On the equivalence of N = 1 brane worlds and geometric singularities with flux

We consider Kaluza Klein reductions of M-theory on the ℤ_N orbifold of the spin bundle over S³ along two different U(1) isometries. The first one gives rise to the familiar "large-N duality" of the N = 1 SU (N) gauge theory in which the UV is realized as the world-volume theory of N D6-branes wrapped on S³, whereas the IR involves N units of RR flux through an S². The second reduction gives an equivalent version of this duality in which the UV is realized geometrically in terms of a ℙ¹ of A_N-1 singularities, with one unit of RR flux through the ℙ¹. The IR is reached via a geometric transition and involves a single D6 brane on a lens space S ³/ℤ_N or, alternatively, a singular background (S ² × ℝ⁴)/ℤ_N, with one unit of RR flux through S² and, localized at the singularities, an action of their stabilizer group in the U(1) RR gauge bundle, so that no massless twisted states occur. We also consider linear σ-model descriptions of these backgrounds.