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A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold

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
Creators:
CreatorsEmailORCID
Rosenstock, Saritarosensts@uci.edu
Weatherall, James Owenweatherj@uci.edu
Keywords: Yang-Mills theory; holonomy maps; principal bundles; loops
Subjects: Specific Sciences > Physics > Classical Physics
Specific Sciences > Physics
Specific Sciences > Physics > Quantum Field Theory
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
Date: 2015
URI: http://philsci-archive.pitt.edu/id/eprint/11410

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