Pedro Vieira, Laboratoire de Physique Théorique de l'École Normale Supérieure
Integrability and AdS/CFT
The AdS/CFT duality  is a conjecture relating string theory in some curved background and a certain four dimensional gauge theory which can be regarded as a toy model for QCD. The perturbative (easy) regime of one of the theories is dual to the strongly coupled (hard) regime of the dual theory. Thus, we might wonder whether it could be possible to use this conjecture as an inspiration/guide to solve each of the theories exactly? A priori this might seem a superhuman task but, since (four years ago) an integrable structure behind these theories was found , the distance towards this goal is being reduced at an amazing pace. I will try to explain the Bethe ansatz formalism (appearing naturally from the Gauge theory side ) and the algebraic curve construction (emerging in the string theory side ) and I will explain the motivation behind the various conjectures  which seem to connect these two formalisms. In this way I will try to transmit where we are (that is, what is proven and what is conjectured) and what are the most likely advances in the short term (that is, in which direction is this science moving).
The following list of references is obviously not extensive:
- The Large $N$ limit of superconformal field theories and supergravity. Juan M. Maldacena hep-th/9711200 Gauge theory correlators from noncritical string theory. S.S. Gubser, Igor R. Klebanov, Alexander M. Polyakov hep-th/9802109 Anti-de Sitter space and holography. Edward Witten hep-th/9802150.
- The Bethe ansatz for $N=4$ superYang-Mills. J.A. Minahan, K. Zarembo hep-th/0212208. Hidden symmetries of the $AdS(5) \times S^5$ superstring. Iosif Bena, Joseph Polchinski, Radu Roiban hep-th/0305116
- The Bethe ansatz for $N=4$ superYang-Mills. J.A. Minahan, K. Zarembo hep-th/0212208 The Dilatation operator of $N=4$ super Yang-Mills theory and integrability. Niklas Beisert hep-th/0407277.
- Classical/quantum integrability in AdS/CFT. V.A. Kazakov, A. Marshakov, J.A. Minahan, K. Zarembo hep-th/0402207. The Algebraic curve of classical superstrings on $AdS(5) \times S^5$. N. Beisert, V.A. Kazakov, K. Sakai, K. Zarembo hep-th/0502226.Transcendentality and Crossing. Niklas Beisert, Burkhard Eden, Matthias Staudacher hep-th/0610251 A Crossing-symmetric phase for $AdS(5) \times S^5$ strings. Niklas Beisert, Rafael Hernandez, Esperanza Lopez hep-th/0609044.