The dilatation operator of N=4 super Yang-Mills theory and integrability

In this work we review recent progress in four-dimensional conformal quantum field theories and scaling dimensions of local operators. Here we consider the example of maximally supersymmetric gauge theory and present techniques to derive, investigate and apply the dilatation operator which measures the scaling dimensions. 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! (C) 2004 Elsevier B.V. All rights reserved.