Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N} = 4$ Gauge Theories

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.