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

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.

Note on the first law withp-form potentials

“…Here, we show that the integration of the surface form k ∂t,0 [δφ, φ] along a path γ in solution space [10,32],…”

“…For additional properties of these surface charges, as the representation theorem of the Lie algebra of reducibility parameters, the reader is referred to the original work [9,10].…”

“…The first aim of this paper is to improve the covariant analysis [7,8] by deriving an expression for the conserved charges taking better care of the form factors. Following the Lagrangian methods based on cohomological results [9,10], our expression for the surface charges will moreover get round the usual ambiguities of covariant phase methods. Several properties of these surface charges will be discussed.…”

Asymptotically anti–de Sitter spacetimes and scalar fields with a logarithmic branch

“…One should expect that a similar expression for Q(ξ) in Eq. (2.7) could also be found using covariant methods as in [10] The algebra of the charges (2.7) is identical to the standard one, namely AdS for D > 3 [6,7], and two copies of the Virasoro algebra with the same central extension for D = 3 [8]. This can be readily obtained following Ref [11], where it is shown that the bracket of two charges provides a realization of the asymptotic symmetry algebra with a possible central extension.…”

BMS charge algebra

“…Our starting point is the covariant approach to surface charges and their algebra developed in [10] (see also [11,12]). In particular, for pure Einstein gravity with or without a cosmological constant, it has been shown in [13] that for the linearized theory, described by h µν around a background g µν , the conserved surface charges are completely classified by the Killing vectors ξ µ of the metric g µν .…”