Showing papers in "Communications in Mathematical Physics in 1997"

The Spectral action principle

Abstract: We propose a new action principle to be associated with a noncommutative space \(\). The universal formula for the spectral action is \(\) where \(\) is a spinor on the Hilbert space, \(\) is a scale and \(\) a positive function. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling constants identical to those of SU(5) as well as the Higgs self-coupling, to be taken at a fixed high energy scale.

Vector Bundles and F Theory

Abstract: To understand in detail duality between heterotic string and F theory compactifications, it is important to understand the construction of holomorphic G bundles over elliptic Calabi-Yau manifolds, for various groups G. In this paper, we develop techniques to describe these bundles, and make several detailed comparisons between the heterotic string and F theory.

Elliptic genera of symmetric products and second quantized strings

Abstract: In this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N-fold symmetric product M N /SN of a manifold M to the partition function of a second quantized string theory on the space M × S 1 . The generating function of these elliptic genera is shown to be (almost) an automorphic form for O(3,2, Z). In the context of D-brane dynamics, this result gives a precise computation of the free energy of a gas of D-strings inside a higher-dimensional brane.

Integrable structure of conformal field theory. ii. q-operator and ddv equation

Abstract: This paper is a direct continuation of [1] where we began the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators ${\bf Q}_{\pm}(\lambda)$ which act in the highest weight Virasoro module and commute for different values of the parameter λ. These operators appear to be the CFT analogs of the Q - matrix of Baxter [2], in particular they satisfy Baxter's famous T- Q equation. We also show that under natural assumptions about analytic properties of the operators as the functions of λ the Baxter's relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV) [3] for the eigenvalues of the Q-operators. We then use the DDV equation to obtain the asymptotic expansions of the Q - operators at large λ; it is remarkable that unlike the expansions of the T operators of [1], the asymptotic series for Q(λ) contains the “dual” nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the Q - operators and the stationary transport properties in the boundary sine-Gordon model. On this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum Hall system.

Stochastic Burgers and KPZ equations from particle systems

Abstract: We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.

q-Gaussian Processes: Non-commutative and Classical Aspects

Abstract: We examine, for −1

Differentiation of SRB States

Abstract: :Let f be a diffeomorphism of a manifold M, and ρf a (generalized) SRB state for f. If supp ρf is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup, and we compute the derivative. When supp ρf is an attractor, the derivative is given by where X is the vector field . This formula can be extended to time dependent situations and also, at least formally, to nonuniformly hyperbolic situations. The above results will find their use in the study of the Onsager reciprocity relations and the fluctuation-dissipation formula of nonequilibrium statistical mechanics.

Quantum Integrable Models and Discrete Classical Hirota Equations

Abstract: The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived.

Non-commutative martingale inequalities

Abstract: We prove the analogue of the classical Burkholder-Gundy inequalites for non-commutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an Lp-martingale via its integrand, and then extend the Ito-Clifford integral theory in L2, developed by Barnett, Streater and Wilde, to Lp for all 1

The Calogero-Sutherland Model and Generalized Classical Polynomials

Abstract: Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schrodinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case.

Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

Abstract: For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B3s with s greater than 1/3. B3s consists of functions that are Lip(s) (i.e., Holder continuous with exponent s) measured in the Lp norm. Here this result is applied to a velocity field that is Lip(α0) except on a set of co-dimension \(\) on which it is Lip($agr;1), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if \(\). Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.

Motion by Mean Curvature from the Ginzburg-Landau Interface Model

Abstract: We consider the scalar field φ t with a reversible stochastic dynamics which is defined by the standard Dirichlet form relative to the Gibbs measure with formal energy . The potential V is even and strictly convex. We prove that under a suitable large scale limit the φ t -field becomes deterministic such that locally its normal velocity is proportional to its mean curvature, except for some anisotropy effects. As an essential input we prove that for every tilt there is a unique shift invariant, ergodic Gibbs measure for the -field.

Le Groupe Quantique Compact Libre U(n)

Abstract: The free analogues of U(n) in Woronowicz' theory [Wo2] are the compact matrix quantum groups \(\) introduced by Wang and Van Daele. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If \(\) we find an embedding \(\), where Ao(F) is the deformation of SU(2) studied in [B2]. We use the representation theory and Powers' method for showing that the reduced algebras Au(F)red are simple, with at most one trace.

Global Existence of Small Solutions to a Relativistic Nonlinear Schrödinger Equation

Abstract: On etudie le probleme de Cauchy associea une equation de Schrodinger non lineaire modelisant une impulsion laser ultra-intense et ultra-courte dans un plasma. Les termes non lineaires nouveaux sont dus aux effets relativistes et a la force ponderomotrice. Nous prouvons l'existence locale et l'unicite de solutions petites lorsque la dimension de l'espace transverse est egale a 2 ou 3, ainsi que l'existence locale pour des donnees initiales de taille arbitraire, en dimension un d'espace transverse.

On homogenization and scaling limit of some gradient perturbations of a massless free field

Abstract: We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2-point correlation functions of these models.

Large N 2D Yang-Mills Theory and Topological String Theory

Abstract: We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A= 0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's “Ω-1 points ”. We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach.

Deformations of W-algebras associated to simple Lie algebras

Abstract: Deformed W-algebra Wq,t(g) associated to an arbitrary simple Lie alge- bra g is defined together with its free field realizations and the screening operators. Explicit formulas are given for generators of Wq,t(g) when g is of classical type. These formulas exhibit a deep connection between Wq,t(g) and the analytic Bethe Ansatz in integrable models associated to quantum affine algebrasUq(b) and Ut( L b). The scaling limit of Wq,t(g) is closely related to affine Toda field theories.

Deformation quantization and Nambu Mechanics

Abstract: Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

Field Theory on a Supersymmetric Lattice

Abstract: A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the differencial calculus of non-commutative geometry. The regulated theory involves only finite number of degrees of freedom and is manifestly supersymmetric.

Exact Ground State Energy of the Strong-Coupling Polaron

Abstract: The polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant, a. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result, E 0~ —Cα2,and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons.

Poisson Structures on the Poincaré Group

Abstract: An introduction to inhomogeneous Poisson groups is given. Poisson inhomogeneous O(p,q) are shown to be coboundary, the generalized classical Yang-Baxter equation having only a one-dimensional right-hand side. Normal forms of the classical r-matrices for the Poincare group (inhomogeneous O(1,3)) are calculated.

The “Two and One–Half Dimensional” Relativistic Vlasov Maxwell System

Abstract: The motion of a collisionless plasma is modeled by solutions to the Vlasov–Maxwell system. The Cauchy problem for the relativistic Vlasov–Maxwell system is studied in the case when the phase space distribution function f = f(t,x,v) depends on the time t, \(\) and \(\). Global existence of classical solutions is obtained for smooth data of unrestricted size. A sufficient condition for global smooth solvability is known from [12]: smooth solutions can break down only if particles of the plasma approach the speed of light. An a priori bound is obtained on the velocity support of the distribution function, from which the result follows.

The Brjuno Functions and Their Regularity Properties

Abstract: We show that various possible versions of the Brjuno function, based on different kinds of continued fraction developments, are all equivalent and we study their regularity (L p, BMO and Holder) properties, through a systematic analysis of the functional equation which they fulfill.

Chirality and Dirac operator on noncommutative sphere

Abstract: We give a derivation of the Dirac operator on the noncommutative 2-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin's quantization. The map to the local frame in noncommutative geometry is also discussed.

The cauchy problem in local spaces for the complex ginzburg-landau equation. ii. contraction methods

Abstract: We continue the study of the initial value problem for the complex Ginzburg—Landau equation (with a > 0, b > 0, g≥ 0) in initiated in a previous paper [I]. We treat the case where the initial data and the solutions belong to local uniform spaces, more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms, but no decay conditions (or arbitrarily weak decay conditions) at infinity in . In [I] we used compactness methods and an extended version of recent local estimates [3] and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces L r for r≥ 2 or H 1. Here we treat the same problem by contraction methods. This allows us in particular to prove that the solutions obtained in [I] are unique under suitable subcriticality conditions, and to obtain for them additional regularity properties and uniform bounds. The method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces.

Determinant representation for dynamical correlation functions of the quantum nonlinear schrodinger equation

Abstract: Painleve analysis of correlation functions of the impenetrable Bose gas by M. Jimbo, T. Miwa, Y. Mori and M. Sato [1] was based on the determinant representation of these correlation functions obtained by A. Lenard [2]. The impenetrable Bose gas is the free fermionic case of the quantum nonlinear Schrodinger equation. In this paper we generalize the Lenard determinant representation for \(\langle \psi (0,0)\psi^{\dagger}(x,t)\rangle\) to the non-free fermionic case. We also include time and temeprature dependence. In forthcoming publications we shall perform the JMMS analysis of this correlationl function. This will give us a completely integrable equation and asymptotic for the quantum correlation function of interacting fermions.

Quantum field theory on spacetimes with a compactly generated Cauchy horizon

Abstract: We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, \(\), with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's ‘Chronology Protection Conjecture’, according to which the laws of physics prevent one from manufacturing a ’time machine‘. Specifically, we prove:

Classification of $N$-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of $Spin(p,q)

Abstract: We classify extended Poincare Lie super algebras and Lie algebras of any signature (p, q), that is Lie super algebras (resp. Z2-graded Lie algebras) \(\), where \(\) is the (generalized) Poincare Lie algebra of the pseudo-Euclidean vector space \(\) of signature (p,q) and \(\) is the spinor \(\)-module extended to a \(\)-module with kernel V. The remaining super commutators \(\) (respectively, commutators \(\)) are defined by an \(\)-equivariant linear mapping $$$$ Denote by \(\) (respectively, \(\)) the vector space of all such Lie super algebras (respectively, Lie algebras), where \(\) and \(\) is the classical signature. The description of \(\) reduces to the construction of all \(\)-invariant bilinear forms on S and to the calculation of three \(\)-valued invariants for some of them.

Vassiliev Knot Invariants and Chern-Simons Perturbation Theory to All Orders

Abstract: At any order, the perturbative expansion of the expectation values of Wilson lines in Chern-Simons theory gives certain integral expressions. We show that they all lead to knot invariants. Moreover these are finite type invariants whose order coincides with the order in the perturbative expansion. Together they combine to give a universal Vassiliev invariant.

On the Classification of Reflexive Polyhedra

Abstract: Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi–Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of these matrices is the existence of a pair of weight systems, indicating a relation to weighted projective spaces. This is the cornerstone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi–Yau compactifications in string theory.