Singularity, Sasaki-Einstein manifold, Log del Pezzo surface and $\mathcal{N}=1$ AdS/CFT correspondence: Part I

TL;DR

This paper explores the geometric and algebraic properties of Sasaki-Einstein manifolds and their associated singularities and surfaces, aiming to expand the understanding and classification of these manifolds within the AdS/CFT correspondence.

Contribution

It introduces a conjecture linking K stability of singularities to the existence of Sasaki-Einstein metrics, facilitating the search for new SE manifolds.

Findings

Increased the known space of SE manifolds significantly.

Established a connection between K stability and the existence of SE metrics.

Proposed a conjecture to reduce K stability checks to finite cases.

Abstract

A five dimensional Sasaki-Einstein (SE) manifold provides a AdS/CFT pair for four dimensional $\mathcal{N}=1$ SCFT, and those pairs are very useful in studying field theory and AdS/CFT correspondence. The space of known SE manifolds is increased significantly in the last decade, and we initiated the study of various field theory properties through the geometric property of these new SE manifolds. There is an associated three dimensional log-terminal singularity $X$ for each SE manifold $L_X$, and for quasi-regular case, there is an associated two dimensional log Del Pezzo surface $(S,\Delta)$. The algebraic geometrical methods are quite useful in extracting interesting physical properties from singularity and log Del Pezzo surface. The necessary and sufficient condition for the existence of SE metric on $L_X$ is related to K stability of $X$. Motivated by dual field theory, we propose a conjecture on how to reduce the check of K stability to possibly finite cases, which hopefully would give us a guideline to find a much larger space of SE metrics.