Abstract:Année de publication : 2012
In this thesis is given a review of the methods of integrability in the context of the AdS/CFT correspondence. We investigate integrable structures on both sides of the AdS/CFT duality using different methods. On the string side of the duality we observe how the supersymmetry and automorphism of the symmetry group organize the model into integrable one. Then, using the consequences of the finite gap method for the integrable system we perform a one– loop quantization procedure which allows us to compute the one–loop spectrum of the model. We illustrate this method by computing the spectrum of a short string. On the gauge side we review the method of the functional Y –system equations for computing the spectrum of the theory in the finite volume. Due to the existence of the two–particle S–matrix it is possible to use the Zamolodchikov’s trick to setup a system of functional equations, which can be later recast as a Hirota equation defined on some domain. In the strong coupling limit these equations can be drastically simplified. This gives us a chance to have an analytic solution of them, which can be compared to the string side computation. These two results are in a perfect agreement.