Quiver Theories and Formulae for Nilpotent Orbits of Exceptional Algebras

TL;DR

This work addresses the closures of nilpotent orbits in Exceptional Lie algebras by framing them as moduli spaces described with Hilbert series and highest-weight generating functions. It extends Coulomb branch quiver constructions to all orbits of Characteristic Height 2 and introduces a Nilpotent Orbit Normalisation (NON) localisation framework to compute normal nilpotent orbits and, where applicable, their normalisations, across Classical and Exceptional groups. By combining unitary monopole formula techniques with localisation methods, the paper derives refined Hilbert series and HWGs for numerous orbits (e.g., F_4 22-dim, E_6 32-dim, and notable orbits in E_7 and E_8), and uncovers dualities between different SU(2) embeddings that yield identical moduli spaces. It also establishes connections between NON and T^*(G/H) theory, and discusses the relationships between various moduli spaces, including non-normal orbits and their normalisations, while outlining computational limits and future directions for extending these constructions to broader classes of moduli spaces and higher-height 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.

Figures (4)

• Figure 1: Exceptional Group Quivers with Characteristic Height 2. Round (blue) nodes denote unitary gauge nodes of the indicated rank. Square (red) nodes denote numbers of flavour nodes. The Characteristics coincide with the numbers of flavour nodes attached to each gauge node. The dimension of a Coulomb branch nilpotent orbit construction equals twice the sums of the ranks of its gauge nodes. Note that the quiver for the $G_2$ 10 dimensional nilpotent orbit is not based on its Characteristic (see discussion).
• Figure 2: $G_2$ Nilpotent Orbit Hasse Diagram. The diagram is derived from Hilbert series inclusion relations, with the yellow node indicating a non-normal nilpotent orbit.
• Figure 3: $F_4$ Nilpotent Orbit Hasse Diagram. The left hand diagram is derived from Hilbert series and HWG inclusion relations. The right hand diagram is taken from the mathematical Literature Adams:jkBaohua-Fu:2015nr. Yellow nodes indicate non-normal nilpotent orbits.
• Figure 4: $E_6$ Nilpotent Orbit Hasse Diagram. The left hand diagram is indicative, being partly derived from unrefined Hilbert series, with arrows indicating inclusion relations and yellow nodes indicating non-normal nilpotent orbits. The right hand diagram is taken from the mathematical Literature.