Abstract
Toric geometry is applied for construction the enhanced gauge groups in F-theory compactified on elliptic Calabi-Yau fourfolds. The Hodge numbers calculated from the polyhedra for the chain H = SU (1), ... ,SU (5), SO(10), E6, E7 determine the number of tensor multiplets, vector multiplets and hypermultiplets of soli- tonic states that appear from singularities of elliptic fibration. Due to duality between the compactification of E8timesE8 heterotic string and the type IIA string compactification on a Calabi-Yau manifold there is a natural sequence of E-group embeddings which gives the matter content of Minimal Supersymmetric Standard Model and the possibility of searching for supersymmetry at the LHC.
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