Tensorial methods and renormalization in Group Field Theories

Abstract

In this thesis, we study the structure of Group Field Theories (GFTs) from the point of view of renormalization theory. Such quantum field theories are found in approaches to quantum gravity related to Loop Quantum Gravity (LQG) on the one hand, and to matrix models and tensor models on the other hand. They model quantum space-time, in the sense that their Feynman amplitudes label triangulations, which can be understood as transition amplitudes between LQG spin network states. The question of renormalizability is crucial if one wants to establish interesting GFTs as well-defined (perturbative) quantum field theories, and in a second step connect them to known infrared gravitational physics. Relying on recently developed tensorial tools, this thesis explores the GFT formalism in two complementary directions. First, new results on the large cut-off expansion of the colored Boulatov-Ooguri models allow to explore further a non-perturbative regime in which infinitely many degrees of freedom contribute. The second set of results provide a new rigorous framework for the renormalization of so-called Tensorial GFTs (TGFTs) with gauge invariance condition. In particular, a non-trivial 3d TGFT with gauge group SU(2) is proven just-renormalizable at the perturbative level, hence opening the way to applications of the formalism to (3d Euclidean) quantum gravity.

PhD thesis, 229 pages, many figures. Partly based on arXiv:1104.5158, arXiv:1203.5082, arXiv:1207.6734 and arXiv:1303.6772

References in corpus (7)

• Quantum Nature of the Big Bang
• LQG vertex with finite Immirzi parameter
• Random tensor models in the large N limit: Uncoloring the colored tensor models
• Perfect discretization of reparametrization invariant path integrals
• Holonomy Spin Foam Models: Definition and Coarse Graining
• Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
• Diffeomorphism symmetry in quantum gravity models

Cited by in corpus (53)

• Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model
• The Tensor Track, III
• Flowing in Group Field Theory Space: a Review
• The Multi-Orientable Random Tensor Model, 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
• Functional Renormalisation Group analysis of a Tensorial Group Field Theory on ${\mathbb{R}}^3$
• Asymptotic safety in three-dimensional SU(2) Group Field Theory: evidence in the local potential approximation
• Closed equations of the two-point functions for tensorial group field theory
• Renormalizable Group Field Theory beyond melonic diagrams: an example in rank four
• Generalized quantum gravity condensates for homogeneous geometries and cosmology
• Tensorial Gross-Neveu models
• Renormalization of an Abelian Tensor Group Field Theory: Solution at Leading Order
• Random Tensors and Quantum Gravity
• Why are tensor field theories asymptotically free?
• SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry
• Nonperturbative renormalization group beyond melonic sector: The Effective Vertex Expansion method for group fields theories
• Constructive tensor field theory: The $T_4^4$ model
• Group Field Theory in dimension four minus epsilon
• Functional renormalization group for the $U(1)$-$T_5^6$ tensorial group field theory with closure constraint
• An elementary introduction to loop quantum gravity
• Constructive Tensor Field Theory: The $T_3^4$ Model
• Topology in colored tensor models via crystallization theory
• Exact Renormalisation Group Equations and Loop Equations for Tensor Models
• Physical states in the canonical tensor model from the perspective of random tensor networks
• Mean-field phase transitions in TGFT quantum gravity
• The Complete Barrett-Crane Model and its Causal Structure
• Phase transitions in TGFT: Functional renormalization group in the cyclic-melonic potential approximation and equivalence to O$(N)$ models
• Progress in the solving nonperturbative renormalization group for tensorial group field theory
• Parametric Representation of Rank d Tensorial Group Field Theory: Abelian Models with Kinetic Term ${\sum }_s|p_s|+\mu$
• Continuous point symmetries in Group Field Theories
• Renormalization of group field theories for quantum gravity: new scaling results and some suggestions
• Flowing in discrete gravity models and Ward identities: A review
• Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom
• Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices
• Nontrivial UV behavior of rank-4 tensor field models for quantum gravity
• Surgery in colored tensor models
• On the large N limit of the Schwinger-Dyson equation of tensor field theory
• QFT with Tensorial and Local Degrees of Freedom: Phase Structure from Functional Renormalization
• Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs
• Scalar Cosmological Perturbations from Quantum Gravitational Entanglement
• Intertwiner Entanglement Excitation and Holonomy Operator
• Stochastic dynamics for Group Field Theories
• Large-$d$ behavior of the Feynman amplitudes for a just-renormalizable tensorial group field theory
• A Solvable Tensor Field Theory
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• A power counting theorem for a $p^{2a}{\varphi }^4$ tensorial group field theory
• Scale invariance beyond criticality within the mean-field analysis of tensorial field theories
• Quantum Gravity, Hydrodynamics and Emergent Cosmology: A Collection of Perspectives
• New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications
• Graph calculus and the disconnected-boundary Schwinger-Dyson equations of quartic tensor field theories
• Quartic Tensor Models
• 3D Quantum Gravity from Holomorphic Blocks