ellipticcovers.lib. A Singular 4 library for Gromov-Witten invariants of elliptic curves: Tropical mirror symmetry for elliptic curves: Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov-Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.
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References in zbMATH (referenced in 11 articles )
Showing results 1 to 11 of 11.
- Böhm, Janko; Goldner, Christoph; Markwig, Hannah: Counts of (tropical) curves in (E \times\mathbbP^1) and Feynman integrals (2022)
- Huang, Min-xin: Boson-fermion correspondence and holomorphic anomaly equation in 2d Yang-Mills theory on torus (2021)
- Li, Si; Zhou, Jie: Regularized integrals on Riemann surfaces and modular forms (2021)
- Böhm, Janko; Bringmann, Kathrin; Buchholz, Arne; Markwig, Hannah: Erratum to: “Tropical mirror symmetry for elliptic curves” (2020)
- Mandel, Travis; Ruddat, Helge: Descendant log Gromov-Witten invariants for toric varieties and tropical curves (2020)
- Markwig, Hannah: Tropical curves and covers and their moduli spaces (2020)
- Okuyama, Kazumi; Sakai, Kazuhiro: Holomorphic anomaly of 2D Yang-Mills theory on a torus revisited (2019)
- Böhm, Janko; Bringmann, Kathrin; Buchholz, Arne; Markwig, Hannah: Tropical mirror symmetry for elliptic curves (2017)
- Rose, Simon C. F.: Introduction to modular forms (2015)
- van Garrel, Michel; Overholser, D. Peter; Ruddat, Helge: Enumerative aspects of the Gross-Siebert program (2015)
- Janko Boehm, Kathrin Bringmann, Arne Buchholz, Hannah Markwig: Tropical mirror symmetry for elliptic curves (2013) arXiv