Extended corner symmetry, charge bracket and Einstein's equations

TL;DR

The paper develops a covariant phase space framework that accommodates flux and anomalies, and defines a Lagrangian-dependent charge bracket extending the Barnich–Troessaert construction to the extended corner symmetry, including normal translations. It proves that, on-shell, this bracket provides a faithful representation of the extended corner algebra and that enforcing this representation is equivalent to Einstein's equations, revealing a locally holographic encoding of bulk dynamics at corners. The authors generalize Noether charges, fluxes, and their splits, explore boundary shifts and anomalies, and apply the formalism to null infinity where the extended corner symmetry reduces to the BMSW group. This work thus bridges edge-mode corner symmetries with bulk dynamics, offering a path toward quantization through local holography and highlighting open questions about edge modes and embedding data.

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.

Figures (2)

• Figure 1: Different topologies of the hypersurface $\Sigma$.
• Figure 2: Foliation leaves $\Sigma_t$ with boundaries $S_t$.