I read that in the limit of large $N$, the CFT on the boundary becomes classical. My question is if in such limit the physics in the bulk also becomes classical, or if we can still have a quantum field theory in the bulk (so that entanglement is possible in the bulk but not in the boundary).
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1$\begingroup$ It sounds like you read imprecise sources. AdS / CFT is concerned with situations where a classical bulk calculation gives the same answer as some (quantum) CFT at large N. $\endgroup$– Connor BehanCommented Sep 1, 2023 at 14:12
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The precise statement is that in AdS/CFT we have a relation between the parameters of the bulk and boundary theories. For example, in the first example of the correspondence Type IIB superstring theory on AdS$_5\times S^5$ with $N$ units of five-form flux has been related to ${\cal N}=4$ SYM theory with gauge group ${\rm SU}(N)$ subject to the relation between parameters: $$g_{YM}^2=2\pi g_s,\quad 2g_{YM}^2N=\frac{R^4}{\ell_s^4},$$
where $g_s$ is the string coupling, $g_{YM}$ is the Yang-Mills coupling, $R$ is the common radius of AdS$_5$ and $S^5$ and $\ell_s$ is the string length. It is also common to introduce $\lambda = g_{YM}^2N$ the t'Hooft coupling.
One can then study the large $N$ limit with either $\lambda$ fixed or taken to be large as well. In the first case $g_s\to 0$ and $\ell_s^2/R^2\neq 0$, so that one gets classical string theory in the bulk. In the second case, $g_s\to 0$ and $\ell_s^2/R^2\to 0$, so that we get classical supergravity, since the string length shrinks to zero compared to the AdS radius.
So what really happens is that the large $N$ limit of the CFT is connected to classical string theory or supergravity in the bulk, but nevertheless, there is no classical limit involved in the CFT side. We just have a quantum CFT at large $N$.
For more details I suggest the book Gauge/Gravity Duality by Martin Ammon and Johanna Erdmenger.
