Higgs fields from symmetric connections—The bundle picture

Abstract

The derivation (Forgács and Manton) of a classical gauge plus Higgs field theory from a purely symmetric gauge theory by dimensional reduction is translated into the pictorial language of symmetric connections on principal bundles. The bundle description provides a better understanding of symmetric gauge theories; its global nature excludes some gauge group-symmetry group combinations, and shows that the reduced gauge group is larger than the centralizer CH as previously thought. In general it is a certain factor group, NK/K, involving both the gauge and symmetry groups, and contains CH as a normal subgroup. The commonly used F2 Lagrangian of symmetric gauge theories will not prescribe the dynamics of fields associated with this larger group, unless these fields are suitably constrained. It is shown that a constrained Kaluza–Klein metric is required to construct the F2 Lagrangian, thus pointing the way to include dynamics for the NK/K group.