The Hilbert Series of the One Instanton Moduli Space

TL;DR

This paper computes a simple, previously unknown form of the Hilbert series for one-instanton moduli spaces in classical gauge groups, enabling extensions to exceptional groups and applications to dualities.

Contribution

It introduces a new simple formula for the Hilbert series of one-instanton moduli spaces, facilitating analysis for exceptional groups without known ADHM constructions.

Findings

Derived a simple form of the Hilbert series for classical groups

Extended the analysis to exceptional groups using new techniques

Applied results to study Argyres-Seiberg dualities

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.