### January 12, 2023

TITLE: Tower Counting for the Weak Gravity Conjecture

ABSTRACT: This talk presents recent advances in our understanding of the Tower Weak Gravity Conjecture (WGC) in string compactifications with minimal supersymmetry. The underlying mathematics involves aspects of the Kahler and enumerative geometry of Calabi-Yau manifolds, in particular modular properties of partition functions of certain D4-D2-D0 bound states. The Tower Weak Gravity Conjecture predicts that any consistent gauge theory coupled to quantum gravity should exhibit an infinite tower of so-called super-extremal particles, i.e. of states whose charge-to-mass ratio exceeds that of an extremal black hole. While BPS states are automatically super-extremal, the Tower WGC is less obvious in those directions in the charge lattice that do not support towers of BPS states.

For time constraints we focus in this talk on M-theory compactifications on Calabi-Yau threefolds, but similar results hold for F-theory compactifications on Calabi-Yau three- or fourfolds. To deduce the presence of super-extremal towers, we first classify all weak coupling limits in M-theory compactifications on Calabi-Yau threefolds, extending an earlier classification of the possible infinite distance limits in the classical Kahler moduli space. We then show that every direction in the charge lattice dual to a gauge group with a weak coupling limit admits a tower of BPS or of superextremal non-BPS states at least asymptotically. To this end we translate the problem into a counting problem for certain D4-D2-D0 bound states and make use of the modular properties of their partition function and results from Noether-Lefschetz theory. From a physics perspective, the asymptotic Tower WGC can be viewed as a consequence of the Emergent String Conjecture.

Slides of Lecture