Proving the Weak Gravity Conjecture in Perturbative String Theory, Part I: The Bosonic String

We present a complete proof of the Weak Gravity Conjecture in any perturbative bosonic string theory in spacetime dimension $D\ge 6$. Our proof works by relating the black hole extremality bound to long range forces, which are more easily calculated on the worldsheet, closing the gaps in partial arguments in the existing literature. We simultaneously establish a strict, sublattice form of the conjecture in the same class of theories. We close by discussing the scope and limitations of our analysis, along with possible extensions including an upcoming generalization of our work to the superstring.