Twistor geometrie, integrability and the AdS/ CFT correspondence

The developments of the last decade of research in mathematical and theoretical high energy physics resulted in many relations between string theories and quantum field theories. One of the most important results in this field is the celebrated AdS/CFT duality, which in its strongest form conjectures that the four-dimensional maximally supersymmetric Yang-Mills theory is equivalent to IIB string theory on AdS-space. More recently a different type of relation between the Yang-Mills theory and a string theory on a twistor space has been proposed. Recent developments have indicated that both theories, the maximally supersymmetric Yang-Mills theory and the string theory on the AdS-space, share some aspects of integrability. Within the Yang-Mills theory these integrable structures are observed as perturbative corrections to the dilatation operator, which are of the form of an integrable spin chain Hamiltonian. The proposed project aims at finding an explicit relation between the integrable structures of the Yang-Mills and string theory. For this, the twistor formulation of the Yang-Mills theory will be used to construct in a non- perturbative way infinitely many conserved charges of the Yang-Mills theory. The algebraic structure and symmetry considerations will help to relate the Yang-Mills theory quantities to the integrable structures of the string theory. A one loop calculation should show that the infinitely many conservation laws of the Yang-Mills theory are anomalous.