Quantum Liouville theory and BTZ black hole entropy

TL;DR

This paper provides a conformal field theory framework using Liouville theory to explicitly compute the entropy of (2+1)-dimensional BTZ black holes, matching the Bekenstein-Hawking entropy in the semi-classical limit.

Contribution

It introduces a novel CFT description of BTZ black hole entropy via boundary Liouville theory and analyzes reducible Verma modules and decoupling states to count microstates.

Findings

Decoupling states have positive-definite norms.

Counting these states reproduces Bekenstein-Hawking entropy.

Results hold when q is a root of unity of odd order.

Abstract

In this paper I give an explicit conformal field theory description of (2+1)-dimensional BTZ black hole entropy. In the boundary Liouville field theory I investigate the reducible Verma modules in the elliptic sector, which correspond to certain irreducible representations of the quantum algebra U_q(sl_2) \odot U_{\hat{q}}(sl_2). I show that there are states that decouple from these reducible Verma modules in a similar fashion to the decoupling of null states in minimal models. Because ofthe nonstandard form of the Ward identity for the two-point correlation functions in quantum Liouville field theory, these decoupling states have positive-definite norms. The explicit counting from these states gives the desired Bekenstein-Hawking entropy in the semi-classical limit when q is a root of unity of odd order.