$H$-covering of a supermanifold

TL;DR

This paper extends the theory of supermanifolds by developing $H$-graded structures for any finitely generated abelian group and introduces the concept of $H$-graded coverings for finite abelian groups, enriching the geometric framework.

Contribution

It introduces a general theory of $H$-graded supermanifolds for arbitrary finitely generated abelian groups and defines $H$-graded coverings for finite abelian groups, expanding existing supergeometry concepts.

Findings

Established the theory of $H$-graded supermanifolds for any finitely generated abelian group.

Defined and analyzed $H$-graded coverings of supermanifolds for finite abelian groups.

Connected representation theory tools with supergeometry structures.

Abstract

We develop the theory of $H$-graded manifolds for any finitely generated abelian group, using tools from representation theory. Furthermore, we introduce and investigate the notion of $H$-graded coverings of supermanifolds in the case where $H$ is a finite abelian group.