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

<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>

Symplectic manifolds

The third chapter introduces the basic notions of symplectic topology, such as symplectic forms, symplectomorphisms, and Lagrangian submanifolds. A fundamental classical construction is Moser isotopy, with its various applications such as Darboux’s theorem and the Lagrangian neighbourhood theorem. The chapter now includes a brief discussion of the Chekanov torus and Luttinger surgery. The last section on contact structures has been significantly expanded.

Poisson Algebra of Differential Forms

We give a natural definition of a Poisson differential algebra. Consistency conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on the differential calculus in a simple canonical form by a coordinate trans-formation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization of the exterior derivative through a certain canonical one-form. All the above are carried out similarly for the case of a complex Poisson differential algebra. The case of one complex dimension is treated in detail and interesting features are noted. Conclusions are made in the last section.