Finite charges from the bulk action

Abstract

Constructing charges in the covariant phase space formalism often leads to formally divergent expressions, even when the fields satisfy physically acceptable fall-off conditions. These expressions can be rendered finite by corner ambiguities in the definition of the presymplectic potential, which in some cases may be motivated by arguments involving boundary Lagrangians. We show that the necessary corner terms are already present in the variation of the bulk action and can be extracted in a straightforward way. Once these corner terms are included in the presymplectic potential, charges derived from an associated codimension-2 form are automatically finite. We illustrate the procedure with examples in two and three dimensions, working in Bondi gauge and obtaining integrable charges. As a by-product, actions are derived for these theories that admit a well-defined variational principle when the fields satisfy boundary conditions on a timelike surface with corners. An interesting feature of our analysis is that the fields are not required to be fully on-shell.

Renormalization of spin-one asymptotic charges in AdSD

“…These divergences can however be renormalized with various techniques, see e.g. [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62], and in this paper we systematically explore two of them in the instructive and yet relatively handy example of Maxwell fields in Anti de Sitter (AdS) backgrounds of any dimensions. These investigations can be relevant for both AdS/CFT and flat holography, as well as to prepare for a corresponding study of de Sitter spacetime, although in the latter case the boundary analysis would require a different interpretation.…”

One-loop partition function of gravity with leaky boundary conditions

“…The breaking term on the right-hand side is due to the violation of the Noether identities. In appendix D, we provide a more explicit expression for the breaking term (see Equation (D.24)) and re-derive this result using the Iyer-Wald approach [74], following the steps presented in [42].…”

“…With these formulae, we are now ready to derive the breaking of the Gauss law in the Iyer-Wald formalism. We follow the steps of [42] by keeping track of the anomalous terms. We start by computing…”

“…In two and three spacetime dimensions, there are no propagating degrees of freedom in pure gravity, which makes the interpretation of leaky boundary conditions less straightforward (see, however, [38][39][40][41][42][43] where they are interpreted in terms of the boundary geometry). Nevertheless, one can implement leaky boundary conditions by simply coupling gravity with a matter field (see, e.g., [44,45] for an example in three-dimensional Einstein-Maxwell theory).…”

The partial Bondi gauge: Further enlarging the asymptotic structure of gravity

“…The charges (4.2) contain the r-divergent contribution (4.3). This divergent contribution can be renormalized with the corner term inherited from the divergent part (2.26) of the radial symplectic potential, or equivalently from θ u by virtue of (2.27) [20,[85][86][87]. Performing an innocent integration by parts on δ, this corner term can be chosen as…”