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I have been told that the AdS/CFT correspondence proof does not rely on the validity of string theory. To be honest I don't know what to make of this. The idea of taking seriously the results of applying the techniques of this correspondence is appealing, but before I head in that direction, I need some help finding any references that actually make clear the fact that such a correspondence is independent of the validity of string theory. I am also curious as to the original explicit derivation of the correspondence. I just want to be able to really learn and truly believe the results myself before considering working applying its results. I'm not contesting results of any kind, I just want to know this stuff well.

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    $\begingroup$ Here you go, for the original 'derivation': arxiv.org/abs/hep-th/9711200 $\endgroup$ – Danu Jan 23 '14 at 20:32
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    $\begingroup$ You might want to look at arxiv.org/abs/1308.1977 by Don Marolf. However, to my knowledge, such a line of reasoning is not well-established (with much lesser evidence than stringy arguments). $\endgroup$ – Siva Jan 23 '14 at 22:29
  • $\begingroup$ An interesting example: Vasiliev theories in $AdS$ are holographically dual to theories on flat space. Minwalla and co. showed that in some particular cases, these Vasiliev theories can be made out as a limiting case of string theory. I don't think it is known whether this holds in general. Then again, string theory has this tendency to crop up in places where you least expect. $\endgroup$ – Siva Jan 27 '14 at 19:37
  • $\begingroup$ I am not sure if higher-spin theories can be embedded into string theory. First of all these dualities exist in any dimension, in particulat in d much greater than 10. Secondly they do not rely on SUSY at all, though one can introduce SUSY versions. Thirdly, Minwalla et al. gave some very general words for $4/3$ duality and string theory, but these are just words and they rely on particular dimension and $N=6$ SUSY and there is no stringy explanation without SUSY even in their version. $\endgroup$ – John Feb 8 '14 at 7:31
  • $\begingroup$ This presentation by Juan seems to distil some of the conceptual basis for the duality youtube.com/watch?v=50-WpB14OMo $\endgroup$ – user37343 Feb 8 '14 at 19:06
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The logic of AdS/CFT is independent of string theory, but one finds that the theories that have sensible AdS duals have a stringy character. You could hope to derive from conformal field theory that quantum gravity in AdS is stringy. A "bottom-up" way to think about AdS/CFT (see here, and references therein, for a recent treatment; also this slightly less recent paper) is that conformal field theories satisfying certain conditions (roughly, having a large-N approximation and having a gap in the spectrum of operator dimensions) have correlation functions of low-dimension operators that can be approximately calculated in a perturbative way. These calculations map precisely onto low-energy effective field theory calculations in AdS space. (This low energy field theory includes gravity, because the map tells you that the stress tensor $T_{\mu \nu}$ in the CFT becomes the metric $g_{\mu \nu}$ in AdS.)

More generally, you can think of any CFT, even if it doesn't admit such a perturbative description, as a quantum gravity theory in AdS space, because they have the same symmetries and you can always map one language to the other. For many CFTs the AdS description will be strongly curved and not very useful.

None of this logic refers explicitly to string theory. But once you think about CFTs with a weakly coupled AdS description, you find that you need some sort of $1/N$ expansion to make sense of the theory; otherwise, you don't have a sensible perturbation theory. Having a $1/N$ expansion essentially means you can talk about a state with a well-defined number of particles in AdS. The particles in AdS are single-trace operators in the field theory: e.g., in a gauge theory, you might have operators that look like ${\rm tr}(\phi^\dagger D_{\mu_1} D_{\mu_2} \ldots D_{\mu_n}\phi)$. The basic logic of AdS/CFT tells you that such "long" operators are massive states in the bulk. Longer and longer traces look like more and more massive excited states. In concrete examples, these turn out to be described by string theory: long operators are long strings.

So the way that I would characterize the situation is this: we have a vast number of examples of AdS/CFT originating in string theory. We also have a general understanding of why AdS spaces and CFTs are equivalent to each other. This general understanding tells us that the theories that can be treated as weakly-coupled gravitational theories in AdS all have states described by stringy-looking traces. I don't think we're at the point yet where we can say that this proves that every quantum gravity theory in AdS is a string theory, but it is strongly suggestive of that (as Lubos said, even some of the theories that don't look like string theory at first turn out to be connected to it).

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  • $\begingroup$ I went through your references a few times, I am pretty sure I still don't understand Mellin amplitudes, but the logic of both papers seems pretty sane $\endgroup$ – user37343 Mar 29 '14 at 11:32
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There is no doubt that the gravitational theory in the AdS space equivalent to the CFT on the boundary is string theory with all objects, effects, and interactions that string theory predicts, and this fact has been reflected in all the AdS/CFT literature since the very first Maldacena paper. This shouldn't be surprising because string theory is the only consistent theory of quantum gravity, at least in $d\gt 3$.

Maldacena derived the two sides of the duality as decoupling limits of the dynamics of D-branes in string theory and/or the dynamics in their near-horizon limit. No other controllable, non-stringy context is known that would allow one to prove that the limits are decoupled.

BMN (Berenstein-Maldacena-Nastase) is the most explicit proof of the presence of excited strings and their interactions (see e.g. this) from the CFT. Detailed knowledge is available for the presence of the stringy wrapped branes (baryonic operators, Witten) and many other objects in the theory. The stringiness of AdS/CFT is also reflected by a class of AdS/CFT examples that involve (conical) Calabi-Yau compactifications of string theory and by the correct $d=10$ or $d=11$ of all the known rigorous supersymmetric examples of the AdS/CFT duality.

There exist examples of AdS/CFT that are not constructed out of the usual critical superstring, for example the minimum "pure gravity in 3D" which is dual to a 2D CFT involving the monster group as a symmetry; or the Vasiliev theory. However, one may view such vacua as examples of irregular compactifications of string theory, too.

There are also many known derivations, especially of the black hole entropy, that only rely on some general algebraic features of the CFT but not on their stringy origin. This fact changes nothing about the fact that a fully consistent theory of quantum gravity in any working AdS/CFT pair has to obey additional consistency conditions which are linked to string/M-theory.

Karl-Henning Rehren's "holography" mentioned in another answer is a vacuous pseudoscience that has nothing to do with the depth of the actual holography in quantum gravity. What is done in that paper is just rewriting fields $\phi(x^0,x^1,x^2,x^3,x^4)$ as $\phi_{x^4}(x^0,x^1,x^2,x^3)$ i.e. rewriting one of the coordinates as an "index" of the fields, and claiming that coordinates may be reduced by one (or any number, for that matter). However, nothing changes about the physical "number of dimensions" by this sleight-of-hand.

Some more comments about the unavoidable stringiness of AdS/CFT here:

http://motls.blogspot.com/2014/01/quantum-gravity-in-adscft-is-inevitably.html?m=1

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    $\begingroup$ Voted down because of the unscientific and misleading sentence "string theory is the only consistent theory of quantum gravity", rather than the more humble version 'string theory is the best candidate for a consistent theory of quantum gravity'. $\endgroup$ – user24155 Jan 24 '14 at 18:31
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    $\begingroup$ It is not a misleading sentence, it is a very important result of modern scientific research. Science is not about being humble; science is about finding the truth about Nature. We are not saying that the Earth's mass is about 99 kilograms just to be humble. $\endgroup$ – Luboš Motl Jan 24 '14 at 18:56
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    $\begingroup$ Science never understands some phenomena "quite fully" but it often understands them vastly better than non-science or less developed science. Whether some scientists are humble is about their personality and has nothing to do with the scientific validity of their claims. Otherwise, your claim that good scientists are "almost always humble" is a historical lie. Almost all great scientists were highly self-confident and assertive, starting from the founders Newton and Galileo and continuing with tons of other - Feynman, Zwicky, I could say hundreds - were self-confident if not "arrogant". $\endgroup$ – Luboš Motl Jan 26 '14 at 7:10
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    $\begingroup$ Otherwise it is logically fallacious for you to say that the proposition "string theory is the unique consistent theory of quantum gravity" is less humble than your proposition about a "best candidate" that boldly suggests that there are other candidates. Claiming that there is another possible consistent theory of quantum gravity is an extraordinary claim that would require extraordinary evidence to be taken seriously. Many people have said such a claim but all of them were quickly shown to be wrong. $\endgroup$ – Luboš Motl Jan 26 '14 at 7:20
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    $\begingroup$ I completely agree with your "Almost all great scientists were highly self-confident and assertive" when speaking about things they understood well, but I still think they were humble when it got to things they didn't comprehend. And when you say "no alternative theory [to string] is mathematically possible", it means string theorists are somehow close to showing that every consistent theory of quantum gravity has a Hilbert space isomorphic to that of string theory; I don't think we are anywhere near proving such statements. $\endgroup$ – user24155 Jan 26 '14 at 22:32
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Ok, it is important to make some observations here otherwise people will believe all sort of nonsense (mainly the nonsense in Lubos Motl's answer). First: Heavy Ion scattering. AdS/CFT does NOT offer an acceptable answer free of ambiguities. Viscosity of the Quark Gluon plasma is not observable. The elliptic flow is observable and the observed values are consistent with lots of ideas about what happens at a fundamental level. We do NOT have an accurate value for the internal viscosity and when facing scaling problems AdS/CFT fails "dramatically". For me this is evidence enough that nature does not take AdS/CFT for granted, quite the opposite... Why, this is the main reason (and yes, it is experimental) to consider AdS/CFT unsuitable for describing reality... another subject that in honest conditions should be largely ignored or the subject for abstract mathematics. Second, string theory and AdS/CFT have some things in common but they are not one the cause of the other despite some proofs regarding one done by the use of the other. A set of axioms is not always unique. You can prove different things using aspects from different domains of science and come up with the same result. This doesn't mean (not even remotely) that the truth lies within the set of axioms you like most. Let me be more clear about this: AdS/CFT is not a duality of nature. Natural space is not AdS and natural field theory is not CFT and it never can be exactly in that way. The duality is allowed to explain some situations when strong interactions are relevant and when CFT is relevant (critical points, etc.). The duality does that in a non-unique and ambiguous way and it fails in giving the right (holographic) scaling by orders of magnitude. It is simply to weak to account for real life data and it is NOT a duality of nature. No, you are not allowed to renormalize in that way so the problem is not easily solved and the fact that the predictions do not scale is fatal. However, if AdS/CFT were a duality of nature then string theory would be non-realistic too. Why? simple: because if the duality were to be exact and fundamental gauge theory on the CFT part would be undistinguishable from some strings in the bulk it is known that two isomorphic theories are not to be considered as different. A perfect string/gauge duality would mean strings lose their significance as "fundamental objects". This happens indeed but not because of the exactness of AdS/CFT... String theory could be useful in some cases as for example dimensional extension and reduction procedures are but strings are NOT fundamental in any form and any assumption that they are is against all experimental evidence.

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    $\begingroup$ As a remark: whenever you hear anyone claiming strange things ask yourself: is the proof including conditions like existence and unicity? Is the experiment unambiguous in both directions? Is he/she hiding something? What is the parameter space of the experiment? How do the parameters correlate to each other? Is the "basic assumption" the only one possible? If your answers are "no" or "I don't know" then simply think more about it before jumping to the conclusions. String theory is certainly not "the only game in town" mainly because it is based on non-unique and not necessary assumptions. $\endgroup$ – user33923 Feb 8 '14 at 9:57

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