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...

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The metric of a spacetime with a parallel plane (pp)-wave can be obtained in a certain limit of the space AdS 5 ×S 5 . According to the AdS/CFT correspondence, the holographic dual of superstring theory on that background should be the analogous limit of N = 4 supersymmetric Yang-Mills theory. In this paper we shall show that, contrary to widespread expectation, non-planar diagrams survive this limiting procedure in the gauge theory. Using matrix model techniques as well as combinatorial reasoning it is demonstrated that a subset of diagrams of arbitrary genus survives and that a non-trivial double scaling limit may be defined. We exactly compute two-and three-point functions of chiral primaries in this limit. We also carefully study certain operators conjectured to correspond to string excitations on the pp-wave background. We find non-planar linear mixing of these proposed operators, requiring their redefinition. Finally, we show that the redefined operators receive non-planar corrections to the planar one-loop anomalous dimension.of the "old matrix models" of non-critical bosonic string theory discovered in [6]. One consequence is that, in order to keep exact orthonormality of the operators (1.2), their correct normalization involves a non-trivial scaling functionAfter establishing that the recipe (1.1) does not fully suppress non-planar diagrams we are immediately led to the following puzzling question: What, then, characterizes the classes of diagrams that are favored, respectively suppressed, in this novel limit? By carefully investigating the pertinent combinatorics we find the following picture: Interpreting the operator TrZ J as a discrete closed string consisting of J "string bits" (see [3] and references therein) non-planar diagrams contributing to this operator correspond to the string splitting into multiple strings in intermediate channels. Taking J large may be interpreted as a continuum limit: The number of string bits diverges and the string becomes long and macroscopic. On the other hand, taking N large acts towards suppressing the string splitting. We then find that the scaling J 2 ∼ N leads to a delicate balance between the two effects such that microscopic strings, made out of only a small number of string bits (small w.r.t. J), are suppressed but macroscopic strings, made out of a large number of bits (i.e. of O(J)), survive.We also look at the three point correlation function of the chiral primaries (1.3), which are protected as well, and find the exact scaling function. As it explicitly encodes information for arbitrary genus, it would be fascinating if its structure could be understood from the string side [7].The present picture is very clear in the case of operators which are protected by supersymmetry, such as (1.3). In the case of unprotected operators, their quantum corrections require further analysis. Interactions involve index loops which produce factors of N. In the 't Hooft limit of Yang-Mills theory, these factors are controlled by making the coupling constant small, g 2 YM ∼ 1...

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.

Correlation functions in perturbative N = 4 supersymmetric Yang-Mills theory are examined in the Berenstein-Maldacena-Nastase (BMN) limit. We demonstrate that non-extremal four-point functions of chiral primary fields are ill-defined in that limit. This lends support to the assertion that only gauge theoretic two-point functions should be compared to pp-wave strings. We further refine the analysis of the recently discovered non-planar corrections to the planar BMN limit. In particular, a full resolution to the genus one operator mixing problem is presented, leading to modifications in the map between BMN operators and string states. We give a perturbative construction of the correct operators and we identify their anomalous dimensions. We also distinguish symmetric, anti-symmetric and singlet operators and find, interestingly, the same torus anomalous dimension for all three. Finally, it is discussed how operator mixing effects modify three point functions at the classical level and, at one loop, allow us to recover conformal invariance.