  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
$E_{\mathbb{T}}\times\mathbb{P}^1_\mathbb{T}$ of so-called leaky degree.
