On 3d Bulk Geometry of Virasoro Coadjoint Orbits: Orbit invariant charges and Virasoro hair on locally AdS3 geometries

TL;DR

This paper provides a detailed analysis of Ba ilde{n}ados geometries in AdS3 gravity, exploring their structure, charges, and the distinction between geometric and non-local Virasoro hair, with implications for dual 2D CFTs.

Contribution

It extends previous work by analyzing orbit invariant charges and Virasoro hair in Ba ilde{n}ados geometries, clarifying their geometric and non-local features.

Findings

Orbit invariant charges are geometric quantities.

Virasoro hairs are characterized by non-local surface integrals.

Multi-BTZ geometries have thermodynamic quantities that are orbit invariants.

Abstract

Expanding upon [arXiv:1404.4472, 1511.06079], we provide further detailed analysis of Ba\~nados geometries, the most general solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions. We analyze in some detail the causal, horizon and boundary structure, and geodesic motion on these geometries, as well as the two class of symplectic charges one can associate with these geometries: charges associated with the exact symmetries and the Virasoro charges. We elaborate further the one-to-one relation between the coadjoint orbits of two copies of Virasoro group and Ba\~nados geometries. We discuss that the information about the Ba\~nados goemetries fall into two categories: "orbit invariant" information and "Virasoro hairs". The former are geometric quantities while the latter are specified by the non-local surface integrals. We elaborate on multi-BTZ geometries which have some number of disconnected pieces at the horizon bifurcation curve. We study multi-BTZ black hole thermodynamics and discuss that the thermodynamic quantities are orbit invariants. We also comment on the implications of our analysis for a 2d CFT dual which could possibly be dual to AdS3 Einstein gravity.