Ambiguity resolution for integrable gravitational charges

TL;DR

The paper analyzes how to define gravitational charges associated to boundary diffeomorphisms within an extended phase-space framework that includes embedding fields. It shows that ambiguity-free, integrable charges can be obtained by applying a covariant-phase-space approach to subregions, yielding corrected diffeomorphism charges and new potential obstructions to integrability. It clarifies how Ciambelli–Leigh–Pai’s (CLP) embedding-field construction relates to Donnelly–Freidel and Wald–Zoupas approaches, demonstrating that different phase-space transformations reproduce different charge algebras, all while maintaining a consistent Barnich–Troessaert bracket. The work highlights that the CLP charges represent a genuine extension with new boundary degrees of freedom, but that obstructions to integrability can arise depending on the flux terms, and discusses implications for subregion holography, entropy, and asymptotic charges. Overall, it provides a unified view of how embedding fields, ambiguity resolution, and boundary conditions shape the algebra and integrability of localized gravitational charges.

Abstract

Recently, Ciambelli, Leigh, and Pai (CLP) [arXiv:2111.13181] have shown that nonzero charges integrating Hamilton's equation can be defined for all diffeomorphisms acting near the boundary of a subregion in a gravitational theory. This is done by extending the phase space to include a set of embedding fields that parameterize the location of the boundary. Because their construction differs from previous works on extended phase spaces by a covariant phase space ambiguity, the question arises as to whether the resulting charges are unambiguously defined. Here, we demonstrate that ambiguity-free charges can be obtained by appealing to the variational principle for the subregion, following recent developments on dealing with boundaries in the covariant phase space. Resolving the ambiguity produces corrections to the diffeomorphism charges, and also generates additional obstructions to integrability of Hamilton's equation. We emphasize the fact that the CLP extended phase space produces nonzero diffeomorphism charges distinguishes it from previous constructions in which diffeomorphisms are pure gauge, since the embedding fields can always be eliminated from the latter by a choice of unitary gauge. Finally, we show that Wald-Zoupas charges, with their characteristic obstruction to integrability, are associated with a modified transformation in the extended phase space, clarifying the reason behind integrability of Hamilton's equation for standard diffeomorphisms.