Abstract
We test various conjectures about quantum gravity for six-dimensional string compactifications in the framework of F-theory. Starting with a gauge theory coupled to gravity, we analyze the limit in Kähler moduli space where the gauge coupling tends to zero while gravity is kept dynamical. We show that such a limit must be located at infinite distance in the moduli space. As expected, the low-energy effective theory breaks down in this limit due to a tower of charged particles becoming massless. These are the excitations of an asymptotically tensionless string, which is shown to coincide with a critical heterotic string compactified to six dimensions.For a more quantitative analysis, we focus on a U(1) gauge symmetry and use a chain of dualities and mirror symmetry to determine the elliptic genus of the nearly tensionless string, which is given in terms of certain meromorphic weak Jacobi forms. Their modular properties in turn allow us to determine the charge-to-mass ratios of certain string excitations near the tensionless limit. We then provide evidence that the tower of asymptotically massless charged states satisfies the (sub-)Lattice Weak Gravity Conjecture, the Completeness Conjecture, and the Swampland Distance Conjecture. Quite remarkably, we find that the number theoretic properties of the elliptic genus conspire with the balance of gravitational and scalar forces of extremal black holes, such as to produce a narrowly tuned charge spectrum of superextremal states. As a byproduct, we show how to compute elliptic genera of both critical and non-critical strings, when refined by Mordell-Weil U(1) symmetries in F-theory.
Highlights
Introduction and summaryQuantum gravitational effects are deeply woven into the fabric of string theory
For a more quantitative analysis, we focus on a U(1) gauge symmetry and use a chain of dualities and mirror symmetry to determine the elliptic genus of the nearly tensionless string, which is given in terms of certain meromorphic weak Jacobi forms
Our analysis has revealed an intricate interplay between various conjectured properties of quantum gravity and the geometric realisation of gauge symmetries in string compactifications
Summary
Quantum gravitational effects are deeply woven into the fabric of string theory. Effective field theories that derive from string theory should reflect these, and in particular must be consistent with any consistency constraints that quantum gravity may impose. These can in turn be fixed by using a non-trivial duality [50, 52] which relates the elliptic genus to the topological string theory on Y3 For given examples, this allows us to determine a characteristic part of the charge/mass spectrum of the nearly tensionless heterotic string explicitly, and to address questions related to the various Quantum Gravity conjectures mentioned above. At least in the limit of vanishing tension, the states of maximal charge per string excitation level satisfy the Sublattice Weak Gravity Conjecture bound of [39], i.e. the charge-to-mass ratio exceeds that of an extremal (non-BPS) Reissner-Nordstrom black hole in six dimensions This condition is required in order for non-BPS extremal black holes to decay, which in turn was argued [5] to be a necessary property of any theory of quantum gravity to evade various entropy bounds.
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