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I am following Hong Liu's MIT 8.821 String Theory and Holographic Duality lectures. He starts discussing the large-$N$ expansion in the context of a hermitian matrix model described by the Lagrangian

$$\mathcal{L}=-\frac{1}{g^2}\text{Tr} \left\{\frac{1}{2}\partial_\mu\Phi\partial^\mu\Phi+\frac{1}{4}\Phi^4\right\},$$

where $g$ is the coupling constant and $\Phi$ is a $N\times N$ hermitian matrix. He shows that the large-$N$ limit of this model only makes sense when we consider the 't Hooft parameter

$$\lambda_\text{'t Hooft}=g^2N,$$

and take the $N\rightarrow\infty$ limit while keeping $\lambda_\text{'t Hooft}$ fixed.

At some point in the lecture, a student asks a question regarding whether $\lambda_\text{'t Hooft}$ should be a small parameter or not, and Liu remarks that it does not matter, and later on in the course he will eventually take the limit $\lambda_\text{'t Hooft}\rightarrow\infty$.

My question is what is the point in considering the limit $\lambda_\text{'t Hooft}\rightarrow\infty$? What does it mean physically? Is it regarding some sort of a weak-strong coupling duality?

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  • $\begingroup$ I don't know about that toy model. But in $\mathcal{N} = 4$ SYM, changing the 'tHooft coupling corresponds to changing $\alpha^\prime$ in the bulk. $\endgroup$ Jun 26 '21 at 12:18
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    $\begingroup$ He discusses the limit in a later lecture. $\endgroup$
    – bolbteppa
    Jun 26 '21 at 13:45
  • $\begingroup$ @bolbteppa Thank you. $\endgroup$
    – 123456
    Jun 26 '21 at 14:24
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As pointed out by @bolbteppa, in the lecture note Chapter3:DualityToolbox Liu answers this question. Taking the limit $N\rightarrow \infty, \lambda_{t'Hooft}\rightarrow\infty$ corresponds to the semi-classical gravity limit. This means that strongly coupled $\mathcal{N}=4$ SYM is dual to classical gravity!

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