We present an optical/near-infrared search for a counterpart to the perplexing radio transient GCRT J1745-3009, a source located ~1 degree from the Galactic Center. Motivated by some similarities to radio bursts from nearby ultracool dwarfs, and by a distance upper limit of 70 pc for the emission to not violate the 1e12 K brightness temperature limit for incoherent radiation, we searched for a nearby star at the position of GCRT J1745-3009. We found only a single marginal candidate, limiting the presence of any late-type star to >1 kpc (spectral types earlier than M9), >200 pc (spectral types L and T0-T4), and >100 pc (spectral types T4-T7), thus severely restricting the possible local counterparts to GCRT J1745-3009. We also exclude any white dwarf within 1 kpc or a supergiant star out to the distance of the Galactic Center as possible counterparts. This implies that GCRT J1745-3009 likely requires a coherent emission process, although whether or not it reflects a new class of sources is unclear.
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