Constructive tensor field theory: the \({T^4_3}\) model
type: Item
instance of: scholarly article
title: Constructive tensor field theory: the \({T^4_3}\) model
zbMATH Open document ID: 1345.81073
author: Thibault Delepouve
author: Vincent Rivasseau
published in: Communications in Mathematical Physics
publication date: 2016-07-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1412.5091
review text: Constructing interacting quantum field theories in a mathematically precise sense is a challenging task. The construction consists of two main steps: a combinatorial step to prove the renormalizability of the theory followed by an analytical step to prove the summability of the corresponding perturbative series providing the \(S\)-matrix elements of the renormalized theory. In this paper the authors construct the partition function (the simplest \(S\)-matrix element) of the simplest interacting superrenormalizable tensor field theory called the \textit{\(T^4_3\) model}. More precisely, this specific theory arises as follows. For the underlying space-time one takes the 3-torus and the two fields in this theory are a section of the trivial complex Hermitian line bundle over the 3-torus and a complex conjugate section. The Lagrangian contains a usual (Laplacian) kinetic term and quartic interaction terms with a common complex coupling constant \(g\). (For the precise formulation of the theory using a Fourier transformed dual picture cf. Section 2 and in particular Equations 2.1, 2.2 and 2.3 in the paper.) The main result is that the carefully defined partition function \(Z(g)\) of the theory exists and is a complex-analytic function of \(g\) if the coupling constant belongs to a small cardioidal region about the origin of the complex plane, moreover \(\log Z(g)\) arises as the Borel sum of a perturbative power series in \(g\) (cf. Theorem 3.1 of the paper). The proof is based on a so-called multiscale loop vertex expansion (MLVE) technique.
reviewed by: Gábor Etesi
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 81T15
Mathematics Subject Classification ID: 81U20
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 6605253
MaRDI profile type: Publication
OpenAlex ID: W1871177843
arXiv ID: 1412.5091
cites work: Colored group field theory
cites work: Colored tensor models -- a review
cites work: A model of three-dimensional lattice gravity
cites work: Q4648180
cites work: Three-dimensional simplicial quantum gravity and generalized matrix models
cites work: Tensor model for gravity and orientability of manifold
cites work: Tensor models and simplicial quantum gravity in \(>2\)-D
cites work: The \(1/N\) expansion of colored tensor models
cites work: The complete \(1/N\) expansion of colored tensor models in arbitrary dimension
cites work: Universality for random tensors
cites work: Intersection theory on the moduli space of curves and the matrix Airy function
cites work: Renormalisation of \(\phi^4\)-theory on noncommutative \(\mathbb R^{4}\) in the matrix base
cites work: Vanishing of beta function of non-commutative \(\Phi_4^4\) theory to all orders
cites work: Q3633693
cites work: Self-dual noncommutative \(\phi^4\)-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
cites work: The tensor track, III
cites work: The tensor theory space
cites work: Functional renormalisation group approach for tensorial group field theory: a rank-3 model
cites work: Q4421439
cites work: Group field theory as the second quantization of loop quantum gravity
cites work: Radiative corrections in the Boulatov-Ooguri tensor model: the 2-point function
cites work: A renormalizable 4-dimensional tensor field theory
cites work: 3D tensor field theory: renormalization and one-loop \(\beta \)-functions
cites work: Renormalizable models in rank \({d \geq 2}\) tensorial group field theory
cites work: Renormalization of tensorial group field theories: abelian \(\mathrm U(1)\) models in four dimensions
cites work: Renormalization of a \(\mathrm{SU}(2)\) tensorial group field theory in three dimensions
cites work: Tensorial Methods and Renormalization in Group Field Theories
cites work: Quantum Einstein gravity
cites work: Two- and four-loop β-functions of rank-4 renormalizable tensor field theories
cites work: Discrete renormalization group for \(\mathrm{SU}(2)\) tensorial group field theory
cites work: Renormalization of an abelian tensor group field theory: solution at leading order
cites work: Q3910965
cites work: Constructive \(\phi^{4}\) field theory without tears
cites work: How to resum Feynman graphs
cites work: Scaling behavior of three-dimensional group field theory
cites work: The \(1/N\) expansion of tensor models beyond perturbation theory
cites work: Universality and Borel summability of arbitrary quartic tensor models
cites work: The multiscale loop vertex expansion
cites work: Corrected loop vertex expansion for Φ24 theory
cites work: Constructive tensorial group field theory II: the ${U(1)-T^4_4}$ model
cites work: Q3809004
cites work: Q3643263
cites work: Q4840505
cites work: Explicit fermionic tree expansions
cites work: An improvement of Watson’s theorem on Borel summability
DOI: 10.1007/S00220-016-2680-1
description: scientific article