Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

TL;DR

This paper classifies irreducible representations of a specific $Z_2^2$-graded supersymmetry algebra and constructs invariant classical actions, revealing how these representations depend on eigenvalues and auxiliary variables.

Contribution

It provides a systematic derivation of irreducible representations of $Z_2^2$-graded ${ m N}=2$ supersymmetry and constructs associated invariant classical actions.

Findings

Irreps are four-dimensional for zero Casimir eigenvalue and eight-dimensional otherwise.

Eight-dimensional irreps reduce to four-dimensional under specific eigenvalue relations.

One Noether charge vanishes when all variables of certain $Z_2^2$-degree are auxiliary.

Abstract

Irreducible representations (irreps) of $\mathbb{Z}_2^2$-graded supersymmetry algebra of ${\cal N}=2$ are obtained by the method of induced representation and they are used to derive $\mathbb{Z}_2^2$-graded supersymmetric classical actions. The irreps are four dimensional for $ \lambda = 0$ where $ \lambda $ is an eigenvalue of the Casimir element, and eight dimensional for $\lambda \neq 0.$ The eight dimensional irreps reduce to four dimensional ones only when $\lambda$ and an eigenvalue of Hamiltonian satisfy a particular relation. The reduced four dimensional irreps are used to define $\mathbb{Z}_2^2$-graded supersymmetry transformations and two types of classical actions invariant under the transformations are presented. It is shown that one of the Noether charges vanishes if all the variables of specific $\mathbb{Z}_2^2$-degree are auxiliary.