$D_n$ Dynkin quiver moduli spaces

TL;DR

This paper classifies and analyzes the moduli space singularity structures of good $D_n$ Dynkin quiver gauge theories in 3d $ ext{N}=4$ supersymmetry, introducing quiver addition and poset methods for comprehensive understanding.

Contribution

It provides a complete characterization of good $D_n$ Dynkin quivers, introduces quiver addition, and uses poset structures to classify and analyze their moduli spaces.

Findings

Classified all good $D_n$ Dynkin quivers.

Determined the singularity structure of their moduli spaces.

Established poset relations and Hasse diagrams for quiver classification.

Abstract

We study $3d$ $\mathcal{N}=4$ quiver gauge theories with gauge nodes forming a $D_n$ Dynkin diagram. The class of good $D_n$ Dynkin quivers is completely characterised and the moduli space singularity structure fully determined for all such theories. The class of good $D_n$ Dynkin quivers is denoted $D_\nu^\mu(n)_p$ where $n \geq 2$ is an integer, $\nu$ and $\mu$ are integer partitions and $p \in \{ \textrm{even}, \textrm{odd}\}$ denotes membership of one of two broad subclasses. A full assessment of which $\mathfrak{so}_{2n}$ nilpotent varieties are realisable as $D_n$ Dynkin quiver moduli spaces is provided. Quiver addition is introduced and is used to give large subclasses of $D_n$ Dynkin quivers poset structure. The partial ordering is determined by inclusion relations for the moduli space branches. The resulting Hasse diagrams are used to both classify $D_n$ Dynkin quivers and determine the moduli space singularity structure for an arbitrary good theory. The poset constructions and local moduli space analyses are complemented throughout by explicit checks utilising moduli space dimension matching.