Kahler Quantization of H3(CY3,R) and the Holomorphic Anomaly

Studying the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis, with phase space being the third cohomology of the Calabi-Yau three-fold, the generators of translation along moduli directions of Calabi-Yau three-fold are constructed. The algebra of these generators is derived which take a simple form in canonical coordinates. Applying the Dirac method of quantization of second class constraint systems, we show that the Schr\"{o}dinger equations corresponding to these generators are equivalent to the holomorphic anomaly equations if one defines the action functional of the quadratic field theory with a proper factor one-half.