Abstract
Abstract. In this paper we are concerned with the monodromy of PicardFuchs differential equations associated with one-parameter families of CalabiYau 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,Z) of finite index and whose levels are related to the geometric invariants of the Calabi-Yau threefolds.
Highlights
Let Mz be a family of Calabi-Yau n-folds parameterized by a complex variable z A P1ðCÞ, and oz be the unique holomorphic di¤erential n-form on Mz
Our computation in the hypergeometric cases shows that the matrix representation of the monodromy around the finite singular point relative to the Frobenius basis at the origin can be expressed completely using the geometric invariants of the associated Calabi-Yau threefolds
We remark that what we show in Theorem 2 is merely the fact that the monodromy groups are contained in the congruence subgroups Gðd1; d2Þ
Summary
Let Mz be a family of Calabi-Yau n-folds parameterized by a complex variable z A P1ðCÞ, and oz be the unique holomorphic di¤erential n-form on Mz (up to a scalar). Our computation in the hypergeometric cases shows that the matrix representation of the monodromy around the finite singular point (di¤erent from the origin) relative to the Frobenius basis at the origin can be expressed completely using the geometric invariants of the associated Calabi-Yau threefolds. This phenomenon is verified numerically in the nonhypergeometric cases.
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More From: Journal für die reine und angewandte Mathematik (Crelles Journal)
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