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":


[Link](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-123/issue-2/Classical-and-quantum-conformal-field-theory/cmp/1104178762.full)




---


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).



What exactly is the relation between string theory and conformal field theory?
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. 


