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

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.