Quiver $\mathcal{D}$-modules and the Riemann-Hilbert correspondence

In this paper, we show that every regular singular $\mathcal{D}$-module in ${\mathbb{C}}^n$ whose singular locus is a normal crossing is isomorphic to a quiver $\mathcal{D}$-module - a $\mathcal{D}$-module whose definition is based on certain representations of the hypercube quiver. To be more precise we give an equivalence of the respective categories. Our definition of quiver $\mathcal{D}$-modules is based on the one of Khoroshkin and Varchenko. To prove the equivalence, we use an equivalence by Galligo, Granger and Maisonobe for regular singular $\mathcal{D}$-modules whose singular locus is a normal crossing which involves the classical Riemann-Hilbert correspondence.