# 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:

1. $\mathcal{N}=4$ SYM and 10D type IIB SUGRA.

2. 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.

1. 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 $AdS_5 \times S^5$?

2. 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 $\alpha^\prime \to 0$!

4. 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 $S^5$ ?

1. 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.
• But people don't talk about the $S^5$ much even calculating in classical SUGRA. – Physics Moron Jul 21 '15 at 21:42
• @pinu: That depends on what you mean. The $S^5$ is actually quite important, since it determines the R-symmetry of the dual gauge theory. – Frederic Brünner Jul 21 '15 at 21:51