Virasoro Orbits, AdS_3 Quantum Gravity and Entropy

TL;DR

The paper analyzes AdS$_3$ gravity through the lens of Virasoro coadjoint orbits to understand the microstructure behind BTZ black hole entropy. It identifies two orbit sectors: the BTZ sector described by two chiral free fields with a background charge, and a second sector with two chiral pairs subject to a constraint; both sectors admit Kähler quantization but only the first yields a modular-invariant spectrum with $c = 1 + 3Q^2$ and weights consistent with Cardy counting. The second sector, however, fails modular invariance, so Cardy’s formula does not apply and the full BH entropy cannot be reproduced without introducing additional degrees of freedom to ensure modular invariance, implying the need for a UV completion (e.g., string theory). The work highlights limits of pure 3D gravity with Brown-Henneaux boundary conditions and points to the necessity of extended microscopic structure to account for black hole 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 Kähler quantization and generate a Hilbert space which, despite of having $Δ_0(\barΔ_0)=0$, 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.