Iterated integrals related to Feynman integrals associated to elliptic curves. (English) Zbl 1478.81015
Bluemlein, Johannes (ed.) et al., Anti-differentiation and the calculation of Feynman amplitudes. Selected papers based on the presentations at the conference, Zeuthen, Germany, October 2020. Cham: Springer. Texts Monogr. Symb. Comput., 519-545 (2021).
Summary: This talk reviews Feynman integrals, which are associated to elliptic curves. The talk will give an introduction into the mathematics behind them, covering the topics of elliptic curves, elliptic integrals, modular forms and the moduli space of \(n\) marked points on a genus one curve. The latter will be important, as elliptic Feynman integrals can be expressed as iterated integrals on the moduli space \({\mathcal M}_{1,n}\), in same way as Feynman integrals which evaluate to multiple polylogarithms can be expressed as iterated integrals on the moduli space \({\mathcal M}_{0,n}\). With the right language, many methods from the genus zero case carry over to the genus one case. In particular we will see in specific examples that the differential equation for elliptic Feynman integrals can be cast into an \(\epsilon \)-form. This allows to systematically obtain a solution order by order in the dimensional regularisation parameter.
For the entire collection see [Zbl 1475.81004].
For the entire collection see [Zbl 1475.81004].
MSC:
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 14H52 | Elliptic curves |
| 11G05 | Elliptic curves over global fields |
| 81T18 | Feynman diagrams |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 11G55 | Polylogarithms and relations with \(K\)-theory |
| 33E05 | Elliptic functions and integrals |
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