I was wondering how the AdS/CFT correspondence fits in the context of integrability. As I understand, the AdS/CFT correspondence postulates a duality between gravity theories and CFT's. If one theory has a strong coupling, the other has weak coupling. AdS/CFT would then be used to study the strongly coupled theory by analyzing the weakly coupled dual theory instead. However, with integrability, (I think) one can directly analyze the strongly coupled theory. Where would AdS/CFT correspondence fit in this situation?
I would recommend the review
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99, 3 (2012), arXiv:1012.3982.
For example, here is an excerpt from p.10:
Methods of integrability provide us with reliable data over the complete range of couplings. We can investigate in practice a gauge theory at strong coupling. There it behaves like a weakly coupled string theory. Likewise a string theory on a highly curved background (equivalent to low tension) behaves like a weakly coupled gauge theory. At intermediate coupling, the results are reminiscent of neither model or of both; this is merely a matter of taste and depends crucially on whether one’s intuition is based on classical or quantum physics. In any case, integrability can give us valuable insights into a truly quantum gauge and/or string theory at intermediate coupling strength.
A crucial benefit of integrability is that the spectral equations include the coupling constant λ in functional form. Whereas standard methods produce an expansion whose higher loop coefficients are exponentially or even factorially hard to compute, here we can directly work at intermediate coupling strength.
Having a weakly coupled gravity dual does not imply integrality. To the contrary black holes in classical gravity are the most chaotic objects there can be! See this article: