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