On symplectically rigid local systems of rank four and Calabi-Yau operators

We classify all ${\text{Sp}}_4(\mathbb{C})$-rigid, quasi-unipotent local systems and show that all of them have geometric origin. Furthermore, we investigate which of those having a maximal unipotent element are induced by fourth order Calabi-Yau operators. Via this approach, we reconstruct all known Calabi-Yau operators inducing a ${\text{Sp}}_4(\mathbb{C})$-rigid monodromy tuple and obtain closed formulae for special solutions of them.