Hopf algebra of ribbon graphs and renormalization

TL;DR

This paper extends the Hopf algebra framework to ribbon graphs, which are relevant for matrix field theories, and explores its implications for renormalization, including examples like a^4 theory and the 1/N expansion.

Contribution

The authors generalize the Hopf algebra structure to ribbon graphs, connecting algebraic methods with surface topology in matrix field theories.

Findings

Hopf algebra structure is applicable to ribbon graphs

Renormalization of a^4 theory is analyzed within this framework

Insights into the 1/N expansion are provided

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.