Tinkertoys for the E 7 theory

We classify the class S theories of type E7. These are four-dimensional N=2 superconformal field theories arising from the compactification of the E7 (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 E7 gauge theory with 3 hypermultiplets in the 56. Using our results, we also verify recent conjectures that the T2 compactification of certain 6d (1, 0) theories can alternatively be realized in class S as fixtures in the E7 or E8 theories.