  We present the theoretical prediction of spontaneous rotating vortex rings in
a parametrically driven quantum fluid of polaritons -- coherent superpositions
of coupled quantum well excitons and microcavity photons. These rings arise not
only in the absence of any rotating drive, but also in the absence of a
trapping potential, in a model known to map quantitatively to experiments. We
begin by proposing a novel parametric pumping scheme for polaritons, with
circular symmetry and radial currents, and characterize the resulting
nonequilibrium condensate. We show that the system is unstable to spontaneous
breaking of circular symmetry via a modulational instability, following which a
vortex ring with large net angular momentum emerges, rotating in one of two
topologically distinct states. Such rings are robust and carry distinctive
experimental signatures, and so they could find applications in the new
generation of polaritonic devices.
