Abstract
In this study, we introduce a new class of rational elliptic 3-folds, which we refer to as “1/2 Calabi-Yau 3-folds”. We construct elliptically fibered Calabi-Yau 3-folds by utilizing these rational elliptic 3-folds. The construction yields a novel approach to build elliptically fibered Calabi-Yau 3-folds of various Mordell-Weil ranks. Our construction of Calabi-Yau 3-folds can be considered as a three-dimensional generalization of the operation of gluing pairs of 1/2 K3 surfaces to yield elliptic K3 surfaces. From one to seven U(1)s form in six-dimensional N = 1 F-theory on the constructed Calabi-Yau 3-folds. Seven tensor multiplets arise in these models.
Highlights
The presence of U(1) aids in explaining some characteristic properties of GUT such as mass hierarchies of quarks and leptons and a suppression of proton decay, and it relates to realization of GUT in the formulation of F-theory
Given that the singularity type of the elliptic Calabi-Yau 3-fold obtained as double cover (3.3) is identical to that of the original 1/2 Calabi-Yau 3-fold as we will show in appendix A, the construction of elliptically fibered Calabi-Yau 3-folds considered in this study can yield 6D F-theory models with seven tensor fields with E6 or E7 gauge group factor7 up to one
We introduced certain rational elliptic 3-folds that are referred to as 1/2 Calabi-Yau 3-folds, and we constructed elliptically fibered Calabi-Yau 3-folds by taking double covers of the 1/2 Calabi-Yau 3-folds
Summary
We provide the summary of the main results obtained in this study. When one considers three quadrics, Q1, Q2, Q3 in P3, they intersect in eight points. Elliptically fibered Calabi-Yau 3-folds of Mordell-Weil ranks at least from one to seven are obtained as a result of taking double covers of 1/2 Calabi-Yau 3-folds of MordellWeil ranks from one to seven. When the generic three quadrics Q1, Q2, Q3 are selected, 1/2 Calabi-Yau 3-folds obtained by blowing up P3 in the eight base points have Mordell-Weil rank seven. Six, five, four, and one U(1) factors arise in 6D N = 1 F-theory models on the Calabi-Yau 3-folds as obtained by considering double covers of the constructed 1/2 Calabi-Yau 3-folds. When the three quadrics Q1, Q2, Q3 are selected to be generic such that the eight intersection points are in a generic configuration, the resulting 1/2 Calabi-Yau 3-fold has the Mordell-Weil rank seven. When 1/2 Calabi-Yau 3-fold has an ADE singularity of rank strictly less than 7, such as E6 and A4, it has a positive Mordell-Weil rank
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