Classification of Boundary Gravitons in AdS$_3$ Gravity

TL;DR

This work reframes the classical phase space of AdS$_3$ gravity as a union of Virasoro coadjoint orbits, with boundary gravitons corresponding to nontrivial improper diffeomorphisms. By equipping each orbit with a Virasoro-invariant symplectic form and mapping gravitational charges to orbit functions, the authors derive energy bounds for BTZ and AdS$_3$ sectors and reveal a family of exotic boundary geometries. They show how AdS$_3$ and BTZ live in distinct orbits, and discuss the possibility of extending certain exotic geometries beyond horizons, guided by isotropy and Killing-vector invariants. The paper also surveys quantization avenues, highlighting geometric quantization and Verma-module perspectives, and outlines how these orbit structures inform the prospects for a quantum theory of AdS$_3$ gravity.

Abstract

We revisit the description of the space of asymptotically AdS3 solutions of pure gravity in three dimensions with a negative cosmological constant as a collection of coadjoint orbits of the Virasoro group. Each orbit corresponds to a set of metrics related by diffeomorphisms which do not approach the identity fast enough at the boundary. Orbits contain more than a single element and this fact manifests the global degrees of freedom of AdS3 gravity, being each element of an orbit what we call boundary graviton. We show how this setup allows to learn features about the classical phase space that otherwise would be quite difficult. Most important are the proof of energy bounds and the characterization of boundary gravitons unrelated to BTZs and AdS3. In addition, it makes manifest the underlying mathematical structure of the space of solutions close to infinity. Notably, because of the existence of a symplectic form in each orbit, being this related with the usual Dirac bracket of the asymptotic charges, this approach is a natural starting point for the quantization of different sectors of AdS3 gravity. We finally discuss previous attempts to quantize coadjoint orbits of the Virasoro group and how this is relevant for the formulation of AdS3 quantum gravity.