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 α′-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.
Highlights
We suggest that having a light superextremal particle in the spectrum is closely tied to a positive shift in charge-to-mass ratio of an extremal black hole, effectively relating the mild and strong form of the Weak Gravity Conjecture (WGC)
Its mild form has been derived via unitarity and causality [11] and the AdS/conformal field theory (CFT) correspondence [47]
Only heuristically, based on the consistency of black hole thermodynamics [10, 48]. It is not transparent how the mild form of the WGC constrains low-energy physics, which is the goal of the swampland program
Summary
General relativity is not renormalizable, we can still consider it as an effective theory. These corrections are controlled by (α )nRn, where Rn is a curvature invariant with 2(n + 1) derivatives, which should be small relative to the scalar curvature R for a well-defined perturbative expansion These α corrections modify the solutions to the theory and in general change the black hole extremality bound, shifting the charge-to-mass ratio Zext,α above which a black hole forms a naked singularity. The positivity bound for the four-derivative terms modifying an electrically-charged Reissner-Nordstrom black hole was calculated in [9] and has been shown to be satisfied in some (heterotic) string theory examples. We take a first step in this direction and provide evidence that in theories with a worldsheet description that is modular invariant, the existence of a light state with Z > 1 is related to having a positive charge-to-mass ratio shift for extremal black holes, that is ∆Z > 0
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