Symmetries of deformed supersymmetric mechanics on Kähler manifolds

Abstract

Based on the systematic Hamiltonian and superfield approaches we construct the deformed $\mathcal{N}=4,8$ supersymmetric mechanics on Kähler 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 Kähler 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 Kähler 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.

arXiv comment20 pages

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Cited by in corpus (3)

• Kähler geometry for $su(1,N|M)$-superconformal mechanics
• Quantum SU$(2|1)$ supersymmetric ${\mathbb{C}}^N$ Smorodinsky--Winternitz system
• SU$(2|1)$ supersymmetric spinning models of chiral superfields