Elliptic fibrations among toric hypersurface Calabi-Yau manifolds and mirror symmetry of fibrations

Abstract

In this thesis, we investigate the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau three folds by (1) constructing explicitly elliptically fibered Calabi-Yau threefolds with large Hodge numbers using Weierstrass model techniques motivated by F-theory, and comparing the Tate-tuned Wierstrass model set with the set of Calabi-Yau threefolds constructed using toric hypersurface methods, and (2) systematically analyzing directly the fibration structure of 4D reflexive polytopes by classifying all the 2D subpolytopes of the 4D polytopes in the Kreuzer and Skarke database of toric Calabi-Yau hypersurfaces. With the classification of the 2D fibers, we then study the mirror symmetry structure of elliptic toric hypersurface Calabi-Yau threefolds. We show that the mirror symmetry of Calabi-Yau manifolds factorizes between the toric fiber and the base: if there exist 2D mirror fibers of a pair of mirror reflexive polytopes, the base and fibration structure of one hypersurface Calabi-Yau determine the base of the other.