I am a bit confused about Maldacena's original decoupling argument. There are two different low energy (i.e, $\alpha^\prime \to 0$) descriptions of the stack of D3-branes:
$\mathcal{N}=4$ SYM and 10D type IIB SUGRA.
Full type IIB superstring in $AdS_5 \times S^5$ and 10D type IIB SUGRA.
Comparing (1) and (2) (actually cancelling 10D SUGRA!) we obtain the celebrated AdS/CFT correspondence. I have the following questions regarding this argument.
If one takes $\alpha^\prime \to 0$ it is same as taking $G_N \to 0$. Then how do the branes backreact to produce non-trivial background namely $AdS_5 \times S^5$?
One arrives at the AdS/CFT correspondence by taking $\alpha^\prime \to 0$, by the above decoupling argument. Then how can one claim that there should be full string theory in $AdS_5 \times S^5$? I understand that any high-energy excitation will be infinitely red-shifted for the observer at infinity. But these are all happening at $\alpha^\prime \to 0$!
Isn't full string theory defined only on asymptotically AdS rather than AdS? (I am not sure about this though.)
Also the radius of the $S^5$ turns out to be same as $AdS_5$ scale, $L$. Now small $L$ means highly fluctuating string i.e., quantum gravity regime and thus notion of this classical backgrounds break down. Then how can one do Kaluza-Klein reduction of the $S^5$ ?