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

TL;DR

This paper explores the moduli space of instantons on ALE spaces using 3d $ ext{N}=4$ field theories, providing a method to compute Hilbert series and relate Coulomb and Higgs branches to instanton moduli spaces.

Contribution

It introduces a new prescription to determine quiver gauge theories' ranks and flavor node positions for instanton moduli spaces on ALE spaces, linking gauge theory data to geometric structures.

Findings

Computed Hilbert series for instanton moduli spaces on ALE spaces.

Established a correspondence between Coulomb/Higgs branches and instanton moduli spaces.

Presented explicit quivers for $SO(2N)$ instantons on ALE spaces.

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 $D_{N}$.