Hilbert Series for Moduli Spaces of Two Instantons

TL;DR

This work derives explicit Hilbert series for the moduli space of two instantons for all simple groups by combining ADHM constructions for classical groups with Hall-Littlewood index methods for exceptional groups. A unifying theme is that the HS can be written as a finite, lattice-generated character expansion under $SU(2)\times G$, where the lattice structure encodes the generators and relations of the moduli space. The authors develop consistency checks via pole structure, palindromic numerators, and HL-limit matches, and provide detailed lattice analyses, including special low-rank cases and universal lattice features. The results enable a systematic understanding of two-instanton moduli across all simple groups and offer a framework for exploring higher-instanton generalizations and interactions.

Abstract

The Hilbert Series (HS) of the moduli space of two G instantons on C^2, where G is a simple gauge group, is studied in detail. For a given G, the moduli space is a singular hyperKahler cone with a symmetry group U(2) \times G, where U(2) is the natural symmetry group of C^2. Holomorphic functions on the moduli space transform in irreducible representations of the symmetry group and hence the Hilbert series admits a character expansion. For cases that G is a classical group (of type A, B, C, or D), there is an ADHM construction which allows us to compute the HS explicitly using a contour integral. For cases that G is of E-type, recent index results allow for an explicit computation of the HS. The character expansion can be expressed as an infinite sum which lives on a Cartesian lattice that is generated by a small number of representations. This structure persists for all G and allows for an explicit expressions of the HS to all simple groups. For cases that G is of type G_2 or F_4, discrete symmetries are enough to evaluate the HS exactly, even though neither ADHM construction nor index is known for these cases.

Figures (24)

• Figure 1: The quiver diagram of a 4d ${\cal N}=2$ gauge theory with the gauge group $U(2)$ and a global symmetry $SU(N)$. The matter content consists of a bifundamental hypermultiplet of $U(2) \times SU(N)$, and an adjoint hypermultiplet of the $U(2)$ gauge group.
• Figure 2: The quiver diagram for the theory described by Figure \ref{['fig:N22SU2inst']}, written in $\mathcal{N}=1$ notation. The superpotential (setting mass terms to zero) is $W =\widetilde{Q}_i \cdot \varphi \cdot Q^{i} + \epsilon^{\alpha \beta} \phi_\alpha \cdot \varphi \cdot \phi_\beta$.
• Figure 3: The Dynkin diagram of $SU(N)$. The labels in black indicate ordering of the nodes; the one with number $n$ can be associated with the representation $[0,\ldots, 0, 1, 0, \ldots, 0]$ of $SU(N)$, with $1$ in the $n$-th position from the left. The labels in blue indicate the indices in the universal lattice $f(0;0,\ldots,0)$. The labels in red, green and pink indicate the indices in the non-universal lattices. Observe that the lattices in (\ref{['gen2SUN']}) occupy only 4 nodes of the Dynkin diagram in a symmetric fashion, two from each end.
• Figure 4: The Dynkin diagram of $SU(2)$. The label indicate the indices in the universal lattice.
• Figure 5: The Dynkin diagram of $SU(3)$. The labels in blue indicate the indices in the universal lattice $f(0;0,\ldots,0)$. The labels in red, green and pink indicate the indices in the non-universal lattices. Note that the projection from (\ref{['gen2SUN']}) to (\ref{['gen2SU3']}) removes the index $n_4$, keeps $k_4$, and acts additively on $k_5$, $k_6$.