Abstract
We examine the minimal constraints imposed by the Weak Gravity Conjecture (WGC) on the particle spectrum of a quantum gravity theory. Towers of super-extremal states have previously been argued to be required for consistency of the WGC under circle reduction. At the same time, there exist classes of theories where no tower of super-extremal particle states below the black hole threshold has been established with current techniques. We resolve this tension by arguing for the existence of a minimal radius for circle reductions of generic quantum gravity theories. Below this threshold, the notion of a circle compactification breaks down, bypassing the need for a tower of super-extremal states to satisfy the WGC after circle reduction. Based on this we propose that if a theory satisfies the WGC at the particle level below the black hole threshold, these states are sufficient for consistency under dimensional reduction, even in absence of a tower of super-extremal particles. Apart from general arguments, we provide independent evidence for this main result in F-, M- and string theory compactifications. According to the Emergent String Conjecture the only exception to the appearance of a minimal radius arises in asymptotically weak-coupling limits for heterotic strings, which aligns with the appearance of a weakly coupled super-extremal tower of particle states. This observation motivates a Minimal Weak Gravity Conjecture which states that towers of super-extremal particles occur if and only if they are required by consistency of the WGC under dimensional reduction.
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Acknowledgments
The authors thank Rafael Álvarez-García, Ivano Basile, Florent Baume, Ralph Blumenhagen, Niccolò Cribiori, Mateo Galdeano, Naomi Gendler, Alvaro Herráez, Dieter Lüst, Fernando Marchesano, Paul-Konstantin Oehlmann, Marco Scalisi, Damian van de Heisteeg, Cumrun Vafa and Xin Wang for helpful discussions. A. M. thanks LMU and MMP for the hospitality during the finalization of this work. C.F.C., A.M. and T.W. are supported in part by Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2121 Quantum Universe 390833306 and by Deutsche Forschungsgemeinschaft through a German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”. M.W. is supported in part by a grant from the Simons Foundation (602883, CV) and also by the NSF grant PHY-2013858.
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Cota, C.F., Mininno, A., Weigand, T. et al. The minimal weak gravity conjecture. J. High Energ. Phys. 2024, 285 (2024). https://doi.org/10.1007/JHEP05(2024)285
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DOI: https://doi.org/10.1007/JHEP05(2024)285