The AdS/CFT Correspondence Conjecture and Topological Censorship

TL;DR

The paper extends topological censorship to $(n+1)$-dimensional asymptotically anti-de Sitter spacetimes to relate the topology of the timelike boundary-at-infinity $\mathcal{I}$ to the interior. By imposing an averaged null energy condition and a generic condition, it proves that causal curves cannot reveal interior topology beyond that of $\mathcal{I}$, leading to a surjective map $\Pi_1(\Sigma_0)\to\Pi_1(V)$ and a homology constraint $H_{n-1}(V;\mathbb{Z})=\mathbb{Z}^k$, with $k$ the number of interior boundaries. In $2+1$ dimensions these results force the interior topology to be either $B^2$ or $I\times S^1$, while in higher dimensions the topology is constrained but not fully determined, indicating a Lorentzian analogue of the Witten–Yau scenario with direct implications for AdS/CFT. Overall, the work strengthens the link between boundary topology and bulk topology in AdS/CFT by showing that boundary data largely governs interior structure, particularly in lower dimensions.

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 $I\times S^1$.