  We study half-BPS line operators in 3d N=4 gauge theories, focusing in
particular on the algebras of local operators at their junctions. It is known
that there are two basic types of such line operators, distinguished by the
SUSY subalgebras that they preserve; the two types can roughly be called
"Wilson lines" and "vortex lines", and are exchanged under 3d mirror symmetry.
We describe a large class of vortex lines that can be characterized by basic
algebraic data, and propose a mathematical scheme to compute the algebras of
local operators at their junctions --- including monopole operators --- in
terms of this data. The computation generalizes mathematical and physical
definitions/analyses of the bulk Coulomb-branch chiral ring. We fully classify
the junctions of half-BPS Wilson lines and of half-BPS vortex lines in abelian
gauge theories with sufficient matter. We also test our computational scheme in
a non-abelian quiver gauge theory, using a 3d-mirror-map of line operators from
work of Assel and Gomis.
