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

Understand

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.