Maldacena's decoupling argument 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: 


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*$\mathcal{N}=4$ 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.


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*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 $S^5$ ?
 A: *

*The limit $\alpha'\rightarrow 0$ is understood in terms of an expansion in $\alpha'$, where the leading order term is given by supergravity. This does not mean that one cannot have solutions with curved spacetime. As $\alpha'=\ell_s^2$, where $\ell_s$ is the string length, this limit tells us what happens if we remove "stringiness", and as it turns out, we end up at pure (super-)gravity. Since string-effects are only apparent at high energies, this corresponds to going to the low-energy regime of the theory. The branes do not backreact on anything, they are solutions to supergravity. This follows from the equivalence of p-branes and D-branes, which was shown by Polchinski.

*This is part of the conjecture: it is not proven, one can just find evidence by doing calculations on both sides of the duality and see whether results match. One problem is that it is hard to go away from the supergravity limit, i.e. treat string theory nonperturbatively. 

*One can define string theory on whatever space is appropriate, as long as it is asymptotically AdS, since that is what is needed for the conjecture to hold. Fields on the gravity side act as sources for CFT operators, and the identification is made by matching partition functions at the boundary.  

*You are right, in the case of small radius, the supergravity approximation is no longer valid and the setup is hard to treat. This is precisely why most calculations are done in the supergravity limit, which happens to be dual to the strongly coupled regime of the gauge theory.

