Cluster Structures and Subfans in Scattering Diagrams
We give more precise statements of the Fock-Goncharov duality conjecture for cluster varieties parametrizing SL₂/PGL₂-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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210597 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Cluster Structures and Subfans in Scattering Diagrams. Yan Zhou. SIGMA 16 (2020), 013, 35 pages |
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Zhou, Yan 2025-12-12T10:36:18Z 2020 Cluster Structures and Subfans in Scattering Diagrams. Yan Zhou. SIGMA 16 (2020), 013, 35 pages 1815-0659 2020 Mathematics Subject Classi cation: 13F60; 14J32; 14J33; 14N35 arXiv:1901.04166 https://nasplib.isofts.kiev.ua/handle/123456789/210597 https://doi.org/10.3842/SIGMA.2020.013 We give more precise statements of the Fock-Goncharov duality conjecture for cluster varieties parametrizing SL₂/PGL₂-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. Linhui Shen first suggested introducing folding into scattering diagrams. I cannot thank him more. I am grateful to my advisor Sean Keel for giving suggestions for simplifying the quotient construction of scattering diagrams. Besides, I benefit from inspiring discussions with Andy Neitzke and Daping Wen and email correspondence with Travis Mandel and Greg Muller. I also want to thank Andy Neitzke for carefully proofreading the draft of this paper and thank anonymous referees for their numerous suggestions for improvement. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cluster Structures and Subfans in Scattering Diagrams Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Cluster Structures and Subfans in Scattering Diagrams |
| spellingShingle |
Cluster Structures and Subfans in Scattering Diagrams Zhou, Yan |
| title_short |
Cluster Structures and Subfans in Scattering Diagrams |
| title_full |
Cluster Structures and Subfans in Scattering Diagrams |
| title_fullStr |
Cluster Structures and Subfans in Scattering Diagrams |
| title_full_unstemmed |
Cluster Structures and Subfans in Scattering Diagrams |
| title_sort |
cluster structures and subfans in scattering diagrams |
| author |
Zhou, Yan |
| author_facet |
Zhou, Yan |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We give more precise statements of the Fock-Goncharov duality conjecture for cluster varieties parametrizing SL₂/PGL₂-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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210597 |
| citation_txt |
Cluster Structures and Subfans in Scattering Diagrams. Yan Zhou. SIGMA 16 (2020), 013, 35 pages |
| work_keys_str_mv |
AT zhouyan clusterstructuresandsubfansinscatteringdiagrams |
| first_indexed |
2025-12-17T12:04:18Z |
| last_indexed |
2025-12-17T12:04:18Z |
| _version_ |
1851756965492948992 |