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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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think know. A 2D CFT assigns a Hilbert space $H$ {\cal H}$to a circle and$$A(X): H^{\otimes {\cal H}^{\otimes n}\rightarrow H^{\otimes {\cal H}^{\otimes m}$$Hilbert space$H${\cal H}$ might be the space of functions on the$H=L^2(Maps(S^1,M))$ {\cal H}=L^2(Maps(S^1,M))$with some suitable restrictions on the maps.of$n$circles is$H^{\otimes {\cal H}^{\otimes n}$. Now we consider$n$strings All this makes a modicum of sense. So$H${\cal H}$ isfunctions on $T^*Map(S^1,M)$ into operators on $H$. {\cal H}$. To myThus, when applied to a vector$\psi_0\in H${\cal H}$, the theory would generate a solution to Schroedinger's equation

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 $M$.
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 $\psi_0\in H$, 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?