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

TL;DR

This work shows that AdS$_3$ gravity with Brown–Henneaux boundaries possesses a bulk phase space of Ba\~nados geometries on which invariant presymplectic structure vanishes on-shell, promoting boundary Virasoro symmetries to bulk symplectic symmetries. The phase space carries two global $U(1)$ Killing charges $J_\pm$ that commute with Virasoro charges and label Virasoro coadjoint orbits, while the dynamics decomposes into independent left/right sectors described by Virasoro coadjoint orbits. In extremal sectors, a decoupling (near-horizon) limit yields a chiral Virasoro symmetry that maps to the bulk symmetries, offering two equivalent routes (Fefferman–Graham and Gaussian null coordinates) to the same near-horizon structure. The results unify holographic (Virasoro) and boundary (Killing) perspectives, clarify thermodynamics via orbit labels, and point to robust generalizations to other boundary conditions and Chern–Simons formulations. Overall, the paper clarifies how bulk symplectic charges and Killing charges organize the AdS$_3$ phase space into Virasoro coadjoint orbits, with clear implications for holography and extremal black hole decoupling limits.

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.