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

19 pages, minor typos corrected and clarifying comments added. The exposition has been improved. Comments welcomed