Generalized N=1 Orientifold Compactifications and the Hitchin   functionals
  The four-dimensional N=1 supergravity theories arising in compactifications of type IIA and type IIB on generalized orientifold backgrounds with background fluxes are discussed. The Kahler potentials are derived for reductions on SU(3) structure orientifolds and shown to consist of the logarithm of the two Hitchin functionals. These are functions of even and odd forms parameterizing the geometry of the internal manifold, the B-field and the dilaton. The superpotentials induced by background fluxes and the non-Calabi-Yau geometry are determined by a reduction of the type IIA and type IIB fermionic actions on SU(3) and generalized SU(3) x SU(3) manifolds. Mirror spaces of Calabi-Yau orientifolds with electric and part of the magnetic NS-NS fluxes are conjectured to be certain SU(3) x SU(3) structure manifolds. Evidence for this identification is provided by comparing the generalized type IIA and type IIB superpotentials. 
Iman Benmachiche, Thomas W. Grimm