Mirror symmetry for the Tate curve via tropical and log corals

Arguez NH. 2022. Mirror symmetry for the Tate curve via tropical and log corals. Journal of the London Mathematical Society.

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Journal Article | Epub ahead of print | English

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Abstract
We introduce tropical corals, balanced trees in a half-space, and show that they correspond to holomorphic polygons capturing the product rule in Lagrangian Floer theory for the elliptic curve. We then prove a correspondence theorem equating counts of tropical corals to punctured log Gromov–Witten invariants of the Tate curve. This implies that the homogeneous coordinate ring of the mirror to the Tate curve is isomorphic to the degree-zero part of symplectic cohomology, confirming a prediction of homological mirror symmetry.
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Date Published
2022-02-05
Journal Title
Journal of the London Mathematical Society
Acknowledgement
This paper is based on my PhD thesis, which would not be possible without the support of my advisor Bernd Siebert. I also thank Dan Abramovich, Mohammed Abouzaid, Mark Gross, Tom Coates and Dimitri Zvonkine for many useful conversations. Finally, I thank the anonymous referees for their many insightful comments and valuable suggestions which have resulted in major improvements to this article. This project has received funding from the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement Number: 682603), and from Fondation Mathématique Jacques Hadamard.
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Arguez NH. Mirror symmetry for the Tate curve via tropical and log corals. Journal of the London Mathematical Society. 2022. doi:10.1112/jlms.12515
Arguez, N. H. (2022). Mirror symmetry for the Tate curve via tropical and log corals. Journal of the London Mathematical Society. London Mathematical Society. https://doi.org/10.1112/jlms.12515
Arguez, Nuroemuer Huelya. “Mirror Symmetry for the Tate Curve via Tropical and Log Corals.” Journal of the London Mathematical Society. London Mathematical Society, 2022. https://doi.org/10.1112/jlms.12515.
N. H. Arguez, “Mirror symmetry for the Tate curve via tropical and log corals,” Journal of the London Mathematical Society. London Mathematical Society, 2022.
Arguez NH. 2022. Mirror symmetry for the Tate curve via tropical and log corals. Journal of the London Mathematical Society.
Arguez, Nuroemuer Huelya. “Mirror Symmetry for the Tate Curve via Tropical and Log Corals.” Journal of the London Mathematical Society, London Mathematical Society, 2022, doi:10.1112/jlms.12515.
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