I have been trying to understand why one should look into $c=9, N=2$ superconformal models like the Gepner models or the Kazama-suzuki models, and I am quite confused.
This is what I understood from the arguments I've seen: (mainly from [1])
The usual way to deal with the $6$ spacetime dimensions which we do not observe in reality is by assuming that we can write the $10$-dimensional spacetime manifold as a product $M_4 \times K$, where $M_4$ is Minkowski space together with the $4$-dimensional superstring action with $4$ free bosons and $4$ free fermions, forming a $c= 4\times \frac{3}{2} = 6$, $N=1$ superconformal field theory, and the six dimensional manifold $K$ is called the internal manifold and has to be compactified. It's field theory must be supersymmetric and have central charge $c=15-6=9$.
In order to ensure $\mathcal{N}=1$ spacetime supersymmetry, one forces both the internal and the external CFTs to actually be $N=2$ SCFTs: the spectral flow plays the part of spacetime supersymmetry generator $Q$. Therefore, if we want to understand the internal manifold, we should study $c=9$, $N=2$ SCFTs.
My problems with this:
We deduced that $c=9$ because we used that the superstring action was just an action with free bosons and free fermions (the RNS action). This action is not $N=2$ supersymmetric (only $N=1$), so by demanding the theory to have $N=2$ SUSY don't we have to throw away all the results we obtained from the RNS superstring? In particular, $c\ne 15$ if $N=2$, because in that case we would have two fermionic superpartners for each boson, not one.
This seems to ensure 10D spacetime SUSY, but don't we also want 4D spacetime SUSY?
What is the connection between this and the Calabi-Yau approach? Is one approach more general than the other? Do these SCFTs somehow correspond to specific CY manifolds?
[1] Greene, B. (1997). String theory on calabi-yau manifolds. arXiv preprint hep-th/9702155.
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