# What is the purpose and meaning of taking the 't Hooft parameter to infinity?

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?

• 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. Jun 26 '21 at 12:18
• He discusses the limit in a later lecture. Jun 26 '21 at 13:45
• @bolbteppa Thank you. Jun 26 '21 at 14:24

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!