Superextensions of Landau models

The paper is a review of recent works on superextensions of the model of non-relativistic quantum charged particle moving in a homogeneous magnetic field on the plane $R^{2}$ (Landau model), and a model of the particle in the field of Dirac monopole on the sphere $S^{2}: SU(2)/U(1)$ (Haldane model). We consider the models on the supersphere $SU(2|1)/U(1|1)$, superflag $SU(2|1)/[U(1)\times U(1)]$ and their planar limits, based upon a geometric interpretation of these models and their bosonic proptotypes as of $d=1$ analogs of nonlinear sigma models of the Wess-Zumino-Novikov-Witten type. While quantizing supersymmetric models, there arise states with the negative norms and, in order to overcome this difficulty, it proves necessary to introduce a non-trivial metrics on the Hilbert space of quantum states. A characteristic feature of the planar models is the presence of hidden dynamical $N=2$ worldline supersymmetry.