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Suppose I have a compactification of string theory to four Minkowski dimensions of the form $M^{1,3}\times X$, where the internal CFT on $X$ is a $c=9$, $(2,2)$ SCFT. For example, let the SCFT on $X$ be described by the IR fixed point of an LG model with the superpotential $W(\Phi)=\Phi^p$. Can someone explain to me (hopefully in details) how can we extract the space-time physics in $M^{1,3}$ from the data of the internal CFT. What is the relation between the world-sheet superpotential $W(\Phi)=\Phi^p$ and the space-time superpotential? What is the space-time interpretation of the chiral ring of $W(\Phi)$? How does the non-perturbative corrections to the space-time superpotential are related to the correction to $W(\Phi)$. I know this is a basic question in string theory and it should be easy to find an answer in Polchinski but I hope someone will clarify things further.

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There is no simple relationship between the worldsheet objects and spacetime objects (such as superpotentials) in this curved case (and many other general cases).

The LG-like models correspond to Calabi-Yau compactifications whose typical radii are comparable to the string length so it is very hard to see the spacetime geometry - and moreover, the mirror symmetric geometry is as easy or as hard to see as the "original" one, whichever is which.

Moreover, much of the world sheet objects are auxiliary and don't survive into spacetime. In particular, the whole world sheet supersymmetry is an auxiliary symmetry - a gauge symmetry - that doesn't really survive in spacetime. Its existence has consequences in spacettime (nonrenormalization theorems etc.) but its detailed building blocks don't have any counterparts because they're partly "unphysical states" from the spacetime viewpoint.

So the way to proceed when identifying the topology of Gepner-like models is to find the Hodge numbers and primary operators, choose a corresponding topology of a Calabi-Yau space that matches the constraints, and verify that this candidate is indeed equivalent.

To see that there can't be any coordinate-by-coordinate map between the world sheet pieces and spacetime physics, note that the central charges of the world sheet CFT are fractional. The relationship between Gepner models and particular Calabi-Yaus is a form of duality, and as any duality, it must be nontrivial to be seen, otherwise it wouldn't be interesting and it wouldn't really be a "duality".

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