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

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.

33 pages. Version 2: Incorporated suggestions from referee

References in corpus (3)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Riemann-Hilbert problems from Donaldson-Thomas theory
• Y-System and Deformed Thermodynamic Bethe Ansatz

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• Stability conditions and Teichmüller space