We analyze symmetries corresponding to separated topological sectors of 3d N= 4gauge 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 from various physical perspectives: in the context of the 3d gauge theory itself, in a 1d gauged quantum mechanics, in 2d q-Todatheory, and in 6d little string theory. We establish the dictionary between all approaches.