This work proposes a bootstrapping with positivity methodology to study random
U(N)D invariant tensors in the large
N limit. As has been done for
U(N) invariant random matrices, we combine the Dyson-Schwinger equations and positivity constraints of moments to approximate the moments of such tensor models. As examples, we bootstrap the quartic and two hexic rank three tensor models. All models studied converge quickly, and for those which have known analytic formulae, they converge to such solutions. We conjecture new explicit formulae for all moments of the rank three quartic model and support this conjecture using bootstrapped results and explicit double-series computations with 'feyntensor'.
This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-
N melonic expansion of tensor models. For the first model, invariant under
U(N)3, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless
U(N)3 symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into
U(N2)×U(N/2)×U(N/2). A matrix model invariant under
U(N)×U(N2), with close properties, is also studied. For the other models, with symmetry groups
U(N)3 and
O(N)5, a non-melonic coupling (the "wheel") with an optimal scaling in
N drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-
N, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension
4/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.
We prove the redundancy of a family (the so-called melonic family) of large-
N moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator
C at large-
N has an expectation value that depends on
C exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.
Abstract In Giombi et al (2018 Phys. Rev. D 98 105005), a prismatic tensor model was introduced. We study here the diagrammatics and the double scaling limit of this model, using the intermediate field method. We explicitly exhibit the next-to-leading order Feynman graphs in the 1 / N expansion. Using appropriate combinatorial tools, we further study the general term of the 1 / N expansion and we compute the two-point function in the double scaling limit. We find that the double scaling limit mechanism implemented here is similar to the one implemented for various quartic tensor models.
In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large
N limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some physical applications of large
N limits, we present a few solvable examples in zero space-time dimension. Using models with fields in the fundamental representation of
O(N),
O(N)2, or
O(N)3 symmetry, we compare their combinatorial properties and highlight a competition between the snail and melon diagrams. We exhibit the different methods used for solving the vector, matrix, and tensor large
N limits. In the latter example we review how the dominance of melonic diagrams follows when a special ``tetrahedral" interaction is introduced. The second part of the lectures is mostly about the fermionic quantum mechanical tensor models, whose large
N limits are similar to that in the Sachdev-Ye-Kitaev (SYK) model. The minimal Majorana model with
O(N)3 symmetry and the tetrahedral Hamiltonian is reviewed in some detail; it is the closest tensor counterpart of the SYK model. Also reviewed are generalizations to complex fermionic tensors, including a model with
SU(N)2×O(N)×U(1) symmetry, which is a tensor counterpart of the complex SYK model. The bosonic large
N tensor models, which are formally tractable in continuous spacetime dimension, are reviewed briefly at the end.