We have discovered 11 ultra-faint ($rlesssim 22.1$) low surface brightness (LSB, central surface brightness $23lesssim mu_rlesssim 26$) dwarf galaxy candidates in one deep Virgo field of just $576$ arcmin$^2$ obtained by the Large Binocular Camera (LBC) at the Large Binocular Telescope (LBT). Their association with the Virgo cluster is supported by their distinct position in the central surface brightness - total magnitude plane with respect to the background galaxies of similar total magnitude. They have typical absolute magnitudes and scale sizes, if at the distance of Virgo, in the range $-13lesssim M_rlesssim -9$ and $250lesssim r_slesssim 850$ pc, respectively. Their colors are consistent with a gradually declining star formation history with a specific star formation rate of the order of $10^{-11}$ yr$^{-1}$, i.e. 10 times lower than that of main sequence star forming galaxies. They are older than the cluster formation age and appear regular in morphology. They represent the faintest extremes of the population of low luminosity LSB dwarfs that has been recently detected in wider surveys of the Virgo cluster. Thanks to the depth of our observations we are able to extend the Virgo luminosity function down to $M_rsim -9.3$ (corresponding to total masses $Msim 10^7$ M$_{odot}$), finding an average faint-end slope $alphasimeq -1.4$. This relatively steep slope puts interesting constraints on the nature of the Dark Matter and in particular on warm Dark Matter (WDM) often invoked to solve the overprediction of the dwarf number density by the standard CDM scenario. We derive a lower limit on the WDM particle mass $>1.5$ keV.
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