1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 $\endgroup$ – Ahmad Mar 6 '18 at 12:36
  • $\begingroup$ Thanks, Ahmad. There is only 1 four form which is just volume form and is not dynamical. $\endgroup$ – Hare Mar 6 '18 at 14:20

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.