Scaling behaviour of three-dimensional group field theory

Abstract

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 Boulatov's 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.

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• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• EPRL/FK Group Field Theory
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• Phase Transition in Dually Weighted Colored Tensor Models
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• Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models
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• Spinfoams in the holomorphic representation
• Generalization of the Bollobás-Riordan polynomial for tensor graphs
• Algebraic structures in quantum gravity
• Critical Overview of Loops and Foams
• Bubbles and jackets: new scaling bounds in topological group field theories
• Bubble divergences: sorting out topology from cell structure
• A Deformed Poincare Invariance for Group Field Theories
• Effective Hamiltonian Constraint from Group Field Theory
• Constructive Tensor Field Theory
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• Constructive tensor field theory: The $T_4^4$ model
• Constructive Tensor Field Theory: The $T_3^4$ Model
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• Quantum fluctuations in the effective relational GFT cosmology
• Spinors and Voros star-product for Group Field Theory: First Contact
• Comment on `Lost in Translation: Topological Singularities in Group Field Theory'
• A simple proof of orientability in colored group field theory
• Discrete and continuum third quantization of Gravity
• Renormalization of group field theories for quantum gravity: new scaling results and some suggestions
• Emergent matter from 3d generalised group field theories
• Ward-Takahashi identities for the colored Boulatov model
• Minimizers of the dynamical Boulatov model
• Classical Group Field Theory
• Spheres are rare
• Translation-Invariant Noncommutative Renormalization
• A Diagrammatic Equation for Oriented Planar Graphs
• Intermediate Field Representation for Positive Matrix and Tensor Interactions
• Constructive Renormalization of the $2$-dimensional Grosse-Wulkenhaar Model
• From Dimensional Reduction of 4d Spin Foam Model to Adding Non-Gravitational Fields to 3d Spin Foam Model
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