Three-dimensional quantum geometry and black holes

TL;DR

This paper reviews three-dimensional quantum gravity, focusing on the CFT-geometry correspondence, classical solutions, boundary degrees of freedom, and approaches to understanding black hole entropy.

Contribution

It provides a detailed analysis of the classical solutions, the CFT-geometry map, and the boundary degrees of freedom in 3D quantum gravity, including recent entropy proposals.

Findings

Classical solutions parametrized by Virasoro operators

Map from states to classical solutions established

Discussion of boundary degrees of freedom and entropy proposals

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.