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 Plebański-Demiański 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 Plebański-Demiański 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.
Highlights
Rank Killing-Yano tensors (KYTs) and it was used in certain backgrounds to construct asymptotic charges [21]
We further discuss the special case of three dimensions (3D) at some length, finding obstructions to asymptotic charges, comparing to results in Topologically Massive Gravity (TMG) and generalizing our current to 3D conformal supergravity in superspace
Where the last equality follows from the fact that the Cotton tensor is traceless. Since both the Cotton tensor and the conformal KYTs (CKYTs) are related to conformal properties of the manifold, one may ask about the transformation properties of Ja under Weyl rescalings of the metric gab = eC gab
Summary
We introduce conserved currents constructed from the Cotton tensor and (C)KYTs. The Cotton tensor is defined in D ≥ 3 dimensions as. Where the last equality follows from the fact that the Cotton tensor is traceless Since both the Cotton tensor and the CKYT are related to conformal properties of the manifold, one may ask about the transformation properties of Ja under Weyl rescalings of the metric gab = eC gab. From (2.11) and (2.12) it follows that 1-form current Ja = Cabckbc transforms as This is clearly not a conformal scaling in general. In 3D the Weyl tensor does not exist which means that the covariant current scales with e−C/2 and that the contravariant current scales with e−3C/2 This holds in arbitrary D for conformally flat metrics or for constant scalings C.
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