Modular fluxes, elliptic genera, and weak gravity conjectures in four dimensions

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-tomass 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. seung.joo.lee, wolfgang.lerche, timo.weigand @cern.ch arXiv:1901.08065v1 [hep-th] 23 Jan 2019 F Example of an emergent critical string in a Type B weak coupling limit 62 Table B.2: GLSM charges of the toric coordinates of the four-fold Y 4 that is an elliptic fibration over the base three-fold B 3 , which by itself is described in Table B.1.Here, each of the generators is described in terms of its intersection numbers with the 10 toric divisors d ρ := {ν ρ = 0}, for ρ = x 0 , · · · , s, where we keep the same ordering as in Table B.2. The (closure of the) Kähler cone K(Y 4 ) is then spanned by the divisor classes, where l (a) J b = δ a b for a, b = 1, . . . , 5. Note that three of the generators J 1 , J 2 , and J 4 of K(Y 4 ) are respectively pull-backs of the generators j 1 , j 2 , and j 3 in (B.6) of K(B 3 ), where, by slight abuse of notation, the base toric divisors and their pull-backs are denoted by the same symbols. Finally, the intersection polynomial is given as I(Y 4 ) = 10J 1 2 J 3 2 + 46J 2 2 J 3 2 + 262J 3 4 + 16J 3 2 J 4 2 + 4J 1 2 J 5 2 + 20J 2 2 J 5 2 + 134J 3 2 J 5 2 + 8J 4 2 J 5 2 + 52J 5 4 + 57J 3 3

The Swampland: Introduction and Review

“…This was tied to the convex hull condition, as discussed in section 3.5.1. In [] tests of the Weak Gravity Conjecture in rich and general Heterotic settings in both six and four dimensions were performed by utilising Heterotic/F‐theory duality.…”

“…Note that this shows that the Weak Gravity Conjecture is really a statement about non‐perturbative quantum gravity, it is violated in perturbative string theory. Studies of the Weak Gravity Conjecture in the context of closed‐string U (1)s were made in []. Note that some of these can also be considered studies of open‐string U (1)s, as considered in section 4.5.2, because at strong coupling the distinction becomes rather blurred.…”

“…Such an analysis was performed in the context of the Weak Gravity Conjecture in []. It was shown that indeed at weak coupling , and infinite tower of states becomes light with a characteristic mass scale of .…”

“…This scenario is illustrated in Figure . In [] the four‐dimensional version of this analysis was performed. A notable point is that in the six‐dimensional case the sub‐Lattice Weak Gravity Conjecture, see section 3.5.6, was satisfied by the tower of states, but in four dimensions it was found that the tower of charged states did not form a lattice.…”

“…Figure illustrating the tower of charged states for a U (1) on a D7‐brane in type IIB as studied in []. Going to weak gauge coupling implies making a cycle large, which for fixed extra‐dimensional volume means a different cycle becomes small.…”