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

We study the Zariski closure of the monodromy group $\mathbf{M}\mathbf{o}\mathbf{n}$ of Lauricella’s hypergeometric function $F_C$. If the identity component ${\mathbf{M}\mathbf{o}\mathbf{n}}^0$ acts irreducibly, then $\overline{$\mathbf{M}\mathbf{o}\mathbf{n}$} \cap $\mathbf{S}\mathbf{L}${2^n}($\mathbb{C}$)$mustbeoneofclassicalgroups$$\mathbf{S}\mathbf{L}${2^n}($\mathbb{C}$), $\mathbf{S}\mathbf{O}${2^n}($\mathbb{C}$)$and$$\mathbf{S}\mathbf{p}${2^n}($\mathbb{C}$)$. We also study Calabi-Yau varieties arising from integral representations of $F_C$.