  We develop a functional integral approach to quantum Liouville field theory
completely independent of the hamiltonian approach. To this end on the sphere
topology we solve the Riemann-Hilbert problem for three singularities of finite
strength and a fourth one infinitesimal, by determining perturbatively the
Poincare' accessory parameters. This provides the semiclassical four point
vertex function with three finite charges and a fourth infinitesimal. Some of
the results are extended to the case of n finite charges and m infinitesimal.
With the same technique we compute the exact Green function on the sphere on
the background of three finite singularities. Turning to the full quantum
problem we address the calculation of the quantum determinant on the background
of three finite charges and of the further perturbative corrections. The zeta
function regularization provides a theory which is not invariant under local
conformal transformations. Instead by employing a regularization suggested in
the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the
correct quantum conformal dimensions from the one loop calculation and we show
explicitly that the two loop corrections do not change such dimensions. We then
apply the method to the case of the pseudosphere with one finite singularity
and compute the exact value for the quantum determinant. Such results are
compared to those of the conformal bootstrap approach finding complete
agreement.
