Principal bundles in the category of $\mathbb{Z}_2^n$-manifolds

TL;DR

This paper extends the concept of principal bundles to the setting of $Z_2^n$-manifolds, a higher graded generalization of supermanifolds, exploring their geometric properties and foundational aspects.

Contribution

It introduces and develops the theory of principal $Z_2^n$-bundles, generalizing classical principal bundle concepts to higher graded supergeometry.

Findings

Principal $Z_2^n$-bundles can be constructed with properties analogous to classical bundles.

Frame bundles of $Z_2^n$-vector bundles serve as canonical examples.

The differential calculus extends to $Z_2^n$-manifolds despite their formal power series structure.

Abstract

We introduce and examine the notion of principal $\mathbb{Z}_2^n$-bundles, i.e., principal bundles in the category of $\mathbb{Z}_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $\mathbb{Z}_2^n$-grading replaces $\mathbb{Z}_2$-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of $\mathbb{Z}_2^n$-manifolds is essentially different than that of supermanifolds, as for $n>1$ we have formal variables of even parity, so local smooth functions are formal power series. On the other hand, a full version of differential calculus is still valid. We show in this paper that the fundamental properties of classical principal bundles can be generalised to the setting of this `higher graded' geometry, with properly defined frame bundles of $\mathbb{Z}_2^n$-vector bundles as canonical examples. However, formulating these concepts and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to $\mathbb{Z}_2^n$-manifolds is therefore included together with basic examples.