A Correspondence between Ricci-flat Kerr and Kaluza-Klein AdS Black Hole

TL;DR

This paper establishes a detailed correspondence between Ricci-flat Kerr metrics and Kaluza-Klein AdS black holes, revealing new mappings and solutions in higher-dimensional gravity and supergravity theories.

Contribution

It introduces an explicit mapping between Ricci-flat Kerr metrics and Kaluza-Klein AdS black holes, including new solutions and a superpotential in four-dimensional Kaluza-Klein AdS gravity.

Findings

Mapped Ricci-flat Kerr metrics to charged Kaluza-Klein AdS black holes.

Connected gravitational instantons to AdS domain walls and interpreted them as M-branes and D3-branes.

Derived a new superpotential in four-dimensional Kaluza-Klein AdS gravity.

Abstract

We establish an explicit correspondence of Einstein gravity on the squashed spheres that are the $U(1)$ bundles over $\mathbb{CP}^m$ to the Kaluza-Klein AdS gravity on the tori. This allows us to map the Ricci-flat Kerr metrics in odd dimensions with all equal angular momenta to charged Kaluza-Klein AdS black holes that can be lifted to become singly rotating M-branes and D3-branes. Furthermore, we find maps between Ricci-flat gravitational instantons to the AdS domain walls. In particular the supersymmetric bolt instantons correspond to domain walls that can be interpreted as distributed M-branes and D3-branes, whilst the non-supersymmetric Taub-NUT solutions yield new domain walls that can be lifted to become solutions in M-theory or type IIB supergravity. The correspondence also inspires us to obtain a new superpotential in the Kaluza-Klein AdS gravity in four dimensions.