${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-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.

arXiv comment23 pages; final version to appear in Eur. Phys. J. C with added comments and references

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• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the quantization
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• Inequivalent quantizations from gradings and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ parabosons
• Is the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded sine-Gordon equation integrable?
• Classification of minimal ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie (super)algebras and some applications
• Minimal bosonization of double-graded supersymmetric quantum mechanics
• Comments of ${\mathbb{Z}}_2^2$-supersymmetry in superfield formalism
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• New aspects of the Z${}_{\text{2}}$ $\times$ Z${}_{\text{2}}$-graded 1D superspace: induced strings and 2D relativistic models
• A classification of lowest weight irreducible modules over ${\mathbb{Z}}_2^2$-graded extension of $osp(1|2)$
• On classical $Z_2\times Z_2$-graded Lie algebras
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
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