Exact conditions for Quasi-normal modes of extremal M5-branes and Exact WKB analysis

We study the quasi-normal modes (QNMs) of a massless scalar perturbation to the extremal M5-branes metric by using the exact WKB analysis. The exact WKB analysis provides two exact QNMs conditions depending on the argument of the complex frequency of the perturbation. The exact conditions show that the discontinuity of the perturbative part of the QNMs leads the non-perturbative part of themselves. We also find a new example of the Seiberg-Witten/gravity correspondence, which helps us to compute the QNMs from our conditions.