Twisting D(2,1; α ) Superspace

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