Effective action of IIB Calabi-Yau compactification

I'm currently reading K. Becker, M. Becker and John. H. Schwartz book on string theory. I have a question about Calabi-Yau compactifications of IIB string theory. In chapter 9 page 403, Why do we don't consider self-dual four-form with two fundamental index and two antifundamental index $C_{ij\bar k\bar l}$. It seems that these should leads to$\ h^{(2,2)}$ scalars in four dimensional world. What am I missing? Does self-duality play a role?

In 4-D, We only see one form gauge field and its corresponding 2-form field strength and of-course Scalars. The main reason is higher forms is dual to scalars in 4-D. Two form field has three form field strength which is dual to one form. So corresponding two form field is just scalars. This operation is done by Hodge star product [https://en.wikipedia.org/wiki/Hodge_star_operator].

Similar thing follows in higher dimensions. In 10-D, Type II-B theory, There exist 4-form field which has self-dual 5-form field strength. We want to compactify this form on the Calabi-Yau 3 folds which have 3 complex or 6 real dimensions.There is a theorem which is great interest to us.

"Given a massless p form gauge field in 10 D, we get $b_{p-q}$ massless q form gauge field in $D=4$. Here $b_{\alpha}$ are Betti's number for the manifold."

So for 4 form gauge field in 10 D, we will get

1. $b_{4}=h_{1,1}$ scalars

2. $b_{3}=2(1+h_{2,1})$ vectors but only half due to self duality.

3. $b_{2}=h_{1,1}$ tensors or two form field which is dual to scalars in $4-D$.

4. $b_{1}=0$ --3-form which is dual to 1-form.

We will get only the half no. of scalars because of self-duality in 10-D.

• Dear Hare, thank you for reply, very clear, I also found some hints in Polchinski's book, chapter 19. Because of the self-duality, scalars which come from $b_4$ and $b_2$ are identical. I think that the four-form in the line no. 5 is not dynamical, because there is no field strength for it in four-dimensions and shouldn't be considered Mar 6 '18 at 12:36
• Thanks, Ahmad. There is only 1 four form which is just volume form and is not dynamical.
– Hkw
Mar 6 '18 at 14:20