LIOUVILLE THEORY - WARD IDENTITIES FOR GENERATING FUNCTIONAL AND MODULAR GEOMETRY

We continue the study of quantum Liouville theory through Polyakov's f unctional integra,1,2 started in Ref. 3. We derive the perturbation ex pansion for Schwinger's generating functional for connected multi-poin t correlation functions involving stress-energy tensor, give the ''dyn amical'' proof of the Virasoro symmetry of the theory and compute the value of the central charge, confirming previous calculation in Ref. 3 . We show that conformal Ward identities for these correlation functio ns contain such basic facts from Kahler geometry of moduli spaces of R iemann surfaces, as relation between accessory parameters for the Fuch aisan uniformization, Liouville action and Eichler intergrals, Kahler potential for the Weil-Petersson metric, and local index theorem. Thes e results affirm the fundamental role that universal Ward identities f or the generating functional play in Friedan-Shenker modular geometry. 4