Related papers: Renormalization and Hopf Algebraic Structure of th…

We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hep-th]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop $\gamma$- and $\beta$-functions of the model…

We consider the Hopf algebra of B-diagrams as an algebra projecting onto the Heisenberg algebra and designed to encode the combinatorics of the bosonic normal-ordering problem. In order to understand and generalize the properties of the…

In this work we apply the Drinfel'd twist of Hopf algebras to the study of deformed quantum theories. We prove that, by carefully considering the role of the central extension, it is indeed possible to construct the universal enveloping…

We obtain the contributions to the renormalization group functions of all the diagrams containing the unique one-loop primitive divergence of a simple supersymmetric Wess--Zumino model, up to more than 200 loops. The asymptotic behavior of…

For our own education, we reconstruct the Hopf algebra of Connes and Moscovici obtained by the action of vector fields on a crossed product of functions by diffeomorphisms. We extend the realization of that Hopf algebra in terms of rooted…

These are the notes of five lectures given at the Summer School {\em Geometric and Topological Methods for Quantum Field Theory}, held in Villa de Leyva (Colombia), July 2--20, 2007. The lectures are meant for graduate or almost graduate…

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for…

We give a simple presentation of the combinatorics of renormalization in perturbative quantum field theory in terms of triangular matrices. The prescription, that may be of calculational value, is derived from first principles, to wit, the…

We demonstrate that the fundamental algebraic structure underlying the Connes-Kreimer Hopf algebra -- the insertion pre-Lie structure on graphs -- corresponds directly to the canonical pre-Lie structure of polynomial vector fields. Using…

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…

We briefly review the r\^ole played by algebraic structures like combinatorial Hopf algebras in the renormalizability of (noncommutative) quantum field theory. After sketching the commutative case, we analyze the noncommutative…

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 consider general renormalizable scalar field theory and derive six-loop beta functions for all parameters in d = 4 dimensions within the $\overline{MS}$-scheme. We do not explicitly compute relevant loop integrals but utilize…

Let $\mathsf{Rep}(H)$ be the category of finite-dimensional representations of a finite-dimensional Hopf algebra $H$. Andruskiewitsch and Mombelli proved in 2007 that each indecomposable exact $\mathsf{Rep}(H)$-module category has form…

Let $H$ be a finite-dimensional connected Hopf algebra over an algebraically closed field $\field$ of characteristic $p>0$. We provide the algebra structure of the associated graded Hopf algebra $\gr H$. Then, we study the case when $H$ is…

We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees,…

In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some…

We consider the perturbative renormalisation of the $\Phi^4_d$ model from Euclidean Quantum Field Theory for any, possibly non-integer dimension $d<4$. The so-called BPHZ renormalisation, named after Bogoliubov, Parasiuk, Hepp and…

Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and…

``Bonsai'' Hopf algebras, introduced here, are generalizations of Connes-Kreimer Hopf algebras, which are motivated by Feynman diagrams and renormalization. We show that we can find operad structure on the set of bonsais. We introduce a new…