Asymptotic dynamics of three-dimensional gravity

TL;DR

This paper reviews how three-dimensional gravity with negative cosmological constant asymptotically reduces to Liouville theory, connecting classical gravity to a conformal field theory at the boundary.

Contribution

It details the boundary conditions and reduction steps from Chern-Simons theory to Liouville theory, clarifying the classical asymptotic dynamics of AdS3 gravity.

Findings

Reduction from Chern-Simons to WZW model established

Further constraints lead to Liouville theory

Discussion on the dual CFT beyond semi-classical approximation

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.