  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).
