Obstructions to the Existence of Sasaki-Einstein Metrics

TL;DR

This paper identifies new simple obstructions to the existence of Sasaki-Einstein and Ricci-flat Kahler cone metrics on certain singularities, impacting the understanding of geometric structures and their physical duals.

Contribution

It introduces two novel obstructions to Ricci-flat Kahler cone metrics and Sasaki-Einstein metrics, with applications to Fano orbifolds and hypersurface singularities.

Findings

Several families of obstructed hypersurface singularities identified.

Obstructions relate to violations of unitarity bounds in AdS/CFT correspondence.

New criteria for the non-existence of certain geometric structures.

Abstract

We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler-Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R-charge of a gauge invariant chiral primary operator violates the unitarity bound.