Stability conditions and cluster varieties from quivers of type $A$

TL;DR

This paper explores the connection between stability conditions on triangulated categories and cluster varieties for type A quivers, establishing a local biholomorphism using classical differential equations.

Contribution

It constructs a local biholomorphism between stability condition spaces and cluster varieties specifically for type A quivers, linking two important geometric structures.

Findings

Established a local biholomorphism for type A quivers.

Connected stability conditions with cluster varieties via differential equations.

Demonstrated the relationship using classical ODE theory.

Abstract

We describe the relationship between two spaces associated to a quiver with potential. The first is a complex manifold parametrizing Bridgeland stability conditions on a triangulated category, and the second is a cluster variety with a natural Poisson structure. For quivers of type $A$, we construct a local biholomorphism from the space of stability conditions to the cluster variety. The existence of this map follows from results of Sibuya in the classical theory of ordinary differential equations.