# How Quintic 3-fold is a Calabi–Yau manifold and has non-vanishing Ricci scalar?

It’s well known that quintic 3-fold is a Calabi-Yau manifold in the complex projective space $$\mathbb{CP}^{n+1}$$ , see for instance:

Now the main feature of Calabi–Yau manifold that it has vanishing Ricci tensor $$R_{\mu\nu}=0$$ , so the Ricci scalar vanishes as well.

But when I was studying the Quintic from this paper:

https://www.staff.science.uu.nl/~beuke106/HypergeometricFunctions/COGP.pdf

I have found in figure 10, the Ricci scalar has been plotted.

So how is that? do I miss something here?

Another question, in general, if I’m studying CY 3-fold in the context of string theory decomposition, for instance, can I take the metric of the quintic driven in that paper, equ. (4.3) , to explore CY’s complex space structure?

• I think you are confusing things between the Calabi-Yau and its moduli space. The complex structure moduli and Kähler moduli of the Calabi-Yau, both respectivly parametrize a manifold, which are Special Kähler manifolds. The plots are for a moduli space. – Sparticle Sep 23 '19 at 23:11
• No. I don’t think I’m confusing or any thing. I know what are you saying very well. So don’t explain for me please the very basics of CY manifold. The question is very specific if you got it , the plot in figure 10 says a plot of the Ricci scalar against.... etc, have you seen that? @Sparticle – S.S. Sep 24 '19 at 1:31

On page 50, it says

we note that for a one-dimensional manifold that is special Kahler, the Ricci scalar is related to the invariant coupling by $$R+4=2\gamma_\text{inv}^2$$ and we present a three-dimensional plot of the Ricci scalar in fig. 10

On page 55, it says

for large $$\psi$$, the Ricci scalar of the moduli space differs from its limiting value by inverse powers of log $$\psi$$ as is evident from fig. 9

The bolding is mine, not theirs, to emphasize that the authors are not talking about the Ricci scalar of the Calabi-Yau manifold itself. As you rightly pointed out, its Ricci scalar vanishes.

Thus I believe that @Sparticle's comment was correct: the plots of a Ricci scalar in figures 9 and 10 are for the Ricci scalar of a moduli space.

• So that forms had been made! I have said in the question already, I may have missed something about, so thank you!! And that’s also new area for me, so I wasn’t see a point to calculate R for only the moduli space . Any ways didn’t mean any offend @Sparticle , @G.smith – S.S. Sep 24 '19 at 16:32