Einstein Manifolds, Abelian Instantons, Bundle Reduction, and the Cosmological Constant

The anti-self-dual projection of the spin connections of certain four-dimensional Einstein manifolds can be Abelian in nature. These configurations signify bundle reductions. By a theorem of Kobayashi and Nomizu such a process is predicated on the existence of a covariantly constant field. It turns out that even without fundamental Higgs fields and other physical matter, gravitational self-interactions can generate this mechanism if the cosmological constant is non-vanishing. This article identifies the order parameter, and clarifies how these Abelian instanton solutions are associated with a Higgs triplet which causes the bundle reduction from SO(3) gauge group to U(1).