F-Theory $E_7$, Heterotic String-Derived Vacua And Flipped $SO(10)$ In Hor\v{a}va-Witten Theory

In this paper, $E_6$ and especially $E_7$ GUT are considered in the F-theory setting in view of the free fermionic construction of the $4D$ heterotic string. In particular, the NAHE-Based LRS model of \cite{Cleaver:2000ds, Cleaver:2002ps} is revisited as an illustration where the starting point was taken to be the $N=4$, $E_7 \times E_7 \times SO(16)$ which through the use of boundary condition basis vectors is reduced to obtain the flipped $SO(10)$ GUT symmetry. We also seek to extend the results of \cite{Faraggi:2002ah} in the case of the flipped $SU(5)$ to home in on the flipped $SO(10)$ vacua from the Hor\v{a}va-Witten theory where the $E_8$ gauge group on the observable sector decomposes as $E_{8}\supset E_{6}\times SU(3)$ with $E_{6}$ being the gauge group of the effective field theory. We find for the $E_{6}$ GUT symmetry, solutions of type A and solutions of type B where the Hirzebruch surfaces are considered for the base contrary to \cite{Faraggi:2002ah} where flipped $SU(5)$ vacua were studied and only solutions of type B were found. Moreover, no solutions are found in the case of the base being the del Pezzo surfaces. Furthermore, this goes hand in hand with the heterotic, low-energy string-derived effective model discussed in \cite{Ashfaque:2016psv,Ashfaque:2016ydg}.