$E_8$ instantons on type-A ALE spaces and supersymmetric field theories

TL;DR

This work provides a concrete bridge between geometry and field theory for E8 instantons on type-A ALE spaces by constructing a complete algorithm that, from a given asymptotic holonomy (encoded as a Kac label), yields the 6d tensor-branch quiver, the 4d type-A class S description of its T^2 reduction, and a 3d star-shaped quiver as the mirror of the T^3 reduction. It connects the Higgs branches of 6d theories to E8-instanton moduli spaces on ALE spaces via a hyperkähler quotient with an SU(k) action, and shows that 4d/3d reductions organize into well-known frameworks (class S and mirror symmetry) with consistent anomaly inflow and central charges. The paper substantiates its construction through a suite of examples (notably k=2, k=4) and discusses subtle θ-angle effects, enhanced flavor symmetries, and Higgsing of SU(k) flavor symmetry, providing a unified picture across dimensions. These results offer a robust toolkit for analyzing E8 instanton moduli spaces in geometric settings and for cross-checking with dual descriptions in lower dimensions.

Abstract

We consider the 6d superconformal field theory realized on M5-branes probing the $E_8$ end-of-the-world brane on the deformed and resolved $\mathbb{C}^2/\mathbb{Z}_k$ singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the $T^2$ compactification, and the star-shaped quiver obtained as the mirror of the $T^3$ compactification.