Modular Constraints on Superconformal Field Theories

TL;DR

The authors constrain the spectra of ${\mathcal N}=(1,1)$ and ${\mathcal N}=(2,2)$ 2D SCFTs by enforcing NS-sector modular invariance under ${\Gamma}_{\theta}$ and solving a modular bootstrap via semi-definite programming. They obtain both charge-independent and charge-dependent bounds, uncovering pronounced twist-gap features (kinks and plateaus) that align with candidate RCFTs and reveal holomorphic factorization at special central charges ${c}$, all while connecting to AdS$_3$/CFT$_2$ via the weak gravity conjecture. Notably, they identify single-character RCFTs at ${c=9}$ and ${c=33/2}$, relate plateau spectra to free fermion theories, and derive a bound ${\Delta} \le {c}/{8}+O(1)$ for charged non-BPS states, along with charge bounds that imply the existence of light charged states. Their WGC analysis disfavors a ${\Delta}/{Q} \lesssim \sqrt{c}$ scaling and supports a linear ${\Delta}$–${c}$ bound for charged sectors in ${\mathcal N}=(2,2)$ theories, offering quantitative tests of holographic gravity in AdS$_3$.

Abstract

We constrain the spectrum of $\mathcal{N}=(1, 1)$ and $\mathcal{N}=(2, 2)$ superconformal field theories in two-dimensions by requiring the NS-NS sector partition function to be invariant under the $Γ_θ$ congruence subgroup of the full modular group $SL(2, \mathbb{Z})$. We employ semi-definite programming to find constraints on the allowed spectrum of operators with or without $U(1)$ charges. Especially, the upper bounds on the twist gap for the non-current primaries exhibit interesting peaks, kinks, and plateau. We identify a number of candidate rational (S)CFTs realized at the numerical boundaries and find that they are realized as the solutions to modular differential equations associated to $Γ_θ$. Some of the candidate theories have been discussed by Höhn in the context of self-dual extremal vertex operator (super)algebra. We also obtain bounds for the charged operators and study their implications to the weak gravity conjecture in AdS$_3$.

Figures (16)

• Figure 1: Numerical upper bounds on the twist gap for the $\mathcal{N}=1$ SCFTs with imposing the conserved currents of $j \ge \frac{1}{2}$, $j \ge 1$ and $j \ge \frac{3}{2}$.
• Figure 2: Numerical upper bounds on the twist gap for the ${\cal N}=2$ SCFTs under various assumptions. The blue line represents the most generic bounds for the ${\cal N}=(2, 2)$ SCFT.
• Figure 3: Domain of the unitary irreducible representation of $c=6$, $\mathcal{N}=2$ super-Virasoro algebra. The orange curve passes through the points that satisfy a relation $2 h - Q^2 + 2 = 0$.
• Figure 4: The spin-independent bound on the lowest primary for the ${\cal N}=(1,1)$ (upper) and ${\cal N}=(2,2)$ (lower) SCFTs. The upper bounds decrease as we increase the parameter $N$ for the linear functional $\alpha$ from 51 to 131. The stringiest bounds represent the extrapolated result at each $c$, to $N \rightarrow \infty$.
• Figure 5: Numerical upper bounds on the twist gap for the $\mathcal{N}=0, 1$ SCFTs with the conserved currents of $j \ge \frac{1}{2}$.