Classification of minimal Abelian Coulomb branches

Obtaining the classification of three-dimensional (3D) <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>4</a:mn></a:math> quivers whose Coulomb branches have an isolated singularity is an essential step in understanding moduli spaces of vacua of supersymmetric field theories with eight 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msup><d:mi mathvariant="double-struck">H</d:mi><d:mi>n</d:mi></d:msup></d:math> by 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi mathvariant="normal">U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>1</g:mn><g:msup><g:mo stretchy="false">)</g:mo><g:mi>r</g:mi></g:msup></g:math> gauge theory, which is sensitive to discrete gauge factors in the mirror theory. This constitutes a confirmation for the decay and fission algorithm.