When computing the first order perturbative corrections to string theory over a curved background, we find the background has to be Ricci-flat if the dilaton is constant and we have no fluxes. Such is the case for Calabi-Yau compactifications. However, to fourth order in perturbation theory, we find nonzero contributions to the beta function. But this can be resolved by perturbative modifications to the background metric which cancels the beta function order by order in perturbation theory.
Does this procedure work for a generic time-varying background which is Ricci-flat to first order in perturbation theory? If not, does that tell us we can't apply first quantized string theory to such backgrounds?