I am a physicist and would like to understand the section 1 of this math paper, which explains how the SYZ conjecture implies topological mirror symmetry. I have some technical problem and would appreciate your help. My question is purely mathematical and nothing scary to most of you, I think.

The setting is the following; let $f:X\rightarrow B$ be a 3-torus $T^3$-fibration of a Calabi-Yau threefold (which has singular fibers). Since $X$ is a real sixfold, $B$ is a real threefold. Under certain technical assumption, using Leray spectral sequence, the author shows that the dual torus fibration gives another Calabi-Yau threefold with mirrored Hodge diamond.

My questions concern the Leray spectral sequence associated with $f$. First I don't quite understand why $E_2$ of the Leray spectral sequence looks like the diagram in the paper; how can we conclude that $$ H^{i}(B,R^{j}f_{*}\mathbb{R})=\mathbb{R} \ (i=0,3), \ \ 0 \ (i=1,2) $$ for $j=0,3$? My understanding is that roughly the direct image sheaves $R^{i}f_{*}\mathbb{R}$ are local system, so on a small open subset we have $R^{i}f_{*}\mathbb{R}\cong H^{i}(T^{3},\mathbb{R})$. How should we understand the computation above?

Secondly why $\pi_{1}(X)=0$ (or rather $H^{1}(X,\mathbb{R})=H^{5}(X,\mathbb{R})=0$) implies that $$ H^{0}(B,R^{1}f_{*}\mathbb{R})=H^{3}(B,R^{2}f_{*}\mathbb{R})=0? $$ I know that the problem is caused by my poor understanding of direct image sheaves. I hope to make my understanding clearer with this example.

I am afraid that my questions are elementary and this may not be a place to ask such a question. Any help, comments and reference suggestions will be appreciated.

Thank you,