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

Free field constructions for the elliptic algebra Aq,p(sl^2){\cal A}_{q,p}(\hat{sl}_2) and Baxter's eight-vertex model

Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these (for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an integral formula for the correlation functions of Baxter's eight-vertex model is obtained. This formula shows different structure compared with the one obtained by Lashkevich and Pugai.Comment: 23 pages. Based on talks given at "MATHPHYS ODYSSEY 2001-Integrable Models and Beyond" at Okayama and Kyoto, February 19-23, 2001, et

Elliptic algebra U_{q,p}(^sl_2): Drinfeld currents and vertex operators

We investigate the structure of the elliptic algebra U_{q,p}(^sl_2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra U_q(^sl_2), which are elliptic analogs of the Drinfeld currents. They enable us to identify U_{q,p}(^sl_2) with the tensor product of U_q(^sl_2) and a Heisenberg algebra generated by P,Q with [Q,P]=1. In terms of these currents, we construct an L operator satisfying the dynamical RLL relation in the presence of the central element c. The vertex operators of Lukyanov and Pugai arise as `intertwiners' of U_{q,p}(^sl_2) for level one representation, in the sense to be elaborated on in the text. We also present vertex operators with higher level/spin in the free field representation.Comment: 49 pages, (AMS-)LaTeX ; added an explanation of integration contours; added comments. To appear in Comm. Math. Phys. Numbering of equations is correcte

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

Fusion of the qq-Vertex Operators and its Application to Solvable Vertex Models

We diagonalize the transfer matrix of the inhomogeneous vertex models of the 6-vertex type in the anti-ferroelectric regime intoducing new types of q-vertex operators. The special cases of those models were used to diagonalize the s-d exchange model\cite{W,A,FW1}. New vertex operators are constructed from the level one vertex operators by the fusion procedure and have the description by bosons. In order to clarify the particle structure we estabish new isomorphisms of crystals. The results are very simple and figure out representation theoretically the ground state degenerations.Comment: 35 page

Free Field Approach to the Dilute A_L Models

We construct a free field realization of vertex operators of the dilute A_L models along with the Felder complex. For L=3, we also study an E_8 structure in terms of the deformed Virasoro currents.Comment: (AMS-)LaTeX(2e), 43page