Explicit construction of Yang-Mills instantons on ALE spaces

We describe the explicit construction of Yang-Mills instantons on Asymptotically Locally Euclidean (ALE) spaces, following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we determine the Abelian instanton connections that are needed for the construction in the non-Abelian case. We compute the partition function of Maxwell theories on ALE manifolds and comment on the issue of electromagnetic duality. We discuss the topological characterization of the instanton bundles as well as the identification of their moduli spaces. We generalize the 't Hooft ansatz to SU(2) instantons on ALE spaces and on other hyper-Kähler manifolds. Specializing to the Eguchi-Hanson gravitational background, we explicitly solve the ADHM equations for SU(2) gauge bundles with second Chern class {1}/{2}, 1 and {3}/{2}.