#
Gauge invariant surface holonomy and monopoles

##
Arthur J. Parzygnat

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.

Published 2015-10-14.

http://www.tac.mta.ca/tac/volumes/30/42/30-42.pdf

TAC Home