Showing papers in "Communications in Mathematical Physics in 1998"

Hopf Algebras, Renormalization and Noncommutative Geometry

Abstract: We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

Instantons on Noncommutative R 4 , and (2; 0) Superconformal Six Dimensional Theory

Abstract: We show that the resolution of moduli space of ideal instantons parameterizes the instantons on noncommutative ℝ4. This moduli space appears to be the Higgs branch of the theory of k D0-branes bound to N D4-branes by the expectation value of the B field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of (2,0) superconformal theories in six dimensions.

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

Abstract: This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether' s theorem Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes

Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem

Abstract: In this paper we solve a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. We show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific Hopf algebra \(\), associated to each integer codimension. This Hopf algebra reduces transverse geometry, to a universal geometry of affine nature. The structure of this Hopf algebra, its relation with the Lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper, in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the Hochschild cohomology of a Hopf algebra carries a highly nontrivial cyclic structure.

Quantum Symmetry Groups of Finite Spaces

Abstract: We determine the quantum automorphism groups of finite spaces. These are compact matrix quantum groups in the sense of Woronowicz.

Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.I. Existence for Euler and Prandtl Equations

Abstract: This is the first of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the projection method is used in the primitive variables, to which the Cauchy-Kowalewski theorem is directly applicable. For the Prandtl equations, Cauchy-Kowalewski is applicable once the diffusion operator in the vertical direction is inverted.

Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space. II. Construction of the Navier-Stokes Solution

Abstract: This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.

On the Algebras of BPS States

Abstract: We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac–Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi–Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry.

On Ruelle's Probability Cascades and an Abstract Cavity Method

Abstract: We construct in this work a Markov process which describes a clustering mechanism through which equivalence classes on ℕ are progressively lumped together. This clustering process gives a new description of Ruelle's continuous probability cascades. It also enables to introduce an abstract cavity method, which mimicks certain features of the cavity method developed by physicists in the context of the Sherrington Kirkpatrick model.

Generalized hermite polynomials and the heat equation for dunkl operators

Abstract: Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ N . The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In the case of the symmetric group S N , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.

Integrable Evolution Equations on Associative Algebras

Abstract: This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found.

Modified Prüfer and EFGP Transforms and the Spectral Analysis of One-Dimensional Schrödinger Operators

Abstract: Using control of the growth of the transfer matrices, wediscuss the spectral analysis of continuum and discrete half-line Schrodinger operators with slowly decaying potentials. Among our results we show if , where W has compact support and , then H has purely a.c. (resp. purely s.c.) spectrum on (0,∞) if ). For λn {-1/2} a n potentials, where a n are independent, identically distributed random variables with E(a n ) = 0, E(a 2 n)=1, and λ < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.

Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation

Abstract: We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov–Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.

New Braided Endomorphisms from Conformal Inclusions

Abstract: Various puzzles about subfactors and integrable lattice models associated with conformal inclusions are resolved in the framework of constructive quantum field theory in two dimensions. In particular, a new class of braided endomorphisms are obtained for a general class of conformal inclusions and their properties are analyzed. The existence of subfactors with principal graphs E 6 or E 8 follows from a rather simple argument in our construction. The fusion graphs of many new examples are given.

The Pointwise Estimates of Diffusion Wave for the Navier–Stokes Systems in Odd Multi-Dimensions

Abstract: We study the dissipation of solutions of the isentropic Navier–Stokes equations in odd multi-dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and shown to exhibit the generalized Huygen's principle. Our approach is based on the detailed analysis of the Green function of the linearized system. This is used to study the coupling of nonlinear diffusion waves.

D-Brane Bound States Redux

Abstract: We study the existence of D-brane bound states at threshold in Type II string theories. In a number of situations, we can reduce the question of existence to quadrature, and the study of a particular limit of the propagator for the system of D-branes. This involves a derivation of an index theorem for a family of non-Fredholm operators. In support of the conjectured relation between compactified eleven-dimensional supergravity and Type IIA string theory, we show that a bound state exists for two coincident zero-branes. This result also provides support for the conjectured description of M-theory as a matrix model. In addition, we provide further evidence that there are no BPS bound states for two and three-branes twice wrapped on Calabi–Yau vanishing cycles.

Smooth Irrotational Flows in the Large to the Euler–Poisson System in R3+1

Abstract: A simple two-fluid model to describe the dynamics of a plasma is the Euler–Poisson system, where the compressible electron fluid interacts with its own electric field against a constant charged ion background. The plasma frequency produced by the electric field plays the role of „mass” term to the linearized system. Based on this „Klein–Gordon” effect, we construct global smooth irrotational flows with small velocity for the electron fluid.

Geometry and Classificatin of Solutions of the Classical Dynamical Yang–Baxter Equation

Abstract: The classical Yang–Baxter equation(CYBE) is an algebraic equation central in the theory of integrable systems. Its nondegenerate solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CYBE was given by Drinfeld and gave rise to the theory of Poisson–Lie groups.

Existence of Gelling Solutions for Coagulation-Fragmentation Equations

Abstract: We study the Smoluchowski coagulation-fragmentation equation, which is an infinite set of non-linear ordinary differential equations describing the evolution of a mono-disperse system of particles in a well stirred solution. Approximating the solutions of the Smoluchowski equations by a sequence of finite Markov chains, we investigate the qualitative behavior of the solutions. We determine a device on the finite chains which can detect the gelation phenomena – the density dropping phenomena. It shows how the gelation phenomena are reflected on the sequence of finite Markov chains. Using this device, we determine various types of gelation kernels and get the bounds of gelation times.

The Pair Correlation Function of Fractional Parts of Polynomials

Abstract: We investigate the pair correlation function of the sequence of fractional parts of αn d , n=1,2,…,N, where d≥ 2 is an integer and α an irrational. We conjecture that for badly approximable α, the normalized spacings between elements of this sequence have Poisson statistics as N?∞. We show that for almost all α (in the sense of measure theory), the pair correlation of this sequence is Poissonian. In the quadratic case d=2, this implies a similar result for the energy levels of the “boxed oscillator” in the high-energy limit. This is a simple integrable system in 2 degrees of freedom studied by Berry and Tabor as an example for their conjecture that the energy levels of generic completely integrable systems have Poisson spacing statistics.

Framed Vertex Operator Algebras, Codes and the Moonshine Module

Abstract: For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ½, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ½ are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.

Affine Weyl Groups, Discrete Dynamical Systems and Painlevé Equations

Abstract: A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painleve type are discussed.

A Conjectural Generating Function for Numbers of Curves on Surfaces

Abstract: I give a conjectural generating function for the numbers of δ-nodal curves in a linear system of dimension δ on an algebraic surface. It reproduces the results of Vainsencher [V] for the case δ &\le; 6 and Kleiman–Piene [K-P] for the case δ &\le; 8. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau–Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso–Harris for the Severi degrees in ℙ2. We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points.

Dynamical localization for discrete and continuous random schrodinger operators

Abstract: We show for a large class of random Schrodinger operators H ο on and on that dynamical localization holds, i.e. that, with probability one, for a suitable energy interval I and for q a positive real, Here ψ is a function of sufficiently rapid decrease, and P I (H ο) is the spectral projector of H ο corresponding to the interval I. The result is obtained through the control of the decay of the eigenfunctions of H ο and covers, in the discrete case, the Anderson tight-binding model with Bernoulli potential (dimension ν = 1) or singular potential (ν > 1), and in the continuous case Anderson as well as random Landau Hamiltonians.

Solutions of the Quantum Dynamical Yang–Baxter Equation and Dynamical Quantum Groups

Abstract: The quantum dynamical Yang–Baxter (QDYB) equation is a useful generalization of the quantum Yang–Baxter (QYB) equation. This generalization was introduced by Gervais, Neveu, and Felder. Unlike the QYB equation, the QDYB equation is not an algebraic but a difference equation, with respect to a matrix function rather than a matrix. The QDYB equation and its quasiclassical analogue (the classical dynamical Yang–Baxter equation) arise in several areas of mathematics and mathematical physics (conformal field theory, integrable systems, representation theory). The most interesting solution of the QDYB equation is the elliptic solution, discovered by Felder. In this paper, we prove the first classification results for solutions of the QDYB equation. These results are parallel to the classification of solutions of the classical dynamical Yang–Baxter equation, obtained in our previous paper. All solutions we found can be obtained from Felder's elliptic solution by a limiting process and gauge transformations. Fifteen years ago the quantum Yang–Baxter equation gave rise to the theory of quantum groups. Namely, it turned out that the language of quantum groups (Hopf algebras) is the adequate algebraic language to talk about solutions of the quantum Yang–Baxter equation. In this paper we propose a similar language, originating from Felder's ideas, which we found to be adequate for the dynamical Yang–Baxter equation. This is the language of dynamical quantum groups (or ?-Hopf algebroids), which is the quantum counterpart of the language of dynamical Poisson groupoids, introduced in our previous paper.

Relative Zeta Functions, Relative Determinants and Scattering Theory

Abstract: We use the method of zeta function regularization to regularize the ratio det A det A0$ of the determinants of two elliptic self-adjoint operators A, A0 satisfying certain natural assumptions This is of interest, especially, if the regularized determinants of the individual operators don't exist as, for example, in the case of elliptic operators on a noncompact manifold

Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and Standard Ordering with Differential Operator Representation

Abstract: In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T*Q by means of the Fedosov procedure using a symplectic torsion-free connection on T*Q homogeneous of degree zero with respect to the Liouville vector field. By a fibrewise equivalence transformation we construct a homogeneous Fedosov star product of standard ordered type equivalent to the homogeneous Fedosov star product of Weyl type. Representations for both star product algebras by differential operators on functions on Q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex-valued functions on T*Q polynomial in the momenta (where an arbitrary fixed torsion-free connection ∇0 on Q is used). Motivated by the flat case T*ℝn another homogeneous star product of Weyl type corresponding to the Weyl ordering prescription is constructed. The example of the cotangent bundle of an arbitrary Lie group is explicitly computed and the star product given by Gutt is rederived in our approach.

Special Quantum Field Theories¶in Eight and Other Dimensions

Abstract: We build nearly topological quantum field theories in various dimensions. We give special attention to the case of eight dimensions for which we first consider theories depending only on Yang–Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU(4) and Spin(7). The choice of SU(4) gives a quantum field theory for a Calabi–Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M- and F-theories. Relations to the eight dimensional supersymmetric Yang–Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg–Witten equations. The latter are thus related to pure Yang–Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang–Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions.

On the Structure of Correlation Functions in the Normal Matrix Model

Abstract: We study the structure of the normal matrix model (NMM). We show that all correlation functions of the model with an axially symmetric potential can be expressed in terms of a single holomorphic function of one variable. This observation is used to demonstrate the exact solvability of the model. The two-point correlation function is calculated in the continuum limit. The answer is proven to be universal, i.e. potential independent up to a change of the scale. In connection with NMM with a general polynomial potential we have developed a two-dimensional free fermion formalism and constructed a family of completely integrable hierarchies of non-linear differential equations, which we call the extended-KP(N) hierarchies. The well-known KP hierarchy is a lower-dimensional reduction of this family. The extended-KP(1) hierarchy contains the (2+1)-dimensional Burgers equations. The partition function of the (N×N) NMM is the τ function of the extended-KP(N) hierarchy which is invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane.

Extensions of Conformal Nets and Superselection Structures

Abstract: Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Mobius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of , showing that they violate 3-regularity for $n > 2. When n≥ 2, we obtain examples of non Mobius-covariant sectors of a 3-regular (non 4-regular) net.