2025/03/12 by Strobl, Thomas, Suszek, Rafał R.
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2503.09886
We present a novel generalisation of principal bundles -- principaloid bundles: These are fibre bundles π:P→ 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→ 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 \Oi\ of B, the connection gives rise to Lie algebroid-valued objects living over bundle trivialisations \Oi× 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× 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.