Quantum integrability and functional equations: applications to the spectral problem of AdS/CFT and two-dimensional sigma models

In this review, a general procedure to represent the integral Bethe Ansatz equations in the form of the Reimann-Hilbert problem is given. This allows us to study integrable spin chains in the thermodynamic limit in a simple way. Based on the functional equations we give the procedure that allows finding the subleading orders in the solution of various integral equations solved to the leading order by the Wiener-Hopf techniques. The integral equations are studied in the context of the AdS/CFT correspondence, where their solution allows for the verification of the integrability conjecture up to two loops of the strong coupling expansion. In the context of the two-dimensional sigma models we analyze the large-order behavior of the asymptotic perturbative expansion. The obtained experience with the functional representation of the integral equations also allowed us to solve explicitly the crossing equations that appear in the AdS/CFT spectral problem. (review)