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

  • * Corresponding author: Andrew James Bruce

    * Corresponding author: Andrew James Bruce 

The Second author is supported by the Polish National Science Centre grant HARMONIA under the contract number 2016/22/M/ST1/00542

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  • 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.

    Mathematics Subject Classification: Primary: 53D05, 53D15, 58A50; Secondary: 14A22, 17B63, 17B75.


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