Comparison between various notions of conserved charges in asymptotically AdS-spacetimes

TL;DR

The paper develops a covariant phase space construction for conserved charges in asymptotically AdS spacetimes using the Wald–Zoupas framework, deriving the Hamiltonian generators H_ξ and showing their equivalence to Ashtekar–Magnon/Das, Henneaux–Teitelboim, and spinor charges. It shows that the counterterm subtraction charges differ only by a boundary-constant offset, which is independent of the bulk solution, and remains compatible with the covariant phase space structure. The analysis extends to matter fields, including BF-bound scalars, where H_ξ retains its form and conservation, with matter contributions canceling in critical variations. A perturbative linear analysis around AdS justifies the chosen boundary conditions and reveals dimension-dependent freedom in asymptotics, guiding possible generalizations. Overall, the work unifies multiple charge definitions in AdS and clarifies the role of boundary data and matter in holographic contexts.

Abstract

We derive hamiltionian generators of asymptotic symmetries for general relativity with asymptotic AdS boundary conditions using the ``covariant phase space'' method of Wald et al. We then compare our results with other definitions that have been proposed in the literature. We find that our definition agrees with that proposed by Ashtekar et al, with the spinor definition, and with the background dependent definition of Henneaux and Teitelboim. Our definition disagrees with the one obtained from the ``counterterm subtraction method,'' but the difference is found to consist only of a ``constant offset'' that is determined entirely in terms of the boundary metric. We finally discuss and justify our boundary conditions by a linear perturbation analysis, and we comment on generalizations of our boundary conditions, as well as inclusion of matter fields.