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SELECTED READINGS FOR ESSAY 1 (II)

SELECTED READINGS FOR ESSAY 1 (II)

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A case study: the AdS/CFT correspondence

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At the end of 1997, Juan Maldacena, at that time a young researcher at Harvard University, proposed what some physicists consider to be one of the main breakthroughs in the history of string theory and even of theoretical physics. Approaching the physics of black holes with the powerful mathematical tools of superstring theory, he conjectured the existence of a deep relationship between pure non-gravitational theories and superstring theories. [source] Even though the proposal was not well understood by everybody, it was welcomed and enjoyed rapid acceptance within the community.

This subject has developed with breathtaking speed: Maldacena’s paper appeared in November 1997, yet by the Strings 98 conference seven months later, more than half the invited speakers chose to speak on this subject.[source]*******You can read this blog for free! Please, do not copy its content.*******

Today, Maldacena’s publication is one of the most well-known papers ever written in theoretical high energy physics.

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The first papers submitted to the electronic preprint library citing Maldacena’s conjecture did not delve deeply into the original proposal; they simply mentioned it in a superficial way. In these papers we find the following assertions: ‘‘It would be interesting to understand the relation between our arguments and those of [reference to Maldacena’s paper],’’ and ‘‘Maybe an argument along the lines of [reference to Maldacena’s paper] can be carried out here as well.’’ Things changed dramatically when Edward Witten published a paper formalizing many of the original ideas put forward by Maldacena. He discovered a precise correspondence (here the term “correspondence” is used for the first time) between string states on the ten-dimensional spacetime, dubbed the bulk, and operators of the particle physics-like model. He also computed some scattering processes. To this end Witten identified the boundary of the ten-dimensional bulk with the space where the non-gravitational particles reside and interact. After this essential contribution, more and more people started to work on this correspondence.

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Here is how Witten referred to Maldacena’s proposal (first lines of the abstract):

Recently, it has been proposed by Maldacena that large N limits of certain conformal field theories in d dimensions can be described in terms of supergravity (and string theory) on the product of d + 1-dimensional AdS space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity.[source] (Italics added.)*******You can read this blog for free! Please, do not copy its content.*******

Note Witten’s prudence when referring to it: “It has been proposed.” A year and a half later Maldacena’s “conjecture” was still a conjecture, that is, nothing exceptional demanded that its scientific status should be upgraded. This was the state of affairs in 1999 when a group of leading string theorists, including Maldacena himself, published a review article on the subject. This report comprises more than two hundred and fifty pages and is still considered one of the most complete accounts on the subject.

So, we conclude that N=4 U(N) Yang-Mills theory could be the same as ten dimensional superstring theory on AdS5 × S5 [reference to Maldacena’s paper]. Here we have presented a very heuristic argument for this equivalence; later we will be more precise and give more evidence for this correspondence.[source] (Italics added.)*******You can read this blog for free! Please, do not copy its content.*******

In the authors’ opinion the correspondence was still in the phase of gathering evidence. It was not yet established as scientific fact. At this point it is worth digressing a moment in order to say a few words about the physics of the correspondence. This will help us to understand its successive evolution towards a higher degree of truthfulness.

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AdS/CFT in its strongest version states that superstring theory in the bulk corresponds to a full quantum non-gravitational theory on the boundary of such volume. But, so far support for it has been provided only in the supergravity approximation: the point-like model where the length of the string is equal to zero (α′ → 0). In addition, since these computations are also very difficult to carry out, the classical limit is necessary. In this last approximation quantum corrections are discarded (only tree diagrams are considered). The theory is said to be weakly coupled. Figure 1 illustrates this process: from strong to weak coupling.

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The consistency of the theory relies on every operation being done on the gravitational theory having a counterpart on the boundary. (Due to this relationship between what happens in the bulk and on its boundary, the correspondence has been called holographic.) Thus, the supergravity and classical limits must have their corresponding procedure in the boundary theory. Experts have found that this relationship is of the type weak ↔ strong. In short, this means that the easier the computations in the gravitational theory, as in the limits above, the harder it is to find the corresponding non-gravitational results on the conformal field theory side. In turn, when the string calculations are difficult, the boundary computations are easier to perform. This explains why the AdS/CFT correspondence is also often called ‘‘AdS/CFT duality.’’

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After these brief observations, we are now ready to evaluate the following extract from MAGOO (as the report by Maldacena and collaborators is sometimes labelled):

One might wonder why the above argument was not a proof rather than a conjecture. It is not a proof because we did not treat the string theory non-perturbatively (not even non-perturbatively in α′). We could also consider different forms of the conjecture. In its weakest form the gravity description would be valid for large gsN [supergravity description with no quantum corrections as explained above], but the full string theory on AdS might not agree with the field theory. A not so weak form would say that the conjecture is valid even for finite gsN, but only in the N → ∞ limit (so that the α′ corrections would agree with the field theory, but the gs corrections may not). The strong form of the conjecture, which is the most interesting one and which we will assume here, is that the two theories are exactly the same for all values of gs and N.[source] (Italics added.)

This passage from MAGOO suggests that in those days many string theorists were not fully convinced of the validity of the correspondence; although something like 1500 papers had already been published on the subject (MAGOO includes 757 references in its bibliography). Despite such wide interest and some important contributions to theoretical physics, the general opinion was that the correspondence was in the process of being proved. In their ‘‘Summary and Discussion’’ the authors concluded saying:

To summarize, the past 18 months have seen much progress in our understanding of string/M theory compactifications on AdS and related spaces, and in our understanding of large N field theories. However, the correspondence is still far from realizing the hopes that it initially raised, and much work still remains to be done.[source] (Italics added.)

So, by May 1999 string theory experts were convinced that the ‘‘simple and powerful observation’’[source] made by Maldacena was in its infancy. At the same time, an optimistic vision was transmitted to young researchers by means of courses and written materials. In a widely used introductory review written by Jens Petersen, which appeared three months earlier than MAGOO, we read:

The Maldacena conjecture [reference to Maldacena’s paper] is a conjecture concerning string theory or M theory on certain backgrounds of the form AdSd × MD-d. … The conjecture asserts that the quantum string- or M-theory on this background is mathematically equivalent — or dual as the word goes — to an ordinary but conformally invariant quantum field theory in a space-time of dimension d-1, which in fact has the interpretation of “the boundary” of AdSd.[source] (Italics in the original.)

Similarly, in the written translation of a couple of lectures delivered during the spring of 1998 at The Abdus Salam International Centre for Theoretical Physics, two leading theoretical physicists wrote: ‘‘Assuming this conjecture, one can derive results for the large ’t Hooft coupling limit of gauge theory, by doing computations in AdS supergravity.’’[source] And concluded, in the last lines, by saying: ‘‘Nevertheless, now that we have a precise and better motivated conjecture for the appropriate string in this case, we can hope that progress along these lines will be made in the near future.’’ Analysis of written publications at that time shows that the veracity of the holographic correspondence was understood by string theoreticians more as a hope rather than as a completed task.

The lines of development for the following three years, from May 1999 to February 2002, were foreseen, and in a certain sense dictated, by Maldacena and collaborators: in chapter 5 they summarized the main results of BTZ black holes and showed how this was related to the boundary theory; in chapter 6, the final one, they focused on QCD-like theories.

Thanks to the great amount of available results on the physics of three-dimensional black holes and a manageable two-dimensional conformal field theory, the AdS3/CFT2 model was for many years the favourite setting for analyzing the correspondence beyond the supergravity limit.

In this paper we study the spectrum of critical bosonic string theory on AdS3 × M with NS-NS backgrounds, where M is a compact space. Understanding string theory on AdS3 is interesting from the point of view of the AdS/CFT correspondence since it enables us to study the correspondence beyond the gravity approximation.[source]

Juan Maldacena and Hirosi Ooguri continued working on this framework for some time, arriving at several remarkable results. Unfortunately, the correspondence was not demonstrated beyond the supergravity approximation as first expected. The other main line of research within AdS/CFT was the construction and description of viable QCD-like theories by means of weak gravitational processes.

A fruitful extension of the basic AdS/CFT correspondence [reference to Maldacena] stems from studying branes at conical singularities [references]. Consider, for instance, a stack of D3-branes placed at the apex of a Ricci-flat 6-d cone Y6 whose base is a 5-d Einstein manifold X5.[source]

Even though the two approaches were different, both were trying to provide evidence for the stronger versions of the correspondence. The AdS3/CFT2 effort wanted to prove the exact correspondence in a special case (classical limit with α′ ≠ 0), and the AdS/QCD attempt tried to find plausible phenomenological results. However, by the end of 2001, after four years of intense work and more than two thousand citations to Maldacena’s original paper, the correspondence was still waiting for a definitive proof.

The Berenstein-Maldacena-Nastase (BMN) conjecture was proposed in February 2002 and it rapidly seized the attention of string theorists working on AdS/CFT. This fervent interest on BMN was reflected by the large number of publications that followed. In the month the paper appeared, a fifth of the publications on AdS/CFT was on BMN or at least mentioned it in the main text. The following month, articles on BMN grabbed half the attention of the research on AdS/CFT. A few months later, up four fifths (June and September 2002) of the citations to Maldacena’s 1997 proposal came from the novel BMN conjecture. This rough count clearly shows that the new conjecture was an essential breakthrough within the field. Moreover, as we will see next, it represented the end of a period and the beginning of a new one. After BMN, the truthfulness of the AdS/CFT correspondence changed: it was nearing a scientific fact.

In this new conjecture Maldacena and collaborators envisaged an alternative setting to verify the correctness of the AdS/CFT correspondence beyond the supergravity limit. The idea was to concentrate on a very special case of the original formulation and see how the standard correspondence between string states and operators matched within this new framework. On the bulk side of the correspondence the spacetime background was changed to parallel plane waves. The new condition, pp-waves, was obtained by taking the Penrose limit of the anti-de Sitter space. In the conformal field theory this corresponded to a truncation of the number of operators. It was believed that this new model could shed light on the full quantum correspondence.

It is interesting to see how Berenstein, Maldacena and Nastase referred to the AdS/CFT correspondence, the basis of their new proposal:

The fact that large N gauge theories have a string theory description was believed for a long time. These strings live in more than four dimensions. One of the surprising aspects of AdS/CFT correspondence is the fact that for N = 4 super Yang Mills these strings move in ten dimensions and are the usual string of type IIB string theory.[source] (Italics added.)

These are the first lines of the paper. Such a presentation suggests that the relationship between gravity and particle physics is a matter of ‘‘fact.’’ They take it for granted. Obviously, there is something paradoxical in all this: what is expected to be proven is at the same time considered true knowledge. But, was this an isolated judgement or rather a belief shared by other string theorists?

Two months after the BMN proposal, Steven Gubser, Igor Klebanov and Alexander Polyakov, collaborating then at Princeton, submitted a paper where it is said:

It was found in [reference to Maldacena, Witten, and a previous article by GKP], developing some earlier findings of [reference to Polyakov], that the desired string theory in this case lives in the space AdS5 × S5 and that there is a unique prescription relating physical quantities in the string and gauge pictures. Many more complicated examples have been analyzed since then, confirming the existence of a dual string picture for various gauge theories.[source] (Italics added.)

Though the authors confess in the next lines that the correspondence has only been ‘‘tested’’ ‘‘mostly in the supergravity limit,’’ as BMN they also presuppose the full validity of the correspondence. Notice the use of the terms ‘‘it was found’’ and ‘‘confirming the existence.’’ The same predisposition is shown in another important paper written by a group of researchers from MIT and Harvard:

More recently the Maldacena conjecture has established a duality between a conformal gauge theory (with a fixed line of couplings) and string theories on an AdS background. However these dualities are well understood only at large values of the gauge coupling [supergravity limit in the bulk].[source] (Italics added.)

A widespread trait among publications following the BMN proposal is that the few lines making explicit reference to the AdS/CFT correspondence are often in the abstract or in the first paragraphs. For instance, the well known article by Joseph Minahan and Konstantin Zarembo begins with a very short discussion on AdS/CFT results and limitations. After the six-line review of AdS/CFT, they move on to the main subject of the paper: BMN.[source] The function of this brief reference to AdS/CFT in the opening to the paper is simply to contextualize the article; a context that everybody must be familiar with, and accept, in order to proceed further. Post-BMN doctoral dissertations show a similar pattern: the correspondence is assumed and chapters once intended to explain it are systematically dropped. A short section or even several citations now replaced the detailed summary. Another confirmation that the correspondence was entering a new state regarding its factuality is that some authors did not even consider relevant the citation of Maldacena’s original paper. From the eight most important papers on AdS/CFT published after BMN, only four of them cited it.

What followed in the next years was a confirmation of the previous analysis. As a sample, let us consider the nine most cited articles on AdS/CFT during that period. Three of the papers deal with phenomenological issues and concentrate on the implications of the AdS/QCD duality; that is, the possibility of using the holographic correspondence to obtain precious information on strongly coupled particle physics processes. As stated in one of these papers: ‘‘Recently, the gravity/gauge, or anti-de Sitter/conformal field theory (AdS/CFT) correspondence [reference to Maldacena] has revived the hope that QCD can be reformulated as a solvable string theory.’’[source] Another three articles focus on a different spacetime background for the correspondence, the Lunin-Maldacena background. This include one written by Oleg Lunin and Juan Maldacena. The other two are by Sergey Frolov and collaborators: ‘‘A relative simplicity of the Lunin-Maldacena supergravity background and the N=1 superconformal theory makes the conjectured duality a new promising arena for studying the AdS/CFT correspondence.’’[source] In contrast to the articles on AdS/QCD, these last three are not phenomenologically motivated; rather they try to prove the correspondence beyond a constraining condition called the BPS limit. It is interesting to notice that Lunin and Maldacena called the new proposal ‘‘conjecture duality,’’ while the original AdS/CFT proposal is simply called ‘‘correspondence.’’ This subtlety differentiation suggests that the latter is in a higher, better consolidated, factual level. Another of the nine papers on AdS/CFT concerns integrable models, a subject seeking a solution to superstring theory on non-trivial backgrounds with RR-fluxes. There are two more papers. One proposes a sort of AdS/CFT correspondence for Sasaki-Einstein backgrounds, and the other is about flux compactifications. Strictly speaking, the last article is not about the correspondence; it simply acknowledges the important contribution of the latter to the renewal of the studies on flux compactifications. And it does it in a single line.

Here concludes our short story of the AdS/CFT correspondence. In it we saw how string theorists treated the conjecture when it was proposed for the first time; how they changed their view in the course of time; and how they communicated it to younger members of the community. We discovered that the AdS/CFT conjecture became a fact at the same time as most of the talks and papers changed the ‘‘recently Maldacena conjectured that …’’ to ‘‘as the AdS/CFT correspondence teaches us …’’ and, finally, to the more impersonal ‘‘as the AdS/CFT establishes.’’ We saw how the sentence ‘‘Maldacena has recently conjectured that …’’ transformed into a single number that pointed to the original paper. Nonetheless this was not imposed, as some interpreters would be incline to declare, by a ‘‘great leader,’’ nor by the ‘‘will of power’’ of an authoritarian group of researchers, nor by mere convention. Instead, it is the end result of several years of long, hard, and exhausting work. I have sustained that the breaking point was the new ‘‘bold’’ conjecture of BMN, a hypothesis that assumed implicitly the correctness of the old AdS/CFT correspondence. After years spent accumulating “evidence,” but without a definitive proof in sight, there was the desire and need within the community to surmount the old correspondence. The research could safely continue only by protecting Maldacena’s conjecture from profanation, namely, elevating it to the factual level of the more authentic mathematical demonstrations. In another context, the historian of science Steven Shapin wrote: ‘‘It was necessary to speak confidently of matters of fact because, as the foundations of proper philosophy, they required protection. And it was proper to speak confidently of matters of fact, because they were not of one’s own making; they were, in the empiricist model, discovered rather than invented.’’[source] To shield the correspondence from attacks was a necessity for the whole community of practitioners. Consequently, more and more discussions on the correspondence were transferred from research papers and PhD theses to graduate and even undergraduate courses. This was the final step towards its final entrance into public lectures and popular science books. Today, the AdS/CFT correspondence pervades the public debate on superstring theory.

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SELECTED READINGS FOR ESSAY 1 (II)

SELECTED READINGS FOR ESSAY 1 (II)

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