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In recent literature, it has been argued that a mild form of the Weak GravityConjecture (WGC) is satisfied by wide classes of effective field theories inwhich higher-derivative corrections can be shown to shift the charge-to-massratios of extremal black holes to larger values. However, this mild form doesnot directly constrain low-energy physics because the black holes satisfyingthe WGC have masses above the cutoff of the effective theory. In this note, wepoint out that in string theory modular invariance can connect a lightsuperextremal state to heavy superextremal states; the latter collapse intoblack holes at small string coupling. In the context of heterotic stringtheory, we show that these states are black holes that have $\alpha'$-exactcharge-to-mass ratios exceeding the classical extremality bound. This suggeststhat modular invariance of the string partition function can be used to relatethe existence of a light superextremal particle to the positive shift incharge-to-mass ratio of extremal black holes.

Weak Gravity Conjecture, Black Hole Entropy, and Modular Invariance