A Note on Einstein Sasaki Metrics in D \ge 7

TL;DR

This paper constructs new non-singular Einstein-Sasaki spaces in dimensions seven and higher by using circle bundles over Einstein-Kahler bases, with conditions ensuring smooth, complete, and compact manifolds.

Contribution

It introduces a method to generate smooth Einstein-Sasaki spaces in higher dimensions via specific bundle constructions and rationality conditions.

Findings

New non-singular Einstein-Sasaki spaces in D≥7 dimensions.

Conditions for smoothness and compactness of the constructed spaces.

Explicit local construction method for these spaces.

Abstract

In this paper, we obtain new non-singular Einstein-Sasaki spaces in dimensions D\ge 7. The local construction involves taking a circle bundle over a (D-1)-dimensional Einstein-Kahler metric that is itself constructed as a complex line bundle over a product of Einstein-Kahler spaces. In general the resulting Einstein-Sasaki spaces are singular, but if parameters in the local solutions satisfy appropriate rationality conditions, the metrics extend smoothly onto complete and non-singular compact manifolds.