Non-perturbative schwinger-dyson equations for 3d \( \mathcal{N} \) = 4 gauge theories

Abstract

We analyze symmetries corresponding to separated topological sectors of 3d 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 3-dimensional Seiberg duality.