  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.
