Geometric Microstates for the Three Dimensional Black Hole?

TL;DR

The paper proposes a geometric program to account for BTZ black hole entropy by quantizing topologically nontrivial behind-horizon geometries in 3D gravity, focusing on chiral gravity and the moduli space of bordered Riemann surfaces. It shows that, at fixed genus, microstate counts derived from intersection theory of moduli spaces fall short of the Bekenstein-Hawking entropy, while summing over all topologies leads to divergences. Two asymptotic conjectures for intersection numbers are presented to estimate state counts, and a speculative nonperturbative resummation suggests possible area-proportional entropy in AdS units, highlighting the need for a nonperturbative understanding to fully recover the horizon area law. The work connects geometric quantization, Chern-Simons theory, and Teichmüller theory to the microstate problem in a concrete 3D setting, offering a framework for further exploration of entropy from purely geometric degrees of freedom.

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.

Figures (5)

• Figure 1: AdS is the solid cylinder whose constant time slices are given by the disk $\Sigma=D_2$ in (1a). The metric (\ref{['metric']}) covers the coordinate patch of AdS inside the green diamond depicted in (1b).
• Figure 2: The constant time slices of the BTZ black hole are the annulus, whose constant time slice is shown in (2a). The Penrose diagram of BTZ is shown in (2b). The $t=0$ slice, shown in blue, is the annulus. The metric (\ref{['metric']}) covers the coordinate patch inside the green diamond. The event horizon is shown in red, and intersects the annulus at the minimum length geodesic of length $L$. Dotted lines indicate past and future singularities.
• Figure 3: A microstate geometry whose constant time slice is the surface $\Sigma_g$ with one hole (3a). The Penrose diagram is sketched in (3b). The blue line is the surface $\Sigma_g$ at the $t=0$ slice. The metric (\ref{['metric']}) covers the coordinate patch inside the green diamond. The event horizon is shown in red, which intersects $\Sigma_g$ at the geodesic of length $L$. The curved line on the left side of 3b) indicates that the interior geometry caps off smoothly. This is only a sketch of the Penrose diagram; since the geometry breaks the $U(1)$ rotation symmetry of AdS$_3$, one cannot draw a two dimensional Penrose diagram inside the horizon.
• Figure 4: The intersection numbers $I_{20,d}$ at genus 20 exhibit exponential scaling with $d$ for $d \gtrsim 10$. The linear fit of $\log(I_{20,d})$ (indicated by the blue line) gives coefficients $B_{20}\approx 600$ and $A_{20}\approx 3 \times10^{118}$ in equation (\ref{['conjecture1']}). Numerical data supplied by P. Zograf.
• Figure 5: The linear behaviour of $\log(I_{g,3g/2}/ (2g)!)$ (indicated by the blue line) with $g$ indicates that the intersection numbers $I_{3g/2,3g/2}$ scale approximately as $(2g)!$, in accordance with (\ref{['conjecture2']}). Numerical data supplied by P. Zograf.