We discuss the spectrum of a string propagating on η-deformed AdS 5 × S 5 by treating its world-sheet theory as an integrable quantum field theory. The exact S-matrix of this field theory is given by a q-deformation of the AdS 5 × S 5 world-sheet S-matrix with real deformation parameter. By considering mirror (double Wick-rotated) versions of these world-sheet theories we give the Thermodynamic Bethe Ansatz description of their exact finite size spectra. Interestingly, this class of models maps onto itself under the mirror transformation. At the level of the string this appears to say that the light-cone worldsheet theories of strings on particular pairs of backgrounds are related by a double Wick rotation, a feature we call 'mirror duality'. We provide a partial check of these statements at the level of the sigma model by considering reduced actions and their corresponding (mirror) giant magnon solutions.

We give an AdS/CFT interpretation to homogeneous Yang-Baxter deformations of the AdS 5 × S 5 superstring as noncommutative deformations of the dual gauge theory, going well beyond the canonical noncommutative case. These homogeneous Yang-Baxter deformations can be of so-called abelian or jordanian type. While abelian deformations have a clear interpretation in string theory and many already had well understood gauge theory duals, jordanian deformations appear novel on both counts. We discuss the symmetry structure of the deformed string from the uniformizing perspective of Drinfeld twists and indicate that this structure can be realized on the gauge theory side by considering theories on various noncommutative spaces. We then conjecture that these are the gauge theory duals of our strings, modulo subtleties involving singularities. We support this conjecture by a brane construction for two jordanian examples, corresponding to noncommutative spaces with [x − , x i ] ∼ x i (i = 1, 2). We also discuss κ-Minkowski type deformations of AdS 5 × S 5 , one of which may be the gravity dual of gauge theory on spacelike κ-Minkowski space. arXiv:1506.01023v5 [hep-th]

Doing a double Wick rotation in the worldsheet theory of the light cone AdS5 × S 5 superstring results in an inequivalent, so-called mirror theory that plays a central role in the field of integrability in AdS/CFT. We show that this mirror theory can be interpreted as the light cone theory of a free string on a different background. This background is related to dS5 × H 5 by a double T duality, and has hidden supersymmetry. The geometry can also be extracted from an integrable deformation of the AdS5 × S 5 sigma model, and we prove the observed mirror duality of these deformed models at the bosonic level as a byproduct. While we focus on AdS5 × S 5 , our results apply more generally.PACS numbers: 11.25.TqIntegrability has and continues to be of central importance in furthering our understanding of the AdS/CFT correspondence [1], giving important insights into finite coupling quantum field theory. Through fruitful interplay between results on both sides of the correspondence, remarkable progress has been made in the spectral problem in particular [2]. Namely, we can describe scaling dimensions in planar N = 4 supersymmetric Yang-Mills theory (SYM) at finite 't Hooft coupling through the corresponding energy levels of a string on AdS 5 × S 5 , and these energy levels can be computed exactly thanks to the integrability of the string [3]. More precisely, under the assumption that integrability persists at the quantum level, the spectrum of the AdS 5 × S 5 superstring can be determined by means of the thermodynamic Bethe ansatz applied to a doubly Wick rotated version of its worldsheet theory [4,5], as put forward in [6] and worked out in [7][8][9][10][11]. Since the light cone gauge fixed AdS 5 × S 5 string is not Lorentz invariant however, this double Wick rotation results in an inequivalent quantum field theory, the socalled mirror theory [7]. The associated mirror transformation also appears extensively in the exact description of polygonal Wilson loops or equivalently planar scattering amplitudes [12][13][14]. Given the central importance of the mirror theory, we would like to elevate it beyond the status of a technical tool. This raises a question that has gone unanswered since the model's introduction in 2007, namely whether the mirror theory itself can arise directly by light cone gauge fixing a free string on some background. Here we show that this is the case, giving an interesting relation between strings on different backgrounds different from other well known dualities. Our results extend to the integrable deformation of the AdS 5 × S 5 superstring of [15], show an intriguing link to de Sitter space, and indicate new backgrounds with hidden supersymmetry in the associated string sigma model. Our construction is based on a simple observation regarding the bosonic light cone string action that directly produces the desired 'mirror' metric for a fairly generic class of backgrounds. The doubly Wick-rotated light cone action is obtained by a formal exchange and inversion of the metric components of the two di...

Interesting deformations of AdS 5 × S 5 such as the gravity dual of noncommutative SYM and Schödinger spacetimes have recently been shown to be integrable. We clarify questions regarding the reality and integrability properties of the associated construction based on R matrices that solve the classical Yang-Baxter equation, and present an overview of manifestly real R matrices associated to the various deformations. We also discuss when these R matrices should correspond to TsT transformations, which not all do, and briefly analyze the symmetries preserved by these deformations, for example finding Schrödinger superalgebras that were previously obtained as subalgebras of psu(2, 2|4). Our results contain a (singular) generalization of an apparently non-TsT deformation of AdS 5 × S 5 , whose status as a string background is an interesting open question.

We introduce a series of articles reviewing various aspects of integrable models relevant to the AdS/CFT correspondence. Topics covered in these reviews are: classical integrability, Yangian symmetry, factorized scattering, the Bethe ansatz, the thermodynamic Bethe ansatz, and integrable structures in (conformal) quantum field theory. In the present article we highlight how these concepts have found application in AdS/CFT, and provide a brief overview of the material contained in this series. -mails: diegobombardelli@gmail.com, alessandra.cagnazzo@desy.de, rouven.frassek@durham.ac.uk, fedor.levkovich@gmail.com, loebbert@physik.hu-berlin.de, stefano.negro@lpt.ens.fr, sfondria@itp.phys.ethz.ch, i.m.szecsenyi@durham.ac.uk, svantongeren@physik.hu-berlin.de, a.torrielli@surrey.ac.uk arXiv:1606.02945v2 [hep-th] Jul 2016In this article we introduce a series of articles reviewing aspects of integrable models. The articles provide a pedagogical introduction to the topic of integrability, with special emphasis on methods relevant in the AdS/CFT correspondence. After a brief motivation regarding the value of general integrable models in the development of theoretical physics, here we discuss the application of the framework of integrability to the AdS/CFT correspondence.We then provide an overview of the material contained in the various reviews, referring back to AdS/CFT applications, and indicating links between the reviews themselves and to the relevant literature. While written with an AdS/CFT background in mind, the methods covered in the reviews themselves have applications throughout the wider field of integrability. IntegrabilityIntegrable models appear throughout theoretical physics, starting from classical mechanics where models such as the Kepler problem can be solved-in the sense of the Liouville theorem-by integration. In general, integrable models show special behaviour due to many underlying symmetries, symmetries due to which they can often be exactly solved. Only a fraction of the physical systems appearing in nature can be described in these terms. Nevertheless, integrable models offer insight into real-world situations through universality, or when used as a theoretical laboratory to develop new ideas. In statistical mechanics for example, many subtleties of the thermodynamic limit have been understood by working out specific models, notably phase transitions in the Lenz-Ising model and the role of boundary conditions in the ice model. In hydrodynamics, the Korteweg-de Vries equation illustrates how a nonlinear partial differential equation can admit stable, wave-like localized solutions: solitons. In condensed matter physics, both integrable quantum spin chains and one-dimensional gases of almost-free particles play a pivotal role. Finally, in quantum field theories (QFTs) in two space-time dimensions, exactly solvable models helped unravel phenomena like dimensional transmutation, as in the case of the chiral Gross-Neveu model, or concepts like bosonisation, as in the case of the sine-Go...