Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 013, 35 pages      arXiv:1901.04166      https://doi.org/10.3842/SIGMA.2020.013
Contribution to the Special Issue on Cluster Algebras

Cluster Structures and Subfans in Scattering Diagrams

Yan Zhou
Beijing International Center for Mathematical Research, Peking University, China

Received July 24, 2019, in final form March 01, 2020; Published online March 11, 2020

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.

Key words: cluster varieties; Donaldson-Thomas transformations; Markov quiver; non-equivalent cluster structures; scattering diagrams; quiver folding.

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