Generalized $\mathcal{N} = 2$ super Landau models

Abstract

We generalize previous results for the superplane Landau model to exhibit an explicit worldline $\mathcal{N} = 2$ supersymmetry for an arbitrary magnetic field on any two-dimensional manifold. Starting from an off-shell $\mathcal{N} = 2$ superfield formalism, we discuss the quantization procedure in the general case characterized by two independent potentials on the manifold and show that the relevant Hamiltonians are factorizable. In the restricted case, when both the Gauss curvature and the magnetic field are constant over the manifold and, as a consequence, the underlying potentials are related, the Hamiltonians admit infinite series of factorization chains implying the integrability of the associated systems. We explicitly determine the spectrum and eigenvectors for the particular model with $\mathbb{C}{\mathbb{P}^1}$ as the bosonic manifold.