Abstract
A Killing-Yano tensor is an antisymmetric tensor obeying a first-order differential constraint similar to that obeyed by a Killing vector. In this article we consider generalisations of such objects, focusing on the conformal case. These generalised conformal Killing-Yano tensors are of mixed symmetry type and obey the constraint that the largest irreducible representation of mathfrak{o} (n) contained in the tensor constructed from the first-derivative applied to such an object should vanish. Such tensors appear naturally in the context of spinning particles having N0 = 1 worldline supersymmetry and in the related problem of higher symmetries of Dirac operators. Generalisations corresponding to extended worldline supersymmetries and to spacetime supersymmetry are discussed.
Highlights
It is straightforward to show that the dual of a CKY p-form is a CKY (n − p)-form, i.e. satisfies the above equation with p replaced by n − p
We define conformal Killing-Yano tensor (CKYT) to be tensors of the type that can be constructed as highest weight representations arising in products of a conformal Killing tensor (CKT) and one or more CKY forms, and we discuss these in some generality in flat spacetime
In this paper we have studied some aspects of conformal Killing-Yano tensors from an algebraic point of view and shown how these tensors naturally lead to invariants of classical spinning particles
Summary
A conformal Killing-Yano p-form obeys the constraint given above in (1.2). In flat ndimensional Euclidean space, in terms of representations of o(n) this means that the largest representation in the product of a p-form multiplied by the derivative one-form must vanish. One can obtain more complicated CKYTs by taking the highest weight in the product of two or more CKY forms and a single CKT (Cartan product) This would lead to tensors Ap1,p2,...,q generalising (2.2) with p1 ≥ p2 ≥ . He considered objects which combine CKTs and CKYs in a natural way These are constructed from the r-fold symmetric product of p-forms, i.e. they are tensors of the type Ka11...a1p,a21...a2p,...,ar1...arp , antisymmetric on each set of p indices and symmetric under the interchange of any two such sets. These tensors are taken to satisfy certain first-order differential constraints which are given explicitly in [16] This definition is designed to reduce to those for CKTs for p = 1 and to those for CKYs for r = 1. These tensors are not irreducible in general but can be decomposed into irreducible components, and the latter will be tensors of the sort discussed above
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