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.
Categories: hep-th math-ph math.CV math.MP