Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds

Abstract

Very loosely, ${\mathbb{Z}}_2^n$-manifolds are `manifolds' with ${\mathbb{Z}}_2^n$-graded coordinates and their sign rule is determined by the scalar product of their ${\mathbb{Z}}_2^n$-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian ${\mathbb{Z}}_2^n$-manifold, i.e., a ${\mathbb{Z}}_2^n$-manifold equipped with a Riemannian metric that may carry non-zero ${\mathbb{Z}}_2^n$-degree. We show that the basic notions and tenets of Riemannian geometry directly generalise to the setting of ${\mathbb{Z}}_2^n$-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

17 pages. Comments welcomed

References in corpus (16)

• Once more on parastatistics
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Cited by in corpus (9)

• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded parastatistics in multiparticle quantum Hamiltonians
• Irreducible representations of ${\mathbb{Z}}_2^2$-graded $\mathcal{N}=2$ supersymmetry algebra and ${\mathbb{Z}}_2^2$-graded supermechanics
• A classification of lowest weight irreducible modules over ${\mathbb{Z}}_2^2$-graded extension of $osp(1|2)$
• 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
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs