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27
A Renormalizable 4dimensional Tensor Field Theory
, 2012
"... We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on U(1)4 is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of spacetime in 4D Euclidean gravity and is the first example of ..."
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Cited by 42 (8 self)
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We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on U(1)4 is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of spacetime in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are fourstranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the φ6 rather than of the φ4 type, since two different φ6type interactions are logdivergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous logdivergent ( φ2)2 term, which can be interpreted as the generation of a scalar matter field out of pure gravity.
Borel summability and the non perturbative 1/N expansion of arbitrary quartic tensor models
, 2014
"... We extend the proof of Borel summability of melonic quartic tensor models to tensor models with arbitrary quartic interactions. This extension requires a new version of the loop vertex expansion using several species of intermediate fields and new bounds based on CauchySchwarz inequalities. The Bo ..."
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Cited by 5 (1 self)
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We extend the proof of Borel summability of melonic quartic tensor models to tensor models with arbitrary quartic interactions. This extension requires a new version of the loop vertex expansion using several species of intermediate fields and new bounds based on CauchySchwarz inequalities. The Borel summability is proven to be uniform as the tensor size becomes large. Furthermore, we show that the 1/N expansion of any quartic tensor model can be performed at the constructive level, that is we show that every cumulant is a sum of explicit terms up to some order plus a rest term which is an analytic function in the coupling constant in a cardioid domain of the complex plane and which is suppressed in 1/N.
Closed equations of the twopoint functions for tensorial group field theory
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Double Scaling in Tensor Models with a Quartic Interaction
"... In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D ..."
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In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the Ddimensional sphere, closely related to the “stacked ” triangulations. For D < 6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the Ddimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp contrast with random matrix models, this double scaling limit is stable. Apart from its unexpected upper critical dimension 6, it displays a singularity at fixed distance from the origin and is clearly the first step in a richer set of yet to be discovered multiscaling limits.
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, 2013
"... The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which explictly computes the Borel sum of Feynman perturbation series. This LVE relies in a crucial way on symmetric tree weights which define a measure on the set of spanning trees of any connected graph. In this paper we gener ..."
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The Loop Vertex Expansion (LVE) is a quantum field theory (QFT) method which explictly computes the Borel sum of Feynman perturbation series. This LVE relies in a crucial way on symmetric tree weights which define a measure on the set of spanning trees of any connected graph. In this paper we generalize this method by defining new tree weights. They depend on the choice of a partition of a set of vertices of the graph, and when the partition is nontrivial, they are no longer symmetric under permutation of vertices. Nevertheless we prove they have the required positivity property to lead to a convergent LVE; in fact we formulate this positivity property precisely for the first time. Our generalized tree weights are inspired by the BrydgesBattleFederbush work on cluster expansions and could be particularly suited to the computation of connected functions in QFT. Several concrete examples are explicitly given.
Coherent states for quantum gravity: towards collective variables
, 2012
"... We investigate the construction of coherent states for quantum theories of connections based on graphs embedded in a spatial manifold, as in loop quantum gravity. We discuss the many subtleties of the construction, mainly related to the diffeomorphism invariance of the theory. Aiming at approximatin ..."
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We investigate the construction of coherent states for quantum theories of connections based on graphs embedded in a spatial manifold, as in loop quantum gravity. We discuss the many subtleties of the construction, mainly related to the diffeomorphism invariance of the theory. Aiming at approximating a continuum geometry in terms of discrete, graphbased data, we focus on coherent states for collective observables characterizing both the intrinsic and extrinsic geometry of the hypersurface, and we argue that one needs to revise accordingly the more local definitions of coherent states considered in the literature so far. In order to clarify the concepts introduced, we work through a concrete example that we hope will be useful to applying coherent state techniques to cosmology. 1
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"... We study the polynomial Abelian or U(1)d Tensorial Group Field Theories equipped with a gauge invariance condition in any dimension d. We prove the just renormalizability at all orders of perturbation of the ϕ46 and ϕ65 random tensor ..."
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We study the polynomial Abelian or U(1)d Tensorial Group Field Theories equipped with a gauge invariance condition in any dimension d. We prove the just renormalizability at all orders of perturbation of the ϕ46 and ϕ65 random tensor