  We prove that, for every 6D supergravity theory that has an F-theory
description, the property of charge completeness for the connected component of
the gauge group (meaning that all charges in the corresponding charge lattice
are realized by massive or massless states in the theory) is equivalent to a
standard assumption made in F-theory for how geometry encodes the global gauge
theory by means of the Mordell-Weil group of the elliptic fibration. This
result also holds in 4D F-theory constructions for the parts of the gauge group
that come from sections and from 7-branes. We find that in many 6D F-theory
models the full charge lattice of the theory is generated by massless charged
states; this occurs for each gauge factor where the associated anomaly
coefficient satisfies a simple positivity condition. We describe many of the
cases where this massless charge sufficiency condition holds, as well as
exceptions where the positivity condition fails, and analyze the related global
structure of the gauge group and associated Mordell-Weil torsion in explicit
F-theory models.
