F-theory duals of M-theory on G₂ manifolds from mirror symmetry

Using mirror pairs (M₃, W₃) in type II superstring compactifications on Calabi–Yau threefolds, we study, geometrically, F-theory duals of M-theory on seven manifolds with G₂ holonomy. We first develop a way of obtaining Landau–Ginzburg (LG) Calabi–Yau threefolds W₃, embedded in four complex-dimensional toric varieties, mirror to the sigma model on toric Calabi–Yau threefolds M₃. This method gives directly the right dimension without introducing non-dynamical variables. Then, using toric geometry tools, we discuss the duality between M-theory on S¹ × M₃/Z₂ with G₂ holonomy and F-theory on elliptically fibred Calabi–Yau fourfolds with SU(4) holonomy, containing W₃ mirror manifolds. Illustrative examples are presented.