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)


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.