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

Abstract

<p style='text-indent:20px;'>Roughly speaking, <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb Z}_2^n $\end{document}</tex-math></inline-formula>-manifolds are 'manifolds' equipped with <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb Z}_2^n $\end{document}</tex-math></inline-formula>-graded commutative coordinates with the sign rule being determined by the scalar product of their <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb Z}_2^n $\end{document}</tex-math></inline-formula>-degrees. We examine the notion of a symplectic <inline-formula><tex-math id="M4">\begin{document}$ {\mathbb Z}_2^n $\end{document}</tex-math></inline-formula>-manifold, i.e., a <inline-formula><tex-math id="M5">\begin{document}$ {\mathbb Z}_2^n $\end{document}</tex-math></inline-formula>-manifold equipped with a symplectic two-form that may carry non-zero <inline-formula><tex-math id="M6">\begin{document}$ {\mathbb Z}_2^n $\end{document}</tex-math></inline-formula>-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.</p>