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I'm beginning to learn AdS/CFT and I have an elementary question. It is said that since it is very hard to calculate 4 point and above correlators in a strongly coupled CFT, we can use instead the AdS/CFT duality to calculate the correlator of the dual field in AdS and take the appropriate boundary limit (that's one way of doing it.)

But we can also use the large $N$ expansion for a strongly coupled field theory. So instead of using the AdS I could directly calculate the correlators in a large $N$ expansion.

So is it that we have 2 different methods for calculating correlators in a strongly coupled field theory? Or are they actually the same method? If they are different, is there a reason to prefer one over the other?

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    $\begingroup$ Can't help but share one of my favourite McGreevy quotes (not talking about AdS/CFT but rather about the Jordan-Wigner transformation): "Possibly Depressing comment. So now you are starting to see that this duality business is actually often a sad story: we thought we could solve two systems (free bosons and free fermions) but since they are really the same system in disguise, it turns out we can only solve one!" Ref: p101 physics.ucsd.edu/~mcgreevy/s14/239a-lectures.pdf $\endgroup$ Jul 29, 2016 at 4:37
  • $\begingroup$ large N expansion is for strongly coupled field theories not for specially for conformal field theory.but ADS-CFT will give you correlator for conformal theory.I dont know whether thooft's large N expansion can be formalised to conformal theories also. $\endgroup$ Jul 30, 2016 at 21:05
  • $\begingroup$ if you use classical gravity in ads then the dual is large N CFT. Specifically 1/N^2 ~ G_N. $\endgroup$ Jul 31, 2016 at 14:00

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Strongly coupled refers to the lambda expansion and not the large N.

At large N, you have only planar diagrams, but you can still have large number of vertices weighted by lambda. So, you cant really sum the lambda expansion.

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AdS-CFT correspondence is so useful because it's a strong-weak duality. We can apply perturbation theory on the weak gravity side and compute different observables at the boundary at the conformal side. I am still new at this field but I can give an example how this works. Consider a massive scalar field in $\mathrm{AdS}_{1+1}$, the bulk-bulk propagator $G(z,x;z',x')$ of the scalar can be computed (see this paper for the details Erbin: scalar propagators) using the on-shell solution of the fields. Interesting thing is that once we have this bulk-bulk correlator, we can use this quantity to compute the boundary-boundary correlator ($k(x,x')$) using $$k(x,x')= 2\nu^2 \lim_{z,z'\to 0} (zz')^{-\Delta^+} G(z,x;z',x') .$$ Here $\nu^2$ and $\Delta_{+}$ are constants dependent of the mass of the scalar field and the radius of curvature of the space. However, how the large N expansion compares to this kind of correspondence in calculating the correlators at the CFT side is beyond me.

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