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?
 A: 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 


*

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

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

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

*$b_{1}=0$ --3-form which is dual to 1-form.
We will get only the half no. of scalars because of self-duality in 10-D.
