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Non-compact hyperk\"ahler spaces arise frequently in gauge theory. The4-dimensional hyperk\"ahler ALE spaces are a special class of non-compacthyperk\"ahler spaces. They are in one-to-one correspondence with the finitesubgroups of SU(2) and have interesting connections with representation theoryand singularity theory, captured by the McKay Correspondence.  The 4-dimensional hyperk\"ahler ALE spaces are first classified by PeterKronheimer via a finite-dimensional hyperk\"ahler reduction. In this paper, wegive a new gauge-theoretic construction of these spaces. More specifically, werealize each 4-dimensional hyperk\"ahler ALE space as a moduli space ofsolutions to a system of equations for a pair consisting of a connection and asection of a vector bundle over an orbifold Riemann surface, modulo a gaugegroup action. The construction given in this paper parallels Kronheimer'soriginal construction and hence can also be thought of as a gauge-theoreticinterpretation of Kronheimer's construction of these spaces.

A New Gauge-Theoretic Construction of 4-Dimensional Hyperk\"ahler ALE Spaces