MG11 
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QUANTUM LIOUVILLE THEORY WITH HEAVY CHARGES

We develop a general technique for solving the Riemann-Hilbert problem in presence of a number of "heavy charges" and a small one thus providing the exact Green functions of Liouville theory for a various of non trivial backgrounds. The non invariant regularization suggested by Zamolodchikov and Zamolodchikov gives the correct quantum dimensions; this is shown to one loop in the sphere topology and for boundary Liouville theory and to all loop on the pseudosphere. The method is applied to give perturbative checks of the one point function derived in the bootstrap approach by Fateev Zamolodchikov and Zamolodchikov for the one point function in boundary Liouville theory and by Zamolodchikov and Zamolodchikov on the pseudosphere, obtaining complete agreement.