  We combine the effect of the electron-electron and electron-phonon
interactions to study the electronic and optical properties of zb-GaN. We show
that only by treating the two effects at the same time it is possible to obtain
an unprecedented agreement of the zero and finite-temperature electronic gaps
and absorption spectra with the experimental results. Compared to the
state-of-the-art results our calculations predict a large effect on the main
absorption peak position and width as well as on the overall absorption
lineshape. These important modifications are traced back to the combined
electron-phonon damping mechanism and non uniform GW level corrections. Our
results demonstrate the importance of treating on equal footing the electron
and phonon mediated correlation effects to obtain an accurate description of
the III-nitrides group physical properties.
