Integrability and algebraic curve in AdS/CFT

Vladimir Kazakov


I will review current progress in solving the superconformal 4D Yang-Mills theory (CFT) with N=4 supersymmetries and its dual, the superstring theory on AdS5 x S5 space. On both sides of duality, the signs of integrability are observed. The matrix of conformal dimensions of operators of the Yang-Mills theory behaves as a hamiltonian of a chain of quantum spins living on the group PSU(2,2|4), integrable at least in the perturbation theory. In a special limit of long operators the complete perturbative solution is described by an explicit algebraic curve. The superstring theory is classically integrable, and we construct its finite gap solution in terms of a similar algebraic curve.The two curves even coincide in a special limit, which is a nontrivial manifestation of the famous AdS/CFT correspondence.