Monodromy of Picard-Fuchs dierential equations for Calabi-Yau threefolds

In this paper we are concerned with the monodromy of Picard-Fuchs dif- ferential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy rel- ative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suit- able change of bases the monodromy groups are contained in certain congruence sub- groups of Spð4;ZÞ of finite index and whose levels are related to the geometric invariants of the Calabi-Yau threefolds.