Virasoro Orbits, AdS_3 Quantum Gravity and Entropy

TL;DR

This paper explores the structure of AdS_3 gravity through Virasoro coadjoint orbits, revealing the need for extra degrees of freedom to correctly account for black hole entropy.

Contribution

It identifies the canonical structure of AdS_3 gravity in terms of Virasoro orbits and highlights the necessity of additional degrees of freedom for entropy calculation.

Findings

Virasoro orbits describe BTZ black holes and point particles.

Quantization of orbits yields a Hilbert space with zero conformal dimension.

Additional degrees of freedom are needed to restore modular invariance and match Bekenstein-Hawking entropy.

Abstract

We analyse the canonical structure of AdS_3 gravity in terms of the coadjoint orbits of the Virasoro group. There is one subset of orbits, associated to BTZ black hole solutions, that can be described by a pair of chiral free fields with a background charge. There is also a second subset of orbits, associated to point-particle solutions, that are described by two pairs of chiral free fields obeying a constraint. All these orbits admit K\"ahler quantization and generate a Hilbert space which, despite of having $\Delta_0(\bar{\Delta}_0)=0$, does not provide the right degeneracy to account for the Bekenstein-Hawking entropy due to the breakdown of modular invariance. Therefore, additional degrees of freedom, reestablishing modular invariance, are necessarily required to properly account for the black hole entropy.