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Films of iron selenide (FeSe) one unit cell thick grown on strontium titanate (SrTiO₃ or STO) substrates have recently shown^,,, superconducting energy gaps opening at temperatures close to the boiling point of liquid nitrogen (77 kelvin), which is a record for the iron-based superconductors. The gap opening temperature usually sets the superconducting transition temperature T_c, as the gap signals the formation of Cooper pairs, the bound electron states responsible for superconductivity. To understand why Cooper pairs form at such high temperatures, we examine the role of the SrTiO₃ substrate. Here we report high-resolution angle-resolved photoemission spectroscopy results that reveal an unexpected characteristic of the single-unit-cell FeSe/SrTiO₃ system: shake-off bands suggesting the presence of bosonic modes, most probably oxygen optical phonons in SrTiO₃ (refs , , ), which couple to the FeSe electrons …