Kinematical Gravitational Charge Algebra

TL;DR

This work reformulates the kinematics of first-order general relativity with boundaries by extending the phase space with edge modes, so that SU(2) gauge and spatial diffeomorphism constraints become locally conserved boundary charges. Through a duality map and carefully added boundary terms, the authors show these charges generate the correct gauge transformations and assemble into a closed ISU(2) (Poincaré) algebra, providing classical support for Poincaré charge networks as a discretization framework. The results illuminate how diffeomorphisms can be interpreted as field-dependent translations and clarify the role of boundary curvature in the charge algebra. The approach opens avenues for incorporating magnetic edge modes and for connecting boundary symmetries to holographic and discrete gravity formalisms.

Abstract

When formulated in terms of connection and coframes, and in the time gauge, the phase space of general relativity consists of a pair of conjugate fields: the flux 2-form and the Ashtekar connection. On this phase-space, one has to impose the Gauss constraints, the vector, and scalar Hamiltonian constraints. These are respectively generating local SU(2) gauge transformations, spatial diffeomorphisms, and time diffeomorphisms. We write the Gauss and space diffeomorphism constraints as conservation laws for a set of boundary charges, representing spin and momenta, respectively. We prove that these kinematical charges generate a local Poincaré ISU(2) symmetry algebra. This gives strong support to the recent proposal of Poincaré charge networks as a new realm for discretized general relativity [Classical Quantum Gravity 36, 195014 (2019)].