  Generic extensions of the standard model predict the existence of ultralight
bosonic degrees of freedom. Several ongoing experiments are aimed at detecting
these particles or constraining their mass range. Here we show that massive
vector fields around rotating black holes can give rise to a strong
superradiant instability which extracts angular momentum from the hole. The
observation of supermassive spinning black holes imposes limits on this
mechanism. We show that current supermassive black hole spin estimates provide
the tightest upper limits on the mass of the photon (mv<4x10^{-20} eV according
to our most conservative estimate), and that spin measurements for the largest
known supermassive black holes could further lower this bound to mv<10^{-22}
eV. Our analysis relies on a novel framework to study perturbations of rotating
Kerr black holes in the slow-rotation regime, that we developed up to second
order in rotation, and that can be extended to other spacetime metrics and
other theories.
