Liouville Type Models in Group Theory Framework. I. Finite-Dimensional Algebras

TL;DR

This paper explores Liouville and Toda models within a group theory framework, analyzing wave functions and their asymptotics, revealing connections with gamma functions, and introducing new types of Whittaker functions.

Contribution

It provides a detailed analysis of finite-dimensional Whittaker functions and introduces new Gauss Whittaker functions within the group theory approach.

Findings

Asymptotics characterized by Harish-Chandra functions as gamma products.

Representation of wave functions as boundary integrals over lower-dimensional theories.

Construction of new Gauss Whittaker functions.

Abstract

In the series of papers we represent the ``Whittaker'' wave functional of $d+1$-dimensional Liouville model as a correlator in $d+0$-dimensional theory of the sine-Gordon type (for $d=0$ and $1$). Asypmtotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple $\Gamma$-function factors over all positive roots of the corresponding algebras (finite-dimensional for $d=0$ and affine for $d=1$). This is in nice correspondence with the recent results on 2- and 3-point correlators in $1+1$ Liouville model, where emergence of peculiar double-periodicity is observed. The Whittaker wave functions of $d+1$-dimensional non-affine ("conformal") Toda type models are given by simple averages in the $d+0$ dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free-field wave functional, which is originally a Gaussian integral over interior of a $d+1$-dimensional disk with given boundary conditions, as a (non-local) quadratic integral over the $d$-dimensional boundary itself. In the present paper we mostly concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions were known, and we present their survey. We also construct new "Gauss" Whittaker functions.