Mirror symmetry in string theory and fractional transformations

A generalized construction of mirror manifolds

There is a large number of different ways of constructing Calabi-Yau manifolds, as well as related non-geometric formulations, relevant in string compactiﬁcations. Showcasing this diversity, we discuss explicit deformation families of discretely distinct Hirzebruch hypersurfaces in (cid:80) n × (cid:80) 1 and identify their toric counterparts in detail. This precise iso-morphism is then used to investigate some of their special divisors of interest, and in particular the secondary deformation family of their Calabi-Yau subspaces. In particular, most of the above so called Hirzebruch scrolls are non-Fano, and their (regular) Calabi-Yau hypersurfaces are Tyurin-degenerate, but admit novel (Laurent) deformations by special rational sections as well as a sweeping generalization of the transposition construction of mirror models. This bi-projective embedding also reveals a novel deformation connection between distinct toric spaces, and so also the various divisors of interest including their Calabi-Yau subspaces.

Mirror symmetry in string theory and fractional transformations

Citations

A generalized construction of mirror manifolds

Hirzebruch Surfaces, Tyurin Degenerations and Toric Mirrors: Bridging Generalized Calabi-Yau Constructions

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