  In this work we explore many directions in the framework of gauge-gravity
dualities. In type IIB theory we give an explicit derivation of the local
metric for five branes wrapped on rigid two-cycles. Our derivation involves
various interplays between warp factors, dualities and fluxes and the final
result confirms our earlier predictions. We also find a novel dipole-like
deformation of the background due to an inherent orientifold projection in the
full global geometry. The supergravity solution for this deformation takes into
account various things like the presence of a non-trivial background topology
and fluxes as well as branes. Considering these, we manage to calculate the
precise local solution using equations of motion. We also show that this
dipole-like deformation has the desired property of decoupling the Kaluza-Klein
modes from the IR gauge theory. Finally, for the heterotic theory we find new
non-Kahler complex manifolds that partake in the full gauge-gravity dualities
and study the mathematical structures of these manifolds including the torsion
classes, Betti numbers and other topological data.
