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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?

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I will write this only as a comment because it would be nice to see a more complete answer but: covariant string theory always has to respect conformal symmetry on the world sheet, otherwise it breaks down. The fact that the Ricci flatness is not the exact condition in spacetime shouldn't be surprising; it's more surprising that Einstein's vacuum equations don't get any corrections from several orders in the perturbation theory. If the conformal symmetry disappeared in covariant strings at higher orders, the theory would become inconsistent because there's no known replacement. –  LuboŇ° Motl Feb 17 '11 at 8:57
By SUSY, one may pretty much show - or at least argue - that the backgrounds can't disappear at higher orders. For general time-dependent i.e. also non-supersymmetric backgrounds, it's tougher. Many things may occur. But the corrections are still small. Also, let me mention that in the light-cone gauge, the world sheet description may be non-conformal. That's the case of the pp-wave background with some Ramond-Ramond background field strength - the world sheet fields in the light-cone gauge world sheet description become massive. –  LuboŇ° Motl Feb 17 '11 at 8:58
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