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I'm a beginner in learning the AdS/CFT correspondence, and I come across two problems that I hope an answer will be given here.

  1. D branes in string theory are usually introduced through the T-duality of open strings where toroidal compactification of dimension is necessary. However, I think that D branes can live in spacetimes where there is no compactification. Am I right?

  2. I think in the AdS/CFT correspondence, the D branes are placed on the boundary of the AdS spacetime, while the $p$-brane is often said to positioned on the horizon. Is that right? If so, why are the $p$-brane and D brane equivalent?**

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  1. D-branes also arise in perturbative string theory on flat backgrounds as hypersurfaces in space on which strings can end due to Dirichlet boundary conditions.

  2. The equivalence of D-branes and p-branes is completely independent from the AdS/CFT correspondence and was discovered before the correspondence by Polchinski in 1995 (see here). D-branes and p-branes are just two different descriptions of the same objects, the first in terms of perturbative string theory, the second in terms of an effective spacetime description after realizing that branes gravitate and thus curve spacetime around them.
    In the AdS/CFT correspondence the D-branes are NOT placed at the boundary of AdS. After taking the low-energy/near horizon limit in the two descriptions (D-branes and p-brane backgrounds) of the same stack of N 3-branes one realizes that in each description there are two decoupled sectors, in the perturbative D-brane description $\mathcal{N}=4$ Super Yang-Mills with gauge group $SU(N)$ and type $IIB$ supergravity on flat space and in the p-branes scenario type $IIB$ string theory on AdS$_5\times$S$^5$ and type $IIB$ supergravity on flat space. Since the two descriptions are equivalent one identifies $\mathcal{N}=4$ Super Yang-Mills with gauge group $SU(N)$ and type $IIB$ string theory on AdS$_5\times$S$^5$. However, the two descriptions are separate, there are no D-branes in the p-brane description and therefore it does not make sense to say that the D-branes are at the boundary of AdS!
    You are probably refering to the fact that one often says that the $\mathcal{N}=4$ Super Yang-Mills theory is located on the boundary of AdS. This comes from a different realization, namely that the boundary of AdS carries a conformal structure which can be identified with the symmetries of the gauge theory (see this paper for details). In addition, the boundary values of fields in the bulk theory source operators in the field theory in the dictionary of the correspondence. Therefore, it makes sense to think of the gauge theory as being located at the boundary. It is not there by construction!

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  • $\begingroup$ Maybe somebody with a broader background in string theory can expand on the first part of the question. $\endgroup$ – physicus Nov 2 '14 at 1:41

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