I know about the $1/N$ expansion for some time. Apart from the fact that as Witten suggests, it can be the correct expansion parameter of QCD Baryons in the $1/N$ Expansion (in a parallel that he draws between QED and QCD coupling constants), I was thinking of the meaning of the topological expansion in terms of the Riemann surfaces.
I don't know what is the meaning of the embedding of the 't Hooft's double-line diagrams on Riemann surfaces and what the Riemann surface really represent.
Moreover, it seems by taking such a limit, one is dealing with "so many fields". What is the physical advantage of such a limit? It reminds me, though perhaps quite irrelevantly of a thermodynamical limit where one deals with a huge number of degrees of freedom, although each component field, entails an infinite number of harmonic oscillators, this might be a second infinite limit.
Or are the physical intentions behind such limit, relating the two-dimensional QFT which is well-studied, to four-dimensional QFT, and that's the meaning of the embedding on Riemann surfaces?
If so, why a certain limit of something related to the internal symmetry group of the interaction, namely the number of color degrees of freedom, causes this dimensional reduction of spacetime?
Does this suggest that in the large $N$ limit, two-dimensional topological perturbation theory is equivalent to fixed topology 4d non-topological perturbation theory?
EDIT: There's also a noticeable mathematical fact, that the mathematical identities used in the large $N$ limit, resemble that of $U(N)$. But $U(N)$ is not simply connected, in fact, $\pi_{1}(SU(N))=0$ while, $\pi_{1}(U(N))=\mathbb{Z}$ which means there are $U(1)=U(N)/SU(N)$ instanton solutions in two dimensional case, but the $SU(N)$ gauge theory has $SU(N)/[SU(N-2)\times U(1)]$instanton solution in $\mathbb{R}^4$ for any $N \ge 2$. Isn't this another hint for the large $N$ limit as a relation between 2d and 4d QFT regarding the topological solutions?
Having in mind the Massive Thirring model equivalence with the Sine-Gordon in two dimensions where topological solutions of one theory are related to non-topological solutions of the other theory, does this mean that the 2D topological perturbation theory relates with the perturbative non-topological expansion in 4D? I was curious if one can find a link between the quantum corrections to the topological solutions in 4D and those of 2D. The problem is that each Riemann surface demands a different topological solution.
In other words, isn't also the 4D quantum corrections to the topological solutions related to the first term of the topological expansion in 2D (the topologically trivial term, i.e. non-topological) in the large $N$ limit?