Classification of 't Hooft instantons over multi-centered gravitational instantons

This work presents a classification of all smooth 't Hooft-Jackiw-Nohl-Rebbi instantons over Gibbons-Hawking spaces. That is, we find all smooth SU(2) Yang-Mills instantons over these spaces which arise by conformal rescalings of the metric with suitable functions. Since the Gibbons-Hawking spaces are hyper-Kahler gravitational instantons, the rescaling functions must be positive harmonic. By using twistor methods we present integral formulae for the kernel of the Laplacian associated to these spaces. These integrals are generalizations of the classical Whittaker-Watson formula. By the aid of these we prove that all 't Hooft instantons have already been found in a recent paper. This result also shows that actually all such smooth 't Hooft-Jackiw-Nohl-Rebbi instantons describe singular magnetic monopoles over the flat three-space with zero magnetic charge moreover the reducible ones generate the the full L^2 cohomology of the Gibbons-Hawking spaces.