Standard and geometric approaches to quantum Liouville theory on the pseudosphere 1

We compare the standard and geometric approaches to quantum Liouville theory on the pseudosphere by performing perturbative calculations of the one and two point functions up to the third order in the coupling constant. The choice of the Hadamard regularization within the geometric approach leads to a discrepancy with the standard approach. On the other hand, we find complete agreement between the results of the standard approach and the bootstrap conjectures for the one point function and the auxiliary two point function. This work is supported in part by M.I.U.R.