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

TL;DR

This work derives the BTZ black hole entropy from two complementary angles within the CS formulation of 2+1D gravity. In the microcanonical ensemble, a boundary Kac-Moody algebra is reduced via a twisted Sugawara construction to a Virasoro algebra whose central charge reproduces the Brown–Henneaux result, yielding the correct density of states at fixed mass and angular momentum. In the grand-canonical ensemble, the Euclidean CS action with carefully chosen boundary terms and a horizon source term produces the correct partition function, with a nontrivial spin-structure twist in the WZW description essential for matching the entropy. Together, the results link horizon and asymptotic boundary degrees of freedom, show a Virasoro structure at any fixed radius, and suggest a path toward holographic-like entropy counting via horizon deformations, with potential implications for higher-dimensional black holes and Liouville reductions.

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.

Figures (1)

• Figure 1: The Euclidean black hole topology.