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

Abstract

Irreducible representations (irreps) of ${\mathbb{Z}}_2^2$-graded supersymmetry algebra of $\mathcal{N}=2$ 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 $\lambda \ne 0.$ 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.

17 pages, no figures

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Cited by in corpus (5)

• New aspects of the Z${}_{\text{2}}$ $\times$ Z${}_{\text{2}}$-graded 1D superspace: induced strings and 2D relativistic models
• Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
• Orthosymplectic $Z_2\times Z_2$-graded Lie superalgebras and parastatistics
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• $\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction