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

We derive the 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 residues of anomalous dimensions gamma of 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. Moreover, in the multi-color limit of the N = 4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. In the next-to-leading approximation the holomorphic separability of the pomeron Hamiltonian is violated, but the corresponding Bethe-Salpeter kernel has the property of a Hermitian separability. The main singularities of anomalous dimensions gamma at j = -r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles. (C) 2003 Elsevier Science B.V. All rights reserved.