Tinkertoys for the E7 Theory

TL;DR

The paper delivers a comprehensive classification of E7-type class-S theories by enumerating all 3-punctured fixtures and their S–duality-connected cylinders, uncovering thousands of interacting sectors and numerous enhanced-symmetry theories. It employs Hall-Littlewood indices to distinguish free, mixed, and interacting fixtures and to detect symmetry enhancements, providing a publicly accessible web resource. The results yield new SCFTs with exceptional global symmetries, explore dual frames for E7+3(56) and E8-conformal-matter constructions, and tie the 4d theories to 6d (1,0) origins via T^2 compactifications, including very Higgsable theories and M5-brane setups. Collectively, these findings expand the landscape of non-Lagrangian SCFTs in 4d and illuminate rich interconnections between class-S, 6d, and geometric constructions.

Abstract

We classify the class $S$ theories of type $E_7$. These are four-dimensional $\mathcal{N}=2$ superconformal field theories arising from the compactification of the $E_7$ $(2,0)$ theory on a punctured Riemann surface, $C$. The classification is given by listing all 3-punctured spheres ("fixtures"), and connecting cylinders, which can arise in a pants-decomposition of $C$. We find exactly 11,000 fixtures with three regular punctures, and an additional 48 with one "irregular puncture" (in the sense used in our previous works). To organize this large number of theories, we have created a web application at https://golem.ph.utexas.edu/class-S/E7/ . Among these theories, we find 10 new ones with a simple exceptional global symmetry group, as well as a new rank-2 SCFT and several new rank-3 SCFTs. As an application, we study the strong-coupling limit of the $E_7$ gauge theory with 3 hypermultiplets in the $56$. Using our results, we also verify recent conjectures that the $T^2$ compactification of certain $6d$ $(1,0)$ theories can alternatively be realized in class $S$ as fixtures in the $E_7$ or $E_8$ theories.