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.

arXiv comment19 pages, minor typos corrected and clarifying comments added. The exposition has been improved. Comments welcomed

References in corpus (27)

• Geometry of Batalin-Vilkovisky quantization
• Once more on parastatistics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie Symmetries of the Lévy-Leblond Equations
• Double-Graded Supersymmetric Quantum Mechanics
• Characteristic classes of gauge systems
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the classical theory
• On a ${\mathbb{Z}}_2^n$-Graded Version of Supersymmetry
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the quantization
• $Q$-Manifolds and Mackenzie Theory
• $\mathcal{N}$-Extension of duble-graded supersymmetric and superconformal quantum mechanics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded supersymmetry: $2$-d sigma models
• Higher Trace and Berezinian of Matrices over a Clifford Algebra
• Splitting theorem for ${\mathbb{Z}}_2^n$-supermanifolds
• ${\mathbb{Z}}_2^n$-Graded extensions of supersymmetric quantum mechanics via Clifford algebras
• Super-de Sitter and alternative super-Poincaré symmetries
• The Graded Differential Geometry of Mixed Symmetry Tensors
• The geodesic flow on a Riemannian supermanifold
• Functional analytic issues in ${\mathbb{Z}}_2^n$-Geometry
• Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds
• Fedosov and Riemannian supermanifolds
• Products in the category of ${\mathbb{Z}}_2^n$-manifolds
• Parabose-Parafermi Supersymmetry
• Irreducible holonomy algebras of Riemannian supermanifolds
• The Schwarz-Voronov Embedding of ${\mathbb{Z}}_2^n$-Manifolds
• Odd Connections on Supermanifolds: Existence and relation with Affine Connections
• Para-Generalization of Peierls Bracket Quantization
• On a geometric framework for Lagrangian supermechanics

Cited by in corpus (4)

• Irreducible representations of ${\mathbb{Z}}_2^2$-graded $\mathcal{N}=2$ supersymmetry algebra and ${\mathbb{Z}}_2^2$-graded supermechanics
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace