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Colored Tensor Models  a Review
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2012
"... Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor mo ..."
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Cited by 43 (6 self)
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Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating twodimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger–Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), nontrivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.
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 (11 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.
3D Tensor Field Theory: Renormalization and Oneloop βfunctions
, 2012
"... We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hepth]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The oneloop γ and βfunctions of the model are also determined. We find that the model with a unique coupling const ..."
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Cited by 29 (4 self)
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We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hepth]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The oneloop γ and βfunctions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave function renormalization is asymptotically free in the UV.
(Broken) Gauge Symmetries and Constraints in Regge Calculus
, 2009
"... We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly mat ..."
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Cited by 20 (17 self)
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We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so–called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints. We will argue that the long standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken.
Lorentzian LQG vertex amplitude
, 2007
"... We generalize a model recently proposed for Euclidean quantum gravity to the case of Lorentzian signature. The main features of the Euclidean model are preserved in the Lorentzian one. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. As in the Euclidean case, ..."
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Cited by 20 (2 self)
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We generalize a model recently proposed for Euclidean quantum gravity to the case of Lorentzian signature. The main features of the Euclidean model are preserved in the Lorentzian one. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. As in the Euclidean case, the model can be obtained from the Lorentzian BarrettCrane model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small BarberoImmirzi parameter limit.
Multiorientable Group Field Theory
 J. Phys. A
, 2012
"... Group Field Theories (GFT) are quantum field theories over group manifolds; they can be seen as a generalization of matrix models. GFT Feynman graphs are tensor graphs generalizing ribbon graphs (or combinatorial maps); these graphs are dual not only to manifolds. In order to simplify the topologica ..."
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Cited by 12 (1 self)
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Group Field Theories (GFT) are quantum field theories over group manifolds; they can be seen as a generalization of matrix models. GFT Feynman graphs are tensor graphs generalizing ribbon graphs (or combinatorial maps); these graphs are dual not only to manifolds. In order to simplify the topological structure of these various singularities, colored GFT was recently introduced and intensively studied since. We propose here a different simplification of GFT, which we call multiorientable GFT. We study the relation between multiorientable GFT Feynman graphs and colorable graphs. We prove that tadfaces and some generalized tadpoles are absent. Some Feynman amplitude computations are performed. A few remarks on the renormalizability of both multiorientable and colorable GFT are made. A generalization from threedimensional to fourdimensional theories is also proposed. Key words: combinatorics, group field theory, Feynman graphs, orientability ar X iv