Stable Envelopes, Vortex Moduli Spaces, and Verma Modules

Abstract: We explicitly construct K-theoretic and elliptic stable envelopes for certain moduli spaces of vortices, and apply this to enumerative geometry of rational curves in these varieties. In particular, we identify the quantum difference equations in equivariant variables with quantum Knizhnik-Zamolodchikov equations, and give their monodromy in terms of geometric elliptic R-matrices. A novel geometric feature in these constructions is that the varieties under study are not holomorphic symplectic, yet nonetheless have representation-theoretic significance. In physics, they originate from 3d supersymmetric gauge theories with $\mathcal{N} = 2$ rather than $\mathcal{N} = 4$ supersymmetry. We discuss an application of the results to the ramified version of the quantum q-Langlands correspondence of Aganagic, Frenkel, and Okounkov.