Algebraic combinatorics and QFT

AbstractWe have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).