Witten's Geometric Quantization of the Moduli of CY Threefolds

In this paper we will use our results about the local deformation theory of Calabi-Yau manifolds to define a Higgs field on the tangent bundle of the moduli space of CY threefolds. Combining this Higgs field with the Levi-Chevitta connection of the Weil-Petersson metrics on the moduli space of three dimensional CY manifolds, we construct a new $Sp(2h^{2,1}% ,\mathbb{R)}$ connection, following the ideas of Cecotti and Vafa. We prove that this new connection is a flat connection. Using this flat connection, we apply the scheme of geometric quantization introduced by Axelrod, Della Pietra and Witten to the tangent bundle of the moduli space of three dimensional CY manifolds. By modifying the calculations of E. Witten done in 1993 to the tangent bundle of the moduli space of CY threefolds, we derive again the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We also introduced a Z structure on the tangent bundle of the moduli space of polarized CY threefolds by using the flat symplectic connection.