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

TL;DR

This paper reviews and compares recent methods for computing partition functions and correlators in $ ext{N}=2$ quiver gauge theories on ALE spaces, highlighting dualities with conformal field theories and new mathematical structures.

Contribution

It provides a rigorous construction of gauge theories on ALE spaces, extends computations to superconformal quivers, and proves the Nekrasov master formula with new insights into instanton charges and Hitchin systems.

Findings

Rigorous construction of gauge theories on ALE spaces.

Extension to superconformal quiver gauge theories.

Proof of the Nekrasov master formula.

Abstract

We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in $\mathcal{N}=2$ gauge theories on ALE spaces based on quiver varieties and the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$, 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 $\mathcal{N}=2$ 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.