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

TL;DR

The authors classify four-dimensional ${ m N}=1$ SCFTs with simple gauge groups that admit a large $N$ limit and flow to nontrivial IR fixed points, revealing a universal tripartite structure (Type I–III) determined by the effective number of rank-2 tensor fields $N_{ ext{rank-2}}$. In the large-$N$ limit, Type II and Type III theories exhibit $a=c$ with $a,c$ scaling as $rac{27}{128} ext{dim}(G)$ and $rac{1}{4} ext{dim}(G)$ respectively, while Type I theories show $a eq c$ and a spectrum with gap $O(1/N)$ or $O(N)$ depending on the limit; Type I theories are dense and not holographic, whereas II/III theories are sparse and holographically plausible. The work uses anomaly cancellation, $a$-maximization with flip fields to handle decoupled operators, and a Veneziano limit analysis to extract universal $R$-charges and central charges across large families, identifying conformal windows and detailed operator spectra. It also discusses the emergence of conformal manifolds under relevant deformations and tests a modified AdS Weak Gravity Conjecture based on a supersymmetric Cardy formula, which holds across all studied theories, while the standard NN-WGC has counterexamples. The results illuminate connections between large-$N$ SCFTs, holography, and swampland constraints, offering a broad framework for understanding spectra, moduli spaces, and gravity duals in 4d ${ m N}=1$ theories. The findings have potential implications for constructing holographic duals and for exploring universal constraints on quantum gravity from CFT data.

Abstract

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 $N$.

Figures (66)

• Figure 1: The charge-to-dimension ratio space with two flavor charges $U(1)_1$ and $U(1)_2$. The normalization of the ratio is set by the ratio of an extremal black hole lying on the dotted unit circle. Solid lines represent the convex hull made by gauge-invariant operators $O_1, \dots, O_4$. (a) An extremal black hole outside the convex hull cannot decay by emitting charged particles. (b) When the convex hull encloses the unit ball, any extremal black holes can decay by emitting particles.
• Figure 2: The central charge ratio for $SU(N)$ theory with 1 Adj + $N_f$ ( ${{\ydiagram{1}}}$ + $\overline{{{\ydiagram{1}}}}$ ). Left: $a/c$ versus $N$ with a fixed $N_f$. Right: $a/c$ versus $N$ with a fixed $\alpha=N_f/N$.
• Figure 3: $a/c$ versus the number of single-trace relevant operators in $SU(N)$ with 1 Adj + $N_f$ ( ${{\ydiagram{1}}}$ + $\overline{{{\ydiagram{1}}}}$ ) with $N_f=1$ and $100<N<600$.
• Figure 4: Testing AdS WGC for $SU(N)$ theory with 1 Adj + $N_f$ ( ${{\ydiagram{1}}}$ + $\overline{{{\ydiagram{1}}}}$ ). The minimum distance from the origin to the convex hull with a fixed $\alpha=N_f/N$. Theories below the solid line of minimum distance 1 do not satisfy the WGC.
• Figure 5: The central charge ratio for $SU(N)$ theory with 1 ${{\ydiagram{2}}}$ + 1 $\overline{{{\ydiagram{2}}}}$ + $N_f$ ( ${{\ydiagram{1}}}$ + $\overline{{{\ydiagram{1}}}}$ ). Left: $a/c$ versus $N$ with a fixed $N_f$. Right: $a/c$ versus $N$ with a fixed $\alpha=N_f/N$.