Black holes, Heun functions and instanton counting

In this thesis we explore and exploit the relationships between black hole physics, the mathematics of Heun’s equation, Liouville conformal field theory and 4d supersymmetric gauge theories. We study the modular properties of a class of degenerate conformal blocks, including the recently introduced irregular or confluent conformal blocks, computing their connection matrices. In a certain limit, these connection matrices descend to the connection matrices for Heun functions, which are a generalization of the hypergeometric functions. We give explicit formulae, computable in terms of the Nekrasov partition functions of a class of 4d supersymmetric gauge theories. The above study is motivated by the frequent appearance of Heun’s equation in physics, one interesting example being the perturbations of black holes. Using the connection formulae that we have obtained, we analytically solve the equations governing the perturbations of the 4d Kerr black hole and of the 5d AdS-Schwarzschild black hole.