# Cluster structures and subfans in scattering diagrams

## TL;DR

This paper refines the Fock-Goncharov duality conjecture for certain cluster varieties, proves these refined statements, and introduces quiver folding to handle complex scattering diagram structures.

## Contribution

It provides precise formulations and proofs of duality conjectures for SL2/PGL2 local systems and introduces quiver folding to manage infinite wall-crossings in scattering diagrams.

## Key findings

- Confirmed duality conjecture for specific cluster varieties.
- Constructed an example with two non-equivalent cluster structures.
- Developed quiver folding method for scattering diagram quotients.

## Abstract

We give more precise statements of Fock-Goncharov duality conjecture for cluster varieties parametrizing ${\rm SL}_{2}/{\rm PGL}_{2}$-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.