${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded supersymmetry: $2$-d sigma models

Abstract

We propose a natural ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded generalisation of $d=2$, $\mathcal{N}=(1,1)$ supersymmetry and construct a ${\mathbb{Z}}_2^2$-space realisation thereof. Due to the grading, the supercharges close with respect to, in the classical language, a commutator rather than an anticommutator. This is then used to build classical (linear and non-linear) sigma models that exhibit this novel supersymmetry via mimicking standard superspace methods. The fields in our models are bosons, right-handed and left-handed Majorana-Weyl spinors, and exotic bosons. The bosons commute with all the fields, the spinors belong to different sectors that cross commute rather than anticommute, while the exotic boson anticommute with the spinors. As a particular example of one of the models, we present a `double-graded' version of supersymmetric sine-Gordon theory.

arXiv comment17 pages and includes two appendices. The analysis of the double-graded sine-Gordon theory has been extended and numerous sign mistakes have been corrected. Comments still welcomed

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• 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
• 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 $osp(1|2)$
• On classical $Z_2\times Z_2$-graded Lie algebras
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• 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
• Integrable ${\mathbb{Z}}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• $\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction
• On the detectability of paraparticles beyond bosons and fermions
• Transmuted spectrum-generating algebras and detectable parastatistics of the Superconformal Quantum Mechanics
• Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1|2)$ and Virasoro algebra
• Braided quantum mechanics and Majorana qubits at third root of unity: a color Heisenberg-Lie (super)algebra framework