By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

In this thesis we consider certain supersymmetric string theories on curved backgrounds containing the three-dimensional anti-de Sitter space, AdS3. On the one hand, these are gravity theories in three dimensions, which makes them simpler than real-world gravity but still rich in interesting phenomena. On the other hand, according to the holographic duality, each of these gravitational theories should be equivalent to a suitable quantum field theory with conformal symmetry in two dimensions, a CFT2. In that low dimension, such conformal field theories enjoy an infinite-dimensional Virasoro symmetry which strongly constrains them, but remain non-trivial. Our study is confined to a particular regime, the 't Hooft limit, in which the strings propagate freely. Free string theory can be described as a non-linear sigma model from the string worldsheet (geometrically, a two-dimensional cylinder) into a target space of the form AdS3xM, where M is a manifold without boundary. For us, M is a product of circles and three-dimensional spheres, namely M=S3xT4 or M=S3xS3xS1, both of which enjoy 16 supersymmetries. For the sake of brevity, our presentation is mostly focused on the massive sector of the former geometry, while more general cases are discussed in the concluding chapter. Our main result is to describe how these theories are integrable at the quantum level, i.e. how their worldsheet scattering matrix factorises and their spectrum can be efficiently computed by Bethe ansatz techniques. This is done both from the point of view of the worldsheet theory and of a dual spin chain picture. We also discuss how this procedure is supported by overwhelming evidence from independent (perturbative or semiclassical) calculations, up to two loops in the worldsheet theory expansion.

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Towards integrability for AdS3/CFT2

Scale invariance of the η-deformed AdS5 × S5 superstring, T-duality and modified type II equations

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The Hidden Symmetries of the ADS 5 × S 5 Superstring

We consider Jordanian deformations of the AdS5 × S5 superstring action. These deformations correspond to non-standard q-deformations. In particular, it is possible to perform a partial deformation, for example, of the AdS5 part only, or of the S5 part only. Then the classical action and the Lax pair are constructed with a linear, twisted and extended R operator. It is shown that the action preserves the κ-symmetry.

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Jordanian deformations of the AdS5 × S5 superstring

We consider Jordanian deformations of the AdS_5xS^5 superstring action. The deformations correspond to non-standard q-deformation. In particular, it is possible to perform partial deformations, for example, only for the S^5 part. Then the classical action and the Lax pair are constructed with a linear, twisted and extended R operator. It is shown that the action preserves the kappa-symmetry.

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Jordanian deformations of the AdS_5xS^5 superstring

We consider γ-deformations of the AdS5×S5 superstring as Yang-Baxter sigma models with classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). An essential point is that the classical r-matrices are composed of Cartan generators only and then generate abelian twists. We present examples of the r-matrices that lead to real γ-deformations of the AdS5×S5 superstring. Finally we discuss a possible classification of integrable deformations and the corresponding gravity solution in terms of solutions of CYBE. This classification may be called the gravity/CYBE correspondence.

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Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence

We derive the gravity duals of noncommutative gauge theories from the Yang-Baxter sigma model description of the AdS5 × S5 superstring with classical r-matrices. The corresponding classical r-matrices are 1) solutions of the classical Yang-Baxter equation (CYBE), 2) skew-symmetric, 3) nilpotent and 4) abelian. Hence these should be called abelian Jordanian deformations. As a result, the gravity duals are shown to be integrable deformations of AdS5 × S5. Then, abelian twists of AdS5 are also investigated. These results provide a support for the gravity/CYBE correspondence proposed in arXiv:1404.1838.

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Takuya Matsumoto

Papers

Jordanian deformations of the AdS_5xS^5 superstring

Jordanian deformations of the AdS5 × S5 superstring

Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence

Integrability of classical strings dual for noncommutative gauge theories

Jordanian deformations of the AdS5×S5 superstring