fukaya category - How should I think about B-fields?

Let me try to add a different point of view on B-fields and mirror symmetry. Ideally in mirror symmetry, given a Calabi-Yau manifold X, you would like to "construct" its mirror X', where the symplectic form on X should give you the complex structure on X'. As already mentioned, classes of symplectic forms have moduli of real dimension $h^{1,1}(X)$ and complex structures on X' have moduli of complex dimension $h^{2,1}(X') = h^{1,1}(X)$. So the kahler class is not enough to determine all complex structures on X'. In the context of the Strominger-Yau-Zaslow conjecture there is a nice interpretation of the B-field. Suppose X = $T^*B / Lambda$, where B is a smooth manifold and $Lambda$ is locally the span over the integers of 1-forms $dy_1$, ..., $dy_n$ (here $y_1$, ..., $y_n$ are coordinates which
change with affine transformations from one chart to the other). Then $X$ has a standard symplectic form. We can consider $X'= TB / Lambda'$, where $Lambda'$ is the dual lattice.
Then X' has a natural complex structure defined as follows. In standard coordinates on TB, given by $(y,x)$ --> $x partial_y$, the complex coordinates on X' are $z_k = e^{2pi i(x_k + i y_k)}$, which are well defined due to the nature of the coordinates x and y. But the above complex coordinates can be twisted locally (on a coordinate patch) by $z_k (b) = e^{2pi i(x_k + b_k + i y_k)}$, where $b = (b_1, ldots, b_n)$ is some local data. But since on overlaps $U_i cap U_j$ the coordinates have to match, we must have $b(i) - b(j) in Lambda$. It turns out that by putting $b_{ij} = b(i) - b(j)$ on overlaps, we get a cohomology class in $H^{1}(B, Lambda)$, this is the B-field. The cohomology group $H^{1}(B, Lambda)$ shoud coincide (in some cases at least) with $H^2(X, R/Z)$, which is what Kevin Lin mentioned. The elliptic curve case (mentioned by Kevin) can be seen from this point of view.

This point of view is also called "mirror symmetry without corrections" and it only approximates what happens in compact Calabi-Yaus. I have learned this in papers by Mark Gross (such as "Special lagrangian fibrations II: geometry") or the book "Calabi-Yau manifolds and related geometries" by Gross, Huybrechts and Joyce.

I would be interested to know how this interpretation connects to the other ones which have been described.