Charge algebra in Al(A)dSn spacetimes

Abstract

The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

3d gravity in Bondi-Weyl gauge: charges, corners, and integrability

“…In this sense, we are therefore forced to interpret b and c as true ambiguities, which we are fixing with physical requirements of finiteness and integrability, but whose covariant geometrical origin remains partly elusive. We note however that in Fefferman-Graham gauge the Compère-Marolf prescription correctly relates the boundary Lagrangians and corner terms used for renormalization [8,40,44]. It seems therefore that the additional ambiguities encountered in the present work are due to the Bondi gauge.…”

“…Consistently, the subalgebra spanned by f, g, h is precisely the one found in Bondi gauge [21]. Moreover, the subalgebra spanned by f, g, k is the algebra of residual symmetries found in Fefferman-Graham gauge [22,40,44]. This therefore begs the question of the more precise relationship between the Fefferman-Graham and Bondi gauge at the level of symmetries and later on at the level of the charges, and also raises the question of whether the Bondi-Weyl gauge has a Fefferman-Graham counterpart.…”

“…In [21] the authors have used the Bondi gauge in three dimensions with a free boundary metric, and found that the Weyl charges were vanishing. This result is surprising because, as mentioned above, non-trivial Weyl charges have been found in [22,44] using the Fefferman-Graham gauge, and also in [23] when studying null boundaries at finite distance.…”

“…They have found vanishing Weyl charges in the Starobinsky/ Fefferman-Graham gauge. In [44] the authors have extended the analysis of [40] to arbitrary spacetime dimensions, and found that the Weyl charges are non-vanishing in the odd-dimensional case. In order to obtain non-vanishing Weyl charges in the four-dimensional case, [45][46][47] have proposed a relaxed condition on the metric of the celestial 2-sphere, and studied the asymptotic symmetries in this new gauge (without however computing the associated charges).…”

The Weyl BMS group and Einstein’s equations

“…The Noetherian split is also instrumental for the derivation, based on first principles [44], of a new canonical bracket of the symmetry charges that generalizes the one introduced by Barnich and Troessaert in [20,21,48]. The new bracket resolves the ambiguity related to the handling of the non-integrable contributions and the presence of 2-cocycles [21,47,[49][50][51][52][53][54][55]. In particular, we show following [44,47] that the bracket provides a faithful and centerless representation of the algebra of vector fields generating the symmetry transformations.…”