  High-energy behavior of amplitudes in a gauge theory can be reformulated in
terms of the evolution of Wilson-line operators. In the leading logarithmic
approximation it is given by the conformally invariant BK equation for the
evolution of color dipoles. In QCD, the next-to-leading order BK equation has
both conformal and non-conformal parts, the latter providing the running of the
coupling constant. To separate the conformally invariant effects from the
running-coupling effects, we calculate the NLO evolution of the color dipoles
in the conformal ${\cal N}$=4 SYM theory. We define the "composite dipole
operators" with the rapidity cutoff preserving conformal invariance. The
resulting M\"obius invariant kernel for these operators agrees with the forward
NLO BFKL calculation of Ref. 1.
