Quantum Liouville theory in the background field formalism I. Compact   Riemann surfaces
  Using Polyakov's functional integral approach with the Liouville action functional defined in \cite{ZT2} and \cite{LTT}, we formulate quantum Liouville theory on a compact Riemann surface X of genus g > 1. For the partition function <X> and for the correlation functions with the stress-energy tensor components $<\prod_{i=1}^{n}T(z_{i})\prod_{k=1}^{l}\bar{T}(\w_{k})X>$, we describe Feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution - the hyperbolic metric on X. Extending analysis in \cite{LT1,LT2,LT-Varenna,LT3}, we define the regularization scheme for any choice of global coordinate on X, and for Schottky and quasi-Fuchsian global coordinates we rigorously prove that one- and two-point correlation functions satisfy conformal Ward identities in all orders of the perturbation theory. Obtained results are interpreted in terms of complex geometry of the projective line bundle $\cE_{c}=\lambda_{H}^{c/2}$ over the moduli space $\mathfrak{M}_{g}$, where c is the central charge and $\lambda_{H}$ is the Hodge line bundle, and provide Friedan-Shenker \cite{FS} complex geometry approach to CFT with the first non-trivial example besides rational models. 
Leon A. Takhtajan, and Lee-Peng Teo