From Modular Forms to Differential Equations for Feynman Integrals

  • Johannes Broedel
  • Claude DuhrEmail author
  • Falko Dulat
  • Brenda Penante
  • Lorenzo Tancredi
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


In these proceedings we discuss a representation for modular forms that is more suitable for their application to the calculation of Feynman integrals in the context of iterated integrals and the differential equation method. In particular, we show that for every modular form we can find a representation in terms of powers of complete elliptic integrals of the first kind multiplied by algebraic functions. We illustrate this result on several examples. In particular, we show how to explicitly rewrite elliptic multiple zeta values as iterated integrals over powers of complete elliptic integrals and rational functions, and we discuss how to use our results in the context of the system of differential equations satisfied by the sunrise and kite integrals.



We would like to thank the “Kolleg Mathematik und Physik Berlin” for supporting the workshop “Elliptic integrals, elliptic functions and modular forms in quantum field theory”. This research was supported by the ERC grant 637019 “MathAm”, and the U.S. Department of Energy (DOE) under contract DE-AC02-76SF00515.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Johannes Broedel
    • 1
  • Claude Duhr
    • 2
    • 3
    Email author
  • Falko Dulat
    • 4
  • Brenda Penante
    • 2
  • Lorenzo Tancredi
    • 2
  1. 1. Institut für Mathematik und Institut für PhysikHumboldt-Universität zu Berlin, IRIS AdlershofBerlinGermany
  2. 2.Theoretical Physics DepartmentCERNGenevaSwitzerland
  3. 3.Center for Cosmology, Particle Physics and Phenomenology (CP3)Université Catholique de LouvainLouvain-La-NeuveBelgium
  4. 4.SLAC National Accelerator LaboratoryStanford UniversityStanfordUSA

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