Abstract
We discuss the AdS/CFT duality from the perspective of
integrable systems and establish a direct relationship between the
dimension of single trace local operators composed of two types of
scalar fields in 
= 4 super Yang-Mills and the energy of
their dual semiclassical string states in AdS5 × S5. The
anomalous dimensions can be computed using a set of Bethe equations,
which for ``long'' operators reduces to a Riemann-Hilbert problem.
We develop a unified approach to the long wavelength Bethe
equations, the classical ferromagnet and the classical string
solutions in the SU(2) sector and present a general
solution, governed by complex curves endowed with meromorphic
differentials with integer periods. Using this solution we compute
the anomalous dimensions of these long operators up to two loops and
demonstrate that they agree with string-theory predictions.
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