Boundaries of Zero Scalar Curvature in the AdS/CFT Correspondence

TL;DR

This paper extends Witten and Yau's results on the topology of Einstein manifolds in the AdS/CFT correspondence to cases where the boundary conformal class has nonnegative scalar curvature, using geodesic geometry methods.

Contribution

It generalizes previous theorems to include zero scalar curvature boundaries and introduces a new proof approach based on geodesic geometry.

Findings

Results hold for nonnegative scalar curvature boundary metrics.

Connectedness and homology properties extend to zero scalar curvature case.

New proof method avoids geometric measure theory, using only geodesic geometry.

Abstract

In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature which admits a conformal compactification with conformal boundary $N^n$. They prove that if the conformal class of the boundary contains a metric of positive scalar curvature, then $M$ and $N$ have several desirable properties: (1) $N$ is connected, (2) the $n$th homology of the compactified $M$ vanishes, and (3) the fundamental group of $M$ is "bounded by" that of $N$. Here it is shown that all of these results extend to the case where the conformal class of the boundary contains a metric of nonnegative scalar curvature. (The case of zero scalar curvature is of interest as it is borderline for the stability of the theory.) The proof method used here is different from, and in some sense dual to, that used by Witten and Yau. While their method involves minimizing the co-dimension one brane action on $M$, and requires the machinery of geometric measure theory, the main arguments presented here use only geodesic geometry.