A Note on Einstein Sasaki Metrics in D \ge 7

TL;DR

This paper develops a method to construct non-singular Einstein–Sasaki spaces in dimensions $D\ge 7$ by building circle bundles over Einstein–Kähler bases, where the base is itself a complex line bundle over products of Einstein–Kähler spaces. In the key seven-dimensional case, the base is a six-dimensional Einstein–Kähler manifold built as a two-dimensional bundle over $S^2\times S^2$, leading to explicit first-order equations and exact forms for the metric functions, with a crucial algebraic constraint linking parameters. Global regularity is achieved only after enforcing two rationality conditions on fiber periods, yielding a countable family of non-singular metrics parameterized by pairs $(\alpha,\gamma)$ and associated constants, including explicit rational solutions for special $(p,q)$ values. The construction further extends to a general class in higher dimensions, where the base is a product of $N$ Einstein–Kähler spaces, and regularity again requires rational relations among integration constants, offering a broad family of smooth Einstein–Sasaki manifolds with potential applications to AdS/CFT and M-theory compactifications.

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.