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We discuss homogeneous Yang–Baxter deformations of integrable sigma models in terms of twist operators. We show that the twist operators behave as the classical analogue of a Drinfeld twist, for all abelian and almost abelian deformations. We also use twist operators to rederive the well-known interpretation of TsT transformations—equivalent to abelian deformations—in terms of twisted boundary conditions. We discuss complications in extending this boundary condition picture to non-abelian deformations. [ABSTRACT FROM AUTHOR]

The η deformation of the superstring depends on a non-split r matrix for the superalgebra . Much of the investigation into this model has considered one particular choice, however there are a number of inequivalent alternatives. This is also true for the bosonic sector of the theory with , the isometry algebra of , admitting one split and three non-split r matrices. In this article we explore these r matrices and the corresponding geometries. We investigate their contraction limits, comment on supergravity backgrounds and demonstrate their relation to gauged-WZW deformations. We then extend the three non-split cases to and compute four separate bosonic two-particle tree-level S-matrices based on inequivalent BMN-type light-cone gauges. The resulting S-matrices, while different, are related by momentum-dependent one-particle changes of basis. [ABSTRACT FROM AUTHOR]

We consider various homogeneous Yang–Baxter deformations of the superstring that can be obtained from the η-deformed superstring and related models by singular boosts. The jordanian deformations we obtain in this way behave similarly to the η-deformed model with regard to supergravity: T dualizing the classical sigma model it is possible to find corresponding solutions of supergravity, which, however, have dilatons that prevent T dualizing back. Hence the backgrounds of these jordanian deformations are not solutions of supergravity. Still, they do satisfy a set of recently found modified supergravity equations which implies that the corresponding sigma models are scale invariant. The abelian models that we obtain by singular boosts do directly correspond to solutions of supergravity. In addition to our main results we consider contraction limits of our main example, which do correspond to supergravity solutions. [ABSTRACT FROM AUTHOR]

We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi–Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing first on the one-dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the chiral Gross–Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular the chiral Gross–Neveu model. We moreover discuss the simplification of TBA equations to Y systems, including the transition back to integral equations given sufficient analyticity data, in simple examples. [ABSTRACT FROM AUTHOR]

We introduce a series of articles reviewing various aspects of integrable models relevant to the anti-de Sitter (AdS)/conformal quantum field theory (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. [ABSTRACT FROM AUTHOR]

This article reviews the application of integrability to the spectral problem of strings on and its deformations. We begin with a pedagogical introduction to integrable field theories culminating in the description of their finite-volume spectra through the thermodynamic Bethe ansatz (TBA). Next, we apply these ideas to the string and in later sections discuss how to account for particular integrable deformations. Through the AdS/CFT correspondence this gives an exact description of anomalous scaling dimensions of single trace operators in planar supersymmetry Yang–Mills theory, its ‘orbifolds’, and β and γ-deformed supersymmetric Yang–Mills theory. We also touch upon some subtleties arising in these deformed theories. Furthermore, we consider complex excited states (bound states) in the sector and give their TBA description. Finally we discuss the TBA for a quantum deformation of the superstring S-matrix, with close relations to among others Pohlmeyer reduced string theory, and briefly indicate more recent developments in this area. [ABSTRACT FROM AUTHOR]