Symplectic and Killing Symmetries of AdS$_3$ Gravity: Holographic vs Boundary Gravitons

TL;DR

This paper explores the symplectic and Killing symmetries in AdS$_3$ gravity, revealing how boundary and bulk symmetries relate, and extends the Virasoro algebra with additional $U(1)$ generators, especially near the horizon.

Contribution

It demonstrates that the set of AdS$_3$ solutions forms a phase space with symplectic symmetries, extending the Virasoro algebra with $U(1)$ charges and analyzing near-horizon limits.

Findings

Symplectic form vanishes on-shell, promoting solutions to a phase space.

Virasoro algebra extends with two commuting $U(1)$ generators.

Near-horizon limit preserves chiral Virasoro symmetries.

Abstract

The set of solutions to the AdS$_3$ Einstein gravity with Brown-Henneaux boundary conditions is known to be a family of metrics labeled by two arbitrary periodic functions, respectively left and right-moving. It turns out that there exists an appropriate presymplectic form which vanishes on-shell. This promotes this set of metrics to a phase space in which the Brown-Henneaux asymptotic symmetries become symplectic symmetries in the bulk of spacetime. Moreover, any element in the phase space admits two global Killing vectors. We show that the conserved charges associated with these Killing vectors commute with the Virasoro symplectic symmetry algebra, extending the Virasoro symmetry algebra with two $U(1)$ generators. We discuss that any element in the phase space falls into the coadjoint orbits of the Virasoro algebras and that each orbit is labeled by the $U(1)$ Killing charges. Upon setting the right-moving function to zero and restricting the choice of orbits, one can take a near-horizon decoupling limit which preserves a chiral half of the symplectic symmetries. Here we show two distinct but equivalent ways in which the chiral Virasoro symplectic symmetries in the near-horizon geometry can be obtained as a limit of the bulk symplectic symmetries.