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 N = 1 (and N = 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 CP¹ 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 N = 1 Super-Yang-Mills (SYM), respectively. Also, in the large N_c duality that relates D6 branes on S³ to an S² with RR flux, we implement the presence of N_f flavors of quarks. We then compactify M-theory on (T^∗ (S³ ) ×S¹ )/Z₂ 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 )/Z₂ . 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.