Showing papers in "Communications in Mathematical Physics in 1994"

Level spacing distributions and the Airy kernel

Abstract: Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.

Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes

Abstract: We develop techniques to compute higher loop string amplitudes for twistedN=2 theories withĉ=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of theN=2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the ‘number’ of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingN=2 theory. Relations withc=1 strings are also pointed out.

Gromov-Witten classes, quantum cohomology, and enumerative geometry

Abstract: The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given.

On foundation of the generalized Nambu mechanics

Abstract: We outline basic principles of a canonical formalism for the Nambu mechanics—a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973. It is based on the notion of a Nambu bracket, which generalizes the Poisson bracket—a “binary” operation on classical observables on the phase space—to the “multiple” operation of higher ordern≧3. Nambu dynamics is described by the phase flow given by Nambu-Hamilton equations of motion—a system of ODE's which involvesn−1 “Hamiltonians.” We introduce the fundamental identity for the Nambu bracket—a generalization of the Jacobi identity—as a consistency condition for the dynamics. We show that Nambu bracket structure defines a hierarchy of infinite families of “subordinated” structures of lower order, including Poisson bracket structure, which satisfy certain matching conditions. The notion of Nambu bracket enables us to define Nambu-Poisson manifolds—phase spaces for the Nambu mechanics, which turn out to be more “rigid” than Poisson manifolds—phase spaces for the Hamiltonian mechanics. We introduce the analog of the action form and the action principle for the Nambu mechanics. In its formulation, dynamics of loops (n−2-dimensional chains for the generaln-ary case) naturally appears. We discuss several approaches to the quantization of Nambu mechanics, based on the deformation theory, path integral formulation and on Nambu-Heisenberg “commutation” relations. In the latter formalism we present an explicit representation of the Nambu-Heisenberg relation in then=3 case. We emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from Hamilton's to Nambu's dynamical picture.

Onsager's conjecture on the energy conservation for solutions of Euler's equation

Abstract: We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Euler's equation.

The behaviour of eigenstates of arithmetic hyperbolic manifolds

Abstract: In this paper we study some problems arising from the theory of Quantum Chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom.

Periodic nonlinear Schrödinger equation and invariant measures

Abstract: In this paper we continue some investigations on the periodic NLSEiu u +iu xx +u|u| p-2 (p≦6) started in [LRS]. We prove that the equation is globally wellposed for a set of data Φ of full normalized Gibbs measrue $$e^{ - \beta H(\phi )} Hd\phi (x),H(\phi ) = \tfrac{1}{2}\int {\left| {\phi '} \right|^2 - \tfrac{1}{p}\int {\left| \phi \right|p} } $$ (after suitableL 2-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [B1] on periodic NLS and KdV type equations.

Level spacing distributions and the Bessel kernel

Abstract: Scaling models of randomN×N hermitian matrices and passing to the limitN→∞ leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others'as well, the kernel one obtains by scaling in the “bulk” of the spectrum is the “sine kernel”\(\frac{{\sin \pi (x - y)}}{{\pi (x - y)}}\). Rescaling the GUE at the “edge” of the spectrum leads to the kernel\(\frac{{Ai(x)Ai'(y) - Ai'(x)Ai(y)}}{{x - y}}\), where Ai is the Airy function. In previous work we found several analogies between properties of this “Airy kernel” and known properties of the sine kernel: a system of partial differential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of intervals; a representation of the Fredholm determinant in terms of a Painleve transcendent in the case of a single interval; and, also in this case, asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the Laguerre or Jacobi ensembles at the edge of the spectrum, namely $$\frac{{J_\alpha (\sqrt x )\sqrt y J'_\alpha (\sqrt y ) - \sqrt x J'_\alpha (\sqrt x )J_\alpha (\sqrt y )}}{{2(x - y)}},$$ , whereJα(z) is the Bessel function of order α. In the cases α=∓1/2 these become, after a variable change, the kernels which arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an asymptotic expansion we derive will generalize ones found by Dyson for the Fredholm determinants of these kernels.

Asymptotic stability of solitary waves

Abstract: We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation $$\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,$$ is asymptotically stable Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, $$\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 $$ In particular, we study the case wheref(u)=up+1/(p+1),p=1, 2, 3 (and 3 0, withf∈C4) The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation) We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part) The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave Asp→4−, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

Fredholm Determinants, Differential Equations and Matrix Models

Abstract: Orthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (ϕ(x)ψ(y)−ψ(x)ϕ(y))/x−y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals\(J = \cup _{j = 1}^m (a_{2j - 1 ,{\text{ }}} a_{2j} )\). The emphasis is on the determinants thought of as functions of the end-pointsak.

Toeplitz quantization of Kähler manifolds ang gl(N), N→∞ limits

Abstract: For general compact Kahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of teh authors and Klimek and Lesniewski obtained for the tours and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finitedimensional matrix algebrasgl(N), N→∞.

Conformal blocks and generalized theta functions

Abstract: LetSU X r be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofL k with the space of conformal blocks defined in terms of representations of the Lie algebrasl r (C((z))). It follows in particular that the dimension ofH 0(SU X r,L k ) is given by the Verlinde formula.

Batalin-Vilkovisky algebras and two-dimensional topological field theories

Abstract: By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator Δ:A⊙→A⊙+1 such that Δ2 = 0, and [Δ,a]−Δa is a graded derivation ofA for alla∈A. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads.

Kinetic formulation of the isentropic gas dynamics and $p$-systems

Abstract: We consider the 2 x 2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called the p-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce new L°° estimates using moments lemma and prove L°° — w* stability for 7 > 3.

Gaudin model, Bethe ansatz and critical level

Abstract: We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensoproducts of Wakimoto modules. This gives explicit formulas for the eigenvectors via bosonic correlation functions. Analogues of the Bethe Ansatz equations naturally appear as equations on the existence of singular vectors in Wakimoto modules. We use this construction to explain the connection between Gaudin's model and correlation functios of WZNW models.

Existence Theorem for Solitary Waves on Lattices

Abstract: In this article we give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian $$H = \sum\limits_{n \in \mathbb{Z}} {\left( {\tfrac{1}{2}p_n^2 + V(q_{n + 1} - q_n )} \right)} ,$$ whereV(·) is the potential energy due to nearest-neighbour interactions. Until now, apart from rare integrable lattices like the Toda latticeV(φ)=ab−1(e−bφ+bφ−1), the only evidence for existence of such solutions has been numerical. Our result in particular recovers existence of solitary waves in the Toda lattice, establishes for the first time existence of solitary waves in the (nonintegrable) cubic and quartic latticesV(φ)= 1/2φ2 + 1/3aφ3,V(φ) = 1/2φ2 + 1/4bφ4, thereby confirming the numerical findings in [1] and shedding new light on the recurrence phenomena in these systems observed first by Fermi, Pasta and Ulam [2], and shows that contrary to widespread belief, the presence of exact solitary waves is not a peculiarity of integrable systems, but “generic” in this class of nonlinear lattices. The approach presented here is new and quite general, and should also be applicable to other forms of lattice equations: the travelling waves are sought as minimisers of a naturally associated variational problem (obtained via Hamilton's principle), and existence of minimisers is then established using modern methods in the calculus of variations (the concentration-compactness principle of P.-L. Lions [3]).

A duality for Hopf algebras and for subfactors. I

Abstract: We provide a duality between subfactors with finite index, or finite dimensional semisimple Hopf algebras, and a class ofC*-categories of endomorphisms.

Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case

Abstract: Various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed. In particular somefinite size conditions are discussed, together with their implications for the Gibbs measures and for the approach to equilibrium of Glauber dynamics inarbitrarily large volumes. It is shown that Dobrushin-Shlosman's theory ofcomplete analyticity and its dynamical counterpart due to Stroock and Zegarlinski, cannot be applied, in general, to the whole one phase region since it requires mixing properties for regions ofarbitrary shape. An alternative approach, based on previous ideas of Oliveri, and Picco, is developed, which allows to establish results on rapid approach to equilibrium deeply inside the one phase region. In particular, in the ferromagnetic case, we considerably improve some previous results by Holley and Aizenman and Holley. Our results are optimal in the sene that, for example, they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed-dimensional Ising model with or without an external field. In part II we extensively consider the general case (not necessarily attractive) and we develop a new method, based on renormalizations group ideas and on an assumption of strong mixing in a finite cube, to prove hypercontractivity of the Markov semigroup of the Glauber dynamics.

Braid group action and quantum affine algebras

Abstract: We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop-like generators of the algebra are obtained which satisfy the relations of Drinfel'd's new realization. Coproduct formulas are given and a PBW type basis is constructed.

Symplectic structures associated to Lie-Poisson groups

Abstract: The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified, and the corresponding symplectic forms are described. Thus the construction of the Kirillov symplectic form is generalized for Lie-Poisson groups.

Higher algebraic structures and quantization

Abstract: We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in ad+1 dimensional topological theory to manifolds of dimension less thand+1. We then “construct” a generalized path integral which ind+1 dimensions reduces to the standard one and ind dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.

Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case

Abstract: We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube Λ 0 , a Logarithmic Sobolev Inequality for the Gibbs state of a discrete spin system. As a consequence we derive the hypercontractivity of the Markov semigroup of the associated Glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes Λ «multiples» of the cube Λ 0

The stochastic Burgers equation

Abstract: We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise. An explicit solution of the latter is constructed through a generalized Feynman-Kac formula. Typical properties of the trajectories are then discussed. A technical result, concerning the regularizing effect of the convolution with the heat kernel, is proved for stochastic integrals.

Charge deficiency, charge transport and comparison of dimensions

Abstract: We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.

Vertex representations for N-toroidal Lie algebras and a generalization of the Virasoro algebra

Abstract: Vertex representations are obtained for toroidal Lie algebras for any number of variables. These representations afford representations of certainn-variable generalizations of the Virasoro algebra that are abelian extensions of the Lie algebra of vector fields on a torus.

Geometric aspects of quantum spin states

Abstract: A number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S+1)-invariant quantum spin-S chains with the interaction −P(o), whereP(o) is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of Neel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the −P(o) spin-S systems with the Potts and the Fortuin-Kasteleyn random-cluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such structure the dichotomy is implied by a topological argument, and the alternatives correspond to whether, or not, the clusters are of finite mean length.

The structure of perturbative quantum gravity on a de Sitter background

Abstract: Classical gravitation on de Sitter space suffers from a linearization instability. One consequence is that the causal response to a spatially localized distribution of positive energy cannot be globally regular. We use this fact to show that no causal Green's function can give the correct linearized response to certain bilocalized distributions, even though these distributions obey the constraints of linearization stability. We avoid the problem by working on the open submanifold spanned by conformal coordinates. The retarded Green's function is first computed in a simple gauge, then the rest of the propagator is inferred by analyticity — up to the usual ambiguity about real, analytic and homogeneous terms. We show that the latter can be chosen so as to give a propagator which does not grow in any direction. The ghost propagator is also given and the interaction vertices are worked out.

Twistless KAM tori

Abstract: A selfcontained proof of the KAM theorem in the Thirring model is discussed.

Dual isomonodromic deformations and moment maps to loop algebras

Abstract: The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve transcendentsPV andVI.

Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

Abstract: The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t +f(u) x =μu xx with the initial datau 0 which tend to the constant statesu ± asx→±∞. Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f′(u ±). Moreover, the rate of asymptotics in time is investigated. For the casef′(u+)