  In the presence of massive bosonic degrees of freedom, rotational
superradiance can trigger an instability that spins down black holes. This
leads to peculiar gravitational-wave signatures and distribution in the
spin-mass plane, which in turn can impose stringent constraints on ultralight
fields. Here, we demonstrate that there is an analogous spindown effect for
conducting stars. We show that rotating stars amplify low frequency
electromagnetic waves, and that this effect is largest when the time scale for
conduction within the star is of the order of a light crossing time. This has
interesting consequences for dark photons, as massive dark photons would cause
stars to spin down due to superradiant instabilities. The time scale of the
spindown depends on the mass of the dark photon, and on the rotation rate,
compactness, and conductivity of the star. Existing measurements of the
spindown rate of pulsars place direct constraints on models of dark sectors.
Our analysis suggests that dark photons of mass $m_V \sim 10^{-12}$ eV are
excluded by pulsar-timing observations. These constraints also exclude
superradiant instabilities triggered by dark photons as an explanation for the
spin limit of observed pulsars.
