Black hole perturbations from supersymmetric gauge theoryand analytic perturbative methods

We study linear perturbations around different black hole geometries. We describe two methods that provide the quantization condition for the quasinormal mode frequencies of the perturbation field. The first method is based on techniques from supersymmetric gauge theory and conformal field theory that allow us to explicitly write the connection coefficients for the differential equation encoding the spectral problem. With a closely related analysis, we study one-loop effective actions of scalar fields in the black hole backgrounds by applying a version of the Gelfand-Yaglom theorem generalized to include regular singularities. In particular, the analytic properties of the final results are made explicit by the contributions of the quasinormal modes. The second method provides a perturbative expansion of the local solutions of the differential equation based on multiple polylogarithmic functions around regular singular points and a newly introduced set of special functions called multiple polyexponential integrals around irregular singularities. The convergence properties of Nekrasov's functions are also considered, because of their relevance to the connection formulae and the physical problems analyzed.