Scattering diagrams, Hall algebras and stability conditions

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