Extremal 1/2 Calabi-Yau 3-folds and six-dimensional F-theory applications

Abstract

We discuss a method for classifying the singularity types of 1/2 Calabi-Yau 3-folds, a family of rational elliptic 3-folds introduced in a previous study in relation to various U(1) factors in 6D F-theory models. A projective dual pair of del Pezzo manifolds recently studied by Mukai is used to analyze the singularity types. In particular, we studied the maximal rank seven singularity types of 1/2 Calabi-Yau 3-folds. The structures of the singular fibers are analyzed using blow-ups. Double covers of the 1/2 Calabi-Yau 3-folds yield elliptic Calabi-Yau 3-folds and applications to six-dimensional N = 1 Ftheory on the Calabi-Yau 3-folds are also discussed. The deduced singular fibers have applications in studying the gauge groups formed in 6D F-theory compactifications. The blow-up methods used to analyze the singular fibers and sections utilized in this research might have applications in studying the U(1) factors and hypermultiplets charged under U(1) in 6D F-theory.

Types of gauge groups in six-dimensional F-theory on double covers of rational elliptic 3-folds

“…in [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. For recent progress in F-theory compactifications where one or more U(1) factors are formed, see, for example, [4,7,8,10,43,12,44,45,46,47,17,48,20,...…”

“…The aim of this study is to build six-dimensional (6D) F-theory models with U(1) factors on Calabi-Yau 3-folds, by applying the general scheme developed in [36,41], to construct a family of elliptic Calabi-Yau 3-folds of positive Mordell-Weil ranks. Calabi-Yau 3-folds with A-type and D-type singularities are obtained in this work, and one to three U(1) gauge group factors are formed in 6D N = 1 F-theory on the obtained Calabi-Yau 3-folds.…”

“…Calabi-Yau 3-folds with A-type and D-type singularities are obtained in this work, and one to three U(1) gauge group factors are formed in 6D N = 1 F-theory on the obtained Calabi-Yau 3-folds. We demonstrated by the construction of 6D models that the techniques introduced in [36,41] apply in studying U(1) gauge symmetries in the F-theory formulation. The singularity types of the constructed Calabi-Yau 3-folds are determined, and they correspond to the non-Abelian gauge groups [3,54] formed on the 7-branes in 6D F-theory on the constructed Calabi-Yau 3-folds.…”

“…To extract information of the non-Abelian gauge groups and matter spectra formed in 6D Ftheory, the singularity types and singular fibers 1 of Calabi-Yau 3-folds also need to be analyzed. A method to classify the singularity types of the Calabi-Yau 3-folds as double covers of the 1/2 Calabi-Yau 3-folds was discussed in [41]. Because the singularity type of the 1/2 Calabi-Yau 3-fold and that of the Calabi-Yau 3-fold as the double cover are identical [36], it suffices to classify the singularity types of the 1/2 Calabi-Yau 3-folds.…”

“…Because the singularity type of the 1/2 Calabi-Yau 3-fold and that of the Calabi-Yau 3-fold as the double cover are identical [36], it suffices to classify the singularity types of the 1/2 Calabi-Yau 3-folds. In addition to this general method of classifying the singularity types, the singularity types of the 1/2 Calabi-Yau 3-folds of rank seven 2 have also been classified, and all types of the 1/2 Calabi-Yau 3-folds with rank seven singularities were explicitly constructed [41]. Six-dimensional N = 1 F-theory on the Calabi-Yau 3-folds, with rank-seven singularity types as their double covers, was also discussed [41].…”