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Classification of minimal Abelian Coulomb branches

Antoine Bourget1,*, Quentin Lamouret1,†, Sinan Moura Soysüren2,3,‡, and Marcus Sperling2,§

  • 1Institut de physique théorique, Université Paris-Saclay, CEA, CNRS, 91191, Gif-sur-Yvette, France
  • 2Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria
  • 3University of Vienna, Vienna Doctoral School in Physics, Boltzmanngasse 5, 1090 Vienna, Austria

  • *Contact author: antoine.bourget@ipht.fr
  • †Contact author: quentin.lamouret@ipht.fr
  • ‡Contact author: sinan.moura.soysueren@univie.ac.at
  • §Contact author: marcus.sperling@univie.ac
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Phys. Rev. D 111, 125020 – Published 25 June, 2025

DOI: https://doi.org/10.1103/PhysRevD.111.125020

Abstract

Obtaining the classification of three-dimensional (3D) 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 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 Hn 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 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.

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Phys. Rev. D 111, 125020– Published 25 June, 2025
  • Received 11 January 2025
  • Accepted 7 May 2025
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