Some insights in the structure of correlation functions in Liouville and Toda field theories

TL;DR

This paper explores the structure of correlation functions in Liouville and Toda field theories, deriving their properties using operator equations, functional methods, and symmetry considerations, and highlighting the role of special functions like Barnes double Gamma functions.

Contribution

It introduces a generalized framework for understanding correlation functions in Liouville and Toda theories, emphasizing the use of zero mode equations and special functions.

Findings

Pole structure of Liouville correlation functions derived from partition functions

Generalized correlation function structure from zero mode functional equations

Use of Barnes double Gamma functions and Weyl symmetry in constructing correlators

Abstract

We discuss some aspects of Liouville field theory, starting from operator equation of motion in presence of two screening charges and re-derive the dual zero mode Schwinger Dyson equations for the two screening charges from the path integral. Using functional methods we show the familiar pole structure of Liouville correlation function using the partition function. Next we discuss a generalized structure of the correlation functions obtained from the zero mode functional equations. From this structure we infer the use of the Barnes double Gamma functions to construct a part of the denominator of the correlators and also use Weyl symmetry of the theory to deduce more information about the rest. We similarly extend these arguments in the case of Toda field theories where we make a general statement about the denominator of the three point function and Sine-Liouvile field theory where we only obtain an infinite product structure.