Aspects of (2+1) dimensional gravity: AdS3 asymptotic dynamics in the framework of Fefferman-Graham-Lee theorems

TL;DR

The paper reframes 2+1 dimensional gravity with negative cosmological constant as a pair of Chern–Simons theories and shows that, under Fefferman–Graham–Lee boundary conditions, the boundary degrees of freedom organize into a Liouville mode on a curved 2D boundary. By solving bulk constraints and using a Gauss decomposition, the authors derive a Liouville action on the boundary determined by the boundary metric, providing a finite, purely boundary description of AdS3 dynamics. This extends the CHD result from flat to curved boundary geometries and clarifies how the subleading metric data encode boundary conformal dynamics. The work highlights both the classical emergence of Liouville dynamics and caveats regarding potential quantum determinants in the variable changes used to reach the boundary theory.

Abstract

Using the Chern-Simon formulation of (2+1) gravity, we derive, for the general asymptotic metrics given by the Fefferman-Graham-Lee theorems, the emergence of the Liouville mode associated to the boundary degrees of freedom of (2+1) dimensional anti de Sitter geometries.