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

TL;DR

This paper classifies symplectically rigid local systems of rank four with geometric origin, identifies those induced by Calabi-Yau operators, and provides explicit solutions for these special cases.

Contribution

It offers a complete classification of $ ext{Sp}_4( ext{C})$-rigid local systems and links them to Calabi-Yau operators, including explicit solution formulas.

Findings

All $ ext{Sp}_4( ext{C})$-rigid, quasi-unipotent local systems have geometric origin.

Identified which systems are induced by fourth order Calabi-Yau operators.

Reconstructed known Calabi-Yau operators and derived closed-form solutions.

Abstract

We classify all $\Sp_4(\mathbb{C})$-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 $\Sp_4(\mathbb{C})$-rigid monodromy tuple and obtain closed formulae for special solutions of them.