Product SCFTs in Class-S

TL;DR

The paper introduces a technique to identify product SCFTs in 4D $ N=2$ theories, revealing that such product structures are rare among class-S fixtures, with only a few exceptions found in extensive classifications.

Contribution

A novel method for counting stress tensor multiplets to diagnose product SCFTs in class-S theories, applied to classify and find rare product structures.

Findings

Product SCFTs are very rare among class-S fixtures.

Only 23 out of 2979 $E_6$ fixtures are product SCFTs.

Discovered a new product SCFT involving $(E_7)_8$ Minahan-Nemeschansky theory.

Abstract

We develop a technique for counting the number of stress tensor multiplets in a 4D $\mathcal{N}=2$ 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 ${(E_7)}_8$ Minahan-Nemeschansky theory and a new rank-2 SCFT.