Abstract This chapter presents the double scaling limit of quartic tensor models. In the first part of this chapter it is shown show that the melonic family in the quartic melonic tensor model can be analytically resumed and that this resummation is encoded in a translation of the intermediate matrix fields to a nontrivial vacuum. The most striking consequence of this translation is that one can now show that the continuum limit of tensor models corresponds to a genuine phase transition (in the field theory sense) associated to a symmetry breaking. In the second part of this chapter the intermediate field representation is used to derive the double scaling limit of the quartic melonic model.
We explore how matrix bootstrap techniques can be used to constrain matrix and tensor models at finite
N, where
N is the dimension of the matrix/tensor, taking a Gaussian model with a quartic interaction as example. For matrix models, we find further evidence that bounds do not depend explicitly on
N, but rather on properties of multi-trace expectation values. For tensor models, the structure of the Schwinger-Dyson equations allow for bounds that vary as a function of
N, admitting a broader scan of the parameter space of the theory. In the latter case, we find novel bounds on the two-point function as a function of the quartic coupling of the theory.
Abstract 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.
We study the spectrum of the large
N quantum field theory of bosonic rank-3 tensors, the quartic interactions of which are such that the perturbative expansion is dominated by the melonic diagrams. We use the Schwinger-Dyson equations to determine the scaling dimensions of the bilinear operators of arbitrary spin. Using the fact that the theory is renormalizable in
d=4, we compare some of these results with the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator ${\ensuremath{\phi}}^{abc}{\ensuremath{\phi}}^{abc}$ is complex for
d<4: the large
N fixed point in $d=4\ensuremath{-}\ensuremath{\epsilon}$ has complex values of the couplings for some of the
O(N)3 invariant operators. We show that a similar phenomenon holds in the
O(N)2 symmetric theory of a matrix field ${\ensuremath{\phi}}^{ab}$, where the double-trace operator has a complex coupling in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. We also study the spectra of bosonic theories of rank-$q\ensuremath{-}1$ tensors with ${\ensuremath{\phi}}^{q}$ interactions. In dimensions
d>1.93, there is a critical value of
q, above which we have not found any complex scaling dimensions. The critical value is a decreasing function of
d, and it becomes 6 in $d\ensuremath{\approx}2.97$. This raises a possibility that the large
N theory of rank-5 tensors with sextic potential has an IR fixed point which is free of perturbative instabilities for
2.97<d<3. This theory may be studied using renormalized perturbation theory in $d=3\ensuremath{-}\ensuremath{\epsilon}$.
This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with
O(N)3 symmetry and long-range propagator. The existence of a non-perturbative fixed point in any
d at large
N is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large
N. We then compute
1/N corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the
F-theorem. The next part of the thesis investigates sextic tensor field theories in rank
3 and
5. In rank
3, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank
5 is the Gaussian one. For the rank
3 model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large
N fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large
N limit. We prove that models with tensors in an irreducible representation of
O(N) or
Sp(N) in rank
5 indeed admit a large
N limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.
Tensor models are natural generalizations of matrix models. The interactions and observables in the case of unitary invariant models are generalizations of matrix traces. Some notable interactions in the literature include the melonic ones, the tetrahedral one as well as the planar ones in rank three, or necklaces in even ranks. Here we introduce generalized melonic interactions which generalize the melonic and necklace interactions. We characterize them as tree-like gluings of quartic interactions. We also completely characterize the Feynman graphs which contribute to the large
N limit. For a subclass of generalized melonic interactions called totally unbalanced interactions, we prove that the large
N limit is Gaussian and therefore the Feynman graphs are in bijection with trees. This result further extends the class of tensor models which fall into the Gaussian universality class. Another key aspect of tensor models with generalized melonic interactions is that they can be written as matrix models without increasing the number of degrees of freedom of the original tensor models. In the case of totally unbalanced interactions, this new matrix model formulation in fact decreases the number of degrees of freedom, meaning that some of the original degrees of freedom are effectively integrated. We then show how the large
N Gaussian behavior can be reproduced using a saddle point analysis on those matrix models.