A companion to "Non-K\"ahler Deformed Conifold, Ultra-Violet Completion and Supersymmetric Constraints in the Baryonic Branch"

In the appropriate limit, a type IIB string theory setup involving D3 branes, wrapped D5 branes, and fluxes on a conifold generally leads to a supergravity background involving a warped version of the conifold with fluxes. We study the supergravity dual of the baryonic branch of the Klebanov Strassler theory by writing down a very general conifold metric--the non-K\"ahler resolved warped-deformed conifold--and a general set of fluxes that satisfy the supergravity equations of motion, and derive the necessary constraints that allow the geometry to be dual to an $\mathcal{N}=1$ supersymmetric gauge theory in $3+1$ dimensions. These backgrounds encompass known solutions, such as the KS, MN and Butti et al. models, but the added layer of generality can lead to a larger class of gauge-gravity dualities. We also present many consistency checks that validate our background matches known cases for certain values in our parameter space. This is a companion paper to arXiv:1805.03676 [hep-th] covering the section 'IR physics, dualities, and supersymmetry'.