Geometric Transition, Large N Dualities and MQCD Dynamics

TL;DR

This work analyzes Vafa's geometric transition within the MQCD/M-theory framework, showing how D5 branes on ${\bf P}^1$ of a resolved conifold are replaced by flux on a deformed conifold, and how, in the small ${\bf P}^1$ limit, the M5 curve decomposes into $N$ plane M5 branes, yielding ${\cal N}=1$ ${U(1)}$ theories on the bulk while preserving information about the original ${SU(N)}$ dynamics via flux and domain-wall data. By employing T-duality and M-theory lifts, the authors connect the IIA/IIB pictures to a large $N$ duality where branes disappear in favor of RR/NS flux, with the glueball superfield $S$ organizing the vacua as $\langle S\rangle = e^{2\pi i k/N}\zeta$. They provide a coherent picture of how confinement, domain walls, and QCD strings emerge in this setup, including the reincorporation of decoupled $U(1)$ factors through the noncompact part of the Jacobian of the associated Riemann surface. The results bridge M-theory brane dynamics, flux backgrounds, and geometric transitions, offering insights into 4D ${\cal N}=1$ Yang-Mills physics and its large $N$ dual description. Significant aspects include the explicit M5 curve transitions $\Sigma_k$ to plane M5 branes, the interpretation of NS/RR flux in the dual frames, and the realization of domain walls and QCD strings within this geometric transition framework.

Abstract

We study Vafa's geometric transition from a brane setup in M-theory. In this transition D5 branes wrapped on P^1 cycles of a resolved conifold disappear and are replaced by fluxes on a deformed conifold. In the limit of small sized P^1, we describe this mechanism as a transition from curved M5 branes to plane M5 branes which replaces SU(N) MQCD by U(1) theories on the bulk. This agrees with the results expected from the geometric transition. We also discuss the reduction to ten dimensions and a brane creation mechanism in the presence of fluxes.