Super Landau Models on Odd Cosets

TL;DR

This paper develops supersymmetric Landau models on fermionic cosets, demonstrating their classical and quantum properties, including unitarity and hidden symmetries, with implications for quantum mechanics on curved superspaces.

Contribution

It constructs novel supersymmetric Landau models on SU(n|1)/U(n) cosets, analyzes their quantum unitarity, and uncovers hidden SU(2|2) symmetry in the n=2 case.

Findings

Quantum models are unitary with positive-definite norms.

The n=2 quantum model exhibits hidden SU(2|2) symmetry.

Planar limit models have distinct SU(2|2) symmetry from other cosets.

Abstract

We construct d=1 sigma models of the Wess-Zumino type on the SU(n|1)/U(n) fermionic cosets. Such models can be regarded as a particular supersymmetric extension (with a target space supersymmetry) of the classical Landau model, when a charged particle possesses only fermionic coordinates. We consider both classical and quantum models, and prove the unitarity of the quantum model by introducing the metric operator on the Hilbert space of the quantum states, such that all their norms become positive-definite. It is remarkable that the quantum n=2 model exhibits hidden SU(2|2) symmetry. We also discuss the planar limit of these models. The Hilbert space in the planar n=2 case is shown to carry SU(2|2) symmetry which is different from that of the SU(2|1)/U(1) model.