Novel symmetries in N=2N=2 supersymmetric quantum mechanical models

• Discrete symmetries of two completely different kinds of N=2N=2 supersymmetric quantum mechanical models have been discussed.
• The discrete symmetries provide physical realizations of Hodge duality.
• The continuous symmetries provide the physical realizations of de Rham cohomological operators.
• Our work sheds a new light on the meaning of the above abstract operators.

We demonstrate the existence of a novel set of discrete symmetries in the context of the N=2N=2 supersymmetric (SUSY) quantum mechanical model with a potential function f(x)f(x) that is a generalization of the potential of the 1D SUSY harmonic oscillator. We perform the same exercise for the motion of a charged particle in the XX–YY plane under the influence of a magnetic field in the ZZ-direction. We derive the underlying algebra of the existing   continuous symmetry transformations (and corresponding conserved charges) and establish its relevance to the algebraic structures of the de Rham cohomological operators of differential geometry. We show that the discrete symmetry transformations of our present general theories correspond to the Hodge duality operation. Ultimately, we conjecture that any arbitrary N=2N=2 SUSY quantum mechanical system can be shown to be a tractable model for the Hodge theory.