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-Nordström 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.
Highlights
It is one of the celebrated properties of string theory that it relates deep properties of quantum gravity and field theory in dimensions lower than ten to the geometry of the space on which the theory is compactified
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
String compactifications offer an outstanding opportunity to test these conjectures and to uncover the mathematical structures of the compactification space to which they point. Among the earliest such conjectures is the Weak Gravity Conjecture (WGC) [1], which asserts that in any gauge theory coupled to quantum gravity there should exist some set of particles whose charge-to-mass ratio exceeds that of an extremal black hole
Summary
It is one of the celebrated properties of string theory that it relates deep properties of quantum gravity and field theory in dimensions lower than ten to the geometry of the space on which the theory is compactified. Both terms necessarily appear at one-loop order, if the elliptic genus is (quasi-)modular and non-zero In this case the Stuckelberg mass is parametrically smaller than the Kaluza-Klein scale in the asymptotic weak coupling limit, and for the purpose of discussing the Weak Gravity Conjecture, we can consider the U(1) gauge field as effectively being massless. The relation to the Weak Gravity Conjecture is analysed in section 6: after showing that — modulo the caveats above — the weak coupling limit gives rise to an asymptotically tensionless, critical heterotic string, we use the (quasi-)modularity properties of its elliptic genus to prove the Weak Gravity Conjecture whenever a limit of this type is available.
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