  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}.
