Séminaire Lotharingien de Combinatoire, B65g (2012), 30 pp.

Adrian Tanasa

Some Combinatorial Aspects of Quantum Field Theory

Abstract. 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 \Phi4 field theory. We then generalize this to ribbon graphs and present the relation of the Bollobás-Riordan polynomial with the parametric representation of some \Phi4 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 \Phi4 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.

Received: January 31, 2011. Revised: February 6, 2012; August 22, 2012. Accepted: September 24, 2012.

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