Product SCFTs in Class-S

TL;DR

The paper addresses whether class-S fixtures can be decomposed as product SCFTs by counting $N=2$ stress-tensor multiplets from limits of the superconformal index. The authors develop a practical method using the Schur index $I_Schur$ and the Hall-Littlewood index $I_{HL}$ (after removing free hypers) to extract the number of stress tensors, and apply it to low-rank $A_N$, $D_N$ and the $E_6$ theories. They find product SCFTs are rare (at most a few percent of fixtures), with a detailed tally in the untwisted and twisted $E_6$ theories: 23 products among 2979 fixtures, 22 known and 1 new. The new product is a rank-3 fixture built from the $(E_7)_8$ Minahan-Nemeschansky theory and a new rank-2 SCFT with $(F_4)_{10} imes U(1)$ global symmetry and $(n_h,n_v)=(32,16)$, illustrating the method's power and revealing a novel entry in the class-S landscape.

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.