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.
Highlights
The relationships between supersymmetric (“SUSY”) quiver gauge theories and the nilpotent orbits of Classical Lie groups were examined in the companion paper [1]
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
While Coulomb branch quiver theory constructions for minimal nilpotent orbits have been known for some time [2,3,4], and while maximal nilpotent orbits correspond to modified Hall Littlewood polynomials transforming in the singlet representation of a group G [5], quiver theory constructions for other nilpotent orbits of Exceptional groups have not been given in the Literature
Summary
The relationships between supersymmetric (“SUSY”) quiver gauge theories and the nilpotent orbits of Classical Lie groups were examined in the companion paper [1] (which elaborates on the motivation for these studies). The purpose of this note is to examine Coulomb branch quiver theory constructions for the nilpotent orbits of Exceptional groups, beyond the minimal nilpotent orbit, and to develop representation theoretic methods for calculating properties of these moduli spaces. The quivers for these constructions can be found by a variety of means; either from affine Dynkin diagrams, or from the canonical data associated with nilpotent orbits via their Characteristics. Hr) for representations based on (modified) Hall-Littlewood polynomials (m)HLG[n], we may use other letters, where this is helpful We define these counting variables to have a complex modulus of less than unity and follow established practice in referring to them as “fugacities”, along with the monomials formed from the products of CSA or root coordinates
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