Counts of (tropical) curves in E x P1 and Feynman integrals

We study generating series of Gromov-Witten invariants of E x 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 minor 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-p-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in E-T x P-T(1) of so-called leaky degree.