Local Forms of Morphisms of Colored Supermanifolds

TL;DR

This paper develops the local differential calculus for colored supermanifolds, extending classical theorems like inverse and implicit function theorems to the $Z_2^n$-graded setting, advancing the mathematical foundation of supergeometry.

Contribution

It provides a detailed account of the $Z_2^n$-differential calculus and proves local theorems such as inverse, implicit, and constant rank theorems for colored supermanifolds, which is a novel extension.

Findings

Extension of differential calculus to $Z_2^n$-supermanifolds

Proof of inverse function theorem in $Z_2^n$-setting

Proof of implicit and constant rank theorems in $Z_2^n$-supergeometry

Abstract

In \cite{Covolo:2016}, \cite{Covolo:2012} and \cite{Poncin:2016}, we introduced the category of colored supermanifolds ($\mathbb{Z}_2^n$-super\-ma\-ni\-folds or just $\mathbb{Z}_2^n$-manifolds ($\mathbb{Z}_2^n=\mathbb{Z}_2\times\ldots\times\mathbb{Z}_2$ ($n$ times))), explicitly described the corresponding $\mathbb{Z}_2^n$-Berezinian and gave first insights into $\mathbb{Z}_2^n$-integration theory. The present paper contains a detailed account of parts of the $\mathbb{Z}_2^n$-differential calculus and of the $\mathbb{Z}_2^n$-variants of the trilogy of local theorems, which consists of the inverse function theorem, the implicit function theorem and the constant rank theorem.