The AdS/CFT Correspondence Conjecture and Topological Censorship

Abstract

In gr-qc/9902061 it was shown that (n+1)-dimensional asymptotically anti-de-Sitter spacetimes obeying natural causality conditions exhibit topological censorship. We use this fact in this paper to derive in arbitrary dimension relations between the topology of the timelike boundary-at-infinity, $\scri$, and that of the spacetime interior to this boundary. We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other. This corollary may be viewed as a Lorentzian analog of the Witten and Yau result hep-th/9910245, but is independent of the scalar curvature of $\scri$. Furthermore, the topology of V', the Cauchy surface (as defined for asymptotically anti-de Sitter spacetime with boundary-at-infinity) for regions exterior to event horizons, is constrained by that of $\scri$. In this paper, we prove a generalization of the homology results in gr-qc/9902061 in arbitrary dimension, that H_{n-1}(V;Z)=Z^k where V is the closure of V' and k is the number of boundaries $Σ_i$ interior to $Σ_0$. As a consequence, V does not contain any wormholes or other compact, non-simply connected topological structures. Finally, for the case of n=2, we show that these constraints and the onto homomorphism of the fundamental groups from which they follow are sufficient to limit the topology of interior of V to either B^2 or $I\times S^1$.