arXiv:1608.07219v1 [hep-th] 25 Aug 2016

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We compute the discrete gauge groups in F-theory models on genus-one fibered ..... [x1 : x2 : x3 : x4] is homogeneous coordinates on P3 and t is an inhomogeneous .... values. Then the assumed section of the Jacobian (18) of the multidegree (3,2 .... [60] represents the attractive K3 surface denoted by S[6 3 6] in this paper.
YITP-16-98

arXiv:1608.07219v1 [hep-th] 25 Aug 2016

Discrete Gauge Groups in F-theory Models on Genus-One Fibered Calabi-Yau 4-folds without Section Yusuke Kimura1 1

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected]

Abstract We compute the discrete gauge groups in F-theory models on genus-one fibered Calabi-Yau 4-folds without a section to the fibration. In general, discrete gauge group arises in genus-one fibration Y without a section. The Tate-Shafarevich group X(J(Y )) of the Jacobian J(Y ) is identified with the discrete gauge group. An n-section of a genusone fibration gives rise to the discrete gauge group Zn . We deduce the discrete gauge group by computing a smallest multisection of a genus-one fibration without a section. The discrete gauge groups Z2 , Z3 and Z4 appear in our models. We also investigate the Mordell-Weil group of the Jacobian of a genus-one fibration without a section.

Contents 1 Introduction

2

2 Review of Models without Section 2.1 K3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fermat Type Hypersurfaces of Bidegree (3,2) in P2 × P1 . . . . . . . . 2.1.2 Double Cover of P1 × P1 ramified over Bidegree (4,4) Curve . . . . . . 2.1.3 Fermat Quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Calabi-Yau 4-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Multidegree (3,2,2,2) Hypersurface in P2 × P1 × P1 × P1 . . . . . . . . 2.2.2 Double Cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Review of Possibility of Cancelling the Tadpole Anomaly . . . . . . . . . . . . 2.3.1 K3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Calabi-Yau 4-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 6 7 8 8 10 13 13 14

3 Complete Intersections of multidegree (2,1,1,1) and (2,1,1,1) Hypersurface in P3 × P1 × P1 × P1 3.1 Defining Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Forms of the Discriminant Components and Gauge Symmetries . . . . . . . . 3.3 Euler Characteristic and Consideration on Tadpole Anomaly . . . . . . . . . . 3.4 Matter Fields and Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . . . 3.5 Mordell-Weil Group of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . .

16 17 17 19 21 23

4 Discrete Symmetries 24 4.1 Discrete Symmetries on K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.1 Fermat Type Hypersurfaces of Bidegree (3,2) in P2 × P1 . . . . . . . . 24 4.1.2 Double Cover of P1 × P1 ramified over Bidegree (4,4) Curve . . . . . . 25 4.1.3 Fermat Quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Discrete Symmetries on Calabi-Yau 4-folds . . . . . . . . . . . . . . . . . . . . 27 4.2.1 Multidegree (3,2,2,2) Hypersurface in P2 × P1 × P1 × P1 . . . . . . . . 27 4.2.2 Double Cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2.3 Complete Intersections of multidegree (2,1,1,1) and (2,1,1,1) Hypersurface in P3 × P1 × P1 × P1 . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 Conclusion

30

1

1

Introduction

F-theory[1, 2, 3] is a non-perturbative extension of type IIB superstring theory. F-theory is compactified on an elliptic Calabi-Yau 4-fold over a base 3-fold, with a generic fiber an elliptic curve. The complex structure of a smooth elliptic fiber possesses SL2 (Z) symmetry, and this symmetry is identified with S-duality of type IIB superstring. The axio-dilaton field τ = C0 + ie−φ , identified with the modular parameter of an elliptic fiber, can have SL2 (Z) monodromy in the framework of F-theory. Physical concepts such as gauge groups and matter spectra are described in geometrical language in F-theory. The structure of (non-Abelian part of) the gauge group is encoded in the singular fibers of the elliptic fibration. The structure of U (1) gauge fields is determined by the rank of Mordell-Weil groups of the elliptic fibration. Elliptic fibers become degenerate over a complex codimension one locus in the base 3-fold, or discriminant locus. The ways an elliptic fiber degenerates over a codimension 1 locus have a good description1 , and in essence they are the same as those in elliptic surfaces. The singular fibers of elliptic surfaces were classified by Kodaira[7], and they are given in terms of ADE-types. Same as an elliptic surface, singular fibers over a codimension 1 locus of an elliptic Calabi-Yau 4-fold are classified by ADE-types. 7-branes are wrapped on the components of the discriminant locus. A fiber type of singular fibers on a component is in correspondence with the structure of the non-Abelian gauge group in the low-energy theory on the 7-branes wrapped on the component. Matter fields arise from the deformation of singularities of a Calabi-Yau manifold. Matters arise from a local rank one enhancement of the singularities of ADE-type[8]. See also [9, 10, 11, 12] for the correspondence between the singularities of a manifold and the associated matter representations. For recent advances on the matters and singularities, see for instance [13, 14, 15]. F-theory models with a section to the fibration have been studied in detail. For the models, see for instance [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Recently, F-theory models without a section was considered in [27, 28]2 . As argued in [28], F-theory models lacking a section to the fibration fit naturally into the moduli of F-theory models, by considering their Jacobian fibrations. The Jacobian fibration J(Y ) of genus-one fibration Y is an elliptic fibration with a section. The axio-dilaton function τ and discriminant locus of the Jacobian fibration J(Y ) and those of genus-one fibration Y without a section are identical. For recent advances of F-theory on genus-one fibration without a section, see also for instance [30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. One of the characteristics of F-theory on genus-one fibration without a section is that discrete gauge group3 appears in a natural way. As discussed in [28], the Tate-Shafarevich group X(J(Y )) of the Jacobian J(Y ) of a Calabi-Yau manifold Y is identified with the 1

For elliptic manifolds of dimension n ≥ 3, the degenerations of fibers over codimension 2 loci or higher do not have a good description, except some tractable special cases. See [4, 5, 6] on this issue. 2 F-theory model without a section was considered before in [29]. 3 For recent progress of the discrete gauge symmetries, for instance see [40, 41, 42, 43, 44, 45, 46, 47, 48, 49].

2

discrete gauge group of F-theory compactification[50]. In general, it is expected that an nsection gives rise to discrete gauge group Zn . So, when a genus-one fibered Calabi-Yau 4-fold Y has an n-section, the discrete gauge group of F-theory compactification on Y includes Zn factor. In this paper, we deduce the discrete gauge groups of F-theory models on genus-one fibrations without a section considered in [37, 38, 39]. The models considered are: K3 surfaces as i)hypersurfaces of bidegree (3,2) in P2 × P1 [37], ii)double covers of P1 × P1 ramified over a bidegree (4,4) curve and iii)Fermat quartic[38], and genus-one fibered Calabi-Yau 4-folds without a section as i)multidegree (3,2,2,2) hypersurface in P2 × P1 × P1 × P1 , and ii)double cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold in [39]. In addition to these models, we also consider complete intersections of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 × P1 × P1 × P1 . Such complete intersections are genus-one fibered CalabiYau 4-folds, lacking a section to the fibration. For this additional model, we consider a class of equations so that the complete intersections are elliptic K3-fibered, and elliptic K3 fibers are Fermat quartic. Then the model admits a bisection, so the discrete group that arises is Z2 . For F-theory compactifications on these complete intersection Calabi-Yau 4-folds, we will deduce the forms of the discriminant components, the gauge groups, Euler characteristic and potential matter spectra and Yukawa couplings in Section 3. We also compute the Mordell-Weil group of the Jacobian of this complete intersection Calabi-Yau 4-fold in Section 3. Similar derivations can be found in [39]. The structure of this paper is as follows: in Section 2, we briefly review the models of genus-one fibered Calabi-Yau 4-folds considered in [37, 38, 39]. In Section 3, we introduce complete intersections of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 × P1 × P1 × P1 as an additional family of genus-one fibered Calabi-Yau 4-folds. In Section 4, we derive the discrete gauge groups of the F-theory models on genus-one fibrations without a section. In Section 5, we derive conclusions.

2

Review of Models without Section

In this section, we give a brief review of models without a section considered in [37, 38, 39]. We also briefly review the possibilities of cancelling the tadpole anomaly for these models. We attach Table 1 showing the correspondence of the types of the singular fibers and the gauge groups, and Figure 1 showing the figures of singular fibers for the convenience.

2.1

K3 Surfaces

We review K3 surfaces constructed in [37, 38]. When paired with a K3 surface, they give genusone fibered Calabi-Yau 4-folds without a section. But, for these direct products K3×K3, all the 7-branes are parallel, and the components of the discriminant locus do not intersect. Consequently, only the matter fields arising on the 7-branes are adjoints and singlets, and chiral matters do not arise.

3

Figure 1: Figures of singular fibers. Each line of a figure represents a P1 component. A figure shows the configuration of P1 components in a singular fiber and how they intersect.

4

fiber type gauge group In

SU (n)

I0∗

SO(8)

III

SU (2)

IV

SU (3)

IV ∗

E6

III ∗

E7

II ∗

E8

Table 1: The correspondence of the types of the singular fibers and the gauge groups.

In this paper, we limit our attention to most enhanced K3 surfaces, i.e. attractive4 K3 surfaces. As observed in [38, 39], when K3 families of hypersurfaces of bidegree (3,2) in P2 × P1 [37], and double covers of P1 × P1 ramified over a bidegree (4,4) curve (given by the specific equations), become attractive, their Jacobians are seen to be extremal, i.e. attractive K3’s with section, with the rank 0 Mordell-Weil groups. Therefore, for such cases F-theory compactifications do not have U (1) gauge field. 2.1.1

Fermat Type Hypersurfaces of Bidegree (3,2) in P2 × P1

Fermat type hypersurfaces of bidegree (3,2) in P2 × P1 (t − α1 )(t − α2 )X 3 + (t − α3 )(t − α4 )Y 3 + (t − α5 )(t − α6 )Z 3 = 0

(1)

was considered in [37]. [X : Y : Z] is homogeneous coordinates on P2 and t is an inhomogeneous coordinate on P1 . These K3 surfaces S are genus-one fibered under the natural projection S   y P1 and a generic member does not admit a section. We outline some of the results in [37] (and some comments made on this K3 hypersurface in [39]). When α1 = α2 , α3 = α4 , α5 = α6 , (2) 4

In mathematics, it is standard to call a K3 surface with the highest Picard number ρ = 20 a singular K3 surface. We follow the convention of the term in [51].

5

the Fermat type hypersurface (t − α1 )2 X 3 + (t − α3 )2 Y 3 + (t − α5 )2 Z 3 = 0

(3)

is an attractive K3. Singular fibers are at t = α1 , α3 , α5 , and each has type IV ∗ so the gauge group on the 7-branes is E6 × E6 × E6 . The complex structure of an attractive K3 is uniquely specified by its transcendental lattice[52]. We denote S[2a b 2c] an attractive K3 whose transcendental lattice has the  by  2a b intersection matrix . b 2c   6 3 The transcendental lattice of the attractive K3 (3) has the intersection matrix . 3 6 So the attractive K3 (3) is denoted by the symbol S[6 3 6] . The Jacobian of the attractive K3 (3) is given by the equation X 3 + Y 3 + (t − α1 )2 (t − α3 )2 (t − α5 )2 Z 3 = 0,

(4) 

 2 1 and this is an extremal K3. Its transcendental lattice has the intersection matrix . So 1 2 the Jacobian (4) is the extremal K3 S[2 1 2] with three IV ∗ fibers. The attractive K3 S[2 1 2] with three IV ∗ fibers has the Mordell-Weil group Z3 [53]. So, the Jacobian J(S[6 3 6] ) = S[2 1 2] given by (4), of the attractive K3 S[6 3 6] (3), has the Mordell-Weil group Z3 . In particular, F-theory compactification on S[6 3 6] ×K3 does not have U (1) gauge field. 2.1.2

Double Cover of P1 × P1 ramified over Bidegree (4,4) Curve

Genus-one fibered K3 surface as a double cover of P1 × P1 ramified over a bidegree (4,4) curve was considered in [38]. We briefly outline some of the results obtained in [38]. The equations for the double covers considered in [38] are of the form τ 2 =(t − α1 )(t − α2 )(t − α3 )(t − α4 ) · x4 + (t − α5 )(t − α6 )(t − α7 )(t − α8 ),

(5)

so each fiber possesses the complex multiplication of order 4. t and x are inhomogeneous coordinates on the P1 ’s. t gives an inhomogeneous coordinates on the base P1 , and the natural projection onto this base P1 gives a genus-one fibration. In this paper, we only consider the case α1 = α2 = α3 , α5 = α6 = α7 , α4 = α8 .

(6)

For this case, the equation of the double cover of P1 × P1 (5) becomes τ 2 = (t − α1 )3 (t − α4 ) · x4 + (t − α5 )3 (t − α4 ),

6

(7)

The most enhanced double cover of P1 × P1 (7) is an attractive K3, with the gauge group E7 × E7 × SO(8). The singular fibers at t = α1 and t = α5 have the type III ∗ , and the singular fiber at t = α4 has the type I0∗ . The corresponding reducible fiber type is E72 D4 . The Jacobian of the attractive K3 (7) is given by 1 τ 2 = x3 − (y − α1 )3 (y − α5 )3 (y − α4 )2 x. (8) 4 This is   an extremal K3 surface whose transcendental lattice has the intersection matrix 2 0 . Therefore, the Jacobian (8) is denoted by S[2 0 2] . 0 2 The attractive K3 with discriminant 4 S[2 0 2] with reducible fiber type E72 D4 has the Mordell-Weil group Z2 [53]. So, the Jacobian (8) has the Mordell-Weil group Z2 . In particular, the Jacobian (8) of the attractive K3 (7) has the Mordell-Weil group of rank 0. So, F-theory compactification on S×K3, where S is the attractive K3 (7), does not have U (1) gauge field. 2.1.3

Fermat Quartic

Another K3 surface considered in [38] is Fermat quartic x4 + y 4 + z 4 + w 4 = 0 ⊂ P 3 .

(9)

Fermat quartic has a presentation as a complete intersection of bidegree (2,1) and (2,1) hypersurfaces in P3 × P1 given by the equations x21 + x23 + 2tx2 x4 = 0 x22 + x24 + 2tx1 x3 = 0.

(10)

[x1 : x2 : x3 : x4 ] is homogeneous coordinates on P3 and t is an inhomogeneous coordinate on P1 . Fermat quartic is  known  to be an attractive K3, and its transcendental lattice has the 8 0 intersection matrix . So Fermat quartic is denoted by S[8 0 8] . With the presentation 0 8 as the complete intersection in P3 × P1 (10), Fermat quartic is genus-one fibered lacking a section to the fibration, and has six I4 fibers5 . As argued in [38], the Jacobian of Fermat quartic with six I4 fibers is given by τ 2 = −t2 λ4 + (t4 + 1)λ2 − t2 .

(11)

This is an extremal K3 with reducible fiber type A63 (i.e. with six I4 fibers). An extremal K3 with reducible fibertype  A63 is uniquely determined, and its transcendental lattice has the 4 0 intersection matrix [55]. Therefore we see that the Jacobian J(S[8 0 8] ) given by (11) 0 4 is the attractive K3 S[4 0 4] with six I4 fibers6 . 5

Fermat quartic with six I4 fiber is well known in mathematics. For instance, see section 8 in [54]. [56] mentions the facts that S[4 0 4] with six I4 fibers is the Jacobian of Fermat quartic S[8 0 8] with six I4 fibers, and Fermat quartic with six I4 fibers does not admit a section. 6

7

According to [53], the attractive K3 S[4 0 4] with reducible fiber type A63 has the MordellWeil group Z4 × Z4 . So, the Jacobian (11) has the Mordell-Weil group Z4 × Z4 . In particular, its rank is 0. Therefore, F-theory compactification on S[8 0 8] ×K3, with the Fermat quartic S[8 0 8] presented as the complete intersection (10), does not have U (1) gauge field.

2.2

Calabi-Yau 4-folds

We now turn to give a brief review of genus-one fibered Calabi-Yau 4-folds lacking a section to the fibration, considered in [39]. These models do not admit rational section. In F-theory compactifications on these models, components of the discriminant loci intersect, and chiral matter fields arise. Yukawa couplings also arise from the interactions of chiral matters. Chiral matters arise on the bulk, and there are also matter fields localised along the matter curves. Models considered in [39] fall into two families: i)multidegree (3,2,2,2) hypersurface in 2 P × P1 × P1 × P1 , and ii)double cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold. By construction, these Calabi-Yau 4-folds are genus-one fibered and elliptic K3-fibered. Below we outline the results on these genus-one fibered Calabi-Yau 4-folds without a section obtained in [39]. 2.2.1

Multidegree (3,2,2,2) Hypersurface in P2 × P1 × P1 × P1

A multidegree (3,2,2,2) hypersurface in P2 × P1 × P1 × P1 is a Calabi-Yau 4-fold, and this is genus-one fibered under the projection ←−−− E

Y   y P1 × P1 × P1 .

A hypersurface Y is also elliptic K3-fibered under the projection onto the last P1 × P1 in P2 × P1 × P1 × P1 : Y   y P1 × P1 . Among the family of multidegree (3,2,2,2) hypersurfaces in P2 × P1 × P1 × P1 , in this paper we limit our consideration to the most enhanced Fermat type hypersurfaces (t − α1 )2 f X 3 + (t − α2 )2 gY 3 + (t − α3 )2 hZ 3 = 0.

(12)

[X : Y : Z] is homogeneous coordinates on P2 , and t is an inhomogeneous coordinate on the first P1 in P2 × P1 × P1 × P1 . αi (i = 1, 2, 3) are points in P1 . f, g, h are bidegree (2,2) polynomials on P1 × P1 , where these P1 ’s are the last two P1 ’s in P2 × P1 × P1 × P1 . 8

Discriminant components are given by the vanishing loci t = αi

(i = 1, 2, 3)

(13)

and f = 0 g = 0 h = 0

(14)

in the base 3-fold P1 × P1 × P1 . Similar to the convention made in [39], the discriminant components given by t = α1,2,3 are denoted by E1,2,3 , and the components {f = 0}, {g = 0}, {h = 0} are denoted by D1 , D2 and D3 respectively. E1,2,3 has the form

and D1,2,3 has the form

Ei ∼ = P1 × P1

(i = 1, 2, 3),

(15)

Di ∼ = P1 × Σ1

(i = 1, 2, 3).

(16)

Σ1 denotes a Riemann surface of genus 1, i.e. an elliptic curve. Ei ’s are parallel to one another, and Ei and Dj intersect in an elliptic curve Σ1 Ei ∩ Dj ∼ = Σ1 .

(17)

Singular fibers on the component Di (i = 1, 2, 3) have the type IV , so the gauge group on the 7-branes wrapped on Di (i = 1, 2, 3) is SU (3). Singular fibers on the discriminant component Ei (i = 1, 2, 3) have the fiber type IV ∗ , so the gauge group on the 7-branes wrapped on Ei (i = 1, 2, 3) is E6 . The gauge groups on the discriminant components are displayed in the Table 2 below. Component Fiber type non-Abel. Gauge Group Ei

IV ∗

E6

Di

IV

SU (3)

Table 2: The types of singular fibers and the corresponding (non-Abelian) gauge groups on the discriminant components of Fermat type hypersurface[39].

The Jacobian of the Fermat type hypersurface (12) is given by X 3 + Y 3 + (t − α1 )2 (t − α3 )2 (t − α2 )2 · f gh · Z 3 = 0.

9

(18)

An elliptic K3 fiber of the Fermat type hypersurface (12) is given by (t − α1 )2 X 3 + (t − α2 )2 Y 3 + (t − α3 )2 Z 3 = 0.

(19)

Note that this is the attractive K3 S[6 3 6] with three IV ∗ fibers, discussed in 2.1.1. This is a genus-one fibered attractive K3 lacking a section. So each elliptic K3 fiber is genus-one fibered but does not have a global section. Therefore, the Fermat type hypersurface (12) does not have a rational section. An elliptic K3 fiber of the Jacobian (18) is given by X 3 + Y 3 + (t − α1 )2 (t − α3 )2 (t − α2 )2 · Z 3 = 0.

(20)

This is the attractive K3 S[2 1 2] with three IV ∗ fibers, which is the Jacobian of the attractive K3 S[6 3 6] with three IV ∗ fibers (19), as in 2.1.1. The attractive K3 S[2 1 2] with three IV ∗ fibers (20) has the Mordell-Weil group Z3 as mentioned in 2.1.1. Both the Jacobians (18) and (20) have the constant sections [1 : −1 : 0], [1 : ω : 0] and [1 : ω : 0]7 , and these three constant sections form the group Z3 . From this, we deduce that the Jacobian (18) of the Fermat type multidegree (3,2,2,2) hypersurface (12) has the Mordell-Weil group Z3 as follows: suppose the Jacobian (18) of the Fermat type hypersurface had a section other than the three constant sections forming the group Z3 mentioned the above. We can consider a specialisation of the multidegree (3,2,2,2) Fermat type hypersurface (12) in P2 × P1 × P1 × P1 to the bidegree (3,2) Fermat type hypersurface (19) in P2 × P1 , i.e. choose a specific point in P1 × P1 so the polynomials f, g, h take specific values. Then the assumed section of the Jacobian (18) of the multidegree (3,2,2,2) Fermat type hypersurface (12) in P2 × P1 × P1 × P1 other than the three constant sections [1 : −1 : 0], [1 : ω : 0] and [1 : ω : 0] specialises to a section of the Jacobian (20) of the bidegree (3,2) Fermat type hypersurface (19). This obtained section of the Jacobian (20) of the bidegree (3,2) Fermat type hypersurface is not any of the three constant sections [1 : −1 : 0], [1 : ω : 0] and [1 : ω : 0]. This contradicts the fact that the Jacobian (20) of the bidegree (3,2) Fermat type hypersurface (19) has the Mordell-Weil group Z3 . Therefore, we conclude that the Jacobian (18) of the Fermat type hypersurface does not have a rational section other than the three constant sections [1 : −1 : 0], [1 : ω : 0] and [1 : ω : 0]. In particular, the Jacobian (18) has the Mordell-Weil group of rank 0, so F-theory compactification on the Fermat type hypersurface (12) does not have U (1) gauge field. 2.2.2

Double Cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold

We next outline the results on double covers of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold obtained in [39]. In this paper, we limit our consideration to the most enhanced double covers of P1 × P1 × 1 P × P1 ramified along a multidegree (4,4,4,4) 3-fold given by the equation τ 2 = f · (t − α1 )3 (t − α2 ) · x4 + g · (t − α3 )3 (t − α2 ). 7

ω denotes a primitive cube root of unity.

10

(21)

x is an inhomogeneous coordinate on the first P1 in the product P1 × P1 × P1 × P1 , and t is an inhomogeneous coordinate on the second P1 in the product P1 × P1 × P1 × P1 . f and g are bidegree (4,4) polynomials on P1 × P1 , where these two P1 ’s are the last two P1 ’s in the product P1 × P1 × P1 × P1 . The discriminant components of the double cover of P1 × P1 × P1 × P1 (21) are given by the vanishing loci t = αi (i = 1, 2, 3) (22) and f = 0 g = 0

(23) (24)

in the base 3-fold P1 × P1 × P1 . Similar to the convention in [39], we denote the components by D1 := {f = 0},

D2 := {g = 0},

Ei := {t = αi } (i = 1, 2, 3).

(25)

The form of the component Ei (i = 1, 2, 3) is

and D1,2 has the form

Ei ∼ = P1 × P1

(i = 1, 2, 3),

(26)

Di ∼ = P1 × Σ9

(i = 1, 2, 3).

(27)

Σ9 denotes a Riemann surface of genus 9. Ei ’s are parallel to one another, and Ei and Dj intersect in Σ9 Ei ∩ Dj ∼ (28) = Σ9 . The singular fibers on the components E1 and E3 have the fiber type III ∗ . The singular fibers on the component E2 have the type I0∗ . So the gauge groups on the 7-branes wrapped on E1 and E3 are E7 . The gauge group on the 7-branes wrapped on the discriminant component E2 is SO(8). The singular fibers on the components D1 and D2 have the fiber type III. So the corresponding gauge groups on the 7-branes wrapped on the components D1 and D2 are SU (2). The gauge groups on the discriminant components are displayed in the Table 3 below. The Jacobian of the double cover of P1 × P1 × P1 × P1 (21) ramified along a multidegree (4,4,4,4) 3-fold is given by the equation 1 τ 2 = x3 − f g · (t − α1 )3 (t − α2 )2 (t − α3 )3 · x. 4

(29)

An elliptic K3 fiber of the double cover of P1 × P1 × P1 × P1 (21) is given by τ 2 = (t − α1 )3 (t − α2 ) · x4 + (t − α3 )3 (t − α2 ). 11

(30)

Component Fiber type non-Abel. Gauge Group E1

III ∗

E7

E2

I0∗

SO(8)

E3

III ∗

E7

Di

III

SU (2)

Table 3: The types of singular fibers and the corresponding (non-Abelian) gauge groups on the discriminant components of double cover of P1 × P1 × P1 × P1 [39].

Note that this is the double cover of P1 × P1 discussed in 2.1.2. So, the elliptic K3 fiber (30) is a genus-one fibered attractive K3 without a section, with two III ∗ fibers and one I0∗ fiber. Since each elliptic K3 fiber does not have a section, the double cover of P1 × P1 × P1 × P1 (21) ramified along a multidegree (4,4,4,4) 3-fold does not have a rational section. An elliptic K3 fiber of the Jacobian (29) is given by 1 τ 2 = x3 − (t − α1 )3 (t − α2 )2 (t − α3 )3 · x. 4

(31)

This is the Jacobian of the double cover (30) of P1 × P1 , as in 2.1.2. So we find that the elliptic K3 fiber (31) of the Jacobian (29) is the attractive K3 S[2 0 2] with two III ∗ fibers and one I0∗ fiber. The Jacobian (31) of the double cover (30) of P1 × P1 has the Mordell-Weil group Z2 as mentioned in 2.1.2. Both the Jacobian (29) and the Jacobian (31) have the sections {τ = 0, x = 0} and {τ = ∞, x = ∞}. These two sections form the group Z2 . From this, we find that the Jacobian (29) of the double cover of P1 × P1 × P1 × P1 (21) ramified along a multidegree (4,4,4,4) 3-fold has the Mordell-Weil group Z2 as follows: suppose the Jacobian (29) had a section other than the sections {τ = 0, x = 0} and {τ = ∞, x = ∞}. We can consider a specialisation of the double cover of P1 ×P1 ×P1 ×P1 (21) to the double cover (30) of P1 ×P1 . Then the assumed section of the Jacobian (29) of the double cover of P1 ×P1 ×P1 ×P1 (21) other than the sections {τ = 0, x = 0} and {τ = ∞, x = ∞} specialises to a section of the Jacobian (31) of the double cover of P1 × P1 (30). This obtained section of the Jacobian (31) is not either of the sections {τ = 0, x = 0} and {τ = ∞, x = ∞}, contradicting the fact that the Jacobian (31) of the double cover of P1 × P1 (30) has the Mordell-Weil group Z2 . Therefore, we conclude that the Jacobian (29) of the double cover of P1 × P1 × P1 × P1 (21) does not have a rational section other than the sections {τ = 0, x = 0} and {τ = ∞, x = ∞}. In particular, the Jacobian (29) has the Mordell-Weil group of rank 0, so F-theory compactification on the double cover of P1 × P1 × P1 × P1 (21) does not have U (1) gauge field.

12

2.3

Review of Possibility of Cancelling the Tadpole Anomaly

In this subsection, we briefly review the possibility of cancelling the tadpole anomaly, considered in [37, 38, 39]. 2.3.1

K3 Surfaces

For a product of attractive K3 surfaces S1 × S2 , the second Chern class c2 (S1 × S2 ) is even. So, a quantisation condition[57] imposed on 4-form flux G for a product of attractive K3’s S1 × S2 is G ∈ H 4 (S1 × S2 , Z). (32) In the presence of 4-form flux G, the tadpole anomaly cancellation condition [58] is Z 1 1 G ∧ G + N3 = χ(K3 × K3) = 24. 2 K3×K3 24

(33)

N3 denotes the number of 3-branes turned on. Flux compactification of M-theory on a product of attractive K3 surfaces S1 × S2 was considered in [59]. 4-form flux G has decomposition G = G0 + G1 ,

(34)

G0 ∈ H 1,1 (S1 , R) ⊗ H 1,1 (S2 , R) G1 ∈ H 2,0 (S1 , C) ⊗ H 0,2 (S2 , C) + h.c.

(35) (36)

where

Under the assumptions G0 = 0

(37)

N3 = 0,

(38)

and [59] obtained all the pairs S[2a b 2c] × S[2d e 2f ] which satisfy the tadpole anomaly cancellation condition (33). Relaxing (38) to N3 ≥ 0, (39) the list of the pairs of attractive K3 surfaces was extended in [60]. Both the lists in [59] and [60] contain only finitely many pairs of attractive K3 surfaces. So the complex structure moduli spaces are the sets of finite discrete points. Therefore, the complex structure moduli in [59, 60] are stabilised. According to Table 2 in 2.2 of [60]8 , for the Fermat type K3 hypersurface S[6 3 6] (3) discussed in 2.1.1, the tadpole anomaly can be cancelled for F-theory compactifications on 8

[60] uses different notation for attractive K3 surfaces. They use [a b c] for the attractive K3  the subscript  2a b surface whose transcendental lattice has the intersection matrix . For example, [3 3 3] in the list of b 2c [60] represents the attractive K3 surface denoted by S[6 3 6] in this paper.

13

S[6 3 6] × S[4 2 4] and on S[6 3 6] × S[2 1 2] , by turning on sufficiently many 3-branes. Similarly, for the Fermat quartic S[8 0 8] (9) discussed in 2.1.3, for F-theory compactification on S[8 0 8] × S[2 0 2] , tadpole anomaly can be cancelled by including sufficiently many 3-branes. As in 2.1.2, the double cover of P1 ×P1 (7) ramified over a bidegree (4,4) has two type III ∗ singular fibers and one type I0∗ fiber.  Theattractive K3 surface S[4 0 4] whose transcendental 4 0 lattice has the intersection matrix is the Kummer surface Km(Ei × Ei ). Following 0 4 the method in [61], we find that the attractive K3 S[4 0 4] admits a genus-one fibration without a section with two singular fibers of type III ∗ and one fiber of type I0∗ . From this observation, we expect that this attractive K3 surface S[4 0 4] with two III ∗ fibers and one I0∗ fiber is the double cover of P1 × P1 (7). According to Table 2 of [60] (the attractive K3 S[4 0 4] also appears in Table 1 of [59]), the tadpole anomaly can be cancelled for F-theory compactifications on the three pairs S[4 0 4] × S[6 0 6] , S[4 0 4] × S[4 0 4] and S[4 0 4] × S[2 0 2] . Therefore, we see that the tadpole anomaly can be cancelled for F-theory compactifications on the double cover (7) times (some appropriately chosen) attractive K3. To summarise, tadpole anomaly can be cancelled for F-theory compactifications on K3 surfaces discussed in 2.1 times some appropriate attractive K3, by including sufficiently many 3-branes. 2.3.2

Calabi-Yau 4-folds

We now review the possibility of cancelling the tadpole anomaly considered in [39]. For a genus-one fibered Calabi-Yau 4-fold Y , let G4 denote a 4-form flux on Y , then a 4-form flux G4 is subject to the quantisation condition[57] 1 G4 + c2 (Y ) ∈ H 4 (Y, Z). 2

(40)

c2 (Y ) denotes the second Chern class of Y . Two conditions need to be imposed on the 4-form flux G4 to preserve supersymmetry in the 4d theory[62]: G4 ∈ H 2,2 (Y, Z), (41) and G4 ∧ J = 0.

(42)

J denotes a Kähler form of Calabi-Yau 4-fold Y . To preserve Lorentz invariance in the 4d theory, one more condition needs to be imposed[63] on the 4-form flux G4 . The condition is that a 4-form flux G4 has one leg in the fiber. Let Y ←−−− E   py B3 14

be a genus-one fibration of a Calabi-Yau 4-fold. (So for the models considered in this paper, the base 3-fold B3 is P1 × P1 × P1 .) For an elliptic fibration with a section p˜, the condition that a 4-form flux G4 has one leg in the fiber, in equation, reads: G4 · p˜−1 (C) · p˜−1 (C 0 ) = 0

(43)

G4 · S0 · p˜−1 (C) = 0

(44)

for any C, C 0 ∈ H 1,1 (B3 ). In the equation (44), S0 represents a rational zero section. Generalisation of the condition that a 4-form flux G4 has one leg in the fiber to genus-one fibration p lacking a global section was proposed in [36]. A genus-one fibration p has some multisection, say an n-section N . Then the condition that a 4-form flux G4 has one leg in the fiber for genus-one fibration, in equation, reads[36]: G4 · p−1 (C) · p−1 (C 0 ) = 0

(45)

ˆ · p−1 (C) = 0 G4 · N

(46)

ˆ denotes some appropriate sum of the n-section N and exceptional for any C, C 0 ∈ H 1,1 (B3 ). N divisors. The following equation needs to be satisfied to cancel tadpole anomaly[58]: 1 χ(Y ) = G4 · G4 + N3 . 24 2

(47)

N3 is the net number of 3-branes turned on, i.e. N3 denotes the number of 3-branes minus anti 3-branes. For a stable compactification, we need N3 ≥ 0,

(48)

to exclude anti 3-branes on Y . Algebraic 2-cycles in Y correspond to (2,2)-forms. So 2-cycles in Y satisfy the condition (41), and they are candidates for 4-form fluxes. It is easy to see that the primitivity condition (42) together with the first equation (45) of the condition that a 4-form flux has one leg in the fiber exclude all the 2-cycles coming from the ambient space P2 × P1 × P1 × P1 for a multidegree (3,2,2,2) hypersurface. Similarly, these two constraints rule out all the 2-cycles coming as pullbacks of P1 × P1 × P1 × P1 for a double cover. So, only the candidate 2-cycles satisfying the constraints (42) and (45) are intrinsic 2-cycles. Most 2-cycles of Calabi-Yau 4-fold Y is intrinsic, but it is in general very difficult to give an explicit description of an intrinsic 2-cylcle for genus-one fibered Calabi-Yau 4-folds considered in this paper. As a consequence, the self-intersection of such intrinsic 2-cycle in Y is very hard to calculate. Therefore, only numerical bounds on the self-intersection of a 4-form flux G4 to cancel the tadpole anomaly are deduced in [39], by computing the Euler characteristics of the genus-one fibered Calabi-Yau 4-folds without a section. 15

We list the Euler characteristics χ(Y ) of the genus-one fibered Calabi-Yau 4-folds computed in [39] in Table 4 below. Note that Euler characteristics are topological invariants, so they do not depend on defining equations. Every multidegree (3,2,2,2) hypersurface in P2 × P1 × P1 × P1 shares the identical Euler characteristic, and the same is true for a double cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold. [39] confirmed that the Euler characteristics of multidegree (3,2,2,2) hypersurface in P2 × P1 × P1 × P1 and double cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold are divisible by 24. CY 4-fold Y

Euler char. χ(Y )

χ(Y ) 24

(3,2,2,2) hypersurface

1584

66

double cover of P1 × P1 × P1 × P1

3744

156

Table 4: Euler characteristics of multidegree (3,2,2,2) hypersurface in P2 × P1 × P1 × P1 and double cover of P1 × P1 × P1 × P1 ramified along a multidegree (4,4,4,4) 3-fold[39].

3

Complete Intersections of multidegree (2,1,1,1) and (2,1,1,1) Hypersurface in P3 × P1 × P1 × P1

In addition to the two families of genus-one fibered Calabi-Yau 4-folds without a section[39] discussed in 2.2, we introduce complete intersections of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 × P1 × P1 × P1 as a new family of Calabi-Yau 4-folds without a section in this paper. This additional family is genus-one fibered and elliptic K3-fibered. We choose a specific form of equations so an elliptic K3 fiber is the Fermat quartic, with presentation as a complete intersection (10) of bidegree (2,1) and (2,1) hypersurfaces in P3 ×P1 . In particular, each elliptic K3 fiber is genus-one fibered but does not admit a global section, so the complete intersection Calabi-Yau 4-fold does not have a rational section. We determine the forms of the discriminant components. We compute the types of singular fibers to deduce the gauge symmetries on the 7-branes wrapped on the components. Similar to the Calabi-Yau 4-folds discussed in 2.3.2, it is very hard to give explicit descriptions of intrinsic 2-cycles for the complete intersection Calabi-Yau 4-folds. As a result, whether the tadpole anomaly can be cancelled remains open. We compute the Euler characteristics of the multidegree (2,1,1,1) and (2,1,1,1) complete intersection Calabi-Yau 4-folds to obtain a bound on the self-intersections of 4-form fluxes. We confirm that the Euler characteristic is divisible by 24. We also compute the potential chiral matters on a bulk of our interest, and potential chiral matter fields localised along the matter curves. We determine the numbers of generations. We also determine the potential Yukawa couplings. Since whether the tadpole anomaly can be cancelled remains open, we can only say that the chiral matters and the Yukawa couplings could arise. We also compute the Mordell-Weil group of the Jacobian. 16

3.1

Defining Equation

Complete intersections of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 ×P1 ×P1 ×P1 are Calabi-Yau 4-folds. The intersection of two degree 2 hypersurfaces in P3 is a genus-one curve, so these Calabi-Yau 4-folds are genus-one fibered under the projection onto a base 3-fold P1 × P1 × P1 . The complete intersection of two bidegree (2,1) hypersurfaces in P3 × P1 is a K3 surface, so they are also elliptic K3-fibered, under the projection onto a base surface P1 × P1 , where these two P1 ’s are the last two P1 ’s in P3 × P1 × P1 × P1 . We in particular consider the complete intersection given by x21 + x23 + 2t · f · x2 x4 = 0 x22 + x24 + 2t · g · x1 x3 = 0.

(49)

[x1 : x2 : x3 : x4 ] is homogeneous coordinates on P3 and t is an inhomogeneous coordinate on the first P1 in P3 × P1 × P1 × P1 . f, g are bidegree (1,1) polynomials on P1 × P1 , where these two P1 ’s are the last two P1 ’s in P3 × P1 × P1 × P1 . The simultaneous vanishing of the two equations in (49) gives a complete intersection. Each elliptic K3 fiber of complete intersections given by (49) is a complete intersection of bidegree (2,1) and (2,1) hypersurfaces in P3 × P1 x21 + x23 + 2tx2 x4 = 0 x22 + x24 + 2tx1 x3 = 0.

(50)

This is the Fermat quartic S[8 0 8] with six I4 fibers discussed in 2.1.3. (Recall that theFermat  8 0 quartic is an attractive K3 whose transcendental lattice has the intersection matrix , 0 8 so the Fermat quartic is denoted by S[8 0 8] .) As in [56, 38], the Fermat quartic S[8 0 8] with six I4 fibers (with genus-one fibration induced from (50)) does not admit a section. Therefore, complete intersections given by (49) do not admit a rational section.

3.2

Forms of the Discriminant Components and Gauge Symmetries

We compute the Jacobian J(Y ) of complete intersection Y given by (49). Similar computation can be found in [38]. We introduce a parameter λ, and obtain the equation built from (49): x21 + x23 + 2t · f · x2 x4 − λ(x22 + x24 + 2t · g · x1 x3 )). We put this equation into a symmetric matrix   1 0 −tgλ 0  0 −λ 0 tf   . −tgλ 0 1 0  0 tf 0 −λ

17

(51)

(52)

The determinant of this 4 × 4 matrix is   1 0 −tgλ 0  0 −λ 0 tf   = −t2 g 2 λ4 + (t4 f 2 g 2 + 1)λ2 − t2 f 2 . det  −tgλ 0 1 0  0 tf 0 −λ

(53)

Equating with τ 2 , we obtain the equation τ 2 = −t2 g 2 λ4 + (t4 f 2 g 2 + 1)λ2 − t2 f 2 .

(54)

This gives a Jacobian J(Y ) of the complete intersection Y given by (49). (J(Y ) (54) is a double cover of P1 × P1 × P1 × P1 .) Complete intersection Y and its Jacobian J(Y ) have exactly the same types of singular fibers at exactly the same locations. So, from Jacobian J(Y ), we can determine the discriminant locus, singular fiber types and corresponding gauge groups. The discriminant of the Jacobian (54) is ∆ = 16f 2 g 2 t4 (f 2 g 2 t4 − 1)4 = 16f 2 g 2 t4 (f gt2 − 1)4 (f gt2 + 1)4

(55)

From the vanishing of the discriminant ∆ (55), we see that the discriminant components are E1 E2 D1 D2 D3 D4 D5 D6

:= := := := := := := :=

{t = 0} {t = ∞} {f = 0} {g = 0} {u = ∞} {v = ∞} {f gt2 = 1} {f gt2 = −1}.

(56)

u and v are inhomogeneous coordinates on the second P1 and the third P1 in P1 × P1 × P1 respectively. So, f, g are bidegree (1,1) polynomials in the two variables u and v. We have Ei ∼ (57) = P1 × P1 (i = 1, 2) and

Di ∼ = P1 × P1

(i = 3, 4).

(58)

(i = 1, 2, j = 3, 4).

(59)

E1 and E2 are parallel. D3 ∩ D4 is P1 . We have Ei ∩ Dj ∼ = P1

18

Bidegree (1,1) curve in P1 × P1 has genus g = (1 − 1) · (1 − 1) = 0, so is a rational curve Σ0 ∼ = P1 . So, Di ∼ (60) = P1 × Σ0 ∼ = P1 × P1 (i = 1, 2), and

Ei ∩ Dj ∼ = Σ0 ∼ = P1

(i, j = 1, 2).

(61)

Two bidegree (1,1) curves in P1 × P1 meet in 2 points, so D1 ∩ D2 is a disjoint sum of 2 P1 ’s. We omit the components D5,6 . We summarise the forms of irreducible components Ei (i = 1, 2) and Di (i = 1, 2, 3, 4) of the discriminant locus and their intersections in Table 5. Component

Topology

Ei (i = 1, 2)

P1 × P1

Di (i = 1, 2, 3, 4)

P1 × P1

Intersections D1 ∩ D2

parallel 2 P1 ’s

D3 ∩ D4

P1

Ei ∩ Dj (i = 1, 2, j = 1, 2, 3, 4)

P1

Table 5: The discriminant components of complete intersection of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 × P1 × P1 × P1 , and their intersections. The components D5 and D6 are omitted.

We compute the types of singular fibers. From the discriminant ∆ (55), we see that singular fibers are multiplicative. So the fiber types are In for some n. From the discriminant ∆ (55), the fiber types can be determined. The fiber type on the components E1 and E2 is I4 . The corresponding gauge groups on the 7-branes wrapped on the components E1 and E2 are SU (4). The fiber type on the components D1 and D2 is I2 . Similarly, the fiber type on the components D3 and D4 is also I2 . The corresponding gauge groups on the 7-branes wrapped on D1 , D2 , D3 and D4 are SU (2). The fiber type on the components D5 and D6 is I4 . The corresponding gauge groups on the 7-branes wrapped on D5 and D6 are SU (4). We summarise the result in Table 6 below.

3.3

Euler Characteristic and Consideration on Tadpole Anomaly

We now compute the Euler characteristic χ(Y ) of complete intersection of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces Y in P3 × P1 × P1 × P1 . We use the exact sequence 0 −−−→ TY −−−→ TP3 ×P1 ×P1 ×P1 |Y −−−→ NY −−−→ 0 19

(62)

Component Fiber type non-Abel. Gauge Group E1,2

I4

SU (4)

D1,2,3,4

I2

SU (2)

D5,6

I4

SU (4)

Table 6: The types of singular fibers and the corresponding gauge groups on the discriminant components of complete intersection.

to obtain the equality c(TY ) =

c(TP3 ×P1 ×P1 ×P1 )|Y . c(NY )

(63)

TY is the tangent bundle of a complete intersection of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces Y , and this naturally embeds into the tangent bundle TP3 ×P1 ×P1 ×P1 of the ambient space P3 × P1 × P1 × P1 . |Y means the restriction to Y . NY is the resultant normal bundle. NY ∼ = O(2, 1, 1, 1)⊕2 ,

(64)

c(NY ) = (1 + 2x + y + z + w)2 .

(65)

c(TP3 ×P1 ×P1 ×P1 )|Y = (1 + 4x + 6x2 + 4x3 )(1 + 2y)(1 + 2z)(1 + 2w)|Y .

(66)

so We have

From the equation (63), we find that the Euler characteristic χ(Y ), which is equal to the top Chern class of c(Y ), is χ(Y ) = 864. (67) This is divisible by 24, and we have 864 χ(Y ) = = 36. 24 24

(68)

We also find the second Chern class c2 (Y ) from the equation (63): c2 (Y ) = (2x2 + 4xy + 4xz + 4xw + 2yz + 2yw + 2zw)|Y .

(69)

This is even, so the quantisation condition for a 4-form flux G4 [57] reduces to G4 ∈ H 4 (Y, Z).

20

(70)

A bound on self-intersection of a 4-form flux to cancel the tadpole anomaly is χ(Y ) 1 1 − G4 · G4 = 36 − G4 · G4 24 2 2 ≥ 0.

N3 =

(71)

Whether tadpole anomaly can be cancelled for complete intersections of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 × P1 × P1 × P1 remains open. Note that the Euler characteristic is a topological invariant, so it does not depend on a defining equation. Every complete intersection of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 × P1 × P1 × P1 has the same Euler characteristic. CY 4-fold Y

Euler char. χ(Y )

χ(Y ) 24

(2,1,1,1) and (2,1,1,1) complete intersection

864

36

Table 7: Euler characteristics for complete intersections of multidegree (2,1,1,1) and (2,1,1,1) hypersurfaces in P3 × P1 × P1 × P1 .

3.4

Matter Fields and Yukawa Couplings

We compute matter spectra on a bulk and matters localised along matter curves. We can only say that matters we compute could arise, since the possibility of cancelling the tadpole anomaly remains open. We also compute Yukawa coupling. As discussed in [64], consider the situation gauge group G on 7-branes breaks to a subgroup Γ so that Γ×H ⊂G (72) is maximal. This corresponds to a deformation of singularity associated with gauge group G, and as a result matter fields arise on 7-branes[8] under the deformation. When Γ × H has a representation (τ, T ), matter fields arise in representation τ of Γ and its generation (i.e. the number of chiral generations minus anti-chiral generations) is given by the following equation[64] Z nτ − nτ ∗ = −

c1 (S)c1 (T ).

(73)

S

In the equation (73), S denotes a discriminant component on which 7-branes are wrapped. T is a bundle transforming in representation T of H. We limit our attention to the case H is U (1). We focus our attention on the bulk component E1 . We abbreviate E1 for E. Let L be a line bundle on the bulk E which preserves supersymmetry for some Kähler class ω. Since

21

E∼ = P1 ×P1 , L ∼ = O(a, b) for integers a, b ∈ Z. As discussed in [64], the line bundle L ∼ = O(a, b) preserves supersymmetry if and only if ab < 0.

(74)

When a matter field has U (1) charge n, the associated line bundle is Ln . When the gauge group SU (4) on the bulk E breaks to SU (3) under SU (4) ⊃ SU (3) × U (1),

(75)

adjoint 15 of SU (4) decomposes as 15 = 80 + 3−4 + 34 + 10 . So chiral matters 3 arise on the bulk E, and its generation is Z n3 − n3 = − c1 (E)c1 (L−4 ) = 4(2x + 2y)(ax + by) E

(76)

(77)

= 8(a + b). We now compute the matter fields localised along a matter curve E ∩ Di ∼ = Σ0 (i = 1, 2). 1 Σ0 denotes a Riemann surface of genus 0, i.e. a rational curve P . Since E ∩ D1,2 is a bidegree (1,1) curve in P1 × P1 , as a divisor it is represented by x + y. so the restriction LΣ0

(ax + by)(x + y) = (a + b)xy, of the line bundle L ∼ = O(a, b) to E ∩ D1,2 = Σ0 is LΣ0 ∼ = OΣ0 (a + b).

(78) (79)

(80)

Matter field 4 localised along a matter curve Σ0 decomposes as 4 = 3−1 + 13 .

(81)

Now by Riemann-Roch theorem, 1/2

n3 = h0 (KΣ0 ⊗ L−1 Σ0 ) = h0 (OΣ0 (−(a + b) − 1)) ( −(a + b) (a + b ≤ 0) = 0 (a + b > 0)

(82)

Similarly, 1/2

n1 = h0 (KΣ0 ⊗ L3Σ0 ) = h0 (OΣ0 (3(a + b) − 1)) ( 3(a + b) (a + b ≥ 0) = 0 (a + b < 0) 22

(83)

So, when a + b > 0 singlets 1 localise along a matter curve Σ0 . When a + b < 0, matter fields 3 localise along a matter curve Σ0 . As discussed in [64], Yukawa coupling arises from the following three cases: i)interaction of three matter fields on a bulk component, ii)interaction of a field on a bulk component and two matter fields localised along a matter curve, and iii)triple intersection of three matter curves meeting in a point. E ∼ = P1 × P1 is a Hirzebruch surface, so three matter fields on the bulk E does not generate Yukawa coupling, as mentioned in [64]. Therefore, Yukawa coupling does not arise from the case i) on the bulk E. We focus our attention on Yukawa couplings arising from the case ii). When a + b > 0, a matter field 3 on the bulk E and two singlets 1 localised along a matter curve Σ0 generate Yukawa coupling 3 · 1 · 1. (84) When a + b < 0, a matter field 3 on the bulk E and two matter fields 3 localised along a matter curve Σ0 generate Yukawa coupling 3 · 3 · 3.

(85)

The results are shown in Table 8 below. Gauge Group SU (4)

a + b Matters on E

# Gen. on E

Matters on Σ0

# Gen. on Σ0

Yukawa

>0

3

8(a + b)

1

3(a + b)

3·1·1