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

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.