Linear differential equations on the Riemann sphere and representations of quivers

TL;DR

This paper generalizes the additive Deligne-Simpson problem to include differential equations with multiple irregular singular points on the Riemann sphere, linking solutions to quiver varieties and moduli space geometry.

Contribution

It extends the classical problem to more complex singularities and establishes a connection between moduli spaces of meromorphic connections and quiver varieties.

Findings

Criteria for the existence of solutions to the generalized problem

Open embeddings of moduli spaces into quiver varieties

Connectedness of the moduli spaces

Abstract

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown.