Supersymmetrizing Landau Models

TL;DR

This paper reviews recent developments in supersymmetric extensions of the Landau problem, highlighting models invariant under ISU(1|1) with hidden N=2 supersymmetry at classical and quantum levels.

Contribution

It introduces minimal superextensions of the Landau model with a focus on supergroup invariance and reveals hidden N=2 supersymmetry in these models.

Findings

Superextensions invariant under ISU(1|1) are constructed.

Hidden N=2 supersymmetry exists at classical and quantum levels.

Off-shell superfield formulation for the superplane Landau model is provided.

Abstract

This is an overview of recent progress in constructing and studying superextensions of the Landau problem of a quantum particle on a plane in the uniform magnetic field, as well as of its Haldane's $S^2$ generalization ({\tt hep-th/0311159, hep-th/0404108, hep-th/0510019, hep-th/0612300}). The main attention is paid to the planar super Landau models which are invariant under the inhomogeneous supergroup ISU(1|1), a contraction of the supergroup SU(2|1), and provide minimal superextensions of the original Landau model. Their common notable feature is the presence of a hidden dynamical worldline N=2 supersymmetry. It exists at the classical and quantum levels and is revealed most naturally while 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 N=2 supersymmetry gets manifest.