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 construct the covariant κ-symmetric superstring action for a type IIB superstring on AdS 5 ⊗ S 5 background. The action is defined as a 2d σ-model on the coset superspace SU (2,2|4) SO(4,1)×SO(5) and is shown to be the unique one that has the correct bosonic and flat space limits. *

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The world volume theory on N regular and M fractional D3-branes at the conifold singularity is a non-conformal N = 1 supersymmetric SU (N + M ) × SU (N ) gauge theory. In previous work the Type IIB supergravity dual of this theory was constructed to leading non-trivial order in M/N : it is the AdS 5 × T 1,1 background with NS-NS and R-R 2-form fields turned on. Far in the UV this dual description was shown to reproduce the logarithmic flow of couplings found in the field theory. In this paper we study the supersymmetric RG flow at all scales. We introduce an ansatz for the 10-d metric and other fields and show that the equations of motion may be derived in first order form from a simple superpotential. This allows us to explicitly solve for the gravity dual of the RG trajectory. 02/00

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We describe in detail the solution of type IIB superstring theory in the maximally supersymmetric plane-wave background with constant null Ramond-Ramond 5-form field strength. The corresponding light-cone Green-Schwarz action found in hep-th/0112044 is quadratic in both bosonic and fermionic coordinates. We obtain the light-cone Hamiltonian and the string representation of the corresponding supersymmetry algebra. The superstring Hamiltonian has a "harmonic-oscillator" form in both the string oscillator and the zero-mode parts and thus has a discrete spectrum. We analyze the structure of the zero-mode sector of the theory, establishing the precise correspondence between the lowest-lying "massless" string states and the type IIB supergravity fluctuation modes in the plane-wave background. The zero-mode spectrum has certain similarity to the supergravity spectrum in AdS 5 × S 5 background of which the plane-wave background is a special limit. We also compare the plane-wave string spectrum with expected form of the light-cone gauge spectrum of the AdS 5 × S 5 superstring. *