Is string theory over a time varying background a conformal field theory to all orders in perturbation theory? 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?
 A: Note: This is a repost of a former upvoted answer which was plagiarized from a MathOverflow answer by Jeff Harvey. It is reproduced below as a community wiki, since visitors of this question apparently found it helpful nevertheless.


One must distinguish between quantum/classical on the string world-sheet and in spacetime.
  Both of your statements are basically correct, but should read something like  "CFT theory is the space of classical solutions to the spacetime equations of string theory" and "Quantization of the
  the world-sheet sigma model of a string theory gives rise to a CFT." 
In a little more detail,
  the sigma-model describing string theory propagation on some manifold M is a 2-dimensional
  quantum field theory which in order to describe a consistent string theory must be a conformal
  field theory. The "classical limit" of this 2-dimensional field theory is a limit in which some
  measure of the curvature of M is small in units of the string tension. To construct a CFT one
  must solve the sigma-model exactly, including world-sheet quantum effects. 
The coupling constants of the sigma-model are fields in spacetime such as the metric $g_{\mu \nu}(X(\sigma))$ on $M$ where $X: \Sigma \rightarrow M$ define the embedding of the string world-sheet  $\Sigma$ into $M$. Now there
  is also a spacetime theory of these fields. You can think of it as a ``string field theory".  At low-energies it can sometimes be usefully approximated by a theory of gravity coupled to some finite number
  of quantum fields, but in full generality it is a theory of an infinite number of quantum fields. Roughly speaking, each operator in the CFT gives rise to a field in spacetime.  The spacetime string field theory lives
  in 10 dimensions for the superstring or 26 dimensions for the bosonic string and it also has a classical limit. The classical limit is $g_s \rightarrow 0$ where $g_s$ is
  a dimensionless coupling constant. It appears in perturbative string theory as a factor which
  weights the  contribution of a Riemann surface by the Euler number of the surface. It can also be
  thought of as the constant (in spacetime) mode of a scalar spacetime field known as the dilaton.
The main point is that there are two notions of classical/quantum in string theory, one involving
  the world-sheet theory, the other the spacetime theory. In order to avoid confusion one must be clear which is being discussed. Unfortunately string theorists often assume it is clear from the context.
In response to the further question about the space of string fields, I would suggest that you have a look at the introductory material in http://arXiv.org/pdf/hep-th/9305026. You may also find http://arXiv.org/pdf/hep-th/0509129 useful. I should add that while string field theory has had some success recently in the description of D-brane states, it is not widely thought to be a completely satisfactory definition of non-perturbative string theory. 

