The Hilbert Series of the One Instanton Moduli Space

TL;DR

This work establishes a unified framework linking one-instanton moduli spaces for classical groups to Higgs branches of Dp-D(p+4) brane gauge theories, revealing that the coherent component is captured by g^Irr_G(t; x) = Σ_k χ[Adj^k] t^{2k}. For classical groups, this follows from the ADHM construction; for exceptional groups, the authors propose a conjecture, tested via sewing relations and Argyres-Seiberg dualities. The paper develops explicit Hilbert series techniques, including Molien-Weyl integrals and refined/unrefined expansions, to count holomorphic functions on the moduli spaces and to extract generator and relation data. It provides detailed results for SU, SO, and Sp instantons, and extends to exceptional groups such as E6, E7, E8, F4, and G2 through branching rules and gluing constructions, offering new checks of dualities and consistency with known dimensions. The insights advance understanding of instanton moduli spaces, their Hilbert series, and their role in string-theoretic realizations and S-dualities in N=2 theories.

Abstract

The moduli space of k G-instantons on R^4 for a classical gauge group G is known to be given by the Higgs branch of a supersymmetric gauge theory that lives on Dp branes probing D(p + 4) branes in Type II theories. For p = 3, these (3 + 1) dimensional gauge theories have N = 2 supersymmetry and can be represented by quiver diagrams. The F and D term equations coincide with the ADHM construction. The Hilbert series of the moduli spaces of one instanton for classical gauge groups is easy to compute and turns out to take a particularly simple form which is previously unknown. This allows for a G invariant character expansion and hence easily generalisable for exceptional gauge groups, where an ADHM construction is not known. The conjectures for exceptional groups are further checked using some new techniques like sewing relations in Hilbert Series. This is applied to Argyres-Seiberg dualities.

Figures (19)

• Figure 4: The ${\cal N}=2$ quiver diagram for the ${\cal N}=4$ SYM theory with gauge group $U(N)$. The loop around the $U(N)$ gauge group denotes an adjoint hypermultiplet.
• Figure 5: The ${\cal N}=1$ quiver diagram of the ${\cal N}=4$ SYM theory. The adjoint field $\Phi$ comes from the ${\cal N}=2$ vector multiplet, whereas the adjoint fields $\phi_1, \phi_2$ come from the ${\cal N}=2$ adjoint hypermultiplet. The superpotential is $W = \mathop{\rm Tr} (\phi_1 \cdot \Phi \cdot \phi_2 - \phi_2 \cdot \Phi \cdot \phi_1) = \mathop{\rm Tr} \left( \Phi \cdot [ \phi_1, \phi_2] \right)$.
• Figure 6: The ${\cal N}=2$ quiver diagram for $k$$SU(N)$ instantons on $\mathbb{C}^2$. The circular node represents the $U(k)$ gauge symmetry and the square node represents the $SU(N)$ flavour symmetry. The line connecting the $SU(N)$ and $U(k)$ groups denotes $kN$ bi-fundamental hypermultiplets, and the loop around the $U(k)$ group denotes the adjoint hypermultiplet.
• Figure 7: Flower quiver; The ${\cal N}=1$ quiver diagram for $k$$SU(N)$ instantons on $\mathbb{C}^2$ with the corresponding superpotential, $W =X_{21} \cdot \Phi \cdot X_{12} + \epsilon_{\alpha \beta} \phi^{(\alpha)} \cdot \Phi \cdot \phi^{(\beta)}$.
• Figure 8: The ${\cal N}=2$ quiver diagram for $k$$SO(N)$ instantons on $\mathbb{C}^2$. The circular node represents the $Sp(k)$ gauge symmetry and the square node represents the $SO(N)$ flavour symmetry. The line connecting the $SO(N)$ and $Sp(k)$ groups denotes $2kN$ half-hypermultiplets, and the loop around the $Sp(k)$ gauge group denotes a hypermultiplet transforming in the (reducible) second rank antisymmetric tensor.