Related papers: Symetries Galoisiennes et Renormalisation

We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory as well as number theory and…

Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is…

We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov's preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus…

We extend the results we obtained in an earlier work. The cocommutative case of rooted ladder trees is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the…

It was recently shown that the renormalization of quantum field theory is organized by the Hopf algebra of decorated rooted trees, whose coproduct identifies the divergences requiring subtraction and whose antipode achieves this. We…

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…

These lecture notes aim to present the algebraic theory of regularity structures as developed in arXiv:1303.5113, arXiv:1610.08468, and arXiv:1711.10239. The main aim of this theory is to build a systematic approach to renormalisation of…

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of B, with the…

We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals. We consider…

We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we…

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. For an extension $L/K$ which is classically Galois with $G=Gal(L/K)$ the Hopf algebras in question are of…

We derive the existence of Hopf subalgebras generated by Green's functions in the Hopf algebra of Feynman graphs of a quantum field theory. This means that the coproduct closes on these Green's functions. It allows us for example to derive…

We formulate the Hopf algebraic approach of Connes and Kreimer to renormalization in perturbative quantum field theory using triangular matrix representation. We give a Rota-Baxter anti-homomorphism from general regularized functionals on…

This is a survey of our results on the relation between perturbative renormalization and motivic Galois theory. The main result is that all quantum field theories share a common universal symmetry realized as a motivic Galois group, whose…

We observe that the Connes--Kreimer Hopf-algebraic approach to perturbative renormalisation works not just for Hopf algebras but more generally for filtered bialgebras $B$ with the property that $B_0$ is spanned by group-like elements (e.g.…

We review the appearance of Hopf algebras in the renormalization of quantum field theories and in the study of diffeomorphisms of the frame bundle important for index computations in noncommutative geometry.

We show that the process of renormalization encapsules a Hopf algebra structure in a natural manner. This sheds light on the recently proposed connection between knots and renormalization theory.

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several…

Let $ L/K $ be a finite separable extension of fields whose Galois closure $ E/K $ has group $ G $. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on $ L/K $ has the form $ E[N]^{G}…

We consider two interacting connected graded Hopf algebras, the former being a comodule-coalgebra on the latter. We show how to define analogues of Connes-Kreimer's renormalization group and Beta function, when the graduation operator is…