Holographic boundary actions in AdS$_3$/CFT$_2$ revisited

TL;DR

This paper reviews the holographic derivation of the stress tensor generating functional in AdS$_3$/CFT$_2$, compares it with flat Liouville theory, and discusses implications for black hole spectral statistics and random matrix theory.

Contribution

It clarifies the relationship between holographic boundary actions and flat Liouville theory, and offers a new interpretation of black hole spectral statistics in holography.

Findings

Holographic derivation of stress tensor functional in AdS$_3$/CFT$_2$ is reviewed.

Flat Liouville theory is shown to be unrelated to the holographic boundary action.

Black hole spectral statistics can be interpreted as averaging over boundary geometries.

Abstract

The generating functional of stress tensor correlation functions in two-dimensional conformal field theory is the nonlocal Polyakov action, or equivalently, the Liouville or Alekseev-Shatashvili action. I review its holographic derivation within the AdS$_3$/CFT$_2$ correspondence, both in metric and Chern-Simons formulations. I also provide a detailed comparison with the well-known Hamiltonian reduction of three-dimensional gravity to a flat Liouville theory, and conclude that the two results are unrelated. In particular, the flat Liouville action is still off-shell with respect to bulk equations of motion, and simply vanishes in case the latter are imposed. The present study also suggests an interesting re-interpretation of the computation of black hole spectral statistics recently performed by Cotler and Jensen as that of an explicit averaging of the partition function over the boundary source geometry, thereby providing potential justification for its agreement with the predictions of a random matrix ensemble.