A large population of ultra-diffuse galaxies (UDGs) was recently discovered in the Coma cluster. Here we present optical spectra of three such UDGs, DF7, DF44 and DF17, which have central surface brightnesses of $mu_g approx 24.4-25.1$ mag arcsec$^{-2}$. The spectra were acquired as part of an ancillary program within the SDSS-IV MaNGA Survey. We stacked 19 fibers in the central regions from larger integral field units (IFUs) per source. With over 13.5 hours of on-source integration we achieved a mean signal-to-noise ratio (S/N) in the optical of $9.5$AA$^{-1}$, $7.9$AA$^{-1}$ and $5.0$AA$^{-1}$, respectively, for DF7, DF44 and DF17. Stellar population models applied to these spectra enable measurements of recession velocities, ages and metallicities. The recession velocities of DF7, DF44 and DF17 are $6599^{+40}_{-25}$km/s, $6402^{+41}_{-39}$km/s and $8315^{+43}_{-43}$km/s, spectroscopically confirming that all of them reside in the Coma cluster. The stellar populations of these three galaxies are old and metal-poor, with ages of $7.9^{+3.6}_{-2.5}$Gyr, $8.9^{+4.3}_{-3.3}$Gyr and $9.1^{+3.9}_{-5.5}$Gyr, and iron abundances of $mathrm{[Fe/H]}$ $-1.0^{+0.3}_{-0.4}$, $-1.3^{+0.4}_{-0.4}$ and $-0.8^{+0.5}_{-0.5}$, respectively. Their stellar masses are $3$-$6times10^8 M_odot$. The UDGs in our sample are as old or older than galaxies at similar stellar mass or velocity dispersion (only DF44 has an independently measured dispersion). They all follow the well-established stellar mass$-$stellar metallicity relation, while DF44 lies below the velocity dispersion-metallicity relation. These results, combined with the fact that UDGs are unusually large for their stellar mass, suggest that stellar mass plays a more important role in setting stellar population properties for these galaxies than either size or surface brightness.
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