### Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries

, 2011

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### arXiv:1104.2474v2 [hep-th] 6 Jul 2011

"... Abstract. This review is meant to be an account of the properties of the infinitedimensional quantum group (specifically, Yangian) symmetry lying behind the integrability of the AdS/CFT spectral problem. In passing, the chance is taken to give a concise anthology of basic facts concerning Yangians ..."

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Abstract. This review is meant to be an account of the properties of the infinitedimensional quantum group (specifically, Yangian) symmetry lying behind the integrability of the AdS/CFT spectral problem. In passing, the chance is taken to give a concise anthology of basic facts concerning Yangians and integrable systems, and to store a series of remarks, observations and proofs the author has collected in a fiveyear span of research on the subject. We hope this exercise will be useful for future attempts to study Yangians in field and string theories, with or without supersymmetry 1 .

### Review of AdS/CFT Integrability, Chapter II.2: Quantum Strings in AdS5×S5

, 2010

"... We review the semiclassical analysis of strings in AdS5×S5 with a focus on the relationship to the underlying integrable structures. We discuss the perturbative calculation of energies for strings with large charges, using the folded string spinning in AdS3 ⊂ AdS5 as our main example. Furthermore, ..."

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We review the semiclassical analysis of strings in AdS5×S5 with a focus on the relationship to the underlying integrable structures. We discuss the perturbative calculation of energies for strings with large charges, using the folded string spinning in AdS3 ⊂ AdS5 as our main example. Furthermore, we review the perturbative light-cone quantization of the string theory and the calculation of the worldsheet S-matrix.

### Imperial-TP-BH-2011-01 Towards the quantum S-matrix of the Pohlmeyer reduced version

"... We investigate the structure of the quantum S-matrix for perturbative excitations of the Pohlmeyer reduced version of the AdS5×S5 superstring following arXiv:0912.2958. The reduced theory is a fermionic extension of a gauged WZW model with an integrable potential. We use as an input the result of th ..."

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We investigate the structure of the quantum S-matrix for perturbative excitations of the Pohlmeyer reduced version of the AdS5×S5 superstring following arXiv:0912.2958. The reduced theory is a fermionic extension of a gauged WZW model with an integrable potential. We use as an input the result of the one-loop perturbative scattering amplitude computation and an analogy with simpler reduced AdSn×Sn theories with n = 2, 3. The reduced AdS2 × S2 theory is equivalent to the N = 2 2-d supersymmetric sine-Gordon model for which the exact quantum S-matrix is known. In the reduced AdS3 × S3 case the one-loop perturbative S-matrix, improved by a contribution of a local counterterm, satisfies the group factorization property and the Yang-Baxter equation, and reveals the existence of a novel quantum-deformed 2-d supersymmetry which is not manifest in the action. The one-loop perturbative S-matrix of the reduced AdS5 × S5 theory has the group factorisation property but does not satisfy the Yang-Baxter equation suggesting some subtlety with the realisation of quantum integrability. As a possible resolution, we propose that the S-matrix of this theory may be identified with the quantum-deformed [psu(2|2)]2 n R2 symmetric R-matrix constructed in arXiv:1002.1097. We conjecture the exact all-order form of this S-matrix and discuss its possible relation to the perturbative S-matrix defined by the path integral. As in the AdS3 × S3 case the symmetry of the S-matrix may be interpreted as an extended quantum-deformed 2-d supersymmetry.

### IMPERIAL-TP-AR-2010-2, arxiv:1012.3985 overview article: arxiv:1012.3982 Review of AdS/CFT Integrability, Chapter I.3: Long-range spin chains

"... Abstract: In this contribution we review long-range integrable spin chains that origi-nate from the recently discovered integrability in the planar AdS/CFT correspondence. We also briefly summarise the theory of generic integrable perturbatively long-range spin chains. ar X iv ..."

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Abstract: In this contribution we review long-range integrable spin chains that origi-nate from the recently discovered integrability in the planar AdS/CFT correspondence. We also briefly summarise the theory of generic integrable perturbatively long-range spin chains. ar X iv

### and

"... Abstract: This review is devoted to the classical integrability of the AdS5 × S5 super-string theory. It starts with a reminder of the corresponding action as a coset model. The symmetries of this action are then reviewed. The classical integrability is then considered from the lagrangian and hamilt ..."

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Abstract: This review is devoted to the classical integrability of the AdS5 × S5 super-string theory. It starts with a reminder of the corresponding action as a coset model. The symmetries of this action are then reviewed. The classical integrability is then considered from the lagrangian and hamiltonian points of view. The second part of this review deals with the gauge fixing of this theory. Finally, some aspects of the pure spinor formulation are also briefly reviewed. ar X iv

### On Yangian Symmetry in Planar N = 4 SYM

"... Planar N = 4 supersymmetric Yang–Mills theory appears to be pertur-batively integrable. This work reviews integrability in terms of a Yan-gian algebra and compares the application to the problems of anoma-lous dimensions and scattering amplitudes. 1 ..."

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Planar N = 4 supersymmetric Yang–Mills theory appears to be pertur-batively integrable. This work reviews integrability in terms of a Yan-gian algebra and compares the application to the problems of anoma-lous dimensions and scattering amplitudes. 1

### UMTG--269, arxiv:1012.3991 overview article: arxiv:1012.3982 Review of AdS/CFT Integrability, Chapter III.2: Exact world-sheet S-matrix

"... Abstract: We review the derivation of the S-matrix for planar N = 4 supersymmet-ric Yang-Mills theory and type IIB superstring theory on an AdS5 × S5 background. After deriving the S-matrix for the su(2) and su(3) sectors at the one-loop level based on coordinate Bethe ansatz, we show how su(2|2) sy ..."

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Abstract: We review the derivation of the S-matrix for planar N = 4 supersymmet-ric Yang-Mills theory and type IIB superstring theory on an AdS5 × S5 background. After deriving the S-matrix for the su(2) and su(3) sectors at the one-loop level based on coordinate Bethe ansatz, we show how su(2|2) symmetry leads to the exact asymp-totic S-matrix up to an overall scalar function. We then briefly review the spectrum of bound states by relating these states to simple poles of the S-matrix. Finally, we review the derivation of the asymptotic Bethe equations, which can be used to determine the asymptotic multiparticle spectrum. ar X iv