  Despite the weak, van-der-Waals interlayer coupling, photoinduced charge
transfer vertically across atomically thin interfaces can occur within
surprisingly fast, sub-50fs timescales. Early theoretical understanding of the
charge transfer is based on a noninteracting picture, neglecting excitonic
effects that dominate the optical properties of such materials. Here, we employ
an ab initio many-body perturbation theory approach which explicitly accounts
for the excitons and phonons in the heterostructure. Our large-scale
first-principles calculations directly probe the role of exciton-phonon
coupling in the charge dynamics of the WS$_2$/MoS$_2$ heterobilayer. We find
that the exciton-phonon interaction induced relaxation time of photo-excited
excitons at the $K$ valley of MoS$_2$ and WS$_2$ is 67 fs and 15 fs at 300 K,
respectively, which sets a lower bound to the intralayer-to-interlayer exciton
transfer time and is consistent with experiment reports. We further show that
electron-hole correlations facilitate novel transfer pathways which are
otherwise inaccessible to non-interacting electrons and holes.
