This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010.arXiv:1012.3982v5 [hep-th] PrefaceSince late 2002 tremendous and rapid progress has been made in exploring planar N = 4 super Yang-Mills theory and free IIB superstrings on the AdS 5 × S 5 background. These two models are claimed to be exactly dual by the AdS/CFT correspondence, and the novel results give full support to the duality. The key to this progress lies in the integrability of the free/planar sector of the AdS/CFT pair of models.Many reviews of integrability in the context of the AdS/CFT correspondence are available in the literature. They cover selected branches of the subject which have appeared over the years. Still it becomes increasingly difficult to maintain an overview of the entire subject, even for experts. Already for several years there has been a clear demand for an up-to-date review to present a global view and summary of the subject, its motivation, techniques, results and implications.Such a review appears to be a daunting task: With around 8 years of development and perhaps up to 1000 scientific articles written, the preparation would represent a major burden on the prospective authors. Therefore, our idea was to prepare a coordinated review collection to fill the gap of a missing global review for AdS/CFT integrability. Coordination consisted in carefully splitting up the subject into a number of coherent topics. These cover most aspects of the subject without overlapping too much. Each topic is reviewed by someone who has made important contributions to it. The collection is aimed at beginning students and at scientists working on different subjects, but also at experts who would like to (re)acquire an overview. Special care was taken to keep the chapters brief (around 20 pages), focused and self-contained in order to enable the interested reader to absorb a selected topic in one go.As the individual chapters will not convey an overview of the subject as a whole, the purpose of the introductory chapter is to assemble the pieces of the puzzle into a bigger picture. It consists of two parts: The first part is a general review of AdS/CFT integrability. It concentrates on setting the scene, outlining the achievements and putting them into context. It tries to provide a qualitative understanding of what integrability is good for and how and why it works. The second part maps out how the topics/chapters fit together and make up the subject. It also contains sketches of the contents of each chapter. This part helps the reader in identifying the chapters (s)he is most interested in.There are reasons for and against combining all the contributions into one article or book. Practical issues however make it advisable to have the chapters appear as autonomous review articles. After all, they are the works of individuals. They are merely tied together by the...

We prove that the validity of the recently proposed dressed, asymptotic Bethe ansatz for the planar AdS/CFT system is indeed limited at weak coupling by operator wrapping effects. This is done by comparing the Bethe ansatz predictions for the four-loop anomalous dimension of finite-spin twist-two operators to BFKL constraints from high-energy scattering amplitudes in N = 4 gauge theory. We find disagreement, which means that the ansatz breaks down for length-two operators at four-loop order. Our method supplies precision tools for multiple all-loop tests of the veracity of any yet-to-be constructed set of exact spectral equations. Finally we present a conjecture for the exact four-loop anomalous dimension of the family of twist-two operators, which includes the Konishi field.

Recently it was established that a certain integrable long-range spin chain describes the dilatation operator of N = 4 gauge theory in the su(2) sector to at least three-loop order, while exhibiting BMN scaling to all orders in perturbation theory. Here we identify this spin chain as an approximation to an integrable short-ranged model of strongly correlated electrons: The Hubbard model.Recently it was discovered that the planar one-loop dilatation operator of supersymmetric N = 4 gauge theory is completely integrable [1,2]. This means that its spectrum may be exactly determined in the form of a set of non-linear Bethe equations. Evidence was found that this integrability is preserved beyond the one-loop approximation, and it was conjectured that the dilatation operator might be integrable to all orders in perturbation theory [3]. Given the usually benign, analytic nature of planar perturbation theory, one may then even hope for the theory's complete large N integrability at all values of the Yang-Mills coupling constant.Deriving the dilatation operator from the field theory, and subsequently demonstrating its integrability, is not easy. The three-loop planar dilatation operator in the maximally compact su(2|3) sector was found by Beisert, up to two unknown constants, by algebraic means in [4]. These constants could later be unequivocally fixed from the results of a solid field theory calculation of Eden, Jarczak and Sokatchev [5]. This basically completely determines the planar dilatation operator in this large sector up to three loops. Its restriction to su(2) agrees with the original conjecture of [3]. Three-loop integrability in su (2) was then demonstrated in [6] by embedding the dilatation operator into an integrable long-range spin chain due to Inozemtsev, and a three-loop Bethe ansatz was derived.The Inozemtsev spin chain exhibited a four-loop breakdown of BMN scaling [8]. This scaling behavior seemed, and still seems, to be a desirable, albeit unproven, property of perturbative gauge theory. Mainly for that reason an alternative long-range spin chain, differing from the Inozemtsev model at and beyond four loops, was conjectured to exist in [7]. Its construction principles were an extension of the ones already laid out in [3]:(1) Structural consistency with general features of Yang-Mills perturbation theory, (2) perturbative integrability and (3) qualitative BMN scaling. The model's Hamiltonian is only known up to five loops, and increases exponentially in complexity with the loop order. In striking contrast, a very compact Bethe ansatz may be conjectured for the model and shown to diagonalize the Hamiltonian to the known, fifth, order. The conjecture reads e ip k L = M j=1 j =k

We study a refined large spin limit for twist operators in the sl(2) sector of AdS/CFT. We derive a novel non-perturbative equation for the generalized two-parameter scaling function associated to this limit, and analyze it at weak coupling. It is expected to smoothly interpolate between weakly coupled gauge theory and string theory at strong coupling.