Tinkertoys for the $E_6$ Theory

Understand

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 $C$.