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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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  • $\begingroup$ mathoverflow.net/questions/1726/… $\endgroup$
    – Aaron Bergman
    Commented Jun 17, 2010 at 18:24
  • $\begingroup$ I don't see any mathematical explanation for "B" there. Just some physics which I don't understand. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jun 17, 2010 at 18:52
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    $\begingroup$ It would be great if you can explain the physics behind it in terms of real math. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Jun 17, 2010 at 18:53
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    $\begingroup$ I recommend that you read Aaron's answer there. He talks about gerbes and twisting the derived category. $\endgroup$
    – Charles Siegel
    Commented Jun 17, 2010 at 20:11
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    $\begingroup$ (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?) $\endgroup$
    – Michael Hardy
    Commented Jun 18, 2010 at 16:51

2 Answers 2

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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.

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This B field is from string theory. When you quantize bosonic, type II or Heterotic string theories, you will find a massless quantum state which is a second rank tensor on spacetime. The antisymmetric part of this tensor is the B field (the symmetric part is the graviton, and the trace part is the daliton). When you compactify superstring theories on Calabi-Yau 3-fold, the moduli space of this Calabi-Yau space is interpreted as massless fields in the four dimensional effective theory. Remeber the B field is massless, which means it shoud be part of the moduli space. This is the "physical" reason why we consider the complexified kahler form.

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