Current Algebra Levels in the $E_8$ Theory

TL;DR

This paper computes the levels of flavor symmetry current algebras in a large class of $E_8$ class-S theories, clarifying previous ambiguities and providing insights into the associated VOAs and Higgs branch flows.

Contribution

It determines the previously unknown levels for 235 out of 244 three-punctured spheres in $E_8$ class-S theories using S-duality and nilpotent Higgsing.

Findings

Levels for 235 three-punctured spheres are computed.

Detailed discussion of Drinfeld-Sokolov reduction and Higgs branch RG flows.

Remaining 9 cases with unknown levels are identified.

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.