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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 ...


I am sorry to disappoint you that cosmological constant is not a dynamical field in usual GR sense. Therefore no quantization can be done. However, there were a lot of interesting attempts to interpret the cosmological constant in other way. One of the famous example is DGP theory of gravity. Roughly a scalar is introduced in the theory such that the ...

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