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

TL;DR

This paper formulates quantum Liouville theory on compact Riemann surfaces of genus greater than one using a background field approach, establishing Feynman rules, regularization schemes, and conformal Ward identities with geometric interpretations.

Contribution

It extends the background field formalism to quantum Liouville theory on higher genus surfaces, providing rigorous proofs of conformal Ward identities and geometric insights.

Findings

Feynman rules for correlation functions are derived.

Conformal Ward identities are proven in all orders of perturbation.

The geometric interpretation relates to the moduli space and line bundles.

Abstract

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.