Scalar Quasinormal modes in Reissner--Nordstr\"om black holes: implications for Weak Gravity Conjecture

Microscopic charged black holes can provide possibilities to test the consistency of the effective field theory (EFT) corrections to Einstein-Maxwell theory. A particularly interesting result is fixing the sign of a certain combination of EFT couplings from the requirement that all charged black holes should be able to evaporate (Weak Gravity Conjecture). In our work, we analysed the EFT corrections to a set of zero-damping quasinormal modes (QNMs) of the scalar wave probe in a nearly extremal Reissner-Nordstr\"om black hole. We review the duality of this setup to the problem of the quantum Seiberg-Witten curve of $N=2$ Super-Yang-Mills theory with three flavors. We provide an analytic result for the EFT corrections to the QNMs obtained from the quantization condition imposed on the Seiberg-Witten cycle. Our main result is that the causality requirement of the gravitational theory formulated for the QNMs translates to the same condition on EFT couplings as the one appearing in the Weak Gravity Conjecture.