Modular Fluxes, Elliptic Genera, and Weak Gravity Conjectures in Four Dimensions

TL;DR

This work extends a quantitative Weak Gravity Conjecture analysis from six to four dimensions by exploiting modular properties of the elliptic genus of emergent heterotic strings in F-theory compactifications. Using mirror symmetry and flux-dependent BPS invariants on elliptic Calabi–Yau four-folds, it demonstrates that in certain U(1) flux backgrounds, a tower of super-extremal states arises in the weak coupling limit, though these states do not necessarily populate a charge sublattice. The authors identify two distinct weak-coupling regimes (a heterotic-type and a non-critical-string type) and show how D-term constraints and Stückelberg masses interplay with modularity to govern the presence and character of the WGC tower. They provide explicit examples with quasi-modular and modular fluxes, derive criteria linking fluxes to modularity, and illuminate how NS5-brane dynamics in the heterotic dual shape the spectrum and WGC realization. Overall, the paper offers a non-perturbative framework for verifying quantum gravity constraints in 4d with N=1 supersymmetry, anchored in flux-controlled elliptic genera and duality with heterotic strings.

Abstract

We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory compactifications with N=1 supersymmetry. Extending our previous work on nearly tensionless heterotic strings in six dimensions, we show that under certain assumptions a tower of asymptotically massless states arises in the limit of vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower contains super-extremal states whose charge-to-mass ratios are larger than those of certain extremal dilatonic Reissner-Nordstrom black holes, precisely as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower of super-extremal states does not always populate a charge sub-lattice. The main tool for our analysis is the elliptic genus of the emergent heterotic string in the chiral N=1 supersymmetric effective theories. This also governs situations where the heterotic string is non-perturbative. We show how it can be computed in terms of BPS invariants on elliptic four-folds, by making use of various dualities and mirror symmetry. Compared to six dimensions, the geometry of the relevant elliptically fibered four-folds is substantially richer than that of the three-folds, and we classify the possibilities for obtaining critical, nearly tensionless heterotic strings. We find that the (quasi-)modular properties of the elliptic genus crucially depend on the choice of flux background. Our general results are illustrated in a detailed example.

Figures (2)

• Figure 1: Shown is the charge-mass spectrum encoded in the fully modular elliptic genus (\ref{['modular_Z']}). The fat red dots indicate super-extremal states, which lie above the solid blue line encoding the charge-to-mass ratio of classical extremal black holes as discussed in Section \ref{['subsec_WGCbounds']}. Note the gaps at charges ${\mathfrak{q}}=4\mathbb{Z}$; in particular the open dots, which would correspond to the possible maximal super-extremal states and which would form a charge sublattice by themselves, are not populated. Qualitatively this picture does not only apply to the present example, but to the generic situation with modular fluxes. We will come back to it in the next section in the context of Weak Gravity Conjectures.
• Figure 2: Same as Figure 1, except for quasi-modular (\ref{['quasimodular_Z']}), and generic non-modular background fluxes, for which there are fewer and fewer cancellations in the spectrum. Note that in the last case, some super-extremal states do appear to form a sublattice, ${\mathfrak{q}}=2k$. However, for other models this is generically not the case.

Theorems & Definitions (8)

• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6