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?
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2 Answers
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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.
and p.11:
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.
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$\begingroup$ Thanks for the paper link. To clarify a bit, say I have a string theory which is known to have a gauge theory dual. If this string theory is analyzed using integrability at strong, intermediate and weak couplings, and conclusions are drawn in each case, does one gain any more information about the string theory by invoking the AdS/CFT correspondence and analyzing the dual gauge theory (using integrability or other techniques)? $\endgroup$– Matrix23Commented Apr 17, 2019 at 11:54
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$\begingroup$ My understanding is that integrability only works well to study certain operators. So I would guess that the usual strong/weak duality might still offer insight into quantities that are harder to access via integrability. But I am certainly not an expert on AdS/CFT, and someone else might have a more precise answer! $\endgroup$ Commented Apr 17, 2019 at 23:59
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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:
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$\begingroup$ In no way does OPs comment suggest that weak coupling implies integrability. Instead he supposes (rightly so) that integrability allows access to strong coupling phenomena. $\endgroup$ Commented Apr 15, 2019 at 18:56
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$\begingroup$ @Herr_Mitesch If you downvoted the answer, I don't agree with it. It was neither wrong or misleading. In many ways (at least to me and I suppose a lot of people) the OP's comment suggest weak coupling implies integrability, the question has "integrablity" in it! If the OP does not, my answer only clarifies the question. Why the hate? $\endgroup$ Commented Apr 15, 2019 at 21:30
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$\begingroup$ No hate at all :-) Clarifications of the question should be comments to the OP. I also think that the OP's use of 'however' implies that he is well aware that weak coupling does not imply integrability but rather that integrability would provide a second tool to study strong coupling regimes without appealing to a strong-weak duality. I also admit that i'm no expert on GR but I don't understand what black holes have to do with weakly coupled gravity. I hope your not mad about the downvote, i felt it to be justified. $\endgroup$ Commented Apr 15, 2019 at 23:04