Moduli Space Reconstruction and Weak Gravity

TL;DR

This paper develops a GV-invariant–driven algorithm to reconstruct the full Kähler moduli space of Calabi–Yau threefolds by identifying geometric phases connected through flops and Weyl reflections. By assembling the hyperextended Kähler cone $ K_{ ext{hyp}}$ from flop/Weyl data and using analytic continuation of the prepotential, it connects GV invariants to global moduli-space structure, including non-toric phases. It also provides a large-scale, GV-based test of the lattice Weak Gravity Conjecture in M-theory compactifications on CY threefold hypersurfaces, finding full consistency in the explored ensemble up to finite GV-degree cutoffs. The results demonstrate that genus-zero GV invariants encode both the chamber structure of the moduli space and the spectrum required by the WGC, enabling robust checks of quantum gravity constraints across diverse geometric phases and paving the way for systematic studies at higher $h^{1,1}$.

Abstract

We present a method to construct the extended Kähler cone of any Calabi-Yau threefold by using Gopakumar-Vafa invariants to identify all geometric phases that are related by flops or Weyl reflections. In this way we obtain the Kähler moduli spaces of all favorable Calabi-Yau threefold hypersurfaces with $h^{1,1} \le 4$, including toric and non-toric phases. In this setting we perform an explicit test of the Weak Gravity Conjecture by using the Gopakumar-Vafa invariants to count BPS states. All of our examples satisfy the tower/sublattice WGC, and in fact they even satisfy the stronger lattice WGC.

Figures (8)

• Figure 1: A cartoon of an extended Kähler cone. Each cone represents the Kähler cone of an individual Calabi-Yau. The cones shaded blue represent Calabi-Yaus obtained as hypersurfaces in a toric variety, whereas those shaded red represent Calabi-Yaus that are not manifestly obtained by performing birational transformations on a toric ambient variety, but can nonetheless be reached by flops of curves in the Calabi-Yaus. One of the main results of this work is a method for assembling extended Kähler cones.
• Figure 2: Mori cone $\mathcal{M}_X$ of a Calabi-Yau threefold $X$, and its integer sites populated by non-vanishing genus zero GV invariants. Depicted in red: a generator of $\mathcal{M}_X$ that is nilpotent and lies outside of $\mathcal{M}_\infty$, shown in yellow.
• Figure 3: Mori cone $\mathcal{M}_{X'}$ of another Calabi-Yau $X'$ related to $X$ by a flop transition, and sites populated by non-vanishing genus zero GV invariants. Depicted in red: the flopped generator of $\mathcal{M}_X$. In this example, $\mathcal{K}$ is the dual of $\mathcal{M}_\infty$.
• Figure 4: Mori cone $\mathcal{M}_X$ of a Calabi-Yau threefold $X$, and its integer sites populated by non-vanishing genus zero GV invariants. Depicted in red: a generator of $\mathcal{M}_X$ that is nilpotent and lies on the boundary of $\mathcal{M}_{\infty}$.
• Figure 5: Non-pointed Mori cone $\mathcal{M}_{X'}$ related to $\mathcal{M}_{X}$ by an unstable Weyl flop, if one assumes no wall-crossing. Depicted in red: the flopped generator of $\mathcal{M}_X$.