ar X iv : h ep - t h / 06 02 20 6 v 1 2 1 Fe b 20 06 IFUP – TH / 2006 – 6 Liouville field

We work out the perturbative expansion of quantum Liouville theory on the pseudosphere starting from the semiclassical limit of a background generated by heavy charges. By solving perturbatively the Riemann-Hilbert problem for the Poincaré accessory parameters, we give in closed form the exact Green function on the background generated by one finite charge. Such a Green function is used to compute the quantum determinants i.e. the one loop corrections to known semiclassical limits thus providing the resummation of infinite classes of standard perturbative graphs. The results obtained for the one point function are compared with the bootstrap formula while those for the two point function are compared with the existing double perturbative expansion and with a degenerate case, finding complete agreement. ∗This work is supported in part by M.I.U.R.