Geometric Microstates for the Three Dimensional Black Hole?

TL;DR

This paper investigates the microstates of three-dimensional black holes by quantizing geometries behind the horizon, revealing a slow growth of microstate entropy and a divergence when summing over topologies, with implications for black hole entropy.

Contribution

It introduces a geometric approach to counting black hole microstates via quantization of moduli spaces of Riemann surfaces, providing new insights into black hole entropy.

Findings

Microstates grow too slowly to match Bekenstein-Hawking entropy.

Summing over topologies results in a divergence.

Conjectures on resolving divergence to match horizon area.

Abstract

We study microstates of the three dimensional black hole obtained by quantizing topologically non-trivial geometries behind the event horizon. In chiral gravity these states are found by quantizing the moduli space of bordered Riemann surfaces. In the semi-classical limit these microstates can be counted using intersection theory on the moduli space of punctured Riemann surfaces. We make a conjecture (supported by numerics) for the asymptotic behaviour of the relevant intersection numbers. The result is that the geometric microstates with fixed topology have an entropy which grows too slowly to account for the semiclassical Bekenstein-Hawking entropy. The sum over topologies, however, leads to a divergence. We conclude with some speculations about how this might be resolved to give an entropy proportional to horizon area.