Boundary charges in gauge theories: using Stokes theorem in the bulk
Glenn Barnich
TL;DR
The paper addresses how to define boundary charges in gauge theories without bulk contributions by constructing closed $n-2$ forms in the full theory from a one-parameter family of solutions that admit reducibility parameters. It shows that these forms interpolate between a background and a target solution, reducing to the linear theory near the boundary and permitting Stokes theorem to relate charges to deep-bulk surface integrals. The approach is illustrated in Yang-Mills theory and general relativity with a cosmological constant, and applied to derive the first law of black hole mechanics for asymptotically AdS spacetimes. Overall, the work generalizes Wald–Iyer Noether-charge methods to non-flat backgrounds and offers a practical framework for computing boundary charges in diverse gauge theories.
Abstract
Boundary charges in gauge theories (like the ADM mass in general relativity) can be understood as integrals of linear conserved n-2 forms of the free theory obtained by linearization around the background. These forms are associated one-to-one to reducibility parameters of this background (like the time-like Killing vector of Minkowski space-time). In this paper, closed n-2 forms in the full interacting theory are constructed in terms of a one parameter family of solutions to the full equations of motion that admits a reducibility parameter. These forms thus allow one to apply Stokes theorem without bulk contributions and, provided appropriate fall-off conditions are satisfied, they reduce asymptotically near the boundary to the conserved n-2 forms of the linearized theory. As an application, the first law of black hole mechanics in asymptotically anti-de Sitter space-times is derived.