Thin monodromy in O(5)

TL;DR

This paper investigates the thinness of certain hypergeometric groups of degree five related to orthogonal groups, confirming thinness in many cases and narrowing down the remaining open cases, with some connections to Calabi-Yau 4-folds.

Contribution

It establishes the thinness of multiple hypergeometric groups of types O(3,2) and O(4,1), advancing understanding of their structure and confirming predictions for most cases.

Findings

12 out of 19 groups of type O(3,2) are thin

9 out of 17 groups of type O(4,1) are thin

Only one case remains unproven for thinness

Abstract

This article studies the orthogonal hypergeometric groups of degree five. We establish the thinness of 12 out of the 19 hypergeometric groups of type O(3,2) from [4, Table 6]. Some of these examples are associated with Calabi-Yau 4-folds. We also establish the thinness of 9 out of the 17 hypergeometric groups of type O(4,1) from [12], where the thinness of 7 other cases was already proven. The O(4,1) type groups were predicted to be all thin and our result leaves just one case open.