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)?