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

Abstract

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 of the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus g > 1.1

Liouville Action and Weil-Petersson Metric on Deformation Spaces, Global Kleinian Reciprocity and Holography

“…In this paper we extend McMullen's results along the lines of [ZT87a,ZT87b] by using homological algebra machinery developed by E. Aldrovandi and the first author in [AT97]. We explicitly construct a smooth function on quasi-Fuchsian deformation space and prove that it is an antiderivative of the 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections.…”

“…We also prove that this function is a Kähler potential of the Weil-Petersson metric on the quasi-Fuchsian deformation space. As it will be explained below, construction of the Liouville action functional is not a trivial issue and it requires homological algebra methods developed in [AT97]. Furthermore, we show that the Liouville action functional satisfies holography principle in string theory (also called AdS/CFT correspondence).…”

On holomorphic factorization in asymptotically AdS 3D gravity

“…Instead, we have to assume that the WZW actions I WZW [g], I WZW [ḡ] can be separately defined. It must be possible to define them using a procedure similar to that used in [39] to define a higher genus analog of the Polyakov action. As is shown in [39], one can define it as an integral of the same integrand as in (5.8) over a fundamental region on H plus a certain set of boundary terms, which make the action independent of a choice of the fundamental region, and make the variational principle well-defined.…”

On uniformization of Burnside’s curve y2=x5−x

“…The group of the function x(τ ) was described earlier by Klein & Fricke [17] and used by Rankin and Burnside himself [3]. At present, this is as far as we known about uniformization of the curve (1).…”

“…By this we mean: 1) determining ode's (2), their solutions, and correct accessory parameters; 2) effective series expansions and numerical computations; 3) inversion problems in fundamental polygons, i. e. search for solutions τ to equations of the type x(τ ) = A; 4) conformal representations; 5) Abelian integrals as functions of the global parameter τ ; 6) addition theorems for these Abelian integrals and relation with the Jacobi inversion problem. In the following sections we shall fill up some of these gaps in the example of the curve (1). The availability of all of these attributes in the case of genus g = 1 provides the great efficiency of elliptic functions and their numerous applications, which cannot be said of the cases g > 1 for reasons of the insufficiently advanced analytic tools.…”