Abstract
The definition of a double scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the colored tensor models. We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent, γnlo = 3/2, signals a departure from the leading order behavior. Both pieces of information provide clues for a non-trivial double scaling limit, for which we put forward some precise conjecture.
Highlights
Growing evidence is being accumulated for the (Tensorial) Group Field Theory ((T)GFT) formalism [1, 2, 3, 4] as a promising overarching framework for a quantum theory of gravity; one that is able to incorporate aspects of several current discrete approaches within a powerful quantum field theory setting
The perturbative sum over Feynman diagrams coincides with the definition of quantum gravity given by the (Euclidean) Dynamical triangulations approach [10], after appropriate identification of their respective parameter sets
When one enriches the combinatorics of tensor models with the group-theoretic data suggested by Loop Quantum Gravity [9], Spin Foam models [16] and simplicial geometry [11], one obtains (Tensorial) Group Field Theories: proper field theories, with richer state spaces and quantum amplitudes, given by simplicial path integrals and spin foam models
Summary
Growing evidence is being accumulated for the (Tensorial) Group Field Theory ((T)GFT) formalism [1, 2, 3, 4] as a promising overarching framework for a quantum theory of gravity; one that is able to incorporate aspects of several current discrete approaches within a powerful quantum field theory setting. These were proposed already in the early ’90s as an attempt to reproduce, in 3d and 4d, the successes of the matrix model formalism in defining both a controllable sum over topologies and a theory of random discrete geometries with a nice continuum limit (given in 2d by Liouville gravity) Such tensor models describe discrete geometry is purely combinatorial terms (the natural notion of distance being the graph distance on each cellular complex). While for higher–dimensional models, one should not expect two parameters to control the full series, the two parameters in the simplest iid model should at least allow one to extract a broader subclass of graphs than just the leading order graphs They should capture better the statistical and topological properties of the sum over complexes and reveal new critical behaviour, as has been achieved in matrix models [58, 59].
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