Holography and the Polyakov action

TL;DR

The paper establishes a precise holographic link between the generating functional for boundary stress-tensor correlators in a $d=2$ CFT and the regularized on-shell action of asymptotically AdS$_3$ gravity, using a Chern–Simons, first-order formulation. By analyzing boundary moduli through conformal and light-cone gauges, it shows that the boundary dependence of the holographic action matches the nonlocal Polyakov action, including the Weyl anomaly, and provides explicit local realizations in Liouville-like and chiral forms. Brown–Henneaux diffeomorphisms are shown to act consistently on the bulk currents and boundary stress tensor, with Schwarzian shifts capturing the conformal anomaly and the position on a Brown–Henneaux orbit. The work clarifies how holographic renormalization encodes the Liouville/Polyakov structure of the boundary theory and discusses global data such as holonomies, suggesting a path to incorporate topology and holonomy into the boundary effective action. Overall, it strengthens the AdS$_3$/CFT$_2$ dictionary by deriving the Polyakov generating functional directly from bulk CS gravity and elucidating the role of boundary moduli and bulk symmetries.

Abstract

In two dimensional conformal field theory the generating functional for correlators of the stress-energy tensor is given by the non-local 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/CFT correspondence. We also verify the consistency of this result with regard to the nontrivial transformation properties of bulk solutions under Brown-Henneaux diffeomorphisms.