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

We argue that existing methods for the perturbative computation of anomalous dimensions and the disentanglement of mixing in N=4 gauge theory can be considerably simplified, systematized and extended by focusing on the theory's dilatation operator. The efficiency of the method is first illustrated at the one-loop level for general non-derivative scalar states. We then go on to derive, for pure scalar states, the two-loop structure of the dilatation operator. This allows us to obtain a host of new results. Among these are an infinite number of previously unknown two-loop anomalous dimensions, novel subtleties concerning 't Hooft's large N expansion due to mixing effects of degenerate single and multiple trace states, two-loop tests of various protected operators, as well as two-loop non-planar results for two-impurity operators in BMN gauge theory. We also put to use the recently discovered integrable spin chain description of the planar one-loop dilatation operator and show that the associated Yang-Baxter equation explains the existence of a hitherto unknown planar "axial" symmetry between infinitely many gauge theory states. We present evidence that this integrability extends to two loops and higher, with intriguing consequences for gauge theory and, perhaps, condensed matter theory. Assuming that the integrability structure extends to more than two loops, we e.g. determine the planar three-loop contribution to the dilatation operator