Abstract In this chapter we analyse in detail the diagrammatics of various Sachdev–Ye–Kitaev-like tensor models: the Gurau–Witten model (in the first section), and the multi-orientable and O(N)3-invariant tensor models, in the rest of the chapter. Various explicit graph theoretical techniques are used. The Feynman graphs obtained through perturbative expansion are stranded graphs where each strand represents the propagation of an index nij, alternating stranded edges of colours i and j. However, it is important to emphasize here that since no twists among the strands are allowed, one can easily represent the Feynman tensor graphs as standard Feynman graphs with additional colours on the edges.
The Sachdev-Ye-Kitaev (SYK) model is a model of q interacting fermions. Gross and Rosenhaus have proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, those flavors are reminiscent of the colors used in random tensor theory. This gives us the opportunity to apply some modern, yet elementary, tools developed in the context of random tensors to one particular instance of such colored SYK models. We illustrate our method by identifying all diagrams which contribute to the leading and next-to-leading orders of the 2-point and 4-point functions in the large N expansion and argue that our method can be further applied if necessary. In the second part, we focus on the recently introduced Gurau-Witten tensor model and also extract the leading and next-to-leading orders of the 2-point and 4-point functions. This analysis turns out to be remarkably more involved than in the colored SYK model.
The Sachdev-Ye-Kitaev (SYK) model is a model of
q interacting fermions whose large N limit is dominated by melonic graphs. In this review we first present a diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs. Gross and Rosenhaus have then proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, these flavors can be seen as reminiscent of the colors used in random tensor theory. Applying modern tools from random tensors to such a colored SYK model, all leading and next-to-leading orders diagrams of the 2-point and 4-point functions in the large
N expansion can be identified. We then study the effect of non-Gaussian average over the random couplings in a complex, colored version of the SYK model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in
N. We then derive the form of the effective action to all orders.
Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large
N limit, where
g2N3 is held fixed. In this limit the perturbative expansion in the quartic coupling constant,
g, is dominated by a special class of ``melon'' diagrams. We study ``uncolored'' models of this type, which contain a single copy of real rank-3 tensor. Its three indices are distinguishable; therefore, the models possess
O(N)3 symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large
N limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large
N limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the
O(N)3 symmetry in our quantum mechanical model removes the nonsinglet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anticommuting tensor, which has $U(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(N)$ symmetry and argue that it is equivalent in the large
N limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension
d. While the quartic interaction is not positive definite, we construct the large
N Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion.
Abstract In this chapter, we first review the Sachdev–Ye–Kitaev (SYK) model, which is a quantum mechanical model of N fermions. The model is a quenched model, which means that the coupling constant is a random tensor with Gaussian distribution. The SYK model is dominated in the large N limit by melonic graphs, in the same way the tensor models presented in the previous three chapters are dominated by melonic graphs. We then present a purely graph theoretical proof of the melonic dominance of the SYK model. It is this property which led E. Witten to relate the SYK model to the coloured tensor model. In the rest of the chapter we deal with the so-called coloured SYK model, which is a particular case of the generalisation of the SYK model introduced by D. Gross and V. Rosenhaus. We first analyse in detail the leading order and next-to-leading order vacuum, two- and four-point Feynman graphs of this model. We then exhibit a thorough asymptotic combinatorial analysis of the Feynman graphs at an arbitrary order in the large N expansion. We end the chapter by an analysis of the effect of non-Gaussian distribution for the coupling of the model.
The rank three tensor model with tetrahedral interaction was shown by Carrozza and Tanasa to admit a
1/N expansion, dominated by melonic diagrams, and double tadpoles decorated with melons at next-to-leading order. This model has generated a renewed interest in tensor models because it has the same large
N limit as the SYK model. In contrast with matrix models, there is no method which would be able to prove the existence of
1/N expansions in arbitrary tensor models. The method used by Carrozza and Tanasa proves the existence of the
1/N expansion using two-dimensional topology, before identifying the leading order and next-to-leading graphs. However, another method was required for complex, rank three tensor models with planar interactions, which is based on flips. The latter are moves which cut two propagators of Feynman graphs and reglue them differently. They allow transforming graphs while tracking their orders in the
1/N expansion. Here we use this method to re-prove the results of Carrozza and Tanasa, thereby proving the existence of the
1/N expansion, the melonic dominance at leading order and the melon-decorated double tadpoles at next-to-leading order, all in one go.