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

Journal of Mathematical Physics 57:102902 (2016)
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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, ], 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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Author Profiles

James Weatherall
University of California, Irvine
Sarita Rosenstock
University of Melbourne

Citations of this work

How to count structure.Thomas William Barrett - 2022 - Noûs 56 (2):295-322.
Regarding the ‘Hole Argument’.James Owen Weatherall - 2018 - British Journal for the Philosophy of Science 69 (2):329-350.
Sophistication about Symmetries.Neil Dewar - 2019 - British Journal for the Philosophy of Science 70 (2):485-521.
Equivalent and Inequivalent Formulations of Classical Mechanics.Thomas William Barrett - 2019 - British Journal for the Philosophy of Science 70 (4):1167-1199.
Regarding the ‘Hole Argument’.James Owen Weatherall - 2016 - British Journal for the Philosophy of Science:axw012.

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