On the Optimal Regularity Implied by the Assumptions of Geometry II: Connections on Vector Bundles
| Autor: | Reintjes, Moritz, Temple, Blake |
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| Rok vydání: | 2021 |
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| Zdroj: | Advances in Theoretical and Mathematical Physics, Vol. 28, Issue 5 (2024), pp. 1425-1486 |
| Druh dokumentu: | Working Paper |
| Popis: | We extend authors' prior results on optimal regularity and Uhlenbeck compactness for affine connections to general connections on vector bundles. This is accomplished by deriving a vector bundle version of the RT-equations, and establishing a new existence theory for these equations. These new RT-equations, non-invariant elliptic equations, provide the gauge transformations which transform the fibre component of a non-optimal connection to optimal regularity, i.e., the connection is one derivative more regular than its curvature in $L^p$. The existence theory handles curvature regularity all the way down to, but not including, $L^1$. Taken together with the affine case, our results extend optimal regularity of Kazden-DeTurck and the compactness theorem of Uhlenbeck, applicable to Riemannian geometry and compact gauge groups, to general connections on vector bundles over non-Riemannian manifolds, allowing for both compact and non-compact gauge groups. In particular, this extends optimal regularity and Uhlenbeck compactness to Yang-Mills connections on vector bundles over Lorentzian manifolds as base space, the setting of General Relativity. Comment: Version 4: Improved local and (new) global results; curvature regularity down to L1. Version 3: More details of proof in Section 5. Inclusion of Theorems 2.6 and 2.7. Version 2: New title; improvements to presentation; slightly weaker regularity assumption for the optimal regularity result, and slightly stronger assumption for Uhlenbeck compactness; otherwise results unchanged |
| Databáze: | arXiv |
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