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A Renormalizable 4-Dimensional Tensor Field Theory

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Abstract

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 space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the \({\phi^6}\) rather than of the \({\phi^4}\) type, since two different \({\phi^6}\)-type interactions are log-divergent, 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 log-divergent \({(\int \phi^2)^2}\) term, which can be interpreted as the generation of a scalar matter field out of pure gravity.

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Authors and Affiliations

  1. Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada

    Joseph Ben Geloun

  2. Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris-Sud XI, 91405, Orsay, France

    Vincent Rivasseau

  3. International Chair in Mathematical Physics and Applications, (ICMPA-UNESCO Chair), University of Abomey-Calavi, 072B.P.50, Cotonou, Rep. of Benin

    Joseph Ben Geloun

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  1. Joseph Ben Geloun
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Correspondence to Joseph Ben Geloun.

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Communicated by A. Connes

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Geloun, J.B., Rivasseau, V. A Renormalizable 4-Dimensional Tensor Field Theory. Commun. Math. Phys. 318, 69–109 (2013). https://doi.org/10.1007/s00220-012-1549-1

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