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.

中文翻译：

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6D SCFT和同构$ \ Gamma_ {ADE} \ to E_8 $的分类

根据字符串对偶性的预测，我们阐明了六维中特定类的超共形场论与$ SU（2）$的离散子组到$ E_8 $的同态之间的对应关系。我们展示了该匹配如何适用于从二十面体二十面体组$ SL（2，5）$到$ E_8 $的同构，更正了数学和物理学文献中的先前错误。我们使用这种对应关系将来自二进制二面体组，二进制四面体组和二进制八面体组的同态性列出为$ E_8 $-这是一个新颖的数学结果。由重归一化群流指定的部分排序表明，这些同态的排序类似于由Hasse图指示的简单Lie代数的幂等轨道的已知排序。