Quantum Liouville Theory in the Background Field Formalism I. Compact Riemann Surfaces - Takhtajan, Leon A.; Teo, Lee-Peng (SPRINGER, 2006)
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 (Pi(=1n)(i) T(zi) Pi(l)(k=1) (T) over bar((w) over bar (k)), 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 E-C = lambda H-c/2 over the moduli space M-g where c is the central charge and lambda (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.