Showing papers in "Communications in Mathematical Physics in 1995"

The quantum structure of spacetime at the Planck scale and quantum fields

Abstract: We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.

Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

Abstract: Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Free products of compact quantum groups

Abstract: We construct and study compact quantum groups from free products ofC *-algebras. In this connection, we discover two mysterious classes of natural compact quantum groups,A u (m) andA o (m). TheA u (m)'s (respectivelyA o (m)'s) are non-isomorphic to each other for differentm's, and are not obtainable by the ordinary quantization method. We also clarify some basic concepts in the theory of compact quantum groups.

Local brst cohomology in the antifield formalism. i: general theorems

Abstract: We establish general theorems on the cohomologyH * (s/d) of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of localp-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown thatH −k (s/d) is isomorphic toH k (δ/d) in negative ghost degree−k (k>0), where δ is the Koszul-Tate differential associated with the stationary surface. The cohomology groupH 1 (δ/d) in form degreen is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether's theorem. More generally, the groupH k (δ/d) in form degreen is isomorphic to the space ofn−k forms that are closed when the equations of motion hold. The groupsH k (δ/d)(k>2) are shown to vanish for standard irreducible gauge theories. The groupH 2 (δ/d) is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groupsH k (s/d) under the introduction of non-minimal variables and of auxiliary fields is also demonstrated. In a companion paper, the general formalism is applied to the calculation ofH k (s/d) in Yang-Mills theory, which is carried out in detail for an arbitrary compact gauge group.

Algebraic index theorem

Abstract: We prove the Atiyah-Singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold.

Ergodicity of the 2-D Navier-Stokes equation under random perturbations

Abstract: A 2-dimensional Navier-Stokes equation perturbed by a sufficiently distributed white noise is considered. Existence of invariant measures is known from previous works. The aim is to prove uniqueness of the invariant measures, strong law of large numbers, and convergence to equilibrium.

Symmetries of quantum spaces. subgroups and quotient spaces of quantum su(2) and so(3) groups

Abstract: We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantumSU(2) andSO(3) groups.

On diffraction by aperiodic structures

Abstract: This paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals. It analyses diffraction in the limit of the infinite system, through an appropriately defined autocorrelation. The main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by Bombieri and Taylor.

Vortex condensation in the Chern-Simons Higgs model: An existence theorem

Abstract: It is shown that there is a critical value of the Chern-Simons coupling parameter so that, below the value, there exists self-dual doubly periodic vortex solutions, and, above the value, the vortices are absent. Solutions of such a nature indicate the existence of dyon condensates carrying quantized electric and magnetic charges.

Mirror manifolds in higher dimension

Abstract: We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We isolate certain simplifying features present only in dimension three, and supply alternative methods that do not rely on these special characteristics and hence can be generalized to other dimensions. Although the moduli spaces for Calabi-Yaud-folds are not “special Kahler manifolds” whend>3, they still have a restricted geometry, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi-Yau manifolds. Our mirror-aided calculations agree with those Chern class calculations in the limited number of cases for which the latter can be carried out with current mathematical tools. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra.

The Dirac operator and gravitation

Abstract: We give a brute-force proof of the fact, announced by Alain Connes, that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein-Hilbert action of general relativity. We show that this also holds for twisted (e. g. by electrodynamics) Dirac operators, and more generally, for Dirac operators pertaining to Clifford connections of general Clifford bundles.

Free field representation for massive integrable models

Abstract: A new approach to massive integrable models is considered. It allows one to find symmetry algebras which define the spaces of local operators and to get general integral representations for form-factors in theSU(2) Thirring and Sine-Gordon models.

Combinatorial quantization of the Hamiltonian Chern-Simons theory I

Abstract: Motivated by a recent paper of Fock and Rosly [6] we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *- operation and a positive inner product.

Local BRST cohomology in the antifield formalism. II. Application to Yang-Mills theory

Abstract: Yang-Mills models with compact gauge group coupled to matter fields are considered. The general tools developed in a companion paper are applied to compute the local cohomology of the BRST differentials modulo the exterior space-time derivatived for all values of the ghost number, in the space of polynomials in the fields, the ghosts, the antifields (=sources for the BRST variations) and their derivatives. New solutions to the consistency conditionssa+db=0 depending non-trivially on the antifields are exhibited. For a semi-simple gauge group, however, these new solutions arise only at ghost number two or higher. Thus at ghost number zero or one, the inclusion of the antifields does not bring in new solutions to the consistency conditionsa+db=0 besides the already known ones. The analysis does not use power counting and is purely cohomological. It can be easily extended to more general actions containing higher derivatives of the curvature or Chern-Simons terms.

Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

Abstract: We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV1,...,Vr in a toric varietyPΣ and the system of differential operators annihilating the special generalized hypergeometric series Φ0 constructed from the fan Σ. Using this generalized hypergeometric series, we propose conjectural mirrorsV′ ofV and the canonicalq-coordinates on the moduli spaces of Calabi-Yau manifolds.

Quantum cohomology of flag manifolds and Toda lattices

Abstract: We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.

Singular continuous spectrum for palindromic schrodinger operators

Abstract: We give new examples of discrete Schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potentialx in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is a z ∈ X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in X. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for all x ∈ X if X derives from a primitive substitution. For potentials defined by circle maps, x_n = 1_J (θ_0 + nα), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational α and every half-open interval J.

with central charge N

Abstract: We study representations of the central extension of the Lie algebra of differential operators on the circle, the algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central chargeN has a canonical structure of an irreducible representation of the with the same central charge and that all irreducible representations of with central chargeN arise in this way. We also establish a duality between “integral” modules of and finite-dimensional irreducible modules ofgl N , and conjecture their fusion rules.

On the support of the Ashtekar-Lewandowski Measure

Abstract: We show that the Ashtekar-Isham extension\(\overline {A/G}\) of the configuration space of Yang-Mills theories\(A/G\) is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices.

Fusion Categories Arising from Semisimple Lie Algebras

Abstract: Using tilting modules we equip certain semisimple categories with a “reduced” tensor product structure The fusion rules for this tensor product are determined via known character formulas for the involved modules

Symplectic structure of the moduli space of flat connection on a Riemann surface

Abstract: We consider the canonical symplectic structure on the moduli space of flat g-connections on a Riemann surface of genus g with a marked points. For a being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove

Stationary states of the nonlinear Dirac equation: a variational approach

Abstract: In this paper we prove the existence of stationary solutions of some nonlinear Dirac equations. We do it by using a general variational technique. This enables us to consider nonlinearities which are not necessarily compatible with symmetry reductions.

Invariants of $3$-manifolds and projective representations of mapping class groups via quantum groups at roots of unity

Abstract: An example of a finite dimensional factorizable ribbon Hopf ℂ-algebra is given by a quotientH=u q (g) of the quantized universal enveloping algebraU q (g) at a root of unityq of odd degree. The mapping class groupM g,1 of a surface of genusg with one hole projectively acts by automorphisms in theH-moduleH *⊗g , ifH * is endowed with the coadjointH-module structure. There exists a projective representation of the mapping class groupM g,n of a surface of genusg withn holes labeled by finite dimensionalH-modulesX 1, ...,X n in the vector space Hom H (X 1 ⊗ ... ⊗X n ,H *⊗g ). An invariant of closed oriented 3-manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most ofu q (g) at roots of unityq of even degree) are described.

Dynamical approximation entropies and topological entropy in operator algebras

Abstract: Dynamical entropy invariants, based on a general approximation approach are introduced for C*-and W*-algebra automorphisms. This includes a noncommutative extension of topological entropy.

On operad structures of moduli spaces and string theory

Abstract: We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory.

Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy

Abstract: In this article we consider the Schrodinger operator inR n ,n≧3, with electric and magnetic potentials which decay exponentially as |x|→∞. We show that the scattering amplitude at fixed positive energy determines the electric potential and the magnetic field.

The Coloured Jones Function

Abstract: The invariantsJK,k of a framed knotK coloured by the irreducibleSU(2)q-module of dimensionk are studied as a function ofk by means of the universalR-matrix. It is shown that whenJK,k is written as a power series inh withq=eh, the coefficient ofhd is an odd polynomial ink of degree at most 2d+1. This coefficient is a Vassiliev invariant ofK. In the second part of the paper it is shown that ask varies, these invariants span ad-dimensional subspace of the space of all Vassiliev invariants of degreed for framed knots. The analogous questions for unframed knots are also studied.

About the regularizing properties of the non-cut-off Kac equation

Abstract: We prove in this work that under suitable assumptions, the solution of the spatially homogeneous non-cut-off Kac equation (or of the spatially homogeneous non cut-off 2D Boltzmann equation with Maxwellian molecules in the radial case) becomes very regular with respect to the velocity variable as soon as the time is strictly positive.

Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials

Abstract: We present an explicit formula of the Virasoro singular vectors in terms of Jack symmetric polynomials. The parametert in the Virasoro central chargec=13-6(t+1/t) is just identified with the deformation parameter α of Jack symmetric polynomialsJγ(α). As a by-product, we obtain an integral representation of Jack symmetric polynomials indexed by the rectangular Young diagrams.

Localization for some continuous random Schrödinger operators

Abstract: We study the spectrum of random Schrodinger operators acting onL2(Rd) of the following type\(H = - \Delta + W + \sum _{x \in \mathbb{Z}^d } t_x V_x \). The\((t_x )_{x \in \mathbb{Z}^d } \) are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.