Scattering diagrams, Hall algebras and stability conditions

TL;DR

This paper constructs and analyzes scattering diagrams linked to quivers with relations, connecting them to stability conditions and Hall algebras, especially in Calabi-Yau cases, revealing deep geometric and algebraic structures.

Contribution

It introduces a new construction of scattering diagrams from quivers with relations and relates their chamber structures to stability conditions in derived categories.

Findings

Chamber structures of scattering diagrams match stability condition chambers.

Construction of scattering diagrams in motivic Hall algebras.

Application of integration map in Calabi-Yau cases.

Abstract

To 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 structure 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.