N=2 quiver gauge theories on A-type ALE spaces

We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in (Formula presented.) gauge theories on ALE spaces based on quiver varieties and the minimal resolution X_k of the A_k-1 toric singularity (Formula presented.) , in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion-free sheaves on a suitable orbifold compactification of X_k. We extend these computations to generic (Formula presented.) superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg–Witten geometry.