Global Theory of Graded Manifolds

TL;DR

This paper develops a rigorous mathematical foundation for graded manifolds, generalizing differential geometry to include functions with graded variables, resolving key definitional issues, and focusing on geometric constructions.

Contribution

It provides a comprehensive and consistent framework for the geometry of graded manifolds, addressing previous definitional challenges and including detailed algebraic and sheaf-theoretic foundations.

Findings

Resolved issues in defining graded manifolds with mixed positive and negative coordinates

Established a sheaf-theoretic approach for global description of graded manifolds

Included detailed exposition of graded algebra and sheaf theory

Abstract

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded variables which can either commute or anticommute, according to their degree. To obtain a consistent global description of graded manifolds, one resorts to sheaves of graded commutative associative algebras on second countable Hausdorff topological spaces, locally isomorphic to a suitable "model space". This paper aims to build robust mathematical foundations of geometry of graded manifolds. Some known issues in their definition are resolved, especially the case where positively and negatively graded coordinates appear together. The focus is on a detailed exposition of standard geometrical constructions rather then on applications. Necessary excerpts from graded algebra and graded sheaf theory are included.