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.

34 pages, 9 figures, Latex; improved style

References in corpus (4)

• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Combinatorics of (perturbative) quantum field theory
• Bogolubov's Recursion and Integrability of Effective Actions

Cited by in corpus (10)

• Hopf algebra approach to Feynman diagram calculations
• Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory
• Can Renormalization Group Flow End in a Big Mess?
• Challenges of Matrix Models
• Comment on the Surface Exponential for Tensor Fields
• Phenomenology of a vector-field-induced (and possibly parity breaking) compensated isocurvature perturbation
• Quantum Statistical Mechanics of the Absolute Galois Group
• Renormalization of Quantum Electrodynamics and Hopf Algebras
• Renormalization of Gauge theories and the Hopf Algebra of Diagrams
• A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory