The paper deals with the analytic theory of the quantum q -deformed Toda chains; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N -particle q -deformed open Toda chain are given as a multiple integral of the Mellin-Barnes type. For the periodic chain the two dual Baxter equations are derived. PrefaceIn the late seventies B. Kostant [1] has discovered a fascinating link between the representation theory of non-compact semisimple Lie groups and the quantum Toda chain. Let G be a real split semisimple Lie group, B = MAN its minimal Borel subgroup, let N and V =N be the corresponding opposite unipotent subgroups. Let χ N , χ V be nondegenerate unitary characters of N and V , respectively. Let H T be the space of smooth functions on G which satisfy the functional equationv ∈ V, n ∈ N.

A 1-matrix model is proposed, which nicely interpolates between doublescaling continuum limits of all multimatrix models. The interpolating partition function is always a KP τ -function and always obeys L −1 -constraint and string equation. Therefore this model can be considered as a natural unification of all models of 2d-gravity (string models) with c ≤ 1.

We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a τ -function of KPhierarchy, subjected to a kind of L −1 -constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to X K+1 , this partition function becomes a τ -function of K-reduced KP-hierarchy, obeying a set of W K -algebra constraints identical to those conjectured in [1] for double-scaling continuum limit of (K − 1)-matrix model. In the case of K = 2 the statement reduces to the early established [2] relation between Kontsevich model and the ordinary 2d quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of 2d quantum gravity with c < 1 preserving the property of integrability and the analogue of string equation.

The Kazakov-Migdal model, if considered as a functional of external fields, can be always represented as an expansion over characters of GL group. The integration over "matter fields" can be interpreted as going over the model (the space of all highest weight representations) of GL. In the case of compact unitary groups the integrals should be substituted by discrete sums over weight lattice. The D = 0 version of the model is the Generalized Kontsevich integral, which in the above-

We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice τ -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 τ -function satisfies the same string equation as the original one. In the case of quadratic potential GKM turns out to describe forced Toda chain hierarchy and, thus, corresponds to a 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 doublescaling continuum limit entirely in terms of GKM, i.e. essentially in terms of finite-fold integrals.

We introduce conformal multi-matrix models (CMM) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with c < 1 matter. We define CMM as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP hierarchy and by definition satisfy the discrete W -constraints. We prove that discrete CMM coincide with the (p, q)-series of 2d gravity models in a well-defined continuum limit, thus demonstrating that they provide a proper generalization of Hermitian one-matrix model.

The integral representations for the eigenfunctions of N particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). Both periodic and open N -particle solutions have essentially the same structure being written as a generalized Fourier transform over the eigenfunctions of the N − 1 particle open Toda chain with the kernels satisfying to the Baxter equations of the second and first order respectively. In the latter case this leads to recurrent relations which result to representation of the Mellin-Barnes type for solutions of an open chain. As byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of GL(N, R) group.

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