Ward-constrained melonic renormalization group flow for the rank-four 
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 tensorial group field theory

Lahoche, Vincent; Samary, Dine Ousmane

doi:10.1103/physrevd.100.086009

Cited by 19 publications

(28 citation statements)

References 87 publications

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“…This aspect of the complex field in TGFT seems to have been overlooked so far in the literature [27][28][29][30][50][51][52][53][54][55][56][57][58][59][60][61]. Under some mild conditions it is straightforward to obtain the 2×2 trace of the inverse of the operator…”

Section: Frg Equation In Tgftmentioning

confidence: 99%

Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

Pithis

Thürigen

2020

J. High Energ. Phys.

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In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank r on U(1) we explicitly calculate beta functions and find equivalence with those of O(N) models but with an effective dimension flowing from r − 1 to zero. In the r − 1 dimensional regime, the equivalence to O(N) models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.

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Section: Frg Equation In Tgftmentioning

confidence: 99%

Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

Pithis

Thürigen

2020

J. High Energ. Phys.

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“…Heading towards a quantum gravity perspective, objects appearing in the Functional Renormalization Group [8] -or Ward-constrained flows [40,41]-are expected to be described in terms of graph-generated functionals studied here. Together with the boundary-completeness of the quartic-melonic models [53,Thm 1], this motivates us to study the geometric nature of the flow from a simple quartic model.…”

Section: Discussionmentioning

confidence: 99%

Graph Calculus and the Disconnected-Boundary Schwinger-Dyson Equations of Quartic Tensor Field Theories

Perez-Sanchez

2020

Math Phys Anal Geom

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Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing equations for all correlation functions with disconnected boundary, thus completing the Schwinger-Dyson pyramid for quartic melonic (‘pillow’-vertices) models in arbitrary rank. We first study finite group actions that are parametrised by graphs and build the graph calculus on a suitable quotient of the monoid algebra $\mathcal {A}[G]$ A [ G ] corresponding to a certain function space $\mathcal {A}$ A and to the free monoid G in finitely many graph variables; a derivative of an element of $\mathcal {A}[G]$ A [ G ] with respect to a graph yields its corresponding group action on $\mathcal {A}$ A . The present result and the graph calculus have three potential applications: the non-perturbative large-N limit of tensor field theories, the solvability of the theory by using methods that generalise the topological recursion to the TFT setting and the study of ‘higher dimensional maps’ via Tutte-like equations. In fact, we also offer a term-by-term comparison between Tutte equations and the present Schwinger-Dyson equations.

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“…We will set such terms to zero, which is self-consistent within our truncation scheme that requires that η cannot be too large. We highlight that the introduction of a regulator that depends on N entails a breaking of the previously referred to orthogonal/unitary symmetry of the model, implying the existence of non-trivial Ward identities that have been explored, e.g., in [49,55].…”

Section: The Modelmentioning

confidence: 99%

Universal critical behavior in tensor models for four-dimensional quantum gravity

Eichhorn

Lumma

Pereira

et al. 2020

J. High Energ. Phys.

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Four-dimensional random geometries can be generated by statistical models with rank-4 tensors as random variables. These are dual to discrete building blocks of random geometries. We discover a potential candidate for a continuum limit in such a model by employing background-independent coarse-graining techniques where the tensor size serves as a pre-geometric notion of scale. A fixed point candidate which features two relevant directions is found. The possible relevance of this result in view of universal results for quantum gravity and a potential connection to the asymptotic-safety program is discussed. I. THE CASE FOR UNIVERSAL, BACKGROUND-INDEPENDENT QUANTUM GRAVITYTo describe physically relevant spacetimes, such as black holes or Friedmann-Lemaitre-Robertson-Walker spacetimes, it is necessary to go beyond General Relativity, where these spacetimes feature singularities. Such singularities are expected to be resolved once quantum fluctuations of spacetime are properly accounted for. Yet, this is a particularly challenging task if it is to be compatible with the background-independence at the heart of our modern understanding of gravity. Backgroundindependence implies that no configuration of spacetime should be singled out a priori from all configurations that enter the path integral. This is incompatible with perturbative techniques around a fixed spacetime, which single out a special background to perturb around, and which do not provide a predictive quantum field theory of gravity. Instead, one is led to introduce an infinite number of independent local counterterms to cancel divergences at each loop order [1][2][3]. This motivates that background independence should be taken seriously in the search for an ultraviolet complete definition of the gravitational path integral. Thus, we aim at an implementation of the path integral without auxiliary geometric background structures 1 . A promising route to construct a background independent path integral consists of making a transition to discrete building blocks, as in dynamical triangulations [5], Regge calculus [6], matrix/tensor models [7-9], spin foams [10] and causal sets [11]. This allows to construct a discrete approximation of all random geometries (and potentially additional configurations with no interpreta- * eichhorn@cp3.sdu.dk † j.lumma@thphys.uni-heidelberg.de ‡ adpjunior@id.uff.br § a.sikandar@thphys.uni-heidelberg.de 1 An alternative route makes use of an auxiliary background structure at the technical level while ensuring the independence of physical results from this background structure, see, e.g., [4].tion as a spacetime geometry) that enter the path integral for quantum gravity. These discrete building blocks are typically not viewed as physical, "fundamental" building blocks of spacetime. Rather, they are auxiliary, unphysical entities, allowing to define a regularized path integral in analogy to lattice gauge theories for non-gravitational quantum field theories. One might object that quantum spacetime might be fundamentally discrete, wh...

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Ward-constrained melonic renormalization group flow for the rank-four ϕ6 tensorial group field theory

Cited by 19 publications

References 87 publications

Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

Graph Calculus and the Disconnected-Boundary Schwinger-Dyson Equations of Quartic Tensor Field Theories

Universal critical behavior in tensor models for four-dimensional quantum gravity

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