Minimal bosonization of supersymmetry

TL;DR

This paper presents a minimal bosonization approach to supersymmetry using a single bosonic degree of freedom, linking it to supersymmetric quantum mechanics and extending it to models like Calogero with dynamical superalgebra symmetries.

Contribution

It introduces a novel minimal bosonization framework for supersymmetry, connecting it to known models and extending to deformed oscillators and superalgebras.

Findings

Bosonized supersymmetric quantum mechanics constructed.

Extension to Vasiliev deformed oscillators and Calogero model demonstrated.

osp(2|2) superalgebra identified as dynamical symmetry.

Abstract

The minimal bosonization of supersymmetry in terms of one bosonic degree of freedom is considered. A nontrivial relationship of the construction to the Witten supersymmetric quantum mechanics is illustrated with the help of the simplest $N=2$ SUSY system realized on the basis of the ordinary (undeformed) bosonic oscillator. It is shown that the generalization of such a construction to the case of Vasiliev deformed bosonic oscillator gives a supersymmetric extension of the 2-body Calogero model in the phase of exact or spontaneously broken $N=2$ SUSY. The construction admits an extension to the case of the OSp(2$\vert$2) supersymmetry, and, as a consequence, $osp(2\vert 2)$ superalgebra is revealed as a dynamical symmetry algebra for the bosonized supersymmetric Calogero model. Realizing the Klein operator as a parity operator, we construct the bosonized Witten supersymmetric quantum mechanics. Here the general case of the corresponding bosonized $N=2$ SUSY is given by an odd function being a superpotential.