Boundary dynamics and the statistical mechanics of the 2+1 dimensional black hole

TL;DR

This paper computes the density of states for the 2+1 dimensional BTZ black hole using quantum gravity and Chern-Simons theory, deriving the entropy and partition function through boundary conformal field theories.

Contribution

It introduces a novel approach relating gauge charge algebras to Virasoro algebra at all black hole radii, and explicitly constructs the horizon deformations and their algebra.

Findings

Bekenstein-Hawking entropy derived from algebraic relations.

Partition function matches expected results with horizon source term.

Explicit form of horizon deformations with Virasoro algebra.

Abstract

We calculate the density of states of the 2+1 dimensional BTZ black hole in the micro- and grand-canonical ensembles. Our starting point is the relation between 2+1 dimensional quantum gravity and quantised Chern-Simons theory. In the micro-canonical ensemble, we find the Bekenstein--Hawking entropy by relating a Kac-Moody algebra of global gauge charges to a Virasoro algebra with a classical central charge via a twisted Sugawara construction. This construction is valid at all values of the black hole radius. At infinity it gives the asymptotic isometries of the black hole, and at the horizon it gives an explicit form for a set of deformations of the horizon whose algebra is the same Virasoro algebra. In the grand-canonical ensemble we define the partition function by using a surface term at infinity that is compatible with fixing the temperature and angular velocity of the black hole. We then compute the partition function directly in a boundary Wess-Zumino-Witten theory, and find that we obtain the correct result only after we include a source term at the horizon that induces a non-trivial spin-structure on the WZW partition function.