Local metrics admitting a principal Killing-Yano tensor with torsion

TL;DR

This paper classifies local metrics with a principal Killing-Yano tensor incorporating torsion, revealing new solutions relevant to supergravity theories and extending known geometric structures.

Contribution

It introduces a classification of metrics with principal Killing-Yano tensors with torsion, providing explicit solutions and connecting to supergravity black hole solutions.

Findings

Derived three types of local metrics in even dimensions.

Constructed explicit metrics covering various supergravity solutions.

Linked generalized Killing-Yano tensors to torsion Killing spinors.

Abstract

In this paper we initiate a classification of local metrics admitting the principal Killing--Yano tensor with a skew-symmetric torsion. It is demonstrated that in such spacetimes rank-2 Killing tensors occur naturally and mutually commute. We reduce the classification problem to that of solving a set of partial differential equations, and we present some solutions to these PDEs. In even dimensions, three types of local metrics are obtained: one of them naturally generalizes the torsionless case while the others occur only when the torsion is present. In odd dimensions, we obtain more varieties of local metrics. The explicit metrics constructed in this paper are not the most general possible admitting the required symmetry, nevertheless, it is demonstrated that they cover a wide variety of solutions of various supergravities, such as the Kerr-Sen black holes of (un-)gauged abelian heterotic supergravity, the Chong-Cvetic-L\"u-Pope black hole solution of five-dimensional minimal supergravity, or the K\"ahler with torsion manifolds. The relation between generalized Killing--Yano tensors and various torsion Killing spinors is also discussed.