Tropical mirror symmetry of elliptic curves

In mirror symmetry, tropical geometry can be used as a bridge to link Gromov-Witten invariants on the one hand, and Feynman integrals on the other hand.  From our preliminary work we can predict that tropical mirror symmetry statements for elliptic curves should relate Gromov-Witten invariants and Feynman integrals on a very fine level. We plan to study descendant Gromov-Witten invariants of elliptic curves in tropical geometry and mirror symmetry and to relate our global approach to mirror symmetry to the local versions studied by Gross-Overholser.