On a ${\mathbb{Z}}_2^n$-Graded Version of Supersymmetry

Abstract

We extend the notion of super-Minkowski space-time to include ${\mathbb{Z}}_2^n$-graded (Majorana) spinor coordinates. Our choice of the grading leads to spinor coordinates that are nilpotent but commute amongst themselves. The mathematical framework we employ is the recently developed category of ${\mathbb{Z}}_2^n$-manifolds understood as locally ringed spaces. The formalism we present resembles $\mathcal{N}$-extended superspace (in the presence of central charges), but with some subtle differences due to the exotic nature of the grading employed.

arXiv comment15 pages. Comments welcomed. Part of this work was reported on at the miniworkshop "Supergeometry and Applications", University of Luxembourg, December 14-15, 2017

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• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded parastatistics in multiparticle quantum Hamiltonians
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the classical theory
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the quantization
• $\mathcal{N}$-Extension of duble-graded supersymmetric and superconformal quantum mechanics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded supersymmetry: $2$-d sigma models
• 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?
• Minimal bosonization of double-graded supersymmetric quantum mechanics
• ${\mathbb{Z}}_2^n$-Graded extensions of supersymmetric quantum mechanics via Clifford algebras
• The $Z_2\times Z_2$-graded general linear Lie superalgebra
• Comments of ${\mathbb{Z}}_2^2$-supersymmetry in superfield formalism
• Irreducible representations of ${\mathbb{Z}}_2^2$-graded $\mathcal{N}=2$ supersymmetry algebra and ${\mathbb{Z}}_2^2$-graded supermechanics
• 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 $osp(1|2)$
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• Double-graded quantum superplane
• Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds
• Local Forms of Morphisms of Colored Supermanifolds
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalizations of infinite dimensional Lie superalgebra of conformal type with complete classification of central extensions
• Odd Connections on Supermanifolds: Existence and relation with Affine Connections
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• The Schwarz-Voronov Embedding of ${\mathbb{Z}}_2^n$-Manifolds
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• Extended supersymmetry with central charges in Dirac action with curved extra dimensions
• $\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction
• Linear ${\mathbb{Z}}_2^n$-Manifolds and Linear Actions
• Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1|2)$ and Virasoro algebra
• Principal bundles in the category of ${\mathbb{Z}}_2^n$-manifolds
• Universal weight systems from a minimal ${\mathbb{Z}}_2^2$-graded Lie algebra