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I am wondering how the Dirac operator can be built in the context of Hichin's generalized geometry.

In particular, I have the following questions:

  • On a spin manifold, is the conventional spin connection replaced by a generalized one that has two coordinate indices?

  • If so, does this generalized spin connection behave as a non-Abelian $\text{SO}(3,1)$ "B-field" (called also Kalb-Ramond field in physics)?

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  • $\begingroup$ For those interested, an answer from physicists to this has been given here on PhysicsOverflow: physicsoverflow.org/27324/… $\endgroup$
    – Dilaton
    Commented Feb 20, 2015 at 8:39

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