Currents and Superpotentials in classical gauge theories: II. Global aspects and the example of Affine gravity

TL;DR

This work develops a covariant, boundary-focused framework to define conserved charges for gauge symmetries in classical field theories, with a detailed application to gravity in the affine $gl(D,\mathbb{R})$ formalism. By expressing the Noether current as $J_\xi = d U_\xi + W_\xi$ and employing a covariant variation formula for $\delta U_\xi$, charges are computed from boundary data without relying on bulk details, yielding holographic charges. For gravity, Dirichlet boundary conditions on the metric produce the Katz–Bičák–Lynden-Bell (KBL) superpotential, while Neumann conditions relate to Komar-like forms; the formalism unifies Palatini and vielbein approaches and clarifies the role of background fields in defining charges. The paper further discusses the KBL superpotential at null infinity, showing how Bondi-type quantities can be recovered with an appropriate background, but noting non-integrability issues highlighted by Wald–Zoupas, which motivates a boundary-condition–driven approach to quasi-local and horizon charges.

Abstract

The conserved charges associated to gauge symmetries are defined at a boundary component of space-time because the corresponding Noether current can be rewritten on-shell as the divergence of a superpotential. However, the latter is afflicted by ambiguities. Regge and Teitelboim found a procedure to lift the arbitrariness in the Hamiltonian framework. An alternative covariant formula was proposed by one of us for an arbitrary variation of the superpotential, it depends only on the equations of motion and on the gauge symmetry under consideration. Here we emphasize that in order to compute the charges, it is enough to stay at a boundary of spacetime, without requiring any hypothesis about the bulk or about other boundary components, so one may speak of holographic charges. It is well known that the asymptotic symmetries that lead to conserved charges are really defined at infinity, but the choice of boundary conditions and surface terms in the action and in the charges is usually determined through integration by parts whereas each component of the boundary should be considered separately. We treat the example of gravity (for any space-time dimension, with or without cosmological constant), formulated as an Affine theory which is a natural generalization of the Palatini and Cartan-Weyl (vielbein) first order formulations. We then show that the superpotential associated to a Dirichlet boundary condition on the metric (the one needed to treat asymptotically flat or AdS spacetimes) is the one proposed by Katz, Biuc{á}k and Lynden-Bell and not that of Komar. We finally discuss the KBL superpotential at null infinity.