Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

6D SCFTs and the Classification of Homomorphisms $Γ_{ADE} \rightarrow E_8$

Darrin D. Frey, Tom Rudelius

TL;DR

This work establishes and tests a concrete correspondence between 6D SCFTs and homomorphisms $\Gamma\to E_8$, as predicted by string dualities. By leveraging refined global-symmetry analyses, it revisits the $\mathbb{Z}_k$ and $SL(2,5)$ cases, corrects misclassifications, and extends the classification to the dicyclic groups $\mathrm{Dic}_{k-2}$ and to $\Gamma_{E_{6,7}}$, showing that Dic$_{k-2}$ homomorphisms are labeled by D-partitions of $2k$ plus an admissible affinizing algebra. The RG-flow structure of these theories maps to a partial ordering on the homomorphisms, mirroring the nilpotent-orbit Hasse diagram, which provides a unifying picture of tensor/Higgs branch flows across the spectrum. The results bridge elliptic Calabi–Yau geometry, M-/F-theory constructions, and finite-group representation theory, and hint at deeper mathematical classifications tied to 6D SCFTs and their dimensional reductions.

Abstract

We elucidate the correspondence between a particular class of superconformal field theories in six dimensions and homomorphisms from discrete subgroups of $SU(2)$ into $E_8$, as predicted from string dualities. We show how this match works for homomorphisms from the binary icosahedral group $SL(2,5)$ into $E_8$, correcting previous errors in both the mathematics and physics literature. We use this correspondence to list the homomorphisms from binary dihedral groups, the binary tetrahedral group, and the binary octahedral group into $E_8$--a novel mathematical result. The partial ordering specified by renormalization group flows suggests an ordering on these homomorphisms similar to the known ordering of nilpotent orbits of a simple Lie algebra dictated by the Hasse diagram.

6D SCFTs and the Classification of Homomorphisms $Γ_{ADE} \rightarrow E_8$

TL;DR

This work establishes and tests a concrete correspondence between 6D SCFTs and homomorphisms , as predicted by string dualities. By leveraging refined global-symmetry analyses, it revisits the and cases, corrects misclassifications, and extends the classification to the dicyclic groups and to , showing that Dic homomorphisms are labeled by D-partitions of plus an admissible affinizing algebra. The RG-flow structure of these theories maps to a partial ordering on the homomorphisms, mirroring the nilpotent-orbit Hasse diagram, which provides a unifying picture of tensor/Higgs branch flows across the spectrum. The results bridge elliptic Calabi–Yau geometry, M-/F-theory constructions, and finite-group representation theory, and hint at deeper mathematical classifications tied to 6D SCFTs and their dimensional reductions.

Abstract

We elucidate the correspondence between a particular class of superconformal field theories in six dimensions and homomorphisms from discrete subgroups of into , as predicted from string dualities. We show how this match works for homomorphisms from the binary icosahedral group into , correcting previous errors in both the mathematics and physics literature. We use this correspondence to list the homomorphisms from binary dihedral groups, the binary tetrahedral group, and the binary octahedral group into --a novel mathematical result. The partial ordering specified by renormalization group flows suggests an ordering on these homomorphisms similar to the known ordering of nilpotent orbits of a simple Lie algebra dictated by the Hasse diagram.

Paper Structure

This paper contains 24 sections, 441 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Classification of 6D SCFTs
  3. Global symmetries of 6D SCFTs
  4. 6D SCFTs and Hom($\Gamma,E_8$)
  5. Review of Hom($\mathbb{Z}_k, E_8$)
  6. Hom($\mathbb{Z}_k,E_8$) and the F-theory Swampland
  7. Hom($\mathbb{Z}_k,E_8$) and the 6D $\theta$ Angle
  8. Reconsideration of Hom($SL(2,5), E_8$)
  9. Classification of Hom(Dic$_{k-2}, E_8$)
  10. Hom(Dic$_{k-2}, E_8$) from Nilpotent Orbits
  11. Caveat: Outer Automorphisms
  12. Summary of Hom(Dic$_{k-2}, E_8$)
  13. Classification of Hom($\Gamma_{E_{6,7}}, E_8$)
  14. Renormalization Group Flows
  15. Conclusions
  16. ...and 9 more sections

Figures (1)

  • Figure 1: RG flow hierarchy for 6D SCFTs representing Hom($\mathbb{Z}_4, E_8$).