The division map of principal bundles with groupoid structure and generalized gauge transformations

TL;DR

This paper extends the concept of the groupoid of generalized gauge transformations to principal bundles with a Lie groupoid structure, introducing a division map framework applicable in this more general setting.

Contribution

It generalizes the framework of gauge transformations from groups to Lie groupoids for principal bundles, developing the division map and related concepts in this broader context.

Findings

Established the division map for principal groupoid bundles.

Translated key concepts from group-based to groupoid-based gauge theories.

Provided a foundation for further studies in generalized gauge transformations.

Abstract

Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $\mathsf{Bun}_{G,M}$ of principal $G$-bundles over a given manifold $M$, I develop in this paper the same ideas for the more general case of {\em principal $\calG$-bundles or principal bundles with structure groupoid $\calG$}, where now $\calG$ is a Lie groupoid in the sense of \cite{Moer2}. Most of the concepts introduced in \cite{C1} can be translated almost verbatim in the framework of principal bundles with structure groupoid $\calG$; in particular, the key r�le for the construction of generalized gauge transformations is again played by (the equivalent in the framework of principal bundles with groupoid structure of) the division map $f_P$. Of great importance are also the generalized conjugation in a groupoid and the concept of (twisted) equivariant maps between groupoid-spaces.