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 ${\mathsf{R}}_{\mathrm{d}\mathrm{i}\mathrm{v}}$ in the vector multiplet moduli space of 4d $\mathcal{N}=2$ 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 ${\mathsf{R}}_{\mathrm{d}\mathrm{i}\mathrm{v}}\lesssim {\gamma }^2$, implying that all these divergences are a consequence of RFT limits. More precisely, along geodesics we find that ${\mathsf{R}}_{\mathrm{d}\mathrm{i}\mathrm{v}}\sim ({\mathrm{\Lambda }}_{\mathrm{w}\mathrm{g}\mathrm{c}}\mathrm{/}{\mathrm{\Lambda }}_g{)}^2$, where ${\mathrm{\Lambda }}_{\mathrm{w}\mathrm{g}\mathrm{c}}\equiv g_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}{M}_{\mathrm{P}\mathrm{l}}$ is the RFT cut-off estimate of the Weak Gravity Conjecture and ${\mathrm{\Lambda }}_g=g_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}^{-2}{\mathrm{\Lambda }}_{\mathrm{R}\mathrm{F}\mathrm{T}}$ the electrostatic energy integrated up to its actual cut-off ${\mathrm{\Lambda }}_{\mathrm{R}\mathrm{F}\mathrm{T}}$.