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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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(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,...

However you could now simply start with a CFT of the right kind and declare that to be your quantum string theory. 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.