We derive the one loop mixing matrix for anomalous dimensions in N = 4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.

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 discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N = 4 super YangMills and the energy of their dual semiclassical string states in AdS 5 × S 5 . The anomalous dimensions can be computed using a set of Bethe equations, which for "long" operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU (2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.

This is the foreword to the special volume on localization techniques in quantum field theory. The summary of individual chapters is given and their interrelation is discussed.

We apply recently developed integrable spin chain and dilatation operator techniques in order to compute the planar one-loop anomalous dimensions for certain operators containing a large number of scalar fields in N = 4 Super Yang-Mills. The first set of operators, belonging to the SO(6) representations [J, L − 2J, J], interpolate smoothly between the BMN case of two impurities (J = 2) and the extreme case where the number of impurities equals half the total number of fields (J = L/2). The result for this particular [J, 0, J] operator is smaller than the anomalous dimension derived by Frolov and Tseytlin [hep-th/0304255] for a semiclassical string configuration which is the dual of a gauge invariant operator in the same representation. We then identify a second set of operators which also belong to [J, L − 2J, J] representations, but which do not have a BMN limit. In this case the anomalous dimension of the [J, 0, J] operator does match the Frolov-Tseytlin prediction. We also show that the fluctuation spectra for this [J, 0, J] operator is consistent with the string prediction. * also at ITEP, Moscow, Russia.

We obtain the elliptic curve corresponding to an N = 2 superconformal field theory which has an E 6 global symmetry at the strong coupling point τ = e πi/3 . We also find the Seiberg-Witten differential λ SW for this theory. This differential has 27 poles corresponding to the fundamental representation of E 6 . The complex conjugate representation has its poles on the other sheet. We also show that the E 6 curve reduces to the D 4 curve of Seiberg andWitten. Finally, we compute the monodromies and use these to compute BPS masses in an F -Theory compactification. 8/96

We obtain the elliptic curve and the Seiberg-Witten differential for an N = 2 superconformal field theory which has an E 8 global symmetry at the strong coupling point τ = e πi/3 .The differential has 120 poles corresponding to half the charged states in the fundamental representation of E 8 , with the other half living on the other sheet. Using this theory, we flow down to E 7 , E 6 and D 4 . A new feature is a λ SW for these theories based on their adjoint representations. We argue that these theories have different physics than those with λ SW built from the fundamental representations.10/96

We study the anomalous dimensions for scalar operators for a threedimensional Chern-Simons theory recently proposed in arXiv:0806.1218. We show that the mixing matrix at two-loop order is that for an integrable Hamiltonian of an SU(4) spin chain with sites alternating between the fundamental and the anti-fundamental representations. We find a set of Bethe equations from which the anomalous dimensions can be determined and give a proposal for the Bethe equations to the full superconformal group of OSp(2, 2|6).