arXiv subjects
Publications and source records attributed to 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 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-Poincare 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 introduce partial duality of hypermaps, which include the classical Euler-Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or $τ$-model), or as three permutations on the set of half-edges (rotation system or $σ$-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality.
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 $ϕ^4_3$. 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 $D$-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 $φ^{4}_{5}$-model (with propagator $1/{\bf p}^{2}$) 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. From our analysis, we prove the just renormalizability at all orders of perturbation of the phi^4_6 and phi^6_5 random tensor models. We also deduce that the phi^4_5 tensor model is super-renormalizable.
We extend the quasi-tree expansion of A. Champanerkar, I. Kofman, and N. Stoltzfus to not necessarily orientable ribbon graphs. We study the duality properties of the Bollobas-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 S. Chmutov.
We define a new topological polynomial extending the Bollobas-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behavior 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 generalise the signed Bollobas-Riordan polynomial of S. Chmutov and I. Pak [Moscow Math. J. 7 (2007), no. 3, 409-418] 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 [J. Combinatorial Theory, Ser. B, 99 (3): 617-638, 2009] 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 Lagrangean. 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 contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.
The first renormalisable quantum field theories on non-commutative space have been found recently. We review this rapidly growing subject.
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.
Very high energy physics needs a coherent description of the four fundamental forces. Non-commutative geometry is a promising mathematical framework which already allowed to unify the general relativity and the standard model, at the classical level, thanks to the spectral action principle. Quantum field theories on non-commutative spaces is a first step towards the quantification of such a model. These theories can't be obtained simply by writing usual field theory on non-commutative spaces. Such attempts exhibit indeed a new type of divergencies, called ultraviolet/infrared mixing, which prevents renormalisability. H. Grosse and R. Wulkenhaar showed, with an example, that a modification of the propagator may restore renormalisability. This thesis aims at studying the generalization of such a method. We studied two different models which allowed to specify certain aspects of non-commutative field theory. In x space, the major technical difficulty is due to oscillations in the interaction part. We generalized the results of T. Filk in order to exploit such oscillations at best. We were then able to distinguish between two mixings, renormalizable or not. We also bring the notion of orientability to light : the orientable non-commutative Gross-Neveu model is renormalizable without any modification of its propagator. The adaptation of multi-scale analysis to the matrix basis emphasized the importance of dual graphs and represents a first step towards a formulation of field theory independent of the underlying space.
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 "psibar i 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 and could be more adapted to the future study of these theories (in particular at the non-perturbative or constructive level).