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Riemann-Hilbert treatment of Liouville theory on the torus

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.