Is the Fermi source 4FGL J1824.2+1231 a transitional millisecond pulsar?

Transitional millisecond pulsars (tMSPs) in tight binary systems represent an important evolutionary link between low-mass X-ray binaries and radio millisecond pulsars. To date, only three confirmed tMSPs and a few candidates have been discovered. Most of them are γ-ray sources. For this reason, searching for multi-wavelength counterparts to unassociated γ-ray sources can help us find new tMSPs. We investigate whether the unassociated γ-ray source 4FGL J1824.2+1231 belongs to the tMSP family. To find the counterpart to 4FGL J1824.2+1231, we used data from the SRG/ and X-ray catalogues and from different optical catalogues. We also performed time-series photometric optical observations of the source with the 2.1 m telescope of the Observatorio Astron'omico Nacional San Pedro M'artir, the 1.5 m telescope of the Maidanak Astronomical Observatory, and the 1.5 m Russian-Turkish telescope. In addition, we carried out optical spectroscopic observations with the Russian-Turkish telescope and used archival spectroscopic data obtained with the telescope. Within the position error ellipse of 4FGL J1824.2+1231, we find only one X-ray source that coincides with an optical object. We consider it a likely multi-wavelength counterpart to 4FGL J1824.2+1231. The source shows strong optical variability and significant proper motion. The latter strongly implies that it is a Galactic source. Double-peaked H and He emission lines are detected in its spectrum with a flat continuum, as often observed in accretion disks of compact binary systems. The X-ray spectrum is well fitted by a power law with a photon index of ∼1.7. The derived intrinsic X-ray-to-γ-ray flux ratio is about 0.2. Assuming the X-ray/optical source is the true counterpart to 4FGL J1824.2+1231, all its properties suggest that it is a tMSP in the sub-luminous disk state.