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A renormalizable 4-dimensional tensor field theory. (English) Zbl 1261.83016
Commun. Math. Phys. 318, No. 1, 69-109 (2013); addendum ibid. 322, No. 3, 957-965 (2013).
Summary: We prove that an integrated version of the Gurau colored tensor model [R. Gurau, Commun. Math. Phys. 304, No. 1, 69–93 (2011; Zbl 1214.81170)] 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.

MSC:
83C45 Quantization of the gravitational field
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
81S40 Path integrals in quantum mechanics
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