Tinkertoys for the $E_6$ Theory

TL;DR

This work completes a systematic classification program for 4D N=2 SCFTs arising from the 6D E6 (2,0) theory by decomposing the compactification surface into three-punctured fixtures connected by cylinders. It extends the class S framework to the E6 case (without M5-brane realization) and develops a comprehensive toolkit—Hitchin data, nilpotent-puncture structures, Casimirs via k-differentials, pole structures, and the superconformal index—to catalog 880 interacting fixtures and 144 mixed/free fixtures, including many with enhanced global symmetry. Key contributions include explicit constructions of E6 and F4 gauge theories from E6 data, a detailed map between puncture data and flavor symmetries with level assignments, and a Twisted Sector analysis that resolves level ambiguities and yields new interacting SCFTs. The paper also connects these 4D theories to F-theory realizations and identifies numerous isomorphisms among theories, providing a rich landscape of strongly coupled fixed points with and without Lagrangian descriptions.

Abstract

Compactifying the 6-dimensional (2,0) superconformal field theory, of type ADE, on a Riemann surface, $C$, with codimension-2 defect operators at points on $C$, yields a 4-dimensional $\mathcal{N}=2$ superconformal field theory. An outstanding problem is to classify the 4D theories one obtains, in this way, and to understand their properties. In this paper, we turn our attention to the $E_6$ (2,0) theory, which (unlike the A- and D-series) has no realization in terms of M5-branes. Classifying the 4D theories amounts to classifying all of the 3-punctured spheres ("fixtures"), and the cylinders that connect them, that can occur in a pants-decomposition of $C$. We find 904 fixtures: 19 corresponding to free hypermultiplets, 825 corresponding to isolated interacting SCFTs (with no known Lagrangian description) and 60 "mixed fixtures", corresponding to a combination of free hypermultiplets and an interacting SCFT. Of the 825 interacting fixtures, we list only the 139 "interesting" ones. As an application, we study the strong coupling limits of the Lagrangian field theories: $E_6$ with 4 hypermultiplets in the $27$ and $F_4$ with 3 hypermultiplets in the $26$.