Supersymmetric three-form flux perturbations onAdS5

We consider warped type IIB supergravity solutions with three-form flux and $\mathcal{N}=1\mathrm{}$ supersymmetry, which arise as the supergravity duals of confining gauge theories. We first work in a perturbation expansion around ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$, as in the work of Polchinski and Strassler, and from the $\mathcal{N}=1\mathrm{}$ conditions and the Bianchi identities recover their first-order solution generalized to an arbitrary $\mathcal{N}=1\mathrm{}$ 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.