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A proof of highly-curved AdS/CFT, edition 2019a

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Since the discovery of the AdS/CFT correspondence in 1997, some physicists (including me) tried to prove it. I am talking about the most famous case with the \(AdS_5\times S^5\) background of type IIB string theory that is described by the boundary CFT in \(d=4\) with the \(\NNN=4\) supersymmetry. And I am talking about some sort of a "direct proof", at least in some regime – there is a lot of circumstantial evidence that Maldacena's duality is correct, of course.

If you "thicken" propagators in a gauge theory Feynman diagram, it starts to look like a piece of a plane – which may be considered a world sheet – cut to pieces. Many things may be done with this 't Hooft picture which was the precursor of holography in the mid 1970s. Well, maybe Nathan wants to add at least one reference to a paper by 't Hooft LOL but I understand what's behind such omissions.

That duality is usually studied for a large gauge theory 't Hooft coupling where the radius of the AdS space and the five-sphere (the radii are equal) is much larger than the 10D Planck scale in the bulk quantum gravitational theory (type IIB string theory). But at some level, the correspondence should be true for a small radius as well, i.e. for the highly curved AdS space that cannot be easily described by a low-energy "classical" gravitational action.

You may Google search my blog for a proof of AdS/CFT – this topic is very old. Also because I am being acknowledged (thanks, Nathan) although I didn't give him any useful input recently, I sort of have to write about (my once co-author's and brilliant physicist's) Nathan Berkovits' new iteration of the proof:
Sketching a Proof of the Maldacena Conjecture at Small Radius
It's still a "sketch" so we don't know whether it will be treated as the "final word" on these proofs sometime in the future.

Fine, so the \(\NNN=4\) gauge theory has some "spacetime" supersymmetry that – if you look from the stringy direction – descends from the usual spacetime supersymmetry of string theory. At least the \(\NNN=1\) subgroup of this supersymmetry algebra is very helpful in "easy", manifest definitions of that supersymmetric gauge theory.

So if you want this gauge theory to be shown equivalent to a string theory and you want a sufficiently straightforward proof, it's good for the string theory description to have some manifest spacetime supersymmetry, too. Therefore, the NSR string isn't very good (NSR used to be our main Czech acronym for West Germany although we were increasingly switching to the modern SRN that we use for Germany today) you need something like the Green-Schwarz superstring. But the normal Green-Schwarz superstring (and also matrix string theory, a non-perturbative version of it) only works fine in the light cone gauge and that gauge isn't really possible for the curved \(AdS_5\times S^5\) space.

So you need a consistent covariant generalization of the Green-Schwarz superstring and Berkovits' pure spinor superstring seems to be the most natural (or only?) good answer we have. It has some spacetime spinor-valued degrees of freedom on the world sheet where the spinor has to be "pure" so one demands the vanishing of some bilinear expressions built from the spinors.

Now, it's been already known to 't Hooft some 45 years ago that the Feynman diagrams of gauge theories may be expanded topologically and they resemble stringy world sheets (see the picture at the top for an illustration) – he was the first one who found the relationships between the topology/genus of the stringy diagram and the orders of the gauge-theoretical Feynman diagrams etc. So the world sheet may be obtained as a "continuum limit" of the Feynman diagrams: many propagators and vertices may be arranged in a "planar" way that locally looks like a two-dimensional world sheet. That's what we normally assume in the low-curvature AdS/CFT: the density of the propagators, faces, and vertices is high.

Here we want to go in the opposite direction (study the limit where the low-order Feynman diagrams dominate) so we need to get the Feynman diagrams out of a smooth string world sheet. How does it happen? You need to cut the holes from the stringy world sheet. It's natural to assume that something special happens at the boundaries between the holes – where the gauge theory propagators live – and for quite some time, I have believed that the world sheet position fields touch the AdS boundary at these special points where the propagators arise (search for "touch" e.g. in the article I just linked to). You know, \(X^\mu(\sigma,\tau)\) – the position of a given point of the world sheet in the spacetime – is quantum-fluctuating and the AdS geometry seems to be such that these fluctuations may go very close to the AdS boundary (or literally touch it). Note that most of the hypervolume of the AdS space is near the boundary and in this sense, it's indeed "rare" to be close enough in the middle.

Nathan's new sketched proof does accept this assumption. He showed that his pure spinor world sheet action is a sum of a BRST-exact and therefore "topological" term; and an antisymmetric \(B\)-term that contains bilinears in the currents generating the super-isometries; plus some ghost-dependent terms (they actually include the \(\lambda\) ghosts in the denominators so be ready for some non-trivial, non-polynomial calculations). Ignoring the \(B\)-term, he claims that the BRST-exact term basically produces the right free spectrum – the propagators in the gauge theory – and the \(B\)-term produces the right Yang-Mills vertices! Those vertices are cubic (think about the \(\NNN=1\) superspace).

It sounds very nice. The closed strings are made of "necklaces" that connect "beads" – and the beads, special points on the closed string, correspond to the free Yang-Mills fields. The "beads" tell you where the world sheet touches the AdS boundary (yes, that's "where" the Yang-Mills fields naturally live, assuming a stringy bulky "where").

This proof is particularly meaningful if the coupling is weak i.e. if the AdS curvature is high. That's a limit that is "not quite geometric" because for "real bulk geometry", we usually want some dimensions in which the curvature is much smaller than the Planck scale. But quantum gravity allows us to discuss these "far from flat space" situations, too. And the more popular, low-curvature AdS environments may be viewed as some sort of extrapolation of the current Berkovits picture. At any rate, it could be a direct proof of the AdS/CFT conjecture.

We don't really "need" a proof of it, Andy Strominger would sometimes tell me. We have lots of very strong evidence that it's surely true. Well, I agree with that. It's still nice to have a set of ideas that demystify "why" it works. One reason is that while this duality and others are great, it's better not to preserve too much incorrect "religious fog" about these dualities' potential unprovability when the reasons why they hold are "transparent enough" assuming the right perspective.

Note that it's been more than 10 years from most of my blog posts about the proof of AdS/CFT. I do believe that in the frantic, enthusiastic epochs, such technical things could be completed within less than a year. But the number of people – and especially young people such as grad students – working on these important stringy problems is arguably smaller than a decade or two decades ago. And the degree of motivation for them to make progress is also lower. I think it's true, it's terribly unfortunate, and it's at least partially a consequence of the anti-theoretical-physics atmosphere in the society.

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