BTZ Black Hole Entropy from a Chern-Simons Matrix Model

TL;DR

This paper introduces a Chern-Simons matrix model as a simplified quantum framework for studying BTZ black hole entropy, revealing exponential degeneracy growth and matching the classical entropy formula.

Contribution

It proposes a novel matrix model capturing key features of BTZ black holes and derives its spectrum and degeneracy, connecting quantum invariants to classical entropy.

Findings

Degeneracy grows exponentially with matrix size N

Spectrum matches BTZ black hole entropy formula

Quantum states behave as spin particles under rotations

Abstract

We examine a Chern-Simons matrix model which we propose as a toy model for studying the quantum nature of black holes in 2+1 gravity. Its dynamics is described by two $N\times N$ matrices, representing the two spatial coordinates. The model possesses an internal SU(N) gauge symmetry, as well as an external rotation symmetry. The latter corresponds to the rotational isometry of the BTZ solution, and does not decouple from SU(N) gauge transformations. The system contains an invariant which is quadratic in the spatial coordinates. We obtain its spectrum and degeneracy, and find that the degeneracy grows exponentially in the large $N$ limit. The usual BTZ black hole entropy formula is recovered upon identifying the quadratic invariant with the square of the black hole horizon radius. The quantum system behaves collectively as an integer (half-integer) spin particle for even (odd) $N$ under $2\pi$-rotations.