Showing papers in "Communications in Mathematical Physics in 1992"

Intersection theory on the moduli space of curves and the matrix Airy function

Abstract: We show that two natural approaches to quantum gravity coincide. This identity is nontrivial and relies on the equivalence of each approach to KdV equations. We also investigate related mathematical problems.

Finitely correlated states on quantum spin chains

Abstract: We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Neel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

The isomonodromy approach to matric models in 2D quantum gravity

Abstract: We consider the double-scaling limit in the hermitian matrix model for 2D quantum gravity associated with the measure exp\(\sum\limits_{j = 1}^N {t_{j^{Z^{2j,} } } N \geqq 3} \) We show that after the appropriate modification of the contour of integration the Cross-Migdal-Douglas-Shenker limit to the Painleve I equation (in the generic case of the pure gravity) is valid and calculate the nonperturbative parameters of the corresponding Painleve function Our approach is based on the WKB-analysis of the L-A pair corresponding to the discrete string equation in the framework of the Inverse Monodromy Method Here we extend our results, which were obtained before for the particular casesN=2,3 Our analysis complements the isomonodromy approach proposed by G Moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

Quantum affine algebras and holonomic difference equations

Abstract: We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the “face” formulation for any type of Lie algebra $$\mathfrak{g}$$ and arbitrary finite-dimensional representations of . We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq→1 these solutions degenerate again into solutions with $$q' = \exp \left( {\frac{{2\pi i}}{{k + g}}} \right)$$ . We also study the simples examples of solutions of our holonomic difference equations associated to $$U_q (\widehat{\mathfrak{s}\mathfrak{l}(2)})$$ and find their expressions in terms of basic (orq−)-hypergeometric series. In the special case of spin −1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.

A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description

Abstract: We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature\(\widetilde\beta \) and prove that, in the limitN→∞,\(\widetilde\beta \)/N→β, αN→1, where β∈(−8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ϱN = ϱNx,x∈Λ converges to a superposition of solutions ϱα of the following Mean Field Equation: $$\left\{ {\begin{array}{*{20}c} {\varrho _{\beta (x) = } \frac{{e^{ - \beta \psi } }}{{\mathop \smallint \limits_\Lambda e^{ - \beta \psi } }}; - \Delta \psi = \varrho _\beta in\Lambda } \\ {\psi |_{\partial \Lambda } = 0.} \\ \end{array} } \right.$$

Floquet solutions for the $1$-dimensional quasi-periodic Schrödinger equation

Abstract: We show that the 1-dimensional Schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a Floquet representation for almost every energyE in the upper part of the spectrum. We prove that the upper part of the spectrum is purely absolutely continuous and that, for a generic potential, it is a Cantor set. We also show that for a small potential these results extend to the whole spectrum.

Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

Abstract: We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.

The dispersionless Lax equations and topological minimal models

Abstract: It is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless Lax equations The exact formulae for the tree level partition functions of An topological minimal models are found The Virasoro constraints for the analogue of the τ-function of the dispersionless Lax equation correspond- ing to these models are proved

On holomorphic factorization of WZW and coset models

Abstract: It is shown how coupling to gauge fields can be used to explain the basic facts concerning holomorphic factorization of the WZW model of two dimensional conformal field theory, which previously have been understood primarily by using conformal field theory Ward identities. We also consider in a similar vein the holomorphic factorization ofG/H coset models. We discuss theG/G model as a topological field theory and comment on a conjecture by Spiegelglas.

Smoothing properties and retarded estimates for some dispersive evolution equations

Abstract: Smoothing properties, in the form of space-time integrability properties, play an important role in the study of dispersive evolution equations. A number of them follow from a combination of general arguments and specific estimates. We present a general formulation which makes the separation between the two types of ingredients as clear as possible, and we illustrate it with the examples of the Schrodinger equation, of the wave equation, and of a class of 1+1 dimensional equations related to the Benjamin-Ono equation. Of special interest for the Cauchy problem are retarded estimates expressed in terms of those properties. We derive a number of such estimates associated with the last example, and we mention briefly an application of those estimates to the Cauchy problem for the generalized Benjamin-Ono equation.

Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation

Abstract: We derive explicit formulas for the Chern-Simons-Witten invariants of lens spaces and torus bundles overS1, for arbitrary values of the levelk. Most of our results are for the groupG=SU(2), though some are for more general compact groups. We explicitly exhibit agreement of the limiting values of these formulas ask→∞ with the semiclassical approximation predicted by the Chern-Simons path integral.

The CPT-theorem in two-dimensional theories of local observables

Abstract: Let ℳ be a von Neumann algebra with cyclic and separating vector Ω, and letU(a) be a continuous unitary representation ofR with positive generator and Ω as fixed point. If these unitaries induce for positive arguments endomorphisms of ℳ then the modular group act as dilatations on the group of unitaries. Using this it will be shown that every theory of local observables in two dimensions, which is covariant under translation only, can be imbedded into a theory of local observables covariant under the whole Poincare group. This theory is also covariant under the CPT-transformation.

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

Abstract: The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einstein's field equations is demonstrated.

Characters and fusion rules for W-algebras via quantized Drinfeld-Sokolov reduction

Abstract: Using the cohomological approach toW-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld-Sokolov reduction.

Special geometry, cubic polynomials and homogeneous quaternionic spaces

Abstract: The existing classification of homogeneous quaternionic spaces is not complete. We study these spaces in the context of certainN=2 supergravity theories, where dimensional reduction induces a mapping betweenspecial real, Kahler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the Kahler and quaternionic manifolds in homogeneous holomorphic functions of second degree. We classify all cubic polynomials that have an invariance group that acts transitively on the real manifold. The corresponding Kahler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by Alekseevskii (and the corresponding special Kahler spaces given by Cecotti), but there is a new class of rank-3 spaces of quaternionic dimension larger than 3. We also point out that some of the rank-4 Alekseevskii spaces were not fully specified and correspond to a finite variety of inequivalent spaces. A simpler version of the equation that underlies the classification of this paper also emerges in the context ofW 3 algebras.

Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains

Abstract: We study the phase diagram ofS=1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of aZ2×Z2 symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Neel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of “diagonally dominant” Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hiddenZ2×Z2 symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.

Statistics of shocks in solutions of inviscid Burgers equation

Abstract: The purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid Burgers equation having a typical realizationb(y) of the Brownian motion as an initial datum. This case was proposed and studied numerically in the companion paper by She, Aurell and Frisch. The description of the statistics is given in terms of the behavior of the convex hull of the random process\(w(y) = \int\limits_0^y {(b(\eta ) + \eta )} d\eta \). The Hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied.

q-Deformed Poincaré algebra

Abstract: Theq-differential calculus for theq-Minkowski space is developed. The algebra of theq-derivatives with theq-Lorentz generators is found giving theq-deformation of the Poincare algebra. The reality structure of theq-Poincare algebra is given. The reality structure of theq-differentials is also found. The real Laplacian is constructed. Finally the comultiplication, counit and antipode for theq-Poincare algebra are obtained making it a Hopf algebra.

Spineurs, opérateurs de dirac et variations de métriques

Abstract: In this article a geometric process to compare spinors for different metrics is constructed. It makes possible the extension to spinor fields of a variant of the Lie derivative (called the metric Lie derivative), giving a geometric approach to a construction originally due to Yvette Kosmann. The comparison of spinor fields for two different Riemannian metrics makes the study of the variation of Dirac operators feasible. For this it is crucial to take into account the fact that the bundle in which the sections acted upon by the Dirac operators take their values is changing. We also give the formulas for the change in the eigenvalues of the Dirac operator. We conclude by giving a few cases in which an eigenvalue is stationary.

Quantum continual measurements and a posteriori collapse on CCR

Abstract: A quantum stochastic model for the Markovian dynamics of an open system under the nondemolition unsharp observation which is continuous in time, is given. A stochastic equation for the posterior evolution of a quantum continuously observed system is derived and the spontaneous collapse (stochastically continuous reduction of the wave packet) is described. The quantum Langevin evolution equation is solved for the case of a quasi-free Hamiltonian in the initial CCR algebra with a linear output channel, and the posterior dynamics corresponding to an initial Gaussian state is found. It is shown for an example of the posterior dynamics of a quantum oscillator that any mixed state under a complete nondemolition measurement collapses exponentially to a pure Gaussian one.

Rapidly decaying solutions of the nonlinear Schrödinger equation

Abstract: We consider global solutions of the nonlinear Schrodinger equation $$iu_t + \Delta u = \lambda |u|^\alpha u, in R^N ,$$ (NLS) where λ∈R and\(0 0, we prove asymptotic completeness for\(\alpha _0 \leqq \alpha< \frac{4}{{N - 2}}\).

The logarithmic Sobolev inequality for discrete spin systems on a lattice

Abstract: For finite range lattice gases with a finite spin space, it is shown that the Dobrushin-Shlosman mixing condition is equivalent to the existence of a logarithmic Sobolev inequality for the associated (unique) Gibbs state. In addition, implications of these considerations for the ergodic properties of the corresponding Glauber dynamics are examined.

Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data

Abstract: We show that global asymptotically flat singularity-free solutions of the spherically symmetric Vlasov-Einstein system exist for all initial data which are sufficiently small in an appropriate sense. At the same time detailed information is obtained concerning the asymptotic behaviour of these solutions. A key element of the proof which is also of intrinsic interest is a local existence theorem with a continuation criterion which says that a solution cannot cease to exist as long as the maximum momentum in the support of the distribution function remains bounded. These results are contrasted with known theorems on spherically symmetric dust solutions.

Infinite dimensional Grassmannian structure of two-dimensional quantum gravity

Abstract: We study the infinite dimensional Grassmannian structure of 2D quantum gravity coupled to minimal conformal matters, and show that there exists a large symmetry, theW 1+∞ symmetry. Using this symmetry structure, we prove that the square root of the partition function, which is a τ function of thep-reduced KP hierarchy, satisfies the vacuum condition of theW 1+∞ algebra. We further show that this condition is reduced to the vacuum condition of theW p algebra when the redundant variables for thep-reduction are eliminated. This mechanism also gives a prescription for extracting theW p algebra from theW 1+∞ algebra.

Tensor products of quantized tilting modules

Abstract: LetUk denote the quantized enveloping algebra corresponding to a finite dimensional simple complex Lie algebraL. Assume that the quantum parameter is a root of unity ink of order at least the Coxeter number forL. Also assume that this order is odd and not divisible by 3 if typeG2 occurs. We demonstrate how one can define a reduced tensor product on the familyF consisting of those finite dimensional simpleUk-modules which are deformations of simpleL and which have non-zero quantum dimension. This together with the work of Reshetikhin-Turaev and Turaev-Wenzl prove that (Uk,F) is a modular Hopf algebra and hence produces invariants of 3-manifolds. Also by recent work of Duurhus, Jakobsen and Nest it leads to a general topological quantum field theory. The method of proof explores quantized analogues of tilting modules for algebraic groups.

Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas

Abstract: This paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one-dimensional barotropic model system for compressible viscous gas. We assume that the initial data tend to constant states atx=±∞, respectively, and the Riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave. If the adiabatic constant γ satisfies 1≦γ≦2, then the solution is proved to tend to the rarefaction wave ast→∞ under no smallness conditions of both the difference of asymptotic values atx=±∞ and the initial data. The proof is given by an elementaryL 2-energy method.

The inviscid Burgers equation with initial data of Brownian type

Abstract: The solutions to Burgers equation, in the limit of vanishing viscosity, are investigated when the initial velocity is a Brownian motion (or fractional Brownian motion) function, i.e. a Gaussian process with scaling exponent 0

A matrix integral solution to two-dimensional W p -gravity

Abstract: Thep th Gel'fand-Dickey equation and the string equation [L, P]=1 have a common solution τ expressible in terms of an integral overn×n Hermitean matrices (for largen), the integrand being a perturbation of a Gaussian, generalizing Kontsevich's integral beyond the KdV-case; it is equivalent to showing that τ is a vacuum vector for aW ∿ + , generated from the coefficients of the vertex operator. This connection is established via a quadratic identity involving the wave function and the vertex operator, which is a disguised differential version of the Fay identity. The latter is also the key to the spectral theory for the two compatible symplectic structures of KdV in terms of the stress-energy tensor associated with the Virasoro algebra. Given a differential operator $$L = D^p + q_2 (t) D^{p - 2} + \cdots + q_p (t), with D = \frac{\partial }{{dx}},t = (t_1 ,t_2 ,t_3 ,...),x \equiv t_1 ,$$ consider the deformation equations1 (0.1) $$\begin{gathered} \frac{{\partial L}}{{\partial t_n }} = [(L^{n/p} )_ + ,L] n = 1,2,...,n + - 0(mod p) \hfill \\ (p - reduced KP - equation) \hfill \\ \end{gathered} $$ ofL, for which there exists a differential operatorP (possibly of infinite order) such that (0.2) $$[L,P] = 1 (string equation).$$ In this note, we give a complete solution to this problem. In section 1 we give a brief survey of useful facts about theI-function τ(t), the wave function Ψ(t,z), solution of ∂Ψ/∂t n=(L n/p) x Ψ andL 1/pΨ=zΨ, and the corresponding infinitedimensional planeV 0 of formal power series inz (for largez) $$V^0 = span \{ \Psi (t,z) for all t \in \mathbb{C}^\infty \} $$ in Sato's Grassmannian. The three theorems below form the core of the paper; their proof will be given in subseuqent sections, each of which lives on its own right.

Bohr-sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula

Abstract: We show how the moduli space of flatSU(2) connections on a two-manifold can be quantized in the real polarization of [15], using the methods of [6] The dimension of the quantization, given by the number of integral fibres of the polarization, matches the Verlinde formula, which is known to give the dimension of the quantization of this space in a Kahler polarization

Generalized Drinfel'd-Sokolov hierarchies

Abstract: A general approach is adopted to the construction of integrable hierarchies of partial differential equations. A series of hierarchies associated to untwisted Kac-Moody algebras, and conjugacy classes of the Weyl group of the underlying finite Lie algebra, is obtained. The generalized KdV hierarchies of V.G. Drinfel'd and V.V. Sokolov are obtained as the special case for the Coxeter element. Various examples of the general formalism are treated in some detail; including the fractional KdV hierarchies.