Sil, S (2024) Topology of weak G-bundles via Coulomb gauges in critical dimensions. In: Communications in Analysis and Geometry, 32 (3). pp. 791-835.
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Abstract
The transition maps for a Sobolev G-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle P is equipped with a Sobolev connection A, then one can associate a topological isomorphism class to the pair (P,A), which is invariant under Sobolev gauge changes and coincides with the usual notions for regular bundles and connections. This is based on a regularity result which says any bundle in the critical dimension in which a Sobolev connection is in Coulomb gauges are actually C0, α for any α < 1. We also show any such pair can be strongly approximated by smooth connections on smooth bundles. Finally, we prove that for sequences (Pν,Aν) with uniformly bounded n/2-Yang-Mills energy, the topology stabilizes if the n/2-th power of the norm of the curvatures are equiintegrable. This implies a criterion to detect topological flatness in Sobolev bundles in critical dimensions via n/2-Yang-Mills energy. © 2024 International Press, Inc.. All rights reserved.
| Item Type: | Journal Article |
|---|---|
| Publication: | Communications in Analysis and Geometry |
| Publisher: | International Press, Inc. |
| Additional Information: | The copyright for this article belongs to the authors. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 27 Nov 2024 11:11 |
| Last Modified: | 27 Nov 2024 11:11 |
| URI: | http://eprints.iisc.ac.in/id/eprint/86987 |
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