Hilbert Series for Moduli Spaces of Two Instantons

TL;DR

This paper computes the Hilbert Series of the moduli space of two G instantons on C^2 for various simple gauge groups, revealing a universal lattice structure and explicit formulas for classical and exceptional groups.

Contribution

It provides explicit Hilbert Series formulas for two instanton moduli spaces across all simple gauge groups, including classical, E-type, G_2, and F_4, using new methods and symmetries.

Findings

Explicit HS formulas for classical groups using ADHM construction.

HS expressions for E-type groups via recent index results.

Exact HS evaluations for G_2 and F_4 using discrete symmetries.

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.