# Confluent conformal blocks and the Teukolsky master equation

## TL;DR

This paper connects quasinormal modes of Kerr black holes to confluent Heun equations, expressing solutions via Riemann-Hilbert problems and Painlevé V transcendent, providing new insights into their spectral properties.

## Contribution

It introduces a novel formulation of Kerr black hole quasinormal modes using Riemann-Hilbert problems and Painlevé V, linking monodromy data to conformal field theory concepts.

## Key findings

- Derived small-frequency expansion for spheroidal harmonic eigenvalues.
- Established monodromy conditions for radial modes.
- Connected accessory parameters to semi-classical conformal operators.

## 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 Painlev\'e V transcendent. We use this formulation to generate the small-frequency expansion 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 semi-classical 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.