Some Combinatorial Aspects of Quantum Field Theory

In this survey we present the appearance of some combinatorial notions in quantum field theory. We first focus on graph polynomials (the Tutte polynomial and its multivariate version) and their relation with the parametric representation of the commutative Φ field theory. We then generalize this to ribbon graphs and present the relation of the Bollobás–Riordan polynomial with the parametric representation of some Φ field theory on the non-commutative Moyal space. We also review the role 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 Φ 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.