Hopf algebra of ribbon graphs and renormalization

Abstract: Connes and Kreimer have discovered a Hopf algebra structure behind renormalization of Feynman integrals. We generalize the Hopf algebra to the case of ribbon graphs, i.e. to the case of theories with matrix fields. The Hopf algebra is naturally defined in terms of surfaces corresponding to ribbon graphs. As an example, we discuss renormalization of $\Phi^4$ theory and the 1/N expansion.