Embeddings and Integrable Charges for Extended Corner Symmetry

TL;DR

This work advances the covariant phase space formalism by incorporating embedding maps for subregion boundaries, introducing a field-space extension that renders the entire extended corner symmetry’s Noether charges integrable. The authors construct an extended pre-symplectic structure that yields charges generating the extended corner algebra via Poisson brackets without central extension, while flux is encoded in boundary terms. This universal extension resolves prior issues where charges were non-integrable or vanishing under certain embeddings, and it broadens the applicability to scenarios like black hole horizons and asymptotic symmetries. The results offer a robust, geometry-driven framework for understanding how subregion gluing and corner degrees of freedom influence gauge charges in diffeomorphism-invariant theories.

Abstract

We revisit the problem of extending the phase space of diffeomorphism-invariant theories to account for embeddings associated with the boundary of sub-regions. We do so by emphasizing the importance of a careful treatment of embeddings in all aspects of the covariant phase space formalism. In so doing we introduce a new notion of the extension of field space associated with the embeddings which has the important feature that the Noether charges associated with all extended corner symmetries are in fact integrable, but not necessarily conserved. We give an intuitive understanding of this description. We then show that the charges give a representation of the extended corner symmetry via the Poisson bracket, without central extension.