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...

Using the thermodynamic Bethe ansatz method we derive an infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT. The Y-system conjectured in [1] for the spectrum of all operators in planar N = 4 SYM theory follows from these equations. In particular, we present the integral TBA type equations for the spectrum of all operators within the sl(2) sector. We prove that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics. We find the analogue of DHM formula for the dressing kernel in the mirror kinematics.

We present a new formalism, alternative to the old thermodynamic-Bethe-ansatz-like approach, for solution of the spectral problem of planar N=4 super Yang-Mills theory. It takes a concise form of a nonlinear matrix Riemann-Hilbert problem in terms of a few Q functions. We demonstrate the formalism for two types of observables--local operators at weak coupling and cusped Wilson lines in a near Bogomol'nyi-Prasad-Sommerfield limit.

We give a derivation of quantum spectral curve (QSC) -a finite set of Riemann-Hilbert equations for exact spectrum of planar N = 4 SYM theory proposed in our recent paper Phys. Rev. Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system -a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

We use Integrability techniques to compute structure constants in N = 4 SYM to leading order. Three closed spin chains, which represent the single trace gauge-invariant operators in N = 4 SYM, are cut into six open chains which are then sewed back together into some nice pants, the three-point function. The algebraic and coordinate Bethe ansatz tools necessary for this task are reviewed. Finally, we discuss the classical limit of our results, anticipating some predictions for quasi-classical string correlators in terms of algebraic curves.

We calculate exactly functional determinants for quantum oscillations about periodic instantons with non-trivial value of the Polyakov line at spatial infinity. Hence, we find the weight or the probability with which calorons with non-trivial holonomy occur in the Yang-Mills partition function. The weight depends on the value of the holonomy, the temperature, ΛQCD, and the separation between the BPS monopoles (or dyons) which constitute the periodic instanton. At large separation between constituent dyons, the quantum measure factorizes into a product of individual dyon measures, times a definite interaction energy. We present an argument that at temperatures below a critical one related to ΛQCD, trivial holonomy is unstable, and that calorons "ionize" into separate dyons.

Using integrability and analyticity properties of the AdS 5 /CFT 4 Y-system we reduce it to a finite set of nonlinear integral equations. The Z 4 symmetry of the underlying coset sigma model, in its quantum version, allows for a deeper insight into the analyticity structure of the corresponding Y-functions and T-functions, as well as for their analyticity friendly parameterization in terms of Wronskian determinants of Q-functions. As a check for the new equations, we reproduce the numerical results for the Konishi operator previously obtained from the original infinite Y-system.

We present a set of functional equations defining the anomalous dimensions of arbitrary local single trace operators in planar N = 4 SYM theory. It takes the form of a Y-system based on the integrability of the dual superstring σ-model on the AdS5 × S 5 background. This Y-system passes some very important tests: it incorporates the full asymptotic Bethe ansatz at large length of operator L, including the dressing factor, and it confirms all recently found wrapping corrections. The recently proposed AdS4/CF T3 duality is also treated in a similar fashion.