${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the classical theory

Abstract

${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics admits four types of particles: ordinary bosons, two classes of fermions (fermions belonging to different classes commute among each other) and exotic bosons. In this paper we construct the basic ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded worldline multiplets (extending the cases of one-dimensional supersymmetry) and compute, based on a general scheme, their invariant classical actions and worldline sigma-models. The four basic multiplets contain two bosons and two fermions. They are $(2,2,0)$, with two propagating bosons and two propagating fermions, $(1,2,1{)}_{[00]}$ (the ordinary boson is propagating, while the exotic boson is an auxiliary field), $(1,2,1{)}_{[11]}$ (the converse case, the exotic boson is propagating, while the ordinary boson is an auxiliary field) and, finally, $(0,2,2)$ with two bosonic auxiliary fields. Classical actions invariant under the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superalgebra are constructed for both single multiplets and interacting multiplets. Furthermore, scale-invariant actions can possess a full ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded conformal invariance spanned by $10$ generators and containing an $sl(2)$ subalgebra.

23 pages; final version to appear in Eur. Phys. J. C with added comments and references

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