# How can we demand $N=2$ supersymmetry on the worldsheet?

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:

1. 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.

2. This seems to ensure 10D spacetime SUSY, but don't we also want 4D spacetime SUSY?

3. 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.

• Related post by OP: physics.stackexchange.com/q/504136/2451 – Qmechanic Oct 4 '19 at 17:51
• Maybe page 13 of this thesis (and references therein) is of (partial) help? livrepository.liverpool.ac.uk/3003839 – Heterotic Oct 5 '19 at 9:16
• If I remember correctly it is explained very well in Gepner's lecture notes. Full citation in the thesis above. – Heterotic Oct 5 '19 at 9:33
• @Heterotic Although this thesis provides a nice summary of what is going on, it does not address my issues, just like the standard references don't (unless I missed it on a first reading!). – Soap Oct 5 '19 at 10:00
• @Heterotic And although in the thesis it is claimed that any general Poincare invariant, Weyl invariant, worldsheet reparametrization invariant action must give rise to a CFT with central charge 26 in the bosonic case, in Gepner's lecture notes he explicitly says that this value for $c=26$ arises because we are dealing with the usual string action with 26 free bosons, each contributing with charge $1$. So I see no reason to say that a general action still gives $c=26$ (and similarly for the superstring). Notice how this is basically the problem 1 I wrote in the question. – Soap Oct 5 '19 at 10:16