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

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