Twisting D(2,1; α) Superspace

We develop a three-dimensional $\mathcal{N}=4$ theory of rigid supersymmetry describing the dynamics of a set of hypermultiplets $(Λ^{αα'\dotα'}_I,\,ϕ^{αA}_I)$ on a curved AdS$_3$ worldvolume background, whose supersymmetry is captured by the supergroup ${\rm D}^2(2,1;\, \boldsymbolα)$. To unveil some remarkable features of this model, we perform two twists, involving the SL$(2,\mathbb R)$ factors of the theory. After the first twist, our spacetime Lagrangian exhibits a Chern-Simons term associated with an odd one-form field, together with a fermionic "gauge-fixing'', in the spirit of the Rozansky-Witten model. The second twist allows to interpret the ${\rm D}^2(2,1;\, \boldsymbolα)$ setup as a framework capable of describing massive Dirac particles. In particular, this yields a generalisation of the Alvarez-Valenzuela-Zanelli model of ''unconventional supersymmetry''. We comment on specific values of the combination $α+1$, which in our model is related to a sort of gauging in the absence of dynamical gauge fields.

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