I will discuss some exact results in supersymmetric field theories, focussing on the localization technique in supersymmetric gauge theories and its interplay with holography.

After briefly reviewing the notion of geometries characterised by SU(3) structures, I will discuss a particular class known (in String Theory) as non-Kahler. I will explain how this arises both in Type II and Heterotic Supergravities. Two explicit constructions of these geometries will be discussed. One is a one-parameter solution corresponding to fivebranes wrapped on the two-sphere of the resolved conifold, that can be thought of as a non-Kahler analog of the conifold. The other is a general construction of one-parameter non-Kahler deformations of Calabi-Yau manifolds with a U(1) isometry, where the non-Abelian Yang-Mills field of the Heterotic is non-trivial. The presentation will be loosely based on the two papers: arXiv:0906.0591 and arXiv:1010.4031.

I will discuss a supergravity solution corresponding to fivebranes wrapped on the S2 of the resolved conifold. By changing a parameter the solution continuously interpolates between the deformed conifold with flux and the resolved conifold with branes. Therefore, it displays a geometric transition, purely in the supergravity context. The solution is a simple example of torsional geometry and may be thought of as a non-Kahler analog of the conifold. I will discuss how one can obtain supersymmetric solutions of type IIB supergravity starting from simpler non-Kahler geometries, by U-dualities or other methods. Applying these transformations to the torsional conifold solution we obtain a solution dual to the baryonic branch of the Klebanov-Strassler theory. Far along the baryonic branch the solution resembles D5 branes wrapping a fuzzy two-sphere in the resolved conifold and this can be matched to a weakly coupled field theory analysis.