Stability scattering diagrams and quiver coverings

Abstract

Given a covering of a quiver (with potential), we show that the associated Bridgeland stability scattering diagrams are related by a restriction operation under the assumption of admitting a nice grading. We apply this to quivers with potential associated to marked surfaces. In combination with recent results of the second and third authors, our findings imply that the bracelets basis for a once-punctured closed surface coincides with the theta basis for the associated stability scattering diagram, and these stability scattering diagrams agree with the corresponding cluster scattering diagrams of Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus.

25 pages; minor changes; include infinite-dimensional cases; add Section 6 where the existence of nice gradings is replaced by a more natural assumption

References in corpus (7)

• Kac-Moody groups and cluster algebras
• Quivers with potentials associated to triangulated surfaces
• Bases for cluster algebras from surfaces
• Donaldson-Thomas theory and cluster algebras
• Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions
• Density of $g$-vector cones from triangulated surfaces
• Cluster structures and subfans in scattering diagrams