Charge Algebra in Al(A)dS$_n$ Spacetimes

TL;DR

This paper derives the gravitational charge algebra in asymptotically (A)dS spacetimes across dimensions, revealing boundary fluxes, anomalies, and connections to known symmetry groups like BMS and Brown-Henneaux.

Contribution

It provides a general framework for charge algebra in (A)dS spacetimes without restrictive boundary conditions, including fluxes, anomalies, and the flat limit contraction.

Findings

Charges are generally non-vanishing and non-conserved.

Weyl rescaling charges are non-zero only in odd dimensions due to anomalies.

In 3D AdS, the algebra reduces to Brown-Henneaux central extension.

Abstract

The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in $n$ dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent $2$-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the $2$-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the $\Lambda$-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in $n$ dimensions.