Construction and Deconstruction of Single Instanton Hilbert Series

TL;DR

This paper investigates the group theoretic structures underlying various methods for constructing Hilbert series of instanton moduli spaces, revealing how characters and polynomials relate to subgroup decompositions and gauge theory moduli spaces.

Contribution

It introduces a unified framework connecting different instanton Hilbert series constructions using characters, Hall-Littlewood polynomials, and Dynkin diagram data.

Findings

Faithful deconstruction of instanton moduli space characters into subgroup components

Derivation of Highest Weight Generating functions for classical and exceptional groups

Identification of root space data with relationships between Coulomb branch theories

Abstract

Many methods exist for the construction of the Hilbert series describing the moduli spaces of instantons. We explore some of the underlying group theoretic relationships between these various constructions, including those based on the Coulomb branches and Higgs branches of SUSY quiver gauge theories, as well as those based on generating functions derivable from the Weyl Character Formula. We show how the character description of the reduced single instanton moduli space of any Classical or Exceptional group can be deconstructed faithfully in terms of characters or modified Hall-Littlewood polynomials of its regular semi-simple subgroups. We derive and utilise Highest Weight Generating functions, both for the characters of Classical or Exceptional groups and for the Hall-Littlewood polynomials of unitary groups. We illustrate how the root space data encoded in extended Dynkin diagrams corresponds to relationships between the Coulomb branches of quiver gauge theories for instanton moduli spaces and those for T(SU(N)) moduli spaces.