Abstract
The notion of a Killing tensor is generalised to a superspace setting. Conserved quantities associated with these are defined for superparticles and Poisson brackets are used to define a supersymmetric version of the Schouten-Nijenhuis bracket. Superconformal Killing tensors in flat superspaces are studied for spacetime dimensions 3,4,5,6 and 10. These tensors are also presented in analytic superspaces and super-twistor spaces for 3,4 and 6 dimensions. Algebraic structures associated with superconformal Killing tensors are also briefly discussed
Highlights
As we have seen, the components of any conformal Killing tensor (CKT) are given by representations of the conformal algebra, so that this product can be described in Lie algebraic terms
The most significant point of the general discussion is that superconformal Killing tensors (SCKTs) can be defined as purely even traceless symmetric tensors subject to the constraint that the smaller of the two spin representations that arise when one differentiates with respect to the spinorial covariant derivative should be set to zero
We discussed these objects in the context of superparticles and in various flat superspaces as well as analytic superspaces and super twistor spaces
Summary
We consider superspaces that extend D-dimensional spacetime by a number of odd coordinates that transform as spinors under Spin(1, D − 1) and which, in addition, may carry a representation of an internal R-symmetry group. We shall assume that conventional constraints corresponding to the dimension one-half and one components of the R-symmetry connection have been imposed, but it will not be necessary to be explicit about these in this paper. Since the only other non-zero terms in the equation contributing to this component involves the derivative of S, we conclude that Haβ has the same form as the right-hand side of (2.11) and that we have a superconformal This result means that we can interpret a SCKV in a simpler way: it can be defined to be a vector field K that generates an infinitesimal diffeomorphism that preserves the odd tangent bundle, i.e. LKX is odd if X is, or, equivalently [Eα, K], Eb = 0. The emphasis throughout the rest of the paper will be on the superconformal case, but the non-conformal case can be studied when the scale transformations are omitted
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