Quiver Theories and Formulae for Nilpotent Orbits of Exceptional Algebras

TL;DR

This paper explores the structure of nilpotent orbits in exceptional Lie algebras using moduli space descriptions, extending Coulomb branch quiver theories, and providing new formulas for their representation content.

Contribution

It extends Coulomb branch quiver constructions to all orbits of characteristic height 2 and introduces a new representation theoretic formula for nilpotent orbits of classical and exceptional groups.

Findings

Extended Coulomb branch quiver theories for all height 2 orbits.

Derived a new localisation-based formula for nilpotent orbits.

Analyzed Hilbert series and generating functions for exceptional orbits.

Abstract

We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different nilpotent orbits and propose candidates for the constructions of some non-normal nilpotent orbits of Exceptional algebras.