Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909709506707456 |
|---|---|
| author | Chatterjee, Saikat Chaudhuri, Adittya Koushik, Praphulla |
| author_facet | Chatterjee, Saikat Chaudhuri, Adittya Koushik, Praphulla |
| contents | In this paper, a notion of a principal $2$-bundle over a Lie groupoid has been introduced. For such principal $2$-bundles, we produced a short exact sequence of VB-groupoids, namely, the Atiyah sequence. Two notions of connection structures viz. strict connections and semi-strict connections on a principal $2$-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism have been introduced. We constructed a class of principal $\mathbb{G}=[G_1\rightrightarrows G_0]$-bundles and connections from a given principal $G_0$-bundle $E_0\rightarrow X_0$ over $[X_1\rightrightarrows X_0]$ with connection. An existence criterion for the connections on a principal $2$-bundle over a proper, étale Lie groupoid is proposed. The action of the $2$-group of gauge transformations on the category of strict and semi-strict connections has been studied. Finally we noted an extended symmetry of the category of semi-strict connections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_13747 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid Chatterjee, Saikat Chaudhuri, Adittya Koushik, Praphulla Differential Geometry Mathematical Physics Category Theory Primary 53C08, Secondary 22A22, 58H05 In this paper, a notion of a principal $2$-bundle over a Lie groupoid has been introduced. For such principal $2$-bundles, we produced a short exact sequence of VB-groupoids, namely, the Atiyah sequence. Two notions of connection structures viz. strict connections and semi-strict connections on a principal $2$-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism have been introduced. We constructed a class of principal $\mathbb{G}=[G_1\rightrightarrows G_0]$-bundles and connections from a given principal $G_0$-bundle $E_0\rightarrow X_0$ over $[X_1\rightrightarrows X_0]$ with connection. An existence criterion for the connections on a principal $2$-bundle over a proper, étale Lie groupoid is proposed. The action of the $2$-group of gauge transformations on the category of strict and semi-strict connections has been studied. Finally we noted an extended symmetry of the category of semi-strict connections. |
| title | Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid |
| topic | Differential Geometry Mathematical Physics Category Theory Primary 53C08, Secondary 22A22, 58H05 |
| url | https://arxiv.org/abs/2107.13747 |