Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions

Carrozza, Sylvain; Oriti, Daniele; Rivasseau, Vincent

doi:10.1007/s00220-014-1954-8

Cited by 110 publications

(175 citation statements)

References 115 publications

(191 reference statements)

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“…We focus on models whose kinetic operator is the Laplace-Beltrami operator on SU(2) 4 , together with a 'mass term'. A motivation for this choice is that the presence of the Laplacian seems to be required by GFT renormalisation [65][66][67][68][69]. The equation (5.21) for the function ξ then becomes (setting the g ′′ I which are arbitrary equal to the identity)…”

Section: Jhep06(2014)013mentioning

confidence: 99%

“…In Lorentzian signature, imposing them as we did for the Barrett-Crane prescription used in section 5.2 means that we now require 66) where X 0 ∈ H 3 . The GFT field then becomes a function on four copies of the homogeneous space SL(2, C)/SU(2), which is 3-dimensional hyperbolic space, or Hom(2) as a group manifold (see appendix C for a discussion of this group and its geometry).…”

Section: Jhep06(2014)013mentioning

confidence: 99%

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Homogeneous cosmologies as group field theory condensates

Gielen

Oriti

Sindoni

2014

J. High Energ. Phys.

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138

256

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We give a general procedure, in the group field theory (GFT) formalism for quantum gravity, for constructing states that describe macroscopic, spatially homogeneous universes. These states are close to coherent (condensate) states used in the description of Bose-Einstein condensates. The condition on such states to be (approximate) solutions to the quantum equations of motion of GFT is used to extract an effective dynamics for homogeneous cosmologies directly from the underlying quantum theory. The resulting description in general gives nonlinear and nonlocal equations for the 'condensate wavefunction' which are analogous to the Gross-Pitaevskii equation in Bose-Einstein condensates. We show the general form of the effective equations for current quantum gravity models, as well as some concrete examples. We identify conditions under which the dynamics becomes linear, admitting an interpretation as a quantum-cosmological Wheeler-DeWitt equation, and give its semiclassical (WKB) approximation in the case of a kinetic term that includes a Laplace-Beltrami operator. For isotropic states, this approximation reproduces the classical Friedmann equation in vacuum with positive spatial curvature. We show how the formalism can be consistently extended from Riemannian signature to Lorentzian signature models, and discuss the addition of matter fields, obtaining the correct coupling of a massless scalar in the Friedmann equation from the most natural extension of the GFT action. We also outline the procedure for extending our condensate states to include cosmological perturbations. Our results form the basis of a general programme for extracting effective cosmological dynamics directly from a microscopic non-perturbative theory of quantum gravity.

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Section: Jhep06(2014)013mentioning

confidence: 99%

Section: Jhep06(2014)013mentioning

confidence: 99%

Homogeneous cosmologies as group field theory condensates

Gielen

Oriti

Sindoni

2014

J. High Energ. Phys.

Self Cite

138

256

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“…Furthermore, GFT renormalization is also one of the two main strategies to define and study the renormalization of spin foam models, the other being through a generalised lattice gauge theory approach [22]. Most work in this direction has concerned a particular class of GFTs, called Tensorial Group Field Theories (TGFT's) [23][24][25][26][27][28][29][30][31][32][33], which incorporate recent advances in the statistical analysis of colored tensor models [34][35][36][37]. In particular, in TGFT framework, fields are endowed with tensorial transformation properties under the action of the group itself.…”

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confidence: 99%

Functional Renormalization Group analysis of a Tensorial Group Field Theory on $\mathbb{R}^3$

Geloun

Martini

Oriti

2015

EPL

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-We study a model of Tensorial Group Field Theory (TGFT) on R 3 from the point of view of the Functional Renormalisation Group. This is the first attempt to apply a renormalisation procedure to a TGFT model defined over a non-compact group manifold. IR divergences (with respect to the metric on R) coming from the non-compactness of the group are regularised via compactification, and a thermodynamic limit is then taken. We identify then IR and UV fixed points of the RG flow and find strong hints of a phase transition of the TGFT system from a symmetric to a broken or condensate phase in the IR.Introduction. -Group Field Theories [1-5] (GFTs) are a particular class of quantum field theories with fields defined over a group manifold and characterised by combinatorially non-local interaction terms. This combinatorial non-locality makes the Feynman diagrams of the theory stranded diagrams dual to cellular complexes (simplicial complexes in the simplest constructions) [1,3]. GFT's historically were born from an attempt to generalise matrix models [6] of 2d gravity to higher dimensions in the form of tensor models [7][8][9]. These models were soon enriched with group theoretic data in such a way that the Feynman amplitudes of specific GFT models coincide with state sum models of topological field theory [9,10]. A connection between GFTs and Loop Quantum Gravity (LQG) [11,12] was immediately pointed out [13] at the level of quantum states, and later enforced at the level of the quantum dynamics. In fact, it was later shown [14] that GFTs provide a formal (complete) definition of spin foams models, a covariant formalism for LQG. For GFTs (and spin foam models) endowed with a discrete geometric interpretation (which requires appropriate group theoretic data and suitable choices of dynamics) there is also a direct link with simplicial quantum gravity path integrals, first manifested in the semiclassical analysis of GFT Feynman (spin foam) amplitudes, where one recovers the Regge action [15], and then shown to be generically manifest in the flux repre-

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“…On the other hand, using (23) in Lemma 1, we can write the following expression for a partially rescaled polynomial U od G ,…”

Section: Definition 6 (Sets Of Faces) For All S ⊂ Gmentioning

confidence: 99%

“…Thus (28) holds. In order to find the second equality for U ev; (ρ) G (29), we use the same decomposition (23) of Lemma 1 and (27).…”

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confidence: 99%

Parametric representation of rank d tensorial group field theory: Abelian models with kinetic term ∑sps+μ

Geloun

Toriumi

2015

Journal of Mathematical Physics

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We consider the parametric representation of the amplitudes of Abelian models in the so-called framework of rank d Tensorial Group Field Theory. These models are called Abelian because their fields live on U (1) D . We concentrate on the case when these models are endowed with particular kinetic terms involving a linear power in momenta. New dimensional regularization and renormalization schemes are introduced for particular models in this class: a rank 3 tensor model, an infinite tower of matrix models φ 2n over U (1), and a matrix model over U (1) 2 . For all divergent amplitudes, we identify a domain of meromorphicity in a strip determined by the real part of the group dimension D. From this point, the ordinary subtraction program is applied and leads to convergent and analytic renormalized integrals. Furthermore, we identify and study in depth the Symanzik polynomials provided by the parametric amplitudes of generic rank d Abelian models. We find that these polynomials do not satisfy the ordinary Tutte's rules (contraction/deletion). By scrutinizing the "face"-structure of these polynomials, we find a generalized polynomial which turns out to be stable only under contraction.

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Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions

Cited by 110 publications

References 115 publications

Homogeneous cosmologies as group field theory condensates

Homogeneous cosmologies as group field theory condensates

Functional Renormalization Group analysis of a Tensorial Group Field Theory on $\mathbb{R}^3$

Parametric representation of rank d tensorial group field theory: Abelian models with kinetic term ∑sps+μ

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