We study the Landau–Ginzburg (LG) theory mirror to two-dimensional (2D) N 2 gauged linear sigma models on toric Calabi–Yau manifolds. We derive and solve new constraint equations for LG elliptic Calabi–Yau superpotentials depending on the physical data of dual linear sigma models. In the Calabi–Yau threefold case, we consider two examples. First, we give the mirror symmetry of the canonical line bundle over the Hirzebruch surfaces F_n. Second, we find a special geometry with the affine so(8) Lie algebra toric data extending the geometry of elliptically fibred K3. This geometry leads to a pure N 1 six-dimensional SO(8) gauge model from the F-theory compactification. For Calabi–Yau fourfolds, we give a new algebraic realization for ADE hypersurfaces.