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In mirror symmetry conjecture, we add what is called "B-field" $B \in H^2(X,\mathbb{R}/\mathbb{Z})$ in the Kähler moduli space so that the Kähler moduli space has enough freedom comparable to the complex moduli space of a mirror manifold. In string theory, one may twist the Lagrangian by this auxiliary 2-form $B$.

The usual Kähler moduli space parametrizes the volume of 2-cycles. Naively $B$ parametrizes the imaginary volume of the cycles, but is this really a useful concept? What is the role of B-field in mathematics?

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  • $\begingroup$ In what sort of mathematics? The $B$-field can have many roles, dpeend on context. $\endgroup$
    – user1504
    Commented Aug 7, 2013 at 23:02
  • $\begingroup$ @userN: I thought the OPs mention of mirror symmetry was specific enough. My thesis was supposed to have applications to mirror symmetry so I read some things about it and could never figure out if the $B$ field played any other role than to complexify the Kahler cone in a managable way. It always seemed like a parameter we add just because we can. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Aug 7, 2013 at 23:06
  • $\begingroup$ @GunnarMagnusson Perhaps the OP is looking for connections to other parts of mathematics? The mirror symmetry B-field is a cohomological shadow of more complicated objects. $\endgroup$
    – user1504
    Commented Aug 7, 2013 at 23:11
  • $\begingroup$ Yes, I am wondering if B-filed plays any role in other fileds of mathematics. It seems to me that it is added just to make the dimension of the moduli spaces the same. $\endgroup$
    – BlakeA
    Commented Aug 7, 2013 at 23:20
  • $\begingroup$ Twisted K-theory, higher gauge theory to name a couple. See also ncatlab.org/nlab/show/Kalb-Ramond+field $\endgroup$
    – David Roberts ♦
    Commented Aug 8, 2013 at 0:21

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