Fabien VIGNES -TOURNERET

We consider a quartic O\((N)\)-vector model. Using the Loop Vertex Expansion, we prove the Borel summability in \(1/N\) along the real axis of the partition function and of the connected correlations of the model. The Borel summability holds uniformly in the coupling constant, as long as the latter belongs to a cardioid like domain of the complex plane, avoiding the negative real axis.

We introduce a collection of new operations on hypermaps, partial duality, which include the classical Euler-Poincaré dualities as particular cases. These operations generalize the partial duality for maps, or ribbon graphs, recently discovered in a connection with knot theory. Partial duality is different from previous studied operations of S. Wilson, G. Jones, L. James, and A. Vince. Combinatorially hypermaps may be described in one of three ways: as three involutions on the set of flags (\(\tau\)-model), or as three permutations on the set of half-edges (\(\sigma\)-model in orientable case), or as edge 3-colored graphs. We express partial duality in each of these models.

We continue the constructive program about tensor field theory through the next natural model, namely the rank five tensor theory with quartic melonic interactions and propagator inverse of the Laplacian on \(U(1)^5\). We make a first step towards its construction by establishing its power counting, identifiying the divergent graphs and performing a careful study of (a slight modification of) its RG flow. Thus we give strong evidence that this just renormalizable tensor field theory is non perturbatively asymptotically free.

Partial duality is a duality of ribbon graphs relative to a subset of their edges generalizing the classical Euler-Poincaré duality. This operation often changes the genus. Recently J. L. Gross, T. Mansour, and T. W. Tucker formulated a conjecture that for any ribbon graph different from plane trees and their partial duals, there is a subset of edges partial duality relative to which does change the genus. A family of counterexamples was found by Qi Yan and Xian'an Jin. In this note we prove that essentially these are the only counterexamples.

We continue our constructive study of tensor field theory through the next natural model, namely the rank four tensor theory with quartic melonic interactions and propagator inverse of the Laplacian on \(U(1)^4\). This superrenormalizable tensor field theory has a power counting quite similar to ordinary \(\phi^4_3\). We control the model via a multiscale loop vertex expansion which has to be pushed quite beyond the one of the \(T^4_3\) model and we establish its Borel summability in the coupling constant. This paper is also a step to prepare the constructive treatment of just renormalizable models, such as the \(T^4_5\) model with quartic melonic interactions.

The \((D+1)\)-colored version of tensor models has been shown to admit a large \(N\)-limit expansion. The leading contributions result from so-called melonic graphs which are dual to the \(D\)-sphere. This is a note about the Schwinger-Dyson equations of the tensorial \(\varphi^4_5\)-model (with propagator \(1/p^2\)) and their melonic approximation. We derive the master equations for two- and four-point correlation functions and discuss their solution.

We study the polynomial Abelian or \(U(1)^d\) Tensorial Group Field Theories equipped with a gauge invariance condition in any dimension \(d\). We prove the just renormalizability at all orders of perturbation of the \(\varphi^4_6\) and \(\varphi^6_5\) random tensor models. We also deduce that the \(\varphi^4_5\) tensor model is super-renormalizable.

We define a new topological polynomial extending the Bollobás-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials occurring in the parametric representation of the non-commutative Grosse-Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.

We extend the quasi-tree expansion of Champanerkar et al. [A. Champanerkar, I. Kofman, N. Stoltzfus, Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial, Bull. London Math. Soc., 43(5):972-984. arXiv:0705.3458.] to not necessarily orientable ribbon graphs. We study the duality properties of the Bollobás–Riordan polynomial in terms of this expansion. As a corollary, we get a “connected state” expansion of the Kauffman bracket of virtual link diagrams. Our proofs use extensively the partial duality of Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory, Ser. B 99(3):617-638, 2009. arXiv:0711.3490.].

We generalise the signed Bollobás-Riordan polynomial of S. Chmutov and I. Pak [S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial, Mos. Math. J. 7 (3):409–418, 2007] to a multivariate signed polynomial \(Z\) and study its properties. We prove the invariance of \(Z\) under the recently defined partial duality of S. Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory, Ser. B 99(3):617-638, 2009. arXiv:0711.3490.] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.

We prove that the self-interacting scalar field on the four-dimensional degenerate Moyal plane is renormalisable to all orders when adding a suitable counterterm to the Lagrangian. Despite the apparent simplicity of the model, it raises several non-trivial questions. Our result is a first step towards the definition of renormalisable quantum field theories on a non-commutative Minkowski space.

We construct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.

We compute at the one-loop order the \(\beta\)-functions for a renormalisable non-commutative analog of the Gross-Neveu model defined on the Moyal plane. The calculation is performed within the so called \(x\)-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The \(\beta\)-function for the non-commutative counterpart of the Thirring model is found to be non vanishing.

We prove that the non-commutative Gross-Neveu model on the two-dimensional Moyal plane is renormalizable to all orders. Despite a remaining UV/IR mixing, renormalizability can be achieved. However, in the massive case, this forces us to introduce an additional counterterm of the form \(\bar\psi\imath\gamma^0\gamma^1\psi\). The massless case is renormalizable without such an addition.

In this paper we provide a new proof that the Grosse-Wulkenhaar non-commutative scalar \(\Phi^4_4\) theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in \(x\) space. We think this proof is simpler. It also allows direct interpretation in terms of the physical positions of the particles and should be more adapted to the future study of these theories (in particular at the non-perturbative or constructive level).

In this paper we provide exact expressions for propagators of non-commutative Bosonic or Fermionic field theories after adding terms of the Grosse-Wulkenhaar type to the usual free action in order to ensure Langmann-Szabo covariance. We emphasize the new Fermionic case and we give in particular all necessary bounds for the multiscale analysis and renormalization of the non-commutative Gross-Neveu model.

In this paper we give a much more efficient proof that the real Euclidean \(\phi^4\)-model on the four-dimensional Moyal plane is renormalisable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalisation proof based on renormalisation group equations for non-local matrix models. On the other hand, our bounds permit a powerful multi-scale analysis of the resulting ribbon graphs. Here, the dual graphs play a particular rôle because the angular momentum conservation is conveniently represented in the dual picture. Choosing a spanning tree in the dual graph according to the scale attribution, we prove that the summation over the loop angular momenta can be performed at no cost so that the power-counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph.

The first renormalisable quantum field theories on non-commutative space have been found recently. We review this rapidly growing subject.

We review the renormalisation group properties of non-commutative quantum field theory.

We review the recent approach of Grosse and Wulkenhaar to the perturbative renormalization of non commutative field theory and suggest a related constructive program. This paper is dedicated to J. Bros on his 65^th birthday.

La physique des très hautes énergies nécessite une description cohérente des quatre forces fondamentales. La géométrie non commutative représente un cadre mathématique prometteur qui a déjà permis d’unifier la relativité générale et le modèle standard, au niveau classique, grâce au principe de l’action spectrale. L’étude des théories quantiques de champs sur des espaces non commutatifs est une première étape vers la quantification de ce modèle. Celles-ci ne sont pas simplement obtenues en récrivant les théories commutatives sur des espaces non commutatifs. En effet, ces tentatives ont révélé un nouveau type de divergences, appelé mélange ultraviolet/infrarouge, qui rend ces modèles non renormalisables. H. Grosse et R. Wulkenhaar ont montré, sur un exemple, qu’une modification du propagateur restaure la renormalisabilité. L’étude de la généralisation de cette méthode est le cadre de cette thèse. Nous avons ainsi étudié deux modèles sur espace de Moyal qui ont permis de préciser certains aspects des théories non commutatives. En espace \(x\), la principale difficulté technique est due aux oscillations de l’interaction. Nous avons donc généralisé les résultats de T. Filk afin d’exploiter au mieux ces oscillations. Nous avons pu ainsi distinguer deux types de mélange, renormalisable ou pas. Nous avons aussi mis en lumière la notion d’orientabilité : le modèle de Gross-Neveu non commutatif orientable est renormalisable sans modification du propagateur. L’adaptation de l’analyse multi-échelles à la base matricielle a souligné l’importance du graphe dual et représente un premier pas vers une formulation des théories de champs indépendante de l’espace sous-jacent.