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Classifying Isolated Symplectic Singularities via 3d $\mathcal{N}=4$ Coulomb Branches

Antoine Bourget, Quentin Lamouret, Sinan Moura Soysüren, Marcus Sperling

TL;DR

The paper addresses the problem of classifying unitary quivers whose $3d \mathcal{N}=4$ Coulomb branches are isolated conical symplectic singularities (ICSSs). It adopts the Decay and Fission conjecture to translate the Coulomb-branch stratification into a combinatorial poset of quiver fission/decay products, enabling a complete classification under this hypothesis. The authors identify three new stable quiver families, with $gb_n$ and $gd_n$ yielding two new ICSS families and $gc_n$ providing a novel realization of a known family $\overline{h}_{n,\sigma}$; they also recover all previously known ICSSs (except certain $D$ and $E$ types) within a uniform framework and compute HWGs/Hilbert series to support the geometry identifications. Additionally, the work extends the framework to generalized $(p,q)$-edges, broadening the scope of stable unitary quivers realizing ICSSs and suggesting avenues to prove the conjecture directly from Coulomb-branch data. Overall, the results offer a uniform, physics-inspired route to cataloging ICSSs arising as $3d$ $\mathcal{N}=4$ Coulomb branches, with potential implications for Higgs-branch physics and geometric representation theory.

Abstract

Based on the Decay and Fission Conjecture, we provide a classification of unitary quivers whose 3d $\mathcal{N}=4$ Coulomb branches exhibit isolated singularities. This yields the complete list of isolated conical symplectic singularities that can arise in this way. In the process, we identify three new families of stable quivers: two giving rise to previously unknown isolated symplectic singularities, and one offering a novel realization of a known family.

Classifying Isolated Symplectic Singularities via 3d $\mathcal{N}=4$ Coulomb Branches

TL;DR

The paper addresses the problem of classifying unitary quivers whose Coulomb branches are isolated conical symplectic singularities (ICSSs). It adopts the Decay and Fission conjecture to translate the Coulomb-branch stratification into a combinatorial poset of quiver fission/decay products, enabling a complete classification under this hypothesis. The authors identify three new stable quiver families, with and yielding two new ICSS families and providing a novel realization of a known family ; they also recover all previously known ICSSs (except certain and types) within a uniform framework and compute HWGs/Hilbert series to support the geometry identifications. Additionally, the work extends the framework to generalized -edges, broadening the scope of stable unitary quivers realizing ICSSs and suggesting avenues to prove the conjecture directly from Coulomb-branch data. Overall, the results offer a uniform, physics-inspired route to cataloging ICSSs arising as Coulomb branches, with potential implications for Higgs-branch physics and geometric representation theory.

Abstract

Based on the Decay and Fission Conjecture, we provide a classification of unitary quivers whose 3d Coulomb branches exhibit isolated singularities. This yields the complete list of isolated conical symplectic singularities that can arise in this way. In the process, we identify three new families of stable quivers: two giving rise to previously unknown isolated symplectic singularities, and one offering a novel realization of a known family.

Paper Structure

This paper contains 15 sections, 9 theorems, 21 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Future Directions.
  3. Outline.
  4. Acknowledgments.
  5. Definitions and Tools
  6. Good Quivers
  7. Moduli Space of Vacua, Coulomb Branch, and Symplectic Leaves
  8. The Decay and Fission Algorithm
  9. Results and Proofs
  10. Simply-laced Quivers
  11. From Quiver Classification to Geometry Classification
  12. Non-simply-laced Quivers
  13. HWG Computations
  14. Generalization to pq-edges
  15. Hilbert Series and Highest Weight Generating Functions

Key Result

Theorem 1

Assuming Conjecture conj:Decay-Fission holds, Table tab:results provides the complete list of unitary quivers (as defined in Definition def:quiver) whose Coulomb branches are ICSSs.

Figures (2)

  • Figure 1: Black: Symplectic singularities can be partially characterized by the (Hasse diagram of the) poset of symplectic leaves, or the Hilbert series of graded dimensions. Green: For symplectic singularities that are realized as 3d $\mathcal{N}=4$ Coulomb branches $\mathcal{M}_C$, these objects can be computed from physics-inspired tools such as the Monopole Formula and the Decay and Fission algorithm. Purple: Symplectic singularities are realized in string theory through two mechanisms: Higgs branches of superconformal field theories (SCFTs), which extend beyond hyper-Kähler quotients, and magnetic quivers derived from brane intersections. Remark: In general, it is not possible to go against the arrows.
  • Figure 2: All simply-laced quivers that are stable. There are the affine Dynkin quivers $A_n^{(1)}$ ($n \geq 2$), $D_n^{(1)}$ ($n \geq 4$), and $E_{6,7,8}^{(1)}$; conventions follow Fuchs:1997jv. The subscript in the name of each quiver is one less than the number of nodes. In addition, there is the 2-vertex quiver $A_{N-1}$ with an edge of multiplicity $N$.

Theorems & Definitions (38)

  • Theorem 1
  • Definition 1
  • Remark
  • Definition 2
  • Remark
  • Definition 3
  • Remark
  • Definition 4
  • Definition 5
  • Remark
  • ...and 28 more