Structure Constants and Conformal Bootstrap in Liouville Field Theory

TL;DR

This paper derives an explicit formula for the three-point function in Liouville field theory, constructs the four-point function, and verifies the conformal bootstrap equations numerically, advancing understanding of the theory's operator algebra.

Contribution

It provides an analytic expression for the structure constants in Liouville theory and confirms the consistency of the operator algebra through numerical bootstrap verification.

Findings

The three-point function matches classical predictions in the classical limit.

The constructed four-point function satisfies conformal bootstrap equations.

The reflection amplitude is explicitly derived from the structure constants.

Abstract

An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants.