| Abstract: | 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 R?de 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  , the multi-Taub–NUT family, and the Riemannian Schwarzschild space. |
