Asymptotic dynamics of three-dimensional gravity

TL;DR

The notes trace how $(2+1)$-dimensional gravity with negative cosmological constant, via a Chern-Simons formulation and carefully chosen boundary conditions, reduces to Liouville theory on the boundary. The two-step reduction—first to a non-chiral $SL(2,\mathbb{R})$ WZW model and then to Liouville theory—establishes a classical AdS$_3$/CFT$_2$ link, with the Brown–Henneaux central charge arising from boundary symmetries. Liouville theory provides an effective boundary CFT description that accounts for BTZ entropy through the Cardy formula in the semiclassical regime, though subtle issues remain about its spectrum and exact equivalence beyond semiclassical limits. The framework highlights how boundary dynamics encode bulk gravitational physics and points to rich directions in holography, boundary conditions, and extensions to non-AdS or higher-spin contexts. All mathematical expressions are presented with explicit $...$ formatting to reflect the precise holographic relations between bulk and boundary theories.

Abstract

These are the lectures notes of the course given at the Eleventh Modave Summer School in Mathematical Physics, 2015, aimed at PhD candidates and junior researchers in theoretical physics. We review in details the result of Coussaert-Henneaux-van Driel showing that the asymptotic dynamics of $(2+1)$- dimensional gravity with negative cosmological constant is described at the classical level by Liouville theory. Boundary conditions implement the asymptotic reduction in two steps: the first set reduces the $SL(2,\mathbb R)\times SL(2,\mathbb R)$ Chern-Simons action, equivalent to the Einstein action, to a non-chiral $SL(2,\mathbb R)$ Wess-Zumino-Witten model, while the second set imposes constraints on the WZW currents that reduce further the action to Liouville theory. We discuss the issues of considering the latter as an effective description of the dual conformal field theory describing AdS$_3$ gravity beyond the semi-classical regime.