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

Böhm, Janko; Goldner, Christoph; Markwig, Hannah

Annales de l’Institut Henri Poincaré D, Tome 9 (2022) no. 1, pp. 121-158.

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

Classification : 14-XX, 11-XX, 81-XX
Keywords: Elliptic fibrations, Feynman integral, tropical geometry, Gromov–Witten invariants, quasimodular forms