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Perhaps this is a naive question, but in my recent (admittedly limited) readings about AdS spaces, I keep wondering why they seem to be such a hotbed for theoretical research (AdS/CFT correspondence, etc.). To my understanding, an AdS space has constant negative curvature in a vacuum, which should yield an attractive universe, not one with accelerating expansion. An AdS space can be thought of as having a negative cosmological constant, while a universe with accelerating expansion would imply that such a constant be positive. Since we observe that our universe's expansion is accelerating, it seems that if anything, we should be seeking to model it as a de Sitter space.

Am I mistaken? What aspects of our universe do AdS spaces attempt to model?

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    $\begingroup$ I have very little knowledge of GR, so I don't know much about this. But from the little that I had heard about AdS, I had exactly the same question/doubt as you. Nice question, I hope somebody answers. $\endgroup$ – Physics Llama Jun 30 '14 at 21:12
  • $\begingroup$ Related: physics.stackexchange.com/q/9732/2451 $\endgroup$ – Qmechanic Jun 30 '14 at 21:17
  • $\begingroup$ I came to this exact thing reading "The Black Hole War". I kept asking myself "where is the cosmic horizon", and when we got to the part where he should have explained this, the acceleration went in the opposite direction. I might still ask something if I can obtain some novelty, but before that I might have to re-read the latter chapters again. The theorists seem to be a mess regarding how dark energy has impacted their work. $\endgroup$ – Alan Rominger Sep 23 '14 at 1:45
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The reason why the AdS/CFT correspondence is interesting is not that AdS space is supposed to describe our universe, which, as you have correctly pointed out, would lead to conflicts with experiments. In the context of the correspondence, a four-dimensional (conformal) field theory is mapped to a string theory living in an $AdS_5\times S^5$ space, although there exist generalizations in which the AdS part is of higher or lower dimension than five.

This duality in principle allows one to carry calculations from one side to the other, making it possible to choose the framework in which the solution to the problem at hand can be found conveniently. One key observation in this context is that the duality can map a strongly coupled theory to a weakly coupled one, circumventing the failure of perturbation series. This is especially interesting with respect to QCD, where a a conventional perturbative low energy description is not possible. Even though an exact holographic dual of QCD is yet to be found, there are theories (for example the Sakai-Sugimoto model) that capture important features of QCD surprisingly well.

One may now ask what is so special about AdS space that allows for such a duality? One way to approach this is to point out the rich symmetry content of this kind of spacetime. The isometry group of Anti-de Sitter space is given by $SO(4,2)$, which is precisely the conformal group in four dimensions.

Regarding de Sitter space: the nature of this spacetime makes it difficult to formulate a correspondence analogous to its positively curved counterpart. See this article for more information.

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    $\begingroup$ Ok, so if I understand correctly, AdS spaces aren't meant to directly model the universe, but rather are used as a tool to make certain calculations in CFT easier? If that's the case, it makes a lot more sense. Thanks! $\endgroup$ – JotThisDown Jul 1 '14 at 14:36
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    $\begingroup$ Yes, that is true. Bear in mind that the correspondence also allows for generalizations to non-conformal theories. $\endgroup$ – Frederic Brünner Jul 1 '14 at 15:15
  • $\begingroup$ There's also a dS/CFT correspondence, which isn't as well-understood. In AdS a black hole's entropy-energy power law is as expected for a CFT, albeit in one fewer dimension. Gravity in dS may be metastable (as of 2007; I don't know whether it's been settled since). See Sec. V. 2. here. $\endgroup$ – J.G. Oct 10 '18 at 6:59

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