Charge completeness and the massless charge lattice in F-theory models of supergravity

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.