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I will present a first-principles derivation of the AdS5/CFT4 T-system up to first non-trivial order in the large 't Hooft coupling expansion. The proof relies on the computation of quantum effects in the fusion of some special line operators, namely the transfer matrices. This computation is done in the pure spinor formalism for the superstring in AdS5xS5. I will also discuss the generalization of this computation to other integrable 2D CFTs that define string theory in AdS backgrounds.

I will show how to solve the AdS/CFT Y-system in terms of a finite set of nonlinear integral equations (FiNLIE). To uniquely define the solution we impose the set of constraints on the Y- and T-functions which can be summarized as: symmetry (PSU(2,2|4) + Z_4) + analyticity + large volume asymptotics. Some of these constraints describe previously unknown properties of the Y-system. As an important check of our approach, we showed that the proposed constraints can be also used to derive the infinite set of the TBA equations. We also successfully checked FiNLIE numerically for the case of Konishi operator.

The TBA approach to the AdS/CFT spectral problem is used to compute scaling dimensions of several operators dual to two-particle states of the l.c. AdS5 x S5 string theory. The implementation of the psu(2,2|4) symmetry in the TBA framework is discussed.

An object which has been under attack from several fronts is the planar S-matrix of N=4 SYM. One approach towards addressing the computation of scattering amplitudes using integrability is by using an analogues of an Operator Product Expansion for these observables. It is a very general expansion that is based on the dual conformal symmetry of the amplitudes or their dual description in terms of null polygon Wilson loops. In this expansion the Wilson loop/amplitude is viewed as a transition amplitude for flux tube excitations. The flux tube in question is the color flux stretched between two fast moving quarks and the excitation are the excitations of that color flux. In the planar limit, it has an holographic description in terms of a two dimensional world sheet, known as the GKP string. For N=4 SYM, the dynamics of the flux excitation is integrable to all loops.