On quivers, spectral networks and black holes.

It was recently found that connection coefficients of the Heun equation can be derived in a 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.