The Statistical Mechanics of the Three-Dimensional Euclidean Black Hole

TL;DR

The paper addresses the origin of black hole entropy in 3D Euclidean gravity by reformulating general relativity as a Chern-Simons theory with a boundary, where the horizon acts as a boundary giving rise to a boundary WZW model. By counting boundary states and performing an analytic continuation to Lorentzian signature, it shows that the leading, semiclassical entropy matches the Bekenstein-Hawking result, with the leading contribution governed by Virasoro zero-modes and a controllable one-loop correction. The approach identifies the microscopic degrees of freedom as ``would-be gauge'' modes that become dynamical at the horizon, offering a concrete microscopic mechanism for black hole entropy in this setting. While directly generalizing to 3+1 dimensions is not straightforward, the results support the broader idea that horizon boundary data encode the quantum gravitational microstates responsible for black hole thermodynamics.

Abstract

In its formulation as a Chern-Simons theory, three-dimensional general relativity induces a Wess-Zumino-Witten action on spatial boundaries. Treating the horizon of the three-dimensional Euclidean black hole as a boundary, I count the states of the resulting WZW model, and show that when analytically continued back to Lorentzian signature, they yield the correct Bekenstein-Hawking entropy. The relevant states can be understood as ``would-be gauge'' degrees of freedom that become dynamical at the horizon.