Embeddings and Integrable Charges for Extended Corner Symmetry

TL;DR

This paper develops a refined phase space framework for diffeomorphism-invariant theories, emphasizing the role of embeddings in boundary regions, leading to integrable corner charges that form a symmetry algebra without central extension.

Contribution

It introduces a new notion of field space extension incorporating embeddings, ensuring integrability of corner charges and their algebraic representation without central extension.

Findings

Corner charges are integrable but not necessarily conserved.

Extended corner symmetry is represented via Poisson brackets.

No central extension appears in the symmetry algebra.

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.