Tropical Mirror Symmetry in Dimension One

Symmetry Integrability and Geometry-Methods and Applications

article

Janko Böhm¹  Christoph Goldner²  Hannah Markwig² 

[1]Fachbereich Mathematik

[2]Universität Tübingen

关键词: mirror symmetry;    elliptic curves;    Feynman integral;    tropical geometry;    Hurwitz numbers;    quasimodular forms;    Fock space.;   

DOI  :  10.3842/SIGMA.2022.046

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown