Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Three dimensional canonical singularity and five dimensional N=1 SCFT

Dan Xie, Shing-Tung Yau

TL;DR

We address the problem of engineering five-dimensional $\mathcal{N}=1$ SCFTs from geometry by proposing that M-theory on a 3-fold canonical singularity $X$ yields a 5d SCFT. The main approach ties flavor symmetry to the ADE structure over one-dimensional singular loci and identifies the Coulomb-branch data with crepant resolutions: the prepotential is cubic, $${\cal F}=\frac{1}{6}(\sum_i \phi_i D_i)^3$, and the full Coulomb branch is the nef-cone fan formed by all flop-related resolutions. The paper provides concrete geometric frameworks for toric, quotient, and hypersurface singularities, detailing how to compute flavor symmetries, Coulomb-branch dimensions, prepotentials, and gauge-theory limits via nef/Mori data and triangulations, often connecting to $(p,q)$-web constructions. This geometric program yields a unifying lens to classify and compute 5d SCFT data, with explicit methods to access non-abelian flavor enhancements, Coulomb-branch chambers, and BPS spectra, and suggests extensions to non-toric and Fano geometries and to circle compactifications that bridge to Seiberg-Witten-type descriptions.

Abstract

We conjecture that every three dimensional canonical singularity defines a five dimensional N=1 SCFT. Flavor symmetry can be found from singularity structure: non-abelian flavor symmetry is read from the singularity type over one dimensional singular locus. The dimension of Coulomb branch is given by the number of compact crepant divisors from a crepant resolution of singularity. The detailed structure of Coulomb branch is described as follows: a): A chamber of Coulomb branch is described by a crepant resolution, and this chamber is given by its Nef cone and the prepotential is computed from triple intersection numbers; b): Crepant resolution is not unique and different resolutions are related by flops; Nef cones from crepant resolutions form a fan which is claimed to be the full Coulomb branch.

Three dimensional canonical singularity and five dimensional N=1 SCFT

TL;DR

We address the problem of engineering five-dimensional SCFTs from geometry by proposing that M-theory on a 3-fold canonical singularity yields a 5d SCFT. The main approach ties flavor symmetry to the ADE structure over one-dimensional singular loci and identifies the Coulomb-branch data with crepant resolutions: the prepotential is cubic, $, and the full Coulomb branch is the nef-cone fan formed by all flop-related resolutions. The paper provides concrete geometric frameworks for toric, quotient, and hypersurface singularities, detailing how to compute flavor symmetries, Coulomb-branch dimensions, prepotentials, and gauge-theory limits via nef/Mori data and triangulations, often connecting to -web constructions. This geometric program yields a unifying lens to classify and compute 5d SCFT data, with explicit methods to access non-abelian flavor enhancements, Coulomb-branch chambers, and BPS spectra, and suggests extensions to non-toric and Fano geometries and to circle compactifications that bridge to Seiberg-Witten-type descriptions.

Abstract

We conjecture that every three dimensional canonical singularity defines a five dimensional N=1 SCFT. Flavor symmetry can be found from singularity structure: non-abelian flavor symmetry is read from the singularity type over one dimensional singular locus. The dimension of Coulomb branch is given by the number of compact crepant divisors from a crepant resolution of singularity. The detailed structure of Coulomb branch is described as follows: a): A chamber of Coulomb branch is described by a crepant resolution, and this chamber is given by its Nef cone and the prepotential is computed from triple intersection numbers; b): Crepant resolution is not unique and different resolutions are related by flops; Nef cones from crepant resolutions form a fan which is claimed to be the full Coulomb branch.

Paper Structure

This paper contains 19 sections, 1 theorem, 77 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Generality of 5d $\mathcal{N}=1$ SCFT
  3. 5d Gauge theory and enhanced flavor symmetry
  4. Canonical singularity and five dimensional $\mathcal{N}=1$ SCFT
  5. Three dimensional canonical singularity
  6. Classification of rational Gorenstein singularity
  7. Classification of terminal singularity
  8. Physics and geometry
  9. Toric singularity
  10. Toric Gorenstein singularity
  11. Flavor symmetry
  12. Partial resolution and gauge theory description
  13. Crepant resolution and Coulomb branch solution
  14. Relation to $(p,q)$ web and gauge theory construction
  15. Classification: lower rank theory
  16. ...and 4 more sections

Key Result

Theorem 2

X is terminal if and only if (up to permutations of $(x,y,z)$ and symmetries of $\mu_r$) $X=C^3/\mu_r$ of type ${1\over r}(a, -a, 1)$ with $a$ coprime to $r$, where $\mu_r$ is the cyclic group of order $r$.

Figures (17)

  • Figure 1: A toric diagram and one of its partial resolution, and we also write down a gauge theory description.
  • Figure 2: Two different resolutions of a lattice polygon can be related by sequence of flops shown above.
  • Figure 3: Quadrilaterals in a triangulation of a polygon $P$.
  • Figure 4: One toric diagram and its crepant resolution.
  • Figure 5: Three toric diagrams which have pure SU(2) gauge theory description along certain locus of Coulomb branch.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Conjecture 1
  • Definition 1
  • Theorem 2
  • Conjecture 2