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

TL;DR

This paper establishes an equivalence between regular singular $ abla$-modules with normal crossing singularities and quiver $ abla$-modules, connecting algebraic and geometric perspectives via the Riemann-Hilbert correspondence.

Contribution

It proves an equivalence of categories between regular singular $ abla$-modules with normal crossing singularities and quiver $ abla$-modules, extending the Riemann-Hilbert correspondence.

Findings

Categorical equivalence between regular singular $ abla$-modules and quiver $ abla$-modules.

Extension of Riemann-Hilbert correspondence to quiver $ abla$-modules.

Based on and generalizing previous work by Galligo, Granger, and Maisonobe.

Abstract

In this paper, we show that every regular singular $\mathscr{D}$-module in $\mathbb{C}^n$ whose singular locus is a normal crossing is isomorphic to a quiver $\mathscr{D}$-module - a $\mathscr{D}$-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 $\mathscr{D}$-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 $\mathscr{D}$-modules whose singular locus is a normal crossing which involves the classical Riemann-Hilbert correspondence.