Explicit Construction of Yang-Mills Instantons on ALE Spaces

TL;DR

The paper addresses explicit self‑dual Yang–Mills instantons on ALE spaces by extending the ADHM framework to the Kronheimer–Nakajima (KN) construction, leveraging hyper‑Kähler quotients and the ADE classification. It develops Γ‑equivariant data to build KN–ADHM data and constructs instanton bundles ${\\cal E}$ as ${\\rm Ker}{\\cal D}^†_{\\Gamma}$ with gauge connections $A_\\mu = U^†\\nabla_\\mu U$, including deformation by moment map parameters $\\zeta$. The authors solve SU(2) instantons on the Eguchi–Hanson background for instanton numbers $\\kappa = \frac{1}{2}, 1, \frac{3}{2}$, and analyze the topology and moduli spaces, showing the moduli spaces are hyper‑Kähler and possess computable dimensions. They also discuss abelian Maxwell theory on ALE spaces, EM duality, and implications for instanton effects in supersymmetric theories, highlighting both the mathematical depth and potential physical applications of KN instantons.

Abstract

We describe the explicit construction of Yang-Mills instantons on ALE spaces, following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we determine the abelian instanton connections which 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-Kahler 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.