We consider black hole linear perturbation theory in a four-dimensional Schwarzschild (anti-)de Sitter background. We describe two methods that provide the quantization condition for the quasinormal mode frequencies of the perturbation field. The first consists of using the Nekrasov-Shatashvili functions, or, equivalently, the classical Virasoro conformal blocks, to obtain the connection coefficients for the differential equation encoding the spectral problem. The second method is based on a perturbative expansion of the local solutions of the differential equation, that involves multiple polylogarithmic functions. We conclude by showing how the two methods shed light on the mathematical structure of the quasinormal mode frequencies, and discussing how they can be generalized to problems in different backgrounds, emphasizing their effectiveness.

Paolo Arnaudo, SISSA