Tinkertoys for the $E_8$ Theory

TL;DR

The paper delivers a comprehensive classification of 4D $\mathcal{N}=2$ SCFTs arising from the $E_8$ (2,0) theory via the class-S construction, producing a vast catalog of 3-punctured- and 4-punctured-sphere fixtures. It employs Hall-Littlewood index calculations to separate free hypermultiplets, identify enhanced global symmetries, and detect product SCFTs, while noting that full Seiberg-Witten constraint data are yet to be completed. The authors report 49,836 regular-puncture fixtures, 50 irregular fixtures, and over a million 4-punctured-sphere gaugings, with an interactive web resource provided for exploration. By linking to MN theories, special-piece phenomena, enhanced $E_8$ symmetry, and 6D conformal-matter compactifications, the work expands the landscape of non-Lagrangian 4D $\mathcal{N}=2$ SCFTs and offers a rich set of dualities and construction tools for string-theoretic realizations and higher-dimensional origins.

Abstract

We construct the 4D N=2 SCFTs of class-S, which stem from the $E_8$ (2,0) theory. There are 49,836 isolated SCFTs which arise as 3-punctured spheres. Of these, 149 are "mixed" (contain free hypermultiplets accompanying the interacting SCFT) and 775 have enhanced global symmetries (beyond the manifest global symmetry associated to the punctures). Among the 49,836 3-punctured spheres we find (after removing any free hypermultiplets which may be present) 29 that are product SCFTs. Turning to 4-punctured spheres, we find 1,025,438 4D SCFTs arising as a gauging (with a simple gauge group) of a pair of 3-punctured spheres. We discuss a number of applications, including recovering several known 4D SCFTs. Our full set of results can be accessed on the Web at https://golem.ph.utexas.edu/class-S/E8/ .