Supersymmetric Three-Form Flux Perturbations on $AdS_5$

We consider warped type IIB supergravity solutions with three-form flux and ${\cal N}=1$ supersymmetry, which arise as the supergravity duals of confining gauge theories. We first work in a perturbation expansion around $AdS_5 \times S^5$, as in the work of Polchinski and Strassler, and from the ${\cal N}=1$ conditions and the Bianchi identities recover their first-order solution generalized to an arbitrary ${\cal N}=1$ superpotential. We find the second order dilaton and axion by the same means. We also find a simple family of exact solutions, which can be obtained from solutions found by Becker and Becker, and which includes the recent Klebanov--Strassler solution.