I was reading a paper by Veronika Hubeny The AdS/CFT correspondence 1. Maldacena chose a D3-brane system to derive his conjecture. So I was wondering, why "D3-brane"? In other words, I need to know the importance of D3-brane system, so that is used in the AdS/CFT correspondence. It would be nice if a reference is recommended if the answer isn't so straightforward. Thanks in advance.
$3$-branes are special in the following sense: only for $p=3$, the black p-brane solution admits a constant dilaton, while it is running for $p\not=3$. In particular, the dilaton $\phi$ diverges at the horizon of extremal $p$-brane solutions for $p\not=3$, which means that the string coupling $g_s=e^\phi$ cannot be kept small. In the Maldacena decoupling argument, two limits are taken:
- A near-horizon limit in the $p$-brane background
- A supergravity limit in which string loop corrections are supressed, i.e. $g_s\ll 1$
For $p\not=3$, the second limit cannot be taken near the horizon since the string coupling diverges. Only for $p=3$ the constant dilaton can also be taken to be small near the horizon.
In summary, only 3-brane solutions admit a simultaneous near-horizon and supergravity limit!
For all the mathematical details, you can refer to pages 16 - 19 in the MAGOO review.