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 ${\alpha }_0\le 0$. Using the moduli space EFT we prove this bound for the $\text{projected}$ ${\Delta }_{\min }(Q)$, 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 ${\alpha }_1\ge 0$. We also compute ${\alpha }_1$ and ${\alpha }_0$ in several new 3d $\mathcal{N}=1$ theories via the $\epsilon$-expansion and large-$N$ methods.