Tropical Mirror Symmetry in Dimension One
We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 2...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| 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/123456789/211622 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Tropical Mirror Symmetry in Dimension One. Janko Böhm, Christoph Goldner and Hannah Markwig. SIGMA 18 (2022), 046, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862543422076747776 |
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| author | Böhm, Janko Goldner, Christoph Markwig, Hannah |
| author_facet | Böhm, Janko Goldner, Christoph Markwig, Hannah |
| citation_txt | Tropical Mirror Symmetry in Dimension One. Janko Böhm, Christoph Goldner and Hannah Markwig. SIGMA 18 (2022), 046, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach.
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| first_indexed | 2026-03-12T21:57:35Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211622 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T21:57:35Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
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| spelling | Böhm, Janko Goldner, Christoph Markwig, Hannah 2026-01-07T13:39:42Z 2022 Tropical Mirror Symmetry in Dimension One. Janko Böhm, Christoph Goldner and Hannah Markwig. SIGMA 18 (2022), 046, 30 pages 1815-0659 2020 Mathematics Subject Classification: 14J33; 14N35; 14T05; 81T18; 11F11; 14H30; 14N10; 14H52; 14H81 arXiv:1809.10659 https://nasplib.isofts.kiev.ua/handle/123456789/211622 https://doi.org/10.3842/SIGMA.2022.046 We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach. We would like tothank Renzo Cavalieri, Elise Goujard, Gerhard Hiss, Martin Möller, and Dhruv Ranganathan for helpful discussions. Gefordert durch die Deutsche Forschungsgemeinschaft (DFG) – Projektnummer 286237555 – TRR195 [Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project ID 286237555 –TRR195]. The authors have been supported by Project I.10 (INST 248/237-1) of TRR 195. Computations have been made with Singular using the ellipticcovers library. Part of this work was completed during the Mittag-Leffler program Tropical geometry, amoebas and polytopes in spring 2018. The authors would like to thank the institute for its hospitality and excellent working conditions. We would like to thank the anonymous referees for their helpful suggestions to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tropical Mirror Symmetry in Dimension One Article published earlier |
| spellingShingle | Tropical Mirror Symmetry in Dimension One Böhm, Janko Goldner, Christoph Markwig, Hannah |
| title | Tropical Mirror Symmetry in Dimension One |
| title_full | Tropical Mirror Symmetry in Dimension One |
| title_fullStr | Tropical Mirror Symmetry in Dimension One |
| title_full_unstemmed | Tropical Mirror Symmetry in Dimension One |
| title_short | Tropical Mirror Symmetry in Dimension One |
| title_sort | tropical mirror symmetry in dimension one |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211622 |
| work_keys_str_mv | AT bohmjanko tropicalmirrorsymmetryindimensionone AT goldnerchristoph tropicalmirrorsymmetryindimensionone AT markwighannah tropicalmirrorsymmetryindimensionone |