Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation for the action functional in terms of the geometry of different fiber spaces over the Teichm\"{u}ller space of compact Riemann surfaces of genus $g>1$.