Some combinatorial aspects of quantum field theory

Abstract

In this short survey we present the appearance of some combinatorial notions in quantum field theory. We first focus on topological graph polynomials (the Tutte polynomial and its multivariate version) and their relation with the parametric representation of the commutative ${\Phi }^4$ field theory. We then generalize this to ribbon graphs and present the relation of the Bollobás-Riordan polynomial with the parametric representation of some ${\Phi }^4$ field theory on the non-commutative Moyal space. We also review the rôle played by the Connes-Kreimer Hopf algebra as the combinatorial backbone of the renormalization process in field theories. We then show how this generalizes to the scalar ${\Phi }^4$ field theory implemented on the non-commutative Moyal space. Finally, some perspectives for the further generalization of these tools to quantum gravity tensor models are briefly sketched.

30 pages, 9 figures. An explicit example of the Bogoliubov subtraction operator's action on the bare Feynman integral of a two-loop graph has been added; the resulting formula has then been compared with the corresponding formula obtained within the Connes-Kreimer algebraic setting. An appendix introducing Grassmann variables and the Grassmann representation of determinants have been added

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