Coulomb Branch and The Moduli Space of Instantons

TL;DR

The paper addresses the problem of characterizing the moduli spaces of pure Yang–Mills instantons on $\mathbb{C}^2$ for any simple gauge group $G$ by computing the Coulomb-branch Hilbert series of 3d $\mathcal{N}=4$ quiver gauge theories associated with over-extended Dynkin diagrams. The main method extends the monopole formula to generalized, potentially non-simply-laced quivers, and encodes the instanton data through dressing factors and GNO charges, yielding explicit results for $k=1$ and $k=2$ and new results for higher $k$ across $G_2$, $B_N$, $C_N$, and $F_4$. The paper also develops an algebraic-geometry perspective, identifying generators and relations for the reduced moduli spaces and explaining the monopole-operator origin of global-symmetry enhancements, including an additional $SU(2)$ factor acting on the reduced moduli space. Overall, the approach provides a uniform, exact framework to study instanton moduli spaces beyond the classical ADHM construction, with potential extensions to metric data and ALE spaces.

Abstract

The moduli space of instantons on C^2 for any simple gauge group is studied using the Coulomb branch of N=4 gauge theories in three dimensions. For a given simple group G, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. The computation includes the cases of non-simply-laced gauge groups G, complementing the ADHM constructions which are not available for exceptional gauge groups. Even though the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, the prescription for computing the Coulomb branch Hilbert series of such diagrams is very simple. For instanton numbers one and two, the results are in agreement with previous works. New results and general features for the moduli spaces of three and higher instanton numbers are reported and discussed in detail.

Figures (1)

• Figure 1: Brane construction for the $C_N$ affine Dynkin diagram with the attached $U(1)$ node. Each type of brane is indicated in the diagram. Here there is one $D3$-brane per interval. The red and black segments indicate D3-branes in correspondence with the simple roots of the $B$-type algebra, which is dual to the $C$-type algebra associated with $ON^+$. The blue dot in the leftmost interval indicates the D5-brane corresponding to the over-extended $U(1)$ node.