Counts of (tropical) curves in E×ℙ 1 and Feynman integrals
Annales de l’Institut Henri Poincaré D, Tome 9 (2022) no. 1, pp. 121-158.
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We study generating series of Gromov–Witten invariants of E×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 ×ℙ 𝕋 1 of so-called \textit{leaky degree}.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/115
Classification : 14-XX, 11-XX, 81-XX
Mots-clés : Elliptic fibrations, Feynman integral, tropical geometry, Gromov–Witten invariants, quasimodular forms 
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     title = {Counts of (tropical) curves in $E \times \mathbb{P}^1$ and {Feynman} integrals},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {121--158},
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     year = {2022},
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     zbl = {1492.14100},
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     url = {http://www.numdam.org/articles/10.4171/aihpd/115/}
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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://www.numdam.org/articles/10.4171/aihpd/115/

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