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5D and 6D SCFTs from $\mathbb{C}^3$ orbifolds

Jiahua Tian, Yi-Nan Wang

TL;DR

The paper develops a unified framework for building 5d SCFTs from orbifold singularities $X=\mathbb{C}^3/\Gamma$ with $\Gamma\subset SU(3)$ by combining the 3d McKay correspondence with explicit geometric resolutions. It shows how to extract the 5d rank and flavor symmetry from group data via Ito–Reid, and how to determine the 1-form symmetry from the McKay quiver, while the resolution data yields the Coulomb-branch structure and potential IR gauge descriptions. It further connects the 5d theories to 6d $(1,0)$ SCFTs through elliptic fibrations and presents concrete examples including a new 6d theory from $\mathbb{C}^3/\Delta(48)$ and various $\Delta(3n^2)$, $\Delta(6n^2)$, and exceptional subgroups. Together, these results illuminate how geometry, group theory, and higher-dimensional SCFTs interrelate, and provide a roadmap for constructing and classifying new theories with precise flavor and 1-form symmetries.

Abstract

We study the orbifold singularities $X=\mathbb{C}^3/Γ$ where $Γ$ is a finite subgroup of $SU(3)$. M-theory on this orbifold singularity gives rise to a 5d SCFT, which is investigated with two methods. The first approach is via 3d McKay correspondence which relates the group theoretic data of $Γ$ to the physical properties of the 5d SCFT. In particular, the 1-form symmetry of the 5d SCFT is read off from the McKay quiver of $Γ$ in an elegant way. The second method is to explicitly resolve the singularity $X$ and study the Coulomb branch information of the 5d SCFT, which is applied to toric, non-toric hypersurface and complete intersection cases. Many new theories are constructed, either with or without an IR quiver gauge theory description. We find that many resolved Calabi-Yau threefolds, $\widetilde{X}$, contain compact exceptional divisors that are singular by themselves. Moreover, for certain cases of $Γ$, the orbifold singularity $\mathbb{C}^3/Γ$ can be embedded in an elliptic model and gives rise to a 6d (1,0) SCFT in the F-theory construction. Such 6d theory is naturally related to the 5d SCFT defined on the same singularity. We find examples of rank-1 6d SCFTs without a gauge group, which are potentially different from the rank-1 E-string theory.

5D and 6D SCFTs from $\mathbb{C}^3$ orbifolds

TL;DR

The paper develops a unified framework for building 5d SCFTs from orbifold singularities with by combining the 3d McKay correspondence with explicit geometric resolutions. It shows how to extract the 5d rank and flavor symmetry from group data via Ito–Reid, and how to determine the 1-form symmetry from the McKay quiver, while the resolution data yields the Coulomb-branch structure and potential IR gauge descriptions. It further connects the 5d theories to 6d SCFTs through elliptic fibrations and presents concrete examples including a new 6d theory from and various , , and exceptional subgroups. Together, these results illuminate how geometry, group theory, and higher-dimensional SCFTs interrelate, and provide a roadmap for constructing and classifying new theories with precise flavor and 1-form symmetries.

Abstract

We study the orbifold singularities where is a finite subgroup of . M-theory on this orbifold singularity gives rise to a 5d SCFT, which is investigated with two methods. The first approach is via 3d McKay correspondence which relates the group theoretic data of to the physical properties of the 5d SCFT. In particular, the 1-form symmetry of the 5d SCFT is read off from the McKay quiver of in an elegant way. The second method is to explicitly resolve the singularity and study the Coulomb branch information of the 5d SCFT, which is applied to toric, non-toric hypersurface and complete intersection cases. Many new theories are constructed, either with or without an IR quiver gauge theory description. We find that many resolved Calabi-Yau threefolds, , contain compact exceptional divisors that are singular by themselves. Moreover, for certain cases of , the orbifold singularity can be embedded in an elliptic model and gives rise to a 6d (1,0) SCFT in the F-theory construction. Such 6d theory is naturally related to the 5d SCFT defined on the same singularity. We find examples of rank-1 6d SCFTs without a gauge group, which are potentially different from the rank-1 E-string theory.

Paper Structure

This paper contains 78 sections, 384 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction and summary
  2. 3d McKay correspondence, resolution and physics
  3. Ito-Reid theorem on $\mathbb{C}^3/\Gamma$ orbifolds
  4. Resolution
  5. Hypersurface
  6. Complete intersection in ambient 5-fold
  7. Applicability of intersection number computations
  8. Codimension-two singularities and flavor symmetry
  9. Codimension-two singularity from group structure of $\Gamma$
  10. 5d flavor symmetry from resolution
  11. McKay quiver and 1-form symmetry of 5d SCFT
  12. Relation between 6d and 5d SCFTs
  13. The finite subgroups of $SU(3)$
  14. Abelian subgroups
  15. $\mathbb{Z}_n$
  16. ...and 63 more sections

Figures (3)

  • Figure 1: The splitting of flavor curve into two $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ curves.
  • Figure 2: The ramification point of codimension-two $D_4$ singularity gives rise to an $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ curve on $S_0$.
  • Figure 3: Before the base blow-up there are two self-intersecting divisors intersect at $w = z = 0$. After the base blow-up at $w = z = 0$ the $w = 0$ line and the $z = 0$ line are separated by the compact curve $\delta = 0$ on the base over which there is a type $II$ elliptic fibration except at the point $\delta = w = 0$.