Orphan Calabi-Yau threefold with arithmetic monodromy group

TL;DR

This paper investigates the monodromy groups of orphan Calabi-Yau threefolds' Picard-Fuchs operators, demonstrating their density in symplectic groups and identifying cases of arithmetic monodromy.

Contribution

It constructs rational symplectic bases for all orphan double octic Picard-Fuchs operators of order 4 and proves the arithmetic nature of one such monodromy group.

Findings

Monodromy groups are dense in Sp(4,Z)

Maximally unipotent elements identified in most cases

One orphan operator has an 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.