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I wouldn't go too far with the entire "first quantization"/"second quantization" thing. You could imagine a free string field theory where the string isn't allowed to interact, but the nice thing about string perturbation theory is that the interactions and the propagation are different aspects of the same thing. This is in contrast to the perturbation theory of quantum field theories where the interactions and the propagators are different things (one is pointlike, the other is lifelike). The CFT (really, a theory of 2D quantum gravity) is what you start with, and you automatically get both the "free" string theory and the interactions. The question of "second quantization" (a term I really hate) is whether or not you can derive the formal power series resulting from adding the various amplitudes associated with Riemann surfaces of varying general as the perturbation expansion of another theory.

To answer your questions about how you go from fields on the worldsheet to fields in spacetime, you quantize the theory on the cylinder, and each vector in the Hilbert space corresponds to a spacetime field (because you can Fourier expand fields on the cylinder, this isn't as crazy as it sounds). However, because you really are doing 2D quantum gravity, you have to deal with the gauge invariances. The nice way to do this is using BRST quantization, and the actually physical fields are the cohomology of the BRST operator acting on the CFT Hilbert space.

This is pretty standard material in a first course on string theory. I don't have them on me, but I'd expect Eric D'Hoker's lectures in the IAS volumes on QFT and strings for mathematicians would do this.

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This is reiterating a lot of what Jeff said, but maybe I can explain from a different perspective.

There are two things going on here (as there always are in string perturbation theory.) The first is the string worldsheet, and the second is what is going on in spacetime.

The string worldsheet is a non-linear sigma model into spacetime. Here, spacetime is a Riemannian manifold (with plenty of other structure depending on the exact string model you're using.) The "nonlinear sigma model" on the string worldsheet (a surface potentially with multiple punctures/boundaries) has a metric (different from the metric on the target manifold) and a map from the worldsheet into the spacetime manifold -- there are other fields in fancier versions of string theory, but I'll neglect them. In string perturbation theory, you integrate over the moduli space of metrics and embeddings. The resulting theory is invariant under conformal transformations, and because metrics in two dimensions don't have a huge amount information in them, an essential part of the theory on the worldsheet ends up being a conformal theory. There are various other conditions which ensure that the CFT gives rise to a full theory of 2D quantum gravity, meaning that you really can integrate over the space of the metrics. If those conditions hold, using the CFT, you can compute string amplitudes corresponding to your punctured Riemann surface. The can be thought of as scattering string in spacetime.

The important thing is that the amplitudes computed above are supposed to be terms in an asymptotic expansion of, er, something. This is why it's called string perturbation theory: in analogy to quantum field theory, combining individual string amplitudes of higher and higher genus in a formal power series (where the parameter is called the "string coupling") is supposed to be an expansion arising from some "nonpertubative" theory. What this theory is in complete generality is still unknown (although we know a lot in various special cases).

We can try to ask what this all looks like from the point of view of spacetime. Now, a basic fact about perturbation theory is that it only really makes sense (or, at least, makes the best sense) when you're perturbing around a solution. All of this is a roundabout way of saying that the spacetime only makes sense when the target manifold and its various structures give rise to a good perturbation expansion which means that the two dimensional theory is a conformal field theory.

This is what people mean when they say that a 2D CFT is a solution to the equations of motion of string theory. In fact, you can drop the requirement that your 2D theory is a "non-linear sigma model", ie, that it has the structure of maps into a manifold. Then you get into the "moduli space" of two dimensional field theories. Which, as far as I know, is completely undefined. But, even in this case (the world of string field theory), the "classical solutions" are the ones where you can define a good perturbation expansion around, and those are the conformal field theories.