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Coulomb and Higgs Branches from Canonical Singularities: Part 0

Cyril Closset, Sakura Schafer-Nameki, Yi-Nan Wang

TL;DR

The paper develops a geometric framework linking 4d and 5d SCFTs from canonical and toric singularities via 3d ${\rm N}=4$ mirror symmetry and S-type gauging, translating Higgs-branch data into Coulomb-branch data of magnetic quivers. By exploiting circle and torus compactifications, it relates the HB of 5d/4d theories to the CB of their magnetic/electric quivers, enabling geometric extraction of HB dimensions, flavor symmetries, and higher-form symmetries. It provides detailed analyses of exemplars including the rank-N $E_8$ theory, rank-two theories, and various Argyres-Douglas constructions, highlighting AS hypermultiplet phase transitions and rank-zero fixed points. The results illuminate how geometry encodes both Lagrangian and non-Lagrangian UV data, reveal enhanced flavor symmetries via magnetic quivers, and offer robust checks through anomaly matching and torsion symmetry analyses. Collectively, the work offers a unifying, geometry-driven approach to understanding moduli spaces and global structures of 4d/5d SCFTs, with concrete computational tools for HB/Cb and higher-form symmetries.

Abstract

Five- and four-dimensional superconformal field theories with eight supercharges arise from canonical threefold singularities in M-theory and Type IIB string theory, respectively. We study their Coulomb and Higgs branches using crepant resolutions and deformations of the singularities. We propose a relation between the resulting moduli spaces, by compactifying the theories to 3d, followed by 3d $\mathcal{N}=4$ mirror symmetry and an $S$-type gauging of an abelian flavor symmetry. In particular, we use this correspondence to determine the Higgs branch of some 5d SCFTs and their magnetic quivers from the geometry. As an application of the general framework, we observe that singularities that engineer Argyres-Douglas theories in Type IIB also give rise to rank-0 5d SCFTs in M-theory. We also compute the higher-form symmetries of the 4d and 5d SCFTs, including the one-form symmetries of generalized Argyres-Douglas theories of type $(G, G')$.

Coulomb and Higgs Branches from Canonical Singularities: Part 0

TL;DR

The paper develops a geometric framework linking 4d and 5d SCFTs from canonical and toric singularities via 3d mirror symmetry and S-type gauging, translating Higgs-branch data into Coulomb-branch data of magnetic quivers. By exploiting circle and torus compactifications, it relates the HB of 5d/4d theories to the CB of their magnetic/electric quivers, enabling geometric extraction of HB dimensions, flavor symmetries, and higher-form symmetries. It provides detailed analyses of exemplars including the rank-N theory, rank-two theories, and various Argyres-Douglas constructions, highlighting AS hypermultiplet phase transitions and rank-zero fixed points. The results illuminate how geometry encodes both Lagrangian and non-Lagrangian UV data, reveal enhanced flavor symmetries via magnetic quivers, and offer robust checks through anomaly matching and torsion symmetry analyses. Collectively, the work offers a unifying, geometry-driven approach to understanding moduli spaces and global structures of 4d/5d SCFTs, with concrete computational tools for HB/Cb and higher-form symmetries.

Abstract

Five- and four-dimensional superconformal field theories with eight supercharges arise from canonical threefold singularities in M-theory and Type IIB string theory, respectively. We study their Coulomb and Higgs branches using crepant resolutions and deformations of the singularities. We propose a relation between the resulting moduli spaces, by compactifying the theories to 3d, followed by 3d mirror symmetry and an -type gauging of an abelian flavor symmetry. In particular, we use this correspondence to determine the Higgs branch of some 5d SCFTs and their magnetic quivers from the geometry. As an application of the general framework, we observe that singularities that engineer Argyres-Douglas theories in Type IIB also give rise to rank-0 5d SCFTs in M-theory. We also compute the higher-form symmetries of the 4d and 5d SCFTs, including the one-form symmetries of generalized Argyres-Douglas theories of type .

Paper Structure

This paper contains 49 sections, 175 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Setup and summary
  3. Comments on rank-0 theories from singularities
  4. 5d/4d correspondence via 3d mirror symmetry
  5. 5d SCFTs from M-theory
  6. 4d SCFTs from Type IIB
  7. Crepant resolution and anomaly matching on the Higgs branch of ${\mathscr{T}_{\mathbf X}^{\rm 4d}}$
  8. Flavor symmetries, conifold transitions and boundary five-manifold
  9. Circle reductions and the electric quiverines
  10. Higgs branches, magnetic quiverines and 3d mirrors
  11. Higher-form symmetries of SCFTs
  12. Three-form/zero-form symmetries of ${\mathcal{T}_{\mathbf X}^{\rm 5d}}$.
  13. One-form symmetries of ${\mathscr{T}_{\mathbf X}^{\rm 4d}}$.
  14. 5d SCFTs with Lagrangian 4d SCFT counterparts
  15. The rank-$N$ $E_8$ theory
  16. ...and 34 more sections

Figures (4)

  • Figure 1: 4d $SU$-type ${\cal N}=2$ gauge-theory quiver for the rank-N $E_8$ theory. The same quiver graph describes the 'electric quiver' ${\text{EQ}^{(4)}}$, seen as a 3d ${\cal N}=4$ theory.
  • Figure 2: The ${\cal N}=2$ SCFT ${\mathscr{T}_{\mathbf X}^{\rm 4d}}$ for the singularity $x_1^2+x_2^5+x_3^{10}+x_3 x_4^3=0$, where each node is an $SU(K)$ gauge group, with $K$ as indicated.
  • Figure 3: Hasse diagram for ${\mathcal{T}_{\mathbf X}^{\rm 5d}}$ defined by $x_1^2+x_2^5+x_3^{10}+x_3 x_4^3=0$, with magnetic quiver \ref{['MQ5 NMNOE8']}.
  • Figure 4: The 4d ${\cal N}=2$ superconformal quiver for the singularity $x_1^2+x_2^5+x_3^5+x_4^5=0$. Here, the links are hypermultiplets in the 'bifundamental' $(\bm{m}, \bm{k})$ of $Sp(m)\times \text{Spin}(k)$.