# Dirac operator in Generalized Geometry

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)?

• For those interested, an answer from physicists to this has been given here on PhysicsOverflow: physicsoverflow.org/27324/… – Dilaton Feb 20 '15 at 8:39