  We present new analytic results on black hole perturbation theory. Our
results are based on a novel relation to four-dimensional $\mathcal{N}=2$
supersymmetric gauge theories. We propose an exact version of Bohr-Sommerfeld
quantization conditions on quasinormal mode frequencies in terms of the
Nekrasov partition function in a particular phase of the $\Omega$-background.
Our quantization conditions also enable us to find exact expressions of
eigenvalues of spin-weighted spheroidal harmonics. We test the validity of our
conjecture by comparing against known numerical results for Kerr black holes as
well as for Schwarzschild black holes. Some extensions are also discussed.
