Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs


Renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.


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I thank M. Borinsky, J. Ben Geloun, A. Hock, D. Kreimer, A. Pithis and R. Wulkenhaar for discussions and comments on the paper, in particular M. Borinsky for very helpful discussions on the various subalgebra structures occuring in Hopf algebras of diagrams. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in two ways, primarily under the project number 418838388 and furthermore under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.


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Thürigen, J. Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs. Math Phys Anal Geom 24, 19 (2021).

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  • Hopf algebra
  • Non-local field theory
  • Renormalization
  • Random geometry

Mathematics Subject Classification (2010)

  • MSC2020: 81T18
  • 81T15
  • 16T05
  • 16T30
  • 81T32
  • 05C10
  • 81V17