition is proved.

√ i arg Fa √ For a ∈ A, we use Fa to denote the square root |Fa |e 2 of Fa . For a1 , a2 , a3 ∈ A, consider the modified pairings Fa 3 ·, ·V a3 . a1 a2 Fa1 Fa2 These pairings give a nondegenerate bilinear form (·, ·)V on V. For any σ ∈ S3 , {σ (Yaa13a2 ;i ) | i = . . . , Naa13a2 } is a basis of σ (Vaa13a2 ). We have: Proposition 3.7. The nondegenerate bilinear form (·, ·)V is invariant with respect to the a

action of S3 on V, that is, for a1 , a2 , a3 ∈ A, σ ∈ S3 , Y1 ∈ Vaa13a2 and Y2 ∈ Va 3a , 1 2

(σ (Y1 ), σ (Y2 ))V = (Y1 , Y2 )V . Equivalently, for a1 , a2 , a3 ∈ A, ⎫ ⎧ Fa1 Fa2 Faσ −1 (3) ⎬ ⎨ ;a3 a3 i = . . . , N σ (Y ) a1 a2 a1 a2 ;i ⎭ ⎩ Fa Faσ −1 (2) Fa3 σ −1 (1) is the dual basis of {σ (Yaa13a2 ;i ) | i = . . . , Naa13a2 }. Proof. The equivalence of the first conclusion and the second conclusion is clear. We first prove the result for σ = σ12 . In this case, we need to show that a

{σ12 (Ya 3a ; j ) | i = . . . , Naa13a2 } is the dual basis of {σ12 (Yaa13a2 ;i ) | i = . . . , Naa13a2 }. 1 2

For i, j = 1, . . . , Naa13a2 , by (3.3), we have ;a

;a

a1 σ12 (Ya a3 ;i ), σ12 (Yaa13a2 ; j ) = F(σ23 (σ12 (Ya a3 ;i )) ⊗ σ12 (Yaa13a2 ; j ); Yea ⊗ Yae a2 ;1 ). 1 ;1 2 1 2 1 2 (3.15)

Full Field Algebras

391

By Proposition 3.4 in [H9], the right-hand side of (3.7) is equal to ;a

a1 )) F(σ132 (σ12 (Yaa13a2 ; j )) ⊗ σ123 (σ23 (σ12 (Ya a3 ;i ))); σ123 (Yae a2 ;1 ) ⊗ σ132 (Yea 1 ;1 2

1 2

=

;a a F(σ23 (Yaa13a2 ; j ) ⊗ Ya a3 ;i ; Yea2 ;1 1 2 2 a

;a

⊗ Yae1 a ;1 ) 1

a

= F(σ23 (Ya 3a ; j ) ⊗ Ya a3 ;i ; Yea2 ;1 ⊗ Yae a ;1 ). 1 2

1 2

(3.16)

1 1

2

By (3.3) again, the right-hand side of (3.16) is equal to a

;a

Ya 3a ; j , Ya a3 ;i = δi j , 1 2

1 2

proving the case of σ = σ12 . Next we prove the result for σ = σ23 . We need to find the relation between the matrices a

σ23 (Ya 3a ;i ), σ23 (Yaa3a ; j )

a2 a1 a3

V

1 2

1 2

and a

Ya 3a ;i , Yaa3a ; j Vaa3a . 1 2

1 2

1 2

By definition, we need to find the relation between the matrices a

a

F(Ya 3a ;i ⊗ σ23 (Yaa13a2 ; j ); Yea3 ;1 ⊗ Yae a 1 2

and

a ;(2)

)

(3.17)

).

(3.18)

1 1 ;1

3

a ;(1)

a2 ⊗ Yae a F(σ23 (Ya 3a ;i ) ⊗ Ya13a2 ; j ; Yea 2 ;1

1 1 ;1

1 2

From (4.9) in [H9] (or (3.6)), we see that the inverse of the matrix (3.17) is a

F(Ya 3e;1 ⊗ Yae

a

2 a2 ;1

3

; Ya 3a ;k ⊗ σ132 (σ23 (Yaa13a2 ; j ))) 1 2

Fa3

.

(3.19)

By Proposition 3.4 in [H9] and the fact Fa = Fa for a ∈ A, (3.19) is equal to a

F(Ya 2e;1 ⊗ Yae 2

3 a3 ;1

a

; σ23 (Yaa13a2 ; j ) ⊗ σ132 (Ya 3a ;k )) 1 2

Fa3 a

a

a3 e 2 3 Fa2 F(Ya2 e;1 ⊗ Ya3 a3 ;1 ; σ23 (Ya1 a2 ; j ) ⊗ σ132 (Ya1 a2 ;k )) = , Fa3 Fa2

which by (4.9) in [H9] (or (3.6)) again is equal to inverse of the matrix (3.17) is equal to (3.17) is equal to

Fa3 Fa2

Fa2 Fa3

Fa2 Fa3

times the inverse of (3.18). So the

times the inverse of (3.18). Thus the matrix

times the matrix (3.18), or equivalently, a

a

(σ23 (Yaa13a2 ; j ), σ23 (Ya 3a ;k ))V = (Yaa13a2 ; j , Ya 3a ;k )V . 1 2

1 2

Since S3 is generated by σ12 and σ23 , the conclusion of the proposition follows.

392


We are ready to construct a full field algebra using the bases of intertwining operators we have chosen. Let

F = ⊕a∈A W a ⊗ W a .

For wa1 ∈ W a1 , wa2 ∈ W a2 , wa1 ∈ W a1 and wa2 ∈ W a2 , we define Y((wa1 ⊗ wa1 ); z, ζ )(wa2 ⊗ wa2 ) N

=

a3

a1 a2

a3 ∈A p=1

;a

Yaa13a2 ; p (wa1 , z)wa2 ⊗ Ya a3 ; p (wa1 , ζ )wa2 . 1 2

Theorem 3.8. The quadruple (F, Y, 1⊗1, ω⊗1, 1⊗ω) is a conformal full field algebra over V ⊗ V . Proof. The identity property, the creation property and the single-valuedness property are clear. We prove the associativity and the skew-symmetry here. We prove associativity first. For a1 , a2 ∈ A, wa1 ∈ W a1 , wa2 ∈ W a2 , wa 1 ∈ (W a1 ) , wa 2 ∈ (W a2 ) , using the associativity of intertwining operators and Proposition 3.6, we have Y((wa1 ⊗ wa 1 ); z 1 , ζ1 )Y((wa2 ⊗ wa 2 ); z 2 , ζ2 ) =

a

a

Na14a5 Na25a3

a3 ,a4 ,a5 ∈A p=1 q=1

(Yaa14a5 ; p (wa1 , z 1 )Yaa25a3 ;q (wa2 , z 2 )) ;a

;a

⊗(Ya a4 ; p (wa 1 , ζ1 )Ya a5 ;q (wa 2 , ζ2 )) 1 5

=

2 3

a Na14a5

a

a

5 7 a a a Na25a3 Na65a3 Na16a2 Na7 a3 Na1 a2

a3 ,a4 ,a5 ,a6 ,a7 ∈A p=1 q=1 m=1 n=1 k=1 l=1 ·F(Yaa14a5 ; p ⊗ Yaa25a3 ;q ; Yaa64a3 ;m ⊗ Yaa16a2 ;n ) · ;a

;a

a

a

·F(Ya a4 ; p ⊗ Ya a5 ;q ; Ya 4a ;k ⊗ Ya 7a ;l ) · 1 5

2 3

7 3

1 2

·((Yaa64a3 ;m (Yaa16a2 ;n (wa1 , z 1 − z 2 )wa2 , z 2 )) ;a

;a

⊗(Ya a4 ;k (Ya a7 ;l (wa 1 , ζ1 − ζ2 )wa 2 , ζ2 ))) 7 3

1 2

a

=

a

5 7 a a Na65a3 Na16a2 Na7 a3 Na1 a2

δa6 a7 δmk δnl a3 ,a4 ,a6 ,a7 ∈A m=1 n=1 k=1 l=1 ·(Yaa64a3 ;m (Yaa16a2 ;n (wa1 , z 1 − z 2 )wa2 , z 2 )) ;a ;a ⊗(Ya a4 ;k (Ya a7 ;l (wa 1 , ζ1 − ζ2 )wa 2 , ζ2 )) 7 3 1 2

·

Full Field Algebras

393 a

=

a

Na65a3 Na16a2

a3 ,a4 ,a6 ∈A m=1 n=1 ;a

(Yaa64a3 ;m (Yaa16a2 ;n (wa1 , z 1 − z 2 )wa2 , z 2 ))

;a

⊗(Ya a4 ;m (Ya a6 ;n (wa 1 , ζ1 − ζ2 )wa 2 , ζ2 )) 6 3

1 2

= Y(Y((wa1 ⊗ wa 1 ), z 1 − z 2 , ζ1 − ζ2 )(wa2 ⊗ wa 2 ), z 2 , ζ2 ). ;a

We now prove the skew-symmetry. By Proposition 3.7, {σ12 (Ya a3 ;i ) | i = 1, . . . , 1 2

Naa13a2 } is the dual basis of {σ12 (Yaa13a2 ;i ) | i = 1, . . . , Naa13a2 }. Thus for a1 , a2 ∈ A, wa1 ∈ W a1 , wa2 ∈ W a2 , wa 1 ∈ (W a1 ) , wa 2 ∈ (W a2 ) , we have Y((wa2 ⊗ wa 2 ); z, ζ )(wa1 ⊗ wa 1 ) N

=

a3

a2 a1

a3 ∈A p=1 N

=

;a

σ12 (Yaa13a2 ; p )(wa2 , z)wa1 ⊗ σ12 (Ya a3 ; p )(wa2 , ζ )wa1 1 2

a3

a1 a2

a

e

−πi (Ya 3a

1 2;p

) z L(−1)

e

a3 ∈A p=1

⊗e

a3 ) a1 a2 ; p

πi (Y

Yaa13a2 ; p (wa1 , eπi z)wa2

;a

eζ L(−1) Ya a3 ; p )(wa1 , e−πi ζ )wa2 1 2

= (e z L(−1) ⊗ eζ L(−1) ) · N

a3

a1 a2

a3 ∈A p=1

;a

Yaa13a2 ; p (wa1 , eπi z)wa2 ⊗ Ya a3 ; p (wa1 , e−πi ζ )wa2 1 2

= (e z L(−1) ⊗ eζ L(−1) )Y((wa1 ⊗ wa 1 ); eπi z, e−πi ζ )(wa2 ⊗ wa 2 ).

Definition 3.9. A nondegenerate bilinear form (·, ·) on a conformal full field algebra (F, m, 1, ω L , ω R ) is said to be invariant if for u, v, w ∈ F, (Y(u; z, z¯ )v, w) = (v, Y(e z L

L (1)+¯z L R

(1)eπi L

L (0)−πi L R (0)

z −2L

L (0)

z¯ −2L

R (0)

u; z −1 , z¯ −1 )w).

The conformal full field algebra F we constructed above has a natural nondegenerate bilinear form (·, ·) F : F ⊗ F → C given by 0 a1 = a2 ((wa1 ⊗ wa 1 ), (wa2 ⊗ wa 2 )) F = Fa1 wa1 , wa2 wa 1 , wa 2 a1 = a2 for a1 , a2 ∈ A, wa1 ∈ W a1 , wa2 ∈ W a2 , wa 1 ∈ (W a1 ) , wa 2 ∈ (W a2 ) . We have:

394


Theorem 3.10. The nondegenerate bilinear form (·, ·) F is invariant. Proof. For a1 , a2 , a3 ∈ A, wa1 ∈ W a1 , wa2 ∈ W a2 , wa3 ∈ W a3 , wa 1 ∈ (W a1 ) , wa 2 ∈ (W a2 ) , wa 3 ∈ (W a3 ) , using Proposition 3.7 for the case σ = σ23 , we have (Y((wa1 ⊗ wa 1 ); z, ζ )(wa2 ⊗ wa 2 ), (wa3 ⊗ wa 3 )) F N

a4

a1 a2

=

a4 ∈A p=1

;a

((Yaa14a2 ; p (wa1 , z)wa2 ⊗ Ya a4 ; p (wa 1 , ζ )wa 2 ), (wa3 ⊗ wa 3 )) F 1 2

a

Na13a2

=

;a

Fa3 Yaa13a2 ; p (wa1 , z)wa2 , wa3 Ya a3 ; p (wa 1 , ζ )wa 2 , wa 3 1 2

p=1 a

Na13a2

=

;a

Fa3 σ23 (σ23 (Yaa13a2 ; p )(wa1 , z)wa2 , wa3 σ23 (σ23 (Ya a3 ; p )(wa 1 , ζ )wa 2 , wa 3 1 2

p=1 a

Na13a2

=

Fa3 wa2 , eπi h a1 σ23 (Yaa13a2 ; p )(e z L(1) e−πi L(0) z −2L(0) wa1 , z −1 )wa3 ·

p=1 ;a

·wa 2 , e−πi h a1 σ23 (Ya a3 ; p )(eζ L(1) eπi L(0) ζ −2L(0) wa 1 , ζ −1 )wa 3 1 2

a

Na13a2

=

Fa2 wa2 , σ23 (Yaa13a2 ; p )(e z L(1) e−πi L(0) z −2L(0) wa1 , z −1 )wa3 ·

p=1

· wa 2 ,

Fa3 ;a σ23 (Ya a3 ; p ) (eζ L(1) eπi L(0) ζ −2L(0) wa 1 , ζ −1 )wa 3 1 2 Fa2

a

Na13a2

=

((wa2 ⊗ wa 2 ), (σ23 (Yaa13a2 ; p )(e z L(1) e−πi L(0) z −2L(0) wa1 , z −1 )wa3

p=1

⊗

Fa3 ;a3 σ23 (Ya a ; p ) (eζ L(1) eπi L(0) ζ −2L(0) wa 1 , ζ −1 )wa 3 )) F 1 2 Fa2 N

=

a3

a1 a4

a4 ∈A p=1

⊗

((wa2 ⊗ wa 2 ), (σ23 (Yaa13a4 ; p )(e z L(1) e−πi L(0) z −2L(0) wa1 , z −1 )wa3

Fa3 ;a σ23 (Ya a3 ; p ) (eζ L(1) eπi L(0) ζ −2L(0) wa 1 , ζ −1 )wa 3 )) F 1 4 Fa2

= ((wa2 ⊗ wa 2 ), Y((e z L(1) e−πi L(0) z −2L(0) wa1 ⊗(eζ L(1) eπi L(0) ζ −2L(0) wa 1 ); z −1 , ζ −1 )(wa3 ⊗ wa 3 )) F , proving the invariance.

Acknowledgement. The first author is partially supported by NSF grant DMS-0401302.

Full Field Algebras

395

References [ABD] [BK] [BPZ] [Bi] [Bo] [DMZ]

[FFFS] [FHL] [FLM] [FFRS] [FRS1] [FRS2] [FRS3] [H1] [H2] [H3]

[H4] [H5] [H6] [H7] [H8] [H9] [H10] [KO] [K] [L] [MS1]

Abe, T., Buhl, G., Dong, C.: Rationality, regularity and c2 -cofiniteness. Trans. Amer. Math. Soc. 356(8), 3391–3402 (2004) Bakalov, B., Kirillov, A., Jr.: Lectures on tensor categories and modular functors, University Lecture Series, Vol. 21, Providence, RI: Amer. Math. Soc., 2001 Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetries in twodimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984) Birman, J.S.: Braids, links, and mapping class groups. Annals of Mathematics Studies, Vol. 82, Princeton, NJ: Princeton University Press, 1974 Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986) Dong, C., Mason, G., Zhu, Y.: Discrete series of the Virasoro algebra and the moonshine module. In: Algebraic Groups and Their Generalizations: Quantum and infinite-dimensional Methods, Proc. 1991 Amer. Math. Soc. Summer Research Institute, ed. by W. J. Haboush, B. J. Parshall, Proc. Symp. Pure Math. 56, Part 2, Providence, RI: Amer. Math. Soc., 1994, pp. 295–316 (1991) Felder, G., Fröhlich, J., Fuchs, J., Schweigert, C.: Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology. Compositio. Math. 131, 189– 237 (2002) Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 104, 593 (1993) Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster. Pure and Appl. Math., Vol. 134, NewYork: Academic Press, 1988 Fjelstad, J., Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators V: Proof of modular invariance and factorisation. Theory and Appl. of Categories 16, 342–433 (2006) Fuchs, J., Runkel, I., Schweigert, C.: Conformal correlation functions, Frobenius algebras and triangulations. Nucl. Phys. B624, 452–468 (2002) Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B646, 353–497 (2002) Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators, IV: Structure constants and correlation functions. Nucl. Phys. B715, 539–638 (2005) Huang, Y.-Z.: A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure. Appl. Alg. 100, 173–216 (1995) Huang, Y.-Z.: Virasoro vertex operator algebras, (nonmeromorphic) operator product expansion and the tensor product theory. J. Alg. 182, 201–234 (1996) Huang, Y.-Z.: Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories. In: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, A. A. Voronov, Contemporary Math., Vol. 202, Providence, RI: Amer. Math. Soc., pp. 335–355, 1997. Huang, Y.-Z.: Genus-zero modular functors and intertwining operator algebras. Internat, J Math. 9, 845–863 (1998) Huang, Y.-Z.: Generalized rationality and a “Jacobi identity” for intertwining operator algebras. Selecta Math. (N.S.) 6, 225–267 (2000) Huang, Y.-Z.: Vertex operator algebras, the verlinde conjecture and modular tensor categories. Proc. Natl. Acad. Sci. USA 102, 5352–5356 (2005) Huang, Y.-Z.: Differential equations and intertwining operators. Comm. Contemp. Math. 7, 375– 400 (2005) Huang, Y.-Z.: Differential equations, duality and modular invariance. Comm. Contemp. Math. 7, 649–706 (2005) Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0406291, 2004 Huang, Y.-Z.: Rigidity and modularity of vertex tensor categories. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0502533, 2005 Kapustin, A., Orlov, D.: Vertex algebras, mirror symmetry, and D-branes: The case of complex tori. Commun. Math. Phys. 233, 79–136 (2003) Kong, L.: A mathematical study of open-closed conformal field theories. Ph.D. thesis, Rutgers University, 2005 Li, H.S.: Some finiteness properties of regular vertex operator algebras. J. Algebra, 212, 495– 514 (1999) Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B212, 451–460 (1988)

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Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989) Moore, G., Seiberg, N.: Naturality in conformal field theory,. Nucl. Phys. B313, 16–40 (1989) Rosellen, M.: OPE-algebras, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2002, Bonner Mathematische Schriften [Bonn Mathematical Publications], Vol. 352, Universität Bonn, Mathematisches Institut, Bonn, 2002 Rosellen, M.: Ope-algebras and their modules. Int. Math. Res. Not. 2005, 433–447 (2005) Segal, G.: The definition of conformal field theory. In: Differential geometrical methods in theoretical physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 250, Dordrecht: Kluwer Acad. Publ., pp. 165–171, 1988 Segal, G.: Two-dimensional conformal field theories and modular functors. In: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Bristol: Hilger, pp. 22–37, 1989 Segal, G.: The definition of conformal field theory, preprint, 1988; also in: Topology, geometry and quantum field theory, ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308, Cambridge: Cambridge University Press, pp. 421–457, 2004 Tsukada, H.: String path integral realization of vertex operator algebras. Mem. Amer. Math. Soc. 91(444), 1991, vi+138pp. Turaev, V.G.: Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Math., Vol. 18, Berlin: Walter de Gruyter, 1994

Communicated by L. Takhtajan

Commun. Math. Phys. 272, 397–442 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0238-y

Communications in


Dynamics of the Quasi-Periodic Schrödinger Cocycle at the Lowest Energy in the Spectrum Kristian Bjerklöv Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4. E-mail: [email protected] Received: 7 February 2006 / Accepted: 12 November 2006 Published online: 3 April 2007 – © Springer-Verlag 2007

Abstract: In this paper we consider the quasi-periodic Schrödinger cocycle over Td (d ≥ 1) and, in particular, its projectivization. In the regime of large coupling constants and Diophantine frequencies, we give an affirmative answer to a question posed by M. Herman [21, p.482] concerning the geometric structure of certain Strange Nonchaotic Attractors which appear in the projective dynamical system. We also show that for some phase, the lowest energy in the spectrum of the associated Schrödinger operator is an eigenvalue with an exponentially decaying eigenfunction. This generalizes [39] to the multi-frequency case (d > 1). 1. Introduction We consider the quasi-periodic Schrödinger cocycle S E : (θ, x) ∈ Td × R2 → (θ + ω, A E (θ )x) ∈ Td × R2 ,

(1.1)

where A E (θ ) =

0 1 2 −1 λ V (θ ) − E

∈ S L(2, R),

E ∈ R.

Here V : Td → R is a continuous function (T = R/Z), d ≥ 1, and the entries of the frequency vector ω ∈ Td are rationally independent with 1. The latter implies that the base dynamics θ → θ + ω on Td is minimal and uniquely ergodic. The real constant λ > 0 is introduced as a coupling constant. It should be thought of as fixed, while E is the parameter. We shall leave the dependence on λ implicit. Research partially supported by STINT (Institutional Grant 2002-2052), The Royal Swedish Academy of Sciences and SVeFUM

398

K. Bjerklöv

The cocycle (1.1) is closely related to the discrete quasi-periodic Schrödinger equation −(u n+1 + u n−1 ) + λ2 V (θ + (n − 1)ω)u n = Eu n , n ∈ Z, (1.2) which has been extensively studied in the mathematics and physics literature. Indeed, (1.2) can be written as the system

un u n+1

= A E (θ + (n − 1)ω)

u n−1 . un

In the setting of the Schrödinger equation, θ ∈ Td is called the phase and E ∈ R the energy. The sequence (λ2 V (θ + (n − 1)ω))n∈Z is called the potential. Let A(θ + (n − 1)ω) · · · A(θ ), n ≥ 1, AnE (θ ) = I, n = 0, A E (θ + nω)−1 · · · A E (θ − ω), n ≤ −1 be the fundamental solution of (1.1), and introduce the (maximal) Lyapunov exponent 1 γ (E) = lim log AnE (θ )op dθ ≥ 0. n→∞ n Td The limit, which exists by subadditivity, is independent of the matrix norm since all norms are equivalent. We have chosen the operator norm. From Oseledets’ theorem (see below) or Kingman’s subadditive ergodic theorem, we have lim

n→∞

1 log |AnE (θ )x| = ±γ (E) Lebesgue-a.e. θ ∈ Td and all x ∈ R2 \ {0}. n

We say that the cocycle S E is uniformly hyperbolic if there exist constants c, β > 0 and a continuous splitting W E+ (θ ) ⊕ W E− (θ ) = R2 such that for all θ , |AnE (θ )x| ≤ c exp(±βn)|x| for all x ∈ W E± (θ ), n ∈ Z. If such a splitting exists, it must be unique, and thus invariant: A E (θ )W E± (θ ) = W E± (θ + ω). If γ (E) > 0 but S E is not uniformly hyperbolic, we say that S E is non-uniformly hyperbolic. Next we introduce the family of Schrödinger operators (Hθ u)n = −(u n+1 + u n−1 ) + λ2 V (θ + (n − 1)ω)u n , acting on the Hilbert space l 2 (Z). These operators are self-adjoint. Since the function V is continuous, and since {θ + nω}n∈Z is dense in Td for all θ by our assumption on ω, it is easy to show that the spectrum of Hθ , which we denote by σ (Hθ ), is independent of θ . This justifies the notation σ (H ) for the spectrum of any of the operators Hθ . One also verifies that σ (H ) ⊂ [λ2 min V − 2, λ2 max V + 2]. There are many connections between the dynamics of S E and spectral properties of Hθ . One fundamental result [22] states that / σ (H ). S E is uniformly hyperbolic if and only if E ∈

Dynamics of the Q.P. Schrödinger Cocycle

399

This implies, in particular, that γ (E) > 0 for all E ∈ / σ (E). Moreover, since the spectrum σ (H ) never is empty, Hθ being self-adjoint, there will always be values of E for which S E is not uniformly hyperbolic. These are the interesting parameters. A general question is whether for a given (θ, E) we have “localization” – a phenomenon where γ (E) > 0 and lim log |AnE (θ )x|/n = ∓γ (E) for some 0 = x ∈ R2 .

n→±∞

For the Schrödinger operator Hθ this means the existence of eigenfunctions which decay exponentially fast at both +∞ and −∞. This is related to so-called Anderson localization [1]. We will show below (see Main Theorem) that this phenomena can happen for E = E 0 = min{E ∈ σ (E)}. In the case when S E is non-uniformly hyperbolic we can still always conclude the existence of a measurable splitting W E+ (θ ) ⊕ W E− (θ ) = R2 . Indeed, from Oseledets’ theorem [33] (see also [36]) it follows that for Lebesgue-a.e. θ ∈ Td , 1 log |AnE (θ )x| = lim n→±∞ n

+γ (E), −γ (E),

x ∈ W E+ (θ ) \ {0} x ∈ W E− (θ ) \ {0}.

The subspaces are invariant: A E (θ )W E± (θ ) = W E± (θ + ω). It is a general fact that

inf θ ∠(W + (θ ), W − (θ )) > 0 inf θ ∠(W + (θ ), W − (θ )) = 0

if S E is uniformly hyperbolic; and if S E is non-uniformly hyperbolic,

where ∠ denotes the angle between two subspaces in R2 . In the non-uniform case there can be certain θ ∈ Td where the two subspaces W E+ (θ ) and W E− (θ ) collapse. These are candidates of θ :s for which one can have the localization phenomenon. Before we proceed describing the setting in which we shall work more precisely, we introduce a quantity which is in fact intimately related to the Lyapunov exponent: the rotation number ρ(E). Let (u n )n∈Z be any non-trivial solution of (1.2), i.e.,

un u n+1

= AnE (θ )

u0 u0 , = 0. u1 u1

Denote by S(N ) the number of changes of sign of this solution in the interval [1, n], adding 1 if u(N ) = 0. Then the limit ρ(E) = lim

N →∞

S(N ) 2N

exists and is independent of θ and (u n )n∈Z [12]. This limit ρ(E) is called the rotation number. It was first introduced in [25] for the continuum Schrödinger equation. A dynamical definition was given in [21]. The rotation number is continuous and increasing in E, it grows exactly on the spectrum σ (E) and it coincides with the Integrated Density of States k(E) (see for example [2]) of the Schrödinger operator, up to a factor 2: 2ρ(E) = k(E).

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1.1. The projective dynamics and Strange Non-chaotic Attractors. The map S E induces in a canonical way a diffeomorphism E on the space Td × P, where P is the projective space (the space of all lines in R2 going through the origin). We shall represent points in P as numbers r ∈ R ∪ {∞}, where r = ∞ represents the line with the direction vector (1, r ) and ∞ the vertical line. Thus E can be expressed as E (θ, r ) = (θ + ω, λ2 V (θ ) − E − 1/r ).

(1.3)

Note that if (u n , u n+1 )T = AnE (θ )(u 0 , u 1 )T and (u 0 , u 1 ) = 0, then nE (θ, u 1 /u 0 ) = (θ + nω, u n+1 /u n ). Since P is isomorphic to the circle S1 = (−π/2, π/2], we can also view E as a diffeomorphism on Td × S1 : E (θ, α) = (θ + ω, arctan(λ2 V (θ ) − E − 1/ tan(α))).

(1.4)

The fiber map α → arctan(λ2 V (θ ) − E − 1/ tan(α)) on S1 is an orientation preserving E is an example of a quasi-periodically forced circle circle homeomorphism. Thus, map. By controlling the dynamics of E , one can recover information about the dynamics of the cocycle S E , and thus be able to describe the solutions of the Schrödinger equation. The map E is the one we will focus on in this paper and all our results will be derived from a detailed analysis of its dynamics. If γ (E) > 0 for some E it follows from the above discussion that there exists an invariant measurable splitting W E± (θ ). In the projective space, the subspaces W E± (θ ) are represented as points l ± E (θ ) ∈ P (the directions of the subspaces). We call them the d projective Oseledets directions. Thus we get two measurable functions l ± E : T → P whose graphs are E -invariant. If the cocycle S E is uniformly hyperbolic, then the functions l ± E are continuous. If S E is non-uniformly hyperbolic then the subspaces W E± (θ ) have to be “very discontinuous”. Indeed, if they were continuous it is easy to verify that S E in fact is uniformly hyperbolic. And if they are discontinuous at some θ , then they must be discontinuous along the dense orbit {θ + nω}n∈Z . The following result by Herman [21] (see also [23]) shows how mixed together the two graphs in fact are in the non-uniform case: If γ (E) > 0, then the diffeomorphism E has exactly two ergodic invariant probability measures µ± , each being the Lebesgue measure on the base Td lifted to the graph of the function θ → l ± E (θ ). Moreover, if S E is non-uniformly hyperbolic then E has a unique non-empty minimal set M E ⊂ Td × P and it satisfies M E = supp(µ+ ) = supp(µ− ). Recall that the set M E being minimal means that it is closed and E −invariant, and the orbit of any point in M E under E is dense in M E , that is, { nE (θ, r )}n∈Z = M E for all (θ, r ) ∈ M E . If γ (E) > 0, it follows from the Oseledets’ theorem that for Lebesgue-a.e.θ ∈ Td , lim log |AnE (θ )x|/n = ±γ (E) for any x ∈ R2 \ (W E+ (θ ) ∪ W E− (θ )).

n→±∞

This implies that the vectors AnE (θ )x have to approach the direction of the subspace W E+ (θ + nω) for all n 0 and that of W E− (θ + nω) for all n 0. In view of the projective map E , this means that the graph of the function l +E will attract “most” dynamics


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under forward iteration, and l − E will attract most of the backward dynamics. Thus, in the case when S E is non-uniformly hyperbolic, the unique minimal set M E attracts both the forward and the backward dynamics. In the case when S E is non-uniformly hyperbolic, the two E -invariant curves (θ, l +E (θ )) and (θ, l +E (θ )) both lie densely in the attractor M and they attract the forward and the backward dynamics, respectively. Thus the dynamics on M is not chaotic (compare with the chaotic dynamics on the Hénon attractor [3]). Still the topological structure of the set M can be very complicated, it can be fractal-like. Due to these two facts, M is sometimes called a Strange Non-chaotic Attractor (SNA), a notion introduced in [18]. These kind of “strange” objects seem to appear frequently in numerous quasi-periodically forced systems, not only in the projective Schrödinger cocycle. To our knowledge, the first examples of systems having an SNA are the ones constructed in [32, 41]. When it was realized that this new kind of attractors can exist, the problem of showing existence and describing finer properties of SNA’s have attracted much interest, both in the mathematics and physics community. Although much numerics have been performed (see [34] for an overview), there are still very few existing rigorous results. Only in some special models are there actual proofs of existence of SNA’s (see for example [7, 27–29, 26, 30] and the references therein). Concerning the projective Schrödinger cocycle E , besides the existence of the set M E , not much is known about what it can look like. In [4] (see also [5] for results regarding the time-continuous Schrödinger equation) the author has shown that when d = 1, M E “often” is the whole space T1 × P. To our knowledge, this is the only rigorous result about finer geometric properties of M E . In the present paper we obtain a description of M E in a case when it is not the whole space (see Main Theorem). This will answer a question posed by M. Herman [21] in the regime of large coupling constants λ. 1.2. Dynamics below the lowest energy in the spectrum. In this paper we shall investigate the behavior of the map E at the lowest energy in the spectrum of the Schrödinger operator, i.e., for E = E 0 , where E 0 := min{E ∈ σ (H )}. This choice implies that the cocycle S E is uniformly hyperbolic for all E < E 0 . Furthermore, the projective Oseledets directions l ± E satisfy the following for all E < E 0 : a) b) c) d)

The functions θ → l ± E (θ ) are continuous; − + (θ ) < ∞; 0 < l (θ ) < l E E ± Td log l E (θ )dθ = ±γ (E); and − + + l− E (θ ) ≤ l E (θ ) < l E (θ ) ≤ l E (θ ) for all θ if E ≤ E < E 0 .

Moreover, since the graphs of l ± E (θ ) are E -invariant, they satisfy the relation ± 2 e) l ± E (θ + ω) = λ V (θ ) − E − 1/l E (θ ).

Indeed, a) follows from the uniform hyperbolicity of S E . Actually, the two functions l ± E will have the same regularity as the function V . The fact that the rotation number ρ(E) is zero for E < E 0 , forces 0 < l ± E (θ ) < ∞ (the curves can not twist around P since that would imply that there is a solution of the Schrödinger equation which change signs in a recurrent manner). By using Birkhoff’s ergodic theorem and the fact that u ± n (θ ) =

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± l± E (θ ) · · · l E (θ +(n−1)ω), u 0 (θ ) = 1 are solutions of the Schrödinger equation we obtain c). Moreover, the uniform hyperbolicity implies that inf θ |l +E (θ ) − l − E (θ )| > 0. Since c) holds, we must have b). The last inequality in d) can be shown by iterating the horizontal line (θ, ∞) forward under E and E , respectively. Since b) holds, and since the fiber map in E preserves the orientation on P, we have that πr ( nE (θ, ∞)) > l +E (θ + nω) for all n ≥ 0. It is easy to verify that the functions l +E,n (θ ) = πr ( nE (θ − nω, ∞)) satisfy l +E ,n (θ ) ≤ l +E,n (θ ) for all n and l +E,m (θ ) ≤ l +E,n (θ ) for all n ≤ m. Since l +E,n (θ ) → l +E (θ ) as n → ∞, the result follows by taking limits. To get the first inequality, we iterate the line (θ, 0) under −1 E . Let us stress that d) is a special case of a monotonicity property first observed by Johnson in [24, Lemma 3.4]. The monotonicity property d) implies that the functions

l˜± (θ ) = lim l ± E (θ ) EE 0

(1.5)

are well defined and that l˜+ (θ ) is upper semi-continuous and l˜− (θ ) is lower semicontinuous. b) and c) together imply that 0 < l˜− (θ ) ≤ l˜+ (θ ) < ∞. Furthermore, from e) we conclude that l˜± (θ + ω) = λ2 V (θ ) − E 0 − 1/l˜± (θ ), so their graphs are E 0 -invariant. There are now two possibilities; either γ (E 0 ) = 0 or γ (E 0 ) > 0. We will focus on the latter case in this paper. In [5] we have constructed exotic potentials for which γ (E) = 0 for all E ∈ σ (H ) in a small neighborhood of E 0 and positive everywhere else. Let us stress that verifying whether the Lyapunov exponent is zero or not generally is a very delicate problem. One of the results in this paper is to give conditions on V, λ and ω under which γ (E 0 ) > 0 (see Main Theorem below). This condition is valid for V ∈ C2 . In the case of real-analytic V it was already known that γ (E) log λ for all energies E ∈ R if λ is sufficiently large [9]. The conditions a) − d) above suggest a way to numerically find E 0 . As already pointed out, the spectrum of the Schrödinger operator lies in the interval [λ2 min V − 2, λ2 max V + 2]. Thus, a first approximation of E 0 could be E = λ2 min V − 2. Take some point (θ, r ) ∈ Td × (0, ∞) and iterate it forward under E . If E < E 0 , the iterates must exponentially fast (since γ (E) > 0) approach the curve (θ, l +E (θ )) (unless we were so unlucky and started with a point on (θ, l − E (θ ))). Plotting these iterates gives a picture + of l E . Iterating backward instead, and plotting, gives a picture of l − E . If these two curves lie in Td × (0, ∞) (so the rotation number ρ(E) = 0) and are separated (which implies that S E is uniformly hyperbolic), we must indeed have that E < E 0 . Increasing E, plotting the new graphs and checking the distance between the curves eventually gives us a value of E for which the two curves “hit” each other (the infimum over θ of the angles between the subspaces W E± (θ ) is zero). We do not claim that this is the smartest way to estimate E 0 , but this is close to what we will do analytically later. In Fig. 1 we√ have made computer simulations in the case where d = 1, V (θ ) = cos(2π θ ), ω = ( 5 − 1)/4 and λ2 = 10. For this choice it is well-known that γ (E) ≥ log(λ2 /2) = log 5 > 0 E (see 1.4) in these for all E (see [21]). Note that we have used the diffeomorphism experiments, instead of E . This is just to get clearer pictures. From this experiment, E = −10.146 would be an approximate value of E 0 .


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Fig. 1. Finding the lowest energy E 0 . Approximations of l +E (upper curve) and l − (E) (lower curve) for E = −11 (left), E = −10.17 (middle) and E = −10.145886 (right).

Note how the curves l ± E “fractalize” as we approach E = E 0 from below. This is because at E = E 0 , both the condition Td log l ± E (θ )dθ = ±γ (E) (and γ (E) > log 5), − + which forces l E (θ ) to be larger than l E (θ ) a.e., and the condition inf θ (l +E (θ )−l − E (θ )) = 0 must hold simultaneously. 1.3. The main problem: A question by M. Herman. Assume that γ (E 0 ) > 0. Then S E 0 is non-uniformly hyperbolic and the map E 0 has a unique minimal set M. Herman [21] shows that it satisfies M ⊂ Td × [1/c, c] for some constant 1 < c < ∞.

(1.6)

Since the base dynamics θ → θ + ω on Td is minimal, and since M is closed and E -invariant, it follows that for all θ ∈ Td there is at least one r in the fiber above θ such that (θ, r ) ∈ M. Since also (1.6) holds, we can define the two functions l + (θ ) = sup{r : (θ, r ) ∈ M} and l − (θ ) = inf{r : (θ, r ) ∈ M}. Note that they satisfy 1/c ≤ l ± (θ ) ≤ c. Furthermore, since the fiber map r → λ2 V (θ )− E 0 − 1/r in E 0 preserves the orientation on P, the bounding curves (θ, l ± (θ )) must be E 0 invariant, as well as the “filled in” set M1 := {(θ, r ) : θ ∈ Td , l − (θ ) ≤ r ≤ l + (θ )}. Moreover, by definition, we have M ⊂ M1 . In [21, p. 482] M. Herman asks the following question: Is M = M1 ?

(Q)

Remark 1. A positive answer to (Q) immediately implies that the set M has positive Lebesgue measure, since γ (E 0 ) > 0 implies that l + (θ ) > l − (θ ) a.e.

1

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Note that a priori there very well could be that for some θ the set M ∩ ({θ } × P) has a complicated fractal structure, instead of being the line segment {θ } × [l − (θ ), l + (θ )]. To the author’s knowledge, no positive nor negative answers to this question exist in the literature under any conditions on V, ω and d. Since the two curves (θ, l + (θ )) are E 0 -invariant, and since there can be at most two curves with this property (we have assumed that γ (E 0 ) > 0), they must coincide with the graphs of l˜± (θ ) defined in (1.5). Thus Fig. 1 (right) gives a hint about the complicated geometry of the set M. 1.4. Statement of results. Before we can state our results we have to put some extra assumptions on V and ω. We stress that the dimension d ≥ 1 can be any positive integer. Let V : Td → R be a C 2 -function with a unique non-degenerate minimum and let the frequency vector ω ∈ Td satisfy the Diophantine Condition inf | < n, ω > − p| >

p∈Z

κ , for all n ∈ Zd \ {0}, |n|τ

(DC)κ,τ

for some constants κ > 0 and τ ≥ d. Here < ·, · > is the standard scalar product in Rd . Note that the Condition on V is generic in the C 2 -topology and that the Diophantine Condition is satisfied for a full Lebesgue measure set of frequency vectors ω. Main Theorem. Assume that V and ω satisfy the assumptions above. There exists a λ0 = λ0 (V, κ, τ ) > 0 such that for all λ > λ0 the following hold: Let E 0 = min{E ∈ σ (H )}. Then i) γ (E 0 ) > log(λ)/2; and ii) The unique minimal set M satisfies M = M1 , i.e., (Q) has a positive answer. Furthermore, iii) There exists a θ ∈ Td and a vector u ∈ l 2 (Z), exponentially decaying at ±∞, such that Hθ u = E 0 u. Remark 2. Result iii) shows that for some phase θ , the energy E 0 is an eigenvalue of the Schrödinger operator Hθ with an exponentially decaying eigenfunction. This is related to so-called Anderson localization. Our result generalizes that of Soshnikov [39] to the case of d > 1. We believe that for d > 1, iii) is new under any regularity assumptions on V . Remark 3. In particular, ii) shows that the set M has positive Lebesgue measure. Remark 4. By symmetry, the statement of Main Theorem also holds for E = E 1 = max{E : E ∈ σ (H )} if we assume that V has a unique non-degenerate maximum. The minimal set M will in this case be in Td × [−λ2 , −1/λ2 ]. Remark 5. We do not know if it is possible to construct examples where γ (E 0 ) > 0 and M = M1 . Our result shows that to produce such examples, if possible, one must take ω Liouville or λ ’small’ or V having degenerate or multiple minima. Remark 6. The techniques used to show the Main Theorem can also be used to prove an equivalent statement for the continuum Schrödinger equation −u (t) + λ2 V (θ + tω)u(t) = Eu(t), t ∈ R. By letting AtE (θ ) ∈ S L(2, R) be the fundamental solution, the


405

main object of study is then the skew-product flow (θ, x) → (θ + tω, AtE (θ )x) and its projectivization. The difference from the discrete case is that the computations will be a bit harder. In [6] the author has developed techniques for the analysis of these continuous (projective) skew-product flows. Remark 7. It is well-known that in order for the quasi-periodic Schrödinger operator to have eigenfunctions, some kind of non-resonance condition on the frequency vector ω , like (DC) above, must hold. If ω is “very” Liouville, we get the so-called Gordon potentials [17] (see also [37]) for which the point spectrum of Hθ is empty for all θ . 1.5. Related results. Concerning the positivity of the Lyapunov exponent γ (E), the most famous result is probably [21]. Here Herman shows that if V is a real nonconstant trigonometric polynomial (any d ≥ 1), then γ (E) log λ for all E provided that the coupling constant λ is sufficiently large. For d = 1 this was later generalized by Sorets-Spencer [40] to real-analytic potentials. Recently the techniques by BourgainGoldstein-Schlag [8, 10, 19] has led to [9] where Bourgain extends, among other things, Herman’s result (for d > 1) to the real-analytic case. For results in the non-analytic case, and d = 1, we mention the works [4, 11, 31]. For d = 1 it has been shown in [13, 16, 31, 38], under various regularity assumptions on V , that the operator Hθ has a pure-point spectrum for a.e. phase θ and the eigenfunctions are exponentially decaying. This result has also been established for d > 1 only in the analytic case [8, 10]. Although Hθ has pure-point point spectrum, it is very difficult to say which energies E are actually eigenvalues. In general it can not be true that for every energy E in σ (H ) there is a phase θ such that E is an eigenvalue of Hθ with an exponentially decaying eigenfunction. Such a result would imply that for a certain dual-model, its cocycles are reducible for all E. But it is known that these cocycles can be non-reducible for certain energies (see [35] and [15] for details). The creation of SNA’s in saddle-node bifurcations, like in our case, have been studied in [28, 29] for other classes of quasi-periodically forced circle maps. Results analogous to ii) in the Main Theorem have been obtained for certain quasi-periodic pinched skew products [27]. They are of the form (θ, x) ∈ T × R → (θ + ω, αg(θ ) f (x)) ∈ T × R, where f is monotonically increasing, f (0) = 0 and g(θ0 ) = 0 for some θ0 . Moreover, g(θ ) must have a ’spike’ at θ0 (so g (θ0 ) does not exist). Since the analysis in [27] very much relies on the fact that one fiber is pinched (αg(θ0 ) f (x)) = 0 for all x) and the fact that the line (θ, 0) is invariant and that g has a ’spike’ at θ0 , it would probably require quite a bit of work to adapt this method to the Schrödinger case. We also do not think that it would be easy to derive ii) from any other known methods in the field, like the works [8, 10, 13, 19, 38, 42]. Finally, we mention that the technique used in this paper is similar to the one we used in [4] and [5]. This kind of dynamical systems approach is close in spirit to the fundamental work by Benedicks and Carleson on the Hénon map [3], and to that by Young [42] where she considers certain classes of quasi-periodic S L(2, R)-cocycles. 1.6. Idea of the proof and some comments. As mentioned above, the key object we study is the map E . The coupling constant λ should be thought of as being extremely large; λ 1. Roughly speaking, what we will do is to, on finer and finer scales, get approximates of the two functions l ± E near the largest “peak” going down in Fig. 1. This peak, as we

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will see, comes from the unique quadratic minimum of the function V . More precisely, up to a certain scale, λ2 V (θ ) − E, where θ is in a small neighborhood of the minimum of V , will be an approximation of l +E , and l − E will be close to constant 0. We can then inductively, by studying approximations of l ± E on smaller and smaller neighborhoods around the “peak”, adjust the parameter E so that in the limit the two graphs “collide” for a certain value E = E 0 . Let us stress that if the two graphs coincide at one point, they have to do it along the whole orbit under E . So the curves can not be continuous. Along this inductive procedure, we will obtain better and better control on the orbits of points (θ, r ) under E for E closer and closer to E = E 0 . In the limit, we will have very good control on E 0 . The value E = E 0 which we find like this will be the lowest energy in the spectrum of the Schrödinger operator. The intersection point, (θ c , r c ), will give the θ c for which Hθ c has an exponentially decaying eigenfunction. An eigenfunction (u n ) is obtained by taking initial condition (u 0 , u 1 ) such that u 1 /u 0 = r c . To verify that ii) in Main Theorem holds, we have to refine the analysis. Our approach will be to assume that statement ii) is not true. This will imply the existence of what we call a “jump point” θ j . This is a point θ j where the function l + (the upper bounding curve of the set M) makes a jump: there is an open ball B such that θ j lies on the boundary of B and l + (θ j ) − l + (θ ) > const > 0 for all θ ∈ B. To reach a contradiction we will, by using the whole inductive machinery we have constructed, show that there in fact have to be points (θ, r ) in the set M such that θ ∈ B and r is arbitrarily close to l + (θ j ). The fact that V has a unique minimum implies that the set of θ where V is close to its minimum is contained in one single small ball. It is this region which “connects the stable and unstable directions” and breaks the uniform hyperbolicity. That we only have one ball is crucial. This will enable us to use the Diophantine condition (DC) to estimate the return time for a point starting in the ball to come back. These return times will give us “time to recover”: each time we go into the “bad region” we lose something, but if we do not go in too often, we will be able to keep the necessary control. The technique we develop in this paper is adapted for analyzing E for the lowest (or, by symmetry, the highest) energy in the spectrum of H . It is not clear to us what would play the role of the “peak” coming from the minimum of V if we, for example, considered the dynamics of E when E approaches the endpoint of another spectral gap of H (other than the trivial ones (−∞, min σ (E)) and (max σ (H ), ∞) which we treat here). It is even far from trivial to locate such a gap. A general conjecture is that in the multi-frequency case (d > 1), the spectrum σ (H ) should be a single interval for all large λ. In the case of d = 1, the spectrum has a tendency to be a Cantor set [20, 38]. If we consider values of E which are not endpoints of spectral gaps, we will not have the nice situation with one ’critical ball’, as in the case we treat here, where we can use the Diophantine condition in a convenient way. One would have to either exclude parameters (as in [4] for example) or refine the techniques to handle the resonances which will appear. In comparison to most techniques used in the field of quasi-periodic Schrödinger equations, our method does not use the fact that E comes from a linear system. We only look at the quasi-periodically forced circle homeomorphism E . In this sense it is possible to extend the techniques used here to more general systems. In [7] we are extending the techniques from [4] to a wider class of quasi-periodically forced circle homeomorphisms. The rest of this paper is organized as follows. In Sect. 2 we introduce some preliminaries. Section. 3 is devoted to the derivation of some formulas which will be used


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frequently. In Sect. 4 we derive two abstract help lemmas. Section 5 contains the inductive lemma on which the proof of Main Theorem is based. In Sect. 6 we include some extra constructions needed for the proof of statement ii) in Main Theorem. Finally, in Sect. 7 we prove Main Theorem. 2. Notation and Basic Definitions and Lemmas 2.1. Notations. Below follows a short list of notations and basic assumptions on V and ω which we shall use in the paper. The coupling constant λ should always be thought of as being “extremely” large. It will depend on v and ω. • The dimension d ≥ 1 is fixed from now on, and ω ∈ Td is assumed to satisfy the Diophantine Condition (DC)κ,τ for some constants κ > 0 and τ ≥ d, which also are fixed from now on. • When d > 1, we write a point θ ∈ Td in coordinate form as θ = (θ 1 , θ 2 , . . . , θ d ). • We assume that the function V : Td → R is C 2 and satisfies the following: The supremum norm V C 0 = 1/2, (2.1) and V has a unique non-degenerate global minimum 0, located at θ = 0 for simplicity, so d

V (x) = V (x 1 , x 2 , . . . , x d ) = ai j x i x j + o(|x|2 ) as x → 0, (2.2) i, j=1

the quadratic form being positive definite. When we write the quadratic form above, we think of V being defined on Rd and that V is 1-periodic in each variable. Since we always can scale with the coupling constant λ and shift with the parameter E, there is no loss of generality to assume that min V = 0 and V C 0 = 1/2. Note that these two assumptions on V imply that the spectrum of the Schrödinger operator lies in the interval [−2, λ2 /2 + 2]. It is a basic fact that the spectrum always lies in [inf λ2 V − 2, sup λ2 V − 2]. • The diffeomorphism E is defined as in (1.3). Note that 2 −1 −1 E (θ, r ) = (θ − ω, (λ V (θ − ω) − E − r ) ).

• We denote by · the standard distance on Td . • By a ball, we always mean an open ball, i.e., a set of the form {x ∈ Td : x − c < R} (when d = 1, the ball is of course an open interval, but still called a ball). • If I ⊂ Td is a ball with radius ε > 0 centered at the point c and if k > 0, we denote by k I the ball with radius kε centered at c. • We define the projections πθ , πr by: πθ (θ, r ) = θ, πr (θ, r ) = r, (θ, r ) ∈ Td × P. • Given θ0 ∈ Td , r0 ∈ P and E we use the notation (θk , rk ) = kE (θ0 , r0 ), k ∈ Z.

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In particular we have θk = θ0 + kω ∈ Td ,

r1 = λ2 V (θ0 ) − E −

1 1 , r2 = λ2 V (θ0 + ω) − E − , . . . ; and r0 λ2 V (θ0 ) − E − r1 0

1 1 r−1 = 2 , r−2 = , .... 1 λ V (θ0 − ω) − E − r0 λ2 V (θ0 − 2ω) − E − 2 λ V (θ0 −ω)−E−r0

Remark 8. Note that a point θk ∈ Td is written as (θk1 , θk2 , . . . , θkd ) in coordinate form.

2.2. The first critical set and the first energy interval. Since V has the unique global minimum 0 at θ = 0, and since it is non-degenerate, we have that for large enough λ, the set {θ ∈ Td : V (θ ) ≤ 10/λ} √ is contained in a ball of diameter c0 / λ, centered at 0, where the constant c0 only depends on V . Definition 2.1 (The first critical set). Define √ I0 = {θ ∈ Td : θ < c0 /(2 λ)}, where c0 = √ c0 (V ) > 0 is the constant mentioned above, that is, I0 is the ball with diameter c0 / λ centered at θ = 0. Thus, for all large λ we have I0 ⊃ {θ ∈ Td : V (θ ) ≤ 10/λ}

(2.3)

V (θ ) > 10/λ for all θ ∈ Td \ I0 .

(2.4)

and

We also define the first energy interval Definition 2.2 (The first energy interval). E−1 := [−1, 1]. This means that we “guess” that E 0 = E 0 (λ) — the smallest energy in the spectrum — lies in this interval. The procedure is later to inductively construct energy intervals E−1 ⊃ E0 ⊃ E1 ⊃ · · · converging to {E 0 }.


409

2.3. Some basic lemmas. The first lemma deals with the situation of iterating outside the critical set I0 . Lemma 2.3. For each E ∈ E−1 the following hold for all large λ > 0, depending on V : / I0 , then 2λ < r1 < λ2 ; a) If r0 ≥ 1/λ and θ0 ∈ b) If |r0 | ≤ λ and θ0 ∈ / I0 + ω, then 1/λ2 < r−1 < 1/(2λ).

Proof. This is just an easy verification, using (2.1) and (2.4).

The next lemma just states that E preserves orientation and that one has some trivial bounds. Lemma 2.4. The following holds for all E ∈ E−1 and all large λ: a) If (θ0 , r0 ), (θ0 , r0 ) are two points such that 0 < r0 < r0 , then r1 < r1 < λ2 . b) If (θ0 , r0 ), (θ0 , r0 ) are two points such that 0 < r0 < r0 and if r−1 > 0, then 0. Then (I + nω) = ∅, I∩ 0

κ 1/τ ε

.

It is also possible to get an upper bound for the return-time. Lemma 2.6. Let I ⊂ Td be a ball in Td with diameter ε > 0 and let θ ∈ Td . Then θ + kω ∈ I for some 0 ≤ k ≤ Cε−(τ +1) . 3

The constant C depends on κ and τ . Proof. See [14, Appendix] and recall that τ ≥ d.

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K. Bjerklöv

n , Fig. 2. The sets A Bn — approximation of the tangency

2.4. The inductive assumptions (C1)n , (C2)n and (C3)n . Next follows two definitions including the inductive assumptions which we will use to prove Main Theorem. The idea of the proof is to find the energy for which one has a “tangency” between the stable and unstable directions (i.e., between l − and l + ). See Fig. 1 (right). Definition 2.7. For given sequences of integers 0 < M0 < M1 < . . . < Mn and balls I0 ⊃ I1 ⊃ · · · ⊃ In , we define the following sets: n = Td \

n

Mj

(I j + mω), −1 = Td ;

j=0 m=−M j

j (E) = M j +1 (A j ), where A j = {(θ, r ) : θ ∈ I j − M j ω, λ ≤ r ≤ λ2 }; and A E −M +1 B j (E) = E j (B j ), where B j = {(θ, r ) : θ ∈ I j + M j ω, 1/λ2 ≤ r ≤ 1/λ}

for j = 0, 1, . . . , n. Note that −1 ⊃ 0 ⊃ · · · ⊃ n . On each scale n, the set n contains the ’good’ θ . n (E) and The two sets A Bn (E) should be thought of as approximations of the unstable and stable direction, respectively, near the lowest “peak” of l +E in Fig. 1. See Fig. 2. On a finer and finer scale we shall try to find the energy which creates the tangency. Definition 2.8. We say that 0 < M0 < . . . < Mn , balls I0 ⊃ · · · ⊃ In and non-empty closed intervals E−1 ⊃ · · · ⊃ En satisfy Condition (C1)n , (C2)n , (C3)n , respectively, if the following holds (Condition (C1)n consists of two parts, one part concerning forward iterations and one backward): Condition (C1)nF . Assume that θ0 ∈ n−1 , E ∈ En−1 and λ ≤ r0 ≤ λ2 . Then for any k = 0, . . . , N : rkak · · · r Na N ≥ λ(1/2+1/2

n+1 )(N −k)+a

N

, any ak , . . . , an ∈ [1, 2];

1/λ ≤ rk ≤ λ ; 2

rk < λ

2

⇒

θk ∈

(2.5) (2.6)

Mj n−1 j=0 m=1

(I j + mω);

(2.7)


411

where N ≥ 0 is the smallest integer such that θ N ∈ In . Condition (C1)nB . Assume that θ0 ∈ n−1 , E ∈ En−1 and 1/λ2 ≤ r0 ≤ 1/λ. Then for any k = 0, . . . , N : −N · · · r−k−k | ≤ λ−(1/2+1/2 |r−N

a

a

n+1 )(N −k)−a

−N

, any a−N , . . . , a−k ∈ [1, 2]; (2.8)

1/λ ≤ r−k ≤ λ ; 2

2

r−k > 1/λ

⇒

θ−k ∈

(2.9) Mj n−1

(I j − mω);

(2.10)

j=0 m=0

where N ≥ 0 is the smallest integer such that θ−N ∈ In + ω. Condition (C2)n . In ± Mn ω ⊂ n−1 and In ± (Mn + 1)ω ⊂ n−1 .

(2.11)

Condition (C3)n . For all E ∈ En we have n (E) = {(θ, r ) : θ ∈ In + ω, ϕn− (θ, E) ≤ r ≤ ϕn+ (θ, E)}, A Bn (E) = {(θ, r ) : θ ∈ In + ω, ψn− (θ, E) ≤ r ≤ ψn+ (θ, E)}, where the functions ϕn± , ψn± : (In + ω) × En → R are C 2 and satisfy the following (see also Fig. 3): For all (θ, E) ∈ (In + ω) × En we have: ϕn± (θ, E) = λ2 V (θ − ω) − E + φn± (θ, E)

(2.12)

0 < φn+ (θ, E) − φn− (θ, E) < 1/λ Mn ;

(2.13)

0 < ψn+ (θ, E) − ψn− (θ, E) < 1/λ Mn ; |∂ E φn± (θ, E)|, |∂ E ψn± (θ, E)| < 1/λ; φn± C 2 , ψn± C 2 < λ3/2 .

(2.14)

and

(n)

Fig. 3. Condition (C3)n . Cases E = E 1

(2.15) (2.16)

(n)

(left) and E = E 2

(right).

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K. Bjerklöv

Moreover, 1 ϕn− (θ, E) > ψn+ (θ, E), for all θ ∈ (In + ω) \ ( In + ω), E ∈ En . 3

(2.17)

Finally, writing En = [E 1(n) , E 2(n) ], 1 In + ω s.t. ϕn− (θ ∗ , E) = ψn+ (θ ∗ , E); (2.18) 3 1 (n) for E = E 2 ∃ a unique point θ ∗ ∈ In + ω s.t. ϕn+ (θ ∗ , E) = ψn− (θ ∗ , E). (2.19) 3

for E = E 1(n) ∃ a unique point θ ∗ ∈

3. Some Formulas Since the formulas below are of great importance for the next sections, we spend some n and time deriving them in detail. They will give us the geometry control of the sets A Bn , given certain estimates on products of iterates rk . Let r0± = r0± (θ, E) be given and define, as always, rk± (θ, E) = πr kE (θ, r0± (θ, E)) , k ∈ Z. Forward(k > 0). In particular (skipping the ±), r1 (θ, E) = λ2 V (θ ) − E −

1 . r0 (θ, E)

Calculating the different derivatives we obtain (recall that we write θ = (θ 1 , θ 2 , . . . , θ j )): ∂ E r1 = −1 +

∂ E r0 ∂ i r0 , ∂θ i r1 = λ2 ∂θ i V + θ 2 , i = 1, 2, . . . , d; r02 r0 ∂ i j r0 (∂ i r0 )(∂θ j r0 ) ∂θ i θ j r1 = λ2 ∂θ i θ j V + θ θ2 − 2 θ . r0 r03

And for the contraction/expansion we have the formula r1+ − r1− =

r0+ − r0− r0+r0−

.

Hence, by induction, we have the expressions (k > 1): ∂ E rk = −1 −

k−1

1

r 2 · · · rl2 l=1 k−1

∂θ i rk = λ2 ∂θ i V + λ2

k−1

+

∂ E r0 rk−1 · · · r02

∂θ i V

r 2 · · · rl2 l=1 k−1

∂θ i θ j rk = λ2 ∂θ i θ j V + λ2

;

∂ i r0 + 2 θ , i = 1, . . . , d; rk−1 · · · r02

(3.1)

(3.2)

k−1

k−1

(∂ i rl )(∂ j rl ) ∂ i j r0 ∂θ i θ j V θ θ + 2θ θ −2 , 2 2 2 2 · · · r2 · r r · · · r r · · · r r l 0 k−1 l k−1 k−1 l l=1 l=0

i, j = 1, . . . , d;

(3.3)


and rk+ − rk− =

413

r0+ − r0−

+ · · · r + )(r − · · · r − ) (rk−1 0 0 k−1

By using (3.2) we can write the terms (∂θ i rl )(∂θ j rl ) 2 ···r 2 ·r rk−1 l l

=

=

ι=i, j

ι=i, j

(∂θ i rl )(∂θ j rl ) 2 ···r 2 ·r rk−1 l l

l−1

∂θ ι V p=1 r 2 ···r 2 l−1 p 2 ···r 2 r rk−1 l l

λ2 ∂θ ι V +λ2

λ2 ∂θ ι V 1/2 rk−1 ···rl ·rl

+ λ2

=

+

r1+ − r1−

+ · · · r + )(r − · · · r − ) (rk−1 1 1 k−1

.

(3.4)

(3.5)

in (3.3) as:

∂θ ι r0 2 ···r 2 rl−1 0

l−1

∂θ ι V p=1 r ···r r 1/2 r 2 ···r 2 k−1 l l p l−1

+

∂θ ι r0 1/2 2 rk−1 ···rl rl rl−1 ···r02

.

Backward. Similarly, r−1 (θ, E) =

1 λ2 V (θ − ω) − E − r0 (θ, E)

and we obtain (k > 1) ∂ E r−k =

k

2 2 2 2 r−k · · · r−l + (∂ E r0 )r−k · · · r−1 ;

(3.6)

l=1

∂θ i r−k = −λ2

k

2 2 2 2 (∂θ i V )r−k · · · r−l + (∂θ i r0 )r−k · · · r−1 ;

(3.7)

l=1

∂θ i θ j r−k = 2

k−1

(∂θ i r−l )(∂θ j r−l ) 2 (∂ i r−k )(∂θ j r−k ) 2 r−k · · · r−l−1 +2 θ r−l r−k

(3.8)

l=1

− λ2 + r−k

− − r−k

=

k

2 2 2 2 (∂θ i θ j V )r−k · · · r−l ) + (∂θ i θ j r0 )r−k · · · r−1 ;

l=1 + + (r0 − r0− )(r−k

− − + · · · r−1 )(r−k · · · r−1 ).

(3.9)

By using formula (3.7) we can express (∂θ i r−l )(∂ j r−l ) 2 2 θ r−k · · · r−l−1 r−l

=

ι=i, j

3/2

2 2 − λ2 · · · r−1 (∂θ ι r0 )r−k · · · r−l−1 r−l r−l+1

l p=1

(3.10) 3/2 2 2 (∂θ ι V )r−k · · · r−l−1 r−l r−l+1 · · · r− p .

4. Two Help Lemmas In this section we derive two lemmas needed for the inductive procedure later. They are abstract in the sense that they give a conclusion under the assumptions that (C1)nB,F hold. They state, roughly, that the two Conditions (C1)nB,F imply a large part of Condition (C3)n . When we say that (C1)nB,F hold, we assume that all the sets Ek , Ik and the integers Mk appearing in the statement are chosen. The geometric picture behind the next lemma is that the box A is mapped into the very thin set M+1 (A), which looks like the graph of the function λ2 V over In ⊂ I0 , which is a parabola.

414

K. Bjerklöv

Lemma 4.1. Assume that (C1)nF holds for some n ≥ 0 and that λ is larger than some M universal constant. Suppose M > 0 satisfy In −Mω ⊂ n−1 and In ∩ m=1 (In −mω) = ∅. Let A = {(θ, r ) : θ ∈ In − Mω, λ ≤ r ≤ λ2 }. Then, for each E ∈ En−1 we have − + := M+1 A E (A) = {(θ, r ) : θ ∈ In + ω, ϕ (θ, E) ≤ r ≤ ϕ (θ, E)},

where the functions ϕ ± : (In + ω) × En−1 → R satisfy the following: ϕ ± (θ, E) = λ2 V (θ − ω) − E + φ ± (θ, E);

(4.1)

|φ ± | ≤ 1/λ; 0 < φ + (θ, E) − φ − (θ, E) < 1/λ M ; |∂ E φ ± (θ, E)| < 1/λ; |∂θ i φ ± | < 2V C 1 , i = 1, . . . , d; ± |∂θ i θ j φ | < 100λV C 2 , i, j = 1, . . . , d.

(4.2) (4.3) (4.4) (4.5) (4.6)

and

Moreover, πr ( kE (A)) ⊂ [1/λ2 , λ2 ] for all k = 0, 1, . . . , M.

(4.7)

Remark 10. Note that (4.7) implies, by the use of Lemma 2.4, that the upper boundary and lower boundary of the set A is mapped onto the upper and lower boundary of M+1 E (A), respectively. Proof. Let r0− (θ, E) ≡ λ and r0+ (θ, E) ≡ λ2 for θ ∈ In − Mω and E ∈ En−1 be the horizontal boundaries of the set A. Then ± ϕ ± (θ, E) := r M+1 (θ − (M + 1)ω, E), θ ∈ In + ω, E ∈ En−1

where, as always, are the boundaries of the set A, rk (θ, E) = πr kE (θ, r0 (θ, E)) , θ ∈ In − Mω, E ∈ En−1 . Since r M+1 = λ2 V − E − 1/r M , we can write ϕ ± (θ, E) = λ2 V (θ − ω) − E + φ ± (θ, E), where φ ± (θ, E) = −

± rM (θ

1 , θ ∈ In + ω, E ∈ En−1 . − (M + 1)ω, E)

(4.8)

Fix E ∈ En−1 . By the assumption on M, N = M is the smallest positive integer such that a point θ0 ∈ In − Mω satisfy θ N ∈ In . Since we have assumed that In − Mω ⊂ n−1 and that (C1)nF holds, we can apply (C1)nF to the points (θ, r0± (θ )), θ ∈ In − Mω, and


415

obtain the following estimates (with N = M), where we have skipped the superscript ±: for all k = 0, 1, . . . , M, rk · · · r M ≥ λ(M−k)/2+1 , 1/λ2 ≤ rk ≤ λ2 ,

(4.9) (4.10)

rk2 · · · r 2j−1r j r j+1 · · · r M ≥ λ(M−k)/2+1 , ∀ k ≤ j ≤ M. 3/2

(4.11)

From (4.9) it immediately follows (by letting k = M and using (4.8)) that |φ ± (θ, E)| ≤ 1/λ,

(4.12)

and statement (4.7) follows from (4.10). Moreover, using (4.9), the two formulas (3.1) and (3.2), and the fact that r0 = r0 (θ, E) is constant, we get: M ∞

1 1 2 ± ≤ |∂ E φ (θ, E)| = < 2 (4.13) 2 2 k+2 λ λ r M · · · rk k=1

k=0

and ∞ M

∂θ i V 1 2 2 ≤ λ |∂θ i φ(θ, E)| = λ V < 2V C 1 . 1 C 2 · · · r2 k+2 λ rM k k=1

(4.14)

k=0

From (4.9) and formula (3.4) we obtain: 0 < φ+ − φ− = ϕ+ − ϕ− =

r0+ − r0− + + )(r − · · · r − ) (r0 · · · r M 0 M

1 satisfy In + Mω ⊂ n−1 and (In + ω) ∩ m=2 (In + mω) = ∅. Let B = {(θ, r ) : θ ∈ In + Mω, 1/λ2 ≤ r ≤ 1/λ}. Then, for all E ∈ En−1 , (B) = {(θ, r ) : θ ∈ In + ω, ψ − (θ, E) ≤ r ≤ ψ + (θ, E)}, B := −M+1 E where the functions ψ ± : (In + ω) × En−1 → R satisfy the following: |ψ ± | ≤ 1/λ; 0 < ψ (θ, E) − ψ (θ, E) < 1/λ M ; |∂ E ψ ± (θ, E)| < 1/λ; ± |∂θ i ψ | < 2V C 1 , i = 1, . . . , d; ± |∂θ i θ j ψ | < 100λV C 2 , i, j = 1, . . . , d. +

−

(4.17) (4.18) (4.19) (4.20) (4.21)

Moreover, 2 2 πr ( −k E (B)) ⊂ [1/λ , λ ] for all k = 0, 1, . . . , M − 1.

(4.22)

Remark 11. The geometric picture in this case is that the box B is mapped by −M+1 E to a very thin box B which is relatively flat. −M+1 The global idea now is to choose energies E so that the sets M+1 (B) E (A) and E intersect and, in the limit, will create a tangency. This is what we inductively will do in the next section.

5. The Induction This section contains the inductive procedure on which the proof of Main Theorem is based.


417

5.1. The Basic Step. Recall that that diam(I0 ) = c0 /λ1/2 (see Definition 2.1). From Lemma 2.5 we know that a point θ0 in I0 will satisfy θk ∈ / I0 for |k| = 1, 2, . . . N , where N = [c1 λ1/(2τ ) ], c1 = (κ/c0 )1/τ . Definition 5.1. We define Note that M0 ≈

√

M0 = [λ1/(4τ ) ].

(5.1)

N . In the rest of this paper M0 = M0 (λ) will be as above.

Lemma 5.2 (Basic Step). There exists a λ1 = λ1 (κ, τ, V ) > 0 such that for all λ > λ1 there is a closed interval of energies E−1 ⊃ E0 = ∅ such that Conditions (C1)0 , (C2)0 and (C3)0 hold. Proof. Assume that λ is sufficiently large, depending on κ, τ and V . Condition (C1)0 follows by repeated use of Lemma 2.3, and (C2)0 is trivial since −1 = Td by definition. We have to verify that (C3)0 holds. Recall the definition of the sets A0 , A0 , B0 and B0 . The definition of M0 implies that I0 ∩ (I0 + mω) = ∅. 0 0. Thus (2.17)0 holds for all E ∈ E−1 . It remains to find an interval E0 ⊂ E−1 so that (2.18)0 and (2.19)0 hold. Let h(θ, E) = ϕ0+ (θ, E) − ψ0− (θ, E) = λ2 V (θ − ω) − E + φ0+ (θ, E) − ψ0− (θ, E) for θ ∈ I0 + ω and E ∈ E−1 . As we saw above, h(θ, E) > λ for all θ ∈ ∂(I0 + ω) and E ∈ E−1 . Moreover, since V (0) = 0 (so in particular ω ∈ I0 + ω) and since (4.2) and (4.17) hold, we have h(ω, E) < 2 for all E ∈ E−1 . Hence, for each E ∈ E−1 , the function h(θ ) = h(θ, E) has a minimum. Furthermore, V has a unique non-degenerate minimum and (2.16)0 holds, so this minimum has to be unique. Denote by m(E) ∈ I0 +ω the point where the minimum is attained and let g(E) = h(m(E), E) for E ∈ E−1 = [−1, 1]. Then g(−1) ≥ 1 − 2/λ > 1/2, since min V = 0, and g(1) ≤ h(ω, 1) ≤ −1 + 2/λ < −1/2. Moreover, since m(E) is the minimum for fixed E, we get g (E) =

d

i=1

∂θ i h(m(E), E)

dm i (E) + ∂ E h(m(E), E) = ∂ E h(m(E), E) < −1 + 2/λ. dE

418

K. Bjerklöv (0)

In the last step we used the estimate (2.15)0 . Consequently there is (a unique) E = E 2 (0) such that g(E 20 ) = 0. This proves (2.19)0 . Similarly we find E 1 such that (2.18)0 holds. Finally, define (0)

(0)

E0 = [E 1 , E 2 ]. 0 ∩ (Recall Fig. 3.) The analysis also shows that A B = ∅ for all E ∈ E0 .

5.2. The Inductive Step. The following lemma contains the inductive step. Lemma 5.3. There exists a λ2 = λ2 (κ, τ, V ) > 0 such that for all λ > λ2 the following hold: Assume that M0 and I0 are as above and that M0 < . . . < Mn , the balls I0 ⊃ · · · ⊃ In = ∅ and the closed intervals E−1 ⊃ E0 ⊃ · · · ⊃ En = ∅ have been constructed (n ≥ 0), satisfying (if n > 0) λ M j−1 /(4τ ) ≤ M j ≤ 2λ M j−1 /(4τ ) , diam(I j ) = c0 /λ

M j−1 /2

Ij ⊂

,

j = 1, . . . , n,

(5.2)

j = 1, . . . , n,

(5.3)

j = 1, . . . n,

(5.4)

1 I j−1 , 2

and that (C1)n , (C2)n and (C3)n hold. Then there exists an integer Mn+1 , λ Mn /(4τ ) ≤ Mn+1 ≤ 2λ Mn /(4τ ) , a ball In+1 ⊂ (1/2)In , satisfying diam(In+1 ) = 1/λ Mn /2 , and a closed interval ∅ = En+1 ⊂ En such that (C1)n+1 , (C2)n+1 and (C3)n+1 hold. Moreover, n , n+1 ∩ n+1 ⊂ A Bn and A Bn+1 = ∅ for all E ∈ En+1 . A Bn+1 ⊂

(a)n

Remark 12. The reason for having the constant c0 included in the estimate of the diameter of the balls I j is just to get formulas that work for all n ≥ 0: diam(I0 ) = c0 λ M−1 /2 , if we define M−1 = 1. Proof of Lemma 5.3. We assume that λ is sufficiently large, depending on κ, τ and V , so that the statements below hold true, independently of n. Since the proof of this lemma is quite lengthy, we will divide it into steps. We start with some general observations which we will use several times in what follows. To treat the case n = 0 smoothly, we define M−1 = 1. Basics. From the facts that In is a ball with diameter c0 λ−Mn−1 /2 and Mn ∈ [λ Mn−1 /(4τ ) , M 2λ n−1 /(4τ ) ], it folLows from Lemma 2.5 that: (In − Mn ω) ∩

Mn m=0

(In + mω) = ∅, and (In + Mn ω) ∩

Mn

(In − mω) = ∅. (5.5)

m=0

If θ ∈ In − Mn ω, then K = Mn is the smallest positive integer such that θ + K ω ∈ In .

(5.6)


419

Fig. 4. Intersection

If θ ∈ In + Mn ω, then K = Mn − 1 is the smallest positive integer such that θ − K ω ∈ In + ω.

(5.7)

Step 1. We begin by defining the critical set In+1 . The idea is that the projection onto n (E) and the base Td of the intersection between A Bn (E) should be in In+1 + ω for E ∈ En , i.e.,

n (E) ∩ πθ A Bn (E) . In+1 + ω ⊃ E∈En

We have the functions ϕn± (θ, E) = λ2 V (θ − ω) − E + φn± (θ, E) and ψn± (θ, E), θ ∈ In + ω, satisfying the conditions in (C3)n , so we need to estimate the diameter of the sets K (E) = {θ ∈ In + ω : ϕn− (θ, E) ≤ ψn+ (θ, E)} for E ∈ En (see Fig. 4). Let h(θ, E) = ϕn+ (θ, E) − λ−Mn − (ψn− (θ, E) + λ−Mn ). Then h(θ, E) ≤ ϕn− (θ, E) − ψn+ (θ, E) by the estimates (2.13 − 2.14)n . This implies that K (E) := {θ ∈ In + ω : h(θ, E) ≤ 0} ⊃ K (E), For each fixed θ ∈ In + ω we have by (2.15)n : ∂ E h(θ, E) < −1 + 2/λ < −1/2. If we write En = [E 1 , E 2 ], we thus have K (E 2 ) ⊃ K (E) for all E ∈ E.

E ∈ En .

420

K. Bjerklöv

The diameter of K (E 2 ) is estimated as follows: We fix E = E 2 and write only h(θ ). By conditions (2.17)n and (2.19)n , we have h(θ ) ≥ −2λ−Mn for θ ∈ In + ω with equality if and only if θ = θ ∗ , θ ∗ ∈ (1/3)In + ω being the point in (2.19)n . Hence h has its minimum at θ = θ ∗ and ∂θ i h(θ ∗ ) = 0 for all i = 1, 2, . . . , d. Moreover, from (2.2), (2.16)n and the definition of h we have ∂θ i θ j h(θ ) = λ2 ai j + O(λ3/2 ) for i, j = 1, 2, . . . , d. Since the quadratic form ai j x i x j is positive definite (since V has a non-degenerate minimum), it now follows that the set K (E 2 ) is contained in a ball Jn+1 of radius const/λ−(Mn /2+1) , centered at θ = θ ∗ . The constant only depends on the ai j . We have thus shown that

n (E) ∩ πθ A Bn (E) ⊂ Jn+1 . E∈En

We now define In+1 to be the ball of radius c0 /λ−Mn /2 , centered at θ = θ ∗ . Then clearly In+1 ⊃ Jn+1 . Since θ ∗ ∈ (1/3)In + ω and diam(In ) = c0 λ−Mn−1 /2 (i.e., it is much larger than the diameter of In+1 ), it is clear that In+1 ⊂ (1/2)In . For later use, we note that the definition of Jn+1 implies ϕn− (θ, E) > ψ + (θ, E) for all θ ∈ (In + ω) \ (Jn+1 + ω) and E ∈ En . In particular,

n (E) (θ, r ) ∈ A (θ, s) ∈ Bn (E) θ∈ / In+1 + ω

⇒

r > s, for E ∈ En .

(5.8)

(5.9)

Step 2. Now we shall verify that Condition (C1)n+1 holds. We begin with the first half, F . (C1)n+1 Take θ0 ∈ n , λ ≤ r0 ≤ λ2 and E ∈ En , and let N > 0 be the smallest positive integer such that θ N ∈ In+1 . We denote by (C1)[T ], T ∈ N, the condition that for any k = 0, 1, . . . , T , rkak · · · r TaT ≥ λ(1/2+1/2

n+2 )(T −k)+a

T

, any ak , . . . , aT ∈ [1, 2],

1/λ ≤ rk ≤ λ , 2

rk < λ

2

⇒

θk ∈

(5.10) (5.11)

Mj n

(I j + mω).

(5.12)

j=0 m=1 F when changing N to T . Thus we have to prove This is exactly the conditions in (C1)n+1 that (C1)[N ] holds. Let 0 < T0 < T1 < . . . < T p = N be the times such that θT j ∈ In . (Note that we can have p = 0.) From the fact that In is a ball of diameter c0 /λ Mn−1 /2 (by assumption) and since (DC)κ,τ holds, it follows from Lemma 2.5 that

T j+1 − T j ≥ c1 λ Mn−1 /(2τ ) , c1 = (κ/c0 )1/τ .

(5.13)


421

Moreover, since θ0 ∈ n we also have T0 > Mn .

(5.14)

Since n ⊂ n−1 , En ⊂ En−1 and λ ≤ r0 ≤ λ2 , and since (C1)nF holds (by assumption), the weaker condition (C1)[T0 ] also holds (recall that T0 is the smallest positive integer such that θT0 ∈ In ). Assume now that we have proved that (C1)[T j ] holds for some j < p. From the definition of T j we have θT j ∈ In and hence θT j −Mn ∈ In − Mn ω. Since (C2)n holds, we have In − Mn ω ⊂ n−1 . Moreover, from (5.5) we get (In − Mn ω) ∩

Mn

(In + mω) = ∅.

n=1

Consequently, (In − Mn ω) ∩

Mj n

(I j + mω) = ∅

j=0 m=1

(so In − Mn ω does not lie in the set appearing in (5.12)). Since T j − Mn ≥ 0 by (5.14), it hence follows from (5.12) that r T j −Mn ≥ λ. Since also r T j −Mn ≤ λ2 by (5.11), we have (θT j −Mn , r T j −Mn ) ∈ An , and thus n . (θT j +1 , r T j +1 ) ∈ A

(5.15)

Next, let sT j +M j = 1/λ. Then (θT j +M j , sT j +M j ) ∈ Bn . From (C2)n we have In + Mn ω ⊂ n−1 . Furthermore, by (5.7), K = Mn − 1 is the smallest positive integer such that θT j +Mn −K ∈ In + ω. Thus, from (C1)nB (which holds by assumption) we get that sT j +m ≥ 1/λ2 , for m = 1, 2, . . . , Mn .

(5.16)

Since (5.15) holds and (θT j +1 , sT j +1 ) ∈ Bn (by the choice of sT j +Mn ), and since θT j +1 ∈ / In+1 + ω (recall the definition of T j ), we obtain from (5.9) that r T j +1 > sT j +1 . Hence, using (5.16) we have r T j +2 − sT j +2 = 1/sT j +1 − 1/r T j +1 > 0, i.e., r T j +2 > sT j +2 . By induction we end up with rk > sk ≥ 1/λ2 , for k = T j + 1, T j + 2, . . . , T j + Mn . (∗) Note that we could have rk < λ for k = T j + 1, . . . T j + Mn , i.e., for θk ∈ mω).

(5.17) Mn

m=1 (In

+

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K. Bjerklöv

Moreover, from the assumption that (C1)[T j ] holds, we get from (5.11) that r T j ≥ 1/λ2 . Hence it follows from Lemma 2.3 that r T j +1 < λ2 . By also applying Lemma 2.3 to the rk in (5.17) we obtain rk < λ2 , for k = T j + 1, T j + 2, . . . , T j + Mn .

(5.18)

Since sT j +Mn = 1/λ, by definition, (5.17) yields r T j +Mn > 1/λ. From (C2)n we have θT j +Mn ∈ In + Mn ω ⊂ n−1 , and if n = 0 we have (5.5). In any case, θT j +Mn ∈ / I0 . Hence Lemma 2.3 gives 2λ < r T j +Mn +1 < λ2 . Thus, since also θT j +Mn +1 ∈ n−1 by (C2)n , we can apply (C1)nF to the point (θT j +Mn +1 , r T j +Mn +1 ) and conclude that for each k = T j + Mn + 1, . . . , T j+1 we have the following (N = T j+1 − (T j + Mn + 1) being the smallest positive integer such that θ(T j +Mn +1)+N ∈ In ): for any ak , . . . , aT j+1 ∈ [1, 2], aT

rkak · · · r T j+1j+1 ≥ λ

(1/2+1/2n+1 )(T j+1 −k)+aT j+1

;

1/λ2 ≤ rk ≤ λ2 ; rk < λ

⇒

θk ∈

(5.19) (5.20)

Mj n−1

(I j + mω).

(5.21)

j=0 m=1

Since (C1)[T j ] holds, combining (5.12)T j , (∗) and (5.21) yields (5.12)T j+1 ; combining (5.11)T j , (5.17), (5.18) and (5.20) gives (5.11)T j+1 . From (5.17) and (5.19) we get for k ∈ [T j + 1, T j + Mn ] and any exponents ak , . . . , aT j+1 ∈ [1, 2]: aT aT j +Mn aT j +Mn +1 aT j+1 r · · · r rkak · · · r T j+1j+1 = rkak · · · r T j +M T +M +1 T n n j j+1 >

1 4(T j +Mn +1−k)

λ

λ

(1/2+1/2n+1 )(T j+1 −T j −Mn −1))+aT j+1

.

We claim that (1/2 + 1/2n+1 )(T j+1 − T j − Mn − 1)) + aT j+1 − 4(T j + Mn + 1 − k) > (1/2 + 1/2n+2 )(T j+1 − k)) + aT j+1 . Indeed, subtracting LHS by RHS gives T j+1 − 4(T j + Mn + 1 − k) − (1/2 + 1/2n+1 )(T j + Mn + 1) + (1/2 + 1/2n+2 )k. 2n+2 This is larger than (put k = T j + 1 and skip the last term above) T j+1 − T j T j+1 − T j − 4Mn − (1/2 + 1/2n+1 )Mn − 1/2n+2 > − 6Mn . 2n+2 2n+2


423

Since (5.13) holds, and since we have (5.2), the above expression is positive for all large λ, independently of n. Thus, aT

rkak · · · r T j+1j+1 ≥ λ

(1/2+1/2n+2 )(T j+1 −k)+aT j+1

, k = T j + 1, . . . T j + Mn .

(5.22)

Now (5.10)T j+1 follows from (5.10)T j ,(5.19) and (5.22). Thus (C1)[T j+1 ] holds. By F , holds. induction we see that (C1)[N ], i.e., (C1)n+1 B The verification of (C1)n+1 is very similar. We only notice the following, which is the analogue of the passage from T j to T j + Mn above (the argument below Eq. (5.15)). If (θ1 , r1 ) ∈ Bn and θ1 ∈ / In+1 + ω, we let s−Mn = λ. Then (θ−Mn , s−Mn ) ∈ An , so n . Hence it follows from (5.9) that r1 < s1 . Moreover, we can apply (C2)nF (θ1 , s1 ) ∈ A to the point (θ−Mn , s−Mn ) and derive 1/λ2 ≤ sk ≤ λ2 for k = −Mn , . . . , 0.

(5.23)

Since 0 < r1 < s1 and 0 < s0 , it follows from Lemma 2.4 that 1/λ2 ≤ r0 < s0 ≤ λ2 . Inductively, using (5.23), we get 1/λ2 ≤ rk < sk ≤ λ2 for k = −Mn , . . . , 0. Step 3. Next we will choose the number Mn+1 and verify that (C2)n+1 holds. For each j = 0, 1, . . . , n, let N j be the positive integer given by Lemma 2.5 when it is applied to I = 3I j . By the inductive estimates (5.3), and the definition of I0 , we get 1/τ κ κ 1/τ M j−1 /(2τ ) Nj = λ = , j = 1, . . . , n. (5.24) diam(3I j ) 3c0 We thus have (3I j ) ∩

((3I j ) + mω) = ∅, for j = 0, 1, . . . , n.

0 0, it follows from (5.30) that r Mn+1 −Mn ≥ λ. Moreover, from (5.29) we have r Mn+1 −Mn ≤ λ2 . Consequently (θ Mn+1 −Mn , r Mn+1 −Mn ) ∈ An . Since also In+1 ⊂ (1/2)In , this gives (5.27). Statement (5.28) is proved analogously. We shall now choose the interval En+1 and verify that (C3)n+1 and the last part of (a)n+1 , i.e., n+1 ∩ A Bn+1 = ∅ for all E ∈ En+1 , (5.33) ± hold. Since (C1)n+1 holds, it follows from Lemmas 4.1 and 4.2 that the functions ϕn+1 ± and ψn+1 satisfy (2.12 − 2.16)n+1 . Recall that we have

In+1 ∩

(In+1 + mω) = ∅

0 ψn+1 (θ ) for all θ ∈ In+1 \ Jn+1 .

(5.36)

Since the balls Jn+1 and In+1 + ω have the same center, and since the diameter of In+1 is a factor λ bigger than the diameter of Jn+1 , we clearly have Jn+1 ⊂ (1/3)In+1 + ω. By (5.36), this gives (2.17)n+1 . Since (2.18 − 2.19)n hold, and since we have the inclusions (5.34),(5.35) and the derivative estimates on ϕn+1 , ψn+1 , we can proceed as in the proof of Lemma 5.2 to find a non-degenerate interval En+1 ⊂ En such that (2.18 − 2.19)n+1 and (5.33) hold. This completes the proof of Lemma 5.3.

426

K. Bjerklöv

Fig. 5. The Inclusion

6. An Extra Construction The inductive construction we made in the previous section is sufficient to derive statements i) and iii) in Main Theorem. In order to prove statement ii), the result about the geometry of the minimal set M, we will need one more ingredient. This could very well have been included in Lemma 5.3, but we have chosen to split it into two parts for the sake of transparency. So far we have the following: For all λ > max{λ1 , λ2 }, we can apply Lemma 5.2 and Lemma 5.3 to inductively construct critical intervals I0 ⊃ I1 ⊃ · · · ⊃ In ⊃ · · ·, energy intervals E−1 ⊃ E0 ⊃ · · · ⊃ En ⊃ · · ·, and integers M0 ≤ M1 ≤ · · · ≤ Mn ≤ · · · so that (C1 − 3)n hold for each n ≥ 0. We have the estimates M0 = [λ1/(4τ ) ] and λ Mn−1 /(4τ ) ≤ Mn ≤ 2λ Mn−1 /(4τ ) , n = 1, 2, . . . ; and diam(In ) = λ−Mn−1 /2 , n = 0, 2, . . . (M−1 = 1). From now on we shall only work with E = E0 ∈

En = ∅. n≥0

(We will later see that E 0 = min σ (H ).) With this choice of E, (C1 − 3)n hold for each n ≥ 0. Note that E 0 depends on λ. In each critical interval In there is a super-critical interval In+1 . We shall now con of different orders. We proceed as follows: struct super-critical intervals In+1 and In+1 Let M0 M0 M0 = and M0 = . 100 100


427

Fig. 6. The additional sets Bn and Bn

Lemma 6.1. There exists a λ3 = λ3 (κ, τ, V ) > 0 such that the following holds for all λ > λ3 . For each n > 0 there exist positive integers 2Mn Mn 2Mn Mn , and Mn ∈ , Mn ∈ 100 100 10000 10000 such that In + Mn ω, In + (Mn + 1)ω ∈ n−1 In + Mn ω, In + (Mn + 1)ω ∈ n−1 . Remark 13. Note that 50Mn ≤ Mn and 50Mn ≤ Mn . Proof. The proof is almost identical to that in Step 3 in the proof of Lemma 5.3. Let N j be as there. Then (In−1 + pω) ∩

n−1

2M j

(I j + mω) = ∅

j=0 m=−2M j

for at most

(4Mn−1 + 1) + (4Mn−2 + 1)

Nn−1 Nn−1 + 1 + . . . + (4M0 + 1) + 1 Nn−1 Nn−2 N0

integers p ∈ [Mn /100, Mn /100 + Nn−1 ] (resp. p ∈ [Mn /10000, Mn /10000 + Nn−1 ]). Since Nn−1 Mn /10000 the result follows. Next we let (analogously to the definition of the sets Bn and Bn )

n = −Mn +1 (B ), where B = {(θ, r ) : θ ∈ In + M ω, 1/λ2 ≤ r ≤ 1/λ}, B n n n E0 and

−M +1 B n = E 0 n (Bn ), where Bn = {(θ, r ) : θ ∈ In + Mn ω, 1/λ2 ≤ r ≤ 1/λ}, n ≥ 0.

Lemma 6.2. There exists a λ4 = λ4 (κ, τ, V ) > 0 such that for all λ > λ4 we have the satisfying following for each n ≥ 0: There are balls In+1 and In+1 diam(In+1 )=

1 1 , diam(In+1 ) = M /2 ; λ Mn /2 λ n

428

K. Bjerklöv In+1 ⊂ In+1 ⊂ In+1 ⊂

1 In ; 2

n and θ1 ∈ / In+1 + ω, then 1/λ < r Mn < λ2 ; If (θ1 , r1 ) ∈ A n and θ1 ∈ If (θ1 , r1 ) ∈ A / In+1 + ω, then 1/λ < r Mn < λ2 .

Proof. We first prove that

n ⊂ Bn ⊂ B B n .

(6.1)

Take (θ0 , r0 ) ∈ Bn , i.e., θ0 ∈ In + Mn ω and ≤ r0 ≤ 1/λ. Since In + Mn ω ⊂ n−1 by (C2)n , we can apply (C1)nB to (θ0 , r0 ) and get 1/λ2

1/λ2 ≤ r−k ≤ λ2 and r−k > 1/λ

⇒ θ−k ∈

Mj n−1

(6.2)

(I j − mω)

(6.3)

j=0 m=0

for k = 0, 1, 2, . . . , Mn − 1 (recall (5.7)). Since In + Mn ω ⊂ n−1 by Lemma 6.1, since n−1 ∩

Mj n−1

(I j − mω) = ∅

j=0 m=0

and since θ−(Mn −Mn ) ∈ In + Mn ω, we have θ−(Mn −Mn ) ∈ /

Mj n−1

(I j − mω).

j=0 m=0

Thus, (6.2) and (6.3), and the fact that Mn < Mn , implies that 1/λ2 ≤ r−(Mn −Mn ) ≤ 1/λ. n . In the same way Bn ⊂ B Hence (θ−(Mn −Mn ) , r−(Mn −Mn ) ) ∈ Bn , and consequently n ⊂ one proves that B B n. By applying Lemma 4.2 with M = Mn , and recalling the definition of Bn , we get n = {(θ, r ) : θ ∈ In + ω, ψ − (θ ) ≤ r ≤ ψ + (θ )}, B where the functions ψ ± satisfy the estimates in Lemma 4.2. Since we have (6.1), we can use (C3)n and proceed as in Step 1 to see that n ∩ A n ) ⊂ (2/5)In + ω, πθ ( B

that the diameter of this set is < const./λ Mn /2+1 , and that this set intersects the ball Jn+1 , defined in Step 1. See Fig. 7. We now define In+1 to be the interval with diameter /2 M n 1/λ n with the same center as Jn+1 . Then In+1 ⊂ (1/2)In . Moreover, if (θ1 , r1 ) ∈ A + + and θ1 ∈ / In+1 , then r1 > ψ (θ1 ) (recall that ψ is the image of the upper boundary of Bn ). As we did in Step 2, we can use (C1)nB and the fact that E preserves orientation to show that 1/λ < r Mn < λ2 (the argument below Eq. (5.15)). Analogously we can proceed with Bn and define In with the required properties.


429

n n ∩ B Fig. 7. The worst case for the size of the projection of A

The next lemma could be formulated much more generally, but we will only need it in the following setting. Lemma 6.3. There exists a λ5 = λ5 (κ, τ, V ) > 0 such that the following holds for all λ > λ5 and all n ≥ 1: Let θ0 ∈ In + Mn ω and let r0+ = λ2 , r0− = 1/λ, ∂θ i r0± = 0 for k = 0, 1, 2, . . . , T . (i = 1, 2, . . . , d) and assume that T ≥ Mn is such that θk ∈ / In+1 Then

0 < r T+ − r T− < λ−Mn /10 and |∂θ i r T | < λ Mn /10 . Proof. The proof is very similar to Step 2 in the inductive lemma above. Since it becomes a bit technical, due to several cases, let us just very briefly sketch the idea. By assumption . This means that the worst case we can have is that during iteration we never enter In+1 ± of the points (r0 , θ0 ), we have the estimate rk ≥ 1/λ2 for k = l, l + 1, . . . , l + Mn , which gives a derivative of size λ4Mn . This is when we go from In + ω to In + Mn ω (see Fig. 6). But after this we will have rl+Mn > 1/λ (we are not in Bn ) and we will get in better shape. We fix n ≥ 1. We also recall that 50Mn ≤ Mn and 50Mn ≤ Mn , which will be used without further comments in the proof. We will often skip the superscripts ± and write only r0 , r1 , r2 , . . .. By Lemma 6.1 we have In + Mn ω ⊂ n−1 . Thus, θ0 ∈ / I0 . Since r0± ≥ 1/λ it follows ± from Lemma 2.3 that λ ≤ r1 ≤ λ2 . Lemma 6.1 also gives In + (Mn + 1)ω ⊂ n−1 , and F hence we can apply (C2)nF and (C2)n−1 to the points (θ1 , r1± ). We will have two cases, the first one being the worst. Case 1 (There exists 0 < T < T such that θT ∈ In ). Let 0 < T0 < T1 < · · · < T p < T be the times such that θT j ∈ In . Then, using the facts that diam(In ) = c0 /λ Mn−1 /2 and Mn < Mn ≤ 2λ Mn−1 /(4τ ) , and recalling that θ0 ∈ In + Mn ω, it follows from Lemma 2.5 that T0 > c1 λ Mn−1 /(2τ ) − (Mn + 1) > Mn

3/2

and T j+1 − T j > c1 λ Mn−1 /(2τ ) , c1 = (κ/c0 )( 1/τ ). and that we never enter I ) By proceeding as in Step 2 we get (recall that In+1 ⊂ In+1 n+1

rk · · · r T p ≥ λ(T p −k)/2+1 for k = 1, 2, . . . , T p .

430

K. Bjerklöv

n . At this stage we thus have (note that r +r − = λ) Moreover, we get (θT p +1 , r T p +1 ) ∈ A 0 0 r T+p +1 − r T−p +1 = and, using formula (3.2),

r0+ − r0−

r0+ · · · r T+p r0− · · · r T−p

< λ2 /λT p +2 = 1/λT p

|∂θ i r TP +1 | < 2V C 1 .

(6.4)

(6.5)

If T = T p + 1 we are done. If not, we continue as follows. By assumption we have . Since (θ n we thus get from Lemma 6.2 that r T p +M > 1/λ. / In+1 θT p ∈ T p +1 , r T p +1 ) ∈ A n Moreover, as we did in Step 2, we can use (C2)nB to derive r T p +1 , r T p +2 , . . . , r T p +Mn ≥ 1/λ2 . Combining this and the estimates (6.4), (6.5) with formulas (3.2) and (3.4), we get

|∂θ i r TP + j | < λ4Mn +10 , and

(6.6)

r T+p + j − r T−p + j < 1/λT p −4Mn −10 ≤ 1/λT0 −4Mn −10

(6.7)

for j = 1, 2, . . . , Mn + 1. If T = T p + j for some j = 1, . . . , Mn + 1 we are finished. If not, we do as follows. Since r T p +Mn > 1/λ and θT p +Mn ∈ In + Mn ω ⊂ n−1 , so in particular θT p +Mn ∈ / I0 , we get from Lemma 2.3 that λ < r T p +Mn +1 < λ2 . By F to the point Lemma 6.2 we have In + (Mn + 1)ω ⊂ n−1 . Thus we can apply (C2)n−1 (θT p +Mn +1 , r T p +Mn +1 ). Let S0 > T p + Mn +1 be the smallest integer such that θ S0 ∈ In−1 . If S0 ≤ T , let S0 < S1 < · · · < Sq−1 ≤ T < Sq be the times such that θ S j ∈ In−1 . From the definition of T p we note that θ S j ∈ / In for j = 0, 1, . . . , q − 1. If S0 > T , let S−1 = T p + Mn + 1 and q = 0. By proceeding as in Step 2 we get

and

rk · · · r Sq−1 ≥ λ(Sq−1 −k)/2+1 for k = T p + Mn + 1, . . . , Sq−1

(6.8)

rk ≥ 1/λ2 for k = T p + Mn + 1, . . . , Sq .

(6.9)

Since diam(In−1 ) = c0 λ−Mn−2 /2 , it follows from Lemma 2.6 that S0 − (T p + Mn + 1), S j − S j−1 ≤ C(λ Mn−2 /2 )(τ +1) < Mn /100. 3

Using (6.6),(6.7) and (6.8) together with formulas (3.2) and (3.4), we get (rough estimates if q > 0) (6.10) |∂θ i r Sq−1 | < λ4Mn +10 and

r S+q−1 − r S−q−1 < 1/λT0 −4Mn −10 .

(6.11)

Since Sq−1 ≤ T < Sq and Sq − Sq−1 < Mn /100, we can use the rough estimates r j ≥ 1/λ2 for j = Sq−1 , . . . , Sq together with (6.10) and (6.11), and the formulas (3.2) and (3.4), to derive

|∂θ i r T | < λ5Mn


431

and

r T+ − r T− < 1/λT0 −5Mn . This finishes the first case. Case 2. ( θk ∈ / In for k = 1, . . . , T ). In this case we proceed as we did in the second part of the one above. Let 0 < S0 < S1 < · · · < Sq−1 < T ≤ Sq be the times such that θ S j ∈ In−1 . Since diam(In−1 ) = λ−Mn−2 /2 , Lemma 2.6 gives the estimates (τ +1)3 S0 , S j − S j−1 , T − Sq−1 ≤ C λ Mn−2 /2 < Mn /100. F Recall that T ≥ Mn , so we clearly have q > 0. By applying (C1)n−1 to the points (θ1 , r1 ) we get, by proceeding as above and as in Step 2:

rk · · · r Sq−1 ≥ λ(Sq−1 −k)/2+1 for k = 1, . . . , Sq−1 and rk ≥ 1/λ2 for k = 1, . . . , Sq .

(6.12)

|∂θ i r Sq−1 +1 | < 2V C 1

(6.13)

This gives and

r S+q−1 +1 − r S−q−1 +1 < 1/λ Sq−1 = 1/λT +(Sq−1 −T ) < 1/λ Mn −Mn /100 .

(6.14)

If T = Sq−1 + 1 we are done. If not, we use the fact that T − Sq−1 < Mn /100 and the estimates (6.12-6.14) to obtain

|∂θ i r T | < λ5Mn /100 ,

r T+ − r T− < 1/λ Mn −Mn /100−5Mn /100 . 7. Proof of Main Theorem In this final section we finish the proof of the main theorem. Proof of Main Theorem. We begin by recalling the connection between E and the Schrödinger equation −(u k+1 + u k−1 ) + λ2 V (θ0 + (k − 1)ω)u k = Eu k .

(7.1)

If we are given (θ0 , r0 ) and if, as always, (θk , rk ) = kE (θ0 , r0 ), then rk = u k+1 /u k , where {u k } is a solution of (7.1) with initial condition u 1 /u 0 = r0 . Hence, uk = r0 · · · rk−1 . u0

(7.2)

432

K. Bjerklöv

Assume that λ > max{λ1 , . . . , λ5 } and so large that the arguments below hold. λ depends on κ, τ, V . From Lemma 5.2 and Lemma 5.3 we inductively get a sequence {Mk }∞ k=0 satisfying M0 = [λ1/(4τ ) ] (7.3) λ Mn−1 /(4τ ) ≤ Mn ≤ 2λ Mn−1 /(4τ ) , n = 1, 2, . . . , a nested sequence I0 ⊃ I1 ⊃ I2 ⊃ · · · of open balls, satisfying diam(In ) = 1/λ Mn−1 /2 , n ≥ 0, M−1 = 1,

(7.4)

and a nested sequence of closed intervals E−1 ⊃ E0 ⊃ · · ·, each En = ∅, such that (C1)n ,(C2)n and (a)n in Lemma 5.3 hold for each n ≥ 0. Fix E = E 0 ∈ ∩∞ k=1 Ek (since clearly the length of the Ek tends to zero, there is a unique point in the intersection). From now on we work only with this energy and we will write = E 0 . From Lemmas 6.1 and 6.2, we get sequences {Mn }, {Mn } such that M0 = [M0 /100], M0 = [M0 /100] and Mn ∈ [Mn /100, 2Mn /100],

Mn ∈ [Mn /10000, 2Mn /10000], n ≥ 1

(7.5)

and balls I0 ⊃ I1 ⊃ I2 ⊃ · · ·, I0 ⊃ I1 ⊃ I2 ⊃ · · · such that In ⊂ In ⊂ In ⊂

1 In−1 . 2

(7.6)

Let ∞

∞ =

n = Td \ n=1

∞

Mj

(I j + mω)

j=0 m=−M j

be the set of ’good’ phases. Since the volume of a d-dimensional ball with diameter δ is cd δ d , where cd only depends on d, we obtain, by using (7.4): |Td \ ∞ | ≤ cd

∞

(2M j + 1) . λd M j−1 /2 j=0

Here | · | denotes the Lebesgue measure on Td . Since (7.3) holds, and since τ ≥ 1, this expression can be made arbitrarily small by choosing λ large. Thus we have |∞ | > 0.

(7.7)

Part 1. We now prove that E 0 = min{E ∈ σ (H )} and that i) and iii) in Main Theorem hold. Take any θ0 ∈ ∞ and λ ≤ r0 ≤ λ2 . For each n ≥ 0, let Nn > 0 be the smallest integer such that θ Nn ∈ In . Since θ0 ∈ ∞ we must have Nn ≥ Mn . From (C1)nF , which holds for each n with E = E 0 , we get r0 · · · r Nn ≥ λ(Nn +1)/2 ,

(7.8)


433

and 1/λ2 ≤ rk ≤ λ2 , for all k ≥ 0. From Oseledets’ theorem we know that for a.e. θ0 ∈ the solution of (7.1) satisfies

Td

(7.9)

and every 0 = (u 0 , u 1 ) ∈ R2 ,

1 log(u 2k + u 2k+1 ) ± γ (E). k→∞ 2k lim

Using (7.2), (7.7) and (7.8), we conclude that γ (E 0 ) ≥ lim sup k→∞

1 log λ log |r0 · · · rk | ≥ . k 2

Moreover, (7.9) implies that the rotation number is zero (cf. [12]) and consequently we have E 0 ≤ min{E ∈ σ (H )}. (7.10) Since (a)n in Lemma 5.3 hold for each n ≥ 0, it follows from our choice of E = E 0 that n ∩ (A Bn ) = ∅. n≥0

Thus there is a point (θ ∗ , r ∗ ) such that n ∩ (θ ∗ , r ∗ ) ∈ A Bn , for all n ≥ 0.

(7.11)

From now on, (θ ∗ , r ∗ ) always denotes this special point. Let (θ1 , r1 ) = (θ ∗ , r ∗ ). Then (θ−Mn , r−Mn ) ∈ An and (θ Mn , r Mn ) ∈ Bn for each n ≥ 0. Since (C2)n holds, we can apply (C1)nF and (C1)nB to the points (θ−Mn , r−Mn ) and (θ Mn , r Mn ), respectively. Then we get |r−Mn · · · r0 | ≥ λ(Mn +1)/2 and |r1 · · · r Mn | ≤ λ−(Mn )/2 for each n ≥ 0. Recalling the relation (7.2) we see that the Schrödinger equation has an exponentially decaying eigenfunction. In particular, E 0 ∈ σ (H ). Combining this with (7.10) we get that E 0 = min{E : E ∈ σ (H )}. Part 2. Here we shall verify that ii) in Main Theorem holds. First we have to derive some preliminary information. We have proved that E 0 ∈ σ (H ) (in fact that E 0 is the lowest energy in the spectrum) and γ (E 0 ) > 0. Thus we know that has a unique non-empty minimal set M (recall the introduction). We first prove Lemma 7.1.

1 2 M ⊂ T × 2,λ . λ

d

(7.12)

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K. Bjerklöv

Proof. Let ∞ = ∞ ∩ (∞ − ω). Since the measure of ∞ can be made arbitrarily close to 1 (|Td | = 1) by choosing λ large, we clearly have |∞ | > 0. Take θ0 ∈ ∞ and let r0 = 1/λ. Since (C1)nB holds for each n we get (as we did above) 1/λ2 ≤ r−k ≤ λ2 , for all k ≥ 0. Moreover, since θ0 ∈ / I0 (recall the definition of the n ) we get from Lemma 2.3 that λ ≤ r1 ≤ λ2 . By definition we have θ1 = θ0 + ω ∈ ∞ and thus we can apply (C1)nF to the point (θ1 , r1 ) for each n to conclude 1/λ2 ≤ rk ≤ λ2 , for all k ≥ 1. Consequently, the closure of the orbit {(θk , rk )}k∈Z is a closed -invariant set that satisfies ∞ 1 2 d (θk , rk ) ⊂ T × 2 , λ . λ k=−∞

Since has a unique minimal set M, we must have (7.12).

We recall the definition of the functions l ± (θ ), introduced in Sect. 1: l + (θ ) = max{r : (θ, r ) ∈ M} and l − (θ ) = min{r : (θ, r ) ∈ M}. From (7.12) we immediately get 1/λ2 ≤ l − (θ ) ≤ l + (θ ) ≤ λ2 , for all θ ∈ Td .

(7.13)

Next, let (θ ∗ , r ∗ ) be as in (7.11). We now prove that in the fiber above θ ∗ , there is only one point in M, namely (θ ∗ , r ∗ ). This is a ’tangent point’. Lemma 7.2. In particular,

l − (θ ∗ ) = r ∗ = l + (θ ∗ ). (θ ∗ , r ∗ )

(7.14)

∈ M.

Proof. Let θ1 = θ ∗ , s1 = l − (θ ∗ ) and r1 = r ∗ . Since (7.13) holds and since preserves orientation (Lemma 2.4), we must have 1/λ2 ≤ sk < rk for all k ∈ Z. From the choice of (θ1 , r1 ) we have, using (7.11) (θ Mn , r Mn ) ∈ Bn , ∀ n ≥ 0. Thus, from (7.15) and the definition of Bn , we also have (θ Mn , s Mn ) ∈ Bn , ∀ n ≥ 0.

(7.15)


435

Consequently, r Mn − s Mn ≤ 1/λ − 1/λ2 < 1/λ. Applying (C1)nB to each of these points gives |s1 · · · s Mn −1 |, |r1 · · · r Mn −1 | ≤ λ−Mn /2 , and hence, using formula (3.9), we have for each n ≥ 0, |l − (θ ∗ ) − r ∗ | = |s1 − r1 | = |s Mn − r Mn ||r1 · · · r Mn −1 s1 · · · s Mn −1 | ≤ 1/λ · λ−Mn , which forces l − (θ ∗ ) = r ∗ . The proof that r ∗ = l + (θ ∗ ) is similar, using the definition of An and Condition (C1)nF . The next lemma states that the set An (recall the definition) contains ’much’ of M. Lemma 7.3. For each n ≥ 0, the set πθ (M ∩ An ) is λ−Mn /5 -dense in In − Mn ω. Remark 14. We say that a subset 1 of a set ⊂ Td is ε-dense in , if for each θ ∈ , any ball with diameter > ε and centered at θ contains a point from 1 . Proof. We will consider the backward orbit of the point (θ ∗ , r ∗ ) ∈ M. Let (θ1 , r1 ) = (θ ∗ , r ∗ ). Recall that θ ∗ ∈ ∩n≥0 (In + ω). Fix n ≥ 0. Since (C1)iF holds for each i, and i for all i, we can apply (C1) F to the point (θ−Mi , r−Mi ) ∈ Ai for all since (θ1 , r1 ) ∈ A i i and derive that for any −Mi ≤ k ≤ 0, rk < λ

⇒ θk ∈

Mj i

(I j + mω).

j=0 m=1

Consequently, for all k ≤ 0 we have rk < λ

⇒ θk ∈

Mj ∞

(I j + mω).

j=0 m=1

Since (C2)n holds, and since (In − Mn ω) ∩ rk < λ and θk ∈ In − Mn ω

Mn

⇒

m=1 (In

θk ∈

+ mω) = ∅ (recall (5.5)), we get

Mj ∞

(I j + mω).

j=n+1 m=1

M j The biggest “hole” in the set (In − Mn ω)\ ∞ j=n+1 m=1 (I j +mω) must have a diameter which is smaller or equal to the sum of the diameters of the small balls, i.e., it must be smaller or equal to ∞

j=n+1

M j diam(I j ).

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From the estimates (7.3) and (7.4), and the fact that τ ≥ 1, we get ∞

M j diam(I j ) ≤

j=n+1

∞

λ M j−1 /(4τ )−M j−1 /2
0 such that the cylinder C = {θ : θ − x < 3ε1 } × [ p − δ, p + δ] satisfies C ∩ M = ∅. Consider now the orbit of the point (θ ∗ , r ∗ ) ∈ M under . Since this orbit is dense in M, M being a minimal set, it must come arbitrarily close to the point (x, l − (x)) ∈ M. Thus there has to be an integer k such that the point (z, s) = k (θ ∗ , r ∗ ) satisfies z − x < ε1 and |s − l − (x)| < δ. Note that s < p − δ, i.e., the point (z, s)


437

Fig. 8. The jump point y ∈ Td .

is under the cylinder C. Note also that (7.14) holds, and that is a diffeomorphism, so we have l − (z) = s = l + (z). Since M is closed, l + (θ ) (and l − (θ )) has to be close to s for all θ sufficiently close to z. Thus, there is an ε2 > 0 such that l + (θ ) < p − δ for all θ − z < ε2 . Let ε = sup{ε2 : l + (θ ) < p − δ for all θ − z < ε2 } > 0. From the fact that z − x < ε1 and l + (x) ≥ p1 > p, we must have that ε ≤ ε1 . Moreover, by definition, on the circle θ − z = ε there must be a y such that l + (y) ≥ p − δ. We have y − x < ε + ε1 < 2ε1 . If l + (y) ≤ p + δ we would have that the point (y, l + (y)) ∈ M lies in the cylinder C, which is a contradiction. Thus l + (y) > p + δ. Summarizing, if we let D = {θ : θ − y < ε}, we have l + (θ ) < p − δ, for all θ ∈ D and at y, which lies on ∂ D, we have l + (y) > p + δ. Making the Contradiction. The idea now is to show that there are points (θ, r ) in M such that θ ∈ D and r > p − δ, so, by definition, l + (θ ) > p − δ at these θ . This then contradicts the construction of D. Fix n ≥ 0 sufficiently large, depending on δ and ε. Let I be the ball centered at y (the jump point) with diameter diam(In )/4. Let t ≥ 0 be the smallest integer such that (I −tω) ⊂ In +ω (note that we could have (I −t ω)∩(In +ω) = ∅ for some 0 ≤ t < t). Thus, I − (t + Mn + 1)ω ⊂ In − Mn ω ⊂ n−1

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by (C2)n . Now let A = {(θ, r ) : θ ∈ I − (t + Mn + 1)ω, λ ≤ r ≤ λ2 } ⊂ An and note that the horizontal boundaries of A is a part of those of An (recall the definition of An ). Since (C1)nF holds, we get from Lemma 4.1 that Mn +1 (A) = {(θ, r ) : θ ∈ I − tω, ϕ − (θ ) ≤ r ≤ ϕ + (θ )}, where

ϕ ± (θ ) = λ2 V (θ − ω) − E 0 + φ ± (θ )

and 0 < φ+ − φ−
0. For each θ0 ∈ I − tω, let r0± = ϕ ± (θ0 ). By construction, −Mn −1 (θ0 , r0+ ) is a point on the + upper boundary of A, so r−M = λ2 . Hence it follows from Lemma 7.5 that n −1 rk+ ≥ l + (θk ) ≥ 1/λ2 ∀k ∈ Z.

(7.18)

In particular, + r0+r1+ · · · rt−1 ≥

1 . λ2t

(7.19)

By Eqs. (7.16) and (7.17), r0+ − r0− < λ M1 n . Since t ≤ 2Mn and Mn ≤ Mn /50 (recall (7.5)), it follows from Lemma 7.4 that also rk− ≥ 1/λ2 for k = 0, 2, . . . , t − 1, so, − r0−r1− · · · rt−1 ≥

1 . λ2t

Thus, rt+ − rt− =

r0+ − r0−

+ )(r − · · · r − ) (r0+ · · · rt−1 0 t−1

p+δ−1/λ Mn /2 − dλ4Mn +4−Mn /10 > p for all θ − y < λ−Mn /10 , (7.23) provided that n was chosen big. By the definition of the set A, it follows from Lemma 7.3 that πθ (A ∩ M) is λ−Mn /5 dense in I − (Mn + 1 + t). Hence πθ ( Mn +1+t (A) ∩ M) is λ−Mn /5 -dense in I , and hence, in particular, in the ball θ − y < λ−Mn /10 , which lies in I . Since y is a boundary point of the ball D, it thus follows from (7.23) that there must be a point (θ, r ) ∈ M such that θ ∈ D and r > p. This is a contradiction to the choice of D. See Fig. 9. Case 2. (t > 2Mn .) Since A ⊂ An by definition, we automatically have Mn +1 (A) ⊂ n . Consider now the set Mn +Mn (A). It follows from Lemma 6.2 that all points in A Mn +Mn (A), except maybe the points over In+1 + Mn ω, lie above level r = 1/λ and

below r = λ2 . Note that πθ Mn +Mn (A) ⊂ In +Mn ω and I +(Mn −1−t)ω ⊂ In +Mn ω. Let S = {(θ, r ) : θ ∈ I + (Mn − 1 − t)ω, 1/λ ≤ r ≤ λ2 }.

See Fig. 10. Since πθ (A ∩ M) is λ−Mn /5 -dense in I − (Mn + 1 + t) (Lemma 7.3) and thus πθ ( Mn +Mn (A) ∩ M) is λ−Mn /5 -dense in I + (Mn − 1 − t)ω, it hence follows from the ))-dense in I + (M − 1 − t)ω. above observation that πθ (S ∩ M) is (λ−Mn /5 + diam(In+1 n Now we will iterate the set S until we get over I . To do this, we will apply Lemma 6.3, which was designed for this application. First we just note that by the definition of t, there

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K. Bjerklöv

Fig. 10. The set S

is no 0 ≤ t < t such that the ball I −t ω is contained in In +ω. If there is 0 ≤ t < t such that (I − t ω) ∩ (In + ω) = ∅, we hence know that I − t ω must contain a boundary point of In + ω. Since diam(I ) =diam(In )/4, we conclude that (I − t ω) ∩ (1/2In + ω) = ∅. ⊂ (1/2)I , we thus have Since In+1 n (I − t ω) ∩ In+1 = ∅ for all 0 ≤ t < t.

We have hence verified that we indeed can apply Lemma 6.3 to iterate the box S. We get

t+1−Mn (S) = {(θ, r ) : θ ∈ I, ψ − (θ ) ≤ r ≤ ψ + (θ )}, where

ψ + − ψ − < λ−Mn /10 and |∂θ i ψ ± | < λ Mn /10 . Since ψ + (θ ) is the image of the upper boundary of the set S, we get as in the previous case that ψ(y) > l + (y) > p + δ. Proceeding as in Case 1, making use of the estimates on ψ together with the observation ))-dense in I , as we that the projection πθ ( t+1−Mn (S) ∩ M) is (λ−Mn /5 + diam(In+1 ) = λ−Mn /2 , we again noticed above, and recalling that Mn ≤ Mn /50 and diam(In+1 find a point (θ, r ) in M such that θ ∈ D and r > p, which is the contradiction. This completes the proof of Main Theorem.

Acknowledgements. I would like to thank the Royal Swedish Academy of Sciences (KVA) for supporting my visit to the Independent University of Moscow and the Steklov Institute where this work was started. I wish to thank Y. Ilyashenko for all his support in Moscow. Finally I thank STINT (Institutional Grant No. 2002-2052) and SVeFUM for partial financial support in Toronto.

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5. Bjerklöv K.: Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent. Geom. Funct. Anal. 16(6), 1183–1200 (2006) 6. Bjerklöv K.: Positive Lypunov exponent and minimality for the continuous 1-d quasi-periodic Schrödinger equation with two basic frequencies. To appear in Ann. Henri Poincaré 7. Bjerklöv K., Jäger T.H.: Strange non-chaotic attractors in quasi-periodically forced circle maps. In preparation. 8. Bourgain J.: Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematics Studies, 158. Princeton, NJ: Princeton University Press 2005 9. Bourgain, J.: Positivity and continuity of the Lyapounov exponent for shifts on Td with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96, 313–355 (2005) 10. Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. of Math. (2) 152(3), 835–879 (2000) 11. Chan, J.: Methods of variations of potentials of quasi-periodic Schrödinger equation. To appear in Geom. Funct. Anal. 12. Delyon, F., Souillard, B.: The rotation number for finite difference operators and its properties. Commun. Math. Phys. 89(3), 415–426 (1983) 13. Eliasson, L.H.: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179(2), 153–196 (1997) 14. Eliasson, L.H.: Ergodic skew-systems on Td × SO(3, R). Ergodic Theory Dynam. Systems 22(5), 1429–1449 (2002) 15. Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146(3), 447–482 (1992) 16. Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one-dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132(1), 5–25 (1990) 17. Gordon, A.: The point spectrum of the one-dimensional Schrödinger operator. (Russian) Usp. Mat. Nauk 31(1)(190), 257–258 (1976) 18. Grebogi, C., Ott, E., Pelikan, S., Yorke J.: Strange attractors that are not chaotic. Phys. D 13(1–2), 261–268 (1984) 19. Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2) 154(1), 155–203 (2001) 20. Goldstein, M., Schlag, W.: On resonances and the formation of gaps in the spectrum of quasi-periodic Schroedinger equations. Manuscript, available at http://arxive.org/list/math.DS/0511392, 2005 21. Herman, M.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3), 453–502 (1983) 22. Johnson, R.: Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients. J. Differ. Eqs. 61(1), 54–78 (1986) 23. Johnson, R.: Ergodic theory and linear differential. equations. J. Differential Equations 28(1), 23–34 (1978) 24. Johnson, R.: The recurrent Hill’s equation. J. Differ. Eqs. 46(2), 165–193 (1982) 25. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84(3), 403–438 (1982) 26. Jorba, A., Núñez, C., Obaya, R., Tatjer, J.: Old and new results on snas on the real line. Preprint. 27. Jäger, T.H.: On the structure of strange nonchaotic attractors in pinched skew products. To appear in Ergodic Theory Dynam. Systems, doi: 10.1017/S0143385706000745, 2006 28. Jäger, T.H.: The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. Manuscript, available at www.inv.uni-erlargen.de/njaeger/n preprints/sna.ps.2006 29. Jäger, T.H.: Existence and structure of strange non-chaotic attractors. Ph.D Thesis, 2005, available at http://www.ini.uni-erlargen.de/njaeger 30. Keller, G.: A note on strange nonchaotic attractors. Fund. Math. 151(2), 139–148 (1996) 31. Klein, S.: Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function. J. Funct. Anal. 218(2), 255–292 (2005) 32. Millionšˇcikov, V.M.: A proof of the existence of nonregular systems of linear differential equations with quasiperiodic coefficients. (Russian) Differencial’nye Uravnenija 5, 1979–1983 (1969) 33. Oseledets, V.I.: A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. (Russian) Trudy Moskov. Mat. Obšˇc. 19, 179–210 (1968) 34. Prasad, A., Negi, S., Ramaswamy, R.: Strange nonchaotic attractors. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 35. Puig, J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19(2), 355–376 (2006) 36. Ruelle, D.: Ergodic theory of differentiable dynamical systems. Inst. Hautes Études Sci. Publ. Math. No. 50, 27–58 (1979)

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Communications in


On the Asymptotic Stability of Bound States in 2D Cubic Schrödinger Equation E. Kirr1 , A. Zarnescu2 1 Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street,

Urbana, IL 61801, USA. E-mail: [email protected]

2 Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA

Received: 27 February 2006 / Accepted: 26 December 2006 Published online: 28 March 2007 – © Springer-Verlag 2007

Abstract: We consider the cubic nonlinear Schrödinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states. 1. Introduction In this paper we study the long time behavior of solutions of the cubic nonlinear Schrödinger equation (NLS) with potential in two space dimensions (2-d): i∂t u(t, x) = [−x + V (x)]u + γ |u|2 u, u(0, x) = u 0 (x),

t > 0, x ∈ R2 ,

(1) (2)

where γ ∈ R−{0}. The equation has important applications in statistical physics, optics and water waves. It describes certain limiting behavior of Bose-Einstein condensates [8, 14] and propagation of time harmonic waves in wave guides [12, 15, 17]. In the latter, t plays the role of the coordinate along the axis of symmetry of the wave guide. It is well known that this nonlinear equation admits periodic in time, localized in space solutions (bound states or solitary waves). They can be obtained via both variational techniques [1, 28, 21] and bifurcation methods [19, 21], see also the next section. Moreover the set of periodic solutions can be organized as a manifold (center manifold). Orbital stability of solitary waves, i.e. stability modulo the group of symmetries u → e−iθ u, was first proved in [21, 33], see also [9, 10, 24]. In this paper we are going to show that the center manifold is in fact a global attractor for all small, localized in space initial data. This means that the solution decomposes

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into a modulation of periodic solutions (motion on the center manifold) and a part that decays in time via a dispersion mechanism (radiative part). For a precise statement of hypotheses and the result see Sect. 3. Asymptotic stability studies of solitary waves were initiated in the work of A. Soffer and M. I. Weinstein [25, 26], see also [2–4, 6, 11]. Center manifold analysis was introduced in [19], see also [32]. The techniques developed in these papers do not apply to our problem. Indeed the weaker L 1 → L ∞ dispersion estimates for Schrödinger operators in 2-d, see (27), compared to 3-d and higher, respectively lack of end point Strichartz estimates in d = 2, prevent the bootstrapping argument in [6, 19, 25, 26], respectively [11], from closing. The technique of virial theorem, used in [2–4] to compensate for the weak dispersion in 1-d, would require at least a quintic nonlinearity in our 2-d case. Finally, in [32], the nonlinearity is localized in space, a feature not present in our case, which allows the author to completely avoid any L 1 → L ∞ estimates. To overcome these difficulties we used Strichartz estimates, fixed point and interpolation techniques to carefully analyze the full, time dependent, non-Hamiltonian, linearized dynamics around solitary waves. We obtained dispersive estimates that are similar with the ones for the time independent, Hamiltonian Schrödinger operator, see Sect. 4. Related results have been proved for the 1-d and 3-d case in [20, 13, 23] but their argument does not extend to the 2-d case. We relied on these estimates to understand the nonlinear dynamics via perturbation techniques. We think that our estimates are also useful in approaching the dynamics around large 2-d solitary waves while the techniques that we develop may be used in lowering the power of nonlinearity needed for the asymptotic stability results in 1-d and 3-d mentioned in the previous paragraph. Note that, in 3-d, the case of a center manifold formed by two distinct branches (ground state and excited state) has been analyzed. Under the assumption that the excited branch is sufficiently far away from the ground state one, in a series of papers [27, 29– 31], the authors show asymptotic stability of the ground states with the exception of a finite dimensional manifold where the solution converges to excited states. We cannot extend such a result to our 2-d problem as of now. The reason is the slow convergence in time towards the center manifold, t −1+ in 2-d compared to t −3/2 in 3-d. This prevents us from even analyzing the projected dynamics on a single branch center manifold, i.e. the evolution of one complex parameter describing the projection of the solution on the center manifold, and obtain, for example, convergence to a periodic orbit as in [4, 19, 26]. However the evolution of this parameter, respectively two parameters in the presence of the excited branch, is given by an ordinary differential equation (ODE), respectively a system of two ODE’s, and the contribution of most of the terms can be determined from our estimates, see the discussion in Sect. 5. We think it is only a matter of time until the remaining ones will be understood. The paper is organized as follows. In the next section we discuss previous results regarding the manifold of periodic solutions that we subsequently need. In Sect. 3 we formulate and prove our main result. As we mentioned before the proof relies on certain estimates for the linear dynamics which we prove in Sect. 4. We conclude with possible extensions and comments in Sect. 5. Notations. H = − + V ; L p = { f : R2 → C | f measurable and R2 | f (x)| p d x < ∞}, f p = R2 | f (x)| p d x)1/ p denotes the standard norm in these spaces; < x >= (1+|x|2 )1/2 , and for σ ∈ R, L 2σ denotes the L 2 space with weight < x >2σ , i.e., the space of functions f (x) such that < x >σ f (x) are square integrable endowed with the norm f (x) L 2σ = < x >σ f (x)2 ;

Asymptotic Stability of Bound States in 2D Cubic Schrödinger Equation

445

f, g = R2 f (x)g(x)d x is the scalar product in L 2 , where z = the complex conjugate of the complex number z; Pc is the projection on the continuous spectrum of H in L 2 ; H n denote the Sobolev spaces of measurable functions having all distributional partial derivatives up to order n in L 2 , · H n denotes the standard norm in this space. 2. Preliminaries. The Center Manifold The center manifold is formed by the collection of periodic solutions for (1): u E (t, x) = e−i Et ψ E (x),

(3)

where E ∈ R and 0 ≡ ψ E ∈ H 2 (R2 ) satisfy the time independent equation: [− + V ]ψ E + γ |ψ E |2 ψ E = Eψ E .

(4)

Clearly the function constantly equal to zero is a solution of (4) but (iii) in the following hypotheses on the potential V allows for a bifurcation with a nontrivial, one parameter family of solutions: (H1) Assume that (i) There exists C > 0 and ρ > 3 such that: |V (x)| ≤ C < x >−ρ ,

f or all x ∈ R2 ;

(ii) 0 is a regular point1 of the spectrum of the linear operator H = − + V acting on L 2 ; (iii) H acting on L 2 has exactly one negative eigenvalue E 0 < 0 with corresponding normalized eigenvector ψ0 . It is well known that ψ0 (x) can be chosen strictly positive and exponentially decaying as |x| → ∞. Conditions (i)-(ii) guarantee the applicability of dispersive estimates of Murata [16] and Schlag [22] to the Schrödinger group e−i H t , see Sect. 4. In particular (i) implies the local well posedness in H 1 of the initial value problem (1-2), see Sect. 3. Condition (iii) guarantees bifurcation of nontrivial solutions of (4) from (E 0 , ψ0 ). In Sect. 5, we discuss the possible effects of relaxing (iii) to allow for finitely many negative eigenvalues. We construct the center manifold by applying the standard bifurcation argument in Banach spaces [18] for (4) at E = E 0 . We follow [19] and decompose the solution of (4) in its projection onto the discrete and continuous part of the spectrum of H: ψ E = aψ0 + h, a = ψ0 , ψ E , h = Pc ψ E . Using the notations f p (a, h) ≡ ψ0 , |aψ0 + h|2 (aψ0 + h), f c (a, h) ≡ Pc |aψ0 + h|2 (aψ0 + h) ,

(5) (6)

1 This condition is somewhat stronger than H ψ = 0 has no solutions satisfying < x >1+ ψ ∈ H 2 , > 0. For an exact definition and more details see [22, Definition 7] and Mµ = {0} in relation (3.1) in [16].

446


and projecting (4) onto ψ0 and its orthogonal complement = Range Pc we get: h = −γ (H − E)−1 f c (a, h), E 0 − E = −γ a −1 f p (a, h).

(7) (8)

Although we are using milder hypothesis on V the argument in the Appendix of [19] can be easily adapted to show that: F(E, a, h) = h + γ (H − E)−1 f c (a, h) is a C 1 function from (−∞, 0) × C × L 2σ ∩ H 2 to L 2σ ∩ H 2 , here C is viewed as a two dimensional real vectorial space. Moreover, F(E 0 , 0, 0) = 0, Dh F(E 0 , 0, 0) = I, ˜ a) and, by the implicit function theorem, there exists δ1 > 0 and the C 1 function h(E, 2 2 from (E 0 − δ1 , E 0 + δ1 ) × {a ∈ C : |a| < δ1 } to L σ ∩ H such that (7) has a unique ˜ solution h = h(E, a) for all h, h L 2σ ∩H 2 < δ1 , E ∈ (E 0 − δ1 , E 0 + δ1 ) and |a| < δ1 . Note that if (a, h) solves (7) then (eiθ a, eiθ h), θ ∈ (0, 2π ) is also a solution, hence by uniqueness we have: a ˜ ˜ h(E, a) = h(E, |a|). (9) |a| Because ψ0 is real valued, we could apply the implicit function theorem to (7) under the restriction a ∈ R and h in the subspace of real valued functions as it is actually done in ˜ [19]. By uniqueness of the solution we deduce that h(E, |a|) is a real valued function. ˜ Replacing now h=h(E, a) in (8) and using (5) and (9) we get the equivalent formulation: ˜ E 0 − E = −γ |a|−1 f p (|a|, h(E, |a|)). (10) To this we can apply again the implicit function theorem by observing that G(E, a) = ˜ E 0 − E + γ |a|−1 f p (|a|, h(E, |a|)) is a C 1 function [19, Appendix] from (E 0 − δ1 , E 0 + δ1 ) × (−δ1 , δ1 ) to R with the properties G(E 0 , 0) = 0, ∂ E G(E 0 , 0) = −1. We obtain the existence of 0 < δ ≤ δ1 , 0 < δ E ≤ δ1 and the C 1 function E˜ : (−δ, δ) → (E 0 − δ E , E 0 + δ E ) such that, for |E − E 0 | < δ E , |a| < δ, the unique solution of (8) ˜ ˜ with h = h(E, a), is given by E = E(|a|). If we now define: h(a) ≡

a ˜ ˜ h( E(|a|), |a|) |a|

we have the following center manifold result: Proposition 2.1. There exist δ E , δ > 0 and the C 1 function h : {a ∈ C : |a| < δ} → L 2σ ∩ H 2 , such that for |E − E 0 | < δ E and ψ E L 2σ ∩H 2 < δ, the eigenvalue problem (4) has a unique solution up to multiplication with eiθ , θ ∈ (0, 2π ), which can be represented as: ψ E = aψ0 + h(a),

ψ0 , h(a) = 0, |a| < δ.


447

Note that differentiating the identity: h(aeiθ ) ≡

aeiθ ˜ ˜ h( E(|a|), |a|) |a|

with respect to θ at θ = 0 we get: Dh|a ia = i h(a),

for all a ∈ C, |a| < δ.

(11)

Since ψ0 (x) is exponentially decaying as |x| → ∞ the proposition implies that ψ E ∈ L 2σ . A regularity argument, see [25], gives a stronger result: Corollary 2.1. For any σ ∈ R, there exists a finite constant Cσ such that: < x >σ ψ E H 2 ≤ Cσ ψ E H 2 . We are now ready to prove our main result. 3. Main Result. The Collapse on the Center Manifold Theorem 3.1. Assume that Hypothesis (H1) is valid and fix σ > 1. Then there exists an ε0 > 0 such that for all initial conditions u 0 (x) satisfying max{u 0 L 2σ , u 0 H 1 } ≤ ε0 , the initial value problem (1)-(2) is globally well-posed in H 1 . Moreover, for all t ∈ R and p0 ≥ 6, we have that: u(t, x) = a(t)ψ0 (x) + h(a(t)) +r (t, x),

(12)

ψ E (t)

r (t) L 2

−σ

≤ C¯ 1 ( p0 )

ε0 , (1 + |t|)1−2/ p0 1−2/ p

r (t) L p

log 1−2/ p0 (2 + |t|) ≤ C¯ 2 ( p, p0 ) ε0 , 2 ≤ p ≤ p0 , (1 + |t|)1−2/ p

where the constants C¯ 1 , C¯ 2 , are independent of ε0 . Before proving the theorem let us note that (12) decomposes the evolution of the solution of (1)-(2) into an evolution on a center manifold ψ E (t) and the “distance” from the center manifold r (t). The estimates on the latter show collapse of solution onto the center manifold. The evolution on the center manifold is determined by Equation (14) below. We discuss it in Sect. 5. Proof of Theorem 3.1. It is well known that under Hypothesis (H1)(i) the initial value problem (1)-(2) is locally well posed in the energy space H 1 and its L 2 norm is conserved, see for example [5, Cor. 4.3.3. at p. 92]. Global well posedness follows via energy estimates from u 0 2 small, see [5, Remark 6.1.3 at p. 165]. In particular we can define a(t) = ψ0 , u(t), for t ∈ R.

448


Cauchy-Schwarz inequality implies |a(t)| ≤ u(t)2 ψ0 2 = u 0 2 ≤ ε0 , for t ∈ R, where we also used conservation of L 2 norm of u. Hence, if we choose ε0 < δ we can define h(a(t)), t ∈ R, see Proposition 2.1. We then obtain (12), where r (t) = u(t) − a(t)ψ0 − h(a(t)), ψ0 , r (t) ≡ 0. The solution is now described by the scalar a(t) ∈ C and r (t) ∈ C(R, H 1 ). To obtain their equations we plug (12) into (1): i

da ∂r da ψ0 + i Dh|a +i = H ψ E + Hr + γ |u|2 u dt dt ∂t = Eψ E + Hr + γ [|u|2 u − |ψ E |2 ψ E ] = Eaψ0 + Eh(a) + Hr (13) +γ [2|ψ E |2 r + ψ E2 r¯ + 2ψ E |r |2 + ψ¯ E r 2 + |r |2 r ],

where we used (4). Projecting now onto ψ0 and taking into account that h, Dh have range orthogonal to ψ0 , while r and ∂r ∂t are orthogonal to ψ0 , we get: i

da = E(|a(t)|)a(t) + γ ψ0 , 2|ψ E |2 r + ψ E2 r¯ + 2ψ E |r |2 + ψ¯ E r 2 + |r |2 r . dt

(14)

The projection of (13) onto the orthogonal complement of ψ0 = range of Pc gives: i Dh|a

∂r da +i = Eh(a) + Hr + γ Pc [2|ψ E |2 r + ψ E2 r¯ + 2ψ E |r |2 + ψ¯ E r 2 + |r |2 r ]. dt ∂t

We then plug in (14) and use the identity Dh|a (−i Ea) = −Ei h(a), see (11), to obtain i

∂r = Hr + γ Pc [2|ψ E |2 r + ψ E2 r¯ + 2ψ E |r |2 + ψ¯ E r 2 + |r |2 r ] ∂t +iγ Dh|a(t) iψ0 , 2|ψ E |2 r + ψ E2 r¯ + 2ψ E |r |2 + ψ¯ E r 2 + |r |2 r .

(15)

In order to obtain the estimates for r (t), we analyze Eq. (15). In the next section we study its linear part: i ∂z = H z + γ Pc [2|ψ E |2 z + ψ E2 z¯ + iγ Dh|a(t) iψ0 , 2|ψ E |2 z + ψ E2 z¯ ] ∂t z(s) = v. Let us denote by (t, s)v the operator which associates to the function v the solution of the above equation: def

(t, s)v = z.

(16)

Using Duhamel’s principle, (15) becomes

t r (t) = (t, 0)r (0) − iγ (t, s)Pc [2ψ E |r (s)|2 + ψ¯ E r 2 (s) + |r |2 r (s)]ds 0

t +γ (t, s)Dh|a(t) iψ0 , 2ψ E |r |2 + ψ¯ E r 2 + |r |2 r ds. (17) 0


449

It is here where we differ essentially from the approach for the 1-d case [2–4] and 3-d case [6, 25, 26, 19]. The right-hand side of our equation contains only nonlinear terms in r. Hence, if we make the ansatz r (t) ∼ (1 + t)−3/4 then the quadratic and cubic terms in (17) decay like (1 + s)−6/4 respectively (1 + s)−9/4 . Both are integrable functions in time, hence, via convolution estimates, the integral term on the right-hand side decays like (t, 0). We have a chance of "closing" the ansatz provided (t, 0) ∼ (1 + t)−3/4 . Contrast this with the case in the above cited papers where a linear term in r is present on the right-hand side. The same argument leads to a loss of 1/4 power decay in the linear term and requires (t, 0) ≡ e−i H t ∼ (1 + t)−1−δ , δ > 0 for closing. This turns out to be impossible in L p norms in 2-d, see (27), while the use of weighted L 2 norms, see (26), for delocalized terms as the cubic term in (17), would require compensation via virial inequalities, see [2], which needs a much higher power nonlinearity than cubic in the delocalized terms2 . In the following we make the above heuristic argument rigorous. Essential in our proof are the estimates for (t, s). Since (t, s) is the propagator for the linearization around a changing nonlinear bound state, its properties may have applications beyond our result, so we chose to study them separately in Sect. 4. Here we show how to control the nonlinear terms. Consider the nonlinear operator in (17):

t (N u)(t) = −iγ (t, τ )Pc [2ψ E |u|2 + ψ¯ E u 2 + |u|2 u]dτ 0

t +γ (t, τ )Dh|a(τ ) iψ0 , 2ψ E |u|2 + ψ¯ E u 2 + |u|2 udτ. 0

In order to apply a contraction mapping argument for (17) in the Banach space: Y = {u : R → L 2−σ ∩ L p ∩ L 2 | sup (1 + |t|) t≥0

sup t≥0

1− 2p

u(t) L 2 < ∞ −σ

(1 + |t|)1−2/ p u(t) L p < ∞, sup u(t) L 2 < ∞} log(2 + |t|) t≥0

endowed with the norm uY = max{sup (1 + |t|) t≥0

1− 2p

u(t) L 2 , sup −σ

t≥0

(1 + |t|)1−2/ p u(t) L p , sup u(t) L 2 }, log(2 + |t|) t≥0

where p ≥ 6 is fixed, it is sufficient to prove the following two properties of N : Lemma 3.1. We have: (i) If u ∈ Y then N u ∈ Y, i.e. N : Y → Y is well defined. (ii) There exists C˜ > 0 such that 2 2 ˜ N u 1 − N u 2 Y ≤ C(u 1 Y + u 2 Y + u 1 Y + u 2 Y )u 1 − u 2 Y .

In particular N is locally Lipschitz. ˜ Moreover, C˜ = C(C, C p , C p,q0 ) where the constants C, C p , C p,q0 are those from the linear estimates for (t, s) (see Theorems 4.1, 4.2 in the next section). 2 Heuristically we arrived at quintic power nonlinearity; hence this technique may be applicable to the quintic Schrödinger in 2-D but definitely not to the cubic one.

450


Note that the lemma finishes the proof of Theorem 3.1. Indeed, if we denote: v = (t, 0)r (0), then vY ≤ C0 r (0) L 2σ , where C0 = max{C, C p , C2 }, see Theorem 4.1. We chose ε0 in the hypotheses of Theorem 3.1, such that

1 ˜ 1 + 2/C − 1 . C0 ε0 < 2 Then, by continuity there exists 0 < Li p < 1 such that:

2 − Li p ˜ vY ≤ 1 + 2Li p/C − 1 . 4

(18)

Let R = LvY /(2 − Li p) and B(v, R) be the closed ball in Y with center v and radius R. A direct calculation shows that the right-hand side of (17): Ku = v + Nu leaves B(v, R) invariant, i.e. K : B(v, R) → B(v, R), and it is a contraction with Lipschitz constant Li p on B(v, R). By the contraction mapping argument, (17) has a unique solution in B(v, R) ⊂ Y. We now have two solutions of (15), one in C(R, H 1 ) from classical well posedness theory and one in C(R, L 2−σ ∩ L 2 ∩ L p ), p ≥ 6 from the above argument. Using uniqueness and the continuous embedding of H 1 in L 2−σ ∩ L 2 ∩ L p , we infer that the two solutions must coincide. Therefore, the time decaying estimates in the lemma hold also for the H 1 solution. The L q , 2 ≤ q ≤ p estimates in the theorem follow from interpolation: 1/q−1/ p

1/2−1/q

p p r (t) L q ≤ r (t) L1/2−1/ r (t) L1/2−1/ . p 2

It remains to prove Lemma 3.1: Proof of Lemma 3.1. Let us observe that it will suffice to show part (ii) and then, using the fact that N (0) ≡ 0, we will have part (i). Let u 1 , u 2 ∈ Y and consider the difference N u 1 − N u 2 , which is

t (N u 1 − N u 2 )(t) = −iγ (t, τ )Pc [2ψ E (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) 0

+ψ¯ E (u 1 − u 2 )(u 1 + u 2 ) +(u 1 − u 2 )|u 1 | + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |)]dτ 2

+γ 0

t

(t, τ )Dh|a(τ ) iψ0 , 2ψ E (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) + ψ¯ E (u 1 − u 2 )(u 1 + u 2 ) + (u 1 − u 2 )|u 1 |2 + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |)dτ.


451

The L 2−σ estimate. We can work under the less restrictive hypothesis: 4 < p < ∞. Let L p be the dual of L p , i.e. 1/ p + 1/ p = 1. We have

t (t, τ ) L 2 →L 2 × N u 1 − N u 2 L 2 ≤ |γ | −σ

σ

0

−σ

× 2ψ E < x > (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) + ψ¯ E < x > (u 1 − u 2 )(u 1 + u 2 ) L 2 dτ σ

σ

A

t

+|γ |

(t, τ ) L p →L 2 × −σ

0

× (u 1 − u 2 )|u 1 |2 + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |) L p dτ B1

B2

t

+|γ | 0

(t, τ ) L 2 →L 2 Dh|a(τ ) L 2σ × σ

−σ

×{|ψ0 , 2ψ E (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) + ψ¯ E (u 1 − u 2 )(u 1 + u 2 )| + F

+ |ψ0 , (u 1 − u 2 )|u 1 | + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |)|}dτ. 2

G

(19) To estimate the term A we observe that < x >σ ψ E (|u 1 |−|u 2 |)(|u 1 |+|u 2 |) L 2 ≤ < x >σ ψ E L α u 1 −u 2 L p |u 1 |+|u 2 | L p (20) with α1 + 2p = 21 . Then

t (t, τ ) L 2 →L 2 A(τ )dτ σ −σ 0

t C ≤ · 3ψ E < x >σ |u 1 − u 2 |(|u 1 | + |u 2 |) L 2 dτ 2 0 (1 + |t − τ |) log (2 + |t − τ |)

t log2 (2 + |τ |) |u 1 | − |u 2 |Y |u 1 | + |u 2 |Y ˜ ≤ 3C C1 · 2 (2 + |t − τ |) (1− 2 ) (1− 2 ) (1 + |t − τ |) log 0 (1 + |τ |) p (1 + |τ |) p ≤ 3C C˜ 1 C˜ 2 (u 1 Y + u 2 Y )

u 1 − u 2 Y , (1 + |t|) log2 (2 + |t|)

where for the first inequality we used Theorem 4.1, part (i). The constants are given by t 2 (2+|τ |) C˜ 1 = supt>0 < x >σ ψ E L α and C˜ 2 = supt>0 (1+|t|) log2 (2+|t|) 0 (1+|t−τlog|) log 2 (2+|t−τ |) · dτ

(1+|τ |)

2− 4p

< ∞, because p > 4.

To estimate the cubic terms B1 , B2 we can not use the term ψ E as before, and this is what forces us to work in the L p space. We have: (u 1 − u 2 )|u 1 |2 L p ≤ u 1 − u 2 L p u 1 2L α , respectively (u 1 − u 2 )(u 2 |u 1 | + u 2 |u 2 |) L p ≤ u 1 − u 2 L p u 2 L α (u 1 L α + u 2 L α ),

452

E. Kirr, A. Zarnescu 2 α

with

+

1 p

=

1 p . Since 4

≤ p we have 2 ≤ α ≤ p. Therefore we can again interpolate:

u i bL p , i = 1, 2, u i L α ≤ u i 1−b L2 where

1 α

=

1−b 2

+ bp . Combining these relations we obtain for B1 : u 1 2b (u 1 − u 2 )|u 1 |2 L p ≤ u 1 − u 2 L p u 1 2(1−b) Lp L2

(21)

respectively, for B2 : u 2 bL p (u 1 − u 2 )(u 2 |u 1 | + u 2 |u 2 |) L p ≤ u 1 − u 2 L p u 2 1−b L2 ×(u 1 1−b u 1 bL p + u 2 1−b u 2 bL p ) L2 L2

(22)

with 1 2(1 − b) 2b 1 + + = . 2 p p p A consequence of this relation and of p < ∞ is: (1 −

2 )(1 + 2b) = 1 + 2/ p > 1, p

(23)

which will play an essential role in what follows. Thus, the estimate for the term containing B1 + B2 is

t

(t, τ )Pc L p →L 2 B1 + B2 L p dτ ≤ C p (u 1 2Y + u 2 2Y )u 1 − u 2 Y × −σ

0

t

×

log(2 + |τ |)(1+2b) |t − τ |

0

1− 2p

·

1 (1− 2p )(1+2b)

(1 + |τ |)

≤ C p C˜ 3 (u 1 2Y + u 2 2Y )

dτ

u 1 − u 2 Y , (1 + |t|)1−2/ p

where for the first inequality we used Theorem 4.1, part (ii), inequalities (21), (22) and the definition of the norm in Y. For the last inequality we used the fact that (1− 2p )(1+2b) > 1 t |)(1+2b) 1 dτ < ∞. (see (23)) with C˜ 3 = supt>0 (1 + |t|)1−2/ p 0 log(2+|τ1− 2 (1− 2 )(1+2b) |t−τ |

For estimating the term containing F we have

p

(1+|τ |)

p

|F| ≤ 3ψ0 L ∞ ψ E L α u 1 − u 2 L p (u 1 L p + u 2 L p ) = 1. Then, the term containing F is estimated as the term containing A with C˜ 1 replaced by C˜ 4 = supτ >0 Dh|a(τ ) L 2σ ψ0 L ∞ ψ E L α . We estimate G as 1 α

+

2 p

|G| ≤ ψ0 < x >σ L α u 1 − u 2 L 2 (u 1 2L p + u 2 2L p + u 1 L p u 2 L p ) −σ


with

t 0

1 α

+

1 2

+

453

= 1. Then the term containing G is estimated as

2 p

(t, τ ) L 2 →L 2 |G|dτ ≤ 3 σ

−σ

t 0

Cψ0 < x >σ L α log2 (2 + |τ |) dτ · (1 + |t − τ |) log2 (2 + |t − τ |) (1 + |τ |)3−6/ p u 1 − u 2 Y ≤ 3C C˜ 5 (u 1 2Y + u 2 2Y ) (1 + |t|) log2 (2 + |t|)

t 2 (2+|τ |) with C˜ 5 = ψ0 < x >σ L α supt>0 (1+|t|) log2 (2+|t|) 0 (1+|t−τ |) loglog 2 (2+|t−τ |)(1+|τ |)3−6/ p dτ < ∞ because p > 3. The L p estimate. With p , q0 , q given by Theorem 4.2, we have

t N u 1 − N u 2 L p ≤ |γ | (t, τ ) L 2σ →L p × 0

× 2ψ E < x >σ (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) + ψ¯ E < x >σ (u 1 − u 2 )(u 1 + u 2 ) L 2 dτ A

+|γ |

t

(t, τ )

0

L p ∩L q0 ∩L q →L p

×

× (u 1 − u 2 )|u 1 |2 + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |) p q0 q dτ L ∩L ∩L B1

+|γ | 0

B2 t

(t, τ ) L 2σ →L p Dh|a(τ ) L 2σ ×

×{|ψ0 , 2ψ E (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) + ψ¯ E (u 1 − u 2 )(u 1 + u 2 )| + F

+ |ψ0 , (u 1 − u 2 )|u 1 |2 + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |)|}dτ. G

(24) The term A can be treated exactly as before and for the term we use Theorem 4.1, part (iii). Since 1 < p , q0 , q ≤ 4/3, we can estimate the B1 , B2 terms in each of the norms L p , L q0 , L q as we did above for their L p norm only. For we use Theorem 4.2, part (iii). The terms F and G are also treated as in the previous case. The convolution integrals in (24) will all decay like (1 + |t|)−(1−2/ p) except the second one which will have a logarithmic correction dominated by log(2 + |t|). The L 2 estimate. We have

t N u 1 − N u 2 L 2 ≤ |γ | (t, τ ) L 2σ →L 2 × 0

× 2ψ E < x > (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) + ψ¯ E < x >σ (u 1 − u 2 )(u 1 + u 2 ) L 2 dτ σ

A

+|γ |

t

0

(t, τ ) L p ∩L 2 →L 2 ×

× (u 1 − u 2 )|u 1 |2 + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |) L p ∩L 2 dτ B1

B2

454


+|γ | 0

t

(t, τ ) L 2σ →L 2 Dh|a(τ ) L 2σ ×

×{|ψ0 , 2ψ E (|u 1 | − |u 2 |)(|u 1 | + |u 2 |) + ψ¯ E (u 1 − u 2 )(u 1 + u 2 )| + F

+ |ψ0 , (u 1 − u 2 )|u 1 |2 + (|u 1 | − |u 2 |)(u 2 |u 1 | + u 2 |u 2 |)|}dτ. G

(25)

We estimate the term A as in (20) while the estimates in L p for B1 , B2 are as in (21) and (22). For their estimate in L 2 norm we use (u 1 − u 2 )|u 1 |2 L 2 ≤ u 1 − u 2 L p u 1 2L α , respectively (u 1 − u 2 )(u 2 |u 1 | + u 2 |u 2 |) L 2 ≤ u 1 − u 2 L p u 2 L α (u 1 L α + u 2 L α ), with

2 α

+

1 p

= 21 . Since 6 ≤ p we have 4 ≤ α ≤ p. Therefore we can again interpolate: u i L α ≤ u i 1−b u i bL p , i = 1, 2, L2

where

1 α

=

1−b 2

+ bp . Combining these relations we obtain for B1 : 2(1−b)

(u 1 − u 2 )|u 1 |2 L 2 ≤ u 1 − u 2 L p u 1 L 2

u 1 2b Lp,

respectively, for B2 : (u 1 − u 2 )(u 2 |u 1 | + u 2 |u 2 |) L 2 ≤ u 1 − u 2 L p u 2 1−b u 2 bL p L2 ×(u 1 1−b u 1 bL p + u 2 1−b u 2 bL p ), L2 L2 with 2(1 − b) 2b 1 1 + + = . 2 p p 2 A consequence of this relation is: (1 −

2 )(1 + 2b) = 2. p

Using now the definition of the norm in Y we will have: B1 + B2 2L ≤ u 1 − u 2 Y (u 1 2Y + u22 )

log2 p/( p−2) (2 + |t|) . (1 + |t|)2

The previous estimates for F and G suffice here as well. Recalling from Theorem 4.1, part (iii) and Theorem 4.2, part (ii), that (t, τ ) L 2σ →L 2 and (t, τ ) L p ∩L 2 →L 2 are bounded, and combining with the estimates above, as well as taking into account the definition of the functional space Y we have that N u 1 − N u 2 L 2 ≤ Cu 1 − u 2 Y [C˜ 6 (u 1 Y + u 2 Y ) + C˜ 7 (u 1 2Y + u 2 2Y )] t log2 (2+|τ |) t log(1+2b) (2+|τ |) dτ 1 and some constant C M > 0 independent of u 0 and t ∈ R, see [16, Theorem 7.6], and Cp e−i H t Pc u 0 L p ≤ 1−2/ p u 0 L p (27) |t| for some constant C p > 0 depending only on p ≥ 2 and p given by p −1 + p −1 = 1. The case p = ∞ in (27) is proven in [22]. The conservation of the L 2 norm, see [5, Corollary 4.3.3], gives the p = 2 case: e−i H t Pc u 0 L 2 = u 0 L 2 . The general result (27) follows from Riesz-Thorin interpolation. We would like to extend these estimates to the linearized dynamics around the center manifold. In other words we consider the linear equation, with initial data at time s, i ∂z = (− + V (x))z +γ Pc [2|ψ E (t)|2 z +ψ E2 (t)¯z ]+iγ Dh|a(t) (iψ0 , 2|ψ E |2 z +ψ E2 z¯ ) ∂t z(s) = v. Note that this is a nonautonomous problem as the bound state ψ E around which we linearize may change with time. By Duhamel’s principle we have: z(t) = e−i H (t−s) Pc v − iγ

t s

e−i H (t−τ ) Pc {[2|ψ E |2 z + ψ E2 z¯ ]

+i Dh|a(τ ) (iψ0 , 2|ψ E |2 z + ψ E2 z¯ )}dτ.

(28)

As in (16) we denote de f

(t, s)v = z(t).

(29)

In the next two theorems we will extend estimates of type (26)-(27) to the operator (t, s) relying on the fact that ψ E (t) is small. It would be useful to find sufficient conditions under which our results generalize to large bound states. Such conditions have been obtained in one or three space dimensions, see [2, 13, 23, 6], unfortunately their techniques cannot be applied in the two space dimension case. We start with estimates in weighted L 2 spaces:

456


Theorem 4.1. There exists ε1 > 0 such that if < x >σ ψ E H 2 < ε1 then there exist constants C, C p > 0 with the property that for any t, s ∈ R the following hold: (i) (t, s) L 2 →L 2 ≤ σ

(ii) (t, s) L p →L 2 ≤ −σ

−σ

C , (1 + |t − s|) log2 (2 + |t − s|)

Cp |t − s|

1− 2p

(iii) (t, s) L 2σ →L p ≤

, for any p ≥ 2 where p −1 + p −1 = 1,

Cp |t − s|

1− 2p

, for any p ≥ 2.

Before proving the theorem let us remark that (i) is a generalization of (26) while (ii) and (iii) are a mixture between (26) and (27). We have used all these estimates in the previous section. They are consequences of contraction principles applied to (28) and involve estimates for convolution operators based on (26) and (27). It will prove much more difficult to remove the weights from the estimates (ii) and (iii), see Theorem 4.2. Proof of Theorem 4.1. Fix s ∈ R. (i) By definition (see (29)), we have (t, s)v = z(t), where z(t) satisfies Eq. (28). We are going to prove the estimate by showing that the linear equation (28) can be solved via contraction principle argument in an appropriate functional space. To this extent let us consider the functional space X 1 := {z ∈ C(R, L 2−σ (R2 ))| sup(1 + |t − s|) log2 (2 + |t − s|)z(t) L 2 < ∞} −σ

t∈R

endowed with the norm z X 1 := sup{(1 + |t − s|) log2 (2 + |t − s|)z(t) L 2 } < ∞. −σ

t∈R

Note that the inhomogeneous term in (28): z 0 (t) = e−i H (t−s) Pc v def

satisfies z 0 ∈ X 1 and

z 0 X 1 ≤ C M v L 2σ

(30)

because of (26). We collect the z dependent part of the right-hand side of (28) in a linear operator L(s) : X 1 → X 1 ,

t [L(s)z](t) = −iγ e−i H (t−τ ) Pc [2|ψ E |2 z+ψ E2 z¯ +i Dh|a(τ ) (iψ0 , 2|ψ E |2 z+ψ E2 z¯ )]dτ. s

(31) In what follows we will show that L is a well defined bounded operator from X 1 to X 1 whose operator norm can be made less then or equal to 1/2 by choosing ε1 in the hypothesis sufficiently small. Consequently I d − L is invertible and the solution of Eq. (28) can be written as z = (I d − L)−1 z 0 . In particular z X 1 ≤ (1 − L)−1 z 0 X 1 ≤ 2z 0 X 1


457

which, in combination with the definition of , the definition of the norm in X 1 and estimate (30), finishes the proof of (i). It remains to prove that L is a well defined bounded operator from X 1 to X 1 whose operator norm can be made less than 1/2 by choosing ε1 in the hypothesis sufficiently small. We have the following estimates:

t L(s)z(t) L 2 ≤ |γ | e−i H (t−τ ) Pc L 2 →L 2 · [3|ψ E |2 (τ )z(τ ) L 2σ −σ

σ

s

σ

−σ

−σ

+Dh|a(τ ) C→L 2σ |ψ0 , 2|ψ E | < x > < x > < x >σ < x >−σ z¯ (τ )|]dτ

t e−i H (t−τ ) Pc L 2 →L 2 · [3|ψ E |2 (τ )z(τ ) L 2σ ≤ |γ | 2

z(τ )+ψ E2

σ

s

−σ

+Dh|a(τ ) C→L 2σ ψ0 L 2 3 < x >σ ψ E2 L ∞ z(τ ) L 2 ]dτ. −σ

On the other hand |ψ E |2 z L 2σ ≤ z L 2 < x >2σ |ψ E |2 L ∞ , and < x >σ ψ E 2L ∞ ≤ ε12 , (32) −σ

where the last inequality is due to the Sobolev imbedding H 2 (R2 ) ⊂ L ∞ (R2 ) and the inequality < x >σ ψ E H 2 ≤ ε1 . Also, from Proposition 2.1, ¯ for |a(τ )| ≤ ε1 < δ. Dh|a(τ ) ≤ C, Using the last three relations, as well as the estimate (26) and the fact that z ∈ X 1 we obtain that L(s) X 1 →X 1 ≤ 3|γ |ε12 sup (1 + |t − s|) log2 (2 + |t − s|) × t>0

t

×

s

CM 1 · dτ ≤ C1 ε12 . 2 2 (1 + |t − τ |) log (2 + |t − τ |) (1 + |τ − s|) log (2 + |τ − s|) I

(33) Indeed, in order to prove the above we will split I into A + B, where

t+s 2 1 1 A= dτ 2 (2 + |t − τ |) (1 + |τ − s|) log2 (2 + |τ − s|) (1 + |t − τ |) log s for which we have the bound |A| ≤

1 | 2 (2 + | t−s |) (1 + | t−s |) log 2 2

t+s 2

s

dτ | (1 + |τ − s|) log2 (2 + |τ − s|) 1 ≤ C2 . 3 (2 + | t−s |) (1 + | t−s |) log 2 2

Observing that A = B and using the last estimate in (33) we obtain that

458


L X 1 →X 1 ≤ C1 ε12 ≤ 1/2 for ε1 small enough. (ii) By the definition of it is sufficient to prove that the solution of (28) satisfies z(t) L 2 ≤ −σ

Cp |t − s|

1− 2p

v L p , for all p ≥ 2 where p −1 + p −1 = 1.

(34)

We will use a similar functional analytic argument as in the proof of (i). Fix p, 2 ≤ p < ∞ and assume v ∈ L p , p −1 + p −1 = 1. We will work in the following functional space: X 2 := {z ∈ C(R, L 2−σ (R2 )| sup z(t) L 2 |t − s|

1− 2p

−σ

t∈R

< ∞}

endowed with the norm z X 2 := sup z(t) L 2 |t − s|

1− 2p

−σ

t∈R

< ∞.

Using the fact that L p → L 2−σ continuously and the estimate (27) we have e−i H (t−s) Pc v ∈ X 2 . In addition, for L defined in the proof of (i), we have sup |t − s|

1− 2p

t>0

L(s)z(t) L 2 ≤ −σ

e−i H (t−τ ) Pc L 2 →L 2 · 2|ψ E |2 (τ )z(τ ) |γ | sup |t − s| σ −σ t>0 s +ψ E2 (τ )¯z (τ ) L 2σ + Dh|a(τ ) C→L 2σ |ψ0 , 2|ψ E |2 z(s) + ψ E2 z¯ (s)| dτ

t Cp 1− 2 ≤ |γ | sup |t − s| p 2 t>0 s (1 + |t − τ |) log (2 + |t − τ |) 2 2σ ∞ ¯ 3(1 + C)ψ L E < x > · dτ < C3 ε12 . 1− 2p |τ − s| 1− 2p

t

(35)

Using now the bounds (32) in (35), for ε1 small enough, we obtain that the norm of L(s) is less or equal to 1/2, i.e. the operator I d − L(t, s) is invertible, which, as in the proof of (i), finishes the proof of estimate (ii). (iii) We already know from part (i) that Eq. (28) has a unique solution in L 2−σ provided v ∈ L 2σ . We are going to show that the right hand side of (28) is in L p . Indeed e−i H (t−s) Pc v L p ≤

Cp |t − s|

1− 2p

v L p ≤

Cp |t − s|

1− 2p

v L 2σ ,

(36)

where the C p ’s in the two inequalities are different, for the first inequality we used (27) while for the second we used the continuous embedding L 2σ → L p , 1 ≤ p ≤ 2. For


459

the remaining terms we combine (27) with z X 1 < ∞ obtained in part (i):

t e−i H (t−τ ) Pc (2|ψ E2 |z(τ ) + ψ E2 z¯ (τ ))dτ L p s

t 3C p ≤ < x >σ ψ E2 α < x >−σ z(τ ) L 2 dτ 2 s |t − τ |1− p

t Cε12 v L 2σ C4 v L 2σ 3C P · ≤ dτ ≤ 2 2 1− 2 (1 + |τ − s|) log (2 + |τ − s|) s |t − τ |1− p |t − s| p

(37)

with α1 + 21 = p1 . Similarly, we have

t e−i H (t−τ ) Pc Dh|a(τ ) iψ0 , 2|ψ E |2 z(s) + ψ E2 z¯ (s)dτ L p s

t

≤ s t

≤ s

|t − τ |

s

C p C¯

t

≤ C p C¯ |t − τ | C p C¯

|t − τ |

1− 2p

1− 2p

|ψ0 , 2|ψ E |2 z(s) + ψ E2 z¯ (s)|dτ

|ψ0 L 2 < x >σ ψ E2 L ∞ < x >−σ z L 2 dτ Cε12 v L 2σ

1− 2p

− s|)(log2 (2 + |τ

(1 + |τ

dτ ≤

− s|)

C5 v L 2σ |t − s|

1− 2p

.

(38)

Plugging (36)-(38) into (28) we get: C( p)

z(t) L p ≤

|t − s|

1− 2p

v L 2σ ,

which by the definition (t, s) = z(t) finishes the proof of part (iii). The next step is to obtain estimates for (t, s) in unweighted L p spaces. They are needed for controlling the cubic term in the operator N of the previous section. Theorem 4.2. Assume that < x >σ ψ E H 2 < ε1 (where ε1 is the one used in Theorem 4.1). Then for all t, s ∈ R the following estimates hold: (i) (t, s) L 1 ∩L q ∩L p →L p ≤

C p,q log(2 + |t − s|) (1 + |t − s|)

1− 2p

,

for all p, q , 2 ≤ p < ∞, 1 < q ≤ 2, p −1 + p −1 = 1; (ii) (t, s)

L 2 ∩L q0 →L 2

≤ Cq0 , for all q0 , 1 < q0
0 and 1 < q0 < 4/3 and for any 2 ≤ p ≤ p0 , 1−2/ p

(t, s)

L q ∩L p ∩L q0 →L p

≤

C p,q0 log(2 + |t − s|) 1−2/ p0 |t − s|

1− 2p

,

460


where 1 1−θ =θ+ q q0 with θ=

θ 1 − 2/ p 1 1−θ . , i.e. = + 1 − 2/ p0 p p0 2

Note that (iii) is similar to the standard estimate for Schrödinger operators (27) except for the logarithmic correction and a smaller domain of definition. We will obtain it by interpolation from (i) and (ii). The proof of (i) will rely on a fixed point technique for Eq. (40) while the proof of (ii) will rely on Strichartz inequalities. It turns out that we need to regularize (28) in order to obtain (i) and (ii). The inhomogeneous term has a nonintegrable singularity at t = s when estimated in L ∞ : e−i H (t−s) Pc v L ∞ ≤ |t − s|−1 v L 1 . Using estimates with integrable singularities at t = s, for example in L p , p < ∞ see (27), would lead to a slower time decay in (i) and eventually will make it impossible to close the estimates for the operator N in the previous section. We avoid this by defining: de f

W (t) = z(t) − e−i H (t−s) Pc v = [ (t, s) − e−i H (t−s) Pc ]v,

(39)

which, by plugging in (28), will satisfy the following "regularized" equation:

t e−i H (t−τ ) Pc [2|ψ E (τ )|2 e−i H (τ −s) Pc v + ψ E2 (τ )ei H (τ −s) Pc v]dτ ¯ W (t) = −iγ s f (t)

+ iγ

t s

e−i H (t−τ ) Pc Dh|a(τ ) ψ0 , 2|ψ E |2 e−i H (τ −s) Pc v(s) + ψ E2 ei H (τ −s) Pc v(s)dτ ¯ f˜(t)

+[L(s)W ](t), (40) where the operator L(s) is defined in (31). Some other new notations are necessary for the sake of easy reference. We will denote by T (t, s) the operator which associates to the initial data at time s, v, the function W (t), so that de f T (t, s)v = W (t), (41) which will be related to the operator (t, s) = z(t) (see (16)) by (t, s) = T (t, s) + e−i H (t−s) Pc .

(42)

For T we can not only extend the estimate in Theorem 4.1 (ii) to the case p = ∞ but also obtain a nonsingular version of it:


461

Lemma 4.1. Assume that < x >σ ψ E H 2 < ε1 (where ε1 is the one used in Theorem 4.1). Then for each 1 < q ≤ 2 there exists the constant Cq > 0, Cq → ∞ as q → 1, such that for all t, s ∈ R we have: T (t, s) L 1 ∩L q →L 2 ≤ −σ

Cq . 1 + |t − s|

Proof of the lemma. Fix q , 1 < q ≤ 2. Consider Eq. (40) with s ∈ R arbitrary and v ∈ L 1 ∩ L q . We are going to show that (40) has a unique solution in C(R, L 2−σ ) satisfying: W (t) L 2 ≤ −σ

Cq max{v L 1 , v L q } 1 + |t − s|

which will be equivalent to the conclusion of the lemma via the definition of T (41). Let us observe that it suffices to prove this estimate only for the forcing term f (t)+ f˜(t) because then we will be able to do the contraction principle in the functional space (in time and space) in which f (t) + f˜(t) will be, and thus obtain the same decay for W as for f (t) + f˜(t). Indeed, this time we will consider the functional space X 3 := {u ∈ C(R, L 2−σ (R2 )| sup u(t) L 2 (1 + |t − s|) < ∞} −σ

t∈R

endowed with the norm u X 3 := sup{u(t) L 2 (1 + |t − s|)} < ∞. −σ

t∈R

We have sup(1 + |t − s|)L(s)u(t) L 2 t>0

−σ

t

≤ |γ | sup(1 + |t − s|) t>0

s σ

e−i H (t−τ ) Pc L 2 →L 2 σ

σ

−σ

×[2 < x > |ψ E | (τ ) < x > < x > 2

−σ

u(τ ) L 2σ

+Dh L 2σ ψ0 L 2 < x >σ ψ E2 L ∞ u(τ − s) L 2 ]dτ −σ

t C7 ψ E2 < x >2σ L ∞ ≤ |γ | sup(1 + |t − s|) < C8 ε12 . (43) 2 t>0 s (1 + |t − τ |)(1 + |τ − s|) log (2 + |τ − s|) Using the bounds (32) in (43) we obtain that for ε1 small enough the norm of L(t, s) in X is less than one, i.e. the operator I d − L(t, s) is invertible. We need now to estimate f (t) + f˜(t): || f (t) + f˜(t)|| L 2 ≤ |γ | −σ

s

t

C M · 3(1 + Dh|a(t) C→ L 2σ ) |ψ E |2 < x >σ e−i H (τ −s) Pc v L 2 (1 + |t − τ |) log2 (2 + |t − τ |)

where we used the estimate (26). Denote C9 = |γ |C M · 3(1 + Dh|a(t) C→ L 2σ ).

dτ, (44)

462


We will split now (44) into two parts to be estimated differently: || f + f˜|| L 2 ≤ −σ

t ...+ ... . s s+1 s+1

I

(45)

II

Then, we have:

s+1

|I| ≤ s

≤

C9 (1+|t −s −1|) log2 (2+|t −s −1|) ≤

C9 |ψ E |2 < x >σ e−i H (τ −s) Pc v L 2 dτ ≤ (1 + |t − τ |) log2 (2 + |t − τ |)

s+1 s

e−i H (τ −s) Pc v L q · |ψ E |2 < x >σ L α ≤

C10 (1 + |t − s − 1|) log2 (2 + |t − s − 1|)

≤fixed constant

s+1

v L q

1

dτ 1− 2 (τ − s) q C11 v L q 1 v L q ≤ ≤ C12 1 + |t − s| (1 + |t − s − 1|) log2 (2 + |t − s − 1|) s

with α1 + q1 = 21 and q1 + q1 = 1. For the second integral we have:

C9 |ψ E |2 < x >σ || L 2 e−i H (τ −s) Pc v L ∞ dτ ≤ (1 + |t − τ |) log2 (2 + |t − τ |) s+1

t C9 |ψ E |2 < x >σ L 2 1 v L 1 dτ · ≤ 2 s+1 (1 + |t − τ |) log (2 + |t − τ |) |τ − s| C13 ≤ ||v|| L 1 . 1 + |t − s|

|II| ≤

t

Let us observe that the last two estimates are for the case when t > s+1. If s < t < s+1 we have

t C9 |ψ E |2 < x >σ e−i H (τ −s) Pc v L 2 ˜ || f + f || L 2 ≤ dτ −σ (1 + |t − τ |) log2 (2 + |t − τ |) s

t ψ E2 < x >σ L α e−i H (τ −s) Pc v L q dτ ≤ C9 (1 + |t − τ |)(log2 (2 + |t − τ |)) s

t 1 dτ vq ≤ Cv L q ≤ C14 2 s (τ − s)1− q with α1 + q1 = 21 . Combining the last three estimates we get the lemma.

We can now proceed with the proof of Theorem 4.2. Proof of Theorem 4.2. (i) Because of estimate (27) and relation (42) it suffices to prove (i) for T (t, s).


463

Consider Eq. (40) with arbitrary s ∈ R and v ∈ L 1 ∩ L q . In the previous lemma we showed that the solution W (t) ∈ L 2−σ . Now we show that it is actually in L p for all 2 ≤ p < ∞. Fix such a p. Then:

t ||W (t)|| L p ≤|| f (t)+ f˜(t)|| L p+|γ | ||e−i H (t−τ ) Pc || L p →L p 3(1+Dh|a(τ ) L p ψ0 L p ) s

≤ || f (t) + f˜(t)|| L p +

s

|||ψ E |2 < x >σ < x >−σ |W (τ )||| L p dτ t

C14 (t − τ )

1− 2p

|||ψ E |2 < x >σ || L α || < x >−σ W (τ )|| L 2 dτ (46)

(with

1 α

+

1 2

=

1 p )

The estimate for f (t)+ f˜(t) is similar, but this time the term |ψ E |2 e−i H (τ −s) Pc v L p is controlled for s + 1 < τ < t by C15 v L 1 |τ − s|

ψ E2 L p e−i H (τ −s) Pc v∞ ≤ and for s ≤ τ < s + 1 by

ψ E2 L α e−i H (τ −s) Pc v L q ≤

C16 (τ − s)

1− q2

,

where α −1 + q −1 = p −1 and q −1 + q −1 = 1. Using now the previous lemma to estimate the term || < x >−σ W (τ )|| L 2 and replacing in (46) we get: ||W (t)|| L p ≤

C17 log(1 + |t − s|) (1 + |t − s|)

1− 2p

max{||v|| L 1 , ||v|| L q }

with 1 < q ≤ 2 which is equivalent to T (t, s) L 1 ∩L q →L p ≤

C17 log(1 + |t − s|)

(47)

1− 2p

(1 + |t − s|)

for all 2 ≤ p < ∞ and 1 < q ≤ 2. This finishes the proof for part (i). (ii) Recalling the equation for W (40), let us observe that we have

t

e s

−i H (t−τ )

Pc (|ψ E |

2

W (τ ) + ψ E2 W¯ (τ ))dτ L 2

≤ CS s

t

ψ E2

≤ CS

t

|ψ E |

s σ

<x>

αL α W (τ )

⎛ ≤ C S ε12 ⎝

t

2

W (τ )αL ρ dτ

−σ

<x>

αL 2 dτ

≤

1 α

α

α (1− q2 )

(1 + |τ − s|)

α

⎞ 1

vqα

0

s

1

dτ ⎠

0

ε12 C18 vq0 , (48)

464


where for the first inequality we used the Strichartz estimate

t (T f )(t) = e−i H (t−τ ) f (τ )dτ : L α (0, T ; L ρ ) → L ∞ (0, T ; L 2 ) s

with (α, ρ) satisfying 2/α = 1 − 2/ρ and α > 2 . For the second inequality we used Hölder’s inequality and for the third one we used (34) combined with (39) and (27). Finally the last inequality holds when α (1 − q20 ) > 1 which happens for q0 > 2α > 4. Also, we have the estimates

t e−i H (t−τ ) γ Pc Dh|a(τ ) iψ0 , 2|ψ E |2 W (τ ) + ψ E2 W¯ (τ )dτ L 2 s

t 1 ≤ C S ( Dh|a(τ ) iψ0 , 2|ψ E |2 W (τ ) + ψ E2 W¯ (τ )αL ρ dτ ) α s

t

≤ C19 s

t

≤ C20

≤

ε12 C21

s

(ψ0 L 2 ψ E2

t

s

|ψ0 , 2|ψ E |

2

W (τ ) + ψ E2 W¯ (τ )|α dτ

α

σ

< x > L ∞ W (τ ) L 2 ) dτ

1 α

1 α

−σ

1 α

1 (1− q2 )α 0

dτ

(1 + |τ − s|)

v

L q0

≤ ε12 C22 v

L q0

,

(49)

where for the first inequality we used Strichartz estimate as before and for the second inequality we use the fact that Dh|a(τ ) is bounded in H 2 and thus in any L p and its norm is small. For the fourth inequality we used the fact that ψ0 L 2 and |ψ E |2 < x >σ L ∞ are bounded and small. Finally the last inequality holds, as before, for q0 > 2α > 4. For f (t) + f˜(t) we’ll need to estimate differently the short time behavior and the long time behavior, namely:

s+1

t f (t) + f˜(t) = ...+ .... s s+1 I

We have:

I L 2 = |γ |

s

s+1

s

s s+1

e−i H (t−τ ) Pc [2|ψ E |2 e−i H (τ −s) Pc v + ψ E2 e+i H (τ −s) Pc v¯

¯ L 2 −Dh(iψ0 , 2|ψ E |2 e−i H (τ −s) v + ψ E2 ei H (τ −s) v)]dτ

≤ C23 ≤ C24

s+1

II

|ψ E |2 e−i H (τ −s) Pc v L 2 + |ψ0 , |ψ E |2 e−i H (τ −s) v|dτ

||ψ E |2 || L α ||e−i H (τ −s) Pc v|| L q0 + ψ0 L 2 |ψ E |2 L α e−i H (τ −s) v L q0 dτ

s+1

≤ C25 s

1 (τ − s)

1− q2

0

dτ ||v||

L q0

≤ C26 ||v||

where we used the fact that the operator e−i H t preserves the L 2 norm, and

1 α

+

1 q0

L q0

,

= 21 .


465

We continue by estimating II:

II L 2 ≤ |γ |

s+1 s

e−i H (t−τ ) Pc [2|ψ E |2 e−i H (τ −s) Pc v + ψ E2 e+i H (τ −s) Pc v¯

¯ L 2 −Dh(iψ0 , 2|ψ E |2 e−i H (τ −s) v + ψ E2 ei H (τ −s) v)]dτ t

1 α ≤ CS |||ψ E |2 e−i H (τ −s) Pc v||qα dτ t

+C S s

≤ C27

t

e

−i H (τ −s)

s+1

1 α

Pc vqα0

0

s+1

≤C28

ψ0 L 2 |ψ E | e 2

t

s+1

Lα

vαL q0 dτ

1 α

1 α

1 (τ − s)

−i H (τ −s)

α (1− q2 ) 0

dτ

vq0 ≤C29 vq0 ,

where for the first inequality we used the fact that the L 2 norm is preserved by the operator e−i H t Pc . For the second inequality we used the Strichartz estimate

t

(T f )(t) =

e−i H (t−τ ) f (τ )dτ : L q0 (0, T ; L α ) → L ∞ (0, T ; L 2 )

s

for the f (t) term. For the f˜(t) term we used similarly the same Strichartz estimate, the fact that Dh q0 is bounded (as it is in any L p norm), and we estimated the scalar prodL

uct by the product ψ0 L 2 |ψ E |2 L α e−i H (τ −s) v L q0 . For the third inequality we used Hölder’s inequality and the fact that ψ E |2 L β , |ψ E |2 L α , ψ0 L 2 ≤ C, ∀t (where 1 = q10 + β1 and 21 = α1 + q10 ). Finally the last inequality holds because α (1 − q20 ) > 1, q 0

as q0 > α2 . Let us observe that we assumed that t > s + 1. If s < t < s + 1, only the estimate for I will suffice, where the upper limit of integration s + 1 should be replaced by t. Combining the estimates for I, II, (48) and (49) we have that W (t) is uniformly bounded in L 2 which, by (41), implies T (t, s)

L q0 →L 2

≤ Cq0 , for all t, s ∈ R.

(50)

Using now (42) and (27) with p = p = 2 we obtain (ii). (iii) We start from (47): T (t, s)

L 1 ∩L q0 →L p0

≤

C p0 ,q0 log(2 + |t − s|) 1− p2

(1 + |t − s|)

0

and (50): T (t, s)

L q0 →L 2

≤ Cq0 , 1 < q0
4. However, the linear terms in r do not decay fast enough to be absolutely integrable. It is possible though that a combination of decay and oscillatory cancellations would render it integrable. We think that it is only a matter of time until a suitable treatment of this term is found. Note that, in the 1-d and 3-d cases, the linear terms in r were absolutely integrable in time, see for example [4, 19]. But these estimates relied on the integrable decay in time of the Schrödinger operator in L ∞ norm in 3-d, respectively on the large power nonlinearity to compensate for the linear growth in time introduced by virial type estimates in 1-d. None would work for our cubic NLS in 2-d. The situation is even more complex and possibly more interesting when the center manifold has more than one branch (more than one connected component). For simplicity, consider the case when Hypothesis (H1) part (iii) is relaxed to allow for two, simple, negative eigenvalues E 0 < E 1 with corresponding normalized eigenvectors ψ0 , ψ1 . In this case the center manifold has two branches ψ E j = a j ψ j + h j (a j ), j = 0, 1, each bifurcating from one eigenvector as described in Sect. 2. The decomposition into the evolution on the center manifold and the one away from it will now be: 1 u(t, x) = (a j (t)ψ j (x) + h j (a j (t))) +rm (t, x). j=0

ψCM (t)


467

The equation for rm (t) remains essentially the same as (15) in Section 3, with ψ E replaced by ψCM and the differential of h replaced by the sum of the differentials of h j , j = 0, 1. However, one has to add to the right hand side of (15) the projection onto the continuous spectrum of the interaction term between the branches: 2|ψ E 0 |2 ψ E 1 + ψ E2 0 ψ¯ E 1 + 2ψ E 0 |ψ E 1 |2 + ψ¯ E 0 ψ E2 1 .

(51)

In principle one could use our techniques and obtain a decay in time for rm (t), hence collapse on the center manifold, provided one makes the ansatz that the term above, or at least its projection onto the continuous spectrum, decays in time. Such an ansatz needs to be supported by the analysis of the motion on the center manifold given now by a system of two ODE’s, one for a0 and one for a1 . Each of the equations will be similar to (14) but the projection of (51) onto ψ0 , respectively ψ1 , has to be added to the right hand side. Note that, in the 3-d case, under the additional assumption 2E 1 − E 0 > 0, it has been shown that the evolution approaches asymptotically a ground state (a periodic solution on the branch bifurcating from ψ0 ) except when the initial data is on a finite dimensional manifold near the excited state branch (the one bifurcating from ψ1 ), see [27, 29–31]. But the authors’ analysis relies heavily on the much better dispersive estimates for Schrödinger operators in 3-d compared to 2-d. The 2-d case remains open. Returning now to the case of one branch center manifold in 2-d, an important question is whether its stability persists under time dependent perturbations. In [7] we showed that this is not the case in 3-d. The slower decay in time of the Schrödinger operator in 2-d compared to 3-d prevents us, yet again, from extending the technique in [7] to the 2-d setting. Acknowledgement. The authors wish to thank M. I. Weinstein, O. Mizrak and the anonymous referee for their helpful comments on the manuscript. E. Kirr was partially supported by NSF grants DMS-0405921 and DMS-0603722.

References 1. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313–345 (1983) 2. Buslaev, V.S., Perel’man, G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. Algebra i Analiz. 4, 63–102 (1992) 3. Buslaev, V.S., Perel’man, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear evolution equations, Vol. 164 Amer. Math. Soc. Transl. Ser. 2, Providence, RI Amer. Math. Soc., 1995, pp. 75–98 4. Buslaev, V.S., Sulem, C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20, 419–475 (2003) 5. Cazenave, T.: Semilinear Schrödinger equations. Vol. 10 of Courant Lecture Notes in Mathematics, New York: New York University Courant Institute of Mathematical Sciences, 2003 6. Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54, 1110–1145 (2001) 7. Cuccagna, S., Kirr, E., Pelinovsky, D.: Parametric resonance of ground states in the nonlinear Schrödinger equation. J. Differ. Eq. 220, 85–120 (2006) 8. Dalfovo, F., Giorgini, S., Pitaevskii, L., Stringari, S.: Theory of bose-einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999) 9. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74, 160–197 (1987) 10. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94, 308–348 (1990) 11. Gustafson, S., Nakanishi, K., Tsai, T.-P.: Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves. Int. Math. Res. Not. 2004, 3559–3584 (2004)

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12. Kohler, W., Papanicolaou, G.C.: Wave propagation in a randomly inhomogenous ocean. In: Wave propagation and underwater acoustics (Workshop, Mystic, CT, 1974), Lecture Notes in Phys., Vol. 70 Berlin: Springer, 1977, pp. 153–223 13. Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Amer. Math. Soc. 19, 815–920 (2006) (electronic) 14. Lieb, E.H., Seiringer, R., Yngvason, J.: A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Commun. Math. Phys. 224, pp. 17–31 (2001) (Dedicated to Joel L. Lebowitz) 15. Marcuse, D.: Theory of Dielectric Optical Waveguides. San Diego, CA: Academic Press, 1974 16. Murata, M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49, 10–56 (1982) 17. Newell, A.C., Moloney, J.V.: Nonlinear optics. In: Advanced Topics in the Interdisciplinary Mathematical Sciences, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA: AddisonWestey, 1992 18. Nirenberg, L.: Topics in nonlinear functional analysis. Vol. 6 of Courant Lecture Notes in Mathematics, New York: New York University Courant Institute of Mathematical Sciences 2001; Chapter 6 by E. Zehnder, Notes by R. A. Artino, Revised reprint of the 1974 original. 19. Pillet, C.-A., Wayne, C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Differ. Eq. 141, 310–326 (1997) 20. Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and timedependent potentials. Invent. Math. 155, 451–513 (2004) 21. Rose, H.A., Weinstein, M.I.: On the bound states of the nonlinear Schrödinger equation with a linear potential. Phys. D 30, 207–218 (1988) 22. Schlag, W.: Dispersive estimates for Schrödinger operators in dimension two. Commun. Math. Phys. 257, 87–117 (2005) 23. Schlag, W.: Stable manifolds for an orbitally unstable nls. To appear in Annals of Math, available at http://www.math.uchicago.edu/ schlag/recent.html, 2004. 24. Shatah, J., Strauss, W.: Instability of nonlinear bound states. Commun. Math. Phys. 100, 173–190 (1985) 25. Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990) 26. Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differ. Eqs. 98, 376–390 (1992) 27. Soffer, A., Weinstein, M.I.: Selection of the ground state for nonlinear Schroedinger equations. http://arXiv.org/abs/nlin/0308020, 2003, submitted to Reviews in Mathematical Physics 28. Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149– 162 (1977) 29. Tsai, T.-P., Yau, H.-T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Commun. Pure Appl. Math. 55, 153–216 (2002) 30. Tsai, T.-P., Yau, H.-T.: Relaxation of excited states in nonlinear Schrödinger equations. Int. Math. Res. Not. 2002, 1629–1673 (2002) 31. Tsai, T.-P., Yau, H.-T.: Stable directions for excited states of nonlinear Schrödinger equations. Commun. Partial Differ. Eq. 27, 2363–2402 (2002) 32. Weder, R.: Center manifold for nonintegrable nonlinear Schrödinger equations on the line. Commun. Math. Phys. 215, 343–356 (2000) 33. Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39, 51–67 (1986) Communicated by P. Constantin


Communications in


Algebro-Geometric Approach in the Theory of Integrable Hydrodynamic Type Systems Maxim V. Pavlov Mathematical Physics Department, P.N. Lebedev Physical Institute, Leninskij Prospekt 53, Moscow, Russia. E-mail: [email protected] Received: 31 March 2006 / Accepted: 8 November 2006 Published online: 3 April 2007 – © Springer-Verlag 2007

Abstract: The algebro-geometric approach for integrability of semi-Hamiltonian hydrodynamic type systems is presented. The class of symmetric hydrodynamic type systems is defined and the calculation of the associated Riemann surfaces is greatly simplified for this class. Many interesting and physically motivated examples are investigated.

Contents 1. 2. 3. 4. 5.

6.

7. 8. 9. 10.

Introduction . . . . . . . . . . . . . . . . . . . . . . . Symmetric Hydrodynamic Type Systems . . . . . . . . Tsarev’s Observations for the Vector NLS . . . . . . . . Generalized Chromatography . . . . . . . . . . . . . . The Generalized Hodograph Method . . . . . . . . . . 5.1 Quasi-symmetric form . . . . . . . . . . . . . . . 5.2 Hamiltonian formalism . . . . . . . . . . . . . . . 5.3 Mirrored curvilinear conjugate coordinate systems 5.4 The chromatography system . . . . . . . . . . . . 5.5 Reciprocal transformations . . . . . . . . . . . . . Homogeneous Hydrodynamic Type Systems . . . . . . 6.1 The Kodama hydrodynamic type system . . . . . . 6.2 Cubic Hamiltonian hydrodynamic type system . . . 6.3 The ideal gas dynamics . . . . . . . . . . . . . . . 6.4 Whitham averaged Sinh-Gordon equation . . . . . Integrable Hydrodynamic Chains . . . . . . . . . . . . Hamiltonian Chromatography System . . . . . . . . . . Exceptional (Linearly Degenerate) Case . . . . . . . . Conclusion and Outlook . . . . . . . . . . . . . . . . .

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470 471 474 475 477 479 482 484 486 489 490 491 492 494 495 496 498 499 501

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M. V. Pavlov

1. Introduction The theory of the integrable hydrodynamic type systems j

u it = υ ij (u)u x ,

i, j = 1, 2, . . . , N

(1)

was initiated by S.P. Novikov, B.A. Dubrovin (see [6]) and S.P. Tsarev (see [33]). The differential-geometric approach has been developed also by E.V. Ferapontov (see [7, 10]), I.M. Krichever (see, [20]), O.I. Mokhov (see [25]) and by the author (see [30]). Also, H. Flaschka, M.G. Forest, D.W. McLaughlin (see [11]), B.A. Dubrovin and I.M. Krichever (see [5, 21]), Yu. Kodama and J. Gibbons (see [16, 18]) used an algebro-geometric approach in the theory of integrable hydrodynamic type systems arising as dispersionless limits of integrable dispersive equations or, more generally, from Whitham’s averaging of their multi-phase solutions. Thus, information such as the Riemann surfaces, quasi-momentum and quasi-energy can be reconstructed – expansion of these then yields the conservation laws and hence the commuting flows. This paper is devoted to the algebro-geometric approach for hydrodynamic type systems whose origin is unknown. For simplicity in this paper we restrict our consideration to symmetric hydrodynamic type systems, because in precisely such cases a generating function of conservation laws is given in advance. In all other cases derivation of a generating function of conservation laws is a separate and complicated computational problem, which will be investigated in detail elsewhere. The next step is a computation of the equation of the Riemann surface. The corresponding linear ODE system can be solved if it is invariant under some Lie group action. For instance, if the symmetric hydrodynamic type system is homogeneous (this is a typical formulation of physically motivated examples), then the corresponding generating function of conservation laws and the equation of the Riemann surface are homogeneous too. Moreover, the generalized hodograph method established by S.P. Tsarev in [33] is based on the concept of Riemann invariants. In this paper we suggest an alternative approach based on a particular conservative form of hydrodynamic type systems. Most physically motivated problems can be given in such a form. The paper is organized in the following order. In the second section a semiHamiltonian (integrability) property for hydrodynamic type systems is reformulated in a conservative form. A method allowing the immediate construction of the generating function of the conservation laws and of the corresponding Riemann surface, is established. In the third section we briefly describe Tsarev’s observations, which simplify many subsequent calculations. In the fourth section the chromatography system as the most interesting and complicated example of symmetric hydrodynamic type systems is investigated. In the fifth section the generalized hodograph method adopted to a conservative form is considered in detail. Several different sub-classes of hydrodynamic type systems are presented. In the sixth section homogeneous hydrodynamic type systems are considered. In such case a computation of the Riemann surface can be found in quadratures. In the seventh section integrable hydrodynamic chains as a natural generalization of the symmetric hydrodynamic type systems are discussed. In the eighth section the Hamiltonian chromatography hydrodynamic type system is considered. The corresponding hydrodynamic chain is found as well as its local Hamiltonian structure. In the ninth section the linearly degenerate case is investigated. In the conclusion the main open problem is considered.

Algebro-Geometric Approach for Integrability of Hydrodynamic Type Systems

471

2. Symmetric Hydrodynamic Type Systems The hydrodynamic type system (1) written in the Riemann invariants rti = µi (r)r xi ,

i = 1, 2, . . . , N

(2)

is integrable by the generalized hodograph method (see [33]) iff the semi-Hamiltonian condition ∂ j µi ∂k µi ∂j k = ∂ , i = j = k (3) k j µ − µi µ − µi is valid identically (here ∂k ≡∂/∂r k ; the semi-Hamiltonian condition also can be written in arbitrary field variables u k , see [29]). Then the hydrodynamic type system (1) has infinitely many conservation laws and commuting flows r yi = wi (r)r xi ,

(4)

parameterized by N arbitrary functions of a single variable. Remark. The characteristic velocities wi (r) of the commuting flows satisfy the linear PDE system (see [33]) ∂k w i =

∂k µi (w k − wi ), µk − µi

i = k,

(5)

which cannot be solved explicitly in the general case, because this is a linear system with variable coefficients. However, in this paper we are able to avoid this problem, because the generalized hodograph method can be formulated via arbitrary field variables (see the beginning of Sect. 6). The algebro-geometric approach is based on the concept of the Riemann surface, where all conservation laws and commuting flows can be found (see below) via the Riemann invariants as well as conservation law densities u k , which appear in this framework in a natural way. The system (5) has the general solution parameterized by N arbitrary functions of a single variable. The generalized hodograph method established by S.P. Tsarev (see [33]) leads to the general solution written in implicit form as an algebraic system x + µi t = wi ,

i = 1, 2, . . . , N

(6)

for the semi-Hamiltonian hydrodynamic type system (2). In this paper several tools useful for constructing commuting flows are suggested in Sect. 5. Many hydrodynamic type systems are known from physical applications, which may be written in the very special conservative form u it = ∂x ψ(u 1 , u 2 , . . . , u N ; u i ),

(7)

where there is a single function ψ(u 1 , u 2 , . . . , u N ; p) which is a symmetric function of its first N arguments. In particular, (except in Sect. 9) we restrict to the case where ψ(u; p) is a nonlinear function of p. We call such systems symmetric hydrodynamic type systems. The theory presented below can be easily extended to more general symmetric classes, for instance on u it = ∂x ψ(u 1 , u 2 , . . . , u N ; υ 1 , υ 2 , . . . , υ M ; u i ), υ it = ∂x g(u 1 , u 2 , . . . , u N ; υ 1 , υ 2 , . . . , υ M ; υ i ),

472

M. V. Pavlov

where N and M are arbitrary integers. Moreover, we shall demonstrate below (in a set of examples) that the symmetricity is not a necessary restriction. However, in the symmetric case the theory is very effective. Let us introduce the matrix Aik given by Aik (u;

p) =

∂ψ ∂ψ | p=u i − ∂p ∂p

δ ik +

∂ψ | k ∂u i p=u

(8)

and formulate the phenomenological algebro-geometric approach. Statement 1. If the symmetric hydrodynamic type system (7) is integrable, then this system has the generating function of conservation laws pt = ∂x ψ(u 1 , u 2 , . . . , u N ; p).

(9)

If the generating function of conservation laws (9) is consistent with the hydrodynamic type system (7), then ∂p ∂ψ = Bik k , (10) i ∂u ∂u where the matrix Bik (u; p) is an inverse matrix to the matrix Aik (see (8)). This is a system of N ODE’s of the first order for every fixed index i, where any of them can be written in the form dy/dz = f (z, y) (i.e. p → y and u i → z, all other field variables u k are “frozen-in constants” in each Eq. (10)). Let us introduce the function λ(u 1 , u 2 , . . . , u N ; p) determined by N linear PDE’s of the first order (10), ∂λ ∂ψ ∂λ = 0. (11) Aik k + i ∂u ∂u ∂ p Definition 1. The function λ(u; p) satisfying (11) is called the equation of the Riemann surface. If the linear PDE system (11) has an integration factor then the equation of the Riemann surface can be found in quadratures ∂λ n ∂ψ k dλ = dp − Bk n du . ∂p ∂u If, for instance, the hydrodynamic type system (7) is homogeneous (see examples below), then functions ψ(u; p) and λ(u; p) are homogeneous too. Then the integration factor can be found by using the Euler theorem λ= p

∂λ ∂λ + uk k . ∂p ∂u

Statement 2. A deformation of the Riemann surface determined by the equation λ(u; p) satisfies the Gibbons equation λt −

∂ψ ∂λ λx = [ pt − ∂x ψ(u; p)]. ∂p ∂p

(12)


473

The Gibbons equation (first given in this form in [13], see also [11]) has three distinguished features: 1. if one fixes λ = const (free parameter), then one obtains (9), 2. if one fixes p = const (free parameter), then one obtains the kinetic equation (a collisionless Vlasov equation) written in so-called Lax form = ∂λ ∂ H − ∂λ ∂ H , λt = {λ, H} ∂x ∂p ∂p ∂x = ψ(u; p). where H 3. if one chooses coordinates which are the Riemann invariants r i (i = 1, 2, . . . , N ) determined by the condition ∂λ/∂ p = 0 (see (11)), then the corresponding hydrodynamic type system (7) can be written in the diagonal form (2) rti =

∂ψ i | ir , ∂ p p= p x

i = 1, 2, . . . , N ,

(13)

where the corresponding values pi can be expressed via these Riemann invariants r k . In this algebro-geometric construction the Riemann invariants are the branch points r i = λ|∂λ/∂ p=0 of the Riemann surface (exactly as it is in the Whitham theory, see [5] and [21]). Remark. The characteristic velocities µk of hydrodynamic type system (1) can be found from algebraic system det |υ ik (u) − µδ ik | = 0. (14) All of them must be distinct if Tsarev’s results are to be applicable (see [33]). Thus, if the hydrodynamic type system (7) is integrable by the generalized hodograph method, then the values pi are given by (see (13)) µi (u) ≡

∂ψ | i, ∂ p p= p

(15)

where characteristic velocities are determined from (14) det Aik (u; p) = 0.

(16)

Suppose the hydrodynamic type system (7) is semi-Hamiltonian. Then such system must have N series of conservation laws, which can be obtained by the substitution of the formal series p (k) = u k + λυ k (u) + λ2 w k (u) + · · · (17) into (9). The compatibility conditions of the first N extra conservation laws ∂ψ | p=u i ∂t υ i (u) = ∂x υ i (u) ∂p with the hydrodynamic type system (7) are equivalent to the semi-Hamiltonian property. Main statement: The symmetric hydrodynamic type system (7) is semi-Hamiltonian iff the compatibility condition ∂i (∂k p) = ∂k (∂i p) is fulfilled. Comment. Computation of this compatibility condition can be made in the coordinates u k (see (7)) or in the Riemann invariants (see (13) and details in the Conclusion). In both cases the nonlinear PDE system being in involution coincides with the integrability condition for hydrodynamic type systems following from existence of N conservation laws and the vanishing of the Haantjes tensor (see [17, 29]).

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M. V. Pavlov

3. Tsarev’s Observations for the Vector NLS Let us consider the dispersionless limit of the vector NLS (see [36]) u it = ∂x

(u i )2 k + η , 2

ηit = ∂x (u i ηi ),

i = 1, 2, . . . , N .

(18)

The equation of the Riemann surface λ=µ+

ηn µ − un

(19)

was derived directly from the spectral problem for the vector NLS (see [36]). A deformation of the Riemann surface is described by the Gibbons equation (see [13]) 2 µ ∂λ µt − ∂ x (20) λt − µλx = + ηn . ∂µ 2 The Zakharov reduction (18) can be written in the diagonal form (see [13]) rti = µi (r)r xi ,

i = 1, 2, . . . , 2N ,

(21)

where 1. the characteristic velocities µi are given by the condition ∂λ/∂µ = 0 (see (19)); 2. the Riemann invariants r i are the branch points λ|µ=µi ≡ λ|∂λ/∂µ=0 of the Riemann surface (19) in accordance with the first such observation made in [11]. Remark. Under the substitution Ak = (u n )k ηn , the Zakharov reduction (18) can be re-written in the form (39), see [36] and also [13] . In this section we call attention to some properties of the Zakharov reduction, observed by S.P. Tsarev (1985) and useful in other symmetric hydrodynamic type systems. The eigenvalue–eigenfunction problem (cf. (16)) is

(u i − µ)δ ik 1 ηi δ ik (u i − µ)δ ik

qi si

= 0.

Thus, qi =

1 k s , µ − ui

si =

ηi sk . (µ − u i )2

(22)

The first Tsarev observation [34] is that the sum of the last N equations yields an expression determining characteristic velocities µk (see (14), (16) and (21)) via the very compact “determinant formula” 1=

ηn , (µ − u n )2

It means that the algebraic equation (23) is nothing but the condition ∂λ/∂µ = 0.

(23)


475

Thus, the derivation of a similar “determinant formula” F(u, µ) = 0 like (23) is equivalent to (16) via means the substitution µ = ∂ψ/∂ p (see (15)); The second Tsarev observation [34] is that (23) can be integrated once with respect to µ. It means that the equation of the Riemann surface (19) can be obtained directly avoiding (10). Indeed, in some cases, the integration of such a “determinant formula” once with respect to the variable p is equivalent to the integration of the linear PDE system (11), yielding immediately the equation of the Riemann surface λ(u, p) (see, for instance, the dispersionless limit of the Yajima–Oikawa system (42)). The third Tsarev observation [34] is that the flat diagonal metric of the Zakharov reduction (21) is given by (see also [33]) gii = res

λ=r i

∂λ ∂µ

2 dµ.

(24)

Indeed, since the Hamiltonian structure of the dispersionless limit of the vector NLS is u it = ∂x

∂h , ∂ηi

ηit = ∂x

∂h , ∂u i

the diagonal metric (in Riemann invariants) g ii =

N ∂r i ∂r i ∂u k ∂ηk k=1

is given by g =2 ii

N k=1

ηk , (µi − u k )3

where (see (19) and cf. (22)) ∂r i ηk = , ∂u k (µi − u k )2

∂r i 1 = i . k ∂η µ − uk

Thus, g ii = ∂ 2 λ/∂µ2 |µ=µi in agreement with (24). Remark. This Tsarev’s formula (24) was proved later in [5] (see also [21]). 4. Generalized Chromatography The chromatography process (see, for instance, [10]) is described by the hydrodynamic type system (u i )α u it = ∂x , i = 1, 2, . . . , N , (25) [1 + γ k (u k )β ]ε where α, β, ε and γ i are constants. All results presented in this section generalize results from [10] (see formulas 1, 3, 4, 5). This hydrodynamic type system has the obvious pair of conservation laws

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M. V. Pavlov

α − βε 1−ε γ k (u k )β−α+1 = (β − α + 1)∂x − −ε , β(1 − ε) ε−βε+α β(α + ε − βε) ε 1−β−α ∂t α = ∂x γ n (u n )α+β−1 , α α+β −1 k β where = 1 + γ k (u ) . If α = β − 1, then the hydrodynamic type system (25) is Hamiltonian (see [6, 30]) ∂t

u it =

∂h 1 ∂x i , γ i ∂u

(26)

where the momentum density is γ k (u k )2 and the Hamiltonian density is h = 1−ε . Suppose the above hydrodynamic type system is integrable for some values of constants α, β, ε and γ i . Then the generating function of conservation laws given by pt = ∂ x

pα ε

should exist. It is easy to check that the compatibility conditions ∂i (∂k p) = ∂k (∂i p) are valid iff α = βε, where (see (10)) −1 ∂p γ n (u n )α+β−1 γ i (u i )β−1 p α α = i α−1 − . (27) ∂u i (u ) − p α−1 (u n )α−1 − p α−1 βε All first derivatives (see (11)) γ i (u i )β−1 ∂λ βε = ϕ(u, p) p , ∂u i (u i )βε−1 − p βε−1 ∂λ γ n (u n )β = ϕ(u, p) 1 − p βε−1 ∂p (u n )βε−1 − p βε−1 are determined up to integration factor ϕ(u, p), which has not been found yet. Then the equation of the Riemann surface λ(u, p) can be found in quadratures ⎛ ⎞ β+1 β n ) βε−1 −1 dw n βε−1 q (w ⎠, dλ = ϕ(u, p) ⎝dp + γn βε − 1 wn − 1 1

if the integration factor is ϕ( p) = p −1−β (here we use the substitutions p = q βε−1 and 1

u n = (w n q) βε−1 ). Then, the integrable hydrodynamic type system (25) u it = ∂x

(u i )βε , [1 + γ k (u k )β ]ε

i = 1, 2, . . . , N

(28)

is connected with the equation of the Riemann surface λ=

p −β 1 + γ k (w k )δ F(1, δ, δ + 1, w k ), β βε − 1

where 2 F1 (a, b, c, z) is a hyper-geometric function and δ = β/(βε − 1). If δ = m/n, where m and n are integers, then the equation of the Riemann surface λ(u, p) can be found in elementary functions.


477

Remark. If β → ∞, then the hydrodynamic type system (28) reduces to eεu , k [1 + γ k eu ]ε i

u it = ∂x

i = 1, 2, . . . , N .

(29)

The corresponding equation of the Riemann surface is λ = e− p +

1 1 1 γ n (w n )1/ε F(1, , + 1, w n ), ε ε ε

where p = ln q and u n = ln(w n q). Moreover, if ε = 1, the above hydrodynamic system has a local Hamiltonian structure (26) with the Hamiltonian density h = ln . The above equation of the Riemann surface reduces to n γ n ln(eu − p − 1). (30) λ = e− p − Remark. If ε → 0, then the hydrodynamic type system (28) reduces to u it = ∂x ln

(u i )β , 1 + (u k )β

i = 1, 2, . . . , N ,

where the constants γ k are removed by appropriate scaling of the field variables u k . The corresponding equation of the Riemann surface is λ=

p −β n δ − (w ) F(1, −β, 1 − β, w n ), β

where u k = w k p. If β = 1, the corresponding hydrodynamic type system (28) has a local Hamiltonian structure (26) with the Hamiltonian density h= u m (ln u m − 1) − 1 + un − 1 . u m ln 1 + 5. The Generalized Hodograph Method If the hydrodynamic type system (1) is semi-Hamiltonian, then the general solution parameterized by N arbitrary functions of a single variable is given in an implicit form by the algebraic system (cf. (6); see [33]) xδ ik + tυ ik (u) = wki (u),

(31)

where wki (u) are characteristic velocities of an arbitrary commuting flow. However, in this paper, we present an approach producing the generating function of conservation laws only. Thus, we need to extend this mechanism to produce the generating function of conservation laws and commuting flows simultaneously. In this paper we restrict our consideration to the three sub-cases connected with Egorov conjugate curvilinear coordinate nets (see [30]), with orthogonal coordinate nets (see [33]) and with so-called “mirrored” conjugate nets (see below). The general case will be considered elsewhere. The first step is a description of N series of conservation laws (see (17)). They can be obtained by expansion in the Bü rmann–Lagrange series (see, for instance, [24]) at the vicinity of each singular point.

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Theorem 1 ([24]). The analytic function y = y1 (x − x0 ) + y2 (x − x0 )2 + y3 (x − x0 )3 + · · · can be inverted (y(x) → x(y)) as the Bü rmann–Lagrange series x = x0 + x1 y + x2 y 2 + x3 y 3 + · · · , whose coefficients are

d n−1 x − x0 n 1 lim , n = 1, 2, .... (32) n! x→x0 d x n−1 y For example, the so-called “waterbag” hydrodynamic type system (see [16, 18]) i 2 (a ) (33) + εk a k ati = ∂x 2 is connected with the equation of the Riemann surface λ= p− εk ln( p − a k ). (34) xn =

The main (so-called “Kruskal”) series of conservation law densities can be obtained by substitution of the Taylor series H0 H1 H2 p =λ− − 2 − 3 − ... (35) λ λ λ into the above expression if ε k = 0. If ε k = 0, then at first, the above equation of the Riemann surface must be replaced on λ− εk ln λ = p − ε k ln( p − a k ), (36) ˜ because the Gibbons equation is invariant under scaling λ → λ(λ). In both cases the Hk are polynomials with respect to the field variables a n . Also, the “waterbag” hydrodynamic type system has N infinite series of conservation laws. First, let us rewrite the equation of the Riemann surface in the form λ = ( p − a i )e− p/εi ( p − a k )εk /εi k=i

for any fixed index i. Then the infinite series of conservation laws (17) (i)

(i)

(i)

p (i) = a i + h 1 (a)λ + h 2 (a)λ2 + h 3 (a)λ3 + · · ·

(37)

can be obtained with the aid of Bürmann–Lagrange series (see [24]), whose coefficients are determined by (see (32)) ⎛ ⎞ n−1 d 1 i ⎝ena /εi (a i − a k )−nεk /εi ⎠ , n = 1, 2, .... h (i) n = n! d(a i )n−1 k=i

Thus, the first conservation law densities are i (i) h 1 = ea /εi (a i − a k )−εk /εi , k=i

h (i) 2 =

e

2a i /ε

εi

⎛

i

⎝1 −

⎞

n=i

ai

εn ⎠ i (a − a k )−2εk /εi , .... − an k=i


479

5.1. Quasi-symmetric form. The dispersionless limit of the vector NLS (18) is a degeneration of the “waterbag” hydrodynamic type system (33). A substitution of the expansion (see, for instance, [2]) u˜ (k) = u k + ηk /ε k + · · · into (34) N

λ=µ−

εk ln

k=1

µ − u˜ k µ − uk

yields (19) if εk → ∞. Let us consider again the generating function of conservation laws (see (20)) µt = ∂ x

µ2 + A0 2

(38)

for the Benney hydrodynamic chain (see [1]) k−1 0 Akt = Ak+1 Ax , x + kA

k = 0, 1, 2, . . . ,

(39)

and substitute N Taylor series (37) µ(i) = a i + λbi + λ2 ci + · · · .

(40)

1. Suppose the function A0 depends on N field variables a k only. Then the corresponding hydrodynamic type system is ati

= ∂x

(a i )2 0 + A (a) . 2

This hydrodynamic type system is integrable iff the function A0 (a) satisfies the nonlinear PDE system (a i − a k )∂ik A0 = ∂k A0 ∂i

(a i − a k )

∂n A0 − ∂i A0 ∂k ∂n A0 , i = k,

∂ jk A0 ∂i j A0 ∂ik A0 k j j i + (a − a ) + (a − a ) = 0, i = j = k, ∂i A0 ∂k A0 ∂ j A 0 ∂k A 0 ∂i A0 ∂ j A0

which is a consequence of the compatibility conditions ∂i (∂k µ) = ∂k (∂i µ), where ∂i µ =

∂i A0 µ − ai

−1 ∂n A 0 − 1 . µ − an

Some particular examples are described above. Remark. The above nonlinear PDE system is nothing but the Gibbons–Tsarev system describing N component hydrodynamic reductions via Riemann invariants (see [15]).

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M. V. Pavlov

2. Suppose the function A0 depends on N field variables a k and M field variables bk (where M must not exceed N ) only. Then corresponding hydrodynamic type system is ati = ∂x

(a i )2 + A0 (a, b) , 2

j

bt = ∂x (a j b j ),

i = 1, 2, . . . , N ,

j = 1, 2, . . . , M.

The simplest such system is given by (18), i.e. M = N and A0 = bn (see [36]). This procedure can be extended to other auxiliary field variables from (40). For instance, the third such sub-case is i 2 (a ) + A0 (a, b, c) , i = 1, 2, . . . , N , ati = ∂x 2 j

bt = ∂x (a j b j ),

j = 1, 2, . . . , M,

1 ctk = ∂x a k ck + (bk )2 , 2

k = 1, 2, . . . , K ,

where K M N . 3. Suppose the function A1 (see (39)) is a function of the field variables a k only. Then the corresponding hydrodynamic type system is A0t = ∂x A1 (a),

ati = ∂x

(a i )2 + A0 , 2

i = 1, 2, . . . , N .

(41)

Suppose the function A2 is a function of the field variables a k only. Then the corresponding hydrodynamic type system is A0t

= ∂x A , 1

A1t

i 2 (a ) (A0 )2 2 i 0 , at = ∂ x = ∂x A (a) + + A , i = 1, 2, . . . , N . 2 2

Thus, in the general case one can consider M first moments Ak and K first sets a i1 , bi2 , ci3 , ... (where the index i k runs through the values from 1 up to Nk , k = 1, 2, . . . , K ) as field variables of the corresponding hydrodynamic type systems. In such a case just the latest moment A M depends on the sets a i1 , bi2 , ci3 , ... Nevertheless, the theory established above still works for these hydrodynamic type systems, because all of them have the same generating function of conservation laws (38). Remark. In the cases above the functions A0 (a, b), A0 (a, b, c), A1 (a), A2 (a) can be determined from the Gibbons–Tsarev system written via appropriate field variables. Example. The dispersionless limit of the vector Yajima–Oikawa system (see [30]) is the hydrodynamic type system (cf. (18) and (41)) A0t

= ∂x

η , n

u kt

= ∂x

(u k )2 0 +A , 2

ηkt = ∂x (u k ηk ), k = 1, 2, . . . , N . (42)


481

The eigenvalue–eigenfunction problem is ⎛

⎞⎛ ⎞ −µ 0 1 r ⎝ 1 (u i − µ)δ ik ⎠ ⎝ q i ⎠ = 0. 0 0 ηi δ ik (u i − µ)δ ik si Thus, qi =

r , µ − ui

si =

ηi r, (µ − u i )2

where 2N + 1 eigenvalues are solutions of the discriminant equation µ=

ηk . (µ − u k )2

In accordance with Tsarev’s observations, let us integrate this expression once. Indeed, λ=

ηk µ2 + A0 + 2 µ − uk

is the equation of the Riemann surface connected with the hydrodynamic type system (42). The factorized form of this equation

λ=

1 2

µ−

uk +

ak

N +1

(µ − a n )

n=1 N

(43)

(µ − u k )

k=1

yields 2N + 1 independent series of conservation laws µ(k) = u k + ληk (u, a) + λw k (u, a) + · · · ,

k = 1, 2, . . . , N ,

µ(n) = a n + λbn (u, a) + λcn (u, a) + · · · ,

n = 1, 2, . . . , N + 1,

whose coefficients can be found with the aid of the Bürmann–Lagrange expansion (see (32)). Remark. While the conservation law densities ηk are coefficients of the first N series, the function A0 is a coefficient of the Kruskal series for (43) at the infinity λ → ∞, µ → ∞, λ=

µ2 A1 + A0 + + ··· , 2 µ

where A1 ≡ ηn (see (41) and (42)).

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M. V. Pavlov

5.2. Hamiltonian formalism. The Hamiltonian formalism of hydrodynamic type systems was established in [6] for a local case, and developed later for a nonlocal case in [7, 9]. The generating function of conservation laws for the symmetric hydrodynamic type systems (7) is given by (9). Suppose for simplicity and without loss of generality that the field variables u k are flat coordinates. Then (7) can be written in the symmetric Hamiltonian form ⎛ ⎞ ∂h ∂h i u t = ∂ x ⎝ε i i + γ i γk k ⎠, ∂u ∂u k=i

where the Hamiltonian density h is an expression symmetric under permutation of indices; εi and γ i are constants. Thus, the generating function of commuting flows is given by the formula ⎛ ⎞ ∂p ∂ p u iτ = ∂x ⎝εi i + γ i γk k ⎠. ∂u ∂u k=i

Then infinitely many particular solutions are given by the generalized hodograph method (31) ⎛ ⎞ ⎛ ⎞ ∞ N (s) ∂hn(s) ∂h ∂h ∂ ∂h ∂ n γ m m⎠ = σ sn k ⎝εi + γi γ m m ⎠, xδ ik + t k ⎝εi i + γ i ∂u ∂u ∂u ∂u ∂u i ∂u m=i

m=i

n=1 s=1

where σ sn are arbitrary constants. Example. The Hamiltonian exponential chromatography system (29) i

u it = ∂x

eu , k 1 + γ k eu

i = 1, 2, . . . , N

leads to the equation of the Riemann surface (30) n γ n ln(eu − p − 1). λ = e− p − The Gibbons equation describing deformations of this Riemann surface is given by ep ep ∂λ λt − λx = pt − ∂ x . ∂p The Kruskal series of conservation law densities can be found by the application of the Bürmann–Lagrange series (32) at the vicinity λ → 0, q → 0, where q = exp(− p). In this case the coefficients qn of the inverse series q = q1 λ + q2 λ2 + q3 λ3 + q4 λ4 + · · · are determined by qn =

d n−1 1 lim n−1 n! q→0 dq

q−

n

q γ k ln(1 − qeu ) k

,

n = 1, 2, ....

Then the Kruskal conservation law densities can be found from p = − ln q, p˜ = p + ln λ = − ln[1 + q2 λ + q3 λ2 + q4 λ3 + · · · ]. For instance, p1 = ln , p2 = −4 γ n e2u . n


483

The Kruskal series can be extended in the opposite direction λ → ∞, q → ∞. First, the equation of the Riemann surface (30) must be replaced by the equation (cf. (36)) k λ− γ k ln λ = s − γ k ln s + γ m u m − e−u , where s = exp(− p) −γ m u m . Then the Kruskal conservation law densities p−k can be found exactly as for the Benney hydrodynamic chain (39) (see (36)) by the substitution (cf. (35)) C1 C2 C3 + 2 + 3 + ··· s =λ+ λ λ λ in the above formula. Then p−k are coefficients of the series 1 C1 C2 C3 p˜ = p + ln λ = ln 1 + γ m um + 2 + 3 + 4 + · · · . λ λ λ λ For instance, p−1 = γ m u m , p−2 = γ m e−u + (γ m u m )2 /2. Any semi-Hamiltonian hydrodynamic type system has N series of conservation laws. The equation of the Riemann surface (30) can be written in the form ⎛ ⎞ −p γ k e i k + ln(eu − p − 1)⎠ , p → ui . λ = (e− p − e−u ) exp ⎝u i − γi γi m

k=i

Then the coefficients qk of the Bürmann–Lagrange series are given by (32) n i d n−1 q − e−u 1 lim qn = , n = 1, 2, . . . , n! q→exp(−u i ) dq n−1 λ(q) where q = exp(− p). Finally, N series of conservation law densities p (i) can be found from the series p˜ (i) = p (i) + ln λ = − ln[q1 + q2 λ + q3 λ2 + q4 λ3 + · · · ]. For instance, the first N nontrivial conservation law densities are i k i (i) p1 = e−u − γ k ln(eu −u − 1).

(44)

k=i

Commuting flows of this Hamiltonian chromatography system are u it k = u it k,n =

1 ∂hk ∂x , γ i ∂u i

k = 0, ±1, ±2, ...,

1 ∂hk,n ∂x , γi ∂u i

k = 1, 2, ...,

n = 1, 2, . . . , N ,

where hk are Kruskal conservation law densities and hk,n are N series of conservation law densities. The first N commuting flows ⎛ ⎞ i un e eu 1 k k −u ⎠ i ⎝ u t 1,k = ∂x γn n −e , i = k , u t 1,k = ∂x k k i γk eu − eu eu − eu n=i

are given by (44).

484

M. V. Pavlov

5.3. Mirrored curvilinear conjugate coordinate systems. If the semi-Hamiltonian property (3) is valid, then the diagonal metric gii can be introduced by the relations ∂k µi = ∂k ln Hi , µk − µi

i = k,

(45)

where gii ≡ Hi2 and Hi are the Lame coefficients. Following G. Darboux (see [4]), let us introduce the rotation coefficients of the conjugate curvilinear coordinate nets β ik =

∂i Hk , Hi

i = k.

(46)

Then any solution H˜ k of the linear problem (see [33]) ∂i H˜ k = β ik H˜ i ,

i = k

(47) wi

are connected determines a commuting flow of (2), whose characteristic velocities with H˜ i by the Combescure transformation wi = H˜ i /Hi . Any solution of the conjugate linear problem ∂i ψ k = β ki ψ i , i = k (48) determines a conservation law density ∂i a = ψ i Hi .

(49)

Definition 2 ([30]). The conjugate curvilinear coordinate net determined by symmetric rotation coefficients β ik = β ki is called a Egorov conjugate curvilinear coordinate net. Theorem 2 ([30]). If the integrable hydrodynamic type system (2) has the pair of conservation laws at = b x , bt = cx , (50) then the corresponding conjugate curvilinear coordinate net is Egorov. Remark ([30]). This pair is unique for each given set of Lame coefficients Hk . Corollary 1 ([30]). Any commuting flow (4) has a similar pair of conservation laws (50) ay = h x , h y = fx , where a is the unique potential of the Egorov metric for all commuting flows. Definition 3. Two conjugate curvilinear coordinate nets determined by the rotation coefficients β ik and β¯ ik satisfying β¯ ik = β ki are called a pair of mirrored conjugate curvilinear coordinate nets. Theorem 3. If one integrable hydrodynamic type system (2) has the conservation law at = b x

(51)

such that another integrable hydrodynamic type system ˜ i (r)r zi r yi = µ

(52)

has the pair of conservation laws a y = cz ,

b y = Bz ,

then the corresponding conjugate curvilinear coordinate nets are mirrored.

(53)


485

Proof. The characteristic velocities υ i of the hydrodynamic type system (2) are (see (49)) (1)

µi =

(1)

Hi ψ i Hi ∂i b , = = Hi ψ i Hi ∂i a

(1)

where Hi is a solution of the linear system (47). Let us consider the hydrodynamic type system (52), whose characteristic velocities µ ˜ i are (see (49)) (1)

µ ˜i =

(1)

(1)

(1)

ψi Hi ψ i H ψ ∂i c ∂i B = = = i (1) i = , ψi Hi ψ i ∂i a ∂i b Hi ψ i

(1)

where ψ i is a solution of the conjugate linear system (48). The rotation coefficients can be found in two steps (45) and (46). Indeed, β¯ ik = β ki . 2 Remark. The construction described above is symmetric. Thus, the second conservation law of the first hydrodynamic type system (2) ct = C x is given by quadratures dc =

(1)

ψ i Hi dr i ,

dC =

(1)

(1)

ψ i Hi dr i .

Corollary 2. Any conservation law Py = Q z of the hydrodynamic type system (52) determines the corresponding commuting flow in the conservative form aτ = Px

(54)

or in the Riemann invariants (see (4)) rτi =

Hi(2) i r Hi x

(55)

of the hydrodynamic type system (2), where Hi(2) is some solution of the linear system (47), ∂i P = Hi(2) ψ i and ∂i Q = Hi(2) ψ i(1) . Thus, if two symmetric hydrodynamic type systems are related by the above link, then the generating function of the second hydrodynamic type system (52) determines the generating function of commuting flows for the first hydrodynamic type system (2).

486

M. V. Pavlov

5.4. The chromatography system. The integrable chromatography system (28) has the pair of conservation laws γ k (u k )β−βε+1 = (βε − β − 1)∂x −ε , ∂t ∂t 1/β = ∂x

1−β−βε βε β γ n (u n )βε+β−1 . βε + β − 1

1. The Egorov sub-case. If βε = −1, then this is the Egorov hydrodynamic type system (see (50)) [1 + γ k (u k )β ]1/β u it = ∂x , i = 1, 2, . . . , N , ui where the potential of the Egorov metric (see [30]) is a= γ k (u k )β+2 . 2. The local Hamiltonian sub-case. If β(1 − ε) = 1, then the chromatography system (28) can be written in the Hamiltonian form (26). If β → ∞, the exponential chromatography system (29) (ε = 1) also has the Hamiltonian form (26), where the Hamiltonian density is h = ln . 3. The nonlocal Hamiltonian sub-case (associated with constant curvature metric, see [9, 26]) ˜ ∂ h u it = ∂x (g¯ ik − u i u k ) k + u i h˜ . ∂u If β = 2, then the conservation law density q =1−

1+

γ k (u k )2

is the momentum density (see [26]) of this nonlocal Hamiltonian structure, with the diagonal matrix elements g¯ik = −γ i δ ik (where δ ik is the Kronecker symbol) and the Hamiltonian density −ε h˜ = − γ k (u k )2ε+1 . 2ε + 1 4. The general (mirrored) sub-case. If two hydrodynamic type systems (28) are related via mirrored conjugate curvilinear coordinate nets, then they must have one common conservation law density (see (51) and (53)). Let us prove that such a conservation law density is a= γ k (u k )β−βε+1 . (56) If another chromatography system (28) u˜ iy = ∂z

˜

(u˜ i )β ε˜ , [1 + γ k (u˜ k )β˜ ]ε˜

i = 1, 2, . . . , N

has the same conservation law density ˜ ˜ a= γ k (u˜ k )β−β ε˜ +1 ,

(57)


487

˜ ε → ε˜ can be found from two other obvious then transformation u → u, ˜ β → β, assumptions (see (51) and (53)) u i = (u˜ i )δ , Result. δ=−

1 , βε

˜ 1/β˜ . −ε =

1 ε˜ = − , β

1 β˜ = − . ε

Since the second chromatography system (57) has the generating function of conservation laws 1 P βε , Py = ∂z [1 + γ k (u k )β ]−1/β then the generating function of commuting flows is given by (54) aτ = ∂x p −βε , where the relationship

(58)

P = p −βε

is obtained by comparing the equations of the Riemann surface (which are equivalent) for both chromatography systems. The integrable chromatography system (28) in the Riemann invariants has the form (13) ( pi )βε−1 i rti = βε rx . ε Since this hydrodynamic type system has two conservation laws pt = ∂ x

p βε , ε

at = (βε − β − 1)∂x −ε ,

one can obtain, respectively ∂i p =

∂i ln p βε , βε−1 i βε−1 p − (p ) β

∂i a =

1 + β − βε i 1−βε (p ) ∂i ln . β

Thus, the generating function of commuting flows (58) in the Riemann invariants (55) rτi = σ

( pi )βε−1 ri p[( pi )βε−1 − p βε−1 ] x

is connected with the Gibbons equation ( p(λ))βε−1 βε λx 1 + β − βε p(ζ )[( p(λ))βε−1 − p βε−1 (ζ )] ∂λ βε ∂τ (ζ ) p(λ) + ∂x = ∂ p(λ) (βε − 1)(1 + β − βε) p(λ) βε p βε−1 (λ) × F 1, σ , σ + 1, βε−1 , p(ζ ) p (ζ )

λτ (ζ ) −

488

M. V. Pavlov

where we use the notation σ =

βε . βε − 1

Then the generating function of commuting flows written via the physical field variables u k (cf. (28)) is u iτ (ζ )

βε ∂x =− (βε − 1)(1 + β − βε)

ui p(ζ )

βε

(u i )βε−1 F 1, σ , σ + 1, βε−1 . p (ζ )

Substituting the formal series ∂τ (ζ ) = ∂t i + ζ ∂t i + ζ 2 ∂t i + · · · and respectively the gener0 1 2 ating function of conservation law densities expanded in the series (37), one can obtain infinitely many generating functions of conservation laws for all commuting flows. For instance, the first N such functions are p βε p βε−1 pt i = ∂ x . F 1, σ , σ + 1, i βε−1 0 ui (u ) Remark. Suppose that some N component hydrodynamic type system contains N − 1 equations (see the above generating function of conservation laws) u kti = ∂x 0

uk ui

βε

(u k )βε−1 F 1, σ , σ + 1, i βε−1 (u )

,

k = i.

Then such hydrodynamic type system is integrable if its N th equation satisfies some extra conditions. This hydrodynamic type system is not symmetric like (7). However, the algebro-geometric approach is still valid. Thus, if any given hydrodynamic type system contains N − 1 equations u kt

uk = ∂x ψ u; 1 , u

k = 1,

then one should substitute the Taylor series (37) in the generating function of conservation laws p pt = ∂x ψ u; 1 . u The N th equation will be obtained by the limit (see such example at the end of the sub-section “Hamiltonian formalism”) u 1t

= ∂x

h 1 (u) lim ψ u; 1 + ε 1 1 ε→0 u

.

All other computations are exactly the same as in the symmetric case.


489

5.5. Reciprocal transformations. The concept of “reciprocal transformation” was introduced by S.A. Chaplygin (see, for instance, [31] and [32]) dz = A(u)d x + B(u)dt,

dy = C(u)d x + D(u)dt,

where two arbitrary conservation laws At = Bx and Ct = Dx preserve the gas dynamic equations changing adiabatic index only. The gas dynamics equations were the first known example in the theory of integrable hydrodynamic type systems. In this sub-section we show that the integrable chromatography system (28) is invariant under a pair of different reciprocal transformations. Thus, the infinite sets of such systems are related by a chain of reciprocal transformations which are described below. 1. The first such reciprocal transformation is very simple dz = dt,

dy = d x.

Then the integrable chromatography system (28) reduces to the chromatography system 1

wiy where

(wi ) βε = ∂z , [1 − γ k (w k )1/ε ]1/β 1

¯ −1/β , u i = (wi ) βε

i = 1, 2, . . . , N , ¯ = −1 .

Thus, two chromatography systems (28) (β, ε, γ k ) and (1/ε, 1/β, −γ k ) are related by the transformation x ↔ t. Remark. The first conservation law ∂t γ k (u k )β−βε+1 = (βε − β − 1)∂x −ε transforms into the second conservation law ¯ β¯ ε¯ 1−β− β¯ ε¯ ¯ 1/β¯ n β¯ ε¯ +β−1 ¯ β ¯ ¯ γ¯ n (w ) ∂y = ∂z β¯ ε¯ + β¯ − 1 and vice versa. 2. The reciprocal transformation 1−β−βε βε 1/β n βε+β−1 β γ n (u ) dt, dz = d x + βε + β − 1

dy = βεdt

connects the chromatography system (28) and another symmetric system i βε (υ ) υi υ iy = ∂z − γ n (υ n )βε+β−1 , i = 1, 2, . . . , N , βε βε + β − 1 where υ i = u i −1/β . However, the chromatography system (28) has the commuting flow i 2−βε (u ) ui i n β−βε+1 u t 1 = ∂x . + γ n (u ) 2 − βε β − βε + 1

(59)

(60)

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M. V. Pavlov

It is easy to verify the compatibility conditions ∂t 1 (u it ) = ∂t (u it 1 ). Then the hydrodynamic type system (59) has the commuting flow υ it = ∂x

(υ i )2−βε , [1 − γ k (υ k )β ]2/β−ε

i = 1, 2, . . . , N

which is the chromatography system (28) again. Thus, two chromatography systems (28) (β, ε, γ k ) and (β, 2/β − ε, −γ k ) are related by the above reciprocal transforma¯ = −1 = 1 − γ n (υ n )β . tion, where Applying both reciprocal transformations iteratively, one can construct a link between the chromatography systems (28) with the distinct indices β and ε. 6. Homogeneous Hydrodynamic Type Systems Another hydrodynamic type system

u it = ∂x (u i )β (u n )γ n

arising in chromatography (see formula 6 in [10], β and γ n are arbitrary constants) is invariant under scaling of field variables u k → cu k (and appropriate scaling of the independent variable t or x). We call such hydrodynamic type systems homogeneous. Many physically interesting hydrodynamic type systems belong to this class. The existence of the corresponding generating function of conservation laws pt = ∂ x p β (u n )γ n yields (see (10)) γ i pβ ∂p = i i β−1 ∂u βu [ p − (u i )β−1 ]

−1 1 γ k (u k )β−1 − 1 . β p β−1 − (u i )β−1

Also, the equation of the Riemann surface λ(u, p) must be invariant under the extended scaling u k → cu k , p → cp. Since (see (11)) −1 1 γ k (u k )β−1 ∂λ ∂λ γ i pβ 1 − , = i i β−1 i β−1 β−1 i β−1 ∂u βu [ p − (u ) ] β p − (u ) ∂p the equation of the Riemann surface λ(u, p) can be found in quadratures ln λ = ln p +

1 (u k )β−1 γ k ln β−1 , (β − 1)(β + γ m ) p − (u k )β−1

where we used the Euler theorem λ = pλ p +

u k λk .

˜ Since the Gibbons equation (12) is invariant under the point transformation λ → λ(λ), the above equation of the Riemann surface λ(u, p) can be written in the form q −γ k λ = q β+ γ m 1− k , w where q = p β−1 , w k = (u k )β−1 .


491

6.1. The Kodama hydrodynamic type system. The Kodama hydrodynamic type system (see [18]) k 1 m k+1−m k k,1 N at = ∂ x a a +δ a , k = 1, 2, . . . , N , 2 m=1

where δ ik is the Kronecker symbol, is a hydrodynamic reduction of the Benney hydro = (N + k − 2)a k ∂k . dynamic chain (39), homogeneous under the Euler operator E Thus, the corresponding generating function of conservation laws is (38) (the generating function of conservation laws for any integrable hydrodynamic chain is unique for all its hydrodynamic reductions). The Kodama hydrodynamic type system can be obtained from the above generating function by substitution of the Taylor series p = a 1 + λa 2 + λ2 a 3 + · · · + λ N −1 a N + λ N h 1 (a) + λ N +1 h 2 (a) + λ N +2 h 3 (a) + · · · . Then A0 ≡ a N and h k (a) are some polynomial conservation law densities. For instance, h1 (a) = a k a N +1−k /2 is the momentum, h2 (a) is the Hamiltonian, where the Kodama hydrodynamic type system has a bi-Hamiltonian structure, and the first of these structures is ∂h2 atk = ∂x N +1−k , k = 1, 2, . . . , N . ∂a Without loss of generality let us restrict our attention to the three component case 2 u υ2 + w , υ t = ∂x (uυ), wt = ∂x uw + . u t = ∂x 2 2 Thus, the Gibbons equation (see (12) and (38)) is 2 p ∂λ λt − pλx = pt − ∂ x +w . ∂p 2 Then one has w( p − u) + υ 2 υ( p − u) λu = λ p , λυ = λp, where

( p − u)3 − 1 λp, λw =

= ( p − u)3 − w( p − u) − υ 2 .

Since the Kodama hydrodynamic type system is homogeneous, the function λ(u, υ, w, p) must be homogeneous. Then λ = (2 p∂ p + 2u∂u + 3υ∂υ + 4w∂w )λ up to some insignificant constant factor (the degree of homogeneity). Thus, the equation of the Riemann surface can be found in quadratures. For instance, ∂ p ln λ =

(p

− u)[2 p 3

− 4up 2

2 . + 2(u 2 + w) p − 2uw + υ 2 ]

Then (cf. [18]) the equation of the Riemann surface for the Kodama hydrodynamic type system is w υ2 λ= p+ . + p − u 2( p − u)2

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Remark. The same procedure can be repeated for any N component Kodama hydrodynamic type system. Moreover, suppose the determinant (14) is computed and written in the factorized form N ( p − p k (a)) k=1

for arbitrary N , then the equation of the Riemann surface is given by rational function (see [18]) N N −1 p − p k (a) Bk (a) λ= dp ≡ p + , (61) p − a1 ( p − a 1 )k k=1

k=1

where Bk are some polynomials with respect to flat coordinates a n . These coefficients Bk can be found by substitution of the above formula in (11). The corresponding linear system is N

∂n B k+1 =

a m+1−n ∂m B k ,

k = 1, 2, . . . , N − 2,

n = 2, 3, . . . , N − 1,

m=n+1

where B 1 ≡ a N and N −1

a m+1−n ∂m B N −1 = 0,

∂ N B n = 0,

n = 2, 3, . . . , N − 1.

m=n+1

Remark. In the symmetric case (7) all derivatives λk = (...)λ p can be found immediately, but in the above case for each N , one must (step by step) compute all above derivatives consequently. And if the given hydrodynamic type system is non-symmetric, if the generating function of conservation laws (as in the above example) is not given a priori, then derivation of the equation of the Riemann surface becomes a very complicated computational problem. Remark. The equation of the Riemann surface (61) can be written in the totally factorized form N λ = ( p − a 1 )1−N ( p − bk ), k=1

where (N − 1)a 1 = bk (a). Substituting the Taylor series (17) p (k) = bk + λck (b) + · · · in (38) yields the Kodama hydrodynamic type system written in the symmetric form k 2 2 (b ) 1 k 2 1 k bt = ∂x . (62) + (b ) − bk 2 2 2(N − 1) 6.2. Cubic Hamiltonian hydrodynamic type system. The cubic Hamiltonian hydrodynamic type system (26) with the Hamiltonian density h=

3 1 γ k (u k )3 + β γ n (u n )2 + ε γ k uk γ k uk 6


is equivalent to the homogeneous hydrodynamic type system (cf. (62)) i 2 2 (a ) i k 2 k at = ∂ x γ ka +α γ k (a ) + δ 2

493

(63)

under the transformation a i = u i + 2βγ k u k . Following the scheme given above (see the Gibbons–Tsarev system in Sub-sect. 5.1), one can verify that the hydrodynamic type system (63) is integrable iff δ=−

2α 2 1 + 2α γ n

⇐⇒

ε=−

2β 2 /3 1 + 2β γ n

and is connected with the equation of the Riemann surface 1+2α γ s 2α m γ ma ( p − a k )−2αγ k . λ= p− 1 + 2α γ n Remark. The mechanical interpretation. Let us consider the classical Hamilton’s system x˙ =

∂H , ∂p

p˙ = −

∂H , ∂x

where the Hamiltonian is H = p 2 /2+V (x, t). We seek the Hamiltonian system x¨ = −Vx possessing the first integral (“energy constant surface”; see [19]) λ = V0 + V1 x˙ + V2 x˙ 2 + · · · + VN −2 x˙ N −2 + VN x˙ N ,

(64)

where Vk (x, t) are some functions and N is arbitrary. Differentiating this equation with respect to t yields the hydrodynamic type system (63) (cf. (62)) k 2 2 1 (u ) − um + u kt = −∂x (u m )2 , 2 2(N + 1) where the field variables u k are coefficients of the polynomial (64) written in the factorized form λ = (x˙ + u m ) (x˙ − u k ) and the “potential energy” is given by the symmetric expression 2 1 m m 2 u V =− + (u ) . 2(N + 1) Let us replace (64) by an arbitrary dependence λ = λ(u; x), ˙ u k (x, t)

where are some functions. Differentiating the above equation with respect to t, one can obtain (see (1), (14 ); cf. (11)) λµ ∂i V (u) = (υ ik (u) − µδ ik )λk , where we use the equation of the Riemann surface λ = λ(u; µ). In the symmetric case (7) u it = ∂x ψ(u 1 , u 2 , . . . , u N ; u i )

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the above system reduces to (see (15), (8); cf. (11)) λµ ∂i V (u) = Aik (u, µ)λk , which must be connected with (11) by the substitution (see (15)) µ=

∂ψ . ∂p

6.3. The ideal gas dynamics. The ideal gas dynamics 2 u υβ + , υ t = ∂x (uυ) u t = ∂x 2 β

(65)

is connected with the equation of the Riemann surface (see [3]) λ=

1 β [ p + βu + υ β p −β ]. 4

(66)

The corresponding Gibbons equation is β+1 p ∂λ pt − ∂ x λt − ( p + u)λx = + up . ∂p β +1 β

(67)

Remark. The equation of the Riemann surface (66) is unique for all values of index β. Indeed, introducing the Riemann invariants r 1,2 = u ±

2 β/2 υ , β

the ideal gas dynamics (65) can be written in the diagonal form rt1,2 =

1 1 β [r + r 2 ± (r 1 − r 2 )]r x1,2 . 2 2

Then the equation of the Riemann surface (66) in terms of the Riemann invariants is λ= where

ν r 1 + r 2 (r 1 − r 2 )2 + + , 4 2 4ν

(68)

ν = pβ .

Remark. Two infinite series of conservation laws can be obtained with the aid of the Bürmann–Lagrange series (see the next section) at the vicinity of two zeros of the equation of the Riemann surface √ √ √ √ [ν + ( r 1 + r 2 )2 ][ν + ( r 1 − r 2 )2 ] 4λ = , ν while the Kruskal series can be derived at the infinity (λ → ∞, ν → ∞) and at the vicinity of another singular point (λ → ∞, ν → 0).


495

6.4. Whitham averaged Sinh-Gordon equation. A one-phase solution of the Sinh–Gordon u xt = sinh u averaged by the Whitham approach, is the two component hydrodynamic type system (see [35]) (69) rti = µi (r)r xi , where the differentials of the quasi-momentum and the quasi-energy (see [12]); respectively, 1 − < λ1 > λ− < λ > dλ, dq = λ dλ dp = λ(r 1 − λ)(r 2 − λ) λ(r 1 − λ)(r 2 − λ) determine the characteristic velocities µ

1,2

±1 dq(λ) 1 2 K(s) 1 − (1 − s ) | 1,2 = √ (r) = , dp(λ) λ=r E(s) r 1r 2

where K(s) and E(s) are complete elliptic integrals of the first and second kind, respectively; s 2 = r 2 /r 1 is elliptic modulus, and 1 < a >≡ T

r 2 0

adλ λ(r 1 − λ)(r 2 − λ)

,

r 2 dλ T= . λ(r 1 − λ)(r 2 − λ) 0

However, this hydrodynamic type system can be put into the framework presented in the previous sub-section. N -phase solutions of the Sinh-Gordon equation are associated with the Riemann surface determined by the hyper-elliptic curve (see [12]) µ2 = λ

2N

(λ − r k ).

k=1

However, only in the one-phase case can all the above formulas be connected with an alternative Riemann surface determined by (68). The next two theorems can be proved by a straightforward calculation. Theorem 4. The Gibbons equation λt −

d q/dν ˜ ∂λ λx = ( p˜ t − q˜ x ) d p/dν ˜ ∂ p˜

connects the averaged (by the Whitham approach) one-phase solution of the SinhGordon equation (69) with the equation of the Riemann surface (68), where √ √ 1 ν 1 2 r + r2 ν + ( r 1 − r 2 )2 < λ > p˜ = + r + r + −√ ln ν, − < λ > ln ν, q˜ = ln √ √ 2 2 ν + ( r 1 + r 2 )2 r 1r 2

with < λ >= r

1

K(s) 1− . E(s)

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M. V. Pavlov

Theorem 5. The generating function of commuting flows in the Riemann invariants i i ii ν + 2P ri rτ (ζ ) = ln ν(ζ ) + g i ν − ν(ζ ) x is connected with the Gibbons equation 3ν 2 (λ) + 4(r 1 + r 2 )ν(λ) + (r 1 − r 2 ) λτ (ζ ) − ln ν(ζ ) + λx (ν(λ) + 2P)(ν(λ) − ν(ζ )) ∂λ ∂τ (ζ ) p(λ) = ˜ − ∂x Q(λ, ζ ) , ∂ p(λ) ˜ where the potential P of the Egorov metric gii = ∂i P is P=

r1 + r2 − < λ >, 2

the Egorov metric is g11 =

E2 (s)/K2 (s) , 2(1 − s 2 )

g22 = −

[1 − s 2 − E(s)/K(s)]2 , 2s 2 (1 − s 2 )

and 1 1 1 Q(λ, ζ ) = 2(r 1 + r 2 ) + ν(λ) + ν(ζ ) + P ln ν(λ) ln ν(ζ ) + [r 1 + r 2 + ν(ζ )] ln ν(λ) 2 2 2 1 + [r 1 + r 2 + ν(λ)] ln ν(ζ ) + 2(λ + ζ ) ln[ν(λ) − ν(ζ )] 2 − 2λ ln ν(ζ ) − 2ζ ln ν(λ). 7. Integrable Hydrodynamic Chains The Gibbons equation for the hydrodynamic type system (60) 2−βε p ∂λ λt 1 − ( p 1−βε + a)λ pt 1 − ∂ x + a˜ p , ˜ x= ∂p 2 − βε

(70)

where a˜ = a/(β − βε + 1) (see (56)), is exactly the same as the Gibbons equation for the ideal gas dynamics (67). Introducing the moments 1 Bk = γ i (u i )(1−βε)(k+1)+β , k = 0, 1, 2..., (1 − βε)(k + 1) + β the hydrodynamic type system (60) i 2−βε ui (u ) u it 1 = ∂x + γ n (u n )β−βε+1 2 − βε β − βε + 1 can be rewritten as the Kupershmidt hydrodynamic chain (see [22]) Btk1 = Bxk+1 + B 0 Bxk + [(1 − βε)(k + 1) + β]B k Bx0 ,

k = 0, 1, 2, . . . ,

(71)


497

connected with the Gibbons equation (70), where a˜ =

B0 . 1 − βε

Suppose the moments B k of the Kupershmidt hydrodynamic chain are some functions of N field variables u k , then the N -component hydrodynamic reduction u it 1 = ∂x

(u i )2−βε B 0 (u) i + u 2 − βε 1 − βε

(72)

is an integrable hydrodynamic type system (7) iff the function B 0 (u) satisfies some nonlinear PDE system, which is a consequence of the compatibility condition ∂i (∂k p) = ∂k (∂i p), where (cf. (27)) −1 ∂k B 0 (u) ∂i B 0 (u) 1 − βε + ∂i p = p i 1−βε . (u ) − p 1−βε (u k )1−βε − p 1−βε

(73)

If B 0 (u) = (1 − βε)γ k (u k )β−βε+1 /(β − βε + 1), then (73) reduces to (27). The compatibility condition ∂i (∂k p) = ∂k (∂i p) results in a nonlinear PDE system on the function B 0 (u) only. A solution is parameterized by N arbitrary functions of a single variable. Thus, any symmetric hydrodynamic type system (7) can be used for a derivation of the corresponding integrable hydrodynamic chain which all have the same Gibbons equation. At the same time, all hydrodynamic reductions can be written in a similar symmetric form, but with another dependence of the r.h.s. functions like B 0 (u) with respect to the field variables u k . If one could solve the corresponding nonlinear PDE system (which is known in the language of Riemann invariants as the Gibbons–Tsarev system, see [15]), then infinitely many symmetric hydrodynamic type systems (7) would be produced. Corollary 3. Let us consider the generating function of conservation laws (see (72)) pt 1 = ∂ x

p α+1 B 0 (u) + p α+1 α

and introduce the parameter α = 1 − βε. Then N + 1 parametric family (N parameters γ k and β for each fixed index α) of hydrodynamic reductions of the hydrodynamic chain (71) Btk1 = Bxk+1 + B 0 Bxk + [α(k + 1) + β]B k Bx0 ,

k = 0, 1, 2, ...

is a set of the hydrodynamic type systems (72) u it 1

= ∂x

(u i )α+1 + α+1

γ k (u k )α+β i u , α+β

which are distinct for every value of index β (all above hydrodynamic chains are equivalent for each fixed index α and for any value of the index β, see details in [28]).

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M. V. Pavlov

Remark. Introducing the moments Ck =

1 γ i (u i )(βε−1)k+β , (βε − 1)k + β

k = 0, 1, 2...,

the hydrodynamic type system (28) can be rewritten as the Kupershmidt hydrodynamic chain (see [23]) Ctk = βε(1+βC 0 )−ε C xk+1 +[(βε −1)(k +1)+β]C k+1 [(1+βC 0 )−ε ]x , k = 0, 1, 2, . . . . (74) In this section we proved that the hydrodynamic type systems (28) and (60) commute with one another, and hence that the corresponding (B and C) hydrodynamic chains commute with each other. All other details can be found in [28]. 8. Hamiltonian Chromatography System Another generalization of the chromatography system (cf. (25) and [10]) is given by the Hamiltonian hydrodynamic type system ati = ∂x

∂h , ∂a i

i = 1, 2, . . . , N ,

where the Hamiltonian density is h( ) and = z k (a k ). If this system is diagonalizable (see (2)) rti = µi (r)r xi , i = 1, 2, . . . , N , then it is integrable (see [6]). Thus, we are looking for the corresponding transformation r i (a). A direct computation yields (the indices of the Riemann invariants r k and characteristic velocities µi are omitted for simplicity below) ∂r ϕ = , ∂z i ζ − Vi

ρ( ) =

Vn , ζ − Vn

where Vk (z k ) = z k /2, (ln h ) ρ( ) = 1/2, ϕ = ρ −1 ( )Vk ∂r/∂z k and µ = ζ h . The Riemann invariants exist iff the compatibility conditions ∂i (∂k r ) = ∂k (∂i r ) are fulfilled, where ∂i ≡ ∂/∂z k . Eliminating ϕ and its first derivatives from the compatibility conditions, one can obtain the integrability condition 2

q j (∂i qk − ∂k qi ) + qk (∂ j qi − ∂i q j ) + qi (∂k q j − ∂ j qk ) = 0 for every three distinct indices i, j, k, where qi = (ζ − Vi )−1 . Taking into account the equality −1 Vi Vi Vi Vm ∂i ζ = + − ρ ( ) , (ζ − Vi )2 ζ − Vi (ζ − Vm )2 the above integrability condition reduces to the single ODE V V = (1 + α)V + βV + γ , 2

(75)

z(a i )

where Vi ≡ V (z i ), z i ≡ and ρ( ) = α + δ (α, β, γ , δ are arbitrary constants). Thus, the Hamiltonian chromatography system ati = ∂x [h ( )z (a i )]

(76)


499

is integrable if h = e (α = 0), h = ln (α = −1/2) and h = ε (for all other values of α; ε is an arbitrary constant). Then the Gibbons equation λt − z ( p)h ( )λx =

∂λ pt − ∂x [z ( p)h ( )] ∂p

is determined by the equation of the Riemann surface λ(a; p), which can be found in quadratures dz dz n 1+4α dλ = z z ( p) ( p) exp β V z ( p) − Vn Vn dp , +2 α + δ − z ( p) − Vn where ζ = z ( p). Remark. Introducing the moments A = k

Vi dz i , k

the Hamiltonian chromatography system (76) can be rewritten as a Hamiltonian hydrodynamic chain k+1 k Fnk (A)Anx , k = 0, 1, 2, . . . , At = n=0

determined by the Hamiltonian density h(A0 ) (h = e A , h = ln A0 , h = (A0 )ε ) and by the Poisson bracket 0

{Ak , An } = [Bk,n ∂x + ∂x Bn,k ]δ(x − x ), where (here we use (75)) Bk,n = [2(1 + α)k + 1]Ak+n+1 + 2βk Ak+n + 2γ k Ak+n−1 . 9. Exceptional (Linearly Degenerate) Case In this paper the algebro-geometric approach for integrability of hydrodynamic type systems is proposed. However, this approach is most effective just in case the symmetric hydrodynamic type systems possess any symmetry operator (see Sect. 6). Then the generating function of conservation laws for the symmetric hydrodynamic type systems can be found immediately; the integration factor in the computation of the Riemann surface can be found just if a corresponding hydrodynamic type system is invariant under some Lie group of symmetries (like homogeneity). Nevertheless, this algebro-geometric approach can be used in all other more general and complicated cases, except possibly hydrodynamic type systems which are hydrodynamic reductions of linear-degenerate hydrodynamic chains (see [8, 27]). These hydrodynamic type systems usually can be written explicitly in the Riemann invariants (see

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M. V. Pavlov

[27]) and the conservation law fluxes of their generating functions of conservation laws are linear functions with respect to conservation law density p (see [27]) pt = ∂x [ pυ(r; λ)].

(77)

For instance, in the limit βε = 1, the integrable chromatography system (28) u it = εi ∂x

[1 +

ui , γ k (u k )1/ε ]ε

i = 1, 2, . . . , N

can be written explicitly in the Riemann invariants (see [10], formula 5; also [27]; the parameters εi and γ k do not affect explicit expressions of characteristic velocities in the Riemann invariants) m ε r i r xi , i = 1, 2, . . . , N . rt = ri The generating function of conservation laws is (see [27]) ε rm pt = ∂ x p . λ In the same limit βε = 1, the hydrodynamic type system (60) γ n (u n )1/ε u it 1 = δ i ∂x u i 1 + ε in the Riemann invariants is (the parameters δ i and γ k do not affect the explicit expressions of characteristic velocities in the Riemann invariants) rti1 = r i − ε r m r xi . The generating function of conservation laws is (see [27]) pt 1 = ∂ x p λ − ε rm . In this section we consider a hydrodynamic type system written explicitly in the Riemann invariants rti = υ i (r)r xi , i = 1, 2, . . . , N , such that the characteristic velocities υ i (r) are determined by the unique function υ(r;λ), υ i (r) = υ(r;λ)|λ=r i .

(78)

Then the semi-Hamiltonian criterion (3) reduces to ∂j

∂jυ ∂i υ = ∂i , υi − υ υj −υ

i = j.

(79)

Suppose this hydrodynamic type system has the generating function of conservation laws (77). Then ∂i υ ∂i ln p = υi − υ


501

and the semi-Hamiltonian criterion (79) is satisfied automatically. Moreover, the generating function of conservation law densities p in such a case can be found explicitly: ∂k υ k p = exp dr . υk − υ Thus, any commuting flow rτi = wi (r)r xi has a similar generating function of conservation laws pτ = ∂x [ pw(r,λ)], where wi (r) = w(r,λ)|λ=r i . Remark. N -phase solutions of the nonlinear equations (like KdV, NLS, Sinh-Gordon) averaged by the Whitham approach, are hydrodynamic type systems which belong to the class presented in this section (78) rti =

dq | i ri , dp λ=r x

where the Abelian holomorphic differentials of the quasi-momentum dp and the quasienergy dq are determined on the Riemann surfaces of genus N (see, for instance, details in [20]). Thus, the corresponding hydrodynamic chains should be linear degenerate. 10. Conclusion and Outlook In this paper we suggest a universal (except for the linearly degenerate case) approach to integrable (semi-Hamiltonian) symmetric hydrodynamic type systems. This approach is very effective because the generating function of conservation laws in each such case is unique; this is a consequence of the construction ((7) → (9)). In all other cases the first step consists of the computation of the generating function of conservation laws. Let us illustrate the complexity of this problem to two component hydrodynamic type systems (for simplicity and without loss of generality we can restrict our consideration to the two component case only). Suppose we have the nonlinear elasticity equation (this is nothing but the ideal gas dynamic written in the Lagrangian coordinates, while (65) is written in the Euler coordinates), which is a commuting flow to the ideal gas dynamics υ y = ux ,

u y = ∂x

υ β−1 . β −1

(80)

This hydrodynamic type system is written in non-symmetric form. Thus, we do not know in advance the generating function of conservation laws in this case. Nevertheless, obviously, we must seek such a generating function in the form p y = ∂x ψ(u, υ, p).

(81)

However, if such a generating function could be found, then this would be a generating function for the whole hydrodynamic chain, not just for the nonlinear elasticity.

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M. V. Pavlov

Thus, one should seek N component hydrodynamic type systems written in the Riemann invariants (13) compatible with the above generating function. Thus, we have ∂i p =

∂ψ ∂ψ ∂u ∂i u + ∂υ ∂i υ . ∂ψ ∂ψ ∂ p | p= pi − ∂ p

Here is a crucial point of the general construction. From the first equation of the nonlinear elasticity equation (80) one can obtain (take into account (13)) ∂ψ | i ∂i υ = ∂i u. ∂ p p= p

(82)

From the second equation one gets ∂ψ β−2 | ∂i υ. i ∂i u = υ ∂ p p= p

(83)

Thus, we have two choices ∂i p =

∂ψ ∂ψ ∂ψ ∂u ∂ p | p= pi + ∂υ ∂ψ ∂ψ ∂ p | p= pi − ∂ p

∂i υ,

∂i p =

∂ψ ∂u

+ υ 2−β ∂ψ ∂υ

∂ψ ∂ p | p= pi ∂i u. ∂ψ ∂ψ ∂ p | p= pi − ∂ p

These two options are no longer equivalent (in comparison with the symmetric case), because we must relax the relation, a consequence of (82) and (83), 2 ∂ψ | = υ β−2 , i ∂ p p= p which is only valid for ideal gas dynamics. Since we consider the generating function of conservation laws (81) for the whole hydrodynamic chain, then depending on this choice, we shall be able to construct two different hydrodynamic chains. Thus, for instance, the nonlinear elasticity equation (80) can be embedded into different Kupershmidt hydrodynamic chains (71) and (74). Example. The dispersionless limit of the Boussinesq system (see (80), β = 3; [14], [27]) υ y = u x , u y = ∂x (υ 2 /2) (84) satisfies the Gibbons equation λy −

υ2 υ2 ∂λ , p λ = + ∂ x y x p3 ∂p 2 p2

where µ = p 3 and the equation of the Riemann surface is (cf. (66)) λ = µ + 3u +

υ3 . µ

(85)

Simultaneously, the simplest reduction of the Benney moment chain (see [14]) is again the dispersionless limit of the Boussinesq system determined by the Gibbons equation µ ˜2 ∂λ λ y − µλ −υ , ˜ x= µ ˜ y − ∂x ∂µ ˜ 2


503

where the equation of the Riemann surface is λ = −µ ˜ 3 + 3υ µ ˜ + 3u.

(86)

The substitution µ ˜ = −( p + υ/ p) in this cubic equation yields exactly (85) λ = p 3 + 3u +

υ3 . p3

Factorizing the cubic polynomial (86) λ = − p( ˜ p˜ − u 1 )( p˜ − u 2 ), where p˜ = µ ˜ + (u 1 + u 2 )/3, the dispersionless limit of the Boussinesq system (84) u 1y =

1 ∂x [(u 1 )2 − 2u 1 u 2 ], 6

u 2y =

1 ∂x [(u 2 )2 − 2u 1 u 2 ] 6

satisfies the Gibbons equation λy −

u1 + u2 p˜ − 3

2 p˜ u1 + u2 ∂λ λx = p˜ y − ∂x − p˜ . ∂ p˜ 2 3

Remark. Generating functions of conservation laws p y = −∂x

υ2 , 2 p2

p˜ y = ∂x

p˜ 2 u1 + u2 − p˜ 2 3

have different sets of conservation law densities Hk , which coincide for the nonlinear elasticity equation (84) up to insignificant factors. In another paper we present a classification of integrable hydrodynamic chains based on the concept of generating functions of conservation laws. For instance, all generating functions of conservation laws (81) can be found. At least two of them are connected with the ideal gas dynamics (65); each two component hydrodynamic type system (1) must be connected with some function ψ(u, υ, p). We cannot suggest a scheme for constructing this function ψ(u 1 , u 2 , . . . , u N ; p) for any a priori given hydrodynamic type system (1) in the general case. However, for any given function ψ(u 1 , u 2 , . . . , u N ; p) we are able to reconstruct a corresponding hydrodynamic type system (1) together with its commuting flows. Acknowledgements. I thank Boris Dubrovin, Eugeni Ferapontov, John Gibbons, Yuji Kodama, Boris Kupershmidt, Sergey Tsarev, Vladimir Zakharov and Mikhail Zhukov for their stimulating and clarifying discussions. I am grateful to the Institute of Mathematics in Taipei (Taiwan) where some part of this work has been done, and especially to Jen-Hsu Chang, Jyh-Hao Lee, Ming-Hien Tu and Derchyi Wu for fruitful discussions.

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References 1. Benney, D.J.: Some properties of long non-linear waves. Stud. Appl. Math. 52, 45–50 (1973) 2. Bogdanov, L.V., Konopelchenko, B.G.: Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type. Phys. Lett. A 330, 448–459 (2004) 3. Brunelli, J.C., Das, A.: A Lax description for polytropic gas dynamics. Phys. Lett. A 235(6), 597–602 (1997) 4. Darboux, G.: Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Paris: GauthierVillars, 1910 5. Dubrovin, B.A.: Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models. Commun. Math. Phys. 145, 195–207 (1992). Dubrovin, B.A.: Geometry of 2D topological field theories. Lecture Notes in Math. 1620, Berlin-Heidelberg Newyork: Springer-Verlag, pp.120–348 (1996) 6. Dubrovin, B.A., Novikov, S.P.: Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method. Soviet Math. Dokl. 27, 665–669 (1983); Dubrovin, B.A., Novikov S.P.: Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory. Russ. Math. Surv. 44(6), 35–124 (1989) 7. Ferapontov, E.V.: Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications. Amer. Math. Soc. Transl. (2) 170, 33–58 (1995) 8. Ferapontov, E.V., Khusnutdinova, K.R.: On integrability of (2+1)-dimensional quasilinear systems. Commun. Math. Phys. 248, 187–206 (2004); Ferapontov, E.V., Khusnutdinova, K.R.: The characterization of 2-component (2+1)-dimensional integrable systems of hydrodynamic type. J. Phys. A: Math. Gen. 37(8), 2949–2963 (2004) 9. Ferapontov, E.V., Mokhov, O.I.: Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature, Russian Math. Surv 45(3), 218–219 (1990) 10. Ferapontov, E.V., Tsarev, S.P.: Systems of hydrodynamic type that arise in gas chromatography. Riemann invariants and exact solutions. (Russian) Mat. Model. 3(2), 82–91 (1991) 11. Flaschka, H., Forest, M.G., McLaughlin, D.W.: Multi-phase averaging and the inverse spectral solution of the Korteweg - de Vries equation. Commun. Pure Appl. Math. 33(6), 739–784 (1980) 12. Forest, M.G., McLaughlin, D.W.: Modulations of sinh-Gordon and sine-Gordon wavetrains. Stud. Appl. Math. 68(1), 11–59 (1983) 13. Gibbons, J.: Collisionless Boltzmann equations and integrable moment equations. Physica D 3, 503–511 (1981) 14. Gibbons, J., Kodama, Yu.: Solving dispersionless Lax equations. In: N. Ercolani et al., editor, Singular limits of dispersive waves, V. 320 of NATO ASI Series B, New York, Plenum 1994 p. 61 15. Gibbons, J., Tsarev, S.P.: Reductions of the Benney equations. Phys. Lett. A 211 19–24 (1996); Gibbons, J., Tsarev, S.P.: Conformal maps and reductions of the Benney equations. Phys. Lett. A 258, 263–271 (1999) 16. Gibbons, J., Yu, L.A.: The initial value problem for reductions of the Benney equations, Inverse Problems 16(3), 605–618 (2000); Yu, L.A.: Waterbag reductions of the dispersionless discrete KP hierarchy. J. Phys. A: Math. Gen. 33, 8127–8138 (2000) 17. Haantjes, J.K: On X m−1 -forming sets of eigenvectors. Indagationes Mathematicae 17, 158–162 (1955) 18. Kodama, Yu.: A method for solving the dispersionless KP equation and its exact solutions. Phys. Lett. A 129(4), 223–226 (1988); Kodama, Yu.: A solution method for the dispersionless KP equation. Prog. Theor. Phys. Supplement. 94, 184 (1988) 19. Kozlov, V.V.: Polynomial integrals of dynamical systems with one-and-a-half degrees of freedom. (Russian) Mat. Zametki 45(4), 46–52 (1989); translation in Math. Notes 45(3–4), 296–300 (1989) 20. Krichever, I.M.: The averaging method for two-dimensional “integrable” equations, Funct. Anal. Appl. 22(3), 200–213 (1988); Krichever, I.M.: Spectral theory of two-dimensional periodic operators and its applications. Russ.1 Math. Surv. 44(2), 145–225 (1989) 21. Krichever, I.M.: The dispersionless equations and topological minimal models. Commun. Math. Phys. 143(2), 415–429 (1992). Krichever, I.M.: The τ -function of the universal Whitham hierarchy, matrix models and topological field theories. Commun. Pure Appl. Math. 47, 437–475 (1994) 22. Kupershmidt, B.A.: Deformations of integrable systems. Proc. Roy. Irish Acad. Sect. A 83(1), 45–74 (1983); Kupershmidt, B.A.: Normal and universal forms in integrable hydrodynamical systems. In: Proceedings of the Berkeley-Ames conference on nonlinear problems in control and fluid dynamics (Berkeley, Calif., 1983), in Lie Groups: Hist., Frontiers and Appl. Ser. B: Systems Inform. Control, II, Brookline, MA: Math Sci Press, 1984 pp. 357–378 23. Kupershmidt, B.A: Hydrodynamic chains of Pavlov class. Phys. Lett. A 356, 115–118 (2006) 24. Lavrentiev, M.A., Shabat, B.V.: Metody teorii funktsi˘ıkompleksnogo peremennogo(Russian) [Methods of the theory of functions of a complex variable] Third corrected edition Izdat. “Nauka”, Moscow (1965) 716 pp; P. Henrici. Topics in computational complex analysis. IV. The Lagrange-Bürmann formula for


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systems of formal power series. Computational aspects of complex analysis (Braunlage, 1982), NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci. 102, Dordrecht; Reidel, 1983, pp.193–215 Mokhov, O.I.: Compatible nonlocal Poisson brackets of hydrodynamic type and related integrable hierarchies. (Russian) Teoret. Mat. Fiz. 132 (1), 60–73 (2002); translation in Theoret. and Math. Phys. 132(1), 942–954 (2002); Mokhov, O.I.: The Liouville canonical form of compatible nonlocal Poisson brackets of hydrodynamic type, and integrable hierarchies. (Russian) Funktsional. Anal. i Prilozhen. 37(2), 28–40 (2003); translation in Funct. Anal. Appl. 37(2), 103–113 (2003) Pavlov, M.V.: Integrable systems and metrics of constant curvature. J Nonlinear Math. Phys. No. 9, Supplement 1, 173–191 (2002) Pavlov, M.V.: Integrable hydrodynamic chains. J. Math. Phys. 44(9), 4134–4156 (2003) Pavlov, M.V. The Kupershmidt hydrodynamic chains and lattices. IMRN, pp 1–43 (2006) Pavlov, M.V., Svinolupov, S.I., Sharipov, R.A.: An invariant criterion for hydrodynamic integrability, Funktsional. Anal. i Prilozhen. 30, 18–29 (1996); translation in Funct. Anal. Appl. 30, 15–22 (1996) Pavlov, M.V., Tsarev, S.P.: Three-Hamiltonian structures of the Egorov hydrodynamic type systems. Funct. Anal. Appl. 37(1), 32–45 (2003) Rogers, C., Shadwick, W.F.: Bäcklund Transformations and their Applications. New York: Academic Press, 1982 Rozhdestvenski, B.L., Yanenko, N.N.: Systems of quasilinear equations and their applications to gas dynamics. Translated from the second Russian edition by J. R. Schulenberger. Translations of Mathematical Monographs 55. Providence, RC Amer. Math. Soc., 1983; Russian ed., Moscow: Nauka, 1968 Tsarev, S.P.: On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. Soviet Math. Dokl. 31, 488–491 (1985); Tsarev, S.P.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Math. USSR Izvestiya 37(2), 397–419 (1991) Tsarev, S.P.: Private communications, 1985 Whitham, G.B.: Linear and Nonlinear Waves. New York: Wiley–Interscience, 1974. Zakharov, V.E.: Benney’s equations and quasi-classical approximation in the inverse problem method. Funct. Anal. Appl. 14(2), 89–98 (1980); V.E. Zakharov,: On the Benney’s Equations. Physica D 3, 193–202 (1981)

Communicated by G.W. Gibbons


Communications in


Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras Dmitry Donin, Boris Khesin Dept. of Mathematics, University of Toronto, Toronto, Ont M5S 2E4, Canada. E-mail: [email protected]; [email protected] Received: 6 April 2006 / Accepted: 23 October 2006 Published online: 3 April 2007 – © Springer-Verlag 2007

Abstract: We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.

1. Introduction The Krichever-Novikov algebras are the (centrally extended) Lie algebras of meromorphic vector fields on a Riemann surface , which are holomorphic away from several fixed points [7, 8], see also [11, 15]. They are natural generalizations of the Virasoro algebra, which corresponds to the case of = CP 1 with two punctures. Central extensions of the corresponding algebras of vector fields on a given Riemann surface are defined by fixing a projective structure (that is a class of coordinates related by projective transformations) and the corresponding Gelfand-Fuchs cocycle, along with the change-of-coordinate rule. In this paper we deal with two generalizations of the Krichever-Novikov (KN) algebras. The first one is the Lie algebras of all meromorphic differential operators and pseudodifferential symbols on a Riemann surface, while the second one is the Lie algebras of meromorphic differential operators and pseudodifferential symbols which are holomorphic away from several fixed points. The main tool which we employ is fixing a reference meromorphic vector field instead of a projective structure on . It turns out that such a choice allows one to write more explicit formulas for the corresponding cocycles, both for the Krichever-Novikov algebra of vector fields and for its generalizations.

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Several features of these algebras of meromorphic symbols make them different from their smooth analogue, the algebra of pseudodifferential symbols with smooth coefficients on the circle. First of all, this is the existence of many invariant traces on the former algebras: one can associate such a trace to every point on the surface. Furthermore, we show that the logarithm log X of any meromorphic pseudodifferential symbol X defines an outer derivation of the Lie algebra of meromorphic symbols. In turn, the combination of invariant traces and outer derivations produces a variety of independent non-trivial 2-cocycles on the Lie algebras of meromorphic pseudodifferential symbols and differential operators, as well as it gives rise to Lie bialgebra structures (see Sect. 2). Note that the above mentioned scheme of generating numerous 2-cocycles in the meromorphic case, which involve log X for any meromorphic pseudodifferential symbol X , provides a natural unifying framework for the existence of two independent cocycles (generated by log ∂/∂ x and log x) in the smooth case, cf. [6, 5]. The second type of algebras under consideration, those of holomorphic differential operators and pseudodifferential symbols, are more direct generalizations of the Krichever-Novikov algebra of holomorphic vector fields on a punctured Riemann surface. For them we prove the density and filtered generalized grading properties, similarly to the corresponding properties of the KN algebras [7, 8]. Furthermore, one can adapt the notion of a local cocycle proposed in [7] to the filtered algebras of (pseudo)differential symbols. It turns out that all logarithmic cocycles become linearly dependent when we confine to local cocycles on holomorphic differential operators. On the other hand, for holomorphic pseudodifferential symbols the local cocycles are shown to form a twodimensional space (see Sect. 3). Finally, for meromorphic differential operators, as well as for holomorphic differential operators on surfaces with trivialized tangent bundle, there exist Lie bialgebra structures and integrable hierarchies mimicking the structures in the smooth case. We deliberately put the exposition in a form which emphasizes the similarities with and differences from the algebras of (pseudo)differential symbols with smooth coefficients on the circle, developed in [3, 5]. In many respects the algebras of holomorphic symbols extended by local 2-cocycles turn out to be similar to their smooth counterparts on the circle. On the other hand, by giving up the condition of locality, one obtains higher-dimensional extensions of the Lie algebras of holomorphic symbols by means of the 2-cocycles related to different paths on the surface. This way one naturally comes to holomorphic analogues of the algebras of “smooth symbols on graphs,” which also have central extensions given by 2-cocycles on different loops in the graphs.

2. Meromorphic Pseudodifferential Symbols on Riemann Surfaces 2.1. The algebras of meromorphic differential and pseudodifferential symbols. Let be a compact Riemann surface and M be the space of meromorphic functions on . Fix a meromorphic vector field v on the surface and denote by D (or Dv ) the operator of Lie derivative L v along the field v. Then D sends the space M to itself, and one can consider the operator algebras generated by it. Definition 2.1. The associative algebras of meromorphic differential operators M D O :=

A=

n k=0

ak D | ak ∈ M k

Krichever–Novikov Algebras of Pseudodifferential Symbols

509

and meromorphic pseudodifferential symbols n k M DS := A = ak D | ak ∈ M k=−∞

are the above spaces of formal polynomials and series in D which are equipped with the multiplication ◦ defined by k Dk ◦ a = (D a)D k− , (2.1) ≥0

makes sense for both positive and where the binomial coefficient k = k(k−1)...(k−+1) ! negative k. This multiplication law naturally extends the Leibnitz rule D◦a = a D+(Da). The algebras M D O and M DS are also Lie algebras with respect to the bracket [A, B] = A ◦ B − B ◦ A . Note that the algebra M D O is both an associative and Lie subalgebra in M DS. (In the sequel, we will write simply X Y instead of X ◦ Y whenever this does not cause an ambiguity.) For different choices of the meromorphic vector field v the corresponding algebras of (pseudo) differential symbols are isomorphic: any other meromorphic field w on can be presented as w = f v for f ∈ M, and then the relation Dw = f Dv delivers the (both associative and Lie) algebra isomorphism. Meromorphic vector fields on embed both into M D O and M DS as Lie subalgebras. Remark 2.2. Equivalently, one can define product of two pseudodifferential symbols the m by the following formula: if A(D) = i=−∞ ai D i and B(D) = nj=−∞ b j D j , for D := L v then ⎛ ⎞ 1 A ◦ B := ⎝ ∂ k A(ξ ) ∂vk B(ξ )⎠ . (2.2) k! ξ k≥0

ξ =D

Here ∂v is the operator of taking the Lie derivative of coefficients of a symbol (i.e. of functions b j ) along v. Note that the right-hand side of this formula expresses the commutative multiplication of functions A(z, ξ ) and B(z, ξ ). Of course, this formula also extends the usual composition of differential operators. 2.2. Outer derivations of pseudodifferential symbols. It turns out that both the associative and Lie algebras of meromorphic pseudodifferential symbols have many outer derivations. Definition 2.3. (cf. [6]) Let v be a meromorphic vector field on and set D := L v . Define the operator log D or, rather, [log D, ·] : M DS → M DS by [log D, a D n ] :=

(−1)k+1 k≥1

k

(D k a)D n−k .

(2.3)

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The above formula is consistent with the Leibnitz formula (2.1): it can be obtained from the latter by regarding k as a complex parameter, say, λ and differentiating in λ at λ = 0: d/dλ|λ=0 D λ = log D. (Below we will be using the notation log D for the derivation and [log D, ·] for explicit formulas.) Proposition 2.4. The operator log D : M DS → M DS defines a derivation of the (both associative and Lie) algebra M DS of meromorphic pseudodifferential symbols for any choice of the meromorphic vector field v. Proof. One readily verifies that for any two symbols A and B, [log D, AB] = [log D, A]B + A[log D, B] , i.e. log D is a derivation of the associative algebra M DS. This also implies that log D is a derivation of the Lie algebra structure. It turns out that one can describe a whole class of derivations in a similar way. Definition 2.5. Associate to any meromorphic pseudodifferential symbol X the derivation log X : M DS → M DS, where the commutator [log X, A] with a symbol A is defined by means of the formula (2.2). Namely, recall that log(v(z)ξ ) can be regarded as a (multivalued) symbol for log Dv , i.e. a multivalued function on T ∗ . Indeed, only the derivatives of this function in ξ or along the field v appear in the formula for the commutator [log D, A] = log D ◦ A − A ◦ log D, where the products in the right-hand-side are defined by formula (2.2). Similarly, one can regard log X (ξ ) as a function on T ∗ and only its derivatives appear in the commutators [log X, A] with any meromorphic symbol A ∈ M DS. Remark 2.6. We note that the formula for [log X, A] involves the inverse X −1 , which is a well-defined element of M DS. Indeed, to find, say, the inverse of a pseudodifferential symbol X we have to solve X ◦ A = 1 with unknown coefficients. If X = f n D n + f n−1 D n−1 + . . . ,

A = an D −n + an−1 D −n−1 + . . . .,

we solve recursively the equations f n an = 1, f n an−1 + f n−1 an + n f n (Dan ) = 0, . . . . Each equation involves only one new unknown a j as compared to preceding ones and hence the series for X −1 = A can be obtained term by term, i.e. its coefficients are meromorphic functions. Example 2.7. To any meromorphic function f ∈ M on we associate the operator log f : M DS → M DS given by [log f, a D n ] := na

D f n−1 f (D 2 f ) − (D f )2 n−2 + n(n − 1)a D + .... D f f2

Note that, while the function log f is not meromorphic and branches at poles and zeros of f , all its derivatives D k (log f ) with k ≥ 1 are meromorphic, and the right-hand side of the above expression is a meromorphic pseudodifferential symbol.


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We shall show that log X for any symbol X is an outer derivation of the Lie algebra M DS, i.e. it represents a nontrivial element of H 1 (M DS, M DS). The latter space is by definition the space of equivalence classes of all derivations modulo inner ones. Theorem 2.8. All derivations defined by log X for any meromorphic pseudodifferential symbol X are outer and equivalent to a linear combination of derivations given by logarithms log Dvi of meromorphic vector fields vi on . This theorem is implied by the following two properties of the log-map. Theorem 2.8 . The map X → log X ∈ H 1 (M DS, M DS) is nonzero and satisfies the properties: a) the derivation log(X ◦ Y ) is equivalent to the sum of derivations log X + log Y , and b) the derivation log(X + Y ) is equivalent to the derivation log X if the degree of the symbol X is greater than the degree of Y . One can see that any derivation log X is equivalent to a linear combination of log f for some meromorphic function and log Dv for one fixed field v. The above properties of derivations log X modulo inner ones are similar to those of tropical calculus.1 We prove this theorem in Sect. 2.5. Conjecture 2.9. All outer derivations of the Lie algebra M DS are equivalent to those given by log X for pseudodifferential symbols X . 2.3. The traces. The Lie algebra M DS has a trace attached to any choice of the “special” points on . All the constructions below will be relying on this choice of the points and we fix such a point (or a collection of points) P ∈ from now on. Definition 2.10. Define the residue map res from M DS to meromorphic 1-forms on D

by setting

res D

n

ak D

k

:= a−1 D˜ −1 .

k=−∞

Here D˜ −1 in the right-hand side is understood as a (globally defined) meromorphic differential on , the pointwise inverse of the meromorphic vector field v. On the algebra M DS we define the trace associated to the point P ∈ by Tr A := res res (A). P

D

Here res f D P

−1

1 f f = res dz = = res P v P h 2πi

γ

f dz, h

where v = h(z)∂/∂z is a local representation of the vector field v at a neighborhood of the point P, while γ is a sufficiently small contour on around P which does not contain poles of f / h other than P. (Here and below we omit the index P in the notation of the trace Tr P .) 1 We are grateful to A. Rosly for drawing our attention to this analogy.

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Proposition 2.11. Both the residue and trace are well-defined operations on the algebra M DS, i.e. they do not depend on the choice of the field v. Furthermore, for any choice of the point(s) P, Tr is an algebraic trace, i.e. Tr [A, B] = 0 for any two pseudo-differential symbols A, B ∈ M DS. In particular, this property allows us to use the notation res or Tr without mentioning D

a specific field v.

Proof. Under the change of a vector field v → w = gv only the terms Dv−1 contribute −1 , which implies that the corresponding 1-form a D −1 to Dw −1 ˜ v and hence the residue operator are well-defined. The algebraic property of the trace is of local nature, since Tr is defined locally near P. One can show that for any two X, Y ∈ M DS the residue of the commutator is a full derivative, i.e. res [X, Y ] = (D f )D −1 D

for some function f defined in a neighborhood of P (see [1], p.11). Then the proposition follows from the fact that a complete derivative has zero residue: Lv f (D f )D −1 = df = 0 = γ γ v γ for a contour γ around P. This proposition allows one to define the pairing ( , ) on M DS, associated with the chosen point P ∈ : (A, B) := Tr (AB) .

(2.4)

This pairing is symmetric, non-degenerate, and invariant due to the proposition above. The pairings associated to different choices of the point(s) P ∈ are in general not related by an algebra automorphism (unless there exists a holomorphic automorphism of the surface sending one choice to the other). Remark 2.12. The existence of the invariant trace(s) on M DS allows one to identify this Lie algebra with (the regular part of) its dual space. This identification relies on the choice of the point P. We also note that both res and Tr vanish on the subalgebra M D O of meromorphic D

purely differential operators. In particular, this subalgebra is isotropic with respect to the above pairing, i.e. ( , ) | M D O = 0. The complementary subalgebra to M D O is the k Lie algebra M I S of meromorphic integral symbols { −1 k=−∞ ak D }, which is isotropic with respect to this pairing as well. 2.4. The logarithmic 2-cocycles. Being in the possession of a variety of outer derivations, as well as of the invariant trace(s), we can now construct many central extensions of the Lie algebra DS. The simple form of the invariant trace allows us to follow the analogous formalism for pseudodifferential symbols on the circle [3, 5, 6]. We start by defining a logarithmic 2-cocycle attached to the given choice of the point P and a meromorphic field v on :


513

Theorem 2.13. (cf. [6]) The bilinear functional cv (A, B) := Tr ([log Dv , A] ◦ B)

(2.5)

is a nontrivial 2-cocycle on M DS and on its subalgebra M D O for any meromorphic field v and any choice of the point P on , where the trace is taken. In particular, the skew-symmetry property of this cocycle follows from the fact that the derivation log Dv preserves the trace functional: Tr ([log Dv , A]) = 0 for all symbols A ∈ M DS. Remark 2.14. The restriction of this 2-cocycle to the algebra of vector fields is the Gelfand-Fuchs 2-cocycle cv (a Dv , bDv ) =

(Dv2 a)(Dv b) 1 res 6 P Dv

on the Lie algebra of meromorphic vector fields on . The restriction of the cocycle (2.5) to the algebra M D O gives the Kac-Peterson 2-cocycle cv (a Dvm , bDvn ) =

(Dvn+1 a)(Dvm b) m!n! res , (m + n + 1)! P Dv

m, n ≥ 0,

on meromorphic differential operators on (see [10]). The above construction can be generalized in the following way. Definition 2.15. Associate the logarithmic 2-cocycle c X (A, B) := Tr ([log X, A] ◦ B) to a meromorphic pseudodifferential symbol X and a point P ∈ (where the trace Tr is taken). Theorem 2.8 . For any meromorphic pseudodifferential symbols X and Y a) the logarithmic 2-cocycle c X Y is equivalent to the sum of the 2-cocycles c X + cY , and b) the logarithmic 2-cocycle c X +Y is equivalent to the logarithmic 2-cocycle c X provided the degree of the symbol X is greater than the degree of Y . Proof. This follows from Theorem 2.8 thanks to the following claim (see e.g. [2]). Let g be a Lie algebra with a symmetric invariant nondegenerate pairing ( , ). Consider a derivation φ : g → g preserving the pairing, i.e. satisfying (φ(a), b) + (φ(b), a) = 0 for any a, b ∈ g, and associate to it the 2-cocycle c(a, b) := (φ(a), b) ∈ H 2 (g) on g. Then the subspace in H 1 (g, g) consisting of invariant derivations (and understood up to coboundary) is isomorphic to the space H 2 (g): the 2-cocycle c is cohomologically nontrivial if and only if the derivation φ is outer. Since the outer derivation log X preserves the pairing (2.4), it defines a nontrivial 2cocycle. The properties of the outer derivations in Theorem 2.8 are equivalent to those of the logarithmic 2-cocycles in Theorem 2.8 .

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Corollary 2.16. (i) For any meromorphic pseudodifferential symbol X the logarithmic 2-cocycle c X is equivalent to a linear combination of the 2-cocycles cvi (A, B) := Tr ([log Dvi , A] ◦ B) associated to meromorphic vector fields vi . (ii) For two meromorphic vector fields v and w related by w = f v the cocycles cv and cw are related by cw = cv + c f , where c f (A, B) := Tr ([log f, A] ◦ B) is the 2-cocycle associated to the meromorphic function f ∈ M, and the equality is understood in H 2 (M DS, C), i.e. modulo a 2-coboundary. Theorem 2.17. All the 2-cocycles cv are nontrivial and non-cohomologous to each other on the algebra M DS for different choices of meromorphic fields v = 0. Equivalently, cocycles c f are all nontrivial for non-constant functions f . Proof. Note that the 2-cocycle cv is nontrivial, since its restriction to the subalgebra of vector fields holomorphic in a punctured neighborhood of the point P already gives the nontrivial Gelfand-Fuchs 2-cocycle. (In other words, the nontriviality of the cocycle cv for a meromorphic vector field v follows from its nontriviality on the Krichever-Novikov subalgebra L of holomorphic vector fields on \ {P, Q}, where Q is any other point on , see the next section.) To show the nontriviality of the cocycle c f for a non-constant function f we use the existence of many traces on M DS. First we choose the point P (and the corresponding trace Tr P ) at a zero of the function f . The corresponding 2-cocycle c f is non-trivial, since so is its restriction to pseudodifferential symbols holomorphic in a punctured neighborhood of P. The latter is obtained by exploiting the nontriviality of the 2-cocycle c (A, B) = Tr ([log z, A]◦ B) on holomorphic pseudodifferential symbols on C∗ , see [5, 2]. Now, by applying the above-mentioned equivalence between derivations and cocycles, we conclude that log f defines an outer derivation of the algebra M DS. Once we know that the derivation is outer, we can use the same equivalence “in the opposite direction” for the point P anywhere on to obtain a nontrivial 2-cocycle from any other invariant trace. Remark 2.18. Note that the cocycle c f for a meromorphic function f vanishes on the subalgebra M D O of meromorphic differential operators: for any purely differential operators X and Y , the expression [log f, X ] ◦ Y is also a meromorphic differential operator (see Example 2.7) and hence its coefficient at Dv−1 is 0. This shows that all the 2-cocycles cv for the same point P ∈ , but for different choices of the meromorphic field v are cohomologous when restricted to the algebra M D O. The choice of a different point P to define the trace may lead to a non-cohomologous 2-cocycle cv . 2.5. Proof of the theorem on outer derivations. In this section we will prove Theorem 2.8 on properties of the derivation log X : M DS → M DS. Proof. For the part a) we rewrite the product X Y of two symbols as X Y = exp(log X ) ◦ exp(log Y ) and use the Campbell-Hausdorff formula: X Y = exp(log X + log Y + R(log X, log Y )) .


515

Note that the remainder term R(log X, log Y ) is a pseudodifferential symbol, since only (iterated) commutators of log X and log Y appear in it, but not the logarithms themselves. (In particular, the commutator [log X, log Y ] defined by the formula (2.2) is a pseudodifferential symbol from M DS.) Hence log X Y = log X + log Y + R(log X, log Y ) . Thus the derivation log X Y is cohomological to the sum of derivations log X + log Y , since the commutation with the symbol R(log X, log Y ) defines an inner derivation of the algebra M DS. This proves a). n To prove the part b) we will show that the derivation log X for X = i=−∞ ai Dvi is n defined by an Dv , the principal term of X . Indeed, rewrite X as X = (an Dvn ) ◦ (1 + Y ),

j where Y = −1 j=−∞ b j Dv is a meromorphic integral symbol. Then due to a), log X is cohomological to log(an Dvn )+log(1+Y ). However, the derivation log(1+Y ) is inner, i.e. log(1 + Y ) is itself a meromorphic pseudodifferential symbol. Indeed, expand log(1 + Y ) in the series: log(1 + Y ) = Y − Y 2 /2 + Y 3 /3 − . . . . The right-hand side is a well-defined meromorphic integral symbol, since so is Y . Thus log X is cohomological to log(an Dvn ), which proves b). 2.6. The double extension of the meromorphic symbols and Manin triples. Definition 2.19. Consider the following double extension of the Lie algebra M DS by means of both the central term and the outer derivation for a fixed meromorphic field v: n M DS = C · log D ⊕ M DS ⊕ C · I = λ log D + ak D k + µ · I , k=−∞

where the commutator of a pseudodifferential symbol with another one or with log D for D = Dv was defined above, while the cocycle direction I commutes with everything else. There is a natural invariant pairing on the Lie algebra M DS, which extends the pairing Tr (A ◦ B) on the non-extended algebra M DS. Namely,

(λ1 log D + A1 + µ1 · I, λ2 log D + A2 + µ2 · I) = Tr (A1 ◦ A2 ) + λ1 · µ2 + λ2 · µ1 . Consider also two subalgebras of the Lie algebra M DS: the subalgebra of centrally extended meromorphic differential operators n k M DO = ak D + µ · I k=0

and the subalgebra of co-centrally extended meromorphic integral symbols −1 k M I S = C · log D ⊕ M I S = λ log D + ak D . k=−∞

Similarly to the case of smooth coefficients, one proves the following

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Theorem 2.20. Both the triples of algebras ( M DS, M D O, M I S) are Manin triples.

(M DS, M D O, M I S)

and

Definition 2.21. A Manin triple (g, g+ , g− ) is a Lie algebra g along with two Lie subalgebras g± ⊂ g and a nondegenerate invariant symmetric form ( , ) on g, such that (a) g = g+ ⊕ g− as a vector space and (b) g+ and g− are isotropic subspaces of g with respect to the inner product ( , ). The existence of a Manin triple means the existence of a Lie bialgebra structure on both g± and allows one to regard each of the subalgebras as dual to the other with respect to the pairing (see the Appendix for the definitions). Corollary 2.22. Both the Lie algebras M I S and M I S are Lie bialgebras, while the groups corresponding to them are Poisson-Lie groups. This makes the meromorphic consideration parallel to the case of smooth pseudodifferential symbols developed in [3, 5]. For holomorphic symbols, however, such a Manin triple exists only in special cases, as we discuss below. 3. Holomorphic Pseudodifferential Symbols on Riemann Surfaces 3.1. The Krichever-Novikov algebra. Let be a Riemann surface of genus g. Fix two generic points P+ and P− on the surface. Consider the Lie algebra L of meromorphic ◦

vector fields on , holomorphic on := \ {P± }. We will call such fields simply ◦ holomorphic (on ). ◦

Definition 3.1. The Lie algebra L of holomorphic on vector fields is called the Krichever-Novikov (KN) algebra. A special basis in L, called the Krichever-Novikov basis, is formed by vector fields ek having a pole of order k at P+ and a pole of order 3g − k − 2 at P− (as usual we refer to a pole of negative order k as to a zero of order −k). (More precisely, this prescription of basis elements works for surfaces of genus g ≥ 2, while for g = 1 one has to alter it for certain small values of k, see [7].) Note that each field ek has g additional zeros elsewhere on \ {P± }, since the degree of the tangent bundle of is 2 − 2g. This algebra was introduced and studied in [7, 8] along with its central extensions. ◦

It generalizes the Virasoro algebra, which corresponds to the case = CP 1 \ {0, ∞}. Below we will be concerned with the case of two punctures P± on , although most of the results below hold for the case of many punctures as well, cf. [11–13, 16, 18]. 3.2. Holomorphic differential operators and pseudodifferential symbols. Denote by O ◦

the sheaf of holomorphic functions on , which are meromorphic at P± . Definition 3.2. The sheaves of holomorphic differential operators and pseudodiffer◦

ential symbols on \ {P± } are defined by assigning to each open set U ⊂ an abelian group (a vector space) of sections


HDO(U ) :=

X=

HDS(U ) :=

X=

n

517

u i D | u i ∈ OU i

i=0 n

and

u i D i | u i ∈ OU ,

i=−∞

respectively, where D := Dv stands for some holomorphic non-vanishing vector field v in U . (Another choice of a non-vanishing field v gives the same spaces of operators and symbols.) The Lie algebras of global sections of the sheaves HDO and HDS are called the Lie algebras of holomorphic differential operators and of pseudodifferential symbols, respectively. We denote these algebras of global sections by H D O and H DS. Note that the definitions of the residue and trace of symbols are local and hence can be defined on holomorphic symbols in the same way as they were defined for meromorphic ◦

ones: res D X is a globally defined holomorphic 1-form on , given in local coordinates by u −1 D˜ −1 , while Tr X := res res X = res u −1 D˜ −1 P+

D

P+

n

for a section X given by X = i=−∞ u i D i in a (punctured) neighborhood of P+ . The algebra H DS has two subalgebras: that of holomorphic differential operators (H D O) and of holomorphic integral symbols (H I S). They consist of those symbols ◦

whose restriction to any open subset U ⊂ are, respectively, holomorphic purely differential operators or holomorphic purely integral symbols. As in the meromorphic case, these holomorphic subalgebras are isotropic with respect to the natural pairing. ◦

The KN-algebra L of holomorphic vector fields on can be naturally viewed as a subalgebra of the algebras of holomorphic differential operators and pseudodifferential symbols (H DS). In turn, the algebra H DS is a subalgebra in the algebra of meromorphic symbols M DS. Remark 3.3. The Lie algebra H D O can be alternatively defined as the universal enveloping algebra H D O = U(O L)/J of the Lie algebra O L quotiented over the ideal J , generated by the elements f ◦ g − f g, f ◦ v − f v, and 1 − 1, where ◦ denotes the multiplication in H D O, 1 is the unit of U, while f, g, 1 ∈ O and v ∈ L, see e.g. [12]. It is easy to see that this definition matches the one above. A convenient way to write some global sections the above sheaves is by fixing a of n holomorphic field v ∈ L. Then the symbols X = i=−∞ u i Dvi with any holomorphic coefficients u i ∈ O define global sections of HDS, provided that for every i < 0 the coefficient u i has zero of order at least i at zeros of v (this way we compensate all the poles of the negative powers Dvi by appropriate zeros of the corresponding coefficients). 3.3. Holomorphic pseudodifferential symbols and the spaces of densities. One can think of holomorphic (pseudo)differential symbols as sequences of holomorphic densities on ◦

. Namely, let K be the canonical line bundle over and consider the tensor power Kn ◦ of K for any n ∈ Z. Denote by F n the space of holomorphic n-densities on , i.e. the

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D. Donin, B. Khesin ◦

space of global meromorphic sections of Kn which are holomorphic on . Note that F 0 is the ring O of holomorphic functions, F 1 is the space of holomorphic differentials on ◦

, and F −1 is the space L of holomorphic vector fields. Holomorphic vector fields act on holomorphic n-densities by the Lie derivative: to v ∈ L and ω ∈ F n one associates L v ω ∈ F n . Explicitly, in local coordinates for v = h(z)∂/∂z and ω = f (z)(dz)n one has ∂h ∂f L v ω = h(z) + n f (z) (dz)n . ∂z ∂z This action turns the space F n of n-densities into an L-module. The following proposition is well-known: Proposition 3.4. Each graded space for the filtration of pseudodifferential symbols by degree is naturally, as an L-module, isomorphic to the corresponding space of holomorphic densities. Namely, H DS n /H DS n−1 ≈ F −n , where H DS n is the space of pseudodifferential symbols of degree n and the isomorphism is given by taking the principal symbol of the pseudodifferential operator. Proof. The action of vector fields from L on pseudodifferential operators of degree n is explicitly given by: [h D, f D n ] = h(D f ) − n f (Dh) D n + (terms of degree < n in D). Thus the action on their principal symbols coincides with the above L-action on (−n)densities F −n , i.e. they satisfy the same change of coordinate rule. Furthermore, taking the principal symbols of the operators of a given degree n is a surjective map onto F −n . One can see this first for differential operators, i.e. for n ≥ 0, where it follows from their description as H D O = U(O L)/J and the PBW theorem. Indeed, one can form a basis in differential operators of degree n from the products f Dei1 ...Dein , where f ∈ F 0 and ei form the KN-basis in L = F −1 . Their principal symbols will be the (commutative) products of the principal symbols of the basis elements, which, by definition, span the space of meromorphic sections of Kn , holomorphic ◦

on , i.e. the space F −n . The surjectivity for negative n, i.e. for principal symbols of integral operators, can be derived by considering natural pairing on densities (F n × F −n−1 → C) and on pseudodifferential symbols (H DS −n × H DS n−1 → C) given by taking at the point P+ the residue for densities and the trace for symbols, respectively. The whole vector space H DS can be treated as the direct limit of the semi-infinite products of the spaces of holomorphic n-densities: H DS ≈ lim kn=−∞ F −n as −→ k → ∞, on which one has a Lie algebra structure.


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3.4. The density of holomorphic symbols in the smooth ones. The algebra H DS of ◦

holomorphic symbols on , as well as the KN-algebra L of holomorphic vector fields, can be regarded as a subalgebra of smooth symbols on the circle S 1 in the following way. In [7] a family of special contours Cτ , τ ∈ R on was constructed as level sets of ◦

some harmonic function on . These contours separate the points P± , and as τ → ±∞ the contours Cτ become circles shrinking to P± . Denote by S 1 ≈ Cτ an arbitrary contour from this family for a sufficiently large τ , thought of as a small circle around P+ . ◦

◦

Consider the natural restriction homomorphism from to S 1 ⊂ . Theorem 3.5 [7, 8]. The restrictions of holomorphic functions, vector fields, and differ◦

entials on to the contour S 1 ≈ Cτ are dense among, respectively, smooth functions, vector fields, and differentials on the circle S 1 . Now we consider the algebra of all smooth pseudodifferential symbols on the circle (with a coordinate x): n d DS(S 1 ) = , f i ∈ C ∞ (S 1 ) . f i (x)∂ i | ∂ := dx i=−∞

The latter is a topological space under the natural topology (on the Laurent series) while the sum, multiplication, and taking the inverse (for a nowhere vanishing highest coefficient f n ) are continuous operations. The following theorem is a natural extension of the one above. Consider the restriction homomorphism H DS → DS(S 1 ) of holomorphic symbols to the smooth ones for the contour S 1 ≈ Cτ ⊂ and denote by H DS | S 1 the corresponding image. Theorem 3.6. The restriction H DS | S 1 of holomorphic symbols is dense in the smooth ones DS(S 1 ). Proof. It suffices to prove that the monomials of the form f (x)∂ i , i ∈ Z, f ∈ C ∞ (S 1 ) can be approximated by holomorphic ones. Write out such a monomial as a product of f (x), ∂ and ∂ −1 . Since the smooth function f (x), the vector field ∂ := ddx and the 1-form ∂ −1 := d x on the circle can be approximated by the restrictions of holomorphic ones [8], the result follows by the continuity of the multiplication in DS(S 1 ). Note that the original density result in [7] for a pair of points P± extends to a collection of points by representing functions, fields, etc. with many poles as sums of the ones with two poles only. 3.5. The property of generalized grading. Definition 3.7. An associative or Lie algebra A is generalized graded (or N -graded) if it admits a decomposition A = ⊕n∈Z An into finite-dimensional subspaces, with the property that there is a constant N such that Ai A j ⊂

N s=−N

for all i, j.

Ai+ j+s ,

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D. Donin, B. Khesin

Similarly, a module M over a generalized graded algebra A is generalized graded, if M = ⊕n∈Z Mn , and there is a constant L such that L

Ai M j ⊂

Mi+ j+s ,

s=−L

for all i, j. ◦

Theorem 3.8 [7, 8]. The KN-algebra L of vector fields holomorphic on is generalized graded. The space F n of holomorphic n-densities for any n is a generalized graded module over L. (n)

(n)

The generalized graded components M j for the module F n are the spaces C · f j , (n)

where the forms f j are uniquely determined by the pole orders at both points P± (as before, we assume the points to be generic): for n = 0 they have the following expansions (n)

fj

± j−g/2+n(g−1)

:= a ± j z±

(1 + O(z ± ))(dz ± )n

in local coordinates z ± of neighborhoods of the points P± . Here the index j runs over the integers Z if g is even and over the half-integers Z + 1/2 if g is odd. (The formulas differ slightly for F 0 = O, see details in [7]. For the space of vector fields L ≈ F −1 this basis { f j(−1) } differs by an index shift from the fields {e j } discussed in Sect. 3.1.) Now we consider the spaces H D O and H DS of holomorphic (pseudo)differential symbols not only as modules over vector fields L, but as Lie algebras. These Lie algebras are not generalized graded, but naturally filtered by the degree of D = Dv . Con(n) sider a basis {F j } (which we construct below) in pseudodifferential symbols H DS, which is compatible with the basis in the forms: the principal symbol of the operator (n) (−n) F j of degree n is the (−n)-form f j . It turns out that the algebras of holomorphic (pseudo)differential symbols have the following analogue of the generalized grading: Theorem 3.9. The Lie algebras H D O and H DS are filtered generalized graded: the (n) pseudodifferential symbols of an appropriate basis {F j } in H DS satisfy (n) [Fi ,

(m) Fj ]

=

∞

N (k)

(n+m−k)

αisj Fi+ j+s

,

k=1 s=−N (k)

for some constants αirj ∈ C, where n, m ∈ Z, the indices i, j, and s are either integers or half-integers according to parity of the genus g, and N (k) is a linear function of k. Proof. First we define a basis for differential operators from H D O ⊂ H DS recursively in degree n (cf. [12], where a similar basis was constructed for H D O). Assume that the genus g is even, so that all the indices are integers (the case of an odd g is similar). Consider the above KN-basis {F j(0) } in differential operators of degree 0, which (1)

constitute 0-densities F 0 , and the KN-basis {F j } in holomorphic vector fields, which are differential operators of degree 1.


521

(n) For a degree n ≥ 2 consider a differential operator F˜ j whose principal symbol is (−n)

the (−n)-density f j

, and which exists due to surjectivity discussed in Proposition (n) 3.4. One can “kill the lower order terms” of the operator F˜ by adding a linear comj

bination of the basis elements in H D O n−1 constructed at the preceding step. (More precisely, due to the filtered structure of H D O only the cone of lower order terms for F˜ j(n) is well-defined by the pole orders of the coefficients of the differential operators at (n) the points P± . By “killing the terms” above we mean confining F˜ to this cone.) We (n)

j (k)

call these adjusted differential operators by F j . Along with {F j } for 0 ≤ k ≤ n − 1 they constitute a basis in H D O n , holomorphic differential operators of degree ≤ n. Finally, for integral symbols (of negative degrees) we choose the basis dual to the one chosen in differential operators, by using the nondegenerate pairing: H I S can be thought of as the dual space H D O ∗ . It is easy to see that this basis in integral symbols of degree −n is also given by the orders of zeros and poles at P± and hence is compatible with the KN-basis in n-forms. (n) Once the basis {F j } is constructed, a straightforward substitution of these symbols into the formula for the symbol commutator and the calculation of orders of poles/zeros at the points P± yield the result. This theorem implies the property of generalized grading for modules of holomorphic densities, established in [7, 12, 16], as modules of principal symbols of holomorphic pseudodifferential operators. 3.6. Cocycles and extensions. Recall first the cocycle construction for the KN-algebra ◦

L of holomorphic vector fields on . A closed contour γ on , not passing through the marked points P± , defines the Gelfand-Fuchs 2-cocycle on the algebra L. Namely, in a fixed projective structure (where admissible coordinates differ by projective transformations) it is defined by the Gelfand-Fuchs integral c( f, g) = f (z)g (z) dz γ

∂ ∂ for vector fields f = f (z) ∂z and g = g(z) ∂z given in such a coordinate system. One can check that this cocycle is well-defined, nontrivial, and represents every cohomology class in the space H 2 (L, C) of 2-cocycles on the algebra L for various contours γ , see [8, 13]. In this variety of 2-cocycles there is a subset of those satisfying the following property of locality.

Definition 3.10 [7]. Let A = ⊕n∈Z An be a generalized graded Lie algebra. A 2-cocycle c on A is called local if there is a nonnegative constant K ∈ Z such that c(Am , An ) = 0 for all |m + n| > K . The central extensions of generalized graded Lie algebras defined by local 2-cocycles are also generalized graded Lie algebras. Theorem 3.11 [7]. The cohomology space of local 2-cocycles of the Krichever-Novikov algebra L is one-dimensional. It is generated by the Gelfand-Fuchs 2-cocycle on any separating contour Cτ on .

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As we discussed above, for large τ one can think of Cτ as a small circle around P+ . Thus the local cocycles on L are defined by the restrictions of vector fields to a small neighborhood of P+ . To describe the logarithmic cocycles on the algebras H D O and H DS of holomor◦

phic (pseudo)differential symbols on we adapt the notion of the cocycle locality to the filtered generalized grading. First we recall the corresponding results for the algebras D O(S 1 ) and DS(S 1 ) of smooth operators and symbols of the circle. Theorem 3.12. (i) The cohomology space of 2-cocycles on the algebra D O(S 1 ) of differential operators on the circle is one-dimensional ([4, 9]). A non-trivial 2-cocycle is defined by the restriction of the logarithmic cocycle Tr ([log ∂, A] ◦ B) to differential operators, ∂ := ∂∂x , and A, B ∈ D O(S 1 ) ([6]). (ii) The cohomology space of 2-cocycles of the algebra DS(S 1 ) of pseudodifferential symbols is two-dimensional ([4, 2]). It is generated by the logarithmic cocycle above and the 2-cocycle Tr ([x, A] ◦ B), where x is the coordinate on the universal covering of S 1 , and A, B ∈ DS(S 1 ) ([5, 6]). Here the trace is Tr A := S 1 res ∂ A for smooth pseudodifferential symbols, which replaces Tr A := res P+ res D A for holomorphic ones. Example 3.13. Consider the Lie algebra of holomorphic symbols on C∗ = CP 1 \{0, ∞}, whose elements are allowed to have poles at 0 and ∞ only, and where we take Dv := z∂/∂z. Two independent outer derivations of the latter algebra are log(z∂/∂z) and log z, the logarithms of a vector field and a function, respectively [6, 5]. The corresponding 2-cocycles are ∂ Tr [log z , A] ◦ B and Tr ([log z, A] ◦ B) . ∂z This algebra can be thought of as a graded version of smooth complex-valued symbols DS(S 1 ) on the circle S 1 = {|z| = 1}: the change of variable z = exp(i x) sends ∂ := ∂/∂ x to i Dv : ∂/∂ x = ∂z/∂ x · ∂/∂z = i exp(i x) ∂/∂z = i z∂/∂z = i Dv . Under this change of variables (and upon restricting the symbols to the circle S 1 ), the derivations [log (i Dv ), .] and −i [ log z, .] for holomorphic symbols in DS(C∗ ) become the derivations [log ∂, .] and [x, .] for smooth symbols in DS(S 1 ). The above theorem describes the 2-cocycles on DS(S 1 ) constructed with the help of the latter outer derivations and the corresponding change in the notion of trace. After having described the smooth case, we adapt the definition of the local 2-cocycle to the filtered generalized graded case of the algebra H DS by allowing the constant K in Definition 3.10 to depend on the filtered component. Definition 3.14. A 2-cocycle on the filtered generalized graded algebra H DS is called (n) (m) local if for any integers i, j there is a number N = N (n +m) such that c(Fi , F j ) = 0 (n)

for the basis pseudodifferential symbols Fi

(m)

and F j

as soon as |i + j| > N .


523

Such a cocycle preserves the property of filtered generalized grading when passing to the corresponding central extension. ◦

◦

Consider a holomorphic vector field v on and a smooth path γ on \ (div v), i.e. a smooth path γ avoiding P± , as well as zeros and poles of v. Associate to v and γ the 2-cocycle cv,γ defined as the following bilinear functional on H DS and H D O: cv,γ (A, B) := res ([log Dv , A] ◦ B) , γ Dv

◦

where we integrate over γ the residue, which is a meromorphic 1-form on (with possible poles at the divisor of v, and hence off γ ). We confine ourselves to considering cocycles of the form cv,γ . Theorem 3.15. (i) The cohomology space of local 2-cocycles of the form cv,γ on the Lie ◦

algebra H D O on is one-dimensional and it is generated by the 2-cocycle cv,P+ (A, B) := Tr ([log Dv , A] ◦ B) for a holomorphic vector field v. The cocycles cv,P+ are local ◦

for any choice of a holomorphic field v on . (ii) The cohomology space of local 2-cocycles cv,γ on the algebra H DS is twodimensional. It is generated by the 2-cocycles cvi ,P+ for two holomorphic vector fields v1 and v2 with different orders of poles/zeros at P+ . Remark 3.16. Alternatively, one can generate the 2-dimensional space of local cocycles for H DS by considering Tr ([log Dv , A] ◦ B) and Tr ([log f, A] ◦ B), where v is any holomorphic vector field, while f is a function with a zero or pole (of any non-zero order) at P+ . The restriction of the latter 2-cocycle to H D O vanishes. Proof. We have adapted the definition of grading and locality in such a way that the locality of a 2-cocycle on the filtered algebras H D O and H DS implies its locality on the subalgebra L. According to the Krichever-Novikov Theorem 3.11 local cocycles on L are given by the integrals over contours Cτ . In turn, the cocycles cv,γ for γ = Cτ for large τ correspond to integration over a simple contour around P+ , and hence reduce to cv,P+ (A, B) := Tr P+ ([log Dv , A] ◦ B) = res res ([log Dv , A] ◦ B) . P+

Dv

To find the dimension of the cohomology space of such cocycles for H D O and H DS we consider the restriction homomorphism to the smooth operators and symbols on the contour. In both cases the image is dense in the latter due to Theorem 3.6. One can see that the cocycle cv,P+ for any v is nontrivial in both H D O and H DS, since it is nontrivial on the subalgebra L. Indeed, upon restriction to the contour S 1 ≈ Cτ it gives the (nontrivial) Gelfand-Fuchs 2-cocycle on V ect (S 1 ). Furthermore, the cohomology space of 2-cocycles for smooth differential operators D O(S 1 ) is 1-dimensional, and hence so is the cohomology space of local 2-cocycles for holomorphic differential operators H D O, due to the density result and continuity of the cocycles. Verification of locality of cv,P+ for any holomorphic field v is a straightforward calculation. This proves part (i). For part (ii) we note that the algebra DS(S 1 ) admits exactly two independent nontrivial 2-cocycles up to equivalence. The density theorem will imply the required

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statement, once we show that there are two linearly independent cocycles of the type cv,P+ . Take 2-cocycles cv,P+ and cw,P+ for two fields v and w of different orders of pole/zero at P+ . Then v = f w for a meromorphic function f on , which is either zero or infinity at P+ . The same argument as in Corollary 2.16 (ii) gives that cv,P+ = cw,P+ + c f,P+ , where c f,P+ (A, B) := Tr P+ ([log f, A] ◦ B). In order to show that the cocycle c f,P+ is nontrivial and independent of cv,P+ , provided that f has a zero or pole at P+ , we again consider the restriction homomorphism to the smooth symbols DS(S 1 ) on the contour S 1 ≈ Cτ . In a local coordinate system around P+ we have f (z) = z k g(z) with holomorphic g(z) such that g(0) = 0 and k = 0. Since log f (z) = k log z + log g(z), the corresponding logarithmic cocycles are related in the same way. Note that Tr P+ ([log g, A] ◦ B) defines a trivial cocycle (i.e. a 2-coboundary) upon restriction to S 1 . Indeed, the function log g(z) is holomorphic at P+ = 0 since g(0) = 0, and its restriction to a small contour around P+ = 0 is univalued. Hence, it defines a smooth univalued function on the contour, and therefore [log g, A] is an inner derivation of the corresponding algebra DS(S 1 ). On the other hand, Tr P+ ([log z, A] ◦ B) upon restriction to S 1 ≈ Cτ defines the second non-trivial cocycle of the algebra DS(S 1 ), see Example 3.13. Hence the cocycle c f,P+ is nontrivial and defines the same cohomology class as k · cz,P+ . This completes the proof of (ii). Conjecture 3.17. Every continuous 2-cocycle on the Lie algebras H DS and H D O is cohomologous to a linear combination of regular 2-cocycles cv,γ for some holomorphic ◦

fields v and a contour γ on . Remark 3.18. The latter is closely related to Conjecture 2.9. Presumably, all the continuous 2-cocycles on H DS and H D O have the form c X,γ (A, B) := γ res D ([log X, A] ◦ B) for holomorphic pseudodifferential symbols X and cycles γ on the surface . In turn, one can reduce the cocycles c X,γ for an arbitrary symbol X to cocycles cv,γ with X = Dv , similarly to the proof of Theorem 2.8. 3.7. Manin triples for holomorphic pseudodifferential symbols. Given the point P+ ∈ and the invariant pairing on H DS associated to the trace at P+ it is straightforward to verify the following proposition. Proposition 3.19. The (non-extended) algebras (H DS, H D O, H I S) form a Manin triple. Although to any holomorphic field v with poles at P± one can associate the central extension of the Lie algebra H DS by the local 2-cocycle cv (A, B) = Tr ([log Dv , A] ◦ B), the double extension of H DS does not necessarily exist. ◦

Confine first to the special case, in which on there exists a holomorphic field ◦ ◦ v without zeros, i.e. to with a trivialized tangent bundle. Such a surface can be obtained from any and any field v on it by choosing the collections of points P± to include all zeros and poles of v. (Example: v = z ∂/∂z in C∗ .) In this case the operator log Dv maps H DS to itself, i.e it is an outer derivation of the latter. Then the construction of the co-central extension H I S = C · log Dv ⊕ H I S and the double extension H DS goes through in the same way as for the meromorphic or smooth cases.


525 ◦

Proposition 3.20. If the holomorphic field v has no zeros on , the Lie algebras ( H DS, H D O, H I S) form a Manin triple. Equivalently, H I S is a Lie bialgebra. In this case the Lie bialgebra on H I S defines a Poisson-Lie structure on the corresponding pseudodifferential symbols of complex degree, just like in the case of C∗ or in the smooth case on the circle. Furthermore, the Poisson structure on this group is the Adler-Gelfand-Dickey quadratic Poisson bracket, while the corresponding Hamiltonian equations are given by the n-KdV and KP hierarchies on Riemann surfaces, following the recipe for the smooth case. We recall the latter consideration from [3, 5] in the Appendix. ◦ Now let v have zeros in . Consider the central extension H D O of the algebra of holomorphic differential operators H D O by the 2-cocycle cv . One can see that now the “regular dual” space to H D O cannot be naturally identified with the vector space C · log Dv ⊕ H I S. Indeed, the coadjoint action of H D O is uniquely defined by the commutator [log Dv , A] wherever v = 0. However, this commutator may have poles at zeros of v, i.e. the space C · log Dv ⊕ H I S does not form a Lie algebra, as it is not closed under commutation. This does not allow one to define a natural Lie bialgebra structure on H D O or H I S. The same type of obstruction arises for the existence of a formal group of symbols of complex degrees on the surface .

4. Appendix 4.1. Poisson–Lie groups, Lie bialgebras, and Manin triples. Definition 4.1. A group G equipped with a Poisson structure η is a Poisson–Lie group if the group product G × G → G is a Poisson morphism (i.e., it takes the natural Poisson structure on the product G × G into the Poisson structure on G itself) and if the map G → G of taking the group inverse is an anti-Poisson morphism (i.e. it changes the sign of the Poisson bracket). Theorem 4.2 [17]. For any connected and simply connected group G with Lie algebra g there is a one-to-one correspondence between Lie bialgebra structures on g and Poisson-Lie structures η on G. This correspondence sends a Poisson-Lie group (G, η) into the Lie bialgebra g tangent to (G, η). By definition, a Lie algebra g is a Lie bialgebra if its dual space g∗ is equipped with a Lie algebra structure such that the map g → g ∧ g dual to the Lie bracket map g∗ ∧ g∗ → g∗ on g∗ is a 1-cocycle on g relative to the adjoint representation of g on g ∧ g. Theorem 4.3 [14]. Consider a Manin triple (g, g+ , g− ). Then g+ is naturally dual to g− and each of g− and g+ is a Lie bialgebra. Conversely, for any Lie bialgebra g one can find a unique Lie algebra structure on g¯ = g ⊕ g∗ such that the triple (¯g, g, g∗ ) is a Manin triple with respect to the natural pairing on g¯ and the corresponding Lie bialgebra structure on g¯ is the given one.

526

D. Donin, B. Khesin

4.2. The Poisson structure and integrable hierarchies on pseudodifferential symbols. Start with the Lie bialgebra of co-centrally extended smooth integral symbols on the circle: −1 k I S = C · log ∂ ⊕ I S = λ log ∂ + u k (x)∂ . k=−∞

(Alternatively, one can start with holomorphic symbols H I S and log D corresponding to a non-vanishing holomorphic field v on a punctured Riemann surface.) The corresponding Lie group consists of monic symbols of arbitrary complex degrees: G IS =

L=∂

λ

−1

1+

u k (x)∂

k

|λ∈C .

k=−∞

The Poisson-Lie structure on this group is given by the generalized quadratic Adler-Gelfand-Dickey bracket. Namely, the degree λ is a Casimir and we can consider the bracket on the hyperplane of symbols {L | λ = const}. The cotangent space to such planes can be identified with the symbols of the form X = ∂ −λ ◦ Y , where Y is a purely differential operator. Then the bracket on {L} is defined by the following Hamiltonian mapping X → V X (L) (from the cotangent space {X } to the tangent space to symbols {L} of fixed degree): V X (L) = (L X )+ L − L(X L)+ , see details in [3, 5]. To obtain dynamical systems, consider the following family of Hamiltonian functions {Hk } on this Poisson-Lie group G I S: Hk (L) :=

λ T r (L k/λ ), k

where L has degree λ = 0. The corresponding Hamiltonian equations with respect to the quadratic Adler-Gelfand-Dickey Poisson structure form the following universal KdV-KP hierarchy: ∂L = [(L k/λ )+ , L], k = 1, 2, . . . , ∂tk see [3, 5]. For λ = 1 this is the standard KP hierarchy of commuting flows. For integer λ = n the restriction of this universal hierarchy to the Poisson submanifolds of purely differential operators of degree n gives the n-KdV hierarchy. Acknowledgements. We are indebted to F. Malikov, I. Krichever and A. Rosly for fruitful discussions. In particular, the definition of the sheaves of holomorphic symbols was proposed to us by F. Malikov. We are also thankful to the anonymous referee for providing us with the reference [12] and useful remarks. B.K. is grateful to the Max-Plank-Institut in Bonn for kind hospitality. The work of B.K. was partially supported by an NSERC research grant.


527

References 1. Dickey, L.A.: Soliton equations and Hamiltonian systems. Adv. Series in Math. Phys. 12, Singapore: World Scientific, 1991, 310pp 2. Dzhumadildaev, A.S.: Derivations and central extensions of the Lie algebra of formal pseudodifferential operators. St. Petersburg Math. J. 6(1), 121–136 (1995) 3. Enriquez, B., Khoroshkin, S., Radul, A., Rosly, A., Rubtsov, V.: Poisson-Lie aspects of classical W-algebras. AMS Translations (2) 167, Providence, RI: AMS, 1995, pp. 37–59 4. Feigin, B.L.: gl(λ) and cohomology of a Lie algebra of differential operators. Russ. Math. Surv. 43(2), 169– 170 (1988) 5. Khesin, B., Zakharevich, I.: Poisson-Lie group of pseudodifferential symbols. Commun. Math. Phys. 171, 475–530 (1995) 6. Kravchenko, O.S., Khesin, B.A.: Central extension of the algebra of pseudodifferential symbols. Funct. Anal. Appl. 25(2), 83–85 (1991) 7. Krichever, I.M., Novikov, S.P.: Algebras of Virasoro type, Riemann surfaces and structure of the theory of solitons. Funct. Anal. Appl. 21(2), 126–142 (1987) 8. Krichever, I.M., Novikov, S.P.: Virasoro type algebras, Riemann surfaces and strings in Minkowski space. Funct. Anal. Appl. 21(4), 294–307 (1987) 9. Li, W.L.: 2-cocyles on the algebra of differential operators. J. Algebra 122(1), 64–80 (1989) 10. Radul, A.O.: A central extension of the Lie algebra of differential operators on a circle and W-algebras. JETP Letters 50(8), 371–373 (1989) 11. Schlichenmaier, M.: Krichever-Novikov algebras for more than two points. Lett. in Math. Phys. 19, 151– 165 (1990) 12. Schlichenmaier, M.: Verallgemeinerte Krichever-Novikov Algebren und deren Darstellungen, PhD-thesis, Universität Mannheim, Juni 1990; Differential operator algebras on compact Riemann surfaces. In: Generalized symmetries in physics (Clausthal, 1993), River Edge, NJ: World Sci. Publ., 1994, pp. 425–434 13. Schlichenmaier, M.: Local cocycles and central extensions for multipoint algebras of Krichever-Novikov type. J. Reine Angew. Math. 559, 53–94 (2003) 14. Semenov-Tyan-Shanskii, M.A.: Dressing transformations and Poisson group actions. Kyoto University, RIMS Publications 21(6), 1237–1260 (1985) 15. Sheinman, O.K.: Affine Lie algebras on Riemann surfaces. Funct. Anal. Appl. 27(4), 266–272 (1993) 16. Sheinman, O.K.: Affine Krichever-Novikov algebras, their representations and applications. In: Geometry, Topology and Mathematical Physics. S. P. Novikov’s Seminar 2002–2003, V. M. Buchstaber, I. M. Krichever, eds. AMS Translations (2) 212, Providence, RI: AMS, 2004, pp. 297–316 17. Lu, J.-H., Weinstein, A.: Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Diff. Geom. 31(2), 501–526 (1990) 18. Wagemann, F.: Some remarks on the cohomology of Krichever-Novikov algebras. Lett. in Math. Phys. 47, 173–177 (1999) Communicated by L. Takhtajan


Communications in


On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations Jean-Yves Chemin1 , Ping Zhang2 1 Laboratoire J.-L. Lions, Case 187, Université Pierre et Marie Curie, 75230 Paris Cedex 05, France.


2 Academy of Mathematics & Systems Science, CAS, Beijing 100080, China.

E-mail: [email protected] Received: 25 April 2006 / Accepted: 30 November 2006 Published online: 3 April 2007 – © Springer-Verlag 2007

Abstract: Corresponding to the wellposedness result [2] for the classical 3-D Navier−1+ 3

Stokes equations (N Sν ) with initial data in the scaling invariant Besov space, B p,∞ p , here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (AN Sν ), where the vertical viscosity is zero. In order to do so, we first introduce the − 21 , 21

Besov-Sobolev type spaces, B4 −1,1 B4 2 2 ,

− 21 , 21

and B4

(T ). Then with initial data in the scaling

we prove the global wellposedness for (AN Sν ) provided the invariant space norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (AN Sν ) with high oscillatory initial data (1.2). 1. Introduction 1.1. Introduction to the anisotropic Navier-Stokes equations. Let us first recall the classical (isotropic) Navier-Stokes system for incompressible fluids in the whole space: ⎧ ⎨ ∂t u + u · ∇u − νu = −∇ p, (N Sν ) div u = 0, ⎩ u| t=0 = u 0 , where u(t, x) denote the fluid velocity and p(t, x) the pressure and x = (x h , x3 ) a point of R3 = R2 × R. In this text, we are going to study a version of the system (N Sν ) where the usual Laplacian is substituted by the Laplacian in the horizontal variables, namely ⎧ ⎨ ∂t u + u · ∇u − νh u = −∇ p, (AN Sν ) div u = 0, ⎩ u| t=0 = u 0 .

530

J.-Y. Chemin, P. Zhang

Systems of this type appear in geophysical fluids (see for instance [5]). In fact, instead of putting the classical viscosity −ν in (N Sν ), meteorologist often modelize turbulent diffusion by putting a viscosity of the form: −νh h −ν3 ∂x23 , where νh and ν3 are empiric constants, and ν3 is usually much smaller than νh . We refer to the book of J. Pedlovsky [13], Chapter 4 for a more complete discussion. And in the particular case of the so-called Ekman layers [7, 10] for rotating fluids, ν3 = νh and is a very small parameter. It has been studied first by J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier in [6] and D. Iftimie in [12] where it is proved that the system (AN Sν ) is locally wellposed for initial data in the anisotropic Sobolev space def 0, 21 +ε def 2 3 2 H = u ∈ L (R ) / u 1 +ε = |ξ3 |1+2ε |u(ξ ˆ h , ξ3 )|2 dξ < +∞ , H˙ 2

R3

for some ε > 0. Moreover, it has also been proved that if the initial data u 0 is small enough in the sense that u 0 εL 2 u 0 1−ε ≤ cν 0, 1 +ε H˙

(1.1)

2

for some sufficiently small constant c, then we have a global wellposedness result. Let us notice that the space in which uniqueness is proved is the space of continuous functions 1 1 with value in H 0, 2 +ε and the horizontal gradient of which belongs to L 2 ([0, T ]; H 0, 2 +ε ). Let us observe that, as classical Navier-Stokes system, the system (AN Sν ) has a scaling. Indeed, if u is a solution of (AN Sν ) on a time interval [0, T ] with initial data u 0 , then the vector field u λ defined by def

u λ (t, x) = λu(λ2 t, λx) is also a solution of (AN Sν ) on the time interval [0, λ−2 T ] with the initial data λu 0 (λx). The smallness condition (1.1) is of course scaling invariant. But the norm · ˙ 1 +ε H2 is not and this norm determines the level of regularity required to have wellposedness. For classical Navier-Stokes system a lot of results of global wellposedness in scaling invariant space are available. The first one is the theorem of Fujita-Kato (see [9]) in which it is proved that the system (N Sν ) is globally wellposed for small initial data in 1 the homogeneous Sobolev spaces H˙ 2 which is the space of tempered distributions u with Fourier transform of which satisfy def u2 1 = |ξ | | u (ξ )|2 dξ < ∞. H˙ 2

R3

M. Paicu proved in [15] a theorem of the same type for the system (AN Sν ) in the 1 case when the initial data u 0 belongs to B 0, 2 (see Definition 1.2 below). As we shall see, this space has a scaling invariant norm. On the other hand, the classical isotropic system (N Sν ), is globally wellposed for small initial data in Besov norms of negative index. Let us first recall the definition of the Besov norms of negative index. Definition 1.1. Let f be a tempered distribution. Then we state, for positive s , and for ( p, q) in [1, ∞]2 , def s t 2e p f B˙ −s = f . t L + dt p,q q L (R ,

t

)

Global Wellposedness to the 3-D Incompressible N-S Equations

531

In [2], M. Cannone, Y. Meyer and F. Planchon proved that: if the initial data satisfies, for some p greater than 3, u 0

−1+ 3

p B˙ p,∞

≤ cν,

for some constant c small enough, then the incompressible Navier-Stokes system is globally wellposed. Let us mention that H. Koch and D. Tataru generalized this theorem to the ∂ B M O norm (see [14]). In particular, this theorem implies for any function φ in the Schwartz space S(R3 ), if we consider the family of initial data u ε0 defined by u ε0 (x) = sin

x

1

ε

(0, −∂3 φ, ∂2 φ),

(1.2)

the system (N Sν ) is globally wellposed for such initial data when ε is small enough. The goal of this work is to prove a theorem of this type for the anisotropic Navier-Stokes system (AN Sν ). 1.2. Statement of the results. Let us begin by the definition of the spaces we are going to work with. It requires an anisotropic version of dyadic decomposition of the Fourier space, let us first recall the following operators of localization in Fourier space, for (k, ) ∈ Z2 , kh a = F −1 (ϕ(2−k |ξh |) a ), and v a = F −1 (ϕ(2− |ξ3 |) a ), h h v v Sk a = k a, and S a = a, k ≤k−1

(1.3)

≤ −1

where Fa and a denote the Fourier transform of the function a, and ϕ a function 3 8

, , such that, for any positive τ , in D 4 3 ϕ(2− j τ ) = 1. j∈Z

Before we present the space we are going to work with, let us first recall the Besov1 Sobolev type space B 0, 2 , which was first introduced by D. Iftimie in [11] to study the well-posedness of classical Navier-Stokes equations. 1

Definition 1.2. We denote by B 0, 2 the space of distributions, which is the completion of S(R3 ) by the following norm: a

B

def 0, 21

=

∈Z

2 2 v a L 2 (R3 ) .

(1.4)

In [15], M. Paicu proved the global wellposedness of (AN Sν ) for small initial data 1 in B 0, 2 (compared with the horizontal viscosity). In order to state Paicu’s Theorem, let us introduce the following space.

532

J.-Y. Chemin, P. Zhang 1

Definition 1.3. We denote by B 0, 2 (T ) the space of distributions, which is the completion of the space C ∞ ([0, T ], S(R3 )) by the norm: def

a

1 B0, 2 (T )

=

∈Z

1 2 2 v a L ∞ (L 2 (R3 )) + ν 2 ∇h v a L 2 (L 2 (R3 )) . T

T

(1.5)

Now let us recall M. Paicu’s theorem. 1

Theorem 1.1. If u 0 ∈ B 0, 2 , then a positive time T exists such that the system (AN Sν ) 1 has a unique solution u in B 0, 2 (T ). Moreover, a constant c exists such that u 0

1

B0, 2

≤ cν =⇒ T = +∞. 1

Let us note that u ε0 defined in (1.2) is not small in this space B 0, 2 no matter how small the parameter ε is. Our main motivation to introduce the following spaces is to find a scaling invariant Besov-Sobolev type space such that in particular u ε0 is small in this space for ε sufficient small. − 1 , 21

Definition 1.4. We denote by B4 2 of S(R3 ) by the following norm: a

def − 21 , 21

B4

=

∈Z

2

2

∞ k= −1

2

−k

the space of distributions, which is the completion

kh v a2L 4 (L 2 ) v h

1 2 j + 2 2 S hj−1 vj a L 2 (R3 ) . (1.6) j∈Z − 21 , 21

Remark. The other motivation for us to construct the space B4 linear equation: ∂t u − ν(∂x21 + ∂x22 )u = f,

on (0, ∞) × R3 .

comes from the

(1.7)

In the anisotropic Littlewood-Paley decomposition of u : the terms kh v u with k ≥ −1 satisfies ∂t kh v u − ν(∂x21 + ∂x22 )kh v u = kh v f. As k ≥ − 1 and according to Lemma 2.1 below, (∂x21 + ∂x22 )kh v u is equivalent to (∂x21 + ∂x22 + ∂x23 )kh v u in this case, and therefore we can play viscosity for these parts. While for the remaining parts in the anisotropic Littlewood-Paley decomposition of u, we regroup them as S hj−1 vj u, j ∈ Z, should be dealt with as solutions of hyperbolic equation, as the vertical derivatives of S hj−1 vj u control its horizontal derivatives and there is no vertical viscosity term in (1.7). − 21 , 21

To study the evolution of (AN Sν ) with initial data in B4 duce the following space, which is smaller than

, we need also to intro-

1 1

− , L 2 ([0, T ]; B4 2 2 ):


533

−1,1

Definition 1.5. We denote by B4 2 2 (T ) the space of distributions, which is the completion of C ∞ ([0, T ], S(R3 )) by the norm: def

a

−1,1 B4 2 2 (T )

=

2

∈Z

+ν +

2

∞ k= −1

1 2

∞

k= −1

2−k kh v a2L ∞ (L 4 (L 2 )) T

T

2

v

h

2k kh v a2L 2 (L 4 (L 2 ))

1

1 2

(1.8)

v

h

j 1 2 2 S hj−1 vj a L ∞ (L 2 (R3 )) + ν 2 ∇h S hj−1 vj a L 2 (L 2 (R3 )) . T

T

j∈Z

In the following section, we shall use Littlewood-Paley theory to study the inner − 21 , 12

relations between B4 paper.

1

(T ) and B 0, 2 (T ). Now, we present the main results of this

− 21 , 21

Theorem 1.2. A constant c exists such that, if u 0 ∈ B4

and u 0

− 21 , 21

B4

≤ cν, then, − 21 , 12

with initial data u 0 , the system (AN Sν ) has a unique global solution u in B4

(∞).

The following proposition, which will be proved in Sect. 2, shows that this theorem can be applied to initial data given by (1.2). def

Proposition 1.1. Let φ ∈ S(R3 ). A constant Cφ exists such that, if φε (x) = ei x1 /ε φ(x), we have, for any positive ε, φε

1

−1,1 B4 2 2

≤ Cφ ε 2 .

Remark. It is easy to observe from Definition 1.2 that φε

B

0, 21

=

∈Z

2 2 v φε L 2 (R3 ) =

∈Z

2 2 v φ L 2 (R3 ) = φ

1

B0, 2

,

which can not be small no matter how small ε is. Repeating the proof of Theorem 1.2, we may conclude the following theorem concerning local wellposedness for large data. − 21 , 12

Theorem 1.3. If u 0 belongs to B4

, then a positive T exists such that the system − 21 , 21

(AN Sν ) has a unique solution in the space B4

(T ).

534


1.3. Structure of the proof. The purpose of the second section is to establish some results about anisotropic Littlewood-Paley theory which will be of constant use in what follows. The third section will be devoted to the proof of the existence of a solution of (AN Sν ). In order to do it, we shall search for a solution of the form

u = u F + w with u F = eνth u hh , u hh = def

def

1

kh v u 0 and w ∈ B 0, 2 (∞). (1.9)

k≥ −1

In the last section, we shall prove the uniqueness in the following way. First, we shall − 21 , 21

establish a regularity theorem claiming that if u ∈ B4 −1,1 with initial data in B4 2 2 , then w

= u − uF ∈ B

0, 21

(T ) is a solution of (AN Sν )

(T ). Therefore, looking at the equa1

tion of w, we shall prove the uniqueness of the solution u in the space u F + B 0, 2 (T ). We should mention that the method introduced by M. Paicu in [15] will play a crucial role in our proof here. Let us complete this section with the notations we are going to use in this paper. Notations. Let A, B be two operators, we denote [A; B] = AB − B A, the commutator between A and B a b; we mean that there is a uniform constant C, which may be different on different lines, such that a ≤ Cb. We shall denote by (a|b) the L 2 (R3 ) p q inner product of a and b. Finally, we denote L rT (L h (L v )) the space L r ([0, T ]; p q L (R x1 × R x2 ; L (R x3 ))). 2. Some Properties of Anisotropic Littlewood-Paley Theory As we shall constantly use the anisotropic Littlewood-Paley theory, and in particular anisotropic Bernstein inequalities. We list them as the following: Lemma 2.1. Let Bh (resp. Bv ) a ball of R2h (resp. Rv ), and Ch (resp. Cv ) a ring of R2h (resp. Rv ); let 1 ≤ p2 ≤ p1 ≤ ∞ and 1 ≤ q2 ≤ q1 ≤ ∞. Then there holds: If the support of a is included in 2k Bh , then ∂xαh a L p1 (L qv1 ) h

2

k |α|+2 p1 − p1 2

1

a L p2 (L qv1 ) . h

If the support of a is included in 2 Bv , then β

∂3 a L p1 (L qv1 ) 2

(β+( q1 − q1 )) 2

h

1

a L p1 (L qv2 ) . h

If the support of a is included in 2k Ch , then a L p1 (L qv1 ) 2−k N sup ∂hα a L p1 (L qv1 ) . |α|=N

h

h

If the support of a is included in 2 Cv , then a L p1 (L qv1 ) 2− N ∂3N a L p1 (L qv1 ) . h

h


535

Proof of Lemma 2.1 . Those inequalities are classical (see for instante [15] or [4]). For the reader’s convenience, let us prove the last one. The scaling allows us to assume that = 0. Let ϕ be a function of D(R \{0}) with value 1 near Cv . We have a (ξh , ξ3 ) =

ϕ (ξ3 ) F(∂3N a). (iξ3 ) N

(2.1)

def If h N = F −1 ϕ (ξ3 )(iξ3 )−N , then we have a(x h , x3 ) = h N (x3 − y3 )a(x h , y3 )dy3 . R

Young’s inequality gives the result.

Let us state two corollaries of this lemma, the proofs of which are obvious and thus omitted. − 21 , 21

1

Corollary 2.1. The space B 0, 2 is continuously embedded in the space B4 −1,1 B4 2 2 (T )

0, 21

for any positive T . Moreover, the space B is B (T ) in 2 ∞ ously embedded in the space L ∞ T (L h (L v )). − 21 , 21

Corollary 2.2. If a belongs to B4

22

∈Z

∈Z

k∈Z

2−k kh v a(0)2L 4 (L 2 ) h

0, 21

(T ) is continu-

(T ), then we have

1 2

a(0)

and

− 21 , 21

B4

v

21 −k h v 2 k h v 2 2 k a L ∞ (L 4 (L 2 )) +ν2 k a L 2 (L 4 (L 2 )) 2 a 2

T

k∈Z

h

v

T

− 21 , 21

Remark. Corollary 2.2 tells us that the space B4 butions which are

−1 B4,22

and so

h

− 21 , 21

B4

v

(T )

.

is included in the space of distri1

2 in the horizontal variable and B2,1 in the vertical one. Let us

−1

notice that this inclusion is strict. If a distribution a on R2 belongs to B4,22 (R2 ) and a 1

2 function b belongs to B2,1 (R), then it can happen that the distribution a ⊗ b on R3 does

− 21 , 21

not belong to B4

.

Proposition 1.1 stated in this preceding section tells us how large the difference is between the norms · 0, 1 and · − 1 , 1 . B

B4

2

2 2

Proof of Proposition 1.1. By definition of the norm · norm is less than or equal to the · 1 norm, φε

−1,1 B4 2 2

≤

− 21 , 21

B4

4 j=1

(ε j)

, we have, as the · 2

536


with

(1) ε =

def

2−

k− 2

2−

k− 2

kh v φε L 4 (L 2 ) ,

(2) ε =

def

v

h

ε2k >1 k≥ −1

kh v φε L 4 (L 2 ) , v

h

ε2k ≤1

k≥ −1

def (3) ε =

j

2 2 S hj−1 vj φε L 2 , and

ε2 j >1

(4) ε =

def

j

2 2 S hj−1 vj φε L 2 .

ε2 j ≤1 (1)

In order to estimate ε , let us notice that

k (1) 2− 2 2 2 sup kh v φε L 4 (L 2 ) ε ≤ 1

≤ ε2

∈Z

v

h

k∈Z

∈Z

ε2k >1

2 2 sup kh v φε L 4 (L 2 ) . h

k∈Z

v

Using Lemma 2.1, we have, by definition of φε , sup kh v φε L 4 (L 2 ) φε L 4 (L 2 ) φ L 4 (L 2 ) , h

k∈Z

v

h

v

h

v

and also sup kh v φε L 4 (L 2 ) 2− ∂3 φε L 4 (L 2 ) 2− ∂3 φ L 4 (L 2 ) . h

k∈Z

v

h

v

h

v

Thus, taking the sum over ≤ N and > N and choosing the best N gives 1 1 1 1 2 2 2 sup kh v φε L 4 (L 2 ) ≤ ε 2 φ L2 4 (L 2 ) ∂3 φ L2 4 (L 2 ) . (1) ε ≤ε ∈Z

v

h

k∈Z

h

v

h

v

The estimate of (2) ε demands the use of oscillations. By integration by parts, we get 1,ε 2,ε kh v φε = φk, + φk, with 1,ε (x) = iεkh v (ei φk,

def

y1 ε

∂1 φ) and

2,ε 2,ε (x) with (x) = −iε2k φ φk, k, y1 def 2,ε (x) = 22k 2 (∂1 φ g )(2k (x h − yh )) h(2 (x3 − y3 ))ei ε φ(y)dy, k,

def

where g (x h ) ∈ S(R2 ), h(x3 ) ∈ S(R) such that F g (ξh ) = ϕ (|ξh |) and F h(ξ3 ) = ϕ (ξ3 ). Using Lemma 2.1, we get y1 k 1,ε 2− 2 2 2 φk, L 4 (L 2 ) ε sup kh v (ei ε ∂1 φ) L 4 (L 2 ) ≤k+1

h

v

∈Z k

ε2 2 ∂1 φ L 2 (R3 ) .

h

v


537

Moreover, we have

k

2− 2

≤k+1

k

2,ε 2 2 φk, L 4 (L 2 ) ≤ ε2 2 h

v

∈Z

2,ε 4 2 . 2 2 φ k, L (L ) v

h

Using Lemma 2.1, we get 2,ε 4 2 φ 4 2 and φ 2,ε 4 2 2− ∂3 φ 4 2 . φ L (L ) L (L ) k, L (L ) k, L (L ) v

h

h

v

v

h

h

v

Again taking the sum over ≤ N and > N and choosing the best N , we get ∈Z

1

1

2,ε 4 2 φ 2 4 2 ∂3 φ 2 4 2 . 2 2 φ k, L (L ) L (L ) L (L ) h

v

Thus it turns out that (2) ε ≤ Cφ ε

v

h

h

k

v

1

2 2 ≤ Cφ ε 2 .

ε2k ≤1 (3)

In order to estimate ε , let us notice that, thanks to Lemma 2.1, we have j (3) 2− 2 S hj−1 vj ∂3 φε L 2 ε ε2 j ≥1

∂3 φε L 2

j

2− 2

ε2 j ≥1 1 2

ε ∂3 φ L 2 . (4)

The estimate about ε demands the use of the oscillations again. By integration by part, we get 2,ε S hj−1 vj φε = φ 1,ε with j + φj

φ 1,ε = iεS hj−1 vj (ei j def

y1 ε

∂1 φ) and

2,ε with −iε2 j φ j y1 def 3 j = 2 h(2 j (x3 − y3 ))ei ε φ(y)dy (∂1 g)(2 j (x h − yh ))

def φ 2,ε = j

2,ε φ j

for some function g in S(R2 ). Using Lemma 2.1, we get j j 1,ε 1 2 2 φ j L 2 ε∂1 φ L 2 2 2 ε 2 ∂1 φ L 2 . ε2 j ≤1

ε2 j ≤1

2,ε L 2 ≤ ∂3 φ L 2 . Thus, we infer Using Lemma 2.1 again, we get 2 j φ j

j

2 2 φ 2,ε j L 2 ε∂3 φ L 2

ε2 j ≤1

This concludes the proof of Proposition 1.1.

ε2 j ≤1

j

1

2 2 Cφ ε 2 .

538


Notations. In that follows, we make the convention that (ck )k∈Z (resp. (d j ) j∈Z ) denotes a generic element of the sphere of 2 (Z) (resp. 1 (Z)). Moreover, (ck, )(k, )∈Z2 denotes a generic element of the sphere of 2 (Z2 ) and (dk, )(k, )∈Z2 denotes a generic sequence indexed by Z2 such that

2 dk,

1 2

= 1.

∈Z k∈Z

Let us notice that we shall often use the following property, the easy proof of which is omitted. Lemma 2.2. Let α be a positive real number and N0 an integer. Then we have 2−α( − j) dk, ck d j . (k, )∈Z2 ≥ j−N0

The following lemma will be of frequent use in this work. It describes some estimates − 21 , 21

of dyadic parts of functions in B4 − 21 , 21

Lemma 2.3. For any a ∈ B4

(T ).

(T ), one has

1

k

Skh v a L ∞ (L 4 (L 2 )) + ν 2 ∇h Skh v a L 2 (L 4 (L 2 )) dk, 2 2 2− 2 a T

v

h

T

h

v

1

k

Skh a L ∞ (L 4 (L ∞ )) + ν 2 ∇h Skh a L 2 (L 4 (L ∞ )) ck 2 2 a T

v

h

T

v

h

− 21 , 21

B4

− 21 , 21

B4

(T )

(T )

.

Proof of Lemma 2.3. By definition of Skh , we have 1

Sk, (a) = Skh v a L ∞ (L 4 (L 2 )) + ν 2 ∇h Skh v a L 2 (L 4 (L 2 )) T h v T h v

1 h v h v 2 k a L ∞ (L 4 (L 2 )) + ν ∇h k a L 2 (L 4 (L 2 )) . ≤ def

h

v

2−

k−k 2

T

k ≤k−1

T

v

h

Noticing that

k

2 2 2− 2 Sk, (a) ≤ 2 2

k ≤k−1

k 2− 2 kh v a L ∞ (L 4 (L 2 )) T

h

v

1 + ν 2 ∇h kh v a L 2 (L 4 (L 2 )) , T

h

v

we get, by applying Cauchy-Schwarz inequality, that

1 2 −k 2 22 2 Sk, (a) ≤ 22 2−k kh v a L ∞ (L 4 (L 2 )) k ∈Z

k∈Z

+ν

1 2

T

∇h kh v a L 2 (L 4 (L 2 )) T h v

By Corollary 2.2, this proves the first inequality.

h

v

2 21

.

and


539

In order to prove the second inequality, we shall prove that, for any (ck )k∈Z , we have def

I (a) =

k

2− 2 Sk ck a

def

− 21 , 21

B4

k∈Z

(2.2)

(T )

1

with Sk = Skh a L ∞ (L 4 (L ∞ )) + ν 2 ∇h Skh a L 2 (L 4 (L ∞ )) . Using again Lemma 2.1, we T h v T h v have

1 Sk 2 2 kh v a L ∞ (L 4 (L 2 )) + ν 2 kh v ∇h a L 2 (L 4 (L 2 )) . T

k ≤k−1 ∈Z

v

h

T

v

h

We deduce I (a)

22

∈Z

2−

k−k 2

k 2− 2 ck kh v a L ∞ (L 4 (L 2 )) T

(k,k )∈Z2 k ≤k−1

v

h

1 + ν 2 kh v ∇h a L 2 (L 4 (L 2 )) . T

v

h

Using Cauchy-Schwarz inequality with the weight 2−

I (a)

2

− k−k 2

ck2

21

×2

∈Z

(k,k )∈Z2 k ≤k−1 −k

22

k−k 2

1k ≤k−1 , we infer

2−

k−k 2

(k,k )∈Z2 k ≤k−1

2 21 1 h v h v k a L ∞ (L 4 (L 2 )) + ν 2 k ∇h a L 2 (L 4 (L 2 )) . T

v

h

T

v

h

From this we deduce that k−k 22 2− 2 2−k kh v a L ∞ (L 4 (L 2 )) I (a) ∈Z

T

(k,k )∈Z2 k ≤k−1

1

+ ν 2 kh v ∇h a L 2 (L 4 (L 2 )) T

22

k ∈Z

∈Z

a

− 21 , 21

B4

(T )

h

h

v

2 21

v

2 21 1 2−k kh v a L ∞ (L 4 (L 2 )) + ν 2 kh v ∇h a L 2 (L 4 (L 2 )) T

h

v

T

h

v

,

which proves (2.2) and thus Lemma 2.3. With Lemma 2.3, we are going to state a result which is very close to Sobolev embedding and will be of constant use in the existence proof of Theorem 1.2.

540

J.-Y. Chemin, P. Zhang − 21 , 21

Lemma 2.4. The space B4 − 21 , 21

a function in B4

(T ), then we have

vj a L 4 (L 4 (L 2 )) T

h

v

(T ) is included in L 4T (L 4h (L ∞ v )). More precisely, let a be

dj ν

1 4

j

2− 2 a

−1,1 B4 2 2 (T )

and a L 4 (L 4 (L ∞ )) T

v

h

1 1

ν4

a

− 21 , 21

B4

(T )

.

Proof of Lemma 2.4 . Let us first notice that vj a2L 4 (L 4 (L 2 )) = (vj a)2 L 2 (L 2 (L 1 )) . T

v

h

T

v

h

Then using Bony’s decomposition in the horizontal variables, we write h h (vj a)2 = Sk−1 vj akh vj a + Sk+2 vj akh vj a. k∈Z

k∈Z

These two terms are estimated exactly in the same way. Applying Hölder inequality, we get k

k

h h Sk−1 vj akh vj a L 2 (L 2 (L 1 )) ≤ 2− 2 Sk−1 vj a L ∞ (L 4 (L 2 )) 2 2 kh vj a L 2 (L 4 (L 2 )) . T

v

h

T

v

h

T

h

v

Using the first inequality of Lemma 2.3 and Corollary 2.2, we infer h vj akh vj a L 2 (L 2 (L 1 )) Sk−1 T h v

2 dk, j

ν

1 2

2− j a2 − 1 , 1 B4

2 2 (T )

.

Taking the sum over k and using Lemma 2.2, we deduce (vj a)2 L 2 (L 2 (L 1 )) T

v

h

d 2j ν

1 2

2− j a2 − 1 , 1 B4

2 2 (T )

,

which is exactly the first inequality of the lemma. Now, using Lemma 2.1, we have j

vj a L 4 (L 4 (L ∞ )) 2 2 vj a L 4 (L 4 (L 2 )) . T

v

h

T

h

v

This proves the whole lemma.

Now let us use Lemma 2.1 to study the free evolution u F to the high horizontal frequency part of the initial data u 0 , as defined in (1.9). In order to do so, let us first recall a lemma from [3] or [4], which describes the action of the semi-group of the heat equation on distributions, the Fourier transform of which are supported in a fixed ring. − 21 , 21

Lemma 2.5. Let u 0 ∈ B4 holds

kh v u F L p (L 4 (L 2 )) T h v

and u F be as in (1.9), α ∈ N3 , 1 ≤ p ≤ ∞. Then, there ⎧ ⎨ dk,

2

1 2 2− p

2− 2 u 0

− 21 , 21

B4

⎩ ν 0, otherwise.

− 21 , 21

Moreover, u F belongs to B4

1 p

k

, for k ≥ − 1,

(2.3)

(∞), and we have

u F

− 21 , 21

B4

(∞)

u 0

− 21 , 21

B4

.

(2.4)


541

Proof of Lemma 2.5. The relations (4) and (5) of the proof of Lemma 2.1 of [3] tell us that kh v u F (t) = 22k g(t, 2k ·) kh v u 0 with g(t, ·) L 1 (R2 ) ≤ Ce−cνt2 . 2k

(2.5)

Here the convolution must be understood as the convolution on R2 . Thus kh v u F (t, x h , ·) L 2v ≤ 22k |g(t, 2k ·)| kh v u 0 (·) L 2v . Using (2.5) and again Lemma 2.1, we get kh v u F (t) L 4 (L 2 ) e−cνt2 kh v u 0 L 4 (L 2 ) 2k

v

h

v

h

e

−cνt22k

k 2

dk, 2 2

− 2

u 0

− 21 , 21

B4

.

By integration, the lemma follows.

From Lemma 2.5, we immediately deduce the following corollary. Corollary 2.3. The following estimates on u F holds. For any ( p, q) in [1, ∞] × [4, ∞], we have kh u F L p (R+ ;L q (L ∞ )) v

h

If in addition

1

ck 2

1

νp

−k 2 1p + q1 −1

u 0

− 21 , 21

B4

.

1 1 1 + > , we have p q 2 vj u F L p (R+ ;L q (L 2 )) v

h

1 1

νp

dj2

− j 2 1p + q1 − 21

u 0

− 21 , 21

B4

.

The following lemma is the end point of the second estimate of Corollary 2.3. Lemma 2.6. Under the assumptions of Lemma 2.5, one has j dj 1 vj u F L 2 (R+ ;L ∞ (L 2v )) √ 2− 2 u 0 − 1 , 1 and u F L 2 (R+ ;L ∞ (R3 )) √ u 0 − 1 , 1 . h 2 2 ν ν B4 B4 2 2

Proof of Lemma 2.6. Trivially there holds vj u F 2L 2 (L ∞ (L 2 )) = (vj u F )2 L 1 (L ∞ (L 1 )) . T

h

v

T

v

h

While by using Bony’s paradifferential decomposition in the horizontal variables, one has h h (vj u F )2 = Sk−1 vj u F kh vj u F + kh vj u F Sk+2 vj u F . (2.6) k∈Z

k∈Z

Now the idea is to take advantage of the smoothing effect on u F on the highest possible horizontal frequencies. Then we get, by applying Hölder inequality, h h Sk−1 vj u F kh vj u F L 1 (L ∞ (L 1 )) 2k Sk−1 vj u F L ∞ (L 4 (L 2 )) kh vj u F L 1 (L 4 (L 2 )) . T

h

v

T

h

v

T

h

v

542


Let us notice that, by (2.3) and the proof of Lemma 2.3, one has k

j

h Sk−1 vj u F L ∞ (L 4 (L 2 )) dk, j 2 2 2− 2 u 0 T

h

v

− 21 , 21

B4

.

Therefore, by using (2.3) once again, we arrive at h Sk−1 vj u F kh vj u F k∈Z

1 L 1T (L ∞ h (L v ))

2 − j 2

dk, j u 0 2 − 1 , 1 . ν B 2 2 k∈Z

4

A similar argument yields a similar estimate for the other term in (2.6), from which, we deduce d 2j

vj u F 2L 2 (L ∞ (L 2 )) T

h

ν

v

2− j u 0 2 − 1 , 1 . B4

2 2

Then from Lemma 2.1 we conclude j 1 2 2 vj u F L 2 (L ∞ (L 2 )) √ u 0 − 1 , 1 . S vj u F L 2 (L ∞ (R3 )) v T h T ν B4 2 2 j ≤ j−1

This completes the proof of the lemma.

Let us finish this section by recalling the isotropic paradifferential decomposition of J.-M. Bony from [1]: let a, b ∈ S (R3 ), then ab = Ta b + Tb a + R(a, b), and ab = Ta b + R(a, b), where Ta b =

S j−1 a j b

R(a, b) =

and

j a j b,

| j− j |≤1

j∈Z

R(a, b) =

(2.7)

j aS j+2 b.

j∈Z

In what follows, instead of using the isotropic version of (2.7), we will constantly use a version of (2.7) only in the horizontal or vertical variables. 3. The Proof of an Existence Theorem The purpose of this section is to prove the following existence theorem. Theorem 3.1. A sufficiently small constant c exists which satisfies the following prop− 21 , 12

erty: if u 0 is in B4

, and u 0

solution in the space

− 21 , 21

B4

≤ cν, then the system (AN Sν ) has a global

1

u F + B 0, 2 (∞), where u F is defined in (1.9).


543

Proof of Theorem 3.1. As announced in the introduction, we shall look for a solution of the form u = u F + w. Let us first establish the equation satisfied by w. Actually by substituting the above formula to (AN Sν ), we obtain ⎧ ⎨ ∂t w + w · ∇w − νh w + w · ∇u F + u F · ∇w = −u F · ∇u F − ∇ p, div w = 0, ( AN S ν ) ⎩ def w|t=0 = u h = u 0 − u hh . Notice that by (1.9), we have u h =

S hj−1 vj u 0 .

(3.1)

j∈Z

Moreover, there holds

vj u h =

| j− j |≤1

and thus vj u h L 2

S hj −1 vj vj u 0

| j− j |≤1

− 21 , 12

This implies that, if u 0 belongs to B4 u h

1

B0, 2

S hj −1 vj u 0 L 2 . 1

, then u h belongs to B 0, 2 and u 0

− 21 , 21

B4

.

(3.2)

We shall use the classical Friedrichs’ regularization method to construct the approximate S ν ). For simplicity, we just outline it here (for the details in this context, solutions to ( AN see [15] or [4]). In order to do so, let us define the sequence of operators (Pn )n∈N by def a , Pn a = F −1 1 B(0,n) and we define ⎧ ∂t wn − νh wn + Pn (wn · ∇wn ) + Pn (wn · ∇u F,n ) + Pn (u F,n · ∇w n ) ⎪ ⎪ ⎪ j j ⎨ = −Pn (u F,n · ∇u F,n ) + Pn ∇−1 ∂ j ∂k (u F,n + wn )(u kF,n + wnk ) S ν,n ) ( AN ⎪ div wn = 0, ⎪ ⎪ ⎩ def wn |t=0 = Pn (u h ) = Pn (u 0 − u hh ). def

where u F,n = (Id −S jn )u F with jn ∼ − log2 n and where −1 ∂ j ∂k is defined precisely by a ). −1 ∂ j ∂k a = F −1 (|ξ |−2 ξ j ξk def

544


Because of properties of L 2 and L 1 functions the Fourier transform of which are S ν,n ) appears to be an ordinary differsupported in the ball B(0, n), the system ( AN ential equation in the space def L 2n = a ∈ L 2 (R3 ) / Supp a ⊂ B(0, n) . This ordinary differential equation is globally wellposed because d wn (t)2L 2 ≤ Cn u F,n (t) L ∞ wn 2L 2 + Cn u F,n (t)2L 4 (L 2 ) wn (t) L 2 dt h v and u F,n belongs to L 2 (R+ ; L ∞ ∩ L 4h (L 2v )). We refer to [4] and [15] for the details. Now, the proof of Theorem 3.1 reduces to the following three propositions, which we shall admit for the time being. − 21 , 21

Proposition 3.1. Let u 0 be in B4

def

T

I j (T ) =

0

1

, and a in B 0, 2 (T ). Then, for any j in Z,

d 2j − j v 2 u 0 2 − 1 , 1 a 0, 1 j (u F · ∇u F )|vj a dt B 2 (T ) ν B 2 2 4

holds, where u F is defined in (1.9). 1

Proposition 3.2. Let a be a divergence free vector in B 0, 2 (T ), and b a vector field 1 of B 0, 2 (T ). Then, for any j in Z, def

J j (T ) =

0

T

d 2j − j v 2 a 0, 1 u 0 − 1 , 1 b 0, 1 j (a · ∇u F )|vj b dt B 2 (T ) B 2 (T ) ν B4 2 2

holds, where u F is defined in (1.9). − 21 , 12

Proposition 3.3. Let a be a divergence free vector in B4 Then, for any j ∈ Z, def

F j (T ) =

0

T

1

(T ), and b ∈ B 0, 2 (T ).

d 2j − j v 2 a − 1 , 1 b2 0, 1 j (a · ∇b)|vj b dt ν B 2 (T ) B4 2 2 (T )

holds. S ν,n ) that Pn wn = wn ; we Conclusion of the Proof of Theorem 3.1. Notice from ( AN v 2 apply the operator j to ( AN S ν,n ) and take the L inner product of the resulting equation with vj wn to get d vj wn (t)2L 2 + 2ν∇h vj wn (t)2L 2 dt = −2(vj (wn · ∇wn )|vj wn ) −2(vj (u F,n · ∇wn )|vj wn ) − 2(vj (wn · ∇u F,n )|vj wn ) −2(vj (u F,n · ∇u F,n )|vj wn ).


545

By integrating the above equation over [0, T ], we get 2 j vj wn 2L ∞ (L 2 ) +2 j+1 ν∇h vj wn 2L 2 (L 2 ) ≤ 2 j vj wn (0)2L 2 +2 T

T

with def

W 1j (T ) = 2 j def

W 2j (T ) = 2 j def

W 3j (T ) = 2 j def

W 4j (T ) = 2 j

0

0

0

0

4

W kj (T )

(3.3)

k=1

T

v ( j (wn (t) · ∇wn (t))|vj wn (t))dt,

T

v ( j (u F,n (t) · ∇wn (t))|vj wn (t))dt,

T

v ( j (wn (t) · ∇u F,n (t))|vj wn (t))dt,

T

v ( j (u F,n (t) · ∇u F,n (t))|vj wn (t))dt.

Proposition 3.3 applied with a = b = wn together with Corollary 2.1 gives W 1j (T )

d 2j ν

wn 3 0, 1 B

2 (T )

.

(3.4)

Thanks to Lemma 2.5, Proposition 3.3 applied with a = u F,n and b = wn implies in particular that u 0

− 21 , 21

B4

W 2j (T ) d 2j

ν

wn 2 0, 1

.

(3.5)

wn 2 0, 1

.

(3.6)

wn

.

(3.7)

B

2 (T )

Proposition 3.2 applied with a = b = wn yields u 0

− 21 , 21

B4

W 3j (T ) d 2j

ν

B

2 (T )

Finally Proposition 3.1 claims that u 0 2 − 1 , 1 B4

W 4j (T ) d 2j

ν

2 2

1

B0, 2 (T )

Plugging estimates (3.4)–(3.7) into (3.3) gives

2 √ 2 j vj wn L ∞ (L 2 ) + 2ν∇h vj wn L 2 (L 2 ) T T C ≤ 2 j vj wn (0)2L 2 + d 2j wn 2 0, 1 + u 0 2 − 1 , 1 wn 0, 1 . B 2 (T ) ν B 2 (T ) B 2 2 4

0, 21

Using (3.2), we get, by definition of B (T ), 1 C wn 0, 1 ≤ 2C0 u 0 − 1 , 1 + √ wn 0, 1 +u 0 − 1 , 1 wn 2 0, 1 . (3.8) B 2 (T ) B 2 (T ) ν B 2 (T ) B4 2 2 B4 2 2

546


Let us define

def

Tn = sup T > 0/wn

≤ 4C0 u 0

1

B0, 2 (T )

.

− 21 , 21

B4

The fact that wn is continuous with value in H N for any integer N implies that Tn is positive. Then, inequality (3.8) implies that, for any n and for any T < Tn , we have wn

1

B0, 2 (T )

Then, if u 0

− 21 , 21

B4

is small enough with respect to the viscosity ν, we get, for any n

− 21 , 21

B4

≤ 2C0 u 0

√ 3 2C(4C0 + 1) C0 + u 0 2 − 1 , 1 . √ ν B4 2 2

and for any T < Tn , wn

5 C0 u 0 − 1 , 1 . 2 B4 2 2

≤

1

B0, 2 (T )

Thus Tn = +∞ for any n. Then, the existence follows from classical compactness methods, the details of which are omitted (see [15] or [4]). Then, Theorem 3.1 is proved, provided of course that we have proved the three propositions 3.1–3.3.

Proof of Propositions 3.1–3.3. Everything will be different for terms involving the horizontal derivative and for terms involving the vertical derivatives. For terms involving the horizontal derivative, the following two lemmas will be crucial. − 21 , 21

Lemma 3.1. Let a be in B4 vj (a∂h b)

1

(T ) and b in B 0, 2 (T ). We have

4 4 L T3 (L h3 (L 2v ))

dj ν

3 4

j

2− 2 a

− 21 , 21

B4

(T )

b

1

B0, 2 (T )

.

−1,1

2 Lemma 3.2. Let (a, b) be in B4 2 2 (T ) . We have vj (ab) L 2 (L 2 ) T

dj ν

1 2

j

2− 2 a

− 21 , 21

B4

(T )

b

− 21 , 21

B4

(T )

.

Proof of Lemma 3.1. Using Bony’s decomposition (2.7) in the vertical variable gives vj (a∂h b) =

| j− j |≤5

vj (S vj −1 avj ∂h b) +

vj (S vj +2 (∂h b)vj a).

j ≥ j−N0

Using Hölder the inequality and then Lemma 2.4, we have vj (S vj −1 avj ∂h b)

4 4 L T3 (L h3 (L 2v ))

S vj −1 a L 4 (L 4 (L ∞ )) vj ∂h b L 2 (L 2 ) T

d j 3

ν4

j 2

h

2− a

v

− 21 , 21

B4

T

(T )

b

1

B0, 2 (T )

.


547

Similarly, one has vj (S vj +2 (∂h b)vj a)

4

4

L T3 (L h3 (L 2v ))

S vj +2 ∂h b L 2 (L 2 (L ∞ )) vj a L 4 (L 4 (L 2 )) T

d j

ν

3 4

v

h

T

j

2− 2 a

− 21 , 21

B4

(T )

b

1

B0, 2 (T )

v

h

.

Then it turns out that j

2 2 vj (a∂h b)

4 4 L T3 (L h3 (L 2v ))

1 ν

3 4

a

−1,1 B4 2 2 (T )

b

B

0, 21

(T )

2−

j− j 2

d j

j ≥ j−N1

which implies the lemma.

Proof of Lemma 3.2. Let us write vj (S vj −1 avj b) + vj (ab) = | j − j|≤5

vj (S vj +2 bvj a).

j ≥ j−N

0

Using again the Hölder inequality and Lemma 2.4, we get vj (S vj −1 avj b) L 2 (L 2 (L 2 )) S vj −1 a L 4 (L 4 (L ∞ )) vj b L 4 (L 4 (L 2 )) T

h

v

T

d j 1

ν2

j 2

h

2− a

v

− 21 , 21

B4

T

(T )

b

− 21 , 21

B4

v

h

(T )

.

Then we conclude as in the previous lemma.

Proof of Proposition 3.1. Let us remark that, thanks to the fact that u F is divergence free, we have T

v I j (T ) = j (u F · ∇u F )|vj a dt ≤ I hj (T ) + I vj (T ), with 0

def

I hj (T ) =

v h j (u F ⊗ u F )|vj ∇h a dt and 0 T

def v I j (T ) = ∂3 vj (u 3F u F )|vj a dt. T

0

Using Lemmas 2.5 and 3.2, we get I hj (T ) ≤ vj (u hF u F ) L 2 (L 2 ) vj (∇h a) L 2 (L 2 ) T

d 2j ν

T

2− j u 0 2 − 1 , 1 a B4

2 2

1

B0, 2 (T )

.

For the term with the vertical derivative, let us write, using Lemma 2.1, I vj (T ) 2 j vj (u 3F u F ) L 1 (L 2 ) vj b L ∞ (L 2 ) . T

T

Using again Bony’s decomposition (2.7) in the vertical variable, we infer vj (u 3F u F ) = vj (S vj −1 u 3F vj u F ) + vj (vj u 3F S vj +2 u F ). | j − j|≤5

j ≥ j−N0

548


Using Bony’s decomposition (2.7) in the horizontal variables, we get, h h Sk−1 S vj −1 u F vj u F = S vj −1 u 3F kh vj u F + kh S vj −1 u 3F Sk+2 vj u F . k≥ j −N0

The two terms of the above sum are estimated exactly along the same lines. As in the proof of Lemma 2.6, we use the smoothing effect on u F on the highest possible horizontal frequencies. Using Hölder inequality, this gives h S vj −1 u 3F kh vj u F L 1 (L 2 ) Sk−1 T

≤2

− k2

k h Sk−1 S vj −1 u F L ∞ (L 4 (L ∞ )) 2 2 kh vj u F L 1 (L 4 (L 2 )) . T h v T h v

Lemma 2.5 gives k

2 2 kh vj u F L 1 (L 4 (L 2 )) T

h

v

j 1 dk, j 2− 2 2−k u 0 − 1 , 1 . ν B4 2 2

Lemma 2.3 claims in particular that k

h 2− 2 Sk−1 S vj −1 u F L ∞ (L 4 (L ∞ )) ck u 0 T

h

v

− 21 , 21

B4

.

Then, using Lemma 2.2, it turns out that S vj −1 u F vj u F L 1 (L 2 ) T

j 1 ck dk, j 2−k 2− 2 u 0 2 − 1 , 1 ν B 2 2 k≥ j −1

4

d j − 3 j 2 2 u 0 2 − 1 , 1 . ν B 2 2 4

We deduce that 3j

2 2 vj (u 3F u F ) L 1 (L 2 ) T

1 u 0 2 − 1 , 1 ν B 2 2 4

d j 2−

3( j − j) 2

.

j ≥ j−N1

This concludes the proof of Proposition 3.1.

Proof of Proposition 3.2. Again, we distinguish the terms with horizontal derivatives from the terms with vertical ones, writing

v j (a · ∇u F )|vj b dt ≤ J jh (T ) + J jv (T ), where 0 T T

def def v h v 3 h v v J j (T ) = j (a · ∇h u F )| j b dt and J j (T ) = j (a ∂3 u F )|vj b dt.

J j (T ) =

0

T

0

Using integration by part gives

vj (a h · ∇h u F )|vj b = − vj (u F divh a h )|vj b − vj (a h ⊗ u F )|∇h vj b .


549

From Lemma 2.4 and Lemma 3.1, one has T

v j (u F divh a h )|vj b dt ≤ vj (u F divh a h ) 0

d 2j

ν

2− j u 0

− 21 , 21

B4

4

4

L T3 (L h3 (L 2v ))

a h

vj b L 4 (L 4 (L 2 ))

1

T

B0, 2 (T )

b

h

1

B0, 2 (T )

v

.

Lemma 3.2 gives T

v h j (a ⊗ u F )|∇h vj b dt ≤ vj (a h ⊗ u F ) L 2 (L 2 ) vj ∇h b L 2 (L 2 ) T

0

d 2j − j

ν

2

u 0

− 21 , 21

B4

T

a h

1

B0, 2 (T )

b

1

B0, 2 (T )

.

Therefore, J jh (T )

d 2j ν

2− j u 0

− 21 , 21

B4

a h

1

B0, 2 (T )

b

1

B0, 2 (T )

.

On the other hand, we get, by using Bony’s decomposition (2.7) in the vertical variables, vj (a 3 ∂3 u F ) = vj (S vj −1 a 3 ∂3 vj u F ) + vj (vj a 3 S vj +2 ∂3 u F ). (3.9) | j − j|≤5

j ≥ j−N0

To deal with the first term, we use the Hölder inequality to get

S vj −1 a 3 ∂3 vj u F L 1 (L 2 (R3 )) 2 j S vj −1 a 3 L ∞ (L 2 (L ∞ )) vj u F L 1 (L ∞ (L 2 )) . T

T

h

v

T

h

v

Corollary 2.1 and Corollary 2.3 applied with p = 1 and q = ∞ imply S vj −1 a 3 ∂3 vj u F L 1 (L 2 (R3 )) T

d j − j 2 2 u 0 − 1 , 1 a 0, 1 , B 2 (T ) ν B4 2 2

from which we infer j dj vj (S vj −1 a 3 ∂3 vj u F ) L 1 (L 2 (R3 )) 2− 2 u 0 − 1 , 1 a 0, 1 . T B 2 (T ) ν B4 2 2 | j − j|≤5

Now, let us estimate the second term of (3.9). The Hölder inequality gives

vj a 3 S vj +2 ∂3 u F L 1 (L 2 (R3 )) 2 j vj a 3 L 2 (L 2 (R3 )) S vj +2 u F L 2 (L ∞ (R3 )) . T

T

T

From Lemma 2.1, we get

vj a 3 L 2 (L 2 (R3 )) 2− j vj ∂3 a 3 L 2 (L 2 (R3 )) . T

T

Using the fact that div a = 0, we have 3 j d j vj a 3 L 2 (L 2 (R3 )) 2− j vj divh a h L 2 (L 2 (R3 )) √ 2− 2 a h 0, 1 . T T B 2 (T ) ν

550


Together with Lemma 2.6, this implies that j ≥ j−N0

vj (vj a 3 S vj +2 ∂3 u F ) L 1 (L 2 (R3 )) T

dj − j 2 2 u 0 − 1 , 1 a h 0, 1 . (3.10) B 2 (T ) ν B4 2 2

This ends the proof of Proposition 3.2.

Proof of Proposition 3.3. Similar to the proof of Proposition 3.2, we separate F j (T ) as

T

F j (T ) = def

F jh (T ) =

0 T 0

v j (a · ∇b)|vj b dt ≤ F jh (T ) + F jv (T ), where T

def v h v 3 j (a · ∇h b)|vj b dt and F jv (T ) = j (a ∂3 b)|vj b dt. 0

Firstly, we apply the Hölder inequality to get F jh (T ) ≤ vj (a h · ∇h b)

4

4

L T3 (L h3 (L 2v ))

vj b L 4 (L 4 (L 2 )) . T

h

v

Then Lemma 3.1 together with Corollary 2.1 and Lemma 2.4 implies that F jh (T )

d 2j ν

2− j a

− 21 , 21

B4

(T )

b2 0, 1 B

2 (T )

. 1

On the other hand, as there is no gain of vertical derivative in both norms of B 0, 2 (T ) − 21 , 21

and B4 (T ), in order to gain this vertical derivative from the assumption that div a = 0, we are going to use the trick from [6]. Using paradifferential decomposition in the vertical variable to vj (a 3 ∂3 b) first, then, by a commutator process, one gets

vj (a 3 ∂3 b) = S vj−1 a 3 ∂3 vj b +

v [vj ; S −1 a 3 ]∂3 v b

| j− |≤5

v (S −1 a3 + | j− |≤5

− S vj−1 a 3 )∂3 vj v b +

≥ j−N0

v vj (v a 3 ∂3 S +2 b),

correspondingly, we decompose F jv (T ) as F jv (T ) = F j1,v + F j2,v + F j3,v + F j4,v . def

In what follows, we will estimate term by term all the terms above. Note that div a = 0 implies that ∂3 a 3 = − divh a h .

(3.11)

Therefore, by integration by parts twice, we get F j1,v =

1 2

0

T

R3

S vj−1 divh a h |vj b|2 d x dt = −

T 0

R3

S vj−1 a h · ∇h vj bvj b d x dt.


551

Lemma 2.4 together with Corollary 2.1 applied gives |F j1,v | ≤ S vj−1 a h L 4 (L 4 (L ∞ )) ∇h vj b L 2 (L 2 ) vj b L 4 (L 4 (L 2 )) T

d 2j ν

2− j a h

v

h

− 21 , 21

B4

T

(T )

b2 0, 1 B

2 (T )

T

h

v

.

To deal with the commutator in F j2,v , we first use the Taylor formula to get F j2,v

=

T

2j h(2 j (x3 − y3 )) R

| j− |≤5 0

1

× 0

v S −1 ∂3 a 3 (x h , τ y3 + (1 − τ )x3 )dτ · (y3 −x3 )∂3 v b(x h , y3 ) dy3 |vj b dt.

Using (3.11) and integration by parts, we rewrite F j2,v as F j2,v

=

T

| j− |≤5 0

1

× 0

v S −1 a h (x h , τ y3 + (1 − τ )x3 ) dτ · ∇h ∂3 v b(x h , y3 ) dy3 |vj b dt

+

¯ j (x3 − y3 )) h(2

R

T

1

× 0

¯ j (x3 − y3 )) h(2 R

| j− |≤5 0

v S −1 a h (x h , τ y3 + (1 − τ )x3 ) dτ · ∂3 v b(x h , y3 ) dy3 |∇h vj b dt,

¯ 3 ) = x3 h(x3 ). Then Young’s inequality together with Corollary 2.1 and where h(x Lemma 2.4 applied yields v |F j2,v | 2 − j S −1 a h L 4 (L 4 (L ∞ )) ∇h v b L 2 (L 2 ) vj b L 4 (L 4 (L 2 )) T

| j− |≤5

v

h

T

+ v b L 4 (L 4 (L 2 )) ∇h vj b L 2 (L 2 ) T

d 2j − j ν

2

h

a h

− 21 , 21

B4

Let us note that |F j3,v |

v

≤

T

h

v

T

(T )

b2 0, 1 B

| j− |≤5 | j− |≤5

0

T

2 (T )

.

v 3 a ∂3 vj v b|vj b dt.

Then to estimate F j3,v , we need to gain two derivatives from v a 3 . In order to do so, we need to use a particular form of (2.1) as v a 3 (x) = g v (2 (x3 − y3 ))∂3 v a 3 (x h , y3 ) dy3 , (3.12) R

552


ϕ (|ξ3 |) . Substituting (3.12) to F j3,v , we use iξ3 (3.11) then integration by parts in the horizontal variables to get T 3,v v v h v v v Fj ≤ g (2 (x3 − y3 )) a (x h , y3 ) dy3 · ∇h ∂3 j b| j b dt

where g v ∈ S(R) such that F(g v )(ξ3 ) =

| j− |≤5 | j− |≤5

R

0

+

T 0

| j− |≤5 | j− |≤5

v v h v v v dt. g (2 (x − y )) a (x , y ) dy ∂ b|∇ b 3 3 h 3 3 3 j h j R

Together with Young’s inequality, Corollary 2.1 and Lemma 2.4 imply |F j3,v | 2 − v a h L 4 (L 4 (L ∞ )) ∇h vj b L 2 (L 2 ) vj b L 4 (L 4 (L 2 )) T

| j− |≤5 | j− |≤5

d 2j ν

2− j a h

−1,1 B4 2 2 (T )

v

h

b2 0, 1 B

2 (T )

T

T

h

v

.

Finally using (3.12) once again, we rewrite F j4,v as

T v 4,v v v v h v j g (2 (x3 − y3 )) a (x h , y3 ) dy3 · ∇h ∂3 S +2 b j b dt Fj = ≥ j−N0

+ 0

R

0

T v vj g v (2 (x3 − y3 ))v a h (x h , y3 ) dy3 ∂3 S +2 b ∇h vj b dt . R

Young’s inequality applied gives v |F j4,v | a h L 4 (L 4 (L 2 )) (∇h S +2 b L 2 (L 2 (L ∞ )) vj b L 4 (L 4 (L 2 )) T

h

v

≥ j−N0 v +S +2 b L 4 (L 4 (L ∞ )) ∇h vj b L 2 (L 2 ) ), T h v T

T

v

h

T

h

v

which together with Corollary 2.1 and Lemma 2.4 implies |F j4,v |

d 2j ν

2− j a h

−1,1 B4 2 2 (T )

b2 0, 1 B

2 (T )

.

This completes the proof of Proposition 3.3.

4. The proof of the Uniqueness The first step in order to prove the uniqueness part of Theorems 1.2 and 1.3 is the proof of the following regularity theorem. −1,1

− 21 , 12

Theorem 4.1. Let u ∈ B4 2 2 (T ) be a solution of (AN Sν ) with initial data u 0 in B4 Then, if u F is defined by (1.9), we have 1

w = u − u F ∈ B 0, 2 (T ).

.


553

Proof of Theorem 4.1. We already observe at the beginning of Sect. 3 that the vector S ν ), which is field w is the solution of the linear problem ( AN ⎧ ⎨ ∂t w − νh w = −u · ∇u − ∇ p, S ν ) div w = 0, ( AN ⎩ w|t=0 = u h , − 21 , 21

where u h is defined in (3.1). As told by Lemma 2.5, u F belongs to B4 does w. This implies that the only thing we have to prove is

(T ), thus so j

1

(Id −S hj−1 )vj w L ∞ (L 2 ) + ν 2 (Id −S hj−1 )vj ∇h w L 2 (L 2 ) d j 2− 2 . T

T

S ν ), and set In order to do so, let us apply the operator (Id −S hj−1 )vj to the system ( AN w j = (Id −S hj−1 )vj w. def

This gives, by the L 2 energy estimate, t w j (t)2L 2 + 2ν ∇h w j (t )2L 2 dt ≤ vj (u h )2L 2 0 t +2 (Id −S hj−1 )vj (u(t ) · ∇u(t )), w j (t ) dt . 0

Then, using the Fourier-Plancherel theorem, we infer t t 2 2 2j ∇h w j (t ) L 2 dt + cν2 w j (t )2L 2 dt w j (t) L 2 + ν 0 0 t v 2 ≤ j (u h ) L 2 + 2 (Id −S hj−1 )vj (u(t ) · ∇u(t )), w j (t ) dt . 0

Let us observe that, thanks to the divergence free condition, we have u · ∇u m = divh (u m u h ) + ∂3 (u m u 3 ). By integration by part, we get,

(Id −S hj−1 )vj divh (u m u h ), w j ≤ (Id −S hj−1 )vj (u m u h ), ∇h w j ≤ vj (u m u h ) L 2 ∇h w j L 2 ≤

ν C ∇h w j 2L 2 + vj (u m u h )2L 2 . 2 ν

(4.1)

By using Lemma 2.1, we have (Id −S hj−1 )vj ∂3 (u m u 3 ), w j ≤ 2 j vj (u m u 3 ) L 2 w j L 2 ≤ cν22 j w j 2L 2 +

C vj (u m u 3 )2L 2 . ν

(4.2)

554


Then, using inequality (3.2) and Lemma 3.2, we deduce √ 1 − 2j 2 w j L ∞ (L 2 ) + ν∇h w j L 2 (L 2 ) d j 2 u 0 − 1 , 1 + 1 u − 1 , 1 . T T B4 2 2 B4 2 2 (T ) ν2 This concludes the proof of Theorem 4.1.

The above theorem implies that, if u j , j = 1, 2, are two solutions of (AN Sν ) in the − 21 , 21

space B4

belongs to B

def

(T ) associated with the same initial data, then the difference δ = u 2 − u 1 0, 21

(T ). Moreover, it satisfies the following system: ⎧ ⎨ ∂t δ − νh δ = Lδ − ∇ p div δ = 0 (AN Sν ) , ⎩ δ|t=0 = 0

where L is the following linear operator: def

Lδ = −δ∇u 1 − u 2 ∇δ. In order to prove uniqueness, we have to prove that δ ≡ 0. Because the existence of the solution to (AN Sν ) is not proved by using Picard’s fixed point method, the uniqueness −1,1

1

can not be given by a contraction in the space B 0, 2 or even B4 2 2 . As pointed out first by D. Iftimie in [12], the system (AN Sν ) is hyperbolic in the vertical direction. Roughly speaking, for a hyperbolic system, the contraction argument can be realized with one less derivative than the existence space. Here of course, the derivative is lost in the vertical direction. The first idea is the introduction of the homogeneous norm, given in the following definition. Definition 4.1. Let s ∈ R, and let us define the following semi norm: def

a H 0,s =

2

2 js

vj a2L 2

1 2

.

j∈Z

Remark. It is obvious that a2

1

0, 2 L∞ ) T (H

+ ν∇h a2

1

L 2T (H 0, 2 )

a2 0, 1 B

2 (T )

.

(4.3)

The norm · 0,− 1 is not very convenient to work with. In particular, it carries on 2 H information about low frequencies which is not necessarily relevant in the proof of an uniqueness theorem which is by definition a local result. Moreover, there is no evidence that δ belongs to such a space. We shall bypass this problem by the introduction of the inhomogeneous version of the above norm. In order to do it, let us introduce the following notations: v vi vi vi vi vij = vj , S vi j = S j if j ≥ 0 and j = S j+1 = 0 if j ≤ −2, and −1 = S0 .

This leads to the following definition of the norm, which we shall use for a contraction argument.


555

Definition 4.2. Let us denote by H the space of tempered distribution such that def − j a2H = 2 vij a2L 2 < ∞. j∈Z

Now the key point is the estimate of def − j (Lδ|δ)H = 2 (vij (Lδ)|vij δ) L 2 . j∈Z 1

We shall follow mainly [15] up to the fact that the solutions u 1 and u 2 are not in B 0, 2 (T ) − 21 , 12

but only in B4

(T ). This imposes the introduction of the following space.

Definition 4.3. Let us denote by Bu the following (semi) norm: def j−k b2Bu = 2 kh vj a2L 4 (L 2 ) . h

k∈Z j∈N

v

Remark. We obviously have b2L ∞ (Bu ) + ν∇h b2L 2 (B ) b2 − 1 , 1 T

T

u

B4

2 2 (T )

.

(4.4)

Let us state (and admit for the time being) the following variation of Lemma 3.2 of [15]. Lemma 4.1. Let a and b be two divergence free vector fields such that a and ∇h a are 1 in H 0, 2 ∩ H, b is in Bu ∩ L 4h (L ∞ v ) with ∇h b ∈ Bu . Let us assume in addition that a2H ≤ 2−16 . Then we have (b · ∇a|a)H + (a · ∇b|a)H ≤ ν ∇h a2 + C(a, b)µ(a2 ) H H 10 def

with µ(r ) = r (1 − log2 r ) log2 (1 − log2 r ) and b2L 4 (L ∞ )

C h v 2 C(a, b) = b L 4 (L ∞ ) 1 + ν ν2 h v

b4Bu C 2 2 2 2 1 + b2Bu 1 + b + ∇ b + a ∇ a h Bu h 1 1 . Bu ν ν2 H 0, 2 H 0, 2 def

Conclusion of the Proof of Theorem 1.3. We postpone the proof of the fact that 2 δ ∈ L∞ T (H) and ∇h δ ∈ L T (H)

(4.5)

which is a low vertical frequency information on δ. Lemma 4.1 implies that, for any t ∈ [0, T ], t def 2 δ(t)H ≤ f (t )µ(δ(t )2H )dt with f (t) = C(u 1 (t), δ(t)) + C(u 2 (t), δ(t)). 0

Lemma 2.4 and assertions (4.3) and (4.4) imply that f ∈ L 1 ([0, T ]). Then the uniqueness follows from the Osgood Lemma (see for instance [8]). Thus Theorems 1.2 and 1.3 are proved, provided of course that we prove Lemma 4.1 and Assertion (4.5).

556


We start the proof of Lemma 4.1 by the following lemma. Lemma 4.2. A constant C exists such that, for any p ∈ [4, ∞[, we have √

vj b L p (L 2 ) ≤ Cc j v

h

2

j

1− 2

p 2− 2 bBp u ∇h bBu p ,

j ≥ 0.

Proof of Lemma 4.2. By definition of · Bu , using Lemma 2.1, we can write, for any p in [4, ∞[, k 1− 2 j−k j p 2 2 vj b L p (L 2 ) ≤ C 2 2 2 kh vj b L 4 (L 2 ) h

v

v

h

k≤N

+C

2

− 2k p

2

j−k 2

k≥N

≤ CbBu

kh vj ∇h b L 4 (L 2 ) h

2

k 1− 2p

v

ck, j + C∇h bBu

k≤N

2

− 2k p c . k, j

k≥N

Using the Cauchy-Schwarz inequality, we deduce

1 2k 1− 2 1 4k 1 j 2 2 2 − 2 p b ck, 2 + ∇ b 2 p 2 2 vj b L p (L 2 ) ≤ C h Bu Bu j h

v

k

≤C

2 ck, j

1 2

bBu 2

k≤N

N 1− 2p

k≥N

+ ∇h bBu

√

p2

− 2N p

k

√ − 2N

N 1− 2p ≤ Cc j bBu 2 + ∇h bBu p 2 p .

Choosing 2 N ∼

∇h bBu gives the lemma.

bBu

Proof of Lemma 4.1. As the proof is analogous, we only estimate the term (b · ∇a|a)H . We first use Bony’s paradifferential decomposition in the vertical variable and in the inhomogeneous context. This gives b · ∇a = Tb ∇a + R(b, ∇a) with def vi S −1 b · ∇vi Tb ∇a = a and

def

R(b, ∇a) =

vi vi b · ∇ S +2 a.

Step 1. The estimate of (Tb ∇a|a)H . As now usual, we shall treat terms involving vertical derivatives in a different way from terms involving horizontal derivatives. This leads to vij (Tb ∇a) = T jh + T jv with def vi S −1 bh · ∇h vi T jh = vij a, and | j− |≤5

def T jv =

vij

| j− |≤5

vi S −1 b3 ∂3 vi a.


557

By definition of the space H, we get, by using anisotropic Hölder estimates, b L 4 (L ∞ ) ∇h vi T jh 4 a L 2 L h3 (L 2v )

v

h

| j− |≤5

j 2

c j 2 b L 4 (L ∞ ) ∇h aH . h

v

We immediately infer that j

|(T jh |vij a) L 2 | c j 2 2 b L 4 (L ∞ ) vij a L 4 (L 2 ) ∇h aH . h

v

h

v

As we have vij a2L 4 (L 2 ) vij a L 2 ∇h vij a L 2 ,

(4.6)

v

h

we get

3

1

2 2 2− j |(T jh |vij a) L 2 | b L 4 (L 2 ) ∇h aH aH . v

h

j

(4.7)

The estimate of (T jv |vij a) L 2 is more delicate. Let us write that T jv =

3

T jv,n with

n=1

def T jv,1 =

T jv,2

3 vi S vi j−1 b ∂3 j a, def vi = [vij ; S −1 b3 ]∂3 vi a and | j− |≤5

def T jv,3 =

vi 3 vi vi (S −1 b3 − S vi j−1 b )∂3 j a.

| j− |≤5

Step 1a. The estimate of j 2− j |(T jv,1 |vij a) L 2 |. In order to estimate T jv,1 , we use once again the trick from [6]. By integration by parts, we obtain 1 (T jv,1 |vij a) L 2 = − S vi ∂3 b3 vij avij a d x. 2 R3 j−1 Using the divergence free condition and integration by parts in the horizontal variables, we infer 1 v,1 vi (T j | j a) L 2 = S vi divh bh vij avij a d x 2 R3 j−1 h vi vi =− S vi j−1 b · ∇h j a j a d x. R3

As for (4.7), we get

3

h

j

1

2 2 2− j |(T jv,1 |vij a) L 2 | b L 4 (L 2 ) ∇h aH aH . v

(4.8)

558


Step 1b. The estimate of j 2− j |(T jv,2 |vij a) L 2 |. In order to estimate the commutator, f on R4 let us use the Taylor formula. For a function f on R3 , we define the function by 1 def f (x h , x3 + τ (y3 − x3 ))dτ. f (x, y3 ) = 0

def

Then, stating h(x3 ) = x3 h(x3 ), one has v,2 vi Tj = h(2 j (x3 − y3 ))(S −1 ∂3 b3 )(x, y3 )∂3 vi a(x h , y3 )dy3 . | j− |≤5 R

Using that b is divergence free and the fact that ∂h f = (∂ h f ), we infer that vi vi h T jv,2 = − h(2 j (x3 − y3 )) divh (S −1 b )(x, y3 )∂3 a(x h , y3 )dy3 . | j− |≤5 R

Then, by integration by parts with respect to the horizontal variable, we get (T jv,2 |vij a) L 2 = | j− |≤5

+

R4

R4

vi vi vi h h(2 j (x3 − y3 ))(S −1 b )(x, y3 )∂3 ∇h a(x h , y3 ) j a(x)d xd y3

vi b h )(x, y )∂ vi a(x , y )∇ vi a(x)d xd y h(2 j (x3 − y3 ))(S 3 3 h 3 h j 3 . −1

As we have b(x h , ·, y3 ) L ∞ ≤ b(x h , ·) L ∞ , we infer v v v,2 vi v ∂3 ∇h vi (T j | j a) L 2 2− j b L 4 (L ∞ a L 2 j a L 4 (L 2v ) v ) h

h

| − j|≤5

vi + ∂3 vi a L 4h (L 2 ) ∇h j a L 2 vi ∇h vi b L 4 (L ∞ ) a L 2 j a L 4 (L 2 ) . h

v

h

| − j|≤5

Using (4.6), we get that 3 1 2 2 2− j |(T jv,2 |vij a) L 2 | b L 4 (L ∞ ) ∇h aH aH . h

j

v

v

(4.9)

Step 1c. The estimate of j 2− j |(T jv,3 |vij a) L 2 |. The estimate of T jv,3 is based on the following observation. For any divergence free vector field u, we have, from (3.12), v u 3 (x) = g v (2 (x3 − y3 ))v ∂3 u 3 (x h , y3 ) dy3 R = − divh g v (2 (x3 − y3 ))v u h (x h , y3 ) dy3 R

v u h . = −2− divh

(4.10)


559

vi − S vi b3 which appear in T v,3 are a sum of the terms vi If j ≥ 7, then the terms S −1 j j−1 with ≥ 0. Thus, if j ≥ 7, we get, using (4.10) and integration by parts in the horizontal variable,

v bh ∇h vj v ∂3 a vij a (T jv,3 |vij a) L 2 = 2− vj ∈( , j) | − j|≤5

+

vj

v h v v vi b j ∂3 a ∇h j a .

Now, following the lines leading to (4.9), we get 3 1 2 2 2− j |(T jv,3 |vij a) L 2 | b L 4 (L ∞ ) ∇h aH aH . h

j≥7

v

(4.11)

If j ≤ 7, let us simply observe that |(T jv,3 |vij a) L 2 | ≤ b L 4 (L ∞ ) ∇h aH aH . h

v

Plugging this inequality with inequalities (4.7), (4.8), (4.9) and (4.11), we get 3

1

2 2 aH + b L 4 (L ∞ ) ∇h aH aH . |(Tb ∇a|a)H | b L 4 (L ∞ ) ∇h aH h

v

h

v

Using (for θ = 1/4 and θ = 1/2), the convexity inequality 1

1

αβ ≤ θ α θ + (1 − θ )β 1−θ ,

(4.12)

we infer

1 ν C ∇h a2H + b2L 4 (L ∞ ) 1 + 2 b2L 4 (L ∞ ) a2H . (4.13) 100 ν ν h v h v Step 2. The estimate of (R(b, ∇a)|a)H . In the estimate of (R(b, ∇a)|a)H , we have to deal with the fact that the sum of the indices of the vertical regularity is 0. Again, let us treat terms involving vertical derivatives in a different way from terms involving horizontal derivatives. This leads to |(Tb ∇a|a)H | ≤

vij R(b, ∇a) = Rhj + Rvj + R0j with def v v bh · ∇h S +2 a, Rhj = vij ≥( j−N0 )+

Rvj = vij def

≥( j−N0 )+

v v b3 S +2 ∂3 a with

R0j = vij (S0v b · ∇ S2v a). def

Let us first estimate R0j . It is obvious that if j is large enough, this term is 0. Thus, we have

3 1 2 2 |(R0j |vij a) L 2 | b L 4 (L ∞ ) ∇h aH aH + ∇h aH aH h v

1 C ν ≤ ∇h a2H + b2L 4 (L ∞ ) 1 + 2 b2L 4 (L ∞ ) a2H . (4.14) 100 ν ν h v h v

560


Step 2a. The estimate of j 2− j |(Rhj |vij a) L 2 |. We estimate Rhj first using high (vertical) regularity of a. Thanks to Lemma 2.1 and 4.2, we get Rhj

j

22

4 L h3 (L 2v )

≥( j−N0 )+

j

22

≥( j−N0 )+

j

22

v v bh ∇h S +2 a

4

L h3 (L 1v )

v v bh L 4 (L 2 ) ∇h S +2 a L 2 h

v

1 1 c 2 bB2 u ∇h bB2 u ∇h aH .

Then using (4.6), we infer that 1

1

1 2

1 2

j

|(Rhj |vij a) L 2 | bB2 u ∇h bB2 u ∇h aH 2 2 vij a L 4 (L 2 ) h

bBu ∇h bBu ∇h aH a

1 2

v

1

1 H 0, 2

∇h a 2 0, 1 . H

(4.15)

2

Now we shall estimate Rhj using only the fact that a and ∇h a belong to H. Following a key idea of [15], let us write that, for any p in [4, ∞[, we have, using Lemma 2.1 and 4.2, Rhj

j

2p p+2 L h (L 2v )

22

≥( j−N0 )+

j

22

≥( j−N0 )+ j

22

c 2

v v bh ∇h S +2 a

2p p+2

Lh

(L 1v )

v v bh L p (L 2 ) ∇h S +2 a L 2 h

√

v

2

1− 2

p bBp u ∇h bBu p ∇h aH .

By interpolation, a constant C exists (independent of p) such that, for any p in [4, ∞[, we have vij a

1− 2

2p p−2 Lh (L 2v )

2

≤ Cvij a L 2 p vij ∇h a Lp 2 .

Thus we get j

|(Rhj |vij a) L 2 | 2 2

√

cj2j

2

1− 2

1− 2

2

p bBp u ∇h bBu p ∇h aH vij a L 2 p vij ∇h a Lp 2

√

2

1− 2

1− 2

1+ 2

p bBp u ∇h bBu p aH p ∇h aH p .

(4.16)


561

Using the estimates (4.15) and (4.16), we infer that, for any positive integer M and for any p in [4, ∞[,

2− j |(Rhj |vij a) L 2 | =

j

2− j |(Rhj |vij a) L 2 | +

j≤M

2− j |(Rhj |vij a) L 2 |

j>M

1 1 1 1 2− j bB2 u ∇h bB2 u ∇h aH a 2 0, 1 ∇h a 2 0, 1 H

j>M

+

cj

√

2

1− 2

H

2

1− 2

2

1+ 2


0≤ j≤M

Using Cauchy-Schwarz inequality, we obtain

1

1

1

1

2− j |(Rhj |vij a) L 2 | 2−M bB2 u ∇h bB2 u ∇h aH a 2 0, 1 ∇h a 2 0, 1 H

j 2 p

1

1− 2p

1− 2p

H

2

2

1+ 2p

+ ( pM) 2 bBu ∇h bBu aH ∇h aH . Using the convexity inequality (4.12) with θ =

1 2

and with θ =

p+2 respectively, we 2p

deduce

2− j |(Rhj |vij a) L 2 | ≤

j

ν C ∇h a2H + 2−2M bBu ∇h bBu ∇h a 0, 1 a 0, 1 H 2 H 2 10 ν C

+ ν

p+2 p−2

p

4

( pM) p−2 bBp−2 ∇h b2Bu a2H . u

Let us assume that M ≥ 16. As p is in [4, ∞[, we can choose p = log2 M. This gives, for any M ≥ 16,

2− j |(Rhj |vij a) L 2 | ≤

j

ν C ∇h a2H + 2−2M bBu ∇h bBu ∇h a 0, 1 a 0, 1 H 2 H 2 10 ν b4Bu

b4Bu 1+ ∇h b2Bu a2H M log2 M. + ν ν2

If aH ≤ 2−16 , then we can choose M such that 2−M ∼ aH . This gives

2− j |(Rhj |vij a) L 2 | ≤

j

def

C1 (a, b) =

ν ∇h a2H + C1 (a, b)µ(a2H ) with 10

(4.17)

b4Bu

b4Bu C 1+ ∇h b2Bu . bBu ∇h bBu ∇h a 0, 1 a 0, 1 + H 2 H 2 ν ν ν2

562


Step 2b. The estimate of j 2− j |(Rvj |vij a) L 2 |. First of all, let us use (4.10). Together with integration by parts in the horizontal variable, this gives v,2 (Rvj |vij a) L 2 = Rv,1 j (a) + R j (a) with

def − vi v h v vi Rv,1 (a) = 2 ( b · ∇ ∂ S a)| a h 3 +2 j j j ≥( j−N0 )+

Rv,2 j (a) =

def

≥( j−N0 )+

v v bh ∂3 S +2 2− vij ( a)|∇h vij a

L2

L2

and

.

1

Once observed that, for any u ∈ H 0, 2 ∩ H, we have

3

∂3 S v u L 2 c 2 2 uH and ∂3 S v u L 2 c 2 2 u

1

H 0, 2

,

(4.18)

following exactly the lines leading to (4.17), we find ν ∇h a2H + C1 (a, b)µ(a2H ). 2− j |Rv,1 j (a)| ≤ 10

(4.19)

j

0, 2 . Using Lemma 2.1, Now let us estimate Rv,2 j (a) by using that a and ∇h a are in H we get 1

j

v v v bh ∂3 S +2 v bh ∂3 S +2 vij ( a) L 2 2 2 a L 2 (L 1 ) h

2

j 2

v

v v bh 4 2 ∂3 S +2 a L 4 (L 2 ) . L h (L v ) h v

Using (4.18), we infer 1

1

v ∂3 S +2 a L 4 (L 2 ) c 2 2 ∇h a 2 0, 1 a 2 0, 1 . v

h

H

H

2

2

Lemma 4.2 applied with p = 4 then leads to

1 1 1 1 j j − bB2 u ∇h bB2 u a 2 0, 1 ∇h a 2 0, 1 2− 2 vij ∇h a L 2 |Rv,2 (a)| 2 2 j H

≥ j−N0 1

1

1

H

2

2

1

bB2 u ∇h bB2 u a 2 0, 1 ∇h a 2 0, 1 ∇h aH . H

H

2

(4.20)

2

Then let us estimate |Rv,2 j (a)| using the fact that a and ∇h a belong to H. Lemma 4.2 applied for any p ∈ [4, ∞[ together with (4.18) gives j

v v v b p 2 ∂3 S +2 a) L 2 2 2 a vij (v bh ∂3 S +2 L (L ) h

j

2 2 d

√

v

2 p

1− 2p

2p p−2

Lh

(L 2v )

1− 2

Thus, we deduce that j |Rv,2 j (a)| c j 2

√

2

2

p p bBu ∇h bBu aH p ∇h aH .

1− 2

1− 2

1+ 2



563

Then using (4.20) and following exactly the lines leading to (4.17), we get ν ∇h a2H + C1 (a, b)µ(a2H ). 2− j |(Rvj |vij a) L 2 | ≤ 10

(4.21)

j

This proves Lemma 4.1.

Proof of Assertion (4.5). Let us write that S0v δ is a solution (with initial value 0) of ∂t S0v δ − νh S0v δ = g1 + g2 + g3 with def v S0 ∂3 (aλ bλ ) g1 = λ∈

def

g2 =

S0v ∂h (cλ (Id −S0v )δ) and

λ∈

g3 = S0v ∂h def

dλ S0v δ,

λ∈ − 21 , 21

where is a finite set of indices and aλ , bλ , cλ and dλ belong to B4 Lemmas 2.1 and 3.2, we get that S0v ∂3 (aλ bλ ) L 2 (L 2 ) 2 j vj (aλ bλ ) L 2 (L 2 ) T

(T ). Using

T

j≤−1

1 1

ν2 1

1

ν2

j 2 2 aλ

aλ

− 21 , 21

B4

j≤−1 − 21 , 21

B4

(T )

bλ

(T )

− 21 , 21

B4

bλ

− 21 , 21

B4

(T )

(T )

.

Thus we have that 1 2 def g1 L 2 (L 2 ) 1 C12 (T ) with C12 (T ) = u 1 − 1 , 1 + u 2 − 1 , 1 . T B4 2 2 (T ) B4 2 2 (T ) 2 ν

(4.22)

We estimate g2 using Lemma 2.4. It claims in particular that 1 − j

1 2 2 δ − 1 , 1 1 δ − 1 , 1 . (Id −S0v )δ L 4 (L 4 (L 2 )) 1 2 2 (T ) T h v B B4 2 2 (T ) ν 4 j≥0 ν4 4 Lemma 2.4 also claims that cλ L 4 (L 4 (L ∞ )) T

1 1

v

h

ν4

cλ

− 21 , 21

B4

(T )

. Then we have

cλ (Id −S0v )δ L 2 (L 2 ) cλ L 4 (L 4 (L ∞ )) (Id −S0v )δ L 4 (L 4 (L 2 )) T

T

1 1

ν2

h

cλ

v

− 21 , 21

B4

T

(T )

δ

− 21 , 21

B4

(T )

h

v

.

This gives that g2 with g2 L 2 (L 2 ) g2 = divh

1

2 C12 (T ).

(4.23) ν The term g3 must be treated with a commutator argument based on the following lemma. T

1 2

564


Lemma 4.3. Let χ be a function of S(R). A constant C exists such that, for any function a in L 2h (L ∞ v ), we have 1

[χ (εx3 ); S0v ]a L 2 ≤ Cε 2 a L 2 (L ∞ ) . h

v

Proof of Lemma 4.3. Taylor’s formula at order one gives Cε (a)(x h , x3 ) = [χ (εx3 ); S0v ]a(x h , x3 ) = ε h(x3 − y3 )χ (ε((1 − τ )x3 + τ y3 )) a(x h , y3 )dy3 dτ. def

R ×[0,1]

Then, using Cauchy-Schwarz inequality for the measure |h(x3 − y3 )|d x3 dy3 dτ on R2 ×[0, 1], let us write that 2 2 2 2 Cε (a)(x h , ·) L 2 ≤ ε a(x h , ·) L ∞ sup |h(x3 − y3 )|ϕ (x3 )d x3 dy3 (H1ε + H2ε ) v v

ϕ L 2 (R) ≤1

R2

(H1ε + H2ε ), ≤ Cε2 a(x h , ·)2L ∞ v where we define H1ε and H2ε by ε def H1 = (χ )2 (ε((1 − τ )x3 + τ y3 )) |h(x3 − y3 )|d x3 dy3 dτ and R2 ×[0, 21 ]

H2ε =

def

R2 ×[ 21 ,1]

(χ )2 (ε((1 − τ )x3 + τ y3 )) |h(x3 − y3 )|d x3 dy3 dτ.

Changing variables xτ = (1 − τ )x3 + τ y3 yτ = y3

in

H1ε

and

xτ = x3 yτ = τ y3 + (1 − τ )x3

in H2ε

gives H1ε

=

H2ε =

R2 ×[0, 21 ] R2 ×[ 21 ,1]

x − y 1 τ τ (χ )2 (εxτ )h d xτ dyτ dτ and 1−τ 1−τ x − y 1 2 τ τ (χ ) (εyτ )h d xτ dyτ dτ. τ τ 1

and the lemma is We immediately infer that Cε (a)(x h , ·) L 2v ≤ Cε 2 a(x h , ·) L ∞ v proved.

def

v a = χ (ε·)S v a. Now let us choose χ ∈ D(R) with value 1 near 0 and let us state S0,ε 0 We get, through a classical L 2 energy estimate and the convexity inequality (4.12) that t t v v v S0,ε δ(t)2L 2 + ν ∇h S0,ε δ(t )2L 2 dt ≤ 2 g1 (t ) L 2 S0,ε δ(t ) L 2 dt 0 0 t 1 t 2 v + g2 (t ) L 2 dt + 2 χ (ε·)g3 (t ), S0,ε δ(t )dt . ν 0 0


565

By definition of g3 , the integrand in the last term of the above equality is a finite sum of terms of the type v δ Dλ = χ (ε·)S0v (dλ S0v δ), ∂h S0,ε

def

− 21 , 12

with dλ ∈ B4

(T ). Writing that Dλ = Dλ1 + Dλ2 with

v v v δ and Dλ2 = S0v (dλ S0,ε δ), ∂h S0,ε δ, Dλ1 = [χ (ε·); S0v ](dλ S0v )δ, ∂h S0,ε

def

def

Lemmas 2.4 and 4.3 imply that t 1 2 |Dλ1 (t )|dt ε 2 C12 (t)∇h S0,ε δ L 2 (L 2 ) t

0

ν C 4 v ≤ ∇h S0,ε δ L 2 (L 2 ) + εC12 (t). t 4 ν

Then let us write that 1

3

v v |Dλ2 (t)| dλ (t) L 4 (L 2 ) S0,ε δ(t) L2 2 ∇h S0,ε δ(t) L2 2 h

v

ν C v v ≤ ∇h S0,ε (t)2L 2 + 3 dλ (t)4L 4 (L ∞ ) S0,ε (t)2L 2 . 4 ν h v Using (4.22) we get v S0,ε δ(t)2L 2 +

ν 2

t

v ∇h S0,ε δ(t )2L 2 dt 0 t C 4 ≤ (ε + 1)C12 (T ) + C g1 (t )2L 2 dt ν 0 t

1 4 v 1 + 3 u 1 L 4 (L ∞ ) + u 2 4L 4 (L ∞ ) S0,ε +C δ(t )2L 2 dt . v v ν h h 0

The Gronwall Lemma together with (4.22) gives ν t v v δ(t)2L 2 + ∇h S0,ε δ(t )2L 2 dt S0,ε 2 0 t

1 C 4 4 4 1 + 3 u 1 L 4 (L ∞ ) + u 2 L 4 (L ∞ ) dt ≤ (ε + 1)C12 (T ) exp C ν ν h v h v 0 and thus by Lemma 2.4

ν t C 1 4 v 2 v 4 S0,ε δ(t) L 2 + ∇h S0,ε δ(t )2L 2 dt ≤ (ε + 1)C12 (T ) exp C 1 + 3 C12 (T ) . 2 0 ν ν Passing to the limit when ε tends to 0 allows to conclude the proof of Assertion (4.5).

Acknowledgements. This work was done when Ping Zhang was visiting the Centre de Mathématiques Laurent Schwartz at École Polytechnique. He would like to thank the Department for its hospitality. Zhang is partially supported by NSF of China under Grant 10525101 and 10421101, and the innovation grant from Chinese Academy of Sciences.

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References 1. Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales de l’École Normale Supérieure 14, 209–246 (1981) 2. Cannone, M., Meyer, Y., Planchon, F.: Solutions autosimilaires des équations de Navier-Stokes, Séminaire “Équations aux Dérivées Partielles de l’École Polytechnique”, Exposé VIII, 1993–1994 3. Chemin, J.-Y.: Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. Journal d’Analyse Mathématique 77, 27–50 (1999) 4. Chemin, J.-Y.: Localization in Fourier space and Navier-Stokes system. Phase Space Analysis of Partial Differential Equations, Proceedings 2004, CRM series, Pisa: Centro Edizioni, Scunla Normale Superiore, pp. 53–136 5. Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics. An introduction to rotating fluids and the Navier-Stokes equations. Oxford Lecture Series in Mathematics and its Applications, 32. The Clarendon Press, Oxford, 2006. 6. Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Fluids with anisotropic viscosity. M2AN Math. Model. Numer. Anal. 34, 315–335 (2000) 7. Ekman, V.-W.: On the influence of the earth’s rotation on ocean currents. Arkiv. Matem. Astr. Fysik (Stockholm) 2(11), 1–52 (1905) 8. Fleet, T.-M.: Differential Analysis. Cambridge:, Cambridge University Press, 1980 9. Fujita, H., Kato, T.: On the Navier-Stokes initial value problem I. Archiv Rat. Mech. Anal. 16, 269– 315 (1964) 10. Grenier, E., Masmoudi, N.: Ekman layers of rotating fluids, the case of well prepared initial data. Commun. Partial Diff. Eqs. 22, 953–975 (1997) 11. Iftimie, D.: The resolution of the Navier-Stokes equations in anisotropic spaces. Rev. Mat. Iberoamericana 15, 1–36 (1999) 12. Iftimie, D.: A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity. SIAM J. Math. Anal. 33, 1483–1493 (2002) 13. Pedlovsky, J.: Geophysical Fluid Dynamics. Berlin-Heidelberg-NewYork: Springer, 1979 14. Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. in Math. 157, 22–35 (2001) 15. Paicu, M.: Équation anisotrope de Navier-Stokes dans des espaces critiques. Rev. Mat. Iberoamericana 21, 179–235 (2005) 16. Vishik, M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Annales de l’École Normale Supérieure 32, 769–812 (1999) Communicated by P. Constantin


Communications in


Approximating Multi-Dimensional Hamiltonian Flows by Billiards A. Rapoport1 , V. Rom-Kedar1 , D. Turaev2 1 Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel.

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

2 Department of Mathematics, Ben-Gurion University, P.O.B. 653, Beèr Sheva 84105, Israel.

E-mail: [email protected] Received: 24 November 2005 / Accepted: 25 August 2006 Published online: 13 April 2007 – © Springer-Verlag 2007

Abstract: The behavior of a point particle traveling with a constant speed in a region D ⊂ R N , undergoing elastic collisions at the regions’s boundary, is known as the billiard problem. Various billiard models serve as approximation to the classical and semiclassical motion in systems with steep potentials (e.g. for studying classical molecular dynamics, cold atom’s motion in dark optical traps and microwave dynamics). Here we develop methodologies for examining the validity and accuracy of this approximation. We consider families of smooth potentials V , that, in the limit → 0, become singular hard-wall potentials of multi-dimensional billiards. We define auxiliary billiard domains that asymptote, as → 0 to the original billiards, and provide, for regular trajectories, asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C r norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials that limit to the multi-dimensional close to ellipsoidal billiards, we predict when the billiard’s separatrix splitting (which appears, for example, in the nearly flat and nearly oblate ellipsoids) persists for various types of potentials.

1. Introduction Imagine a point particle travelling freely (without friction) on a table, undergoing elastic collisions with the edges of the table. This model resembles a game of billiards, but it looks much simpler - we have only one ball, which is a dimensionless point particle. There is no friction and the table has no pockets. The shape of the table determines the nature of the motion (see [26] and references therein) – it can be ordered (integrable, e.g. in ellipsoidal tables), ergodic (e.g. in generic polygons), mixing and possessing K and B-properties (in dispersing-Sinai [50] tables or focusing-Bunimovich tables [7]), or of a mixed nature (most billiard shapes). While the above dynamical classification of two-dimensional billiards is well established, the statistical properties of such systems

568

A. Rapoport, V. Rom-Kedar, D. Turaev

(e.g. the rate of the decay of correlations) is still widely open and is non-trivial even for mixing systems, like Sinai and Bunimovich billiards, see [36, 60] and references therein. While two-dimensional billiards had been extensively studied, much less in known regarding the dynamics in higher dimensional billiards; motivated by Boltzmann ergodic hypothesis, the geometrical and dynamical properties of a hard sphere gas had been the focuss of a series of works; indeed the motion of N rigid d-dimensional balls in a d-dimensional box (d = 2 or 3) corresponds to a billiard problem in an n-dimensional domain, where n = N × d and the domain’s boundary is formed by a union of cylinders [13, 31, 49]. The corresponding billiards are called semi-dispersing, and their ergodic properties had been extensively studied [27–30, 45–48]. On the other extreme, there exist several studies of integrable billiards in ellipsoids and their perturbations (see below), and several studies of ergodic and/or hyperbolic multi-dimensional billiards [8–11, 39, 59] (the nature of the singularity set in the higher dimensional case is a delicate issue [2]). Multi-dimensional billiards with mixed phase space were also studied in [11, 59] and in [61]. In [40] a semiclassical study of the three-dimensional Sinai billiard had been conducted. Recently, the hyperbolicity of certain finite range multi-dimensional spherically symmetric billiard potentials was proved [4] (see below). In the context of Physics, the billiard description is usually used to model a more complicated system for which a particle is moving approximately inertially, and then is reflected by a steep potential. The reduction to the billiard problem simplifies the analysis tremendously, often allowing to describe completely the dynamics in a given geometry. Numerous applications of this idea appear in the physics literature; it works as an idealized model for the motion of charged particles in a steep potential, a model which is often used to examine the relation between classical and quantized systems (see [22, 52] and references therein); this approximation was utilized to describe the dynamics of the motion of cold atoms in dark optical traps (see [24] and references therein); this model has been suggested as a first step for substantiating the basic assumption of statistical mechanics – the ergodic hypothesis of Boltzmann ([31, 49–51, 53, 54]). The opposite point of view may be taken when one is interested in studying numerically the hard wall system in a complicated geometry (e.g. apply ideas of [37] to [38]) – then designing the “correct” limiting smooth Hamiltonian may simplify the complexity of the programming. The first strategy for studying the effects of the soft potentials was to introduce finite range axis-symmetric potentials (potentials which identically vanish away from a set of circular scatterers and are axisymmetric at the scatterers). For the two-dimensional case [1, 3, 17, 19, 25, 32, 33, 35, 49], it was shown that a modified billiard map may be defined, and several works have utilized this modified map to prove ergodicity of some configurations [3, 19, 32, 33, 49], or to prove that other configurations may possess stability islands [1, 17]. More recently [4], it was shown that a similar strategy may be employed in arbitrary finite dimensions, and, moreover, that such an approach together with a careful study of the cone properties of the multi-dimensional modified billiard map, may be utilized to prove hyperbolicty of such a flow (under suitable conditions on the spherically symmetric finite range potentials and the spacing between the scatterers). The more general problem of studying the limiting process of making a steep twodimensional potential steeper up to the hard-wall limit can be approached in a variety of ways. In [37] approach based on generalized functions was proposed. In [56] we developed a different paradigm for studying this problem. We first formulated a set of conditions on general smooth steep potentials in two-dimensional domains (C r -smooth, not necessarily of finite range, nor axis-symmetric) which are sufficient for proving that

Approximating Multi-Dimensional Hamiltonian Flows by Billiards

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regular reflections of the billiard flow and of the smooth flow are close in the C r topology. This statement, which may appear first as a mathematical exercise, is quite powerful. It allows to prove immediately the persistence of various kinds of billiard orbits in the smooth flows (see [56] and Theorem 5 in Sect. 3.4) and to investigate the behavior near singular orbits (e.g. orbits which are tangent to the boundary) by combining several Poincaré maps, see for example [12, 43, 57]. The first part of this paper (see Theorems 1-2) is a generalization of this result to the multi-dimensional case in the most general geometrical setting. Thus, it appears that the Physicists’ approach, of approximating the smooth flow by a billiard has some mathematical justification. How good is this approximation? Can this approach be used to obtain an asymptotic expansion to the smooth solutions? The second part of this paper answers these questions. We propose an approximation scheme, with a constructive twist - we show that the best zero-order approximation should be a billiard map in a slightly distorted domain. We provide the scaling of the width of the corresponding boundary layer with the steepness parameter and with the number of derivatives one insists on approximating. Furthermore, the next order correction is explicitly found, supplying a modified billiard map (reminiscent of the shifted billiard map of [4, 17, 49]) which may be further studied. We believe this part is the most significant part of the paper as it supplies a constructive tool to study the difference between the smooth flow and the billiard flow. In the last part of this paper we demonstrate how these tools may be used to instantly extend novel results (that were obtained for billiards) to the steep potential setting; it is well known that the billiard map is integrable inside an ellipsoid [26]. Moreover, the Birkhoff-Poritski conjecture claims that in 2 dimensions among all the convex smooth concave billiard tables only ellipses are integrable [55]. In [58] this conjecture was generalized to higher dimensions. Delshams et al ([15, 14] see references therein) studied the effect of small entire symmetric perturbations to the ellipsoid shape on the integrability. They proved that in some cases (nearly flat and nearly oblate) the separatrices of a simple periodic orbit split; thus, they proved a local version of the Birkhoff conjecture in the two-dimensional setting, and provided several non-integrable models in the n-dimensional case. Here, we show that a simple combination of their results with ours, extends their result to the smooth case - namely it shows that the Hamiltonian flow, in a sufficiently steep potential which asymptotically vanishes in a shape which is a small perturbation of an ellipsoid, is chaotic. Furthermore, we quantify, for a given perturbation of the ellipsoidal shape, what “sufficiently steep” means for exponential, Gaussian and power-law potentials. These results may give the impression that the smooth flow and the billiard flow are indeed very similar, and so a Scientist’s dream of greatly simplifying a complicated system is realized here. In the discussion we go back to this point - as usual dreams never materialize in full. In particular, we discuss there the possibility of having non-hyperbolic orbits and even effective stability islands in the case of steep repelling multi-dimensional potentials. The paper is ordered as follows; in Sect. 2 we define and describe the billiard flow and billiard map. In Sect. 3 we study the smooth Hamiltonian flow; we first prove that if the potential satisfies some natural conditions, the smooth regular reflections will limit smoothly to the billiard’s regular reflections (Theorems 1, 2). Then, we define a natural Poincaré section on which a generalized billiard map may be defined for the smooth flow. Next, we derive the correction term to the zeroth order billiard approximation (Theorem 3) and calculate it for three model potentials (exponential, Gaussian and power-law). We

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end this section by stating its immediate implication - a persistence theorem for various types of trajectories (Theorem 5). In Sect. 4 we apply these results to the perturbed ellipsoidal billiard. We end the paper with a short summary and discussion. The appendices contain most of the proofs, whereas in the body of the paper we usually only indicate their main steps. 2. Billiards in d Dimensions 2.1. The billiard flow. Consider a billiard flow as the motion of a point mass in a compact domain D ⊂ Rd or Td . Assume that the boundary ∂ D consists of a finite number of C r +1 smooth (r ≥ 1) (d − 1) -dimensional submanifolds: ∂ D = 1 ∪ 2 ∪ ... ∪ n ,

i = 1 . . . n.

(1)

The boundaries of these submanifolds, when these exist, form the corner set of ∂ D: ∗ = ∂1 ∪ ∂2 ∪ ... ∪ ∂n , i = 1 . . . n.

(2)

The moving particle has a position q ∈ D and a momentum vector p ∈ Rd which are functions of time. If q ∈ int (D), then the particle moves freely with a constant velocity according to the rule1 : q˙ = p . (3) · p=0 Equation (3) is Hamiltonian with the Hamiltonian function (hereafter p 2 = p, p) H (q, p) =

p2 . 2

(4)

The particle moves at a constant speed and bounces of ∂ D according to the usual elastic reflection law : the angle of incidence is equal to the angle of reflection. This means that the outgoing vector pout is related to the incoming vector pin by pout = pin − 2 pin , n(q)n(q),

(5)

where n(q) is the inward unit normal vector to the boundary ∂ D at the point q, see [13]. To use the reflection rule (5), we need the normal vector n(q) to be defined, hence the rule cannot be applied at points q ∈ ∗ , where such a vector fails to exist2 . Definition 1. The domain D is called the configuration space of the billiard system. The phase space of the system is P = D × S d−1 , where S d−1 is a (d −1)-dimensional unit sphere (we set H = 21 ) of velocity vectors. So the elements of P are ρ ≡ (q, p). 1 We assume that the particle has mass one (otherwise one may rescale time). 2 To be precise, one may define n(q) by continuity at points of ∗ , but this might give more than one

normal vector n(q), hence the dynamics would be multiply defined for a generic corner. We adopt the standard convention that the reflection is not defined at any q ∈ ∗ .


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Denote the time t map of the billiard flow as bt : ρ0 → ρt .

(6)

We do not consider reflections at the points of the corner set, so ρt = bt ρ0 implies here that the distance between any point on the trajectory connecting q0 with qt and the set ∗ is bounded away from zero. A point ρ ∈ P is called an inner point if q ∈ / ∂ D and a collision point if q ∈ ∂ D \ ∗ . Obviously, if ρ0 and ρt = bt ρ0 are inner points, then ρt depends continuously on ρ0 and t. If ρt is a (non-tangent) collision point then the velocity vector undergoes a jump. Thus, in this case both bt−0 and bt+0 are defined. The −1 is the reflection law (5) (augmented by qout = qin ). map R◦ = bt+0 bt−0 If the piece of the trajectory which connects q0 with qt does not have tangencies with the boundary, then ρt depends C r -smoothly on ρ0 . It is well-known [50, 56] that the map bt loses smoothness at any point q0 whose trajectory is tangent to the boundary at least once on the interval (0, t). Clearly a tangency may occur only if the boundary is concave in the direction of motion at the point of tangency. Consider hereafter only non-degenerate tangencies, namely assume that the curvature in the direction of motion does not vanish. Choose local coordinates q = (x, y) in such a way that the origin corresponds to the collision point, the y-axis is normal to the boundary and looking inside the billiard region D, and the x-coordinates (x ∈ Rd−1 ) correspond to the directions tangent to the boundary. If Q(x, y) = 0 is the equation of the boundary in these coordinates, then Q y (0, 0) = 0 and Q x (0, 0) = 0. We choose the convention that Q y (0, 0) > 0. Obviously, the tangent trajectory is characterized by the condition p y = 0, where ( px , p y ) are the components of the momentum p. The vector px = x˙ indicates the direction of motion of the tangent trajectory. It is easy to check that the tangency is non-degenerate if and only if pxT Q x x (0, 0) px > 0.

(7)

If the billiard boundary is strictly concave (strictly dispersing), then all the tangencies are non-degenerate. On the other hand, if the billiard’s boundary has saddle points (or if the billiard is semi-dispersing), then there always exist directions for which this non-degeneracy assumption fails. Let x = (x1 , . . . , xd−1 ) with x1 corresponding to the direction of motion (i.e. px = (1, 0, . . . , 0)). Then, the boundary surface near the point of non-degenerate tangency is described by the following equation: y = −αx12 + O(z 2 , x1 z),

α > 0,

where we denote z = (x2 , . . . , xd−1 ). It is easy to see now that for a non-degenerate tangency, for a small τ the map bτ of the line ρ0 = (x0 = (−τ/2, 0, . . . , 0), y0 ≤ 0, p0x = (1, 0, . . . , 0), p0y = 0) is given by √ ρτ = ((τ/2, 0, . . . , 0) + O(y0 ), 2τ −αy0 + O(y0 ), √ (1, 0, . . . , 0) + O(y0 ), 4 −αy0 + O(y0 )). As we see, the billiard flow loses smoothness indeed (it has a square-root singularity in the limit y → −0) near the tangent trajectory. See Fig. 1.

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Fig. 1. Singularity near a tangent trajectory. For better visualization we present a slanted hyperplane which is divided into 2 parts: bτ has a square-root singularity on the boundary between A R and A S

2.2. The billiard map. It is standard in dynamical system theory to reduce the study of flows to maps by constructing a cross-section. The latter is a hypersurface transverse to the flow. For the flow bt , such a hypersurface in phase space P can be naturally constructed with the help of the boundary of D, i.e. the natural cross-section S corresponds exactly to the collision points of the flow with the domain’s boundary: S = {ρ = (q, p) ∈ P : q ∈ ∂ D, p, n(q) ≥ 0}.

(8)

This is a (2d − 2)-dimensional submanifold in P. Any trajectory of the flow bt crosses S every time it reflects at ∂ D. This defines the Poincaré map B : S → S such that Bρ = bτ◦ (ρ)+0 ρ,

(9)

where τ◦ (ρ) = min{t > 0 : bt+0 ρ ∈ S}. Definition 2. The map B is called the billiard map. We propose to represent the billiard map as a composition of a free-flight and a reflection: B = R◦ ◦ F◦ , where the free-flight map is given by F◦ (q, p) = bτ◦ (ρ)−0 (q, p),

(10)

and the reflection law is given by R◦ (q, p) = (q, p − 2 p, n(q)n(q)). The billiard map B is a C r −diffeomorphism at all points ρ ∈ S \ such that Bρ ∈ S \, where is the singular set cor ner s = {(q, p) ∈ S : p, n(q) = 0} ∪ {(q, p) ∈ S : q ∈ ∗ }, = tangencies (11) and B is

C0

at the non-degenerate tangent trajectories.


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3. Smooth Hamiltonian Approximation 3.1. Setup and conditions on the potential. Consider the family of Hamiltonian systems associated with: H=

p2 + V (q; ), 2

(12)

where the C r +1 -smooth potential V (q; ) tends to zero inside a region D as → 0, and it tends to infinity (or to a constant larger than the fixed considered energy level, say H = 21 ) outside. Formally, the billiard flow in D may be expressed as a limiting Hamiltonian system of the form: Hb = where

Vb (q) =

p2 + Vb (q), 2

0 q ∈ int (D) . +∞ q ∈ / D

(13)

(14)

Let us formulate conditions under which this simplified billiard motion approximates the smooth Hamiltonian flow. In the two-dimensional case these conditions were introduced in [56]. Condition I. For any fixed (independent of ) compact region K ⊂ int (D) the potential V (q; ) diminishes along with all its derivatives as → 0: lim V (q; )|q∈K C r +1 = 0.

→0

(15)

The growth of the potential near the boundary for sufficiently small values needs to be treated more carefully. We assume that the level sets of V may be realized by some finite function near the boundary. Namely, let N ( ∗ ) denote the fixed (independent of ) neighborhood of the corner set and N (i ) denote the fixed neighborhood of the boundary component i (in the R d topology); define N˜ i = N (i )\N ( ∗ ) (we assume that N˜ i ∩ N˜ j = ∅ when i = j). Assume that for all small ≥ 0 there exists a pattern function Q(q; ) : N˜ i → R1 i

which is C r +1 with respect to q in each of the neighborhoods N˜ i and it depends continuously on (in the C r +1 -topology, so it has, along with all derivatives, a proper limit as → 0). See Fig. 2. Further assume that in each of the neighborhoods N˜ i the following is fulfilled. Condition IIa. The billiard boundary is composed of level surfaces of Q(q; 0)3 : Q(q; = 0)|q∈i ∩ N˜ i ≡ Q i = constant.

(16)

In the neighborhood N˜ i of the boundary component i (so Q(q; ) is close to Q i ), define a barrier function Wi (Q; ), which is C r +1 in Q, continuous in and does not depend explicitly on q, and assume that there exists 0 such that 3 This is the Q(x, y; 0) defined in Sect. 2.1.

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Fig. 2. Level sets of a pattern function Q(q; ). A bold line is a trajectory of the Hamiltonian flow near the boundary; a solid is a billiard trajectory

Condition IIb. For all ∈ (0, 0 ] the potential level sets in N˜ i are identical to the pattern function level sets and thus: V (q; )|q∈ N˜ i ≡ Wi (Q(q; ) − Q i ; ),

(17)

and Condition IIc. For all ∈ (0, 0 ], ∇V does not vanish in the finite neighborhoods of the boundary surfaces, N˜ i , thus: ∇ Q|q∈ N˜ i = 0

(18)

d Wi (Q − Q i ; ) = 0. dQ

(19)

and for all Q(q; )|q∈ N˜ i

Now, the rapid growth of the potential across the boundary may be described in terms of the barrier functions alone. Note that by (18), the pattern function Q is monotone across i ∩ N˜ i , so either Q > Q i corresponds to the points near i inside D and Q < Q i corresponds to the outside, or vice versa. To fix the notation, we will adopt the first convention. Condition III. There exists a constant (may be infinite) E > 0 such that as → +0 the barrier function increases from zero to E across the boundary i : 0, Q > Q i lim W (Q; ) = . (20) E, Q < Q i →+0 By (19), for small , Q could be considered as a function of W and near the boundary: Q = Q i + Q i (W ; ). Condition IV states that for small a finite change in W corresponds to a small change in Q: Condition IV. As → +0, for any fixed W1 and W2 such that 0 < W1 < W2 < E, for each boundary component i , the function Q i (W ; ) tends to zero uniformly on the interval [W1 , W2 ] along with all its (r + 1) derivatives.


575

1 0.9 ε=0.1

0.8

W(Q;ε)

0.7 0.6 ε=0.01

0.5 0.4 0.3 0.2 0.1 ε=0.001

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Q

Fig. 3. Gaussian potential given near the boundary by W (Q; ) = e−

Q2

satisfies Conditions I-IV

Figure 3 shows the geometric interpretation of the pattern function and a typical dependence of the barrier function on Q and . The use of the pattern and barrier functions essentially reduces the d-dimensional Hamiltonian dynamics in arbitrary geometry to a one-dimensional dynamics, thus allowing direct asymptotic integration of the smooth problem. This is the main tool, introduced first in [56] for the two-dimensional case, which enables us to deal with arbitrary geometry and dimension. 3.2. C0 and Cr - closeness theorems. Theorem 1. Let the potential V (q; ) in (12) satisfy Conditions I-IV. Let h t be the Hamiltonian flow defined by (12) on an energy surface H = H ∗ < E (with positive H ∗ ), and bt be the billiard flow in D. Let ρ0 and ρT = bT ρ0 be two inner phase points. Assume that on the time interval [0, T ] the billiard trajectory of ρ0 has a finite number of collisions, and all of them are either regular reflections or non-degenerate tangencies. Then h t ρ −→ bt ρ, uniformly for all ρ close to ρ0 and all t close to T . →0

Theorem 2. In addition to the conditions of Theorem 1, assume that the billiard trajectory of ρ0 has no tangencies to the boundary on the time interval [0, T ]. Then h t −→ bt →0 in the C r -topology in a small neighborhood of ρ0 , and for all t close to T . Recall that we speak here about trajectories which do not visit corner points, i.e. all the collisions are on ∪i i ∩ N˜ i . The proof of the theorems follows closely their proof for the two-dimensional case in [56]. However, as minor corrections needed to be introduced, we presented the proof in the supplement [42]. Most of it deals with the analysis of the equations of motion in the boundary layers N˜ i . Informally, the logic behind Conditions I-IV is as follows. Condition I implies that the particle moves with almost constant velocity (along a straight line) in the interior of D until it reaches a thin layer near the boundary where V runs

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from zero to large values (a smaller corresponds to a thinner boundary layer). Note that the boundary layer can not be fully penetrated by the particle. Indeed, as in all mechanical Hamiltonians, the energy level defines the region of allowed motion: for a fixed energy level H = H ∗ < E, all trajectories stay in the region V (q; ) ≤ H ∗ . It follows from Condition III that for any such H ∗ , the region of allowed motion approaches D as → 0. Thus, by Condition III, if the particle enters the layer near a boundary surface (note that points from ∗ are not considered in this paper), it has, in principle, two possibilities. First, it may be reflected and then exit the boundary layer near the point it entered. The other possibility, which we want to avoid, is that the particle sticks to the boundary and travels along it far from the entrance point. It is shown in the proof of Theorem 1 that Condition IV guarantees that if the reflection is regular, or if it is tangent and the tangency is non-degenerate, then the travel distance along the boundary vanishes asymptotically with . The case of degenerate tangencies is important but it is not studied here; notice that degenerate tangencies cannot occur in the strictly dispersing case, yet these are unavoidable if the boundary has directional curvatures of opposite signs (saddle points) or in the semi-dispersing case. Once we know that the time spent by the particle near the boundary is small, we can see that Condition II guarantees that the reflection will be of the right character, namely the smooth reflection is C 0 -close to that of the billiard. Indeed, Condition II implies that the reaction force is normal to the boundary, hence, as the time of collision is small and the position of the particle does not change much during this time, the direction of the force stays nearly constant during the collision. Thus, only the normal component of the momentum is changing sign while the tangent components are nearly preserved. Computations along these lines provide a proof of Theorem 1. Proving Theorem 2, i.e. the C r -closeness, makes a substantial use of Condition IV. Let us explain in more detail the difference between the C 0 and C r topologies in this context. Take the same initial condition (q0 , p0 ) for a billiard orbit and for an orbit of the Hamiltonian system (12) (the Hamiltonian orbit will be called the smooth orbit). Consider a time interval t for which the billiard orbit collides with the boundary only once. In these notations ϕin is the angle between p0 (the momentum at the point q0 ) and the normal to the boundary at the collision point, ϕout is the angle between pt (the velocity vector at the point qt ) and the normal. Define the incidence and reflection angles (ϕin () and ϕout ()) for the smooth trajectory in the same way. Theorem 1 implies the correct reflection law for smooth trajectories: ϕin () + ϕout () ≈ 0

(21)

for sufficiently small . However, ϕin + ϕout is a function of the initial conditions, so a non-trivial question is when it is close to zero along with all its derivatives. In Theorem 2 we prove that Condition IV is sufficient for guaranteeing the correct reflection law in the C r -topology in the case of non-tangent collision (near tangent trajectories the derivatives of the smooth flow cannot converge to those of the billiard because the billiard flow is singular there, see Fig. 1). Hereafter, we will fix the energy level of the Hamiltonian flow to H ∗ = 21 . Notice that the analysis may be applied to systems with steep potentials which do not depend explicitly on (or do not degenerate as → 0) in the limit of sufficiently high energy: the reduction to the setting (12) which we consider here may be achieved by a scaling of time.


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Fig. 4. Free flight between boundaries i and j . A smooth trajectory is marked by a bold line and an auxiliary billiard trajectory is marked by a solid line

3.3. Asymptotic expansion for regular reflections. It follows from the proof of Theorem 2 that the behavior of smooth trajectories close to billiard trajectories of regular reflections can be described by an analogue of the billiard map. More precisely, one can construct a cross-section S in the phase space of the Hamiltonian flow, close to the “natural” cross-section S where the billiard map B is defined; the trajectories of the Hamiltonian flow which are close to regular billiard trajectories define the Poincaré map on S , and this map is C r -close to B. Let us explain this in more details. It is convenient to consider an auxiliary billiard in the modified domain D , defined as follows. For each i, take any νi () → +0 such that the function (inverse barrier) Q i (W ; ) tends to zero along with all its derivatives, uniformly for 21 ≥ W ≥ νi . We will use the notation (r )

Mi (ν; ) =

(l)

sup |Q i (W ; )|. ν ≤ W ≤ 21 0≤l ≤r +1

(22)

Condition IV implies that M approaches zero as → 0 for any fixed ν > 0, hence the same holds true for any sufficiently slowly tending to zero ν(), i.e. the required νi () exist. Let ηi () = Q i (νi ; ) and consider the billiard in the domain D which is bounded by the surfaces i : Q(q; )|q∈ N˜ i = Q i + ηi () (see Fig. 4). For sufficiently small , the surface i is a smooth surface which is close to i and is completely contained in N˜ i (its boundaries belong to N ( ∗ )). Indeed, recall that the boundaries i of the original billiard table D are given by the level sets Q(q; 0) = Q i and that ηi () is small, so the new billiard is close to the original one. In particular, for regular reflections, the billiard map B of the auxiliary billiard tends to the original billiard map B along with all its derivatives. It is established in the proof of Theorem 2 that for any choice of νi ’s tending to zero, the condition q ∈ ∂ D defines a cross-section in the phase space of

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the smooth Hamiltonian flow; trajectories which are close to the billiard trajectories of regular reflection, i.e. those which intersect ∂ D at an angle bounded away from zero, define the map F : (q ∈ ∂ D , p looking inwards D ) → (q ∈ ∂ D , p looking outwards D ), namely F (q, p) = h τ (q, p)

(23)

and this map is close to the free-flight map F◦ (see Sect. 2.2) of the billiard in D : F◦ (q, p) = bτ◦ −0 (q, p),

(24)

where τ (q, p) is the time the smooth Hamiltonian orbit of (q, p) needs to reach ∂ D , and τ◦ (q, p) denotes the same for the billiard orbit. Note that we cannot claim the closeness of the time τ maps for the smooth Hamiltonian and billiard flows everywhere in D , still we claim that the maps (23) and (24) are close; we will return to this later. Outside D , the overall effect of the motion of smooth orbits is close to that of a billiard reflection. Namely, as it is proved in Theorem 2, once νi is chosen such that (r ) Mi (νi , ) → 0, the smooth trajectories which enter the region Wi (Q; ) ≥ νi at a bounded away from zero angle to the boundary, spend in this region a small interval of time (denoted by τc (qin , pin )) after which they return to the boundary Wi (Q; ) = νi (namely to Q(q; ) = Q i + ηi ()). Thus, these orbits define the map R : (qin ∈ ∂ D , pin looking outwards D ) → (qout ∈ ∂ D , pout looking inwards D ).

It follows from the proof of Theorem 2 that the map R is close to the standard reflection law R◦ from the boundary ∂ D : R◦ (q, p) = (q, p − 2n(q) n(q), p) ,

(25)

where n(q) is the unit normal vector to the boundary ∂ D at the point q. See Fig. 5. Note that the smooth reflection law R corresponds to a non-zero (though small) collision time τc (q, p), unlike the billiard reflection R◦ which happens instantaneously. Summarizing, from the proof of Theorem 2 we extract that on the cross-section S = {ρ = (q, p) : q ∈ ∂ D , p, n(q) > 0}

(26)

= R ◦ F

(27)

the Poincaré map

is defined for the smooth Hamiltonian flow (for regular orbits - orbits which intersect ∂ D at an angle bounded away from zero), and this map is C r -close to the billiard map B = R◦ ◦ F◦ . As the billiard map B is close to the original billiard map B, we obtain the closeness of the Poincaré map to B as well. However, when developing asymptotic expansions for , it is convenient to use the map B (rather than B) as the zeroth order approximation for . Then, the next term in the asymptotic may be explicitly found (see below) and the whole asymptotic expansion may be similarly developed. We start with the estimates for the “free flight” segment of the motion, i.e. for the smooth Hamiltonian trajectories inside D . For every boundary surface i , choose some δi () → 0 such that the surfaces Q(q; )|q∈ N˜ i = Q i +δi () together with ∂ N ( ∗ ) bound


579

Fig. 5. Reflection from the boundary i . A smooth trajectory is marked by a bold line. An auxiliary billiard trajectory only changes its direction according to the law (25)

⊂ D Fig. 6. The partition of the domain D into regions: Dint

inside D in which the potential V tends to zero uniformly along with the region Dint all its derivatives. See Fig. 6. Let

m (r ) (δ; ) =

sup ∂ l V (q; ). q ∈ Dint 1≤l ≤r +1

(28)

According to Condition I, m (r ) approaches zero as → 0 for any fixed δ of the appropriate signs, therefore the same holds true for any choice of sufficiently slowly tending to the flow of the smooth Hamiltonian zero δi (). As m (r ) → 0, it follows that within Dint r trajectories is C -close to the free flight, i.e. to the billiard flow. In other words, the time

580


is O (m (r ) )-close to the time τ τ map h τ (q, p) = (qτ , pτ ) of the smooth flow in Dint Cr map of the billiard flow

bτ (q, p) =

q + pτ . p

(29)

Note that on the boundary of D we have, by construction, Qi (W ; ) → 0, i.e. we have W (Q; ) → 0. Thus, we have → ∞, while on the boundary of Dint i a boundary layer D \Dint of a non-zero width |δi () − ηi ()| in which the gradient of the potential rapidly decreases. The speed with which the value of Q(q(t); ) changes within this boundary layer is bounded away from zero (see the proof of Theorem 2), so the time the orbit needs to penetrate it is O(δi ). Within this boundary layer the time τ map (q, p) → (qτ , pτ ) of the smooth flow is not necessarily close to the time τ map of the billiard flow (29). However, it is shown in the proof of Theorem 2, that the maps from one surface Q = const to any other such surface within the boundary layer are C r -close for the two flows. This, obviously, implies the closeness of the maps F and F◦ (because the corresponding cross-section is the surface of the kind Q = const indeed). In Appendix 7.1 we show that by an appropriate change of coordinates in each of , in D \D , and outside D ), the equations the three regions we consider (inside Dint int of motion may be written as differential equations integrated over a finite interval with r a right-hand side which tends to zero in the C -topology as → 0. Thus, not only do we obtain error estimates for the zeroth order approximation, we also find a method for obtaining higher order corrections using Picard iterations; the asymptotic behavior of the right-hand side of the equations leads to a contractivity constant which asymptotically vanishes and thus the Picard iteration scheme provides asymptotic for the solutions (each new iteration provides a better asymptotic). In this way we prove in Appendix 7.2 the following Wi (Q; )

Lemma 1. Let q be an inner point of D, and p be such that the first hit of the billiard orbit of (q, p) with the boundary is at i \N ( ∗ ) and is non-tangent. Let the potential V (q; ) (r ) satisfy Conditions I-IV, and choose δi ’s and νi ’s such that δi (), νi (), m (r ) (), Mi () → 0 as → 0. Then, for sufficiently small ≤ 0 , the orbit of the smooth flow hits the cross-section {q ∈ i } = {Q(q; ) = Q i + ηi ()} at the point (qτ , pτ ) such that q + pτ + OC r (m (r ) + νi ) qτ = τ = q + pτ + 0 ∇V (q + ps; )(s − τ )ds + OC r −1 ((m (r ) + νi )2 ), pτ = =

p−

τ 0

p + OC r (m (r ) + νi ) ∇V (q + ps; )ds + OC r −1 ((m (r ) + νi )2 ),

(30)

where τ = τ (q, p) denotes the travel time to the boundary of D (so Q(qτ ; ) = Q i + ηi ()): τ (q, p) = = τ◦ (q, p) +

τ◦ (q, p) + OC r (m (r ) + νi ) τ◦ ∇ Q, 0 ∇V (q+ ps;)(τ◦ −s)ds + OC r −1 ((m (r ) ∇ Q, p

+ νi )2 ),

(31)

∇ Q is evaluated at the (auxiliary) billiard collision point q + pτ◦ ( p, q), and τ◦ ( p, q) is the time the billiard orbit of (q, p) needs to reach i .


581

Now, let us estimate the free-flight map F of the Hamiltonian flow. If q ∈ j and p, n(q) is positive and bounded away from zero, and if the straight line issued from q in the direction of p first intersects ∂ D (say, the surface i ) transversely as well (in our notations this can be expressed as the condition that p, n(q + pτ◦ (q, p)) is negative and bounded away from zero), then the orbits of the Hamiltonian flow define the map F from a small neighborhood of (q, p) on the cross-section {q ∈ j } in phase space into a small neighborhood of the point (q + pτ◦ (q, p), p) on the cross-section {q ∈ i }, see Fig. 4. Take an inner point (q1 , p1 ) on the smooth Hamiltonian trajectory of (q, p). By construction (see (23)), τ (q, p) = τ (q1 , − p1 ) + τ (q1 , p1 ), (q, p) = h −τ (q1 ,− p1 ) (q1 , p1 ) and F (q, p) = h τ (q1 , p1 ) (q1 , p1 ). As q1 is bounded away from the billiard boundary, we can plug (30) and (31) in these relations, which gives us the following Lemma 2. Let the potential V (q; ) satisfy Conditions I-IV , and choose δi ’s and νi ’s such that δi (), νi (), m (r ) (), Mi(r ) () → 0 as → 0. Given a c > 0 and sufficiently small ≤ 0 , consider the set of initial condition (q, p) such that q ∈ j for some j, the segment q + pτ with τ ∈ [0, τ◦ ( p, q)] connects j with i for some i and lies inside D so that q + pτ◦ (q, p) ∈ i , p, n(q) > c and p, n(q + pτ◦ (q, p)) < −c. Then, the free flight map F : (q, p) → (qτ , pτ ) for the smooth Hamiltonian flow is OC r (m (r ) + νi + ν j )-close to the free flight map F◦ of the billiard in D and is given by τ qτ = q + pτ + 0 ∇V (q + ps; )(s − τ )ds + OC r −1 ((m (r ) + νi + ν j )2 ), τ (32) pτ = p − 0 ∇V (q + ps; )ds + OC r −1 ((m (r ) + νi + ν j )2 ). The flight time τ (q, p) is OC r (m (r ) + νi + ν j )-close to τ◦ ( p, q) and is uniquely defined by the condition Q(qτ ; ) = Q i + ηi () (cf.(31)): τ ∇ Q, 0 ◦ ∇V (q + ps; )(τ◦ − s)ds + OC r −1 ((m (r ) + νi + ν j )2 ), τ (q, p) = τ◦ (q, p) + ∇ Q, p

(33)

where ∇ Q is taken at the billiard collision point q + pτ◦ ( p, q), where Q i (q + pτ◦ ( p, q); ) = Q i + ηi (). This could be written as F = F◦ + OC r (m (r ) + νi + ν j ) = F◦ + F1 + OC r −1 ((m (r ) + νi + ν j )2 ), where F1 = OC r (m (r ) + νi + ν j ) and F◦ is defined by (24). Note that the above estimates hold true for any choice of δi ’s such that m (r ) → 0. Therefore, one may take δi ’s tending to zero as slow as needed in order to ensure as good estimates as possible for the error terms in (32),(33).

582


Next we estimate the reflection law R for the smooth orbit. Consider a point q ∈ i and let the momentum p be directed outside D (i.e. towards the boundary) at a bounded from zero angle with i . As we explained, the smooth trajectory of (q, p) spends a small time τc (q, p) outside D and then returns to i with the momentum directed strictly inside D . Let p y and px denote the components of momentum, respectively, normal and tangential to the boundary i at the point q: p y = n(q), p,

px = p − p y n(q).

(34)

We assume that the unit normal to i at the point q, n(q), is oriented inside D , so p y < 0 at the initial point. Denote by Q y (q; ) the derivative of Q in the direction of n(q): Q y (q; ) := ∇ Q(q; ), n(q). Recall that the surface i is a level set of the pattern function Q(q; ), and thus we may study how the normal n(q) changes as one moves along the level set i (in the tangential plane) and as one moves to nearby level sets (in the normal direction). Let K (q; ) denote the derivative of n(q) in the directions tangent to i , and let l(q; ) denote the derivative of n(q) in the direction of n(q). Obviously, Q y is a scalar, K is a matrix, and l is a vector tangent to i at the point q. Note that Q y = 0 by virtue of Condition IIc. Define the integrals: I1 = I1 (q, p) = 2 I2 = I2 (q, p) =

− py

0− p 2 0 y

1− px2 −s 2 ; )ds, 2 1− px2 −s 2 Q i ( 2 ; )s 2 ds,

Qi (

(35)

and the vector J :

I2 (q, p) J (q, p) = − l(q; ) + I1 (q, p)K (q; ) px /Q y (q; ). py

(36)

Notice that J is a vector tangent to i at the point q and that by (22), (r )

(r )

I1,2 = OC r (Mi ), J = OC r −1 (Mi ).

(37)

In Appendix 7.3 we prove the following Lemma 3. Let the potential V (q; ) satisfy Conditions I-IV , and choose δi ’s and νi ’s (r ) such that δi (), νi (), m (r ) (), Mi () → 0 as → 0. Consider a point q ∈ i and assume p, n(q) < −c < 0. Then, for sufficiently small ≤ 0 the collision time of the smooth Hamiltonian flow is estimated by τc (q, p) = OC r (Mi(r ) ) = −

1 I1 (q, p) + OC r −1 ((Mi(r ) )2 ). Q y (q; )

(38)

The reflection map R : (q, p) → (q, ¯ p) ¯ is given by: q¯ = q + OC r (Mi(r ) ) = q + px τc (q, p) + OC r −1 ((Mi(r ) )2 ), (r )

p¯ = p − 2n(q) p y + OC r (Mi ) = p − 2n(q) p y − p y J (q, p) − n(q) px , J (q, p) (r )

+OC r −1 ((Mi )2 ).

(39)


583

Fig. 7. Billiard map B (solid). Billiard map of the auxiliary billiard B (dashed). Poincaré map for the smooth Hamiltonian flow (bold). The first approximation of B + 1 (dash-dotted)

As we see from this lemma (see also (37)), (r )

(r )

R = R◦ + OC r (Mi ) = R◦ + R1 + OC r −1 ((Mi )2 ), where R1 = OC r −1 (Mi(r ) ) and R◦ is defined by (25). Thus, the smooth reflection law is (r )

OC r (Mi )-close to the billiard reflection law (25) and we have obtained explicit expression to the next order correction term. These estimates are obviously non-uniform in c they must break when the collision is nearly tangent. Combining the above lemmas we establish: Theorem 3. Let the potential V (q; ) satisfy Conditions I-IV , and choose δi ’s and νi ’s (r ) such that δi (), νi (), m (r ) (), Mi () → 0 as → 0. Then, on the cross-section S (see (26)) near orbits of regular reflections4 , for all sufficiently small ≤ 0 the Poincaré map of the smooth Hamiltonian flow is defined, and it is O(m (r ) + ν + M (r ) )-close in the C r -topology to the billiard map B = R◦ ◦ F◦ in the auxiliary billiard table D (see Fig. 7). Furthermore, =R ◦F =B +OC r (m (r ) + ν +M (r ) ) = (R◦ +R1 )◦(F◦ +F1 )+OC r −1 ((m (r ) +ν+M (r ) )2 ) =: B + 1 + OC r −1 ((m (r ) + ν + M (r ) )2 )

(40)

(r )

(where ν = maxi νi , M (r ) = maxi Mi ,1 = OC r −1 (m (r ) + ν + M (r ) ), and the first order corrections F1 and R1 are explicitly calculated in Lemmas 2 and 3). 4 That is, given any constant C > 0, near the points (q, p) ∈ S such that n(q), p ≥ C and |n(q), ¯ p| ¯ ≥C where (q, ¯ p) ¯ = B (q, p).

584


Theorem 4. Under the same conditions as in Theorem 3, given a finite T and a regular billiard trajectory in [0, T ], the time t map of the smooth Hamiltonian flow and of the corresponding auxiliary billiard are O(ν + m (r ) + M (r ) )-close in the C r -topology for all t ∈ T \TR , where TR is the finite collection of impact intervals each of them of length O(|δ| + M (r ) ). 3.3.1. Error estimates for some model potentials. Now we can estimate the deviation of the smooth Hamiltonian trajectories from the regular (non-tangent, non-corner) billiard ones for various concrete potentials V (q; ). To make a general estimate possible, we assume that the behavior of the potential near the boundary dominates the estimate; we say that V (q; ) is boundary dominated, if V (q; ) and its derivatives are smaller in the (i.e. in the region bounded by the surfaces Q(q; ) = Q + δ ()) than interior of Dint i i on the boundary of this domain. This means that for boundary dominated potentials m (r ) (δ; ) =

sup sup ∂ l V (q; ) = ∂ l V (q; ). q ∈ Dint q ∈ ∂ Dint 1≤l ≤r +1 1≤l ≤r +1

(41)

By the definition of the pattern function Q, near a given boundary i , ≡ Wi (Q(q; ) − Q i ; ) = Wi (δi ; ). V (q; ) q∈∂ Dint

Q=Q i +δi

Since Q(q; ) is bounded with its derivatives, we conclude that there exists a constant C such that (l)

m (r ) (δ; ) = C max max |Wi (δi ; )|. i

1≤l≤r +1

(42)

Thus, for boundary dominated potentials, one can estimate the differences h t − bt and − B in terms of the barrier functions alone. Remark 1. The boundary dominance condition is a natural condition which is introduced as a matter of convenience. Its introduction allows to explicitly estimate the errors for general geometries with given potential growth near the boundaries, without the need of specifying the potentials in the full domain. When it is not satisfied (e.g. if in a specific application the potential becomes mildly rougher in the domain’s interior as is decreased), one needs to compute m (r ) (δ; ) explicitly and then proceed as below. The corresponding estimates given by Theorems 4 and 3 hold true for every choice of ν and δ such that δ(), ν(), m r (δ(); ), Mr (ν(); ) → 0 as → 0 (for simplicity of notation we assume hereafter that the barrier function W is the same for all boundary surfaces i , and thus suppress the dependence on i). To obtain the best estimates, we have to find ν() and δ() which minimize the expression ν + M (r ) (ν; ) + m (r ) (δ; ). In this way, we first find ν() which minimizes ν + M (r ) (ν; ). Since M (r ) is a decreasing function of ν (see (22)), the sought ν() solves the equation ν = M (r ) (ν; ).

(43)

After ν is determined, we may try to make δ() go to zero so slow that the corresponding value of m (r ) (see (42)) will be asymptotically equal to ν(). Then, this ν() (given by (43)) estimates the deviation between regular billiard and smooth trajectories.


585

Notice that the significance of ν() is three-fold. First, it determines the optimal auxiliary billiard which supplies the best approximation to the smooth Hamiltonian flow (see Lemma 3). Second, it estimates the accuracy of this approximation. Third, it determines, via the relation m (r ) (δ) = ν, the width |δ()| + ν() of the boundary layer in which the billiard and the Hamiltonian flows are not close (Theorem 4). Let us proceed to examples. Proposition 1. Consider the boundary dominated potential V (q; ) corresponding to Q 1 the barrier function W (Q) = e− for small Q. Then, δ() = −(r + 1 + r +2 ) ln √ (r ) (r ) and ν(), m (), M () = O( r +2 ) supply adequate bounds for Theorem √ 3 to apply near regular billiard trajectories. Hence, the smooth Hamiltonian flow is O( r +2 )-close in the C r -topology to the billiard flow within the auxiliary billiard defined by the level set Q(q; √ ) = Q i + η() = Q i + O( ln ). The corresponding Poincaré map is r +2 OC r ( )-close to the auxiliary billiard map B . The impact intervals lengths are √ O( r +2 ). Q

δ

Proof. Since W (l) (Q; ) = (−)−l e− , we obtain that m (r ) (δ; ) = O( −(r +1) e− ) (since the potential is boundary dominated, we may use (42)). The inverse to W (Q; ) is given by Q(W ; ) = − ln W , so Q(l) (W ; ) = (−1)l (l − 1)!W −l , and M (r ) (ν, ) = O(ν −(r +1) ) (see (22)). Plugging this in (43), we find √ ν() = r +2 . (44) 1 By choosing δ() = −(r + 1 + r +2 ) ln , we obtain m (r ) (δ, ) ∼ ν(), so for ν given by (44) we have that ν + M (r ) + m (r ) = O(ν), and the proposition now follows immediately from Theorems 3 and 4 (the value of η() = O( ln ) is given by η = Q(ν; )).

Proposition 2. Let the boundary dominated potential V (q; ) correspond to the bar2 1 − Q for small Q. Then, δ() = − 21 (r + 1 + r +2 ) ln and rier function W (Q) = e (r ) (r ) ν(), m (), M () = O( 2(r +2) | ln | ) supply adequate bounds for Theorem 3 to apply near regular billiard trajectories. Hence, near the regular billiard trajectories, the r 2(r +2) smooth Hamiltonian flow is O(ν()) = O( | ln | )-close in the C -topology to the

billiard flow √ within the auxiliary billiard defined by the level set Q(q; ) = Q i + η() = Q i + O( | ln |). The corresponding Poincaré map is OC r (ν())-close to the auxiliary billiard map B . The impact intervals are of the length O(ν()).

Q2 √ Proof. It is easy to see that W (l) (Q; ) = O(( Q )l e− ) for Q , hence m (r ) (δ; ) = √ δ2 O(( δ )r +1 e− ). From Q(W ; ) = − ln W we obtain M (r ) (ν; ) = O( | ln ν| ν −(r +1) ). Plugging this in (43), we indeed find

(r ) 2(r +2) ), ν() = M (ν; ) = O( | ln |

1 as required. By choosing δ() ∼ − 21 (r + 1 + r +2 ) ln , we obtain m (r ) (δ; ) ∼ ν(), so the rest follows directly from Theorems 3 and 4.

586


Proposition 3. Let the boundary dominated potential V (q; ) correspond to the bar√ 1 rier function W (Q) = ( Q )α . Then, ν(), m (r ) (), M (r ) () = O( r +2+ α ) and δ() = α(r +2)1

ν r +1+α supply adequate bounds for Theorem 3 to apply near regular billiard trajectories. Hence, near the regular billiard trajectories, the smooth Hamiltonian flow is √ 1 O(ν()) = O( r +2+ α )-close in the C r -topology to the billiard flow within the auxiliary billiard defined by the level set Q(q; ) = Q i + η() = Q i + O(ν r +2 ). The corresponding Poincaré map is OC r (ν())-close to the auxiliary billiard map B . The impact α(r +2)

intervals are O(ν()) when α ≥ 1, and O(ν() α+r +1 ) when α ≤ 1. α

α

Proof. As above, using W (l) (Q; ) = O( Ql+α ) we obtain that m (r ) (δ; ) = O( δr +1+α ),

we find Q(l) (W ; ) = O( W l+1/α ) and thus M (r ) (ν; ) = √ 1 O( ν r +1+1/α ). It follows that ν() = O( r +2+ α ) solves ν = M (r ) (ν; ). Now η() =

and since Q(W ; ) =

, W 1/α

α(r +2)

Q(ν) = ν 1/α = O(ν r +2 ). By taking δ() = ν r +1+α , we ensure that m (r ) (δ, ) ∼ ν(). The length of the impact intervals is now given by O(ν + δ).

These three propositions are summarized by the following table (see the next section for the interpretation of the last column): Table 3.3.1. Potential Boundary layer width Approximation error Length of impact intervals Transverse homoclinics W (Q; ) η() Q e− Q2 e−

O(| ln |) √ O( | ln |)

)α (Q

r +2+1/α

r +2

m (r ) + ν + M (r ) √ O( r +2 ) O( 2(r +2) | ln | ) 1√ O( r +2+ α )

T R = O(|δ| + M (r ) ) √ O( r +2 ) O( 2(r +2) | ln | ) 1√ O( r +2+ α ) α ≥ 1

α(r +2) 1) (r +1+α)(r +2+ α

α≤1

g (γ ) γ 3+κ , κ > 0 γ 6+κ , κ > 0 1

γ 3+ α +κ , κ > 0

Note that the asymptotic for the deviation of the smooth trajectories from the billiard ones and for the length of the impact intervals depend strongly on r , i.e. on the number of derivatives (with respect to initial conditions) which we want to control. 3.4. Persistence of periodic and homoclinic orbits. The closeness of the billiard and smooth flows after one reflection leads, using standard results, to persistence of regular periodic and homoclinic orbits. For completeness we state these results explicitly: Theorem 5. Consider a Hamiltonian system with a potential V (q, ) satisfying Conditions I-IV in a billiard table D. Let P b (t) denote a non-parabolic, non-singular periodic orbit of a period T for the billiard flow. Then, for any choice of ν(), δ() such that ν(), δ(), m (1) (), M (1) () → 0 as → 0, for sufficiently small , the smooth Hamiltonian flow has a uniquely defined periodic orbit P (t) of period T = T + O(ν + m (1) + M (1) ), which stays O(ν + m (1) + M (1) )-close to P b for all t outside of collision intervals (finitely many of them in a period) of length O(|δ| + M (1) ). Away from the collision intervals, the local Poincaré map near P is OC r (ν + m (r ) + M (r ) )-close to the local


587

Fig. 8. P b is a billiard periodic orbit (solid). P is a periodic orbit of the smooth Hamiltonian flow (bold)

Poincaré map near P b . In particular, if P b is hyperbolic, then P is also hyperbolic and, inside D , the stable and unstable manifolds of P approximate OC r (ν + m (r ) + M (r ) )closely the stable and unstable manifolds of P b on any compact, forward-invariant or, respectively, backward-invariant piece bounded away from the singularity set in the billiard’s phase space; furthermore, any transverse regular homoclinic orbit to P b is, for sufficiently small , inherited by P as well. Proof. As P b is a regular periodic orbit, i.e. it makes only regular reflections from the boundary (a finite number of them on the period), it follows from Theorem 3 that a Poincaré map for the smooth Hamiltonian flow near P b is O(ν + m (1) + M (1) )-close in C 1 topology to the Poincaré map of the auxiliary billiard D , while the latter is O(η())close to the Poincaré map for the original billiard D. Moreover, from (22) it follows that η() ≤ M (0) ≤ M (1) and we can conclude that a Poincaré map for the smooth Hamiltonian flow near P b is O(ν + m (1) + M (1) )-close in C 1 topology to the Poincaré map for the original billiard D. Since, by assumption, P b (t) is non-parabolic, the corresponding fixed point of the Poincaré map persists for sufficiently small by virtue of the implicit function theorem (the closeness of the corresponding continuous-time orbits is given by Theorem 4). The continuous dependence of the invariant manifolds on in the hyperbolic case follows from the continuous dependence of the Poincaré map on at all ≥ 0 (Theorem 3), and implies the persistence of transverse homoclinics immediately. Indeed, the formulation regarding the closeness of compact pieces of the global stable and unstable manifolds may be easily verified by applying finite time extensions of the local stable and unstable manifolds. Note that a similar persistence result holds true for topologically transverse homoclinic orbits.

588


More generally, one may claim (by the shadowing lemma) the persistence of compact uniformly hyperbolic sets composed of regular billiard orbits. Note that the accuracy of the approximation of smooth orbits (periodic and aperiodic) by the billiard ones, does not depend on the orbit (e.g. is independent of its period) and is given by the maximal deviation for each reflection (times a constant). This holds true for any compact set of regular orbits of a strictly dispersing billiard flow (since such billiards are uniformly hyperbolic); see for example a nice application by Chen [12]. In some cases, to establish the existence of transverse or topologically transverse homoclinic orbits in a family of billiard flows bt (γ ) in Dγ , one uses higher dimensional generalizations of the Poincaré-Melnikov integral (see Sect. 4). In particular, with the near integrable setting, the “splitting distance” between the manifolds near the transverse homoclinic orbit may be proportional to an unfolding parameter γ . The above theorem implies that if g = g (γ ) is chosen so that ν(g , γ ) + m (1) (δ(g , γ ); g , γ ) + M (1) (ν(g , γ ); g , γ )) = o(γ ) and g (γ ) → 0 as γ → 0 then, for sufficiently small γ , transverse homoclinic orbits appear in the smooth flow for all ∈ (0, g (γ )). The value of g (γ ) for the three types of potentials considered here are listed in the last column of Table 3.3.1. In the next section we use this remark and [14] to establish that transverse homoclinic orbits appear in families of smooth billiard potentials that limit to the ellipsoidal billiard. 4. Application to Ellipsoidal Billiards with Potential Consider the billiard motion in an ellipsoid D = {q ∈ Rn : q, A−2 q ≤ 1},

(45)

A = diag(d1 , . . . , dn ) d1 ≥ . . . ≥ dn ≥ 0. The ellipsoid is called generic if all the above inequalities are strict. A well known result of Birkhoff [5] is that the billiard motion in an ellipsoid is integrable, and the mathematical theory which may be invoked to describe and generalize this result is still under development - see [20] and references therein. Delshams et al [14] and recently Bolotin et al [6] (see also references therein) investigate when small non-quadratic symmetric perturbations to the ellipsoidal shape change the integrability property. In this series of works the authors prove the persistence of some symmetric homoclinic orbits, and for specific cases they prove that these orbits are transverse homoclinic orbits of the perturbed billiard, thus proving that integrability is destroyed. Here, we show that using the machinery we developed we can immediately extend their work to the smooth billiardpotential case (notice that in [6] some results are extended to billiards with a C 2 -small Hamiltonian perturbation in the domain’s interior, however the billiard potentials which we consider do not fall into this category - near the boundary they correspond to a large perturbation even in the C 1 -norm). We will first explain the relevant main results of Delshams et al, then supply the corresponding proposition for the smooth case (consequences of Theorem 2, or more specifically of Theorem 5) and then the corresponding quantitative estimates for specific potentials (which follows from Propositions 1-3 and are summarized in Table 3.3.1). 4.1. The billiard in a perturbed ellipsoid. Consider the simplest unstable periodic orbit in an ellipsoidal billiard - the orbit along the diameter of the ellipsoid joining the vertices (−d1 , 0, . . . , 0) and (d1 , 0, . . . , 0). Denote the set formed by the two-periodic points


589

associated with the diameter by P b = {ρ+ , ρ− } ρ± = {q± , p∓ } q± = (±d1 , 0, . . . , 0) p± = (±1, 0, . . . , 0). (46) These points correspond to isolated two-periodic hyperbolic orbits of the Billiard map B and the corresponding periodic orbit Ptb = bt (ρ+ ) of the billiard flow. The (n − 1)-dimensional (n-dimensional for the flow) stable and unstable manifolds of this periodic orbit coincide; in 2-dimensions there are 4 separatrices connecting {ρ+ , ρ− } whereas the topology of the separatrices in the higher dimensional case is non-trivial - it is well described by CW complexes for the 3 dimensional case and by hierarchal structure of separatrix submanifolds in the higher dimensional case (see [14]). Of specific interest are the symmetric homoclinic orbits - it is established in [14] that in the generic 2 dimensional case there are exactly 4 homoclinic orbits which are x−symmetric (symmetric, in the configuration space, to reflections about the x-axis) and 4 which are y−symmetric. In the generic 3 dimensional case, in addition to the 16 planar symmetric orbits (8 in each of the symmetry planes- x y and x z) there are 16 additional symmetric spatial orbits - 8 are symmetric with respect to reflection about the x z plane and 8 are y axial. In the case there are 2n+1 spatial symmetric orbits. Denote by n dimensional P b−hom =

Pib−hom

∞

one of these symmetric homoclinic orbits of the billiard

i=−∞ b−hom so Pi+1

map in the ellipsoid, = B Pib−hom and Ptb−hom = bt (P0b−hom ) denotes the corresponding continuous orbit of the billiard flow. Given a ς such that 0 < ς dn , define the local cross-sections of the billiard map by: − = {(q, p)|q ∈ ∂ D, q1 + d1 < ς, 1 − p1 < ς }, + = {(q, p)|q ∈ ∂ D, d1 − q1 < ς, p1 + 1 < ς }, so, in particular, ρ± ∈ ± and ± ⊂ S, where S is the natural cross-section on which the billiard map is defined (see Sect. 2.2). It follows that only a finite number of points in P b−hom do not fall into ± , and that for any given geometry there exist a finite ς such that P b−hom \{P b−hom ∩ ± } = ∅ for all the symmetric orbits. See Fig. 9. Thus, it is possible to choose P0b−hom and a local cross-section 0 such that P0b−hom ∈ 0 ⊂ {S\{ + ∪ − }}. Notice that for the ellipsoid all the reflections are regular, and furthermore, for the symmetric homoclinic orbits, if d1 is finite and dn is positive then all the reflection angles of P b−hom are strictly bounded away from π/2. Now, consider a symmetric perturbation of the ellipsoid Q of the form: Dγ = {q ∈ Rn : q, A−2 q ≤ 1 + γ (

q12 d12

,...,

qn2 )}, dn2

(47)

where the hypersurface Dγ ⊂ Rn is symmetric with regard to all the coordinate axis of Rn and the function : Rn → R is either a general entire function, such that (0, . . . , 0) = 0 or of a specific form (e.g. quadratic). By using symmetry arguments, Delshams et al [14] prove that for a generic billiard the above mentioned symmetric homoclinic orbits persist under such symmetric perturbations. Furthermore, analyzing the asymptotic properties of the symplectic discrete version of the Poincaré-Melnikov potential (the high-dimensional analog [14, 16] of the planar Poincaré-Melnikov integral [21, 15]), they prove that for sufficiently small perturbations (small γ ) the n-dimensional symmetric homoclinic orbits are transverse in the following four cases:

590


Fig. 9. Billiard trajectory giving rise to a y-symmetric homoclinic orbit (solid)

1. In two-dimensions, for narrow ellipses (β1 =

d22 d12

1), for any analytic small enough

symmetric perturbation. 2 2 2. In two-dimensions, in the non-circular case (β1 = 1), for ( dx 2 , dy 2 ) = 3. In the three-dimensional case, for nearly flat ellipses (β2 = tions of the form: (

x2 d12

y2

, d2 , 2

(or of some specific list).

z2 d32

)=

z2 d32

y2

R( d 2 , 2

z2 d32

1 d32 d12

2

1), for perturba-

), where R is a generic polynomial

4. In the three-dimensional case, for nearly oblate ellipses (β1 = 2

2

2

1

2

3

perturbation ( dx 2 , dy 2 , dz 2 ) =

y4 . d24

z2 y2 . d32 d22

d22 d12

1), for the

To establish these results, the Poincaré-Melnikov potential is calculated for each of these cases, and it is shown that it has non-degenerate critical points at the corresponding symmetric trajectories. It follows that P b−hom −γ persists and the change in the splitting b−hom −γ distance between the separatrices W u and W s near P0 is proportional to γ , the b−hom −γ perturbation amplitude, so that near P0 at the local cross-section 0−γ , d(Wγs , Wγu ) = M(τ )γ + O(γ 2 ),

(48)

where τ ∈ R n−1 denotes some parametrization along W and M(τ ) (the gradient of the Poincaré-Melnikov potential) has simple zeroes at the parameter values corresponding to any of the spatial symmetric homoclinic orbits P0b−hom . 4.2. Smooth potential in a near ellipsoidal region. Let us now consider a two parameter family of smooth potentials V (q; γ , ) which limit, as → 0 to the billiard flow in the perturbed ellipsoid family Dγ ; namely, consider the family of Hamiltonian flows: H (, γ ) =

p2 + V (q; γ , ), 2

(49)


591

Fig. 10. Perturbation of a billiard flow inside a perturbed ellipsoid family Dγ

where V (q; γ , ) satisfies conditions I-IV for all γ values. In the four cases mentioned above, the flow limits, as → 0, to an integrable billiard motion inside the ellipsoid D when γ = 0 and, for γ = 0, to a non-integrable billiard motion inside the perturbed ellipsoid Dγ . See Fig. 10. Applying Theorem 5 to an interior transverse local return map near 0−γ , and noticing that all homoclinic orbits of the billiard flow in Dγ are regular orbits, we immediately establish: Proposition 4. Consider the Hamiltonian flow (49), where V (q; γ , ) is a billiard potential limiting to the billiard flow in Dγ (V (q; γ , ) satisfies Conditions I-IV for all ∈ (0, 0 ] for all γ values). Let the function g (γ ) satisfy ν(g , γ ) + m (1) (δ(g , γ ); g , γ ) + M (1) (ν(g , γ ); g , γ )) = o(γ ) and g (γ ) → 0 as γ → 0. Then, for each of the above cases 1-4, for sufficiently small γ > 0, the smooth flow has transverse homoclinic orbits which limit the billiard’s transverse homoclinic orbits for all 0 < < g (γ ). Indeed, for sufficiently small γ > 0, g (γ ) < 0 and Eq. (48) is valid, and thus the homoclinic billiard orbit P b−hom −γ is transverse, so the above theorem follows immediately from Theorem 5 and the discussion after it. Based on this proposition and Propositions 1-3 we conclude (see Table 3.3.1): Proposition 5. Consider the Hamiltonian flow (49), where V (q; γ , ) is a billiard potential limiting the billiard flow in Dγ (V (q; γ , ) satisfies Conditions I-IV for all ∈ (0, 0 ] for all γ values). Further assume that the potential V (q; γ , ) is boundary dominated and is given near the boundary of Dγ by W (Q; ), so that (41) holds for the corresponding δ values which are specified below. Then, for each of the above cases 1-4, for

592


sufficiently small γ > 0, the smooth flow has transverse homoclinic orbits which limit the billiard’s transverse homoclinic orbits and thus is non-integrable for all 0 < < g (γ ), where Q

• For W (Q; ) = e− : δ = O(− ln ) and g (γ ) = γ 3+κ , κ > 0. √ Q2 • For W (Q; ) = e− : δ = O( − ln ) and g (γ ) = γ 6+κ , κ > 0. √ 1 1 • For W (Q; ) = ( Q )α : δ = O( 3+ α ) and g (γ ) = γ 3+ α +κ , κ > 0. The existence of transverse homoclinic orbits implies non-integrability; there are regions which are mapped onto themselves in the shape of a horseshoe, and thus there are invariant sets on which the motion is chaotic (conjugate to a Bernoulli shift). In particular, it is thus established that the topological entropy of the flow is positive. Whether the metric entropy is positive is, as usual in these near-integrable settings, unknown (see for example [21]). 5. Discussion The paper includes three main results: • Theorems 1-2 deal with the smooth convergence of flows in steep potentials to the billiard’s flow in the multi-dimensional case. These results, which are a natural extension of [56], provide a powerful theoretical tool for proving the persistence of various billiard trajectories in the smooth systems, and vice versa. Several issues are yet to be addressed in this higher dimensional setting. First, the study of corners and regular tangencies (extending [43, 57] to higher dimensions) is yet to be developed. Second, for geometries which are not strictly dispersing, the unavoidable emergence of degenerate tangencies, which is inherently a higher dimensional phenomena, is yet to be addressed. • Theorems 3-4 provide the first order corrections for approximating the smooth flows by billiards for regular reflections. Theorem 3 proposes the appropriate zeroth order billiard geometry which best approximates the steep billiard and a simple formula for computing the first order correction terms, thus allowing to study the effect of smoothing. The smooth flow and the billiard flow do not match in a boundary layer - the width of it and the time spent in it are specified in Theorem 4. Propositions 1-3, as summarized in Table 3.3.1, supply the estimates for the boundary layer width and the accuracy of the auxiliary billiard approximation for some typical potentials (exponential, Gaussian and power-law). All these results are novel for any dimension, and propose a new approach for studying problems with relatively steep potentials. A plethora of questions regarding the differences between the smooth and hard wall systems can now be rigorously analyzed. • Theorem 5 and Proposition 5 are two examples for applications of the above results. The C 1 estimates of the error terms lead naturally to the persistence Theorem 5. Applying these results to the billiards studied in [14], we prove that the motion in steep potentials in various deformed ellipsoids is non-integrable for an open interval of the steepness parameter, and we provide a lower bound for this interval length for the above mentioned typical potentials. While the analysis of higher dimensional Hamiltonian systems is highly non-trivial, we demonstrate here that some results which are obtained for maps may be immediately extended to the smooth steep case. We note that the same statement works in the opposite direction. Furthermore, one


593

may use the first order corrections developed in Theorem 3 and Propositions 1-3 to study the possible appearance of non-integrability due to the introduction of smooth potentials. These results may give the impression that the smooth flow and the billiard flow are very similar. While in this work we emphasize the closeness of the two flows, it is important to bear in mind that this is not the case in general. This observation applies to the local behavior near solutions which are not structurally stable and is especially important when dealing with asymptotic properties such as ergodicity, as discussed below. Let us first remark about the local behavior. As in the two-dimensional settings, we expect that singular orbits of the billiard give rise to various types of orbits in the smooth setting. The larger the dimension of the system, the larger is the variety of orbits which may emerge from these singularities. Moreover, in this higher dimensional setting, even though our theory implies that regular elliptic or partially-elliptic (non-parabolic) periodic orbits persist, the motion near them (and their stability) may change due to resonances. Global properties of the phase space are even more sensitive to small changes. If the billiard periodic orbit is hyperbolic, while it and its local stable and unstable manifolds persist (see for example Theorem 5 ), their global structure in the smooth case may be quite different; first of all, integrability of one of the systems does not imply integrability of the other (for example, it may be possible to use the correction terms computed in Sect. 3.3 to establish that the smooth flow has separatrix splitting even when the billiard is integrable). Second, if the billiard flow has singularities, the global manifolds of a hyperbolic billiard orbit may have discontinuities and singularities whereas the global manifolds of the smooth orbit are smooth (see for example [56]). Finally, the most celebrated global property one is interested in is ergodicity and mixing. In [18] it was shown that when two particles with a finite range potential move on a two-dimensional torus a stable periodic orbit may emerge. In [43, 56, 57] we proved that in the two-dimensional case (C r -smooth potentials, not necessarily finite range, not necessarily symmetric), near singular trajectories (tangent trajectories or corner trajectories) islands of stability are born in the smooth flow for arbitrarily steep potentials. Furthermore, the scaling of the island’s size with the steepness parameter for the general two-dimensional setting was found analytically. Thus there is a fundamental difference in the ergodic properties of hard-wall potentials as compared to smooth potentials. Although these results only apply to two-particle systems, they raise the possibility that systems with large numbers of particles interacting by smooth potentials could also be non-ergodic. Recently, we provided numerical evidence for the appearance of islands of effective stability for arbitrary steep potentials which are close to particular strictly dispersing billiards (not the N − particle case) in three dimensions [41] and in n dimensions [42]. The tools developed here may be essential for studying these and other prototype examples theoretically. In particular, one would hope that beyond an existence proof, these tools may assist in finding the scaling of the islands with the dimension n. Acknowledgements. We would like to thank Ilana Zhurubin for her help in preparing the figures. This research is supported by the Israel Science Foundation (Grant no. 926/04) and by the Minerva Foundation.

7. Appendix 7.1. Picard iteration for equations with small right-hand side. Before we proceed to the proof of Lemmas 1 and 3, we recall the main tool of their proofs - the Picard iteration

594


scheme for equations with small right-hand side (see e.g. [23] and [44]). Consider the differential equation v˙ = ψ(v, µ, t, ),

(50)

where ψ is a C r -smooth function of v and µ, continuous with respect to t and . Assume that for t ∈ [0, L()] and bounded (v, µ) we have a function J () such that J ()L() → 0 and ψC r ≤ J ().

(51)

Then, according to the contraction mapping principle, the Picard iterations vn where

vn+1 (t) = v0 +

t

ψ(vn (s), µ, s, )ds

(52)

0

converge to the solution of (50) starting at t = 0 with initial condition v(0) = v0 on the interval t ∈ [0, L()], in the C r -norm as a function of v0 and µ:

vn (t; v0 , µ) →C r v(t; v0 , µ) = v0 +

t

ψ(v(s; v0 , µ), µ, s, )ds.

0

One can show by induction that vn (t) − v0 C r = O(L()J ()) uniformly for all n. Then it follows that v(t; v0 , µ) = v0 + OC r (L()J ()).

(53)

Furthermore, we now show that v(t; v0 , µ) = vn (t; v0 , µ) + OC r −1 ((L()J ())n+1 )

(54)

(such kind of estimates are, in fact, a standard tool in the averaging theory). In order to prove (54), we will use induction in n. At n = 0 we have an even better result than (54) (see (53)). Now note that

t

v(t) − vn+1 (t) =

(ψ(v(s), µ, s, ) − ψ(vn (s), µ, s, )ds

t 1 = ψv (vn (s) + z(v(s) − vn (s)), µ, s, )dz · (v(s) − vn (s))ds. 0

0

0

It follows immediately that v−vn+1 C r −1 = O(L()ψv C r −1 ) · O(v−vn C r −1 ) = O(L()J ()) · v−vn C r −1 , and (54) indeed holds true by induction.


595

7.2. Proof of Lemma 1. The “free flight” (the motion inside D ) is composed of motion and the motion in the layer N = D \D . We show that in each of these in Dint < int regions the equations may be brought to the form (50), (51). We will first consider the . Recall that the equations of motion for the smooth orbit are flight inside Dint q˙ = p p˙ = −∇V (q; ).

(55)

Let us make the following change of coordinates q(t) ˜ := q(t) − p(t)t.

(56)

q˙˜ = ∇V (q˜ + pt; )t, p˙ = −∇V (q˜ + pt; ),

(57)

Then (55) takes the form

must be finite as with initial data (q(0), ˜ p(0)) = (q0 , p0 ). Since the time spent in Dint r it is C −close to the billiard’s travel time in Dint which is finite here, and using (28), we have ∇V (q˜ + pt; )t r ψC = r = O(m (r ) (δ(); )). −∇V (q˜ + pt; ) C

Thus, system (57) does satisfy (51) with L = O(1), J = O(m (r ) ). It follows then from (53) that p(t) = p0 + OC r (m (r ) ).

(58)

Furthermore, by applying one Picard iteration (52), we obtain from (54) the following estimate for p(t):

t p(t) = p0 − ∇V (q0 + p0 s; )ds + OC r −1 ((m (r ) )2 ). (59) 0

By integrating the equation q˙ = p, we also obtain from (58) that q(t) = q0 + p0 t + OC r (m (r ) ).

(60)

Next, we show that the equations in the layer N< = {W : W (Q; ) ≤ ν} can be brought to the form (50), (51) as well. Recall (see the proof of Theorem 2 in the supplement [42]) that Q˙ = ∇ Q, p is bounded away from zero in N< , hence Q can be taken as a new independent variable (it changes in the interval η ≤ Q − Q i ≤ δ). Now the time t is considered as a function of Q and of the initial conditions (q(tδ ), p(tδ ) (where tδ is the moment the trajectory enters N< ). We show in the proof of Theorem 2 that t is a smooth function of the initial conditions, with all the derivatives bounded. So, in N< , we rewrite (57) as d q˜ ∇ Q(q˜ + pt; ) = W (Q; ) t, dQ ∇ Q(q˜ + pt; ), p dp ∇ Q(q˜ + pt; ) = −W (Q; ) . dQ ∇ Q(q˜ + pt; ), p

596


As W is a monotone function of Q in this layer (i.e. W (Q; ) = 0), we can take W as a new independent variable, so the equations of motion take the form d q˜ dW dp dW

∇ Q(q+ ˜ pt;) ∇ Q(q+ ˜ pt;), p t, Q(q+ ˜ pt;) − ∇∇Q( q+ ˜ pt;), p .

= =

(61)

Since all the derivatives of t with respect to the initial conditions are bounded, we may consider (61) as the system of type (50), (51) with J = O(1), and L = O(ν) (recall that the value of W changes monotonically from W0 = W (δ, ) to ν). Thus, by applying one Picard iteration (52), we obtain from (54) that

p(W ) = p(W0 ) −

W W (δ,)

˜ 0 ) + p(W0 )t; ) ∇ Q(q(W dW + OC r −1 (ν 2 ). ∇ Q(q(W ˜ 0 ) + p(W0 )t; ), p(W0 )

From (53) we also obtain p(W ) = p(W0 ) + OC r (ν). Note that OC r −1 (ν 2 ) and OC r (ν) refer here to the derivatives (with respect to the initial conditions) of p at constant W or, equivalently, at constant Q. Returning to the original time variable, these equations yield

p(t) = p(tδ ) + OC r (ν) = p(tδ ) −

t tδ

∇V (q(tδ ) + p(tδ )(s − tδ ); )ds + OC r −1 (ν 2 ).

Using expressions (58),(59) for p(tδ ) and (60) and q(tδ ), we finally obtain p(t) = p0 + OC r (ν + m (r ) ) = p0 −

0

t

∇V (q0 + p0 s; )ds + OC r −1 ((m (r ) + ν)2 ) (62)

for all t such that q(t) ∈ D , in complete agreement with the claim of the lemma (as we mentioned, the OC r −1 (·) and OC r (·) terms refer to the derivatives at constant Q). The corresponding expression for q(t) (see (30)) is obtained by integrating the equation q˙ = p. The expression (31) for the flight time τ is immediately found from the relation W (Q(q(τ ); ); ) = ν or, equivalently, Q(q(t); ) = Q i + η (recall that Q˙ is bounded away from zero in the layer N< ). 7.3. Proof of Lemma 3. Here we compute the reflection map R : (qin , pin ) :→ (qout , pout ) defined by the smooth trajectories within the most inner layer N> : {W ≥ ν}. We put the origin of the coordinate system at the point qin (corresponding to q at Fig. 5) and rotate the axes with so that the y-axis coincides with the inward normal (denoted by n(q) in Fig. 5) to the surface Q(q; ) = Q(qin ; ) at the point qin and the x-coordinates span the corresponding tangent plane. It is easy to see that in the notations of Lemma 3 we have (the explicit dependence on is suppressed for brevity, and the normal direction is suppressed) K (qin ) = Q x x (qin )/Q y (qin ),

l(qin ) = Q x y (qin )/Q y (qin ).

(63)


597

By (12) and Condition II, near the boundary, the equations of motion have the form: ∂H ∂H = −W (Q; )Q x , = px p˙x = − ∂ px ∂x ∂H ∂H y˙ = = −W (Q; )Q y . = p y p˙y = − ∂ py ∂y

x˙ =

(64) (65)

Note that Q y (qin ) = 0, because the y-direction is normal to the level surface of Q at the point qin . It is shown in the proof of Theorem 2 that the orbit may spend only a small time in N> , so it stays close to qin all the time it stays in N> . It follows that Q y (qin ) stays dp bounded away from zero. Hence, dty is bounded away from zero in (65). Therefore, we may use p y as the new independent variable. Equations (64), (65) are then rewritten as p dq = −Q (W ; ) , dp y Qy

dt 1 = −Q (W ; ) , dp y Qy

dpx Qx = , dp y Qy

(66)

where W =H−

1 2 p . 2

(67)

In order to bring the equations of motion to the required form with the small right-hand side, we make the additional transformation px → p˜ = px −

Q x (q) py . Q y (q)

(68)

Note that Q x (qin ; ) = 0, hence (see (63)) p˜ = px − K (qin )(x − xin ) p y − l(qin )(y − yin ) p y + O((q − qin )2 ).

(69)

In particular p(t ˜ in ) = px,in .

(70)

After the transformation, taking into account (67), Eqs. (66) take the form dq 1 1 p = −Q ( − p 2 ) , dp y 2 2 Qy dt 1 1 1 = −Q ( − p 2 ) , dp y 2 2 Qy p Qx d p˜ 1 1 d = Q ( − p 2 ) py . dp y 2 2 dq Q y Q y

(71) (72) (73)

Since Q (W ; ) is small in the inner layer, these equations belong to the class (50), (51), with J = O(M (r ) ) (see (22)) and L = O(1) (the change in p y is bounded by the energy constraint). Thus, by (53), we obtain (see (70)) (q, t, p) ˜ = (qin , tin , px,in ) + OC r (M (r ) ).

(74)

Recall that W (qout ) = W (qin ). Therefore, by energy conservation, 2 2 px,in + p 2y,in = px,out + p 2y,out ,

(75)

598


so (74) implies p y,out = − p y,in + OC r (M (r ) ).

(76)

By (74), and by using Q x (qin , ) = 0, Eqs. (71) may be written up to OC r −1 ((M (r ) )2 )terms as ( px,in , p y ) dq 1 2 = −Q ( (1 − px,in − p 2y )) , dp y 2 Q y (qin ) 1 1 dt 2 , = −Q ( (1 − px,in − p 2y )) dp y 2 Q y (qin ) py d p˜ 1 2 . = Q ( (1 − px,in − p 2y ))(K (qin ) px,in + l(qin ) p y )) dp y 2 Q y (qin )

(77) (78) (79)

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Communications in


A TQFT Associated to the LMO Invariant of Three-Dimensional Manifolds Dorin Cheptea1,2 , Thang T. Q. Le3 1 Center for the Topology and Quantization of Moduli Spaces, Department of Mathematical Sciences,

University of Aarhus, Ny Munkegade, Bygning 1530, 8000 Aarhus C, Denmark. E-mail: [email protected] 2 Institute of Mathematics of the Romanian Academy, P.O.Box 1-764, Bucharest, 014700, Romania 3 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA. E-mail: [email protected] Received: 6 February 2006 / Accepted: 14 November 2006 Published online: 18 April 2007 – © Springer-Verlag 2007

Abstract: We construct a Topological Quantum Field Theory associated to the universal finite-type invariant of 3-dimensional manifolds, as a functor from a category of 3-dimensional manifolds with parametrized boundary, satisfying some additional conditions, to an algebraic-combinatorial category. This is built together with its truncations with respect to a natural grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The TQFT(s) induce(s) a (series of) representation(s) of a subgroup Lg of the Mapping Class Group that contains the Torelli group. The N = 1 truncation is a TQFT for the Casson-Walker-Lescop invariant. A TQFT for the LMO invariant was constructed by Murakami and Ohtsuki [21], but it has complicated anomaly. One of the results of the present paper is to show that their anomaly operator reflects only: 1) the way we define the gluing between two cobordisms, i.e. if we regard #g (S 1 × S 2 ) or S 3 as the simplest manifold(s), and 2) that they consider the un-normalized invariant. Our construction of TQFT allows us to associate to the LMO invariant an infinite-dimensional linear representation of the Torelli group, in fact of a larger Lagrangian subgroup of the Mapping Class Group. The new results of this paper include an isomorphism (Proposition 2.2), reducing the study of the LMO invariant of 3-dimensional manifolds with parametrized boundary to that of finite-type invariants of string-links; the construction (from truncations; Theorem 2.9) of a composition operation on chord diagrams to correspond to the gluing of cobordisms; a limit property (Lemmas 3.7, 3.8); non-degeneracy of the TQFT. The natural truncation induces a TQFT for the Walker-Lescop extension of the Casson invariant, and we can identify Morita’s representation as its first non-trivial part. It remains, however, to interpret our TQFT as some sort of “perturbative expansion around 0” of the Reshetikhin-Turaev TQFT. The results of this article were obtained when the authors were at the Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA.

602

D. Cheptea, T. T. Q. Le

This paper is organized as follows. In Sect. 1 we recall the topological categories Q, Z introduced in [6], and the pertaining results that we will need subsequently in this paper. The categories A and A≤N of chord diagrams are explained in Sect. 2. We use a simpler (then in [21]) definition of Z on elementary pseudo-quasi-tangles1 , and an even associator instead of the Knizhnik-Zamolodchikov one. In Sect. 3 we formally construct the anomaly-free (by [6]) truncated and full TQFTs. One of the results of that section is showing that the completion of the algebraic image of cobordisms with one boundary component of genus g is precisely the whole space A(↑g ) of chord diagrams on g vertical lines. This means that the induced representation can (in principle) be used for combinatorial calculations in addressing topological questions about three-dimensional manifolds and the Mapping Class Group. We also finish the proof of Theorem 2.9 there. Functoriality, and other results regarding the TQFT are gathered in Theorem 3.2. In Sect. 4 we restrict to the case N = 1 to get a TQFT for the Casson-Walker-Lescop invariant. Also there we describe (algebraically) the map that sends the invariant of a manifold with parametrized boundary to the invariant of the closed manifold obtained from the former by the natural procedure that we call below filling. Let Hg denote a fixed “standard” handlebody of genus g, and let H g denote S 3 − Hg . The TQFT construction in [21] and the Reshetikhin-Turaev TQFT for quantum invariants are based on the classical convention under which the identical gluing Hg ∪id Hg = #g (S 1 × S 2 ). However for invariants that are primarily of homology spheres this convention is not natural. The LMO invariant is very strong for Z- and Q-homology spheres, but is much weaker if the Betti number is higher. In order to address this issue in the present paper we will regard g = ∂ Hg = −∂ H g ⊂ R3 ⊂ S 3 as the standard surface of genus g used to parametrize boundary components of cobordisms, and in order to obtain the composition-cobordism M2 ∪ f M1 we will always glue the “top” boundary component of M1 to the “bottom” boundary component of M2 along an orientationpreserving homeomorphism f , as explained in Subsect. 1.1. Hence we shall always have H g ∪id Hg = S 3 for all g, as opposed to Murakami-Ohtsuki’s and ReshetikhinTuraev’s Hg ∪id Hg = #g (S 1 × S 2 ). Remarks. By fixing a homeomorphism between H g and Hg , one fixes a Heegaard homeomorphism that Morita [20] calls ιg . If one then takes, what we call below, the filling (g × I, ϕ, id), one obtains the manifold denoted by Morita Wϕ .

0.1. Chord diagrams. (For details see [2, 4].) An open chord diagram is a vertex-oriented uni-trivalent graph, i.e. a graph with univalent and trivalent vertices together with a cyclic order of the edges incident to the trivalent vertices. Self-loops and multiple edges are allowed. A univalent vertex is also called a leg, and a trivalent vertex is also called an internal vertex. In planar pictures, the orientation of the edges incident to a vertex is the counterclockwise orientation, unless otherwise stated; the pictures can not be perfect since not every graph is planar, therefore when reading pictures one should keep in mind that four-valent vertices do not exist. The degree of an open chord diagram is half the number of all vertices. Suppose is a compact oriented 1-manifold (possibly with boundary) and X a finite set of asterisks. A chord diagram with support ∪ X is a vertex-oriented uni-trivalent graph D together with a decomposition D = ∪ E, where E is an open chord diagram 1 “pseudo” stands for the presence of 3-valent vertices.

A TQFT Associated to the LMO Invariant of Three-Dimensional Manifolds

603

with some legs labeled by elements of X , such that D is the result of gluing all nonlabeled legs of E to distinct interior points of (the 3-valent vertices resulted from gluing, which will not have an associated cyclic order of adjacent edges, are called external vertices). Repetition of labels is allowed and not all labels have to be used. The degree of D is, by definition, the degree of E. is also called the skeleton of D, and in pictures is represented by bold lines. Often the components of , as well as different asterisks in X , are distinguished in pictures by labels. By a graph we will mean a uni-trivalent graph, with all edges oriented, and with a cyclic order of edges incident to trivalent vertices prescribed. Self-loops and multiple edges are allowed. The connected components of the graph have to be always ordered (and this order has to be preserved by a homeomorphism). Additionally we may label (color) some subgraphs within each connected component. One should think of a graph as a generalization of the notion of an oriented compact 1-manifold. One can repeat the definition of the previous paragraph to obtain the notion of a chord diagram with support a graph; the graph is then the skeleton of the chord diagram. In the definition of the degree the vertices of the support are not counted. Examples of graphs : • the oriented manifold which is the union of g ∈ N∗ copies of [0, 1], each copy labeled (colored) by a distinct element of a finite abstract ordered set X of asterisks. This special graph will be denoted ↑ X , and in planar pictures will be represented by vertical lines 6 6… 6 . g times - -i- …--i. The order of the edges • the chain graph2 suggestively denoted -i adjacent to each vertex is everywhere counterclockwise with respect to its standard embedding in R2 ⊂ R3 , the subgraphs -i are labeled 1 through g from left to right. Let us denote this special graph by g . Note that g standardly embedded in R2 ⊂ R3 ⊂ S 3 has a preferred (the blackboard) framing, indicated by a planar surface with g as core. For g = 1, set 1 = -i, one oriented edge (loop), no vertices. • it is convenient to set 0 = one point as a chain graph. Chord diagrams on 0 automatically can have only internal vertices.

1. The Topological Category Definition 1.1 (see also [6]). 1) Two triplets (K , G 1 , G 2 ) and (L , H1 , H2 ), consisting each of a framed oriented link and two disjoint embedded Framed chain graphs in S 3 , are equivalent (notation ∼ =) if there is a PL-homeomorphism φ : S 3 → S 3 which sends K to L, G 1 to H1 , and G 2 to H2 . Here ∅ is also considered a framed oriented link in S 3 . We call G 1 the bottom, and G 2 the top of the triplet. 2) Let M be a compact oriented 3-manifold with boundary ∂ M = (−S1 ) ∪ S2 , and suppose that parametrizations f i : gi → Si are fixed.3 Such (M, f 1 , f 2 ) will be called a (parametrized) (2+1)-cobordism, S1 will be called its bottom, and S2 – its top. The cobordisms (M, f 1 , f 2 ) and (N , h 1 , h 2 ) are equivalent (homeomorphic) if there is a PL-homeomorphism F : M → N such that F ◦ f i = h i , i = 1, 2. 3) Let Hg be a fixed neighborhood of g in S 3 . g := ∂ Hg ⊂ S 3 is the standard oriented surface of genus g. Denote H g = S 3 − Hg . Clearly ∂ H g = −g . Always glue 2 We borrow this terminology from [21]. 3 For simplicity restrict to 3-manifolds with ≤ 2 boundary components, one “bottom”, and one “top”. Note,

however, that Propositions 1.2, 1.3 and 1.4 can be easily generalized.

604


b

G'

G2

...

c .. .

... G

G'1 Wg

.. .

horizontal lines are removed when cobordisms are composed

a

.. .

Tg

Fig. 1. a: The triplet Wg := (∅, G, G ) represents the cobordism (g × [0, 1], p1 , p2 ); b: The horizontal lines (blue) of G 1 and G 2 (The shaded boxes represent symbolically the rest of the triplets); c: The framed tangle Tg ⊂ B(0, 2) − B(0, 1).

cobordisms along orientation-preserving homeomorphisms of boundary components, as specified by the parametrizations: (M2 , f 2 , f 2 ) ◦ (M1 , f 1 , f 1 ) = (M2 ∪ f2 ◦( f )−1 1 ∪M1 , f 1 , f 2 ). When g = 0 we assume that g is a point, and Hg is a ball (resp. for H g ). Given a parametrized cobordism (M, f 1 , f 2 ), use the maps f i to glue the standard handlebody Hg1 to the bottom, and the standard anti-handlebody H g2 to the top of := H g2 ∪ f2 M ∪ f1 Hg1 the filling of (M, f 1 , f 2 ). (M, f 1 , f 2 ), and call the result M Remarks. In classical notation our manifold M2 ∪ f2 ◦( f )−1 ∪M1 is M2 ∪− f2 (g × 1 [0, 1]) ∪ι−1 (g × [0, 1]) ∪−( f )−1 M1 , where we can fix a Morita’s ιg by fixing a g 1 homeomorphism between H g and Hg , and where the − signs respect the fact that the “bottom” boundary component has orientation opposite to the one induced from the 3-manifold. 1.1. Surgery description of gluing cobordisms. Let G denote set of equivalence classes of triplets (L , G 1 , G 2 ) in S 3 . Let C denote the set of equivalence classes of 3-cobordisms, with connected non-empty bottom and top. Proposition 1.2. [6] 1) The map κ : G → C that associates to every equivalence class of triplets (L , G 1 , G 2 ) the equivalence class of cobordisms (M, f 1 , f 2 ), obtained by doing surgery on L ⊂ S 3 , removing tubular neighborhoods N1 , N2 of each G 1 , G 2 , and recording the parametrizations of the two obtained boundary components, is welldefined and surjective. If one glues according to these parametrizations a standard handlebody to −∂ N1 and a standard anti-handlebody to ∂ N2 , then one obtains SL3 . 2) Let a first Kirby move on a triplet be the cancellation / insertion of a O±1 separated by an S 2 from anything else, and an extended second Kirby move be a slide over a link component of an arc, either from another link component or from a chain graph. Then, if one factors G by the extended Kirby moves and changes of orientations of link components, the induced map κ is a bijection.

For example, to represent the identity cobordism (g × [0, 1], p1 , p2 ), where pi : g → g ×(i −1) ⊂ S 3 are two copies of the standard embedding of g in S 3 , ∀g ≥ 1, we can choose the framed graph Wg = (G, G ) shown in Fig. 1a. Let , respectively , generically denote the bottom, respectively the top, of a triplet. Call the union of the lower half-circles and the horizontal segments of , the horizontal line of . Similarly, call the union of the upper half-circles and the horizontal segments of , the horizontal line of . (See Fig. 1b.)


605

Proposition 1.3. [6] Let (M1 , f 1 , f 1 ) and (M2 , f 2 , f 2 ) be two 3-cobordisms with connected non-empty bottoms and tops, represented by triplets (L 1 , G 1 , G 1 ) and (L 2 , G 2 , G 2 ), and suppose genus(G 1 ) = genus(G 2 ) = g. Remove a 3-ball neighborhood of the horizontal line of G 1 ⊂ S 3 , and identify the remain with B(0, 1). Remove a 3-ball neighborhood of the horizontal line of G 2 ⊂ S 3 , and identify the remain with S 3 − B(0, 2). Glue the framed tangle Tg ⊂ B(0, 2) − B(0, 1) shown in Fig. 1c to the ends of the remains of G 1 in B(0, 1) and G 2 in S 3 − B(0, 2), strictly preserving the order of the points, so that the composition of these framed tangles is a smooth framed oriented link L 0 in S 3 = (S 3 − B(0, 2)) ∪ (B(0, 2) − B(0, 1)) ∪ (B(0, 1)). Then κ(L 1 ∪ L 0 ∪ L 2 , G 1 , G 2 ) = (M2 ∪ f2 ◦( f )−1 M1 , f 1 , f 2 ), 1

(1.1)

where the framed graphs G 1 , G 2 in this formula are determined in the obvious way by the original G 1 , G 2 in the two copies of S 3 . Hence, any triplet representing (M2 ∪ f2 ◦( f )−1 1 M1 , f 1 , f 2 ) is equivalent to (L 1 ∪ L 0 ∪ L 2 , G 1 , G 2 ) by extended Kirby moves and changes of orientations of link components. Suppose G is an arbitrary embedding of the chain graph g in S 3 , then H1 (S 3 − G, Z) ∼ = Zg , with free generators the meridians of the circle components. Proposition 1.4. [6] Suppose M is a connected compact oriented 3-manifold with two distinguished boundary components ∂ M = (−S1 ) ∪ S2 , let f 1 , f 2 be parametrizations of these surfaces, and let i : ∂ M → M be the inclusion. The following conditions are equivalent: F) = 0, (1) H1 ( M, (2) H1 (M, F) = i ∗ H1 (∂ M, F)/( f 1∗ H1 (Hg1 , F) + f 2∗ H1 (H g2 , F)) . They imply: (3) 2 · rank H1 (M; F) = rank H1 (∂ M; F).

This proposition holds true for F = Z, Q or Z p . In condition (2) above, H1 (Hg , F) and H1 (H g , F) denote (by abuse of notation) the subspaces of H1 (g , F) generated by the longitudes, resp. by the meridians. A 3-cobordism satisfying the equivalent conditions (1), (2) of Proposition 1.4 will be called an F-cobordism. Note that in this definition we allow one or both Si to be empty, although from the point of our TQFT the case of empty top and/or bottom is undistinguished from the case when that component is S 2 . 1.2. Description of the categories Q ⊃ Z. Objects in each of these are natural numbers. The morphisms between g1 and g2 are equivalence (homeomorphism) classes of connected F-cobordisms with bottom S1 of genus g1 and top S2 of genus g2 , satisfying the F-doubly-Lagrangian condition: f 1∗ L a ⊇ f 2∗ L a and f 1∗ L b ⊆ f 2∗ L b ,

(1.2)

where L a = ker(incl∗ : H1 (g , F) → H1 (Hg , F)), and L b = ker(incl∗ : H1 (g , F) → H1 (H g , F)), and F = Z (for Z) or Q (for Q). The composition-morphism of two cobordisms (M2 , f 2 , f 2 ) and (M1 , f 1 , f 1 ) is the equivalence class of the 3-cobordism (M2 ∪ f2 ◦( f )−1 M1 , f 1 , f 2 ). 1 In general condition (1.2) over Z is stronger than (1.2) over Q. It may hold with strict inclusion. [6]

606


Proposition 1.5. [6] The composition of two morphisms (say, class of M and class of N ) in category Q (respectively Z) is again a morphism in the category Q (respectively Z).

Let us restrict to 3-cobordisms M of the form (g × [0, 1], f × 0, f × 1), with f, f ∈ Aut (g ), i.e. the parametrization of the top differs by that of the bottom by the automorphism w = ( f )−1 ◦ f . The equivalence classes of this cobordism depends only on the isotopy class of w (i.e. we don’t need to specify both f , f ). The equivalence class of M = (g × [0, 1], f × 0, f × 1) is a Z-doubly-Lagrangian cobordism iff it is a Q-doubly-Lagrangian cobordism iff it satisfies L a = w∗ (L a ) and L b = w∗ (L b ). In is always a Z-homology sphere. [6] particular, M These cobordisms form a category, denoted by L. The composition of two cobordisms (g × I, f 2 × 0, f 2 × 1) ∼ = (g × I, w2 × 0, id × 1) and (g × I, f 1 × 0, f 1 × 1) ∼ = (g × I, w1 × 0, id × 1) is the 3-cobordism (g × I, f 2 ◦ ( f 1 )−1 ◦ f 1 × 0, f 2 × 1) ∼ = (g × I, ( f 2 )−1 ◦ f 2 ◦ ( f 1 )−1 ◦ f 1 × 0, id × 1) ∼ = (g × I, (w2 ◦ w1 ) × 0, id × 1). Definition 1.6. [6] Call the Lagrangian subgroup of the Mapping Class Group, the subgroup consisting of isotopy classes of elements w ∈ Aut (g ), such that w∗ (L a ) = L a and w∗ (L b ) = L b (over Q or over Z, is equivalent by the above). Denote it by Lg . The TQFT of the LMO invariant (Sect. 3 below) induces a representation of Lg . This subgroup of MC G(g) is big enough to be interesting, it contains the Torelli group. Its A 0 , image under the action on homology is the group of matrices of the form 0 (A T )−1 where A ∈ G L(g, Z). 2. The Algebraic-Combinatorial Category Murakami and Ohtsuki have extended the Kontsevich integral to an invariant Z (G) of oriented framed trivalent graphs G in S 3 ([21, Theorem 1.4]). A framed graph G ⊂ S 3 is represented as a plane projection (with implicit blackboard framing), then decomposed into elementary pseudo-quasi-tangles, and Z is defined for each piece (see [21], Fig. 2 for the exact definition of Z ). It is easy to observe that in order to verify the independence of Z of the decomposition into pseudo-quasi-tangles and the invariance under extended Reidemeister moves for trivalent graphs, one is forced to introduce relations that “move” (in the sense of Proposition 2.1) a box-diagram over a trivalent vertex to a box-diagram. Hence the branching relations (Fig. 4 here, Fig. 1 in [21]). These relations are necessary to impose regardless of the definition of Z for the neighborhood of a trivalent vertex, and obviously regardless of what associator is used. From the extended Z , Murakami and Ohtsuki [21] derived an invariant of oriented 3-manifolds with boundary, along the same lines the Z L M O is constructed [17] from the Kontsevich integral of framed links. 2.1. The modules of chord diagrams. Let be a graph; we will be mainly interested in the cases = a 1-manifold and = a chain graph. Let A() be the formal series completion with respect to the degree of the Q-vector space freely generated by the set of homeomorphism classes of chord diagrams with support , without self-loops and


_

=

STU

Convention 1:

607

=

and

( Consequence: LHS ( STU-left ) = - RHS ( STU-left ) )

Convention 2:

STU-right

=

STU-left

=

_ _ =

and

_

Fig. 2. The two conventions for chord diagrams

a

... ... =

c1

+ c2

+ ... + c n

... ...

i

...

i

...

Σ

... ...

... ...

=

b i

c = -1

c i= 1

ci = 1

i

i

i Fig. 3. The box-diagram

= Fig. 4. The 8 branching relations (all but the vertical edge are bold): one for each possible orientation of the 3 bold edges

univalent vertices, modulo AS, IHX, STU and branching relations (which are homogeneous with respect to the degree).4 We will use the following box-diagram notation for the formal sum of chord diagrams, as shown in Fig. 3a. There outside the drawn part the diagrams are identical, the vertical edge is dashed, the horizontal edges are arbitrary. If the horizontal edge i is dashed, then ci = 1, if it is bold, then ci is as shown in Fig. 3b.5 The branching relations, introduced in [21, Fig. 1], are shown in Fig. 4 using this box-notation. 4 There are essentially two conventions in defining STU and AS relations, and drawing certain elements of A(), as shown in Fig. 2. Note that in Convention 1, which is the one that we use (as well as [17, 21]), AS relations refer only to internal vertices, and no cyclic order of edges adjacent to external trivalent vertices is defined. In this convention, as a consequence, the LHS of STU-left is equal to minus the RHS of STU-left. Using the second convention, the definition of a chord diagram has to be changed to account for the cyclic order of edges adjacent to external trivalent vertices. The two A(), from the two conventions, are canonically isomorphic; in fact only the meaning of some diagrams as elements of A() is changed by adding a − sign. 5 For Convention 2 all coefficients c = 1. Then the box-diagrams in the two conventions correspond i precisely one to the other via the canonical isomorphism between the conventions.

608


a

b _

...

... ...

...

...

...

...

=_ ...

...

...

...

...

...

=

Fig. 5. a: The box-STU relation (each term in the RHS contains a double sum over the horizontal edges), b: The box-AS relation

= 0

= 0

=

Fig. 6. Invariance over “elementary pseudo-tangles” (the dashed/bold type of horizontal edges is arbitrary, the vertical edge is dashed)

Similarly, let A(∅) denote the formal series completion with respect to the degree of the Q-module freely generated by the set of homeomorphism classes of open chord diagrams without self-loops and univalent vertices, modulo AS and IHX relations. For a chord diagram D, denote [D] the corresponding element of A(). A() and A(∅) are co-algebras with respect to the decomposition of the dashed part of a diagram in connected components (the elements represented by diagrams that have non-empty connected dashed part are defined to be primitive).6 A(∅) is an algebra with respect to disjoint union, and together with (completed) comultiplication forms a Hopf algebra. Note that A() is an A(∅)-module with respect to the disjoint union. Proposition 2.1. (a) The box-STU and box-AS relations, schematically shown in Fig. 5 hold in A(). (b) The three relations in Fig. 6 hold in A(). (c) The box-STU and box-AS relations can be “moved” over any trivalent vertex of , using only branching relations (see Fig. 7 for an example).7 Proof. (a) Let xi denote the horizontal edges. Let [D Y ], [D I I ], [D X ] denote the three terms of the box-STU relation. Note that the brackets are also part of the notation, D Y means the box-diagram, which is not a chord diagram. Let [DxYi ] denote the element of A() corresponding to the chord diagram obtained from D Y by replacing the box with a prolongation of the vertical edge until the edge xi . With similar notations [DxIiIx j ] and [DxXk xl ], note that for i = j, [DxIiIx j ] = [DxXj xi ]. Hence:

ci c j [DxIiIx j ] − c j ci [DxXj xi ] RHS = xi

xj

xj

xi

6 One can check (e.g. by induction on the number of internal vertices of chord diagrams) that this comultiplication is well-defined (remember the presence of STU relations). 7 One can reformulate this statement: Every IHX (respectively AS) relation on-the-left-of-the-trivalentvertex is a consequence of branching relations and IHX (respectively AS) relations on-the-right-of-thetrivalent-vertex.


branching relations

_

=

_

_

+

_

=

+

609

+

=

+

+

Fig. 7. “Moving” a box-STU relation over a trivalent vertex

=

ci2 [DxIiIx j ] +

=

i= j

ci c j [DxIiIx j ] −

i= j

i= j

ci2 [DxIiIx j ]−

i= j

i= j

ci2 [DxXj xi ] =

ci2 [DxXj xi ] −

c j ci [DxXj xi ]

i= j

([DxIiIxi ] − [DxXi xi ]) = ci [DxYi ] = L H S, i

i

where in the equality before the last we have used an IHX, STU, or Convention-1 form of STU-left for each xi . The proof of the box-AS relation is elementary, using AS relations and the definition of coefficients ci . (b) Consider all dashed/bold possibilities for the edges. The relations then follow from the AS, IHX, STU and branching relations. (c) Every box-diagram is a sum of box-diagrams with small boxes. For the later follow the calculation shown in Fig. 7, for the box-STU case. The box-AS case is obvious.

Note that this proposition for the case of being a 1-manifold is part of [24, Prop. 1.4]. The “formal series completion” (i.e. the topology is given by8 distance( p, q) ≤ 21n ⇔ p − q has no terms of degree < n) is algebraically nothing else but the direct product over i ∈ N of the vector spaces generated by diagrams of a fixed order i. AS, IXH, STU and branching relations are homogeneous with respect to the degree. For every i, the degree i part Ai is defined as Di /Ri , where Di is the Q-module freely generated by the chord diagrams of degree i (without factoring through relations), and Ri is the Q-module freely generated by the relations involving onlydiagrams of order i. By the universal property of the direct product A = i∈N Ai ∼ = i∈N Di / i∈N Ri = D/R, i.e. factoring and taking completion commute. We will not use anywhere below the next proposition that A ∼ = D/R, our object is always A. …- not i i --i Proposition 2.2. Denote [g] = {1, . . . , g}. Let φ :↑g → - [g] = g be the embedding of ↑g onto the upper half-circles of -i--i- …--i, sending the arrow labeled i to the i th upper half-circle of g , preserving orientation. Then it extends to an isomorphism of Q-vector spaces φ∗ : A(↑g ) → A( g ). Proof. Fix an arbitrary degree i of chord diagrams. Then φ induces a homomorphism of vector spaces φ∗ : Di (↑g ) → Di ( g ), under which Ri (↑g ) is sent exactly to the set of AS, IHX and STU relations in g that involve only diagrams with support in φ(↑g ). For simplicity of notation, let us denote φ∗ Di (↑g ) by Di (↑g ), and φ∗ Ri (↑g ) by Ri (↑g ). 8 The only reason we choose 1 instead of 1 is Lemma 3.7. See the remark after it. n 2n

610


...

= 0

... Fig. 8. Relation 9: the ambiguity of “moving” a dashed end off the horizontal line.

Replace each external trivalent vertex in g − φ(↑g ) of a chord diagram by a small box (and add a sign to it, the coefficient ci ), then “move”, using the branching relations, one by one all boxes off g − φ(↑g ). This assigns to an arbitrary chord diagram with support in g a diagram with boxes (with a ± sign) with support in φ(↑g ). It depends on the choice of the sequence of trivalent vertices over which boxes are “moved” in g . Observe, however, that different such choices result in diagrams with boxes, representing elements of Di (↑g ) that differ one from the other by a sum (with coefficients ±1) of relations depicted in Fig. 8. Let us call them Relations 9 as reference to Fig. 9 in [21]. By linearity, this defines a homomorphism of Q-vector spaces α : Di ( g ) → Di (↑g )/R9, which when restricted to Di (↑g ) → Di (↑g )/R9 is the canonical quotient map. Here R9 is the Q-vector subspace of Di (↑g ) generated by the set of Relations 9. Proposition 2.1(b) implies that Relations 9 are true in Ai (↑g ), i.e. R9 ⊂ Ri (↑g ). Let β : Di (↑g )/R9 → Di (↑g )/Ri (↑g ) be the canonical projection. Let us observe that for every branching relation R, α(R) = 0. Therefore (β ◦ α)(R) = 0, so if we denote by Bi the Q-vector subspace of Di ( g ) generated by the set of branching relations, then Bi ∩ Di (↑g ) ⊂ Ri (↑g ). On the other hand, any IHX, AS, STU or branching relation on g is, by Proposition 2.1, a sum of IHX, AS and STU relations on φ(↑g ), plus a sum of branching relations. Indeed, an IHX or AS relation refers only to a neighborhood outside g − φ(↑g ), hence the “moving” procedure can be applied simultaneously to all terms of the relation; while a STU relation is, up to sign, a box-STU relation, therefore using Proposition 2.1(c) can be “moved” to a box-STU relation with support in g − φ(↑g ), the later being a consequence of Ri (↑g ) by Proposition 2.1(a). The difference between the start and the end of each step of a “moving” procedure is, of cause, an element of Bi . Hence Ri ( g ) = Ri (↑g ) + Bi . The two established relations imply Ri ( g ) ∩ Di (↑g ) ⊂ Ri (↑g ). Since the opposite inclusion is obvious, Ri ( g ) ∩ Di (↑g ) = Ri (↑g ). Then, by the second isomorphism theorem for vector spaces, Di (↑g )/Ri (↑g ) ∼ = Di ( g )/Ri ( g ). Composing with φ∗ from the first paragraph, we obtain φ∗ : Ai (↑g ) → Ai ( g ), for every i ≥ 0. Moreover the induced φ∗ : A(↑g ) → A( g ) preserves the topology.

Remarks. This proposition still holds if has two or more connected components, but we can “eliminate” the horizontal line of only one component. If we “eliminate” more than one horizontal line, the corresponding φ∗ is still well-defined and surjective. 2.2. The algebra structure of A(↑g ). Let Ac (↑g ) be the Q-vector subspace of A(↑g ) generated by formal series of diagrams on ↑g with no components of the dashed graph


y'

y'

y' _

= x'

611

x'

x'

Fig. 9. Two consecutive external vertices from connected components x and y can be interchanged up to ± a diagram with dashed graph having one component less

disconnected from the support. Viewing each chord diagram as a union of the connected components of the dashed graph that do not meet the support with the part that meets the support, we get A(↑g ) = A(∅) ⊗Q Ac (↑g ). Let a(↑g ) be the Q-vector subspace of Ac (↑g ) generated by formal series of diagrams on ↑g with non-empty and connected dashed graph (and connected to the support). a(↑g ) is precisely the set of primitive elements of Ac (↑g ). A similar notation a() for any abstract graph is self-evident. Ac (↑g ) is an algebra with respect to justaposition of the bold vertical arrows. Denote this associative, generally (if g > 1) non-commutative operation •. In fact Ac (↑g ) is a co-commutative Hopf algebra [24, Prop. 1.5]. Recall the following “common knowledge”: Proposition 2.3. 1) a(↑g ) is a Lie algebra over Q with respect to the operation (x, y) → x • y − y • x. 2) Let I be the topological ideal of Ac (↑g ) generated by a(↑g ). Then exp : I → 1+ I ∞ ∞ x n n and log : 1 + I → I , defined by exp(x) = n=0 n! and log(1 + x) = n=1 (−1)n+1 xn , where the product is the operation •, satisfy exp ◦ log = id1+ I . In I and log ◦ exp = id particular, exp and log are bijections. 3) exp is a bijection from a(↑g ) ⊂ I to the set of group-like elements in 1 + I. 4) If α, β ∈ a(↑g ), then exp(α) • exp(β) = exp(γ ) for some γ ∈ a(↑g ). Moreover, γ is given by the Campbell-Hausdorff formula. 5) I coincides with the set of formal series of chord diagrams of degree ≥ 1. Proof. 1) The statement is sufficient to prove for x, y = diagrams with connected dashed graph. Using STU relations, as shown in Fig. 9, we can interchange two consecutive external vertices, one from x, the other from y, on any bold arrow, up to ± a diagram with connected dashed graph. Therefore, iteratively we can interchange all external vertices of x, with all external vertices of y, obtaining x • y − y • x = a sum of diagrams with connected dashed graph. 2), 3) and 4) are classical statements. The proofs in [22, Theorem 7.2, Corollary 7.3 and Theorem 7.4] apply môt-a-môt. For 3) and 4) note that if γ is primitive, then γ ∈ a(↑g ). 5) Since the set of formal series of chord diagrams of degree ≥ 1 is an ideal containing a(↑g ), and is closed topologically, I certainly belongs to it. Conversely, pick an arbitrary connected component y of the dashed graph of a chord diagram. Observe that using the “‘trick”’ in Fig. 9, up to ± a sum of diagrams with the number of connected components of the dashed graph less by 1, y can be assumed to have all external vertices below all the other external vertices of the diagram. Hence an induction on the number of connected

612


components of the dashed graph shows that any chord diagram of degree ≥ 1 is a sum of terms of type ±z 1 • z 2 • . . . • z k , k ≥ 1, with z i a diagram in a(↑g ). We conclude that the set of finite sums of chord diagrams of degree ≥ 1 is contained in I . Hence so is its completion.

This proposition also holds if we replace Ac (↑g ) by A(↑g ), Q by A(∅), and a(↑g ) by a(↑g ) + a(∅) · 1, where a(∅) is the set of primitive elements of A(∅). 2.3. The LMO invariant for closed manifolds and extending the maps ιn . In [17] from the Kontsevich integral an invariant of oriented framed links L was constructed, which does not change under Kirby-1,2 moves and change of orientation of components of L. We recall it here, together with the maps ιñ , necessary to extend it to an invariant of unions of embedded framed chain graphs in S 3 . ◦

Let A(∅) be the formal series completion of the Q-vector space generated by the homeomorphism classes of open chord diagrams without univalent vertices (but allowing dashed self-loops - these are set of degree 0) modulo AS and IHX relations. Let B(X ) be the formal power series completion of the Q-vector space generated by the homeomorphism classes of open chord diagrams without dashed self-loops, with the univalent vertices colored by elements of X , modulo AS and IHX relations. not Denote [m] = {1, . . . , m}, and let = m S 1 , where each component is colo be the subspace of B([m] ∪ {∗}), gered by a different element of [m]. Let B([m]) → B([m]), nerated by the diagrams with one ∗-colored vertex, and f i : B([m]) f i := average o f the diagrams obtained by attaching the ∗ −ver tex near B([m]) all i − ver tices. We can define a map ϕ : C(m) := → A(), ϕ := average of the diagrams obtained by attaching i-colored vertices to the i th copy of S 1 in , ∀i. (One checks that the definition on diagrams extends over relations to a map between formal series completions.) This map is in fact an isomorphism of Q-modules (and coalgebras). For details, please consult [17, 24]. (If = S 1 , C(1) and A(S 1 ) are algebras, but ϕ is not an algebra homomorphism.) ◦

For every n ≥ 0 define a map κn : C(m) →A(∅); κn (K ) = 0, if ∃i such that the number of i-colored vertices is not 2n, κn (K ) = sum o f all ways o f attaching ◦

i − color ed ver tices in pair s, ∀i, otherwise. Let On be the ideal of A(∅) generated ◦ by i+ 2n. (A(∅) is an algebra with respect to disjoint union.) It can be shown that ◦ as modules (and even as algebras) A(∅) /On ∼ = A(∅). Now, let ιn = qn ◦ κn ◦ ϕ −1 : ◦

◦

A() → C(m) →A(∅)→A(∅) /On ∼ = A(∅), where qn is the quotient map. Let κn∗ : B(X {∗}) → B(X ) be defined as κn , but only involving ∗-colored vertices (see [17, 24] for details). Let Pn = Im(κn∗ ). The map κn∗ passes to the quotient from the definition on C(m), and hence we get a submodule Pn of C(m). The relations Pn also ◦

◦

commute with ϕ. Define the quotient map jn :A(∅) /On →A(∅) /On , Pn+1 of graded modules. It is isomorphism in degree ≤ n, and is the main ingredient in showing that deg≤n ιn ( Zˇ (L)) is invariant under the second Kirby move [17, 24]. This construction can be extended for = g1 ( m S 1 ) g2 , disjoint union of ◦ two chain graphs and m copies of S 1 , i.e. ιn : A( m S 1 ) →A(∅) /On ∼ = A(∅) can be extended (meaning that for g1 = g2 = 0, ιñ acts exactly as ιn ) to a map:


613

◦

ιñ = qñ ◦ κñ ◦ ϕ˜ −1 : A( g1 ( m S 1 ) g2 ) →A( g1 g2 ) /On ∼ = A( g1 g2 ), (2.1) where the corresponding homomorphism ϕ˜ −1 refers only to all present circle components of . Here, to define the preimage of ϕ˜ : C( g1 , [m], g2 ) → A( g1 , m S 1 , g2 ) we consider absolutely analogous chord diagrams with support the disjoint union of two chain graphs g1 , g2 , and points indexed by elements of [m] (it is convenient not to call these points vertices), κ˜ n is extended in the same manner, and qñ is just the quotient map. ϕ˜ is an isomorphism with the proof of Sect. 2 of [17]. Moreover, the similarly constructed map jñ is an isomorphism in degree ≤ n. Namely, and this is exactly the statement of Lemma 2.3 of [21], if = g1 ( m S 1 ) g2 , then ◦ ◦ ∼ = jñ : deg≤n (A( g1 g2 ) ∼ =A( g1 g2 ) /On ) −→ deg≤n (A( g1 g2 ) /On , Pn+1 ). To check this fact it is enough to follow the proof of Lemma 3.3 in [17] or Proposition 4.4 in [24]. Let Z (L) be the usual Kontsevich integral of the (oriented) framed link L, ν = Z (zer o− f ramed unknot). Denote Zˇ (L) := Z (L)⊗ν |L| , meaning we take the “connected sum” of Z (L) on each of its component with ν. Like Z (L), Zˇ (L) is also group-like of the form 1 + (ter ms o f degr ee ≥ 1).9 Let σ± be the number of positive, resp. negative, eigenvalues of the linking matrix of L. Denote O+1 , resp. O−1 , the unknot with +1, resp. −1, framing, and SL3 the 3-manifold obtained by surgery on the framed link L in S 3 . Recall the definition of the LMO invariant for oriented closed 3-manifolds M ≡ SL3 :

n (SL3 )

:= deg≤n

ιn ( Zˇ (L)) ˇ −1 ))σ− ιn ( Zˇ (O+1 ))σ+ · ιn (Z(O

and Z lmo (M) :=

degn n (M),

(2.2)

(2.3)

n≥0

and for Q-homology spheres also: Z L M O (M) :=

d(M)−n degn n (M),

(2.4)

n≥0

where d(M) = |det (lk(L))|, which is 0 if H1 (SL3 , Q) = 0 and |H1 (M, Z)| otherwise. We use the convention |det (lk(∅))| = 1. Then we have deg≤n n+1 (SL3 ) = d(M) · n (SL3 ), hence we can write deg≤n Z L M O (M) = d(M)−n n (M). More precisely, the following holds [24, Prop. 4.5]: deg≤n [ιn+1 Zˇ (L)] = (−1)|L| det (lk(L))deg≤n [ιn Zˇ (L)],

(2.5)

and therefore we can define: 9 It can be shown by induction that then for |L| = 1 the formal graded series log(element) is a primitive element of A( S 1 ), and has no part of degree 0, hence it is a formal power series of chord diagrams with connected dashed part. More precisely, a statement similar to Proposition 2.3 holds.

614


c+ = lim (−1)n deg≤n [ιn Zˇ (O+1 )],

(2.6)

c− = lim deg≤n [ιn Zˇ (O−1 )].

(2.7)

n→∞ n→∞

These elements of A(∅) are canonical constants in the theory of LMO invariant. Equation (2.5) implies

(−1) N σ+ Zˇ (L) LMO deg≤N Z (M) = · deg≤N ι N . (2.8) σ d(M) N c+σ+ c−− 2.4. The definition of Z on elementary pseudo-quasi-tangles. To extend n (SL3 ) ∈ A(∅) to invariants n (L , G) ∈ A(), where is G as an abstract graph, we will extend now Z (L) to Z (L ∪ G). However we shall do this differently from Murakami and Ohtsuki [21], see Fig. 1 there, who use the Knizhnik-Zamolodchikov associator. We will use the even associator. Let G be an embedded framed graph in S 3 . Fix a plane projection such that G is given the blackboard framing. This projection of G can be decomposed into elementary tangles10 :

@

, @@

@ , , , @

,

,

@

and . @ @ We need only to specify the definition of Z on the first two, since on the others we know it from the link case. Let be an abstract (disjoint union of) chain graph(s), and e be with edge e erased. Suppose e is also a chain graph. A similar notation e G for a framed graph G is self-evident. Define the map (e) : A() → A(e ), (e) (D) = 0, if D has an external vertex on the removed edge, and (e) (D) = D, otherwise. To verify well-definedness of (e) it is enough to check its invariance under branching relations of diagrams on . There are 3 diagrams involved in a branching relation. Suppose v is a trivalent vertex of , and e1 , e2 , e3 the edges adjacent to v. Edge e cannot be repeated twice among e1 , e2 , e3 , since then e would not be a (union of) chain graph(s). Therefore we can assume e = e1 , e = e2 , e = e3 . It is easy to check that then one of the three diagrams in the relation is sent to 0 by (e) , while the other two are sent to diagrams that form an AS relation in A(e ). If e is an edge of G, denote by Se G the graph obtained from G by reversing the orientation of the edge e (without changing the framing). If is the underlying abstract graph of G, denote by Se the underlying abstract graph of Se G. Let S(e) : A() → A(Se ) be the linear map which sends every diagram D in A() to the diagram obtained from D by reversing the orientation of e, multiplied by (−1)m , where m is the number of vertices of D on the edge e. @ We define Z for the elementary tangles to satisfy the following @ and two conditions (compare with [21, Prop. 1.5]): @ ,

(1) Z (Se G) = S(e) Z (G), for any embedded framed graph G and edge e. (2) Z (e G) = (e) Z (G), for any (disjoint union of) embedded chain graph(s) G and edge e, such that e G is still a (disjoint union of) embedded chain graph(s). 10 The words quasi and pseudo are left out for simplicity of language.


Moreover, we seek to define Z (

@

) of the form

615

ai @ ci

bi (for all possible

8 orientations). By Condition (2) above we must have a = b = c−1 , and hence also Z ( ) = m . But Z ( ) = ν 1/2 , therefore we must require a = b = c−1 = a 2 √ √ √ 4 −1 4 4 ν ν ν @ @ . (These . Similarly Z ( @ ) = √ ν 1/4 , i.e. Z ( ) = √ @√ 4 −1 4 4 ν ν ν formulas are each for the 8 possible orientations.)

Theorem 2.4. 11 Z (G) is an isotopy invariant of embedded framed chain graphs. Proof. In large part, this is mostly a repetition of the proofs of the statements in [21, Sect. 1], hence we only sketch here the details that are not identical. First, one shows that Z (G) is invariant under isotopies of the plane. If such isotopies fix a neighborhood of each trivalent vertex, the result is known from the link case. If isotopies move such neighborhoods “as a whole”, the result follows using branching relations [21, Lemma 1.2.] Finally, it is sufficient to show Z (

@

) = Z(

@ @ ) and Z (

Z(

) =

@ ). Secondly, one shows that Z (G) is invariant under extended Reidemeister moves. This is also easily achieved from results known from the link case and the branching relations [21, Lemma 1.4.] To prove the two remaining relations, note that in [16, p. 8] it is proved (using an even √ ν −1 √ √

( ν) −1 ν √ √ @ 4 associator) that Z ( . Therefore Z ( )= )= √ ν 4ν

( ν) @ √ 4 −1 ν 11 This statement is considered known, but a complete proof was missing from the literature.

616


√ 4 −1 ν √ 4 ν −1 √ 4 ν −1 √ @

( ν) branching r elations √ = √ @√ = Z( =√ = = √ 4 @ ν 4 ν 4 4 ν ν ν √ 4 −1 √ 4 −1 ν ν

@ ).

Similarly Z ( @

) = Z(

@ ).

The above properties (1) and (2) we have now for granted (compare to [21, Prop. 1.5]). It then follows directly from their definitions in Sect. 3.1 that τ N and τ also enjoy properties (1) and (2). Conjecture 2.5. 12 If G is a chain graph, then this definition of Z (G) using an even associator coincides with the definition in [21], which uses a KZ associator. Remarks. If we use an even associator it is easy to see that Z ( )= 6. . . 6 6 6 ... ), where φ∗ is the isomorphism from Proposition 2.2. φ∗ ( νm νm Definition 2.6 [21]. For any L ∪ G → S 3 , let Zˇ (L ∪ G) = Z (L ∪ G) ⊗ (ν ⊗|L| ). 2.5. The composition ∗ of chord diagrams. Denote A( g1 , g2 ) := A( g1 g2 ), where the order ( g1 , g2 ) is specified. g1 is the union of its horizontal line and the upper half-circles, g2 is the union of its horizontal line and lower half-circles. As remarked after Proposition 2.2, every element of A( g1 , g2 ) can be represented as a formal series (with rational coefficients) of chord diagrams whose external vertices don’t meet one horizontal line. For α ∈ A( g1 , g2 ), β ∈ A( g2 , g3 ) represented by single chord diagrams x, respectively y, let α ∗ β denote the element of A( g1 , g2 S 1 , g3 ), represented by the diagram obtained by attaching φ∗−1 y (the horizontal line of the first graph removed) on top of φ∗−1 x (the horizontal line of the second graph removed). For g = 0 set ∗ to be the disjoint union. Extend ∗ by linearity to formal power series of chord diagrams. Note that ∗ is associative. Every chord diagram D is a disjoint union D1 D2 , where every connected component of D1 intersects the support, and every connected component of D2 does not. Define the vacuum degree vdeg(D) := deg(D2 ). Since all relations are vdeg-homogeneous, the vacuum-degree N part vdeg N (α) is well-defined ∀N ∈ N, ∀α ∈ A( g1 , g2 ), 12 The results of this paper are equally true for any associator for which Theorem 2.4 holds. We have been able to obtain only partial results toward the proof of this conjecture with direct methods. It follows, however, from results in [7].


617

and is in general a series. For any N ∈ N define A≤N (∅) = A(∅)/(vdeg > N ). Let A≤N +1 (∅) → A≤N (∅) be the forgetful map. The sequence . . . → A≤N +1 (∅) → A≤N (∅) → . . . → A≤1 (∅) → A≤0 (∅) = Q has inverse limit A(∅). Similarly, let A≤N ( g1 , g2 ) = A( g1 , g2 )/(vdeg > N ). Then for any , and ∀N , A≤N () = A≤N (∅) ⊗ Ac (), hence A( g1 , g2 ) is the inverse limit of the sequence: . . . → A≤N +1 ( g1 , g2 ) → A≤N ( g1 , g2 ) → . . . → A≤1 ( g1 , g2 ) → A≤0 ( g1 , g2 ) = Ac ( g1 , g2 ). A≤N (∅) ≤N A () or

is an algebra, and A≤N () is a A≤N (∅)-module: ∀α ∈ A≤N (∅), ∀β ∈ A≤N (∅), α · β := vdeg≤N (α β), where is the disjoint union of chord diagrams. The multiplication • in A(↑g ) induces one on A≤N (↑g ), making it an algeA(), induced by summing over all ways of splitting chord bra. : A() → A()⊗ diagrams into connected components, preserves vacuum degree parts, hence induces a A≤N ()) ⊂ A≤N ()⊗ A≤N (). co-multiplication : A≤N () → vdeg≤N (A≤N ()⊗ But is not an algebra homomorphism when =↑g . Call an element of β ∈ A≤N () α). primitive if β = β ⊗ 1 + 1 ⊗ β. Call α ∈ A≤N () group-like if α = vdeg≤N (α ⊗ Since A≤N (↑g ) = A≤N (∅) ⊗Q Ac (↑g ), Proposition 2.3 holds if we replace A(↑g ), a(↑g ) and I by their vacuum degree ≤ N truncations A≤N (↑g ), a≤N (↑g ) and I ≤N , provided we use the above notions of primitive and group-like. By Proposition 2.2, it then also holds for A≤N ( g ). Let: Z (Tg ) ⊗ (ν 1/2 )⊗2g ∈ A(↑g , ↑g ), g g c+ · c−

Z (Tg ) ⊗ (ν 1/2 )⊗2g = vdeg≤N ∈ A≤N (↑g , ↑g ), g g c+ · c−

zg = z gN

(2.9) (2.10)

where Tg is the q-tangle from Fig. 1c with the non-associative structure (. . . (((••)(••)) (••)) . . .), ν = Z (O) ∈ A( -i) is the Kontsevich integral of the zero-framed unknot, and ⊗ means taking the connected sum of chord diagrams on each of the 2g components, c+ , c− have been defined in Sect. 2.3. Note that Z (Tg ) ⊗ (ν 1/2 )⊗2g = vdeg0 (Z (Tg ) ⊗ (ν 1/2 )⊗2g ). For g = 0 define z 0 = z 0N = 1. Proposition 2.7. 1) Let ∗ be the gluing operation defined above, let ιÑ : A( g1 ( 2g2 S 1 ) g3 ) → A( g1 g3 ) be the A(∅)-linear map defined by (2.1), which refers exactly to all present circle components (in this case 2g2 ). Then13 N (α, β) := vdeg≤N ((−1)g N ιÑ (α ∗ z gN2 ∗ β))

(2.11)

defines a A≤N (∅)-bilinear form A≤N ( g1 , g2 ) ⊗ A≤N ( g2 , g3 ) → A≤N ( g1 , g3 ). 2) For any N and any respective elements α, β, γ we have N ( N (α, β), γ ) = N (α, N (β, γ )). 13 For elements belonging to the space of chord diagrams on specific 2g arrows which connect 2g points on a “bottom line” with 2g points on a “top line”, as is z g , one can similarly define ∗.

618


Proof. 1) Let α ∈ A≤N ( g1 , g2 ), β ∈ A≤N ( g2 , g3 ), z gN2 ∈ A≤N ( g2 , g2 ). N (α, β) is well-defined. Indeed, it can be calculated in two steps: first ϕ˜ −1 (α ∗ z gN2 ∗ β), then apply q˜ N ◦ κ˜ N . Suppose α = 0 ∈ A≤N ( g1 , g2 ). Then, since ϕ˜ is an isomorphism, vdeg≤N ϕ˜ −1 (α ∗ z gN2 ∗ β) = 0 (i.e. if we factor in C( g1 , [m], g2 ) by the subspace spanned by diagrams with vacuum degree > N ). Hence vdeg≤N ιÑ = 0 ∈ A≤N ( g1 , g2 ), and similarly for β. A≤N (∅)-bilinearity of (α, β) → vdeg≤N ιÑ α ∗ z gN2 ∗ β is obvious. α ∗ z gN2 ∗ β

2) follows from the fact that ∗ is associative, and ιÑ (˜ι N (α ∗ z gN1 ∗ β) ∗ z gN2 ∗ γ ) = ιÑ (α ∗ z gN1 ∗ ιÑ (β ∗ z gN2 ∗ γ )) = ιÑ (α ∗ z gN1 ∗ β ∗ z gN2 ∗ γ ).

Note that when g1 = 0, N becomes a A≤N (∅)-linear map: N : A≤N ( g2 ) ⊗ A≤N ( g2 , g3 ) → A≤N ( g3 ). Hence every element in A≤N ( g2 , g3 ) defines a A≤N (∅)-linear map from A≤N ( g2 ) to A≤N ( g3 ), and the induced map Ñ : A≤N ( g2 , g3 ) → A≤N ( g2 )∗ ⊗ A≤N ( g3 ) is A≤N (∅)-linear. Using the isomorphism φ∗ : A≤N (↑gi ) → A( gi ) we obtain an A≤N (∅)-linear map also denoted Ñ : A≤N ( g1 , g2 ) → A≤N (↑g1 )∗ ⊗ A≤N (↑g2 ). In fact the image of Ñ lies in the space of continuous operators, since we will show in Sect. 3 that the pairing N is continuous. Extend g → ( g , ∅) to A≤N ( g ) → A≤N ( g , ∅), and compose with Ñ to obtain a A≤N (∅)-linear map ( N )∗ : A≤N ( g ) → A≤N ( g )∗ . Namely ( N )∗ (β)(α) = N (α, β), ∀α, β ∈ A≤N ( g ). Similarly, there is a map ( N )∗ : A≤N (↑g ) → A≤N (↑g )∗ . The second part of the above proposition shows that Ñ ( N (β, γ )) = Ñ (γ ) ◦ Ñ (β) for any corresponding β, γ , i.e. the following diagram is commutative: A≤N ( g1 , g2 ) ⊗ A≤N ( g2 , g3 ) ↓ Ñ ⊗ Ñ A≤N( g1 )∗ ⊗A≤N ( g2 )⊗A≤N ( g2 )∗ ⊗A≤N ( g3 )

N

−→ evaluation

−→

A≤N ( g1 , g3 ) ↓ Ñ A≤N ( g1 )∗ ⊗A≤N ( g3 )

Remarks. If we were to use Knizhnik-Zamolodchikov or any other associator, in the definition of , between α, z g and β, we would have to insert an element A (and its horizontal reflection) from the space of chord diagrams on 2g arrows alternatively oriented downward and upward, such that φ∗−1 Z ( ) ∗ A = ν ⊗g ; and similarly 6. . . 6 for β. For the even associator, A can be taken 1, i.e. it can be omitted. Conjecture 2.5 above claims that one can take A = 1 for any associator. 2.6. The categories A≤N and A. Let A≤N be the category with objects A≤N (↑g ) ≡ A≤N ( 0 , ↑g ), g ≥ 0, and morphisms the set of A≤N (∅)-homomorphisms between these modules. Similarly define the category A. The isomorphism φ∗ of Proposition 2.2 identifies A(g ) and A(↑g ). The following two statements are proved in Sect. 3. Proposition 2.8. Let N be the bilinear form defined by the previous proposition, and Ñ be the induced map A≤N ( g1 , g2 ) → A≤N (↑g1 )∗ ⊗ A≤N (↑g2 ) = H om(A≤N (↑g1 ), A≤N (↑g2 )). Denote wg := Z (Wg ) = vdeg≤N Z (Wg ) ∈ A≤N ( g , g ), where Wg is the


619

embedded framed graph in Fig. 1a, with the first g corresponding to the lower of the two chain graphs in the picture, and the second - to the upper. For g = 0, set wg = 1. Then Ñ (wg ) is the identity operator on A≤N (↑g ). Theorem 2.9. 1) There is a (unique) A(∅)-bilinear form : A( g1 , g2 )⊗A( g2 , g3 ) → A( g1 , g3 ), such that (vdeg≤N ) ◦ = N . 2) Let ˜ be the induced map A( g1 , g2 ) → A(↑g1 )∗ ⊗A(↑g2 ) = H om A(∅) (A(↑g1 ), A(↑g2 )), and denote as before wg = Z (Wg ) ∈ A( g , g ), where Wg is shown in Fig. 1a. ˜ g ) is the identity operator on A(↑g ). Then (w 3. The TQFT Now we construct the truncated (with respect to the vacuum degree14 ) and full TQFTs. We will show that they are non-degenerate and anomaly-free. All cobordisms in the category Q are connected, hence it does not make sense to require multiplicativity or self-duality. A TQFT based on Q has to satisfy four axioms [6], similar to those of Atiyah [1]: Naturality: Axiom (A1) in [6], or (III.1.4.1) in [23]. Functoriality: Axiom (A2) in [6], or (A4) in [21], or (III.1.4.3) in [23]. Normalization: Axiom (A3) in [6], or (III.1.4.4) in [23]. pseudo-Hermitianity: Axiom (A4) in [6], or (A2) in [21], or (III.5.2.2) in [23]. Assign to every oriented closed surface of genus g ≥ 0 the A(∅)-vector space A( g ) ∼ = A(↑g ), and, in the case of truncations, ∀N ≥ 0, the A≤N (∅)-vector space A≤N ( g ) ∼ = A≤N (↑g ). Naturality is therefore trivial. The other axioms, together with the nondegeneracy property15 are stated in Theorem 3.2 below. Let L ∪ G → S 3 be an arbitrary embedding of a link and a (union of) chain graph(s) in S 3 . Let σ+ , σ− be the number of positive, respectively negative eigenvalues of lk(L), the linking matrix of L, let g be the number of circle components of G, and let be G as an abstract graph. For every n ∈ N define:

ιñ ( Zˇ (L ∪ G)) n (L , G) := vdeg≤n ∈ A≤n () ⊂ A(). ˇ −1 ))σ− ιn ( Zˇ (O+1 ))σ+ · ιn (Z(O Here Zˇ (L ∪ G) is an isotopy invariant (see [21, Theorem 1.4], where KZ associator is used, or Sect. 2.4, where an even associator is used) of L ∪ G ⊂ S 3 . Note that once Z (L ∪ G) has been shown well-defined, it does not matter which associator we use. Proposition 3.1. 1) The following relation holds in A() for any16 chain graph G and link L: vdeg≤n ιñ+1 Zˇ (L , G) = (−1)|L| det (lk(L))vdeg≤n ιñ Zˇ (L , G) , (3.1) 14 Since the map ι˜ , which we had to introduce if we want to have invariance under Kirby moves for chain N graphs, decreases the total degree of a diagram by 2g N , and since ιÑ must be applied every time we glue two cobordisms, one does not expect the theory to truncate with respect to the total degree of chord diagrams. 15 We show (∀N and non-truncated, and ∀g) that the closure of the subspace spanned by the images of cobordisms is all the respective space of chord diagrams. 16 possibly with several connected components

620


and therefore: vdeg≤n n+1 (L , G) = d(SL3 ) · n (L , G). 2)

1 (L , G) d(SL3 ) n

is a group-like element of A≤n () ⊂ A() of the form 1+ higher

order terms.

Proof. 1) Follow the proof of Proposition 4.5 in [24]. 2) First note that Z of any elementary pseudo-quasi-tangle is group-like of the desired form. Indeed, if one uses a KZ associator, the elements a, b in [21, p. 503] used in the definition of Z for the vicinity of trivalent vertices are clearly so. If one uses an even associator, then this follows from the fact that ν = ν ⊗ ν and ν = 1 + h.o.t. Hence Zˇ (L ∪ G) is group-like of the form 1 + h.o.t. for any L ∪ G → S 3 (compare with [17, Subsect. 1.4]). That commutes with ιñ follows from the fact that commutes with ϕ, ˜ and an explicit calculation of ◦ (qñ ◦ κ˜ n ) and (qñ ◦ κ˜ n ) ⊗ (qñ ◦ κ˜ n ) ◦ for any diagram with 2n legs of each color 1, . . . , |L|, just as in the case G = ∅ [24, 17]. Similarly it then follows that d(S1 3 ) n (L , G) has the form 1 + h.o.t. (compare with [17, Lemma 4.7]). L

Let (M, f 1 , f 2 ) be an arbitrary Q-cobordism (in particular, a morphism in the category Q between g1 and g2 ). Let (L , G 1 , G 2 ) be such that κ(L , G 1 , G 2 ) = (M, f 1 , f 2 ). By Proposition 2.1 in [21], the ambiguity in this choice is a finite sequence of KI and extended KII moves, and change of orientation of a link component. Define:

ιñ ( Zˇ (L ∪ G 1 ∪ G 2 )) 1 · vdegn τ (M, f 1 , f 2 ) = ˇ −1 ))σ− |det (lk(L))|n ιn ( Zˇ (O+1 ))σ+ · ιn (Z(O n≥0

ιñ ( Zˇ (L ∪ G 1 ∪ G 2 )) (−1)σ+ n (2.5)

= · vdegn ∈ A( g1 , g2 ), σ− σ+ |det (lk(L))|n c · c + − n≥0 (3.2) where ιñ refers to the circle components of chord diagrams, all coming here from the components of the link L, c+ , c− have been defined in Sect. 2.3, vdegn means taking the vacuum degree n part, and for simplicity we suppose that G 1 , G 2 as abstract graphs are g1 , g2 . We use the convention det (lk(∅)) = 1. Also, let:

ιÑ ( Zˇ (L ∪ G 1 ∪ G 2 )) (−1)σ+ N N · vdeg≤N τ (M, f 1 , f 2 ) = ∈ A≤N ( g1 , g2 ), σ N c+σ+ · c−− d( M) (3.3) = |H1 ( M, Z)|. Note that τ N (M, f 1 , f 2 ) = vdeg≤N τ (M, f 1 , f 2 ). The where d( M) same proof as in the case of closed manifolds (Sect. 3 in [17], or Sect. 4 in [24], or [21, Prop. 2.4]), shows17 that τ (M, f 1 , f 2 ), being invariant under KI and extended KII moves, and change of orientation of a link component, is independent of the choice of the triplet (L , G 1 , G 2 ) for (M, f 1 , f 2 ). Hence τ (M, f 1 , f 2 ) ∈ A( g1 , g2 ) and τ N (M, f 1 , f 2 ) ∈ A≤N ( g1 , g2 ) are invariants of Q-cobordisms (in particular of 17 looking at every (total) degree part


621

morphisms in the category Q). As we have seen in Sect. 2.5, Ñ (τ N (M, f 1 , f 2 )) is then a A≤N (∅)-homomorphism from A≤N (↑g1 ) to A≤N (↑g2 ). It is obvious to similarly define (3.2) and (3.3) for cobordisms (M, ∅, f ) with is a only one connected and parametrized boundary component, as long as M Q-homology sphere18 . Note that we think of the boundary as the top of the 3-cobordism. We associate to (M, ∅, f ) an element τ (M, ∅, f ) ∈ φ∗−1

A( g )

φ∗−1

∼ = A(↑g ), and an element

τ N (M, ∅, f ) ∈ A≤N ( g ) ∼ = A≤N (↑g ), ∀N . Theorem 3.2. 1) Let A≤N τ (↑g ), respectively Aτ (↑g ), be the Q-vector subspace of A≤N (↑g ), respectively A(↑g ), generated by all φ∗−1 τ N (M, ∅, f ), respectively by all is a Z-homology sphere. Then the completion of A≤N φ∗−1 τ (M, ∅, f ), such that M τ (↑g ) is A≤N (↑g ), and the completion of Aτ (↑g ) is A(↑g ). g g g g 2) Let g, g ∈ N, and let A≤N τ ( , ), respectively Aτ ( , ), be the Q-vector subspace of A≤N ( g , g ), respectively A( g , g ), generated by all τ N (M, f, f ), respectively by all τ (M, f, f ), which belong to the category Z. Then the completion of g g ≤N ( g , g ), and the completion of A ( g , g ) is A( g , g ). A≤N τ τ ( , ) is A 3) Let (M1 , f 1 , f 1 ) and (M2 , f 2 , f 2 ) be two Q-cobordisms, such that their composition is also a Q-cobordism. Then, for any N ∈ N, the following gluing formula without anomaly holds: τ N (M2 ∪ f2 ◦( f )−1 M1 , f 1 , f 2 ) = N (τ N (M1 , f 1 , f 1 ), τ N (M2 , f 2 , f 2 )). 1

(3.4)

4) Let (M1 , f 1 , f 1 ) and (M2 , f 2 , f 2 ) be two Q-cobordisms, such that their composition is also a Q-cobordism. Then the following gluing formula without anomaly holds: τ (M2 ∪ f2 ◦( f )−1 M1 , f 1 , f 2 ) = (τ (M1 , f 1 , f 1 ), τ (M2 , f 2 , f 2 )). 1

(3.5)

5) Let ( × [0, 1], ( × 0, p1 ), ( × 1, p2 )) be the identity 3-cobordism of genus g (see Sect. 1.1), then the following holds: τ N (g × [0, 1], (g × 0, p1 ), (g × 1, p2 )) = idA≤N ( g , g ) , τ (g × [0, 1], (g × 0, p1 ), (g × 1, p2 )) = idA( g , g ) .

(3.6) (3.7)

6) For every g, g ∈ N, there exist antimorphisms (maps linear in 0-supergrading and antilinear in 1-supergrading) · : A( g ) → A( g ) and · : A( g , g ) → A( g , g ), such that for every cobordism M between g and g : τ (−M) = τ (M) and τ N (−M) = τ N (M).

(3.8)

It is known [11, Prop. 13.1], that for every chord diagram ξ ∈ A(↑g ) of degree m, with connected dashed graph, there exist string links L ± , with lk(L ± ) = (0), such that Z (L ± ) = 1 ± ξ + o(m + 1). There is a very intuitive topological realization of L ± from Habiro’s calculus of claspers [14, 16]. 18 which is equivalent to the equivalence class of the cobordism (M − {a ball}, S 2 → ∂(the ball), f ) being in the category Q

622


Lemma 3.3. For every n ≥ 0 and every chord diagram ξ on ↑g , deg(ξ ) ≤ n, there exist string links L 1 , . . . , L k , with zero linking matrix, and positive integers a1 , . . . , ak , such k that ai Z (L i ) = ξ + o(n + 1). i=1

Proof. Induction on n. For n = 0, Z (trivial string link) = 1 ∈ A(↑g ). For n = 1, ξ must have connected dashed graph, hence the claim follows from the mentioned result of Habegger and Masbaum, because Z (trivial string link) = 1. For general n, suppose ξ has degree m. We prove the statement first for m = n, then for m = n − 1, . . . , 1 (m = 0 is obvious). For arbitrary m, by the same argument as in the proof of Proposition 2.3.5) we can assume ξ = ±ξ1 • . . . • ξk , ξi have connected dashed graph and degree ≥ 1. If k > 1, by the induction hypothesis there exist α ij ∈ Z and string links L ij , i1 such that j a ij Z (L ij ) = ξi + o(n), ∀i. Therefore a1 · · · akik · Z (L i11 • . . . • L ikk ) = i 1 ,...,i k

i 1 ,...,i k

a1i1 Z (L i11 )•. . .•akik Z (L ikk ) = (ξ1 + o(n))•. . .•(ξk + o(n)) = ξ1 •. . .•ξk +o(n +1). i

Moreover, since all L jj have zero linking matrix, so does L i11 • . . . • L ikk . If k = 1, by the Habegger-Masbaum result, there is a string link L, such that Z (L) − Z (trivial string link) = ξi + o(m + 1). Therefore the statement for m follows from the fact that it holds for m + 1 (express in the later formula the degree m + 1 terms of o(m + 1)). If k = 1 and m = n, it is precisely the Habegger-Masbaum result. Note that all coefficients ai appearing throughout the proof can be arranged positive or negative as we wish [14, 16], hence the ones in the statement can be ensured positive.

It is known [16 Theorem 4.5; see also 12] that for any connected trivalent graph D of degree n there exist Z-homology 3-spheres M ± such that Z L M O (M ± ) = 1 ± D + o(n + 1) ∈ A(∅). (This is proved there for Z lmo , but it is obviously then true for Z L M O .) Since Z L M O (S 3 ) = 1 ∈ A(∅), with a proof absolutely similar to the one above, we have: Lemma 3.4. For any n ≥ 0 and any chord diagram ξ ∈ A(∅), deg(ξ ) ≤ n, there exist k Z-homology spheres M1 , . . . , Mk and positive integers b1 , . . . , bk such that bi Z L M O i=1 (Mi ) = ξ +o(n +1). In particular the set { i bi Z L M O |Mi Z-homology sphere, bi ∈ N∗ } is dense in A(∅). Proof of Theorem 3.2.1). Let ↓↑2g denote the graph with 2g edges oriented alternatively down- and upward. Lemma 3.3 is clearly true for ξ ∈ A(↓↑2g ) as well. Therefore for every n ≥ 0 and every β ∈ A(↓↑2g ) there exist string links L i and ai ∈ Q such ⊗g that ai · Z (L i ) = β • ↓ ⊗ν −1 + o(n + 1). Using the operation ∗ defined in ... ) on top of, and Z ( . . . ) below each side of Sect. 2.5, attach Z ( this equality, to obtain the existence of embedded framed graphs G i and ai ∈ Q, such that ⊗g ... )∗ β • ↓ ⊗ν −1 ∗ Z ( . . . )+o(n +1) = ai · Z (G i ) = Z ( i …i i + o(n + 1), where β → β : A(↓↑2g ) → A(- β - ) is the map induced by inclusion. The later is well-defined, since at the level of D(↓↑ 2g ) AS, IHX and STU i …- i i relations are sent to the same type relations on - - . It is clearly surjective by Proposition 2.2. Therefore for every n ≥ 0 and α ∈ A(↑g ) there exist G i and ai ∈ Q such that ai · Z (G i ) = φ∗ (α) + o(n + 1).


623

Let N = 0. Then by (3.3) we have τ 0 (M, ∅, f ) = ι˜0 Zˇ (L ∪ G) whenever κ(L , G) = (M, ∅, f ). Since ιÑ refer only to link components, for every embedded framed graph G we obtain τ N (κ(∅, G)) = τ 0 (κ(∅, G)) = ι˜0 Zˇ (G) = Zˇ (G) = Z (G). Together with the conclusion of the previous paragraph this shows that, for every n ≥ 0, and every i = S 3 , and α ∈ A≤0 (↑g ) = Ac (↑g ), there exist cobordisms (Mi , ∅, f i ) having M ai ∈ Q, such that

ai · τ 0 (Mi , ∅, f i ) = φ∗ (α) + o(n + 1). (3.9) i

This proves the theorem for N = 0. Let N > 1. Recall that A≤N (↑g ) = A≤N (∅) ⊗Q Ac (↑g ). Therefore the statement is enough to prove for ξ · α, ξ ∈ A≤N (∅) and α ∈ Ac (↑g ). By Lemma 3.4 there exist Z-homology spheres Mi and bi ∈ N∗ , such that bi Z L M O (Mi ) = ξ + o(N + 1). Then, by the previous paragraph, for every n ≥ 0, there exist cobordisms (M j , ∅, f j ) having L M O (M ) · 0 j = S 3 , and ai ∈ Q, such that vdeg≤N M b Z a i i i j j τ (M j , ∅, f j ) = ξ · φ∗ (α) + o(n + 1). Therefore:

φ∗ (ξ · α) = ξ · φ∗ (α) = bi a j vdeg≤N Z L M O (Mi ) · τ 0 (M j , ∅, f j ) + o(n + 1) =

i, j

bi a j τ (Mi )τ 0 (M j , ∅, f j ) N

i, j

+o(n + 1) = =

bi a j τ

N

bi a j τ N (SL3 i )τ 0 (κ(∅, G j )) + o(n + 1)

i, j

SL3 i #κ(∅, G j ) + o(n + 1),

i, j

where bi a j ∈ Q, Mi = SL3 i , (M j , ∅, f j ) = κ(∅, j ) and SL3 i #κ(∅, G j ) = κ(L i G j ), whose filling is SL3 i , a Z-homology sphere. This proves the theorem for arbitrary N . Since for any cobordism (M, ∅, f ) we have vdeg≤N τ (M, ∅, f ) = τ N (M, ∅, f ), by taking N = n, we can see that, for every n ≥ 0 and any α ∈ A(↑g ), there exist ci ∈ Q, and i Z-homology spheres, such that φ∗ (α) = ci vdeg≤n τ (Mi , ∅, f i )+ (Mi , ∅, f i ) with M i o(n + 1) = ci τ (Mi , ∅, f i ) + o(n + 1).

i

Let (L , G, G ) be a triplet and (M, f, f ) = κ(L , G, G ). We can talk about linking number between a link component K and a circle U of a chain graph, as well as between two circles U and V of chain graphs: lk(K , U ) = lk(U, K ) is defined to be the linking number between K , and the knot obtained from the graph by deleting all but the circle component U , and similarly for lk(U, V ). The linking matrix of a triplet is then: ⎛ ⎞ ⎛ ⎞ lk(L) lk(L , G) lk(L , G ) A BT CT lk(L , G, G ) = ⎝ lk(G, L) lk(G, G) lk(G, G ) ⎠ = ⎝ B D E T ⎠ , (3.10) lk(G , L) lk(G , G) lk(G , G ) C E F where A, D, F are symmetric matrices. In [6] it has been shown that the doublyLagrangian condition can be expressed: D = B A−1 B T , F = C A−1 C T

(3.11)

624


(for Q-cobordisms this in particular means that the entries on the left-hand side, a priori in Z det1 A , must be in Z), and for the case F = Q, additionally B A−1 C T ∈ Mg1 ×g2 (Z). In the case L = ∅, the doubly-Lagrangian condition can be expressed: D = F = 0.

(3.12)

Proof of Theorem 3.2.2). The proof of part 1) does not use the fact that in Lemma 3.3 the links L i have zero linking matrix, because for connected chain graphs the Lagrangian condition is trivial. Since a Z-cobordism (M, f, f ) is doubly-Lagrangian iff any representing triplet satisfies condition (3.12), the same proof as in part 1) shows that, for ∈ Q, and doubly-Lagrangian every n ≥ 0, and every α ∈ Ac ( g , g ), there exist ai i = S 3 , such that i ai · τ 0 (Mi , f i , f ) = α + Z-cobordisms (Mi , f i , f i ) having M i o(n + 1), proving the theorem for N = 0. This identity replaces (3.9) in the proof of part 1), and the argument can be continued to prove part 2) for any N , and for non-truncated τ.

Lemma 3.5. [6, Lemma 5] Let (M1 , f 1 , f 1 ) and (M2 , f 2 , f 2 ) be two 3-cobordisms. Suppose (M1 , f 1 , f 1 ) = κ(L 1 , G 1 , G 1 ), (M2 , f 2 , f 2 ) = κ(L 2 , G 2 , G 2 ), and (M2 ∪ f2 ◦( f )−1 1 M1 , f 1 , f 2 ) = κ(L 1 ∪ L 0 ∪ L 2 , G 1 , G 2 ), the later triplet obtained from the previous two by the construction described in Proposition 1.3. Denote σ+1 = sign + (lk(L 1 )), σ+2 = sign + (lk(L 2 )), σ+ = sign + (lk(L 1 ∪ L 0 ∪ L 2 )), and let g be the genus of the connected closed surface along which is this splitting. Then the integer s(M, M1 , M2 ) = σ+1 + σ+2 + g − σ+ is an invariant of the decomposition M = M2 ∪ f2 ◦( f )−1 M1 ), i.e. it 1 does not depend on the choice of triplets representing the 3-cobordisms M1 and M2 .

Lemma 3.6. Let (M1 , f 1 , f 1 ) and (M2 , f 2 , f 2 ) be two Q-cobordisms, such that their composition is also a Q − cobor dism. Denote d = |H1 ( M2 ∪ f 2 ◦( f 1 )−1 M1 , Z)|, d1 = 1 , Z)|, d2 = |H1 ( M 2 , Z)|. Suppose that these cobordisms are glued along a |H1 ( M surface of genus g. Then: τ (M2 N

M1 , f 1 ,

f 2 ◦( f 1 )−1

f 2 )

σ+1 +σ+2 +g−σ+ c+ d1 d2 N N = (−1) vdeg≤N c− d · N (τ N (M1 , f 1 , f 1 ), τ N (M2 , f 2 , f 2 )),

(3.13)

where (−1) N · vdeg≤N (c+ /c− ) ∈ A≤N (∅), the multiplication by scalars is thought in the category A≤N , and σ+1 + σ+2 + g − σ+ is an integer. Proof. Let (L 1 , G 1 , G 1 ), (L 2 , G 2 , G 2 , ), and (L 1 ∪ L 0 ∪ L 2 , G 1 , G 2 ) represent the 3cobordisms M1 , M2 , and M2 ◦ M1 , as in the previous lemma, and let (σ+ , σ− ), (σ+1 , σ−1 ), respectively (σ+2 , σ−2 ) be the signatures of lk(L 1 ∪ L 0 ∪ L 2 ), lk(L 1 ), resp. lk(L 2 ). Then, locally abbreviating (when space requires) vdeg≤N (c+ ) and vdeg≤N (c− ) to c+ and c− : ⎛ ⎞ τ N ⎝ M2 M1 , f 1 , f 2 ⎠ f 2 ◦( f 1 )−1

(−1)σ+ N = · vdeg≤N dN

ιÑ Zˇ (L 1 ∪ L 0 ∪ L 2 ∪ G 1 ∪ G 2 ) σ

c+σ+ · c−−


625

1 2 c+ σ+ +σ+ +g−σ+ d1N d2N = (−1) N · · c− dN ⎛ ⎞ ιÑ (˜ι N Zˇ (L 1 , G 1 , G 1 ) ∗ (Z (Tg ) ⊗ (ν 1/2 )⊗2g ) ∗ ιÑ Zˇ (L 2 , G 2 , G 2 )) ⎠ × vdeg≤N ⎝ 1N 2N σ+1 σ−1 σ+2 σ−2 g g N N σ σ σ N + + + (−1) (−1) (−1) d1 d2 · c+ c− · c+ c− · c+ c− σ+1 +σ+2 +g−σ+ N N d d c+ = (−1) N · 1 N2 · c− d ⎛ ⎞ ιÑ Zˇ (L 1 , G 1 , G 1 ) ιÑ Zˇ (L 2 , G 2 , G 2 ) Z (Tg ) ⊗ (ν 1/2 )⊗2g ⎠ × vdeg≤N ιÑ ⎝ ∗ ∗ 1 2 g N cg cg 1 N N σ+1 σ− 2 N N σ+2 σ− (−1) σ σ + − + + (−1) d1 c+ c− (−1) d2 c+ c− σ+1 +σ+2 +g−σ+ N N d1 d2 c+ = (−1) N vdeg≤N · · N (τ N (M1 ), τ N (M2 )), c− dN where we have used that σ+ + σ− = σ1+ + σ1− + σ2+ + σ2− + 2 · g. Observe that in the second equality, when “braking” Zˇ into three, on each component of L 0 a ν 1/2 “goes” to Z of G 1 or G 2 , and another ν 1/2 goes to z g . In fact, the two middle expressions are written for the even associator. For any other associator we would insert between the ∗’s the element A mentioned in the remark at the end of Sect. 2.5.

Proof of Theorem 3.2.3). We use the notations of the previous lemma. Observe that (3.11) implies: ⎛

0 A BT ⎜ B B A−1 B T −I lk(L 1 ∪ L 0 ∪ L 2 ) = ⎜ ⎝0 −I DC −1 D T 0 0 DT

⎞ 0 0⎟ ⎟, D⎠ C

(3.14)

where A = lk(L 1 ) ∈ M|L 1 |×|L 1 | (Z), C = lk(L 2 ) ∈ M|L 2 |×|L 2 | (Z), B = lk(G 1 , L 1 ) ∈ Mg×|L 1 | (Z), D = lk(G 2 , L 2 ) ∈ Mg×|L 2 | (Z), B A−1 B T , DC −1 D T ∈ Mg×g (Z). In [6] it has been shown that the signature of this linking matrix is (σ+1 +σ+2 + g, σ−1 +σ−2 + g), where (σ+1 , σ−1 ), respectively (σ+2 , σ−2 ) is the signature of lk(L 1 ), respectively lk(L 2 ), and that the following holds: det(lk(L 1 ∪ L 0 ∪ L 2 )) = (−1)g · det(lk(L 1 )) · det(lk(L 2 )). Substituting into (3.13), we obtain the functoriality of τ N .

(3.15)

Lemma 3.7. Let β ∈ A( g2 , g3 ), and let αn , n ∈ N be a sequence of elements of A( g1 , g2 ), such that for every n, deg≤n (αn ) = deg≤n (αm ), ∀m > n. Then, both sides of the following equality are well-defined, and the equality holds: N (lim αn , β) = lim N (αn , β). n

n

A similar property holds for the rôle of two arguments of N reversed.

(3.16)

626


Proof. The existence of α := lim αn ∈ A( g1 , g2 ) follows directly from the fact that n

we defined the topology on A( g1 , g2 ) such that dist ( p, q) < 21n if and only if p − q has degree > n. Then, since deg>n+g N (α) does not contribute to deg≤n N (α, β), we have: lim deg≤n N (α, β) = lim deg≤n N (deg≤n+g N α, β) = lim N (deg≤n+g N α, β) n

n

n

= lim N (deg≤m α, β) = N (lim deg≤m α, β) = N (α, β). m

m

The existence of the third limit and the second equality follow from a standard Cauchysequences argument. The fourth equality is true since lim commutes with ∗, ι N and m

vdeg≤N . On the other hand N (αn , β) and N (α, β) agree in degree ≤ n − 2g N . Hence lim N (αn , β) = lim deg≤n N (α, β). Putting the two together we obtain (3.16).

Remark. If in the statement of this lemma we assume that lim αn exists, which is the case n

whenever the result is used in this paper, then we can relax the topology: distance( p, q) ≤ 1 n ⇔ p − q has no terms of degree < n. A −I Elementary remark. The signature of a symmetric 2g × 2g-matrix with −I 0 integer, respectively real entries is (g, g). The determinant of such a matrix is (−1)g . Proof of Proposition 2.8. Note that wg = τ (g × [0, 1], (g × 0, p1 ), (g × 1, p2 )). Using the gluing formula (3.13), for any Q-cobordism (M, f 1 , f 2 ), τ N ((g × [0, 1]) σ+1 +σ+2 +g−σ+ N ∪ p1 ◦( f 2 )−1 M, f 1 , p2 ) = (−1) N vdeg≤N cc−+ · d1dd2 · N (τ N = S 3 , and the linking matrix of L is lk(L), (M, f 1 , f 2 ), wg ). If L is a link such that M ⎛L ⎞ ⎛ ⎞ lk(L) ∗ 0 lk(L) 0 0 ∗ −I ⎠ ∼ ⎝ 0 ∗ −I ⎠. then the linking matrix of the link L ∪ L 0 is ⎝ ∗ 0 −I 0 0 −I 0 Using the above remark, σ+ = σ+1 + g, σ− = σ−1 + g, σ+2 = σ−2 = 0, d2 = lk(∅) = 1, d1 = d. Observe that ((g × [0, 1]) ∪ p1 ◦( f2 )−1 M, f 1 , p2 ) ∼ = (M, f 1 , f 2 ). Hence N (τ N (M, f 1 , f 2 ), wg ) = τ N (M, f 1 , f 2 ). In particular, this holds if (M, f 1 , f 2 ) is a Z-cobordism with bottom homeomorphic to S 2 , and hence also for any (M, ∅, f ) such is a Z-homology sphere. The statement now follows from Part 1) of Theorem that M 3.2 and Lemma 3.7.

Proof of Theorem 2.9. 1) By construction, the inverse limits lim A≤N (∅) = A(∅) ∞←N

and lim A≤N ( g1 , g2 ) = A( g1 , g2 ). Let us show that the following diagram is ∞←N

commutative for every N ∈ N: A≤N +1 ( g1 , g2 ) ⊗ A≤N +1 ( g2 , g3 ) ↓ ≤N +1

vdeg≤N

A≤N ( g1 , g2 ) ⊗ A≤N ( g2 , g3 ), ↓ N (3.17)

A≤N +1 ( g1 , g3 )

vdeg≤N

A≤N ( g1 , g3 ),

→

→

where the horizontal arrows are the maps that forget the degrees N + 1 parts. Let α = τ N +1 (M1 ), β = τ N +1 (M2 ) for some Q-cobordisms M1 and M2 from category Q. Then, as previously observed vdeg≤N τ N +1 (Mi ) = τ N (Mi ), i = 1, 2, i.e. vdeg≤N α =


627

τ N (M1 ), vdeg≤N β = τ N (M2 ). By the gluing formula (3.4) we then have τ N +1 (M2 ∪ M1 ) = ≤N +1 (α, β) and τ N (M2 ∪ M1 ) = N (vdeg≤N α, vdeg≤N β). Now again, using vdeg≤N τ ≤N +1 (M2 ∪ M1 ) = τ N (M2 ∪ M1 ), we get vdeg≤N ≤N +1 (α, β) = N (vdeg≤N α, vdeg≤N β). Hence the diagram (3.17) is commutative for α, β as above. By Part 2) of Theorem 3.2 and Lemma 3.7, the diagram is then commutative for arbitrary α, β. Therefore there exists a well-defined A(∅)-bilinear A( g2 , g3 ) → A( g1 , g3 ), map : A( g1 , g2 ) ⊗ A( g2 , g3 ) → A( g1 , g2 )⊗ such that when restricting to the vacuum degree ≤ N parts, one obtains the map N . ˜ g ) = lim Ñ (wgN ). By Proposition 2.8 the operators 2) By the proof of 1), (w ∞←N

Ñ (wgN ) are identities, hence so is the limit.

Proof of Theorem 3.2.4). Theorem 2.9.1) shows that N are the vdeg≤N -truncations of . The result then follows from (3.4).

Proof of Theorem 3.2.5). Proposition 2.8 proves the first formula, the normalization of the truncated TQFTs Q → A≤N and Z → A≤N . Theorem 2.9.2) in particular implies the second formula, the normalization of the non-truncated TQFTs Q → A and Z → A.

Lemma 3.8 (The continuity of ). Let β ∈ A( g2 , g3 ), and let αn , n ∈ N be a sequence of elements of A( g1 , g2 ), such that for every n, deg≤n (αn ) = deg≤n (αm ), ∀m > n. Then, both sides of the following equality are well-defined, and the equality holds: (lim αn , β) = lim (αn , β). n

n

(3.18)

A similar property holds for the rôle of two arguments of reversed. Proof. N are the vdeg≤N -truncations of . Apply (3.16) and pass to the limit (keeping, for example n = (2g + 1)N ).

Proof of Theorem 3.2.6). The conjugation in A(∅) can be extended to A( g ) and A( g , g ) as follows. For an arbitrary chord diagram D, define D = D, if vdeg(D) = even, and D = −D, if vdeg(D) = odd. They induce maps also for vdeg≤N -truncations. Let (L , G) be the embedding of a link and (disjoint union of) chain graph(s) in S 3 , and let (L , G) denote its mirror image. Then ιÑ Zˇ (L , G) = (−1)|L|N ιÑ Zˇ (L , G) (compare with [17, Prop. 5.2]). This is true for the Murakami-Ohtsuki extension of Z because a and b from [21], and hence Z (vicinity of a trivalent vertex) are “mirrors” of themselves, which is easy to check. For the extension of Z using an even associator (Sect. 2.4), this property is obvious. In the proof of [17, Prop. 5.2] it is shown that c− = c+ . Hence for any N ∈ N:

ιÑ ( Zˇ (L , G)) (−1)σ+ N N τ (−M) = vdeg≤N [≤N ] σ− dN (c+[≤N ] )σ+ · (c− ) ⎛ ⎞ σ N + ˇ ιÑ ( Z (L , G)) (−1) ⎠ = τ N (M). = vdeg≤N ⎝ σ σ− [≤N ] + dN (c ) · (c[≤N ] ) +

−

Therefore also τ (−M) = τ (M), for any Q-cobordism M. Using this formula and (3.5), it follows that for α = τ (M1 ), β = τ (M2 ), where Mi are 3-cobordisms in the category Q,

628


we have (α, β) = (τ (M1 ), τ (M2 )) = τ (M2 ◦ M1 ) = τ (−(M2 ◦ M1 )) = τ ((−M1 ) ◦ (−M2 )) = (τ (−M2 ), τ (−M1 )) = (τ (M2 ), τ (M1 )) = (β, α). By Theorem 3.2.2) and the continuity (3.18) of , (α, β) = (β, α) holds for arbitrary α, β. In particular, it remains true if we add vdeg≤N . The property is, therefore, verified for truncated and non-truncated TQFTs.

We have shown that τ : Q → A, τ N : Q → A≤N are functors, and the TQFTs are non-degenerate. The full TQFT induces a linear representation Lg → G L A(∅) (A( g )). The truncated TQFTs induce linear representations Lg → G L A≤N (∅) (A≤N ( g )). It is known [8] that any ZHS can be obtained as filling of a parametrized 3-cobordism (g × I, w, id) for some g ≥ 0 and some w ∈ Tg , the Torelli group of genus g. Furthermore [20], it even suffices to consider only w ∈ Kg , the kernel of the Johnson homomorphism, or topologically, the subgroup of Tg generated by Dehn twists on bounding simple closed curves. Our TQFTs, of course, induce linear representations of both these subgroups of Lg . The group Lg has not been studied before, no explicit set of generators, less so one of relations, is known. Note, that Theorem 3.2.1) and 2) address the so-called realization problem for links, string links, three-dimensional manifolds, and chain graphs, by showing (see also 3.3, 3.4) that Z (links), τ (closed 3-manifolds), Z (string links), and τ (3-manifolds with boundary), in the closure, generate the corresponding combinatorial spaces of chord diagrams: A( -i. . . -i), A(∅), A(↑g ), and A( g1 , g2 ). Without proving Theorem 3.2.1) and 2), even partial results of this sort were hard to obtain, as can be seen from the following Proposition 3.9. For every N ≥ 0 and every Q-cobordism (M, f 1 , f 2 ) from g to g, Ñ (τ N (M, f 1 , f 2 )) sends the A≤N (∅)-submodule of A≤N (↑g ) generated by exp(α), ∀α ∈ vdeg≤N a(↑g ), to itself. Proof. By Proposition 3.1.3) τ N (M, f 1 , f 2 ) is group-like. Observe that commutes with ∗, and repeating the argument from the proof of 3.1.3) for ιÑ in the definition of N , we can see that Ñ (τ N (M, f 1 , f 2 )) takes a group-like element of A≤N ( g1 ) of the form 1 + h.o.t. to a group-like element of A≤N ( g2 ) of the form 1 + h.o.t. Now apply Proposition 2.3.5) and 3) for the truncated case. Hence it sends the A≤N (∅)-submodule of A≤N (↑g ) generated by exp(α), α ∈ vdeg≤N a(↑g ) to itself.

Remarks. 1) This construction of TQFT can be done also in the language of the Aarhus integral. 2) A combinatorial formula for the pairing : A( g1 , g2 ) ⊗ A( g2 , g3 ) → A( g1 , g3 ) is given in [7]. In [17], Le, Murakami and Ohtsuki have introduced the chord-KII move to mirror the second Kirby move for links, which then allowed them to define Z L M O . However, it is well-known that handle canceling can not be obtained solely by Kirby-2, and would require in addition Kirby-1. But no corresponding chord-KI move exists, the invariance of Z L M O under Kirby-1 is achieved via normalization. Therefore there is no a priori reason to suspect that a chord-canceling-handle relation is true for arbitrary chord diagrams. Proposition 3.10. (Chord-handle canceling) The chord-handle-canceling relation, schematically depicted in Fig. 10 holds for arbitrary β ∈ A(↑g ). (The upper part of each Fi should be read as Z (drawn tangle).) Proof. For arbitrary β, F1 differs from F2 by a chord-KII move. (An argument similar to the one in [17, Prop. 3.2] works.) But now F2 = (β, wg ) = β = F3 .


...

...

β F1

...

...

...

629

...

β

β

F2

F3

Fig. 10. Chord-handle-canceling relation F1 = F3

4. A TQFT for the Casson-Walker-Lescop Invariant The term of degree one of Z L M O of a 3-manifold is (−1)b1 (M) λ(M) 2 θ , where b1 (M) is the first Betti number, λ(M) is the Casson invariant (in Walker-Lescop extension), and θ is the (only) open chord diagram of degree 1, which looks like the symbol θ [17]. Recall the definition and basic properties of the Casson invariant. Let K be a knot in an oriented Z-homology 3-sphere M, and K (t) = a0 +a1 (t +t −1 )+a2 (t 2 +t −2 )+... be its Alexander polynomial normalized such that K (1) = 1. Denote λ (K ) = 21 (1) = n 2 an . n

Theorem 4.1 (Casson). There is an integer-valued invariant λ for oriented integer homology 3-spheres such that: (1) (2) (3) (4) (5)

λ mod 2 is the Rohlin invariant, λ(M) = 0 for any homotopy 3-sphere, λ(−M) = −λ(M), λ(M1 #M2 ) = λ(M1 ) + λ(M2 ), If K is a knot in an oriented integer homology 3-sphere M, and M(K , n1 ) denotes the integer homology 3-sphere obtained from M by a n1 -surgery on K , then λ(M(K , n1 )) = λ(M) + nλ (K ).

Property (5) from this theorem, for n = ±1, and normalization λ(S 3 ) = 0 determine λ uniquely, since any integer homology 3-sphere can be obtained from S 3 by a succession of ±1-surgeries on knots. λ was extended to rational homology 3-spheres by Walker, and corresponding properties (4) and (5) were given by Lescop [18]: (4 ) (5 )

λ(M1 #M2 ) = |H1 (M2 , Z)|λ(M1 ) + |H1 (M1 , Z)|λ(M2 ), λ(M(L ,

where M(L ,

p|L| p1 q1 , . . . , q|L| ))

=

p

|H1 (M(L , q 1 ,..., q |L| ),Z)| p 1

|L|

|H1 (M,Z)|

λ(M) + F M (L ,

p|L| p1 q1 , . . . , q|L| ) is the manifold obtained from

p|L| p1 q1 , . . . , q|L| ),

M by performing rational surp

|L| gery with indicated coefficients on the components of the link L, and F M (L , qp11 , . . . , q|L| ) is a certain function on the set of surgery presentations in M, in fact a function of the linking matrix, homology, and Alexander polynomial [18].

4.1. The vacuum-degree N ≤ 1 truncation. of the TQFT defined in Sect. 3 is a TQFT for the Casson-Walker-Lescop invariant. Indeed, let R = A≤1 (∅) = ({r + sθ |r, s ∈

630


r s |r, s ∈ Q . Observe [17], that c+ = 0r θ θ 1 − 16 + h.o.t., c− = 1 + 16 + h.o.t., hence c+ c− = 1+terms of degree ≥ 2, and Z (Tg )⊗(ν 1/2 )⊗2g = (−1)g Z (Tg ) ⊗ (ν 1/2 )⊗2g . If α ∈ therefore z 1g = (−1)g·1 vdeg≤1 c+ c−

Q}, [≤ 1]-multiplication) ∼ = Q[θ ]/(θ 2 ) ∼ =

A≤1 ( g1 , g2 ) and β ∈ A≤1 ( g2 , g2 3 ), then:

ι1 (α ∗ z 1g1 ∗ β) . 1 (α, β) = vdeg≤1

(4.1)

If g = 0, 1 is the disjoint union. A formula for Z (T1 ) and Z (W1 ) is given for example in [9]. Using an even associator it is easy to write down z 11 and w1 explicitly in low degrees. Let κ(L , G 1 , G 2 ) = (M, f 1 , f 2 ). Then, keeping in mind that c+ c− = 1+terms of degree ≥ 2, denote c(M, f 1 , f 2 ) := τ 1 (M, f 1 , f 2 ) =

(−1)σ+ vdeg≤1 ι1 ( Zˇ (L , G 1 , G 2 )) , d(M)

(4.2)

Z)| and σ+ = sign + (L), is an invariant of 3-cobordisms where d(M) = |H1 ( M, of category Q. In particular, for cobordisms between S 2 and S 2 , c(M, id S 2 , id S 2 ) = (−1)σ+ ≤1 ι ( Zˇ (L))) = vdeg Z L M O (M) = 1 + λ(M) θ , where we have identi1 ≤1 d(M) vdeg ( 2 ≤1 0 0 ≤1 fied A ( , ) ≡ R = A (∅). The filling of the composition of two 3-cobordisms between S 2 and S 2 is the connected sum of the fillings. Hence c(M2 ∪ M1 , S 2 , S 2 ) = c(M1 , S 2 , S 2 )c(M2 , S 2 , S 2 ) implies property (4 ) of the Casson invariant (the generalized version). By results of Sect. 3, the following axioms of TQFT hold: c(M2 ∪ f2 ◦( f )−1 M1 , f 1 , f 2 ) = 1 (c(M1 , f 1 , f 1 ), c(M2 , f 2 , f 2 )), 1

c(g × [0, 1], p1 , p2 ) = idA≤1 ( g ) ,

(4.3) (4.4)

c(−M, − f 2 , − f 1 ) = c(M, f 1 , f 2 ),

(4.5)

where the notations are obvious. R, A≤1 ( 0 , g ), and A≤1 ( g1 , g2 ) are Z2 -graded by the vacuum degree. In particular (4.5) implies property (3) of the CWL invariant. Unfortunately, explicit calculations for c(M, f 1 , f 2 ), as expected, are hard to do. We show below that the induced representation Lg → G L R (A≤1 ( g )) descends to Morita’s homomorphism λ∗ : Kg → Z. (λ∗ extends to Lg , but fails to be a homomorphism there.) Proposition 4.2. 1) Let B be the completion of the Q-vector subspace of A( g1 ( m S 1 ) g2 ) generated by finite sums of chord diagrams which intersect g1 g2 . Then p : A( g1 ( m S 1 ) g2 ) → A( m S 1 ), the natural map “erase g1 and g2 from a chord diagram”, if it does not intersect g1 g2 , and set = 0, otherwise, is well-defined, and the following sequence is short exact: p

0 → B → A( g1 ( m S 1 ) g2 ) → A( m S 1 ) → 0. We will denote also by p the induced maps on the vacuum degree ≤ N parts. They have the same property.


631

2) Denote by r the maps similar to p from 1), corresponding to the case m = 0. Then the following diagram is commutative: ιÑ

A( g1 ( m S 1 ) g2 ) −→ A( g1 g2 ) −→ A≤N ( g1 g2 ) ↓p

↓r

A( m S 1 )

ιN

−→

↓r

A(∅)

A≤N (∅)

−→

3) For every embedding L ∪ G → S 3 , such that G as an abstract graph is g1 g2 , p Zˇ (L ∪ G) = Zˇ (L). 4) For every embedding L ∪ G → S 3 , such that G as an abstract graph is g1 g2 , ∪ G)). In particular and every N ≥ 1, p(τ N (κ(L ∪ G))) = deg≤N Z L M O (κ(L λ( M) (if N = 1), p(c(M, f 1 , f 2 )) = 1 + 2 θ . 5) If ϕ1 , ϕ2 ∈ Kg , then p(c(g × I, ϕ2 ◦ ϕ1 , id)) = p(c(g × I, ϕ1 , id)) p(c(g × I, ϕ2 , id)). Proof. 1) The following argument works for every fixed degree, and since all relations are homogeneous, we can use the universality property of the direct product as mentioned after Proposition 2.1 to obtain the result. Consider the corresponding diagram before introducing relations: p

0 → B → D( g1 ( m S 1 ) g2 ) → D( m S 1 ) → 0. The terms of any relation for diagrams on g1 ( m S 1 ) g2 , either all intersect g1 g2 , or none does. Hence, if we denote by R1 the Q-vector space generated by relations of the first type, by R2 - the space generated by relation of the second type, and by R - the one generated by all relations, then R/R1 ∼ = R2 . All in all we get a diagram: 0 → R1 ↓

−→

−→

R

−→ 0

R2

↓

↓ p

0 → B → D( g1 ( m S 1 ) g2 ) → D( m S 1 ) → 0 ↓

↓ i

↓ p

0 → B → A( g1 ( m S 1 ) g2 ) → A( m S 1 ) → 0 where all columns and the first two rows are short exact. The arrows i and p in the third row are then induced and make the diagram commutative. They clearly are the maps described in the statement. The exactness in the third row follows from the exactness in the second. 2) Let α ∈ A( g1 ( m S 1 ) g2 ) and β be such that ϕ(β) ˜ = α. Recall that ι Ñ = qÑ ◦ κÑ ◦ ϕ˜ −1 . A chord diagram x from the expression of β connects to g1 g2 , if and only if its image via ϕ is a sum y of chord diagrams expressing α, all connected to g1 g2 . Again, using the fact that the terms in any relation either all connect, or

632


all do not, p(y) = 0 implies (in fact ⇔) that qÑ ◦ κÑ (x) connects to g1 g2 , i.e. r (qÑ ◦ κÑ ◦ ϕ˜ −1 (y)) = r (qÑ ◦ κÑ (x)) = 0. Now, if we decompose β = β1 + β2 such that all terms in β1 connect to g1 g2 , and all terms in β2 do not, the result follows: (r ◦ ι Ñ )(α) = r (ϕ(β1 ) + ϕ(β2 )) = r (qÑ ◦ κÑ (x1 ) + qÑ ◦ κÑ (x2 )) = qÑ ◦ κÑ (x2 ) = ι Ñ (ϕ(β2 )) = ι Ñ (ϕ( p(β))) = ι Ñ ( p(ϕ(β))) = (ι Ñ ◦ p)(α). 3) Decompose L ∪ G into elementary pseudo-quasi-tangles. Observe that for every one, except ’s and @@ ’s, possibly with multiple strands, Z either returns diagrams, which are either all in B, or all have no intersection between the dashed graph and g1 g2 . Thus, suppressing G for these elementary tangles corresponds precisely to applying p. The remaining cases. Observe, first, that one can “lift L above G”, leaving only some “fingers” from L attached to G. To see this, from a generic plane projection of L ∪ G on R2 ⊂ R3 obtain an isotopic embedding of G ∪ L in R3 , such that G is in an ε-neighborhood of the plane {z = 0} ∈ R3 , and L, except for some fingers that correspond to intersections between G and L in the original plane projection, lies in an ε-neighborhood of the plane {z = 1} ∈ R3 . Hence, by “opening the two-page book”, we can find such a tangle decomposition, that all occurring associator-tangles are of one of the following three types: (A) refer only to G or only to L; (B) a single middle strand, which comes from L, the left-most strand (with “big” multiplicity) comes from G, the right-most strand (also with “big” multiplicity) comes from L. Moreover, if such an associator-tangle occurs, its inverse (on the same strands) will occur “soon”; (C) one of the left-most two strands is a single strand coming from L, all other strands come from G. We will assume that he associator is horizontal, i.e. it is a formal series in two noncommuting variables r12 , r23 , which correspond to a dashed line joining these indicated strands [17]. (A) If all strands are from G or none are from G, then all terms of ±1 = Z (tangle), connect, respectively do not connect, to g1 g2 . (C) If two of the three stands come from G, ±1 = Z (tangle) will have all terms connected to g1 g2 . Then, eliminating G corresponds precisely to replacing this tangle-associator by the single strand from L, i.e. corresponds to applying p in this case. (B) If exactly one (multiple) strand comes from G, this corresponds to setting one of the two non-commutative variables r12 , r23 zero. But, as mentioned, such tangles occur in pairs with their opposite. Hence, both and 321 = −1 occur. Setting one of r12 , r23 zero in this case leaves a series, and its inverse (elementary exercise). Thus, eliminating G corresponds again to applying p. 4) Recall the definitions of τ N (3.3) and Z L M O (2.2). Apply p and use the result of part 3). Then, use the commutativity of the diagram from part 2) to obtain the desired relation. 5) Applying p to (4.3), p(c(g × I, ϕ2 ◦ ϕ1 , id)) = p(1 (c(g × I, ϕ1 , id), c(g × λ(Wϕ2 ◦ϕ1 ) θ , p(c(g × 2 λ(W ) I, ϕ2 , id)) = 1 + 2ϕ2 θ , where Wϕi = λ(Wϕ1 ) λ(W ) 1 + 2ϕ2 θ in R, because λ∗ : 2 θ

I, ϕ2 , id))). Using part 4), p(c(g × I, ϕ2 ◦ ϕ1 , id)) = 1 + λ(W )

I, ϕ1 , id)) = 1 + 2ϕ1 θ , and p(c(g × λ(Wϕ2 ◦ϕ1 ) (g × I, ϕi , id). But 1 + θ = 1+ 2

Kg → Z, λ∗ (ϕ) := λ(Wϕ ) satisfies λ∗ (ϕ2 ◦ ϕ1 ) = λ(ϕ1 ) + λ(ϕ2 ) by [20].


633

Remarks. Using expression (4.1) for , and observing that p commutes with ι˜1 , and with vdeg≤1 by Proposition 4.2.2), we can re-write p(1 (c(g × I, ϕ1 , id), c(g × I, ϕ2 , id))) as vdeg≤1 ι˜1 ( p(c(g × I, ϕ1 , id) ∗ z 1g ∗ c(g × I, ϕ2 , id))). Expressing c(g × I, ϕi , id) by (4.2), and keeping of in mind the definition of p and properties 1 ˇ ˇ ι˜1 , we get to having to apply p on Z (L 1 , G 1 , G 1 ) ∗ z g ∗ Z (L 2 , G 2 , G 2 ) , respecti vely to apply p on Zˇ (L i , G i , G i ) , i = 1, 2. On the other hand, it is possible to show directly that ι˜1 p Zˇ (L 1 , G 1 , G 1 ) ∗ z 1g ∗ Zˇ (L 2 , G 2 , G 2 ) = p Zˇ (L 1 , G 1 , G 1 ) · p Zˇ (L 2 , G 2 , G 2 ) , for suitably chosen L i in the triplets. This gives another proof of Proposition 4.2.5). Thus the fact that λ∗ : Kg → Z is a homomorphism follows from the Kontsevich integral. References 1. Atiyah, M.: Topological Quantum Field Theories. Publications Mathématiques IHES 68, 175–186 (1988) 2. Bar-Natan, D.: On the Vassiliev knot invariants. Topology 34, 423–472 (1995) 3. Bar-Natan, D., Lawrence, R.: A rational surgery formula for the LMO invariant. Israel J. Math. 140, 29– 60 (2004) 4. Bar-Natan, D., Le, T., Thurston, D.: Two applications of elementary knot theory to Lie algebras and Vassiliev invariants. Geom. Topol. 7, 1–31 (2003) 5. Blanchet, C., Habegger, H., Masbaum, G., Vogel, P.: Topological quantum field theories derived from the Kauffman bracket. Topology 34(4), 883–927 (1995) 6. Cheptea, D., Le, T.: 3-cobordisms with their rational homology on the boundary. Preprint, available at http://arxiv.org/math/0602097, 2006 7. Cheptea, D., Habiro, K., Massuyeau, G.: A functorial LMO invariant for Lagrangian cobordisms. Preprint, available at http://arxiv.org/math/0701277, 2007 8. Fomenko, A., Matveev, S.: Algorithmic and computer methods for three-manifolds. Dordrecht: Kluwer Academic Publishers, 1997 9. Gille, C.: On the Le-Murakami-Ohtsuki invariant in degree 2 for several classes of 3-manifolds. J Knot Theory Ramifications 12(1), 17–45 (2003) 10. Gompf, R.E., Stipsicz, A.I.: 4-manifolds and Kirby calculus. Graduate Studies in Mathematics 20, Providence, RI: Amer. Math. Soc., 1999 11. Habegger, N., Masbaum, G.: The Kontsevich integral and Milnor’s invariants. Topology 39, 1253– 1289 (2000) 12. Habegger, N., Orr, K.: Finite type three manifold invariants -realization and vanishing. J. Knot Theory Ramifications 8(8), 1001–1007 (1999) 13. Habegger, N., Orr, K.: Milnor link invariants and quantum 3-manifold invariants. Comment. Math. Helv. 74(2), 322–344 (1999) 14. Habiro, K.: Claspers and finite-type invariants of links. Geom. and Top. 4, 1–83 (2000) 15. Le, T.T.Q.: An invariant of integral homology 3-spheres which is universal for all finite type invariants. AMS Translation series 2, 179, 75–100 (1997) 16. Le, T.T.Q.: The LMO invariant, “Invariants de noeuds at de variétés de dimension 3”. In: Proc. of École d’été de Mathématiques, Grenoble: Institut Fourier, 1999 17. Le, T.T.Q., Murakami, J., Ohtsuki, T.: On a universal perturbative invariant of 3-manifolds. Topology 37(3), 539–574 (1998) 18. Lescop, C.: Global surgery formula for the Casson-Walker invariant. Princeton, NJ: Princeton University Press, 1996 19. Matveev, S.V.: Generalized surgery of three-dimensional manifolds and representations of homology spheres. Matematicheskie Zametki 42(2), 268–278 (1986) 20. Morita, S.: Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I. Topology 28(3), 305–323 (1989) 21. Murakami, J., Ohtsuki, T.: Topological Quantum Field Theory for the Universal Quantum Invariant. Commun. Math. Phys. 188, 501–520 (1997) 22. Serre, J-P.: Lie Algebras and Lie Groups, 2nd ed., Lecture Notes in Mathematics 1500, New York: Springer, 1992 23. Turaev, V.: Quantum Invariants of Knots and 3-Manifolds. Berlin: Walter de Gruyter, 1994

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24. Vogel, P.: Invariants de type fini. In: “Nouveaux Invariants en Géométrie et en Topologie”, publié par D. Bennequin, M. Audin, J. Morgan, P. Vogel, Panoramas et Synthèses 11, Paris: Société Mathématique de France, 2001 pp. 99–128 Communicated by L. Takhtajan


Communications in


Quantum Conjugacy Classes of Simple Matrix Groups A. Mudrov1,2 1 Department of Mathematics, University of York, York, YO10 5DD, UK 2 St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191011 St. Petersburg, Russia.

E-mail: [email protected] Received: 30 March 2006 / Accepted: 5 September 2006 Published online: 5 April 2007 – © Springer-Verlag 2007

Dedicated to the memory of Joseph Donin Abstract: Let G be a simple complex classical group and g its Lie algebra. Let U(g) be the Drinfeld-Jimbo quantization of the universal enveloping algebra U(g). We construct an explicit U(g)-equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Drinfeld-Jimbo quantum group . . . . . . . . . . . . . . . . . . . . . . 2.1 Quantized universal enveloping algebra . . . . . . . . . . . . . . . 2.2 Defining representations of classical matrix groups . . . . . . . . . 2.3 Parabolic subalgebras in U(g) . . . . . . . . . . . . . . . . . . . . 3. Generalized Verma modules over U(g) . . . . . . . . . . . . . . . . . . 3.1 Upper and lower (generalized) Verma modules . . . . . . . . . . . . 3.2 Pairing between upper and lower generalized Verma modules . . . . 3.3 Tensor product of finite dimensional and generalized Verma modules 4. Properties of the universal RE Matrix . . . . . . . . . . . . . . . . . . . 4.1 Characteristic polynomial for RE matrix . . . . . . . . . . . . . . . 4.2 Minimal polynomial for RE matrix . . . . . . . . . . . . . . . . . . 4.3 A construction of central elements . . . . . . . . . . . . . . . . . . 5. On quantization of affine algebraic varieties . . . . . . . . . . . . . . . . 5.1 A flatness criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Quantization of simple algebraic groups . . . . . . . . . . . . . . . . . . 6.1 Simple groups as Poisson Lie manifolds . . . . . . . . . . . . . . .

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This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center “Group Theoretic Methods in the study of Algebraic Varieties” of the Israel Science foundation, by the EPSRC grant C511166, and by the RFBR grant no. 06-01-00451.

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6.2 Quantization of the STS bracket on the group . . . . 6.3 Embedding of C[G] in U(g) . . . . . . . . . . . . 7. Center of the algebra C[G] . . . . . . . . . . . . . . . . 7.1 The case of classical series . . . . . . . . . . . . . . 8. Quantization of conjugacy classes . . . . . . . . . . . . . 8.1 Non-exceptional classes . . . . . . . . . . . . . . . . 8.2 The quantization theorem . . . . . . . . . . . . . . . 8.3 Ideals of quantized non-exceptional conjugacy classes 8.3.1 The U gl(n) -case. . . . . . . . . . . . . . . . 8.3.2 The U so(2n + 1) -case. . . . . . . . . . . . . 8.3.3 The U sp(n) -case. . . . . . . . . . . . . . . . 8.3.4 The U so(2n) -case. . . . . . . . . . . . . . . A. More on central characters . . . . . . . . . . . . . . . . .

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1. Introduction Deformation quantization of a Poisson structure on a smooth manifold is a classic problem of mathematical physics. Especially interesting is a quantization that is equivariant with respect to an action of a group and, even more generally, a quantum group. Recently, significant progress in this field was triggered by a discovered connection between equivariant quantization and the theory of the dynamical Yang-Baxter equation, [DM1]. Namely, a star product quantization of semisimple coadjoint orbits and conjugacy classes (with a Levi subgroup as the stabilizer) of simple Lie groups was constructed in [EE, EEM] in terms of the universal dynamical twist. On the other hand, semisimple coadjoint orbits and conjugacy classes are affine algebraic varieties. Therefore their quantization may be sought for in terms of generators and relations, as a deformed ring of polynomial functions. This is an alternative approach as compared to the star product and it has certain advantages because the solution is presented by “finite data”. In the present paper, we construct an equivariant deformation quantization of semisimple conjugacy classes of simple classical algebraic groups with Levi subgroups as the stabilizers. More precisely, let G be a complex simple algebraic group from the series A, B, C, and D. Fix the standard, or Drinfeld-Jimbo, factorizable quasitriangular Lie bialgebra structure on g = Lie G. Equipped with the corresponding Drinfeld-Sklyanin (DS) bracket, G becomes a Poisson Lie group. Consider the adjoint action of G on itself. The Semenov-Tyan-Shansky (STS) Poisson bracket makes G a Poisson Lie manifold over G. The symplectic leaves of this Poisson structure are exactly the conjugacy classes. Let U(g) be the Drinfeld-Jimbo quantum group. We construct a U(g)-equivariant quantization of the ring of polynomial functions along the STS bracket on almost all conjugacy classes with Levi subgroups as the stabilizers. Here “almost all” means “all” for special linear and symplectic groups. For the orthogonal groups our construction covers the classes of matrices with eigenvalues λ subject to the condition λ2 = 1 ⇒ λ = 1. Those are exactly the classes that are isomorphic to semisimple G-orbits in g∗ (via the Cayley transformation). The quantization is given explicitly, in terms of deformed ideals of classes in a quantized ring of polynomial functions on the group (or in the so-called reflection equation algebra). Simultaneously, the quantized classes are realized as subalgebras of operators on generalized Verma modules.

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Let us describe our approach in more detail. Our main tool is the flatness criterion formulated in Proposition 5.2. We consider a pair of U(g)-module algebras and an equivariant homomorphism : S → T. We require that T should have no -torsion and S be a direct sum of U(g)-modules with C[[]]-finite isotypic components. Suppose we are able to construct an ideal J lying in the kernel of and suppose the classical limit of J (more precisely its image J0 in the zero fiber S0 = S/S) is maximal proper g-invariant (for example, the ideal of a g-orbit in Spec S0 ). Then J coincides with the kernel of , and the image of is a flat deformation of S0 /J0 . In our construction, T is the algebra of C[[]]-linear endomorphisms of a generalized Verma module. To construct the algebra S, we consider a certain quotient of the reflection equation algebra corresponding to U(g). That is a quantization of the coordinate ring C[G], which we denote by C[G] (this should not lead to a confusion with the standard quantization of C[G] along the Drinfeld-Sklyanin bracket, because the latter algebra is not involved in our consideration). The algebra C[G] is embedded in U(g) and that embedding induces an equivariant homomorphism to T. We cannot take just C[G] for the role of S, because the isotypic components of C[G] are not C[[]]-finite. To fix the situation, we use the fact that the isotypic components are finitely generated over the center, which coincides with the subalgebra of U(g)-invariants. The homomorphism C[G] → T induces a central character χ , and we put S to be the quotient of C[G] over the ideal generated by ker χ . To construct the ideal J , we show that a generalized Verma module annihilates the entries of a polynomial p(Q) in the matrix ||Q ij || of generators of the algebra C[G]. We set the ideal J ⊂ ker ⊂ S to be generated by the p(Q)ij . This matches our conditions, because the classical ideal of a non-exceptional conjugacy class is generated in C[G] by the entries of the minimal polynomial and the kernel of the subalgebra of invariants. Thus the quantized ideal of a non-exceptional conjugacy class is presented as a quotient of the algebra C[G] by the ideal generated by p(Q)ij over ker χ . Let us describe the setup of the paper. In Sect. 2 we recall the definition of the Drinfeld-Jimbo quantum group and some of its important subalgebras. In Sect. 3 we study tensor products of finite dimensional and generalized Verma modules over U(g). This will serve as a basis for the calculations of Sect. 4. In Sect. 4 we study properties of a fundamental object of our theory, the universal reflection equation matrix Q = R21 R (expressed through the universal R-matrix of U(g)). In particular, we study its action on the tensor product of a finite dimensional and generalized Verma modules. We determine the spectrum Q, compute the minimal polynomial for Q and q-traces of the matrix powers Q . In Sect. 5 we develop a method of quantization of affine homogeneous varieties. In Sect. 6 we recall some results of [M] on quantization of simple algebraic groups and construct an embedding of the quantized affine coordinate ring C[G] in U(g). In Sect. 7 we recall some facts concerning the center of C[G]. In Sect. 8 we give the quantization of the conjugacy classes. Appendix contains auxiliary information about invariants of U(g). 2. Drinfeld-Jimbo quantum group In the present paper, we work over the ring C[[]] of formal power series in . Given a C[[]]-module E we denote by E 0 its quotient E/E.

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By a deformation of a complex vector space E 0 we mean a free C[[h]]-module E such that E/E E 0 . Deformation of an associative algebra A0 is a C[[h]]-algebra A such that A/A A0 as associative C-algebras. The term deformation quantization or simply quantization is reserved for deformation of commutative algebras. Quantized universal enveloping algebras are understood as C[[]]-algebras, [Dr1]. We will work with the standard or Drinfeld-Jimbo quantization of simple Lie algebras. We will assume -adic completion of the Cartan subalgebra only. That is possible thanks to the existence of the Poincaré-Birkhoff-Witt base over the Cartan subalgebra. Thus defined, U(g) is a quasitriangular Hopf algebra in a weaker sense. However that is sufficient for our purposes, because we will deal with h-diagonalizable U(g)-modules. Although the quasitriangular structure requires completion of tensor products, we will use this structure in the situation when one of the modules is C[[]]-finite. This does not lead out of U(g). We assume that U(g)-modules are equipped with -adic topology and the action of U(g) is continuous. However we do not require the modules to be complete. We will work with U(g)-modules that are direct sums of C[[]]-finite weight spaces. Each weight space is complete, being C[[]]-finite. Thus our point of view is self-consistent, because we assume -adic completion for the Cartan subalgebra only. 2.1. Quantized universal enveloping algebra. Let g be a complex semisimple Lie algebra and h its Cartan subalgebra. Let R denote the root system of g with a fixed subsystem of positive roots R+ ⊂ R. By ⊂ R+ we denote the subset of simple roots. Let (., .) denote the Killing form on g. We will use the same notation for the invariant scalar product on h∗ that is induced by the Killing form restricted to h. By W we denote the Weyl group of g. For every λ ∈ h∗ we denote by h λ its image under the isomorphism h∗ h implemented by the Killing form. In other words, λ(h) = (h λ , h) for all h ∈ h. We put ρ = 21 α∈R+ α, the half-sum of the positive roots. Its dual h ρ ∈ h is the unique solution of the system of linear equations αi (h ρ ) = 21 (αi , αi ), i = 1, . . . , rk g. Denote by U(h) the -adic completion of the algebra U(h) ⊗ C[[]]. Define U(g) as a C[[h]]-algebra generated by the elements e±α , α ∈ , over U(h). These generators are subject to the following relations, [Dr1, Ji]: [h αi , e±α j ] = ±(αi , α j )e±α j , [eαi , e−α j ] = δi j

1−ai j

k=0

(−1)

k

1 − ai j k

qi

1−a −k

e±αi i j

q h αi − q −h αi qi − qi−1

,

k e±α j e±α = 0, i

2(αi ,α j ) (αi ,αi ) ,

where ai j = i, j = 1, . . . , rk g, is the Cartan matrix, q := e, qi := e 2 (αi ,αi ) , and [n]q ! q n − q −n n , [n]q ! = [1]q · [2]q . . . [n]q , [n]q = = . k q [k]q ![n − k]q ! q − q −1 The coproduct , antipode γ , and counit in U(g) are given by (eα ) = eα ⊗ 1 + q h α ⊗ eα , (e−α ) = e−α ⊗ q −h α + 1 ⊗ e−α , (h α ) = h α ⊗ 1 + 1 ⊗ h α , −h γ (eα ) = −eα q α , γ (e−α ) = −q h α e−α , γ (h α ) = −h α ,

(h) = (e±α ) = 0,


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for all α ∈ . The correspondence e−α → eα q h α , eα → q −h α e−α , h α → h α , for all α ∈ , extends to an involutive anti-algebra and coalgebra transformation of U(g) denoted by ω. The C[[]]-adic completion of U(g) is isomorphic to U(g)[[]] as an associative algebra, [Dr3]. The subalgebras in U(g) generated over U(h) by {e+α }α∈ and by {e−α }α∈ , respectively, are Hopf algebras. They are quantized universal enveloping algebras of the positive and negative Borel subalgebras b± and denoted further by U(b± ). The elements {e±α }α∈ ⊂ U(g) are called quantum Chevalley generators. The Chevalley generators can be extended to a system of quantum Cartan-Weyl generators {e±α }α∈R+ via the so-called q-commutators, see [KhT] and references therein. The Carrα tan-Weyl base admits an ordering, with respect to which the monomials > α∈R+ e−α < sα α∈R+ eα form a Poincaré-Birkhoff-Witt (PBW) base of U(g) as a U(h)-module, [KhT]. Denote by U (b± ) the ideals in U(b± ) generated by {e±α }α∈ . The PBW monomials of positive and, respectively, negative weights with respect to h form bases for U (b± ) over U(h). The universal R-matrix belongs to the completed tensor square of U(g). It has the structure ˆ (b+ ), R = q h modU (b− )⊗U

(1)

where h ∈ h ⊗ h is the inverse to the Killing form (the canonical element) restricted to h. More precisely, the R-matrix can be presented as the product of the Cartan factor q h and a series in the Cartan-Weyl generators, [KhT]. The universal R-matrix (1) is a “quantization” of the classical r-matrix 1 r− = (α, α)(e−α ⊗ eα − eα ⊗ e−α ) (2) 2 + α∈R

called the standard or Drinfeld-Jimbo r-matrix. Here eα ∈ g are the root vectors nor2 malized to (e−α , eα ) = (α,α) with respect to the Killing form. For α ∈ , they are the classical limits of the quantum Chevalley generators. 2.2. Defining representations of classical matrix groups. By classical Lie algebras (resp. matrix groups) we mean the simple complex Lie algebras of the types g = sl(n), so(2n + 1), sp(n), and so(2n), for n > 1. We reserve the notation V0 for the simple (defining) g-module of dimension N = n, 2n + 1, 2n, and 2n, respectively. We choose the following realization of orthogonal algebras and symplectic algebras. The algebra so(N ) leaves invariant the skew-diagonal unit matrix; the symplectic algebra preserves the skew-diagonal matrix with +1 above the center and −1 below. We assume that h, n+ , and n− are realized in End(V0 ) by, respectively, diagonal, upper- and lower triangular matrices. Below we collect some facts about the defining representation which we will use in our exposition. Let ei j be the standard matrix base in End(V0 ). Define linear functionals n {εi }i=1 on h setting εi (eii ) = 1 for g = sl(n) and εi (eii − ei i ) = 1 otherwise. Here i = N + 1 − i. The positive and simple roots of g can be written in terms of εi as R+ R+ R+ R+

= {εi − ε j }i< j , = {ε1 − ε2 , . . . , εn−1 − εn }, g = sl(n), = {εi ± ε j , εi }i< j , = {ε1 − ε2 , . . . , εn−1 − εn , εn }, g = so(2n + 1), = {εi ± ε j , 2εi }i< j , = {ε1 − ε2 , . . . , εn−1 − εn , 2εn }, g = sp(n), = {εi ± ε j }i< j , = {ε1 − ε2 , . . . , εn−1 − εn , εn−1 + εn }, g = so(2n).

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Our choice of functionals {εi } coincides with [VO] excepting n g = sl(n). In this latter case {εi } are linearly dependent and satisfy the condition i=1 εi = 0. The half-sum of positive roots is expressed in terms of {εi } by ⎧ n−1 ⎪ ⎪ 2 for g = sl(n), n ⎨ 1 n − 2 for g = so(2n + 1), ρ= ρi εi , ρi = ρ1 − (i − 1), ρ1 = ⎪ n for g = sp(n), ⎪ i=1 ⎩ n − 1 for g = so(2n). n n The set of weights of the defining representation is {εi }i=1 for sl(n), {0} ∪ {±εi }i=1 for n so(2n + 1), and {±εi }i=1 for sp(n) and so(2n).

2.3. Parabolic subalgebras in U(g). An element ξ ∈ g is called semisimple if the operator ad ξ ∈ End(g) is diagonalizable. A semisimple element belongs to a Cartan subalgebra, and all the Cartan subalgebras are conjugated; so one can assume that ξ ∈ h. Let g = n− ⊕ h ⊕ n+ be the triangular decomposition relative to R+ . A Levi subalgebra in g is defined as the centralizer of a semisimple element. It is a reductive Lie algebra of rank rk g. Let l be a Levi subalgebra in g and p± = l + n± ⊂ g be the par± abolic subalgebras. Denote by n± l the nillradicals in p . The triangular decomposition − − + g = nl ⊕l⊕nl induces decomposition U(g) = U(nl )U(l)U(n+l ), which has a quantum analog, [JT]. The elements {eα , e−α }α∈l generate over U(h) a Hopf subalgebra U(l) in U(g). This subalgebra is a quantized universal enveloping algebra of the Levi subalgebra l ⊂ g. It can be represented as U(l0 )U(c), where l0 = [l, l] is the semisimple part of l and c ⊂ l is the center. Also, U(p± ) := U(l)U(b± ) are Hopf subalgebras. They are quantized universal enveloping algebras of the parabolic subalgebras in g. This fact follows from the existence of the PBW U(h)-base. Let Z+ denote the set of non-negative integers. Consider in U(b+ ) the sum of weight spaces with weights from Z+ (g − l). It is an algebra and a deformation of U(h + n+l ), due to the existence of the PBW base in U(h + n+l ). Let us denote this algebra by U(h + n+l ). According to [Ke] (see also [Rad, JT]), there is a subalgebra in U(h + n+l ), denoted further by U(n+l ), such that U(b+l )U(n+l ) = U(b+ ). Here b+l = l ∩ b+ is the positive Borel subalgebra in l. The algebra U(n+l ) is U(l)-invariant with respect to the adjoint action, and there exists a smash product decomposition U(p+ ) = U(l) U(n+l ). (n+ )

(3) U(n+ ).

Proposition 2.1. The algebra U l is a deformation of l Proof. Decomposition (3) induces the decomposition U(h + n+l ) = U(h) U(n+l ) ⊂ U(p). Since U(h + n+l ) is a free U(h)-module generated by the PBW base, U(n+l ) is isomorphic to U(n+l ) ⊗ C[[]] as a C[[]]-module. By construction, the algebra U(n+l ) is generated by ad(u)(eα ), where α ∈ g − l and u ∈ U(l). This implies the proposition. ) the ω-image of U(n+l ), where ω is the quantum Chevalley invoDenote by U(n− l + lution. Since ω U(p ) = U(p− ) and U(l) is ω-stable, (3) induces decomposition U(p− ) = U(n− l ) U(l), through ω. The algebra U(g) admits the triangular decomposition + U(g) = U(n− l )U(l)U(nl ),

(4)


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which is a deformation of the classical one, [JT]. Note that U(n± l ) are not Hopf algebras. Thanks to the triangular decomposition (4), the algebra U(g) can be represented as a direct sum (e−α U(g) + U(g)eα ). U(l) ⊕ α∈g −l

By Pl : U(g) → U(l) we denote the projection along the second summand. 3. Generalized Verma modules over U(g) In the present section we study the tensor product of finite dimensional and generalized Verma modules over U(g). This will be the basis for our further considerations, as the quantized conjugacy classes will be realized via operators on generalized Verma modules. The key result of this section is decomposition (8), which will be used in Sect. 4 for computing characteristics of the quantized classes. 3.1. Upper and lower (generalized) Verma modules. Let Uq (g) be the quantum group ±h α

in the sense of Lusztig, [L]. It is a C(q)-Hopf algebra generated by {e±αi , qi i }αi ∈ . The algebra Uq (g) contains a C[q, q −1 ]-Hopf subalgebra U˘q (g) such that Uq (g) ±h α U˘q (g) ⊗C[q,q −1 ] C(q). It is generated by {e±αi , qi i , [eαi , e−αi ]}αi ∈ , see e.g. [DCK]. Clearly the algebra U˘(g) := U˘q (g) ⊗C[q,q −1 ] C[[]], where C[q, q −1 ] is embedded in C[[]] via q → e, is dense in U(g) in the -adic topology. Remark that U˘(g) is U(g)-invariant with respect to the adjoint action. There is a one-to-one correspondence between finite-dimensional g-modules and finite-dimensional Uq (g)-modules with q Z -valued weights. Every such module is isomorphic to W˘ ⊗C[q,q −1 ] C(q), where W˘ is a U˘q (g)-module, free and finite over C[q, q −1 ], [Jan2]. The specialization W˘ mod(q − 1) gives a finite dimensional g-module. Therefore W˘ extends to a U(g)-module, free and finite over C[[]]. We will call such U(g)-modules finite dimensional. They are deformations of g-modules, diagonalizable over U(h), and have the same weight structure. This correspondence between finite dimensional gmodules and U(g)-modules is additive. That is, every finite dimensional U(g)-module is a direct sum of “simple” modules, i. e. those whose classical limit is simple. Every submodule of a finite dimensional U(g)-module is h-diagonalizable with q Z -valued weights; hence it is again finite dimensional. We will also deal with U(g)modules that are finitely generated over C[[]] but not free. We always assume that such modules are quotients of finite dimensional and call them just C[[]]-finite. A highest weight U(g)-module is generated by a weight vector annihilated by U (n+ ). Similarly, a lowest weight module is generated by a weight vector annihilated by U (n− ). Finite dimensional U(g)-modules have highest and lowest weights simultaneously. Their highest weights are integral dominant. Finite dimensional U(g)-modules with highest weights are deformations of irreducible finite dimensional g-modules. They are substitutes for irreducibles, since a U(g)-module is almost never irreducible in the usual sense (multiplication by is a morphism). For reductive l, the highest weight U(l)-modules are defined similarly to the semisimple case. Let A be a representation of U(l) with highest weight. It extends to a

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representation of U(p) for p = p+ , by setting it zero on U (n+l ). A generalized Verma g module over U(g) is the induced module Mp,A = U(g) ⊗U (p) A =: Indp A, cf. [J1]. By Cλ we will denote the one dimensional U(l)-module defined by a character λ : l → C. The module Mp,Cλ will be denoted simply by Mp,λ and the Verma module Mb,λ by Mλ . Every U(g)-module with highest weight λ is a quotient of Mλ . The generalized Verma module Mp,A induced from a U(l)-module A with highest weight λ is a highest weight module, so it is a quotient of Mλ as well. Lemma 3.1. Let A be a finite dimensional U(l)-module. Then the U(g)-module Mp,A is a deformation of the classical generalized Verma module over U(g). Proof. It follows from (4) that Mp,A is a free U(n− l )-module generated by 1 ⊗U (p) A and hence it is C[[]]-free. Since the decomposition (4) is a deformation of the classical triangular decomposition, Mp,A is a deformation of the corresponding generalized Verma module over U(g). A lowest weight representation of U(l) can be extended to a representation of U(p− ) + by setting it trivial on U(n− l ). This is possible, due to (3). Similarly to Mp ,A , the module Mp− ,A := U(g) ⊗U (p− ) A is introduced, where A is taken to be a lowest weight U(l)-module. Proposition 3.2. Let A be a lowest and B a highest weight U(l)-modules. Then the g U(g)-module M p− ,A ⊗ M p+ ,B is isomorphic to Indl (A ⊗ B) := U(g) ⊗U (l) (A ⊗ B). Proof. Consider the map U(g) ⊗ A ⊗ B → U(g) ⊗ A ⊗ U(g) ⊗ B defined by u ⊗ a ⊗ b → u (1) ⊗ a ⊗ u (2) ⊗ b, where u (1) ⊗ u (2) is the standard symbolic notation for the coproduct (u). This map induces a homomorphism g

l,A,B : U(g) ⊗U (l) (A ⊗ B) → M p− ,A ⊗ M p+ ,B

(5)

of U(g)-modules. We claim that this map is an isomorphism. First let us prove the statement assuming l = h, A = Cµ , and B = Cν . Introduce the grading in U(n± ) by weight height setting dege±α = 1 for α ∈ . The gradings in g U(n± ) induce a double grading in Indh Cµ ⊗ Cν and M p− ,µ ⊗ M p+ ,ν , which can be identified as graded spaces. Each homogeneous component has finite rank over C[[]]. Let vµ and vν be the generators of M p− ,µ and M p+ ,ν , respectively. Take u ± ∈ U(n± ) g to be monomials in {e±α }α∈ and compute the map l,Cµ ,Cν : u − u + (vµ ⊗ vν ) → (u − )(1) (u + )(1) vµ ⊗ (u − )(2) (u + )(2) vν = (u − )(1) u + vµ ⊗ (u − )(2) vν = cq u + vµ ⊗ u − vν + w + vµ ⊗ w − vν .

Here cq ∈ C[[]] is invertible and the elements w ± ∈ U(n± ) belong to subspaces of g degree < deg u ± . This computation shows that l,Cµ ,Cν is a triangular operator (relative to the double grading) with invertible diagonal. Therefore it is an isomorphism. The above consideration also proves that the map (5) is an epimorphism for general l, as A and B are quotients of M p− ,µ and M p+ ,ν , respectively. We must check that (5) is g injective. The map l,A,B is surjective modulo as a U(g)-morphism. By dimensional arguments based on the bi-grading, we conclude that this U(g)-morphism is an isomorg phism. Therefore l,A,B is an isomorphism by the obvious deformation arguments.


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Remark 3.3. Concluding this section we remark that the triangular decomposition and the generalized Verma modules can be naturally defined for the algebra U˘(g). We will use this observation in Subsect. 8.2, where the induction is made from the character λ/2, λ ∈ c∗ . The U˘(g)-action on Mp,λ/2 does not extend to an action of U(g). However the natural U˘(g)-action on End(Mp,λ/2) extends to an action of U(h), and that is what we need for our construction. 3.2. Pairing between upper and lower generalized Verma modules. Denote by c the center of l and by cr eg ⊂ c the subset {λ ∈ c∗ | (λ, α ∨ ) = 0, ∀α ∈ g − l}. Clearly cr eg is a dense open set in c. The coadjoint g-module g∗ is canonically identified with g via the Killing form. Then the dual space c∗ is identified with the orthogonal complement to the annihilator of c in h∗ ; so c∗ ⊂ h∗ under this convention. 2 Given a root α ∈ R, let α ∨ denote the dual root α ∨ = (α,α) α. We call a weight λ ∈ c∗ generic if (λ, α ∨ ) ∈ Z for all α ∈ g − l. Clearly the set c∗gen of generic weights is a dense open subset in cr∗eg . With every finite dimensional irreducible l-module A one can associate a weight λ A ∈ c∗ such that ha = λ A (h)a for all h ∈ c and a ∈ A. We call A generic if λ A ∈ c∗gen . An arbitrary finite dimensional l-module is called generic if its every irreducible submodule is generic. This terminology extends to the corresponding U(l)-modules. There exists a U(g)-equivariant pairing between M p− ,A∗ and M p+ ,A . The construction goes as follows. Consider a bilinear U(g)-equivariant (g) ⊗ A∗ ) ⊗ map (U (U(g) ⊗ A) → C[[]] defined by u 1 ⊗ ξ ⊗ u 2 ⊗ x → ξ Pl γ (u 1 )u 2 x , where Pl is introduced in Subsect. 2.3. This map is equivariant by construction and factors through a bilinear equivariant map M p− ,A∗ ⊗ M p+ ,A → C[[]], as required. Proposition 3.4. Let A be a generic finite dimensional U(l)-module. Then the equivariant pairing between Mp+ ,A and Mp− ,A∗ is nondegenerate. Proof. Without loss of generality, we may assume that A0 = A/A is irreducible. Since −c∗gen = c∗gen , the modules A and A∗ are generic simultaneously. It follows from [Jan1], Satz 3, that the U(g)-module Mp+ ,A /Mp+ ,A is irreducible for λ ∈ c∗gen . Clearly the same is true for Mp− ,A∗ /Mp− ,A∗ . Therefore the pairing in question is nondegenerate modulo (being U(g)-equivariant and not identically zero). With respect to the pairing, the weight spaces of weights µ and ν are orthogonal unless µ + ν = 0. Since the weight spaces in Mp+ ,A and Mp− ,A∗ are C[[]]-finite, non-degeneracy of the pairing follows from non-degeneracy modulo .

3.3. Tensor product of finite dimensional and generalized Verma modules. We call a U(g)-module a weight module if it is h-diagonalizable and its weight spaces are finite and free over C[[]]. For any weight module let (U ) denote the set of weights of U and U [µ] the weight ∗ space for µ ∈ (U of infinite ). The dual module U of∗linear functionals on U consists formal sums f = µ f µ , where f µ ∈ U [µ] . The action of U(g) on U ∗ is defined to be (x f )(u) = f (γ (x)u), for f ∈ U ∗ , u ∈ U , and x ∈ U(g). Define the restricted dual U ◦ as a natural U(g)-submodule in U ∗ by setting U ◦ = ⊕µ∈ (U ) U [µ]∗ ⊂ U ∗ (only finite sums admitted).

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Lemma 3.5. Let U1 , U2 be weight U(g)-modules and W1 , W2 finite dimensional U(g)modules. Then HomU (g) (W1 ⊗ U1 , W2 ⊗ U2 ) HomU (g) (U2◦ ⊗ U1 , W2 ⊗ W1∗ ).

Proof. Clear.

Proposition 3.4 asserts that Mp− ,A∗ Mp◦,A for generic A. Lemma 3.6. Let W be a finite dimensional U(g)-module, A and B finite dimensional U(l)-modules, and λ ∈ c∗gen . Then the following C[[]]-linear isomorphisms take place: HomU (g) (Mp,A⊗Cλ , Mp,B⊗Cλ ) HomU (g) (Mp,A⊗Cλ , W ⊗ Mp,λ ) HomU (g) (W ⊗ Mp,λ , Mp,A⊗Cλ ) HomU (g) (W ⊗ Mp,λ , W ⊗ Mp,λ )

HomU (l) (A, B), HomU (l) (A, W ), HomU (l) (W, A), HomU (l) (W, W ).

(6)

Proof. The proof is based on Propositions 3.2 and 3.4 and can be conducted similarly as in [DM1] for the classical case of U(g)-modules. For instance, let us check the first isomorphism. By Proposition 3.4, the module Mp− ,C∗λ ⊗B ∗ is isomorphic to the restricted dual Mp◦,B⊗Cλ . Therefore HomU (g) (Mp,A⊗Cλ , Mp,B⊗Cλ ) HomU (g) (Mp− ,C∗λ ⊗B ∗ ⊗ Mp+ ,A⊗Cλ , C[[]]), (7) by Lemma 3.5. Here C[[]] is the trivial U(g)-module. According to Lemma 3.2, the tensor product of lower and upper generalized Verma modules is induced from the U(l)module C∗λ ⊗ B ∗ ⊗ A ⊗ Cλ B ∗ ⊗ A. Applying the Frobenius reciprocity, we continue (7) with HomU (l) (B ∗ ⊗ A, C[[]]) HomU (l) (A, B), as required. Proposition 3.7. Let W be a finite dimensional U(g)-module and λ ∈ c∗gen . Then W ⊗ Mp,λ admits the direct sum decomposition W ⊗ Mp,λ = ⊕ A0 Mp,A⊗Cλ ,

(8)

where summation is taken over the simple l-modules with multiplicities entering W0 . Proof. The isomorphisms (6) hold modulo . Moreover, they commute with taking quotients mod . First let us prove the classical mod analog of decomposition (8), retaining the same notation for the Verma modules over U(g). We assume in (8) a fixed decomposition within each isotypic A0 -component of W0 . Let j A0 : A0 → W0 be the l-equivariant injections such that A j A0 = id W0 . Let jˆ A0 be their lifts Mp,A0 ⊗Cλ → W ⊗ Mp,A0 ⊗Cλ . For generic λ ∈ c∗gen all the modules Mp,A0 ⊗Cλ are irreducible. Therefore all jˆ A0 are

A0

jˆ A0

linearly independent, and the g-equivariant map ⊕ A0 Mp,A0 ⊗Cλ −→ W0 ⊗ Mp,λ is an embedding. Then it is an embedding of h-modules W0 ⊗ U(n− l ) ⊕ A0 A0 ⊗

jˆ A0

− U(n− l ) −→ W0 ⊗ U(nl ) and therefore an isomorphism. This proves the statement modulo . We can choose j A ∈ HomU (l) (A, W ) to be deformations of morphisms j A0 splitting the U(l)-module W into the direct sum of highest weight submodules. Take the U(g)morphisms jˆ A : Mp,A⊗Cλ → W ⊗ Mp,λ corresponding to j A ∈ HomU (l) (A, W ) under the second isomorphism from (6). Since isomorphisms (6) commute with taking quoA0

jˆ A

A0 tients mod , we have jˆ A = jˆ A0 mod . Consider the morphism ⊕ A0 Mp,A⊗Cλ −→ W ⊗ Mp,λ of U(g)-modules. Restricting consideration to weight spaces we conclude that A0 jˆ A is an isomorphism because it is so modulo .


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4. Properties of the universal RE Matrix In the present section we recall general properties of the universal reflection equation (RE) matrix Q = R21 R and study its action on tensor products of finite dimensional and (generalized) Verma modules. In particular, we compute the spectrum of the matrix powers Q . We also evaluate a polynomial in Q whose entries lie in the annihilator of the generalized Verma module. This information will be used in Sect. 8 for construction of quantum orbits. Recall from [Dr2] that the element Q = R21 R can be presented as Q = (υ)(υ −1 ⊗ −1 −1 2 υ ), where1 υ = γ (R−1 1 )R2 = γ (R1 )R2 . Conjugation with υ implements the −1 2 squared antipode υxυ = γ (x) for all x ∈ U(g). On the other hand, the squared antipode can be written as the conjugation γ 2 (x) = q −2h ρ xq 2h ρ , x ∈ U(g). It follows from here that υ = q −2h ρ z, where z is some invertible element from the center of U(g). Therefore we can write Q = (z)(z −1 ⊗ z −1 ), q −2h ρ

h∗

(9)

χλ

is a group-like element. Let λ ∈ and be the corresponding central charas acter of U(g). It is easy to compute the value χ λ (z) via a U(g)-module W with highest weight, using the structure of R-matrix (1). Let wλ be the highest weight vector in W . Since U (n+ )wλ = 0, we have υwλ = q (λ,λ) wλ = χ λ (z)q −2(ρ,λ) wλ . From this we find χ λ (z) = q (λ,λ)+2(ρ,λ) .

(10)

The element Q satisfies the identities R21 Q13 R12 Q23 = Q23 R21 Q13 R12 , ( ⊗ id)(Q) = R−1 12 Q13 R12 Q23 ,

(11) (12)

of which the first may be called the universal reflection equation, cf. [DKM]. Equation (12) is the key identity of the fusion procedure for solutions to the RE, [DKM]. 4.1. Characteristic polynomial for RE matrix. Given a finite dimensional representation (W, πW ) let QW denote the element (πW ⊗ id)(Q) ∈ End(W ) ⊗ U(g). Proposition 4.1. Let W be a highest weight U(g)-module with the multiset of weights (W ). Then i) there exists a polynomial p of degree #(W ) in one variable with coefficients in the center of U(g) such that p(QW ) = 0, ii) the spectrum of the operator QW on W ⊗ Mλ is 2(λ+ρ,ν )−2(ρ,ν)+(ν ,ν )−(ν,ν) i i i , (13) q ν ∈(W ) i

where ν is the highest weight of W . Proof. Notice that a symmetric function in the eigenvalues (13) is invariant under the action of the Weyl group. Then i) follows from ii) through the Harish-Chandra homomorphism, [Jan2]. So let us check ii). As a C[[]]-module, W ⊗ Mλ is isomorphic to W ⊗U(n− ). Under this isomorphism, the subspace of weight β ∈ h∗ in W ⊗ U(n− ) goes to the subspace of weight β + λ in W ⊗ Mλ . Since Q is invariant, it preserves the weight spaces in W ⊗ Mλ , which have finite rank over C[[]]. Now observe that for generic λ ii) follows from Proposition 3.7. This implies ii) for arbitrary λ, since restriction of QW to any weight space has polynomial dependence on λ in every order in . 1 Here and further on we use the symbolic notation R = R ⊗ R with suppressed summation. 1 2

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4.2. Minimal polynomial for RE matrix. In this subsection we determine the spectrum of Q on the tensor product of finite dimensional and generalized Verma modules over U(g). Theorem 4.2. Let λ ∈ c∗gen be a generic character of a Levi subalgebra in l ⊂ g. Let W be a finite dimensional U(g)-module and let l(W ) = {υl = νil } ⊂ (W ) denote the multiset of highest weights of simple l-modules entering W0 . The operator Q is diagonalizable on W ⊗ Mp,λ and has eigenvalues 2(λ+ρ,υ )−2(ρ,ν)+(υ ,υ )−(ν,ν) l l l . (14) q υ ∈ (W ) l

l

Proof. For generic λ the module W ⊗ Mp,λ splits into the direct sum of highest weight modules, by Proposition 3.7. The operator Q is proportional to (z) on W ⊗ Mp,λ . Thus we conclude that Q is diagonalizable because (z) is diagonalizable. For every U(l)-module A ⊂ W with highest weight µ the summand Mp,A⊗Cλ in (8) is a U(g)-module of highest weight λ + µ. Hence (z) and therefore Q act as scalar multipliers on Mp,A⊗Cλ . Their eigenvalues are computed using (10). 4.3. A construction of central elements. Assume that g is an arbitrary complex simple Lie algebra. Let (W, πW ) be a finite dimensional U(g)-module. Let X ∈ End(W ) ⊗ U(g) be an invariant matrix, i. e. commuting with πW (x (1) ) ⊗ x (2) for all x ∈ U(g). It is known that the q-trace defined by Trq (X ) := Tr πW (q 2h ρ )X (15) is ad-invariant and hence belongs to the center of U(g). The annihilator of the Verma module with the highest weight λ is generated by the kernel of a central character χ λ . Let us compute the values χ λ QW . Define a map d : h∗ → C[[]] setting d(λ) :=

q (λ+ρ,α) − q −(λ+ρ,α) , q = e . q (ρ,α) − q −(ρ,α)

(16)

α∈R+

For a finite dimensional U(g)-module W with the highest weight λ, the Weyl character formula [N] gives d(λ) = Trq (id W ), the q-dimension of W . Proposition 4.3. Let W be a finite dimensional U(g)-module with the multiset of weights (W ). Then for any λ ∈ h∗ , d(λ + νi ) , (17) χ λ Trq (QW ) = xνi d(λ) νi ∈(W )

where xνi are the eigenvalues of QW given by (13). Proof. We adapt a proof from [GZB], because it is suitable for any U(g)-invariant operator. Let us check the statement for special λ first. Namely, we suppose that λ is integral dominant and ν, the highest weight of W , is subordinate to λ. The weight ν is called subordinate to λ if λ + νi are dominant integral for all νi ∈ (W ). For ν fixed we denote by D+ν the set of such λ that ν is subordinate to λ. According to [K], a polynomial


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function on h∗ is determined by its values on D+ν . Both sides of (17) are polynomials in λ in every order in , thus it suffices to compare them on D+ν only. For λ ∈ D+ν let us compute the central character in question on the finite dimensional module U with the highest weight λ. In this case, the module W ⊗ U splits into the direct sum of U(g)-modules of highest weights νi + λ for νi ∈ (W ). An invariant operator on W ⊗ U decomposes into a sum over invariant projectors {Pi }νi ∈(W ) to the highest weight submodules, so it suffices to compute the q-trace of these projectors. The operator (Trq ⊗ id)(Pi ) is scalar on U . Taking q-trace over U gives (Trq ⊗ id)(Pi ) = (Trq ⊗ Trq )(Pi )/(id ⊗ Trq )(idU ) = d(λ + νi )/d(λ), since the q-trace is multiplicative with respect to the tensor product. This proves the statement for λ ∈ D+ν and therefore for all λ ∈ h∗ . 5. On quantization of affine algebraic varieties In this section we develop a machinery for equivariant quantization. Throughout the section we assume that g is semisimple. Moreover, here we admit an arbitrary, i. e. quasitriangular, triangular, or even trivial, quantization of U(g). In what follows, we use some standard facts from commutative algebra, such as the Nakayama lemma. The reader can find the “list” of facts we rely on in [M]. 5.1. A flatness criterion. Recall that an associative algebra and (left) U(g)-module A is called a U(g)-module algebra if its multiplication is compatible with the U(g)-action. That is, for all h ∈ U(g) and all a, b ∈ A, h (ab) = (h (1) a)(h (2) b), where h (1) ⊗ h (2) = (h). Definition 5.1. A deformation A of a U(g)-module algebra A0 is called equivariant if A is a U(g)-module algebra and the action of U(g) on A coincides modulo with the action of U(g) on A0 . For every U(g)-module E and a finite dimensional highest weight U(g)-module W there exists a natural morphism W ⊗ HomU (g) (W, E) → E of U(g)-modules. We call the image of this morphism the isotypic W -component of E. We call a U(g)-module admissible if it is a direct sum of its isotypic components and each of them is finitely generated over C[[]]. It can be shown that submodules and quotient modules of an admissible module are again admissible. The following proposition gives a flatness criterion for quotient algebras. Informally, if one constructs an ideal J that lies in the kernel of a certain homomorphism S → T and has the “right classical limit”, then J equals the entire kernel. Proposition 5.2 (deformation method). Let S be an admissible and T be C[[]]-torsion free U(g)-module algebras; let : S → T be a non-zero equivariant homomor phism. Suppose ker contains an invariant ideal J such that the image J0 of J0 in S0 is a maximal g-invariant ideal. Then i) im is a C[[]]-free deformation of S0 /J0 , ii) ker = J. Proof. First of all, the U(g)-module im is admissible and torsion free; hence it is C[[]]-free.

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Since im is free, we have the direct sum decomposition S ker ⊕ im of C[[]]-modules. Hence S0 (ker )0 ⊕ (im )0 and (ker )0 ⊃ J0 . By assumption, J0 is a maximal g-invariant ideal, hence either (ker )0 = S0 or (ker )0 = J0 . The first option is impossible. Indeed, then (im )0 = 0 and im = 0 since im is C[[]]-free. In this case ker = S, and the map would have been zero. Therefore (im )0 = S0 /J0 and i) is proven. We have an embedding J → ker and an epimorphism J0 → (ker )0 = J0 . Applying the Nakayama lemma to each isotypic component (they are C[[]]-finite), we prove ii).

Remark 5.3. We emphasize that J0 is assumed to be not just J0 = J/J but its image in S0 . It is essential to distinguish between J0 and J0 because the functor mod is not left exact. Eventually, J0 and J0 coincide in our situation. However that is not a priori obvious and follows from the proof. In practice, J is often defined via a system of generators. Then J0 is generated by their images in S0 , so it is even easier to control J0 than J0 . 6. Quantization of simple algebraic groups In the present section, we construct quantization of a special Poisson bracket on a simple algebraic group. The quantized ring of polynomial functions is realized as a quotient of the so-called reflection equation algebra and simultaneously as a subalgebra in the quantized universal enveloping algebra. 6.1. Simple groups as Poisson Lie manifolds. Let g be a complex simple Lie algebra and G be a connected Lie group corresponding to g. An element g ∈ G is called semisimple if it belongs to a maximal torus in G. Let r be a classical r -matrix defining a factorizable Lie bialgebra structure on g. Denote by r− and its skew and symmetric parts, respectively. We assume r to be normalized so that is the inverse (canonical element) of the Killing form on g. Given an element ξ ∈ g let ξ l and ξ r denote, respectively the left- and right invariant vector fields on G generated by ξ : (ξ l f )(g) =

d d f (getξ )|t=0 , (ξ r f )(g) = f (etξ g)|t=0 dt dt

for every smooth function f on G. The Semenov-Tyan-Shansky (STS) Poisson structure [STS] on the group G is defined by the bivector field l,l r,r r,l l,r ad,ad r− + r− − r− − r− + l,l − r,r + r,l − l,r = r− + (r,l − l,r ).

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Here ξ ad := ξ l − ξ r , ξ ∈ g. Consider the group G as a G-space with respect to the adjoint action. The STS Poisson structure makes G a Poisson Lie manifold over the Poisson Lie group G endowed with the bracket r l,l − r r,r . In fact, G is a Poisson Lie manifold not only over G, but over a Poisson Lie group corresponding to the double Lie bialgebra Dg.


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Theorem 6.1 ([AM]). Symplectic leaves of the STS Poisson structure coincide with conjugacy classes in G. Let us compute the restriction of the STS bracket to the class C g of a semisimple element g ∈ G. The Lie algebra g splits into the direct sum g = l ⊕ m of vector spaces, where l is the eigenspace of Ad(g) corresponding to the eigenvalue 1 and m is the Ad(g)-invariant subspace where Ad(g) − id is invertible. This decomposition is orthogonal with respect to the Killing form. The tangent space to C g at the point g is identified with m, while l is the Lie algebra of the centralizer of g. Let {ξµ } ⊂ m be a base of eigenvectors of Ad(g) labelled by the eigenvalues of Ad(g). We have (ξµ , ξν ) = 0 unless µν = 1 and assume the normalization (ξµ , ξµ−1 ) = 1. One can check that the restriction of the STS bracket to the tangent space at the point g is the bivector µ+1 rm∧m + ξµ ⊗ ξµ−1 ∈ m ∧ m, µ−1 µ where the first term is the projection of r to m ∧ m. The second term is correctly defined since Ad(g) − id is invertible on m. 6.2. Quantization of the STS bracket on the group. In this subsection we describe quantization of the STS bracket on classical matrix groups in terms of generators and relations. We do not assume G to be connected. The DS and STS brackets can be defined on G if is G-invariant, e.g. G = O(N ) rather than S O(N ). By C[G] we denote the U(Dg)-equivariant deformation (quantization) of the affine ring C[G] along the STS bracket. The reader should not be confused with this notation, because this is the only quantization of C[G] that appears in the present paper. The algebra C[G] in the form of a star product was constructed in [DM3]. Below we give a description of C[G] in terms of generators and relations, as a quotient of the so-called reflection equation algebra. Let (V, π ) be the defining representation of U(g) and R the image of R in End(V ⊗2 ). Denote by K the quotient of the tensor algebra of End(V ∗ ) by the quadratic relations R21 K 1 R12 K 2 = K 2 R21 K 1 R12 .

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||K ij ||

is the matrix of the generators forming the standard matrix base in Here K = ∗ End(V ). The algebra K is called a reflection equation (RE) algebra, [KSkl]. It is a C[[]]-flat deformation of the polynomial ring C[End(V )] only for g = sl(n). The U(g)-equivariant quantization of C[G] can be realized as a quotient of K. Below we describe the corresponding ideals only for G orthogonal and symplectic. That can also be done for the case G = S L(n), but then the algebra K is good enough for our purposes, so it is even more convenient to work with K rather than C[S L(n)]. Assume g to be orthogonal or symplectic and G = O(N ) or Sp(n). Let P ∈ End(V ⊗ V ) be the permutation and let κ be the one-dimensional projector constructed out of the matrix P R, [FRT]. Proposition 6.2. The algebra C[G] is isomorphic to the quotient of K by the U(g)invariant ideal of relations R −1 K 1 R K 2 κ = κ = κ R −1 K 1 R K 2 . A proof that this quotient is C[[]]-free is given in [M].

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6.3. Embedding of C[G] in U(g). Recall that a (left) Yetter-Drinfeld (YD) module over U(g) is simultaneously a left U(g)-module, a left U(g)-comodule, and these two structures are compatible in a certain way, [Y]. A YD algebra over U(g) is a U(g)-module algebra, U(g)-comodule algebra and a YD module with respect to these structures. An example of YD module algebra is U(g) considered as the adjoint module and comodule via the coproduct. Let us describe a method of producing YD modules (module algebras) out of U(Dg)modules, where Dg is the double of g. For a factorizable semisimple Lie bialgebra g R

the algebra U(Dg) is isomorphic to the twisted tensor square U(g) ⊗ U(g), [RS]. The ˆ of U(Dg) is expressed through the universal R-matrix R of U(g) universal R-matrix R ˆ = R−1 R−1 R24 R23 . It is easy to see that the right component of Rˆ lies, by the formula R 41 31 R

in fact, in U(g) ⊂ U(g) ⊗ U(g) (the diagonal embedding). Then any U(Dg)-module (module algebra) becomes a YD module (YD module algebra) when equipped with the ˆ2 ⊗R ˆ 1 a (recall that we use the convention R ˆ =R ˆ1 ⊗R ˆ 2 ). U(g)-coaction δ(a) = R The U(g)-action is induced through the embedding of : U(g) → U(Dg). The algebra C[G] is a U(Dg)-algebra and hence a YD module algebra, by the above construction, [DM3]. As a C[[]]-module, it coincides with the FRT dual to U(g). The Hopf pairing between the FRT dual and U(g) induces a pairing between C[G] and U(g). By means of this pairing, the universal RE matrix Q implements a U(g)-algebra homomorphism C[G] → U(g), a → a, Q1 Q2 .

(21)

It is easy to check using the explicit form of the R matrix that C[G] lies in U˘(g) ⊂ U(g). The map (21) is, in fact, a homomorphism of YD algebras. Proposition 6.3. Suppose that G is connected. Then the map (21) is embedding. Proof. The proof easily follows from Proposition 5.2 after a slight adaptation to YD module algebras. The algebra C[G] decomposes into the direct sum ⊕W0 W ⊗ W ∗ taken over the simple finite dimensional g-modules. Each summand is a YD module and its quotient mod is Dg = g ⊕ g-irreducible with multiplicity one. We have a pair of YD module algebras S = C[G] and T = U(g). The former is admissible (as a YD-module), while the latter has no -torsion. Set to be the map (21). Its image and kernel are YD modules and are free over C[[]]. Therefore S decomposes into the direct sum ker ⊕ im . The rest of the proof is readily adapted from the proof of Proposition 5.2 if one observes that YD modules become Dg-modules in the quasi-classical limit (the first order in ). Put J = 0. In our case Dg is isomorphic to the direct sum g ⊕ g as a Lie algebra. Since G is connected, the algebra S0 = C[G] has no non-zero Dg-invariant proper ideals. Thus we conclude that ker = J = 0. 7. Center of the algebra C[G] Let G be a simple complex algebraic group. If G is simply connected, then C[G] is a free module over the subalgebra of invariants I (G), [R]. More precisely, there exists a submodule E0 ⊂ C[G] such that the multiplication map I (G) ⊗ E0 → C[G] is an isomorphism of vector spaces. Each isotypic component in E0 enters with finite multiplicity. This fact has a quantum analog. An elementary proof of the following theorem is given in [M].


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Theorem 7.1. Let G be a simple complex connected algebraic group and let C[G] be the U(Dg)-equivariant quantization of C[G] along the STS bracket. Then i) the subalgebra I(G) of U(g)-invariants coincides with the center of C[G], ii) I(G) I (G) ⊗ C[[]] as C-algebras. Let Gˆ be the simply connected covering of G. Then ˆ is a free I(G)-module ˆ ˆ Each iii) C[G] generated by a U(g)-submodule E ⊂ C[G]. isotypic component in E is C[[]]-finite. Theorem 7.1 i) implies, in particular, that the center of C[G] is the intersection of C[G] with the center of U(g). If G is not simply connected, Theorem 7.1 iii) will be true if the classical algebra of invariants I (G) is polynomial. That is the case, e.g. for G = S O(2n + 1), however not so for G = S O(2n). We will use Theorem 7.1 for quantization of conjugacy classes in Subsect. 8.2. Important for us will be the following corollary. Proposition 7.2. Let G be a simple complex algebraic group. Suppose λ is a character of I(G), i. e. a unital homomorphism to C[[]]. Denote by Jλ the ideal in C[G] generated by ker λ. Then the quotient C[G]/Jλ is an admissible U(g)-module. Proof. The group G is a quotient of its universal covering group Gˆ over a finite central ˆ The affine ring C[G] is embedded in C[G] ˆ as a subalgebra of Z -insubgroup Z ⊂ G. variants with respect to the regular action. The group Z naturally acts on the quantized ˆ (which is the result of a Z -invariant twist of C[G]), ˆ and the subalgebra algebra C[G] of Z -invariants is exactly C[G]. Accordingly, I(G) is the subalgebra of Z -invariants ˆ The latter is finite over I(G), since the group Z is finite. in I(G). Let λ be a character of I(G). Denote by Jλ and Jˆλ the ideals generated by ker λ in ˆ respectively. Theorem 7.1 implies that Jλ = Jˆλ ∩ C[G]. Therefore C[G] and C[G], ˆ Jˆλ . The quotient I(G)/ ˆ ˆ = Jˆλ ∩ I(G) the quotient C[G]/Jλ is embedded in C[G]/ ˆ ker λ is C[[]]-finite and hence the module C[G]/ ˆ Jˆλ has finite isotypic comI(G)/ ˆ ˆ ponents, by Theorem 7.1. So C[G]/ Jλ is admissible and therefore its submodule C[G]/Jλ is admissible as well. Let us emphasize that the quotient C[G]/Jλ is C[[]]-free if the classical subalgebra of invariants is a polynomial algebra, cf. the remark after Theorem 7.1. Otherwise it may have -torsion. 7.1. The case of classical series. The case of G = S O(2n) differs from other classical matrix groups. Let us focus on a simpler case of the A, B, C series first. Denote by τ , = 1, 2 . . . , the central elements Trq (QV ) ∈ U(g), where V is the defining representation. Proposition 7.3. The elements τ , = 1, . . . , N , generate a dense subalgebra in the center of C[G] for G being S L(n), S O(2n + 1), or Sp(n). Proof. The center in C[G] is isomorphic to the classical subalgebra of invariants extended by C[[]], as stated in Theorem 7.1. Modulo , it is generated by the classical limits τ0 = τ mod . The elements τ0 can be expressed through τ as -series that converge in the -adic topology.

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we assume G = S O(2n). Let V be the defining In the rest of the subsection n U so(2n) -module and {±εi }i=1 = (V ) be the set of weights. In the classical limit, the n th exterior power of V0 splits into two submod irreducible n−1 εi , [VO]. Therefore the U so(2n) -module ∧qn V ules with highest weights ±εn + i=1 (q-anti-symmetrized) is a direct sum of two modules (W± , π± ) with the highest weights n−1 εi , respectively. Let p± : V ⊗n → W± be the intertwining projectors. The ±εn + i=1 n element ( ⊗id)(Q) is expressed through Q and the numerical matrix R = (π ⊗π )(R) in the defining representation, by virtue of (12). Hence QW± can be explicitly expressed through the matrix QV whose entries generate C[G] within U(g). In the same fashion, we can define the invariant matrices K ± ∈ End(W± ) ⊗ K expressing them through K , R, and p± by the same formulas. The matrices QW± are obtained from K ± via the projection K → C[G]. Define a central element τ − of U so(2n) by setting τ − := Trq (QW+ ) − Trq (QW− ).

(22)

Similarly we introduce the central elements Trq (K + ) − Trq (K − ) ∈ K. n Proposition 7.4. Let λ ∈ h∗ be a weight. Then χ λ (τ − ) = i=1 (q 2(λ+ρ,εi ) −q −2(λ+ρ,εi ) ). Proof. By Corollary A.2, the central element Trq (QW± ) acts on a module with high2(h +h ) ρ λ ) . This scalar becomes est weight λ as multiplication by the scalar Tr π± (q the group character ch± (t) (a class function) associated with W± upon the substitution n q 2(λ+ρ,εi ) → ti , where {ti }i=1 are the coordinate functions on the maximal torus in S O(2n). It is known [We] that the ring of characters of the group S O(2n) is isomorphic to C[t1 , . . . , tn , t1−1 , . . . , tn−1 ]W S O(2n) . The group W S O(2n) acts by permutations of the pairs (ti , ti−1 ) and even number of inversions σi : ti ↔ ti−1 . The inversion σi is the restriction to the maximal torus of an automorphism of the group S O(2n). It is factored by (σi σn ) ∈ W S O(2n) and σn , where the latter corresponds to the non-trivial automorphism αn−1 ↔ αn of the Dynkin diagram. The automorphism σn swaps the modules W± , and therefore the function ch+ (t) − ch− (t) changes n (ti − ti−1 ). In fact, it equals sign under σi . Hence ch+ (t) − ch− (t) is divisible by i=1 n −1 i=1 (ti − ti ). This is verified in a standard way by comparing the highest and lowest terms with respect to a natural lexicographic ordering in C[t1 , . . . , tn , t1−1 , . . . , tn−1 ] (and elementary analysis of the weight structure of n V0 ). As a corollary, we obtain the following. Proposition 7.5. The elements τ − and τ , = 1, . . . , N , generate a dense subalgebra in the center of C[S O(2n)]. Proof. Similar to Proposition 7.3.

8. Quantization of conjugacy classes 8.1. Non-exceptional classes. Semisimple conjugacy classes are the only closed classes of a simple algebraic group, [S]. Among them we select classes of special type and call them non-exceptional (as well as the corresponding elements of the group). For


653

the special linear group those are all semisimple classes; all of them can be realized as coadjoint orbits. For G orthogonal and symplectic, non-exceptional classes consist of orthogonal and symplectic matrices with no eigenvalues +1 and −1 simultaneously. Those are precisely the classes which are isomorphic to coadjoint orbits via the Cayley transformation X → (1 ∓ X )(1 ± X )−1 . Thus non-exceptional classes have the Levi subgroups as stabilizers. For the special linear and symplectic groups all classes with Levi stabilizers are non-exceptional. We can assume λ2 = 1 ⇒ λ = 1. The other option λ = −1 is reduced to the first one. It is possible only for G = S O(2n) or G = Sp(n), in which cases the mapping g → −g is an automorphism of the Poisson Lie manifold G. Let g ∈ G be a semisimple element. It satisfies a matrix polynomial equation p(g) = 0 and defines a character χ g of the subalgebra of invariants in C[G]. Theorem 8.1. The defining ideal N (C g ) of a non-exceptional conjugacy class C g g is generated by the kernel of χ g and by the entries of the minimal matrix polynomial for g. A proof of this theorem will be given elsewhere. It is based on the following facts. a) Non-exceptional classes are isomorphic to semisimple coadjoint orbits via the Cayley transformation. b) The defining ideals of semisimple coadjoint orbits can be obtained as classical limits of annihilators of generalized Verma modules, [DGS]. c) It was shown in [G] (see also [J2]) that for certain weights the annihilator of the generalized Verma module is generated by a copy of the adjoint module in U(g) and the kernel of the central character. This is sufficient to describe the defining ideals for all semisimple coadjoint orbits and hence for non-exceptional conjugacy classes.

8.2. The quantization theorem. In this subsection G is a simple complex algebraic group from the classical series and g its Lie algebra. Fix a non-exceptional element g in the maximal torus T ⊂ G. Let l be the Lie algebra of the centralizer of g, which is a Levi subgroup. Take h λ ∈ h such that eh λ = g and λ ∈ cr∗eg is a regular character of l. The generalized Verma modules can be naturally considered over the ring of polynomials C[c∗ ], where c is the center of l. In this case, the parabolic induction is performed from the representation of U˘(l) in C[c∗ ][[]]. Propositions 4.1 and 4.2 are valid for such modules, since they are valid for the generic element of c∗ . Let (ci ) be coordinate functions generating C[c∗ ] and (λi ) the coordinates of a regular element λ ∈ cr∗eg . The homomorphism C[c∗ ][[]] → C(()), ci → λi /(2), defines a C[[]]-valued character of U˘(l) (but not of U(l)). Let M˘ λ/2 and M˘ p,λ/2 denote the corresponding Verma and generalized Verma modules over U˘(g). For these modules Propositions 4.1 and 4.2 hold true as well. It is easy to see that the U˘(g)-action on End( M˘ p,λ/2) extends to an action of U(g). Recall that (V, π ) stands for the defining representation of U(g). Also recall from Proposition 6.3 that the entries of the matrix QV = π(Q1 ) ⊗ Q2 ∈ End(V ) ⊗ U(g) generate the subalgebra C[G] ⊂ U(g), which is the equivariant quantization of C[G]. Theorem 8.2. For non-exceptional λ ∈ c∗ ⊂ h∗ the image of C[G] in End( M˘ p,λ/2) is a quantization of the ring of regular functions on the conjugacy class Cexp(h λ ) ⊂ G. The ideal of the quantized class is generated by the entries of the minimal polynomial in QV over the kernel of a character of I(G).

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Proof. The composite map C[G] → U˘(g) → End( M˘ p,λ/2)

(23)

is a U(g)-equivariant algebra homomorphism. When restricted to the center in C[G], this map defines a character of I(G). Indeed, a central element from C[G] acts on M˘ p,λ/2 as multiplication by a scalar. That scalar is a polynomial in the eigenvalues of the matrix QV , as follows from Proposition A.1. The eigenvalues are given by (13), where λ should be replaced by λ/2. This proves that the representation in M˘ p,λ/2 defines a character of the center of U˘(g) and therefore a character of I(G). This character is a g deformation of the I (G)-character χ g , and will be denoted by χ . The matrix QV satisfies a polynomial equation for a polynomial p with simple roots. The roots of p are given by Theorem 4.2, where again λ should be replaced by λ/2. These roots are regular in and go over to the eigenvalues e(λ,εi ) of the matrix g = eh λ ∈ G, as → 0 . Next we are going to apply Proposition 5.2. Put S to be the quotient of C[G] over g the ideal generated by ker χ . The algebra S is admissible, due to Proposition 7.2. Put T = End( M˘ p,λ/2). The C[[]]-module T has no torsion, as a space of endomorphisms of a torsion free module. By construction, the map (23) factors through an equivariant algebra map : S → T. Put J to be the ideal in S generated by the entries of p(QV ) projected to S. By con struction, J ⊂ ker . The image J0 of J0 in S0 = C[G]/ I (G) − χ g is a maximal ginvariant ideal, because it is the image of the maximal g-invariant ideal N (C g ) ⊂ C[G]. Thus the conditions of Proposition 5.2 are satisfied. Therefore S/J is the quantization of C[C g ] and ker = J. In terms of C[G], the defining ideal of C[C g ] is generated g by the kernel of χ and the entries of p(QV ). Next we compute the quantized ideals of non-exceptional conjugacy classes.

8.3. Ideals of quantized non-exceptional conjugacy classes. First of all, we specialize formula (17) for g being a simple matrix Lie algebra and V the defining representation of U(g). Since every weight has multiplicity one, we rewrite (17) as χ λ (τ ) =

νi ∈(V )

xνi

q (λ+ρ+νi ,α) − q −(λ+ρ+νi ,α) . q (λ+ρ,α) − q −(λ+ρ,α)

(24)

α∈R+

Here xνi , νi ∈ (V ), are the roots of the characteristic polynomial for Q considered as an operator on V ⊗ Mλ . They are related to the highest weight λ by xνi = q 2(λ+ρ,νi )−2(ρ,ν) , νi = 0 and x0 = q −2(ρ,ν)−(ν,ν) . Perform the following substitution in (24). Set x0 := q −2n for g = so(2n + 1), xi = xνi for all g, and xi = x−νi = xi−1 q −4(ρ,ν) for g orthogonal and symplectic. Here i ranges from 1 to n = rk g. The eigenvalue x0 is present only for g = so(2n + 1). Below we use the convention i = N + 1 − i, cf. Subsect. 2.2. Under the adopted enumeration of weights of V the eigenvalues xi and xi , i = 1, . . . , N , correspond to the weights of the opposite signs. As a result of this substitution, we obtain the functions


()

ϑsl(n) (x) =

n

xi

i=1 ()

ϑso(2n+1) (x) = ()

ϑsp(n) (x) =

n

655

n q xi − x j q¯ , xi − x j j=1 j =i

i=1

n q xi − x0 q xi − x j q¯ q xi − x j q¯ + i ↔ i + x0 , xi − x0 q j=1 xi − x j xi − x j

n

q2x

xi

j =i

xi

n − x q¯ 2 q xi − x j q¯ q xi − x j q¯ + i ↔ i , xi − xi x − x x − x i j i j j=1

i=1

() ϑso(2n) (x) =

i

i

j =i

n

xi

i=1

n q xi − x j q¯ q xi − x j q¯ + i ↔ i , xi − x j xi − x j j=1 j =i

− (x) = q 2n(ρ,ν) ϑso(2n)

n

xi − xi ,

i=1 ()

where q¯ := q −1 . Remark that ϑg are in fact polynomial in xi , xi , x0 . k n i = n and n i > 0. Define the vector Fix a pair (µ, n) ∈ Ck × Zk+ such that i=1 x(µ) := µ1 , . . . , µ1 , . . . , µk , . . . , µk ∈ Cn (25) n1

nk

and xq (µ) obtained from x(µ) through replacing the “constant” string (µi , . . . , µi ) by “quantum” (µi , µi q −2 , . . . , µi q −2(n i −1) ) for each i = 1, . . . , k . Define the functions () () ϑg (n, µ) via the substitution x = xq (µ), x0 = q −2n , to ϑg (x). Recall from Subsect. 2.2 that weights of the defining representation can be expressed in terms of the orthogonal set {εi }. Next we specialize different types of Levi subalgebras in g. It is convenient to consider gl(n) instead of sl(n). The Levi subalgebras in gl(n) k gl(n ). The irreducible l-submodules in the defining g-module have the form l = ⊕i=1 i V0 are labelled with highest weights (26) ε1 , εn 1 +1 , . . . , εn 1 +...+n k−1 +1 . The Levi subalgebras in the orthogonal and symplectic algebras can be presented k l , where l = gl(n ) for i = 1, . . . , k − 1 and l equals either gl(n ) or as l = ⊕i=1 i i i k k so(2n k + 1), sp(n k ), so(2n k ) for g being so(2n + 1), sp(n), so(2n), respectively. That is, the latter option corresponds to lk of the same type as g. In the case lk = gl(n k ) the irreducible l-submodules in V0 are labelled with the highest weights (27) ε1 , εn 1 +1 , . . . , εn 1 +...+n k−1 +1 ; −εn 1 +...+n k , . . . , −εn 1 +n 2 , −εn 1 , and the zero weight for so(2n + 1). Each simple l-module enters with its dual. For lk of the same type as g the irreducible l-modules in the vector representation V0 are labelled with the highest weights ε1 , εn 1 +1 , . . . , εn 1 +...+n k−2 +1 ; εn−n k +1 ; −εn 1 +...+n k−1 , . . . , −εn 1 +n 2 , −εn 1 . (28)

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We assume that the components µi of µ are non-zero and pairwise distinct. For g orthogonal and symplectic we also assume µi = µ−1 j for i = j and, unless otherwise 2 stated, µi = 1. Below Q = QV is the matrix whose entries generate C[G] ⊂ U(g). n gi eii ∈ G L(n) to be the diagonal matrix with 8.3.1. The U gl(n) -case. Put g = i=1 gi = x(µ)i . The quantized ideal N(C g ) ⊂ C[S L(n)] of the class C g is generated by the relations k

()

(Q − µi ) = 0, Trq (Q ) = ϑsl(n) (n, µ).

i=1 ()

This reproduces the results of [DM2], where the functions ϑsl(n) (n, µ) are written out in a manifestly polynomial form. In the present form, the central characters of the RE algebras are calculated for the general case of Hecke symmetries in [GS]. 2n+1 8.3.2. The U so(2n + 1) -case. 1. The case lk = gl(n k ). Set g = i=1 gi eii ∈ for S O(2n + 1) with gi = x(µ)i for i = 1, . . . , n, gn+1 = 1, and gi = x(µ)i−1 i = n + 2, . . . , 2n + 1. The quantized ideal N(C g ) ⊂ C[S O(2n + 1)] of the class C g is generated by the relations (Q − q −2n )

k

()

(Q − µi )(Q − µi−1 q −4n+2n i ) = 0, Tr q (Q ) = ϑso(2n+1) (n, µ).

i=1

2. The case lk = so(2n k + 1). Set g as in the case lk = gl(n k ) with µk = 1. The quantized ideal N(C g ) ⊂ C[S O(2n + 1)] of the class C g is generated by the relations (Q − µk )

k−1

()

(Q − µi )(Q − µi−1 q −4n+2n i ) = 0, Trq (Q ) = ϑso(2n+1) (n, µ),

i=1

where µk = q 2(n k −n) . 2n gi eii ∈ Sp(n) with 8.3.3. The U sp(n) -case. 1. The case lk = gl(n k ). Set g = i=1 for i = n + 1, . . . , 2n. The quantized gi = x(µ)i for i = 1, . . . , n, and gi = x(µ)i−1 ideal N(C g ) ⊂ C[Sp(n)] of the class C g is generated by the relations k () (Q − µi )(Q − µi−1 q −4n+2(n i −1) ) = 0, Tr q (Q ) = ϑsp(n) (n, µ). i=1

2. The case lk = sp(n k ). Set g as in the case lk = gl(n k ) with µk = 1. The quantized ideal N(C g ) ⊂ C[Sp(n)] of the class C g is generated by the relations (Q − µk )

k−1

()

(Q − µi )(Q − µi−1 q −4n+2(n i −1) ) = 0, Trq (Q ) = ϑsp(n) (n, µ),

i=1

where µk = q 2(n k −n) .


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2n 8.3.4. The U so(2n) -case. 1. The case lk = gl(n k ). Set g = i=1 gi eii ∈ S O(2n) −1 with gi = x(µ)i for i = 1, . . . , n, and gi = x(µ)i for i = n +1, . . . , 2n. The quantized ideal N(C g ) ⊂ C[S O(2n)] of the class C g is generated by the relations k

i=1 (Q

− µi )(Q − µi−1 q −4n+2(n i +1) ) = 0,

(29)

()

Trq (Q W+ ) − Trq (Q W− ) = ϑ − (n, µ), Trq (Q ) = ϑso(2n) (n, µ). Without specializing the value of Trq (Q W+ ) − Trq (Q W− ) we get a quantization of the intersection of an S L(2n)-class with the group S O(2n). That intersection is an O(2n)class consisting of two isomorphic (as Poisson Lie manifolds) S O(2n)-classes. 2. The case lk = so(2n k ). Set g as in the case lk = gl(n k ) with µk = 1. The quantized ideal N(C g ) ⊂ C[S O(2n)] of the class C g is generated by the relations (Q − µk )

k−1

() (Q − µi )(Q − µi−1 q −4n+2(n i +1) ) = 0, Trq (Q ) = ϑso(2n) (n, µ),

i=1

where µk = q 2(n k −n) . Remark 8.3. To express the quantized ideals in terms of the generators {K ij } of the reflection equation algebra K, one should replace the matrix Q by K in the above formulas and impose additional relations (20) in case g is orthogonal or symplectic. If g = sl(n), one can consider relations of Subsect. 8.3.1 as those in K. There are no additional relations needed, and the eigenvalues may take arbitrary pairwise distinct values. This quantization has been studied in a two parameter setting in [DM2]. Remark 8.4. The quantization constructed in the present paper is relative to the DrinfeldJimbo quantum group. It is possible to extend it for an arbitrary factorizable quantization of U(g). The key fact is that any quantization of U(g) is a twist the Drinfeld-Jimbo quantum group. The corresponding twist of C[C g ] is isomorphic to a quotient of the corresponding reflection equation algebra. The quantized ideal of the class will be given by the same formulas as in the Drinfeld-Jimbo case, upon appropriate re-definition of the matrix R, the projector κ in the formula (20), and the q-trace. The details for the special case of the two-parameter quantization for g = gl(n) can be found in [MO]. A. More on central characters In this appendix we derive some formulas for central characters of the Drinfeld-Jimbo quantum groups. The coproduct on the Cartan-Weyl generators reads (eα ) = eα ⊗ 1 + q h α ⊗ eα + . . . , (e−α ) = e−α ⊗ q −h α + 1 ⊗ e−α + . . . .

(30)

The omitted terms have lower root vectors in each tensor component. R = q h R (e−α ⊗ eα ) = q h mod U (n− ) ⊗ U (n+ ),

(31)

where R (e−α ⊗ eα ) is a series in e−α ⊗ eα , α ∈ R+ , and h ∈ h ⊗ h is the inverse to the Killing form (the canonical element) restricted to h.

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Let W be a finite dimensional U(g)-module. Let X ∈ End(W ) ⊗ U(g) be an invariant matrix, i. e. commuting with πW (x (1) ) ⊗ x (2) for all x ∈ U(g). It is known that the q-trace defined by Trq (X n ) := Tr(πW (q 2h ρ )X n ) belongs to the center of U(g). Below we derive a formula for χ λ Trq (QW ) , where χ λ is a central character U(g) corresponding to λ. For every weight λ ∈ h∗ define a linear endomorphism θλ of U(g) setting θλ (x) := q 2h λ +2h ρ −2(ρ,ν) R1 xR2 , where, as earlier, we use the notation R = R1 ⊗ R2 . This endomorphism restricts to the subspace of h-invariants in U(g), since R is of zero weight. Proposition A.1. Let (W, πW ) be a finite dimensional U(g)-module with the highest weight ν. Then χ λ Trq (QW ) = q −(ν,ν) (Tr W ◦ πW ) θλ◦ (q 2h ρ ) vλ . (32) Proof. We have χ λ Trq (QW ) = Trq (π ⊗ id) (z ) q − (λ,λ)+2(ρ,λ)+(ν,ν)+2(ρ,ν) . Using induction on , we find

z = (q 2h ρ υ) = q 2 h ρ υ = q 2 h ρ γ 2 (R1 ) . . . γ 2 (R1 ) R2 . . . R2 .

Let vλ be the highest weight vector of Mλ . Consider a linear map ψλ : U(g) → C[[]] defined by xvλ = ψλ (x)vλ + lower weight terms. Then χ λ QW = q −(ν,ν) (Tr ◦ π )(ϒ), where we set ϒ = q −(λ,λ)−2(ρ,λ)−2(ρ,ν) q 2h ρ (id ⊗ ψλ ) (z ) ∈ U(g). Using the expression (31) for the universal R-matrix and formulas for comultiplication on generators, we compute ϒ to be q −2(ρ,ν) q 2h ρ q 2 h ρ {q h λ q h λ γ 2 (R1 )q −h λ } . . . {q h λ q h λ γ 2 (R1 )q −h λ } {q h λ R2 } . . . {q h λ R2 } .

We can push the factor q h λ standing before each of R2 to the left till it meets the factor q −h λ following the corresponding component γ 2 j (R1 ). Since R is of zero weight, q h λ commutes with the factors in between. Now recall that γ 2 is given by conjugation with q −2h ρ . Taking this into account, we obtain ϒ = q −2(ρ,ν) {q 2h λ +2h ρ R1 } . . . {q 2h λ +2h ρ R1 } q 2h ρ {R2 } . . . {R2 } = θ ◦ (q 2h ρ ).

This makes the proof immediate.


659

Corollary A.2. Let W be a finite dimensional U(g)-module. Then χ λ Trq (QW ) = Tr πW (q 2(h λ +h ρ ) ) . Proof. Notice that R1 q 2h ρ R2 equals z.

Acknowledgements. The author is grateful to the Max-Planck Institute for Mathematics in Bonn for its hospitality and excellent research conditions. He thanks J. Bernstein for his interest in this work, and for helpful discussions and valuable remarks.

References [AM] [DCK] [Dr1] [Dr2] [Dr3] [DGS] [DKM] [DM1] [DM2] [DM3] [EE] [EEM] [F] [FRT] [G] [GS] [GZB] [Jan1] [Jan2] [Ji] [J1] [J2] [JT] [K] [Ke] [KhT] [KSkl]

Alekseev, A. Malkin A.Z.: Symplectic structures associated to Lie-Poisson groups. Commun. Math. Phys. 162, 147–173 (1994) De Concini, C., Kac, V.: Representation of quantum groups at roots of 1. In: Operator algebras, unitary representations, enveloping algebras, and invariant theory. Prog. Math. 92, BaselBoston:Birkhäuser, 1990 pp. 471–506 Drinfeld, V.: Quantum Groups. In: Proc. Int. Congress of Mathematicians, Berkeley, 1986, ed. A. V. Gleason, Providence, RI: AMS, 1987 pp. 798–820 Drinfeld, V.: Almost cocommutative Hopf algebras. Leningrad Math. J. 1, # 2, 321–342 (1990) Drinfeld, V.: Quasi-Hopf Algebras. Leningrad Math. J. 1, 1419–1457 (1990) Donin, J., Gurevich, D., Shnider, S.: Quantization of function algebras on semisimple orbits in g∗ . http://arxiv.org/list/q-alg/9607008, 1996 Donin, J., Kulish, P.P., Mudrov, A.: On a universal solution to reflection equation. Lett. Math. Phys. 63, #3, 179–194 (2003) Donin, J., Mudrov, A.: Dynamical Yang-Baxter equation and quantum vector bundles. Commun. Math. Phys. 254, 719–760 (2005) Donin, J., Mudrov, A.: Explicit equivariant quantization on coadjoint orbits of G L(n, C). Lett. Math. Phys. 62, 17–32 (2002) Donin, J., Mudrov, A.: Reflection Equation, Twist, and Equivariant Quantization. Isr. J. Math. 136, 11–28 (2003) Enriquez, B., Etingof, P.: Quantization of classical dynamical r-matrices with nonabelian base. Commun. Math. Phys. 254, 603–650 (2005) Enriquez, B., Etingof, P., Marshall, I.: Quantization of some Poisson-Lie dynamical r-matrices and Poisson homogeneous spaces. http://arxiv.org/list/math.QA/0403283, 2004 Fiore, G.: Quantum group covariant (anti) symmetrizers, ε-tensors, vielbein, Hodge map adn Laplacian. J. Phys. A Math. Gen. 37, 9175–9193 (2004) Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1, 193–226 (1990) Gupta, R.K.: Copies of the adjoint representation inside induced ideals. Preprint, 1985, Paris Gurevich, D., Saponov, P.: Geometry of non-commutative orbits related to Hecke symmetries. http:// arxiv.org/list/math.QA/0411579, 2004 Gould, M., Zhang, R., Braken, A.: Generalized Gel’fand invariants and characteristic identities for Quantum Groups. J. Math. Phys. 32, 2298–2303 (1991) Jantzen, J.C.: Kontravariante formen und Induzierten Darstellungen halbeinfacher Lie-Algebren. Math. Ann. 226, 53–65 (1977) Jantzen, J.C.: Lectures on quantum groups. Grad. Stud. in Math. 6, Providence, RI: Amer. Math. Soc. 1996 Jimbo, M.: A q-difference analogue of u(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985) Joseph, A.: Quantum groups and their primitive ideals. Berlin: Springer-Verlag, 1995 Joseph, A.: A criterion for an ideal to be induced. J. Algebra 110, 480–497 (1987) Joseph, A., Todoroic, D.: On the quantum KPRV determinants for semisimple and affine Lie algebras. Alg. Rep. Theor. 5, 57–99 (2002) Kostant, B.: On the tensor product of a finite and an infinite dimensional representation. J. Funct. Anal. 20, 257–285 (1975) Kébé, M.: O-algèbres quantiques. C. R. Acad. Sci. Paris Sr. I Math. 322, 1–4 (1996) Khoroshkin, S., Tolstoy, V.: Universal r -matrix for quantized (super) algebras. Commun. Math. Phys. 141, 599–617 (1991) Kulish, P.P., Sklyanin, E.K.: Algebraic structure related to the reflection equation. J. Phys. A 25, 5963–6975 (1992)

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Lusztig, G.: Introduction to quantum groups. Prog. Math. 110, Boston, MA:Bikhäuser, 1993. Mudrov, A.: On quantization of Semenov-Tian-Shansky Poisson bracket on simple algebraic groups. Alg. \& Anal. 5, # 5, 156–172 (2006) Mudrov, A., Ostapenko, V.: Quantization of orbit bundles in gl(n)∗ . http://arxiv.org/pdf/math. QA/0612397 Naimark, M.: Teoriya predstavlenii grupp. (Russian) [Representation theory of groups], Moscow, 1976 Richardson, R.: An application of the Serre conjecture to semisimple algebraic groups. Lect. Notes in Math. 848, New York:Springer, 141–151 (1981) Radford, D.: The structure of Hopf algebras with a projection. J. Alg. 92, 322–347 (1985) Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: Quantum R-Matrices and factorization problem. J. Geom. Phys. 5, 533–550 (1988) Springer, T.: Conjugacy classes in algebraic groups. Lect. Notes in Math. 1185, Berlin:Springer, pp. 175–209 (1984) Semenov-Tian-Shansky, M.: Poisson-Lie Groups, Quantum Duality Principle, and the Quantum Double. Contemp. Math. 175, 219–248 (1994) Vinberg, E., Onishchik, A.: Seminar po gruppam Li i algebraicheskim gruppam. (Russian) [A seminar on Lie groups and algebraic groups], Moscow, 1988 Weyl, H.: The classical groups. Their Invariants and representations, Princeton, NJ:Princeton Univ. Press, 1966 Yetter, D.: Quantum groups and representations of monoidal categories. Math. Proc. Cambridge Philos. Soc. 108, # 2, 261–290 (1990)

[MO] [N] [R] [Rad] [RS] [S] [STS] [VO] [We] [Y]

Communicated by L. Takhtajan


Communications in


Combinatorial Point for Fused Loop Models P. Zinn-Justin Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626 du CNRS, Université Paris-Sud, Bâtiment 100, F-91405 Orsay Cedex, France. E-mail: [email protected] Received: 11 April 2006 / Accepted: 5 September 2006 Published online: 23 March 2007 – © Springer-Verlag 2007

Abstract: Integrable loop models associated with higher representations (spin /2) of Uq (sl(2)) are investigated at the point q = −e±iπ/(+2) . The ground state eigenvalue and eigenvectors are described. Introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries.

1. Introduction The present work is part of an ongoing project to understand the combinatorial properties of integrable models at special points where a (generalized) stochasticity property is satisfied. The project was started in [1], based on the observations and conjectures found in [2, 3]. The original model under consideration was the XXZ spin chain with (twisted) periodic boundary conditions at the special point = −1/2, or equivalently a statistical model of non-crossing loops with weight 1 per loop (somewhat improperly called the with q = −e±iπ/3 ), which can “O(1)” model, since it is really based on Uq (sl(2)) be reformulated as a Markov process on configurations of arches. Among the various conjectured properties of the ground state eigenvector, a “sum rule” formulated in [2], namely that the sum of components of the properly normalized ground state eigenvector is equal to the number of alternating sign matrices, was proved in [1]. Since then, a number of generalizations have been considered: (i) models based on a different algebra, either the ortho/symplectic series which corresponds to models of crossing loops [4, 5], or higher rank An [6], which can be described as paths in Weyl chambers. Note that in the latter case the stochasticity property must be slightly modified: it becomes the existence of a (known) fixed left eigenvector of the transfer matrix. This idea will reappear in the present work. (ii) models with other boundary conditions [7, 8], which will not be discussed here. Supported by ANR program “GIMP” ANR-05-BLAN-0029-01, European networks “ENIGMA” MRTCT-2004-5652, “ENRAGE” MRTN-CT-2004-005616, and ESF program “MISGAM”.

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There is yet another direction of generalization: the use of higher representations. Indeed all models considered so far were based on fundamental representations (spin 1/2 for A1 ). We thus study here integrable models based on A1 , but representations of spin /2. There is a reasonable way to formulate these in terms of loops, using the fusion procedure (see Sect. 2). One interesting feature is that the resulting models are closer in their formulation to the original O(1) loop model, and we can hope for a richer combinatorial structure in the spirit of the full “Razumov–Stroganov conjecture” [3]. The present work remains indeed very close to that of [1]. It is concerned with the study of the ground state eigenvector and of the properties of its entries in an appropriate basis. In fact, many arguments are direct generalizations of those of [1] – though proofs are sometimes clarified and simplified. There are however some new ideas. In particular, as already mentioned a key technical feature is the existence of a common left eigenvector for the whole family of operators from which one builds the transfer matrix or the Hamiltonian. Here we give an “explanation” of this phenomenon: it is related to the degeneration of a natural “scalar product” on the space of states. Indeed asking for this scalar product to have rank 1 fixes the special value of the parameter q to be q = −e±iπ/(+2) , which generalizes the value = q+1/q = −1/2 for spin 1/2. This will be 2 explained in Sects. 3.1 and 3.2. Section 3.3 deals with the ground state eigenvector for the inhomogeneous integrable transfer matrix, the polynomial character of its components in terms of the spectral parameters and other properties, while Sect. 3.4 describes the computation of the sum rule, both following the general setup of [1]. In the latter, we shall be forced to rely on a conjecture concerning the degree of the polynomial eigenvector: although in the special case = 1 this conjecture was proved in [1], the general proof is beyond the scope of the present paper. 2. Definition of the Model In this section we define the space of states and the Hamiltonian, or Transfer Matrix, acting on it. In order to do that it is convenient to introduce a larger space, corresponding to the case = 1, and then use fusion. This has the advantage that it gives us a natural “combinatorial basis” to work with; however the situation, as we shall see, remains more subtle than in the case = 1, because a projection operation is needed; in many cases, this means that results that are “obvious graphically” must be additionally shown to be compatible with the projection. 2.1. Link Patterns and Temperley–Lieb algebra. Let n be a positive integer, and L2n be the set of link patterns of size n, which are defined as non-crossing (planar) pairings of 2n points. We want to imagine link patterns as living inside a disk, with the 2n endpoints on the boundary; but it is sometimes more practical to unfold them to the traditional depiction on a half-plane, see Fig. 1. The number of such link patterns is known to be the Catalan number cn = (2n)!/(n!(n + 1)!). We view L2n as a subset of the involutions of {1, . . . , 2n} without fixed points, by setting α(i) = j if i and j are paired by α ∈ L2n . Let H2n = C[L2n ]. For i = 1, . . . , 2n − 1, we define ei to be the operator on H2n by defining its action on the canonical basis |α, α ∈ L2n : τ |α if α(i) = i + 1 , ei |α = |c−1 ◦ α ◦ c otherwise, c cycle (i, i + 1, α(i + 1), α(i))

Combinatorial Point for Fused Loop Models

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12 11 10 13

9

14

8

1

7 2

6 3

1 2 3 4 5 6 7 8 9 1011121314

5

4

Fig. 1. A link pattern

where τ is a complex parameter, which for convenience we rewrite as τ = −q − q −1 , q ∈ C× . We shall provide an alternative graphical rule below. The ei , i = 1, . . . , 2n −1, form a representation of the usual Temperley–Lieb algebra T L 2n (τ ). By definition T L L (τ ) is the algebra with generators ei , i = 1, . . . , L − 1 and relations ei2 = τ ei ,

ei ei±1 ei = ei ,

ei e j = e j ei ,

j = i − 1, i + 1 .

(2.1)

It is well-known that the Temperley–Lieb algebra T L L (τ ) can be viewed itself as the space of linear combinations of non-crossing pairings of points on strips of size L, see the example below, multiplication being juxtaposition of strips, with the additional prescription that each time a closed loop is formed, one can erase it at the price of multiplying by τ . In particular, the generators ei correspond to the strip with two little arches connecting sites i and i + 1 on the top and bottom rows. We conclude that the dimension of T L L (τ ) is c L , so that for L = 2n it is c2n = (4n)!/((2n)!(2n + 1)!). The action on link patterns is once again juxtaposition of the strip and of the link pattern (in the unfolded depiction), with the weight τ #loops for erased loops. Since cn2 < c2n , this representation is not faithful; however, when there is no possible confusion, we shall by abuse of language identify Temperley–Lieb algebra elements and the corresponding operators on H2n . Example. T L4 = 1 =

, e1 = 1

2

3

4

, e2 = 1

e3 =

2

3

4

, e1 e2 = 1

2

3

4

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, 1

4

, e2 e1 e3 =

, e2 e1 e3 e2 = 3

, 1

, e3 e2 e1 = 1

4

, e1 e3 = 1

e1 e2 e3 =

3

4

, e3 e2 = 1

2

, e2 e1 = 1

e2 e3 =

, 1

2

3

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, 1

2

3

4

. 1

2

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In what follows, we shall sometimes need an extra operator e2n , defined just like the other ei , but reconnecting the points 2n and 1. The ei , i = 1, . . . , 2n satisfy the same types of relations (2.1) as before, but assuming periodic indices: 2n + 1 ≡ 1. These are

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1

2

3

4

5

6

Fig. 2. Gluing two link patterns together. Here two loops are formed

defining relations of the “periodic” Temperley–Lieb algebra T L 2n (τ ). Clearly, its elements can be represented as certain non-crossing pairings on an annular strip (acting in the obvious way on link patterns in the circular depiction), but in practice are more complex to handle. Fortunately in most circumstances we shall need to use only some subset of consecutive generators – ei , ei+1 , . . . , ei+L−1 , or ei , . . . , e2n , e1 , . . . , ei−2n+L−1 – forming a representation of the usual (non-periodic) T L L (τ ). 2.2. Bilinear form. There is an important pairing ·|· of link patterns which extends into a symmetric bilinear form on H2n . It consists of taking a mirror image of one link pattern, gluing it to the other and assigning it the usual weight τ #loops , see Fig. 2. There is also an anti-automorphism ∗ of the periodic Temperley–Lieb algebra defined by ei ∗ = ei (noting that the defining relations of T L 2n (τ ) are invariant with respect to writing words in ei in the reverse order); graphically, it associates to an operator its mirror image, and therefore we have the identity α|x∗ |β = β|x|α

x∈ T L 2n (τ ) .

(2.2)

Define gαβ = α|β; the determinant of the matrix g was computed in [9]. In particular, it is non-zero when q (that enters into the loop weight τ = −q − q −1 ) is generic, i.e. not a root of unity (see also [10]). However, in what follows we shall be particularly interested in the situation q 2(+2) = 1, in which g is singular for n large enough, and , which requires some the mapping |α → α|· is not an isomorphism from H2n to H2n care in handling bra-ket expressions. In particular, a remark is in order: in the “strip” description of the Temperley–Lieb algebra T L 2n (β), it is clear that any operator |αβ|· belongs to the Temperley–Lieb algebra (they are those pairings of 2 × 2n points with no “up–down” pairings); therefore, for q generic the mapping from T L 2n (τ ) to the space of operators L(H2n ) is surjective. It is however in general not surjective any more for q root of unity; this is consistent with the fact that there is no notion of adjoint operator with respect to the bilinear form for an arbitrary operator on H2n (which ∗ provides for Temperley–Lieb elements), a point that will become crucial in Sect. 3.1. 2.3. Projection. Fix now a positive integer , and assume that n = lm. For each subset Si = {l(i − 1) + 1, . . . , li}, i = 1, . . . , 2m, of consecutive points, we define a local projector pi ; it is uniquely characterized by (i) (ii)

pi |α = 0 if ∃ j, k ∈ Si such that α( j) = k. pi is in the subalgebra generated by the ek , k = l(i − 1) + 1, . . . , li − 1.


(iii)

665

pi2 = pi (normalization).

The details of their construction and their main properties are listed in Appendix A. Here we give the key formula which is the recurrence definition: start with p (1) = 1 and p (k+1) (e j , . . . , e j+k−1 ) = p (k) (e j , . . . , e j+k−2 )(1−µk (τ )e j+k−1 ) p (k) (e j , . . . , e j+k−2 ), (2.3) where µk (τ ) = Uk−1 (τ )/Uk (τ ) and Uk is the Chebyshev polynomial of the second kind. Then pi := p () (e(i−1)+1 , . . . , ei−1 ). In particular we note that at the zeroes of the Chebyshev polynomials U j (τ ), 1 ≤ j ≤ − 1, that is if q 2 j = 1 for some 1 ≤ j ≤ , the pi are undefined; we therefore exclude from now on these roots of unity. 2m pi , and The pi form a family of commuting orthogonal projectors; define P = i=1 H,2m = P(H2n ). Furthermore, define L,2m = { α ∈ L2n : ∀i, j ∈ Si α( j) ∈ Si }, that is, the set of link patterns with no arches within one of the subsets Si . Example.

5

5 6

6

4

L2,6 =

, 1

1

1

3

4

1

3

,

1

3

3

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3

,

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.

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It is crucial to observe that the |α, α ∈ L,2m , do not belong to H,2m . However, if we define |α ˜ := P|α, α ∈ L,2m , we can state: Proposition 1. The |α ˜ form a basis of H,2m . Proof. Clearly, P|α = 0 if α ∈ L,2m . Therefore dim H,2m ≤ #L,2m , and it suffices to show that the |α ˜ are ˜ is of independent. But this is obvious in view of the fact that |α the form |α ˜ = |α + β∈L,2m c(α, β)|β for all α ∈ L,2m . ( j)

We can now introduce a set of local operators, the ei , j = 0, . . . , , acting on the two subsets Si and Si+1 for i = 1, . . . , 2m (with Sm+1 ≡ S1 ). They are defined by ( j) ei = P ei ei−1 ei+1 · · · ei− j+1 · · · ei+ j−1 · · · ei−1 ei+1 ei P, but best understood graphically, see Fig. 3. ( j) The ei satisfy a certain algebra which we do not need to describe entirely. However we need the following results:

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( j)

Fig. 3. Definition of ei . The two subsets of vertices are Si and Si+1 . The circled p’s are the local projection operators pi and pi+1

Fig. 4. Graphical equality of Lemma 2. p refers to p () in the l.h.s. and to p (− j) in the r.h.s ( j)

Lemma 1. The image of ei is included in the span of the |α ˜ such that (at least) there are j arches between Si and Si+1 , i.e. α(i) = i + 1, …, α(i − j + 1) = i + j. ( j)

Proof. This is obvious graphically, since ei mentioned in the lemma, then projects.

reconnects precisely these pairs of points

Lemma 2. The equality of Fig. 4 holds. Consequently, (a) Consider a link pattern α such that Si and Si+1 are fully connected, i.e. α(i) = i + 1, …, α((i − 1) + 1) = (i + 1). Then ( j)

ei |α ˜ =

U (τ ) |α ˜ . U− j (τ )

(2.4)

(b) Equivalently, ( j) ()

ei ei

() ( j)

= ei ei

=

U (τ ) () e . U− j (τ ) i

(2.5)

Proof. The proof is by induction on . Consider the l.h.s. of Fig. 4 and replace p () with its definition by recurrence from Appendix A, choosing to apply p (−1) to the − j open lines and to j − 1 closed lines, excluding the innermost closed line: we obtain two terms which are both precisely of the same form as the l.h.s., but with − 1 lines among which j − 1 close, and coming with coefficients τ (one closed loop) and −µ−1 (τ ). Applying the induction hypothesis we find the coefficient of proportionality to be (τ − U−2 (τ )/U−1 (τ ))U−1 (τ )/U− j (τ ) = (τ U−1 (τ ) − U−2 (τ ))/U− j (τ ) = U (τ )/U− j (τ ). The proof of Eqs. (2.4) and (2.5) is a simple application of this formula, noting that when there are series of projections one can coalesce them into a single projection. This second lemma is particularly important; we provide on Fig. 5 two more graphical corollaries of it.


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(b)

(a)

Fig. 5. Equalities obtained from Lemma 2. On the two figures p = p ()

2.4. Fusion. Let us briefly describe the fusion mechanism. Since this is standard material, we shall not prove the following facts. Start with the = 1 R-matrix ri (z, w) =

z−w qz − q −1 w + ei . −1 qw − q z qw − q −1 z

(2.6)

We now fuse 2 R-matrices1 into a single operator R by q −k z − q k w Ri (z, w) = ri (q −+1 z, q −1 w) q k z − q −k w k=1

ri−1 (q −+3 z, q −1 w)ri+1 (q −+1 z, q −3 w) ··· r(i−1)+1 (q −1 z, q −1 w) · · · r(i+1)−1 (q −+1 z, q −+1 w) ··· ri−1 (q −1 z, q −+3 w)ri+1 (q −3 z, q −+1 w) ri (q −1 z, q −+1 w) P .

(2.7)

Due to the choice of arguments of the R-matrices, Ri leaves H,2m stable. We shall need a more explicit form of Ri . This is possible, using the local operators ( j) ei introduced previously: ⎛ ⎞ −

j q k z − q −k w ⎠ e( j) , Ri (z, w) = aj ⎝ (2.8) i k− −k q z−q w j=0

where a j = a− j =

k=0

j

U−k (τ ) k=1 Uk−1 (τ ) .

Ri (z, w) has poles when w/z = q −2 , . . . , q −2 and is non-invertible when w/z = q 2 , . . . , q 2 ; for other values of z/w it satisfies the unitarity equation Ri (z, w)Ri (w, z) = 1 .

(2.9)

Next we define the fully inhomogeneous transfer matrix T (z|z 1 , . . . , z 2m ). This requires to extend slightly the space H,2m into H,2m+1 where the additional “auxiliary” 1 Note that to define a transfer matrix, one could fuse only R-matrices, keeping a single line for the “auxiliary space”. However we need the doubly fused R-matrix to write the form of the Yang–Baxter equation that we need, and to obtain the Hamiltonian.

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Fig. 6. The transfer matrix ( = 3, 2m = 4). Each × grid is the fused R-matrix

lines are drawn horizontally. One also defines the “partial trace” traux which to an operator on H,2m+1 associates an operator on H,2m obtained by reconnecting together the incoming and outgoing auxiliary lines,2 including a weight of τ = −q − q −1 by closed loop. Then the transfer matrix corresponds to the auxiliary line crossing all other lines then reconnecting itself (see Fig. 6) T (t|z 1 , . . . , z 2m ) = traux R2m (z 2m , t) · · · R2 (z 2 , t)R1 (z 1 , t).

(2.10)

The transfer matrix satisfies two forms of the Yang–Baxter equation. The first one is the well-known “RT T ” form, which implies the commutation relation [T (t), T (t )] = 0, where all z i are fixed. The second one simply reads: T (t|z 1 , . . . , z i , z i+1 , . . . , z 2m )Ri (z i , z i+1 ) = Ri (z i , z i+1 )T (t|z 1 , . . . , z i+1 , z i , . . . , z 2m )

(2.11)

for i = 1, . . . , 2m (with z 2m+1 ≡ z 1 ). We also need the “scattering matrices” Ti := T (z i |z 1 , . . . , z 2m ), 1 ≤ i ≤ 2m. Using the fact that Ri (z, z) = 1, one finds Ti = Ri (z i , z i+1 ) . . . R2m−1 (z i , z 2m )R2m (z i , z 1 )R1 (z i , z 2 ) . . . Ri−1 (z i , z i−1 )ρ, (2.12) where ρ is the rotation of link patterns: (ρα)(i) = α(i − ) + (modulo 2n), which sends Si to Si+1 . One can also define the Hamiltonian. Consider the homogeneous situation z i = 1. Then it is natural to expand T around t = 1 to obtain commuting operators that are ( j) expressed as sums of local operators (the ei ). Explicitly, expanding at first order, we find that T (t) commutes with H := (q − q −1 ) T (1)−1

∂ 1 ( j) T (t)|t=1 + cst = ei ∂t U j−1 (τ ) 2m

(2.13)

i=1 j=1

(the constant term in the expansion has been cancelled for convenience). 2 Making the auxiliary line horizontal conceals the fact that this operation induces a rotation of the link pattern, since the auxiliary line has changed its position from left to right relative to the rest of the lines.


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2.5. Cell depiction. Finally, there is yet another graphical depiction of link patterns in L,2m : since vertices in the same subset Si are never connected to each other, one can simply coalesce them into a single vertex: the result is a division of the disk into 2dimensional cells such that edges come out of each of the 2m vertices on the boundary. Example. At = 2, cells can be conveniently drawn using the natural bicoloration of cells according to whether they touch the exterior circle at vertices or edges (see also 5

5

6

4

6

4

=

below the discussion of exterior vs interior cells): 1

3

. 1

2

3 2

Note that if one straightens edges to produce polygons, one can obtain 2-gons (or worse, several 2-gons that sit on top of each other); it is therefore possible to work with polygonal cells on condition that such singular configurations be included. For future use, we now define the following notion: a link pattern α ∈ L,2m is said to be -admissible if all its cells have an even number of edges. When there is no ambiguity we shall simply say “admissible”, noting that this an abuse of language since admissibility is an -dependent property: some edges disappear when vertices are merged. Call L,2m the set of -admissible link patterns. Example. 5

L2,6 =

⎧ 6 ⎨

5 4

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5 4

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5 4

,

2 5 6

6

6

4

1

3

, 1

3 2

, 2

⎫ ⎬ . ⎭

3 2

We also need a simple fact about admissible link patterns. Call r (i) the remainder of the division of i − 1 by . Lemma 3. If α is an -admissible link pattern, then r (i) + r (α(i)) = − 1 for 1 ≤ i ≤ 2m. Proof. Induction on α(i) − i (mod 2m) ∈ {1, . . . , 2m − 1}. If α(i) = i + 1: α ∈ L,2m forbids any arches inside a given subset of vertices, therefore r (i) = − 1, r (i + 1) = 0. If α(i) − i > 1: call k = r (i + 1). Consider the cell with edge (i, α(i) such that all its other vertices are between i and α(i) moving counterclockwise around the circle.

670

P. Zinn-Justin

The idea is to use the induction hypothesis for all these other vertices. Two cases have to be distinguished: either (i) k = 0, in which case the values of r at vertices (in the sense of the original depiction) of the cell follow a pattern: 0, − 1, 0, etc. and we obtain immediately r (i) = − 1, r (α(i)) = 0; or (ii) k > 0. In this case, the values of r at vertices of the cell are of the form k, − 1 − k, − k, k − 1, k, etc, being careful that these vertices are coalesced into pairs {k − 1, k} and { − 1 − k, − k} to form the actual vertices of the cell. But since α is admissible, the cell has an even number of edges, and when we reach α(i) we get the value − k, so that r (i) = k − 1, r (α(i)) = − k. In the course of the proof, we have found that one can associate to each cell c of an admissible link pattern a pair of integers {k(c), − k(c)}: conventionally we choose k(c) to be the smaller of the two. Graphically, k(c) is the “distance” from the cell to the boundary, defined as the minimum number of edges one needs to cross to reach the exterior circle (excluding the circle itself). Following the subdivision in the proof, we call exterior (resp. interior) a cell c such that k(c) = 0 (resp. k(c) > 0). An exterior cell touches the circle at every other edge, whereas an interior cell touches it at vertices only. In practice exterior cells play no role in what follows, as will become clear, and on the pictures they will be left uncolored. Note that in the case = 2 this notion coincides with the natural bicoloration of cells. In Appendix B, -admissible link patterns are enumerated, and it is found that #L,2m =

(( + 1)m)! . (m + 1)!m!

(2.14)

3. Combinatorial Point We now investigate the special value q = −e±iπ/(+2) , that is τ = −q − q −1 = √ √ √ π = 1, 2, 1+2 5 , 3, . . .. 2 cos +2 3.1. Degeneration of the bilinear form. Define the matrix of the bilinear form in the subspace H,2m : ˜ = α|P|β g˜ αβ := α| ˜ β

α, β ∈ L,2m .

Theorem 1. The rank of the matrix g˜ is one. Proof. As many reasonings in this paper, the proof is best understood pictorially. It makes use of Lemma 2, with the additional assumption that q = −e±iπ/(+2) , which implies that U j (τ ) = U− j (τ ). Figure 4(a) therefore implies the equality of Fig. 7(a), which itself can be rewritten as Fig. 7(b), noting that any link pattern in L,4 is of the form of Fig. 7(a) for some j – and for any other link pattern in L4 , both l.h.s. and r.h.s. are zero. Consider now the braket α|P|β. Using repeatedly the identity of Fig. 7(b), we obtain Fig. 7(c), that is ˜ = α|00| ˜ α| ˜ β ˜ β,

(3.1)


671

(a)

(b)

(c)

Fig. 7. Graphical proof of Thm. 1. L stands for any linear combination of link patterns

where 0 denotes the link pattern which fully connects S2i−1 and S2i , as in the r.h.s. of 9

Fig. 7(c) (e.g.

8

7

10

6

1

5 2

3

). Thus, g˜ = v ⊗ v, v the linear form 0|· on H,2m which

4

is non-zero since g˜ 00 = 0|P|0 = 1 (Fig. 4(b)).

Remark. There is another link pattern 0 related to 0 by rotation, which connects S2i and S2i+1 (2m + 1 ≡ 1). The argument above works equally well with 0 |. We can in fact provide an explicit formula for g, ˜ of the form g˜ αβ = vα vβ , α, β ∈ L,2m : Proposition 2. ˜ = vα = 0|α

0

cell c Uk(c) (τ )

−l(c)/2+1

if α is non-admissible , if α is admissible

(3.2)

where the product can be restricted to interior cells only, l(c) is the number of edges of cell c (note that 2-gons do not contribute), and k(c) is the distance from the cell to the boundary as defined in Sect. 2.3. Proof. Induction on m. m = 1 is trivial. For a given link pattern α, we shall pick a certain pair of subsets Si , Si+1 and reconnect them with a projection: this is one step in the pairing with 0| (or 0 |, depending on the parity of i), and we can then use the induction hypothesis. For any link pattern, it is easy to check that one of these two situations must arise (graphically, that there exists a cell which has no “nested” cells):

672

P. Zinn-Justin

Fig. 8. Case (ii) of the proof of Prop. 2

(i) Either there are two subsets Si and Si+1 which are fully connected to each other. These correspond to 2-gons which should not contribute to vα . Indeed, applying Fig. 5(b), the loops, once closed with a projection, contribute U (τ ) = 1 and can be removed, leading to the step m − 1. (ii) Or there is a subset Si such that j lines connect it to Si−1 and − j lines connect it to Si+1 . We reconnect Si and Si+1 and apply Lemma 2 (Fig. 4). In the process some 2-gons are erased, and the only other (interior) cell that is affected is the one directly above, see Fig. 8, which we denote by c. One checks that c loses 2 edges. If c had 3 edges to begin with, it becomes a cell with 1 edge, i.e. there is a connection inside a subset and the resulting link pattern does not belong to L,2(m−1) , so vα = 0. If c had a higher odd number of edges the resulting link pattern is not admissible and by induction vα is again zero. Finally, if the number of edges of c is even, we note that min( j, − j) = k(c) and the contribution 1/U j (τ ) plus the induction hypothesis reproduce Eq. (3.2) (whether α is admissible or not).

9

Example. Consider the diagram α =

6

1

5

there are two 4-gons at distance 1, so vα = U1 α=

8

6

1

5

vα = U1

3 (τ )−1 U

3

∈ L3,10 . It is admissible, and

4

(τ )−2

= ((1 +

√ 5)/2)−2 .

7

10

2

7

10

2

9

8

∈ L4,10 is also admissible, there are 4-gons at distance 1 and 2, so

4

2 (τ )

−1

√ = 1/(2 3).

Remark. For = 2, 3, since the only non-trivial U j (τ ) are equal to τ , one can simplify the formula for admissible link patterns to: vα = τ −m+#connected components of cells .

3.2. Common left eigenvector. Consider now any operator x of the (periodic) Temperley– Lieb algebra, projected onto H,2m , that is x = P x P. As explained in Sect. 2.2, it possesses a mirror symmetric x∗ . Let us write in components the identity (2.2) expressing β β ˜ and x∗ |α ˜ then g˜ αγ x γ = g˜ βγ x γα , where ˜ = β x∗α |β, this fact: if x|α ˜ = β x α |β ∗β


673

summation over repeated indices is implied, or, choosing any β such that vβ = 0, γ

vγ x γα =

vγ x∗β vβ

vα .

(3.3)

In other words, v is a left eigenvector of x (and of x∗ by exchanging their roles). What we have found is that the right-representation of P T L 2n (τ )P on H,2m possesses a onedimensional stable subspace; and therefore also that the left-representation on H,2m is decomposable (but not reducible, as it turns out) with a stable subspace of codimension one (the kernel of v). Note that an advantage of defining the left eigenvector v from the bilinear form is that it provides a convenient natural normalization of v. Lemma 4. Eigenvalues of various operators for the left eigenvector v: 1 v, U j (τ ) v Ri (z, w) = v, v T (t|z 1 , . . . , z 2m ) = v, v H = 2mτ v. ( j)

v ei

=

j = 0, . . . , ,

(3.4) ( j)

Proof. Since we already know that v is a left eigenvector of ei , we only need to ( j) compute v ei |α, ˜ where α is a given admissible link pattern; we choose it as in the hypotheses of Lemma 2 (for example, either |0 or |0 works). We conclude directly that U (τ )/U− j (τ ) is the eigenvalue for v, which is the announced result using U j (τ ) = U− j (τ ) at q = −e±iπ/(+2) . The other formulae follow by direct computation. Lemma 5. The eigenvalue 1 of T (t|z 1 , . . . , z 2m ) is simple for generic values of the parameters. Note that the set of degeneracies of the eigenvalue 1 is a closed subvariety of the space of parameters. Thus, finding one point where the eigenvalue is simple is enough to show the lemma. There are a variety of ways to find such a point, none of which is particularly simple. One can for example consider the limit z 1 z 2 · · · z 2m , in which all eigenvalues can be computed explicitly. The calculations are too cumbersome and will not be reproduced here.

3.3. Polynomial eigenvector. We have found in the previous section that the transfer matrix T (t|z 1 , . . . , z 2m ) possesses the eigenvalue 1, with left eigenvector v; what about the corresponding right eigenvector? The latter, which we denote by | = α α |α, ˜ depends on the parameters z 1 , . . . , z 2m (but not on t). Being the solution of a degenerate linear system of equations whose coefficients are rational fractions, it can be normalized in such a way that its components α are coprime polynomials in the variables z 1 , . . . , z 2m . Furthermore, all equations being homogeneous, the α are homogeneous polynomials of the same degree deg |. We now formulate a key result: Proposition 3. |(z 1 , . . . , z i , z i+1 , . . . , z 2m ) = Ri (z i , z i+1 )|(z 1 , . . . , z i+1 , z i , . . . , z 2m ).

674

P. Zinn-Justin

Proof. Equation (2.11) shows that Ri (z i , z i+1 )|(z i+1 , z i ) is an eigenvector of T (z 1 , . . . , z 2m ) with the eigenvalue 1 (Lemma 4). Since this eigenvalue is simple (Lemma 5), the l.h.s. and r.h.s. of Prop. 3 must be proportional: Ri (z i , z i+1 )|(z i+1 , z i ) = F(z 1 , . . . , z 2m )|(z i , z i+1 ), F rational fraction. More precisely,

(q −k z i − q k z i+1 )Ri (z i , z i+1 )|(z i+1 , z i ) = P(z 1 , . . . , z 2m )|(z i , z i+1 ),

(3.5)

k=1

where P(z 1 , . . . , z 2m ) = F(z 1 , . . . , z 2m ) k=1 (q −k z i − q k z i+1 ) is a polynomial since the α are coprime and the l.h.s. is already polynomial. Iterating this equation leads to P(z i , z i+1 )P(z i+1 , z i ) =

(q −k z i − q k z i+1 )(q −k z i+1 − q k z i ),

(3.6)

k=1

−k kz , i.e. P and therefore F are functions of only two variables and F(z, w) = k∈K qq −k w−q z−q k w where K is some subset of {1, . . . , }. To fix F, consider Ti | (with Ti given by Eq. (2.12)): on the one hand, since Ti is simply the transfer matrix at a special choice of parameter t, we know that it has eigenvector with eigenvalue 1 (Lemma 4): T | = |; on the other hand, applying Eq. (3.5) repeatedly, we find that Ti | = i j=i F(z i , z j )|; we easily conclude from this that F = 1. Proposition 4. Suppose z i+1 = q 2k z i , 1 ≤ k ≤ . Then α (z 1 , . . . , z 2m ) = 0 unless α is such that there are (at least) − k + 1 arches between Si and Si+1 . Proof. Apply Proposition 3 with Ri (z i , q 2k z i ) replaced with its expression (2.8). As (−k+1) soon as − j ≥ k, the product is zero, so that Ri is a linear combination of ei , …, () ei . The proposition is then a direct application of Lemma 1. Equivalently, since the components α are polynomials, z i+1 − q 2k z i |α . We now state a broad generalization of Prop. 4: Theorem 2. Suppose z j = q 2k z i , 1 ≤ k ≤ , j = i. Then α (z 1 , . . . , z 2m ) = 0 unless #{ p, q ∈ Si, j : p < α( p) = q} ≥ − k + 1. Here Si, j denotes the set of vertices between i and j in a cyclic way, that is Si, j = {(i − 1) + 1, . . . , j} if i < j, {(i − 1) + 1, . . . , 2m, 1, . . . , j} if j < i. Note that this theorem is a generalization of (part of) Theorem 1 of [1], and the proof is completely analogous. We present here a briefer version of it. Proposition 3 shows that |(. . . , z i , z i+1 , . . . , z j , . . .) and |(. . . , z i , z j , z i+1 , . . .) are related by a product of R-matrices from i + 1 to j − 1 (these R-matrices have poles, but are well-defined for generic values of the other z’s). According to Prop. 4, the only non-zero components of |(. . . , z i , z j , z i+1 , . . .) possess − k + 1 arches between subsets Si and Si+1 . Now observe that the action of any Temperley–Lieb generator cannot decrease the number of arches within any range containing the 2 sites on which it is acting. Therefore multiplication by a product of R-matrices (which are themselves linear combinations of products of Temperley–Lieb generators acting somewhere between Si and S j ) does not decrease the number of arches between Si and S j . One can go further and look for cancellation conditions for the whole of |:


675

7

6

8

5

1

4 3

2

Fig. 9. Base link pattern of L3,8

Proposition 5. Assume that z i+1 = q 2k z i and z i+2 = q 2k z i+1 so that z i , z i+1 , z i+2 are in “cyclic order”, that is 1 ≤ k, k and k + k ≤ + 1. Then |(z 1 , . . . , z 2m ) = 0. Proof. Apply twice Prop. 4. For a component α to be non-zero, α should have − k + 1 arches between Si and Si+1 , and − k + 1 arches between Si+1 and Si+2 ; thus the number of lines emerging from Si+1 should be 2 − (k + k ) + 2 ≥ + 1, which is impossible. Once again we can generalize this result to

Theorem 3. Assume that z i = q 2k z i and z i = q 2k z i so that z i , z i , z i are in cyclic order, and i, i , i are also in cyclic order (i < i < i or i < i < i or i < i < i ). Then |(z 1 , . . . , z 2m ) = 0. We use exactly the same process as to go from Prop. 4 to Thm. 2. We note that |(. . . , z i , . . . , z i , . . . , z i , . . .) is related to |(. . . , z i , z i , z i , . . .) by a product of R-matrices, paying attention to the fact that none of these R-matrices are singular for generic values of the other parameters (this is where we use the fact that i, i , i are in cyclic order). Then we apply Prop. 5. Let us now consider what we call the base link pattern δ ∈ L,2m defined by δ(i) = 2n + 1 − i, 1 ≤ i ≤ 2n, see Fig. 9. Theorem 2 implies that δ =

(q k z i − q −k z j ),

(3.7)

1≤i< j≤m k=1 or m+1≤i< j≤2m

where is a polynomial to be determined. Thus, deg | = deg δ ≥ lm(m −1). Based on experience with similar models [1, 4, 6] in which one can prove a “minimal degree property”, as well as extensive computer investigations, it is reasonable to formulate the Conjecture 1. deg | = m(m − 1). One should be able to prove this conjecture either by ad hoc methods, as in e.g. [1], or by a detailed analysis of the underlying representation theory on the space of polynomials, as suggested by the work [11]. This is not the purpose of the present work, and we proceed assuming Conjecture 1. To fix the normalization of | we set

= (−1)m(m−1)/2 in Eq. (3.7), so that the homogeneous value δ (1, . . . , 1) = m(m−1) +2 is positive. 2 sin(π/(+2))

676

P. Zinn-Justin

Proposition 6. Assuming Conjecture 1, each component α is of degree at most (m −1) in each variable. Proof. The proof is strictly identical to that of Thm. 4 of [1], and will be sketched only. Using reflection covariance of the model, it is easy to see that 2m

z id sα

i=1

1 z 2m

,...,

1 z1

= α (z 1 , . . . , z 2m ),

(3.8)

where s is the reflection of link patterns: (sα)(i) = 2m + 1 − α(2m + 1 − i), and d is the maximum degree of the components α in each variable. Equating the total degrees in all variables on both sides of Eq. (3.8), we find 2md − m(m − 1) = m(m − 1), and therefore d = (m − 1). We are now in a position to resolve the following natural question, which is to ask what one can say about the non-zero components when z j = q 2k z i . Here we answer this question in the simplest situation: Proposition 7. Suppose z i+1 = q 2 z i . Consider the embedding ϕi of L,2(m−1) into L,2m which inserts 2 sites at Si , Si+1 and arches between Si and Si+1 . Then, assuming Conj. 1, ϕi (α) (z 1 , . . . , z i+1 = q 2 z i , . . . , z 2m ) = q 2(m−1)

(z i − q 2k z j )α (z 1 , . . . , z i−1 , z i+2 , . . . , z 2m )

(3.9)

j=i,i+1 k=1

for all α ∈ L,2(m−1) , where it is understood that on the r.h.s. is the eigenvector at size m − 1. Proof. First we recall (cf. proof of Prop. 4) that Ri (z i , q 2 z i ) is proportional to ei() , the projector onto the span of the image of ϕi , so that according to Eq. (2.11), T (t|z 1 , . . . , z i , z i+1 = q 2 z i , . . . , z 2m ) leaves this subspace invariant. This alone is sufficient to show that T (t|z 1 , . . . , z i , z i+1 = q 2 z i , . . . , z 2m )ϕi ∝ ϕi T (t|z 1 , . . . , z i−1 , z i+2 , . . . , z 2m ), but we need to compute the proportionality factor explicitly. The latter is given by evaluating Fig. 10. Since the result is proportional to the projector p, one can close the outgoing lines, replace the R-matrices with their expressions (2.8) and then apply repeatedly Lemma 2. Simplifying Eq. (2.8) at q = −eiπ/(+2) , we find that the term j, j in the z i −q −1 t) double sum produces a contribution (q − j−1(zzi −t)(q j+1 t)(q − j z −q j t) times the same for j i −q i with z i replaced with z i+1 (noting in particular that the factors 1/(U j U j ) produced by Lemma 2 compensate a j a j ). Finally we find that the coefficient of proportionality is (z i −t)(q z i −q −1 t) j=0 (q − j−1 z i −q j+1 t)(q − j z i −q j t) = 1 times the same sum with z i replaced with z i+1 . Thus, T (t|z 1 , . . . , z i , z i+1 = q 2 z i , . . . , z 2m )ϕi = ϕi T (t|z 1 , . . . , z i−1 , z i+2 , . . . , z 2m ). (3.10) Lemma 5 then implies that the l.h.s. and r.h.s. of Eq. (3.9) are proportional, up to a rational function of the z i which is independent of α. To fix the proportionality factor, we consider the base link pattern, but rotated i times in such a way that there are arches


677

p p R(z i ,t)

R(z i+1,t)

p

Fig. 10. Contribution of R(z i , t)R(z i+1 , t) to the sector where Si and Si+1 are fully connected to each other

between Si and Si+1 . When we remove the arches between Si and Si+1 this is again the rotated base link pattern but at size 2(m −1). We can therefore compare their expressions (Eq. (3.7) with = (−1)m(m−1)/2 ; this is the only place where we use Conj. 1) and collect the extra factors at size 2m. The theorem can be easily generalized to z j = q 2 z i , along the lines of Thm. 6 of [1], but this will not be needed here. In the case z j = q 2k z i , k > 1, the situation is more subtle: the recursion would lead to a new type of “mixed” loop model with 2(m − 1) usual subsets of vertices and one special site which would have only 2(k − 1) vertices fused together. We do not pursue here this direction. Example. We provide the full analysis of the case 2m = 4. As has already been mentioned in the course of the proof of Thm. 1, a state | j in L,4 is indexed by an integer j, 0 ≤ j ≤ , in such a way that there are − j arches between S1 and S2 and between S3 and S4 , and j arches between S2 and S3 and between S4 and S1 (note that |δ = |0 = |). We immediately conclude from Prop. 4 that

j (z 1 , z 2 , z 3 , z 4 ) = j

j

(q k z 1 − q −k z 2 )(q k z 3 − q −k z 4 )

k=1

×

− j

(q k z 2 − q −k z 3 )(q k z 4 − q −k z 1 ),

(3.11)

k=1

where the j are constants if we assume Conjecture 1 on the degree 2. In order to determine them we consider the homogeneous situation, i.e. the Hamiltonian H . It is not too j hard to compute off-diagonal elements of the matrix of H : H k = 2U1+| j−k| (τ ), j = k, and in particular to conclude that it is a symmetric matrix. Therefore j must be propor 2 +2

j /U j (τ )2 , tional to v j = 1/U j (τ ). We compute j (1, . . . , 1) = (−1) 2 sin(π/(+2)) and using Eq. (3.7) to fix the normalization ( = = (−1) ), we find j = (−1) U j (τ ). Note that for m > 2, Thm. 5 is not sufficient to determine up to a constant the entries α , since they are in general not fully factorizable as products of z j − q 2k z i .

3.4. Sum rule. A very natural object is the pairing of the left eigenvector and of the right eigenvector: we denote it by Z (z 1 , . . . , z 2m ) := 0|(z 1 , . . . , z 2m ).

678

P. Zinn-Justin

Proposition 8. Z (z 1 , . . . , z 2m ) is a symmetric function of its arguments. Proof. Start from Z (z 1 , . . . , z i+1 , z i , . . . , z 2m ) = 0|(z 1 , . . . , z i+1 , z i , . . . , z 2m ). Applying Prop. 3, it is equal to 0|Ri (z i+1 , z i )|(z 1 , . . . , z 2m ). On the other hand, from Lemma 4, 0|Ri (z i+1 , z i )|· = v Ri (z i+1 , z i ) = v. Thus, Z (z 1 , . . . , z i+1 , z i , . . . , z 2m ) = Z (z 1 , . . . , z 2m ), which proves the proposition. Theorem 4. Assuming Conjecture 1, Z (z 1 , . . . , z 2m ) is the Schur function sY,m (z 1 , . . . , z 2m ) associated to the Young diagram ((m − 1), (m − 1), . . . , 2, 2, , ). Proof. In fact, we really claim that sY,m (z 1 , . . . , z 2m ) is, up to multiplication by a scalar, the only symmetric polynomial of degree (at most) (m − 1) in each variable, which vanishes when the conditions of Thm. 3 are met. This clearly implies the theorem (up to a multiplicative constant) due to Prop. 8, Prop. 6 and Thm. 3. First, we show that sY,m does satisfy these conditions. It is symmetric by definition, and its degree in each variable is the width of its Young diagram that is (m − 1). It can be expressed as det 1≤i, j≤2m (z hj i ) sY,m (z 1 , . . . , z 2m ) = , 1≤i< j≤2m (z i − z j )

(3.12)

where the h j are the shifted lengths of the rows of Y,m , that is h 2i−1 = (i − 1)( + 2) ≡ 0 mod + 2, h 2i = (i − 1)( + 2) + 1 ≡ 1 mod + 2, i = 1, . . . , m. Assume now that z 2 = q 2k z 1 , z 3 = q 2k z 2 . Isolate the three first columns of the determinant in the h numerator of Eq. (3.12): the odd rows are of the form z 1 2i−1 (1, 1, 1) whereas the even h 2i rows are of the form z 1 (1, q 2k , q 2(k+k ) ). Thus, we have two series of m proportional rows: this proves that the 3 × 2m matrix is of rank 2, and that the full 2m × 2m matrix is singular. If the z’s are distinct the denominator of Eq. (3.12) is non-zero and we conclude that sY,m vanishes. Next, we show that it is the only such polynomial by induction. The step m = 0 is trivial. At step m, consider a symmetric polynomial Z (z 1 , . . . , z 2m ) in 2m variables, of degree (m − 1) in each variable, which vanishes when the conditions of Thm. 3 are met. Note that since Z is symmetric, the conditions can be in fact extended to arbitrary distinct integers (i, i , i ). Setting z j = q 2k z i , 1 ≤ k ≤ + 1, i = j, they therefore imply the following factorization: +1 2p Z |z j =q 2k zi = (z h − q z i ) W (z 1 , . . . , zˆ i , . . . , zˆ j , . . . , z 2m ), (3.13) h=i, j p=1 p=k

where W is a symmetric polynomial of the 2m − 2 variables z h , h = i, j, of degree (m −2) in each, which still vanishes when the conditions of Thm. 3 are met. W does not depend on z i because the 2(m −1) prefactors exhaust the degree of z i and z j combined. The induction hypothesis implies that W = const sY,m−1 (z 1 , . . . , zˆ i , . . . , zˆ j , . . . , z 2m ). The constant is independent of i or j by symmetry; and of k, as one can check by taking z i → ∞ (indeed in the limit z i , z j → ∞, Z must be proportional to (z i z j )(m−1) , which fixes the relative normalization of Z |z j =q 2k zi for varying k). Z , as a function of a given z j , is thus specified at ( + 1)(2m − 1) points by Eq. (3.13); this is enough to determine


679

Table 1. First few values of Z (1, . . . , 1). m 1 2 1 2 3 4 5

3

4

5

6

1 6 189 30618 25332021 106698472452 1 20 6720 36900864 3280676585472 4702058148658151424 1 50 103125 8507812500 27783325195312500 3574209022521972656250000 1 105 945945 707814508401 43505367274327463505 218541150429748620278689395225 1 196 6117748 29406803321896 21520945685492367246132 2385377935975138162776292257847164

uniquely a polynomial of degree (m − 1). Therefore Z = const sY,m (z 1 , . . . , z 2m ), which concludes the induction. Finally, one fixes the constant by another induction using Prop. 7 (Eq. (3.9)). Note the obvious Corollary. | = sY2,m . A final remark concerns the homogeneous situation where all z i are equal. In this case one can evaluate explicitly the Schur function, i.e. the dimension of the corresponding sl(2m) representation: Z (1, . . . , 1) = (( + 2)i)m(m−1)

m ( + 2)( j − i) + 1 . j −i +m

(3.14)

i, j=1

4. Conclusion This paper has tried to demonstrate the power of the methods devised in [1] and subsequent papers by applying it to the case of fused A1 models. A special point has been found for each such model – which is nothing but the point at which the central charge of the infrared fixed point vanishes. We call this point “combinatorial” because one can hope that the properties it possesses have interesting combinatorial meaning. Some of them have been described in the paper: existence of a left eigenvector with a simple form in the basis that we have built; simple sum rule. However, many questions remain open. First and foremost, one would like to have a generalized Razumov–Stroganov [3] conjecture for these fused models. In the present case, it would correspond to identifying each component of the ground state eigenvector of the Hamiltonian with the τ -enumeration of some combinatorial objects. By τ -enumeration we mean that the enumeration should be somehow weighted with τ to take into account the fact that the components belong to Z[τ ] (in the unfused case they are integers). For example, note that at = 2 we do know an interpretation of the sum of all components: up to a missing factor 2m (which can be naturally introduced in the normalisation of v), it is the 2-enumeration of Quarter-Turn Symmetric Alterating Sign Matrices (QTSASM) [12, 13]. The introduction of spectral parameters and the appearance of the Schur function of Thm. 4 also arise in this context. One should explore how this connection can be extended at the level of each component. Note that in the ASM literature, 1−, √ 2− and 3−enumerations are often considered. In our language, these are really 1−, 2−, √ 3−enumerations, which correspond to = 1, 2, 4. Also, many additional properties should be obtainable, along the lines of the abundant literature on the unfused case. For example we propose here the following

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Conjecture 2. Denote (m) ≡ (z 1 = · · · = z 2m = 1), Z (m) ≡ Z (z 1 = · · · = z 2m = 1). Then 0 (m) is the largest entry of |(m) (where we recall that 0 is the pattern that fully connects S2i−1 and S2i ), and 0 (m) 0 (m − 1) = . δ (m) δ (m − 1) In other words, if is normalized in such a way that the base link pattern has entry 1, the largest entry at size m is the (weighted) sum at size m − 1. Equally interesting is the study of the space of polynomials spanned by the components of the ground state eigenvector and the related representation theory, following the philosophy of [11]. One should emphasize the difficulty of such a task, because it involves separating the action on polynomials from the action on link patterns – even in the case of the Birman–Wenzel–Murakami algebra (BWM) this difficulty appears [4, 14], and for us BWM is only the simplest fused case (corresponding to = 2). Closely related is the extension of this work to a generic value of q by introducing an appropriate quantum Knizhnik–Zamolodchikov (qKZ) equation. Clearly all arguments of Sect. 3.3 depend only on polynomiality of the ground state eigenvector and on Prop. 3, which is one of the equations of the qKZ system. A natural conjecture is that at q generic will appear precisely the qKZ equation at level , which would be of greater interest than the level 1 (“free boson”) qKZ of the unfused model. One should also be able to combine the ideas of [6] and of the present work to study fused higher rank models; it is easy to guess the kind of properties they will possess at the point q = −e±iπ/(k+) , k dual Coxeter number. One could also consider fused models with other boundary conditions (open boundary conditions, etc., as in [7, 8]). Finally, it would be interesting to find some relation between our formulae, and in particular the sum rule, with the recent work [15] which generalizes the domain wall boundary conditions of the six-vertex model (relevant to the sum rule of the unfused loop model) to fused models. Appendix A Projection Operator Following [16], we define recurrently the projectors p (k) by p (1) = 1 and p (k+1) (e j , . . . , e j+k−1 ) = p (k) (e j , . . . , e j+k−2 )(1 − µk (τ )e j+k−1 ) p (k) ×(e j , . . . , e j+k−2 ),

(A.1)

where µk (τ ) = Uk−1 (τ )/Uk (τ ) and Uk is the Chebyshev polynomial of the second kind. The projectors used in this paper are simply pi := p () (e(i−1)+1 , . . . , ei−1 ). By recursion on k one can prove the following facts: (k) (k)2 = (a) ( p (k) (e j , . . . , e j+k−2 )e j+k−1 )2 = µ−1 k p (e j , . . . , e j+k−2 )e j+k−1 and p (k) p . (b) p (k) is -symmetric. (c) p (k) (e j , . . . , e j+k−2 )em = em p (k) (e j , . . . , e j+k−2 ) = 0 for m = j, . . . , j + k − 2. (use (a) for m = j + k − 2). (d) p (k) is “left-right symmetric”, that is it also satisfies

p (k+1) (e j , . . . , e j+k−1 ) = p (k) (e j+1 , . . . , e j+k−1 )(1 − µk (τ )e j ) p (k) ×(e j+1 , . . . , e j+k−1 ). (A.2)


(e)

681

p (k) p (k ) = p (k) p (k ) = p (k) when k ≥ k and the arguments of p (k ) are a subset of those of p (k) (use Eqs. (A.1) and (A.2). Note that property (i) of Sect. (2.3) is a direct consequence of (c).

Appendix B. Enumeration of Admissible States Call W,m the set of Lukacievicz words of length ( + 1)m taking values in {, −1}, that is ⎧ ⎫ j (+1)m ⎨ ⎬

wi ≥ 0 ∀ j < ( + 1)m, wi = 0 . (B.1) W,m = w ∈ {, −1}(+1)m : ⎩ ⎭ i=1

i=1

These words describe rooted planar trees with arity + 1, and it is well-known that #W,m =

(( + 1)m)! . (lm + 1)!m!

(B.2)

We shall therefore describe a bijection between L,2m and W,m . Start from a link pattern α. As an intermediate step it is convenient to rewrite it as a Dyck word w (the case = 1 of the Lukacievicz words above). Considering the link pattern as unfolded in the half-plane, we associate to each vertex where an arch starts (resp. ends) a +1 (resp. −1). This is in fact the bijection in the case = 1. We shall now restrict ourselves to -admissible link patterns. The goal is to transform the word w by condensing groups of “+1” into a single “”. We read the word w from left to right, in sequences of letters. Since α ∈ L,2m , these sequences can only be k “−1” followed by − k “+1”, 0 ≤ k ≤ . We distinguish three cases: (i) k = 0: if there are only “+1”, replace them with a single “”. (ii) k = : if there are only “−1”, leave them intact. (iii) 0 < k < : the k “−1” are left intact. As to the − k “+1”, two situations arise. Either (iiia) they have not been flagged yet, in which case they are replaced with a single “”. Say the first +1 of the sequence is at position i. Find the first j−1 position j for which p=i+1 w p < 0. According to Lemma 3, we know that r (i) + r ( j − 1) = − 1, and that w j and all its successors are +1 (there are − r (i) = − k of them). We flag them. Or (iiib) they have been flagged, in which case we ignore them. It is easy to show that the resulting word is indeed in W,m . In particular, the -admissibility ensures that sequences with k “+1” with 0 < k < always come in pairs, the second one being flagged. Inversely, start from a word w ∈ W,m . Read it from left to right. Each time we come across a “” at position i (all modifications to the left being taken into account in the position), with r (i) = k, we replace it using the following rule: (i) k = 0: we simply replace it with a sequence of “+1”. (ii) k > 0: we replace it with −k “+1”. Then we look for the first position j such that j−1 p=i+1 w p < 0 (being careful that the sum starts with − k − 1 newly created “+1”). We insert k extra “+1” between positions j and j + 1.

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This will clearly produce a Dyck word, and it is not hard to check that the corresponding link pattern is -admissible. The two operations described above being clearly inverse of each other, we conclude that they are bijections. Example. These are the words associated to L2,6 , with the same ordering as in Sect. 2.5: ⎫ ⎧ 2 −1 −1 2 −1 −1 2 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 −1 −1 2 −1 2 −1 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 −1 −1 2 2 −1 −1 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 −1 2 −1 −1 −1 2 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 −1 −1 −1 −1 2 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 2 −1 2 −1 −1 2 −1 −1 −1 ⎪ W2,3 = ⎪ ⎪ ⎪ 2 −1 2 −1 2 −1 −1 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 −1 −1 −1 −1 −1 ⎪ ⎪ ⎪ 2 −1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 −1 −1 −1 2 −1 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 −1 −1 2 −1 −1 −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 −1 2 −1 −1 −1 −1 −1 ⎪ ⎪ ⎭ ⎩ 2 2 2 −1 −1 −1 −1 −1 −1 References 1. Di Francesco, P., Zinn-Justin, P.: Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule. E. J. Combi. 12(1), R6 (2005) 2. Batchelor, M.T., Gier, J. de, Nienhuis, B.: The quantum symmetric XXZ chain at = −1/2, alternating sign matrices and plane partitions. J. Phys. A 34, L265–L270 (2001) 3. Razumov, A.V., Stroganov, Yu.G.: Combinatorial nature of ground state vector of O(1) loop model. Theor. Math. Phys. 138, 333–337 (2004); Teor. Mat. Fiz. 138, 395–400 (2004) 4. Di Francesco, P., Zinn-Justin, P.: Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties. Commun. Math. Phys. 262, 459–487 (2006) 5. Knutson, A., Zinn-Justin, P.: A scheme related to the Brauer loop model. http://arxiv.org/list/math.AG/ 0503224 6. Di Francesco, P., Zinn-Justin, P.: Quantum Knizhnik–Zamolodchikov equation, generalized Razumov– Stroganov sum rules and extended Joseph polynomials. J. Phys. A 38, L815–L822 (2005) 7. Di Francesco, P.: Inhomogeneous loop models with open boundaries. J. Phys. A 38(27), 6091–6120 (2005); Boundary qKZ equation and generalized Razumov–Stroganov sum rules for open IRF models. J. Stat. Mech. P11003 (2005), available at http://arxiv.org/list/math-ph/0509011, 2005 8. Di Francesco, P., Zinn-Justin, P.: From Orbital Varieties to Alternating Sign Matrices. Extended abstract for FPSAC’06, http://arxiv.org/list/math-ph/0512047, 2005 9. Di Francesco, P., Golinelli, O., Guitter, E.: Meanders and the Temperley–Lieb algebra. Commun. Math. Phys. 186, 1–59 (1997) 10. Ko, K.H., Smolinksky, L.: A combinatorial matrix in 3-manifold theory. Pacific J. Math. 149, 319– 336 (1991) 11. Pasquier, V.: Quantum incompressibility and Razumov Stroganov type conjectures. http://arxiv.org/list/ cond-mat/0506075, 2005 12. Kuperberg, G.: Symmetry classes of alternating-sign matrices under one roof. Ann. of Math. (2) 156(3), 835–866 (2002) 13. Okada, S.: Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups. http://arxiv.org/list/math.CO/0408234, 2004 14. Pasquier, V.: Incompressible representations of the Birman-Wenzl-Murakami algebra. http://arxiv.org/list/ math.QA/0507364, 2005 15. Caradoc, A., Foda, O., Kitanine, N.: Higher spin vertex models with domain wall boundary conditions. J. Stat. Mech. P03012 (2006) 16. Martin, P.: Potts model and related problems in statistical mechanics. World Scientific, Singapore (1991) Communicated by L. Takhtajan


Communications in


Dyson’s Constants in the Asymptotics of the Determinants of Wiener-Hopf-Hankel Operators with the Sine Kernel Torsten Ehrhardt Department of Mathematics, University of California, Santa Cruz, CA-95065, USA. E-mail: [email protected] Received: 28 April 2006 / Accepted: 6 November 2006 Published online: 18 April 2007 – © Springer-Verlag 2007 sin(x+y) Abstract: Let K α± stand for the integral operators with the sine kernels sin(x−y) π(x−y) ± π(x+y) acting on L 2 [0, α]. Dyson conjectured that the asymptotics of the Fredholm determinants of I − K α± are given by

log det(I − K α± ) = −

α log α log 2 log 2 3 α2 ∓ − + ± + ζ (−1) + o(1), 4 2 8 24 4 2

as α → ∞. In this paper we are going to give a proof of these two asymptotic formulas. 1. Introduction In random matrix theory one is interested in the three Fredholm determinants det(I − K α ), det(I − K α+ ), det(I − K α− ), where K α is the integral operator on L 2 [0, α] with the sine kernel k(x, y) =

sin(x − y) , π(x − y)

(1)

and K α± are the integral operators on L 2 [0, α] with the Wiener-Hopf-Hankel sine kernels k ± (x, y) =

sin(x − y) sin(x + y) ± . π(x − y) π(x + y)

(2)

These determinants are related to the probabilities E β (n; α) that in the bulk scaling limit of the three classical Gaussian ensembles of random matrices an interval of length α contains precisely n eigenvalues. It is customary to associate the parameter β = 2 with the Gaussian Unitary Ensemble, β = 1 with the Gaussian Orthogonal Ensemble, and

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β = 4 with the Gaussian Symplectic Ensemble. The basic relationship between these probabilities and the Fredholm determinants is given by E 2 (0; α) = det(I − K α ),

E 1 (0; α) = det(I − K α+ ),

and E 4 (0; α) =

1 − + det(I − K 2α ) + det(I − K 2α ) , 2

while expressions for E β (n; α) with n ≥ 1 also exist [15, 4]. A problem which has been open for a long time was the rigorous derivation of the asymptotics of these determinants as α → ∞. Dyson [9] was able to give a heuristic derivation and conjectured that log det(I − K 2α ) = −

α2 log α log 2 − + + 3ζ (−1) + o(1), 2 4 12

α → ∞,

(3)

where ζ stands for the Riemann zeta function. It is known [15] that det(I − K α+ ) =

∞

(1 − λ2n (α)),

n=0

det(I − K α− ) =

∞

(1 − λ2n+1 (α)),

(4)

n=0

where λn (α) are the decreasingly ordered eigenvalues of the operator K 2α . Using (3) and a non-rigorous derivation of the asymptotics of the quotient ∞ det(I − K α+ ) 1 − λ2n (α) , = − 1 − λ2n+1 (α) det(I − K α ) n=0

(5)

which was given by des Cloiseaux and Mehta [8], Dyson obtained the asymptotics formulas log det(I − K α± ) = −

α log α log 2 log 2 3 α2 ∓ − + ± + ζ (−1) + o(1), 4 2 8 24 4 2

(6)

as α → ∞. Recently the asymptotic formula (3) was proved independently by Krasovsky [14] and the author [10] using different methods. Yet another proof was given by Deift, Its, Krasovsky, and Zhou [6]. The proofs [14, 6] are based on the Riemann-Hilbert method, while the proof [10] is based on determinant identities and the asymptotics of WienerHopf-Hankel determinants with certain Fisher-Hartwig symbols [3]. The goal of this paper is to give a proof of (6). In contrast to Dyson’s derivation we will not rely on (3) and (5). In fact, we will use methods similar to those of [10]. As a consequence of (4), the asymptotic formulas (6) then imply the asymptotic formula (3). Hence the results of the present paper give a fourth derivation of (3). As was pointed out to the author by A. Its, another proof of (6), which is based on the Riemann-Hilbert method, can very likely be accomplished. It would rely on (3) and (4) and involve a (rigorous) derivation of the asymptotics of (5) based on observations made in [7, p. 205/206].

Dyson’s Constants in the Asymptotics of the Determinants of Operators with Sine Kernel

685

Let us conclude this introduction with some remarks on what else is known about the Fredholm determinants under consideration. It was shown by Jimbo, Miwa, Môri, and Sato [13] (see also [17]) that the function d log det(I − K α ) dα satisfies a Painlevé V equation. Widom [20, 21] was able to identify the highest term in the asymptotics of σ (α) as α → ∞. Knowing these asymptotics one can derive a complete asymptotic expansion for σ (α). By integration it follows that the asymptotics of det(I − K 2α ) are given by σ (α) = α

C2n log α α2 − +C + + O(α 2N +2 ), α → ∞, 2 4 α 2n N

log det(I − K 2α ) = −

(7)

n=1

with constants C2n that can be computed recursively. However, the constant C cannot be obtained in this way, and its rigorous identification was done - as mentioned above only in [14, 10, 6]. The asymptotic formula (7) was obtained in [7] as well; also, in the earlier work by B. Suleimanov [16] a rigorous derivation of the leading term of the asymptotics of the derivative of σ (α) was obtained. In a similar way, it turns out that the functions σ± (α) = α

d log det(I − K α± ) dα

satisfy a Painlevé III equation [18, 19]. Moreover, the operators K α± are related to special cases of integral operators K ν,α on L 2 [0, α] with Bessel kernel, √ √ √ √ √ √ Jν ( x) y Jν ( y) − x Jν ( x)Jν ( y) , ν > −1. kν (x, y) = 2(x − y) In fact, det(I − K α± ) = det(I − K ∓1/2,α 2 ). In the Bessel case, functions defined similarly to σ± (x) satisfy also a Painlevé III equation. The determinants det(I − K ν,α ) are the probabilities that no eigenvalues lie in an interval of length α for the Laguerre or Jacobi random matrix ensembles in the hard edge scaling limit. It is also interesting to observe that the following identity between det(I − K α± ) and det(I − K α ) exists (see, e.g., [17]): α 1 1 d2 log det(I − K α± ) = log det(I − K 2α ) ∓ − 2 log det(I − K 2x ) d x. (8) 2 2 0 dx Using this formula it is possible to derive from (7) a complete asymptotic expansion for log det(I − K α± ) at infinitiy with the exception of the constant, which remains undetermined due to the integration. Thus, once (7) had been proved, the only open problem was to identify the constant terms in (3) and (6). Let us shortly outline how the paper is organized. In the following section we will fix the basic notation and make some additional comments about the idea of the proof. We will follow essentially the same lines as in [10]. In fact, the proof is even somewhat simpler since some technical results are not needed here (namely, Prop. 4.2 and Prop. 4.9 of [10]). The auxiliary results which are needed here are either the same as or analogous to those of [10]. In Sect. 3 we will prove a formula involving Hankel determinants and in Sect. 4 we will finally prove the asymptotic formula (6).

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2. Basic Notation and Some Remarks We start with introducing some notation. We will denote the real line by R, the positive real half-axis by R+ , and the complex unit circle by T. By L p (M) we will denote the Lebesgue spaces (1 ≤ p ≤ ∞), where in our cases M is any of the above sets or a finite subinterval of R. The n × n Toeplitz and Hankel matrices are defined by Tn (a) = (a j−k )n−1 j,k=0 , where a ∈ L 1 (T) and ak =

1 2π

2π

Hn (a) = (a j+k+1 )n−1 j,k=0 ,

a(eiθ )e−ikθ dθ,

(9)

k ∈ Z,

0

are its Fourier coefficients. We will also need differently defined n × n Hankel matrices Hn [b] = (b j+k+1 )n−1 j,k=0 ,

(10)

where the numbers bk are the (scaled) moments of a function b ∈ L 1 [−1, 1], i.e., 1 1 bk = b(x)(2x)k−1 d x, k ≥ 1. π −1 For a ∈ L ∞ (T) the Toeplitz and Hankel operators are bounded linear operators acting on the Hardy space H 2 (T) = f ∈ L 2 (T) : f k = 0 for all k < 0 by

H (a) = P M(a)J P| H 2 (T) , (11) T (a) = P M(a)P| H 2 (T) , ∞ k where P : k=−∞ f k t k → ∞ k=0 f k t stands for the Riesz projection, J : f (t) → −1 −1 t f (t ) stands for a flip operator, and M(a) : f (t) → a(t) f (t) stands for the multiplication operator. (These last three operators are acting on L 2 (T).) Finally, introduce the projections Pn :

f k t k ∈ H 2 (T) →

k≥0

n−1

f k t k ∈ H 2 (T),

(12)

k=0

the image of which can be naturally identified with Cn . Using this we can make the identifications Pn T (a)Pn ∼ = Tn (a), Pn H (a)Pn ∼ = Hn (a). We will also need the notion of a trace class operator acting on a Hilbert space H . This is a compact operator A such that the series of its singular sn (A) (i.e., the eigenvalues of (A∗ A)1/2 counted according to their algebraic multiplicities) converges. The class of all trace class operators can be made to a Banach space by introducing the norm A 1 = sn (A). (13) n≥1

This class is also a two-sided ideal in the algebra of all bounded linear operators on H . The importance of trace class operators is that for such operators A, the operator trace


687

“trace(A)” and the operator determinant “det(I + A)” can be defined as generalizations of the matrix trace and matrix determinant. More detailed information on this subject can be found, e.g., in [12]. For a ∈ L ∞ (R), let MR (a) : f (x) → a(x) f (x) stand for the multiplication operator acting on L 2 (R). The convolution operator W0 (a) (or, “two-sided” Wiener-Hopf operator) is defined by W0 (a) = F MR (a)F −1 , where F stands for the Fourier transform on L 2 (R). The continuous analogues of Toeplitz and Hankel operators are operators defined W (a) = + W0 (a) + | L 2 (R+ ) ,

(14)

HR (a) = + W0 (a) Jˆ + | L 2 (R+ ) ,

(15)

where ( Jˆ f )(x) = f (−x), and + = MR (χR+ ) is the projection operator on the positive real half axis. The operator W (a) is usually called a Wiener-Hopf operator, and we will refer to HR (a) as a Hankel operator, too. (The notation will avoid a possible confusion between HR (a) and H (a).) One can show that if a ∈ L 1 (R), then W (a) and HR (a) are integral operators on L 2 (R) with the kernel a(x ˆ − y) and a(x ˆ + y), respectively, where ∞ 1 a(ξ ˆ )= e−i xξ a(x) d x 2π −∞ stands for the Fourier transform of a. For α > 0 we will define the projection operator α : f (t) ∈ L 2 (R+ ) → χ[0,α] (x) f (x) ∈ L 2 (R+ ).

(16)

The image of this operator can be identified with L 2 [0, α]. With this notation the integral operators K α and K α± can now be seen to be truncated Wiener-Hopf and Wiener-Hopf-Hankel operators, K α = α W (χ ) α | L 2 [0,α] ,

K α± = α (W (χ ) ± HR (χ )) α | L 2 [0,α] ,

where χ stands for the characteristic function of the interval [−1, 1]. Notice that K α and K α± are trace class operators and that

det(I − K α ) = det α W (1 − χ ) α , (17)

det(I − K α± ) = det α W (1 − χ ) ± HR (1 − χ ) α .

(18)

For determinants of Wiener-Hopf operators (and, more recently, also for determinants of Wiener-Hopf-Hankel operators) results describing the asymptotics as α → ∞ exist under the condition that the underlying symbol is sufficiently well behaved. These results are known as Achiezer-Kac formulas (if the symbol has no singularities) and as FisherHartwig type formulas (if the symbol has a finite number of certain types of singularities). An overview about this topic can be found in [5]. In our case the symbol is the characteristic function 1 − χ vanishing on the interval [−1, 1], a state of affairs which is not covered by the just mentioned cases and which renders the situation completely non-trivial.

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The main idea of the proof given in this paper is to relate the Fredholm determinants det(I − K α± ) to the determinants of different operators for which Fisher-Hartwig type formulas can be applied. Let us introduce the functions 2 β x −i β x uˆ β (x) = , vˆβ (x) = , x ∈ R, β ∈ C, (19) x +i 1 + x2 where these functions are supposed to be continuous on R \ {0} and to have their values approaching 1 as x → ±∞. Then we are going to prove that

α2 α ± det(I − K α ) = exp − (20) ∓ det α (I ± HR (uˆ ∓1/2 ))−1 α . 4 2 It is now illuminating to point out that the determinants on the right-hand side can be identified with determinants of truncated Wiener-Hopf-Hankel operators with FisherHartwig symbols. In fact, it is proved in [3] that

det α (W (vˆβ ) + HR (vˆβ )) α = e−αβ det α (I + HR (uˆ −β ))−1 α 1 3 < Re β < , 2 2

det α (W (vˆβ ) − HR (vˆβ )) α = e−αβ det α (I − HR (uˆ −β ))−1 α if −

if −

1 1 < Re β < . 2 2

However, we will avoid making use of these formulas for two reasons. First of all, the determinants on the right-hand side of (20) are those occurring primarily in the proof, and their asymptotics are computed also in [3]. Secondly, the left-hand side of the last formula is, as it stands, not defined for β = −1/2. (It can be defined by analytic continuation in β because the right-hand side makes sense for −3/2 < Re β < 1/2.) 3. A Hankel Determinant Formula In this section we are going to prove two formulas of the kind

det Hn [b] = G n det Pn (I + H (ψ))−1 Pn , where b ∈ L 1 [−1, 1] is a (sufficiently smooth) continuous and nonvanishing function on [−1, 1] multiplied in one case with the function (1 + x)1/2 and in another case with (1 − x)−1/2 . The function ψ and the constant G depend on b. A formula of the same type was proved in [10]. However, the conditions on the function b and the form of the function ψ were different. Before we state the result we have to introduce more notation. Let W stand for the Wiener algebra, i.e., the set of all functions in L 1 (T) whose Fourier series are absolutely convergent. Moreover, let W± = { a ∈ W : an = 0 for all ± n < 0 } ,

(21)

be two Banach subalgebras of W, where an stand for the Fourier coefficients of a. Notice that a ∈ W+ if and only if a˜ ∈ W− , where a(t) ˜ := a(t −1 ), t ∈ T. Finally, we denote


689

by GW and GW± the group of invertible elements in the Banach algebras W and W± , respectively. A function a ∈ W is said to admit a canonical Wiener-Hopf factorization in W if it can be written in the form a(t) = a− (t)a+ (t),

t ∈ T,

(22)

where a± ∈ GW± . It is easy to see that a ∈ W admits a canonical Wiener-Hopf factorization in W if and only if a ∈ GW and if the winding number of a is zero (see, e.g., [5]) . Moreover, this condition is equivalent to the existence of a logarithm log a which belongs to W. If this is fulfilled, then one can unambiguously define the geometric mean of a by 1 2π

G[a] := exp log a(eiθ ) dθ . (23) 2π 0 The following result (which is not yet what we ultimately want) is cited from [10, Thm. 4.5]. The invertibility statement is taken from [10, Prop. 4.3]. Recall that a function a on T is called even if a˜ = a, where a(t) ˜ := a(t −1 ). Theorem 3.1. Let a ∈ GW be an even function which possesses a canonical WienerHopf factorization a(t) = a− (t)a+ (t). Define ψ(t) = a˜ + (t)a+−1 (t), and let b ∈ L 1 (T) be 1 + cos θ iθ b(cos θ ) = a(e ) . (24) 1 − cos θ Then I + H (ψ) is invertible on H 2 (T) and

det Hn [b] = G[a]n det Pn (I + H (ψ))−1 Pn .

(25)

In order to be able to state the desired result we introduce (for τ ∈ T and β ∈ C) the functions u β,τ (eiθ ) = exp(iβ(θ − θ0 − π )), 0 < θ − θ0 < 2π, τ = eiθ0 .

(26)

These functions are continuous on T\{τ } and have a jump discontinuity at t = τ whose size is determined by β. The promised formulas are now given in the following theorem. Notice that the difference between Theorem 3.1 and Theorem 3.2 (as well as to Thm. 4.6 of [10]) is in the conditions on the underlying functions. Theorem 3.2. Let c ∈ GW be an even function which possesses a canonical WienerHopf factorization c(t) = c− (t)c+ (t). Define b+ , b− ∈ L 1 [−1, 1] and ψ + , ψ − ∈ L ∞ (T) by √ ψ + (eiθ ) = c˜+ (eiθ )c+−1 (eiθ )u −1/2,1 (eiθ ), b+ (cos θ ) = c(eiθ ) 2 + 2 cos θ , c(eiθ ) b− (cos θ ) = √ , 2 − 2 cos θ

ψ − (eiθ ) = c˜+ (eiθ )c+−1 (eiθ )u 1/2,−1 (eiθ ).

Then the operators I + H (ψ ± ) are invertible on H 2 (T) and

det Hn [b± ] = G[c]n det Pn (I + H (ψ ± ))−1 Pn .

(27)

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For the proof of this theorem we will apply some auxiliary results, which are stated in [10] in connection with Thm. 4.6 and which we are not going to restate here. However, we will recall the following notation, which is used here and later on. For r ∈ [0, 1) and τ ∈ T let G r,τ be the following operator acting on L ∞ (T): t +r . (28) G r,τ : a(t) → b(t) = a τ 1 + rt Proof of Theorem 3.2. The first problem is to verify the invertibility of I + H (ψ ± ). In the special case c+ ≡ 1, i.e., for I + H (u −1/2,1 ) and I + H (u 1/2,−1 ), this was done in [3, Thm. 3.6]. (Notice that I + H (u 1/2,−1 ) is similar to I − H (u 1/2,1 ).) The proof in the case where c+ ≡ 1 can be done in the same way as in [3, Sect. 3.2] or [10, Prop. 4.2]. We refrain from copying the proof with the little modifications necessary and make instead only the following remarks. The proof in [3] consists of two parts. First one determines the essential spectrum of the Hankel operators. Since ψ ± have their discontinuities at the same places as u ∓1/2,±1 and the one-sided limits there are also the same, the essential spectrum of H (ψ ± ) is the same as that of H (u ∓1/2,±1 ). The second step is to determine the kernel of the operators I + H (u ∓1/2,±1 ). (Passing to the adjoints gives similarly information about the cokernel.) The crucial point in [3] is to write, e.g., u β,1 = ξ−β ηβ with ξ−β = (1 − t −1 )−β , ηβ (t) = (1 − t)β , t ∈ T. What one uses about these functions are the facts that ξ−β (t) = 1/ηβ (t −1 ), that they and their inverses belong to certain Hardy spaces. In our case, one has to write, e.g., ψ + = (c˜+ ξ1/2 ) · (c+−1 η−1/2 ). The factors in this product have the just mentioned properties, too. Hence the proof works in the same way. The proof of (27) will be carried out by an approximation argument and with the help of Theorem 3.1. For r ∈ [0, 1) consider the even functions

±1/2 ar± (t) = c(t) (1 ∓ r t)(1 ∓ r t −1 ) , t ∈ T. Clearly, ar± ∈ GW. The functions br± defined in terms of ar± by formula (24) evaluate to

±1/2 1 + x ± ± ∓1/2 2 br (x) = b (x)(2 ± 2x) 1 + r ∓ 2r x 1−x ∓1/2 2 ∓ 2x , x ∈ (−1, 1). = b± (x) 1 + r 2 ∓ 2r x Then the functions br± converge to b± in the norm of L 1 [−1, 1]. Hence, if we fix n, det Hn [b± ] = lim det Hn [br± ]. r →1

It is now easily seen that the canonical Wiener-Hopf factorization of ar± is given by ± ± = ar,− (t)ar,+ (t) with the factors

ar± (t)

± ar,+ (t) = c+ (t)(1 ∓ r t)±1/2 ,

± ar,− (t) = c− (t)(1 ∓ r t −1 )±1/2 .

Notice also that G[ar ] = G[c]. If we define ± ± ψr± (t) := a˜ r,+ (t)(ar,+ (t))−1 = c˜+ (t)c+−1 (t)

1 ∓ rt 1 ∓ r t −1

∓1/2

,


691

we can apply Theorem 3.1 and conclude that

det Hn [br± ] = G[c]n det Pn (I + H (ψr± ))−1 Pn . It follows that

det Hn [b± ] = G[c]n lim det Pn (I + H (ψr± ))−1 Pn . r →1

Next define fr± (t) :=

1 ∓ rt 1 ∓ r t −1

∓1/2

and observe that fr± → u ∓1/2,±1 in measure as r → 1. Hence also ψr± → ψ ± in measure. Because the sequence ψr± is bounded in the L ∞ -norm it follows that H (ψr± ) converges strongly to H (ψ ± ) on H 2 (T) (see, e.g., Lemma 4.7 of [10]). In order to obtain that (I + H (ψr± ))−1 → (I + H (ψ ± ))−1

(29)

strongly on H 2 (T), it is necessary and sufficient that the following stability condition, sup (I + H (ψr± ))−1 < ∞, r ∈[r0 ,1)

is satisfied (see, e.g., Lemma 4.8 of [10]). Here r0 is some number in [0, 1). Stability criteria for such a type of operator sequences were established in [11] (see Sects. 4.1, 4.2, and 5.2 therein), and we are going to apply the corresponding results. First of all, there exist certain mappings 0 and τ , τ ∈ T, which are defined by 0 [ψr ] := µ- lim ψr , r →1

0 [ψr ] := µ- lim G r,τ ψr . r →1

Here µ-lim stands for the limit in measure. It is now easy to see that these mappings evaluate as follows, 0 [ fr± ] = u ∓1/2,±1 ,

τ [ fr± ] = u ∓1/2,±1 (τ ),

if τ = ±1, and ±1 [ fr± ] = µ- lim G r,±1 fr± = µ- lim r →1

r →1

1 + rt 1 + r t −1

±1/2

= u ±1/2,−1

if τ = ±1. Because of ψr± = c˜+ c+−1 fr± it follows immediately that 0 [ψr± ] = c˜+ c+−1 u ∓1/2,±1 , ±1 [ψr± ] = u ±1/2,−1 , τ [ψr± ] = constant function,

τ ∈ T\{±1}.

The stability criterion in [11] (Thm. 4.2 and Thm. 4.3) says that I + H (ψr± ) is stable if and only if each of the following operators is invertible:

692

(i) (ii) (iii) (iv)

T. Ehrhardt

0 [I + H (ψr± )] = I + H (0 [ψr± ]) = I + H (ψ ± ), ±1 [I + H (ψr± )] = I ± H (±1 [ψr± ]) = I ± H (u ±1/2,−1 ), ∓1 [I + H (ψr± )] = I ∓ H (∓ [ψr± ]) = I , τ [I + H (ψr± )] = 0 M(τ [ψr± ]) I 0 P 0 0I P 0 I 0 + = 0I 0 Q I 0 0 Q 0I [ψr± ]) 0 M(τ¯ (τ ∈ T, Im(τ ) > 0).

The invertibility is obvious for (iii) and (iv). As to (i) and (ii) the invertibility has been stated at the beginning of the proof. Notice that I ± H (U±1/2,−1 ) is similar to I ∓ H (u ±1/2,1 ). We can thus conclude that the sequence I + H (ψr± ) is stable and the strong convergence (29) follows. Hence the matrices Pn (I + H (ψr± ))−1 Pn converge to Pn (I + H (ψ ± ))−1 Pn as r → 1. This implies that their determinants also converge and proves the assertion. 4. Proof of the Asymptotic Formula In order to prove the asymptotic formula (6), we are going to discretize the underlying Wiener-Hopf-plus-Hankel operators I − K α± . This will give us Toeplitz-plus-Hankel operators. Let χα denote the characteristic function of the subarc {eiθ : α < θ < 2π −α} of T. Proposition 4.1. For each α > 0 we have

det(I − K α± ) = lim det Tn (χ αn ) ± Hn (χ αn ) . n→∞

(30)

Proof. The operator K α± is the integral operator on L 2 [0, α] with the kernel K (x − y) ± x K (x + y), where K (x) = sin π x . Consider the n × n matrices A± n Bn±

α( j − k) α α( j + k + 1) n−1 α K ± K = , n n n n j,k=0 n−1 1 1 α( j − k + ξ − η) α( j + k + ξ + η) α K ±K dξ dη = . n 0 0 n n j,k=0

± −2 The entries of A± n − Bn can be estimated uniformly by O(n ) using the mean value ± ± theorem.√Hence the Hilbert-Schmidt norm of An − Bn is O(n −1 ), and the trace norm is O(1/ n). The rest of the proof can be completed in the same way as in [10,

Prop. 5.1] by show α α ) = det T (χ ) ± H (χ ) . ing that det(I − K α± ) = det(In − Bn± ) and det(I − A± n n n n n

After discretizing, the next goal is to reduce the Toeplitz-plus-Hankel determinants to Hankel determinants. For this purpose we use an exact identity which is stated in the following result cited from [2, Thm. 2.3].


693

Proposition 4.2. Let a ∈ L 1 (T) be an even function, and let b ∈ L 1 [−1, 1] be given by 1 + cos θ iθ . (31) b(cos θ ) = a(e ) 1 − cos θ

Then det Tn (a) + Hn (a) = det Hn [b]. Notice that the assumption b ∈ L 1 [−1, 1] implies that a ∈ L 1 (T). Applying the previous result yields the following. Proposition 4.3. For each α > 0 and n ∈ N we have n 2

±n/2 α,n + 1 ± α α det Tn (χ n ) ± Hn (χ n ) = (µα,n ) det Hn [bα,n ], 2 where + bα,n (x) =

2 + 2x , 2 1 + µα,n − 2µα,n x

− (x) = bα,n

1 + µ2α,n + 2µα,n x , 2 − 2x

(32)

(33)

and α,n and µα,n are numbers (unambiguosly) defined by α,n = cos

α , n

1 + µ2α,n 3 − α,n = , 2µα,n 1 + α,n

0 < µα,n < 1.

(34)

Proof. In the plus-case, we apply Proposition 4.2 with + a(e ) = χ αn (e ), b(x) = bˆα,n (x) := χ[−1,α,n ] (x) iθ

iθ

1+x . 1−x

In the minus-case, we apply this proposition with − a(e ) = χ (−e ), b(x) = bˆα,n (x) := χ[−α,n ,1] (x) iθ

α n

iθ

1+x . 1−x

Hence we obtain (by using the general formula det(Tn ( f ) + Hn ( f )) = det(Tn ( fˆ) − Hn ( fˆ)) with fˆ(t) = f (−t) in the minus-case)

± ]. det Tn (χ αn ) ± Hn (χ αn ) = det Hn [bˆα,n ± ] are the moments [bˆ ± ] The entries of Hn [bˆα,n α,n 1+ j+k , 0 ≤ j, k ≤ n − 1. Computing them yields 1 α,n 1 + x + [bˆα,n (2x)k−1 d x ]k = π −1 1−x 1 − α,n k−1 1 α,n + 1 k 1 1 + y 2y − 2 dy = 3−α,n π 2 1 + α,n −1 −y 1+α,n

√ µα,n α,n + 1 k 1 + 1 − α,n k−1 bα,n (y) 2y − 2 dy = π 2 1 + α,n −1

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and

1

1+x (2x)k−1 d x 1 − x −α,n k 1 3−α,n + y 1 α,n + 1 1 − α,n k−1 1+α,n = 2y + 2 dy π 2 1−y 1 + α,n −1 1 α,n + 1 k 1 − 1 − α,n k−1 = √ bα,n (y) 2y + 2 dy. π µα,n 2 1 + α,n −1

− ]k = [bˆα,n

1 π

Hence ± Hn [bˆα,n ]

= (µα,n )

±1/2

α,n + 1 2

j+k+1

1 π

1 −1

n−1 ± bα,n (y)(2y

∓ 2τα,n )

j+k

,

dy j,k=0

with certain τα,n . One can pull out certain diagonal matrices from the left and the right, 2 which give the terms (µα,n )±n/2 ((1+α,n )/2)n after taking the determinant. The remaining matrix can be written as

1 π

1 −1

± bα,n (y)(2y ∓ 2τα,n ) j (2y ∓ 2τα,n )k dy

n−1 . j,k=0

After expanding (2y ∓ 2τα,n ) j and (2y ∓ 2τα,n )k into two binomial series it is easily ± ] multiplied from the left and right seen that the previous matrix is the matrix Hn [bα,n with triangular matrices having ones on the diagonal. This implies the desired assertion. In the following result and also later on we use the functions t ∓ µα,n ∓1/2 ± (t) := ∓ ψα,n 1 ∓ µα,n t

(35)

with the sequence µα,n defined by (34). Proposition 4.4. For each α > 0 we have

α2

α ± ∓ lim det Pn (I + H (ψα,n ))−1 Pn . lim det Tn (χ αn ) ± Tn (χ αn ) = exp − n→∞ 8 2 n→∞ (36) Proof. The asymptotics (as n → ∞) of the numbers appearing in (32) of Proposition 4.3 are given by 1 + α,n α2 = 1 − 2 + O(n −4 ), 2 4n

µα,n = 1 −

α + O(n −2 ). n

Hence using this proposition it follows that

α2 α ± lim det Tn (χ αn ) ± Hn (χ αn ) = exp − ∓ lim det Hn [bα,n ]. n→∞ 8 2 n→∞


Next we introduce

695

∓1/2 c(eiθ ) = (1 ∓ µα,n t)(1 ∓ µα,n t −1 ) ,

and we are going to employ Theorem 3.2 . It can be verified easily that G[c] = 1 and that c(t) = c− (t)c+ (t) is a canonical Wiener-Hopf factorization of c where c+ (t) = (1 ∓ µα,n t)∓1/2 and c− (t) = (1 ∓ µα,n t −1 )∓1/2 . Moreover, 1 ∓ µα,n t ±1/2 −1 c˜+ (t)c+ (t) = . 1 ± µα,n t −1 The functions b± and ψ ± defined in Theorem 3.2 now evaluate to ± b± (x) = (1 + µ2α,n ∓ 2µα,n x)∓1/2 (2 ± 2x)±1/2 = bα,n (x), 1 ∓ µα,n t ±1/2 ± ± (∓t)∓1/2 = ψα,n (t). ψ (t) = 1 ∓ µα,n t −1

Combining all this we obtain from Theorem 3.2 that

± ± ] = det Pn (I + H (ψα,n ))−1 Pn , det Hn [bα,n which concludes the proof. The next step is to identify the limit on the right-hand side of (36). For this purpose we resort to an auxiliary result, which was stated in [10] (with a slight change of notation). In order to make the reference correct, we allow (for the time being) µα,n ∈ [0, 1) to be an arbitrary sequence and define the functions 1−t t + µα,n n , h α,n (t) = h α (t) = exp −α . (37) 1+t 1 + µα,n t ± as being defined by (35) with this arbitrary Moreover, we also consider the functions ψα,n sequence.

Proposition 4.5. Let α > 0 be fixed, and consider (35) and (37). Assume that µα,n = 1 −

α + O(n −2 ) as n → ∞. n

(38)

Then the following is true: ± ) are unitarily equivalent to the operators ±H (u (i) The operators H (ψα,n ∓1/2,1 ). (ii) The operator ± Pn (I + H (ψα,n ))−1 Pn − Pn

is unitarily equivalent to operators An = H (h α,n )(I ± H (u ∓1/2,1 ))−1 H (h α,n ) − H (h α,n )2 , which are trace class operators and converge as n → ∞ in the trace norm to A = H (h α )(I ± H (u ∓1/2,1 ))−1 H (h α ) − H (h α )2 .

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Proof. These results are proved in [10, Prop. 4.12] with a change of the condition on the sequence µα,n . This change is consistent with the different notation for h α . In fact, one has only to replace α by 2α. Moreover, instead of the functions ψα± , functions ψα± − 1 occur, which do not change the Hankel operators. The fact that the operators I ± H (u ∓1/2,1 ) are invertible has already been stated in the proof of Theorem 3.2 (see also [10, Prop. 4.1] or [3, Thm. 3.6]). In the following proposition we identify the limit of the determinant appearing in the right hand side of (36). We return to the specific definition of µα,n given in (34) and to the definitions (35) and (37) in terms of this sequence. Proposition 4.6. For each α > 0 we have

± lim det Pn (I + H (ψα,n ))−1 Pn = det H (h α )(I ± H (u ∓1/2,1 ))−1 H (h α ) . n→∞

(39)

Proof. Proceeding as in [10, Prop. 5.5] we notice that H (h α )2 is a projection operator. (The slight change in notation, α → 2α, does not affect the statements made here.) Hence, in the same way it is established that

det H (h α )(I ± H (u ∓1/2,1 ))−1 H (h α )

= det I + H (h α )(I ± H (u ∓1/2,1 ))−1 H (h α ) − H (h α )2 , where the determinant on the left-hand side is of that an operator acting on the image of H (h α )2 , while the right-hand side corresponds to an operator acting on L 2 (R+ ). Similarily, the determinant on the left-hand side of (39) can be written as

± ± ))−1Pn = det I + Pn (I + H (ψα,n ))−1 Pn − Pn det Pn (I + H (ψα,n

= det I + H (h α,n )(I ± H (u ∓1/2,1 ))−1 H (h α,n )− H (h α,n )2 . As stated in the proof of Proposition 4.4, the sequence µα,n has the asymptotics (38). By applying the previous proposition the desired assertion follows. We are now finally able to identify the determinants det(I − K α± ). Recall in this connection the definition (19) of the functions uˆ β . Theorem 4.7. For each α > 0 we have

α α2 ± det(I − K α ) = exp − ∓ det α (I ± HR (uˆ ∓1/2 ))−1 α . 4 2

(40)

Proof. From Propositions 4.1, 4.4 and 4.6 it follows that

α α2 ± det(I − K α ) = exp − ∓ det H (h α )(I ± H (u ∓1/2,1 ))−1 H (h α ) . 4 2 As noted in the proof of [10, Thm. 5.6], there exists a unitary S from H 2 (T)

transform 1+i x onto L 2 (R+ ) such that HR (a) = S H (b)S ∗ with a(x) = b 1−i x . In the specific case we obtain HR (uˆ ±1/2 ) = S H (u ±1/2,1 )S ∗ ,

HR (ei xα ) = S H (h α )S ∗ .

This together with the remark that H (ei xα )2 = α implies (40).


697

Finally we are using a result of [3] in order to establish the promised asymptotic formula. Recall that ζ stands for the Riemann zeta function. Theorem 4.8. The following asymptotic formula holds as α → ∞: log det(I − K α± ) = −

α2 α log α log 2 log 2 3 ∓ − + ± + ζ (−1) + o(1). 4 2 8 24 4 2

Proof. In Sect. 3.6 of [3] it has been proved that

det α (I ± HR (uˆ ∓1/2 ))−1 α ∼ α −1/8 π 1/4 2±1/4 G(1/2),

α → ∞,

(41)

(42)

where G(z) stands for the Barnes G-function [1]. Notice that G(3/2) = G(1/2)(1/2), (1/2) = π 1/2 , and G(1) = 1. This together with the previous theorem implies that log det(I − K α± ) = −

α log α log π log 2 α2 ∓ − + ± + log G(1/2) + o(1). 2 2 8 4 4

(43)

Finally observe that log G(1/2) = −

log π 3 log 2 + ζ (−1) + , 4 2 24

which follows from a formula for G(1/2) in terms of Glaisher’s constant A = exp(−ζ (−1) + 1/12) given in [1, p. 290]. References 1. Barnes, E.B.: The theory of the G-function. Quart. J. Pure Appl. Math. XXXI, 264–313 (1900) 2. Basor, E.L., Ehrhardt, T.: Some identities for determinants of structured matrices. Linear Algebra Appl. 343/344, 5–19 (2002) 3. Basor, E.L., Ehrhardt, T.: On the asymptotics of certain Wiener-Hopf-plus-Hankel determinants. New York J. Math. 11, 171–203 (2005) 4. Basor, E.L., Tracy, C.A., Widom, H.: Asymptotics of level-spacing distributions for random matrices. Phys. Rev. Lett. 69(1), 5–8 (1992) 5. Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators. 2nd ed., Berlin: Springer, 2006 6. Deift, P., Its, A., Krasovksy, I., Zhou, X.: The Widom-Dyson constant for the gap probability in random matrix theory. http://arxiv.org/list/math.FA/0601535, 2006 7. Deift, P., Its, A., Zhou, X.: The Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. of Math. (2) 146(1), 149–235 (1997) 8. des Cloiseau, J., Mehta, M.L.: Asymptotic behavior of spacing distributions for the eigenvalues of random matrices. J. Math. Phys. 14, 1648–1650 (1973) 9. Dyson, F.J.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, 171–183 (1976) 10. Ehrhardt, T.: Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel. Commun. Math. Phys. 262, 317–341 (2006) 11. Ehrhardt, T., Silbermann, B.: Approximate identities and stability of discrete convolution operators with flip. Operator Theory: Adv. Appl. 110, 103–132 (1999) 12. Gohberg, I., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators in Hilbert space. Trans. Math. Monographs 18, Providence, R.I.: Amer. Math. 1969 13. Jimbo, M., Miwa, T., Môri, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. 1(D)(1), 80–158 (1980)

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14. Krasovsky, I.V.: Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. (25), 1249–1272 (2004) 15. Mehta, M.L.: Random Matrices. 6th edition, San Diego: Academic Press, 1994 16. Suleimanov, B.: On asymptotics of regular solutions for a special kind of Painlevé V equation. In: Appendix 1 of: Its, A.R., Novokshenov, V.Yu.: The isomonodromic deformation method in the theory of Painlevé equations. Lect. Notes in Math. 1191, Berlin: Springer, 1986, pp. 230–260 17. Tracy, C.A., Widom, H.: Introduction to Random Matrices. In: Geometric and quantum aspects of integrable systems (Scheveningen, 1992), Lecture Notes in Physics 424, Berlin: Springer, 1993, pp. 103–130 18. Tracy, C.A., Widom, H.: Level-spacing distributions and the Bessel kernel. Commun. Math. Phys. 161, 289–310 (1994) 19. Tracy, C.A., Widom, H.: Fredholm Determinants, Differential Equations and Matrix Models. Commun. Math. Phys. 163, 33–72 (1994) 20. Widom, H.: The asymptotics of a continuous analogue of orthogonal polynomials. J. Appr. Theory 77, 51– 64 (1994) 21. Widom, H.: Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals. Commun. Math. Phys. 171, 159–180 (1995) Communicated by J.L. Lebowitz

Commun. Math. Phys. 272, 699–750 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0177-z

Communications in


Algebraic Supersymmetry: A Case Study Detlev Buchholz1 , Hendrik Grundling2 1 Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, Göttingen,

D-37077, Germany. E-mail: [email protected]

2 School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia.

E-mail: [email protected] Received: 4 May 2006 / Accepted: 30 June 2006 Published online: 13 March 2007 – © Springer-Verlag 2007

Dedicated to Daniel Kastler on the occasion of his 80th birthday Abstract: The treatment of supersymmetry is known to cause difficulties in the C∗ –algebraic framework of relativistic quantum field theory; several no–go theorems indicate that super–derivations and super–KMS functionals must be quite singular objects in a C∗ –algebraic setting. In order to clarify the situation, a simple supersymmetric chiral field theory of a free Fermi and Bose field defined on R is analyzed. It is shown that a meaningful C∗ –version of this model can be based on the tensor product of a CAR– algebra and a novel version of a CCR–algebra, the “resolvent algebra”. The elements of this resolvent algebra serve as mollifiers for the super–derivation. Within this model, unbounded (yet locally bounded) graded KMS–functionals are constructed and proven to be supersymmetric. From these KMS–functionals, Chern characters are obtained by generalizing formulae of Kastler and of Jaffe, Lesniewski and Osterwalder. The characters are used to define cyclic cocycles in the sense of Connes’ noncommutative geometry which are “locally entire”. 1. Introduction Graded (super) derivations occur in many parts of physics: supersymmetry, BRS-constraint reduction and cyclic homology, to name a few. To adequately model these in a C*-algebra setting involves notorious domain problems. Kishimoto and Nakamura [16] showed, for example, that apparently natural domain assumptions on the supersymmetry graded derivations lead to an empty theory. Similarly, supersymmetric KMS–functionals underlying the construction of cyclic cocycles as in [15, 11] cannot exist in the case of infinitely extended systems [3]. These obstructions may explain why a general C∗ -algebraic framework for supersymmetry has not yet emerged. It thus seems worthwhile to explore representative examples in more detail in order to identify the pertinent structures. In the present article we aim to develop tools to define and analyze in a C*-algebra setting a simple but, with regard to the mathematical problems under investigation,

700

D. Buchholz, H. Grundling

generic supersymmetric quantum field theory. It is the model of a chiral Fermi– and Bose–field, defined on the light ray R. As the construction of this model is easily accomplished in the Wightman setting of (unbounded) quantum fields, we can concentrate here on the specific problems arising in the passage to a C∗ –framework. Although the model has formally the structure of a tensor product of a CAR–algebra and a CCR–algebra, the adequate formulation of its C∗ –version requires some care. It turns out that the standard Weyl algebra description of the CCR part is not suitable for the formulation of supersymmetry. We therefore introduce a more viable variant of the CCR–algebra, the resolvent algebra, which formally may be thought of as being generated by the resolvents of the underlying Bose–field. These resolvents act as mollifiers for the super–derivation and allow one to define it on a domain which is weakly dense in the underlying C∗ –algebra in all representations of interest. The resolvents also lead to a mollified version of the fundamental relation of supersymmetry, relating the square of the super–derivation and the generator of time translations. These rather weak variants of supersymmetry turn out to be sufficient for the further analysis. Having clarified the C∗ –algebraic formulation of supersymmetry, one has the necessary tools for the analysis of the supersymmetric KMS–functionals in this model. Again, these functionals are easily constructed in the Wightman setting. Yet, as follows from general arguments [3], they cannot be extended continuously to the full underlying C∗ –algebra. In fact, one does not have any a priori information on their domains of definition. In the present model, the restrictions of the supersymmetric KMS–functionals to any local subalgebra of the underlying C∗ –algebra turn out to be bounded. Thus these functionals are densely defined, but their domain of definition does not contain any non– trivial analytic elements with regard to the dynamics, as is required in the construction of cyclic cocycles given in [15, 11]. Nevertheless, by relying on techniques from the theory of analytic functions of several complex variables, it is possible to define cyclic cocycles in the present model as well. The restrictions of these cocycles to any fixed local subalgebra of the underlying C∗ –algebra turn out to be entire in the sense of Connes [5]. So the present field–theoretic model allows for a satisfactory C∗ –algebraic formulation of supersymmetry and the analysis of its consequences. There are three observations which are of interest going beyond the present model: First, a C∗ –algebraic formulation of supersymmetry has to rely on the concept of mollifiers or, complementary, of unbounded operators affiliated with the underlying C∗ –algebra [9]. Second, there is growing evidence that supersymmetric KMS–functionals, although being unbounded, are locally bounded, in accordance with the heuristic considerations in [3]. And third, although these functionals generically do not have analytic elements in their domain of definition, they can still be used to define local versions of Connes’ entire cyclic cocycles by relying on techniques from complex analysis. Based on these insights, a proper C∗ –algebraic framework for the formulation of supersymmetry and the analysis of its consequences in quantum field theory seems within reach. We hope to return to this problem elsewhere. The plan of our paper is as follows. We will state our results in the body of the paper, and defer almost all the proofs to the Appendix. In Sect. 2 we present in the Wightman framework the basic supersymmetry model which we wish to analyze; in Sect. 3 we prepare for its analysis in a C*–setting by considering algebraic mollifying relations for the quantum fields, which leads to the study of the C*–algebras generated by the resolvents of the fields. In Sect. 4 we use these tools to present the C*–algebraic framework of the model. In Sect. 5 we define (unbounded) graded KMS–functionals on the model

Algebraic Supersymmetry: A Case Study

701

and prove basic properties for them, including their supersymmetry invariance and their local boundedness. In Sect. 6 we use these KMS–functionals to define a Chern character formula (generalizing the construction in [15, 11]), from which we obtain a (locally) entire cyclic cocycle in the sense of Connes. This can then be taken as input to an index theory for supersymmetric quantum field theories, of the type proposed by Longo [18]. 2. The Model We begin by presenting here our model in the Wightman framework, which we would like to model in a C*-algebra setting. It is the simplest example for supersymmetry on noncompact spacetime, in that we have one dimension, one boson and one fermion. We assume chiral fields, so there is only one space-time dimension, R. The Fermi field is given by the Clifford operators c( f ) = c( f )∗ , where f ∈ S(R, R) and c( f ), c(g) = ( f, g) := f g d x. The boson field is j ( f ) = j ( f )∗ , where f ∈ S(R, R) and j ( f ), j (g) = iσ ( f, g) := i f g d x. The Z2 –grading automorphism γ comes from the Fermi field by γ c( f ) = −c( f ) , γ j( f ) = j( f )

and defines even and odd parts of the polynomial field algebra by A± = A ± γ (A) 2. The heuristic supercharge Q := c(x) j (x) d x defines the supersymmetry generator δ as the graded derivation: δ(A) := [Q, A+ ] + {Q, A− }, which satisfies δ(AB) = δ(A)B + γ (A)δ(B). Note that on the generating elements of the field algebra we have: δ c( f ) = j ( f ) , δ j ( f ) = ic( f ). (1) Time evolution is given by translation, i.e. αt (c( f )) := c( f t ),

αt ( j ( f )) := j ( f t ),

where f t (x) := f (x − t), x ∈ R. The generator of time evolution is the derivation: δ0 (c( f )) = ic( f ) ,

δ0 ( j ( f )) = i j ( f ).

(2)

The supersymmetry relation is valid on the field algebra: δ 2 = δ0 .

(3)

Our problem is to realize this structure in a C*-algebra setting. Some problems already arise from the relation δ((c( f )) = j ( f ), in which δ takes a bounded operator to an unbounded one. We will deal with this issue in the next section. A deeper source of problems will come from the theorems of Kishimoto and Nakamura [16] which will make it hard to realize the supersymmetry relation (3) on a dense domain.

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3. On Mollifiers and Resolvent Algebras Here we develop tools to handle the unboundedness of the range elements of δ. Recall that a selfadjoint operator A on a Hilbert space H is affiliated with a C*-algebra A ⊂ B(H) if the resolvent (iλ11 − A)−1 ∈ A for some λ ∈ R\0 (hence for all λ ∈ R\0). This notion is used by Georgescu [9] e.a. (and is weaker than the one used by Woronowicz [25]) and it implies the usual one, i.e. that A commutes with all unitaries commuting with A (but not conversely). Observe that A(iλ11 − A)−1 = (iλ11 − A)−1 A = iλ(iλ11 − A)−1 − 11 ∈ A. Thus the resolvent (iλ11 − A)−1 = M acts as a “mollifier” for A, i.e. M A and AM are bounded and in A, and M is invertible such that M −1 M A = A = AM M −1 . This suggests that as AM and M A in A carries the information of A in bounded form, we can “forget” the original representation, and study the affiliated A abstractly through these elements. We want to apply this idea to a representation of the bosonic fields j ( f ) = j ( f )∗ , f ∈ S(R), where j ( f ), j (g) = iσ ( f, g) := i f g d x on some common dense invariant core D ⊂ H of the selfadjoint fields j ( f ). It seems natural to look for mollifiers in the Weyl algebra

(S, σ ) = C ∗ exp(i j ( f )) f ∈ S(R) ,

(abstractly (S, σ ) is the C*-algebra generated by a set of unitaries δ f f ∈ S(R) such that δ ∗f = δ− f and δ f δg = e−iσ ( f,g)/2 δ f +g ). Unfortunately this is not possible because: Proposition 3.1. The Weyl algebra (S, σ ) contains no nonzero element M such that j ( f )M is bounded for some f ∈ S(R)\0. Thus (S, σ ) contains no mollifier for any nonzero j ( f ), and j ( f ) is not affiliated with (S, σ ). Proof. Assume that M ∈ (S, σ ) is nonzero such that j ( f )M is bounded for some nonzero f ∈ S(R). Let U (t) := exp(it j ( f )), and denote the spectral resolution of j ( f ) by j ( f ) = λ d P(λ), then itλ (U (t) − 11)M = (e − 1)d P(λ)M (eitλ − 1) = |t| d P(λ) λ d P(λ )M tλ ≤ C|t| j ( f )M −→ 0 as t → 0, where we used the bound | e x−1 | < C for some constant C. Let J ⊂ (S, σ ) consist of all elements M such that (U (t) − 11)M → 0 as t → 0. This is clearly a norm-closed linear space, and by the inequality (U (t) − 11)M A ≤ (U (t) − 11)M × A it is also a right ideal. To see that it is a two sided ideal note that (U (t) − 11)ei j (g) M = (U (t)eitσ ( f,g) − 11)M ix


703

still converges to 0 as t → 0, and use the fact that (S, σ ) is the norm closure of the span of ei j (g) g ∈ S(R) . But (S, σ ) is simple, hence J M must be zero.

Our solution is to abandon the Weyl algebra as the appropriate C*-algebra to model the bosonic fields j ( f ), and instead to choose the unital C*-algebra generated by the resolvents:

C ∗ 11, R(λ, f ) λ ∈ R\0, f ∈ S(R)\0 , where R(λ, f ) := (iλ11 − j ( f ))−1 . Then by construction all j ( f ) are affiliated to this C*-algebra and it contains mollifiers R(λ, f ) for all of them. The above discussion took place in a concrete setting, i.e. represented on a Hilbert space, and we would like to abstract this. Just as the Weyl algebra can be abstractly defined by the Weyl relations, we now want to abstractly define the C*-algebra of resolvents (of the j ( f )) by generators and relations. Definition 3.2. Given a symplectic

σ ), we define R0 to be the universal unital space (X, *-algebra generated by the set R(λ, f ) λ ∈ R\0, f ∈ X \0 and the relations R(λ, f )∗ = R(−λ, f ),

1 1 R(λ, f ) = R 1, f , λ λ

(4) (5)

R(λ, f ) − R(µ, f ) = i(µ − λ)R(λ, f )R(µ, f ), R(λ, f ), R(µ, g) = iσ ( f, g) R(λ, f ) R(µ, g)2 R(λ, f ),

(7)

R(λ, f )R(µ, g) = R(λ + µ, f + g)[R(λ, f ) + R(µ, g) +iσ ( f, g)R(λ, f )2 R(µ, g)],

(8)

(6)

where λ, µ ∈ R\0 and f, g ∈ X \0, and for (8) werequire λ+µ

= 0 and f +g = 0.That is, start with the free unital *-algebra generated by R(λ, f ) λ∈R\0, f ∈ X \0 and factor out by the ideal generated by the relations (4) to (8) to obtain the *-algebra R0 . Remark 3.3. (i) The *-algebra R0 is nontrivial, because it has nontrivial representations. For instance, in a Fock representation of the CCRs over (X, σ ) we have the CCR-fields ϕ( f ) from which we can define π(R(λ, f )) = (iλ11 − ϕ( f ))−1 to obtain a representation of R0 . (ii) Obviously (4) encodes the selfadjointness of j ( f ), (5) encodes j (λ f ) = λj ( f ), (6) encodes that R(λ, f ) is a resolvent, (7) encodes the canonical commutation relations and (8) encodes additivity j ( f + g) = j ( f ) + j (g). Moreover, the identity was added explicitly, we do not have that R(1, 0) = −i11 , in fact R(1, 0) is undefined. To define our resolvent C*-algebra, we need to decide on which C*-seminorm to define on R0 . The obvious choice is the enveloping C*-norm, however for the purpose of our model, it is more convenient to use a different norm, which we now define. We will say that a state ω on the Weyl algebra (X, σ ) is strongly regular if the functions Rn (λ1 , . . . , λn ) → ω δλ1 f1 · · · δλn fn are smooth for all f 1 , . . . , f n ∈ X and all n ∈ N. Of special importance is that the GNS-representation of a strongly regular state has a common dense invariant domain

704


for all the generators j ( f ) of the one parameter groups λ → πω (δ λ f ) (this domain is obtained by applying the polynomial algebra of the Weyl operators πω (δ f ) f ∈ X to the cyclic GNS-vector). Some important classes of states, e.g. quasi-free states are strongly regular. Denote by π S the direct sum of the GNS-representations of all strongly regular states, then as the resolvents of the fields are in π S (X, σ ) , we can extend π S to a representation of R0 by the Laplace transform: ∞ e−λt π S (δ−t f ) dt, λ > 0. (9) π S (R(λ, f )) := −i 0

We define our resolvent algebra R(X, σ ) as the abstract C*-algebra generated by π S (R0 ), i.e. we factor R0 by Ker π S and complete w.r.t. the operator norm of π S . We state some elementary properties of R(X, σ ). Theorem 3.4. Let (X, σ ) be a given nondegenerate symplectic space, and define R(X, σ ) as above. Then for all λ, µ ∈ R\0 and f, g ∈ X \0 we have: (i) (ii) (iii)

[R(λ, f ),R(µ, f )] = 0. Substitute µ = −λ to see that R(λ, f ) is normal. R(λ, f ) = |λ|−1 . R(λ, f ) is analytic in λ. Explicitly, the series expansion: R(λ, f ) =

∞ (λ0 − λ)n R(λ0 , f )n+1 i n ,

λ, λ0 = 0

(Von Neumann series)

n=0

converges in norm whenever |λ0 − λ| < |λ0 |. (iv) R(λ, t f ) is norm continuous in t ∈ R\0. (v) R(λ, f )R(µ, g)2 R(λ, f ) = R(µ, g)R(λ, f )2 R(µ, g). (vi) Let T ∈ Sp(X, σ ) be a symplectic transformation. Then α R(λ, f ) := R(λ, T f ) defines a unique automorphism α ∈ Aut R(X, σ ). Note that the von Neumann series for R(λ, f ) converges for any z ∈ C with |z − λ0 | < |λ0 |, i.e. on a disk which stays off the imaginary axis. Using different λ0 s we can thus define R(z, f ) for any complex z not on the imaginary axis and deduce the properties in the definition for these from the series. Thus we obtain also resolvents R(z, f ) for complex z in R(X, σ ). Any operator family Rλ satisfying the resolvent equation (6) is called by Hille a pseudo-resolvent (cf. p. 215 in [26]), and for such a family we know (cf. Theorem 1, p. 216 in [26]) that: • All Rλ have a common range and a common null space. • A pseudo resolvent Rλ is the resolvent for an operator B iff Ker Rλ = {0}, and in this case Dom B = Ran Rλ for all λ. Thus we define: Definition 3.5. A regular representation π ∈ Rep R(X, σ ) is a Hilbert space representation such that Ker π R(1, f ) = {0} ∀ f ∈ S(R, R)\0. We denote the collection of regular representations by Reg.


705

Obviously many regular representations are known, e.g. π S and the Fock representation. Given a π ∈ Rep R(X, σ ) with Ker π R(1, f ) = {0}, we can define a field operator by −1 jπ ( f ) := i11 − π R(1, f ) , with domain Dom jπ ( f ) = Ran π R(1, f ) . Thus for π ∈ Reg, all the field operators jπ ( f ), f ∈ S(R, R) are defined, and we have the resolvents π(R(λ, f )) = (iλ11 − jπ ( f ))−1 . Theorem R(X, σ ) be and let π ∈ Rep R(X, σ ) satisfy 3.6. Let as above, Ker π R(1, f ) = {0} = Ker π R(1, h) for given f, h ∈ X . Then (i) jπ ( f ) is selfadjoint, and π(R(λ, f ))Dom jπ (h) ⊆ Dom jπ (h). (ii) lim iλπ(R(λ, f ))ψ = ψ for all ψ ∈ Hπ . λ→∞

(iii) lim iπ(R(1, s f ))ψ = ψ for all ψ ∈ Hπ . s→0 (iv) The space D := π R(1, f )R(1, h) Hπ is a joint dense domain for jπ ( f ) and jπ (h) and we have: [ jπ ( f ), jπ (h)] = iσ ( f, h) on D. (v) jπ (λ f + h) = λjπ ( f ) + jπ (h) for all λ ∈ R on D. (vi) jπ ( f )π(R(λ, f )) = π(R(λ, f )) jπ ( f ) = iλπ(R(λ, f )) − 11 on Dom jπ ( f ). (vii) jπ ( f ), π(R(λ, h)) = iσ (h, f )π(R(λ, h)2 ) on Dom jπ ( f ). (viii) Denote W ( f ) := exp(i jπ ( f )), then W ( f )W (h) = eiσ ( f,h) W (h)W ( f ), W ( f )π R(λ, h) W ( f )∗ = π R(λ + iσ ( f, h), h) .

Moreover W ( f )D ⊆ D ⊇ W (h)D, hence D := π R(1, f )R(1, h) Hπ is a common core for jπ ( f ) and jπ (h). A distinguished regular representation of R(X, σ ) is of course the defining strongly regular representation π S . By definition R(X, σ ) is faithfully represented in it, and moreover, there is a common dense invariant domain D0 for all the field operators jπ S ( f ), f ∈ X . This domain can be enlarged to a dense invariant domain DT for both the resolvents and the fields simply by applying all polynomials in jπ S ( f ) and π S (R(λ, f )) to D0, which makes sense, because from (i) above all resolvents preserve the joint domain

Dom jπ S ( f ) f ∈ X . Thus we can form the *-algebra of (unbounded) operators

E0 := ∗–alg jπ S ( f ), π S (R(λ, f )) f ∈ X, λ ∈ R\0

on DT . Then E0 contains of course the *-algebra π S (R0 ) generated by resolvents alone, which is dense in R(X, σ ). We will need these *-algebras E0 ⊃ π S (R0 ) below, and will generally not indicate the faithful representation π S w.r.t. which they are defined. Note that for any strongly regular state ω, its cyclic GNS-vector is in the domain of all jπω ( f ), hence ω extends to define a functional on E0 . Thus we give a meaning to all expressions of the form ω j ( f 1 ) · · · j ( f n )R(λ1 , g1 ) · · · R(λk , gk ) := ω , jπω ( f 1 ) · · · jπω ( f n )πω R(λ1 , g1 ) · · · R(λk , gk ) ω

706


as above. A very important class of states on (X, σ ) are the quasifree states, which we will need below. They are given by ω(δ f ) = exp − 21 f | f ω , f ∈ X, where · | · ω is a (possibly semi–definite) scalar product on the complex linear space X + i X satisfying f |gω − g| f ω = iσ ( f, g),

f, g ∈ X.

Any quasifree state is also regular in the strong sense. By a routine computation one can represent the expectation values of products of Weyl operators in a quasifree state in the form ω(δ f1 · · · δ fn ) = exp − f k | fl ω − 21 fl | fl ω . k 0, ω(R(λ1 , f 1 ) · · · R(λn , f n )) ∞ ∞ = (−i)n dt1 . . . dtn e− k tk λk ω(δt1 f1 · · · δtn fn )

0

0

∞

= (−i)

n 0

dt1 . . .

0

∞

dtn exp − tk λk − tk tl f k | fl ω − k

k 0. The relation (10) should be regarded as the definition of quasifree states on the resolvent algebra. In our calculations below, we will frequently need the following differentiablility of quasifree states: Proposition 3.7. Let ω be a quasifree state as above, and let xk ∈ R → f k (xk ) ∈ X , k = 1, . . . n be paths in X for which the functions xk , xl −→ f k (xk )| fl (xl )ω , k, l = 1, . . . n, are smooth. Then ∂ ω R(λ1 , f 1 (x1 )) · · · R(λn , f n (xn )) ∂ xr =−

r −1 ∂2 ∂ f k (xk )| fr (xr ) ω ω R(λ1 , f 1 (x1 )) · · · R(λn , f n (xn )) ∂ xr ∂λr ∂λk k=1

− 21 −

∂2 ∂ fr (xr )| fr (xr ) ω ω R(λ , f (x )) · · · R(λ , f (x )) 1 1 1 n n n ∂ xr ∂λr2 n

k=r +1

∂2 ∂ fr (xr )| f k (xk ) ω ω R(λ1 , f 1 (x1 )) · · · R(λn , f n (xn )) ∂ xr ∂λr ∂λk

and all the partial derivatives involved in this formula exist.


707

4. C*-algebra Formulation of Supersymmetry Here we want to write our model of Sect. 2 in a C*-algebra framework. However, to motivate our choices made below, let us recall a theorem of Kishimoto and Nakamura [16]: Theorem 4.1. Let A be a C*-algebra with Z2 –grading γ , let α : R → Aut A be a pointwise continuous action with generator δ0 having a smooth domain C ∞ (δ0 ) := ∞ Dom (δ0n ). Let δ be a closable graded derivation with Dom (δ) ⊃ C ∞ (δ0 ), δ ◦ αt = n=1

αt ◦ δ for all t, and δ 2 = δ0 on C ∞ (δ0 ). Then δ is bounded.

Thus it will be hard to obtain the supersymmetry relation on natural dense domains. For the fermion field, let H = L 2 (R) and define CAR(H) in Araki’s self-dual form (cf. [1]) as follows. On K := H ⊕ H define an antiunitary involution by

(h 1 ⊕h 2 ) := h 2 ⊕ h 1. Then CAR(H) is the unique simple C*–algebra with generators (k) k ∈ K such that k → (k) is antilinear, (k)∗ = ( k), and (k1 ), (k2 )∗ = (k1 , k2 )11,

ki ∈ K.

The correspondence with the heuristic creators and annihilators of fermions is given by (h 1 ⊕ h 2 ) = a(h 1 ) + a ∗ (h 2 ), where a(h) = a(x) h(x) d x, a ∗ (h) = a ∗ (x) h(x) d x. To obtain the Clifford operators c( f ) = c( f )∗ , f ∈ S(R, R) we take c( f ) := ( f ⊕ √ f )/ 2, in which case we have c( f ) = c( f )∗ and {c( f ), c(g)} = ( f, g) = f g d√x. c( f ) := i( f ⊕ − f )/ 2 Let Cliff S(R) := C ∗ c( f ) f ∈ S(R) and notice that also satisfies the Clifford relations, hence generates another copy of Cliff S(R) in CAR(H), and together these two Clifford algebras generate all of CAR(H). In fact, since the c( f ) and c(g) anticommute, we have that CAR(H) ∼ = Cliff S(R) ⊕ S(R) . Conversely, if we are given a real pre-Hilbert space X with complexified completion Y and a projection P and antiunitary involution such that P = 11 − P and these ∼ preserve X , then we have an isomorphism Cliff(X √ ) = CAR(Y ) given by (x) = c(P x) − ic( P x) + c(P x) + ic((11 − P)x) / 2. the bosonic part we take the resolvent algebra R(S(R), σ ), where σ ( f, g) := For f g d x, and so the full C*-algebra in which we want to define our model is A := Cliff S(R) ⊗ R(S(R), σ ), where the tensor norm is unique because the CAR-algebra is nuclear. The grading automorphism γ is the identity on R(S(R), σ ), and γ ((k)) = −(k) for all k on the CAR-part. Next, we want to define on some suitable domain in A the supersymmetry graded derivation δ corresponding to the relations (1). First, considering δ j ( f ) = ic( f ), since δ is a derivation on the bosonic part, it is natural to define δ(R(λ, f )) := ic( f ) R(λ, f )2 ∈ A.

(11)

708


However due to the unbounded rhs of δ c( f ) = j ( f ) we cannot define δ directly on the c( f ), so we need to multiply by mollifiers. Define ζ ( f ) := c( f )R(1, f ) , then

δ(ζ ( f )) := i R(1, f ) − 11 + ic( f )c( f )R(1, f )2 ∈ A,

(12) (13)

where we made use of the graded derivation property, the relations (1) and j ( f )R(λ, f ) = iλR(λ, f ) − 11. Next, we would like to extend δ as a graded derivation to the *-algebra generated by these basic objects:

D S := *–alg 11, R(λ, f ), ζ ( f ) λ ∈ R\0, f ∈ S(R)\0 ⊂ A. Observe that D S is not norm-dense in A, however due to Theorem 3.6(ii) applied to R(λ, f )c( f ) = ζ ( f /λ), it will be strong operator dense in A in any regular representation. Note that δ does not preserve D S , it takes its image in the norm dense *-algebra

A0 := *–alg 11, R(λ, f ), c( f ) λ ∈ R\0, f ∈ S(R)\0 ⊂ A. To see that δ extends as a graded derivation to D S , we proceed as follows. Let π0 be any representation of Cliff S(R) , then π0 ⊗ π S is a faithful representation of A and there is a common dense invariant domain D := Hπ0 ⊗ DT for all π0 (c( f )) ⊗ 11, 11 ⊗ jπ S ( f ) and 11 ⊗ π S (R(λ, f )), where DT denotes the domain of E0 defined at the end of Sect. 3. (Henceforth we will not indicate tensoring by 11 nor the representations π0 , π S when the context makes clear what is meant.) Let

E := ∗–algebra c( f ), j ( f ), R(λ, f ) f ∈ S(R), λ ∈ R\0 ⊃ A0 so we have the *-algebras R0 ⊂ E0 ⊂ E on D. Define on the generating elements of E a map δ, setting δ( j ( f )) = ic( f ), δ(R(λ, f )) = ic( f )R(λ, f )2 , δ(c( f )) = j ( f ). We will see that this map extends to a graded derivation on E. For the proof it suffices to show that δ is linear and satisfies the graded Leibniz rule on any finite polynomial involving operators j ( f ), R(λ, f ) and c( f ), i.e. in each instance only a finite number of test functions f and real parameters λ are involved. We will take advantage of this fact as follows. Let X s ⊂ S(R) be any finite–dimensional subspace and consider the subalgebra E(X s ) ⊂ E generated by the elements j ( f ), R(λ, f ) and c( f ) with λ ∈ R\0, f ∈ X s . We extend X s to a space X s ⊂ S(R) by adding to the elements of X s also their first derivatives. Picking in X s some (finite) orthonormal basis {h n } with regard to the scalar product (·, ·), we have for any f ∈ X the “completeness relations” s n (h n , f ) h n = f, n (h n , f ) h n = f . Next, we define an operator Q s ∈ E, setting c(h n ) j (h n ). Qs = n

As Q s is of fermionic (odd) type, we can consistently define with the help of it a graded derivation δ s on E, setting for even and odd elements E ± ∈ E, respectively, δ s (E + ) = [Q s , E + ] = Q s E + − E + Q s , δ s (E − ) = {Q s , E − } = Q s E − + E − Q s .


709

Computing the action of δ s on the even elements j ( f ), R(λ, f ) and odd elements c( f ), where λ ∈ R\0, f ∈ X s , we obtain from the basic relations in E by some elementary algebraic manipulations: δ s ( j ( f )) = i c(h n ) (h n , f ) = ic( f ) , n δ s (R(λ, f )) = i c(h n ) (h n , f ) R(λ, f )2 = ic( f )R(λ, f )2 , n δ s (c( f )) = (h n , f ) j (h n ) = j ( f ). n

Thus we conclude that the action of δ on the generating elements of E(X s ) coincides with the action of the graded derivation δ s . As the choice of the subspace X s was arbitrary, it follows that δ extends to a graded derivation on the whole polynomial algebra E. The final step consists in showing that the action of δ on the generating elements R(λ, f ), ζ ( f ) of D S coincides with the action of the graded derivation δ. But this follows immediately from the relations given above. Thus δ extends to a graded derivation with domain D S , and range in A0 . Uniqueness is clear from the graded derivation property, so we have proven: Theorem 4.2. There is a unique graded derivation δ : D S → A satisfying relations (11) and (13). Next we need to define the time evolution derivation δ0 in this C*-setting. From Eqs. (2) this suggests that we define on E a *-derivation δ 0 satisfying: δ 0 ( j ( f )) = i j ( f ), δ 0 (R(λ, f )) = i R(λ, f ) j ( f )R(λ, f ), δ 0 (c( f )) = i c( f ), and then proceed to the corresponding mollified relations in A. For the proof that δ 0 extends to a *-derivation on E, we proceed as in the discussion of the superderivation: We pick any finite dimensional subspace X s ⊂ S(R), consider the corresponding subalgebra E(X s ) ⊂ E and choose in the extended space X s ⊂ S(R), containing the elements of X s and their first derivatives, some orthonormal basis {h n }. In addition to the completeness relations mentioned above we will also make use of n (h n , f ) h n = f for f ∈ Xs . We consider now the symmetric operator in E 1 Hs = ic(h n )c(h n ) + j (h n ) j (h n ) . n 2 Putting δ 0 s ( · ) = [Hs , · ], it induces a *–derivation on E. Its action on the generating elements of E(X s ) can easily be computed: δ 0 s ( j ( f )) = j (h n ) i(h n , f ) = i j ( f ), n δ 0 s (R(λ, f )) = i(h n , f ) R(λ, f ) j (h n )R(λ, f ) = i R(λ, f ) j ( f )R(λ, f ), n

1 δ 0 s (c( f )) = i (h n , f ) c(h n ) + (h n , f ) c(h n ) = i c( f ). n 2

710


Thus we conclude as in the preceding discussion that the action of δ 0 on the generating elements of E(X s ) coincides with the action of the derivation δ 0 s . As X s was arbitrary, it follows that δ 0 extends to a derivation on the whole polynomial algebra E. 2 The supersymmetry relation δ = δ 0 can now be verified on the generating elements of E and thus holds on the whole algebra E. The question now is how one should define the time evolution δ0 and the square δ 2 on the C*-algebra A from the unbounded versions in E. Since δ : D S → A0 , its square δ 2 does not make sense on D S . Note however that for every A ∈ A0 there is a monomial M ∈ D S of resolvents R(λ, f ) such that AM ∈ D S M A. (By Theorem 3.6(ii) we know that in regular representations we can let these mollifiers M go to 11 in the strong operator topology.) Definition 4.3. For each A ∈ D S let M A ∈ D S be a monomial of resolvents R(λ, f ) such that M A δ(A) ∈ D S . Define M A δ 2 (A) := δ M A δ(A) − δ(M A )δ(A) ∈ A0 . 2

Note that this definition coincides with M A δ (A) in E, however the definition above involves only bounded quantities, so it can be defined independently in the C*-setting on D S . Of course we then have the mollified SUSY–relations M A δ 2 (A) = M A δ 0 (A) for all A ∈ D S from the unbounded SUSY relation in E. This is not however acceptable for a bounded SUSY–relation until we have demonstrated the connection of M A δ 0 (A) with the time evolution. The time evolution α : R → Aut A is just translation, as this is a chiral theory αt (c( f )) := c( f t ) , αt (R(λ, f )) = R(λ, f t ). The desired connection

d

M A αt (A)

(14) 0 dt exists only in specific regular representations on suitable domains, and for these one will then have supersymmetry. In many applications, one only needs the supersymmetry weakly, i.e.

d

ω(B M A δ 0 (A)C) = −i ω(B M A αt (A)C) = ω(B M A δ 2 (A)C) 0 dt for A, B, C in a suitable domain and ω a distinguished functional. We will verify this relation explicitly below for the functionals used in our constructions. M A δ 0 (A) = −i

5. Graded KMS–Functionals Graded KMS–functionals are used in supersymmetric theories to calculate cyclic cocycles [11, 15], and here we want to develop this theory in the current context for our simple supersymmetric model as a first application of it. Definition 5.1. Let A be a unital C*-algebra with a grading automorphism γ ∈ Aut A, γ 2 = ι, and a (pointwise continuous) action α : R → Aut A such that αt ◦ γ = γ ◦ αt for all t. Then a graded KMS–functional is a (possibly unbounded) functional ϕ on A such that


711

(i) Dom ϕ is a unital dense *–subalgebra of A such that γ (Dom ϕ) ⊆ Dom ϕ ⊇ αt (Dom ϕ) ∀ t. (ii) For all A, B ∈ Dom ϕ there is a continuous complex function FA,B : S → C on the strip S := R + i[0, 1] which is analytic on the interior of S and satisfying on the boundary: FA,B (t) = ϕ (A αt (B)) ∀ t, FA,B (t + i) = ϕ (αt (B)γ (A)) ∀ t ∈ R. (iii) For A, B ∈ Dom ϕ we have |FA,B (t + is)| < C(1 + |t|) N ∀ t ∈ R, s ∈ (0, 1) and some C ∈ R+ and N ∈ N depending on A and B. Example 5.2. Below for our model, we will define on A = Cliff S(R) ⊗ R(S(R), σ ) a functional ϕ = ψ ⊗ω, with Dom ϕ = A0 , where ψ and ω are quasi-free with two-point functions

2 p

ω( j 2 ( f )) = f ( p) dp, − p 1−e p p ˆ p) dp, f ( p) g( ψ(c( f )c(g)) = lim+ − p 2 ε→0 1−e p + ε2 and we will verify that ϕ = ψ ⊗ ω is a graded KMS–functional. Note that ω is a state on R(S(R), σ ), but ψ is unbounded and nonpositive. It does however satisfy supersymmetry, in that ϕ ◦ δ = 0, and Eq. (14) holds weakly. The motivation for using graded KMS–functionals comes from several sources: • Physicists used graded KMS–functionals to construct supersymmetric field theories in a thermal background [8, 10]. • Jaffe e.a. [11] and Kastler [15] used graded KMS–functionals to construct cyclic cocycles in Connes’ cyclic cohomology. The reasons why one has to use nonpositive unbounded KMS–functionals for field theories on noncompact spacetime are as follows. First, there is the theorem of Buchholz and Ojima [4] that supersymmetry breaks down in spatially homogeneous KMS–states, and second there is the theorem of Buchholz and Longo [3] that if ϕ is a bounded graded KMS–functional of A with time evolution α : R → Aut A, and if there are βn ∈ Aut A such that lim ϕ (C[A, βn (B)]) = 0 ∀ A, B, C ∈ A

n→∞

then α = ι. On noncompact spaces, the translations will produce the βn in a local field theory. Thus in local field theories on noncompact spaces, we are inevitably led to unbounded graded KMS–functionals for supersymmetry. First, we would like to establish a few general properties of graded KMS–functionals.

712


Proposition 5.3. Given a graded KMS-functional ϕ defined w.r.t. the data (A, γ , α), then (i) ϕ is α–invariant. (ii) ϕ is γ –invariant. Moreover if a functional ϕ satisfies the graded KMS–property on a subset Y ⊂ Dom ϕ ⊂ A, then it also satisfies the graded KMS–property on Span Y . Proposition 5.4. Let A = C ⊗ B where C and B are unital C*-algebras with C nuclear. Let σ : R → Aut C and β : R → Aut B be dynamical systems, and let γ ∈ Aut C be a grading automorphism, γ 2 = ι. Let ω ∈ S(B) be a KMS–state on B w.r.t. β, and let ψ be a graded KMS–functional on C w.r.t. σ . Define a functional ϕ := ψ ⊗ ω with

Dom ϕ := Span C ⊗ B C ∈ Dom ψ, B ∈ B by ϕ(C ⊗ B) := ψ(C)ω(B). Then ϕ is a graded KMS–functional w.r.t. the grading γ ⊗ ι, and the C*-dynamical system σ ⊗ β : R → Aut (C ⊗ B). Thus for our model, as A = Cliff S(R) ⊗ R(S(R), σ ), it suffices to define a graded KMS–functional ψ on Cliff S(R) and a KMS–state ω on R(S(R), σ ) from which we can then construct the graded KMS–functional ϕ := ψ ⊗ ω. We start by defining the KMS–state ω on R(S(R), σ ). Theorem 5.5. (i) There is a quasi–free state on (S, σ ) defined by ω(δ f ) := exp[−s( f, f )/2], f ∈ S(R, R), where s( f, g) :=

p g ( p) dp = f |gω . f ( p) 1 − e− p

(15)

(ii) This quasi–free state ω on (S, σ ) extends to a KMS–functional on πω ((S, σ )) , hence on R(S(R), σ ), where it is defined by Eq. (10). The time evolution used for the KMS–condition is translation of test functions f . Next, we would

like to define a graded KMS–functional ψ on Cliff S(R) with Dom ψ = *-alg c( f ) f ∈ S(R) . By the last part of Proposition 5.3, it suffices to define ψ and check its KMS-properties on the monomials c( f 1 ) · · · c( f n ). Recall that a quasi–free functional on the Clifford algebra is uniquely defined by its two point functional and the relations: ψ(c( f 1 ) · · · c( f 2k+1 )) = 0,

(16)

ψ(c( f 1 ) · · · c( f 2k )) = (−1)

k 2

k (−1) P ψ c( f P( j) ) c( f P(k+ j) ) , P

(17)

j=1

where k ∈ N and P is any permutation of {1, 2, . . . , 2k} such that P(1) < · · · < P(k) and P( j) < P(k + j) for j = 1, . . . , k, (cf. p. 89 in [21]). Using this formula, we define


713

a quasi–free functional ψ with two point function p p θ ( f, g) := ψ(c( f )c(g)) = lim+ ˆ p) dp f ( p) g( − p 2 ε→0 1−e p + ε2 −ε ∞ 1 = lim+ + ˆ p) dp f ( p) g( ε→0 1 − e− p −∞ ε −ε ∞ 1 p = lim+ + ˆ p) dp f ( p) g( ε→0 p 1 − e− p −∞ ε 1 (G), (18) =P p where P 1p denotes the distribution consisting of the Cauchy Principal Part integral of ˆ p) which is differentiable everywhere and of fast decay 1/ p, and G( p) := 1−ep− p f ( p) g( since 1−ep− p is differentiable and of linear growth, and f, g are real-valued Schwartz functions. Thus θ ( f, g) is well-defined for all f, g ∈ S(R). We mention as an aside that the quasifreeness of a graded KMS–functional ψ on the Clifford algebra and the formula for its two–point function are a consequence of the graded KMS–condition, as can be shown by similar arguments as in [23]. This quadratic form θ is unbounded and not positive definite, because (1 − e− p )−1 is unbounded and not positive. θ has the following useful properties. Theorem 5.6. Let f, g ∈ S(R), then (i) θ ( f, g) = (g, (P + T ) f ), where P = 2π × projection onto positive spectrum of D := id/d x, and T is an unbounded operator given explicitly by ∞ (g, T f ) = 2i d x dy f (x) g(y) (x − y) dp ln(1 − e− p ) cos( p(x − y)). 0

Moreover PJ T PJ is trace–class and selfadjoint for all compact intervals J ⊂ R, where PJ is the projection onto L 2 (J ) ⊂ L 2 (R). (ii) For z ∈ S = R + i[0, 1] define G(z) := θ ( f, gz ) := lim+ ε→0

−∞

∞

−ε

+

ε

ei pz ˆ p) dp. f ( p) g( 1 − e− p

Then G is continuous on S, analytic on its interior, and satisfies

G(t + is) ≤ A + B|t| for t ∈ R, s ∈ [0, 1] and constants A, B. Using these properties, one can now establish that: Theorem 5.7. The quasi–free functional

ψ with two point functional θ and domain Dom ψ = *-alg c( f ) f ∈ S(R) , is a graded KMS–functional on Cliff S(R) where time evolution is given by translation.

714


Thus by Proposition 5.4 we have a KMS–functional ϕ = ψ ⊗ ω on A with domain

Dom ϕ = Span C ⊗ R C ∈ Dom ψ, R ∈ R(S(R), σ ) . What makes this KMS–functional interesting, is that it satisfies supersymmetry, i.e. Theorem 5.8. For the quasifree functional ϕ above, we have that (i) A0 ⊂ Dom ϕ, (ii) ϕ(δ(A)) = 0 for all A ∈ D S ,

d (iii) ϕ B M A δ 0 (A)C = −i dt ϕ B M A αt (A)C = ϕ B M A δ 2 (A)C for all A ∈ 0 D S and B, C ∈ A0 , where M A is a monomial of resolvents R(λ, f ) such that M A δ(A) ∈ D S (as in Definition 4.3). Whilst the functional ϕ is unbounded, it is locally bounded in the sense of the theorem below. For the local algebras, let J ⊂ R be a bounded interval and define

A(J ) := C∗ c( f ), R(λ, f ) supp f ⊆ J, f ∈ S(R), λ ∈ R\0 ,

A0 (J ) := *-alg c( f ), R(λ, f ) supp f ⊆ J, f ∈ S(R), λ ∈ R\0 , hence A0 (J ) is a dense *-algebra of A(J ), and A0 (J ) ⊂ Dom ϕ. Then: Theorem 5.9. For the quasifree functional ϕ above, and a bounded interval J ⊂ R we have that ϕ A0 (J ) ≤ exp(K |J |2 ), where K is a constant (independent of J ), and |J | is the length of J . Thus ϕ is bounded on all the local algebras A0 (J ). 6. The JLO–Cocycle From an assumed supersymmetry structure on a C*-algebra and a KMS-functional, Jaffe, Lesniewski and Osterwalder [11] and Kastler [15] constructed with a Chern character formula an entire cyclic cocycle in the sense of Connes [5]. Their assumed supersymmetry assumptions are too restrictive to include quantum field theories on noncompact spacetimes. Here we want to show that we can adapt the JLO cocycle formula to produce a well-defined (locally) entire cyclic cocycle for our model, using the KMS-functional in the preceding section. We first need to make sense of the Chern character formula: τn (a0 , . . . , an ) := i n ϕ a0 αis1 δγ (a1 ) αis2 δ(a2 ) αis3 δγ (a3 ) · · · σn

· · · αisn δγ n (an ) ds1 · · · dsn ,

ai ∈ D S ,

(19)

where n := n mod 2, ϕ is the graded KMS–functional above w.r.t. the data γ , α, δ above, and

σn := s ∈ Rn 0 ≤ s1 ≤ s2 ≤ · · · ≤ sn ≤ 1 .


715

Since only α : R → Aut A is given, the expressions αis , s ∈ R in the formula are undefined, and need to be interpreted. Let b0 , . . . , bn ∈ A0 ⊂ Dom ϕ, then using the KMS–property of ϕ, the function Q(t1 , . . . , tn ) := ϕ b0 αt1 b1 αt2 (b2 · · · αtn (bn ) · · · = ϕ b0 αt1 (b1 ) · · · αt1 +···+tn (bn )

can be analytically extended in each variable t j into the strip z j 0 ≤ Im z j ≤ 1 keeping the other variables real. This produces n functions Q j : T j → C, where

T j := z ∈ Cn z j ∈ R + i[0, 1], z k ∈ R for k = j such that Q i Rn = Q j Rn for all i, j. The sets T j are flat tubes, i.e. of the form TB = Rn + i B, where the basis B ⊂ Rn is of dimension less than n. To continue, we now need the Flat Tube Theorem [2]: Theorem 6.1. Let TB1 , TB2 ⊂ Cn be two flat tubes whose bases B1 , B2 are convex, with closures which contain 0 and are star-shaped w.r.t. 0. Let F1 , F2 be any two functions analytic in TB1 , TB2 respectively, with continuous boundary values on Rn and such that F1 Rn = F2 Rn . Then there is a unique function F extending F1 and F2 analytically into the tube TB (where B hull of B1 ∪ B2 ) and with continuous 1 ∪ B2 is the convex 1 ∪B2 n boundary values on R . We have that TB = TBλ where Bλ := (1−λ)B1 +λB2 . 1 ∪B2 0≤λ≤1

Using this inductively, we can extend Q by analytic continuation into the tube Tn := Rn + in where

n is the convex hull of the unit intervals on the axes, i.e. the simplex n := s ∈ Rn 0 ≤ si ∀ i, s1 + · · · + sn ≤ 1 . So we will interpret ϕ b0 αz 1 (b1 ) · · · αz 1 +···+z n (bn ) := Q(z 1 , . . . , z n ) for z ∈ Tn to be this unique analytic continuation. The change of variables w1 := z 1 , w2 := z 1 + z 2 , . . . , wn := z 1 + · · · + z n defines an invertible complex linear map W : Rn + in → Rn + iσn , so both W and W −1 are analytic, and hence Q ◦ W −1 )(w1 , . . . , wn ) =: ϕ b0 αw1 (b1 ) · · · αwn (bn ) is analytic on Rn + iσn . In particular ϕ b0 αir1 (b1 ) · · · αir1 +···+irn (bn ) dr1 · · · drn n = ϕ b0 αis1 (b1 ) · · · αisn (bn ) ds1 · · · dsn σn

by the change of variables s1 = r1 , s2 = r1 + r2 , . . . , sn = r1 + · · · + rn . By the substitutions a0 = b0 , b1 = δγ (a1 ), . . . , bn = δγ n (an ) into this formula, we arrive at a consistent interpretation of the Chern character formula (19). Let us recall from [11, 12, 5] the definition of an entire cyclic cocycle. Definition 6.2. Equip D S with the Sobolev norm a∗ = a + δa, and for any *–algebra D ⊆ D S let C n (D) denote the space of (n + 1)–linear functionals on D which are continuous w.r.t. the norm · ∗ , and let · ∗ denote also the norm on C n (D) w.r.t. the norm · ∗ on D. Define the space of cochains C(D) to be the space of sequences ρ = (ρ0 , ρ1 , . . .), where ρn ∈ C n (D) which satisfy the entire analyticity condition: 1/n

lim n 1/2 ρn ∗

n→∞

= 0.

(20)

716


The entire cyclic cohomology is defined by a coboundary operator ∂ = b + B on C(D) for operators b : C n (D) → C n+1 (D), i.e.

B : C n+1 (D) → C n (D),

(∂ρ)n (a0 , . . . , an ) = (bρn−1 )(a0 , . . . , an ) + (Bρn+1 )(a0 , . . . , an ),

(21)

where b and B are given by: (bρn )(a0 , . . . , an+1 ) =

n (−1) j ρn (a0 , . . . , a j a j+1 , . . . , an+1 ) j=0

an+1 · a0 , a1 , . . . , an ), +(−1)n+1 ρn ( (Bρn )(a0 , . . . , an−1 ) = ρn (11, a0 , . . . , an−1 ) + (−1)

n−1

(22)

ρn (a0 , . . . , an−1 , 11)

n−1 + (−1)(n−1) j ρn 11, γ (an− j ), . . . , γ (an−1 ), a0 , . . . , an− j−1 j=1

+ (−1)n−1 ρn γ (an− j ), . . . , γ (an−1 ), a0 , . . . , an− j−1 , 11 ,

where

(23)

γ (a) if ρn ∈ C+n (D) a := n (D) , a if ρn ∈ C−

n (D) the odd part. The entire cyclic where C+n (D) denotes the even part under γ , and C− cocycles are those ρ ∈ C(D) for which ∂ρ = 0, i.e.

(bρn−1 )(a0 , . . . , an ) = −(Bρn+1 )(a0 , . . . , an ) ,

n = 1, 2, . . . .

(24)

Below we will use the even part of τ to define an entire cyclic cocycle. Theorem 6.3. For τn defined in Equation (19) we have that A Bn e a0 ∗ · · · an ∗ n!

for all ai ∈ Dk := *–alg 11, R(1, f ), ζ ( f ) supp( f ) ⊆ [−k, k] ⊂ D S and where a∗ := a + δa as above, and for some constants A and B which depend on k > 0 but are independent of n. Thus condition (20) holds for τ , i.e. |τn (a0 , . . . , an )| ≤

1/n

lim n 1/2 τn ∗

n→∞

= 0.

Using this, we can now prove that: Theorem 6.4. The sequence τ := (τ0 , 0, −τ2 , 0, τ4 , 0, −τ6 , . . .) ∈ C(Dk ) defines an entire cyclic cocycle for each k > 0, i.e. (bτn−1 )(a0 , . . . , an ) = (Bτn+1 )(a0 , . . . , an ) , and the entire analyticity condition holds.

n = 1, 3, 5, . . . ,


717

It is possible to have taken the choice ∂ = b−B for the cyclic coboundary operator above; this is in fact done in [13], and would have led to the cyclic cocycle (τ0 , 0, τ2 , 0, τ4 , 0, . . .) instead of τ above. Note that whilst we have obtained entire cyclic cocycles on each compact set [−k, k] these do not define an entire cyclic cocycle on Dcomp := *–alg 11, R(1, f ), ζ ( f )| supp( f ) is compact ⊂ D S because one can choose a sequence {a0 , a1 , . . .} with a j ∈ Dk j , where k j grows sufficiently fast so that through the dependencies of the constants A and B in Theorem 6.3 on k j the entire analytic condition fails. One expects to use an inductive limit argument, to define an index on Dcomp from the indices on the Dk . From this cyclic cocycle we can calculate an index for this quantum field theory, but its physical significance is presently unclear, though one would expect it to remain stable under deformations. This type of index is discussed in more detail in Longo [18]. 7. Conclusions In this paper we have explored how supersymmetric quantum fields can be treated in a C*-algebra setting, avoiding the obstructions found by Kishimoto and Nakamura [16] and by Buchholz and Longo [3]. We did this in detail for a simple one–dimensional model. In order to establish a reasonable domain of definition for the super–derivation, we found it necessary to analyze a notion of “mollifiers” for the quantum fields and to introduce a corresponding C*-algebra, the resolvent algebra. The full algebra A defining the model can then be taken as the tensor product of this resolvent algebra and the familiar CAR–algebra. The super–derivation is defined on a subalgebra which is weakly dense in A in all representations of physical interest; alternatively, one can define it on a norm dense subalgebra of A with range in a *–algebra E of bounded and unbounded operators which are affiliated with A in the sense of [9]. Similarly, the basic supersymmetry relation can either be formulated in a mollified form on some weakly dense domain or, alternatively, as a relation between maps which have been extended to the *–algebra E. These findings reveal some basic features of supersymmetric quantum field theories which have to be taken into account in a general C*-framework covering such theories. The tools developed here should also be useful in other areas of quantum field theory where one needs to use graded derivations, e.g. in BRS–constraint theory. We also exhibited in the present model graded KMS–functionals for arbitrary positive temperatures which are supersymmetric. In accordance with the general results in [3], these functionals are unbounded. Yet their restrictions to any local subalgebra of the underlying C*-algebra A are bounded. It is an interesting question whether these functionals are also locally normal with respect to the vacuum representation of the theory, as one would heuristically expect. The KMS–functionals were then employed to define cyclic cocycles. In view of the fact that the domain of definition of these functionals does not contain analytic elements with regard to the time evolution, the strategy outlined in [15, 11] could not be applied here. That these cocycles can be constructed, nevertheless, is due to the fact that the functionals inherit sufficiently strong analyticity properties from the KMS–condition which allow one to perform the necessary complex integrations. Moreover, the resulting cocycles are entire on all local algebras. These functionals may thus be taken as an input for a quantum index theory as suggested by Longo [18]. Such an index should be stable under deformations, and one

718


can easily think of possible deformations of our model, e.g. deform the supersymmetry generator Q by an appropriate function M: Q M := M(x) j (x) c(x) d x so

δ M (c( f )) = j (M f ), thus:

δ M ( j ( f )) = c(M f ),

δ 2M =: δ0M

defines the new generator for time evolution. 8. Appendix In this appendix we give proofs of the statements in the main body of the text. Proof of Theorem 3.4. (i) By (6) we have that i(µ − λ)R(λ, f )R(µ, f ) = R(λ, f ) − R(µ, f ) = − R(µ, f )− R(λ, f ) = i(µ−λ)R(µ, f )R(λ, f ), i.e. [R(λ, f ), R(µ, f )] = 0. (ii) The Fock representation is a subrepresentation of π S , and the resolvents of the fields ϕ( f ) give the Fock representation induced on R(X, σ ), i.e. π(R(λ, f )) = (iλ − ϕ( f ))−1 . Since this is nonzero, R(λ, f ) = 0 for all nonzero f and λ. Now by R(λ, f )∗ = R(−λ, f ) we get 2|λ|R(λ, f )2 = 2λR(λ, f )R(λ, f )∗ = R(λ, f ) − R(λ, f )∗ ≤ 2R(λ, f ). Thus by R(λ, f ) = 0 we find R(λ, f ) ≤ 1/|λ|. Now R(λ, f ) ≥ π(R(λ, f )) = (iλ − ϕ( f ))−1 =

sup

t∈σ (ϕ( f ))

1 1

,

= iλ − t |λ|

using the fact that the spectrum σ (ϕ( f )) = R. Thus R(λ, f ) = 1/|λ|. (iii) Rearrange Eq. (6) to get: R(λ, f ) 11 − i(λ0 − λ)R(λ0 , f ) = R(λ0 , f ).

Now by (ii) above, if λ0 − λ < λ0 , then i(λ0 − λ)R(λ0 , f ) < 1, and hence −1 11 − i(λ0 − λ)R(λ0 , f ) exists, and is given by a norm convergent power series in i(λ0 − λ)R(λ0 , f ). That is, we have that ∞ −1 R(λ, f ) = R(λ0 , f ) 11 − i(λ0 − λ)R(λ0 , f ) = (λ0 − λ)n R(λ0 , f )n+1 i n n=0

when λ0 − λ < λ0 , as claimed. (iv) From Eq. (5) we get R(λ, t f ) =

1 t λ R(1, λ

f ) = 1t R( λt , f ) so

R(λ, s f ) − R(λ, t f ) = 1s R( λs , f ) − 1t R( λt , f ) = 1s R( λs , f ) − R( λt , f ) + 1s − 1t R( λt , f ) 1 1 λ 1 1 λ λ = iλ s t − s R( s , f )R( t , f ) + s − t R( t , f ).


719

Thus R(λ, s f ) − R(λ, t f ) ≤ 1s − 1t 2 λt from which continuity away from zero is clear. (v) This follows directly from Eq. (7) by interchanging λ and f with µ and g resp. (vi) Recall that we have a faithful (strongly regular) representation π S of R(X, σ ) which is an extension of a regular representation of the Weyl algebra (X, σ ) such

that π S (X, σ ) ⊃ π S (R(X, σ )) and such that each π S (R(λ, f )) is the resolvent of the generator jπ S ( f ) of the one-parameter group t → π S (δt f ). Let T ∈ Sp(X, σ ), then it defines an automorphism of (X, σ ) by αT (δ f ) = δT f which preserves the set of strongly regular states, and in fact defines a bijection on the set of strongly regular states by ω → ω ◦ αT . Now π S is the direct sum of the GNS–representations of all the strongly regular states, and hence π S ◦ αT is just π S , where its direct summands have been permuted. Such a permutation of direct summands can be done by conjugation of a unitary, thus π S is unitarily equivalent to π S ◦ αT , and so we can extend αT by unitary conju

gation to π S (X, σ ) . By Eq. (9) we get that αT (π S (R(λ, f ))) = π S (R(λ, T f )), and hence αT preserves R(X, σ ), so defines an automorphism on it. Proof of Theorem 3.6. (i) Observe that by Theorem 1, p. 216 of Yosida [26], we deduce from Ker π(R(1, f )) = {0} that π(R(λ, f )) is the resolvent of jπ ( f ), i.e. we have now for all λ = 0 that jπ ( f ) = iλ11 − π(R(λ, f )))−1 . Then jπ (t f ) = i11 − π(R(1, t f ))−1 = i11 − tπ(R( 1t , f ))−1 −1 = t jπ ( f ). = t i 1t 11 − π R( 1t , f ) Thus

∗ ∗ jπ ( f )∗ = i11 − π(R(1, f ))−1 ⊇ −i11 − π(R(1, f ))−1 = −i11 − π(R(1, f )∗ )−1 = −i11 − π(R(−1, f ))−1 = −i11 + π(R(1, − f ))−1 = − jπ (− f ) = jπ ( f ),

and hence jπ ( f ) is symmetric. To see that it is selfadjoint note that: −1 Ran ( jπ ( f ) ± i11) = Ran −π R(±1, f ) = Dom π R(±1, f ) = Hπ , hence the deficiency spaces (Ran ( jπ ( f ) ± i11))⊥ = {0} and so jπ ( f ) is selfadjoint. For the domain claim, recall that Dom jπ ( f ) = Ran π R(1, f ) . So π(R(λ, f ))Dom jπ (h) = π(R(λ, f ))π R(1, h) Hπ = π R(1, h)R(λ, f ) + iσ ( f, h)R(1, h)R(λ, f )2 R(1, h) Hπ

⊆ π (R(1, h)) Hπ = Dom jπ (h).

(ii) Let jπ ( f ) = λd P(λ) be the spectral resolution of jπ ( f ). Then π(R(µ, f )) = 1 iµ−λ d P(λ), hence iµ d P(λ)ψ ∀ ψ ∈ Hπ . iµπ(R(µ, f ))ψ = iµ − λ

720


iµ

Since iµ−λ

< 1 (for µ ∈ R\0) the integrand is dominated by 1 which is an L 1 –function iµ µ→∞ iµ−λ

with respect to d P(λ), and as we have pointwise that lim

= 1, we can apply the

dominated convergence theorem to get that lim iµπ(R(µ, f ))ψ =

µ→∞

d P(λ)ψ = ψ.

i (iii) iπ(R(1, s f ))ψ = i−sλ d P(λ)ψ → ψ as s → 0 by the same argument as in (ii). (iv) Let D := π R(1, f )R(1, h) Hπ , then by definition D ⊆ Ran π(R(1, f )) = 2 Dom jπ ( f ). Moreover π R(1, f )R(1, h) Hπ = π R(1, h)[R(1, f ) + iσ ( f, h)R(1, f ) × R(1, h)] Hπ ⊆ Ran π(R(1, h)) = Dom jπ (h), i.e. D ⊆ Dom jπ ( f ) ∩ Dom jπ (h). That D is dense, follows from (iii) of this theorem, using lim lim π R(1, s f )R(1, th)ψ = −ψ s→0 t→0

for all ψ ∈ Hπ , as well as s R(1, s f ) = R(1/s, f ) and the fact mentioned before (cf. Theorem 1, p. 216 in [26]) that all π(R(λ, f )) have the same range for f fixed. Let ψ ∈ D, i.e. ψ = π R(1, f )R(1, h) ϕ for some ϕ ∈ Hπ . Then π R(1, h)R(1, f ) jπ ( f ), jπ (h) ψ = π R(1, h)R(1, f ) π(R(1, f ))−1 , π(R(1, h))−1 π R(1, f )R(1, h) ϕ = π R(1, f )R(1, h) − R(1, h)R(1, f ) ϕ = iσ ( f, h)π R(1, h)R(1, f )2 R(1, h) ϕ = iσ ( f, h)π R(1, h)R(1, f ) ψ. Since Ker π R(1, h)R(1, f ) = {0} it follows that jπ ( f ), jπ (h) = iσ ( f, h) on D. (v) From Eq. (5) we have that π(R(λ, f )) = (11iλ − jπ ( f ))−1 = λ1 π R(1, λ1 f ) =

1 λ

−1 · i11 − jπ ( λ1 f )

and hence that jπ ( f ) = λ jπ ( λ1 f ) , i.e. jπ (λ f ) = λjπ ( f ) for all λ ∈ R\0. In Eq. (8): π (R(λ, f )R(µ, g)) = π R(λ + µ, f + g)[R(λ, f ) + R(µ, g) + iσ ( f, g)R(λ, f )2 R(µ, g)] multiply on the left by i(λ+µ)11− jπ ( f +g) and apply to (iµ11 − jπ (g))(iλ11 − jπ ( f ))ψ, ψ ∈ D to get i(λ + µ)11 − jπ ( f + g) ψ = ((iµ11 − jπ (g)) + (iλ11 − jπ ( f ))) ψ, making use of [(iµ11 − jπ (g)), (iλ11 − jπ ( f ))] ψ = iσ (g, f )ψ. Thus jπ ( f + g) = jπ ( f ) + jπ (g) on D. (vi) From the spectral resolution for jπ ( f ) we have trivially that on Dom jπ ( f ), λ d P(λ) = iµπ(R(µ, f )) − 11. jπ ( f )π(R(µ, f )) = π(R(µ, f )) jπ ( f ) = iµ − λ


721

(vii) Let ψ ∈ Dom jπ ( f ) = Ran π(R(λ, f )), i.e. ψ = π(R(λ, f ))ϕ for some ϕ ∈ Hπ . Then π(R(λ, f )) jπ ( f ), π(R(λ, g)) ψ = π(R(λ, f )) jπ ( f ), π(R(λ, g)) π(R(λ, f ))ϕ = π R(λ, g), R(λ, f ) ϕ = iσ (g, f )π R(λ, f )R(λ, g)2 R(λ, f ) ϕ = iσ (g, f )π R(λ, f )R(λ, g)2 ψ. Since Ker π(R(λ, f )) = {0}, it follows that

jπ ( f ), π(R(λ, g)) = iσ (g, f )π R(λ, g)2

on Dom jπ ( f ).

:= Span χ (viii) We first prove the second equality. Let ψ, ϕ ∈ D [−a,a] jπ ( f ) Hπ a > 0 , where χ[−a,a] indicates the characteristic function of n[−a, a], and note that D is a dense n subspace. Since jπ ( f ) χ[−a,a] jπ ( f ) Hπ ≤ a , n ∈ N, we can use the exponential series, i.e. n ∞ i jπ ( f ) ψ W ( f )ψ := exp i jπ ( f ) ψ = n!

∀ ψ ∈ D.

n=0

By the usual rearrangement of series we then have ∞ n 1 ∗ ϕ, ad i jπ ( f ) π(R(λ, h)) ψ ϕ, W ( f )π R(λ, h) W ( f ) ψ = n! n=0

Using part (vii) we have for all ϕ, ψ ∈ D.

thus so

ad i jπ ( f ) π(R(λ, h)k ) = k σ ( f, h)π R(λ, h)k+1 , n ad i jπ ( f ) π(R(λ, h)) = n! σ ( f, h)n π R(λ, h)n+1 ,

∞ ϕ, W (t f )π R(λ, h) W (t f )∗ ψ = t n σ ( f, h)n ϕ, π(R(λ, h)n+1 )ψ n=0

= (ϕ, π(R(λ + itσ ( f, h), h))ψ)

whenever tσ ( f, h) < |λ|, and where we made use of the Von Neumann series (Theo is dense, rem 3.4(iii)) in the last it

D step. Since the ∗operators involved are bounded and

follows that W (t f )π R(λ, h) W (t f ) = π(R(λ + itσ ( f, h), h)) for tσ ( f, h) < |λ|. By analyticity in λ this can be extended to complex λ such that λ ∈ iR. Using the group property of t → W (t f ) we then obtain for λ ∈ R\0 that W ( f )π R(λ, h) W ( f )∗ = π(R(λ + iσ ( f, h), h)).

(25)

722


To prove the first equation, let us write W (h) in terms of resolvents. Note that lim (1 − n→∞

−n = 1, it/n)−n = eit , t ∈ R and so by the bound: sup (1 − it/n)−n = sup 1 + t 2 /n 2 t∈R

t∈R

it follows from spectral theory (cf. Theorem VIII.5(d), p. 262 in [22]) that W (h) = ei jπ (h) = lim (1 − i jπ (h)/n)−n = lim π (i R(1, −h/n))n n→∞

n→∞

in strong operator topology. Apply Eq. (25) to this to get n W ( f )W (h)W ( f )∗ = s-lim π i R(1 + iσ ( f, − hn ), − hn ) n→∞

−n = s-lim 11 − i σ ( f, h) + jπ (h) n n→∞

= exp[iσ ( f, h) + i jπ (h)] = eiσ ( f,h) W (h) as required. Now W ( f )D = W ( f )π R(λ, f )R(µ, h) Hπ = π R(λ, f )R(µ + iσ ( f, h), h) Hπ = D, hence we conclude that D is a core for jπ ( f ) (cf. Theorem VIII.11, p. 269 in [22]). Proof of Proposition 3.7. Recall Eq. (10) ω(R(λ1 , f 1 ) · · · R(λn , f n )) ∞ ∞ = (−i)n dt1 . . . dtn exp − tk λ k − tk tl f k | fl ω − 0

0

k

1 2

k 0. Then ∂λ∂r ∂λk F(x)

≤ tr tk exp − 21 l tl2 fl | fl ω which is integrable w.r.t. the remaining variables, and likewise we also get a dominating function for the first derivatives. Thus by dominated convergence (uniformly in the λ–variables) we can take the partial derivatives in λi through the integral in the remaining variables. Thus we get ∂ ω R(λ1 , f 1 (x1 )) · · · R(λn , f n (xn )) ∂ xr =−

r −1 ∂2 ∂ f k (xk )| fr (xr ) ω ω R(λ1 , f 1 (x1 )) · · · R(λn , f n (xn )) ∂ xr ∂λr ∂λk k=1

− 21 −

∂2 ∂ fr (xr )| fr (xr ) ω ω R(λ , f (x )) · · · R(λ , f (x )) 1 1 1 n n n ∂ xr ∂λr2 n

k=r +1

∂2 ∂ fr (xr )| f k (xk ) ω ω R(λ1 , f 1 (x1 )) · · · R(λn , f n (xn )) . ∂ xr ∂λr ∂λk

Proof of Proposition 5.3. To prove this theorem, we first need to establish the following lemma.

724


Lemma 8.1. For the strip S := R +i[0, 1[⊂ C, let F : S → C be a continuous function, analytic on the interior of S, which satisfies for some C > 0 and λ ∈ C, |λ| = 1 the conditions:

F(t + is) ≤ C(1 + |t|) N ∀ t ∈ R, s ∈ (0, 1) F(t + i) = λ F(t) ∀ t ∈ R.

and

Then F = 0 if λ = 1 and F = constant if λ = 1. Proof. Note that C is covered by the strips Sn := S + in. We define G : C → C by G(z) := λn F(z − in) whenever z ∈ Sn . This is consistent, because on the joining lines R + in = Sn ∩ Sn−1 we have that G(t + in) = λn F(t) = λn−1 F(t + i). By the continuity of F on S0 = S, G is continuous. Now G is analytic on the interior of each Sn and continuous on the boundary, i.e. continuous on the lines R + in and analytic on either side of them. So it follows from a well-known theorem of analytic continuation that G is analytic on the lines R + in (cf. [20], p. 183) hence entire. Moreover G(z + i) = λG(z) for all z. Let be a closed anticlockwise circle of radius R centered at the fixed point z 0 . If z ∈ ∩ Sn then

G(z) = λn F(z − in) = F(z − in)

N N ≤ C 1 + |Re(z − in)| = C 1 + |Re(z)| N N ≤ C 1 + R + |Re(z 0 ) | ≤ C 1 + R + |z 0 |

N which is independent of n, i.e. |G(z)| ≤ C 1 + R + |z 0 | for all z ∈ . Applying this to the Cauchy integral formula: G(z) k! G (k) (z 0 ) = dz we find: 2πi γ (z − z 0 )k+1

N

G(z)

1 + R + |z 0 |

(k)

.

G (z 0 ) ≤ k!R sup k+1 ≤ k! C Rk z∈ R When k > N +1 this goes to zero as R → ∞, hence G (k) (z 0 ) = 0 for all k > N +1. This is true for all z 0 ∈ C, so G is a polynomial. However only a constant polynomial can satisfy G(z + i) = λG(z) (or else it has infinitely many zeroes), hence G is a constant, and if λ = 1, the only possible constant is zero.

(i) The KMS-condition for A = 11 reads F11,B (t) = ϕ αt (B) = F11,B (t + i). Thus by Lemma 8.1 it follows that F11,B is constant, hence that ϕ αt (B) is independent of t. (ii) Let γ (A) = −A ∈ Dom ϕ, then FA,11 (t + i) = ϕ(γ (A)) = −ϕ(A) = −FA,11 (t) so by Lemma 8.1 we have 0 = FA,11 = ϕ(A). For any B ∈ Dom ϕ decompose B = B+ + B− into γ -even and odd parts, then we get ϕ(B) = ϕ(B+ ) = ϕ(γ (B)). Finally, let a functional ϕ satisfy the graded KMS-condition on a set Y ⊂ Dom ϕ. Let A, B ∈ Span Y , i.e. A = λi Ai , B = µ j B j for Ai , B j ∈ Y , and λi , µ j ∈ C. Then i

FA,B (t) := ϕ Aαt (B) =

j

i, j

λi µ j ϕ Ai αt (B j ) = λi µ j FAi ,B j (t). i, j


725

Since ϕ is γ –KMS on Y the FAi ,B j are γ –KMS functions. Thus FA,B is continous on S, analytic on its interior, and FA,B (t + i) =

λi µ j FAi ,B j (t + i) =

i, j

λi µ j ϕ αt (B j )γ (Ai )

i, j

= ϕ αt (B)γ (A) , |FA,B (t + is)| ≤ |λi µ j | |FAi ,B j (t + is)| i, j

≤

|λi µ j | Ci j (1 + |t|) Ni j ≤ C(1 + |t|) N ,

i, j

where C :=

|λi µ j | Ci j and N := max(Ni j ). So ϕ is γ –KMS on Span Y . ij

i, j

Proof of Proposition Dom ψ and B are

dense *–algebras, it follows that 5.4. Since Dom ϕ := Span C ⊗ B C ∈ Dom ψ, B ∈ B is a dense *-algebra of A = C ⊗ B, and that it is invariant w.r.t. γ ⊗ ι and σ ⊗ β. Thus by the

last part of Proposi tion 5.3 it suffices to verify the KMS–property on C ⊗ B C ∈ Dom ψ, B ∈ B . Let Ai := Ci ⊗ Bi , i = 1, 2 for Ci ∈ Dom ψ ⊂ C, and Bi ∈ B. Consider for t ∈ R the function FA1 ,A2 (t) := ϕ (A1 (σ ⊗ β)t (A2 )) = ϕ (C1 ⊗ B1 )(σt (C2 ) ⊗ βt (B2 )) = ψ C1 σt (C2 ) ω B1 βt (B2 ) = FCψ ,C (t) FBω ,B (t), 1

2

1

2

where FCψ ,C and FBω ,B are the KMS–functions of ψ and ω resp. Thus, using their 1 2 1 2 analytic properties, it follows that FA1 ,A2 extends to an function on the strip S = R + i[0, 1], analytic on its interior and continuous on the boundary, given by FA1 ,A2 (z) = FCψ ,C (z) FBω ,B (z). Moreover 1

2

1

2

FA1 ,A2 (t + i) = FCψ ,C (t + i) FBω ,B (t + i) = ψ σt (C2 ) γ (C1 ) ω βt (B2 ) B1 1

2

1

2

= ϕ (σt (C2 )γ (C1 ) ⊗ βt (B2 ) B1 ) = ϕ ((σ ⊗ β)t (A2 )(γ ⊗ ι)(A1 )) , and thus FA1 ,A2 will be a (γ ⊗ ι)–KMS function if the tempered growth property also holds. We have FCψ ,C (t + is) ≤ K (1 + |t|) N for a constant K and N ∈ N depending

1 2 on Ci . Now as ω is a state we have from the KMS–property that FBω ,B (t + is) ≤ 1 2 B1 B2 for s = 0, 1. So by the maximum modulus principle (apply it after

first map

ping S to the unit disk by the Schwartz mapping principle) it follows that FBω ,B (z) ≤ 1 2 B1 B2 for all z ∈ S. Thus for t ∈ R, s ∈ [0, 1] we have

F

A1 ,A2 (t

+ is) ≤ B1 B2 K (1 + |t|) N

and so the tempered growth property holds for FA1 ,A2 . Thus ϕ is a (γ ⊗ ι)–KMS functional.

726


Proof of Theorem 5.5. (i) To prove there is a quasi–free state on (S, σ ) defined by

2 ω(δ f ) := exp[−s( f, f )/2], f ∈ S(R, R), where s( f, f ) := 1−ep− p f ( p) dp, it suffices to show that |σ ( f, h)|2 ≤ 4 s( f, f ) s(h, h) by [19],

2

2

p f ( p) f ( p) h( p) dp |σ ( f, h)|2 = i p h( p) dp ≤ = 2 ≤4

∞

0 ∞

2 p f ( p) h( p) dp

2 p f ( p) dp

0

∞

f (−p) = f ( p) h( p) is even by as p f ( p)

2 k h(k) dk

by Cauchy–Schwartz

0

∞ p

2 k

2 0 −p −k 1 − e 1−e 0 0 ∞ ∞

2 p

2 k

2 we have p

730


n

p ln(1 − e− p ) = p n e− p + e−2 p /2 + e−3 p /3 + · · · ≤ p n e− p 1 + e− p /2 + e−2 p /3 + 1 k n − p which is integrable.] Thus by dominated con· · · ≤ p n e− p ∞ k=0 ( 2 ) ≤ 2 p e ∞ vergence the function t → 0 dp ln(1 − e− p ) cos pt is smooth. Thus the kernel K of T is smooth. If J is a compact interval then PJ T PJ has kernel χ J (x)K (x − y)χ J (y) which is smooth and bounded on J . Thus PJ T PJ is trace–class by Theorem 1, p. 128 in Lang [17]. We also have an explicit proof that PJ T PJ is trace–class below in the proof of Theorem 5.9. Selfadjointness now follows from the fact that θ ( f, f ) is real by its formula. This proves (i). To prove (ii), note that we already proved above that G(z) is well-defined for z ∈ R + i[0, 1). To prove that it is well defined on all of S, it is only necessary to prove integrability for the high p part of the integral. For this

2i

e− p

sin p(z + y − x)

− p 1−e 1

∞

e− p i pz

−i pz =

e dp g ( p) − e f ( p) f ( p) g ( p)

−p 1 − e 1 ∞

e− p − ps ≤ e f ( p) g ( p)

dp + e ps − p 1−e 1 ∞

2 2

≤ dp f g, f ( p) g ( p) ≤ −p 1 − e 1 − e−1 1 ∞

dp

dx

dy f (x) g(y)

(32)

where z = t + is ∈ S, and so G(z) is well-defined for z ∈ S. To establish the stated inequality for z = s +it ∈ S, consider

|F( p)| =

Eq. (31). Now e−sp χ[0,∞) ( p) implies F = 1, so F(D = 1 and hence (g, F(D) f, ) ≤ f g. The high p part of the integral in (31) has an estimate (32), so for the low p integral we have for its integrand the inequality (30) above, so that

e− p

sin p(z + y − x)

1 − e− p 0

1 p e− p

≤ cosh 1 · d x dy f (x) g(y) 1 + |t| + |x − y| dp 1 − e− p 0 1

dp

dx

dy f (x) g(y)

= C + |t|E

and E. Combining this with (32) and the estimate for

constants C

for some finite

(g, F(D) f ) , we obtain G(t + is) ≤ A + B|t| for constants A, B as desired. It remains to prove that G is continuous on S and analytic in its interior. Now

d

p e− p 1 −sp e− p

f (x) g(y) ≤ f (x) g(y) sin p(z + y − x) (e + esp )

dz 1 − e− p 1 − e− p 2 which is L 1 for all s ∈ (0, 1). Thus the last integral in (31) is analytic on the interior of S. That (g, F(D) f ) is analytic in z follows from spectral theory, hence G is analytic on the interior of S. For continuity on S, we already have that (g, F(D) f ) is continuous in z, and by inequalities (32) and (30) we can get L 1 estimates to ensure that G is continuous on S.


731

Proof of Theorem 5.7. We now show that the quasi–free ψ with two point functional functional θ is a graded KMS–functional on Cliff S(R) . Its domain

Dom ψ = *-alg c( f ) f ∈ S(R) is clearly a unital dense *-algebra of Cliff S(R) which is invariant w.r.t. both the grading γ and the time evolution αt , so part (i) of Definition 5.1 is satisfied. For the KMS-condition (ii), it suffices to check it for the monomials c( f 1 ) · · · c( f k ). Let A = c( f 1 ) · · · c( f k ) and B = c(g1 ) · · · c(gm ), where k + m = 2n. Then from (16) and (17) we get FA,B (t) := ψ A αt (B) = ψ c( f 1 ) · · · c( f k )c(Tt g1 ) · · · c(Tt gm ) = (−1)( 2 ) n

n (−1) P θ h P( j) , h P(n+ j) , P

(33)

j=1

where h 1 = f 1 , . . . , h k = f k , h k+1 = Tt g1 , . . . , h 2n = Tt gm and Tt f := f t is translation by t. Since P( j) < P(n + j) always, the terms θ h P( j) , h P(n+ j) can only be one of the types or θ Tt f i , Tt g j if k < n, or θ fi , g j if k > n. θ f i , Tt g j Since by the formula for θ we have θ Tt f, Tt g = θ f, g , the last two types are the same and constant in t. For the first type, we get by definition functions G(t) = θ f, Tt g as in Theorem 5.6(ii), which we therefore know extend analytically to the strip S. Thus since Eq. (33) expresses FA,B (t) as a polynomial of constant functions and functions of the form G, it follows that FA,B (t) extends to a continuous function on S which is analytic on its interior. Now −ε ∞ ei p(t+i) + ˆ p) dp G(t + i) = lim+ f ( p) g( ε→0 1 − e− p −∞ ε −ε ∞ ei pt = lim+ + ˆ p) dp f ( p) g( ε→0 ep − 1 −∞ ε −ε ∞ e−i pt = lim+ + ˆ p) dp f (− p) g(− ε→0 e− p − 1 −∞ ε −ε ∞ e−i pt = − lim+ + f ( p) g( ˆ p) dp ε→0 1 − e− p −∞ ε = −θ (Tt g, f ) = ψ αt (c(g))γ (c( f )) , (34) which is the graded KMS-condition for Fc( f ),c(g) (t) = ψ c( f )αt (c(g)) . The terms θ (T the ones which occur in the corresponding expression (33) for t g, f ) are exactly ψ αt (B) γ (A) so the graded KMS-condition for FA,B follows from the one for G, Eq. (34). It remains to prove the growth condition (iii) of Definition 5.1. We already have

G(t + is) ≤ a + b|t| for t ∈ R, s ∈ [0, 1] by Theorem 5.6(ii). So from formula (34) we get that for t + is ∈ S :

F (t + is) ≤ (a1 + b1 |t|) · · · (an + bn |t|) ≤ C (1 + |t|)n A,B for suitable constants ai , bi and C. Thus ψ is a graded KMS-functional.

732


Proof of Theorem 5.8. By construction Dom ϕ contains the *-algebra generated by all c( f ), f ∈ S(R) as well as R(S(R), σ ) and so it will certainly contain the *-algebra generated by 11 and all c( f ), R(λ, f ), which is A0 . So (i) is trivially true. Next, for (ii), we need to prove the SUSY-invariance of ϕ, and for this, we need the following lemma. Lemma 8.2. For all g, f i ∈ S(R)\0 and λi ∈ R\0 we have ϕ R(λ1 , f 1 ) · · · R(λn , f n ) j (g) R(λn+1 , f n+1 ) · · · R(λm , f m ) =

n

s( f k , g) ϕ R(λ1 , f 1 ) · · · R(λk , f k )2 · · · R(λn , f n ) R(λn+1 , f n+1 ) · · · R(λm , f m )

k=1 m + s(g, f k ) ϕ R(λ1 , f 1 ) · · · R(λn , f n ) R(λn+1 , f n+1 ) · · · R(λk , f k )2 · · · R(λm , f m ) , k=n+1

where ϕ is a strongly regular state on R(S(R), σ ) so these expressions make sense on E0 . Proof. Recall by Theorem 5.5 that ϕ is a quasi–free state on (S, σ ) defined by ϕ(δ f ) := exp[−s( f, f )/2] , f ∈ S(R, R), where s is given in Eq. (15). Since the maps t, r → s(r f, tg) are smooth, we can apply Proposition 3.7 to ϕ w.r.t. the maps t → t f . d From the two relations dt R(λ, t f ) = j ( f )R(λ, t f )2 and lim λiπϕ (R(λ, t f ))ψ = ψ, t→0

we get

ϕ R(λ1 , f 1 ) · · · R(λn , f n ) j (g) R(λn+1 , f n+1 ) · · · R(λm , f m ) ∂ ϕ R(λ1 , f 1 ) · · · R(λn , f n ) R(µ, tg) R(λn+1 , f n+1 ) · · · R(λm , f m ) t→0 ∂t n ∂2 d = −µ2 lim − ϕ R(λ1 , f 1 ) · · · s( f k , tg) t→0 dt ∂µ∂λk k=1 · · · R(λn , f n ) R(µ, tg) R(λn+1 , f n+1 ) · · · R(λm , f m ) = −µ2 lim

d ∂2 ϕ R(λ , f ) · · · R(λ , f ) R(µ, tg) R(λ , f ) · · · R(λ , f ) s(tg, tg) 1 1 n n n+1 n+1 m m dt ∂µ2

− 21 −

m ∂2 d ϕ R(λ1 , f 1 ) · · · s(tg, f k ) dt ∂µ∂λk

k=n+1

· · · R(λn , f n ) R(µ, tg) R(λn+1 , f n+1 ) · · · R(λm , f m )

(By Proposition 3.7) =

n

s( f k , g) ϕ R(λ1 , f 1 ) · · · R(λk , f k )2 · · · R(λn , f n ) R(λn+1 , f n+1 ) · · · R(λm , f m )

k=1

+

m

s(g, f k ) ϕ R(λ1 , f 1 ) · · · R(λn , f n ) R(λn+1 , f n+1 ) · · · R(λk , f k )2 · · · R(λm , f m ) ,

k=n+1


where we used

d dλ R(λ,

733

f ) = −i R(λ, f )2 .

Next we need to show that ϕ◦δ is zero on D S , i.e. that it vanishes on all the monomials: ζ ( f 1 ) · · · ζ ( f n )R(λ1 , g1 ) · · · R(λm , gm ) = c( f 1 ) · · · c( f n )R(1, f 1 ) · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm ). Recall that δ is a restriction to D S of a graded derivation on E defined by δ( j ( f )) = ic( f ),

δ(R(λ, f )) = ic( f )R(λ, f )2 ,

and

δ(c( f )) = j ( f ).

So we calculate: ϕ ◦ δ (c( f 1 ) · · · c( f n )R(1, f 1 ) · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm )) =

n

k (−1)k+1 ϕ c( f 1 ) · · · ∨ · · · c( f n ) j ( f k ) R(1, f 1 ) · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm )

k=1

+ i(−1)n

n ϕ c( f 1 ) · · · c( f n )c( f )R(1, f 1 ) · · · R(1, f )2 · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm )

=1

+ i(−1)n

m

ϕ c( f 1 ) · · · c( f n )c(g p )R(1, f 1 ) · · · · · · R(1, f n )R(λ1 , g1 ) · · · R(λ p , g p )2 · · · R(λm , gm )

p=1

=

n

k (−1)k+1 ϕ c( f 1 ) · · · ∨ · · · c( f n )

k=1

n × s( f k , fr ) ϕ R(1, f 1 ) · · · R(1, fr )2 · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm ) r =1

+

m

s( f k , gt ) ϕ R(1, f 1 ) · · · R(1, f n )R(λ1 , g1 ) · · · R(λt , gt )2 · · · R(λm , gm )

t=1

+i(−1)n

n ϕ c( f 1 ) · · · c( f n )c( f ) ϕ R(1, f 1 ) · · · R(1, f )2 · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm )

=1

m + i(−1)n ϕ c( f 1 ) · · · c( f n )c(g p ) ϕ R(1, f 1 ) · · · R(1, f n )R(λ1 , g1 ) · · · R(λ p , g p )2 · · · R(λm , gm ) , p=1

where we made use of Lemma 8.2. Note that as ϕ is quasifree, n must be odd for the last expression to be nonzero, and also: n−1 k ϕ c( f 1 ) · · · c( f n ) = (−1)k+1 ϕ c( f k ) c( f n ) ϕ c( f 1 ) · · · ∨ · · · c( f n−1 ) . k=1

734


So we get: ϕ ◦ δ (c( f 1 ) · · · c( f n )R(1, f 1 ) · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm )) n n k k+1 ∨ = s( f k , fr ) − iϕ c( f k )c( fr ) (−1) ϕ c( f 1 ) · · · · · · c( f n ) k=1

r =1

× ϕ R(1, f 1 ) · · · R(1, fr )2 · · · R(1, f n )R(λ1 , g1 ) · · · R(λm , gm ) +

m s( f k , g p ) − iϕ c( f k ) c(g p ) p=1

× ϕ R(1, f 1 ) · · · R(1, f n )R(λ1 , g1 ) · · · R(λ p , g p )2 · · · R(λm , gm ) . However for the two-point functions we have: −ε ∞ −i p ϕ c( f ) c(g ) = lim+ + ˆ p) dp f ( p) g( ε→0 1 − e− p −∞ ε ∞ p ˆ p) dp = −is( f, g), = −i f ( p) g( −p −∞ 1 − e and so we get ϕ ◦ δ = 0 as desired. Finally, for (iii) we need to prove that

d ϕ B M A δ 0 (A)C = −i ϕ B M A αt (A)C

(35) 0 dt for all A ∈ D S and B, C ∈ A0 . Since αt and δ 0 do not mix Cliff S(R) and R(S(R), σ ) and ϕ has a product structure, it suffices to verify (35) on the CAR and CCR parts separately. First, on the Clifford algebra we have δ 0 (c( f )) = ic( f ), and by the derivative property we only need to check (35) for A = c( f ). However, ϕ is quasifree so it suffices to check for the two-point functions that

d ϕ c(g) αt (c( f )) = iϕ c(g)δ 0 (c( f )) = −ϕ c(g) c( f ) 0 dt for all f, g ∈ S(R). The differentiability of G(t) = ϕ c(g) αt (c( f )) was proven above in Theorem 5.6. So:

−ε ∞ ei pt

d d

ϕ c( f ) αt (c(g)) = lim+ + g( ˆ p) dp f ( p)

0 0 dt dt ε→0 1 − e− p −∞ ε −ε ∞ i p = lim+ + ˆ p) dp f ( p) g( ε→0 1 − e− p −∞ ε = −ϕ c( f ) c(g ) as required. Next, we need to check (35) on the resolvent algebra. By the derivative property, it suffices to do this for A = R(λ, f ), and by linearity for the remaining terms


735

being monomials of resolvents. That is, we need to prove that

d

ϕ R(λ1 , f 1 ) · · · R(λn , f n )R(µ, Tt g) R(λn+1 , f n+1 ) · · · R(λm , f m )

0 dt = i ϕ R(λ1 , f 1 ) · · · R(λn , f n )δ 0 R(µ, g) R(λn+1 , f n+1 ) · · · R(λm , f m ) , (36) where δ 0 R(µ, g) = i R(µ, g) j (g ) R(µ, g). This is an expression of the form of Proposition 3.7, so to apply this, we need to check that the functions t → s( f, Tt g) are smooth (note that s(Tt f, Tt g) = s( f, g)), and this is an easy verification. In fact, s(·, ·) is clearly a distribution in each entry as it is an expectation value ( f, Ag), where A is multiplication by a smooth function which is polynomially bounded. Applying Proposition 3.7 we get:

d

ϕ R(λ1 , f 1 ) · · · R(λn , f n )R(µ, Tt g) R(λn+1 , f n+1 ) · · · R(λm , f m )

t=0 dt =−

n

d ∂2

s( f k , Tt g) ϕ R(λ1 , f 1 )) · · · R(λn , f n )R(µ, Tt g) R(λn+1 , f n+1 ) · · · R(λm , f m )

t=0 dt ∂µ∂λk

k=1

− 21

d ∂2

s(Tt g, Tt g) ϕ R(λ1 , f 1 ) · · · R(λn , f n )R(µ, Tt g) R(λn+1 , f n+1 ) · · · R(λm , f m )

t=0 dt ∂µ2 m

−

k=n+1

=−

n

d ∂2

s(Tt g, f k ) ϕ R(λ1 , f 1 ) · · · R(λn , f n )R(µ, Tt g) R(λn+1 , f n+1 ) · · · R(λm , f m )

t=0 dt ∂µ∂λk

s( f k , g ) ϕ R(λ1 , f 1 )) · · · R(λk , f k )2 · · · R(λn , f n )R(µ, g)2 R(λn+1 , f n+1 ) · · · R(λm , f m )

k=1

s(g , f k ) ϕ R(λ1 , f 1 ) · · · R(λn , f n ) R(µ, g)2 R(λn+1 , f n+1 ) · · · R(λk , f k )2 · · · R(λm , f m ) ,

m

−

k=n+1

p i p2 d d g dp = −s( f, g ), where we used dt s( f, Tt g) 0 = dt g dp 0 = 1−e f ei pt −p f 1−e− p as well as s(Tt f, Tt g) = s( f, g). On the other hand, for the right-hand side of Eq. (36) we have from Lemma 8.2, that i ϕ R(λ1 , f 1 ) · · · R(λn , f n )δ 0 R(µ, g) R(λn+1 , f n+1 ) · · · R(λm , f m ) = −ϕ R(λ1 , f 1 ) · · · R(λn , f n )R(µ, g) j (g ) R(µ, g) R(λn+1 , f n+1 ) · · · R(λm , f m ) =−

n

s( f k , g ) ϕ R(λ1 , f 1 ) · · · R(λk , f k )2 · · · R(λn , f n ) R(µ, g)2 R(λn+1 , f n+1 ) · · · R(λm , f m )

k=1

−

m

s(g , f k ) ϕ R(λ1 , f 1 ) · · · R(λn , f n ) R(µ, g)2 R(λn+1 , f n+1 ) · · · R(λk , f k )2 · · · R(λm , f m )

k=n+1

− s(g, g ) ϕ R(λ1 , f 1 ) · · · R(λn , f n ) R(µ, g)3 R(λn+1 , f n+1 ) · · · R(λm , f m ) − s(g , g) ϕ R(λ1 , f 1 ) · · · R(λn , f n ) R(µ, g)3 R(λn+1 , f n+1 ) · · · R(λm , f m ) .

However, s(g, g ) = −s(g , g) and so the last two terms cancel and hence we have proven (36).

736


Proof of Theorem 5.9. Recall that

A0 (J ) := *-alg c( f ), R(λ, f ) supp f ⊆ J, f ∈ S(R), λ ∈ R\0 = C(J ) ⊗ R0 (J ),

where C(J ) := *–alg c( f ) supp f ⊆ J, f ∈ S(R)

and R0 (J ) := *–alg R(λ, f ) supp f ⊆ J, f ∈ S(R), λ ∈ R\0 . Since ϕ=ψ ⊗ ω is a product functional, and ω is a state, we have that ϕ A0 (J ) = ψ C(J ), and so this is what we need to estimate. Without loss of generality we may assume J to be a closed interval, and also symmetrical about the origin (since ϕ is invariant w.r.t. translations). Recall from Theorem 5.6 that ψ(c( f )c(g)) = θ ( f, g) = (g, (P + T ) f ), where P is a projection (after a normalisation) and PJ T PJ is trace-class and selfadjoint for all compact intervals J , and so this is the case for c( f ), c(g) ∈ C(J ). We will need to use the isomorphism of the Clifford algebra C(J ) with a self–dual CAR–algebra explicitly. First observe that C(J ) = Cliff L 2 (J, R) by continuity of c( f ). Since J is symmetrical about the origin, we can define : L 2 (J, R) → L 2 (J, R) by ( f )( p) := f (− p). Then ( f )(x) = f (−x) and P = (11 − P) , and T = −T by the explicit formula for T . Define for f ∈ L 2 (J, R), ( f ) :=

√1 2

c(P f ) − ic( P f ) + c(P f ) + ic (11 − P) f ,

and observe that ( f ) = ( f )∗ and {( f ), (g)} = ( f, g)11, which establishes the isomorphism. By requiring complex linearity for ( f ), we get ( f ) + i(g) =: ( f + ig), hence get in fact an isomorphism of C(J ) with CAR(L 2 (J, C)). Note that by ( f ) = ( f )∗ the involution has to extend to L 2 (J, C) =: K in a conjugate linear way. The image of C(J ) under the isomorphism is the dense *-algebra CAR0 generated by all ( f ), f ∈ K, and this is the domain of ψ. With to the decomposition K = PK ⊕ (11 − P)K f ⊕ g we decompose respect B so by T = T ∗ we get A = A∗ , D = D ∗ and B = C ∗ , and these operators T = CA D are trace–class because T is. From the relation T = −T we then find that A = −D A B 2 and B ∗ = −B, hence T = −B −A . Since T preserves the original real space L (J, R) we have A f = A f and B f = B f . Then the two-point function of ψ on CAR0 is for f ⊕ g, h ⊕ k ∈ L 2 (J, R): ψ ( f ⊕ g) (h ⊕ k) = 21 ψ c( f ⊕ 0) − i c(0 ⊕ f ) + c(g ⊕ 0) + i c(0 ⊕ g) · c(h ⊕ 0) − i c(0 ⊕ h) + c(k ⊕ 0) + i c(0 ⊕ k) = 21 (h, f ) + (k, f ) + (h, g) + (k, g) + (h, A f ) − i(k, B f ) + i(h, Bg) + (k, Ag)

I I −i B A + ( f ⊕ g) . = (h ⊕ k), 21 I I A iB


737

This expression is complex linear in both entries, so we can extend it by linearity to K to get for all f, g ∈ K that ψ ( f ) (g) = (g), R + Q f , where (37)

I I −i B A R := 21 and Q := . I I A iB + S := R + Q, then S is a bounded selfadjoint operator which satisfies S = S. As S0 := 21 0I 0I has eigenvalues ± 21 and Q is trace class, Define an operator S by

1 2

S = S0 + Q has discrete spectrum with the only possible accumulation points ± 21 (cf. Theorem 9.6 [24]). We now need: Lemma 8.3. (i) E([− 21 , 21 ]c ) · (2|S| − 11) on L 2 (J ) is trace class where E is the spectral resolution of S. Moreover we have for the trace–norms · 1 that E([− 1 , 1 ]c ) · (2|S| − 11) ≤ b PJ T PJ 1 2 2 1 for a positive constant b (independent of J ). (ii) Let {e j | j ∈ J} be an orthonormal system of eigenvectors of S corresponding to the eigenvalues s j ∈ [− 21 , 21 ]c , and exhausting these eigenspaces. For each j ∈ J, let C j be the two–dimensional abelian *-algebras generated by (e j )∗ (e j ), and

let C0 := *–alg ( f ) f ∈ E [− 21 , 21 ] K . Then ψ C(J ) = ψ j = 2|s j | < ∞, (38) j∈{0}∪J

j∈J

where ψ j denotes the restriction of ψ to C j . Proof. (i) Let E 0 ( · ) be the spectral resolution corresponding to S0 . For any s ∈ (0, 21 ) which is not in the spectrum σ (S) of S, since S and S0 are bounded, we obtain by spectral calculus that −1 E((s, ∞)) − E 0 ((s, ∞)) = (2πi) dz ((z − S)−1 − (z − S0 )−1 ) C

= (2πi)−1

dz (z − S0 )−1 Q(z − S)−1 ,

(39)

C

where C is a suitable closed path in C, e.g. a large anticlockwise simple contour with σ (S) ∩ [s, ∞) and σ (S0 ) ∩ [s, ∞) in its interior, and crossing the real axis only at s and some t > s. So from (39) we conclude that E((s, ∞)) − E 0 ((s, ∞)) is a trace class operator. Taking into account that E 0 ((s, ∞))(2S0 − 1) = 0 we obtain E((s, ∞))(2S − 1) = E((s, ∞)) − E 0 ((s, ∞)) (2S − 1) + E 0 ((s, ∞)) 2Q, (40) showing that E((s, ∞))(2S − 1) and hence a fortiori E(( 21 , ∞)) (2S − 1) is trace class. A similar argument establishes that E((−∞, − 21 ))(2S + 1) is trace class, and hence E([− 21 , 21 ]c ) · (2|S| − 11) is trace class. From Eq. (40) we get that E((s, ∞))(2S − 1)1 ≤ E((s, ∞)) − E 0 ((s, ∞)) (2S − 1)1 + E 0 ((s, ∞)) 2Q1 ≤ E((s, ∞)) − E 0 ((s, ∞)) (2S − 1)1 + 2Q1 (41)

738


since all the terms in (40) are trace class, and AQ1 ≤ A · Q1 for A bounded. Let 1 1 0 I P0 be the projection onto the eigenspace of S0 with eigenvalue 2 (since S0 = 2 I 0 this is just the space of even functions w.r.t. the decomposition associated with P ). Then by substituting (z − S0 )−1 = (z − 21 )−1 P0 + (z + 21 )−1 (11 − P0 )

and

(z − S)−1 (2S − 11) = (2z − 1)(z − S)−1 − 211 into (39)×(2S − 11) we get that E((s, ∞)) − E 0 ((s, ∞)) (2S − 11) 1 P0 Q dz (z − 21 )−1 2z−1 − 2 + (1 1 − P )Q dz (z + 21 )−1 2z−1 = 2πi 0 z−S z−S − 2 C

=

1 2πi

P0 Q C

C

dz 2 (z − S)−1 − 2P0 Q +

1 2πi (11 −

P0 )Q

dz C

2z−1 z+1/2

z−S

= P0 Q 2E((s, ∞)) − 2P0 Q + (11 − P0 )Q f (S),

2z−1 where f (z) := z+1/2 χ H (z) and H := z ∈ C Re(z) > s . Now f (S) ≤ f (s, ∞)∞ = 2, and so E((s, ∞)) − E 0 ((s, ∞)) (2S − 11) ≤ 6Q1 ≤ aPJ T PJ 1 1

for a constant a > 0, where we obtain the last inequality Q1 ≤ const.PJ T PJ 1 from the decomposition in (37) from which P Q P = −i B = −i P(PJ T PJ )(11 − P), (11 − P)Q P = P(PJ T PJ )P etc. Thus by (41) we get E(( 21 , ∞))(2S − 1)1 = E( 21 , ∞)E(s, ∞)(2S − 1)1 ≤ E(s, ∞)(2S − 1)1 ≤ aPJ T PJ 1 . A similar argument establishes that E(−∞, − 21 )(2S + 1)1 ≤ aPJ T PJ 1 and hence that E([− 21 , 21 ]c ) · (2|S| − 11)1 ≤ b PJ T PJ 1 for a positive constant b. (ii) Recall that S has a purely discrete spectrum in [− 21 , 21 ]c , so choose an orthonor mal system e j ∈ K j ∈ J ⊆ N of eigenvectors of S, corresponding to eigenvalues s j ∈ [− 21 , 21 ]c and exhausting these eigenspaces (some s j will coincide for higher multiplicities). Let E j be! the one–dimensional orthogonal projection onto e j , j ∈ J, and let λ = (λ1 , λ2 , . . . ) ∈ j∈J {−1, 1}. We define on K the unitaries V (λ) := E([− 21 , 21 ]) + λj E j. j∈J

Since V (λ) commutes with , these unitaries induce an action γ : Aut CAR(K) given by

!

j∈J {−1, 1}

→

γλ (( f )) := (V (λ) f ). Since V (λ)2 = 11, we can decompose CAR(K) into odd and even parts w.r.t. each γλ . Moreover, since V (λ) commutes with S we have that ψ ◦ γλ = ψ and so ψ must vanish


739

on the odd respect to each γλ. Let C ⊂ CAR part of CAR0 with 0 be the *–algebra generated by (e j ) j ∈ J ∪ ( f ) f ∈ E [− 21 , 21 ] K , then C is mapped to itself by all γλ . Since the two-point functional θ is bounded on L 2 (J ), it suffices to calculate the norm of ψ on C, and in fact on the intersection of all the even parts of C with respect to γλ , and we denote this *-algebra by ε(C). It is produced by a projection ε which we can consider as the projection defined on C by averaging over the action of γλ on C. Since for each A ∈ C only a finite number of j’s are involved, these averages will again be in the *-algebra C. Now we only need to consider monomials in the (e j ) and (e j )∗ which are even in each index j. In a given monomial (e j1 ) · · · (e jn ) ∈ ε(C) if we collect all (even number of) terms with the same j together, we can then simplify it with the relations

2

2 2 (e j )2 = e j |e j 11 and (e j )∗ (e j ) = (e j )∗ (e j ) − 41 ( e j , e j ) 11. Thus ε(C) is generated by the two-dimensional abelian *-algebras C j := *–alg{(e j )∗ (e j )}

1 1 and C0 := *–alg ( f ) f ∈ E [− 2 , 2 ] K . Since for i = j we have ε (ei )∗ (ei ), (e j )∗ (e j ) = ε ei , e j (e j )(ei ) + ei , e j (e j )∗ (ei )∗ = 0, it follows that all the Ci commute, and in fact we have the (incomplete) tensor product decomposition " ε(C) = Cj. j∈{0}∪J

Moreover, ψ is a product functional on this tensor product. Hence its norm, if it exists, is given by ψ = ψ j , j∈{0}∪J

where ψ j denotes the restriction of ψ to C j . Now ψ0 is by construction a state on C0 because 21 + S is positive on E [− 21 , 21 ] K, hence the two-point function is positive and so ψ0 = 1. Since for a, b ∈ C, ψ(a11 + b (e j )∗ (e j )) = a + b ( 21 + s j ) and a11 + b (e j )∗ (e j ) = max{|a|, |a + b|} one obtains ψ j = 2|s j |, j ∈ J. Thus since 2|s j | > 1, a necessary and sufficient condition for the existence of ψ is j∈J (2|s j | −1) < ∞. However, this is guaranteed by part (i).

Using this lemma, we can now prove: Lemma 8.4. We have ϕ A0 (J ) ≤ exp bPJ T PJ 1 , where b is a positive constant (independent of J ) and · 1 denotes the trace–norm.

740


Proof. Recall from Eq. (38) that ϕ A0 (J ) = ψ j = 2|s j | = (1 + t j ) < ∞, j∈{0}∪J

j∈J

j∈J

where t j := 2|s j | − 1. Now ln(1 + x) ≤ x for x ≥ 0, so ln

N

(1 + t j ) =

j=1

N

ln(1 + t j ) ≤

j=1

N

tj

hence

(1 + t j ) ≤ exp

tj .

j∈J

j=1

j∈J

Now tj = (2|s j | − 1) = E([− 21 , 21 ]c ) · (2|S| − 11)1 ≤ b PJ T PJ 1 for a j∈J

j∈J

constant b > 0 by Lemma 8.3(i). Combining these claims prove the lemma.

To conclude the proof of the theorem, we need to estimate PJ T PJ 1 . Recall from Theorem 5.6 that ∞ dp ln(1−e− p ) cos p(x − y) χ J (y) PJ T PJ f (x) := 2i χ J (x) dy f (y) (x−y) 0

= i χ J (x)

dy f (y) (x − y)

∞ −∞

dp ln(1 − e−| p| ) ei p(x−y) χ J (y)

= const. χ J (X ) X, ln(1 − e−|P| ) χ J (X ) f (x),

(42)

where X is the multiplication operator (X f )(x) = x f (x), and P is as usual i ddx , and the constant incorporates the 2π factors from the Fourier transforms. Now D := − ln(1 − e−|P| ) is a positive operator, so write trivially χ J (X ) X D χ J (X ) = χ J (X ) X D 1/2 × D 1/2 χ J (X ) , then we show that both factors are Hilbert–Schmidt. Now χ J (X ) X D 1/2 f (x) = dy K (x, y), where K (x, y) = χ J (x) x so

χ (X ) X D 1/2 2 = J 2

1/2 i p(x−y) dp − ln(1 − e−| p| ) e ,

2

d x d y K (x, y) =

dx x

2

dp − ln(1 − e−| p| )

J

≤ const.|J | , 3

using the integrability of ln(1 − e−| p| ). Likewise, we get 1/2 (D χ (X )2 = d x dp − ln(1 − e−| p| ) ≤ const.|J |. J 2 J

Thus χ J (X ) X D χ J (X ) is trace class, and as PJ T PJ is, so is χ J (X ) D X χ J (X ). For their trace norms we find χ (X ) X D χ (X ) ≤ χ (X ) X D 1/2 · (D 1/2 χ (X ) ≤ const.|J |2 J J J J 1 2 2 2 and likewise χ J (X ) D X χ J (X ) 1 ≤ const.|J | . Thus by (42) we get that PJ T PJ ≤ const.χ (X ) X D χ (X ) + const.χ (X ) D X χ (X ) ≤ K |J |2 J J J J 1 1 1


741

for a constant K (independent of J ). Now from Lemma 8.4 we get the claim of the theorem, i.e. that ϕ A0 (J ) ≤ exp K |J |2 . Proof of Theorem 6.3. Fix a compact interval J = [−k, k] and let ai ∈ A0 (J ) for all i, then as αt (ai ) ∈ A0 (J + t) we have αt0 (a0 ) · · · αtn (an ) ∈ A0 ([−M, M]), where M := k + sup |ti |. Thus by Theorem 5.9 we get i

2

ϕ αt0 (a0 ) · · · αtn (an ) ≤ e4K M a0 · · · an . Now M 2 = k 2 + sup ti2 + 2k sup |ti | ≤ k 2 + ti2 + 2k sup(1 + ti2 ) i

≤ k2 +

i

i

i

ti2 + 2k 1 + ti2 ) = k 2 + 2k + (1 + 2k) ti2 ,

i

i

hence ϕ αt0 (a0 ) · · · αtn (an ) ≤ A exp B ti2 a0 · · · an

i

i

for suitable constants A and B depending only on k (but not on n). Now let t0 = 0, t1 = s1 , t2 = s1 + s2 , . . . , tn = s1 + · · · + sn , and define for all si ∈ R: n Fao ,...an (s1 , . . . , sn ) := exp − B (s1 + · · · + sk )2 · ϕ a0 αs1 (a1 ) · · · αs1 +···+sn (an ) . k=1

Then we have Fao ,...an (s1 , . . . , sn ) ≤ A a0 · · · an , and by the KMS–property of ϕ, the function F ao ,...an can be analytically continued in each variable s j into the strip S j := {z j ∈ Cn Im z j ∈ [0, 1]}, keeping the other variables real. This produces functions Fa( j),...a analytic in the flat tubes T j := Rn−1 × S j , and hence by using the Flat o n Tube Theorem 6.1 inductively, we obtain an analytic continuation of Fao ,...an into the tube

Tn := Rn + in , where n := s ∈ Rn 0 ≤ si ∀ i, s1 + · · · + sn ≤ 1 , coinciding with all Fa( j),...a on T j . We want to obtain a bound for thisanalytic function F. We start o

n

by finding bounds for the F ( j) . Let G(s1 , . . . , sn ) := ϕ a0 αs1 (a1 ) · · · αs1 +···+sn (an )

which has analytic extensions to each T j , and by the definition of KMS–functionals we know that G(s1 , . . . , s j +ir j , . . . , sn ) ≤ C(1+|s j |) N , where C and N are independent of s j and r j ∈ [0, 1]. Now Fao ,...an (s1 , . . . , s j + ir j , . . . , sn ) = n G(s1 , . . . , s j + ir j , . . . , sn ) exp Br 2j (n + 1 − j) − B (s1 + · · · + sk )2 + iθ , k=1

(43) where θ is real. Thus from the exponential damping factor in s j we conclude that Fa( j),...a o n is bounded. By the maximum modulus principle (applied after first mapping S j to a unit disk by the Schwartz mapping principle), the bound of |Fa( j),...a | is attained on o n the boundary of S j (this also follows from the Phragmen Lindelöf theorem, cf. p. 138

742


in So on the real

([6]).

part of the boundary of S j we have already from above that

F j) (s1 , . . . , sn ) ≤ A a0 · · · an and by the KMS–condition and translation ao ,...an invariance of ϕ we have on the other part

G(s1 , . . . , s j + i, . . . , sn ) =

ϕ αs1 +···s j (a j ) · · · αs1 +···+sn (an )a0 αs1 (a1 ) · · · αs1 +···+s j−1 (a j−1 )

n ≤ A exp B (s1 + · · · + sk )2 · a0 · · · an ,

hence by (43):

k=1

( j)

F (s , . . . , s j + i, . . . , sn ) ≤ A e Bn a0 · · · an and thus as e Bn > 1, ao ,...an 1

( j)

F (s1 , . . . , z j , . . . , sn ) ≤ A e Bn a0 · · · an =: C for all z j ∈ S j . a ,...a o

n

Now define Hα (z 1 , . . . , z n ) := [Fao ,...an (z 1 , . . . , z n ) − eiα C]−1 , where α ∈ [0, 2π ] for (z 1 , . . . , z n ) ∈ Tn . Then by the estimates above for |Fa( j),...a |, each map z j → o n Hα (s1 , . . . , z j , . . . , sn ) is analytic on the strip S j , and thus by the Flat Tube Theorem 6.1, Hα has a unique extension as an analytic function to Tn , and hence cannot have any singularities in Tn , i.e. Fao ,...an (z 1 , . . . , z n ) = eiα C for all α. By continuity of F, the image set Fao ,...an (Tn ) must be connected. By assumption, this set has some points inside the circle |z| = C, hence the entire image set is inside the circle |z| = C, i.e.

Bn

F a0 · · · an ∀ (z 1 , . . . , z n ) ∈ Tn and ai ∈ A0 (J ). ao ,...an (z 1 , . . . , z n ) ≤A e Consider now the Chern character formula (19): τn (a0 , . . . , an ) := i n ϕ a0 αis1 δγ (a1 ) αis2 δ(a2 ) αis3 δγ (a3 ) σn

ai ∈ Dc , · · · αisn δγ n (an ) ds1 · · · dsn , ϕ b0 αir1 (b1 ) · · · αir1 +···+irn (bn ) dr1 · · · drn , = i n n

where we made a change of variables s1 = r1 , s2 = r1 + r2 , . . . , sn = r1 + · · · + rn and substitutions a0 = b0 , b1 = δγ (a1 ), . . . , bn = δγ n (an ) as in Sect. 6, making use of the Flat Tube Theorem. (Note that bi ∈ A0 (J ) ∀ i for some J.) In fact, from the uniqueness part of the extensions to Tn we have that on Tn , # n $ 2 ϕ b0 αz 1 (b1 ) · · · αz 1 +···+z n (bn ) = exp B (z 1 + · · · + z n ) · Fbo ,...bn (z 1 , . . . , z n ), k=1

and so for (z 1 , . . . , z n ) = i(r1 , . . . , rn ) ∈ in we have n

(r1 +· · ·+rn )2 + Bn A b0 · · · bn ,

ϕ b0 αir1 (b1 ) · · · αir1 +···+irn (bn ) ≤ exp −B k=1

hence

τn (a0 , . . . , an ) ≤ A e Bn b0 · · · bn ≤ A e Bn a0 ∗ · · · an ∗ , n! n!


743

where we used first, that the volume of |n | = 1/n!, and second, that b j ≤ a j ∗ because b j = δγ (a j ) = −γ δ(a j ) for j > 0. Thus τn ∗ ≤ A e Bn n! and hence it is 1/n clear that lim n 1/2 τ ∗ ≤ e B lim n 1/2 (A/n!)1/n = 0 by Stirling’s formula, which n→∞ n→∞ concludes the proof. Proof of Theorem 6.4. By Theorem 6.3 we already have the entireness condition for τ so it is only necessary to prove the cocycle condition for ai ∈ Dc : (bτn−1 )(a0 , . . . , an ) = (Bτn+1 )(a0 , . . . , an ) ,

n = 1, 3, 5, , . . .

with b and B given by Eqs. (22) and (23). We will roughly follow the technique used in [12], but due to the different analytic properties of our model, we will need to go explicitly through the steps. In order to manipulate the expressions involved with Eq. (24), we need the results in the following lemma. Lemma 8.5. Let bi ∈ A0 , then: (i) for all (s1 , . . . , sn ) ∈ σn we have ϕ b0 αis1 (b1 ) · · · αisn (bn ) = ϕ γ (bn ) αi(1−sn ) (b0 ) αi(1−sn +s1 ) (b1 ) · · · αi(1−sn +sn−1 ) (bn−1 ) .

ϕ b0 αis1 (b1 ) · · · αisn (bn ) ds1 · · · dsn σn = ϕ γ (bn ) αis1 (b0 ) · · · αisn (bn−1 ) ds1 · · · dsn . σn (iii) The functions (t1 , . . . , tn ) → ϕ b0 αt1 (b1 ) · · · δ αtk (bk ) · · · αtn (bn ) and (t1 , . . . , tn ) → ϕ b0 αt1 (b1 ) · · · γ αtk (bk ) · · · αtn (bn ) both have analytic continuations to Rn + iσn , and for these we have (ii)

ϕ b0 αis1 (b1 ) · · · δ αisk (bk ) · · · αisn (bn ) = ϕ b0 αis1 (b1 ) · · · αisk δ(bk ) · · · αisn (bn ) and ϕ b0 αis1 (b1 ) · · · γ αisk (bk ) · · · αisn (bn ) = ϕ b0 αis1 (b1 ) · · · αisk γ (bk ) · · · αisn (bn ) . (iv) For j = 2, . . . , n we have:

∂ ϕ b0 αis1 (b1 ) · · · αisn+1 (bn+1 ) ds1 · · · dsn+1 σn+1 ∂s j ϕ b0 αis1 (b1 ) · · · αis j (b j b j+1 ) · · · αisn (bn+1 ) = σn

−ϕ b0 αis1 (b1 ) · · · αis j−1 (b j−1 b j ) · · · αisn (bn+1 ) ds1 · · · dsn , (44)

744


∂ ϕ b0 αis1 (b1 ) · · · αisn+1 (bn+1 ) ds1 · · · dsn+1 σn+1 ∂s1 ϕ b0 αis1 (b1 b2 ) αis2 (b3 ) · · · αisn (bn+1 ) = σn − ϕ b0 b1 αis1 (b2 ) · · · αisn (bn+1 ) ds1 · · · dsn ,

(45)

∂ ϕ b0 αis1 (b1 ) · · · αisn+1 (bn+1 ) ds1 · · · dsn+1 σn+1 ∂sn+1 ϕ γ (bn+1 ) b0 αis1 (b1 ) · · · αisn (bn ) = σn

− ϕ b0 αis1 (b1 ) · · · αisn (bn bn+1 ) ds1 · · · dsn .

(46)

Proof. (i) Recall that the left-hand side is defined by the analytic extension of the function Ft1 ··· ,tn (b0 , · · · , bn ) := ϕ b0 αt1 (b1) · · · αtn (bn ) to the tubeRn + iσn by the KMS–con-

dition and Flat Tube theorem, so ϕ b0 αis1 (b1 ) · · · αisn (bn ) := Fis1 ··· ,isn (b0 , · · · , bn ). By the invariance ϕ ◦ αt = ϕ we have Ft1 ··· ,tn (b0 , · · · , bn ) = ϕ αt (b0 ) αt+t1 (b1 ) · · · αt+tn (bn ) = Ft, t+t1 ··· , t+tn (11, b0 , · · · , bn )=Ft+t1 ··· , t+tn (αt (b0 ), b1 , · · · , bn ).

The latter function has an analytic continuation in the variables (t + t1 , . . . , t + tn ) to Rn + iσn and from the former function it also has an analytic extension in t to the strip R + i[0, 1]. Thus by the flat tube theorem we get a unique analytic extension to all of Rn+1 + iσn+1 . Put t j = is j , where s ∈ σn and t = i(1 − sn ), then Fis1 ··· ,isn (b0 , · · · , bn ) = Fi(1−sn ), i(1−sn +s1 ),··· , i(1−sn +sn−1 ), i (11, b0 , · · · , bn ) , which is justified because we have that the variables r = (r1 , · · · , rn ) := (1 − sn , 1 − sn + s1 , . . . , 1 − sn + sn−1 ) ∈ σn . Now the function Fir1 ,··· , irn , i (11, b0 , · · · , bn ) is obtained from the analytic extension of Ft1 ,··· , tn , v (11, b0 , · · · , bn ) , ti , v ∈ R to Rn+1 + iσn+1 . By the KMS–condition: Ft1 ,··· , tn , i (11, b0 , · · · , bn ) = ϕ bn γ αt1 (b0 ) · · · αtn (bn−1 ) = ϕ γ (bn ) αt1 (b0 ) · · · αtn (bn−1 ) = Ft1 ,··· , tn (γ (bn ), b0 , · · · , bn−1 ). Thus by uniqueness of the analytic continuations we have Fis1 ··· ,isn (b0 , · · · , bn ) = Fir1 ,··· , irn , i (11, b0 , · · · , bn )=Fir1 ,··· , irn (γ (bn ), b0 , · · · , bn−1 ) which is the statement (i) of the lemma.


745

(ii) By part (i) we have: ϕ b0 αis1 (b1 ) · · · αisn (bn ) ds1 · · · dsn σn

= =

σn

σn

ϕ γ (bn ) αi(1−sn ) (b0 ) αi(1−sn +s1 ) (b1 ) · · · αi(1−sn +sn−1 ) (bn−1 ) ds1 · · · dsn ϕ γ (bn ) αir1 (b0 ) · · · αirn (bn−1 ) dr1 · · · drn ,

making use of the change of variables s → r above (with Jacobian = 1), and the fact that r ∈ σn iffs ∈ σn . (iii) Since ϕ b0 αt1 (b1 ) · · · δ αtk (bk ) · · · αtn (bn ) = ϕ b0 αt1 (b1 ) · · · αtk δ(bk ) · · · αtn (bn ) and the latter obviously has an analytic extension to Rn + σn , the claim follows. Likewise for the other one. (iv) For 2 ≤ j ≤ n we have s j+1 ∂ ϕ b0 αis1 (b1 ) · · · αisn+1 (bn+1 ) ds j s j−1 ∂s j = ϕ b0 αis1 (b1 ) · · · αis j−1 (b j−1 ) αis j+1 (b j ) αis j+1 (b j+1 ) · · · αisn+1 (bn+1 ) − ϕ b0 αis1 (b1 ) · · · αis j−1 (b j−1 ) αis j−1 (b j ) αis j+1 (b j+1 ) · · · αisn+1 (bn+1 ) = ϕ b0 αis1 (b1 ) · · · αis j−1 (b j−1 ) αis j+1 (b j b j+1 ) · · · αisn+1 (bn+1 ) − ϕ b0 αis1 (b1 ) · · · αis j−1 (b j−1 b j ) αis j+1 (b j+1 ) · · · αisn+1 (bn+1 ) from which Eq. (44) follows by a change of label of the integration variables. For Eq. (45) we substitute j = 1, s j−1 = 0 into the last equation. For Eq. (46) we substitute j = n + 1, s j+1 = 1 into the last equation to get ∂ ϕ b0 αis1 (b1 ) · · · αisn+1 (bn+1 ) ds1 · · · dsn+1 σn+1 ∂sn+1 ϕ b0 αis1 (b1 ) · · · αisn (bn ) αi (bn+1 ) = σn

− ϕ b0 αis1 (b1 ) · · · αisn (bn bn+1 ) ds1 · · · dsn ϕ γ (bn+1 ) b0 αis1 (b1 ) · · · αisn (bn ) = σn

− ϕ b0 αis1 (b1 ) · · · αisn (bn bn+1 ) ds1 · · · dsn , making use of part (i) for the KMS–condition.

Let us begin with the right-hand side of our desired Eq. (24). From the definition (23) we have for ai ∈ Dc via δ(11) = 0 that:

746


Bτn+1 (a0 , . . . , an ) = i n+1 +

n

σn+1

ds1 · · · dsn+1 ϕ αis1 (δγ a0 ) · · · αisn+1 (δγ n+1 an )

(−1)n j ϕ αis1 (δγ 2 an+1− j ) · · ·

j=1

· · · αis j (δγ j+1 an ) αis j+1 (δγ j+1 a0 ) · · · αisn+1 (δγ n+1 an− j ) .

We can now use Lemma 8.5(ii) in all the terms on the right- hand side to bring the factor with a0 to the front: n+1 ds1 · · · dsn+1 ϕ δγ (a0 ) αis1 (δγ 2 a1 ) · · · Bτn+1 (a0 , . . . , an ) = i σn+1

n · · · αisn (δγ n+1 an ) αisn+1 (11) + (−1)n j ϕ δγ j+1 (a0 ) αis1 (δγ j+2 a1 ) · · · j=1

· · · αisn− j (δγ n+1 an− j ) αisn− j+1 (11) αisn− j+2 (γ δγ 2 an+1− j ) · · · αisn+1 (γ δγ j+1 an )

.

Now substitute ϕ → ϕ ◦ γ j+1 in the last term, and use γ ◦ δ=−δ ◦ γ and Lemma 8.5(iii): n+1 ds1 · · · dsn+1 (−1)n+1 ϕ δ(a0 ) αis1 (δγ a1 ) · · · αisn (δγ n an ) αisn+1 (11) =i σn+1

+

n

(−1)n j ϕ (−1) j+1 δ(a0 ) (−1) j+1 αis1 (δγ a1 ) · · ·

j=1

· · · (−1) j+1 αisn− j (δγ n− j an− j ) αisn− j+1 (11) (−1) j αisn− j+2 (δγ j an+1− j ) · · · · · · (−1) j αisn+1 (δγ an ) ds1 · · · dsn+1 ϕ δ(a0 ) αis1 (δγ a1 ) · · · αisn (δγ n an ) αisn+1 (11) = i n+1 (−1)n+1 σn+1

+

n

ϕ δ(a0 ) αis1 (δγ a1 ) · · ·

j=1

· · · αisn− j (δγ n− j an− j ) αisn− j+1 (11) αisn− j+2 (δγ j an+1− j ) · · · αisn+1 (δγ an )

.

Now recall that τ := (τ0 , 0, −τ2 , 0, τ4 , . . .) ∈ C(Dc ), and hence we may assume that n is odd in the preceding expression (if n is even, B τn+1 = 0 ). Thus n ds1 · · · dsn+1 ϕ δ(a0 ) αis1 (δγ a1 ) · · · Bτn+1 (a0 , . . . , an ) = σn+1

j=0

· · · αisn− j (δγ n− j an− j ) αisn− j+1 (11) αisn− j+2 (δγ n+1− j an+1− j ) · · · αisn+1 (δγ n an )

.

Since αisn− j+1 (11) = 11, we can do the integrals w.r.t. sn− j+1 , and so using 0 ≤ s1 ≤ s2 ≤ · · · ≤ sn+1 ≤ 1 and a relabelling of variables, we get


Bτn+1 (a0 , . . . , an ) =

σn

ds1 · · · dsn

n−1 (sn− j+1 − sn− j ) ϕ δ(a0 ) αis1 (δγ a1 ) · · · j=1

· · · αisn− j (δγ n− j an− j ) αisn− j+1 (δγ n+1− j an+1− j ) · · · αisn (δγ n an ) + (s1 + 1 − sn ) ϕ δ(a0 ) αis1 (δγ a1 ) · · · αisn (δγ n an )

=

747

σn

ds1 · · · dsn ϕ δ(a0 ) αis1 (δγ a1 ) · · · αisn (δγ n an ) .

(47)

Next, we turn our attention to the left hand- side of our desired Eq. (24). Observe first that we have τn γ a0 , . . . , γ an = (−1)n τn a0 , . . . , an because ϕ γ (a0 ) αis1 (δγ (γ a1 )) · · · αisn (δγ n (γ a1 )) = (−1)n ϕ a0 αis1 (δγ a1 ) · · · αisn (δγ n an ) , since δ ◦ γ = −γ ◦ δ, ϕ ◦ γ = ϕ and by Lemma 8.5(iii). Thus τ ◦γ = τ , and so we have a = γ a in definition (22). An application of definition (22) to the left-hand side of Eq. (24) yields:

b τn−1 (a0 , . . . , an ) = i n−1 · · · αis

n−1

σn−1

ds1 · · · dsn−1

n−1

(−1) j ϕ a0 αis (δγ a1 ) · · · αis (δγ j (a j a j+1 )) · · ·

j=0

(δγ n−1 an ) + (−1)n ϕ (γ an ) a0 αis (δγ a1 ) · · · αis 1

1

n−1

j

(δγ n−1 an−1 ) .

(48)

We examine the terms in this sum more closely: j = 0:

σn−1

ds1 · · · dsn−1 ϕ a0 a1 αis1 (δγ a2 ) · · · αisn−1 (δγ n−1 an ) ,

j = 1: −

σn−1

ds1 · · · dsn−1 ϕ a0 αis1 (δγ (a1 a2 )) αis2 (δa3 ) · · · αisn−1 (δγ n−1 an )

=−

=−

σn−1

ds1 · · · dsn−1 ϕ a0 αis1 (δγ (a1 ) γ a2 + a1 δγ a2 αis2 (δa3 ) · · · · · · αisn−1 (δγ n−1 an )

σn−1

ds1 · · · dsn−1 ϕ a0 αis1 (δγ (a1 ) γ a2 αis2 (δa3 ) · · · αisn−1 (δγ n−1 an )

+ ϕ a0 a1 αis1 (δγ a2 ) αis2 (δa3 ) · · · αisn−1 (δγ n−1 an ) ∂ − ds1 · · · dsn ϕ a0 αis1 (a1 ) αis2 (δγ a2 ) αis3 (δa3 ) · · · αisn (δγ n+1 an ) , ∂s1 σn where we made use of Eq. (45) in the last step. Notice that we get a cancellation between the middle term and the j = 0 term in the sum. For 1 < j ≤ n − 1 we have the terms:

748

D. Buchholz, H. Grundling (−1) j

= (−1) j

σn−1

σn−1

· · · αis

ds1 · · · dsn−1 ϕ a0 αis (δγ a1 ) · · · αis δγ j (a j a j+1 ) · · · αis (δγ n−1 an ) 1 n−1 j ds1 · · · dsn−1 ϕ a0 αis (δγ a1 ) · · · αis (δγ j a j )γ j a j+1 + γ j+1 (a j )δγ j a j+1 · · · 1

n−1

= (−1) j

j

(δγ n−1 an )

σn−1

ds1 · · · dsn−1 ϕ a0 αis (δγ a1 ) · · · αis (δγ j a j )γ j a j+1 · · · αis 1

j

n−1

(δγ n−1 an )

+ ϕ a0 αis (δγ a1 ) · · · αis (δγ j−1 a j−1 )γ j+1 a j · · · αis (δγ n−1 an ) 1 n−1 j−1 ∂ + (−1) j ds1 · · · dsn ϕ a0 αis (δγ a1 ) · · · αis (δγ j+1 a j ) · · · αis (δγ n−1 an ) , n 1 j ∂s σn j

where we made use of Eq. (44). Thus we get for Eq. (48), taking into account cancellations between subsequent terms in the sum, that

b τn−1 (a0 , . . . , an ) = i n−1 (−1)n−1

σn−1

n−1 (−1) j + i n−1 j=1

+ i n−1 (−1)n

σn−1

ds1 · · · dsn−1 ϕ a0 αis1 (δγ a1 ) · · · αisn−1 (δγ n−1 an−1 ) γ n−1 an

∂ ϕ a0 αis1 (δγ a1 ) · · · αis j (δγ j+1 a j ) · · · αisn (δγ n−1 an ) ∂s j σn ds1 · · · dsn−1 ϕ (γ an ) a0 αis1 (δγ a1 ) · · · αisn−1 (δγ n−1 an−1 ) ds1 · · · dsn

n ∂ = (−1) j ds1 · · · dsn ϕ a0 αis1 (δγ a1 ) · · · αis j (δγ j+1 a j ) · · · αisn (δγ n−1 an ) , ∂s j σn j=1

where we made use of Eq. (46) and used the fact that since τ = (τ0 , 0, −τ2 , 0, τ4 , . . .), we may take n to be odd. Then b τn−1 (a0 , . . . , an ) n ∂ j+n = (−1) ds1 · · · dsn ϕ (γ a0 ) αis1 (δa1 ) · · · αis j (δγ j a j ) · · · αisn (δγ n an ) ∂s j σn j=1

n ∂ = ds1 · · · dsn ϕ (γ a0 ) αis1(γ δγ a1 ) · · · αis j−1(γ δγ j−1 a j−1 ) αis j(δγ j a j ) · · · ∂s j σn j=1 · · · αisn (δγ n an ) . (49)

To make further progress, we need the following lemma. Lemma 8.6. Let ai ∈ D S and (s1 , . . . , sn ) ∈ σn , then ϕ δ(a0 ) αis1 (δa1 ) · · · αisn (δan ) n ∂ = ϕ (γ a0 ) αis1 (γ δa1 ) · · · αis j−1 (γ δa j−1 ) αis j (a j ) αis j+1 (δa j+1 ) · · · αisn (δan ) . ∂s j j=1

Proof. A close examination of the proof of Theorem 5.8 shows that we actually proved that


749

2 d ϕ B αt (A)C = i ϕ Bδ 0 (A)C = i ϕ Bδ (A)C 0 dt for all A ∈ D S and B, C ∈ A0 where the right-hand side makes sense because ϕ is strongly regular on R S(R), σ , hence is well defined on E0 . Now from the graded product rule for δ on E0 we get δ(b0 ) B1 · · · Bn = δ(b0 B1 · · · Bn ) −

n

γ (b0 B1 · · · B j−1 ) δ(B j ) B j+1 · · · Bn

j=1

for b0 ∈ D S , B1 , . . . , Bn ∈ A0 . Let Bi = δ(bi ) for bi ∈ D S , then δ(b0 ) δ(b1 ) · · · δ(bn ) n 2 γ b0 δ(b1 ) · · · δ(b j−1 ) δ (b j ) δ(b j+1 ) · · · δ(bn ). = δ b0 δ(b1 ) · · · δ(bn ) − j=1

Hence, using ϕ ◦ δ = 0 we get: n 2 ϕ δ(b0 ) δ(b1 ) · · · δ(bn ) = − ϕ γ b0 δ(b1 ) · · · δ(b j−1 ) δ (b j ) δ(b j+1 ) · · · δ(bn ) j=1

d

ϕ γ b0 δ(b1 ) · · · δ(b j−1 ) αt (b j ) δ(b j+1 ) · · · δ(bn ) . 0 dt n

=i

j=1

Now make the replacements b0 → a0 , bi → αti (ai ), i = 1, . . . , n for ai ∈ D S and use the fact that αt ◦ δ = δ ◦ αt to find that: ϕ δ(a0 ) αt1 (δ a1 ) · · · αtn (δ an ) = i

n ∂ ϕ γ (a0 ) αt1 (γ δ a1 ) · · · αt j−1 (γ δ a j−1 ) αt j (a j ) αt j+1 (δ a j+1 ) · · · αtn (δ an ) , ∂t j j=1

where we replaced δ by δ because it is now evaluated on D S only. Now by the KMS– condition, analyticity, flat tube theorem and a complex linear change of variables, we find as in Sect. 6 that the functions (t1 , . . . , tn ) → ϕ δ(a0 ) αt1 (δ a1 ) · · · αtn (δ an ) , (t1 , . . . , tn )→ϕ γ (a0 ) αt1 (γ δ a1 ) · · · αt j−1 (γ δ a j−1 ) αt j (a j ) αt j+1 (δ a j+1 ) · · · αtn (δ an ) extend analytically to the flat tube Tn := Rn + iσ n such that ϕ δ(a0 ) αz 1 (δ a1 ) · · · αz n (δ an ) =i

n ∂ ϕ γ (a0 ) αz 1 (γ δ a1 )· · ·αz j−1 (γ δ a j−1 ) αz j (a j ) αz j+1 (δ a j+1 )· · ·αz n (δ an ) . ∂z j j=1

In the case that z k = isk where (s1 , . . . , sn ) ∈ σ n , we can use ∂ ∂z k = −i ∂ ∂sk to obtain from the last equation the statement of the lemma.

750


Application of the lemma to Eq. (49) then produces ds1 · · · dsn ϕ δ(a0 ) αis1 (δγ a1 ) · · · αisn (δγ n an ) b τn−1 (a0 , . . . , an ) = σn

= Bτn+1 (a0 , . . . , an )

by Eq. (47) and hence τ is a cyclic cocycle. Acknowledgements. We gratefully acknowledge discussions with Arthur Jaffe, Roberto Longo and Hajime Moriya on various aspects of supersymmetry. DB wishes to thank the Department of Mathematics of the University of New South Wales and HG wishes to thank the Institute for Theoretical Physics of the University of Göttingen for hospitality and financial support which facilitated this research. The work was also supported in part by the FRG grant PS01583.

References 1. Araki, H.: Bogoliubov automorphisms and Fock representations of canonical anticommutation relations. Contemp. Math. 62, 23–141 (1987) 2. Bros, J., Buchholz, D.: Towards a relativistic KMS-condition. Nucl. Phys. B 429, 291–318 (1994) 3. Buchholz, D., Longo R.: Graded KMS-functionals and the breakdown of supersymmetry. Adv. Theor. Math. Phys. 3, 615–626 (2000) [Addendum: ibid. 6, 1909–1910 (2000)] 4. Buchholz, D., Ojima, I.: Spontaneous collapse of supersymmetry. Nucl. Phys. B 498, 228–242 (1997) 5. Connes, A.: Entire cyclic cohomology of Banach algebras and characters of θ –summable Fredholm modules. K-Theory 1, 519–548 (1988) 6. Conway, J.W.: Functions of one complex variable. Berlin Heidelberg-New York Springer 1978 7. Folland, G.: Real Analysis. New York: John Wiley & Sons, 2000 8. Fuchs, J.: Thermal and superthermal properties of supersymmetric field theories. Nucl. Phys. B 246, 279–301 (1984) 9. Damak, M., Georgescu, V.: Self-adjoint operators affiliated to C*-algebras. Rev. Math. Phys. 16, 257–280 (2004) 10. Van Hove, L.: Supersymmetry and positive temperature for simple systems. Nucl. Phys. B 207, 15–28 (1982) 11. Jaffe, A., Lesniewski, A., Osterwalder, K.: On super-KMS functionals and entire cyclic cohomology. K-theory 2, no. 6, 675–682 (1989) 12. Jaffe, A., Lesniewski, A., Osterwalder, K.: Quantum K-theory. Commun. Math. Phys. 118, 1–14 (1988) 13. Jaffe, A., Osterwalder, K.: Ward identities for non–commutative geometry. Commun. Math. Phys. 132, 119–130 (1990) 14. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras I. New York: Academic Press, 1983 15. Kastler, D.: Cyclic cocycles from graded KMS functionals. Commun. Math. Phys. 121, 345–350 (1989) 16. Kishimoto, A., Nakamura, H.: Super-derivations. Commun. Math. Phys. 159, 15–27 (1994) 17. Lang, S.: SL2 (R). Reading, Mass.-London-Amsterdam: Addison-Wesley Publishing Co., 1975 18. Longo, R.: Notes for a quantum index theorem. . Commun. Math. Phys. 222, 45–96 (2001) 19. Manuceau, J., Verbeure, A.: Quasi-free states of the CCR–algebra and Bogoliubov transformations. . Commun. Math. Phys. 9, 293–302 (1968) 20. Nehari, Z.: Conformal mapping. Reprinting of the 1952 edition. New York: Dover Publications, Inc., 1975 21. Plymen, R., Robinson, P.: Spinors in Hilbert space. Cambridge: Cambridge University Press, 1994 22. Reed, M., Simon, B.: Methods of mathematical physics I: Functional analysis. New York-London-Sydney: Academic Press, 1980 23. Rocca, F., Sirugue, M., Testard, D.: On a class of equilibrium states under the Kubo-Martin-Schwinger boundary condition. . I. Fermions. Commun. Math. Phys. 13, 317–334 (1969) 24. Weidmann, J.: Linear operators in Hilbert spaces. New York-Heidelberg-Berlin: Springer-Verlag, 1980 25. Woronowicz, S.: C*-algebras generated by unbounded elements. Rev. Math. Phys. 7, 481–521 (1995) 26. Yosida, K.: Functional Analysis. Berlin-Heidelberg-New York: Springer-Verlag, 1980 Communicated by Y. Kawahigashi


Communications in


On the Strong Coupling Limit of the Faddeev-Hopf Model J. M. Speight, M. Svensson School of Mathematics, University of Leeds, Leeds, LS2 9JT, U.K. E-mail: [email protected]; [email protected] Received: 18 May 2006 / Accepted: 13 November 2006 Published online: 27 March 2007 – © Springer-Verlag 2007

Abstract: The variational calculus for the Faddeev-Hopf model on a general Riemannian domain, with general Kähler target space, is studied in the strong coupling limit. In this limit, the model has key similarities with pure Yang-Mills theory, namely conformal invariance in dimension 4 and an infinite dimensional symmetry group. The first and second variation formulae are calculated and several examples of stable solutions are obtained. In particular, it is proved that all immersive solutions are stable. Topological lower energy bounds are found in dimensions 2 and 4. An explicit description of the spectral behaviour of the Hopf map S 3 → S 2 is given, and a conjecture of Ward concerning the stability of this map in the full Faddeev-Hopf model is proved.

1. Introduction Theoretical physics has long been a rich source of geometrically interesting and natural variational problems. The Yang-Mills equations, of deep significance for the differential topology of 4-manifolds [5], and the Yang-Mills-Higgs equations, which have led to interesting results in hyperkähler geometry [1], both originated in elementary particle physics. Harmonic map theory, while not originating in theoretical physics, has found many applications in high energy and condensed matter physics, with physicists frequently contributing genuinely new insights. The purpose of this paper is to present a systematic study of a variational problem arising in the so-called Faddeev-Hopf (or Faddeev-Skyrme) model [7], originally proposed as a model of quark confinement (among other phenomena) in high energy physics. Let M be some Riemannian manifold, representing physical space, and N a Kähler manifold, the target space, with Kähler form ω. The model has a single field φ : M → N , the energy functional (or action functional, in the case where M is euclideanized spacetime The second author was supported by the Swedish Research Council (623-2004-2262).

752

J. M. Speight, M. Svensson

after Wick rotation) being 1 E (φ) = 2

(|dφ|2 + α|φ ∗ ω|2 ), M

α ≥ 0 being a coupling constant. The model of original interest has M = R3 , N = S 2 . The weak coupling limit of this model, α = 0 has of course been intensively studied: it is the harmonic map problem. This is conformally invariant if M has dimension 2. By contrast, we shall study the strong coupling limit, α → ∞, or more precisely, the variational problem for the energy functional 1 E(φ) = lim α −1 E (φ) = |φ ∗ ω|2 . α→∞ 2 M This does not seem to have received systematic study in either the theoretical physics or differential geometry communities. It has been studied in the specific case M = R × S 3 (with a Lorentzian metric, actually) and N = S 2 , C or the hyperbolic plane by de Carli and Ferreira [3]. It has some important similarities with pure Yang-Mills theory. It is invariant under an infinite dimensional group of symmetries, the group of symplectic diffeomorphisms of N , rather as Yang-Mills theory is invariant under gauge transformations. It is also, as we will demonstrate, conformally invariant if M has dimension 4. Both these facts were known to de Carli and Ferreira in the specific context they studied. The most interesting situation physically is when M = S 4 , interpreted as the conformal compactification of R4 . Nontrivial solutions in this case may receive the physical interpretation of instantons in the strong coupling limit of the Faddeev-Hopf model in (3 + 1) dimensions, just as critical points of the Yang-Mills functional on S 4 are interpreted as pure gauge-theory instantons. Such solutions have profound effects on the quantized version of the field theory [16, Ch. 10]. Our motivation for studying this variational problem is twofold. First, simple curiosity prompts us to ask what the geometric character of the variational calculus for this functional is. We will see that both the first and second variation formulae can be given elegant and natural geometric formulations from which strong results quickly follow. For example, we will show that all immersive solutions are stable, and that there are no non-vacuum (i.e., E > 0) immersive solutions in the case M = S 4 , for any choice of target space. Second, we hope that studying one term in the Faddeev-Hopf model in isolation will give valuable insight into the finite coupling model. Indeed, we will identify a large class of critical points of E which are also harmonic maps, and hence critical points of E for all α. In particular, we are able to prove a stability conjecture of Ward concerning the full Faddeev-Hopf model on M = S 3 [20]. The functional E also arises as one term in the so-called baby Skyrme models studied by Zakrzewski and collaborators [14], and our results should find applications in these models too. The rest of the paper is structured as follows. In Sect. 2 we carefully define the functional E, prove that it is conformally invariant in dimension 4, derive the first variation formula (Euler-Lagrange equation) for φ and construct some interesting explicit solutions. In Sect. 3 we consider submersive solutions in particular, identifying a large class of critical submersions which are also harmonic. In Sect. 4 we obtain topological lower bounds on E when M has dimension 2 or 4, and derive the second variation formula in the general case. The results are used to prove the stability of several interesting solutions. Finally, in Sect. 5 the variational calculus for the projection G → G/K onto a Hermitian symmetric space is developed in general, and the results used to show that the Hopf map S 3 → S 2 in particular is stable. A proof of Ward’s conjecture quickly follows from this.

Strong Coupling Limit of the Faddeev-Hopf Model

753

2. The First Variation In this section we assume that (M m , g) is a compact, oriented Riemannian manifold of dimension m. For any vector bundle E over M, we denote by (E) the space of sections of E. Recall that the metric g on M induces a (pointwise) metric on the bundle of p-forms on M, defined by α, β =

1 p!

m

α(ei1 , . . . , ei p )β(ei1 , . . . , ei p ),

i 1 ,...,i p =1

where e1 , . . . , em is a local orthonormal frame on M. By using the Hodge ∗-operator, we get the relation α ∧ ∗β = α, β ∗1 for any two p-forms α and β; here ∗1 is the volume element on M. Integrating this inner product over M gives a global L 2 -product α, β ∗1 = α ∧ ∗β (α, β ∈ (∧ p T ∗ M), α, β L 2 = M

M

with corresponding norm α 2L 2 = α, α L 2 . With respect to this L 2 -product, the exterior differentiation operator d : (∧ p T ∗ M) → (∧ p+1 T ∗ M) has the adjoint operator δ : (∧ p T ∗ M) → (∧ p−1 T ∗ M), δα = (−1)m+mp+1 ∗ d ∗ α. We will also be using the musical isomorphisms on M which are defined as follows: : (T M) → (T ∗ M), X = g(X, ·), = −1 . Let (N n , h, J ) be a compact Kähler manifold of real dimension n and with Kähler form ω = h(J ·, ·). For a smooth map φ : M → N we define the energy functional 1 ∗ 2 1 E(φ) = φ ω L 2 = φ ∗ ω ∧ ∗φ ∗ ω. 2 2 M Any map φ for which E(φ) = 0, the minimum possible, will be called a vacuum solution or vacuum of the theory. Clearly φ is a vacuum if and only if φ ∗ ω = 0 everywhere, that is, if φ is isotropic. We begin our investigation of E(φ) by verifying that it is, like the Yang-Mills functional, invariant under conformal changes of g if M has dimension 4. Proposition 2.1. Assume that φ : M → N is a map from a 4-dimensional Riemannian manifold to a Kähler manifold. Then the functional E(φ), and therefore also the EulerLagrange equation for φ, is invariant under conformal changes of the metric on M.

754


Proof. Assume that g is a Riemannian metric on M and that g˜ = λ2 g, where λ is some positive function on M. Denote by ·, ·g˜ the metric induced by g˜ on 2-forms and let ∗˜ 1 be the corresponding volume element. Then ∗˜ 1 = λ4 ∗1 and α, αg˜ = λ−4 α, α, for any 2-form α on M. Hence the form α, α ∗1 remains unchanged.

Next we derive the Euler-Lagrange equation for E(φ) in the general case. Proposition 2.2. For a smooth variation φt : M → N of φ with variational vector field X ∈ (φ −1 T N ), we have d E(φt ) t=0 = ω(X, dφ( δφ ∗ ω)) ∗1 . dt M For the proof, let us recall the following simple result. Lemma 2.3 (Homotopy Lemma). Let M and N be two manifolds and φt : M → N a smooth family of maps. For any closed 2-form η on N we have ∂ ∗ ∂φt φt η = d φt∗ ι( )η . ∂t ∂t Here ι denotes the interior product. For a proof of this lemma see, e.g., [6, p. 49]. ∂ commutes with the Hodge ∗-operator on ∂t ∗ ∗ ∧ T M. By the Homotopy Lemma we have

Proof of Proposition 2.2. It is obvious that

1 ∂ 1 ∂ 1 ∂ φ ∗ ω ∧ ∗φt∗ ω = ( φt∗ ω) ∧ ∗φt∗ ω + φt∗ ω ∧ ∗( φt∗ ω) 2 ∂t t=0 t 2 ∂t 2 ∂t 1 1 = dφ ∗ ι X ω ∧ ∗φt∗ ω + φ ∗ ω ∧ ∗φ ∗ ι X ω 2 2 = dφ ∗ ι X ω, φ ∗ ω ∗1 . Therefore ∂ E(φt )t=0 = ∂t

φ ∗ ι X ω, δφ ∗ ω ∗1, M

and φ ∗ ι X ω, δφ ∗ ω = φ ∗ ι X ω( δφ ∗ ω) = ω(X, dφ( δφ ∗ ω)). This proves the proposition.

Corollary 2.4. The map φ : M → N is a critical point for the functional if and only if δφ ∗ ω ∈ ker dφ everywhere on M.


755

Example 2.5. Assume that M = N and φ : N → N is the identity map. Then φ ∗ ω = ω, and δω = 0. Hence φ is a critical point for the functional. Remark 2.6. Assume that φ is smooth and is immersive on a dense set, that is, for all x in a dense subset of M, the differential dφx : Tx M → Tφ(x) N is injective. By the corollary, if φ is a critical point of the functional, then the 1-form δφ ∗ ω vanishes almost everywhere, and hence vanishes everywhere by continuity. Hence the 2-form φ ∗ ω is co-closed. Since it is obviously closed, we see that an almost everywhere immersive map is a critical point of the functional if and only if φ ∗ ω is a harmonic 2-form. In particular, when H 2 (M, R) = 0, the only immersive critical points defined on M are isotropic immersions, i.e., maps for which φ ∗ ω = 0. As previously remarked, such a map has E(φ) = 0, and hence is a vacuum solution of the field theory. The set of vacuum solutions of this theory is unusually rich. Example 2.7. The map φ : S 4 → CP 4 , defined as the 2-fold covering by S 4 of RP 4 followed by the natural embedding of RP 4 to CP 4 is clearly isotropic, hence a vacuum. An interesting question is whether this vacuum is path connected, through vacua, to the trivial vacuum φ = constant. One suspects not. Certainly the set of vacua may fail to be path connected. Consider the zero section of T S n , equipped with the Stenzel metric [18]. This is manifestly an isotropic immersion i : S n → T S n , hence a vacuum. Given any smooth map φ : S n → S n , the composition i ◦ φ is still isotropic, hence a vacuum. But if the degree of φ is not unity, then i and i ◦ φ are not even homotopic, much less path connected through isotropic maps. Hence the set of vacua in the case M = S n and N = T S n is not path connected. Of primary physical interest, given their physical interpretation as instantons, are smooth anisotropic critical points on M = S 4 which minimize E(φ) within their homotopy class. In particular, one would like a smooth anisotropic minimizer in the nontrivial class of π4 (S 2 ), a pure Faddeev-Hopf instanton. In fact, it remains an open question whether smooth anisotropic critical points exist on S 4 at all, for any choice of target space. A standard starting point for finding special solutions is the use of symmetry reduction. A fundamental difficulty in exploiting symmetry reduction is raised by Remark 2.6: symmetry reductions of the variational problem on S 4 tend to produce only maps which are either trivial or immersive. The best we have managed is a smooth solution mapping the punctured hemisphere into CP 2 , two copies of which can be glued together to give a continuous map S 4 → CP 2 which, away from the poles and the equator of S 4 , is smooth and satisfies the field equation, and has finite total energy. This map is constructed in the next example. Example 2.8. Let M = S 4 and N = CP 2 . The twice punctured 4-sphere is conformally equivalent to the cylinder R × SU(2) given the metric g = dt 2 + σ12 + σ22 + σ32 , where σi are the usual left-invariant one forms on SU(2). So we may seek critical points on R × SU(2) satisfying appropriate boundary conditions as |t| → ∞. Let I = (a, b) ⊂ R be any open interval and Q be the Banach manifold of, for example,

756


C 2 maps I × SU(2) → CP 2 . Then E : Q → R is C 1 , and there is a natural action of SU(2) × V4 on Q given by (U,1) U φ(t, X ) → φ(t, U X ) 1 ⎡ ⎤ 1 −1 (1,P1 ) X ⎣ −1 ⎦ X φ(t, U X ) φ(t, X ) → 1 1 1 (1,P2 )

φ(t, X ) → φ(t, X ), where P1 , P2 generate the Viergruppe V4 = {1, P1 , P2 , P1 P2 }. The set Q 0 of fixed points of this action consists of maps of the form X [α(t) : 0 : 1], φ(t, X ) = 1 where α : I → S 1 = R∪{∞}. Since SU(2)× V4 is compact, we may apply the principle of symmetric criticality [13] to deduce that any critical point of E| Q 0 is automatically a critical point of E. Routine calculation shows that E| Q 0 (α) = π 2

I

2α 2 α˙ 2 α4 + dt, (1 + α 2 )4 2(1 + α 2 )2

where˙denotes differentiation with respect to t. This may be thought of as the action of a one-dimensional Lagrangian mechanical system. By invariance under t translation, all solutions α(t) conserve the quantity H=

α 2 α˙ 2 α4 − . (1 + α 2 )4 4(1 + α 2 )2

If we wish φ to extend to S 4 we should insist that α, α˙ → 0 as t → ∞, so only solutions with H = 0 are of interest. The H = 0 level curve in the (α, α) ˙ plane is 4α˙ 2 = α 2 (1 + α 2 )2 , whence one finds solutions α+ : (0, ∞) → R and α− : (−∞, 0) → R, α+ (t) = √

1 et − 1

,

α− (t) = √

−1 e−t − 1

.

Gluing these together gives a continuous map S 4 → CP 2 , t |t| X [1 : 0 : φ(t, X ) = e − 1], 1 |t| of total energy E = π 2 , which away from t = ±∞ and 0 × SU(2), is smooth and solves the field equation. Clearly this solution fails to be globally smooth. Remark 2.9. One can find global, smooth, anisotropic, finite energy critical maps on R4 if one equips it with a metric outside the Euclidean conformal class. An example is given in the next section, Example 3.2.


757

Remark 2.10. De Carli and Ferreira have constructed an ingenious symmetry reduction in the case M = R × S 3 (Lorentzian in their original version, but the same reduction works in the Riemannian case) and N = S 2 , by imposing invariance under a T 2 × Z2 group of symmetries [3]. This reduces the field equation, not to a nonlinear ODE as in the example above, but to a linear elliptic PDE for a single real function on R2 . Unfortunately, only solutions which remain bounded on the whole of R2 give globally well defined maps φ on S 4 , and no such nontrivial solutions exist in the Riemannian case. Remark 2.11. By Derrick’s Theorem [4] our variational problem cannot have any finite energy anisotropic solutions on domain M = Rn for any dimension other than n = 4. This is no longer true if we add a potential term to E. Let V : Y → R be a smooth non-negative function with a minimum value of 0, attained at y0 ∈ Y (and perhaps elsewhere). Then the energy functional 1 ∗ 2 |φ ω| + V ◦ φ , E(φ) = Rn 2 subject to the boundary condition φ → y0 as |x| → ∞, evades Derrick’s Theorem if n = 2 or 3. The Euler-Lagrange equation is −J δ(φ ∗ ω) + (grad V ) ◦ φ = 0. The (ostensibly more physical) case n = 3 does not appear to have been studied previously. The case n = 2 has been studied in detail for Y = C with rotationally invariant potential (V = V (|φ|)) by Piette, Tchrakian and Zakrzewski [15], and for Y = S 2 with a specific choice of potential motivated by condensed matter physics by Gisiger and Paranjape [8]. Piette et al find a Bogomol’nyi type topological lower bound on E, attained by solutions of a first order PDE system, and show, furthermore, that all finite energy solutions of (2.1) satisfy this first order system. They go on to give a method for generating the general solution to this system which works for a wide class of potentials: after a suitable change of coordinates on Y it turns out that φ is a solution if and only if it is an area preserving map C → C. It follows that there are no smooth solutions with | deg(φ)| > 1 (indeed, the explicit solutions of higher degree written down in [15] are not differentiable at the origin). The situation for their model is thus reminiscent of the sine-Gordon model on R in that there exist static unit solitons, but no static multisolitons. The explicit solutions found by Gisiger and Paranjape [8] in their model also fail to be globally differentiable (but here they are singular on closed curves in R2 , not just at a single point, and this holds even for degree ±1). So evading Derrick’s theorem certainly does not guarantee the existence of globally smooth soliton solutions, even where there is nontrivial topology and a “competition” between concentrating and expanding terms in the energy functional. Note that after adding a potential, (2.1) remains non-elliptic. We suspect that this underlying non-ellipticity is the root difficulty of this variational problem. 3. Critical Submersions In light of Corollary 2.4 and Remark 2.6 it is natural to seek critical maps in the case where the dimension of M exceeds that of N . In particular, there exists a large number of interesting critical submersions. Such solutions automatically have non-vanishing energy, so are not vacua. We begin with the simple example of projection on a (possibly) warped product, then reformulate the first variation formula in a way better suited to submersions.

758


Example 3.1. Assume that (P, k) and (N , h) are two compact Riemannian manifolds and that f : P → R is a positive, smooth function. The warped product of (P, k) and (N , h) by f is the manifold P × N with the Riemannian metric g = k + f 2 h. Assume further that (N , h, ω) is Kähler. Then the projection map φ : P × N → N onto the second coordinate is critical. To prove this, let ∗ P and ∗ N be the Hodge star operators on P and N , respectively, so that the volume form on P is ∗ P 1. Then, as is easily seen, ∗φ ∗ ω = f 2(n−2) ∗ P 1 ∧ ∗ N ω =

f 2(n−2) ∗ P 1 ∧ ωn−1 , (n − 1)!

where n = dimC N . As f is a function only on P, d f ∧ ∗ P 1 = 0. Furthermore, ∗ P 1 and ωn−1 are obviously closed. Hence δφ ∗ ω = 0. Example 3.2. One can use projection on a product to construct global, smooth, finite energy solutions on M = R4 if one equips it with a metric outside the Euclidean conformal class. For example, let g = (1 + x32 + x42 )2 (d x12 + d x22 ) + (1 + x12 + x22 )2 (d x32 + d x42 ). Then (R4 , g) is complete, has infinite volume and is Ricci positive with bounded scalar 2 × S 2 , where S 2 is the punctured unit curvature. It is conformally equivalent to S× × × 2 projected from the puncture, the sphere. In terms of a stereographic coordinate on S× equivalence is x ≡ (x1 + i x2 , x3 + i x4 ). Hence, by Proposition 2.1 and Example 3.1, the projection map R4 x → x1 + i x2 ∈ S 2 is critical and has energy 8π 2 . To proceed further in our analysis of critical submersions, let us denote by ∇ both the Levi-Civita connexion on T M and on T N . Recall that the connexion on T N induces a connexion ∇ φ on φ −1 T N . This connexion, together with the Levi-Civita connexion on T M, induces a connexion on Hom(T M, φ −1 T N ), which we also denote by ∇. The second fundamental form of φ is the covariant derivative of dφ: φ

∇dφ(X, Y ) = ∇ X dφ(Y ) − dφ(∇ X Y )

(X, Y ∈ (T M)).

The map φ is said to be totally geodesic if its second fundamental form vanishes. The tension field of φ is the trace of the second fundamental form: τ (φ) = trace ∇dφ =

m

∇dφ(ei , ei ).

i=1

The map φ is said to be a harmonic map if its tension field vanishes. To simplify our calculations, let us fix a point x ∈ M and an orthonormal frame m which is normal at x, i.e., {ei }i=1 ∇ei e j (x) = 0


759

for all i, j. Assuming that all calculations take place at the point x ∈ M, we can rewrite the Euler-Lagrange equations in the following way: dφ( δφ ∗ ω) =

m

dφ(δφ ∗ ω(e j )e j )

j=1

= −dφ

m

∇ei (φ ∗ ω(ei , e j )e j )

i, j=1

m ei ω(dφ(ei ), dφ(e j )) dφ(e j ) =− i, j=1

=

m m j=1

ω(∇dφ(ei , e j ), dφ(ei )) − ω(τ (φ), dφ(e j )) dφ(e j ).

i=1

We thus define S(φ) =

m m j=1

ω(∇dφ(ei , e j ), dφ(ei )) − ω(τ (φ), dφ(e j )) dφ(e j );

(3.1)

i=1

by our calculation, the map φ is a critical point if and only if S(φ) = 0. Example 3.3. Assume that φ is a Riemannian submersion; thus, at each point x ∈ M, the differential dφx : Tx M → Tφ(x) N maps the space (ker dφx )⊥ ⊆ Tx M isometrically onto Tφ(x) N . We first demonstrate that ∇dφ(X, Y ) = 0

(X, Y ∈ (ker dφx )⊥ ).

Any vector field X on N can be written as dφ( Xˆ ) for some vector field Xˆ on M taking values in (ker dφ)⊥ , and we can always find a local frame for (ker dφ)⊥ of vector fields of the form Xˆ . Thus, it is enough to show that ∇dφ( Xˆ , Yˆ ) = 0

(X, Y ∈ (T N )).

Denoting by ∇ M and ∇ N the Levi-Civita connexions on M and N , respectively, we have for X, Y, Z ∈ (T N ), 1 ˆ ˆ ˆ X g(Y , Z ) + Yˆ g( Xˆ , Zˆ ) − Zˆ g( Xˆ , Yˆ ) g(∇ M Yˆ , Zˆ ) = Xˆ 2 +g([ Xˆ , Yˆ ], Zˆ ) + g([ Zˆ , Xˆ ], Yˆ ) − g([Yˆ , Zˆ ], Xˆ ) 1 = X h(Y, Z ) + Y h(X, Z ) − Z h(X, Y ) 2 +h([X, Y ], Z ) + h([Z , X ], Y ) − h([Y, Z ], X ) N ˆ = h(∇ XN Y, Z ) = g(∇ X Y , Z ).

760


Thus, ˆ ∇dφ( Xˆ , Yˆ ) = ∇ XN Y − dφ(∇ M ˆ Y) = 0 X

(X, Y ∈ (T N )).

Locally, we can choose an orthonormal frame e1 , . . . , em for T M with the property that e1 , . . . , em−n is a local frame for ker dφ and em−n+1 , . . . , em is a local frame for (ker dφ)⊥ . Then S(φ) = −

m

ω(τ (φ), dφ(e j ))dφ(e j ).

j=m−n+1

Thus, φ is critical if and only if φ is harmonic. In fact, using the same local frame for T M gives m−n

τ (φ) = −dφ(

∇e j e j ) = −dφ(H ),

j=1

where H is the mean curvature vector of the fibres of φ. We conclude that a Riemannian submersion is a critical point if and only if it has minimal fibres, and thus is a harmonic morphism, see [2]. Note that such a map, being harmonic, is automatically a critical point of the full Faddeev-Hopf functional for every value of the coupling, not just the infinite coupling limit. For example, the natural projection φ : S 2n+1 → CP n , φ(z) = [z]

(z ∈ S 2n+1 ⊂ Cn+1 )

is a Riemannian submersion with minimal, even totally geodesic, fibres. 4. The Second Variation and Stability In this section we calculate the second variation of the energy functional. Assume that φ : M → N is a critical point of the functional. We define the Hessian of E at φ as Hφ (X, Y ) =

∂ 2 E(φs,t ); ∂s∂t s=t=0

here φs,t is a 2-parameter variation of φ with X = ∂t φs,t |s=t=0 and Y = ∂s φs,t |s=t=0 . Clearly Hφ is a symmetric, bi-linear form on (φ −1 T N ). The map φ is said to be stable if Hφ (X, X ) ≥ 0

(X ∈ (φ −1 T N ));

the index of φ is the dimension of the largest subspace on which Hφ is negative. Clearly, any map which minimizes the energy within its homotopy class is a stable critical point. In some situations it is easy to give lower bounds for the energy.


761

Proposition 4.1. Let M = M 2 be a surface. Then 2 1 φ∗ω . E(φ) ≥ 2Vol(M) M Note that the right-hand side is a homotopy invariant. If φ attains this lower bound then φ is either isotropic (so E(φ) = 0) or has no critical points. Proof. Since M is a surface, φ ∗ ω = f ∗1 for some function f on M. By the CauchySchwarz inequality we have | f | ∗1 ≤ f L 2 Vol(M), M

with equality if and only if f is a constant. Hence 2 1 1 E(φ) = f 2L 2 ≥ | f | ∗1 2 2Vol(M) M 2 2 1 1 ≥ f ∗1 = φ∗ω , 2Vol(M) M 2Vol(M) M and the right-hand side is a homotopy invariant. This lower bound is attained if and only if f is a constant. If f = 0 then φ ∗ ω = 0 so φ is isotropic. If f = 0 then φ has no critical points.

Example 4.2. Assume that φ : M 2 → S 2 is anisotropic and attains the bound. Then φ is necessarily a covering map; since S 2 is simply connected we must have M = S 2 and φ a diffeomorphism. But then M φ ∗ ω = ±Vol(M), so E(φ) = 21 Vol(M). Thus f ≡ ±1, so φ is, up to an orientation reversing isometry, a symplectomorphism of S 2 . Hence the set of maps attaining the bound consists of the orbit of Id : S 2 → S 2 under the group of symplectomorphisms of S 2 and the image of this orbit under reflexion. Example 4.3. Let M = N = T 2 = C/(Z + iZ). Then the bound is attained in every homotopy class by (L ∈ Mat2 (Z)).

φ(x, y) = (x, y)L Proposition 4.4. Let M =

M 4.

Then

E(φ) ≥

1 | 2

φ ∗ (ω ∧ ω) | M

with equality if and only if φ ∗ ω is (anti-)self-dual. Note that the right-hand side is a homotopy invariant. Proof. Since dim M = 4, the Hodge ∗-operator is an involution on the bundle of 2-forms. Thus any 2-form α can be decomposed as α = α+ + α− , where ∗α+ = α+ and ∗α− = −α− . Since α+ and α− are mutually orthogonal we get 2 2 2 2 2 α L 2 = α+ L 2 + α− L 2 ≥ | α+ L 2 − α− L 2 | = | α ∧ α |, M

with equality if and only if α+ = 0 (α is anti-self-dual) or α− = 0 (α is self-dual). The proposition follows once we apply this to α = φ ∗ ω.

762


Remark 4.5. The proposition tells us nothing useful if H 2 (M; R) = 0 or if dim N = 2. In the first case φ ∗ ω is necessarily exact, and hence so is φ ∗ ω ∧ φ ∗ ω = φ ∗ (ω ∧ ω), so we just recover the trivial fact that E(φ) ≥ 0. In the second case ω ∧ ω = 0, so again we deduce only E(φ) ≥ 0. Example 4.6. In the case M = N = T 4 , the bound is attained in each homotopy class, by linear maps with integer coefficients, as in Example 4.3. Let us now find an explicit formula for the Hessian of a critical point. Note that the metric h on T N induces a metric, also denoted by h, on φ −1 T N . Proposition 4.7. Assume that φ is a critical point of the energy functional. Then the Hessian of φ is given by Hφ (X, Y ) = h(X, Lφ Y ) ∗1 (X, Y ∈ (φ −1 T N )), M

where

φ Lφ Y = −J ∇ Z φ Y + dφ( δdφ ∗ ιY ω) and Z φ = δφ ∗ ω.

Remark 4.8. Note that Z φ is the vector field on M which must lie pointwise in ker dφ given that φ is critical, by Corollary 2.4. Proof of Proposition 4.7. Let X s = ∂t φs,t |t=0 . Then, using the Homotopy Lemma and a calculation similar to that of the previous section, 1 ∗ ∂ ∂ 2 ∗ φs,t ω ∧ ∗φs,t ω= dφ ∗ ι X ω ∧ ∗φs∗ ω (4.1) s=t=0 ∂s∂t 2 ∂s s=0 s s ∂ dφ ∗ ι X ω ∧ ∗φ ∗ ω + dφ ∗ ι X ω ∧ ∗dφ ∗ ιY ω. = ∂s s=0 s s When integrating over M, the second term on the right becomes dφ ∗ ι X ω, dφ ∗ ιY ω L 2 = φ ∗ ι X ω, δdφ ∗ ιY ω L 2 . Now note that, using a local orthonormal frame e1 , . . . , em for M, we get φ ∗ ι X ω, δdφ ∗ ιY ω =

m

φ ∗ ι X ω(ei )δdφ ∗ ιY ω(ei ) = ω(X, dφ( δdφ ∗ ιY ω)).

i=1

Let us now look at the first term on the right-hand side of (4.1). Pointwise we have

m ∂ ∂ ∗ ∗ dφ ∗ ι X ω(ei ) δφ ∗ ω(ei ) dφ ι ω, δφ ω = X s s s s s=0 s=0 ∂s ∂s i=1

∂ φ ∗ ι X ω (Z φ ) = ∂s s=0 s s ∂ = ω(X s , dφs (Z φ )) ∂s s=0 φ φ = ω(∇∂ss X s s=0 , dφ(Z φ )) + ω(X, ∇∂ss dφs (Z φ )s=0 ).


763

The first term on the right vanishes since we assume φ to be a critical point. The second term becomes φ φ φ ω(X, ∇ s dφs (Z φ ) ) = ω(X, ∇ s dφs (∂s ) ) = ω(X, ∇ Y ), ∂s

Zφ

s=0

Zφ

s=0

where we used the fact that φ

φ

φ

∇∂ss dφs (Z φ ) = ∇ Zsφ dφs (∂s ) + dφs ([∂s , Z φ ]) = ∇ Zsφ dφs (∂s ), since [∂s , Z φ ] = 0.

Corollary 4.9. Let φ : M → N be a critical point of the energy functional. Then φ ω(Y, ∇ Z φ Y ) ∗1 + dφ ∗ ιY ω 2L 2 (Y ∈ (φ −1 T N )). Hφ (Y, Y ) = M

In particular, φ is stable if Z φ vanishes. Proof. Take a local orthonormal frame e1 , . . . , em for M. Then h(J Y, dφ( δdφ ∗ ιY ω)) =

m

h(J Y, dφ(ei ))δdφ ∗ ιY ω(ei )

i=1

=

m

φ ∗ ιY ω(ei )δdφ ∗ ιY ω(ei )

i=1

= φ ∗ ιY ω, δdφ ∗ ιY ω. When integrated, this becomes dφ ∗ ιY ω 2L 2 , and the proof follows.

Example 4.10. Assume that φ : M → N is a critical immersion. According to Corollary 2.4, Z φ = 0, so φ is stable. In particular, the identity map of any compact Kähler manifold is stable. In the case dim N = 2 or 4, we have the stronger information that Id : N → N globally minimizes E within its homotopy class, by Propositions 4.1 and 4.4. In fact, we can obtain some idea of spec LId , the eigenvalue spectrum of LId , from the following result. Note that LId = J −1 −1 δd J, so that Y is an eigensection of LId with eigenvalue λ if and only if J Y is an eigenform of δd with eigenvalue λ. Proposition 4.11. Denote by spec δd and spec dδ the eigenvalue spectra of the operators δd : (T ∗ M) → (T ∗ M) and dδ : (∧3 T ∗ M) → (∧3 T ∗ M), respectively. Then spec 2 = spec δd ∪ spec dδ, where 2 denotes the Laplacian on 2-forms on M. Hence spec LId = spec δd ⊆ spec 2 in general. Furthermore, if dim M = 2 or dim M = 4, then spec 2 = spec δd = spec LId .

(4.2)

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Proof. Since the Kähler form on M is harmonic, 0 ∈ spec 2 . Assume that λ ∈ spec δd. To show that λ ∈ spec 2 , we may thus assume that λ = 0. Then there is a 1-form α = 0 on M with δdα = λα. Then we must have dα = 0 and so dδdα = λdα, implying that 2 dα = λdα. Hence spec δd ⊆ spec 2 . The proof that spec dδ ⊆ spec 2 is similar. Conversely, assume λ ∈ spec 2 . As 0 ∈ spec δd ∪ spec dδ, we may assume that λ = 0. Then there is a 2-form ξ = 0 with 2 ξ = λξ.

(4.3)

By the Hodge decomposition we can write ξ = ξ H + dα + δβ, where ξ H is a harmonic 2-form, α a 1-form and β a 3-form. By the Hodge decomposition of α and β we see that we may assume that α is coexact and β exact. From Eq. (4.3) it quickly follows that ξ H = 0, dδdα = λdα, δdδβ = λδβ. The second of these equations implies that the 1-form δdα −λα is closed; by assumption it is also coexact, and so it must vanish. If α = 0 we thus have λ ∈ spec δd. On the other hand, the third equation implies that the 3-form dδβ − λβ is coclosed; by assumption it is also exact, so it must vanish. If β = 0 we thus have λ ∈ spec dδ. Since at least one of α and β is non-zero, we must have λ ∈ spec δd ∪ spec dδ. The last statement is obvious when dim M = 2 since any 3-form vanishes. When dim M = 4, the action of dδ on 3-forms is equivalent to the action of δd on 1-forms under the Hodge isomorphism.

It is interesting to compare this with the behaviour of Id : (M, g) → (M, g) in harmonic map theory, where Id is not stable in general (though it is stable if M is Kähler) [10, 17]. The analogous operator to LId is the Jacobi operator JId , whose spectral properties depend crucially on the Ricci curvature of M. There is a formula similar to (4.2), spec JId = spec 1 −

2s , dim M


765

where s is the scalar curvature of M, but it holds only in the case that (M, g) is Einstein. More generally there is no simple relationship between spec JId and spec p . Analytically, JId is elliptic, and so has finite-dimensional kernel. By contrast, ker LId is the space of symplectic vector fields (those Y for which ιY ω is closed), which has infinite dimension. Clearly dim ker Lφ = ∞ for all critical maps due to the invariance of E(φ) under symplectic diffeomorphisms of N . In fact we will see in the next section an example of a critical map φ (the Hopf map S 3 → S 2 ) for which every eigenspace of Lφ has infinite dimension. Example 4.12. In Example 3.1 we proved that the projection of a warped product φ : P ×f N → N is critical when N is a Kähler manifold. This is not an immersion. However, we showed that Z φ = 0, so such a map is always stable nonetheless. The same is true of the critical projection (R4 , g) → S 2 of Example 3.2. 5. The Hopf Map In this section we prove that the Hopf map S 3 → S 2 is stable and calculate the spectrum of its Hessian. We then apply this to prove a conjecture of Ward regarding the full Faddeev-Hopf model. We begin by introducing some Lie group and Lie algebra technicalities regarding symmetric and Hermitian symmetric spaces. We stringently follow the conventions used in [9], to which the reader is referred for definitions and fundamental results on symmetric spaces. Assume that G is a compact, connected, simple Lie group and that K is a compact subgroup of G such that G/K is an irreducible Hermitian symmetric space of compact type. On the Lie algebra level we have the standard orthogonal decomposition g = k + p, where k is the Lie algebra of K and p an Ad K -invariant subspace with the property that [p, p] ⊆ k. It is well known that the Hermitian structure on G/K is induced by the adjoint action of an element in the centre of k; in accordance with earlier notation, we denote this element by J . We provide G with the Riemannian metric induced by the negative of the Killing form (or a suitable multiple thereof), and give G/K the metric which turns the homogeneous projection φ : G → G/K , g → g · o into a Riemannian submersion; here o denotes the identity coset in G/K . The fibres of φ are clearly minimal, even totally geodesic; according to Example 3.3, φ is a critical point of the functional. For simplicity, we denote by ·, · the negative of the Killing form on g. The pullback bundle φ −1 T G/K is isomorphic to the trivial bundle G × p by the map G × p (g, X ) →

d φ(g) exp t X · o ∈ Tφ(g) G/K ; dt t=0

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the metric on φ −1 T G/K corresponds under this isomorphism to the metric on G × p induced by the restriction of ·, · to p. Similarly, we can identify T G with the trivial bundle G × g by left translation, and this gives the following commutative diagram: G×g G×p

∼ =

∼ =

/ TG

dφ

/ φ −1 T G/K

The map on the left, which we thus identify with dφ, is induced by orthogonal projection g = k + p → p. With this identification in mind, we think of sections of T G as functions on G with values in g and sections of φ −1 T G/K as functions on G with values in p: (T G) ∼ = C ∞ (G, g), (φ −1 T G/K ) ∼ = C ∞ (G, p). The sections of T G are of course also derivations: for any vector space V and smooth function f : G → V , an element X ∈ C ∞ (G, g) acts on f as X ( f )(g) = d f (X )(g) =

d f (g exp(t X (g)) dt t=0

(g ∈ G).

The Levi-Civita connexion on T G corresponds to the connexion 1 ∇ X Y = dY (X ) + [X, Y ] 2

(X, Y ∈ C ∞ (G, g)),

and the pullback of the Levi-Civita connexion on T G/K to φ −1 T G/K corresponds to the connexion φ

∇ X Y = dY (X ) + [X, Y ]

(X ∈ C ∞ (G, g), Y ∈ C ∞ (G, p)).

Proposition 5.1. Choose an orthonormal basis {ek }m k=1 for g such that e1 , . . . , em−n is a basis for k and em−n+1 , . . . , em a basis for p. For the homogeneous projection φ, the second variation takes the form

Lφ Y = −J

λ 3 J (Y ) − J ek ek (Y ) + J [ea (Y ), ea ] 2 2 a=1 k=1 m 1 + ω(er es (Y ) − [er , es ](Y ), er )es . 2 m

n

−

r,s=m−n+1

Here λ is the eigenvalue of the Casimir operator associated to the adjoint representation of g: −

m [ek , [ek , X ]] = λX k=1

(X ∈ g).


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Proof. We begin by calculating Z φ . By Corollary 2.4 we know that δφ ∗ ω(X ) = 0 for X ∈ p, and for X ∈ k we have δφ ∗ ω(X ) =

m

− ek (φ ∗ ω(ek , X )) + φ ∗ ω(∇ek ek , X ) + φ ∗ ω(ek , ∇ek X )

k=1

= = = =

m

ω(dφ(ek ), dφ(∇ek X ))

k=1 m r =m−n+1 m

1 2 1 2

1 ω(dφ(er ), dφ( [er , X ])) 2

r =m−n+1 m

ω(er , [er , X ]) [J er , er ], X .

r =m−n+1

Thus, Zφ =

1 2

m

[J er , er ] =

r =m−n+1

1 2

m r =m−n+1

1 λ [ek , [ek , J ]] = − J, 2 2 m

[er , [er , J ]] =

k=1

where we have used the fact that J belongs to the centre of k. Next we look at φ ∗ ιY ω. Let A, B ∈ g. Then dφ ∗ ιY ω(A, B) = A(ω(Y, dφ(B)) − B(ω(Y, dφ(A)) − ω(Y, dφ([A, B])) = ω(dY (A), Bp) − ω(dY (B), Ap) − ω(Y, [A, B]p). Thus, for X ∈ p, δdφ ∗ ιY ω(X ) =

m

− ek (dφ ∗ ιY ω(ek , X )) + dφ ∗ ιY ω(∇ek ek , X )

k=1

+ dφ ∗ ιω(ek , ∇ek X ) =

m

− ek (ω(dY (ek ), X ) − ω(dY (X ), (ek )p) − ω(Y, [ek , X ]p)

k=1

+ ω(dY (ek ), (∇ek X )p) − ω(dY (∇ek X ), (ek )p) − ω(Y, [ek , ∇ek X ]p) =

m k=1

1 − ω(ek ek (Y ), X ) − ω(Y, [ek , [ek , X ]]p) 2

m−n 3 ω(ea (Y ), [ea , X ]) 2 a=1 1 + ω(er X (Y ), er ) − ω([er , X ](Y ), er ) 2

+

r =m−n+1

768


= −J

m

ek ek (Y ) +

a=1

k=1 m

+

m−n λ 3 J [ea (Y ), ea ], X JY + 2 2

r =m−n+1

1 ω((er X − [er , X ])(Y ), er ). 2

Hence dφ( δdφ ∗ ιY ω) = −J

m

ek ek (Y ) +

k=1 m

+

r,s=m−n+1

m−n 3 λ JY + J [ea (Y ), ea ] 2 2 a=1

1 ω((er es − [er , es ])(Y ), er )es . 2

We thus arrive at

Lφ (Y ) = −J +

m m−n λ 3 J (Y ) − J ek ek (Y ) + J [ea (Y ), ea ] 2 2 a=1 k=1 m 1 ω((er es − [er , es ])(Y ), er )es .

2

−

r,s=m−n+1

The Hopf map is by definition the map φ : S 3 ⊂ C2 → CP 1 , φ(z 1 , z 2 ) = [z 1 , z 2 ]. By the identifications S3 ∼ = SU(2), CP 1 ∼ = SU(2)/S(U(1) × U(1)), we get the alternative definition of the Hopf map as the homogeneous projection φ : SU(2) → SU(2)/S(U(1) × U(1)). Using the previous result and some representation theory, we shall prove the following result: 1 Theorem 5.2. The Hopf map is stable; the Hessian has eigenvalues (n 2 + 2n) and 4 1 (n − 2k)2 , k = 0, . . . , n, n = 1, 2, . . . . Each eigenspace is of infinite dimension. 4 It is well known that, as a harmonic map, the Hopf map is unstable, see [19]. Returning to the full Faddeev-Hopf model for maps SU(2) → S(U(1) × U(1)), 1 E(φ) = (|dφ|2 + α|φ ∗ ω|2 ) ∗1, 2 SU(2) we can give precise information on the stability of the Hopf map for this functional, thus proving a conjecture stated by Ward in [20]. The stability properties turn out to be exactly analogous to those of the identity map in the Skyrme model on S 3 [12].


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Theorem 5.3. The Hopf map is an unstable critical point of the Faddeev-Hopf energy functional if α < 1 and a stable critical point if α ≥ 1. The remaining part of this section is devoted to the proof of these two results, beginning with Theorem 5.2. The Lie algebra su(2) of SU(2) has a basis

i 01 1 0 1 i 1 0 ϑ1 = , ϑ2 = , ϑ3 = , 2 10 2 −1 0 2 0 −1 and we choose the inner product ·, · on su(2) which makes this an orthonormal basis; then ·, · is a multiple of the Killing form. The isotropy subalgebra k = s(u(1) × u(1)) is one-dimensional and spanned by ϑ3 , which also acts as J on p = span{ϑ1 , ϑ2 }. It is easy to see that, for the adjoint representation of su(2), the Casimir operator is just multiplication by 2, i.e., λ = 2. To calculate Lφ , let f ∈ C ∞ (G, R). Then

3 Lφ ( f ϑ1 ) = −J − ϑ3 ( f )ϑ1 − (ϑ12 + ϑ22 + ϑ32 )( f )[ϑ3 , ϑ1 ] + ϑ3 ( f )[ϑ3 , [ϑ1 , ϑ3 ]] 2 1 + (ϑ2 ϑ1 − [ϑ2 , ϑ1 ])( f )[ϑ3 , ϑ1 ], ϑ2 ϑ1 + ϑ22 ( f )[ϑ3 , ϑ1 ], ϑ2 ϑ2 2

3 = −J − ϑ3 ( f )ϑ1 + (ϑ12 + ϑ22 + ϑ32 )( f )ϑ2 + ϑ3 ( f )ϑ1 2 1 −ϑ2 ϑ1 ( f )ϑ1 + ϑ3 ( f )ϑ1 − ϑ22 ( f )ϑ2 2

= −J (ϑ3 ( f ) − ϑ2 ϑ1 ( f ))ϑ1 + (ϑ12 + ϑ32 )( f )ϑ2 = (−ϑ12 − ϑ32 )( f )ϑ1 + (ϑ3 − ϑ2 ϑ1 )( f )ϑ2 . A similar calculation gives Lφ ( f ϑ2 ) = (−ϑ3 − ϑ1 ϑ2 )( f )ϑ1 + (−ϑ22 − ϑ32 )( f )ϑ2 . Hence we can express the differential operator Lφ as a matrix using the basis {ϑ1 , ϑ2 } for p:

2 −ϑ1 − ϑ32 −ϑ3 − ϑ1 ϑ2 . Lφ = ϑ3 − ϑ2 ϑ1 −ϑ22 − ϑ32 To calculate the spectrum of Lφ we recall the Peter-Weyl theorem [11, p. 17]. According to this, L 2 (SU(2), R) is the orthogonal sum of the finite-dimensional subspaces spanned by matrix elements for the (finite-dimensional) irreducible unitary representations of SU(2). Furthermore, these subspaces are invariant under Lφ . To calculate the spectrum of Lφ it is therefore enough to calculate the spectrum of Lφ when restricted to these subspaces. Let us therefore momentarily digress for a study of the irreducible representations of SU(2). As SU(2) is the compact real form of SL2 (C), all irreducible representations of SU(2) are obtained by restriction of the irreducible representations of SL2 (C). For a basis of sl2 (C), let

1 0 00 01 . , H= , Y = X= 0 −1 10 00

770


Denote by V = C2 the standard representation of SL2 (C) and by V (n) = Symn (V ) the n th symmetric power of V , n = 1, 2, . . . . These are precisely the irreducible, finitedimensional representations of SL2 (C), and therefore also of SU(2). To study the action of su(2) on V (n) , recall that there is a highest weight vector v ∈ V (n) ; let vk = Y k v

(k = 0, 1, . . . , n).

We adopt the convention that vk = 0 for k < 0 and k > n. Then, if v is suitably chosen, H vk = (n − 2k)vk , X vk = k(n − k + 1)vk−1 , Y vk = vk+1 . Furthermore, {vk }nk=0 is a basis of V (n) . A simple calculation gives that ϑ1 vk = ϑ2 vk = ϑ3 vk = (−ϑ12 − ϑ22 )vk = (−ϑ12 − ϑ22 − ϑ32 )vk =

i k(n − k + 1)vk−1 + vk+1 , 2 1 k(n − k + 1)vk−1 − vk+1 , 2 i (n − 2k)vk , 2 1 (2kn − 2k 2 + n)vk , 2 1 2 (n + 2n)vk . 4

Let us now return to Lφ and study its action on V (n) ⊗ p, where we think of ϑk as acting by the representation. Since ϑ3 = [ϑ2 , ϑ1 ], we can rewrite Lφ as

ϑ22 −ϑ2 ϑ1 . (5.1) Lφ = (−ϑ12 − ϑ22 − ϑ32 )I d + −ϑ1 ϑ2 ϑ12 Let us denote by A(n) the second operator, as acting on V (n) ⊗ p. Then, by our earlier calculations, 1 Lφ = (n 2 + 2n)I d + A(n) . (5.2) 4 To find the eigenvalues of Lφ on V (n) ⊗ p, we must find the eigenvalues of A(n) . So assume that α ⊗ ϑ1 + β ⊗ ϑ2 ∼ = (α, β) is an eigenvector with eigenvalue λ. Then ϑ22 α − ϑ2 ϑ1 β = λα =⇒ (ϑ12 + ϑ22 )(ϑ2 α − ϑ1 β) = λ(ϑ2 α − ϑ1 β). −ϑ1 ϑ2 α + ϑ12 β = λβ Again, by our earlier calculations, we see that the only possibility is that either λ = 0, or 1 λ = λk = − (2kn − 2k 2 + n) 2

(k = 0, . . . , n).

1 1 ϑ2 vk and β = − ϑ1 vk , then it is easy to see that α ⊗ ϑ1 + λk λk β ⊗ ϑ2 is an eigenvector for A(n) with eigenvalue λk . As the linear map

Furthermore, when α =

V (n) ⊗ p → V (n) , (α, β) → ϑ2 α − ϑ1 β


771

is surjective, the dimension of its kernel equals dim V (n) = n + 1. Hence we conclude 1 that Lφ , acting on V (n) ⊗ p, has the eigenvalue (n 2 + 2n) of multiplicity n + 1, and 4 the eigenvalues 1 2 1 1 (n + 2n) − (2kn − 2k 2 + n) = (n − 2k)2 4 2 4

(k = 0, . . . , n),

each of multiplicity 1. To calculate the spectrum of Lφ , this time acting as a differential operator on the space spanned by the matrix elements for the representation V (n) ⊗ p of SU(2), choose some SU(2)-invariant inner product (·, ·) on V (n) and define the functions πkl (g) = (gvk , vl )

(g ∈ SU(2), k, l = 0, . . . , n).

Then, by the invariance of (·, ·), ϑ(πkl )(g) = (gϑvk , vl )

(g ∈ SU(2), ϑ ∈ su(2), k, l = 0, . . . , n).

It follows that, for k = 0, . . . , n, 1 1 ϑ2 (πkl ) ⊗ ϑ1 − ⊗ ϑ1 (πkl ) ⊗ ϑ2 λk λk

(l = 0, . . . , n)

1 is an eigenfunction of Lφ with eigenvalue (n−2k)2 ; the corresponding eigenspace is of 4 dimension n+1. Furthermore, it is clear that the kernel of A(n) , acting on span{πkl }nk,l=0 ⊗ p, is of dimension (n + 1)2 ; the action of Lφ on ker A(n) is therefore diagonal with eigen1 value (n 2 + 2n). 4 Thus Lφ has the eigenvalue spectrum claimed. Further, we have shown that Lφ is non-negative on an L 2 orthogonal collection of finite-dimensional spaces spanning L 2 . Hence ·, Lφ · is non-negative on L 2 , so φ is stable. This completes the proof of Theorem 5.2. Proof of Theorem 5.3. Following Urakawa [19], we denote by D the Jacobi operator of the Hopf map φ with respect to the Dirichlet energy. The Hessian of φ with respect to the full Faddeev-Hopf functional 1 E(φ) = (|dφ|2 + α|φ ∗ ω|2 ) ∗1 2 SU(2) is then obviously SU(2)

h((D + αLφ )X, Y ) ∗1

(X, Y ∈ C ∞ (SU(2), p)),

where, as before, p = span{ϑ1 , ϑ2 } ⊂ su(2).

772


We begin by studying the action of D + αLφ on the spaces V (n) ⊗ p. From [19] and Theorem 5.2 it follows that D + αLφ is positive semi-definite for n = 1 and all α ≥ 0. According to [19, Corollary 8.12] and (5.1) and (5.2) we have on V (1) ⊗ p,

0 ϑ3 2 2 2 + α A(1) D + αLφ = (1 + α)(−ϑ1 − ϑ2 − ϑ3 )I d − 2 −ϑ3 0

3 −2ϑ3 − αϑ2 ϑ1 αϑ22 = (1 + α)I d + . 2ϑ3 − αϑ1 ϑ2 αϑ12 4 ∼ C2 . In the basis {e1 ⊗ ϑ1 , e1 ⊗ ϑ2 , e2 ⊗ Let {e1 , e2 } be the standard basis for V (1) = (1) ϑ1 , e2 ⊗ ϑ2 } for V ⊗ p we can express the last term as the matrix ⎛ ⎞ −α/4 −i − iα/4 0 0 0 0 ⎜i + iα/4 −α/4 ⎟ . ⎝ 0 0 −α/4 −i − iα/4⎠ 0 0 i + iα/4 −α/4 α − 1, each with multiplicity 2. On the eigenspace 2 corresponding to the eigenvalue 1, the operator D + αLφ is obviously positive semiα definite for all α, while on the eigenspace corresponding to the eigenvalue − − 1, we 2 have

This matrix has eigenvalues 1 and −

D + αLφ =

α α−1 3 (1 + α)I d − ( + 1)I d = I d. 4 2 4

We thus conclude that for α ≥ 1, D + αLφ is positive semi-definite on V (n) ⊗ p for all n, while for α < 1, it is negative definite on a 2-dimensional subspace of V (1) ⊗ p. Hence, the operator D + αLφ , acting on the space spanned by the matrix elements for the representation V (n) ⊗ p of SU(2), is positive semi-definite for all n if α ≥ 1, while it is negative definite on a 4-dimensional subspace if α < 1. Theorem 5.3 now follows from the Peter-Weyl Theorem.

References 1. Atiyah, M.F., Hitchin, N.J.: The Geometry and Dynamics of Magnetic Monopoles. Princeton, NJ: Princeton University Press, 1988 2. Baird, P., Wood, J.C.: Harmonic morphisms between Riemannian manifolds. London Math. Soc. Monogr. No. 29, Oxford: Oxford Univ. Press, 2003 3. De Carli, E., Ferreira, L.A.: A model for Hopfions on the space-time S 3 × R. J. Math. Phys. 46, 012703 (2005) 4. Derrick, G.H.: Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252–1254 (1964) 5. Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford: Oxford University Press, 1990 6. Eells, J., Lemaire, L.: Selected topics in harmonic maps. CBMS Regional Conference Series in Mathematics, Providence, RI: Amer. Math. Soc. 1983 7. Faddeev, L.D., Niemi, A.J.: Stable knot-like structures in classical field theory. Nature 387, 58–61 (1997) 8. Gisiger, T., Paranjape, M.B.: Solitons in a baby-Skyrme model with invariance under area-preserving diffeomorphisms. Phys. Rev. D 55, 7731–7738 (1997) 9. Helgason, S.: Differential geometry, Lie groups and symmetric spaces. London-New York: Academic Press, 1978


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10. Lichnerowicz, A.: Applications harmoniques et variétés kähleriennes. Symp. Math. III, Bologna, 341–402 (1970) 11. Knapp, A.W.: Representation Theory of Semisimple Groups. Princeton, NJ: Princeton University Press, 1986 12. Manton, N.S.: Geometry of Skyrmions. Commun. Math. Phys. 111, 469–478 (1987) 13. Palais, R.S.: The Principle of Symmetric Criticality. Commun. Math. Phys. 69, 19–30 (1979) 14. Piette, B.M.A.G., Schroers, B.J., Zakrzewski, W.J.: Dynamics of Baby Skyrmions. Nucl. Phys. B 439, 205–238 (1995) 15. Piette, B., Tchrakian, D.H., Zakrzewski, W.J.: A class of two dimensional models with extended structure solutions. Z. Phys. C 54, 497–502 (1992) 16. Rajaraman R.: Solitons and Instantons. Amsterdam: North-Holland, 1989 17. Smith, R.T.: The second variation formula for harmonic mappings. Proc. Amer. Math. Soc. 47, 229– 236 (1975) 18. Stenzel, M.B.: Ricci-flat metrics on the complexification of a compact rank one symmetric space. Manuscripta Mathematica 80, 151–163 (1993) 19. Urakawa, H.: Stability of harmoinc maps and eigenvalues of the Laplacian. Trans. Amer. Math. Soc. 301, 557–589 (1987) 20. Ward, R.S.: Hopf solitons on S 3 and R 3. Nonlinearity 12, 241–246 (1999) Communicated by G.W. Gibbons


Communications in


Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution L. Molinet1 , J. C. Saut2 , N. Tzvetkov3 1 L.A.G.A., Institut Galilée, Université Paris 13, 93430 Villetaneuse, France 2 Université de Paris-Sud, UMR de Mathématiques, Bât. 425, 91405 Orsay Cedex, France.


3 Département de Mathématiques, Université Lille I, 59 655 Villeneuve d’Ascq Cedex, France

Received: 22 May 2006 / Accepted: 13 October 2006 Published online: 14 April 2007 – © Springer-Verlag 2007

Abstract: We prove that the Cauchy problem for the KP-I equation is globally wellposed for initial data which are localized perturbations (of arbitrary size) of a nonlocalized (i.e. not decaying in all directions) traveling wave solution (e.g. the KdV line solitary wave or the Zaitsev solitary waves which are localized in x and y periodic or conversely). 1. Introduction We study here the initial value problem for the Kadomtsev-Petviashvili (KP-I) equation (u t + u x x x + uu x )x − u yy = 0,

(1)

where u = u(t, x, y), (x, y) ∈ R2 , t ∈ R, with initial data u(0, x, y) = φ(x, y) + ψc (x, y),

(2)

where ψc is the profile1 of a non-localized (i.e. not decaying in all spatial directions) traveling wave of the KP-I equation moving with speed c = 0. This ψc could be for instance the line soliton of the Korteweg- de Vries (KdV) equation √c x −2 . (3) ψc (x, y) = 3c cosh 2 In (3) the KdV soliton is of course considered as a two dimensional (constant in y) object. Another possibility to see (3) as a solution of KP-I is to consider (1) posed on R×T. Global solutions of (1) for data on R×T, including data close to (3) were recently constructed in a work by Ionescu-Kenig [12]. In (2), the function ψc may also be the 1 This means that ψ(x − ct, y) solves (1).

776

L. Molinet, J. C. Saut, N. Tzvetkov

profile of the Zaitsev [36] traveling waves (see also [30]) which is localized in x and periodic in y: 1 − β cosh(αx) cos(δy) ψc (x, y) = 12α 2 , (4) (cosh(αx) − β cos(δy))2 where (α, β) ∈]0, ∞[×] − 1, 1[, and the propagation speed is given by c = α2

4 − β2 . 1 − β2

Let us observe that the transform α → iα, δ → iδ, c → ic produces solutions of (1) which are periodic in x and localized in y. The profiles of these solutions are also admissible in (2), under the assumption |β| > 1. Notice that for β = 0, (4) coincides with (3). The global well-posedness of (1)-(2) with data given by (3) which will be proved in this paper can be viewed as a preliminary step towards the rigorous mathematical justification of the (conjectured) nonlinear instability of the KdV soliton with respect to transversal perturbations governed by the KP-I flow. This question is, as far as we know, still an open problem (see however [1] for a linear analysis of the instability and [10] for a linear instability analysis in the framework of the full Euler system). The instability scenario of the line soliton seems to be a symmetry breaking phenomenon: the line soliton should evolve towards the Zaitsev solitary wave (4). Note that Haragus and Pego [11] have shown that this solution is the only one close to the line soliton which is periodic in y and decays to zero as x → ∞. The question of solving (1) together with the initial data (2) when ψ is the profile of the KdV line soliton, has been recently addressed by Fokas and Pogrobkov [8], by the inverse scattering transform (IST) techniques. However, the Cauchy problem is not rigorously solved in [8] and it is unlikely that it could be solved for an arbitrary large data φ using IST since the Cauchy problem with purely localized data has been solved by IST techniques only for small initial data (see [35, 38]). On the other hand, PDE techniques have been recently fruitfully used to obtain the global solvability of the KP equation with arbitrary large initial data, starting with the pioneering paper of Bourgain [4] on the KP-II equation (that is (1) with +u yy instead of −u yy ). In [4], the global well-posedness of the KP-II equation for data in H s (R2 ), s ≥ 0 is established. The result in [4] is obtained by performing the Picard iteration scheme to an equivalent integral equation in the Fourier transform restriction spaces of Bourgain. The situation for the KP-I equation turned out to be more delicate. We showed in [27] that the Picard iteration scheme can not be applied in the context of the KP-I equation as far as one considers initial data in Sobolev spaces. Sobolev spaces are natural, since the conservation laws for the KP-I equation control Sobolev type norms. In [22], a quite flexible method is introduced that allows to incorporate the dispersive effects in a context of a compactness method for proving the well-posedness. The work [22] was in turn inspired by the considerations in [5] in the context of the Nonlinear Schrödinger equation on a compact manifold which is solved in [5] as a semi-linear problem (i.e. by the Picard iteration scheme). The main point in [22] is to realize that the idea of [5] can also be used in the context of a quasi-linear problem. The method of [22] turned out to be useful in the context of the KP-I equation (and some other models

KP-I on the Background of a Non-Localized Solution

777

such as the Schrödinger maps [15, 19]) and the first global well-posedness result for the KP-I equation has been obtained by the authors of the present paper in [26]. This result has been improved (i.e. the space of the allowed initial data is larger) by Kenig [17]. Together with the idea of [22], a new commutator estimate for the KP-I equation is used in [17]. The main point in Kenig’s result is the proof that KP-I is locally well-posed for data in the space {u ∈ L 2 (R2 ) : ∂x−1 u y ∈ L 2 (R2 ), |Dx |s u ∈ L 2 (R2 ), s > 3/2}. All papers [4, 26] and [17] consider the KP equations in spaces of “localized” (zero at infinity) functions. The main goal of this paper is to prove that, for a large class of ψc , the Cauchy problem (1), (2) is globally well-posed for data φ ∈ Z , where Z := {u ∈ L 2 (R2 ) : ∂x−2 u yy ∈ L 2 (R2 ), u x x ∈ L 2 (R2 )} (notice that u ∈ Z implies u y ∈ L 2 (R2 ) and ∂x−1 u y ∈ L 2 (R2 )). More precisely, we have the following result. Theorem 1. Let ψc (x − ct, y) be a solution of the KP-I equation such that ψc : R2 −→ R is bounded with all its derivatives2 . Then for every φ ∈ Z there exists a unique global solution u of (1) with initial data (2) satisfying for all T > 0, [u(t, x, y) − ψc (x − ct, y)] ∈ C([0, T ]; Z ), ∂x [u(t, x, y) − ψc (x − ct, y)] ∈ L 1T L ∞ xy . Furthermore, for all T > 0, the map φ → u is continuous from Z to C([0, T ]; Z )). Since the KP-I equation is time reversible, a similar statement to Theorem 1 holds for negative times as well. Let us notice that the assumptions on ψc in Theorem 1 are clearly satisfied by the line or the Zaitsev solitary wave. In the proof of Theorem 1, we write the solution u of (1), (2) as u(t, x, y) = ψc (x − ct, y) + v(t, x, y), where v is localized. This v satisfies the equation (vt + vx x x + vvx + ∂x (ψc v))x − v yy = 0, v(0, x, y) = φ(x, y).

(5)

Our strategy is then to adapt the proof of [26, 17]. Starting from the local well-posedness result, we implement a compactness method based on “almost conservation laws”. New terms occur with respect to [26] but they are controlled since ψ and its derivatives are bounded. It is of importance for our analysis that Eq. (5) does not contain a source term. We refer to the work by Gallo [9] and the references therein, where non-vanishing at infinity solutions to one dimensional dispersive models are constructed. Let us notice that the framework considered in Theorem 1 is also a convenient one for a rigorous study of the interaction of a line and lump solitary waves (see e.g. [6]). The rest of this paper is organized as follows. In the next section, using a compactness method, we prove a basic well-posedness result for (5). In Sect. 3, inspired by the formal KP-I conservation laws, we provide bounds for some Sobolev type norms of the 2 The bounds can of course depend on the propagation speed c.

778


local solutions. These bounds are however not sufficient to get global solutions. For that reason, in Sect. 4 we prove a Strichartz type bound. This bound is then used in Sect. 5 to get a first global well-posedness result. In Sects. 6 and 7 we extend the well-posedness to the class Z introduced above. The last section is devoted to the “usual” KP-I equation. We show how the estimates of Sect. 4 can be used to give a slight improvement of the Kenig well-posedness result [17]. s (R2 ), s > 2 2. Local Well-Posedness for Data in H− 1

In this section, we prove a basic local well-posedness result for (5). The proof follows a standard compactness method. We however need to work in Sobolev spaces of integer indexes, in order to make the commutator estimates work related to the term ∂x (ψc v). By H s (R2 ), s ∈ R, we denote the classical Sobolev spaces. The local existence result s (R2 ) equipped with the norm for (6) will be obtained in the spaces H−1 u (ξ, η) L 2 , u H s (R2 ) = (1 + |ξ |−1 ) |ξ | + |η| s −1 ξη 1

s (R2 ) are where · = (1+|·|2 ) 2 and u denotes the Fourier transform of u. The spaces H−1 r,∞ adapted to the specific structure of the KP type equations. In the estimates W -norms of ψc will appear defined for any integer r ≥ 0 by ψc W r,∞ = ∂xα1 ∂ yα2 ψc L ∞ . xy 0≤|α|=|(α1 ,α2 )|≤r

Consider the “integrated” equation (5) u t + u x x x − ∂x−1 u yy + uu x + ∂x (ψu) = 0,

(6)

u(0, x, y) = φ(x, y).

(7)

with initial data

For the solutions we study in this section, (5) may be substituted by (6). For conciseness we skip the c of ψc in (6) and we suppose that c = 1 in the sequel. Of course, the case c = 1 can be treated in exactly the same manner. We have the following local well-posedness result for (6). s (R2 ) there exists Proposition 1. Let s > 2 be an integer. Then for every φ ∈ H−1 T (1 + φ H s )−1 and a unique solution u to (6) on the time interval [0, T ] satisfying s u ∈ C([0, T ]; H s (R2 )) , u ∈ L ∞ ([0, T ]; H−1 (R2 )) .

In addition, for t ∈ [0, T ],

u(t, ·) H s (R2 ) ≤ Cφ H s (R2 ) exp c∇x,y u L 1 L ∞ + cT ψW s,∞ . T

xy

σ (R2 ) where σ > s is an integer then Moreover if φ ∈ H−1 σ (R2 )). u ∈ C([0, T ]; H−1

Finally, the map φ → u φ is continuous from H s (R2 ) to C([0, T ]; H s (R2 )).

(8)


779

Proof of Proposition 1. The process is very classical (see [13] for a closely related result). For ε > 0 we look at the regularized equation u εt + ε2 u εt = −u εx x x + ∂x−1 u εyy − u ε u εx − ∂x (ψu ε ),

(9)

where = ∂x2 + ∂ y2 is the Laplace operator. Equation (9) with initial condition φε = (1 −

√

ε)−1 φ

can be rewritten under the form t ε ε u (t) = L (t)φε − L ε (t − τ )(1 + ε2 )−1 u ε (τ )u εx (τ ) + ∂x (ψ(τ )u ε (τ ) dτ , (10) 0

where

L ε (t) := exp − t (1 + ε2 )−1 (∂x3 − ∂x−1 ∂ y2 ) .

Notice that, thanks to the regularization effect of (1 + ε2 )−1 , for ε = 0, the map u −→ (1 + ε2 )−1 u u x + ∂x (ψu is locally Lipschitz from H s (R2 ) to H s (R2 ), provided s > 2. The operator L ε (t) is clearly bounded on H s (R2 ) and therefore by the Cauchy-Lipschitz-Picard theorem there is a unique local solution u ε ∈ C([0, T ]; H s (R2 ))

(11)

of (10) with T (1 + φε H s )−1 ≥ (1 + φ H s )−1 . Thanks to the perfect derivative structure of the integral term in (10), we also obtain that s u ε ∈ C([0, T ]; H−1 (R2 )). √ Notice also that, thanks to the assumption s > 2, (1 − ε)u ε belongs to H s (R2 ). We next study the convergence of u ε as ε → 0. For that purpose, we establish a priori bounds, independent3 of ε on u ε (t, ·) H s on time intervals of size of order (1 + φ H s )−1 . Multiplying (9) with u ε , after an integration by parts, we obtain that d ε u (t, ·)2L 2 + εu ε (t, ·)2L 2 ψx L ∞ u ε (t, ·)2L 2 . (12) dt

Let us recall a classical commutator estimate (see e.g. [16]). Lemma 1. Let be the Laplace operator on Rn , n ≥ 1. Denote by J s the operator (1 − )s/2 . Then for every s > 0, [J s , f ]g L 2 (Rn ) ∇ f L ∞ (Rn ) J s−1 g L 2 (Rn ) + J s f L 2 (Rn ) g L ∞ (Rn ) . 3 Notice that the bounds on u ε (t, ·) s resulting from the Cauchy-Lipschitz theorem applied to (9) are H unfortunately very poor (depending on ε).

780


Using Lemma 1, we obtain for fixed y, [∂xs , u(x, ·)]u x (x, ·) L 2x u x (x, ·) L ∞ Jxs u(x, ·) L 2x , x where Jxs = (1 − ∂x2 )s/2 . Squaring, integrating over y and integrating by parts, it yields

∂xs (uu x )∂xs u u x L ∞ Jxs u2L 2 . (13)

xy R2

xy

On the other hand, by Leibniz rule and integration by parts (recall that s is an integer), we get

∂xs (ψu)x ∂xs u ψx W s−1,∞ Jxs u2L 2 . (14)

R2

xy

Therefore applying ∂xs to (9), multiplying it with ∂xs u ε gives d s ε ∂x u (t, ·)2L 2 + ε∂xs u ε (t, ·)2L 2 dt

u εx (t, ·) L ∞ + ψx W s−1,∞ Jxs u ε (t, ·)2L 2 (R2 ) . (15)

Next we estimate the y derivatives. In the same way, using Lemma 1, we obtain that for a fixed x, [∂ ys , u(x, ·)]u x (x, ·) L 2y s−1 s ∞ u x (x, ·) L ∞ J (16) + u (x, ·) u (x, ·) 2 + J y u(x, ·) L 2 , y L x L y y y y y where Jys = (1 − ∂ y2 )s/2 . Squaring (16), integration over x and an integration by parts yield

∂ ys (uu x )∂ ys u ∇x,y u L ∞ u2H s (R2 ) .

xy R2

On the other hand, by Leibniz rule and integration by parts,

∂ ys (ψu)x ∂ ys u ψW s,∞ u2H s (R2 ) .

R2

Therefore applying ∂ ys to (9) and multiplying it by ∂ ys u ε gives d s ε ∂ y u (t, ·)2L 2 + ε∂ ys u ε (t, ·)2L 2 dt

∇x,y u ε (t, ·) L ∞ + ψW s,∞ u ε (t, ·)2H s (R2 ) . (17)

Since for s integer u H s ≈ u L 2 + ∂xs u L 2 + ∂ ys u L 2 , combining (12), (15) and (17) gives d ε u (t, ·)2H s (R2 ) + εu ε (t, ·)2H s (R2 ) dt

∇x,y u ε (t, ·) L ∞ + ψW s,∞ u ε (t, ·)2H s (R2 ) ,


781

and therefore using that φε 2H s (R2 ) + εφε 2H s (R2 ) ≤ φ2H s (R2 ) by the Gronwall lemma for every T > 0 on the time of existence of u ε , u ε L ∞ H s (R2 ) ≤ φ H s (R2 ) exp c∇x,y u ε L 1 L ∞ + cT ψW s,∞ ,

(18)

which is the key inequality. Since s > 2, using the Sobolev embedding, we get T ∇x,y u ε (τ, ·) L ∞ dτ ≤ C T u ε (t, ·) L ∞ H s (R2 ) .

(19)

T

T

xy

T

0

Using (18), (19) and the continuity of u ε (t) with respect to time (see (11)), we obtain that there exists C > 0 such that if T (1 + φ H s (R2 ) )−1 , then

T

0

and

∇x,y u ε (τ, ·) L ∞ (R2 ) dτ ≤ C

(20)

u ε L ∞ H s (R2 ) ≤ Cφ H s (R2 ) .

(21)

T

We next estimate the anti-derivatives of u ε . Let v ε := ∂x−1 u ε . Then, using (10), we obtain that v ε solves the equation t 1 v ε (t) = L ε (t)∂x−1 φε − L ε (t − τ )(1 + ε2 )−1 (u ε (τ ))2 + ψ(τ )u ε (τ ) dτ . 2 0 Therefore, since s > 2, using the Leibniz rule and the Sobolev inequality, we get the bound v ε L ∞ H s (R2 ) ≤ ∂x−1 φ H s (R2 ) +C T u ε L ∞ H s (R2 ) u ε L ∞ H s (R2 ) +ψW s,∞ . (22) T

T

T

Coming back to Eq. (9), we infer from (22) that the sequence (∂t (u ε )) is bounded in s−2005 (R2 ). Therefore from the Aubin-Lions compactness a weaker norm, say in L ∞ T H theorem (see e.g. [23]), we obtain that u ε converges, up to a subsequence, to some limit 2 ((0, T ) × R2 ) which satisfy (8) and u in the space L loc T ∇x,y u(τ, ·) L ∞ (R2 ) dτ ≤ C . (23) 0

Thanks to (22), we obtain that (up to a subsequence) ∂x−1 u ε converges in D ((0, T )×R2 ) to a limit which can be identified as ∂x−1 u. By writing the nonlinearity uu x as 21 ∂x (u 2 ), passing into a limit in Eq. (9) as ε → 0, we obtain that the function u satisfy Eq. (6) in the distributional sense. Moreover thanks to (21) and (22), we obtain that s u ∈ L ∞ ([0, T ]; H−1 (R2 )) . s (R2 ) to φ as ε → 0 In addition, thanks to the Lebesgue theorem, φε converges in H−1 σ 2 and thus u satisfies the initial condition (7). Next, if φ ∈ H−1 (R ) with σ ≥ s, σ ∈ N,

782


then as above, we get the estimate (8) with σ instead of s which, in view of (23), yields the propagation of the H σ (R2 ) regularity. We next estimate the anti-derivatives of u in H σ by invoking (22) (with σ instead of s) in the limit ε → 0. The uniqueness is straightforward from the Gronwall lemma and (23). The continuity of the flow map and the fact that the solution is a continuous curve in H s (R2 ) can be obtained by the Bona-Smith approximation argument [3]. We do not give the details of this construction in this section since a completely analogous discussion will be performed later in this paper. Let us next state a corollary of Proposition 1. s (R2 ) the local solution Proposition 2. Let s > 2 be an integer. Then for every φ ∈ H−1 constructed in Proposition 1 can be extended to a maximal existence interval [0, T [ such that either T = ∞ or

lim ∇x,y u L 1 L ∞ = ∞.

t→T

t

xy

Proof. It suffices to iterate the result of Proposition 1 by invoking (8) at each iteration step. s (R2 ), It results from Proposition 2 that the key quantity for the global existence in H−1 s > 2 is ∇x,y u(t, ·) L ∞ .

3. A Priori Estimates Using Conservation Laws In this section we control the growth of some quantities directly related to the conservation laws of KP-I. Recall that the solutions obtained in Proposition 1 satisfy u t + u x x x + uu x + ∂x (ψu) − ∂x−1 u yy = 0.

(24)

In [37], it is shown that the KP-I equation has a Lax pair representation. This in turn provides an algebraic procedure generating an infinite sequence of conservation laws. More precisely, if u is a formal solution of the KP-I equation then d χn = 0, dt where χ1 = u, χ2 = ∂x u + i∂x−1 ∂ y u and for n ≥ 3, χn =

n−2

χk χn−1−k + ∂x χn−1 + i∂x−1 ∂ y χn−1 .

k=1

For n = 3, we find the conservation of the L 2 norm, n = 5 corresponds to the energy functional giving the Hamiltonian structure of the KP-I equation. As we noticed in [26], there is a serious analytical obstruction to give sense of χ9 as far as R2 is considered as a spatial domain. Inspired by the above discussion, we define the following functionals: 1 1 1 M(u) = u 2 , E(u) = u 2x + (∂x−1 u y )2 − u3 2 R2 2 R2 6 R2 R2


and F ψ (u) =

783

3 5 5 u 2x x + 5 u 2y + (∂x−2 u yy )2 − u 2 ∂x−2 u yy 2 R2 6 R2 6 R2 R2 5 5 5 − u (∂x−1 u y )2 + u2 u x x + u4 6 R2 4 R2 24 R2 5 5 −2 − ψ u ∂x u yy − ψ (∂x−1 u y )2 . 3 R2 6 R2

Recall that the functionals M and E correspond to the momentum and energy conservations respectively while the functional F ψ is motivated by the higher order conservation laws for the KP-I equation associated to χ7 . Let us notice however that the functional F ψ (·) contains two supplementary terms involving ψ with respect to the corresponding conservation law of the KP-I equation. ∞ (R2 ) the intersection of all H s (R2 ). The next proposition gives We denote by H−1 −1 bounds on the quantities M, E and F ψ for data in spaces where the local well-posedness of the previous section holds. ∞ (R2 )) Proposition 3. For every R > 0 there exists C > 0 such that if u ∈ L ∞ ([0, T ]; H−1 ∞ 2 is a solution to (6) corresponding to an initial data φ ∈ Z ∩ H−1 (R ), φ Z ≤ R then E(u(t)) and F ψ (u(t)) are well-defined and ∀t ∈ [0, T ],

M(u(t)) ≤ C exp(C t)M(φ), |E(u(t))| ≤ C exp(C t)|E(φ)| + g1 (t),

(25) (26)

|F ψ (u(t))| ≤ C exp(C t)|F(φ)| + g2 (t),

(27)

where g1 , g2 : R+ → R+ are continuous bijections depending only on R. Proof of Proposition 3. Before entering into the proof of Proposition 3, we state an anisotropic Sobolev inequality which will be used in the proof. ∞ (R2 ), Lemma 2. For 2 ≤ p ≤ 6 there exists C > 0 such that for every u ∈ H−1 6− p

p−2

p−2

u L p (R2 ) ≤ Cu L22p(R2 ) u x L 2p(R2 ) ∂x−1 u y L22p(R2 ) .

(28)

Proof. We refer to [2] for a proof of (28) and for a systematic study of anisotropic Sobolev embeddings. For a sake of completeness, here we reproduce the proof of (28) given in [33]. Inequality (28) clearly holds for p = 2. By convexity, it suffices thus to prove it for p = 6. Following the Gagliardo-Nirenberg proof of the Sobolev embedding, we write x 2 u x (z, y)u(z, y)dz, u (x, y) = 2 −∞

and therefore using the Cauchy-Schwarz inequality in the z integration, we obtain that for a fixed y, 4 ∞ ∞ 2 sup |u(x, y)| ≤ 4 u x (z, y)dz u 2 (z, y)dz . x∈R

−∞

−∞

784


Therefore by writing u 6 = u 4 u 2 , we get ∞ ∞ u 6 (x, y)d xd y ≤ 4 u 2x (z, y)dz R2

−∞

−∞

∞ −∞

u 2 (z, y)dz

2

dy .

(29)

Next, using Fubini theorem and an integration by parts, we obtain ∞ ∞ y 2 u (z, y)dz = 2 u(z, w)u y (z, w)dwdz −∞ −∞ −∞ y ∞ =2 u(z, w)u y (z, w)dzdw −∞ −∞ y ∞ = −2 u x (z, w) ∂x−1 u y (z, w)dzdw . −∞ −∞

An application of the Cauchy-Schwarz inequality now gives ∞ 2 sup u 2 (z, y)dz ≤ 4u x 2L 2 ∂x−1 u y 2L 2 . xy

−∞

y∈R

xy

Coming back to (29) yields u 6 (x, y)d xd y ≤ 16u x 4L 2 ∂x−1 u y 2L 2 R2

xy

xy

which is (28) for p = 6. This completes the proof of Lemma 2.

Let us return to the proof of Proposition 3. Note that since u satisfies (6), one has u t ∈ C([0, T ]; H s (R2 )), ∀ s ∈ N . First, taking the L 2 -scalar product of (6) with u, one obtains 1 d 1 2 u = ψuu x = − ψx u 2 2 dt R2 2 R2 R2 and thus

u(t) L 2 exp(Ctψx L ∞ ) φ L 2 .

(30)

We will use, following [24], an exterior regularization of (6) by a sequence of smooth functions ϕ ε that cut the low frequencies. More precisely, let ϕ ε be defined via its Fourier transform as 1 if ε < |ξ | < 1ε and ε < |η| < 1ε , (31) ϕˆε (ξ, η) := 0 otherwise. s (R2 ), Note that thanks to the Lebesgue dominated convergence theorem, if u ∈ H−1 s (R2 ). Thanks to the Sobolev embedding similar s ∈ R then ϕ ε ∗ u converges to u in H−1 p 2 statements hold for u ∈ L (R ), 2 ≤ p ≤ ∞. Taking the convolution of (6) with ϕ ε , one gets

ϕ ε ∗ u t + ϕ ε ∗ u x x x + ϕ ε ∗ ∂x (ψu + u 2 /2) − ϕ ε ∗ ∂x−1 u yy = 0. Setting u ε = ϕ ε ∗ u,

(32)


785

multiplying (32) by 1 −ϕ ε ∗ u x x + ∂x−2 (ϕ ε ∗ u yy ) − (ϕ ε ∗ u 2 ), 2 and integrating in R2 , one obtains that 1 d (u ε )3 ε 2 −1 ε 2 = (u x ) + (∂x u y ) − 2 2 dt R2 3 R2 R 1 ϕ ε ∗ u 2 − (u ε )2 u εt + (ϕxε ∗ (ψu))u εx x 2 2 R2 R 1 ε −2 ε − (ϕx ∗ (ψu))∂x u yy + (ϕ ε ∗ (ψu))(ϕ ε ∗ u 2 ) 2 R2 x R2 := I + I I + I I I + I V .

(33)

Our aim is to pass to the limit ε → 0. Let us first estimate I . This argument is very typical for the present analysis and a similar situation will appear frequently in the rest of the proof of Proposition 3. Using Eq. (32) and the Cauchy-Schwartz inequality, we can write |I | u εt L 2 (R2 ) ϕ ε ∗ u 2 − (u ε )2 L 2 (R2 ) u H 3 (R2 ) + u2H 3 (R2 ) ϕ ε ∗ u 2 − (u ε )2 L 2 (R2 ) . −1

−1

Next, we write using the triangle inequality and the Sobolev inequality, ϕ ε ∗ u 2 − (u ε )2 L 2 (R2 ) ≤ ϕ ε ∗ u 2 − u 2 L 2 (R2 ) + u 2 − (u ε )2 L 2 (R2 ) ≤ ϕ ε ∗ u 2 − u 2 L 2 (R2 ) + u − u ε L 2 (u L ∞ + u ε L ∞ ) ϕ ε ∗ u 2 − u 2 L 2 (R2 ) + u H 2 (R2 ) u − u ε L 2 . ∞ (R2 ), we can apply the Lebesgue dominated convergence theorem to Since u ∈ H−1 conclude that

lim ϕ ε ∗ u 2 − (u ε )2 L 2 (R2 ) = 0 .

ε→0

Therefore the term I tends to zero as ε tends to zero. Next, we write (ψu ε )x u εx x + [ϕ ε ∗ (ψu)x − (ψu ε )x ]u εx x R2 R2 ε ε ε ε = ψx u u x x + ψu x u x x + [ϕ ε ∗ (ψu)x − (ψu ε )x ]u εx x R2 R2 R2 3 1 =− ψx (u εx )2 + [ϕ ε ∗ (ψu)x − (ψu ε )x ]u εx x ψx x x (u ε )2 + 2 R2 2 R2

II =

786


and

and

R2

(ψu

)x ∂x−2 u εyy

− [ϕ ε ∗ (ψu)x − (ψu ε )x ]∂x−2 u εyy R2 =− ψu εy ∂x−1 u εy − ψ y u ε ∂x−1 u εy − [ϕ ε ∗ (ψu) y − (ψu ε ) y ]∂x−1 u εy 2 2 2 R R R 1 −1 ε 2 ε −1 ε = ψx (∂x u y ) − ψ y u ∂x u y − [ϕ ε ∗ (ψu) y − (ψu ε ) y ]∂x−1 u εy 2 R2 R2 R2

III = −

ε

1 1 ε ε 2 (ψu )x (u ) + [(ϕ ε ∗ (ψu))(ϕ ε ∗ u 2 ) − (ψu ε )x (u ε )2 ] IV = 2 R2 2 R2 x 1 1 = ψx (u ε )3 + [(ϕ ε ∗ (ψu))(ϕ ε ∗ u 2 ) − (ψu ε )x (u ε )2 ]. 3 R2 2 R2 x

Similarly to the analysis for I , thanks to the Lebesgue theorem, all commutator type terms involved in I I , I I I , I V tend to 0 as ε → 0. Using Lemma 2, we get the bound 1 1 1 3/2 1/2 |u|3 ≤ Cu L 2 u x L 2 ∂x−1 u y L 2 ≤ u x 2L 2 + ∂x−1 u y 2L 2 + Cu6L 2 . 6 R2 4 4 We therefore obtain that |E(u)| ≥

1 1 u x 2L 2 + ∂x−1 u y 2L 2 − Cu6L 2 . 4 4

Integrating (33) on (0, t) for t ∈ (0, T ] gives t (ψx L ∞ + ψ y L ∞ )(|E(u ε (τ ))| + u ε (τ )6L 2 ) |E(u ε (t)) − E(u ε (0))| 0 t +(ψx x x L ∞ + ψ y L ∞ )|u ε (τ )|2L 2 dτ + |Aε (τ )|dτ, 0

where limε→0 Aε (τ ) = 0 for every τ ∈ [0, t] (Aε (τ ) corresponds to I and the commutator terms involved in I I , I I I , I V ). Passing to the limit ε → 0, we infer that t (ψx L ∞ + ψ y L ∞ )(|E(u(τ ))| + u(τ )6L 2 ) |E(u(t))| − |E(φ)| 0 +(ψx x x L ∞ + ψ y L ∞ )|u(τ )|2L 2 dτ . By the Gronwall lemma and (30), it follows that |E(u(t))| exp(C t)|E(φ)| + g(t), where g(t) is an increasing bijection of R+ which depends only on φ L 2 . We now turn to the bound on F ψ (u(t)). It is worth noticing that 5 5 ψu ∂x−2 u yy − ψ(∂x−1 u y )2 , F ψ (u) = F(u) − 3 R2 6 R2 where F is the corresponding conservation law of the KP-I equation (see [26]). The introduction of two additional terms in F ψ is the main new idea in this paper. Indeed,


787

if we multiply (32) by the multiplier used in [26], we obtain terms which can not be treated as remainders (see A1 and A2 below). The term −

5 ψu ∂x−2 u yy 3 R2

is introduced in the definition of F ψ in order to cancel such “bad” remainders. The second additional term in the definition of F ψ is needed for the proof of the continuous dependence of the flow map on the space Z . As in [26], a difficulty in the sequel comes from the fact that the variational derivative (F ψ ) (v) contains a term c∂x−2 ∂ yy (v 2 + 2ψv). Recall that the formal derivation of the conservation laws consists in multiplying the KP-I equation with F (u) where u is a solution. This procedure meets a difficulty since ∂x−2 acts only on functions with zero x mean value which is not a priori the case of v 2 + 2ψv. We overcome this difficulty by introducing the functional F ψ,ε defined by 5 F ψ,ε (u(t)) := F ψ (u ε (t)) + (u ε (t))2 ∂x−2 u εyy (t) − (ϕ ε ∗ u 2 (t))∂x−2 u εyy (t) 2 6 R2 R 5 + ψu ε (t) ∂x−2 u εyy (t) − ϕ ε ∗ (ψu(t))∂x−2 u εyy (t) 3 R2 R2 (recall that u ε (t) := ϕ ε ∗ u(t)). We are now in position to state the following lemma. Lemma 3. Under the assumptions of Proposition 3, d ψ,ε 5 5 F (u(t)) = − ψx (∂x−2 u εyy )u εx x + ψ(∂x−2 u εyy )((u ε )2 + 2ψu ε )x dt 3 R2 6 R2 5 5 ε −2 ε + ψu (∂x u yy ) + ψ y (∂x−1 u εy )(∂x−2 u εyy ) 3 R2 3 R2 ε (u), +G ψ (u ε ) + (∂x−2 u εyy )ε (u) + (34) R2

R2

where G ψ is a continuous functional on X = {v ∈ S (R2 ) : v L 2 + ∂x−1 v y L 2 + vx x L 2 + v y L 2 < ∞ }. Moreover sup

t∈[0,T ] R2

|ε (u(t))|2 −→ 0 and ε→0

sup

t∈[0,T ] R2

ε (u(t))| −→ 0. | ε→0

(35)

788


Proof of Lemma 3. After a direct computation, using (32) and the definition of F ψ,ε , we obtain the identity d ψ,ε 5 d 5 d ε −2 ε F (u) = − (ϕ ∗ (ψu))(∂x u yy ) − ψ(∂x−1 u εy )2 (36) dt 3 dt R2 6 dt R2 + (A + B)(u εx x x + ϕ ε ∗ (uu x ) − ∂x−1 u εyy ) 2 R 5 + u ε (∂x−1 u εy )(u εx x y + ϕ ε ∗ (uu y ) − ∂x−2 u εyyy ) 3 R2 5 + [u ε u εt − ϕ ε ∗ (uu t )]∂x−2 u εyy 3 R2 + A(ϕ ε ∗ ∂x (ψu)) + B(ϕ ε ∗ ∂x (ψu)) 2 2 R R 5 ε −1 ε ε + u (∂x u y )(ϕ ∗ (ψu) y ) 3 R2 := I + I I + I I I + I V + V + V I + V I I + V I I I, where 5 5 5 A = − ∂x−4 u ε4y + ∂x−2 ϕε,yy ∗ u 2 + u ε ∂x−2 u εyy 3 6 3 and B=

5 −1 ε 2 5 ε 3 5 5 (∂ u ) − (u ) − 3u ε4x + 10u εyy − u ε u εx x − (u ε )2x x . 6 x y 6 2 4

Next one can check that (see [26, Lemma 1] for a similar computation) −2 ε 1 1ε (u), III + IV + V = (∂x u yy )ε (u) + R2

R2

where 5 5 1ε (u) = − [ϕ ε ∗ (uu t ) − u ε u εt ] + [ϕ ε ∗ (uu x ) − u ε u εx ]u ε , 3 3 and 1ε (u) = ϕ ε ∗ (uu x ) − u ε u εx × 5 5 25 5 5 × (∂x−1 u εy )2 − (u ε )3 − 3u ε4x + u εyy − u ε u εx x − (u ε )2x x 6 6 3 2 4 5 ε ε ε ε −1 ε + ϕ ∗ (uu y ) − u u y u ∂x u y . 3 Using the Lebesgue dominated convergence theorem, we obtain that 1ε (u)| −→ 0, t ∈ [0, T ]. |1ε (u)|2 −→ 0, | R2

ε→0

R2

ε→0


789

Let us next compute the five other terms in the right-hand-side of (36) one by one: 5 VI = A(ϕ ε ∗ ∂x (ψu)) = (ϕ ε ∗ (ψu))∂x−3 u ε4y 3 R2 R2 5 ε − (ϕ ε ∗ (ψu))(∂x−1 ϕ yy ∗ u2) 6 R2 5 + (ϕ ε ∗ (ψu))u ε (∂x−2 u εyy ) 3 R2 x := A1 + A2 + A3 . Next

5 d 5 ε −2 ε I =− ∂x−2 ∂ y2 ϕ ε ∗ (ψu) u εt (ϕ ∗ (ψu))(∂x u yy ) = − 3 dt R2 3 R2 5 − ψ(∂x−2 u εyy )u εt 3 R2 5 − ψt u ε (∂x−2 u εyy ) 3 R2 5 − [(ϕ ε ∗ (ψu)t − (ψu ε )t ]∂x−2 u εyy 3 R2 := C1 + C2 + C3 + C4

with

5 5 −2 2 ε ε C1 = ∂x−2 ∂ y2 ϕ ε ∗ (ψu) ∂x−1 u εyy (∂x ∂ y ϕ ∗ (ψu))u 3x − 3 R2 3 R2 5 5 −2 2 ε ε + (∂ ∂ ϕ ∗ (ψu))(ϕ ∗ (uu x )) + (∂ −2 ∂ 2 ϕ ε ∗ (ψu))(ϕ ε ∗ ∂x (ψu)) 3 R2 x y 3 R2 x y 5 = (∂ 2 ϕ ε ∗ (ψu))u εx − A1 − A2 + 0 . 3 R2 y

As mentioned before, here is the crucial cancellation, thanks to the first additional term in F ψ . Next 5 5 5 2 ε ε 2 ε ε ∂ y2 ϕ ε ∗ (ψu) − ∂ y2 (ψu ε ) u εx (∂ y ϕ ∗ (ψu))u x = ∂ y (ψu )u x + 3 R2 3 R2 3 2 R 5 5 = ψx (u εy )2 + (ψ yy u ε + ψ y u εy )u εx 6 R2 3 R2 5 ∂ y2 ϕ ε ∗ (ψu) − ∂ y2 (ψu ε ) u εx . + 3 R2 Therefore,

5 5 5 ε 2 ε ε ψx (u y ) + ψ yy u u x + ψ y u εy u εx V I + C1 = 6 R2 3 R2 3 R2 5 + (ϕ ε ∗ (ψu))u ε (∂x−2 u εyy ) 3 R2 x 5 + ∂ y2 ϕ ε ∗ (ψu) − ∂ y2 (ψu ε ) u εx . 3 R2

790


On the other hand, 5 5 ψ(∂x−2 u εyy )u ε3x − ψ(∂x−2 u εyy )∂x−1 u εyy 3 R2 3 R2 5 + ψ(∂x−2 u εyy ) ϕxε ∗ ((u ε )2 + 2ψu ε ) 6 R2 := C21 + C22 + C23 .

C2 =

First, C22 =

5 ψx (∂x−2 u εyy )2 . 6 R2

Next, C21 = = =

=

5 5 −1 ε ε − ψ(∂x u yy )u x x − ψx (∂x−2 u εyy )u εx x 3 R2 3 R2 5 5 5 ψu εyy u εx + ψx (∂x−1 u εyy )u εx − ψx (∂x−2 u εyy )u εx x 3 R2 3 R2 3 R2 5 5 ψx (u εy )2 − ψx u εyy u ε 6 R2 3 R2 5 5 5 −1 ε ε −2 ε ε − ψx x (∂x u yy )u − ψx (∂x u yy )u x x − ψ y u εy u εx 3 R2 3 R2 3 R2 15 5 5 ε 2 −1 ε 2 ψx (u y ) − ψx x x (∂x u y ) − ψx (∂x−2 u εyy )u εx x 6 R2 6 R2 3 R2 5 5 5 − ψ y u εy u εx − ψx yy (u ε )2 + ψx x y (∂x−1 u εy ) u ε . 3 R2 6 R2 3 R2

Since ψt = −ψx , C3 =

5 ψx u ε (∂x−2 u εyy ) . 3 R2

Now 5 5 d −1 ε 2 ψ(∂x u y ) = − ψt (∂x−1 u εy )2 − 6 dt R2 6 R2 5 + ψ(∂x−1 u εy )u εx x y 3 R2 5 + ψ(∂x−1 u εy )(ϕ ε ∗ (uu y )) 3 R2 5 + ψ(∂x−1 u εy )(ϕ ε ∗ (ψu) y ) 3 R2 5 − ψ(∂x−1 u εy )(∂x−2 u ε3y ) 3 R2 := D1 + D2 + D3 + D4 + D5 .


791

Since ψt = −ψx and X → L ∞ (R2 ), by involving two commutators, we infer that ε 2ε (u), D1 + D3 + D4 = G 1 (u ) + R2

where G 1 is a continuous functional on X and 2ε (u)| −→ 0, t ∈ [0, T ]. | R2

ε→0

On the other hand, 5 5 ψx (∂x−1 u εy )u εx y − ψu εy u εx y 3 R2 3 R2 5 5 5 −1 ε ε ε 2 = ψx x (∂x u y )u y + ψx (u y ) + ψx (u εy )2 3 R2 3 R2 6 R2 15 5 ε 2 = ψx (u y ) − ψ3x (∂x−1 u εy )2 6 R2 6 R2

D2 = −

and

5 5 ψ y (∂x−1 u εy )(∂x−2 u εyy ) + ψ(∂x−1 u εyy )(∂x−2 u εyy ) 3 R2 3 R2 5 5 −1 ε −2 ε = ψ y (∂x u y )(∂x u yy ) − ψx (∂x−2 u εyy )2 . 3 R2 6 R2

D5 =

Note that the last term above canceled with C22 . Summarizing, we infer that 5 5 −2 ε ε I + II +VI = − ψx (∂x u yy )u x x + ψ(∂x−2 u εyy )((u ε )2 + 2ψu ε )x 3 R2 6 R2 5 5 + (ϕxε ∗ (ψu))u ε (∂x−2 u εyy ) + ψx u ε (∂x−2 u εyy ) 3 R2 3 R2 5 + ψ y (∂x−1 u εy )(∂x−2 u εyy ) + G 2 (u ε ) 3 R2 −2 ε 2 3ε (u), + (∂x u yy )ε (u) + R2

R2

where G 2 is a continuous functional on X and 3ε (u)| −→ 0, t ∈ [0, T ]. |2ε (u)|2 −→ 0, | R2

ε→0

ε→0

R2

Since clearly V I I I = G 3 (u ε ) +

R2

4ε (u),

where G 3 is a continuous functional on X and 4ε (u)| −→ 0, t ∈ [0, T ], | R2

ε→0

792


it remains to estimate V I I . We notice that 5 5 VII = − (∂x−1 u εy )2 (ϕxε ∗ (ψu)) − (u ε )3 (ϕxε ∗ (ψu)) 6 R2 6 R2 ε ε −3 u 4x (ϕx ∗ (ψu)) − 10 u εy (ϕ ε ∗ (ψu) y ) R2 R2 5 5 − u ε u εx x (ϕxε ∗ (ψu)) − ∂ 2 (u ε )2 (ϕxε ∗ (ψu)) . 2 R2 4 R2 x We now observe that we can write ε

V I I = G 4 (u ) +

R2

4ε (u),

where G 4 is continuous on X and 4ε (u)| −→ 0, t ∈ [0, T ]. | ε→0

R2

For example, −3 u ε4x (ϕxε ∗ (ψu)) = −3 u ε4x (ψx u ε + ψu εx ) 2 2 R R −3 [ϕ ε ∗ (ψu)x − (ψu ε )x ]u ε4x R2 =3 u ε3x (2ψx u εx + ψx x u ε + ψu εx x ) R2 −3 [ϕ ε ∗ (ψu)x − (ψu ε )x ]u ε4x 2 R 15 9 =− ψx |u εx x |2 + ψ3x |u εx |2 − 3 ψ3x u ε u εx x 2 R2 2 R2 R2 −3 [ϕ ε ∗ (ψu)x − (ψu ε )x ]u ε4x . R2

All other terms in the representation of V I I can be treated similarly. This achieves the proof of Lemma 3. Now, since

5

5

[ϕ ε ∗ (u 2 + 2ψu)]∂x−2 u εyy ≤ C(u4L 4 + u2L 2 ψ2L ∞ ) + ∂x−2 u εyy 2L 2 ,

6 R2 12 there exists a constant C > 0 such that for ε small enough, F ψ,ε (u(t)) ≥

5 −2 ε ∂ u (t)2L 2 − C, ∀t ∈ [0, T ] . 24 x yy

(37)

We thus deduce from Lemma 3 and (37) that d ψ,ε F (u(t)) ψW 3,∞ |F ψ,ε (u(t))| + Rψ (u ε ) + ε (t), dt

(38)


793

where Rψ is continuous on X and |ε |1 → 0 as ε → 0 uniformly for t ∈ [0, T ]. Here we used that thanks to Lemma 2, ∂x−1 u εy L 4 ∂x−1 u εy L 2 u εy L 2 ∂x−2 u εyy L 2 1/4

1/2

1/4

and u εx L 4 u εx L 2 u εx x L 2 u εy L 2 , 1/4

1/2

1/4

and thus,

ε ε −2 ε

ψu x u (∂x u yy ) ψ L ∞ u ε L 4 u εx L 4 ∂x−2 u εyy L 2 R2

|F ψ,ε (u)| + Rψ (u ε ),

and in the same way

ψ(∂x−1 u εy )u ε u εy ψ L ∞ u ε L 4 u εx L 4 ∂x−1 u εy L 4 u εy L 4 R2

|F ψ,ε (u)| + Rψ (u ε ).

Hence, |F ψ,ε (u(t))| exp(Ct)|F ψ,ε (φ)| + exp(Ct).

(39)

Letting ε tend to 0, F ψ,ε (φ) → F ψ (φ) < ∞ and thus sup

t∈[0,T ], ε>0

F ψ,ε (u(t)) 1.

From (37), one infers sup

t∈[0,T ], ε>0

ε

ϕ ∗ ∂x−2 u yy (t) 1 2

and thus ∂x−2 u yy ∈ L ∞ (0, T ; L 2 (R2 )) , by the Lebesgue theorem. It is then easy to check that F ψ, (u(t)) −→ F ψ (u(t)), t ∈ [0, T ], ε→0

and thus (27) follows from (39). This completes the proof of Proposition 3.

∞ (R2 )), ∂ −2 u ε It follows from (37) and the Lebesgue theorem that, for u ∈ C([0, T ]; H−1 x yy tends to ∂x−2 u yy in L ∞ ([0, T ]; L 2 (R2 )). Therefore we can pass to the limit in ε in Lemma 3 which leads to the next proposition.

794


∞ (R2 )) then Proposition 4. Let u ∈ C([0, T ]; H−1

u(t) Z g(t) ,

(40)

where g is an increasing continuous one to one mapping of R+ only depending on u(0) Z . Moreover, 5 t ψx (∂x−2 u yy )u x x 3 0 R2 5 t + ψ(∂x−2 u yy )((u)2 + 2ψu)x 6 0 R2 5 t 5 + ψu(∂x−2 u yy ) + (ψu)x u(∂x−2 u yy ) 3 0 R2 3 R2 t 5 t + ψ y (∂x−1 u y )(∂x−2 u yy ) + G ψ (u) , (41) 3 0 R2 0

F ψ (u(t)) − F ψ (u(0)) = −

where G ψ is continuous on X . 4. Dispersive Estimates Let us first extend some linear dispersive estimates established in [28] for anti-derivatives in x of the free group of KP. A similar argument was used in [25]. Let U (t) := exp(−t (∂x3 − ∂x−1 ∂ y2 )) be the unitary group on H s (R2 ) defining the free KP-I evolution. Let Dx be the Fourier multiplier with symbol |ξ |. Then we have the following Strichartz inequality for the free KP-I evolution. Lemma 4. Let T > 0. Then for every 0 ≤ ε ≤ 1/2, we have the estimates ) − εδ(r 2

Dx and

t

T

− εδ(r ) Dx 2 U (t

0

U (t)φ L q L r φ L 2 ,

− t )F(t )dt

q

L T L rx y

F L 1 L 2 ,

provided r ∈ [2, ∞], and 0≤

2 = (1 − ε/3) δ(r ) < 1 q

with δ(r ) := 1 −

(42)

xy

2 . r

T

xy

(43)


795

Proof. Since U (t) and Dx commute, using the Minkowski inequality, we obtain that (43) follows from (42). Let us now turn to the proof of (42). For r = 2, (42) follows from the unitarity of U (t) on L 2 (R2 ). It suffices thus to prove it for r = ∞. Let us set 3 2 G(x, y, t) := eit (xξ +yη) eit (ξ −η /ξ ) dξ dη . R2

Then U (t)φ = G(·, ·, t) φ , where denotes the convolution with respect to the spatial variables x and y. Integrating a gaussian integral with respect to η (see [28]), we get for 0 ≤ ε ≤ 1/2,

∞ 1

3

−1/2 2 −ε ei(tξ +xξ ) dξ . |t| sup |ξ | Dx−ε G(x, y, t) L ∞

xy x∈R

0

Next using the Van der Corput lemma as in [20], we infer that Dx−ε G(x, y, t) L ∞ |t|−1+ε/3 , xy and thus

Dx−ε U (t) φ L ∞ |t|−1+ε/3 φ L 1x y , xy

(44)

which is the key dispersive inequality. Let us now perform the duality argument which provides (42) for r = ∞ as a consequence of the dispersive inequality (44). We first fix the q corresponding to r = ∞, i.e. 2 ε =1− . q 3 Set −ε/2

A := Dx

U (t) .

Our goal is to show that A is bounded from L 2x y to L T L ∞ x y . Denote by A the operator formally adjoint to A. Then T

AA ( f ) = Dx−ε U (t − t ) f (t )dt . q

0

Using the dispersive estimate (44) and the Hardy-Littlewood inequality in time, we infer that (45) A A ( f ) L q L ∞ f q 1 , T

xy

LT Lxy

where q is the conjugate of q, i.e. 1 1 + = 1. q q q

1 1 Recall that L ∞ x y can be seen as the dual space of L x y . It is not true that L T L x y can be q ∞ seen as the dual of L T L x y but however, it is easy to see that for q < ∞, we can write

A f L q L ∞ = T

xy

g

sup

≤1 q L T L 1x y

| A f , g |,

(46)

796


where ·, · denotes the L 2T L 2x y inner product. Let us next write by the Cauchy-Schwarz inequality | A f , g | = |( f , A g)| ≤ f L 2x y A g L 2x y , (47) where (·, ·) denotes the L 2x y inner product. Next, using (45), we obtain A g2L 2 = (A g , A g) = A A (g) , g ≤ A A (g) L q L ∞ g T

xy

xy

q

L T L 1x y

≤ Cg2 q

L T L 1x y

.

Coming back to (46) and (47) ends the proof of (42) for r = ∞. This completes the proof of Lemma 4. We next state the crucial dispersive estimate. ∞ (R2 )) be a solution of Proposition 5. Let v ∈ C([0, T ]; H−1

vt + vx x x − ∂x−1 v yy = Fx .

(48)

Then for every ε > 0 there exists Cε such that 1/2+ε

v L 1 L ∞ ≤ Cε (1 + T )Jx T

1/2+ε

v L ∞ L 2x y + Cε Jx T

xy

F L 1 L 2 T

xy

.

(49)

Proof. We consider a Littlewood-Paley decomposition in the x-variable v= vλ , λ −dyadic

where vλ := λ v and the Fourier multipliers λ are defined as follows: ξ ˆ ξ, η), λ = 2k , k ≥ 1, ϕ( λ ) v(t, λ v(t, ξ, η) := χ (ξ ) v(t, ˆ ξ, η), λ = 1, where the nonnegative functions χ ∈ C0∞ (R) and ϕ ∈ C0∞ (R∗ ) are defined as in [22]. For λ ≥ 2 fixed we write a natural splitting [0, T ] = Ij, j

where I j = [a j , b j ] are with disjoint interiors and |I j | ≤ λ−1 . Clearly, we can suppose that the number of the intervals I j is bounded by C(1 + T )λ. Using the Hölder inequality in time, we can write 1 ε vλ L 1 L ∞ vλ L 1 L ∞ λ− 2 − 6 vλ L qε L ∞ , xy

T

Ij

j

xy

Ij

j

xy

where 1/qε = 1/2 − ε/6. Next, we apply the Duhamel formula in each I j to obtain on I j , t −ε/2 ε/2 vλ (t) = Dx U (t − a j )Dx vλ (a j ) − Dx−ε U (t − t )[λ Dxε ∂x F](t ) dt . aj

Using the Strichartz estimates established in Lemma 4, it yields ε/2

vλ L qε L ∞ Dx vλ (a j ) L 2 + λ Dx1+ε F L 1 Ij

xy

Ij

L 2x y

.


797

Therefore vλ L 1

Ij

L∞ xy

λ−1/2+ε/3 vλ (a j ) L 2 + λ1/2+5ε/6 λ F L 1

Ij

L 2x y ,

and summing over j, vλ L 1 L ∞ λ−1/2+ε/3 T

vλ L ∞ L 2x y + λ1/2+5ε/6 λ F L 1 L 2 T

xy

T

xy

j

λ1/2+ε/3 (1 + T )vλ L ∞ L 2x y + λ1/2+5ε/6 λ F L 1 L 2 . (50) T

xy

T

Moreover, again by Duhamel formula and Strichartz estimates 1 v L 1 L ∞ (1 + T ) 1 v(0) L 2x y + 1 F L 1 L 2 . T

T

(51)

xy

Hence, by Minkowski and Bernstein inequalities and (50)-(51), for any 0 < α, ε 0, J s ( f g) L p (Rn ) J s f L p (Rn ) g L ∞ (Rn ) + f L ∞ (Rn ) J s g L p (Rn ) , and for 0 < s < 1, J s ( f g) L p (R) J s f L ∞ (R) g L p (R) + f L ∞ (R) J s g L p (R) . Proposition 5 yields the following estimates on smooth solutions to (24). ∞ (R2 )) Lemma 6. For every 0 < ε 0 such that for every λ ≥ 2, λλ u L ∞ ≤ Cu x L ∞ . xy xy Therefore using the triangle inequality and the Bernstein inequality, we obtain that Jxs u L ∞ ≤ Jxs 1 u L ∞ + Jxs λ u L ∞ xy xy xy u L ∞ + xy

λ≥2

λ λ u L ∞ xy s

λ≥2

u L ∞ + xy

λs−1 u x L ∞ xy

λ≥2

u L ∞ + u x L ∞ . xy xy

This completes the proof of Lemma 7.

Let us now return to the proof of Proposition 6. Noticing that u L ∞ is controlled by u Z , using (52) with ε = 1/2 and Proposition 4, we obtain the existence of an increasing continuous function4 g1 from R+ to R+ such that u x L 1 L ∞ ≤ g1 (T ) T

xy

4 The important point is that this function does not go to infinity in finite time.

(54)

800


as long as u(t) exists. Notice that, thanks to Lemma 7, estimate (54) also provides a bound for Jxs u L 1 L ∞ , 0 < s < 1 . T

xy

Next, using (13) and (14) and the Gronwall lemma, we obtain that for every integer s, (55) ∂xs u(t, ·) L 2 ≤ ∂xs φ L 2 exp u x L 1 L ∞ + tψW s,∞ . t

x,y

Therefore using Proposition 4 to bound u(t, ·) L 2 we obtain that there exists an increasing continuous function g2 from R+ to R+ such that Jx4 u(t, ·) L 2x y ≤ g2 (t)

(56)

as long as u(t) exists. Noticing that 1/2+ε

Jx

u y 2L 2 = u y L 2x y + |ξ |1/2+ε η u(ξ, ˆ η)2L 2

ξη

= u y L 2x y + |ξ |

3/2+ε

|ξ |

−1

η u(ξ, ˆ η)2L 2

ξη

u y L 2x y + Jx4 u L 2x y ∂x−2 u yy L 2x y , we deduce from (56), (53), (54), Lemma 7 and Proposition 4 that there exists an increasing continuous function g3 from R+ to R+ such that u y L 1 L ∞ g3 (t). T

xy

Therefore ∇x,y u L 1 L ∞ can not go to infinity in finite time T . Thanks to the well-poT xy sedness Proposition 1 we deduce that u can be extended on the whole real axis which completes the proof of Proposition 6. 6. Well-Posedness in H 2,0 (R2 ) The next two sections are devoted to the global well-posedness of (5) in Z , i.e. we remove ∞ (R2 ) of Proposition 6. A considerable part of the analysis will the condition φ ∈ H−1 be devoted to the continuous dependence with respect to time and the initial data. For s ∈ R, we denote by H s,0 (R2 ) the anisotropic Sobolev spaces equipped with the norm u H s,0 (R2 ) = Jxs u L 2 (R2 ) . For an integer s ≥ 0 an equivalent norm in H s,0 (R2 ) is given by u L 2 (R2 ) + ∂xs u L 2 (R2 ) . We have the following well-posedness result. Proposition 7. Let φ ∈ H 2,0 (R2 ). Then there exists a unique positive T depending only on φ H 2,0 and a unique solution u of (5) with initial data φ on the time interval [0, T ] satisfying u ∈ C([0, T ]; H 2,0 (R2 )), u x ∈ L 1T L ∞ (57) xy . Furthermore, the map φ → u φ is continuous from H 2,0 (R2 ) to C([0, T ]; H 2,0 (R2 )).


801

6.1. Uniqueness. The uniqueness follows from the next lemma. Lemma 8. Let u, v be two solutions of (5) in the class defined by (57). Then u − v L ∞ L 2x y ≤ exp Cu x L 1 L ∞ + Cvx L 1 L ∞ + C T ψx L ∞ u(0) − v(0) L 2x y . T

T

xy

T

xy

Proof. We set w := u − v . Further we define u ε := ϕ ε ∗ u, v ε := ϕ ε ∗ v, w ε := ϕ ε ∗ w , where ϕ ε is defined by (31). Then w ε solves the equation 1 wtε + wxε x x − ∂x−1 w εyy + ∂x w ε (u + v) + ∂x (ψw ε )− 2 1 1 − ∂x ( (u + v) + ψ)w ε + ϕxε ∗ ( (u + v) + ψ)w = 0 . 2 2 Taking the L 2 scalar product of the last equation with w ε gives 1 d 1 ε 2 ε 2 w L 2 (R2 ) = − (u x + vx )(w ) − ψx (w ε )2 2 dt 2 R2 R2 1 ∂x [(u + v + 2ψ)w ε ] − ϕxε ∗ [(u + v + 2ψ)w] w ε . + 2 R2 Thus, by the Gronwall Lemma, wε 2L ∞ L 2 exp Cu x L 1 L ∞ + Cvx L 1 L ∞ + C T ψx L ∞ w ε (0)2L 2 T xy T xy T xy

+ ∂x [(u + v + 2ψ)w ε ] − ϕxε ∗ [(u + v + 2ψ)w] 1 2 w ε L ∞ L 2x y . T LT Lxy

Passing to the limit in ε → 0, it follows from the Lebesgue theorem and the assumptions on u and v that the commutator term in the right-hand side of the last inequality tends to 0. Thus w2L ∞ L 2 exp Cu x L 1 L ∞ + Cvx L 1 L ∞ + C T ψx L ∞ w(0)2L 2 , T

xy

T

xy

which completes the proof of Lemma 8.

T

xy

xy

(58)

802


6.2. Existence. Let φ ∈ H 2,0 (R2 ). We set φε := ϕ ε ∗ φ , ∞ ∩ Z and that φ → φ in where ϕ ε are defined by (31). It is clear that φε ∈ H−1 ε 2,0 2 H (R ). By Proposition 6, the emanating solution u ε is global in time and belongs to ∞ ). We will show that there exists T = T (φ C(R+ ; H−1 H 2,0 ) > 0 such that the sequence 2,0 regularity we do not control the L ∞ . On the level of H {∂x u ε } is bounded on L 1T L ∞ xy xy norm. Using a Littlewood-Paley decomposition in the x variable and Lemma 7, we can however write u ε L 1 L ∞ 1 u ε L 1 L ∞ + u ε L 1 L ∞ T

xy

xy

T

1 u ε L 1 L ∞ + xy

T

xy

T

λ≥2

λ−1 ∂x u ε L 1 L ∞ T

λ≥2

xy

1 u ε L 1 L ∞ + ∂x u ε L 1 L ∞ . xy

T

xy

T

Next, as in (51), the Duhamel formula and Strichartz estimates of Proposition 4 yield for 0 ≤ T ≤ 1, 1 u ε L 1 L ∞ φε L 2x y + ∂x u ε L 1 L ∞ u ε L ∞ L 2x y + ψ L ∞ u ε L ∞ L 2x y . T

xy

T

xy

T

T

L2

Therefore, thanks to the control (25) of Proposition 3 we obtained that there exists a constant C depending on bounds on ψ and its derivatives in L ∞ but independent of φ such that for 0 ≤ T ≤ 1, u ε L 1 L ∞ φε L 2x y + (1 + φε L 2x y )∂x u ε L 1 L ∞ . xy

T

T

xy

Using (13) and (14) with s = 2 we obtain that there exists a constant C depending on bounds on ψ and its derivatives in L ∞ but independent of φ such that d 2 ∂x u ε (t, ·)2L 2 ≤ C ∂x u ε (t, ·) L ∞ + C Jx2 u ε (t, ·)2L 2 (R2 ) , (59) dt and therefore thanks to the L 2 bound on u ε provided by Proposition 3 and the Gronwall lemma, we obtain that Jx2 u ε L ∞ L 2x y Jx2 φε L 2x y exp(C∂x u ε L 1 L ∞ + C T ). T

T

xy

Let us set f (T ) := u ε L 1 L ∞ + ∂x u ε L 1 L ∞ . T

xy

T

xy

We deduce from the last estimates and (52) that for 0 ≤ T ≤ 1, f (T ) (1 + φε L 2x y )∂x u ε L 1 L ∞ + φε L 2x y T

xy

Jx2 φε L 2x y (1 + φε L 2x y ) exp(C f (T )) . (60) We notice that for α > 0 small enough the continuous map g : y −→ y − α exp(C y)


803

satisfies g(0) < 0 and that g(y0 ) = 0 for some 0 < y0 < 1. Since T −→ f (T ) is continuous and satisfies f (0) = 0 and (60), we deduce that for Jx2 φε L 2x y (1 + φε L 2x y ) small enough, 0 ≤ f (T ) ≤ 1 for all T ∈ [0, 1]. Therefore, if Jx2 φ L 2 1 then for 0 ≤ T ≤ 1, (61) u ε L 1 L ∞ + ∂x u ε L 1 L ∞ ≤ 1 . T

xy

T

xy

We next use a scaling argument to show that a bound of type (61) holds for Jx2 φ L 2 of an arbitrary size. Notice that u(t, x, y) is a solution of (u t + u x x x + uu x + ∂x (ψu))x − u yy = 0 with φ(x, y) as initial data if and only if for every β ∈ R, u β (t, x, y) := β 2 u(β 3 t, βx, β 2 y) is a solution to (∂t u β + ∂x3 u β + u β ∂x u β + ∂x (ψβ u β ))x − ∂ y2 u β = 0, with initial data φβ (x, y) = β 2 φ(βx, β 2 y) and ψβ (t, x, y) = β 2 ψ(βx − β 3 t, β 2 y) instead of ψ(x − t, y). One can check that for 0 < β ≤ 1, Jx2 φβ L 2x y β 1/2 Jx2 φ L 2x y , and for s ∈ N, Jxs ψβ L ∞ β 2 Jxs ψ L ∞ . xy xy Let us set u ε,β (t, x, y) := β 2 u ε (β 3 t, βx, β 2 y) . In view of the above discussion for β ∼ (1 + Jx2 φε L 2x y )−2 −→ (1 + Jx2 φ L 2x y )−2 ε→0

one has the bound

1 0

u ε,β (t) L ∞ + ∂x u ε,β (t) L ∞ dt ≤ 1 xy xy

804


which leads to

β3 0

u ε (t) L ∞ + ∂x u ε (t) L ∞ dt ≤ 1/β 3 . xy xy

We thus obtain for T ∼ (1 + Jx2 φ L 2x y )−6 and ε small enough, u ε L 1 L ∞ + ∂x u ε L 1 L ∞ (1 + Jx2 φ L 2x y )6 , T

xy

T

xy

(62)

which is the substitute of (61) in the case of an arbitrary initial data. Therefore we deduce 2,0 (R2 ) and {∂ u } is bounded in L 1 L ∞ . It then follows that {u ε } is bounded in L ∞ x ε T H T xy from Lemma 8 that u ε − u ε L ∞ L 2x y → 0 T

as ε, ε → 0. Hence, there exists u ∈ C([0, T ]; L 2 (R2 )) ∩ Cw ([0, T ]; H 2,0 (R2 )) 2 2 with u x ∈ L 1T L ∞ x y such that u ε converges to u in C([0, T ]; L (R )). It is clear that u satisfies (5) at least in a weak (distributional for instance) sense.

6.3. Continuity with respect to time. Now the continuity of u with values in H 2,0 (R2 ) as well as the continuity of the flow-map in H 2,0 (R2 ) will follow from the Bona-Smith argument. Note that here we are not in the classical situation since u does not belong to L ∞ (0, T ; H 2+ (R2 )). Let u be a fixed solution of (5) with initial data φ ∈ H 2,0 (R2 ). Recall that u ε is the solution to (5) with φε = ϕ ε ∗ φ as initial data. We will show that {u ε } is in fact a Cauchy sequence in C([0, T ]; H 2,0 (R2 )) which will prove that u ∈ C([0, T ]; H 2,0 (R2 )) . First by straightforward calculations in Fourier space, one can show that for φ ∈ H 2,0 (R2 ), 0 < ε < 1 and r ≥ 0,

and

ϕ ε ∗ φ H 2+r,0 (R2 ) ε−r φ H 2,0 (R2 )

(63)

ϕ ε ∗ φ − φ H 2−r,0 (R2 ) o εr φ H 2,0 (R2 )

(64)

as ε → 0. For 0 < ε2 < ε1 < 1, we set w := u ε1 − u ε2 . It follows from the estimates of the previous subsection applied to u ε1 and u ε2 that w L 1 L ∞ + wx L 1 L ∞ ≤ C, T

xy

T

xy

and that for any s ∈ N, s ≥ 2, Jxs u ε L ∞ L 2x y Jxs u ε (0) L 2x y ε2−s . T

The issue is to show that Jx2 w L ∞ L 2x y is not only bounded but that it tends to zero as T ε1 tends to zero. Using Lemma 8, we obtain that for 0 ≤ T ≤ 1, w L ∞ L 2x y w(0) L 2x y o ε12 . (65) T


805

Next, we observe that w solves the equation wt + wx x x − ∂x−1 w yy + ∂x (ψw) − wwx + u ε1 ∂x w + w∂x u ε1 = 0 .

(66)

Note that since ε2 < ε1 , we privilege u ε1 to u ε2 when writing Eq. (66). We next apply ∂x2 to (66) and multiply it with ∂x2 w. Using (13) and (14) and the Gronwall lemma, we obtain that Jx2 w2L ∞ L 2 exp C∂x u ε1 L 1 L ∞ + C∂x w L 1 L ∞ + C T ψW 3,∞ T xy T xy T xy Jx2 w(0)2L 2 + wx 2L 1 L ∞ Jx2 u ε1 2L ∞ L 2 + w2L 1 L ∞ Jx3 u ε1 2L ∞ L 2 . (67) T

T

xy

xy

T

xy

T

xy

On the other hand, according to (66), (49), (65), Lemma 7 and (62) applied to u ε1 and u ε2 , one infers that for 0 ≤ T ≤ 1, 1/2+ w L 1 L ∞ 1 + u ε1 + u ε2 L 1 L ∞ + T ψW 1,∞ Jx w L ∞ L 2x y T T xy T xy 1 + + 1 + Dx2 (u ε1 + u ε2 ) L 1 L ∞ w L ∞ L 2x y T

T

xy

3 2−

ε1 ,

(68)

where in the last step we used that, by (65) and interpolation argument, for 0 0 fixed, the sequence of functions t → (∂x−2 u εyy (t), ϕα ∗ v) L 2 is equi-continuous on [0, T ] and thus ∂x−2 u εyy → ∂x−2 u yy in Cw ([0, T ]; L 2 (R2 )). Hence, for any t ∈ [0, T ], ε 2 −2 ε (u (t)) ∂x u yy (t) → u 2 (t)∂x−2 u yy (t), R2

R2

R2

|∂x−2 u yy (t)|2 ≤ lim inf ε→0

R2

|∂x−2 u εyy (t)|2 .

Thus passing in the limit in ε in (41), we obtain F ψ (u(t)) ≤ lim inf F ψ (u ε (t)) ε→0 5 t ψx (∂x−2 u yy ) u x x = F ψ (φ) − 3 0 R2 5 t + ψ(∂x−2 u yy )((u)2 + 2ψu)x 6 0 R2 5 t 5 t + ψu(∂x−2 u yy ) + (ψu)x u(∂x−2 u yy ) 3 0 R2 3 0 R2 t 5 t + ψ y (∂x−1 u y )(∂x−2 u yy ) + G(u) . 3 0 R2 0 Taking u(t) as initial data and reversing time, we get the reverse inequality and thus 5 t F ψ (u(t)) = F ψ (φ) − ψx (∂x−2 u yy ) u x x 3 0 R2 5 t 5 t + ψ(∂x−2 u yy )((u)2 + 2ψu)x + ψu(∂x−2 u yy ) 6 0 R2 3 0 R2 t 5 t 5 t −2 −1 −2 + (ψu)x u(∂x u yy ) + ψ y (∂x u y )(∂x u yy ) + G(u). (74) 3 0 R2 3 0 R2 0 It follows that t → F ψ (u(t)) is continuous on [0, T ]. Now, since ∂x−2 u yy ∈ Cw ([0, T ]; L 2 (R2 )), t → u 2 ∂x−2 u yy ∈ C([0, T ]) , R2

and thus the continuity of t → F ψ (u(t)) forces t → C([0, T ]). It follows that

R2

t → ∂x−2 u yy ∈ C([0, T ]; L 2 (R2 ))) , which proves that u(t) describes a continuous curve in Z .

|∂x−2 u yy |2 to belong to

808


Now, let {φn } ⊂ Z such that φn → φ in Z . Since F ψ is continuous on Z , F ψ (φn ) → ψ F (φ). Moreover, using that the emanating solutions u n converges to u in C([0, T ]; X ) and that ∂x−2 u nyy converges to ∂x−2 u yy in Cw ([0, T ]; L 2 (R2 )), we infer from (74) that F ψ (u n (t)) − F ψ (φn ) → F ψ (u(t)) − F ψ (φ), ∀t ∈ [0, T ]. This convergence clearly forces |∂x−2 u nyy (t)|2 → R2

R2

|∂x−2 u yy (t)|2 , ∀t ∈ [0, T ],

which permits to conclude that u n → u in C([0, T ]; Z ). We end this section by an important remark. Notice that our result requires a control on ∇x,y u L 1 L ∞ in order to have the basic global well-posedness theorem of Sect. 4. T xy But once this global in time result is established, further improvements of the local well-posedness theory in the spaces H s,0 (R2 ), or in the spaces considered in [17], only requires a control on u x L 1 L ∞ in terms of Jxs u L ∞ L 2x y . T

xy

T

8. Well-Posedness of the KP-I Equation in H s,0 (R2 ), s > 3/2 The aim of this section is to extend Kenig’s local well-posedness result by showing that the KP-I equation is locally well-posed for initial data in the space H s,0 (R2 ) := {u ∈ L 2 (R2 ) : |Dx |s u ∈ L 2 (R2 )} with s > 3/2 , that is, no y derivative is needed. Consider thus the KP-I equation (u t + u x x x + uu x )x − u yy = 0

(75)

with initial data

u(0, x, y) = φ ∈ H s,0 (R2 ) . (76) Thanks to the estimates established in Sect. 4, we have the following modest extension of Kenig’s result [17].

Theorem 2. The Cauchy problem (75)-(76) is locally well-posed in H s,0 (R2 ) for s > 3/2. ∞ (R2 ) solution to the KP-I equation. Then thanks to Proposition 5, we Let u be a H−1 obtain that for T ≤ 1, 3 +ε 3 +ε u L 1 L ∞ + u x L 1 L ∞ ≤ Cε Jx2 u L ∞ L 2x y + u L 1 L ∞ Jx2 u L ∞ L 2x y . (77) T

xy

T

T

xy

Jxs

T

T

xy

Jxs u

Applying to (75) and multiplying it with gives after applying the Kato-Ponce commutator estimate in x, d J s u(t, ·)2L 2 u x (t, ·) L ∞ Jxs u(t, ·)2L 2 , dt x and the Gronwall lemma gives the bound Jxs u L ∞ L 2x y ≤ Jxs φ L 2 exp(Cu x L 1 L ∞ ) . T

T

xy

(78)

Bounds (77) and (78) enable one to perform a compactness argument as we did in the proof of Proposition 1 which shows that the flow map of (75)-(76) can be extended to a map on H s,0 (R2 ), s > 3/2 with a life span depending only on a bound on Jxs φ L 2 . The continuity of the trajectory in H s,0 (R2 ) as well as the continuity of the flow-map can be derived as in Sects. 6.3-6.4


809

Added in Proof: After this work was completed, we were informed of the paper [39] where Zakharov proved, using an explicit construction via Inverse Scattering theory, the nonlinear instability of the line soliton to the KP-I equation with respect to periodic in y perturbations. Recently, Rousset and Tzvetkov [40] have proven this result by a general method, applicable to nonintegrable equations.

References 1. Alexander, J.C., Pego, R.L., Sachs, R.L.: On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation. Phys. Lett. A 226, 187–192 (1997) 2. Besov, O., Ilin, V., Nikolski, S.: Integral representation of functions and embedding theorems. New York: J. Wiley, 1978 3. Bona, J.L., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278 1287, 555–601 (1975) 4. Bourgain, J.: On the Cauchy problem for the Kadomtsev-Petviashvili equation. GAFA 3, 315–341 (1993) 5. Burq, N., Gérard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math. 126, 569–605 (2004) 6. Freeman, N.C.: Soliton interaction in two dimensions. Adv. Appl. Math. 20, 1–37 (1980) 7. Fokas, A.S., Sung, L.Y.: On the solvability of N -waves, Davey-Stewartson and Kadomtsev-Petviashvili equations. Inverse Problems 8, 673–708 (1992) 8. Fokas, A.S., Pogorobkov, A.K.: Inverse scattering transform for the KP-I equation on the background of a one line soliton. Nonlinearity 16, 771–783 (2003) 9. Gallo, C.: Korteweg-de Vries and Benjamin-Ono equations on Zhidkov spaces. Adv. Differ. Eqs. 10, 277– 308 (2005) 10. Groves, M.D., Haragus, M., Sun, S.M.: A dimension breaking phenomenon in the theory of steady gravity-capillary water waves . Phil. Trans. Roy. Soc. Lond. A 360, 2337–2358 (2002) 11. Haragus, M., Pego, R.L.: Travelling waves of the KP equations with transverse modulations. C.R. Acad. Sci. Paris 328, 227–232 (1999) 12. Ionescu, A., Kenig, C.: Local and global well-posedness of periodic KP-I equations. To appear in Proc. IAS workshop, March 2004 Annals of Math. Studies, Princeton Univ. Press. 13. Iório, R.J.Jr., Nunes, W.V.L.: On equations of KP-type. Proc. Roy. Soc. Edinburgh A 128, 725–743 (1998) 14. Isaza, P., Mejia, J.: Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices. Comm. PDE 26, 1027–1057 (2001) 15. Kato, J.: Existence and uniqueness of the solution to the modified Schrödinger map. Math. Res. Lett. 12, 171–186 (2005) 16. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations . Comm. Pure Appl. Math. 41, 891–907 (1988) 17. Kenig, C.: On the local and global well-posedness for the KP-I equation. Annales IHP Analyse Non linéaire 21, 827–838 (2004) 18. Kenig, C., Koenig, K.: On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Letters 10, 879–895 (2003) 19. Kenig, C., Nahmud, A.: The Cauchy problem for the Hyperbolic-Elliptic Ishimori system and Schrödinger maps. Nonlinearity 18, 1987–2009 (2005) 20. Kenig, C.E., Ponce, G., Vega, L.: Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40(1), 33–69 (1991) 21. Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle. Comm. and Pure Pure and Appl. Math. 46, 527–620 (1993) 22. Koch, H., Tzvetkov, N.: Local well-posedness of the Benjamin-Ono equation in H s (R). I.M.R.N. 26, 1449–1464 (2003) 23. Lions, J.-L.: Quelques méthodes de résolution des équations aux dérivées partielles non linéaires. Paris:Dunod (1969) 24. Molinet, L.: On the asymptotic behavior of solutions to the (generalized) Kadomtsev-Petvishvili-Burgers equation. J. Diff. Eq. 152, 30–74 (1999) 25. Molinet, L., Ribaud, F.: The global Cauchy probem in Bourgain’s type spaces for a dispersive dissipative semilinear equation. SIAM J. Math. Anal. 33, 1269–1296 (2002) 26. Molinet, L., Saut, J.C., Tzvetkov, N.: Global well-posedness for the KP-I equation. Math. Annalen 324, 255–275 (2002). Correction: Math. Ann. 328 707–710 (2004) 27. Molinet, L., Saut, J.C., Tzvetkov, N.: Well-posedness and ill-posedness for the Kadomtsev-Petviashvili-I equation. Duke Math. J. 115(2), 353–384 (2002)

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28. Saut, J.C.: Remarks on the generalized Kadomtsev-Petviashvili equations. Indiana Univ. Math. J. 42, 1017–1029 (1993) 29. Schwarz, M.Jr.: Periodic solutions of Kadomtsev-Petviashvili. Adv. Math. 66, 217–233 (1987) 30. Tajeri, M., Murakami, Y.: The periodic soliton resonance: solutions of the Kadomtsev-Petviashvili equation with positive dispersion. Phys. Lett. A 143, 217–220 (1990) 31. Takaoka, H.: Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete Contin. Dynam. Systems 6, 483–499 (2000) 32. Takaoka, H., Tzvetkov, N.: On the local regularity of Kadomtsev-Petviashvili-II equation. IMRN 8, 77– 114 (2001) 33. Tom, M.M.: On a generalized Kadomtsev-Petviashvili equation. In: Mathematical Problems in the Theory of water waves, Contem. Math. 200, Providence, RI: Aru. Math. Soc., 1996, pp. 193–210 34. Tzvetkov, N.: Global low regularity solutions for Kadomtsev-Petviashvili equation. Diff. Int. Eq. 13, 1289–1320 (2000) 35. Wickerhauser, M.V.: Inverse scattering for the heat equation and evolutions in (2+1) variables. Commun. Math. Phys. 108, 67–89 (1987) 36. Zaitsev, A.A.: Formation of stationary waves by superposition of solitons. Sov. Phys. Dokl. 28(9), 720– 722 (1983) 37. Zakharov, V., Schulman, E.: Degenerative dispersion laws, motion invariants and kinetic equations. Physica D 1, 192–202 (1980) 38. Zhou, X.: Inverse scattering transform for the time dependent Schrödinger equation with applications to the KP-I equation. Commun. Math. Phys. 128, 551–564 (1990) 39. Zakharov, V.E.: Instability and nonlinear oscillations of solutions. JETP Lett. 22, 172–173 (1975) 40. Rousset, F., Tzvetkov, N.: Transverse nonlinear instability for two-dimensional dispersive models, anxiv:math.AP/061249413 7 Feb 2007 Communicated by P. Constantin


Communications in


Parametric Representation of Noncommutative Field Theory Razvan Gurau, Vincent Rivasseau Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405 Orsay Cedex, France. E-mail: [email protected]; [email protected]; [email protected] Received: 9 June 2006 / Accepted: 23 August 2006 Published online: 3 April 2007 – © Springer-Verlag 2007

Abstract: In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable φ44 quantum field theory on the Moyal non commutative R4 space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual “Kirchoff” or “Symanzik” polynomials of commutative field theory, but contain richer topological information. I. Introduction Non-commutative field theories (for a general review see [1]) deserve a thorough and systematic investigation. Indeed they may be relevant for physics beyond the standard model. They are certainly effective models for certain limits of string theory [2, 3]. What is often less emphasized is that they can also describe effective physics in our ordinary standard world but with non-local interactions, such as the physics of the quantum Hall effect [4]. In joint work with J. Magnen and F. Vignes-Tourneret [5], we provided recently a new proof that the Grosse-Wulkenhaar scalar 4 theory on the Moyal space R4 , hereafter called NC44 , is renormalizable to all orders in perturbation theory using direct space multiscale analysis. The Grosse-Wulkenhaar breakthrough [6, 7] found that the right propagator in noncommutative field theory is not the ordinary commutative propagator, but has to be modified to obey Langmann-Szabo duality [8, 7]. Grosse and Wulkenhaar added an harmonic potential which can be interpreted as a piece of the covariant Laplacian in a constant magnetic field. They computed the corresponding “vulcanized” propagator in the “matrix base” which transforms the Moyal product into a matrix product. They use this representation to prove perturbative renormalizability of the theory up to some estimates which were finally proven in [9]. Work supported by ANR grant NT05-3-43374 “GenoPhy”.

812

R. Gurau, V. Rivasseau

Our direct space method builds upon the previous work of Filk [10] who introduced clever simplifications, also called “Filk moves”, to treat the combination of oscillations and δ functions which characterize non-commutative interactions. Minwalla, van Raamsdonk and Seiberg [11] computed a Schwinger parametric representation for the “not-vulcanized” 44 non-commutative theory. Subsequently Chepelev and Roiban computed also such a Schwinger parametric representation for this theory in [12] and used it in [13] to analyze power counting. These works however remained inconclusive, since they worked with the vertex but not the right propagator of NC44 , where ultraviolet/infrared mixing prevents one from obtaining a finite renormalized perturbation series. We have also been unable to find up to now the proofs for the formulas in [11, 12], which in fact disagree. The parametric representation introduced in this work is completely different from the ones of [12] or [11], since it corresponds to the renormalizable vulcanized theory. It no longer involves direct polynomials in the Schwinger parameters but new polynomials of hyperbolic functions of these Schwinger parameters. This is because the propagator of NC44 is based on the Mehler kernel rather than on the ordinary heat kernel. These hyperbolic polynomials contain richer topological information than in ordinary commutative field theory. Based on ribbon graphs, they contain information about their invariants, such as the genus of the surface on which these graphs live. This new parametric representation is a compact tool for the study of non-commutative field theory which has the advantages (positivity, exact power counting) but not the drawbacks (awkwardness of the propagator) of the matrix base representation. It can be used as a starting point to work out the renormalization of the model directly in parametric space, as can be done in the commutative case [14]. It is also a good starting point to define the regularization and minimal dimensional renormalization scheme of NC44 . This dimensional scheme in the ordinary field theory case better preserves continuous symmetries such as gauge symmetries, hence played a historic role in the proof of ‘t Hooft and Veltman that non-Abelian gauge theories on commutative R4 are renormalizable. It is also used extensively in the works of Kreimer and Connes [15, 16] which recast the recursive BPHZ forest formula of perturbative renormalization into a Hopf algebra structure and relate it to a new class of Riemann-Hilbert problems; here the motivations to use dimensional renormalization rather than e.g. subtraction at zero momentum come at least in part from number theory rather than from physics. Following these works, renormalizability has also attracted considerable interest in recent years as a pure mathematical structure. The renormalization group “ambiguity” reminds mathematicians of the Galois group ambiguity for roots of algebraic equations [17]. Finding new renormalizable theories may therefore be important for the future of pure mathematics as well as for physics. This paper is organized as follows. In Sect. II we introduce notations and define our new polynomials HU and H V which generalize the Symanzik polynomials U and V of commutative field theory. In Sect. III we prove the basic positivity property of the first polynomial HU and compute leading ultraviolet terms which allow to recover the right power counting in the parametric representation, introducing a technical trick which we call the “third Filk move”.1 Section IV establishes the positivity properties and computes such leading terms for the second polynomial H V , the one which gives the dependence in the external arguments. Finally examples of these polynomials for various graphs are given in Sect. V. 1 For technical reasons exact power counting was not fully established in [5].

Parametric Representation of Noncommutative Field Theory

813

II. Hyperbolic Polynomials II.1. Notations.. The NC44 theory is defined on R4 equipped with the associative and non-commutative Moyal product 1 d 4k (1) d 4 y a(x+ θ ·k) b(x+y) eik·y . (a b)(x) = (2π )4 2 The renormalizable action functional introduced in [7] is 1 2 1 λ S[φ] = d 4 x ∂µ φ ∂ µ φ + (x˜µ φ) (x˜ µ φ) + µ20 φ φ + φ φ φ φ (x), 2 2 2 4! (2) where the Euclidean metric is used. In what follows the mass µ0 does not play any role so we put it to zero.2 In four dimensional x-space the propagator is [20] αl αl 2 dαl 2 e− 8 coth( 2 )ul − 8 tanh( 2 )vl , (3) [2π sinh(αl )] D/2 l

and the (cyclically invariant) vertex is: V (x1 , x2 , x3 , x4 ) = δ(x1 − x2 + x3 − x4 )ei xθ −1 y

i+ j+1 x θ −1 x i j 1≤i< j≤4 (−1)

,

(4)

2 θ (x 1 y2

where we note ≡ − x2 y1 + x3 y4 − x4 y3 ). Permutational symmetry of the fields at all vertices, which characterizes commutative field theory, is replaced by the more restricted cyclic symmetry. Hence the ordinary Feynman graphs of 44 really become ribbon graphs in NC44 . For such a ribbon graph G, we call n, L, N , F, and B respectively the number of vertices, of internal lines, of external half-lines, of faces and of faces broken by some external half-lines. The Euler characteristic is 2 − 2g = n − L + F, where g is the genus of the graph. To each graph G is associated a dual graph G of the same genus by exchanging faces and vertices. In ordinary commutative field theory, in order to obtain Symanzik’s polynomials it is not convenient to solve the momentum conservation at the vertices through a momentum routing, because this is not canonical. It is better to express these δ functions through their Fourier transform. After integration over internal variables, the amplitude of an amputated graph G with external momenta p is, up to a normalization, in space time dimension D (of course the main case of interest in this paper is D = 4): ∞ e−VG ( p,α)/UG (α) 2 A G ( p) = δ p (e−m αl dαl ). (5) D/2 UG (α) 0 l

The first and second Symanzik polynomials UG and VG are UG = αl , T l∈T

VG =

T2 l∈T2

⎛ αl ⎝

(6)

⎞2 pi ⎠ ,

(7)

i∈E(T2 )

2 This does not lead in this model to any infrared divergences. Beware that our definition of is different from the one of [7] by a factor 4θ −1 .

814


where the first sum is over spanning trees T of G and the second sum is over two trees T2 , i.e. forests separating the graph in exactlytwo connected components E(T2 ) and F(T2 ); the corresponding Euclidean invariant ( i∈E(T2 ) pi )2 is, by momentum conservation, also equal to ( i∈F(T2 ) pi )2 . The topological formulas (6) and (7) are a field-theoretic instance of the tree-matrix theorem of Kirchoff et al; for a recent review of this kind of theorems see [18]. In the non-commutative case, momentum routing is replaced by position routing [5]. However this position routing is again non-canonical, depending on the choice of a particular tree. Therefore to compute the parametric representation we prefer to perform a new level of Fourier transform: we represent the “position conservation” rules as integrals over new “hypermomenta” pv associated to each of the vertices: δ(x1 − x2 + x3 − x4 ) = =

2π 2π

D D

v

v

v

v

eipv (x1 −x2 +x3 −x4 ) dpv v

v

v

v

epv σ (x1 −x2 +x3 −x4 ) dpv ,

(8)

where σ is the D by D matrix defined by D/2 matrices σ y on the diagonal (we assume D even): ⎛ ⎞

σy · · · 0 0 −i . (9) σ = ⎝· · · · · · · · ·⎠ where σ y = i 0 0 · · · σy We note that ıθ −1 = σθ so that the vertex oscillation can be represented with the same σ matrix. The explicit introduction of the σ matrix in Eq. (8) will allow us to treat the phase and δ functions of the vertices in an unified way and ultimately to compute the hyperbolical polynomials. There is here a subtle difference with the commutative case. The first commutative polynomial (6) is not the determinant of the quadratic form in the internal position variables, since this determinant vanishes by translation invariance. It is rather the determinant of the quadratic form integrated over all internal positions of the graph save one (remark the overall momentum conservation in (5)). This is a canonical object which does not depend on the choice of the particular vertex whose position is not integrated (this can be seen explicitly on the form (6), which depends only on G). In the non-commutative case translation invariance is lost. This allows to define the amplitude of a graph as a function of the external positions by integrating over all internal positions and hypermomenta, since the corresponding determinant no longer vanishes. In this way one can define canonical polynomials HUG and H VG which only depend on the ribbon graph G. But in practice it is often more convenient (for instance for renormalization or for understanding the limit towards the commutative case) to define the amplitude of a graph by integrating all the internal positions and hypermomenta save one, pv¯ ; this helps to factorize an overall approximate “position conservation” for the whole graph. However precisely because there is no translation invariance, the corresponding polynomials HUG,v¯ and H VG,v¯ explicitly depend on the “rooted graph” G, v, ¯ i.e. on the choice of v¯ (although their leading ultraviolet terms do not depend on this choice, see below). Consider a graph G with n vertices, N external positions and a set L of 2n − N /2 internal lines or propagators. Each vertex in NCφ 4 is made of four “corners”, bearing


815

either a halfline or an external field, numbered as 1, 2, 3, 4 in the cyclic order given by the Moyal product. To each such corner is associated a position, noted xi . The set I of internal corners has 4n − N elements, labeled usually as i, j, ...; the set E of external corners has N elements labeled as e, e , .... A line l of the graph joins two corners in I , with positions (xil , x lj ) (which in general do not belong to the same vertex). The amplitude of such a NCφ 4 graph G is then given, up to some inessential (-dependent) normalization K , by: dαl AG ({xe }) = K d xi dpv D/2 v l sinh αl i∈I − coth( αl )(x l −x l )2 − tanh( αl )(x l +x l )2 i j i j 2 4 2 × e 4 l

×

i+ j+1 x v θ −1 x v +p σ (x v −x v +x v −x v ) v 1≤i< j≤4 (−1) 1 2 3 4 i j

i

e2

(10)

v

or by AG,v¯ ({xe }, pv¯ ) = K ×

D/2 sinh αl

l

e− 4

l

×

dαl

i

e2

coth(

αl 2

i∈I

d xi

dpv

v=v¯

)(xil −x lj )2 − 4 tanh(

αl 2

)(xil +x lj )2

i+ j+1 x v θ −1 x v +p σ (x v −x v +x v −x v ) v 1≤i< j≤4 (−1) 1 2 3 4 i j

.

(11)

v

II.2. Definition of HU and H V . The fundamental observation is that the integrals do perform being Gaussian, the result is a Gaussian in the external variables divided by a determinant. This gives the definition of our hyperbolic parametric representation. We introduce the notations cl = coth( α2l ) = 1/tl and tl = tanh( α2l ). Using sinh αl = 2tl /(1 − tl2 ) we obtain ∞ H V (t,x ) − G e [dαl (1 − tl2 ) D/2 ]HUG (t)−D/2 e HUG (t) , (12) AG ({xe }) = K 0

AG,v¯ ({xe }, pv¯ ) = K

0

l ∞

[dαl (1 − tl2 ) D/2 ]HUG,v¯ (t)−D/2 e

−

H VG,v¯ (t,xe , pv¯ ) HUG,v¯ (t)

, (13)

l

where K and K are some new inessential normalization constants (which absorb in particular the factors 2 from sinh αl = 2tl /(1 − tl2 )); HUG (t) or HUG,v¯ (t) are polynomials in the t variables (there are no c’s because they are compensated by the t’s coming from sinh αl = 2tl /(1 − tl2 )) and H VG (t, xe ) or H VG,v¯ (t, xe , pv¯ ) are quadratic forms in the external variables xe or (xe , pv¯ ) whose coefficients are polynomials in the t variables (again there are no c’s because they are compensated by the t’s which were included in the definition of HU , see below the difference between (61) and (62)). There is a subtlety here. Overall approximate “position conservation” holds only for orientable graphs in the sense of [5]. Hyperbolic polynomials for non-orientable graphs are well defined through formulas (12)–(13) but they are significantly harder to compute

816


than in the orientable case. Their amplitudes are also smaller in the ultraviolet, and in particular do not require any renormalization. Also many interesting non-commutative ¯ 2 models of [5] and the theories such as the LSZ models [19], the more general (φφ) most natural Gross-Neveu models [20, 21] do not have any non-orientable subgraphs. So for simplicity we shall restrict ourselves in this paper to examples of hyperbolic polynomials for orientable graphs; and when identifying leading pieces under global scaling in the hyperbolic polynomials, something necessary for renormalization, we also limit ourselves to the orientable case. We now proceed to the computation of these hyperbolic polynomials. II.3. Short and long variables. This terminology was introduced in [5]. We order each line l joining corners l = (i, j) (which in general do not belong to the same vertex), in an arbitrary way such that it exits i and enters j. We define the incidence matrix between lines and corners li to be 1 if l enters in v, −1 if it exits at i and 0 otherwise. Also we define ηli = | li |. We note the property: ( li l j + ηli ηl j ) = 2δi j . (14) l

We now define the short variables u and the long variables v as ηli xi li xi ηli vl + li u l vl = . √ , ul = √ ; xi = √ 2 2 2 i i l

(15)

The √ Jacobian of this change of coordinates is 1. Moreover, in√order to avoid unpleasant 2 factors we rescale √ the external positions to hold x¯e = 2xe and the internal hypermomenta p¯ v = pv / 2. Note that if the graph is orientable we can choose li to be (−1)i+1 , so that the incidence matrix is consistent with the cyclic order at the vertices (halflines alternatively enter and go out). The integral in the new variables is: D/2 αl αl 2 1 − tl2 2 dαl du l dvl dpv e− 2 coth( 2 )ul − 2 tanh( 2 )vl tl l

×

v

×e

i 4

i 4

e

l

v=v¯

l

i+ j+1 (η v + u )θ −1 (η v + u ) li l li l l j l l j l i< j; (−1) i+1 i, j∈v e p¯v σ i∈v (−1) (ηli vl + li ul )

v −1 x¯ ]+ i −1 x¯ + e+1 x¯ [ ω(i,e)(η v + u )θ x ¯ θ e 4 e li l li l e e i =e e<e e v e∈v p¯ v σ (−1)

,

(16)

where ω(i, e) = 1 if i < e and ω(i, e) = −1 if i > e. When we write i ∈ v, it means that the corner i belongs to v. From now on we forget the bar over the rescaled variables. We also concentrate on the computation of HUG,v¯ in (13); we indicate alongside the necessary modifications for HUG in (12). We introduce the condensed notations: D/2 1 − t2 t AG = dα d xd pe− 2 X G X , (17) t where X = xe pv¯ u v p , G =

M P Pt Q

.

(18)


817

Gaussian integration gives, up to inessential constants: AG =

1−t t

2 D/2

dα √

1 e det Q

− 2

xe p¯

[M−P Q −1 P t ]

xe p¯ .

(19)

All we have to do now to get HU and H V is to compute the determinant and the minors of the matrix Q for an arbitrary graph. III. The First Hyperbolic Polynomial HU We define I D to be the identity matrix in D dimensions. We also put d = 2L + n − 1 so that Q can then be written as: Q = A ⊗ ID + B ⊗ σ

(20)

with A a d by d symmetric matrix (accounting for the contribution of the propagators in the Gaussian) and B an antisymmetric matrix (accounting for the oscillation part in the Gaussian). Note that the symplectic pairs decouple completely so that det Q = [det(A ⊗ I2 + B ⊗ σ y )] D/2 . Lemma III.1. For any two n × n matrices A and B let R = A ⊗ I2 + B ⊗ σ y . Then: det R = (−1)n det(A + B) det(A − B)

(21)

and: R −1 =

[(A + B)−1 + (A − B)−1 ] [(A + B)−1 − (A − B)−1 ] ⊗ I2 + ⊗ σ y . (22) 2 2

Proof. We express the determinant as a Grassmann-Berezin integral: = det(A ⊗ I2 + B ⊗ σ y ) =

d ψ¯ k1 dψk1 d ψ¯ k2 dψk2 e

¯1 − ψ i

ψ¯ i2

⎛ ⎞ ψ1 (ai j ⊗I2 +bi j ⊗σ )⎝ 2j ⎠ ψj

k

=

¯1

¯2

¯1

¯2

d ψ¯ k1 dψk1 d ψ¯ k2 dψk2 e−[ai j (ψi ψ j +ψi ψ j )+ibi j (−ψi ψ j +ψi ψ j )] . 1

2

2

1

(23)

k

We perform a change of variables of Jacobian −1 to: χi1 =

ψi1 + iψi2 2 ψ 1 − iψ 2 ; χi = i √ i . √ 2 2

(24)

As χ¯ i1 χ 1j =

1 1 1 (ψ¯ ψ − i ψ¯ i2 ψ 1j + i ψ¯ i1 ψ 2j + ψ¯ i2 ψ 2j ), 2 i j

(25)

818


we see that: = (−1)

n

d χ¯ k1 dχk1 d χ¯ k2 dχk2 e−[ai j (χ¯i χ j +χ¯i χ j )−bi j (χ¯i χ j −χ¯i χ j )] . 1 1

2 2

1 1

2 2

(26)

k

Separating the terms in χ¯ 1 χ 1 and χ¯ 2 χ 2 proves (21). The inverse matrix is divided into 2 × 2 blocs with indices i j, according to the values a, b = 1, 2, ¯ d ψ¯ 1 dψ 1 d ψ¯ 2 dψ 2 ψia ψ¯bj e−ψ Rψ −1 ab (R )i j = . (27) d ψ¯ 1 dψ 1 d ψ¯ 2 dψ 2 e−ψ¯ Rψ The four elements of the blocs are given by (taking into account that the integral decouples so that all the crossed terms are zero): (χi1 + χi2 )(χ¯ 1j + χ¯ 2j )

1 1 1 (χ χ¯ + χi2 χ¯ 2j ), 2 2 i j (χi1 + χi2 )(χ¯ 1j − χ¯ 2j ) i ψi1 ψ¯ 2j = = (χi1 χ¯ 1j − χi2 χ¯ 2j ), 2(−i) 2 1 2 1 2 (χi − χi )(χ¯ j + χ¯ j ) −i 1 1 = (χ χ¯ − χi2 χ¯ 2j ), ψi2 ψ¯ 1j = 2i 2 i j (χi1 − χi2 )(χ¯ 1j − χ¯ 2j ) 1 2 2 ¯ = (χi1 χ¯ 1j + χi2 χ¯ 2j ). ψi ψ j = 2(−i)i 2 ψi1 ψ¯ 1j =

=

(28)

Equation (22) follows then easily.

Returning to our initial problem we remark that the matrix A is the symmetric part coming from the propagator, and the oscillating part, when symmetrized, leads naturally to an antisymmetric matrix B times the antisymmetric σ y , so that in our case det Q = [det(A + B)(A − B)] D/2 = [det(A + B)(At + B t )] D/2 = [det(A + B)] D .

(29)

The propagator part is: ⎛

⎞ S 0 0 A = ⎝0 T 0⎠, 0 0 0

(30)

where S and T are the two diagonal L by L matrices with diagonal elements cl = coth( α2l ) = 1/tl , and tl = tanh( α2l ), and the last lines and columns of zeroes reflect the purely oscillating nature of the hypermomenta integrals. The hypermomenta oscillations are (in the case of (13)): ⎞ ⎛ (−1)i+1 li ⎟ ⎜ i∈v ⎟. (31) Clv = ⎜ ⎠ ⎝ (−1)i+1 ηli i∈v


819

Remark that the elements of C are integers which can take only values 0 or ±1. It is easy to check that for a connected graph G the rank of the matrix C is maximal, namely n − 1. Picking a tree of G proves that this is even true for the L by n lower part of C, corresponding to the long variables v only. To generalize to (12), we simply need to add another column to C, the one corresponding to pv¯ . The rank of the extended 2L by n matrix C¯ is then n, but the rank of the restriction of C¯ to its lower part corresponding to the long variables v is either n − 1 or n depending on whether the graph is orientable or not [5]. This has important consequences for power counting. The determinant of the quadratic form is the square of the determinant of the matrix A + B, where ⎛ ⎞ 1 E C⎟ ⎜ (32) B = ⎝ 4θ ⎠ . t −C 0 We can make explicit the oscillation between the u, v variables as the 2L by 2L

uu uvpart E E represents the vertices’ oscillations. One can matrix E. This matrix E = E vu E vv check vv El,l (−1)i+ j+1 ω(i, j)ηli ηl j , = v i= j; i, j∈v

uu El,l

=

(−1)i+ j+1 ω(i, j) li l j ,

v i= j; i, j∈v

uv El,l =

(−1)i+ j+1 ω(i, j) li ηl j ,

(33)

v i= j; i, j∈v

where we recall that ω(i, j) = 1 if i < j and ω(i, j) = −1 if i > j; moreover vu = −E uv . Remark that the matrix elements of E are integers and can in fact only El,l l ,l take values 0, ±1, ±2. Moreover El,l is zero if l and l do not hook to any common vertex; it can take value ±2 only if the two lines hook to at most two vertices in total, which is not generic, at least for large graphs. Lemma III.2. Let A = (ai δi j )i, j∈{1,...,N } be diagonal and B = (bi j )i, j∈{1,...,N } be such that bii = 0 (we need not require B antisymmetric). We have: det(A + B) = det(B Kˆ ) ai , (34) K ⊂{1,...,N }

i∈K

where B Kˆ is the matrix obtained from B by deleting the lines and columns with indices in K . Proof. The proof is straightforward. We have: σ (aiσ (i) + biσ (i) ) det(A + B) = σ ∈SN

=

i∈{1,...,N }

K ⊂{1,...,N } i∈K

and the lemma follows.

ai

k∈{1,...,N }\K σ ∈SN σ (i)=i ∀i∈K

(σ )bkσ (k)

(35)

820


In our case the matrix B =

1 4θ E −C t

C 0

is antisymmetric. Remark that the matrix

E C B = −C t 0

(36)

has integer coefficients. Moreover A in (30) has zero diagonal in the n − 1 by n − 1 lower right corner corresponding to hypermomenta. Exploiting these facts we can develop det(A + B) into Pffafians to get: Lemma III.3. With A and B given by (30) and (32) det(A + B) = (4θ )|I |+|J |+n−1−2L n 2I J cl tl I ⊂{1...L},J ⊂{L+1...2L}, n+|I |+|J | odd

l∈I

(37)

l ∈J

with n I J = Pf(B ˆ ˆ ), the Pffafian of the oscillation matrix with deleted lines and colIJ umns I among the first L indices (corresponding to short variables u) and J among the next L indices (corresponding to long variables v). Proof. Since A has the form (30), the part K of the previous lemma has to be the disjoint union of two sets I and J respectively corresponding to short and long variables. Once these sets are deleted from the matrix B we obtain a matrix B Iˆ Jˆ which has size 2L − |I | − |J | + n − 1. This matrix is antisymmetric, so its determinant is the square of the corresponding triangular Pfaffian. The Pfaffian of such a matrix is zero unless its size 2L − |I | − |J | + n − 1 = 2 p is even, in which case it is a sum, with signs, over the pairings of the 2 p lines into p pairs of the products of the corresponding matrix elements. Now from the particular form of matrix B which has a lower right block 0, we know that any pairing of the n − 1 hypermomentum variables must be with an u or v variable. Hence any pairing contributing to the Pfaffian has necessarily n − 1 terms of the C type, hence [(2L − |I | − |J |) − (n − 1)]/2 terms of the E/4θ type, hence Pf(B Iˆ Jˆ ) =

1

Pf(B Iˆ Jˆ ) .

(38)

Pf 2 (B ˆ ˆ ),

(39)

(4θ ) L−(n+|I |+|J |−1)/2

Therefore det(B Iˆ Jˆ ) =

1 (4θ )2L−n−|I |−|J |+1

IJ

hence the lemma holds, with n I,J = Pf(B ˆ ˆ ) which must be an integer since any Pfaffian IJ with integer entries is integer.

We have thus expressed the determinant of Q as a sums of positive terms. Recalling that det Q = (det(A + B)) D , the amplitude AG,v¯ (0) with external arguments xe and pv¯ put to 0 is nothing but (up to an inessential normalization) AG,v¯ (0) =

0

∞

[det(A + B)]

−D/2

D/2 1 − t2 l dαl . tl l

(40)


821

Putting s = (4θ )−1 , we use the relation 2 − 2g = n − L + F and define the integer k I,J = |I | + |J | − L − F + 1 to get: s 2g−k I,J n 2I,J tl tl . (41) HUG,v¯ (t) = I,J

l∈ I

l ∈J

This is our precise definition of the normalization of the polynomial HUG,v¯ in the variables tl introduced in (13). This normalization is adapted so that the limit s → 0 will give back the ordinary Symanzik polynomial at leading order as t’s go to 0 (the ultraviolet limit), as shown in the next section. To get the polynomial HUG in (12), we proceed exactly in the same way, replacing C ¯ and obtain that it is also a polynomial in the variables tl with positive coefficients, by C, which (up to the factors in 4θ ) are squares of integers. But we will see that the leading terms studied in the next section will be quite different in this case. III.1. Leading terms in the first polynomial HU . By leading terms, we mean terms which have the smallest global degree in the t variables, since we are interested in power counting in the “ultraviolet” regime where all t’s are scaled to 0. Such terms are obtained by taking |I | maximal and |J | minimal in (37). We shall compute the leading terms corresponding to I = [1, . . . , L], hence taking all the cl elements of the diagonal and J minimal so that the remaining minor is non zero.3 Below we prove that such terms have |J | = F − 1. This explains the normalization in (41). To analyze such leading terms we generalize the method of Filk’s moves [10], defining three distinct topological operations on a ribbon graph. The first one is a regular “first Filk move”, namely reduction of a tree line of the graph. This amounts to glue the two end vertices of the line (of coordination p and q) to get a “fatter” vertex of coordination p + q − 2. The new graph thus obtained has one vertex less and one line less. Since 2 − 2g = n − L + F, this operation conserves the genus. On Fig. 1 the contraction of the central line of the Sunshine Graph (also pictured on Fig. 7) is shown. In the dual graph this operation deletes the direct tree line, as shown on Fig. 2. Iterating this operation maximally we can always reduce a spanning tree in the direct graph, with n − 1 lines, obtaining a rosette. We recall that a rosette is simply a ribbon graph with a single vertex. The rosettes we consider all have a root (i.e. an external line on v), ¯ and a cyclic ordering to turn around, e.g. counterclockwise. We always draw the

Fig. 1. The First Filk Move on the Sunshine Graph

3 These are not the only globally leading terms; there are terms whose global scaling is equivalent, for example the t32 term in (84)-(85). By the positivity of HU they can certainly not deteriorate the power counting established in this section, but only improve it in certain particular “Hepp sectors”.

822


Fig. 2. The First Filk Move on the dual of the Sunshine Graph

Fig. 3. A Rooted Rosette

rooted rosette so that no line arches above the root. This defines uniquely a numbering of the halflines (see Fig. 3, where the arrows represent the former line orientations4 ). The second topological operation is the reduction of a tree line in the dual graph, exactly like the previous operation. Therefore it deletes this line in the direct graph. The resulting direct graph again keeps the same genus (remember that the genus of a graph is the same as the one of its dual). Iterating these two operations maximally we can always reduce completely a direct tree with n − 1 lines and a dual tree with f − 1 lines. We end up with a graph which we call a superrosette, which has only one vertex and one face (therefore its dual has one vertex and one face and is also a superrosette) (see Fig. 4). The third operation is a genus reduction on a rosette. We define a nice crossing in a rosette to be a pair of lines such that the end point of the first is the successor in the rosette of the starting point of the other (in the natural cyclic order of the rosette). This ensures that the two lines have a common “internal face”. When there are crossings in the rosette, it is easy to check that there exists at least one such nice crossing, for instance lines 2-5 and 4-8 in Fig. 3. The genus reduction consists in deleting the lines of a nice crossing and interchanging all the halflines encompassed by the first line with those encompassed by the second line, 4 For an orientable graph these arrows are compatible with the numbering, in the sense that halflines with even numbers enter the rosette and halflines with odd numbers exit the rosette.


823

Fig. 4. A SuperRosette

Fig. 5. The Third Filk Move

see Fig. 5. This operation which we call the “third Filk Move”5 decreases the number of lines by two, glues again the faces in a coherent way, and decreases the genus by one. We need then to compute the determinant of B matrices corresponding to reduced graphs of the type: ⎛ ⎞ ω(i, j)ηli (−1)i+ j+1 ηl j (−1)i+1 ηli ⎜ i= j ⎟ i∈v ⎜ ⎟ . (42) ⎝ ⎠ i (−1) ηli 0 i∈v

As the graph is orientable and up to a possible overall sign we can cast the matrix into the form: ⎞ ⎛ ω(i, j) li l j li ⎜ i= j ⎟ i∈v ⎜ ⎟ . (43) ⎝ ⎠ − li 0 i∈v

We claim: Lemma III.4. The above determinant is: • 0 if the graph has more than one face, • 22g if the graph has exactly one face. 5 The second Filk move is the trivial simplification of non-crossing lines in the rosette.

824


Proof. We reduce a tree. At each step we have a big vertex (the “rosette in the making”) V and a small vertex v bound to V by a line l = (i, j) which we contract. We then have at each step a Pfaffian dχl dψv e−B , where ⎛ ⎞ B = χl ⎝ ω(i, j) li l j + ω(i, j) li l j ⎠ χl l,l

+

i= j;i, j∈V

i= j;i, j∈v

χl lv ψv .

(44)

l,v

At each step we use a permutation to put l at the first place in the matrix. Note that this permutation has nothing to do with the ordering of the halflines. Moreover, notice that by the change of variables ψ¯ v = −ψv − 4χ1v one can cyclically change the ordering of the half lines on the small vertex v. We use this property to always fix the halfline j on the small vertex v to be the smallest in the ordering. The terms containing χl or ψv at each step are: ⎛ ⎞ ⎝ Bl = χl ω(i, p) li l p + ω( j, k) l j l k ⎠ χl l

i= p; p∈V

+ χl l j ψv +

j=k;k∈v

χl l k ψv .

(45)

l k∈v;k= j

We perform the triangular change of variables: l k χl , χ¯l = χl + l j

(46)

l k∈v;k= j

ψ¯v = l j ψv +

⎛ ⎝

l

ω(i, p) li l p +

i= p; p∈V

⎞

ω( j, k) l j l k ⎠ χl .

j=k;k∈v

Under this change of variable: χl l k ψv Bl = χl ψ¯v + l k∈v;k= j

= χ¯l ψ¯v − l j

l

+

l

−

l

l

l

χl ⎝

l

ω(i, p) li l k l p

k∈v;k= j p∈V ;i= p

⎞

ω( j, r ) l j l k l r ⎠ χl

k∈v;k= j r ∈v; j=r

= χ¯l ψ¯v +

⎛

⎛

χl ⎝

⎛

χl ⎝

k∈v;k= j p∈V ;i= p

k∈v;k= j r ∈v; j=r

⎞ ω(i, p) l k l p ⎠ χl ⎞

ω( j, r ) l k l r ⎠ χl ,

(47)


825

as l is orientable li l j = −1 and l2j = 1; notice also that ω( j, r ) = 1 as we have set the partial ordering on the small vertex v to start by j. The term in the first line of Eq. (47) corresponds exactly to a new big vertex V˜ , where the ordered halflines k of the small vertex v replace the halfline i ∈ l. The second term is zero, being the square of a grassmann number. We continue this procedure until we have reduced a complete tree in our graph. The remaining Pfaffian is of the form: B =

⎛

χl ⎝

ll1 ;l ∩l1

χl χl2 −

l t3 the subgraph formed by the lines 1, 2, 4 has two broken faces. This is the sign of a power counting improvement due to the additional broken face in that sector. To exploit it, we have just to integrate over the variables of line 3 in that sector, using the second polynomial H VG ,v for the triangle subgraph G made of lines 1, 2, 4.


835

Finally the canonical HUG polynomial is: HUG = (4 + 16s 2 )(t12 t3 + t1 t32 + t22 t3 + t2 t32 + t12 t4 + t1 t42 + t22 t4 + t2 t42 +t32 t4 + t3 t42 + t12 t2 t32 + t1 t22 t32 + t12 t2 t42 + t1 t22 t42 + t12 t22 t32 t4 +t12 t22 t3 t42 ) + (8 + 32s 2 )(t1 t2 t3 + t1 t2 t4 + t1 t2 t32 t4 + t1 t2 t3 t42 ) +(12 + 64s 4 )(t1 t3 t4 + t2 t3 t4 + t12 t2 t3 t4 + t1 t22 t3 t4 ) .

(86)

Acknowledgement. We are indebted to A. Abdesselam for his inspiring reference [18] and his help on the track to the Pfaffian analysis of Sect. III-IV. We also thank M. Disertori, J. Magnen and F. Vignes-Tourneret for useful discussions during preparation of this work.

References 1. Douglas, M., Nekrasov, N.: Noncommutative field theory. Rev. Mod. Phy. 73, 9771029 (2001) 2. Connes, A., Douglas, M.R., Schwarz, A.: Noncommutative Geometry and Matrix Theory: Compactification on Tori. JHEP 9802, 003 (1998) 3. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999) 4. Susskind, L.: The Quantum Hall Fluid, Non-Commutative Chern Simons Theory. http://arxiv.org/list/ hep-th/0101029, 2001 5. Gurau, R., Magnen, J., Rivasseau, V., Vignes-Tourneret, F.: Renormalization of Non Commutative 44 Field Theory in Direct Space. Commun. Math. Phys. 267, 515–542 (2006) 6. Grosse, H., Wulkenhaar, R.: Power-counting theorem for non-local matrix models and renormalization. Commun. Math. Phys. 254, 91–127 (2005) 7. Grosse, H., Wulkenhaar, R.: Renormalization of φ 4 -theory on noncommutative R4 in the matrix base. Commun. Math. Phys. 256, 305–374 (2005) 8. Langmann, E., Szabo, R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168 (2002) 9. Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalization of noncommutative phi 4-theory by multi-scale analysis. Commun. Math. Phys. 262, 565 (2006) 10. Filk, T.: Divergencies in a field theory on quantum space. Phys. Lett. B 376, 53–58 (1996) 11. Minwalla, S., Van Raamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. JHEP 02, 020 (2000) 12. Chepelev, I., Roiban, R.: Convergence theorem for non-commutative Feynman graphs and renormalization. JHEP 0103, 001 (2001) 13. Chepelev, I., Roiban, R.: Renormalization of quantum field theories on noncommutative Rd , 1. Scalars. JHEP 0005, 037 (2000) 14. Bergere, M., Lam, Y.: Bogoliubov-Parasiuk Theorem in the α Parametric Representation. J. Math. Phys. 17, 1546 (1976) 15. Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303 (1998) 16. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann Hilbert problem, I and II. Commun. Math. Phys. 210, 249 and 216–249 (2000) 17. Connes, A., Marcolli, M.: From Physics to Number Theory via Non-Commutative Geometry Part II: Renormalization, the Riemann Hilbert correspondence and motivic Galois theory. In: Frontiers in Number Theory, Physics and Geometry. Berlin Heidelberg-New York: Springer-Verlag, 2006, pp. 269–350 18. Abdelmalek, A.: Calculus and Theorems of the Matrix-Tree Type. Adv. in Applied Math. 33, 51–70 (2004) 19. Langmann, E., Szabo, R.J., Zarembo, K.: Exact solution of quantum field theory on noncommutative phase spaces. JHEP 0401, 017 (2004) 20. Gurau, R., Rivasseau, V., Vignes-Tourneret, F.: Propagators for Noncommutative Field Theories. to appear in Ann. Henri Poincaré, [arXiv:hep-th/0512071]. 21. Vignes-Tourneret, F: Renormalization of the Orientable Non-commutative Gross-Neveu Model http://arXiv.org/list/math-ph/0606069, 2006 to appear in Annales Henri Poincaré Communicated by A. Connes


Communications in


N-Complexes as Functors, Amplitude Cohomology and Fusion Rules Claude Cibils1 , Andrea Solotar2 , Robert Wisbauer3 1 Institut de Mathématiques et de Modélisation de Montpellier I3M, UMR 5149 Université de Montpellier 2,

F-34095 Montpellier cedex 5, France. E-mail: [email protected]

2 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires

Ciudad Universitaria, Pabellón 1 1428, Buenos Aires, Argentina. E-mail: [email protected]

3 Mathematical Institute, Heinrich-Heine-University Universitaetsstrasse 1 D-40225 Duesseldorf, Germany.

E-mail: [email protected] Received: 18 June 2006 / Accepted: 5 October 2006 Published online: 3 March 2007 – © Springer-Verlag 2007

Abstract: We consider N-complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated N-complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive N-complexes is proved to be isomorphic to an Ext functor of an indecomposable N-complex inside the abelian functor category. Finally we show that for the monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other words the fusion rules for N-complexes can be determined. 1. Introduction Let N be a positive integer and let k be a field. In this paper we will consider N-complexes of vector spaces as linear functors (or modules) over a k-category, see the definitions at the beginning of Sect. 2. Recall first that a usual k-algebra is deduced from any finite object k-category through the direct sum of its vector spaces of morphisms. Modules over this algebra are precisely k-functors from the starting category, with values in the category of k-vector spaces. Consequently if the starting category has an infinite number of objects, linear functors with values in vector spaces are called modules over the category, as much as modules over an algebra are appropriate algebra morphisms. An N-complex as considered by M. Kapranov in [19] is a Z-graded vector space equipped with linear maps d of degree 1 verifying d N = 0. The amplitude (or generalized) cohomology are the vector spaces Ker d a / Im d N−a for each amplitude a between This work has been supported by the projects PICT 08280 (ANPCyT), UBACYTX169, PIP-CONICET 5099 and the German Academic Exchange Service (DAAD). The second author is a research member of CONICET (Argentina) and a Regular Associate of ICTP Associate Scheme.

838

C. Cibils, A. Solotar, R. Wisbauer

1 and N − 1. Note that we use the terminology amplitude cohomology in order to give a graphic idea of this theory and in order to clearly distinguish it from classical cohomology theories. M. Dubois-Violette has shown in [9] a key result, namely that for N-complexes arising from cosimplicial modules through the choice of an element q ∈ k such that 1 + q + · · · + q N−1 = 0, amplitude cohomology can be computed using the classical cohomology provided the truncated sums 1+q +· · ·+q n are invertible for 1 ≤ n ≤ N−1. As a consequence he obtains in a unified way that Hochschild cohomology at roots of unity or in non-zero characteristic is zero or isomorphic to classical Hochschild cohomology (see also [20]) and the result proven in 1947 by Spanier [26], namely that Mayer [22] amplitude cohomology can be computed by means of classical simplicial cohomology. Note that N-complexes are useful for different approaches, as Yang-Mills algebras [8], Young symmetry of tensor fields [13, 14] as well as for studying homogeneous algebras and Koszul properties, see [1, 2, 16, 23, 24] or for analysing cyclic homology at roots of unity [28]. A comprehensive description of the use of N-complexes in these various settings is given in the course by M. Dubois-Violette at the Institut Henri Poincaré, [12]. We first make clear an obvious fact, namely that an N-complex is a module over a specific k-category presented as a free k-category modulo the N-truncation ideal. This way we obtain a Krull-Schmidt theorem for N-complexes. The list of indecomposables is well-known, in particular projective and injective N-complexes coincide. This fact enables us to enlarge Kapranov’s acyclicity Theorem in terms of injectives. More precisely, for each amplitude a verifying 1 ≤ a ≤ N − 1 a classic 2-complex is associated to each N-complex. We prove first in this paper that an N-complex is acyclic for a given amplitude if and only if the N-complex is projective (injective), which in turn is equivalent to acyclicity for any amplitude. In [15, 9] a basic result is obtained for amplitude cohomology for N ≥ 3 which has no counterpart in the classical situation N = 2, namely hexagons arising from amplitude cohomologies are exact. This gap between the classical and the new theory is confirmed by a result we obtain in this paper: amplitude cohomology does not discriminate arbitrary N-complexes without projective summands for N ≥ 3, despite the fact that for N = 2 it is well known that usual cohomology is a complete invariant up to a projective direct summand. Nevertheless we prove that left truncated N-complexes sharing the same amplitude cohomology are isomorphic up to a projective (or equivalently injective) direct summand. We also prove that amplitude cohomology for positive N-complexes coincides with an Ext functor in the category of N-complexes. In other words, for each given amplitude there exists an indecomposable module such that the amplitude cohomologies of a positive N-complex are actually extensions of a particular degree between the indecomposable and the given positive N-complex. We use the characterisation of Ext functors and the description of injective positive N-complexes. In this process the fact that for positive N-complexes, projectives no longer coincide with injectives requires special care. We underline the fact that various indecomposable modules are used in order to show that amplitude cohomology of positive N-complexes is an Ext functor. This variability makes the result compatible with the non-classical exact hexagons [15, 9] of amplitude cohomologies quoted above. M. Dubois-Violette has studied in [11] (see Appendix A) the monoidal structure of N-complexes in terms of the coproduct of the Taft algebra, see also [12]. J. Bichon in [3] has studied the monoidal structure of N-complexes, considering them as comodules,

N-Complexes as Functors, Amplitude Cohomology and Fusion Rules

839

see also the work by R. Boltje [4] and A. Tikaradze [27]. We recall in this paper that the k-category we consider is the universal cover of the Taft Hopf algebra Uq+ (sl2 ). As such, there exists a tensor product of modules (i.e. N-complexes) for each non-trivial Nth -root of unity (see also [4, 5]). Using Gunnlaugsdottir’s axiomatisation of ClebschGordan’s formula [18] and amplitude cohomology we show that this formula is valid for N-complexes, determining this way the corresponding fusion rules. 2. N-Complexes and Categories Let C be a small category over a field k. The set of objects is denoted C0 . Given x, y in C0 , the k-vector space of morphisms from x to y in C is denoted y Cx . Recall that composition of morphisms is k-bilinear. In this way, each x Cx is a k-algebra and each y Cx is a y C y -x Cx – bimodule. For instance let be a k-algebra and let E be a complete finite system of orthogonal idempotents in , that is e∈E e = 1, e f = f e = 0 if f = e and e2 = e, for all e, f ∈ E. The associated category C,E has a set of objects E and morphisms f C,E e = f e. Conversely any finite object set category C provides an associative algebra through the matrix construction. Both procedures are mutually inverse. In this context linear functors F : C,E → Modk coincide with left -modules. Consequently for any arbitrary linear category C, left modules are defined as k-functors F : C → Modk . In other words, a left C-module is a set of k-vector spaces {x M}x∈C0 equipped with “left oriented” actions that is, linear maps y Cx

⊗k x M → y M

verifying the usual associativity constraint. Notice that right modules are similar; they are given by a collection of k-vector spaces {Mx }x∈C0 and “right oriented”’ actions. From now on a module will mean a left module. Free k-categories are defined as follows: let E be an arbitrary set and let V = { y Vx }x,y∈E be a set of k-vector spaces. The free category F E (V ) has a set of objects E and a set of morphisms from x to y, the direct sum of tensor products of vector spaces relying x to y: ( y Vxn ⊗ · · · ⊗ x2 Vx1 ⊗ x1 Vx ). y (F E (V ))x = n≥0

x1 ,...,xn ∈E

For instance, let E = Z and let i+1 Vi = k while j Vi = 0 otherwise. This data can be presented by the double infinite quiver having Z as a set of vertices and an arrow from i to i + 1 for each i ∈ Z. The corresponding free category L has one dimensional vector space morphisms from i to j if and only if i ≤ j, namely j Li

= j V j−1 ⊗ · · · ⊗ i+2 Vi+1 ⊗ i+1 Vi .

Otherwise j Li = 0. A module over L is precisely a graded vector space {i M}i∈Z together with linear maps di : i M → i+1 M. This fact makes use of the evident universal property characterizing free linear categories. On the other hand we recall from [19] the definition of an N-complex: it consists of a graded vector space {i M}i∈Z and linear maps di : i M → i+1 M verifying that di+N−1 ◦ · · · ◦ di = 0 for each i ∈ Z.

840


In order to view an N-complex as a module over a k-linear category we have to consider a quotient of L. Recall that an ideal I of a k-category C is a collection of sub-vector spaces y I x of each morphism space y Cx , such that the image of the composition map z C y ⊗ y I x is contained in z I x and y I x ⊗ x Cu is contained in y Iu for each choice of objects. Quotient k-categories exist in the same way that algebra quotients exist. Returning to the free category L, consider the truncation ideal IN given by the entire j Li in case j ≥ i +N and 0 otherwise. Then LN := L/IN has one dimensional morphisms from i to j if and only if i ≤ j ≤ i + N − 1. Clearly N-complexes coincide with LN -modules. We have obtained the following Theorem 2.1. The categories of N-complexes and of LN -modules are isomorphic. An important point is that LN is a locally bounded category, which means that the direct sum of morphism spaces starting (or ending) at each given object is finite dimensional. More precisely: ⎡ ⎤

⎣ ⎦ ∀x0 , y0 ∈ (LN )0 , dim k y (LN )x0 = N = dim k y0 (LN )x . y∈Z

x∈Z

It is known that for locally bounded categories the Krull-Schmidt theorem holds, for instance see the work by C. Sáenz [25]. We infer that each N-complex of finite dimensional vector spaces is isomorphic to a direct sum of indecomposable ones in an essentially unique way, meaning that given two decompositions, the multiplicities of isomorphic indecomposable N-complexes coincide. Moreover, indecomposable N-complexes are well known, they correspond to “short segments” in the quiver: the complete list of indecomposable modules is given by {Mil }i∈Z,0≤l≤N−1 , where i denotes the beginning of the module, i + l its end and l its length. More precisely, i (Mil ) = i+1 (Mil ) = · · · = i+l (Mil ) = k while j (Mil ) = 0 for other indices j. The action of di , di+1 , . . . , di+l−1 is the identity and d j acts as zero if the index j is different. The corresponding N-complex is concentrated in the segment [i, i + l]. Note that the simple N-complexes are {Mi0 }i∈Z and that each Mil is uniserial, which means that Mil has a unique filtration l−2 l−1 0 ⊂ · · · ⊂ Mi+2 ⊂ Mi+1 ⊂ Mil 0 ⊂ Mi+l

such that each submodule is maximal in the following one. Summarizing the preceding discussion, we have the following Proposition 2.2. Let M be an N-complex of finite dimensional vector spaces. Then M

nli Mil i∈Z, 0≤l≤N−1

for a unique finite set of positive integers nli . Indecomposable projective and injective LN -modules are also well known, we now recall them briefly. Note from [17] that projective functors are direct sums of representable functors. Clearly − (LN )i = MiN−1 .


841

In order to study injectives notice first that for a locally bounded k-category, right and left modules are in duality: the dual of a left module is a right module which has the dual vector spaces at each object, the right actions are obtained by dualising the left actions. Projectives and injectives correspond under this duality. Right projective modules are direct sums of i (LN )− as above, clearly (i (LN )− )∗ MiN−1 . This way we have provided the main steps of the proof of the following Proposition 2.3. Let Mil be an indecomposable N-complex, i ∈ Z and l ≤ N − 1. Then Mil is projective if and only if l = N − 1, which in turn is equivalent for Mil to be injective. Corollary 2.4. Let M = i∈Z,0≤l≤N−1 nli Mil be an N-complex. Then M is projective if and only if nli = 0 for l ≤ N − 2, which in turn is equivalent for M to be injective. 3. Amplitude Cohomology Let M be an N-complex. For each amplitude a between 1 and N − 1, at each object i we have Im d N−a ⊆ Ker d a . More precisely we define as in [19] (AH )ia (M) := Ker(di+a−1 ◦ · · · ◦ di )/ Im(di−1 ◦ · · · ◦ di−N+a ), and we call this bi-graded vector space the amplitude cohomology of the N-complex. As remarked in the Introduction, M. Dubois-Violette in [9] has shown the depth of this theory; he calls it generalised cohomology. As a fundamental example we compute amplitude cohomology for indecomposable N-complexes Mil . In the following picture the amplitude is to be read vertically while the degree of the cohomology is to be read horizontally. A black dot means one dimensional cohomology, while an empty dot stands for zero cohomology. a

a N-1

N-1

l+1 l (N-1)-(j-i)

N-l (N-1)-l

(l+1)-(j-i)

1

1 0

k i

k i+1

k j

k i+l-1

k i+l

0

842


From this easy computation we notice that for a non-projective (equivalently noninjective) indecomposable module Mil (0 ≤ l ≤ N−2) and any amplitude a there exists a j degree j such that (AH )a (Mil ) = 0. Concerning projective or injective indecomposable j modules MiN−1 we notice that (AH )a (MiN−1 ) = 0 for any degree j and any amplitude a. These facts are summarized as follows: Proposition 3.1. Let M be an indecomposable N-complex. Then M is projective (or equivalently injective) in the category of N-complexes if and only if its amplitude cohomology vanishes at some amplitude a which in turn is equivalent to its vanishing at any amplitude. Remark 3.2. From the very definition of amplitude cohomology one can check that for a fixed amplitude a we obtain a linear functor (AH )a∗ from mod LN to the category of graded vector spaces. Moreover (AH )a∗ is additive, in particular: (AH )a∗ (M ⊕ M ) = (AH )a∗ (M) ⊕ (AH )a∗ (M ). This leads to the following result, which provides a larger frame to the acyclicity result of M. Kapranov [19]. See also the short proof of Kapranov’s acyclicity result by M. Dubois-Violette in Lemma 3 of [9] obtained as a direct consequence of a key result of this paper, namely the exactitude of amplitude cohomology hexagons. Theorem 3.3. Let M be an N-complex of finite dimensional vector spaces. Then (AH )a∗ (M) = 0 for some a if and only if M is projective (or equivalently injective). Moreover, in this case (AH )a∗ (M) = 0 for any amplitude a ∈ [1, N − 1]. In order to understand the preceding result in a more conceptual framework we will consider the stable category of N-complexes, mod LN . More precisely, let I be the ideal of mod LN consisting of morphisms which factor through a projective N-complex. The quotient category mod LN /I is denoted mod LN . Clearly all projectives become isomorphic to zero in mod LN . Of course this construction is well known and applies for any module category. We have in fact proven the following Theorem 3.4. For any amplitude a there is a well-defined functor (AH )a∗ : mod LN → gr (k), where gr (k) is the category of graded k-vector spaces. Our next purpose is to investigate how far amplitude cohomology distinguishes N-complexes. First we recall that in the classical case (N = 2), cohomology is a complete invariant of the stable category. Proposition 3.5. Let M and M be 2-complexes of finite dimensional vector spaces without projective direct summands. If H ∗ (M) H ∗ (M ), then M M . Proof. Indecomposable 2-complexes are either simple or projective. We assume that M has no projective direct summands; this is equivalent for M to be semisimple, in other words M is a graded vector space with zero differentials. Consequently H i (M) = i M for all i.


843

The following example shows that the favorable situation for N = 2 is no longer valid for N ≥ 3. Example 3.6. Consider M the 3-complex which is the direct sum of all simple modules, in other words, i M = k and di = 0. Then for any degree i we have (AH )i1 (M) = k

and

(AH )i2 (M) = k.

Let M be the direct sum of all the length one indecomposable 3-complexes, M = Mi1 . i∈Z

Recall that the amplitude cohomology of Mi1 is given by 1 (AH )i2 (Mi1 ) = k and (AH )i+1 1 (Mi ) = k,

while all other amplitude cohomologies vanish. Summing up provides (AH )i2 (M ) = k and (AH )i1 (M ) = k, for all i. However it is clear that M and M are not isomorphic. Notice that both M and M are free of projective direct summands. As quoted in the introduction the preceding example confirms that amplitude cohomology is a theory with different behaviour than the classical one. This fact has been previously noticed by M. Dubois-Violette in [9], for instance when dealing with nonclassical exact hexagons of amplitude cohomologies. At the opposite, we will obtain in the following that for either left or right truncated N-complexes amplitude cohomology is a complete invariant up to projectives. More precisely, let M be an N-complex which is zero at small enough objects, namely i M = 0 for i ≤ b, for some b which may depend on M. Of course this is equivalent to the fact that for the Krull-Schmidt decomposition M=

N−1

nli Mil

i∈Z l=0

there exists a minimal i 0 , in the sense that nli = 0 if i < i 0 and nli0 = 0 for some l. Proposition 3.7. Let M be a non-projective N-complex which is zero at small enough objects. Let l0 be the smallest length of an indecomposable factor of M starting at the minimal starting object i 0 . Then (AH )ia (M) = 0 for all i ≤ i 0 − 1 and (AH )ia0 (M) = 0 for a ≤ l0 . Moreover dim k (AH )li00 +1 (M) = nli00 . Proof. The fundamental computation we made of amplitude cohomology for indecomposable N-complexes shows the following: the smallest degree affording non-vanishing amplitude cohomology provides the starting vertex of an indecomposable nonprojective module. Moreover, at this degree the smallest value of the amplitude affording non-zero cohomology is l + 1, where l is the length of the indecomposable. In other words amplitude cohomology determines the multiplicity of the smallest indecomposable direct summand of a left-truncated N-complex. Of course smallest concerns the lexicographical order between indecomposables, namely Mil ≤ M rj in case i < j or in case i = j and l ≤ r .

844


Theorem 3.8. Let M be an N-complex which is zero at small enough objects and which does not have projective direct summands. The dimensions of its amplitude cohomology determine the multiplicities of each indecomposable direct summand. Proof. The proposition above shows that the multiplicity of the smallest indecomposable direct summand is determined by the amplitude cohomology (essentially this multiplicity is provided by the smallest non-zero amplitude cohomology, where amplitude cohomology is also ordered by lexicographical order). We factor out this smallest direct summand X from M and we notice that the multiplicities of other indecomposable factors remain unchanged. Moreover, factoring out the amplitude cohomology of X provides the amplitude cohomology of the new module. It’s smallest indecomposable summand comes strictly after X in the lexicographical order. Through this inductive procedure, multiplicities of indecomposable summands can be determined completely. In other words: if two left-truncated N-complexes of finite dimensional vector spaces share the same amplitude cohomology, then the multiplicities of their indecomposable direct factors coincide for each couple (i, n). Remark 3.9. Clearly the above theorem is also true for N-complexes which are zero for large enough objects, that is right-truncated N-complexes. 4. Amplitude Cohomology is Ext An N-complex M is called positive in case i M = 0 for i ≤ −1. In this section we will prove that amplitude cohomology of positive N-complexes of finite dimensional vector spaces coincides with an Ext functor in this category. First we provide a description of injective positive N-complexes as modules. Notice 0 that positive N-complexes are functors on the full subcategory LN of LN provided by 0 the positive integer objects. Alternatively, LN is the quotient of the free k-category generated by the quiver having positive integer vertices and an arrow from i to i + 1 for each object, by the truncation ideal given by morphisms of length greater than N. Theorem 4.1. The complete list up to isomorphism of injective positive indecomposable N-complexes is

M0l MiN−1 . l=0,...,N−1

i≥1

Proof. As we stated before, injective modules are duals of projective right modules. The 0 indecomposable ones are representable functors i0 (LN )− , for i 0 ≥ 0. 0 ∗ 0 ∗ i0 ) = M if i Clearly for each i 0 we have (i0 LN 0 ≤ N − 1 while (i 0 LN − ) = 0 −

otherwise. MiN−1 0 −(N−1) In order to show that amplitude cohomology is an instance of an Ext, we need to have functors sending short exact sequences of positive N-complexes into long exact sequences: this will enable to use the axiomatic characterization of Ext. For this purpose we recall the following standard consideration about N-complexes which provides several classical 2-complexes associated to a given N-complex, by contraction. More precisely fix an integer e as an initial condition and an amplitude of contraction a (which provides also a coamplitude of contraction b = N − a).


845

The contraction Ce,a M of an N-complex is the following 2-complex, which has e M in degree 0 and alternating a th and bth composition differentials: db

da

db

da

· · · → e−b M → e M → e+a M → e+N M → . . . . Of course the usual cohomology of this complex provides amplitude cohomology: Lemma 4.2. In the above situation, N +a H 2i (Ce,a M) = (AH )ae+i N (M) and H 2i+1 (Ce,a M) = (AH )e+i (M). b

Notice that in order to avoid repetitions and in order to set H 0 as the first positive degree amplitude cohomology, we must restrict the range of the initial condition. More precisely, for a given amplitude contraction a the initial condition e verifies 0 ≤ e < b, where b is the coamplitude verifying a + b = N. Indeed, if e ≥ b, set e = e − b and a = b. Then b = a and 0 ≤ e < b . Remark 4.3. An exact sequence of N-complexes provides an exact sequence of contracted complexes at any initial condition e and any amplitude a. ∗ We focus now on the functor H ∗ (Ce,a −), which for simplicity we shall denote He,a ∗ sends a short exact sequence of N-complexes from now on. We already know that He,a ∗ is the usual cohomology. Our next purpose is twointo a long exact sequence, since He,a ∗ fold. First we assert that He,a vanishes in positive degrees when evaluated on injectives of the category of positive N-complexes. Then we will show that it is representable in degree 0.

Proposition 4.4. In positive degrees we have: ∗ ∗ He,a (M0l ) = 0 for l ≤ N − 1, and He,a (MiN−1 ) = 0 for i ≥ 1.

Proof. Concerning indecomposable modules of length N − 1, they are already injective in the entire category of N-complexes. We have noticed that all their amplitude cohomologies vanish. Consider now M0l , with l ≤ N − 1. In non-zero even degree 2i the amplitude cohomology to be considered is in degree e + i N , which is larger than l since i = 0 and 2i (M l ) = 0. N > l. Hence He,a 0 In odd degree 2i +1 the amplitude cohomology to be considered is in degree e+i N +a. 2i+1 (M l ) = 0 for i = 0. As before, in case i = 0 this degree is larger than l, then He,a 0 l l 1 It remains to consider the case i = 0, namely He,a (M0 ) = (AH )e+a N−a (M0 ). From the picture we have drawn for amplitude cohomology in the previous section, we infer that in degree e + a the cohomology is not zero only for amplitudes inside the closed interval [l + 1 − (e + a), N − 1 − (e + a)]. We are concerned by the amplitude N − a which is 1 (M l ) = 0. larger than N − a − e − 1, hence He,a 0 Proposition 4.5. Let a ∈ [1, N − 1] be an amplitude and let e ∈ [0, N − 1 − a] be an 0 (−) = (AH )e (−) is a representable functor given by the initial condition. Then He,a a indecomposable N-complex Mea−1 . More precisely, (AH )ae (X ) = HomL0 (Mea−1 , X ). N

846


Proof. We will verify this formula for an arbitrary indecomposable positive N-complex X = Mil . The morphism spaces between indecomposable N-complexes are easy 0 . Non-zero morphisms to determine using diagrams through the defining quiver of LN from an indecomposable M to an indecomposable M exist if and only if M starts during M and M ends together with or after M . Then we have: k if e ∈ [i, i + l] and e + a − 1 ≥ i + l a−1 l . HomL0 (Me , Mi ) = 0 otherwise N Considering amplitude cohomology and the fundamental computation we have made, we first notice that (AH )ae (Mil ) has a chance to be non-zero only when the degree e belongs to the indecomposable, namely e ∈ [i, i + l]. This situation already coincides with the first condition for non-vanishing of Hom. Next, for a given e as before, the precise conditions that the amplitude a must verify in order to obtain k as amplitude cohomology is (l + 1) − (e − i) ≤ a ≤ (N − 1) − (e − i). The second inequality holds since the initial condition e belongs to [0, N − 1 − a] and i ≥ 0. The first inequality is precisely e + a − 1 ≥ i + l. As we wrote before it is well known (see for instance [21]) that a functor sending naturally short exact sequences into long exact sequences, vanishing on injectives and being representable in degree 0 is isomorphic to the corresponding Ext functor. Then we have the following: 0 Theorem 4.6. Let LN be the category of positive N-complexes of finite dimensional j vector spaces and let AHa (M) be the amplitude cohomology of an N-module M with amplitude a in degree j. Let b = N − a be the coamplitude. Let j = qN + e be the euclidean division with 0 ≤ e ≤ N − 1. Then for e < b we have: j

AHa (M) = Ext

2q

L0 N

(Mea−1 , M),

and for e ≥ b we have: j

AHa (M) = Ext

2q+1

L0 N

b−1 (Me−b , M).

5. Monoidal Structure and Clebsch-Gordan Formula The k-category LN is the universal cover of the associative algebra Uq+ (sl2 ), where q is a non-trivial Nth root of unity, see [5] and also [7]. More precisely, let C =< t > be the infinite cyclic group and let C act on (LN )0 = Z by t.i = i + N . This is a free action on the objects while the action on morphisms is obtained by translation: namely the action of t on the generator of i+1 Vi is the generator of i+1+N Vi+N . Since the action of C is free on the objects, the categorical quotient exists, see for instance [6]. The category LN /C has a set of objects Z/N. This category LN /C has a finite number of objects, hence we may consider its matrix algebra a(LN /C) obtained as


847

the direct sum of all its morphism spaces equipped with matrix multiplication. In other words, a(LN /C) is the path algebra of the crown quiver having Z/N as a set of vertices and an arrow from i¯ to i¯ + 1 for each i¯ ∈ Z/N, truncated by the two-sided ideal of paths of length greater or equal to N. As described in [5] this truncated path algebra bears a comultiplication, an antipode and a counit providing a Hopf algebra isomorphic to the Taft algebra, also known as the positive part Uq+ (sl2 ) of the quantum group Uq (sl2 ). The monoidal structure obtained for the Uq+ (sl2 )-modules can be lifted to LN -modules providing the monoidal structure on N-complexes introduced by M. Kapranov [19] and studied by J. Bichon [3] and A. Tikaradze [27]. We recall the formula: let M and M be N-complexes. Then M ⊗ M is the N-complex given by ( j M ⊗ r M ) i (M ⊗ M ) = j+r =i

and di (m j ⊗ m r ) = m j ⊗ dr m r + q r d j m j ⊗ m r . Notice that in general i (M ⊗ M ) is not finite dimensional. Proposition 5.1. Let M and M be N-complexes of finite dimensional vector spaces. Then M ⊗ M is a direct sum of indecomposable N-complexes of finite dimensional vector spaces, each indecomposable appearing a finite number of times. Proof. Using the Krull-Schmidt Theorem we have M= nli Mil and M = i∈Z, 0≤l≤N−1

n il Mil .

i∈Z, 0≤l≤N−1

The tensor product Mil ⊗ M rj consists of a finite number of non-zero vector spaces which are finite dimensional. It follows from the Clebsch-Gordan formula that we prove below that for a given indecomposable N-complex Mlu , there is only a finite number of couples of indecomposable modules sharing Mlu as an indecomposable factor. Then each indecomposable module appears a finite number of times in M ⊗ M . The following result is a Clebsch-Gordan formula for indecomposable N-complexes, see also the work by R. Boltje, Chap. III [4]. The fusion rules, i.e. the positive coefficients arising from the decomposition of the tensor product of two indecomposables, can be determined as follows. Theorem 5.2. Let q be a non-trivial Nth root of unity and Miu and let M vj be indecomposable N-complexes. Then, if u + v ≤ N − 1 we have Miu ⊗ M vj =

min(u,v)

u+v−2l Mi+ j+l ,

l=0

if u + v = e + N − 1 with e ≥ 0 we have Miu ⊗ M vj =

e l=0

N−1 Mi+ j+l ⊕

min(u,v) l=e+1

u+v−2l Mi+ j+l .

848


Proof. Using Gunnlaugsdottir’s axiomatization [18], p.188, it is enough to prove the following: 0 Mi0 ⊗ M 0j = Mi+ j,

M01 ⊗ M uj = M u+1 ⊕ M u−1 j j+1 for u < N − 1, M01 ⊗ M N−1 = M N−1 ⊕ M N−1 j j j+1 . The first fact is trivial. The second can be worked out using amplitude cohomology, which characterizes truncated N-complexes. Indeed the algorithm we have described in 3.8 enables us first to determine the fusion rule for M01 ⊗ M uj (u < N − 1), that is to determine the non-projective indecomposable direct summands. More precisely, since u < N − 1, the smallest non-vanishing amplitude cohomology degree is j, with smallest amplitude u + 2, providing M u+1 as a direct factor. The remaining amplitude j cohomology corresponds to M u−1 j+1 . A dimension computation shows that in this case there are no remaining projective summands. On the converse, the third case is an example of vanishing cohomology. In fact, since M N−1 is projective, it is known at the Hopf algebra level that X ⊗ M N−1 is projective. j j A direct dimension computation between projectives shows that the formula holds. References 1. Berger, R., Dubois-Violette, M., Wambst, M.: Homogeneous algebras. J. Algebra 261(1), 172–185 (2003) 2. Berger, R., Marconnet, N.: Koszul and Gorenstein Properties for Homogeneous Algebras. Alg. Rep. Theory 9, 67–97 (2006) 3. Bichon, J.: N-complexes et algèbres de Hopf. C. R. Math. Acad. Sci. Paris 337, 441–444 (2003) 4. Boltje, R.: Kategorien von verallgemeinerten Komplexen und ihre Beschreibung durch Hopf Algebren, Diplomarbeit, Universität München, 1985 5. Cibils, C.: A quiver quantum group. Commun. Math. Phys. 157, 459–477 (1993) 6. Cibils, C., Redondo, M.J.: Cartan-Leray spectral sequence for Galois coverings of categories. J. Algebra 284, 310–325 (2005) 7. Cibils, C., Rosso, M.: Hopf quivers. J. Algebra 254, 241–251 (2002) 8. Connes, A., Dubois-Violette, M.: Yang-Mills algebra. Lett. Math. Phys. 61, 149–158 (2002) 9. Dubois-Violette, M.: d N = 0: generalized homology. K -Theory 14, 371–404 (1998) 10. Dubois-Violette, M.: Generalized homologies for d N = 0 and graded q-differential algebras. In: Secondary calculus and cohomological physics (Moscow, 1997), Contemp. Math., 219, Providence, RI: Amer. Math. Soc., 1998, pp. 69–79 11. Dubois-Violette, M.: Lectures on differentials, generalized differentials and on some examples related to theoretical physics. In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Contemp. Math., 294, Providence, RI: Amer. Math. Soc., 2002, pp. 59–94 12. Dubois-Violette, M.: Résumé et transparents du cours “N-COMPLEXES”, for the semester “K-theory and noncommutative geometry”, Institut Henri Poincaré, Paris, mars 2004. Available at http://qcd.th.u-psud.fr/page_perso/MDV/articles/COURS_IHP.pdf 13. Dubois-Violette, M., Henneaux, M.: Generalized cohomology for irreducible tensor fields of mixed Young symmetry type. Lett. Math. Phys. 49, 245–252 (1999) 14. Dubois-Violette, M., Henneaux, M.: Tensor fields of mixed Young symmetry type and N-complexes. Commun. Math. Phys. 226, 393–418 (2002) 15. Dubois-Violette, M., Kerner, R.: Universal q-differential calculus and q-analog of homological algebra. Acta Math. Univ. Comenian. 65, 175–188 (1996) 16. Dubois-Violette, M., Popov, T.: Homogeneous algebras, statistics and combinatorics. Lett. Math. Phys. 61, 159–170 (2002) 17. Freyd, P: Abelian categories. New York: Harper and Row 1964 18. Gunnlaugsdottir, E.: Monoidal structure of the category of u q+ -modules. Linear Algebra Appl. 365, 183– 199 (2003)


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19. Kapranov, M.: On the q-analog of homological algebra. Preprint, Cornell University, 1991, available at http://arxiv.org/list/q-alg/9611005, 1996 20. Kassel, C., Wambst, M.: Algèbre homologique des N-complexes et homologie de Hochschild aux racines de l’unité . Publ. Res. Inst. Math. Sci. 34, 91–114 (1998) 21. Mac Lane, S.: Homology, New York: Springer-Verlag, 1963 22. Mayer, W.: A new homology theory. I, II. Ann. of Math. 43, 370–380, 594–605 (1942) 23. Martínez-Villa, R., Saorín, M.: Koszul duality for N-Koszul algebras. Colloq. Math. 103, 155–168 (2005) 24. Martínez-Villa, R., Saorín, M.: A duality theorem for generalized Koszul algebras. http://arxiv.org/list/math.RA/0511157, 2005 25. Sáenz, C.: Descomposición en inescindibles para módulos sobre anillos y categorías asociadas. Tesis para obtener el título de matemático, UNAM, Mexico, 1988 26. Spanier, E.H.: The Mayer homology theory, Bull. Amer. Math. Soc. 55, 102–112 (1949) 27. Tikaradze, A.: Homological constructions on N-complexes. J. Pure Appl. Algebra 176, 213–222 (2002) 28. Wambst, M.: Homologie cyclique et homologie simpliciale aux racines de l’unité. K -Theory 23, 377– 397 (2001) Communicated by A. Connes

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