E. Aldrovandi et La. Takhtajan, GENERATING FUNCTIONAL IN CFT AND EFFECTIVE ACTION FOR 2-DIMENSIONAL QUANTUM-GRAVITY ON HIGHER-GENUS RIEMANN SURFACES, Communications in Mathematical Physics, 188(1), 1997, pp. 29-67
We formulate and solve the analog of the universal Conformal Ward Iden
tity for the stress-energy tensor on a compact Riemann surface of genu
s g > 1, and present a rigorous invariant formulation of the chiral se
ctor in the induced two-dimensional gravity on higher genus Riemann su
rfaces. Our construction of the action functional uses various double
complexes naturally associated with a Riemann surface, with computatio
ns that are quite similar to descent calculations in BRST cohomology t
heory. We also provide an interpretation of the action functional in t
erms of the geometry of different fiber spaces over the Teichmuller sp
ace of compact Riemann surfaces of genus g > 1.