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)
  Copy   BIBTEX

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.

Other Versions

No versions found

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 100,497

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Analytics

Added to PP
2015-09-07

Downloads
83 (#249,363)

6 months
23 (#129,576)

Historical graph of downloads
How can I increase my downloads?

Author Profiles

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

Citations of this work

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

View all 24 citations / Add more citations