Liouville Theory: Ward Identities for Generating Functional and Modular Geometry

摘要

We continue the study of quantum Liouville theory through Polyakov'sfunctional integral \cite{Pol1,Pol2}, started in \cite{T1}. We derive theperturbation expansion for Schwinger's generating functional for connectedmulti-point correlation functions involving stress-energy tensor, give the``dynamical'' proof of the Virasoro symmetry of the theory and compute thevalue of the central charge, confirming previous calculation in \cite{T1}. Weshow that conformal Ward identities for these correlation functions containsuch 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 theWeil-Petersson metric, and local index theorem. These results affirm thefundamental role, that universal Ward identities for the generating functionalplay in Friedan-Shenker modular geometry \cite{FS}.