The Moduli Space Curvature and the Weak Gravity Conjecture

Abstract

We unveil a remarkable interplay between rigid field theories (RFTs), charge-to-mass ratios ${\hat{I}}^3$ and scalar curvature divergences ${\mathsf{R}}_{\mathrm{div}}$ 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 ${\hat{I}}^3$ 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 $\mathsf{R}_{\rm div} \lesssim γ^2$, implying that all these divergences are a consequence of RFT limits. More precisely, along geodesics we find that ${\mathsf{R}}_{\mathrm{div}}\sim (\hat{I}{›}_{\mathrm{wgc}}/\hat{I}{›}_g{)}^2$, where $\hat{I}{›}_{\mathrm{wgc}}\equiv g_{\mathrm{rigid}}{M}_{\mathrm{Pl}}$ is the RFT cut-off estimate of the Weak Gravity Conjecture and $\hat{I}{›}_g=g_{\mathrm{rigid}}^{-2}\hat{I}{›}_{\mathrm{RFT}}$ the electrostatic energy integrated up to its actual cut-off $\hat{I}{›}_{\mathrm{RFT}}$.

8 pages; v2: minor modifications and clarifications added