Riemann-Hilbert treatment of Liouville theory on the torus

TL;DR

This paper develops a perturbative approach to classical Liouville theory on the torus, providing exact solutions for special cases and formulas for deformations, with applications to Green functions and hypergeometric functions.

Contribution

It introduces a perturbative method for Liouville theory on the torus and derives exact solutions for the square case using hypergeometric functions.

Findings

Exact solutions for the square torus case.

Formulas for torus deformations and Heun parameter computation.

Explicit Green function expressions on the square background.

Abstract

We apply a perturbative technique to study classical Liouville theory on the torus. After mapping the problem on the cut-plane we give the perturbative treatment for a weak source. When the torus reduces to the square the problem is exactly soluble by means of a quadratic transformation in terms of hypergeometric functions. We give general formulas for the deformation of a torus and apply them to the case of the deformation of the square. One can compute the Heun parameter to first order and express the solution in terms of quadratures. In addition we give in terms of quadratures of hypergeometric functions the exact symmetric Green function on the square on the background generated by a one point source of arbitrary strength.