Principaloid bundles

Abstract

We present a novel generalisation of principal bundles -- principaloid bundles: These are fibre bundles $π:P\to B$ where the typical fibre is the arrow manifold $G$ of a Lie groupoid $G\rightrightarrows M$ and the structure group is reduced to the latter's group of bisections. Each such bundle canonically comes with a bundle map $D:P\to F$ to another fibre bundle $F$ over the base $B$, with typical fibre $M$. Examples of principaloid bundles include ordinary principal $\underline G$-bundles, obtained for $G:=\underline G\rightrightarrows\bullet$, bundles associated to them, obtained for action groupoids $G:=\underline G\ltimes M$, and general fibre bundles if $G$ is a pair groupoid. While $π$ is far from being a principal $G$-bundle, we prove that $D$ is one. Connections on the principaloid bundle $π$ are thus required to be $G$-invariant Ehresmann connections. In the three examples mentioned above, this reproduces the usual types of connection for each of them. In a local description over a trivialising cover $\{O_i\}$ of $B$, the connection gives rise to Lie algebroid-valued objects living over bundle trivialisations $\{O_i\times M\}$ of $F$. Their behaviour under bundle automorphisms, including gauge transformations, is studied in detail. Finally, we construct the Atiyah-Ehresmann groupoid ${\rm At}(P)\rightrightarrows F$ which governs symmetries of $P$, this time mapping distinct $D$-fibres to one another in general. It is a fibre-bundle object in the category of Lie groupoids, with typical fibre $G\rightrightarrows M$ and base $B\times B\rightrightarrows B$. We show that those of its bisections which project to bisections of its base are in a one-to-one correspondence with automorphisms of $π$.