# Weak Gravity Conjecture, Black Hole Entropy, and Modular Invariance

## TL;DR

This paper explores how modular invariance in string theory links light superextremal particles to heavy black holes, providing insights into the Weak Gravity Conjecture and black hole extremality bounds.

## Contribution

It demonstrates that modular invariance connects light superextremal states to black holes with charge-to-mass ratios exceeding classical bounds in heterotic string theory.

## Key findings

- Modular invariance relates light particles to heavy black holes.
- Black holes in heterotic string theory exceed classical extremality bounds.
- String partition function symmetry constrains charge-to-mass ratios.

## Abstract

In recent literature, it has been argued that a mild form of the Weak Gravity Conjecture (WGC) is satisfied by wide classes of effective field theories in which higher-derivative corrections can be shown to shift the charge-to-mass ratios of extremal black holes to larger values. However, this mild form does not directly constrain low-energy physics because the black holes satisfying the WGC have masses above the cutoff of the effective theory. In this note, we point out that in string theory modular invariance can connect a light superextremal state to heavy superextremal states; the latter collapse into black holes at small string coupling. In the context of heterotic string theory, we show that these states are black holes that have $\alpha'$-exact charge-to-mass ratios exceeding the classical extremality bound. This suggests that modular invariance of the string partition function can be used to relate the existence of a light superextremal particle to the positive shift in charge-to-mass ratio of extremal black holes.