General Kerr-NUT-AdS Metrics in All Dimensions

TL;DR

This work realises a universal construction of Kerr-NUT-AdS metrics in all dimensions $D$ by introducing a Jacobi-like diagonalisation that replaces constrained latitude coordinates with unconstrained $y_\alpha$, enabling an immediate inclusion of $[D/2]-1$ NUT charges. The resulting general Kerr-NUT-AdS solutions in odd and even dimensions reveal a consistent parameter structure, discrete inversion symmetries that map over-rotation to under-rotation, and rich BPS limits that yield Einstein-Sasaki (odd $D$) and Einstein-Kähler (odd Euclideanised) geometries, with explicit $D=6$ and $D=7$ examples illustrating the formalism. The approach produces compact, Plebański-like presentations across dimensions and passes checks against the Einstein equations up to $D\leq 15$, broadening the landscape of higher-dimensional AdS black holes with NUT charges and their supersymmetric and special-holonomy limits. These results have potential implications for holography, supersymmetric geometries, and the global structure of higher-dimensional rotating spacetimes.

Abstract

The Kerr-AdS metric in dimension D has cohomogeneity [D/2]; the metric components depend on the radial coordinate r and [D/2] latitude variables μ_i that are subject to the constraint \sum_i μ_i^2=1. We find a coordinate reparameterisation in which the μ_i variables are replaced by [D/2]-1 unconstrained coordinates y_α, and having the remarkable property that the Kerr-AdS metric becomes diagonal in the coordinate differentials dy_α. The coordinates r and y_αnow appear in a very symmetrical way in the metric, leading to an immediate generalisation in which we can introduce [D/2]-1 NUT parameters. We find that (D-5)/2 are non-trivial in odd dimensions, whilst (D-2)/2 are non-trivial in even dimensions. This gives the most general Kerr-NUT-AdS metric in $D$ dimensions. We find that in all dimensions D\ge4 there exist discrete symmetries that involve inverting a rotation parameter through the AdS radius. These symmetries imply that Kerr-NUT-AdS metrics with over-rotating parameters are equivalent to under-rotating metrics. We also consider the BPS limit of the Kerr-NUT-AdS metrics, and thereby obtain, in odd dimensions and after Euclideanisation, new families of Einstein-Sasaki metrics.