On Sasakian-Einstein Geometry

TL;DR

This paper explores the structure and conditions of quasi-regular Sasakian-Einstein orbifolds and manifolds, including their algebraic properties and cohomology, with relevance to supersymmetric conformal field theories.

Contribution

It introduces a natural multiplication on the space of quasi-regular Sasakian-Einstein orbifolds and provides criteria for the product of two such manifolds to be smooth and Sasakian-Einstein.

Findings

The space of quasi-regular Sasakian-Einstein orbifolds admits a natural multiplication.

Necessary and sufficient conditions are established for the product of two Sasakian-Einstein manifolds to be smooth.

Cohomology rings are computed in various cases using spectral sequence arguments.

Abstract

We discuss Sasakian-Einstein geometry under a quasi-regularity assumption. It is shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it. Furthermore, necessary and sufficient conditions are given for the `product' of two Sasakian-Einstein manifolds to be a smooth Sasakian-Einstein manifold. Using spectral sequence arguments we work out the cohomology ring in many cases of interest. This type of geometry has recently become of interest in the physics of supersymmetric conformal field theories.