New Einstein-Sasaki and Einstein Spaces from Kerr-de Sitter

TL;DR

This paper constructs new Einstein-Sasaki and Einstein spaces in odd dimensions by taking BPS limits of Kerr-de Sitter metrics, resulting in smooth, compact manifolds with specific topological and geometric properties.

Contribution

It introduces a systematic method to generate new Einstein-Sasaki and Einstein spaces from Kerr-de Sitter metrics in all odd dimensions, expanding the known classes of such manifolds.

Findings

Constructed new Einstein-Sasaki spaces L^{p,q,r_1,...,r_{n-1}} in all odd dimensions.

Derived complete, non-singular Einstein spaces mbda^{p,q,r_1,...,r_n} that are not Sasakian.

Identified conditions for parameters ensuring smoothness and compactness of the manifolds.

Abstract

In this paper, which is an elaboration of our results in hep-th/0504225, we construct new Einstein-Sasaki spaces L^{p,q,r_1,...,r_{n-1}} in all odd dimensions D=2n+1\ge 5. They arise by taking certain BPS limits of the Euclideanised Kerr-de Sitter metrics. This yields local Einstein-Sasaki metrics of cohomogeneity n, with toric U(1)^{n+1} principal orbits, and n real non-trivial parameters. By studying the structure of the degenerate orbits we show that for appropriate choices of the parameters, characterised by the (n+1) coprime integers (p,q,r_1,...,r_{n-1}), the local metrics extend smoothly onto complete and non-singular compact Einstein-Sasaki manifolds L^{p,q,r_1,...,r_{n-1}}. We also construct new complete and non-singular compact Einstein spaces \Lambda^{p,q,r_1,...,r_n} in D=2n+1 that are not Sasakian, by choosing parameters appropriately in the Euclideanised Kerr-de Sitter metrics when no BPS limit is taken.