$\mathcal{N}$-Extension of duble-graded supersymmetric and superconformal quantum mechanics

Abstract

In the recent paper, Bruce and Duplij introduced a double-graded version of supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic nature of $\mathcal{N}=1$ SQM to a ${\mathbb{Z}}_2^2$-graded superalgebra. In this work, we propose an extension of Bruce-Duplij model to higher values of $\mathcal{N}.$ Furthermore, it is shown that our construction of double-graded SQM is a special case of the method which converts a given Lie superalgebra to a ${\mathbb{Z}}_2^2$-graded superalgebra. By employing this method one may convert a model of superconformal mechanics to its double-graded version. The simplest example of $\mathcal{N}=1$ double-graded superconformal mechanics is studied in some detail.

arXiv comment15 pages, 2 figures, new reference, error corrections

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• 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
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• Graded colour Lie superalgebras for solving Lévy-Leblond equations
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
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• Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1|2)$ and Virasoro algebra
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