RUS  ENG
Full version
JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2020 Volume 16, 013, 35 pp. (Mi sigma1550)

This article is cited in 2 papers

Cluster Structures and Subfans in Scattering Diagrams

Yan Zhou

Beijing International Center for Mathematical Research, Peking University, China

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.

Keywords: cluster varieties, Donaldson–Thomas transformations, Markov quiver, non-equivalent cluster structures, scattering diagrams, quiver folding.

MSC: 13F60; 14J32; 14J33; 14N35

Received: July 24, 2019; in final form March 1, 2020; Published online March 11, 2020

Language: English

DOI: 10.3842/SIGMA.2020.013



Bibliographic databases:
ArXiv: 1901.04166


© Steklov Math. Inst. of RAS, 2025