Towers of Operators in CFTs and Convexity Bounds at Large Charge

Abstract

In arXiv:2406.19441, it was shown that for 3d CFTs with a moduli space along which a $U(1)$ symmetry is spontaneously broken, the minimum scaling dimension at large charge $Q$ scales as $\Delta_{\min}(Q)=\alpha_1 Q+\alpha_0+O(1/Q)$. Motivated by the holographic swampland program, we study possible bounds on the coefficients $\alpha_i$. For $\alpha_0$, the weak gravity conjecture motivates the CFT charge convexity conjecture, which requires the bound $\alpha_0\leq 0$. Using the moduli space EFT we prove this bound for the $\textit{projected}$ $\Delta_{\min}(Q)$, obtained by fixing a single charge $Q$ and minimizing the dimension while allowing all other charges to vary. This provides a WGC-motivated bound that is explicitly provable using CFT methods. On the other hand, we show that $\alpha_1$ admits no universal bound apart from the trivial bound $\alpha_1\geq 0$. We also compute $\alpha_1$ and $\alpha_0$ in several new 3d $\mathcal{N}=1$ theories via the $\epsilon$-expansion and large-$N$ methods.