We predict that a photon condensate inside a dye-filled microcavity forms long-lived spatial structures that resemble vortices when incoherently excited by a focused pump orbiting around the cavity axis. The finely structured density of the condensates have a discrete rotational symmetry that is controlled by the orbital frequency of the pump spot and is phase-coherent over its full spatial extent despite the absence of any effective photon-photon interactions.
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