The paramount object in generalized gomplex geometry is the Courant algebroid $TM\oplus T^\star M$, where the manifold $M$ is called background geometry I think (I am not sure). More generally this Courant algebroid can be twisted by a closed 3-form $H$ on $M$, which I suspect is called a flux (I am not sure too). In fact P. Severa shown that every exact Courant algebroid arise as such a twist of the standard Courant algebroid $TM\oplus T^\star M$. I think these objects arise in Physics when one tries to make compactifications with fluxes in String Theory, which are more realistic ones. They also appear in the AdS-/CFT correspondence (see Generalized Complex Geometry and Theoretical Physics). Please correct me if I am saying something wrong.
Could somebody sum up the way these forms $H$ appear in Physics in mathematical terms? However my true question is the following: could somebody provide concrete examples of both manifolds $M$ and forms $H$ such that, when combined together into the twisted Courant algebroid $(TM\oplus T^\star M,H)$ they are relevant to String Theory?