Renormalization of an Abelian Tensor Group Field Theory: Solution at Leading Order

Abstract

We study a just renormalizable tensorial group field theory of rank six with quartic melonic interactions and Abelian group U(1). We introduce the formalism of the intermediate field, which allows a precise characterization of the leading order Feynman graphs. We define the renormalization of the model, compute its (perturbative) renormalization group flow and write its expansion in terms of effective couplings. We then establish closed equations for the two point and four point functions at leading (melonic) order. Using the effective expansion and its uniform exponential bounds we prove that these equations admit a unique solution at small renormalized coupling.

arXiv comment37 pages, 14 figures

References in corpus (70)

• Background Independent Quantum Gravity: A Status Report
• 2D Gravity and Random Matrices
• The Spin Foam Approach to Quantum Gravity
• Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base
• Critical behavior of colored tensor models in the large N limit
• Topological Lattice Models in Four Dimensions
• Nonperturbative Quantum Gravity
• Colored Tensor Models - a Review
• A Model of Three-Dimensional Lattice Gravity
• Group Field Theory: An overview
• Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space
• The complete 1/N expansion of colored tensor models in arbitrary dimension
• Colored Group Field Theory
• Random tensor models in the large N limit: Uncoloring the colored tensor models
• Spacetime as a Feynman diagram: the connection formulation
• Group field theory with non-commutative metric variables
• The 1/N expansion of colored tensor models in arbitrary dimension
• Cosmology from Group Field Theory Formalism for Quantum Gravity
• The 1/N expansion of colored tensor models
• Homogeneous cosmologies as group field theory condensates
• Melons are branched polymers
• The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis
• Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
• Double Scaling in Tensor Models with a Quartic Interaction
• Renormalization of an SU(2) Tensorial Group Field Theory in Three Dimensions
• Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model
• Tensorial methods and renormalization in Group Field Theories
• The Tensor Track, III
• Coarse graining methods for spin net and spin foam models
• Ten questions on Group Field Theory (and their tentative answers)
• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• Just Renormalizable TGFT's on U(1)^d with Gauge Invariance
• Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions
• Lost in Translation: Topological Singularities in Group Field Theory
• Some classes of renormalizable tensor models
• Zakopane lectures on loop gravity
• A new vacuum for Loop Quantum Gravity
• Can Spacetime be a Condensate?
• Radiative corrections in the Boulatov-Ooguri tensor model: The 2-point function
• The 1/N expansion of multi-orientable random tensor models
• Holonomy Spin Foam Models: Definition and Coarse Graining
• Group field theories for all loop quantum gravity
• Constructive Matrix Theory
• Quantum cosmology from quantum gravity condensates: cosmological variables and lattice-refined dynamics
• Group field theory as the 2nd quantization of Loop Quantum Gravity
• Melonic phase transition in group field theory
• Towards a double-scaling limit for tensor models: probing sub-dominant orders
• The Double Scaling Limit in Arbitrary Dimensions: A Toy Model
• Self-dual noncommutative ϕ^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
• Closed equations of the two-point functions for tensorial group field theory
• Coarse graining of spin net models: dynamics of intertwiners
• Two and four-loop $\beta$-functions of rank 4 renormalizable tensor field theories
• Progress in solving a noncommutative quantum field theory in four dimensions
• Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory
• Discrete Renormalization Group for SU(2) Tensorial Group Field Theory
• Beta functions of $U(1{)}^d$ gauge invariant just renormalizable tensor models
• Addendum to "A Renormalizable 4-Dimensional Tensor Field Theory"
• Loop Quantum Gravity Vacuum with Nondegenerate Geometry
• Holography and the scale-invariance of density fluctuations
• The Tensor Theory Space
• Constructive Tensor Field Theory
• Dynamical Quantum Geometry (DQG Programme)
• Constructive Tensor Field Theory: The $T_3^4$ Model
• An analysis of the intermediate field theory of $T^4$ tensor model
• Spacetime condensation in (2+1)-dimensional CDT from a Horava-Lifshitz minisuperspace model
• The group field theory approach to quantum gravity: some recent results
• Perturbing a quantum gravity condensate
• Solvable 4D noncommutative QFT: phase transitions and quest for reflection positivity
• Effective equations for GFT condensates from fidelity
• Next-to-leading order in the large $N$ expansion of the multi-orientable random tensor model

Cited by in corpus (33)

• Chemical tagging with APOGEE: Discovery of a large population of N-rich stars in the inner Galaxy
• Flowing in Group Field Theory Space: a Review
• Functional Renormalisation Group analysis of Tensorial Group Field Theories on ${\mathbb{R}}^d$
• Functional Renormalization Group Approach for Tensorial Group Field Theory: A Rank-6 Model with Closure Constraint
• Renormalizable Group Field Theory beyond melonic diagrams: an example in rank four
• Functional Renormalization Group analysis of rank 3 tensorial group field theory: The full quartic invariant truncation
• Why are tensor field theories asymptotically free?
• Nonperturbative renormalization group beyond melonic sector: The Effective Vertex Expansion method for group fields theories
• Constructive Tensor Field Theory
• Large $N$ Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension $d\ge 2$
• Functional renormalization group for the $U(1)$-$T_5^6$ tensorial group field theory with closure constraint
• Constructive tensor field theory: The $T_4^4$ model
• An elementary introduction to loop quantum gravity
• Constructive Tensor Field Theory: The $T_3^4$ Model
• Exact Renormalisation Group Equations and Loop Equations for Tensor Models
• Symmetry breaking in tensor models
• Progress in the solving nonperturbative renormalization group for tensorial group field theory
• Ward-constrained melonic renormalization group flow for the rank-four ${\varphi }^6$ tensorial group field theory
• Unitary symmetry constraints on tensorial group field theory renormalization group flow
• Phase Transition in Tensor Models
• Renormalization of a tensorial field theory on the homogeneous space SU(2)/U(1)
• Reliability of the local truncations for the random tensor models renormalization group flow
• Flowing in discrete gravity models and Ward identities: A review
• Polchinski's exact renormalisation group for tensorial theories: Gaussian universality and power counting
• Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices
• Renormalization and Hopf Algebraic Structure of the 5-Dimensional Quartic Tensor Field Theory
• Nontrivial UV behavior of rank-4 tensor field models for quantum gravity
• No Ward-Takahashi identity violation for an Abelian tensorial group field theories with closure constraint
• Stochastic dynamics for Group Field Theories
• Large-$d$ behavior of the Feynman amplitudes for a just-renormalizable tensorial group field theory
• An $OSp$ extension of Canonical Tensor Model
• Anomalous higher order Ward identities in tensorial group field theories without closure constraint
• Constraint algebra of general relativity from a formal continuum limit of canonical tensor model