Mirror Symmetry for Truncated Cluster Varieties
In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on the work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cl...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211732 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Mirror Symmetry for Truncated Cluster Varieties. Benjamin Gammage and Ian Le. SIGMA 18 (2022), 055, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862626365822468096 |
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| author | Gammage, Benjamin Le, Ian |
| author_facet | Gammage, Benjamin Le, Ian |
| citation_txt | Mirror Symmetry for Truncated Cluster Varieties. Benjamin Gammage and Ian Le. SIGMA 18 (2022), 055, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on the work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cluster variety, we construct a symplectic manifold and prove homological mirror symmetry for the resulting pair. We also describe how our construction can be obtained from toric geometry, and we relate our construction to various aspects of cluster theory, which are known to symplectic geometers.
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| first_indexed | 2026-03-14T15:55:03Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211732 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T15:55:03Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
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| spelling | Gammage, Benjamin Le, Ian 2026-01-09T12:54:59Z 2022 Mirror Symmetry for Truncated Cluster Varieties. Benjamin Gammage and Ian Le. SIGMA 18 (2022), 055, 30 pages 1815-0659 2020 Mathematics Subject Classification: 53D37; 13F60 arXiv:2103.12232 https://nasplib.isofts.kiev.ua/handle/123456789/211732 https://doi.org/10.3842/SIGMA.2022.055 In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on the work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cluster variety, we construct a symplectic manifold and prove homological mirror symmetry for the resulting pair. We also describe how our construction can be obtained from toric geometry, and we relate our construction to various aspects of cluster theory, which are known to symplectic geometers. The authors are grateful to Denis Auroux, for explaining the construction of the K¨ahler potential in Lemma5.7; Roger Casals and Dmitry Tonkonog, for explanations about nodal trades; Andrew Hanlon, for discussions about the GHK construction; Vivek Shende and Harold Williams, for discussions about the papers [28, 27]; Harold again, for several helpful comments on a draft of this paper; and Daniel Pomerleano, for a correction to Section 5 and for the content of Remark 5.16. BG is supported by an NSF postdoctoral fellowship, DMS-2001897. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Mirror Symmetry for Truncated Cluster Varieties Article published earlier |
| spellingShingle | Mirror Symmetry for Truncated Cluster Varieties Gammage, Benjamin Le, Ian |
| title | Mirror Symmetry for Truncated Cluster Varieties |
| title_full | Mirror Symmetry for Truncated Cluster Varieties |
| title_fullStr | Mirror Symmetry for Truncated Cluster Varieties |
| title_full_unstemmed | Mirror Symmetry for Truncated Cluster Varieties |
| title_short | Mirror Symmetry for Truncated Cluster Varieties |
| title_sort | mirror symmetry for truncated cluster varieties |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211732 |
| work_keys_str_mv | AT gammagebenjamin mirrorsymmetryfortruncatedclustervarieties AT leian mirrorsymmetryfortruncatedclustervarieties |