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

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.

57 pages. Version 2: Some definitions reworded prior to journal submission. Version 3: Incorporated suggestions from referee

References in corpus (4)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The quantum dilogarithm and representations quantum cluster varieties
• On the wall-crossing formula for quadratic differentials
• Opers and TBA

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