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
2010 MSC: Primary 53C29; Secondary 70S15
Theory and Applications of Categories, Vol. 30, 2015, No. 42, pp 1319-1428.