Orphan Calabi-Yau threefold with arithmetic monodromy group

Abstract

We study monodromy groups of Picard-Fuchs operators of one-parameter families of Calabi-Yau threefolds without a point of Maximal Unipotent Monodromy (\emph{orphan operators}). We construct rational symplectic bases for the monodromy action for all orphan double octic Picard-Fuchs operators of order $4$. As a consequence we show that monodromy groups of all double octic orphan operators are dense in $\mathrm{Sp(4,\mathbb{Z})}$ and identify maximally unipotent elements in all of them, except one. Finally, we prove that the monodromy group of one of these orphan operators is arithmetic.