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 form an abelian group.
Keywords: Surface holonomy, gauge theory, 2-groups, crossed modules, higher-dimensional algebra, monopoles, gauge-invariance, non-abelian 2-bundles, iterated integrals
@article{TAC_2015_30_a41,
author = {Arthur J. Parzygnat},
title = {Gauge invariant surface holonomy and monopoles},
journal = {Theory and applications of categories},
pages = {1319--1428},
year = {2015},
volume = {30},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2015_30_a41/}
}
Arthur J. Parzygnat. Gauge invariant surface holonomy and monopoles. Theory and applications of categories, Tome 30 (2015), pp. 1319-1428. http://geodesic.mathdoc.fr/item/TAC_2015_30_a41/