Scattering fans

TL;DR

This paper explores the geometric structure of scattering diagrams, showing they form complete fans and relating cluster scattering fans to mutation fans, with conjectures on their equivalence.

Contribution

It establishes that consistent scattering diagrams with minimal support form complete fans and relates cluster scattering fans to mutation fans, proposing a conjecture for finite mutation types.

Findings

Scattering diagrams with minimal support form complete fans.

Cluster scattering fans refine mutation fans for exchange matrices.

Conjecture: equality of these fans characterizes finite mutation type matrices.

Abstract

Scattering diagrams arose in the context of mirror symmetry, Donaldson-Thomas theory, and integrable systems. We show that a consistent scattering diagram with minimal support cuts the ambient space into a complete fan. A special class of scattering diagrams, the cluster scattering diagrams, are closely related to cluster algebras. We show that the cluster scattering fan associated to an exchange matrix $B$ refines the mutation fan for $B$ (a complete fan that encodes the geometry of mutations of $B$). We conjecture that, when $B$ is $n\times n$ for $n>2$, these two fans coincide if and only if $B$ is of finite mutation type.