Heterotic String/F-theory Duality from Mirror Symmetry

We use local mirror symmetry in type IIA string compactifications on Calabi– Yau n + 1 folds Xn+1 to construct vector bundles on (possibly singular) elliptically fibered Calabi–Yau n-folds Zn. The interpretation of these data as valid classical solutions of the heterotic string compactified on Zn proves Ftheory/heterotic duality at the classical level. Toric geometry is used to establish a systematic dictionary that assigns to each given toric n+ 1-fold Xn+1 a toric n fold Zn together with a specific family of sheafs on it. This allows for a systematic construction of phenomenologically interesting d = 4 N = 1 heterotic vacua, e.g. on deformations of the tangent bundle, with grand unified and SU(3)×SU(2) gauge groups. As another application we find non-perturbative gauge enhancements of the heterotic string on singular Calabi–Yau manifolds and new non-perturbative dualities relating heterotic compactifications on different manifolds.