F-GUTs with Mordell-Weil U(1)'s

Understand

In this note we study the constraints on F-theory GUTs with extra $U(1)$'s in the context of elliptic fibrations with rational sections.

• We consider the simplest case of one abelian factor (Mordell-Weil rank one) and investigate the conditions that are induced on the coefficients of its Tate form.
• Converting the equation representing the generic hypersurface $P_{112}$ to this Tate's form we find that the presence of a U(1), already in this local description, is consistent with the exceptional ${\cal E}_6$ and ${\cal E}_7$ non-abelian singularities.
• We briefly comment on a viable ${\cal E}_6\times U(1)$ effective F-theory model.