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In [2] (abstract [here]) (https://inspirehep.net/record/245643?ln=en) they say that, when compactifying any superstring theory, the six dimensional internal manifold must have $N=2$ supersymmetry with central charge $c=9$.
I have seen this stated before (e.g. in [1](abstract here)), but did not find any references justifying it.


[1] Gepner, Doron. "Space-time supersymmetry in compactified string theory and superconformal models." Current Physics–Sources and Comments. Vol. 4. Elsevier, 1989. 381-402. (Page 771) [2] Kazama, Yoichi, and Hisao Suzuki. "New N= 2 superconformal field theories and superstring compactification." Nuclear Physics B 321.1 (1989): 232-268. (Page 233)

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The critical dimension of the superstring is $d=10$, fixed by the conformal anomaly. The total central charge for the world sheet conformal theory is therefore \begin{equation} c=d\left(1_\text{scalar}+\frac{1}{2}_\text{fermion}\right)=10 \times\frac{3}{2}=15. \end{equation} From the spacetime point of view we want to compactify on a manifold \begin{equation} \mathcal{M}_{10}=\mathbb{R}^{1,3} \times M_6 \end{equation} which amounts to studying a tensor product of conformal $N=2$ theories, in which central charges add. So the $4$ scalar fields and their superpartners in $\mathbb{R}^{1,3}$ contribute \begin{equation} c_{\mathbb{R}^{1,3}}=4\times 3/2=6 \end{equation} and the remaining central charge of $15-6=9$ has to come from degrees of freedom in the internal manifold. That is, the theory of the internal manifold has to be described by a $N=2$ conformal theory of central charge 9.

Comment on $N=2$: If you start out writing down a generic worldsheet sigma-model for an internal space it is in general neither supersymmetric nor conformal. It can be shown that $N=2$ (or $N=(2,2)$ depending on notation) symmetry of the sigma model, corresponds to the internal space being a complex Kähler manifold. On top of that, conformal invariance requires it to be Ricci flat. These two conditions describe a Calabi-Yau manifold.

In other words, if you want to preserve SUSY by compactifying on a Calabi-Yau manifold, from the spacetime point of view, it corresponds to a superconformal $N=2$ theory on the worldsheet.

For technical details I would refer to Brian Greene's lectures (https://arxiv.org/pdf/hep-th/9702155.pdf) page 49, and references therein.

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  • $\begingroup$ Thanks, I get why $c=9$ now, but why $N=2$ SUSY on the worksheet? $\endgroup$ – Soap Sep 22 '19 at 16:33
  • $\begingroup$ I added a comment to my answer. $\endgroup$ – Sparticle Sep 22 '19 at 20:08
  • $\begingroup$ Two last questions: 1. From your comment, $N=2$ SUSY on the worldsheet implies that the internal manifold is Calabi-Yau. Do you know if the converse statement is also true? (Calabi-Yau implies $N=2$?) 2. Accepting that we need wordsheet $N=2$ SUSY, how does this mean that compactification "amounts to studying a tensor product of conformal $N=2$ theories"? $\endgroup$ – Soap Sep 23 '19 at 10:17
  • $\begingroup$ Since the theories are $N=2$ superconformal, should,t you get $c_{\mathbb{R}^{1,3}} = 4\times (1+1/2+1/2)$ (because you have two fermions for each boson)? $\endgroup$ – Soap Oct 2 '19 at 12:57

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