Principaloid bundles
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910872624955392 |
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| author | Strobl, Thomas Suszek, Rafał R. |
| author_facet | Strobl, Thomas Suszek, Rafał R. |
| contents | 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 $π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09886 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Principaloid bundles Strobl, Thomas Suszek, Rafał R. Differential Geometry High Energy Physics - Theory Mathematical Physics 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 $π$. |
| title | Principaloid bundles |
| topic | Differential Geometry High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2503.09886 |