6D SCFTs and the Classification of Homomorphisms ${\Gamma }_{ADE}\to E_8$

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.

27+48 pages, minor revisions v2

Cited by in corpus (2)

• Revisiting the classifications of 6d SCFTs and LSTs
• General Prescription for Global U(1)'s in 6D SCFTs