Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits

TL;DR

$The paper develops a unified framework to realize classical group nilpotent orbit moduli spaces as Higgs and Coulomb branch quiver varieties, encoding their structure via Hilbert series and highest weight Hall-Littlewood decompositions. For A-series, it presents a complete Higgs-branch analysis and explicit 3d mirror/Coulomb-branch constructions, while for BCD-series it provides Higgs-branch constructions with alternating O/USp quivers and introduces twisted-affine Dynkin diagram-based Coulomb branches, aided by the monopole formula. It shows that these moduli spaces are HyperKähler, palindromic in most cases, and decomposable into irreps or mHL polynomials, with numerous dualities and HKQ relations connecting different quivers to the same orbit. The work analyzes nilpotent orbits up to rank 4, confirms known or conjectured dualities, and lays groundwork for higher-rank extensions and potential exceptional-group generalizations, highlighting systematic methods to obtain HWGs and mHL HWGs. Overall, the study advances a quiver-based, representation-theoretic blueprint for classifying and constructing classical nilpotent orbit moduli spaces with precise Hilbert-series data and symmetry decompositions. $

Abstract

We approach the topic of Classical group nilpotent orbits from the perspective of their moduli spaces, described in terms of Hilbert series and generating functions. We review the established Higgs and Coulomb branch quiver theory constructions for A series nilpotent orbits. We present systematic constructions for BCD series nilpotent orbits on the Higgs branches of quiver theories defined by canonical partitions; this paper collects earlier work into a systematic framework, filling in gaps and providing a complete treatment. We find new Coulomb branch constructions for above minimal nilpotent orbits, including some based upon twisted affine Dynkin diagrams. We also discuss aspects of 3d mirror symmetry between these Higgs and Coulomb branch constructions and explore dualities and other relationships, such as HyperKahler quotients, between quivers. We analyse all Classical group nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials.

Figures (11)

• Figure 1: Unitary Linear Quiver. Square (red) nodes denote flavour nodes. Round (blue) nodes denote gauge nodes. The links represent pairs of bifundamental fields transforming in the fundamental or antifundamental representations. The quiver is ordered such that $N_f > N_1 > N_i > \ldots > N_{max}$.
• Figure 2: Orthogonal Linear Quiver. Square (red) nodes denote flavour nodes. Round (blue) nodes denote gauge nodes. The links represent bifundamental fields transforming in the vector/fundamental representations. The quiver is ordered such that $N_f > N_1 > N_i > \ldots > N_{max}$.
• Figure 3: Symplectic Linear Quiver. Square (red) nodes denote flavour nodes. Round (blue) nodes denote gauge nodes. The links represent bifundamental fields transforming in the vector/fundamental representations. The quiver is ordered such that $N_f > N_1 > N_i > \ldots > N_{max}$.
• Figure 4: Quivers for $A$ Series Minimal and Maximal Nilpotent Orbits. Square (red) nodes denote flavour nodes. Round (blue) nodes denote gauge nodes. The links represent pairs of bifundamental chiral scalars transforming in the fundamental and antifundamental representations. The minimal nilpotent orbit, with two nodes, corresponds to the reduced single instanton moduli space of $SU(N)$. The maximal nilpotent orbit, with $N$ nodes, corresponds to the modified Hall Littlewood polynomial $mHL^{A}_{[0,\ldots, 0]}$ of $SU(N)$.
• Figure 5: Mirror Dual Quivers for $A$ Series Nilpotent Orbits. Round (blue) nodes denote unitary gauge nodes of the indicated rank. Square (red) nodes denote numbers of uncharged flavour nodes.