  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.
