On special geometry of the moduli space of string vacua with fluxes

In this paper we construct a special geometry over the moduli space of type II string vacua with both NS and RR fluxes turning on. Depending on what fluxes are turning on we divide into three cases of moduli space of generalized structures. They are respectively generalized Calabi-Yau structures, generalized Calabi-Yau metric structures and N = 1 generalized string vacua. It is found that the dd lemma can be established for all three cases. With the help of the dd lemma we identify the moduli space locally as a subspace of dH cohomologies. This leads naturally to the special geometry of the moduli space. It has a flat symplectic structure and a Kähler metric with the Hitchin functional (modified if RR fluxes are included) the Kähler potential. Our work is based on previous works of Hitchin and recent works of Graña-Louis-Waldram, Goto, Gualtieri, Yi Li and Tomasiello. The special geometry is useful in flux compactifications of type II string theories.