Interfaces and Quantum Algebras, I: Stable Envelopes

The stable envelopes of Okounkov et al. realize some representations of quantum algebras associated to quivers, using geometry. We relate these geometric considerations to quantum field theory. The main ingredients are the supersymmetric interfaces in gauge theories with four supercharges, relation of supersymmetric vacua to generalized cohomology theories, and Berry connections. We mainly consider softly broken compactified three dimensional $\mathcal{N} =4$ theories. The companion papers will discuss applications of this construction to symplectic duality, Bethe/gauge correspondence, generalizations to higher dimensional theories, and other topics.