Counts of (tropical) curves in $E\times {\mathbb{P}}^1$ and Feynman integrals

Abstract

We study generating series of Gromov-Witten invariants of $E\times {\mathbb{P}}^1$ and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a pearl chain. The individual summands are --- just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals --- complex analytic path integrals involving a product of propagators (equal to the Weierstrass-$\wp$-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in $E_{\mathbb{T}}\times {\mathbb{P}}_{\mathbb{T}}^1$ of so-called leaky degree.

26 pages, 12 figures. We thank an anonymous referee for pointing out a mistake which we fixed in this version