Killing-Yano Cotton Currents

TL;DR

This paper explores conserved currents derived from the Cotton tensor and conformal Killing-Yano tensors, providing explicit examples, mathematical identities, and comparisons across different dimensions and gravity theories.

Contribution

It introduces a new construction of conserved currents from Cotton tensors and KYTs, including explicit calculations for specific metrics and a comparison with 3D gravity.

Findings

Charges proportional to electric and magnetic charges in Plebański-Demiański metric

Identification of conformal KYTs in specific coordinates

Development of potentials and identities for currents and tensors

Abstract

We discuss conserved currents constructed from the Cotton tensor and (conformal) Killing-Yano tensors (KYTs). We consider the corresponding charges generally and then exemplify with the four-dimensional Pleba\'nski-Demia\'nski metric where they are proportional to the sum of the squares of the electric and the magnetic charges. As part of the derivation, we also find the two conformal Killing-Yano tensors of the Pleba\'nski-Demia\'nski metric in the recently introduced coordinates of Podolsky and Vratny. The construction of asymptotic charges for the Cotton current is elucidated and compared to the three-dimensional construction in Topologically Massive Gravity. For the three-dimensional case, we also give a conformal superspace multiplet that contains the Cotton current in the bosonic sector. In a mathematical section, we derive potentials for the currents, find identities for conformal KYTs and for KYTs in torsionful backgrounds.