Interfaces and Quantum Algebras, I: Stable Envelopes

TL;DR

This paper explores the geometric realization of quantum algebra representations via stable envelopes and connects these ideas to supersymmetric gauge theories, supersymmetric interfaces, and Berry connections, with implications for symplectic duality and Bethe/gauge correspondence.

Contribution

It establishes a novel link between geometric stable envelopes and supersymmetric gauge theories, expanding the understanding of quantum algebra representations in a physical context.

Findings

Relation of stable envelopes to supersymmetric interfaces

Connection between vacua and generalized cohomology

Role of Berry connections in gauge theories

Abstract

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.