Super-Landau models: Unitarity and hidden symmetries

We review a recent progress in constructing and studying superextensions of the Landau problem of a quantum particle moving on the two-sphere S ². These are the superspherical and superflag Landau models describing a nonrelativistic particle on the supermanifolds SU(2|1)/U(1|1) and SU(2|1)/[U(1) x U(1)]. Though in both models, under the “naive” definition of the inner product, there are negative-norm states, all norms can be made positive-definite by introducing a nontrivial “metric” in the space of states. So both models are unitary. The superflag model is shown to be quantum-equivalent to the supersphere model for the special choice of the intrinsic parameters. We also address the planar limit of the superspherical model, in which there arises the hidden world-line N =2 supersymmetry. An analog of this phenomenon in the superflag model is a dynamical enhancement of its SU(2|1) symmetry to SU(2|2).