${\mathbb{Z}}_2^n$-Supergeometry I: Manifolds and Morphisms

Abstract

In Physics and in Mathematics ${\mathbb{Z}}_2^n$-gradings, $n\ge 2$, do appear quite frequently. The corresponding sign rules are determined by the `scalar product' of the involved ${\mathbb{Z}}_2^n$-degrees. The present paper is the first of a series on ${\mathbb{Z}}_2^n$-Supergeometry. The new theory 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 (the parity is the parity of the total degree). It is based on the hierarchy: ` ${\mathbb{Z}}_2^0$-Supergeometry (classical differential Geometry) contains the germ of ${\mathbb{Z}}_2^1$-Supergeometry (standard Supergeometry), which in turn contains the sprout of ${\mathbb{Z}}_2^2$-Supergeometry, etc.' The ${\mathbb{Z}}_2^n$-supergeometric viewpoint provides deeper insight and simplified solutions; interesting relations with Quantum Field Theory and Quantum Mechanics are expected. In this article, we define ${\mathbb{Z}}_2^n$-supermanifolds and provide examples in the atlas, the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the category of ${\mathbb{Z}}_2^n$-supermanifolds is closed with respect to the tangent and cotangent functors. The fundamental theorem describing supermorphisms in terms of coordinates is extended to the ${\mathbb{Z}}_2^n$-context.

29 pages, added references, more context in the introduction and in section 3.2

References in corpus (2)

• Higher vector bundles and multi-graded symplectic manifolds
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Cited by in corpus (8)

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• Introduction to graded geometry
• Atiyah classes and dg-Lie algebroids for matched pairs
• Polarisation of Graded Bundles
• The Frobenius theorem for ${\mathbb{Z}}_2^n$-supermanifolds
• ${\mathbb{Z}}_2^n$-Supergeometry II: Batchelor-Gawedzki Theorem
• $N=1$ fermionic D3-branes in RNS formulation I. $C^{\infty }$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\hat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace
• On the Construction of ${\mathbb{Z}}_2^n-$Supergrassmannians as Homogeneous ${\mathbb{Z}}_2^n-$Superspaces