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Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds. (English) Zbl 1153.34055

Summary: We are concerned with the monodromy of Picard-Fuchs differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative 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 suitable change of bases the monodromy groups are contained in certain congruence subgroups of Sp\((4, \mathbb Z)\) of finite index and whose levels are related to the geometric invariants of the Calabi-Yau threefolds.

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)
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