The category of ${\mathbb{Z}}_2^n$-supermanifolds

Abstract

In Physics and in Mathematics ${\mathbb{Z}}_2^n$-gradings, $n>1$, appear in various fields. The corresponding sign rule is determined by the `scalar product' of the involved ${\mathbb{Z}}_2^n$-degrees. The ${\mathbb{Z}}_2^n$-Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. In this article we develop the foundations of the theory: we define ${\mathbb{Z}}_2^n$-supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of ${\mathbb{Z}}_2^{\bullet }$-supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any $n$-fold vector bundle has a canonical `superization' to a ${\mathbb{Z}}_2^n$-supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the ${\mathbb{Z}}_2^n$-context.

18 pages. arXiv admin note: substantial text overlap with arXiv:1408.2755. Added references, added concluding remarks. To appear in Journal of Mathematical Physics

References in corpus (4)

• $Q$-Manifolds and Mackenzie Theory
• Spinor-Vector Duality in Heterotic String Orbifolds
• Graded Bundles in the Category of Lie Groupoids
• Lie superalgebras of differential operators

Cited by in corpus (2)

• Introduction to graded geometry
• Polarisation of Graded Bundles