Quiver D-Modules and the Riemann-Hilbert Correspondence

In this thesis, we show that every regular singular holonomic D-module in Cn whose singular locus is a normal crossing is isomorphic to a quiver D-module -- a 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. Additionally in dimension 1, we recompute the list of basic indecomposable regular singular holonomic D-modules by Boutet de Monvel using indecomposable quiver representations and our equivalence of categories.

Our definition of quiver D-modules is based on the one of Khoroshkin and Varchenko in the case of a normal crossing hyperplane arrangement. To prove the equivalence of categories, we use an equivalence by Galligo, Granger and Maisonobe for regular singular holonomic D-modules whose singular locus is a normal crossing which involves the classical Riemann-Hilbert correspondence.