Hilbert Series for Moduli Spaces of Instantons on C^2/Z_n

TL;DR

This work provides a comprehensive Hilbert-Series-based framework to classify and study moduli spaces of classical G-instantons on C^2/Z_n, realized as Higgs branches of Kronheimer–Nakajima quivers arising from D-brane configurations on A-type ALE spaces. It systematically constructs KN quivers for SU, SO, and Sp groups, derives their superpotentials and F-term constraints, and computes both Molien-integral and localization-based Hilbert series to extract dimensions, generators, and relations of the instanton moduli spaces. The authors perform extensive consistency checks across quiver descriptions for isomorphic gauge groups, demonstrating HS factorization properties and the absence/presence of factorization for ALE vs. flat cases, and they map pure-instanton sectors and brane hybrids (O/S, SA, OA, etc.) to explicit HS data. The results advance both the geometric understanding of instanton moduli on orbifolds and the string-theoretic realization of these spaces via brane systems, with potential applications to Nekrasov function analyses and ALF/ALE bundle constructions. Overall, the paper provides a detailed, checkable catalog of Hilbert-series data for instanton moduli across a wide class of gauge groups and orbifold types, tying together algebraic, geometric, and brane-physics perspectives.

Abstract

We study chiral gauge-invariant operators on moduli spaces of G instantons for any classical group G on A-type ALE spaces using Hilbert Series (HS). Moduli spaces of instantons on an ALE space can be realized as Higgs branches of certain quiver gauge theories which appear as world-volume theories on Dp branes in a Dp-D(p+4) system with the D(p+4) branes (with or without O(p+4) planes) wrapping the ALE space. We study in detail a list of quiver gauge theories which are related to G-instantons of arbitrary ranks and instanton numbers on a generic A_{n-1} ALE space and discuss the corresponding brane configurations. For a large class of theories, we explicitly compute the Higgs branch HS which reveals various algebraic/geometric aspects of the moduli space such as the dimension of the space, generators of the moduli space and relations connecting them. In a large number of examples involving lower rank instantons, we demonstrate that HS for equivalent instantons of isomorphic gauge groups but very different quiver descriptions do indeed agree, as expected.

Figures (21)

• Figure 1: The Kronheimer-Nakajima quivers for $SU(N)$ instantons on $\mathbb{C}^2/\mathbb{Z}_n$. Each node denotes the unitary group with the labelled rank. The circular nodes denote gauge groups and the square nodes denote the flavour symmetries.
• Figure 2: The quiver for $SU(N)$ instantons on $\mathbb{C}^2/\mathbb{Z}_2$: $\bm{k}=(k_1,k_2)$, $\bm{N} =(N_1,N_2)$ with $N=N_1+N_2$. The square nodes represent the flavor symmetries, whereas the circular nodes represent gauge symmetries. Each line between the groups $U(r_1)$ and $U(r_2)$ represent $r_1 r_2$ hypermultiplets in the bi-fundamental representation.
• Figure 3: Quiver diagram in 4d ${\cal N}=1$ notation for $SU(N)$ instantons on $\mathbb{C}^2/\mathbb{Z}_2$: $\bm{k}=(k_1,k_2)$, $\bm{N} =(N_1,N_2)$ with $N=N_1+N_2$. The superpotential is given by (\ref{['WSUNC2Z2']}).
• Figure 4: The $\mathcal{N}=1$ quiver for $SU(N)$ instantons on $\mathbb{C}^2/\mathbb{Z}_n$.
• Figure 28: The $O/O$ quiver for $SO(N)$ instantons on $\mathbb{C}^2/\mathbb{Z}_2$. Here, $N_1+N_2=N$. The line between $Sp(k_1)$ and $Sp(k_2)$ gauge groups denote $4k_1k_2$ hypermultiplets (whose scalar components have $8k_1k_2$ complex degrees of freedom), and each line connecting the square node and the circular node denotes $2kN_1$ and $2kN_2$ half-hypermultiplets respectively.