The Dark Side of Double-Tensor Multiplets

We explore the properties of a set of free double-tensor multiplets in $N=2$ supersymmetry, focusing on their behavior within rigid superspace. These multiplets can be obtained from hypermultiplets by Hodge-dualizing half of their scalars, and feature an off-shell matching of bosonic and fermionic degrees of freedom. Despite this fact, the supersymmetry algebra results to close only on-shell. Our analysis is conducted both in superspace, using the geometric (rheonomic) approach, and in spacetime, comparing how our results are obtained in the two approaches. Notably, the cohomology of superspace requires that the scalars Hodge-dual to the antisymmetric tensors crucially contribute to the superspace description of the tensors super-field strengths. This shows an inherent non-locality of the theory, already in the free case, which however does not forbid a Lagrangian description.