Abstract
The Coulomb phase of a quantum field theory, when present, illuminates the
analysis of its line operators and one-form symmetries. For 4d $\mathcal{N}=2$
field theories the low energy physics of this phase is encoded in the special
K\"ahler geometry of the moduli space of Coulomb vacua. We clarify how the
information on the allowed line operator charges and one-form symmetries is
encoded in the special K\"ahler structure. We point out the important
difference between the lattice of charged states and the homology lattice of
the abelian variety fibered over the moduli space, which, when principally
polarized, is naturally identified with a choice of the lattice of mutually
local line operators. This observation illuminates how the distinct S-duality
orbits of global forms of $\mathcal{N}=4$ theories are encoded geometrically.