Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces

Using Polyakov's functional integral approach and the Liouville action functional defined in [ZT87c] and [TT03a], we formulate quantum Liouville theory on a compact Riemann surface X of genus g > 1. For the partition function «X» and correlation functions with the stress-energy tensor components «Π_i=1ⁿ T(z_i) Π_k=1^l T̄(w̄_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 [Tak93, Tak94, Tak96a, Tak96b], we define the regularization scheme for any choice of the global coordinate on X. For the 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 script E sign_c = λ_H^c/2 over the moduli space fraktur M sign _g, where c is the central charge and λ _H is the Hodge line bundle, and provide the Friedan-Shenker [FS87] complex geometry approach to CFT with the first non-trivial example besides rational models.