2024/12/17 by Andrew James Bruce, Bruce, Andrew James, Janusz Grabowski +1
Mathematics · #14A22 #18F15 #58A50 #58D19 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2412.12652
openalex publication_date 2024/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and examine the notion of principal ℤ2n-bundles, i.e., principal bundles in the category of ℤ2n-manifolds. The latter are higher graded extensions of supermanifolds in which a ℤ2n-grading replaces ℤ2-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of ℤ2n-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 ℤ2n-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 ℤ2n-manifolds is therefore included together with basic examples.