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.
Highlights
Kinematical sector, namely to transformations that do not move the corner [2,3,4,5,6,7,8,9,10]
We develop the covariant phase space formalism allowing for non-vanishing flux, anomalies, and field dependence in the vector field generators
We 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
Summary
The natural arena to develop the covariant phase space formalism is the jet bundle, where fields and their derivatives can be viewed as section of a fiber bundle over the base manifold provided by the spacetime M. On the jet bundle the horizontal derivative provides a notion of spacetime differential, which we denote by d, and the vertical derivative a notion of field space differential, which we denote by δ. The crucial element is a Lagrangian top form L, which defines the physics of the fields and the symmetries of spacetime, whose deep relation is the subject of our investigation
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