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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    $\begingroup$ 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. $\endgroup$ Feb 17, 2011 at 8:57
  • $\begingroup$ 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. $\endgroup$ Feb 17, 2011 at 8:58

1 Answer 1


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.


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