Geometric Transitions, Brane Dynamics and Gauge Theories

TL;DR

This paper explores the relationship between geometric transitions in string theory and brane dynamics, providing insights into gauge theories, confinement, and dualities in various string and M-theory compactifications.

Contribution

It introduces new geometric transition frameworks connecting brane configurations with gauge theory phenomena, including confinement and dualities, in type IIA, IIB, and M-theory contexts.

Findings

Duality between CP^1 singularities with RR flux and D6 branes on Lens spaces.

Implementation of N_f flavors in large N_c duality involving D6 branes on S^3.

Transitions describing confinement and Higgs mechanisms via S^3 and S^2 cycles.

Abstract

We consider the interplay between brane constructions and type IIA, IIB or M-theory geometries on Calabi-Yau (CY) and G_2 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 CP^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 N=1 Super-Yang-Mills (SYM), respectively. Also, in the large N_c duality that relates D6 branes on S^3 to an S^2 with RR flux, we implement the presence of $N_f$ flavors of quarks. We then compactify M-theory on T^*(S^3) X S^1)/Z_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 X S^1)/Z_2. We describe transitions where disconnected S^3's approach and connect each other before they vanish. These effects correspond to non-Abelian Higgs mechanism and confinement. The similar transitions involving S^2 X S^1's are also considered. Finally, we present transitions at finite distance in moduli space, where the first Betti number b_1 of 3-cycles of singularities changes.