The Dilatation Operator of N=4 Super Yang-Mills Theory and Integrability

The dilatation generator measures the scaling dimensions of local operators in a conformal field theory. In this thesis we consider the example of maximally supersymmetric gauge theory in four dimensions and develop and extend techniques to derive, investigate and apply the dilatation operator. These tools are perfectly suited for precision tests of the dynamical AdS/CFT conjecture. In particular, they have been successfully applied in the context of strings on plane waves and spinning strings. We construct the dilatation operator by purely algebraic means: Relying on the symmetry algebra and structural properties of Feynman diagrams we are able to bypass involved, higher-loop field theory computations. In this way we obtain the complete one-loop dilatation operator and the planar, three-loop deformation in an interesting subsector. These results allow us to address the issue of integrability within a planar four-dimensional gauge theory: We prove that the complete dilatation generator is integrable at one-loop and present the corresponding Bethe ansatz. We furthermore argue that integrability extends to three-loops and beyond. Assuming that it holds indeed, we finally construct a novel spin chain model at five-loops and propose a Bethe ansatz which might be valid at arbitrary loop-order! We illustrate the use of our technology in several examples and also present two key applications for the AdS/CFT correspondence: We derive the plane-waves string field theory Hamiltonian from the dilatation operator and compute the energy shift on the torus. Furthermore, we use the Bethe ansatz to find scaling dimensions of operators with large quantum numbers. A comparison to the energy of spinning strings shows an intricate functional agreement.