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

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.

22 pages, editorial changes in one section