Asymptotically Anti-de Sitter Space-times: Conserved Quantities

TL;DR

Problem: establish covariant definitions of conserved charges for asymptotically anti-de Sitter spacetimes in d>=4 and test their relation to ADS/CFT-inspired boundary data. Approach: develop a conformal completion framework, derive an electric Weyl boundary tensor E_{ab}, and define charges Q_ξ[C] that satisfy a flux balance; compare these to the counter-term charges from holographic renormalization. Findings: all Q_ξ vanish for pure AdS; in d=4 the two definitions agree, while in d>=5 they differ by a finite, nontrivial amount Δ_{ab}, with Δ_{ab} not generally trace-free; this reveals potential limits of current ADS/CFT dictionaries. Implications: nonperturbative gravity results constrain how bulk-boundary maps can be formulated, and point to the need for a covariant phase-space or finite-action approach to a fully consistent dictionary.

Abstract

Asymptotically anti-de Sitter space-times are considered in a general dimension $d\ge 4$. As one might expect, the boundary conditions at infinity ensure that the asymptotic symmetry group is the anti-de Sitter group (although there is an interesting subtlety if d=4). Asymptotic field equations imply that, associated with each generator $ξ$ of this group, there is a quantity $Q_ξ$ which satisfies the expected `balance equation' if there is flux of physical matter fields across the boundary $\I$ at infinity and is absolutely conserved in absence of this flux. Irrespective of the dimension d, all these quantities vanish if the space-time under considerations is (globally) anti-de Sitter. Furthermore, this result is required by a general covariance argument. However, it contradicts some of the recent findings based on the conjectured ADS/CFT duality. This and other features of our analysis suggest that, if a consistent dictionary between gravity and conformal field theories does exist in fully non-perturbative regimes, it would have to be more subtle than the one used currently.