On the Liouville three-point function

TL;DR

This paper investigates the three-point function in quantum Liouville theory, deriving functional equations and confirming the proposed expression's uniqueness and correctness through symmetry considerations.

Contribution

It derives functional equations for Liouville three-point functions and proves the uniqueness of the proposed solution using crossing symmetry and locality.

Findings

The proposed three-point function satisfies the derived functional equations.

The solution's uniqueness is established under certain assumptions.

The analysis confirms the validity of the Zamolodchikovs' expression.

Abstract

The recently proposed expression for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered. By exploiting locality or crossing symmetry in the case of those four-point functions, which may be expressed in terms of hypergeometric functions, a set of functional equations is found for the general three point function. It is shown that the expression proposed by the Zamolodchikovs solves these functional equations and that under certain assumptions the solution is unique.