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

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