Tropical Mirror Symmetry in Dimension One

TL;DR

This paper proves a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of elliptic curves, linking tropical geometry, Feynman integrals, and operator methods to deepen understanding of mirror symmetry.

Contribution

It generalizes previous tropical mirror symmetry results to descendant invariants of elliptic curves and connects tropical and physical operator approaches.

Findings

Tropical mirror symmetry holds on a fine level for elliptic curves.

Generating series of invariants equal Feynman integrals.

Bijection between graph covers and monomials established.

Abstract

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 [B\"ohm 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.