Quasicomplex $\mathcal{N}=2$, $d=1$ Supersymmetric Sigma Models

We derive and discuss a new type of $\mathcal{N}=2$ supersymmetric quantum mechanical sigma models which appear when the superfield action of the ($\mathbf{1, 2, 1}$) multiplets is modified by adding an imaginary antisymmetric tensor to the target space metric, thus completing the latter to a non-symmetric Hermitian metric. These models are not equivalent to the standard de Rham sigma models, but are related to them through a certain special similarity transformation of the supercharges. On the other hand, they can be obtained by a Hamiltonian reduction from the complex supersymmetric $\mathcal{N}=2$ sigma models built on the multiplets ($\mathbf{2, 2, 0}$) and describing the Dolbeault complex on the manifolds with proper isometries. We study in detail the extremal two-dimensional case, when the target space metric is defined solely by the antisymmetric tensor, and show that the corresponding quantum systems reveal a hidden $\mathcal{N}=4$ supersymmetry.