Holography and the Polyakov action

In two-dimensional conformal field theory the generating functional for correlators of the stress-energy tensor is given by the nonlocal Polyakov action associated with the background geometry. We study this functional holographically by calculating the regularized on-shell action of asymptotically AdS gravity in three dimensions, associated with a specified (but arbitrary) boundary metric. This procedure is simplified by making use of the Chern-Simons formulation, and a corresponding first-order expansion of the bulk dreibein, rather than the metric expansion of Fefferman and Graham. The dependence of the resulting functional on local moduli of the boundary metric agrees precisely with the Polyakov action, in accord with the AdS/conformal field theory correspondence. We also verify the consistency of this result with regard to the nontrivial transformation properties of bulk solutions under Brown-Henneaux diffeomorphisms