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






KEK-TH-2201



3.0cm




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

1.2cm

Yusuke Kimura^1 
0.6cm
^1KEK Theory Center, Institute of Particle and Nuclear Studies, KEK, 
 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
0.4cm
E-mail: kimurayu@post.kek.jp

2cm
In this study, we analyze gauge groups in six-dimensional N=1 F-theory models. We construct elliptic Calabi–Yau 3-folds possessing various singularity types as double covers of “1/2 Calabi–Yau 3-folds”, a class of rational elliptic 3-folds, by applying the method discussed in a previous study to classify the singularity types of the 1/2 Calabi–Yau 3-folds. One to three U(1) factors are formed in six-dimensional F-theory on the constructed Calabi–Yau 3-folds. The singularity types of the constructed Calabi–Yau 3-folds corresponding to the non-Abelian gauge group factors in six-dimensional F-theory are deduced. The singularity types of the Calabi–Yau 3-folds constructed in this work consist of A- and D-type singularities.  







§ INTRODUCTION

U(1) gauge symmetry plays an important role in realizing a grand unified theory (GUT)  since the presence of a U(1) gauge group explains some characteristic properties of a GUT, such as a suppression of the proton decay, and the mass hierarchy of leptons and quarks. The presence of a U(1) gauge group relates to the realization of a GUT in F-theory model building. 
F-theory <cit.> provides a framework that extends the Type IIB  superstrings to nonperturbative regimes. F-theory is compactified on spaces that have a torus fibration, wherein the modular parameter of tori as fibers is identified with axio-dilaton, enabling the axio-dilaton to possess an SL(2,) monodromy. 
A genus-one fibration is said to have a global section when one can choose a point in each fiber and the chosen point can be moved throughout the base. When an elliptic fibration has a global section, the set of global sections that the fibration possesses forms a group, which is referred to as the “Mordell–Weil group.” The number of U(1) factors formed in F-theory, compactified on an elliptic fibration with a section, is equal to the rank of the Mordell–Weil group of that fibration, as discussed in <cit.>. Models of F-theory compactified on elliptic fibrations admitting a global section have been studied, e.g. in <cit.>. For recent progress in F-theory compactifications where one or more U(1) factors are formed, see, for example, <cit.>. 




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 <cit.>, 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. We demonstrated by the construction of 6D models that the techniques introduced in <cit.> 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 <cit.> formed on the 7-branes in 6D F-theory on the constructed Calabi–Yau 3-folds.
A certain class of rational elliptic 3-folds was introduced <cit.> to generate elliptic Calabi–Yau 3-folds of various Mordell–Weil ranks, referred to as “1/2 Calabi–Yau 3-folds.” Such rational elliptic 3-folds were introduced to build 6D N=1 F-theory models with varying numbers of U(1) factors. Because taking double covers of the 1/2 Calabi–Yau 3-folds yield elliptic Calabi–Yau 3-folds <cit.>, 1/2 Calabi–Yau 3-folds can be regarded as “building blocks” of Calabi–Yau 3-folds. 
Up to seven U(1) factors form in 6D F-theory on the Calabi–Yau 3-folds built as double covers of 1/2 Calabi–Yau 3-folds <cit.>; explicit examples of 6D F-theory models with U(1) have been constructed <cit.>. 
To extract information of the non-Abelian gauge groups and matter spectra formed in 6D F-theory, the singularity types and singular fibers [The types of the singular fibers of the elliptic surfaces were classified in <cit.>, and methods to determine the types of the singular fibers were discussed in <cit.>.] 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 <cit.>. 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 <cit.>, 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 [Seven is the highest rank of all the singularity types of the 1/2 Calabi–Yau 3-folds <cit.>.] have also been classified, and all types of the 1/2 Calabi–Yau 3-folds with rank seven singularities were explicitly constructed <cit.>. Six-dimensional N=1 F-theory on the Calabi–Yau 3-folds, with rank-seven singularity types as their double covers, was also discussed <cit.>. 
On this note, we build Calabi–Yau 3-folds of singularity ranks six and lower as double covers of 1/2 Calabi–Yau 3-folds, discussing 6D F-theory on the resulting Calabi–Yau 3-folds. The U(1) gauge group forms in the resulting 6D F-theory models. The singularity types of the constructed Calabi–Yau 3-folds are determined and the number of U(1) factors formed in 6D F-theory are also deduced. The 6D F-theory models on Calabi–Yau 3-folds built as double covers of 1/2 Calabi–Yau 3-folds constructed in this work are new; the 6D F-theory models explicitly constructed in <cit.> do not include the models we discuss. Our analysis in this study demonstrates that, to a degree, the methods discussed in <cit.> have applications in studying U(1) gauge groups and non-Abelian gauge groups formed in 6D F-theory models. We also discuss the structures of singular fibers of 1/2 Calabi–Yau 3-folds and Calabi–Yau 3-folds as their double covers. These can be used to investigate the non-Abelian gauge symmetries formed on 7-branes. 




Concretely, we construct the 1/2 Calabi–Yau 3-folds with the singularity types: D_4A_1^2, D_5A_1, A_3A_2A_1, A_3A_1^3, A_5A_1, A_1^5, A_2A_1^3, A_3A_1^2, A_1^4. The ranks of these singularity types range from 4 to 6. As the rank of the singularity type and the Mordell–Weil rank of any 1/2 Calabi–Yau 3-fold add to seven <cit.>, the 1/2 Calabi–Yau 3-folds constructed in this work have Mordell–Weil ranks ranging from 1 to 3. One to three U(1) factors form in 6D F-theory compactifications on the Calabi–Yau 3-folds as their double covers. 




A possible relation of 6D N=1 F-theory on the Calabi–Yau 3-folds constructed as double covers of 1/2 Calabi–Yau 3-folds to the swampland conditions was mentioned in <cit.>. The authors of <cit.> discussed the notion of the swampland, and reviews of recent studies on the swampland criteria are given in <cit.>. 
Possible combinations of distinct gauge symmetries and matter spectra that can form in 6D quantum gravity theories with N=1 supersymmetry were discussed in <cit.>.

The structures of elliptic fibrations of 3-folds were analyzed in <cit.>.
An emphasis was placed on local model buildings <cit.> in recent model buildings in F-theory. Global aspects of the compactification geometry, however, need to be analyzed to discuss the issues of gravity and problems pertaining to the early universe, including inflation. In this study, the structures of the elliptic Calabi–Yau 3-folds are studied from a global viewpoint. 




This paper is structured as follows. In section <ref>, applying the discussion <cit.> of the equivalence of the singularity types of the 1/2 Calabi–Yau 3-folds and the plane quartic curves as a consequence of the method in <cit.>, we describe the construction of 1/2 Calabi–Yau 3-folds of various singularity types. Taking double covers of the 1/2 Calabi–Yau 3-folds, we construct elliptic Calabi–Yau 3-folds in section <ref>. F-theory applications are also discussed. U(1) gauge groups are formed in 6D F-theory compactifications on the constructed Calabi–Yau 3-folds. The structures of the singular fibers are analyzed. We state our concluding remarks in section <ref>. 



§ CONSTRUCTION OF 1/2 CALABI–YAU 3-FOLDS WITH VARIOUS SINGULARITY TYPES





 §.§ Method of construction


We construct 1/2 Calabi–Yau 3-folds of positive Mordell–Weil ranks with various singularity types. F-theory on the Calabi–Yau 3-folds constructed as their double covers have U(1) factors, as we discuss in section <ref>. The 1/2 Calabi–Yau 3-folds were constructed as a blow-up of ^3 at the base points of three quadrics, as introduced in <cit.>. Taking the ratio [Q_1: Q_2: Q_3] of the three quadrics Q_1, Q_2, Q_3 gives a projection onto the base surface ^2, yielding an elliptic fibration. 
A method to classify the singularity types of the 1/2 Calabi–Yau 3-folds was introduced in <cit.>. Utilizing method discussed in <cit.> reveals that the singularity types of the 1/2 Calabi–Yau 3-folds are identical to those of the plane quartic curves <cit.>. The classification of the singularity types of the plane quartic curves is provided in <cit.>. We construct 1/2 Calabi–Yau 3-folds with singularity types, D_4A_1^2, D_5A_1, A_3A_2A_1, A_3A_1^3, A_5A_1, A_1^5, A_2A_1^3, A_3A_1^2, A_1^4, by considering the duals of plane quartic curves with identical singularities. The curves with these singularities are shown in Figures <ref>, <ref>, <ref>, <ref>. The equations of the three quadrics yielding 1/2 Calabi–Yau 3-folds with such singularity types are deduced in sections <ref> through <ref>.









We used 1/2 Calabi–Yau 3-folds, the construction of which is described in section <ref> through section <ref>, to build elliptic Calabi–Yau 3-folds, discussed in section <ref>, on which F-theory provides 6D N=1 theories.  



 §.§ D_4A_1^2 singularity type


A quartic curve in ^2 with D_4A_1^2 singularity was realized in <cit.> as a conic and two lines meeting in a common point. The curve is presented in Figure <ref> (left). We construct the 1/2 Calabi–Yau 3-fold with D_4A_1^2 singularity type as dual of the plane quartic curve. We use [λ:μ:ν] to denote ^2 into which a quartic curve is embedded. Then the quartic curve with D_4A_1^2 singularity is given by the following equation:

    (μν-λ^2) λ  (λ-μ)=0.

The common point, [0:0:1], of the conic μν-λ^2=0 and the two lines, λ=0 and λ-μ=0, yields D_4 singularity. The other intersection point, [0:1:0], of the conic and line λ=0, and the other intersection, [1:1:1], of the conic and line λ-μ=0 yield two A_1 singularities. 
By an argument similar to that given in <cit.>, the equations of the three quadrics, the blow-up of ^3 at the base points of which yields the 1/2 Calabi–Yau 3-fold with D_4A_1^2 singularity type, can be deduced from the determinantal representation of the quartic curve (<ref>). 
The determinantal representation of the curve (<ref>) is given as:

    [   μ   λ   0   0;   λ   ν   0   0;   0   0   λ   0;   0   0   0 λ-μ ],

and the equations of the three quadrics Q_1, Q_2, Q_3 are deduced from the determinantal representation (<ref>) as:

    Q_1=     z^2+2xy+w^2 
    
    Q_2=     x^2-w^2 
    
    Q_3=     y^2.

We used [x:y:z:w] to denote the homogeneous coordinates of ^3. We denote the coordinates of the base surface ^2 of a 1/2 Calabi–Yau 3-fold as [a:b:c]. Then, because the curve c=0 in the base is dual to the D_4 singularity point [0:0:1] of the quartic curve (<ref>), type I_0^* fibers lie over the discriminant curve c=0. We use l_1 to denote the curve c=0. We use l_2, l_3 to denote the curves in the base that are dual to the two A_1 singularities of the quartic curve (<ref>). Type I_2 fibers lie over the curves l_2 and l_3. 
We denote the conic μν-λ^2=0 by C, and its dual curve as C^* in the base surface of the 1/2 Calabi–Yau 3-fold, then the discriminant of the 1/2 Calabi–Yau 3-fold is given as follows:

    Δ∼ l_1^6· l_2^2 · l_3^2· C^*.




 §.§ D_5A_1 singularity type


A quartic curve in ^2 with D_5A_1 singularity is reducible into a cubic with a cusp [<cit.> discussed a determinantal representation of the cuspidal cubic.] and a line through the cusp <cit.>. We construct the 1/2 Calabi–Yau 3-fold with D_5A_1 singularity type as dual of the plane quartic curve. The quartic curve with D_5A_1 singularity is given as follows:

    (λ^3+μν^2)(λ-ν)=0.

The cusp is at [λ:μ:ν]=[0:1:0], and this yields D_5 singularity. The other intersection point of the cuspidal cubic λ^3+μν^2=0 and the line λ-ν=0, [1:-1:1] yields A_1 singularity. 
The determinantal representation of the quartic curve (<ref>) is given as:

    [  -μ   0   λ   0;   0  -λ   ν   0;   λ   ν   0   0;   0   0   0 λ-ν ],

and the equations of the three quadrics are deduced from the determinantal representation (<ref>) as:

    Q_1=     -y^2+2xz+w^2 
    
    Q_2=     -x^2 
    
    Q_3=     2yz-w^2.

The curve in the base surface ^2 dual to the D_5 singularity point [0:1:0] of the quartic curve (<ref>) is given by b=0; We use l_1 to denote the curve b=0. We denote the curve dual to the A_1 singularity of the quartic curve at [1:-1:1] by l_2, then type I_2 fibers lie over the curve l_2. We denote the cuspidal cubic by B, and we use B^* to denote its dual in the base surface of the 1/2 Calabi–Yau 3-fold. The discriminant of the 1/2 Calabi–Yau 3-fold with D_5A_1 singularity type is given as follows:

    Δ∼ l_1^7· l_2^2 · B^*.




 §.§ A_3A_2A_1 singularity type


A plane quartic curve with A_3A_2A_1 singularity is reducible into a cuspidal cubic and a tangent line <cit.>. The curve is presented in Figure <ref> (left). We construct the 1/2 Calabi–Yau 3-fold with A_3A_2A_1 singularity type as dual of the plane quartic curve. The quartic curve with A_3A_2A_1 singularity is given as follows:

    (λ^3+μν^2)(3λ+μ+2ν)=0.

The cusp is at [λ:μ:ν]=[0:1:0] yielding A_2 singularity. The point [λ:μ:ν]=[1:-1:-1] where the line 3λ+μ+2ν=0 is tangent to the cuspidal cubic yields A_3 singularity. The other intersection point [2:-8:1] of the tangent line and cuspidal cubic yields A_1 singularity. 
The determinantal representation of the quartic curve (<ref>) is given as:

    [      -μ       0       λ       0;       0      -λ       ν       0;       λ       ν       0       0;       0       0       0 3λ+μ+2ν ],

and the equations of the three quadrics are deduced from the determinantal representation (<ref>) as:

    Q_1=     -y^2+2xz+3w^2 
    
    Q_2=     -x^2+w^2 
    
    Q_3=     2yz+2w^2.

The curve dual to the tangent point is denoted as l_1, then the singular fibers over l_1 have type I_4. The curve in the base surface ^2 dual to the cusp [0:1:0] is given by b=0; type I_3 fibers lie over this curve. We denote this curve as l_2. The curve dual to intersection point [λ:μ:ν]=[2:-8:1] is denoted as l_3, and type I_2 fibers lie over the curve l_3. The discriminant of the 1/2 Calabi–Yau 3-fold with A_3A_2A_1 singularity type is given as follows:

    Δ∼ l_1^4· l_2^3· l_3^2 · B^*.




 §.§ A_3A_1^3 singularity type


A quartic curve in ^2 with A_3A_1^3 singularity is reducible into a conic and a tangent line and another line <cit.>. The curve is presented in Figure <ref> (right). We construct the 1/2 Calabi–Yau 3-fold with A_3A_1^3 singularity type as dual of the plane quartic curve. The quartic curve with A_3A_1^3 singularity is given as follows:

    (-λ^2+μν) μ(μ-ν)=0.

The line μ=0 is tangent to the conic μν-λ^2=0 at [0:0:1], yielding A_3 singularity. Another line μ-ν=0 intersects with the conic in two points, [1:1:1], [1:-1:-1], and these yield two A_1 singularities. Two lines μ=0 and μ-ν=0 intersect in point [1:0:0] yielding an A_1 singularity.
The determinantal representation of the quartic curve (<ref>) is given as:

    [   μ   λ   0   0;   λ   ν   0   0;   0   0   μ   0;   0   0   0 μ-ν ],

and the equations of the three quadrics are deduced from the determinantal representation (<ref>) as:

    Q_1=     2xy 
    
    Q_2=     x^2+z^2+w^2 
    
    Q_3=     y^2-w^2.

The curve c=0 in the base ^2 dual to the tangent point [0:0:1] is denoted as l_1, then the singular fibers over the curve l_1 have type I_4. The curves in the base surface ^2 dual to the three A_1 singularities are denoted as l_2, l_3, l_4, respectively. The conic μν-λ^2=0 is denoted as C. The discriminant of the 1/2 Calabi–Yau 3-fold with A_3A_1^3 singularity type is given as follows:

    Δ∼ l_1^4· l_2^2· l_3^2 · l_4^2 · C^*.




 §.§ A_5A_1 singularity type


A plane quartic curve with A_5A_1 singularity is reducible into a nodal cubic and a tangent line at a flex <cit.>. We construct the 1/2 Calabi–Yau 3-fold with A_5A_1 singularity type as dual of the plane quartic curve. The quartic curve with A_5A_1 singularity is given as follows:

    (λ^3+λ^2ν-μ^2ν)ν =0.

The node at [λ:μ:ν]=[0:0:1] yields A_1 singularity. The flex [λ:μ:ν]=[0:1:0] where the line ν=0 is tangent to the nodal cubic yields A_5 singularity.
The determinantal representation of the quartic curve (<ref>) is given as:

    [  ν  0  λ  0;  0 -λ  μ  0;  λ  μ -λ  0;  0  0  0  ν ],

and the equations of the three quadrics are deduced from the determinantal representation (<ref>) as:

    Q_1=     2xz-z^2-y^2 
    
    Q_2=     2yz 
    
    Q_3=     x^2+w^2.





 §.§ A_1^5 singularity type


A plane quartic curve with A_1^5 singularity was realized in <cit.> as a conic and two lines in a general position. We construct the 1/2 Calabi–Yau 3-fold with A_1^5 singularity type as dual of the plane quartic curve. Then the quartic curve with A_1^5 singularity is given by the following equation:

    (λ^2-μν) λ(μ-ν)=0.

The line λ=0 intersects with the conic λ^2-μν=0 in two points, [0:1:0], [0:0:1]. The line μ-ν=0 intersects with the conic in two points, [1:1:1], [1:-1:-1]. The two lines intersect in a point [0:1:1]. The five intersection points yield five A_1 singularities. 
The determinantal representation of the curve (<ref>) is given as:

    [   μ   λ   0   0;   λ   ν   0   0;   0   0   λ   0;   0   0   0 ν-μ ],

and the equations of the three quadrics Q_1, Q_2, Q_3 are deduced from the determinantal representation (<ref>) as:

    Q_1=     z^2+2xy 
    
    Q_2=     x^2-w^2 
    
    Q_3=     y^2+w^2.

We denote the five curves in the base ^2 dual to the five intersection points [0:1:0], [0:0:1], [1:1:1], [1:-1:-1], [0:1:1] as l_1, l_2, l_3, l_4, l_5, respectively. The conic λ^2-μν=0 is denoted as C. The singular fibers over each of the five curves, l_1, …, l_5, have type I_2. Then, the discriminant of the 1/2 Calabi–Yau 3-fold with A_1^5 singularity type is given as follows:

    Δ∼ l_1^2· l_2^2 · l_3^2· l_4^2 · l_5^2 · C^*.




 §.§ A_2A_1^3 singularity type


A plane quartic curve with A_2A_1^3 singularity is reducible into a cuspidal cubic and a line in a general position <cit.>. The curve is presented in Figure <ref> (left). We construct the 1/2 Calabi–Yau 3-fold with A_2A_1^3 singularity type as dual of the plane quartic curve. The quartic curve with A_2A_1^3 singularity is given as follows:

    (λ^3+μν^2)(λ+μ)=0.

The cusp is at [λ:μ:ν]=[0:1:0] yielding A_2 singularity. Line λ+μ and the cuspidal cubic intersect in three points, [0:0:1], [1:-1:-1], [1:-1:1]. The three intersection points yield three A_1 singularities.
The determinantal representation of the quartic curve (<ref>) is given as:

    [  -μ   0   λ   0;   0  -λ   ν   0;   λ   ν   0   0;   0   0   0 λ+μ ],

and the equations of the three quadrics are deduced from the determinantal representation (<ref>) as:

    Q_1=     -y^2+2xz+w^2 
    
    Q_2=     -x^2+w^2 
    
    Q_3=     2yz.

Curves in the base ^2 dual to the three intersection points [0:0:1], [1:-1:-1], [1:-1:1] of the quartic (<ref>) are denoted as l_1, l_2, l_3, respectively. The curve in the base surface ^2 dual to the cusp [0:1:0] is given by b=0, and this dual curve is denotes as l_4. Type I_2 fibers lie over the curves l_1, l_2, l_3, and type I_3 fibers lie over curve l_4. The cuspidal cubic is denoted as B. The discriminant of the 1/2 Calabi–Yau 3-fold with A_2A_1^3 singularity type is given as follows:

    Δ∼ l_1^2· l_2^2 · l_3^2· l_4^3 · B^*.




 §.§ A_3A_1^2 singularity type


A quartic curve in ^2 with A_3A_1^2 singularity is reducible into two conics meeting in three points, one of which is a tangent point <cit.>. The curve is presented in Figure <ref> (middle). We construct the 1/2 Calabi–Yau 3-fold with A_3A_1^2 singularity type as dual of the plane quartic curve. The quartic curve with A_3A_1^2 singularity is given as follows:

    (ν(4μ+ν)-λ^2) (λν-(μ+ν)^2 )=0.

The tangent point of the two conics, ν(4μ+ν)-λ^2=0 and λν-(μ+ν)^2=0, is at [λ:μ:ν]=[1:0:1], and this yields A_3 singularity. The other two intersection points yield A_1 singularities. 
The determinantal representation of the quartic curve (<ref>) is given as:

    [ 4μ+ν    λ    0    0;    λ    ν    0    0;    0    0    λ  μ+ν;    0    0  μ+ν    ν ],

and the equations of the three quadrics are deduced from the determinantal representation (<ref>) as:

    Q_1=     2xy+z^2 
    
    Q_2=     4x^2+2zw 
    
    Q_3=     x^2+y^2+w^2+2zw.

The curve in the base surface ^2 dual to the A_3 singularity point [1:0:1] of the quartic curve (<ref>) is denoted as l_1, and type I_4 fibers lie over this curve. We denote the curves dual to two A_1 singularities of the quartic curve by l_2 and l_3, then type I_2 fibers lie over the curves l_2, l_3. We denote the two conics by C_1 and C_2, and we use C^*_1, C^*_2 to denote their duals in the base surface of the 1/2 Calabi–Yau 3-fold. The discriminant of the 1/2 Calabi–Yau 3-fold with A_3A_1^2 singularity type is given as follows:

    Δ∼ l_1^4· l_2^2 · l_3^2 · C^*_1 · C^*_2.





 §.§ A_1^4 singularity type


A plane quartic curve with A_1^4 singularity is reducible into two conics in a general position <cit.>. We construct the 1/2 Calabi–Yau 3-fold with A_1^4 singularity type as dual of the plane quartic curve. The quartic curve with A_1^4 singularity is given as follows:

    (μν-λ^2)(λν-μ^2)=0.

The two conics μν-λ^2=0 and λν-μ^2=0 intersect in four points. The four intersection points yield four A_1 singularities.
The determinantal representation of the quartic curve (<ref>) is given as:

    [ μ λ 0 0; λ ν 0 0; 0 0 λ μ; 0 0 μ ν ],

and the equations of the three quadrics are deduced from the determinantal representation (<ref>) as:

    Q_1=     2xy+z^2 
    
    Q_2=     x^2+2zw 
    
    Q_3=     y^2+w^2.

Curves in the base ^2 dual to the four intersection points of the quartic (<ref>) are denoted as l_1, l_2, l_3, l_4, respectively. Type I_2 fibers lie over the curves l_1, l_2, l_3, l_4. The two conics are denoted as C_1 and C_2. The discriminant of the 1/2 Calabi–Yau 3-fold with A_1^4 singularity type is given as follows:

    Δ∼ l_1^2· l_2^2 · l_3^2· l_4^2 · C_1^* · C_2^*.




§ 6D F-THEORY MODELS AND GAUGE GROUPS


We discuss applications to 6D F-theory compactifications. Taking double covers of 1/2 Calabi–Yau 3-folds built in sections <ref> - <ref> yields elliptic Calabi–Yau 3-folds [The double cover must be ramified over a quartic polynomial in the variables of the three quadrics, Q_1, Q_2, Q_3, to satisfy the Calabi–Yau condition. Detail of this can be found in <cit.>.]. F-theory compactifications on the resulting Calabi–Yau 3-folds yield 6D N=1 theories. The base surface of the Calabi–Yau 3-folds as double covers of 1/2 Calabi–Yau 3-folds is isomorphic to del Pezzo surface of degree two <cit.> [We choose a convention to call a del Pezzo surface that is obtained by blowing up ^2 at seven points of a general position as del Pezzo surface of degree two.]; therefore, seven tensor fields arise in 6D N=1 F-theory on the Calabi–Yau 3-folds as double covers of 1/2 Calabi–Yau 3-folds <cit.>. 
The singularity types of 1/2 Calabi–Yau 3-folds constructed in sections <ref> through <ref> and those of the Calabi–Yau 3-folds as their double covers are identical <cit.>, thus they determine the types of the non-Abelian gauge groups <cit.> formed in 6D F-theory on the Calabi–Yau 3-folds obtained as the double covers. The singularity types corresponding to the non-Abelian gauge groups formed on the 7-branes in 6D F-theory on the Calabi–Yau 3-folds as double covers of 1/2 Calabi–Yau 3-folds constructed in sections <ref> through <ref> are given as follows: D_4A_1^2, D_5A_1, A_3A_2A_1, A_3A_1^3, A_5A_1, A_1^5, A_2A_1^3, A_3A_1^2, A_1^4. 
SU(2) gauge group factors form in 6D F-theory on the Calabi–Yau 3-folds as double covers of 1/2 Calabi–Yau 3-folds whose singularity types include A_1, as constructed in sections <ref> through <ref>. For example, SU(2)^3 forms in F-theory on the Calabi–Yau 3-fold with A_3A_1^3 singularity type, as double cover of 1/2 Calabi–Yau 3-fold constructed in section <ref>.
The sum of the rank of the singularity type and the Mordell–Weil rank of any 1/2 Calabi–Yau 3-fold is seven <cit.>. The 1/2 Calabi–Yau 3-folds built in sections <ref> through <ref> have singularity types of rank 6, therefore, their Mordell–Weil rank is 1. The 1/2 Calabi–Yau 3-folds built in sections <ref> through <ref> have rank 5 singularity types, therefore, their Mordell–Weil rank is 2. The 1/2 Calabi–Yau 3-fold built in section <ref> has rank 4 singularity, therefore, the Mordell–Weil rank is 3. 
The Mordell–Weil rank of Calabi–Yau 3-fold as double cover of an 1/2 Calabi–Yau 3-fold is greater than or equal to the Mordell–Weil rank of the original 1/2 Calabi–Yau 3-fold <cit.>. Calabi–Yau 3-folds constructed by taking double covers of 1/2 Calabi–Yau 3-folds built in sections <ref> - <ref> have Mordell–Weil ranks greater than or equal to 1. 6D N=1 F-theory compactifications on these Calabi–Yau 3-folds have at least one U(1) factor. A similar reasoning applied to 6D F-theory on the Calabi–Yau 3-folds constructed by taking double covers of 1/2 Calabi–Yau 3-folds built in sections <ref> - <ref> shows that at least two U(1) factors form in the theories. As per reasoning similar to these, at least three U(1) factors form in 6D F-theory on the Calabi–Yau 3-folds constructed by taking double covers of 1/2 Calabi–Yau 3-folds built in section <ref>. 
The structure of type I_3 fibers corresponding to A_2 singularities can be seen after conducting blow-ups in our constructions in section <ref>. The singular fiber corresponding to an A_2 singularity is given by two conic meeting in two points, where one and only one of the two meeting points is a base point of the three quadrics. When this base point is blown up, restricted to the singular fiber ^1 arises from the blown-up base point. Then the singular fiber consists of three ^1s, each pair of which meet in one point, and the structure of type I_3 fiber becomes clear. 
We use the 1/2 Calabi–Yau 3-fold constructed in section <ref> as a sample to demonstrate this point. The singular fibers corresponding to A_2 singularity are the fibers over the curve l_2, which is dual to the A_2 singularity point [0:1:0] of the quartic curve (<ref>). Thus, the equation of the singular fibers is given as follows:

    -x^2+w^2=     0 
    
    c(-y^2+2xz+3w^2)-a(2yz+2w^2)=     0,

where [a:c] parameterizes the curve l_2. We will see that the structure of type I_3 fiber can be seen after blow-ups.
Before blow-ups, from the equation (<ref>) an argument similar to that given in <cit.> one finds two conics intersecting in two points, where conics are contained in the hyperplanes x-w=0 and x+w=0. The conics intersect along the intersection of the hyperplanes x-w=0 and x+w=0. Therefore, the intersection points lie along the locus x=w=0, and they are the solutions to the following equation:

    cy^2+2ayz=0.

Thus, {x=y=w=0} and {x=w=0,  cy+2az=0} yield two intersection points of the two conics. 
Three quadrics (<ref>) have five base points: [0:0:1:0], [1:1:-1:1], [-1:-1:1:1], [2:-4:1:2], [-2:4:-1:2]. Therefore, we find that the two intersection points of the conics contain one of the base points: [0:0:1:0]. When the base points are blown up, because intersection point [0:0:1:0] is blown up, the blow-up separates the two intersecting conics at the intersection point [0:0:1:0]. (The other intersection point remains unchanged.) ^1 arises from the blown-up intersection point, and the structure of type I_3 fiber becomes evident after the blow-up. The situation is given in Figure <ref>. 



The singularity type of 1/2 Calabi–Yau 3-fold constructed in section <ref> includes D_4. By an argument similar to that given in <cit.>, we can see the structure of type I_0^* fiber corresponding to D_4 singularity after conducting blow-ups at base points of the three quadrics for the 1/2 Calabi–Yau 3-fold in section <ref>. A double conic blown up at four base points yields this fiber type, and the situation of the appearance of a type I_0^* fiber in 1/2 Calabi–Yau 3-fold in section <ref> is analogous to the situation of the 1/2 Calabi–Yau 3-fold with D_4A_1^3 singularity discussed in <cit.>. The situations of the A_3 singularities of 1/2 Calabi–Yau 3-folds constructed in sections <ref>, <ref>, and <ref> are analogous to the analysis of the 1/2 Calabi–Yau 3-fold with A_3^2A_1 singularity studied in <cit.>. Analyses similar to that given in <cit.> also reveal that the structures of type I_4 fibers can be seen after the blow-ups of the base points of three quadrics for the constructions in sections <ref>, <ref>, and <ref>, whose singularity types include A_3. As mentioned in <cit.>, the structure of type I_6 fiber is expected to be seen after multiple stages of blow-ups when a 1/2 Calabi–Yau 3-fold has an A_5 singularity as constructed in section <ref>. 
The structures of the singular fibers of the 1/2 Calabi–Yau 3-folds are determined via the blow-ups as we have seen. The fiber types of the Calabi–Yau 3-folds as their double covers remain invariant; therefore, the fiber types of the Calabi–Yau 3-folds as the double covers can also be deduced in our constructions. However, whether the types of the singular fibers are split/semi-split/non-split need to be determined to deduce the precise non-Abelian gauge group <cit.>. As pointed out in <cit.>, even when the equations of the three quadrics yielding a 1/2 Calabi–Yau 3-fold are determined, deducing the Weierstrass equation of the Calabi–Yau 3-fold as the double cover from the equations of the three quadrics is technically difficult. Owing to this situation, determining whether a singular fiber is split/non-split/semi-split is generally a hard problem for Calabi–Yau 3-folds constructed as double covers of 1/2 Calabi–Yau 3-folds.
The discriminants of the 1/2 Calabi–Yau 3-folds are also deduced in sections <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, <ref>. The discriminants of the Calabi–Yau 3-folds constructed as double covers of 1/2 Calabi–Yau 3-folds can be determined from those of the 1/2 Calabi–Yau 3-folds <cit.>. Thus, the locations of the matter fields in 6D F-theory localized at the intersections of the 7-branes can be obtained from the discriminants of the Calabi–Yau 3-folds in our constructions. Determining the matter spectra by studying the collisions of singular fibers at the intersections of the 7-branes can be a likely target of future study. 



§ CONCLUDING REMARKS


In this note, we built various elliptic Calabi–Yau 3-folds of positive Mordell–Weil ranks as double covers of 1/2 Calabi–Yau 3-folds of various singularity types via utilizing a general method discussed in <cit.>. F-theory compactifications on the Calabi–Yau 3-folds yielded 6D N=1 models wherein one to three U(1) gauge group factors are formed [Examples of 6D F-theory models on Calabi–Yau 3-folds as double covers of 1/2 Calabi–Yau 3-folds with four to six U(1) factors are constructed in <cit.>. Seven U(1) factors form in 6D F-theory on Calabi–Yau 3-folds constructed as double covers of 1/2 Calabi–Yau 3-folds, when the three quadrics are generically chosen <cit.>.]. 6D F-theory compactifications on the elliptic Calabi–Yau 3-folds with the singularity types, D_4A_1^2, D_5A_1, A_3A_2A_1, A_3A_1^3, A_5A_1, as double covers of 1/2 Calabi–Yau 3-folds constructed in sections <ref> - <ref>, have at least one U(1) factor; 6D F-theory on Calabi–Yau 3-folds with the singularity types, A_1^5, A_2A_1^3, A_3A_1^2, as double covers of 1/2 Calabi–Yau 3-folds that we constructed in sections <ref> - <ref>, have at least two U(1) factors; and at least three U(1) factors form in 6D F-theory on the elliptic Calabi–Yau 3-folds with the singularity type A_1^4 as double cover of 1/2 Calabi–Yau 3-fold as constructed in section <ref>. 
Our studies here applied the method in <cit.> to investigate gauge groups formed in 6D F-theory on the Calabi–Yau 3-folds built as the double covers of 1/2 Calabi–Yau 3-folds. A similar analysis can be applied to 1/2 Calabi–Yau 3-folds possessing other singularity types and Calabi–Yau 3-folds as double covers of such 1/2 Calabi–Yau 3-folds. This particularly applies to construct 6D N=1 F-theory models with four or more U(1) factors. To analyze such situations, one can construct 1/2 Calabi–Yau 3-folds with singularity types of rank three or less, and consider 6D F-theory on the elliptic Calabi–Yau 3-folds constructed as their double covers. 
The structures of the singular fibers are analyzed through the blow-ups as discussed in section <ref>. There is a chance that the matter spectra can be deduced if the structures of the singular fibers at the intersections of the 7-branes can be analyzed as mentioned in <cit.>, and it might be interesting if this is achieved and the hypermultiplets at the intersections of the 7-branes are determined. This is a likely target for future studies.




§ ACKNOWLEDGMENTS


We would like to thank Shigeru Mukai for discussions. 


99

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