Cluster Structures and Subfans in Scattering Diagrams

Abstract

We give more precise statements of the Fock-Goncharov duality conjecture for cluster varieties parametrizing SL₂/PGL₂-local systems on the once-punctured torus. Then we prove these statements. Along the way, using distinct subfans in the scattering diagram, we produce an example of a cluster variety with two non-equivalent cluster structures. To overcome the technical difficulty of infinite non-cluster wall-crossing in the scattering diagram, we introduce quiver folding into the machinery of scattering diagrams and give a quotient construction of scattering diagrams.