# Tinkertoys for the E7 Theory

## TL;DR

This paper classifies a large class of four-dimensional $ ext{N}=2$ superconformal field theories derived from the $E_7$ $(2,0)$ theory, identifying new theories, and verifying conjectures about their relations to six-dimensional theories.

## Contribution

It provides a comprehensive classification of $E_7$ class $S$ theories, including new theories and a web application for their organization.

## Key findings

- 11,000 fixtures with three regular punctures identified
- 48 fixtures with one irregular puncture found
- New theories with exceptional global symmetries discovered

## 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.