Abstract
Tensor models generalize the matrix-model approach to 2-dimensional quantum gravity to higher dimensions. Some models allowing a 1/N expansion have been explored, most of them generating branched-polymer geometries. Recently, enhancements yielding an additional 2d pure gravity (planar) phase and an intermediate regime of proliferating baby universes have been found. It remains an open issue to find models escaping these lower-dimensionality universality classes.Here we analyse the dominant regime and critical behaviour of a range of new models which are candidates for such effective geometries, in particular interactions based on the utility graph K3,3. We find that, upon proper enhancement, the two-phase structure of a branched-polymer and a 2d gravity regime is the common case in U(N)-invariant rank D=4 tensor models of small orders. Not only the well known so-called necklace interactions but also K3,3-type interactions turn out as the source for the planar regime. We give a systematic account of the enhancement scaling, the counting of leading-order diagrams and the multi-critical behaviour of a wide range of interactions, in particular for all order-6 interactions of rank 3 and 4. These findings support the claim of universality of such mixtures of branched-polymer and planar diagrams at criticality. In particular, this hints at the necessity to consider new ingredients, or interactions of higher order and rank, in order to obtain higher dimensional continuum geometry from tensor models.
Highlights
Tensor models [1,2,3], which generalize matrix models [4] to higher dimensions, were introduced as a non-perturbative approach to quantum gravity, and an analytical tool to explore random geometries in dimension higher than two
The accurate scaling sB leading to a defined and non-trivial 1/N expansion is deduced from the number of faces (7) of leading-order diagrams, and the critical exponent γ is given by the critical behaviour of the resulting leading-order two-point function, expected to be an indicator for the continuum limit as explained
The usual three regimes appear in this case, as we show in Subsection III C: for two vanishing couplings, this reduces to a usual matrix model, in the universality class of pure 2d gravity (γ = −1/2); for g2 = g3 = g4, there is a destructive interference between the three kind of matrices which reduces the critical behaviour to a square root singularity, with exponent γ = 1/2
Summary
Tensor models [1,2,3], which generalize matrix models [4] to higher dimensions, were introduced as a non-perturbative approach to quantum gravity, and an analytical tool to explore random geometries in dimension higher than two. All models considered here belong to the universality classes of branched polymers (γ = 1/2), of 2d pure gravity (planar diagrams, with γ = −1/2), or to an intermediate transitional regime of “proliferating baby universes”, with critical exponent γ = 1/3, through various combinations of enhanced interactions This last regime was discovered in matrix models by considering modified Einstein-Hilbert actions with higher curvature terms, resulting into multi-trace interactions [33,34,35,36,37]. The bijection with embedded diagrams shows clearly that the influence of non-planarities in the interactions on the structure of leading order diagrams is not strong enough by itself to change their large scale properties, and the corresponding continuum limit This could be taken as an argument that U (N )D-invariant tensor models do not cover new regimes.
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