On gauge theory and parallel transport in principal 2-bundles over Lie groupoids
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910681258786816 |
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| author | Chaudhuri, Adittya |
| author_facet | Chaudhuri, Adittya |
| contents | We investigate an interplay between some ideas in traditional gauge theory and certain concepts in fibered categories. We accomplish this by introducing a notion of a principal Lie 2-group bundle over a Lie groupoid and studying its connection structures, gauge transformations, and parallel transport.
We obtain a Lie 2-group torsor version of the one-one correspondence between fibered categories and pseudofunctors. This results in a classification of our principal 2-bundles based on their underlying fibration structures. This allows us to extend a class of our principal 2-bundles to be defined over differentiable stacks presented by the base Lie groupoids. We construct a short exact sequence of VB-groupoids, namely, the 'Atiyah sequence' associated to our principal 2-bundles. Splitting and splitting up to a natural isomorphism of our Atiyah sequence, respectively, gives us notions of 'strict connections' and 'semi-strict connections' on our principal 2-bundles. We describe such connections in terms of Lie 2-algebra valued 1-forms on the total Lie groupoids. The underlying fibration structure of our 2-bundle provides an existence criterion for strict and semi-strict connections. We study the action of the 2-group of gauge transformations on the groupoid of strict and semi-strict connections, and interestingly, we observe an extended symmetry of semi-strict connections. We demonstrate an interrelationship between `differential geometric connection-induced horizontal path lifting property in traditional principal bundles' and the `category theoretic cartesian lifting of morphisms in fibered categories' by developing a theory of connection-induced parallel transport along a particular class of Haefliger paths in the base Lie groupoid of our principle 2-bundles. Finally, we employ our results to introduce a notion of parallel transport along Haefliger paths in the setup of VB-groupoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On gauge theory and parallel transport in principal 2-bundles over Lie groupoids Chaudhuri, Adittya Differential Geometry Mathematical Physics Category Theory 53C08, 22A22, 58H05 We investigate an interplay between some ideas in traditional gauge theory and certain concepts in fibered categories. We accomplish this by introducing a notion of a principal Lie 2-group bundle over a Lie groupoid and studying its connection structures, gauge transformations, and parallel transport. We obtain a Lie 2-group torsor version of the one-one correspondence between fibered categories and pseudofunctors. This results in a classification of our principal 2-bundles based on their underlying fibration structures. This allows us to extend a class of our principal 2-bundles to be defined over differentiable stacks presented by the base Lie groupoids. We construct a short exact sequence of VB-groupoids, namely, the 'Atiyah sequence' associated to our principal 2-bundles. Splitting and splitting up to a natural isomorphism of our Atiyah sequence, respectively, gives us notions of 'strict connections' and 'semi-strict connections' on our principal 2-bundles. We describe such connections in terms of Lie 2-algebra valued 1-forms on the total Lie groupoids. The underlying fibration structure of our 2-bundle provides an existence criterion for strict and semi-strict connections. We study the action of the 2-group of gauge transformations on the groupoid of strict and semi-strict connections, and interestingly, we observe an extended symmetry of semi-strict connections. We demonstrate an interrelationship between `differential geometric connection-induced horizontal path lifting property in traditional principal bundles' and the `category theoretic cartesian lifting of morphisms in fibered categories' by developing a theory of connection-induced parallel transport along a particular class of Haefliger paths in the base Lie groupoid of our principle 2-bundles. Finally, we employ our results to introduce a notion of parallel transport along Haefliger paths in the setup of VB-groupoids. |
| title | On gauge theory and parallel transport in principal 2-bundles over Lie groupoids |
| topic | Differential Geometry Mathematical Physics Category Theory 53C08, 22A22, 58H05 |
| url | https://arxiv.org/abs/2411.00814 |