Towards natural and realistic $E_7$ GUTs in F-theory

We consider phenomenological aspects of a natural class of Standard Model-like supersymmetric F-theory vacua realized through flux breaking of rigid $E_7$ gauge factors. Three generations of Standard Model matter are realized in many of these vacua. We further find that many other Standard Model-like features are naturally compatible with these constructions. For example, dimension-4 and 5 terms associated with proton decay are ubiquitously suppressed. Many of these features are due to the group theoretical structure of $E_7$ and associated F-theory geometry. In particular, a set of approximate global symmetries descends from the $E_7$ group, leading to exponential suppression of undesired couplings.