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.