Liouville Theory: Ward Identities for Generating Functional and Modular Geometry

TL;DR

This paper advances the understanding of quantum Liouville theory by deriving Ward identities, confirming Virasoro symmetry, and linking conformal correlators to Kähler geometry of moduli spaces.

Contribution

It provides a perturbation expansion for the generating functional, proves Virasoro symmetry, and connects Ward identities to Kähler geometry of Riemann surface moduli spaces.

Findings

Confirmed the central charge value in Liouville theory.

Derived Ward identities linking correlation functions to moduli space geometry.

Established the role of Ward identities in modular geometry of Riemann surfaces.

Abstract

We continue the study of quantum Liouville theory through Polyakov's functional integral \cite{Pol1,Pol2}, started in \cite{T1}. We derive the perturbation expansion for Schwinger's generating functional for connected multi-point correlation functions involving stress-energy tensor, give the ``dynamical'' proof of the Virasoro symmetry of the theory and compute the value of the central charge, confirming previous calculation in \cite{T1}. We show that conformal Ward identities for these correlation functions contain such basic facts from K\"{a}hler geometry of moduli spaces of Riemann surfaces, as relation between accessory parameters for the Fuchsian uniformization, Liouville action and Eichler integrals, K\"{a}hler potential for the Weil-Petersson metric, and local index theorem. These results affirm the fundamental role, that universal Ward identities for the generating functional play in Friedan-Shenker modular geometry \cite{FS}.