Tropical mirror symmetry for elliptic curves

Abstract

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.Comment: 25 pages + Appendix, 16 figures, a gap in Thm 3.2 was discovered and closed by Goujard-Moeller, J. Reine Angew. Math., 201

Counting Feynman-like graphs: Quasimodularity and Siegel–Veech weight

“…down to the correspondence theorem for tropical Hurwitz numbers of torus covers proved in [BBBM13], as we explain in Section 8.…”

Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited

“…3 As demonstrated in [34], one can easily evaluate the Feynman diagram at least in the small Q expansion. One difference from [34] is that here we have the non-holomorphic piece S in the propagator. Consequently, the result is not simply a quasi modular form, but a polynomial in S whose coefficients are given by quasi modular forms.…”

“…Here we simply mention that what we need to do is to take the symmetric average of all the possible orderings of |x i | for the position x i of vertices in (2.13). Third, the symmetry factor is simply given by the order of the automorphism group of the graph Γ [34]. This means that the contribution of the Feynman diagram Γ to the free energy is to be normalized as 1…”

Descendant log Gromov-Witten invariants for toric varieties and tropical curves

“…Remark 5.17. We can easily modify our main theorem to work for elliptic curves, and then we recover the correspondence theorem of [BBBM17] as a corollary of that. This is essentially [CJMR16, Thm.…”