Riemann-Hilbert treatment of Liouville theory on the torus: The general case

TL;DR

This paper extends Liouville theory on the torus to asymmetric charge distributions using Fuchsian differential equations, providing new solutions and exact Green functions for specific geometries.

Contribution

It introduces a group theoretic approach to solve monodromy conditions for asymmetric charges and applies differential equations on Riemann surfaces to compute Green functions.

Findings

Successfully generalizes Liouville theory to asymmetric charge distributions.

Derives exact Green functions for square and rhombus geometries.

Provides a method to satisfy monodromy conditions using Heun parameters.

Abstract

We extend the previous treatment of Liouville theory on the torus, to the general case in which the distribution of charges is not necessarily symmetric. This requires the concept of Fuchsian differential equation on Riemann surfaces. We show through a group theoretic argument that the Heun parameter and a weight constant are sufficient to satisfy all monodromy conditions. We then apply the technique of differential equation on a Riemann surface to the two point function on the torus in which one source is arbitrary and the other small. As a byproduct we give in terms of quadratures the exact Green function on the square and on the rhombus with opening angle 2 pi/6 in the background of the field generated by an arbitrary charge.