Tropical Mirror Symmetry in Dimension One

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2022
• Hauptverfasser: Böhm, Janko, Goldner, Christoph, Markwig, Hannah
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2022
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/211622
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Tropical Mirror Symmetry in Dimension One. Janko Böhm, Christoph Goldner and Hannah Markwig. SIGMA 18 (2022), 046, 30 pages

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.