Gauge invariant surface holonomy and monopoles

Abstract

There are few known computable examples of non-abelian surface holonomy. In this paper, we give several examples whose structure 2-groups are covering 2-groups and show that the surface holonomies can be computed via a simple formula in terms of paths of 1-dimensional holonomies inspired by earlier work of Chan Hong-Mo and Tsou Sheung Tsun on magnetic monopoles. As a consequence of our work and that of Schreiber and Waldorf, this formula gives a rigorous meaning to non-abelian magnetic flux for magnetic monopoles. In the process, we discuss gauge covariance of surface holonomies for spheres for any 2-group, therefore generalizing the notion of the reduced group introduced by Schreiber and Waldorf. Using these ideas, we also prove that magnetic monopoles have an abelian group structure.

99 pages, 31 figures (2 are new), v2 is published version, updates include: several points clarified, added Defn 2.33 and 3.37 for markings, statement of smoothness removed from Thm 2.39 and 3.41, proof of Thm 4.13 corrected, proof of Lem 3.46 has been enhanced, appendix on 2-categories removed, index of notation added

References in corpus (1)

• Higher-Dimensional Algebra V: 2-Groups

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