Towards a double scaling limit for tensor models: probing sub-dominant orders

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, γ NLO = 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.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. size N. These were proposed already in the early 1990s as an attempt to reproduce, in d 3 and d 4 , 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 d 2 by Liouville gravity). Such tensor models describe discrete geometry in purely combinatorial terms (the natural notion of distance being the graph distance on each cellular complex). Their Feynman amplitudes can thus be understood in terms of the Regge action for discrete gravity evaluated on equilateral triangulations. Moreover, the perturbative sum over Feynman diagrams coincides with the definition of quantum gravity given by the (Euclidean) dynamical triangulations (EDTs) 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 (TGFTs) proper field theories, with richer state spaces (with generic states being superpositions of spin networks) and quantum amplitudes, given by simplicial path integrals and spin foam models. It is these richer field theories, building up on the understanding of quantum geometry obtained in loop quantum gravity, that we believe offer the most promising candidates for a complete quantum theory of gravity. Actually, with the appropriate data and constructions [8,13], TGFTs provide what can be argued to be the best fundamental definition of covariant loop quantum gravity dynamics, adapted to a simplicial context. In particular, TGFTs provide loop gravity and spin foams, as well as dynamical triangulations, with powerful, analytic field theoretic tools, suited to study of non-perturbative physics, the dynamics of many degrees of freedom, and the extraction of effective continuum geometry.While a main motivation for TGFTs is quantum gravity, this is not their only reason of interest. TGFTs can be seen, more generally, as a new class of quantum field theories, posing interesting mathematical challenges, in particular from the axiomatic and renormaliza...

Conformality of 1/N corrections in Sachdev-Ye-Kitaev-like models

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Towards phase transitions between discrete and continuum quantum spacetime from the renormalization group

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