Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons

We investigate Yang-Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel-Hunsicker-Mazzeo. First referring to the codimension 2 singularity removal theorem of Sibner-Sibner and Rade we prove that given a smooth, finite energy, self-dual SU(2) connection over a complete ALF space, its energy is congruent to a Chern-Simons invariant of the boundary three-manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly. Then we introduce framed moduli spaces of self-dual connections over Ricci flat ALF spaces. We prove that the moduli space of smooth, irreducible, rapidly decaying self-dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold. We calculate its dimension by a variant of the Gromov-Lawson relative index theorem. As an application, we study Yang-Mills instantons over the flat R-3 x S-1, the multi-Taub-NUT family, and the Riemannian Schwarzschild space.