Large N Universality of 4d N=1 SCFTs with Simple Gauge Groups

We classify four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories with a simple gauge group admitting a large $N$ limit that flow to non-trivial superconformal fixed points in the infrared. We focus on the cases where the large $N$ limit can be taken while keeping the flavor symmetry fixed so that the putative holographic dual has a fixed gauge group. We find that they can be classified into three types -- Type I, Type II, and Type III -- exhibiting universal behavior. Type I theories have $a \neq c$ in the large $N$ limit and scale linearly in $N$; the gap of scaling dimensions among BPS operators behaves as $1/N$. Type II theories have $a=c$ in the large $N$ limit, and satisfy $a \simeq c \simeq \frac{27}{128} \dim G$, and Type III theories have $a \simeq c \simeq \frac{1}{4} \dim G$. For Type II and Type III theories, the gap of scaling dimensions stays $O(1)$ in the large $N$ limit. We enumerate relevant and marginal operators of these theories and find that non-trivial conformal manifolds emerge upon relevant deformations. Moreover, we find that a modified version of the AdS Weak Gravity Conjecture, based on the supersymmetric Cardy formula, holds for all of these theories, even for finite $N$.