Review of AdS/CFT Integrability, Chapter II.3: Sigma Model, Gauge Fixing

This review is devoted to the classical integrability of the AdS ₅ × S⁵ superstring 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.     

Review of AdS/CFT Integrability, Chapter III.3: The Dressing Factor

We review the construction of the AdS/CFT dressing factor, its analytic properties, and several checks of its validity.     

Review of AdS/CFT Integrability,Chapter VI.1: Superconformal Symmetry

Letters in Mathematical Physics - Aspects of the D = 4, $${\mathcal N=4}$$ superconformal symmetry relevant to the AdS/CFT duality and integrability are reviewed. These include...     

Review of AdS/CFT Integrability. Chapter VI.2: Yangian Algebra

Letters in Mathematical Physics - We review the study of Hopf algebras, classical and quantum R-matrices, infinite-dimensional Yangian symmetries and their representations in the context of...     

Review of AdS/CFT Integrability, Chapter I.3: Long-Range Spin Chains

In this contribution, we review long-range integrable spin chains that originate 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.     

Review of AdS/CFT Integrability,Chapter V.2: Dual Superconformal Symmetry

Letters in Mathematical Physics - Scattering amplitudes in planar $${\mathcal{N}=4}$$ super Yang–Mills theory reveal a remarkable symmetry structure. In addition to the superconformal...     

Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz

The aim of the chapter is to introduce in a pedagogical manner the concept of Thermodynamic Bethe Ansatz designed to calculate the energy levels of finite volume integrable systems and to review how it is applied in the planar AdS/CFT setting.     

Review of AdS/CFT Integrability, Chapter III.4: Twist States and the Cusp Anomalous Dimension

We review the computation of the anomalous dimension of twist operators in the planar limit of ${\mathcal{N}=4}$ SYM using the asymptotic Bethe ansatz and demonstrate how this quantity is obtained at weak, strong and intermediate values of the coupling constant. The anomalous dimension of twist operators in the limit of large Lorentz spin played a major role in the construction as well as in many tests of the asymptotic Bethe equations; this aspect of the story is emphasized.     

Review of AdS/CFT Integrability, Chapter I.2: The Spectrum from Perturbative Gauge Theory

We review the constructions and tests of the dilatation operator and of the spectrum of composite operators in the flavour SU(2) subsector of ${\mathcal{N}=4}$ SYM in the planar limit by explicit Feynman graph calculations with emphasis on analyses beyond one loop. From four loops on, the dilatation operator determines the spectrum only in the asymptotic regime, i.e. to a loop order which is strictly smaller than the number of elementary fields of the composite operators. We review also the calculations which take a first step beyond this limitation by including the leading wrapping corrections.     

Review of AdS/CFT Integrability,Chapter V.3: Scattering Amplitudes at Strong Coupling

Letters in Mathematical Physics - We review the computation of scattering amplitudes of planar maximally super-symmetric Yang–Mills at strong coupling. By using the AdS/CFT duality the...     

Review of AdS/CFT Integrability,Chapter V.1: Scattering Amplitudes: A Brief Introduction

Letters in Mathematical Physics - We review current efficient techniques for the construction of multi-leg and multi-loop on-shell scattering amplitudes in supersymmetric gauge theories. Examples...     

Review of AdS/CFT Integrability: An Overview

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.     

Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and {\fancyscript{N} <4 } SYM

There is a growing amount of evidence that QCD (and four-dimensional gauge theories in general) possesses a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians but is only revealed on the quantum level. In this review we consider the scale dependence of local gauge invariant operators and high-energy (Regge) behavior of scattering amplitudes to explain that the effective QCD dynamics in both cases is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations.     

Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in {\mathcal{N}=4} Super Yang-Mills

In this chapter of Review of AdS/CFT Integrability we introduce ${\mathcal{N}=4}$ Super Yang-Mills. We discuss the global superalagebra PSU(2, 2|4) and its action on gauge invariant operators. We then discuss the computation of the correlators of certain gauge invariant operators, the so-called single trace operators in the large N limit. We show that interactions in the gauge theory lead to mixing of the operators. We compute this mixing at the one-loop level and show that the problem maps to a one-dimensional spin chain with nearest neighbor interactions. For operators in the SU(2) sector we show that the spin chain is the ferromagnetic Heisenberg spin chain whose eigenvalues are determined by the Bethe equations.     

The Hopf superalgebra of AdS/CFT

We describe the unconventional infinite-dimensional Hopf superalgebra related to the integrable S-matrix of the AdS/CFT correspondence, and discuss its typical and atypical representations.     

The Infinite Curvature Limit of AdS/CFT

Some kinematical speculations on the infinite curvature limit of the conjectured duality of Maldacena between 10-dimensional strings living in AdS ₅ × S ₅ and an ordinary 4-dimensional quantum field theory, namely super Yang–Mills with gauge group SU (N), are given.     

Algebraic Curve for the SO(6) Sector of AdS/CFT

We construct the general algebraic curve of degree four solving the classical sigma model on ℝ×S⁵. Up to two loops it coincides with the algebraic curve for the dual sector of scalar operators in SYM, also constructed here. Membre de l'Institut Universitaire de France