Cluster structures and subfans in scattering diagrams

Abstract

We give more precise statements of Fock-Goncharov duality conjecture for cluster varieties parametrizing ${\mathrm{SL}}_2/{\mathrm{PGL}}_2$-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.

Final version, a contribution to the Special Issue on Cluster Algebras of SIGMA. Many typos corrected. The proof of Theorem 2.20 cleaned up. The definition of a cluster structure of a log Calabi-Yau variety revised