Witten Geometric Quantization of the Moduli of CY Threefolds

In this paper we study two different topics. The first topic is the applications of the geometric quantization scheme of Witten introduced in [2] and [15] to the problem of the quantum background independence in string theory. The second topic is the introduction of a Z structure on the tangent space of the moduli space of polarized CY threefolds M(M). Based on the existence of a Z structure on the tangent space of the moduli space of polarized CY threefolds we associate an algebraic integrable structure on the tangent bundle of M(M). In both cases it is crucial to construct a flat Sp(2h,R) connection on the tangent bundle of the moduli space M(M) of polarized CY threefolds. In this paper we define a Higgs field on the tangent bundle of the moduli space of CY threefolds. Combining this Higgs field with the Levi-Civita connection of the Weil-Petersson metrics on the moduli space of three dimensional CY manifolds, we construct a new Sp(2h,R) connection, following the ideas of Cecotti and Vafa. 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 to realize Witten program in [35] of solving the problem of background quantum independence for topological string field theories. By modifying the calculations of E. Witten done in 1993 to the tangent bundle of the moduli space of CY threefolds, we derive the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa as flat projective connection.