Deformed $\mathcal{N}=8$ mechanics of $(\mathbf{8},\mathbf{8},\mathbf{0})$ multiplets

We construct new models of curved' SU$(4|1)$ supersymmetric mechanics based on two versions of the off-shell multiplet ${\bf(8,8,0)}$ which are mirror’ to each other. The worldline realizations of the supergroup SU$(4\mathrm{\mid }1)$ are treated as a deformation of flat $\mathcal{N}=8$, $d=1$ supersymmetry. Using SU$(4\mathrm{\mid }1)$ chiral superfields, we derive invariant actions for the first-type $(\mathbf{8},\mathbf{8},\mathbf{0})$ multiplet, which parametrizes special K"ahler manifolds. Since we are not aware of a manifestly SU$(4\mathrm{\mid }1)$ covariant superfield formalism for the second-type $(\mathbf{8},\mathbf{8},\mathbf{0})$ multiplet, we perform a general construction of SU$(4\mathrm{\mid }1)$ invariant actions for both multiplet types in terms of SU$(2\mathrm{\mid }1)$ superfields. An important class of such actions enjoys superconformal OSp$(8\mathrm{\mid }2)$ invariance. We also build off-shell actions for the SU$(4\mathrm{\mid }1)$ multiplets $(\mathbf{6},\mathbf{8},\mathbf{2})$ and $(\mathbf{7},\mathbf{8},\mathbf{1})$ through appropriate substitutions for the component fields in the $(\mathbf{8},\mathbf{8},\mathbf{0})$ actions. The $(\mathbf{6},\mathbf{8},\mathbf{2})$ actions are shown to respect superconformal SU$(4\mathrm{\mid }1,1)$ invariance.