Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. In this talk, we give a gauge-theoretic construction of these spaces, inspired by Kronheimer’s original construction via a finite-dimensional hyperkähler reduction. In the gauge-theoretic construction, we realize each ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a hyperkähler gauge group action. Time permitting, we will discuss an application of the gauge-theoretic construction to give a new proof of the McKay correspondence. Let Gamma be a finite subgroup of SU(2). The McKay correspondence states that the McKay quiver of Gamma is isomorphic to the graph of the minimal resolution of C^2/Gamma. The main approach here is Morse-theoretic, inspired by Atiyah-Bott-Kirwan theory and Chern-Simons theory, as we aim to identify the critical sets of a Morse function coming from the norm square of some moment map with certain flat connections that induce Gamma-representations as in the McKay correspondence.
