Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations

The Hamiltonian approach to the theory of dual isomonodromic deformations is developed within the framework of rational classical R-matrix structures on loop algebras. Particular solutions to the isomonodromic deformation equations appearing in the computation of correlation functions in integrable quantum field theory models are constructed through the Riemann-Hilbert problem method. The corresponding $\tau$-functions are shown to be given by the Fredholm determinant of a special class of integral operators.Comment: LaTeX 13pgs (requires lamuphys.sty). Text of talk given at workshop: Supersymmetric and Integrable Systems, University of Illinois, Chicago Circle, June 12-14, 1997. To appear in: Springer Lecture notes in Physic

Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models

We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice $\tau$-function and discuss various implications of non-vanishing "negative"- and "zero"-time variables: the appear to modify the original GKM action by negative-power and logarithmic contributions respectively. It is shown that so deformed $\tau$-function satisfies the same string equation as the original one. In the case of quadratic potential GKM turns out to describe {\it forced} Toda chain hierarchy and, thus, corresponds to a {\it discrete} matrix model, with the role of the matrix size played by the zero-time (at integer positive points). This relation allows one to discuss the double-scaling continuum limit entirely in terms of GKM, $i.e.$ essentially in terms of {\it finite}-fold integrals.Comment: 46

The Monodromy Matrices of the XXZ Model in the Infinite Volume Limit

We consider the XXZ model in the infinite volume limit with spin half quantum space and higher spin auxiliary space. Using perturbation theory arguments, we relate the half infinite transfer matrices of this class of models to certain $U_q(\hat{sl_2})$ intertwiners introduced by Nakayashiki. We construct the monodromy matrices, and show that the one with spin one auxiliary space gives rise to the L operator.Comment: 19 page

New matrix model solutions to the Kac-Schwarz problem

We examine the Kac-Schwarz problem of specification of point in Grassmannian in the restricted case of gap-one first-order differential Kac-Schwarz operators. While the pair of constraints satisfying $[{\cal K}_1,W] = 1$ always leads to Kontsevich type models, in the case of $[{\cal K}_1,W] = W$ the corresponding KP $\tau$-functions are represented as more sophisticated matrix integrals.Comment: 19 pages, latex, no figures, contribution to the proceedings of the 29th International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, German

Form factors of the XXZ model and the affine quantum group symmetry

We present new expressions of form factors of the XXZ model which satisfy Smirnov's three axioms. These new form factors are obtained by acting the affine quantum group $U_q (\hat{\frak s \frak l_2})$ to the known ones obtained in our previous works. We also find the relations among all the new and known form factors, i.e., all other form factors can be expressed as kind of descendents of a special one.Comment: 11 pages, latex; Some explanation is adde

Bilinear structure and Schlesinger transforms of the qq-PIII_{\rm III} and qq-PVI_{\rm VI} equations

We show that the recently derived ($q$-) discrete form of the Painlev\'e VI equation can be related to the discrete P$_{\rm III}$, in particular if one uses the full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$.Comment: 10 pages, Plain Te

Random Words, Toeplitz Determinants and Integrable Systems. I

It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane.Comment: 15 pages, no figure

Comultiplication in ABCD algebra and scalar products of Bethe wave functions

The representation of scalar products of Bethe wave functions in terms of the Dual Fields, proven by A.G.Izergin and V.E.Korepin in 1987, plays an important role in the theory of completely integrable models. The proof in \cite{Izergin87} and \cite{Korepin87} is based on the explicit expression for the "senior" coefficient which was guessed in \cite{Izergin87} and then proven to satisfy some recurrent relations, which determine it unambiguously. In this paper we present an alternative proof based on the direct computation.Comment: 9 page

Annihilation poles of a Smirnov-type integral formula for solutions to quantum Knizhnik--Zamolodchikov equation

We consider the recently obtained integral representation of quantum Knizhnik-Zamolodchikov equation of level 0. We obtain the condition for the integral kernel such that these solutions satisfy three axioms for form factor \'{a} la Smirnov. We discuss the relation between this integral representation and the form factor of XXZ spin chain.Comment: 14 pages, latex, no figures