Product SCFTs in class-S

We develop a technique for counting the number of stress tensor multiplets in a 4D N = 2 SCFT. This provides a simple diagnostic for when an isolated (non-Lagrangian) SCFT is a product of two (or more) such theories. In class-S, the basic building blocks are the isolated SCFTs arising from the compactification of a 6D (2,0) theory on a 3-punctured sphere (fixture). We apply our technique to determine when a fixture is a product SCFT. The answer is that this phenomenon is surprisingly rare. In the low-rank A(N-1), D-N theories and the E-6 theory studied by the first author and his collaborators, it occurs less than 1% of the time. Of the 2979 fixtures in the (untwisted and twisted) E-6 theory, only 23 are product SCFTs. Of these, 22 were known to the original authors. The new one is a product of the (E7)(8) Minahan-Nemeschansky theory and a new rank-2 SCFT.

Automated summary

The study developed a technique for counting stress tensor multiplets in 4D N = 2 SCFT and identifying when an isolated SCFT is a product of multiple theories. It was found that the phenomenon of SCFT being a product of theories is surprisingly rare in certain cases.