Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

$\mathbb{Z}_2^n$-Supergeometry I: Manifolds and Morphisms (1408.2755v2)

Published 12 Aug 2014 in math.DG, math-ph, math.AG, math.MP, and math.QA

Abstract: In Physics and in Mathematics $\mathbb{Z}_2n$-gradings, $n \geq 2$, 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: $\mathbb{Z}_20$-Supergeometry (classical differential Geometry) contains the germ of $\mathbb{Z}_21$-Supergeometry (standard Supergeometry), which in turn contains the sprout of $\mathbb{Z}_22$-Supergeometry, etc.' The $\mathbb{Z}_2n$-supergeometric viewpoint provides deeper insight and simplified solutions; interesting relations with Quantum Field Theory and Quantum Mechanics are expected. In this article, we define $\mathbb{Z}_2n$-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 $\mathbb{Z}_2n$-supermanifolds is closed with respect to the tangent and cotangent functors. The fundamental theorem describing supermorphisms in terms of coordinates is extended to the $\mathbb{Z}_2n$-context.

Summary

We haven't generated a summary for this paper yet.