Scattering diagrams, Hall algebras and stability conditions

With any quiver with relations, we associate a consistent scattering diagram taking in the motivic Hall algebra of its category of representations. We show that the structure of this scattering diagram coincides with the natural chamber struc- in an open subset of the space of stability conditions on the associated triangulated . In the three-dimensional Calabi–Yau situation, when the relations arise from potential, we can apply an integration map to give a consistent scattering diagram values in a tropical vertex group.