Asymptotically Anti-de Sitter Space-times: Conserved Quantities

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 $\xi$ of this group, there is a quantity $Q_\xi$ 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.