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

Böhm, Janko; Goldner, Christoph; Markwig, Hannah

doi:10.4171/aihpd/115

Geodesic

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Counts of (tropical) curves in

E \times 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.

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Résumé

We study generating series of Gromov–Witten invariants of $E \times ℙ^{1}$ 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 $𝑝𝑒𝑎𝑟𝑙𝑐ℎ𝑎𝑖𝑛$ . 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- $℘$ -function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in $E_{𝕋} \times ℙ_{𝕋}^{1}$ of so-called \textit{leaky degree}.

Accepté le : 2020-09-21
Publié le : 2022-04-11

MR Zbl

DOI : 10.4171/aihpd/115

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

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Böhm, Janko; Goldner, Christoph; Markwig, Hannah. Counts of (tropical) curves in $E \times \mathbb{P}^1$ and Feynman integrals. Annales de l’Institut Henri Poincaré D, Tome 9 (2022) no. 1, pp. 121-158. doi : 10.4171/aihpd/115. http://geodesic.mathdoc.fr/articles/10.4171/aihpd/115/

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