Supersymmetrizing Landau models

We review recent progress in constructing and studying superextensions of the Landau problem of a quantum particle on a plane in a uniform magnetic field and also its Haldane $S^2$ generalization. We focus on the planar super Landau models that are invariant under the inhomogeneous supergroup $ISU(1\,|\,1)$, a contraction of the supergroup $SU(2\,|\,1)$, and are minimal superextensions of the original Landau model. Their significant common feature is the presence of a hidden dynamical worldline $\mathcal N=2$ supersymmetry, which exists at both the classical and quantum levels and is revealed most naturally in passing to the new invariant inner products in the space of quantum states in order to make the norms of all states positive. For one of the planar models, the superplane Landau model, we present an off-shell worldline superfield formulation in which the $\mathcal N=2$ supersymmetry becomes explicit.