Theory of connections on graded principal bundles

TL;DR

This paper develops the theory of connections on graded principal bundles within graded manifold theory, establishing conditions for triviality, defining graded connections and curvature, and relating these to graded distributions and Lie superalgebras.

Contribution

It provides a comprehensive framework for graded connections on principal bundles, including triviality criteria, definitions of curvature, and their geometric and algebraic relations.

Findings

A graded principal bundle is trivial iff it admits a global graded section.

The sheaf of vertical derivations matches the graded distribution from the structure group.

Curvature determines the involutivity of the horizontal distribution.

Abstract

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection; we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection.