Classification of Minimal Abelian Coulomb Branches

TL;DR

This paper classifies all Abelian 3d $ abla$=4 quivers with Coulomb branches having isolated singularities, revealing their geometries as quotients of hyperbolic spaces, and introduces methods for computing their 3d mirrors.

Contribution

It provides a complete classification of Abelian Coulomb branch quivers with isolated singularities and develops a new method for computing 3d mirrors considering discrete gauge factors.

Findings

Coulomb branches are quotients of $ abla^n$ by $ ext{U}(1)$ or cyclic groups.

Two proofs of classification: decay/fission algorithm and explicit mirror symmetry computations.

A new method for computing 3d mirrors that accounts for discrete gauge factors.

Abstract

Obtaining the classification of 3d $\mathcal{N}=4$ quivers whose Coulomb branches have an isolated singularity is an essential step in understanding moduli spaces of vacua of supersymmetric field theories with 8 supercharges in any dimension. In this work, we derive a full classification for such Abelian quivers with arbitrary charges, and identify all possible Coulomb branch geometries as quotients of $\mathbb{H}^n$ by $\mathrm{U}(1)$ or a finite cyclic group. We give two proofs, one which uses the decay and fission algorithm, and another one relying only on explicit computations involving 3d mirror symmetry. In the process, we put forward a method for computing the 3d mirror of any $\mathrm{U}(1)^r$ gauge theory, which is sensitive to discrete gauge factors in the mirror theory. This constitutes a confirmation for the decay and fission algorithm.