Classification of large N superconformal gauge theories with a dense spectrum

TL;DR

The paper provides a comprehensive classification of large-$N$ ${ m N}=1$ supersymmetric gauge theories with simple gauge groups that flow to superconformal fixed points, under the constraints of no superpotential and fixed flavor symmetry. It introduces a concrete criterion, based on Dynkin indices and the dual Coxeter number, to distinguish dense spectra (with $a,c=O(N)$) from sparse ones (with $a,c=O(N^2)$), and demonstrates this across ${ m SU}(N)$, ${ m SO}(N)$, and ${ m Sp}(N)$ theories with various rank-2 tensors and fundamentals. Through $a$-maximization and careful handling of decoupled operators via flip fields, the authors map out the operator spectra, revealing either a single dense band or multiple bands separated by gaps, and identify eight dense theories in total. They further test the AdS version of the Weak Gravity Conjecture using convex-hull conditions, finding it satisfied for large $N$ even in the absence of a weakly coupled gravity dual, which suggests the WGC extends beyond semiclassical holography. The results offer a bridge to Argyres–Douglas-like dense spectra and motivate future work on quivers, higher dimensions, and potential holographic interpretations of these exotic CFTs.

Abstract

We classify the large $N$ limits of four-dimensional supersymmetric gauge theories with simple gauge groups that flow to superconformal fixed points. We restrict ourselves to the ones without a superpotential and with a fixed flavor symmetry. We find 35 classes in total, with 8 having a dense spectrum of chiral gauge-invariant operators. The central charges $a$ and $c$ for the dense theories grow linearly in $N$ in contrast to the $N^2$ growth for the theories with a sparse spectrum. The difference between the central charges $a-c$ can have both signs, and it does not vanish in the large $N$ limit for the dense theories. We find that there can be multiple bands separated by a gap, or a discrete spectrum above the band. We also find a criterion on the matter content for the fixed point theory to possess either a dense or sparse spectrum. We discover a few curious aspects regarding supersymmetric RG flows and $a$-maximization along the way. For all the theories with the dense spectrum, the AdS version of the Weak Gravity Conjecture (including the convex hull condition for the cases with multiple $U(1)$'s) holds for large enough $N$ even though they do not have weakly-coupled gravity duals.

Figures (50)

• Figure 1: Illustration of the sparse and dense spectrum of large $N$ theories. Here we show 3 possible scenarios. The left one depicts the scaling dimension of the single trace gauge-invariant operators for the sparse case. The spacing between the operator dimensions scales as ${\cal O}(1)$ at large $N$. We find two distinct cases for the dense theory. One can have a dense band of low-lying operators and discrete spectrum of heavy operators. The other case comes with multiple bands with an ${\cal O}(N)$ gap between the bands. For the theories with a dense spectrum, the spacing between the operator dimensions in a band scales as ${\cal O}(1/N)$.
• Figure 2: Plot of $a/c$ vs $N$ for the $SU(N)$ theory with 1 adjoint and $N_f=1$. The orange curve fits the plot with $a/c\sim0.994757\, -0.111888/N$.
• Figure 3: Dimensions of single-trace gauge-invariant operators in $SU(N)$ + 1 Adj + 1 ($\hbox{\yng(1)}$ + $\overline{\hbox{\yng(1)}})$ theory. They form a band between $1<\Delta<3$. The baryon operator is rather heavy to be seen in this plot.
• Figure 4: The figure depicts the vector space of charge-to-dimension ratios. The linear combination of lightest meson, baryon, anti-baryon and their conjugate states fill a convex hexagon. It should include the unit circle to satisfy the convex hull condition.
• Figure 5: Checking the Weak Gravity Conjecture for $SU(N)$ with 1 adjoint and $N_f=1$. Plot of distances from the origin to the two boundary lines of convex hexagon vs $N$.