Boundary charges in gauge theories: using Stokes theorem in the bulk

TL;DR

This paper develops a method to construct boundary charges in gauge theories using Stokes theorem, linking linearized and full theories, and applies it to derive the first law of black hole mechanics in AdS space.

Contribution

It introduces a way to build closed n-2 forms in the full interacting theory from solutions with reducibility parameters, enabling boundary charge calculations without bulk contributions.

Findings

Constructed closed n-2 forms in the full theory from solutions with reducibility parameters.

Demonstrated the reduction of these forms to linearized conserved forms asymptotically.

Derived the first law of black hole mechanics in asymptotically AdS space-times.

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.