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

TL;DR

This work extends the holographic charge construction to the most general asymptotically locally (A)dS spacetimes in Starobinsky/Fefferman–Graham gauge, without imposing strong boundary conditions. By holographically renormalizing the symplectic structure, it yields finite, generally non-integrable and non-conserved boundary charges, with Weyl charges present only in odd dimensions due to Weyl anomalies. The charge algebra is analyzed via the Barnich–Troessaert bracket, revealing a field-dependent 2-cocycle that vanishes in odd dimensions and reduces to the Brown–Henneaux central charge in AdS$_3$ Dirichlet cases; leaky boundary conditions give rise to a Lambda–BMS$_{d+1}$ algebroid, contracting to BMS in the flat limit. These results underscore the role of boundary flux and fluctuating boundary data as an intrinsic feature of holography with nontrivial boundary dynamics, and they connect AdS/CFT-type holography to flat-space asymptotics through a controlled limiting procedure.

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 $Λ$-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in $n$ dimensions.

Figures (1)

• Figure 1: Conservative vs. leaky boundary conditions in Al(A)dS spacetimes.