Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory

TL;DR

This paper introduces a novel Hopf algebra structure tailored for the combinatorics of the perturbatively renormalizable Ben Geloun-Rivasseau tensor field theory, highlighting its unique topological and combinatorial features.

Contribution

It defines a new Hopf algebra framework specific to tensorial Feynman graphs, differing from classical models due to tensor-specific combinatorial and topological properties.

Findings

The Hopf algebra captures the combinatorics of tensor field theory graphs.

It accounts for the conservation of divergence degrees in insertions.

The structure differs significantly from Connes-Kreimer algebra.

Abstract

The Ben Geloun-Rivasseau quantum field theoretical model is the first tensor model shown to be perturbatively renormalizable. We define here an appropriate Hopf algebra describing the combinatorics of this new tensorial renormalization. The structure we propose is significantly different from the previously defined Connes-Kreimer combinatorial Hopf algebras due to the involved combinatorial and topological properties of the tensorial Feynman graphs. In particular, the 2- and 4-point function insertions must be defined to be non-trivial only if the superficial divergence degree of the associated Feynman integral is conserved.