Three-dimensional quantum geometry and black holes

TL;DR

The paper argues that in 3D gravity with a negative cosmological constant, all local dynamics vanish and physics resides in boundary degrees of freedom. By formulating gravity as a Chern-Simons theory with AdS boundary conditions, it identifies two main boundary structures—a Kac-Moody affine sector and a Virasoro sector—generated by A and \bar{A} fields whose boundary values encode the bulk geometry. The metric emerges as an operator-valued map from Virasoro and Kac-Moody data, enabling a quantum spacetime picture where a state-to-solution correspondence exists and the Bekenstein-Hawking entropy can be recovered through Cardy counting in a boundary CFT with central charge $c=3l/2G$. The work also connects the boundary theories to WZW/Liouville actions and discusses diffeomorphism vs gauge, unitarity, and the BTZ black hole as a natural realization of these boundary degrees of freedom. Overall, it provides a concrete framework for a quantum geometry of BTZ black holes rooted in the Brown-Henneaux symmetry and boundary conformal dynamics.

Abstract

We review some aspects of three-dimensional quantum gravity with emphasis in the `CFT -> Geometry' map that follows from the Brown-Henneaux conformal algebra. The general solution to the classical equations of motion with anti-de Sitter boundary conditions is displayed. This solution is parametrized by two functions which become Virasoro operators after quantisation. A map from the space of states to the space of classical solutions is exhibited. Some recent proposals to understand the Bekenstein-Hawking entropy are reviewed in this context. The origin of the boundary degrees of freedom arising in 2+1 gravity is analysed in detail using a Hamiltonian Chern-Simons formalism.