The moduli space of instantons on an ALE space from 3d $\mathcal{N}=4$ field theories

TL;DR

The paper develops a unified 3d ${\cal N}=4$ framework to study instanton moduli spaces on ALE spaces, using the Coulomb branch of flavoured affine Dynkin quivers to compute Hilbert series for $G$-instantons on ${\mathbb C}^2/{\mathbb Z}_n$ and linking to $PU(n)$ instantons on ${\mathbb C}^2/\hat G$ via Higgs branches for simply laced groups. It introduces a generalised quiver based on the over-extended Dynkin diagram of $G$ with ranks $U(k a_i^{\vee})$ and fundamental flavours $n_i$, satisfying $n=\sum_i a_i^{\vee} n_i$, to realise instantons for both simply and non-simply laced groups; quaternionic dimensions of Coulomb/Higgs branches are shown to match the expected instanton moduli space dimensions, e.g., $\dim_{\mathbb H}\text{Coulomb}=k h^{\vee}_G$ and $\dim_{\mathbb H}\text{Higgs}=kn$. The authors validate the construction through explicit examples (flavoured $D_4$, $B_3$, and $G_2$ diagrams) and extend the discussion to instantons on smooth ALE spaces, illustrating how blow-up/FI parameters appear in the field theory and identifying several mirror-symmetric pairs. These results provide practical tools for computing moduli spaces across a broad class of gauge groups and orbifold types, with consistency checks against known Kronheimer–Nakajima and Tachikawa-type constructions. The work advances the understanding of ADE/non-ADE instanton moduli via 3d Coulomb/Higgs dynamics and opens directions for handling more exotic monodromies and non-unitary cases.

Abstract

The moduli space of instantons on an ALE space is studied using the moduli space of $\mathcal{N}=4$ field theories in three dimensions. For instantons in a simple gauge group $G$ on $\mathbb{C}^2/\mathbb{Z}_n$, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the affine Dynkin diagram of $G$ with flavour nodes of unitary groups attached to various nodes of the Dynkin diagram. We provide a simple prescription to determine the ranks and the positions of these flavour nodes from the order of the orbifold $n$ and from the residual subgroup of $G$ that is left unbroken by the monodromy of the gauge field at infinity. For $G$ a simply laced group of type $A$, $D$ or $E$, the Higgs branch of such a quiver describes the moduli space of instantons in projective unitary group $PU(n) \cong U(n)/U(1)$ on orbifold $\mathbb{C}^2/\hat{G}$, where $\hat{G}$ is the discrete group that is in McKay correspondence to $G$. Moreover, we present the quiver whose Coulomb branch describes the moduli space of $SO(2N)$ instantons on a smooth ALE space of type $A_{2n-1}$ and whose Higgs branch describes the moduli space of $PU(2n)$ instantons on a smooth ALE space of type $D_{N}$.