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 〈n i=1 T (zi ) ∏l k=1 T̄ (w̄k)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 oneand 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 Ec = λ H over the moduli space Mg , where c is the central charge and λH 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.