Obstructions to the Existence of Sasaki-Einstein Metrics

TL;DR

Gauntlett, Martelli, Sparks, and Yau identify two practical obstructions to the existence of Sasaki–Einstein (Ricci-flat Kähler cone) metrics on links of isolated Gorenstein singularities: Bishop's bound on the normalized volume V(ξ) and Lichnerowicz's bound on the ξ-charge of holomorphic functions, which corresponds to a unitarity bound in AdS/CFT. They develop formulae based on the holomorphic index-character to compute these obstructions for isolated quasi-homogeneous hypersurface singularities, and apply them to families of 3-fold and ADE 4-fold examples, showing many previously-studied singularities cannot admit Sasaki–Einstein links for canonical Reeb actions. The AdS/CFT interpretation clarifies how Lichnerowicz violations map to unitarity violations in dual field theories, providing a cross-check with a–maximisation results and central charge bounds. The work delivers concrete obstruction criteria, analyzes several explicit families (including Brieskorn–Pham and weighted actions), and highlights open cases (notably X_3 in the 3-fold class) for future geometric and field-theoretic exploration.

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.

Figures (1)

• Figure 1: Quiver diagram of the $A_1$ orbifold gauge theory.