  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.
