On exactly marginal deformations of N = 4 SYM and type-IIB supergravity on AdS₅ × S⁵

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 AdS₅ × S⁵ 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).