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

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