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.

arXiv commentFinal 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

References in corpus (7)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Introduction to Cluster Algebras. Chapters 1-3
• Quiver mutation and combinatorial DT-invariants
• Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
• A tropical view on Landau-Ginzburg models
• Weyl groups and cluster structures of families of log Calabi-Yau surfaces

Cited by in corpus (5)

• Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces
• Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams
• Generalized Cluster Structures Related to the Drinfeld Double of $G{L}_n$
• Stability scattering diagrams and quiver coverings
• Cluster algebras and their bases