Confluent conformal blocks and the Teukolsky master equation

Abstract

Quasinormal modes of usual, four-dimensional, Kerr black holes are described by certain solutions of a confluent Heun differential equation. In this work, we express these solutions in terms of the connection matrices for a Riemann-Hilbert problem, which was recently solved in terms of the Painlevé V transcendent. We use this formulation to generate small-frequency expansions for the angular spheroidal harmonic eigenvalue and derive conditions on the monodromy properties for the radial modes. Using exponentiation, we relate the accessory parameter to a semiclassical conformal description and discuss the properties of the operators involved. For the radial equation, while the operators at the horizons have Liouville momenta proportional to the entropy intake, we find that spatial infinity is described by a Whittaker operator.