Picard-Vessiot groups of Lauricella's hypergeometric systems $E_C$ and Calabi-Yau varieties arising integral representations

TL;DR

This paper investigates the monodromy groups of Lauricella's hypergeometric functions, classifies their Zariski closures, and explores associated Calabi-Yau varieties from integral representations.

Contribution

It characterizes the Zariski closure of monodromy groups for Lauricella's functions and connects these to classical groups and Calabi-Yau varieties.

Findings

Monodromy group closures are classical groups under certain conditions

Classification of Zariski closures as SL, SO, or Sp groups

Analysis of Calabi-Yau varieties from integral representations

Abstract

We study the Zariski closure of the monodromy group $\mathbf{Mon}$ of Lauricella's hypergeometric function $F_C$. If the identity component $\mathbf{Mon}^0$ acts irreducibly, then $\overline{\mathbf{Mon}} \cap \mathbf{SL}_{2^n}(\mathbb{C})$ must be one of classical groups $\mathbf{SL}_{2^n}(\mathbb{C}), \mathbf{SO}_{2^n}(\mathbb{C})$ and $\mathbf{Sp}_{2^n}(\mathbb{C})$. We also study Calabi-Yau varieties arising from integral representations of $F_C$.