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

The category of $\mathbb{Z}_2^n$-supermanifolds (1602.03312v2)

Published 10 Feb 2016 in math.DG, math-ph, and math.MP

Abstract: In Physics and in Mathematics $\mathbb{Z}_2n$-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 canonicalsuperization' to a $\mathbb{Z}_2n$-supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the $\mathbb{Z}_2n$-context.

Summary

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