Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 1

Algebraic interplay between renormalization and monodromy

Pages: 87 – 191



Dirk Kreimer (Institute of Physics and Institute of Mathematics, Humboldt University, Berlin, Germany)

Karen Yeats (Faculty of Mathematics, University of Waterloo, Ontario, Canada)


We investigate combinatorial and algebraic aspects of the interplay between renormalization and monodromies for Feynman amplitudes. We clarify how extraction of subgraphs from a Feynman graph interacts with putting edges onshell or with contracting them to obtain reduced graphs. Graph by graph this leads to a study of cointeracting bialgebras. One bialgebra comes from extraction of subgraphs and hence is needed for renormalization. The other bialgebra is an incidence bialgebra for edges put either on- or offshell. It is hence related to the monodromies of the multivalued function to which a renormalized graph evaluates. Summing over infinite series of graphs, consequences for Green functions are derived using combinatorial Dyson–Schwinger equations.

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Published 13 July 2023