Double scaling limit for the O(N)3-invariant tensor model

We study the double scaling limit of the O(N)3-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the 1/N expansion as a finite sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the 1/N expansion. This leads to a double scaling limit which picks up contributions from all orders in the 1/N expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of the Feynman graphs, as well as an analytic analysis of the singularities of the relevant generating series.

Exploiting Spatial Information to Enhance DTI Segmentations via Spatial Fuzzy c-Means with Covariance Matrix Data and Non-Euclidean Metrics

A diffusion tensor models the covariance of the Brownian motion of water at a voxel and is required to be symmetric and positive semi-definite. Therefore, image processing approaches, designed for linear entities, are not effective for diffusion tensor data manipulation, and the existence of artefacts in diffusion tensor imaging acquisition makes diffusion tensor data segmentation even more challenging. In this study, we develop a spatial fuzzy c-means clustering method for diffusion tensor data that effectively segments diffusion tensor images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts. To retain the symmetry and positive semi-definiteness of diffusion tensors, the log and root Euclidean metrics are used to estimate the mean diffusion tensor for each cluster. The method exploits spatial contextual information and provides uncertainty information in segmentation decisions by calculating the membership values for assigning a diffusion tensor at one voxel to different clusters. A regularisation model that allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures is also proposed. Experiments on simulated images and real brain datasets from healthy and Spinocerebellar ataxia 2 subjects showed that the new method was more effective than conventional segmentation methods.

The dimensional reduction of linearized spin-2 theories invariant under transverse diffeomorphisms

Here we perform the Kaluza–Klein dimensional reduction from $$D+1$$ D + 1 to D dimensions of massless Lagrangians described by a symmetric rank-2 tensor and invariant under transverse differmorphisms (TDiff). They include the linearized Einstein–Hilbert theory, linearized unimodular gravity and scalar tensor models. We obtain simple expressions in terms of gauge invariant field combinations and show that unitarity is preserved in all cases. After fixing a gauge, the reduced model becomes a massive scalar tensor theory. We show that the diffeomorphism (Diff) symmetry, instead of TDiff, is a general feature of the massless sector of consistent massive scalar tensor models. We discuss some subtleties when eliminating Stückelberg fields directly at action level as gauge conditions. A non local connection between the massless sector of the scalar tensor theory and the pure tensor TDiff model leads to a parametrization of the non conserved source which naturally separates spin-0 and spin-2 contributions in the pure tensor theory. The case of curved backgrounds is also investigated. If we truncate the non minimal couplings to linear terms in the curvature, vector and scalar constraints require Einstein spaces as in the Diff and WTDiff (Weyl plus Diff) cases. We prove that our linearized massive scalar tensor models admit those curved background extensions.

Instability of complex CFTs with operators in the principal series

We prove the instability of d-dimensional conformal field theories (CFTs) having in the operator-product expansion of two fundamental fields a primary operator of scaling dimension h = $$ \frac{d}{2} $$ d 2 + i r, with non-vanishing r ∈ ℝ. From an AdS/CFT point of view, this corresponds to a well-known tachyonic instability, associated to a violation of the Breitenlohner-Freedman bound in AdSd+1; we derive it here directly for generic d-dimensional CFTs that can be obtained as limits of multiscalar quantum field theories, by applying the harmonic analysis for the Euclidean conformal group to perturbations of the conformal solution in the two-particle irreducible (2PI) effective action. Some explicit examples are discussed, such as melonic tensor models and the biscalar fishnet model.

From random matrices to random tensors

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

Random tensor models—the U(N)D-invariant model

In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.