Connectedness Of The Boundary In The AdS/CFT Correspondence

TL;DR

This paper proves that under certain geometric conditions in the AdS/CFT correspondence, the boundary must be connected, resolving some existing puzzles in the theoretical framework.

Contribution

It establishes a topological result linking the geometry of Einstein manifolds with the connectedness of their conformal boundary in AdS/CFT.

Findings

The boundary $N$ must be connected under the given conditions.

The homology group $H_n(M;Z)$ vanishes for the manifold.

The results clarify the geometric structure in AdS/CFT correspondence.

Abstract

Let $M$ be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary $N$ of positive scalar curvature. We show that under these conditions, $H_n(M;Z)=0$ and in particular $N$ must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence.