Liouville field theory with heavy charges. II. The conformal boundary case

TL;DR

This paper develops a method for calculating functional integrals with fixed geometric constraints in Liouville field theory, confirming the bootstrap formula through explicit one-loop computations of boundary vertex functions.

Contribution

It introduces a regularization technique for Green functions and applies it to compute boundary vertex functions, providing the first one-loop validation of the bootstrap formula in this context.

Findings

Successful regularization of Green functions for boundary conditions

Explicit one-loop calculation of boundary vertex functions

Validation of the bootstrap formula in boundary Liouville theory

Abstract

We develop a general technique for computing functional integrals with fixed area and boundary length constraints. The correct quantum dimensions for the vertex functions are recovered by properly regularizing the Green function. Explicit computation is given for the one point function providing the first one loop check of the bootstrap formula.