Quantum Liouville theory with heavy charges

TL;DR

This paper introduces a technique to solve the Riemann-Hilbert problem in Liouville theory with heavy charges, providing exact Green functions and confirming quantum dimensions through perturbative checks across various backgrounds.

Contribution

It develops a general method for solving the Riemann-Hilbert problem in Liouville theory with heavy charges, enabling exact Green functions and validation of quantum dimensions.

Findings

Exact Green functions for Liouville theory in non-trivial backgrounds.

Validation of quantum dimensions using non-invariant regularization.

Agreement with bootstrap approach results in boundary Liouville theory.

Abstract

We develop a general technique for solving the Riemann-Hilbert problem in presence of a number of heavy charges and a small one thus providing the exact Green functions of Liouville theory for various non trivial backgrounds. The non invariant regularization suggested by Zamolodchikov and Zamolodchikov gives the correct quantum dimensions; this is shown to one loop in the sphere topology and for boundary Liouville theory and to all loop on the pseudosphere. The method is also applied to give perturbative checks of the one point functions derived in the bootstrap approach by Fateev Zamolodchikov and Zamolodchikov in boundary Liouville theory and by Zamolodchikov and Zamolodchikov on the pseudosphere, obtaining complete agreement.