A constructive approach to generalized principal connections

TL;DR

This paper develops a geometric framework for generalized principal connections, extending classical concepts to broader fiber bundle contexts, with implications for generalized gauge theories.

Contribution

It introduces generalized principal bundle coordinates, transformation laws, and characterizations, linking them to classical principal connections and fiber bundles.

Findings

Any Lie group fiber bundle is a generalized principal bundle.

Generalized principal connections are associated with Lie group fiber bundle connections.

The framework recovers known results and aids in understanding generalized gauge theories.

Abstract

We address the recently introduced notions of generalized principal bundle and generalized principal connection by keeping track of global geometric properties through local coordinate transformation laws. This approach leads us to introduce generalized principal bundle coordinates and to find their transformation laws. Besides, we show that any Lie group fiber bundle (and hence, in particular, any vector bundle) is a generalized principal bundle and we give a proof of the fact that any Lie group fiber bundle with connected typical fiber is an associated bundle to a suitable principal bundle. Moreover, we present a direct way to characterize Lie group fiber bundle connections and generalized principal connections in terms of horizontal lifts and of local conditions. Finally, we recover in our setting some already known results, including that generalized principal connections are associated only to Lie group fiber bundle connections and that they reduce to usual principal connections on standard principal bundles. Our results are needed in order to understand how generalized principal connections might fit in the fiber bundle treatment of classical field theories, aiming towards a notion of generalized gauge theory.