Current Algebra Levels in the $E_8$ Theory

Abstract: The class-S theories of type $E_8$ were analyzed in \cite{Chacaltana:2018vhp}. The basic building blocks consisted of 49,836 isolated SCFTs (three-punctured spheres). In 244 cases, there were undetermined levels for the flavour symmetry current algebra. Here, we rectify that omission. Using S-duality and nilpotent Higgsing we compute the levels of 235 of the 244. There remain 9 three-punctured spheres with unknown levels. Along the way, we provide a detailed discussion of the Drinfeld-Sokolov reduction of the VOA, which captures various features of the Higgs branch RG flows arising from giving a VEV to a nilpotent moment map.