Obstructions to the Existence of Sasaki–Einstein Metrics

We describe two simple obstructions to the existence of Ricci–flat Kähler 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 Kähler–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.