  We present a class of models in which the coupling of the photon to an
ultralight scalar field that has a time-dependent vacuum expectation value
causes the fine structure constant to oscillate in time. The scalar field is
assumed to constitute all or part of the observed dark matter. Its mass is
protected against radiative corrections by a discrete $\mathbb{Z}_N$ exchange
symmetry that relates the Standard Model to several copies to itself. The
abundance of dark matter is set by the misalignment mechanism. We show that the
oscillations in the fine structure constant are large enough to be observed in
current and near-future experiments.
