State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture

TL;DR

This paper explores the relationship between BPS spectra in string theories and Calabi-Yau geometries, extending the dictionary to non-Abelian gauge groups, and provides a mathematical proof of the non-Abelian weak gravity conjecture.

Contribution

It introduces a method to derive supersymmetric indices from Calabi-Yau geometry and proves the non-Abelian weak gravity conjecture in this context.

Findings

Reconstruction of Noether-Lefschetz generators as vector-valued modular forms.

Establishment of an isomorphism with lattice Jacobi forms.

Mathematical proof of the non-Abelian weak gravity conjecture.

Abstract

We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on $K3$ fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued lattice Jacobi forms, which relates them to the elliptic genera and twisted-twined elliptic genera of six- and five-dimensional Heterotic strings. The meromorphic Jacobi forms generate the dimensions of the refined cohomology of the Hilbert schemes of symmetric products of the fiber and allow us to refine the BPS indices in the fiber and therefore to obtain, conjecturally, actual state counts. Using the properties of the vector-valued lattice Jacobi forms we also provide a mathematical proof of the non-Abelian weak gravity conjecture for F-/M-theory compactified on this general class of fibered Calabi-Yau 3-folds.