Vincent Rivasseau

From perturbative to constructive renormalization

The last decade has seen striking progress in the subject of renormalization in quantum field theory. The old subject of perturbative renormalization has been revived by the use of powerful methods such as multiscale decompositions; precise estimates have been added to the initial theorems on finiteness of renormalized perturbation theory, with new results on its large order asymptotics. Furthermore, constructive field theory has reached one of its major goals, the mathematically rigorous construction of some renormalizable quantum field theories. For these models one can in particular investigate rigorously the phenomenon of asymptotic freedom, which plays a key role in our current understanding of the interaction among elementary particles. However, until this book, there has been no pedagogical synthesis of these new developments. Vincent Rivasseau, who has been actively involved in them, now describes them for a wider audience. There are, in fact, common concepts at the heart of the progress on perturbative and constructive techniques. Exploiting these similarities, the author uses perturbative renormalization, which is the more widely known and conceptually simpler of the two cases, to explain the less familiar but more mathematically meaningful constructive renormalization.

CRITICAL BEHAVIOR OF COLORED TENSOR MODELS IN THE LARGE N LIMIT

Colored tensor models have been recently shown to admit a large N expansion, whose leading order encodes a sum over a class of colored triangulations of the D-sphere. The present paper investigates in details this leading order. We show that the relevant triangula- tions proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers. Tensor models (1-3) and group field theories (4-7) are the natural generalization of matrix models (8, 9) implementing in a consistent way the sum over random triangulations in dimensions higher than two. They are notoriously hard to control analytically and one usually resorts to numerical simulations (10-12). Progress has recently been made in the analytic control of tensor models with the advent of the 1/N expansion (13-15) of colored (16-18) tensor models. This expansion synthetizes several alternative evaluations of graph amplitudes in tensor models (19-27) and provides a straightforward generalization of the familiar genus expansion of matrix models (28, 29) in arbitrary dimension. The coloring of the fields allows one to address previously inaccessible questions in tensor models like the implementation of the diffeomorphism symmetry (27, 30) in the Boulatov model or the identification of embedded matrix models (31). The symmetries of tensor models have recently been studied using n-ary algebras (32, 33). This paper is the first in a long series of studies of the implications of the 1/N expansion in colored tensor models. We present here a complete analysis of the leading order in the large N limit in arbitrary dimensions, indexed by graphs of spherical topology (14). To perform the study of this leading order one needs to address the following two questions • What is the combinatorics of the Feynman graphs contributing to the leading order, i.e. the higher dimensional extension of the notion of planar graphs? Unlike in matrix models, where planarity and spherical topology are trivially related, this question is non trivial in tensor models. In particular not all triangulations of the sphere contribute to the leading order. • Is the series of the leading order summable with a non zero radius of convergence? If this is the case, then, in the large N limit, the model exhibits a critical behavior whose critical exponents one needs to compute.

The 1/N expansion of colored tensor models in arbitrary dimension

In this paper we extend the 1/N expansion introduced in Gurau R., Ann. Henri Poincare, 12 (2011) 829, to group field theories in arbitrary dimension and prove that only graphs corresponding to spheres SD contribute to the leading order in the large-N limit.

Random tensor models in the large N limit: Uncoloring the colored tensor models

Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a 1/N expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. non-symmetric, complex tensor. Colors appear in this setting as a canonical book-keeping device and not as a fundamental feature. In the large N limit, we exhibit a set of Virasoro constraints satisfied by the free energy and an infinite family of multicritical behaviors with entropy exponents m = 1 1/m.

Vanishing of Beta Function of Non Commutative Phi**4(4) Theory to all orders

Abstract The simplest non-commutative renormalizable field theory, the ϕ 4 model on four-dimensional Moyal space with harmonic potential is asymptotically safe up to three loops, as shown by H. Grosse and R. Wulkenhaar, M. Disertori and V. Rivasseau. We extend this result to all orders.

Renormalisation of Noncommutative ϕ 4-Theory by Multi-Scale Analysis

In this paper we give a much more efficient proof that the real Euclidean ϕ4-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.

Non-Commutative Renormalization

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 70th birthday.

A Translation-invariant renormalizable non-commutative scalar model

In this paper we propose a translation-invariant scalar model on the Moyal space. We prove that this model does not suffer from the UV/IR mixing and we establish its renormalizability to all orders in perturbation theory.

Scaling behaviour of three-dimensional group field theory

Group field theory is a generalization of matrix models, with triangulated pseudomanifolds as Feynman diagrams and state sum invariants as Feynman amplitudes. In this paper, we consider Boulatovs three-dimensional model and its Freidel-Louapre positive regularization (hereafter the BFL model) with a ‘ultraviolet cutoff, and study rigorously their scaling behavior in the large cutoff limit. We prove an optimal bound on large order Feynman amplitudes, which shows that the BFL model is perturbatively more divergent than the former. We then upgrade this result to the constructive level, using, in a self-contained way, the modern tools of constructive field theory: we construct the Borel sum of the BFL perturbative series via a convergent ‘cactus expansion, and establish the ‘ultraviolet scaling of its Borel radius. Our method shows how the ‘sum over trian- gulations in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory.

Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions

We tackle the issue of renormalizability for Tensorial Group Field Theories (TGFT) including gauge invariance conditions, with the rigorous tool of multi-scale analysis, to prepare the ground for applications to quantum gravity models. In the process, we define the appropriate generalization of some key QFT notions, including connectedness, locality and contraction of (high) subgraphs. We also define a new notion of Wick ordering, corresponding to the subtraction of (maximal) melonic tadpoles. We then consider the simplest examples of dynamical 4-dimensional TGFT with gauge invariance conditions for the Abelian U(1) case. We prove that they are super-renormalizable for any polynomial interaction.