Extended corner symmetry, charge bracket and Einstein's equations

TL;DR

This paper extends the covariant phase space formalism to include fluxes and anomalies, constructs a generalized charge bracket, and links corner symmetry algebra to Einstein's equations, revealing a holographic encoding of bulk dynamics.

Contribution

It introduces a generalized charge bracket that accounts for anomalies and fluxes, and demonstrates its role in representing extended corner symmetry algebra and encoding Einstein's equations.

Findings

The charge bracket generalizes previous formulations to include anomalies.

The extended corner symmetry algebra is represented by the charge bracket.

Einstein's equations are encoded in the corner symmetry structure.

Abstract

We develop the covariant phase space formalism allowing for non-vanishing flux, anomalies and field dependence in the vector field generators. We construct a charge bracket that generalizes the one introduced by Barnich and Troessaert and includes contributions from the Lagrangian and its anomaly. This bracket is uniquely determined by the choice of Lagrangian representative of the theory. We then extend the notion of corner symmetry algebra to include the surface translation symmetries and prove that the charge bracket provides a canonical representation of the extended corner symmetry algebra. This representation property is shown to be equivalent to the projection of the gravitational equations of motion on the corner, providing us with an encoding of the bulk dynamics in a locally holographic manner.