Different purposes for using the Large-$N$ Expansion

I've started studying the Large-$$N$$ expansion and there seems to be several different reasons for using it.

• In the context of the SYK model, the limit is useful because it reorganizes the Feynman diagrams in a self-similar way such that the expression for the Green's function becomes a auto-consistency equation. This can be seen in any review on the topic (see for example  )

• In the original paper from t'Hooft  the whole point seems to be that this reorganization of diagrams makes an holographic duality with strings clear (I'm completely unfamiliar with Holography and String Theory so I might be totally wrong)

• Lastly, in relation to quarks but not related to holography, many Large-$$N$$ reviews (see for example ) introduce auxiliary bilinear fields and argue that, in the large-$$N$$ limit, their mean values simplify like

$$\begin{equation} \langle \phi_a (x) \phi_a (y) \rangle \xrightarrow{N\rightarrow \infty} \langle \phi_a (x) \rangle \langle \phi_a (y) \rangle + \frac{1}{N}\dots \end{equation}$$

I was wondering if anyone more knowledgeable on the subject could explain the different approaches, how they relate and in which fields are they used.

From the practical point of view, the underlying principle of any large $$N$$ analysis is the same: it makes it possible to describe the theory of interest as a classical theory using which one can compute interesting physical observables in the original theory. This is essentially what you are saying in your third point and it is also apparent in other examples you have mentioned. The saddle point equations in this classical description often capture the Schwinger Dyson equations that you would otherwise compute using the leading Feynman diagrams. For example, you can think of SYK model in terms of the collective fields $$G(\tau_1,\tau_2)$$ and $$\Sigma(\tau_1,\tau_2)$$ (please see the original papers for this description, like this one) and the equations of motions of the classical theory in terms of these variables are just a different interpretation of the Schwinger-Dyson equations you have referred to as auto-consistency equation. Same is true for the Gross-Neveau model where you can re-write the theory in terms of classical variables (effective action of the theory) and the saddle point equations are basically the same as the sum over all the cactus diagrams in terms of original fermion degrees of freedom (see Problem 11.3 in Peskin–Schroeder).
Note that 't Hooft's original observation was that one could use $$N$$ as an parameter to organize diagrams in a theory. It is not necessary that the diagrams will organize themselves as strings. In fact, for tensor models like the SYK model we don't know if such an organisation in terms of strings is possible. In fact, his work emphasized something rather important in large $$N$$ theories: that it might be possible to reorganize your diagrams but that doesn't imply you can sum them all. This is also emphasized in the introduction of Witten's paper. There are only so many examples that we know of where all the diagrams can be summed exactly to compute physical quantities. And in all such examples, it is usually possible to write the theory in terms of some classical variables. Most of the times these classical theories are non-local, but are still useful because one only needs to do perturbative calculations in this theory.
Lastly, the important vision of AdS/CFT correspondence is in fact to classify a large class of large-$$N$$ theories that do indeed have a local, classical description in terms of a gravitational theory.