Right-handed neutrinos and $U\left(1\right)_{X}$ symmetry-breaking

In Ref. [1] we proposed a model for Heterotic $F$-theory duality with Wilson line symmetry-breaking and a $4+1$ split of the $F$-theory spectral divisor. One goal of this note is to call attention to the existence of right-handed neutrinos in our $F$-theory model. As pointed out in Section 4 of Ref. [2] such existence may be evidence for the $U\left(1\right)_{X}$-symmetry that remains after the Higgsing of $E_{8}$ via \[ E_{8}\Rightarrow SU\left(5\right)_{gauge}\oplus\left[SU\left(4\right)\oplus U\left(1\right)_{X}\right]_{Higgs} \] occasioned by the $4+1$ split of the spectral divisor. In addition, as a result of the $\mathbb{Z}_{2}$-action that supports the Wilson line we argue that the $U\left(1\right)_{X}$-symmetry is, in fact, broken to $\mathbb{Z}_{2}$-matter parity. Finally we identify co-dimension $3$ singularities which determine Yukawa couplings for the MSSM matter fields.


The geometric model
The Tate form for the Calabi-Yau fourfold W 4 /B 3 in our set-up is given by the equation (1.1) x 3 + a 4 zwx 2 + a 2 z 3 w 2 x + a 0 z 5 w 3 1 wy 2 − a 5 wx + a 3 z 2 w 2 y 1 = 0 over a Fano threefold B 3 = P [u0,v0] × D 2 where D 2 is a special del Pezzo surface with a Z 4 -symmetry. The pair W 4 /B 3 admits an equivariant Z 2 -action with respect to which a j , z, y the skew eigenspace. Parametrizing y 2 − x 3 = 0 by the pull-back of the Tate form to y 2 − x 3 = 0 is given by a 5 t 5 + a 4 t 4 z + a 3 t 3 z 2 + a 2 t 2 z 3 + a 0 z 5 = a 5 t 4 + a 54 t 3 z − a 20 t 2 z 2 − a 0 z 3 (t + z) (t − z) = 0 The subvariety D (4) + D (1) ⊆ y 2 − x 3 = 0 is called the spectral variety. We make the assumption that where Q/B 3 has affine fiber coordinate ϑ 0 . The branch locus ∆ is then given by the equation so that the equations of the images of the two components are In [1] we show that C  (1) . Here the divisors (ζ) and (τ ) given by the two sections each project to the locus {ϑ 0 = ∞} ⊆ Q and C and W (2) 4 ofW 4 /Q are induced via fibered product from modifications Q (1) and Q (2) of Q so it will be convenient to describe these modifications in terms of their effect on the branch locus ∆ ⊆ Q, modifications that we denote as ∆ (1) and ∆ (2) =∆.
1.5.1. Resolution of codimension-one singularities of ∆. Partial resolution of singularities of ∆ by blowing up smooth loci in Q. The proper transform ∆ (1) of ∆ in the first modification Q (1) of Q is given by the equation (2) =Q given by the equation However ∆ (2) still has a singular locus, namely a smooth one-dimensional locus that becomes a smooth curve C 1.5.2. Resolution of codimension-two singularities of ∆ (2) . It becomes important at this point that is a smooth K3-surface on which the Z 2 -action is free, that the imbedded image Σ   GUT can be variously written as Z 14 = ϑ 1 Z 0 − a 420 = 0 A careful study of the first modification in [1] shows that the first lifting S that is, along the entire locus {a 420 = a 5 = Z 14 = 0} .
So we obtain the isomorphic lifting S GUT ⊆ W 4 . Finally, since S GUT maps into Z 14 = 0 and so never meets the singular curve C Furthermore (1.9) a 5 = ϑ 1 Z 0 + ϑ 2 Z 23 .
1.6. Matter and Higgs curves in the spectral divisorC (4) Higgs ⊆W 4 . We have let C Higgs = C (4) Higgs ⊆Q = Q (2) denote the proper (also the total) transform of (1.5) and C Higgs ⊆W 4 denote the proper (also the total) transform ofC Higgs .
The matter curves are given by Higgs . The Higgs curve is given by Higgs .
All three curves lie in the surfaceS GUT given in D 4 . The identity can be used to give alternative formulas for the matter and Higgs curves by substituting a 5 − ϑ 2 Z 23 for ϑ 1 Z 0 in (1.12) or for ϑ 1Z0 Z 23 in (1.10).

Configuration of components ofW
is a branched double cover we will first describe the three components of Q (2) × B3 S GUT . These are Z 14 = 0 that is the proper transform of Q × B3 S GUT , {Z 0 = 0}that is the proper transform of the exceptional locus of the blow-up of the locus {ϑ 0 = z = 0} in Q, and {Z 23 = 0}, the exceptional locus of the blow-up of the locus {a 5 − ϑ 1 Z 0 = Z 14 = 0} in Q (1) . Over a general p ∈ S GUT , the configuration is a chain If a 5 (p) = 0, the proper transform of Z 0 (p) ⊆ Q (1) (p) does not equal its total transform that becomes Z 0 (p) Z 23 (p) = 0 ⊆ Q (2) (p). Thus from (1.9) we have giving the functional equation So the chain configuration (2.1) changes, namely it becomes Also when a 5 = 0, (1.8) becomes Therefore by (2.2) the branch locus ∆ (2) =∆ does not intersect Z 14 (p) = 0 and it intersects Z 0 (p) = 0 only if additionally a 4 = 0, in which case its inter- So if in addition a 4 (p) = 0 this intersection is given by We view near w 2 =Z 0 = Z 23 = 0 as a family of affine quadric surfaces in the coordinates w 2 ,Z 0 , Z 23 parametrized by a 4 becoming the quadric cone splits into two disjoint components that we denote as D 1 (p) (meeting the section (ζ)) and D 4 (p) (meeting the section (τ )).
∆ ∩ {Z 23 (p) = 0} is given by the equation consists of two components that cross over the surface ϑ 1 ϑ 2 + a 4Z14 = Z 23 = 0 ⊆ Q. We have designated these two components as D 2 (p) and D 3 (p). Furthermore Thus is an irreducible rational curve meeting D 1 (p) and D 4 (p), each in a single point. By (1.9), {Z 0 (p) = 0} does not meet {Z 23 (p) = 0} unless a 5 (p) = 0, so over a general p ∈S GUT , the components of the fiber ofW 4 /B 3 configure themselves as the extended Dynkin diagram of SU (5).

2.1.2.
Over a general point p ∈ Σ (4) and the chain configuration becomes  inserts an exceptional P 1 , one via the specialization D 2 (p) D ′ 2 (p) + E 23 (p) and the other via the specialization Both configure the fiber ofW 4 /B 3 over p as the extended Dynkin diagram of SU (6).
corresponding to the extended Dykin diagram of SU (6).
5 , both of the two above splittings occur simultaneously yielding a configuration corresponding to the extended Dykin diagram of SU (7).
The situation over Σ (4) 10 := {z = a 5 = 0} ⊆ B 3 is more complex. {Z 0 (p) = 0} contains the center of the second modification, that is, Thus the fiber ofW 4 /B 3 is a tree of P 1 's whose intersection configuration is that of the extended Dynkin diagram of SO (12).

U (1) X -charges
In our model in [1] the U (1) X -section is derived from the line bundle where (τ ) is the divisor associated with the section τ and (ζ) is the divisor associated with the section ζ. Lifting (ζ) − (τ ) to the divisor ζ − (τ ) in the desingularizatioñ W 4 we obtain its U (1) X -action by normalizing it so as to have zero intersection with all components of a general fiber over S GUT . Since, for p ∈ S GUT , the section (ζ) meets D 1 (p) and the section (τ ) meets D 4 (p), we obtain the divisor This divisor is orthogonal to all four of the D j (p) and therefore supports the U (1) Xaction.
3.1. U (1) X -charge on Σ has one intersection with D 2 and zero intersection with all other roots and with (ζ) and (τ ) since the two sections intersect D 1 (p) and D 4 (p) respectively. D 03 (p) has one intersection with D 3 and zero intersection with all other roots and with (ζ) and (τ ).
On the other hand, over the matter curve Σ (41) 5 , D 0 (p) again becomes reducible but this time {Z 0 (p) = 0} does not, rather the branch-points of D 0 (p) over {Z 0 (p) = 0} come together to give where D 01 (p) has one intersection with D 1 and zero intersection with all other roots and D 40 (p) has one intersection with D 4 and zero intersection with all other roots. D 01 (p) also misses (ζ) and (τ ) even though D 01 (p) and (ζ) both meet D 1 (p). We designate D 01 (p) as the new root and so the matter curve Σ Finally over the Higgs curve Σ the new root E 23 (p) arises from a splitting of D 2 (p) or D 3 (p) (depending on which of the two small resolutions of the nodal locus of W Here we can choose either small resolution since in either we have a curve with charge +2 and a curve −2, either of which can be designated as the 'new root' as needed. We summarize charges as follows: Since the cubic (3.5) lies in the spectral divisor, it lifts into D 40 and meets no other component of the fiber as long as a 5 =0. We denote the closure of this locus in D 40 as 4 . If additionally a 5 (p) = 0 then, by allowably general choices of the forms a j , a 4 (p) =0 and solutions to (3.5) specialize to Thus Γ 0 meets {Z 14 = 0} simply over a 5 (p) = 0. Now the transform of (3.5) in Q (2) Also on {Z 0 = 0} we can use the second form of the equation to obtain a curve we designate asΓ 0 . NowΓ 0 is disjoint from the support of [U (1) X ] except over the point {z (p) = a 5 (p) = a 420 (p) = 0} where it meets the remaining components over Z 14 Z 23 = 0 , namelỹ where p ∈ {z = a 420 = a 5 = 0} ⊆ S GUT andΓ 0 meets D 4 simply above the point a 5 = a 420 =Z 0 =Z 14 = 0 ⊆ Q (2) . We have the following table of intersection numbers with (3.1): Recall now that the choice in the last entry in the table depends on the choice of small resolution of C The fact thatΓ is reducible will force a coupling (1Γ, 1Γ, 1Λ) involving a curve with U (1) X -charge +10. To construct this curve, consider the curveΛ 0 inW 4 given by where we recall from §6.2.1 of [3] that this is the lifting Λ 0 of an anti-canonical elliptic curve Λ of B 2 into S GUT . Now the divisor associated to the normal bundle to (τ ) inW 4 is the pullback S GUT ⊆ B 3 so, by the projection formula, Therefore, recalling that, Charges associated to the U (1) X -action are then given by the following table:

Yukawa couplings
(1.2) then implies that at those points we also have a 3 = 0 since our second section required −a 420 = a 53 . So we conclude that over the twelve points {a 5 = a 420 = z = 0} ⊆ B 3 both matter curves, the Higgs curve and the curve that supports the right-handed neutrinos all intersect.
We next recall from §6.2.1 of [3] that the curve {a 5 = z = 0} ⊆ B 3 contains two disjoint rational curves given by {u 0 v 0 = 0} that are interchanged under the Z 2 -action and each meets the residual curve in two points {a 5 = a 4 = z = 0}. It is over these crossing points of components that we find the 'top' (10 M , 10 M , 5 H )couplings associated with the U (1) X -action. Furthermore, since a 3 also vanishes when a 5 = a 420 = 0 allowing 'bottom' (10 M ,5 M ,5 H )-couplings associated with the U (1) X action over {a 5 = a 0 = 0}. Next points where the neutrino curve meets thē The U (1) X -charge c acts on the representations (3.2) as the character u → u c . Under the Z 2 -action, the section τ is interchanged with the inherited section ζ, so the line bundle O W4 ((τ ) − (ζ)) is carried to its inverse. Roots are sent to their negatives by the reversal of the choice of positive Weyl chamber. Therefore Since Z 2 -action on the maximal torus of the complex algebraic group E C 8 is by complex conjugation, the only possible non-zero U (1) X -charges c that are Z 2 -invariant are those such that u c = u −c . Therefore requiring that (5.1) u c = u −c implies u 2c = 1, that is u c = e mπi = ±1. Said otherwise, the requirement (5.1) breaks U (1) X -symmery to Z 2 -symmetry. That is, as mentioned in (5.1) above, descent to the Z 2 -quotient breaks U (1) Xsymmetry to Z 2 -symmetry. Thus the right-handed neutrinos can obtain a Majorana mass at the compactification scale.
Furthermore, the Z 2 -action moves the spectral divisor C (4) Higgs ⊆W 4 to the opposite component ofC (4) Higgs ×QW 4 , namely to the component whose intersection withW 4 × B3 S GUT lies in D 1 . But this 'opposite' component is, in fact, the spectral divisor with respect to the opposite choice of positive Weyl chamber and its associated opposite (or 'flopped') Brieskorn-Grothendieck equivariant crepant resolution as explained in [4]. SinceΓ andΛ are defined entirely with respect to their relationship with the spectral variety for their respective Brieskorn-Grothendieck equivariant crepant resolutions, bothΓ andΛ will be taken to themselves under the Z 2 -action.
In summary, the (4 + 1)-spectral equation breaks E 8 -symmetry to SU (5) gauge × U (1) X ⊆ SO (10) and the Z 2 -action breaks U (1) X , leaving only SU (5) gauge × Z 2 -symmetry before wrapping the Wilson line. In particular, the right-handed neutrinos discussed above can in principle obtain a large Majorana mass near the GUT/compactification scale. Thus the theory of [1] contains three families of quarks and leptons and one pair of Higgs doublets (after GUT symmetry breaking via the Wilson line). Note, the theory also includes a complete MSSM twin sector. The Z 2 -symmetry is identified as matter parity. However, in addition as explained in [3], the theory has an asymptotic Z R 4 -symmetry which forbids dimension 4 and 5 baryon-violating operators as well as the Higgs µ term. Finally the theory has non-trivial Yukawa couplings located at co-dimension 3 singularities.
Further analysis is necessary to determine the 3 × 3 Yukawa coupling matrices. We also want to analyze the MSSM twin world in order to address the question of relative scales between the visible and twin sectors and to determine whether or not there are any possible portals to the twin sector. Finally the issue of supersymmetry breaking must be addressed.