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

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We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in N=2 gauge theories on ALE spaces based on quiver varieties and the minimal resolution Xₖ of the Aₖ₋₁ toric singularity C²/Zₖ, 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 Xₖ. We extend these computations to generic N=2 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.