Recent advances in AdS/CFT integrability

Abstract

Integrability has long played a fascinating role in the theoretical exploration of classical and quantum models. Over the last two decades, this was applied with remarkable success to unveiling the structure of certain gauge and string theories and their relation through the AdS/CFT correspondence. Integrability first emerged in the context of the holographic duality between N = 4 supersymmetric Yang-Mills theory (SYM) and strings in AdS 5 × S 5 , proving particularly fruitful and adding to the hope that integrable methods may be useful in solving non-trivial examples of higher dimensional field theories. The description of the exact spectrum of planar anomalous dimensions in N = 4 SYM is a particularly striking example of the power of integrability-based methods to provide insight into non-perturbative regimes of (conformal) gauge theories. These theories display a much richer structure than most previously-studied exactly solvable theories, such as e.g. minimal models, Wess-Zumino-Novikov-Witten models, or two-dimensional relativistic integrable models.Research in AdS/CFT integrability continues to be very active, addressing an ever wider range of physical observables, extending integrable methods to different theories and developing novel mathematical techniques. This activity necessitates up-to-date reviews and pedagogical introductions to the state-of-the-art which is the goal of this special issue.The topics covered in this issue are all the focus of ongoing research and, in addition to their individual interest, underline both the depth and breadth of this subject. They are: (a) The deep connection between quantum integrable models and the theory of ordinary differential equations, the ODE/IM correspondence, and the application to the theory of embedded surfaces in higher-dimensional manifolds by Patrick E Dorey, Clare Dunning, Stefano Negro and Roberto Tateo. (b) The computation of one-point functions in defect conformal field theories by means of integrable spin-chain methods by Marius de Leeuw. (c) The use of classical integrability to compute semi-classical three-point functions of strings in AdS spaces by Shota Komatsu. (d) The application of integrable methods to compute Wilson loops in N = 4 SYM theory by Hagen Münkler. (e) The AdS/CFT 'quantum spectral curve', which is a new powerful method for the exact computation of planar anomalous dimensions in N = 4 SYM by means of a Riemann-Hilbert problem by Fedor Levkovich-Maslyuk. (f) Four dimensional N = 2 super-conformal field theories, their representation theory and the construction of spin chains for the computation of anomalous dimensions by Elli Pomoni.