  N=4 supersymmetric Yang-Mills theory with gauge group SU(n) (n>=3) is
believed to have two exactly marginal deformations which break the
supersymmetry to N=1. We discuss the construction of the string theory dual to
these deformations, in the supergravity approximation, in a perturbation series
around the AdS_5 * S^5 solution. We construct explicitly the deformed solution
at second order in the deformation. We show that deformations which are
marginal but not exactly marginal lead to a non-conformal solution with a
logarithmically running coupling constant. Surprisingly, at third order in the
deformation we find the same beta functions for the couplings in field theory
and in supergravity, suggesting that the leading order beta functions (or
anomalous dimensions) do not depend on the gauge coupling (the coefficient is
not renormalized).
