Theta bases and log Gromov-Witten invariants of cluster varieties
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Abstract:
Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of “theta functions” on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of “contractible” tropical curves when showing that the relevant log curves are torically transverse.References
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Additional Information
- Travis Mandel
- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
- MR Author ID: 982182
- ORCID: 0000-0003-3127-4429
- Email: tmandel@ou.edu
- Received by editor(s): August 20, 2019
- Received by editor(s) in revised form: July 21, 2020
- Published electronically: May 18, 2021
- Additional Notes: The author was supported by the National Science Foundation RTG Grant DMS-1246989, and later by the Starter Grant “Categorified Donaldson-Thomas Theory” no. 759967 of the European Research Council.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 5433-5471
- MSC (2020): Primary 14J33; Secondary 14N35, 13F60
- DOI: https://doi.org/10.1090/tran/8398
- MathSciNet review: 4293777