Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces

TL;DR

This paper explores the relationship between stability conditions and cluster varieties associated with surfaces, providing a geometric construction that solves Riemann-Hilbert problems in Donaldson-Thomas theory.

Contribution

It constructs a natural map linking stability conditions and cluster varieties for surface-related quivers, enabling solutions to complex Riemann-Hilbert problems.

Findings

Established a map from stability conditions to cluster varieties

Provided solutions to Riemann-Hilbert problems in Donaldson-Thomas theory

Linked geometric structures to complex analysis problems

Abstract

We consider two interesting spaces associated to a quiver with potential: a space of stability conditions and a cluster variety. In the case where the quiver with potential arises from an ideal triangulation of a marked bordered surface, we construct a natural map from a dense subset of the space of stability conditions to the cluster variety. Using this construction, we give solutions to a family of Riemann-Hilbert problems arising in Donaldson-Thomas theory.