Geometric transitions, brane dynamics and gauge theories

We consider the interplay between brane constructions and type IIA, IIB or M-theory geometries on Calabi-Yau (CY) and G2 holonomy manifolds. This is related to = 1 (and = 2) gauge theories in four dimensions. We first discuss simple geometric transitions corresponding to brane set ups involving orthogonal (or parallel) Neveu-Schwarz branes that approach each other. This is related to confinement and Seiberg duality in SQCD. In particular, we argue that in type IIA, a 1 of singularities and one unit of Ramond-Ramond (RR) flux is dual to a D6 brane wrapped on a Lens space, describing the UV and IR of = 1 Super-Yang-Mills (SYM), respectively. Also, in the large Nc duality that relates D6 branes on S3 to an S2 with RR flux, we implement the presence of Nf flavors of quarks. We then compactify M-theory on (T*(S3) × S1)/2 and observe that one phase describes SO(4) SYM in the UV and two others describe confinement. Moreover, we consider compact 7-spaces (CY × S1)/2. We describe transitions where disconnected S3's approach and connect each other before they vanish. These effects correspond to non-abelian Higgs mechanism and confinement. The similar transitions involving S2 × S1's are also considered. Finally, we present transitions at finite distance in moduli space, where the first Betti number b1 of 3-cycles of singularities changes.