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arXiv · 2103.00249

Symplectic $\mathbb{Z}_2^n$-manifolds

Andrew James Bruce·Janusz Grabowski
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Abstract

Roughly speaking, $\mathbb{Z}_2^n$-manifolds are `manifolds' equipped with $\mathbb{Z}_2^n$-graded commutative coordinates with the sign rule being determined by the scalar product of their $\mathbb{Z}_2^n$-degrees. We examine the notion of a symplectic $\mathbb{Z}_2^n$-manifold, i.e., a $\mathbb{Z}_2^n$-manifold equipped with a symplectic two-form that may carry non-zero $\mathbb{Z}_2^n$-degree. We show that the basic notions and results of symplectic geometry generalise to the `higher graded' setting, including a generalisation of Darboux's theorem.

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BibTeXRIS

Andrew James Bruce, Janusz Grabowski. 2021-02-27. Symplectic $\mathbb{Z}_2^n$-manifolds. https://doi.org/10.3934/jgm.2021020

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