complexified kahler form

In mirror symmetry one usually considers a complexified kahler form $B+iw$ instead of kahler form $w$ itself.(Or their moduli)

Here is the question:

What does $B$ correspond to? what kind of information it has?.... I would also like to know about the moduli of complexified kahler structures on a Calabi-Yau 3-fold. Is it a cone? If yes, what are the walls ? does it have a natural metric?

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mathoverflow.net/questions/1726/… – Aaron Bergman Jun 17 2010 at 18:24
I don't see any mathematical explanation for "B" there. Just some physics which I don't understand. – Mohammad F.Tehrani Jun 17 2010 at 18:52
It would be great if you can explain the physics behind it in terms of real math. – Mohammad F.Tehrani Jun 17 2010 at 18:53
I recommend that you read Aaron's answer there. He talks about gerbes and twisting the derived category. – Charles Siegel Jun 17 2010 at 20:11
(If I proposed that either Kähler or Kaehler is the correct spelling, and those two are in a sense the same, whereas Kahler is different from those and represents a different pronunciation, would that be a hopeless cause?) – Michael Hardy Jun 18 2010 at 16:51
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Since B-fields are being discussed in the other question, i'll try to answer your question on the complexified Kahler moduli. The set of Kahler classes of a Kahler manifold is an open cone in $H^{1,1}(X, C) \cap H^2(X, R)$. There are results on the structure of this cone which roughly say that it is rational polyhedral away from some set (I think!). In the construction of the complexified Kahler cone you have to take a "framing" of this cone, in the end you get something isomorphic to $(\Delta^{\ast})^{h^{1,1}}$, where $\Delta^*$ is the punctured disc in $C$. This matches the moduli of complex structures of the mirror, near a large complex structure limit point. These things are discussed in the book "Calabi-Yau manifolds and related geometries", by Gross, Huybrechts and Joyce.