# What is the physical meaning of the large $N$ expansion?

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?

• We don’t dimensionally reduce space time, the string still moves in 4d (the string world sheet is embedded in 4d, just like the particle word lines of ordinary perturbation theory are embedded in 4d) Aug 28, 2022 at 19:13
• And can you please say why the Riemann surface, in this case, is literally the same as the worldsheets of string theory? It sounds quite nontrivial. Nonetheless, saying that the worldsheet can be embedded in the 4d space doesn't answer the question I guess. As far as my minor knowledge of string theory suggests, even there, the worldsheets have their own intrinsic metric and QFT(CFT) and the embedding is a secondary matter where 10 or 22dimensional spaces are the only consistent ones. This means the 2 dimensionality of the Riemann surface after the large $N$ limit is not yet justified @Thomas Aug 28, 2022 at 23:11
• Note that there is no proof that large N non-abelian gauge theory corresponds to a string theory. The hypothesis that it does rests on the observation that Feynman diagrams at large N with g^2N fixed provide an expansion where the power of 1/N is governed by the genus of the surface traced out by the diagram This looks like perturbative string theory, which is also a genus expansion. Aug 29, 2022 at 1:05
• However this is not a proof, and the would-be string theory must be a non-critical theory (the bosonic string, which would corresponds to pure gauge theory is only consistent in D=26), because the theory lives in D=4. Aug 29, 2022 at 1:07
• The only concrete realization of the "large N=string theory" idea is AdS/CFT, which works in a very different way from what people initially imagined (the string theory lives in D=10, and the theory is not equivalent but dual to YM). Aug 29, 2022 at 1:09

The following paper: Large N as a thermodynamic limit shows that the Large $$N$$ limit is correctly interpreted as the thermodynamic limit and not the classical limit that Witten (THE 1 / N EXPANSION IN ATOMIC AND PARTICLE PHYSICS) and Coleman (1/N) advocated.