I am following the BBS book and I am currently studying chapter 9. I have a question - or better yet a point of confusion - regarding the derivation of the massless four-dimensional spectra when considering the IIA/B compactified on a ${\rm CY}_3$. Some context follows in order to present my query in a complete way.

We are considering a compactification of the IIA/B of the form $\mathcal{M}_{10} = \mathcal{M}_{4} \times {\rm CY}_3$. Since the spacetime metric is a sum of a four-dimensional and a six-dimensional piece the Laplacian also assumes the same form, namely $\Delta_{10} = \Delta_{4} + \Delta_{6}$. Then, the number of massless modes in the four-dimensional space is given by the number of the zero modes of the six-dimensional Laplacian. The number of these zero-modes of interest is given by the Betti number.

Let me be a bit more explicit and consider the ten-dimensional two-form field (this is the example considered on page 386). In what follows I am sticking to the notation in the book, that is the splitting of the indices is $M=(\mu,m)$. The two-form field can be written as

$\begin{equation} B_{MN} = B_{\mu \nu} \oplus B_{\mu n} \oplus B_{m n} \end{equation}$

Counting and interpretation.

From the four-dimensional point of view, the first term of the above relation is a two-form, the second one is a gauge field (one-form) and the final term is just a scalar (zero-form). From the six-dimensional point of view, the first term is a zero-form and the associated Betti number is the $b_0$. In the CY3 case, we have $b_0=1$. The second term is a one-form in the six-dimensional picture and thus the related Betti number is $b_1=0$. The final term is a two-form and we have $b_2=h^{1,1}$.

Therefore, the number of massless states in the four-dimensional theory is one massless two-form, no massless gauge field and $h^{1,1}$ scalars.

The above situation is an example that I understand.

My question:

My confusion lies when the authors consider the CY3 compactification of type IIA/B theories. Let's just take IIB on $\mathcal{M}_{4} \times $ CY$3$ to be concrete. Exercise 9.13 from the book on page 403 is precisely that. Let me present one case that confuses me. Consider the ${\rm SU}(3)$ covariant splitting of the indices - following the book - $M=(\mu,i,\overline{i})$. The metric is decomposed as

$\begin{equation} G_{MN} = G_{\mu \nu} \oplus G_{ij} \oplus G_{i \overline{j}} \end{equation}$

In the result of the exercise, it is stated that the first of the above is associated to $1$ which is the $b_0$ and makes sense, the third is related to the $h^{1,1}$ which is the result of $b_1$ and also makes sense but the term $G_{ij}$ is said to be related to $h^{2,1}$. The only Betti number on a CY3 related to that Hodge number is $b_{3}$. This is precisely what does not make any sense to me. The term $G_{ij}$ has two indices on CY3 and is thus a two-form so I was expecting that we would seek the $b_2$ number. Of course, I have similar questions with the indices in the rest of the $p$-forms in that exercise; I just wanted to give a simple example.

Can someone explain what I am missing or misunderstood?

I am certain that the book has no typo as the result that is presented has a nice interpretation in the context of mirror symmetry.