Exact solution of Kerr black hole perturbations via CFT$_2$ and instanton counting

We give the exact solution to the connection problem of the confluent Heun equation, by making use of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the AGT correspondence. Under the appropriate dictionary, this provides the exact connection coefficients of the radial and angular parts of the Teukolsky equation for the Kerr black hole, which we use to extract the finite frequency greybody factor, quasinormal modes and Love numbers. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein equation in other interesting gravitational backgrounds.