A Novel Generalisation of Supersymmetry: Quantum $\mathbb{Z}_2^2$-Oscillators and their `superisation'

TL;DR

This paper introduces a simple toy model of a $ Z_2^2$-supersymmetric quantum system, extending traditional supersymmetry with an additional $ Z_2^2$-grading, and demonstrates its structure via Klein's construction.

Contribution

It presents a novel $ Z_2^2$-supersymmetric quantum model and explains its relation to $N=2$ supersymmetry with an extra grading structure.

Findings

The model exhibits $ Z_2^2$-graded supersymmetry.

It can be understood as an $N=2$ supersymmetric system with an additional $ Z_2^2$-grading.

The construction clarifies how to incorporate extra grading into supersymmetric quantum systems.

Abstract

We propose a very simple toy model of a $\mathbb{Z}_2^2$-supersymmetric quantum system and show, via Klein's construction, how to understand the system as being an $N=2$ supersymmetric system with an extra $\mathbb{Z}_2^2$-grading. That is, the commutation/anticommutation rules are defined via the standard boson/fermion rules, but the system still has an underlying $\mathbb{Z}_2^2$-grading that needs to be taken into account.