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

TL;DR

The paper addresses the problem of constructing explicit, complete Einstein-Sasaki spaces in all odd dimensions by applying a Euclidean Kerr-de Sitter/BPS limit, yielding local cohomogeneity-$n$ metrics with toric $U(1)^{n+1}$ symmetry. The main approach combines BPS limits, Euclideanisation, and a careful analysis of degenerate orbits to impose coprime-integer regularity conditions, producing the infinite families $L^{p,q,r_1,...,r_{n-1}}$; it also identifies non-Sasakian Einstein spaces $\Lambda^{p,q,r_1,...,r_n}$ by avoiding the BPS limit. Key results include the explicit five-dimensional spaces $L^{p,q,r}$ with topology $S^2\times S^3$ and volume formulas, the extension to higher odd dimensions with cohomogeneity $n$, and the demonstration of Bishop bounds and curvature properties that confirm smoothness and non-trivial geometry. The work has significant implications for exact backgrounds in string/M-theory and AdS/CFT, including rational central charges linked to spacetime volumes. It also clarifies symmetry relations (such as $u\leftrightarrow -u$) and the conditions under which non-Sasakian spaces remain non-singular, shaping future explorations of explicit Einstein geometries in higher dimensions.

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 Λ^{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.