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Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions

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

TL;DR

The paper develops a geometric framework linking canonical isolated hypersurface singularities to 4d and 5d SCFTs via deformations and crepant resolutions, emphasizing those with smooth Calabi–Yau resolutions. It provides practical tools, including Mathematica code, to compute the Milnor spectrum, central charges, higher-form symmetries, and resolution data, and uses these to classify smoothable models up to rank $r\le4$ (and a focused study up to $r\le10$ in a subset). The work highlights deep connections between topology (via Newton polytopes and link manifolds), defect groups, and the Coulomb/Higgs branches across dimensions, and presents explicit examples such as the AD$[E_7,E_7]$ model and various $D_p(G)$ trinions. It also establishes conjectures relating smoothness, integral CB spectra, and the finiteness of smoothable models, offering a roadmap for exploring fully smooth models and their 4d/5d dual descriptions. The compiled data and code enhance the toolkit for constructing and analyzing geometrically engineered SCFTs across dimensions, with implications for understanding moduli spaces, dualities, and higher-form symmetries in string/M-theory contexts.

Abstract

Compactification of M-theory and of IIB string theory on threefold canonical singularities gives rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. The resolutions and deformations of the singularities encode salient features of the SCFTs and of their moduli spaces. In this paper, we build on Part 0 of this series (arXiv:2007.15600) and further explore the physics of SCFTs arising from isolated hypersurface singularities. We study in detail these canonical isolated hypersurface singularities that admit a smooth Calabi-Yau (crepant) resolution. Their 5d and 4d physics is discussed and their 3d reduction and mirrors (the magnetic quivers) are determined in many cases. As an explorative tool, we provide a Mathematica code which computes key quantities for any canonical isolated hypersurface singularity, including the 5d rank, the 4d Coulomb branch spectrum and central charges, higher-form symmetries in 4d and 5d, and crepant resolutions.

Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions

TL;DR

The paper develops a geometric framework linking canonical isolated hypersurface singularities to 4d and 5d SCFTs via deformations and crepant resolutions, emphasizing those with smooth Calabi–Yau resolutions. It provides practical tools, including Mathematica code, to compute the Milnor spectrum, central charges, higher-form symmetries, and resolution data, and uses these to classify smoothable models up to rank (and a focused study up to in a subset). The work highlights deep connections between topology (via Newton polytopes and link manifolds), defect groups, and the Coulomb/Higgs branches across dimensions, and presents explicit examples such as the AD model and various trinions. It also establishes conjectures relating smoothness, integral CB spectra, and the finiteness of smoothable models, offering a roadmap for exploring fully smooth models and their 4d/5d dual descriptions. The compiled data and code enhance the toolkit for constructing and analyzing geometrically engineered SCFTs across dimensions, with implications for understanding moduli spaces, dualities, and higher-form symmetries in string/M-theory contexts.

Abstract

Compactification of M-theory and of IIB string theory on threefold canonical singularities gives rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. The resolutions and deformations of the singularities encode salient features of the SCFTs and of their moduli spaces. In this paper, we build on Part 0 of this series (arXiv:2007.15600) and further explore the physics of SCFTs arising from isolated hypersurface singularities. We study in detail these canonical isolated hypersurface singularities that admit a smooth Calabi-Yau (crepant) resolution. Their 5d and 4d physics is discussed and their 3d reduction and mirrors (the magnetic quivers) are determined in many cases. As an explorative tool, we provide a Mathematica code which computes key quantities for any canonical isolated hypersurface singularity, including the 5d rank, the 4d Coulomb branch spectrum and central charges, higher-form symmetries in 4d and 5d, and crepant resolutions.
Paper Structure (68 sections, 165 equations, 2 figures, 10 tables)

This paper contains 68 sections, 165 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Conjecture (1):
  3. Conjecture (2):
  4. Hypersurface Singularities and SCFTs: A Practical Summary
  5. Deformation and 4d Coulomb Branch Spectrum
  6. Canonical singularity condition.
  7. Milnor ring and multiplicity.
  8. The spectrum of ${\mathbf X}$.
  9. The Coulomb branch spectrum of ${\mathscr{T}_{\mathbf X}^{\rm 4d}}$.
  10. Central charges.
  11. Crepant Resolution and 5d Coulomb Branch
  12. Topology of the crepant resolution $\widetilde{{\mathbf X}}$.
  13. Newton polytope and weightings.
  14. Computation of $r$ and $b_3$.
  15. Resolution sequence.
  16. ...and 53 more sections

Figures (2)

  • Figure 1: Resolved AD$[E_7, E_7]$ geometry: compact surfaces are $S_i$ and gluing curves are links between these, with self-intersection numbers shown in the boxes. The LHS is the model after resolution of the IHS, and the RHS after the flop.
  • Figure 2: Hasse diagram for the AD$[D_{2n}, D_{2n}]$ singularity, which has the IR gauge theory description $SU(n)_{\pm 3/2}+(2n+1)\bm{F}$.