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We consider N=1 superconformal field theories in three dimensions possessing a conserved current multiplet F(α1α2α3α4), which we refer to as the superspin-2 current multiplet. At the component level it contains a conserved spin-2 current different from the energy-momentum tensor and a conserved fermionic higher-spin current of spin 5/2. Using a superspace formulation, we calculate correlation functions involving F, focusing particularly on the three-point function FFF . After imposing the constraints arising from conservation equations and invariance under permutation of superspace points, we find that the parity-even and parity-odd sectors of this three-point function are each fixed up to a single coefficient. The presence of the parity-odd contribution is rather nontrivial, as there is an apparent tension between supersymmetry and the existence of parity-odd structures.
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