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Maybe it would be helpful for me to summarize the little bit I think know. A 2D CFT assigns a Hilbert space ${\cal H}$ to a circle and an operator $$A(X): {\cal H}^{\otimes n}\rightarrow {\cal H}^{\otimes m}$$ to a Riemann surface $X$ with $n$ incoming boundaries and $m$ outgoing boundaries. This data is subject to natural conditions arising from the sewing of surfaces.

Here is how I understand the relation to string theory. The Hilbert space ${\cal H}$ might be the space of functions on the configuration space of a string sitting in a manifold $M$. So ${\cal H}=L^2(Maps(S^1,M))$ with some suitable restrictions on the maps. It is natural then that the functions on the configuration space of $n$ circles is ${\cal H}^{\otimes n}$. Now we consider $n$ strings evolving into $m$ strings. There are many ways to do this, one for each Riemann surface $X$ as above. When $X$ is fixed, $A(X)$ is the evolution operator, usually described in terms of some path integral over maps from $X$ into $M$ involving a conformally invariant functional.

All this makes a modicum of sense. So ${\cal H}$ is the Hilbert space arising from quantization of the cotangent bundle of $Map(S^1,M)$, while $A$ describes time evolution. So in this sense, such a conformal field theory appears to be the quantization of the classical string. I guess what is missing in my description up to here is the prescription for turning functions on $T^*Map(S^1,M)$ into operators on ${\cal H}$. To my deficient understanding, maybe this situation corresponds to having quantized only the Hamiltonian.

Now, what I was really wondering was this: When I was in graduate school, I remember frequently hearing the phrase:

CFT theory is the space of classical solutions to string theory.

Does this make some sense? And if so, what does it mean? This phrase has been hindering my understanding of conformal field theory ever since, making me feel like my grasp of physics is all wrong. According to the paragraphs above, my naive formulation would have been:

Quantization of a string theory gives rise to a CFT.

What is wrong with this naive point of view? If you could provide some enlightenment on this, you'll have resolved a long-standing cognitive itch in the back of my mind.

Thanks in advance.


Added:

As Jose suggests, I could simply be remembering incorrectly, or misunderstood before what I heard. That, in fact, is what I had hoped to be the case. But read, for example, the first page of Moore and Seiberg's famous paper "Classical and quantum conformal field theory":

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104178762


Added again:

To quote Moore and Seiberg more precisely, the second sentence of the paper reads 'Conformal field theories are classical solutions of the string equations of motion.' Now, I might attempt to understand this as follows. When the Riemann surface is $$S^1\times [0,t]$$(with the conformal structure induced by the standard metric) one interprets $$A(S^1\times [0,t])$$ as $$e^{itH}.$$ Thus, when applied to a vector $\psi_0\in {\cal H}$, the theory would generate a solution to Schroedinger's equation $$\frac{d}{dt}\psi =iH \psi$$ with initial condition $\psi_0$ as $t$ varies. So one might think of the various $A(X)$ as $X$ varies as being 'generalized solutions' to Schroedinger's equation for a quantized string. I suppose I could get used to such an idea (if correct). But then, the question remains: why do they (and others) say classical solutions? Is there some kind of second quantization in mind with this usage?


Added, 11 October:

Even at the risk of boring the experts, I will have one more go. Jeff Harvey seems to indicate the following. We can think of $Map(X, M)$ as the fields in a non-linear sigma model on $X$, provisionally thought of as 1+1 dimensional spacetime. However, it seems that one can also associate to the situation a space of fields on $M$ (the string fields?). If we denote by ${\cal F}$ this space of fields, it seems that there is a functional $S$ on ${\cal F}$ with the property that the extrema of $S$ (the 'string equations of motion') can be interpreted as the $A(X)$. From this perspective, my main question might then be 'what is ${\cal F}$?' Since I think of fields on $M$ as being sections of some bundle on $M$, I can't see how to get such a thing out of maps from $X$ to $M$.

Thank you very much for your patience with these ignorant questions.


Final addition, 11 October:

Thanks to the kind guidance of Jose, Aaron, and especially Jeff, I think I have some kind of an understanding of the situation. I will attempt to summarize it now, superficial as my knowledge obviously is. I don't wish to waste more of the experts' time on this question. However, I am hoping that truly egregious errors will offend their sensibilities enough to elicit at least a cry of outrage, enabling me to improve my poor understanding. I apologize in advance for putting down even more statements that are either trivial or wrong.

As far as I can tell, the sense of Moore and Seiberg's sentence is as in my second addition: it is referring to second quantization. Recall that in this process, the single particle wave functions become the classical fields, and Schroedinger's equation is the classical equation of motion. Now the truly elementary point that I was missing (as I feared), is that

quantization of a 'single particle' string theory cannot give you a conformal field theory.

At most, a single string will propagate though space, giving us exactly the operators $A(S^1\times [0,t])$. If we want operators $$A(X):{\cal H}^{\otimes n}\rightarrow {\cal H}^{\otimes m}$$ corresponding to a Riemann surface with many boundaries, then we are already requiring a theory where particle numbers can change, that is, a quantum field theory coming from second quantization. WIth such a theory in place, of course, the $A(S^1\times [0,t])$ are exactly the solutions to the classical equations of motion, while the general $A(X)$ can be viewed either as 'generalized classical solutions' (I hope this expression is reasonable) or contributions to a perturbation series, as in the field theory of a point particle. So this, I think. already answers my original question. To repeat, because of the changing 'particle number'

the operators of conformal field theory cannot be the quantization of a 'single particle' theory. They must be construed as classical evolution operators of some kind of quantum (string) field theory.

The part I'm still far, far from understanding even superficially is this: The classical fields in the case of strings would be something like functions on $Map(S^1, M)$. I haven't the vaguest idea of how to get from this to fields on spacetime. The difficulty surrounding this issue seems to be discussed in the beginning pages of Zwiebach's paper referred to by Jeff, which is quite heavy reading for a pure mathematician like me. Some mention is made of infinitely many fields arising from the situation (alluded to also by Jeff), which perhaps is some way of turning the data of a function on loop space to fields on space(-time).

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I hope you are misremembering the phrase from your grad school days, because that is not correct. Your second formulation is much more on target. What you may have heard in grad school is that CFTs (of a certain kind) are the perturbative string backgrounds, which are morally like solutions of string theory, but I would not call them "classicai". There is a separate notion of classical solutions of string theory, which means just that: take the classical equations which come from varying the sigma model action and solve them. –  José Figueroa-O'Farrill Oct 10 '11 at 1:06
    
One problem with your naive formulation is that you have CFTs which do not arise out of quantising the sigma model. These are the so-called "strings without strings": CFTs for which a geometric interpretation in terms of string propagation on some geometry is not obvious. Extracting geometry from CFTs is quite tricky. You can do it in cases where the CFTs come in families with a parameter you can take a "classical limit" in. There was some work by Roggenkamp and Wendland a few years back which succeeded in extracting a geometric picture from such a family of CFTs. –  José Figueroa-O'Farrill Oct 10 '11 at 1:11
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3 Answers

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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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Thank you very much for your nice answer, which I do not doubt is the correct one. Sadly (for me), it confirms the original fear that my grasp of basic physics is woefully inadequate. –  Minhyong Kim Oct 10 '11 at 15:49
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If string theory is basic physics then arithmetic algebraic geometry is basic math. ;) –  Jeff Harvey Oct 10 '11 at 17:23
    
Jeff: but in the context of "strings without strings" that I mentioned, how do you get a classical solution of the spacetime fields from an abstract CFT? –  José Figueroa-O'Farrill Oct 10 '11 at 18:28
    
Jose, I certainly agree there are subtleties that I didn't try to address. At least for open string theory my limited understanding is that one can construct Witten's OSFT from an abstract CFT in such a way that the classical equations of motion are trivially satisfied. p.s. I apologize for my Tex incompetence in not being able to figure out how to put the correct accent on your name. –  Jeff Harvey Oct 10 '11 at 18:54
    
Heh. What I meant was that it's probably very basic deficiencies that keep me from understanding the advanced stuff. –  Minhyong Kim Oct 11 '11 at 0:09
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(Rather than a long series of comments, perhaps I should post an answer, even though it will not be a very comprehensive answer due to lack of time right now.)

String propagating on a manifold (with some extra data depending on which string theory you are looking at) is classically described by a sigma model. There are few sigma models you can actually quantise exactly, but there are some you can. For example, you can quantise string propagating on Minkowski spacetime provided the dimension is right. The dimension depends on the type of string theory you are quantising: 26 dimensions for the bosonic string, 10 dimensions for the NSR strings,... For those sigma models which can be quantised, you find that you get a CFT with a certain value of the central charge and a certain chiral algebra: Virasoro, $N=1$ Virasoro,... The Hilbert space of the quantised string theory is then identified with the (relative) semi-infinite cohomology of the chiral algebra with values in the module given by the CFT. In the Physics literature this is called the BRST cohomology.

However you could now simply start with a CFT of the right kind and declare that to be your quantum string theory upon taking the (relative) semi-infinite cohomology. So for instance, any CFT with central charge (26,26) gives rise to a consistent bosonic string background. Of course not all such CFTs need arise out of quantising a bosonic string sigma model. Similar statements hold for the other string theories.

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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.

Added 10/12:

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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Well, I'm obviously using terms I don't understand. But in this case, I don't see why 'second quantization' is so objectionable a description. After all, aren't we exactly trying to view the QM wavefunction of a classical particle(=string) as a classical field, which then needs to be quantized. This is quite different, I think, from the usage of the term in, say, QED, where it really is rather inappropriate. (Since the classical electromagnetic field is not the quantization of anything.) –  Minhyong Kim Oct 13 '11 at 17:22
    
I just don't like second quantization as a way of thinking about field theory, but that's really besides the point here which is the difference between one and two dimensional manifolds. When you are looking at quantum gravity in one dimension, there's just not all that much going on. In two dimensions, things are a lot more interesting and there is the connection to 2D CFTs. You don't only quantize on the cylinder and not get a CFT; you have a 2D theory of quantum gravity that makes sense on all 2D manifolds. cont'd –  Aaron Bergman Oct 14 '11 at 0:13
    
This is the magic of moving from one dimension to two dimension, from particles to strings. Instead of having to put in your interactions by hand, you get them for free. They're completely determined by your choice of theory of 2D quantum gravity. Admittedly this isn't too far from what you're saying, but I think it's a better way to think about it. The CFT comes first, and you get an interacting theory automagically. –  Aaron Bergman Oct 14 '11 at 0:27
    
Thanks for your further elaboration. Maybe I will bother you again sometime about this one to 2D distinction. For now, I'll try to peruse D'Hoker's notes you mention. I guess I should also modify my remark about QED. It seems to me that quantizing the EM field is not second quantization in any sense, and I'm guessing (out of pure ignorance) that the term came about because the Hilbert space looks like Fock space. But quantizing a spinor does seem to be second quantization. –  Minhyong Kim Oct 16 '11 at 11:51
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