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We study quantum corrections in four-dimensional theories with $N=1$supersymmetry in the context of Quantum Gravity Conjectures. According to theEmergent String Conjecture, infinite distance limits in quantum gravity eitherlead to decompactification of the theory or result in a weakly coupled stringtheory. We verify this conjecture in the framework of $N=1$ supersymmetricF-theory compactifications to four dimensions including perturbative $\alpha'$as well as non-perturbative corrections. After proving uniqueness of theemergent critical string at the classical level, we show that quantumcorrections obstruct precisely those limits in which the scale of the emergentcritical string would lie parametrically below the Kaluza-Klein scale. Limitsin which the tension of the asymptotically tensionless string sits at theKaluza-Klein scale, by contrast, are not obstructed. In the second part of thepaper we study the effect of quantum corrections for the Weak GravityConjecture away from the strict weak coupling limit. We propose that gaugethreshold corrections and mass renormalisation effects modify thesuper-extremality bound in four dimensions. For the infinite distance limits inF-theory the classical super-extremality bound is generically satisfied by asublattice of states in the tower of excitations of an emergent heteroticstring. By matching the F-theory $\alpha'$ corrections to gauge thresholdcorrections of the dual heterotic theory we predict how the masses of thistower must be renormalised in order for the Weak Gravity Conjecture to hold atthe quantum level.

Quantum Corrections in 4d N=1 Infinite Distance Limits and the Weak Gravity Conjecture