It was recently found that connection coefficients of the Heun equation can be derived in closed form using crossing symmetry in two-dimensional Liouville theory via the Nekrasov-Shatashvili functions. In this work, we systematize this approach to second-order linear ODEs of Fuchsian type, which arise in the description of N = 2, four-dimensional quiver gauge theories. After presenting the general procedure, we focus on the specific case of Fuchsian equations with five regular singularities and present some applications to black hole perturbation theory. First, we consider a massive scalar perturbation of the Schwarzschild black hole in AdS7. Next, we analyze vector type perturbations of the Reissner-Nordström-AdS5 black hole. We also discuss the implications of our results in the context of the AdS/CFT correspondence and present explicit results in the large spin limit, where we make connection with the light-cone bootstrap. Furthermore, using the spectral network technology, we identify the region of the moduli space in Seiberg-Witten theory that is relevant for the study of black hole quasinormal modes. Our results suggest that, in some cases, this region corresponds to the strong-coupling regime, highlighting the potential applicability of the conformal GMN TBA framework to address scenarios where the gravitational dictionary implies that the instanton counting parameters are not parametrically small.
