HWG for Coulomb branch of $3d$ Sicilian theory mirrors

TL;DR

This paper investigates a special subclass of star-shaped quivers in 3d N=4 theories where the Coulomb branch's symmetry enhancement can be characterized by a simple Highest Weight Generating Function, aiding in understanding their geometric structure.

Contribution

It identifies a subclass of star-shaped quivers with a notably simple HWG form, either as a product of simple poles or a PE, and tests the Coulomb branch formula on these cases.

Findings

HWG takes a simple rational form for this subclass

The subclass includes freely generated and complete intersection cases

Provides a framework for classifying star-shaped quivers by HWG simplicity

Abstract

Certain star shaped quivers exhibit a pattern of symmetry enhancement on the Coulomb branch of $3d$ $\mathcal{N}=4$ supersymmetric gauge theories. This paper studies a subclass of theories where such global symmetry enhancement occurs through a computation of the Highest Weight Generating Function (HWG) and of the corresponding Hilbert Series (HS), providing a further test of the Coulomb branch formula. This special subclass has a feature in which the HWG takes a particularly simple form, as a simple rational function which is either a product of simple poles (termed freely generated) or a simple PE (termed complete intersection). Out of all possible star shaped quivers, this is a particularly simple subclass. The present study motivates a further study of identifying all star shaped quivers for which their HWG is of this simple form.