A Large N Type IIB Duality via a Spin(7) Geometric Transition as an F-theory Flop

The geometric features and toric descriptions of two different 8-dimensional $Spin(7)$ manifolds constructed via distinct resolutions of the cone over an $SU(3)/U(1)$ base, reveals that the geometry of the $Spin(7)$ conifold transition considered by Gukov et al. in \cite{gst}, is effected by a transition in its 6-dimensional submanifold which is isomorphic to a resolved or deformed Calabi-Yau 3-fold. This allows for a natural extension of the Gopakumar-Vafa large $N$ superstring duality of \cite{gv, v}; IIB superstring theory compactified on the $Spin(7)$ manifold with $N$ space-filling D5-branes wrapping an even-dimensional supersymmetric cycle, can be argued to undergo a large $N$ geometric transition at low energy to a $\it dual$ geometry with no branes but with certain units of 3-form fluxes through appropriate 3-cycles. For small $\it{or}$ large string coupling in a non-trivial axion field background, this large $N$ type IIB duality can be lifted to a purely geometric $\mathbb {RP}^5$ flop without D5-branes and 3-form fluxes via an F-theoretic description. The orientable, 10-dimensional, non-compact, Ricci-flat, $spin^c$ manifold undergoing the $\it{smooth}$ $\mathbb {RP}^5$ flop possesses an extended $SU(5) \odot {\mathbb Z_2}$ holonomy group, thus preserving 1/32 of the maximal supersymmetry, consistent with the resulting $\mathcal N =(1,0)$ supersymmetric pure SU(N) theory in 1+1 dimensions.