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

TL;DR

This paper extends symplectic geometry concepts to $Z_2^n$-manifolds, introducing symplectic forms with non-zero degrees and generalizing fundamental results like Darboux's theorem to this higher graded setting.

Contribution

It introduces the notion of symplectic $Z_2^n$-manifolds and generalizes key symplectic geometry results, including Darboux's theorem, to this new framework.

Findings

Generalization of symplectic geometry to $Z_2^n$-graded manifolds

Existence of Darboux's theorem in the higher graded setting

Development of foundational notions for symplectic $Z_2^n$-manifolds

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.