Multi-valued Feynman Graphs and Scattering Theory

  • Dirk KreimerEmail author
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


We outline ideas to connect the analytic structure of Feynman amplitudes to the structure of Karen Vogtmann’s and Marc Culler’s Outer Space. We focus on the role of cubical chain complexes in this context, and also investigate the bordification problem in the example of the 3-edge banana graph.



It is a pleasure to thank Spencer Bloch for a long-standing collaboration and an uncountable number of discussions. Also, I enjoy to thank David Broadhurst, Karen Vogtmann and Marko Berghoff for discussions, and the audiences at this ‘elliptic conference’, and at the Les Houches workshop on ‘structures in local quantum field theory’ for a stimulating atmosphere. And thanks to Johannes Blümlein for initiating this KMPB conference at DESY-Zeuthen!


  1. 1.
    E. Panzer, Feynman Integrals and Hyperlogarithms, Ph.D. thesis, arXiv:1506.07243 [math-ph]
  2. 2.
    M. Culler, K. Vogtmann, Moduli of graphs and automorphisms of free groups. Inven. Math. 84(1), 91–119 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Hatcher, K. Vogtmann, Rational homology of \(\rm OUT(F_n)\). Math. Res. Lett. 5, 759–780 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    K.-U. Bux, P. Smillie, K. Vogtmann, On the bordification of outer space, arXiv:1709.01296
  5. 5.
    J. Conant, A. Hatcher, M. Kassabov, K. Vogtmann, Assembling homology classes in automorphism groups of free groups. Comment. Math. Helv. 91, 751–806 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Bloch, D. Kreimer, Cutkosky rules and outer space, arXiv:1512.01705
  7. 7.
    S. Bloch, D. Kreimer, Feynman amplitudes and Landau singularities for 1-loop graphs. Commun. Number Theory Phys. 4, 709–753 (2010)., arXiv:1007.0338 [hep-th]MathSciNetCrossRefGoogle Scholar
  8. 8.
    S. Bloch, D. Kreimer, Mixed hodge structures and renormalization in physics. Commun. Number Theory Phys. 2, 637 (2008)., arXiv:0804.4399 [hep-th]MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Berghoff, Feynman amplitudes on moduli spaces of graphs, arXiv:1709.00545
  10. 10.
    R.C. Hwa, V.L. Teplitz, Homology and Feynman Integrals (Benjamin Inc., 1966)Google Scholar
  11. 11.
    F. Brown, D. Kreimer, Angles, scales and parametric renormalization. Lett. Math. Phys. 103, 933 (2013)., arXiv:1112.1180 [hep-th]MathSciNetCrossRefGoogle Scholar
  12. 12.
    F. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [math.AG]
  13. 13.
    F. Brown, The massless higher-loop two-point function. Commun. Math. Phys. 287, 925 (2009)., arXiv:0804.1660 [math.AG]MathSciNetCrossRefGoogle Scholar
  14. 14.
    E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales. JHEP 1403, 071 (2014)., arXiv:1401.4361 [hep-th]
  15. 15.
    M. Mühlbauer, Moduli Spaces of Colored Feynman graphs, Master Thesis,
  16. 16.
    S. Abreu, R. Britto, C. Duhr, E. Gardi, The diagrammatic coaction and the algebraic structure of cut Feynman integrals, arXiv:1803.05894
  17. 17.
    D. Kreimer, The core Hopf algebra. Clay Math. Proc. 11, 313–322 (2010), arXiv:0902.1223
  18. 18.
    A. Bley, Cutkosky Cuts at Core Hopf Algebra, Master thesis,
  19. 19.
    M. Borinsky, Algebraic lattices in QFT renormalization. Lett. Math. Phys. 106(7), 879–911 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series, arXiv:1803.10256 [hep-th]
  21. 21.
    F. Brown, Feynman amplitudes, coaction principle, and cosmic Galois group. Commun. Number Theory Phys. 11, 453 (2017)., arXiv:1512.06409 [math-ph]MathSciNetCrossRefGoogle Scholar
  22. 22.
    S. Bloch, M. Kerr, P. Vanhove, Local mirror symmetry and the sunset Feynman integral. Adv. Theor. Math. Phys. 21, 1373 (2017)., arXiv:1601.08181 [hep-th]MathSciNetCrossRefGoogle Scholar
  23. 23.
    L. Adams, C. Bogner, S. Weinzierl, The sunrise integral and elliptic polylogarithms. PoS LL 2016, 033 (2016)., arXiv:1606.09457 [hep-ph]

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute for Mathematics and Institute for PhysicsHumboldt UniversityBerlinGermany

Personalised recommendations