Rosenstock, Sarita and Weatherall, James Owen (2015) A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold. [Preprint]
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Abstract
A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation.
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| Item Type: | Preprint | |||||||||
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| Keywords: | Yang-Mills theory; holonomy maps; principal bundles; loops | |||||||||
| Subjects: | Specific Sciences > Physics > Classical Physics Specific Sciences > Physics Specific Sciences > Physics > Quantum Field Theory |
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| Depositing User: | James Owen Weatherall | |||||||||
| Date Deposited: | 10 Apr 2015 14:19 | |||||||||
| Last Modified: | 13 Feb 2016 18:48 | |||||||||
| Item ID: | 11410 | |||||||||
| Subjects: | Specific Sciences > Physics > Classical Physics Specific Sciences > Physics Specific Sciences > Physics > Quantum Field Theory |
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| Date: | 2015 | |||||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/11410 |
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- A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold. (deposited 10 Apr 2015 14:19) [Currently Displayed]
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