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

الأصل · EN

Using Polyakov's functional integral approach with the Liouville action functional defined in ZT2 and 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 <∏ᵢ₌₁ⁿT(zᵢ)∏ₖ₌₁ˡT(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 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 =λₕᶜ/² over the moduli space Mg, where c is the central charge and λₕ is the Hodge line bundle, and provide Friedan-Shenker FS complex geometry approach to CFT with the first non-trivial example besides rational models.