Symmetries of deformed supersymmetric mechanics on K\"ahler manifolds

TL;DR

This paper constructs and analyzes deformed supersymmetric mechanics on K"ahler manifolds with magnetic fields, revealing how initial symmetries deform and are inherited by supersymmetric extensions, including models like oscillators and superintegrable systems.

Contribution

It introduces a systematic method to build deformed $ abla$ supersymmetric systems on K"ahler manifolds with magnetic interactions, extending known models and analyzing their symmetries.

Findings

Deformation of flat supersymmetries to $SU(2|1)$ and $SU(4|1)$ due to magnetic field.

Inheritance of kinematical and hidden symmetries in supersymmetric models.

Construction of supersymmetric oscillators and superintegrable systems on complex spaces.

Abstract

Based on the systematic Hamiltonian and superfield approaches we construct the deformed $\mathcal{N}=4,8$ supersymmetric mechanics on K\"ahler manifolds interacting with constant magnetic field, and study their symmetries. At first we construct the deformed $\mathcal{N}=4,8$ supersymmetric Landau problem via minimal coupling of standard (undeformed) $\mathcal{N}=4,8$ supersymmetric free particle systems on K\"ahler manifold with constant magnetic field. We show that the initial "flat" supersymmetries are necessarily deformed to $SU(2|1)$ and $SU(4|1)$ supersymmetries, with the magnetic field playing the role of deformation parameter, and that the resulting systems inherit all the kinematical symmetries of the initial ones. Then we construct $SU(2|1)$ supersymmetric K\"ahler oscillators and find that they include, as particular cases, the harmonic oscillator models on complex Euclidian and complex projective spaces, as well as superintegrable deformations thereof, viz. $\mathbb{C}^N$-Smorodinsky-Winternitz and $\mathbb{CP}^N$-Rosochatius systems. We show that the supersymmetric extensions proposed inherit all the kinematical symmetries of the initial bosonic models. They also inherit, at least in the case of $\mathbb{C}^N$ systems, hidden (non-kinematical) symmetries. The superfield formulation of these supersymmetric systems is presented, based on the worldline $SU(2|1)$ and $SU(4|1)$ superspace formalisms.