Home Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors
Article
Licensed
Unlicensed Requires Authentication

Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

  • Hülya Argüz ORCID logo and Pierrick Bousseau ORCID logo EMAIL logo
Published/Copyright: July 22, 2023

Abstract

Cluster varieties come in pairs: for any 𝒳 cluster variety there is an associated Fock–Goncharov dual 𝒜 cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the 𝒳 cluster variety is a degeneration of the Fock–Goncharov dual 𝒜 cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.

Award Identifier / Grant number: DMS-2302116

Award Identifier / Grant number: DMS-2302117

Funding statement: The research of Hülya Argüz was partially supported by the NSF grant DMS-2302116. The research of Pierrick Bousseau was partially supported by the NSF grant DMS-2302117.

Acknowledgements

We thank Mark Gross and Tom Coates for many useful discussions related to the extensions of Gross–Siebert mirror families.

References

[1] D. Abramovich and Q. Chen, Stable logarithmic maps to Deligne–Faltings pairs II, Asian J. Math. 18 (2014), no. 3, 465–488. 10.4310/AJM.2014.v18.n3.a5Search in Google Scholar

[2] D. Abramovich, Q. Chen, M. Gross and B. Siebert, Punctured logarithmic maps, preprint (2020), https://arxiv.org/abs/2009.07720. Search in Google Scholar

[3] H. Argüz, Equations of mirrors to log Calabi–Yau pairs via the heart of canonical wall structures, Math. Proc. Cambridge Philos. Soc. (2023), 10.1017/S030500412300021X. 10.1017/S030500412300021XSearch in Google Scholar

[4] H. Argüz and M. Gross, The higher-dimensional tropical vertex, Geom. Topol. 26 (2022), no. 5, 2135–2235. 10.2140/gt.2022.26.2135Search in Google Scholar

[5] S. Bardwell-Evans, M.-W. M. Cheung, H. Hong and Y.-S. Lin, Scattering diagrams from holomorphic discs in log Calabi–Yau surfaces, preprint (2021), https://arxiv.org/abs/2110.15234. Search in Google Scholar

[6] L. Bossinger, B. Frías-Medina, T. Magee and A. Nájera Chávez, Toric degenerations of cluster varieties and cluster duality, Compos. Math. 156 (2020), no. 10, 2149–2206. 10.1112/S0010437X2000740XSearch in Google Scholar

[7] B. Davison and T. Mandel, Strong positivity for quantum theta bases of quantum cluster algebras, Invent. Math. 226 (2021), no. 3, 725–843. 10.1007/s00222-021-01061-1Search in Google Scholar

[8] V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211. 10.1007/s10240-006-0039-4Search in Google Scholar

[9] V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865–930. 10.24033/asens.2112Search in Google Scholar

[10] S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. 10.1090/S0894-0347-01-00385-XSearch in Google Scholar

[11] B. Gammage and I. Le, Mirror symmetry for truncated cluster varieties, SIGMA Symmetry Integrability Geom. Methods Appl. 18 (2022), Paper No. 055. 10.3842/SIGMA.2022.055Search in Google Scholar

[12] A. Goncharov and L. Shen, Geometry of canonical bases and mirror symmetry, Invent. Math. 202 (2015), no. 2, 487–633. 10.1007/s00222-014-0568-2Search in Google Scholar

[13] A. Goncharov and L. Shen, Quantum geometry of moduli spaces of local systems and representation theory, preprint (2019), https://arxiv.org/abs/1904.10491. Search in Google Scholar

[14] M. Gross, P. Hacking and S. Keel, Birational geometry of cluster algebras, Algebr. Geom. 2 (2015), no. 2, 137–175. 10.14231/AG-2015-007Search in Google Scholar

[15] M. Gross, P. Hacking and S. Keel, Mirror symmetry for log Calabi–Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168. 10.1007/s10240-015-0073-1Search in Google Scholar

[16] M. Gross, P. Hacking, S. Keel and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 497–608. 10.1090/jams/890Search in Google Scholar

[17] M. Gross, P. Hacking and B. Siebert, Theta functions on varieties with effective anti-canonical class, Mem. Amer. Math. Soc. 278 (2022), no. 1367, 1–103. 10.1090/memo/1367Search in Google Scholar

[18] M. Gross and B. Siebert, From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), no. 3, 1301–1428. 10.4007/annals.2011.174.3.1Search in Google Scholar

[19] M. Gross and B. Siebert, Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. 10.1090/S0894-0347-2012-00757-7Search in Google Scholar

[20] M. Gross and B. Siebert, Intrinsic mirror symmetry, preprint (2019), https://arxiv.org/abs/1909.07649. Search in Google Scholar

[21] M. Gross and B. Siebert, The canonical wall structure and intrinsic mirror symmetry, Invent. Math. 229 (2022), no. 3, 1101–1202. 10.1007/s00222-022-01126-9Search in Google Scholar

[22] P. Hacking and S. Keel, Mirror symmetry and cluster algebras, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Scientific, Hackensack (2018), 671–697. 10.1142/9789813272880_0073Search in Google Scholar

[23] P. Hacking, S. Keel and T. Y. Yu, Secondary fan, theta functions and moduli of Calabi–Yau pairs, preprint (2020), https://arxiv.org/abs/2008.02299. Search in Google Scholar

[24] S. Johnston, Comparison of non-archimedean and logarithmic mirror constructions via the Frobenius structure theorem, preprint (2022), https://arxiv.org/abs/2204.00940. Search in Google Scholar

[25] S. Keel and T. Y. Yu, The Frobenius structure theorem for affine log Calabi–Yau varieties containing a torus, preprint (2019), https://arxiv.org/abs/1908.09861. Search in Google Scholar

[26] M. Kontsevich and Y. Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math. 244, Birkhäuser, Boston (2006), 321–385. 10.1007/0-8176-4467-9_9Search in Google Scholar

[27] T. Mandel, Scattering diagrams, theta functions, and refined tropical curve counts, J. Lond. Math. Soc. (2) 104 (2021), no. 5, 2299–2334. 10.1112/jlms.12498Search in Google Scholar

[28] T. Mandel, Theta bases and log Gromov–Witten invariants of cluster varieties, Trans. Amer. Math. Soc. 374 (2021), no. 8, 5433–5471. 10.1090/tran/8398Search in Google Scholar

[29] D. R. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Journées de géométrie algébrique d’Orsay, Astérisque 218, Société Mathématique de France, Paris (1993), 243–271. Search in Google Scholar

[30] L. Mou, Scattering diagrams for generalized cluster algebras, preprint (2021), https://arxiv.org/abs/2110.02416. Search in Google Scholar

Received: 2022-08-03
Revised: 2023-05-24
Published Online: 2023-07-22
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 18.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2023-0043/html
Scroll to top button