Using the thermodynamic Bethe ansatz method we derive an infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT. The Y-system conjectured in [1] for the spectrum of all operators in planar N = 4 SYM theory follows from these equations. In particular, we present the integral TBA type equations for the spectrum of all operators within the sl(2) sector. We prove that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics. We find the analogue of DHM formula for the dressing kernel in the mirror kinematics.

This work provides a detailed review of recent developments in the field of AdS/CFT correspondence for a particular subject of the correspondence between type IIB superstring on AdS5 × S5 and the N = 4 super Yang-Mills theory. Through analyzing the bound states of Bethe roots for the corresponding P SU (2,2|4) spin chain, it is shown how the lattice system of functional equations, called Y-system, appears. K e y w o r d s: AdS/CFT, integrable systems, spin chains.