  We analyze symmetries corresponding to separated topological sectors of 3d
${\cal} N=4$ gauge theories with Higgs vacua, compactified on a circle. The
symmetries are encoded in Schwinger-Dyson identities satisfied by correlation
functions of a certain gauge-invariant operator, the "vortex character." Such a
character observable is realized as the vortex partition function of the 3d
gauge theory, in the presence of a 1/2-BPS line defect. The character enjoys a
double refinement, interpreted as a deformation of the usual characters of
finite-dimensional representations of quantum affine algebras. We derive and
interpret the Schwinger-Dyson identities for the 3d theory from various
physical perspectives: in the 3d gauge theory itself, in a 1d gauged quantum
mechanics, in 2d $q$-Toda theory, and in 6d little string theory. We establish
the dictionary between all approaches. Lastly, we comment on the transformation
properties of the vortex character under the action of three-dimensional
Seiberg duality.
