Principal bundles in the category of Z2n-manifolds

We introduce and examine the notion of principal $Z_2^n$-bundles, i.e., principal bundles in the category of $Z_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $Z_2^n$-grading replaces $Z_2$-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of $Z_2^n$-manifolds is essentially different than that of supermanifolds, as for n > 1 we have formal variables of even parity, so local smooth functions are power series in formal variables. 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 $Z_2^n$-vector bundles as canonical examples. Additionally, we propose a new approach to the concept of a vector bundle in this setting. However, formulating these ideas and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to $Z_2^n$-manifolds is therefore included together with basic examples.