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

Roughly speaking, Zn 2 -manifolds are ‘manifolds’ equipped with Zn 2 -graded commutative coordinates with the sign rule being determined by the scalar product of their Zn 2 -degrees. We examine the notion of a symplectic Zn 2 -manifold, i.e., a Zn 2 -manifold equipped with a symplectic two-form that may carry non-zero Zn 2 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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