The Moduli Space Curvature and the Weak Gravity Conjecture

We unveil a remarkable interplay between rigid field theories (RFTs), charge-to-mass ratios $\gamma$ and scalar curvature divergences $\mathsf{R}_{\rm div}$ in the vector multiplet moduli space of 4d ${\cal N}=2$ supergravities, obtained upon compactifying type II string theory on Calabi--Yau threefolds. We show that the condition to obtain an RFT that decouples from gravity implies a divergence in the $\gamma$ of (would-be) BPS particles charged under the rigid theory, and vice-versa. For weak coupling limits, where the scalar curvature diverges, we argue that such BPS particles exist and that $\mathsf{R}_{\rm div} \lesssim \gamma^2$, implying that all these divergences are a consequence of RFT limits. More precisely, along geodesics we find that $\mathsf{R}_{\rm div} \sim (\Lambda_{\rm wgc}/\Lambda_g)^2$, where $\Lambda_{\rm wgc} \equiv g_{\rm rigid} M_{\rm Pl}$ is the RFT cut-off estimate of the Weak Gravity Conjecture and $\Lambda_g = g_{\rm rigid}^{-2} \Lambda_{\rm RFT}$ the electrostatic energy integrated up to its actual cut-off $\Lambda_{\rm RFT}$.