DGLAP and BFKL equations in the N = 4 supersymmetric gauge theory

We derive DGLAP and BFKL evolution equations in the N = 4 supersymmetric gauge theory in the next-to-leading approximation. The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin |n|. Its analytic continuation to negative |n| in the leading logarithmic approximation allows us to obtain the residues of the anomalous dimensions of the twist-2 operators in the non-physical points j = 0,−1, ... from the BFKL equation in an agreement with their direct calculation from the DGLAP equation in the same approximation. As a result, in the multi-colour limit the BFKL and DGLAP dynamics for an arbitrary number of particles in N = 4 SUSY turns out to be integrable. In the next-to-leading approximation the holomorphic separability of the BFKL kernel is violated, but the corresponding Bethe-Salpeter kernel has the property of the hermitian separability. We investigate also in this approximation possible relations between the DGLAP and BFKL equations.