Principal bundles in the category of $\mathbb{Z}_2^n$-manifolds
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913996962004992 |
|---|---|
| author | Bruce, Andrew James Grabowski, Janusz |
| author_facet | Bruce, Andrew James Grabowski, Janusz |
| contents | We introduce and examine the notion of principal $\mathbb{Z}_2^n$-bundles, i.e., principal bundles in the category of $\mathbb{Z}_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $\mathbb{Z}_2^n$-grading replaces $\mathbb{Z}_2$-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of $\mathbb{Z}_2^n$-manifolds is essentially different than that of supermanifolds, as for $n>1$ we have formal variables of even parity, so local smooth functions are formal power series. On the other hand, a full version of differential calculus is still valid. We show in this paper that the fundamental properties of classical principal bundles can be generalised to the setting of this `higher graded' geometry, with properly defined frame bundles of $\mathbb{Z}_2^n$-vector bundles as canonical examples. However, formulating these concepts and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to $\mathbb{Z}_2^n$-manifolds is therefore included together with basic examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Principal bundles in the category of $\mathbb{Z}_2^n$-manifolds Bruce, Andrew James Grabowski, Janusz Differential Geometry Mathematical Physics 14A22, 18F15, 58A50, 58D19 We introduce and examine the notion of principal $\mathbb{Z}_2^n$-bundles, i.e., principal bundles in the category of $\mathbb{Z}_2^n$-manifolds. The latter are higher graded extensions of supermanifolds in which a $\mathbb{Z}_2^n$-grading replaces $\mathbb{Z}_2$-grading. These extensions have opened up new areas of research of great interest in both physics and mathematics. In principle, the geometry of $\mathbb{Z}_2^n$-manifolds is essentially different than that of supermanifolds, as for $n>1$ we have formal variables of even parity, so local smooth functions are formal power series. On the other hand, a full version of differential calculus is still valid. We show in this paper that the fundamental properties of classical principal bundles can be generalised to the setting of this `higher graded' geometry, with properly defined frame bundles of $\mathbb{Z}_2^n$-vector bundles as canonical examples. However, formulating these concepts and proving these results relies on many technical upshots established in earlier papers. A comprehensive introduction to $\mathbb{Z}_2^n$-manifolds is therefore included together with basic examples. |
| title | Principal bundles in the category of $\mathbb{Z}_2^n$-manifolds |
| topic | Differential Geometry Mathematical Physics 14A22, 18F15, 58A50, 58D19 |
| url | https://arxiv.org/abs/2412.12652 |