Just Renormalizable TGFT’s on U(1) d with Gauge Invariance

Samary, Dine Ousmane; Vignes-Tourneret, Fabien

doi:10.1007/s00220-014-1930-3

Cited by 71 publications

(125 citation statements)

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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.…”

mentioning

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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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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“…Reciprocally, take f ∈ F • (G, S)/S, then, by definition ∃f 0 ∈ F • (G, S) such that f 0 /S = f and f 0 is not empty, since by definition there must exist l ∈ L(S) and l ∈ f 0 . Note also that (24) does not depend on the type of contraction.…”

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

confidence: 85%

“…-To prove (24), one must notice that we can associate with each element f ∈ F • (G, S) a line l f in G/S which is not touched by the (hard) contraction of S. This line ensures the oneto-one correspondence between an element in F • (G, S) and an element in F • (G, S)/S after (hard) contraction. Indeed, take f ∈ F • (G, S), and ∃l f ∈ L(G/S) such that l f ∈ f .…”

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

confidence: 99%

“…One must use the bijection relation (24) in Lemma 1 to map F int (G; S) to F int (G; S)/S and we can write U od/ev G;G/S := f ∈F int /S A od/ev f . We insert (36), (37) and (39) in (35), and get the expansion…”

Section: Lemma 3 (Factorization Of Amentioning

confidence: 99%

“…In particular, the number of internal faces F int (G) must be dissected. This number F int (G) of a connected graph G has been worked out in [24]. We have for a connected graph G and in our case:…”

Section: Meromorphic Structure Of the Regularized Amplitudesmentioning

confidence: 99%

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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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The tensor track, III

Rivasseau

2013

Fortschritte der Physik

100

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We provide an informal up-to-date review of the tensor track approach to quantum gravity. In a long introduction we describe in simple terms the motivations for this approach. Then the many recent advances are summarized, with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion, Osterwalder-Schrader positivity...) which, while important for the tensor track program, are not detailed in the usual quantum gravity literature. We list open questions in the conclusion and provide a rather extended bibliography.

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Just Renormalizable TGFT’s on U(1) d with Gauge Invariance

Cited by 71 publications

References 30 publications

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

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

The tensor track, III

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