Stable Envelopes, Vortex Moduli Spaces, and Verma Modules

TL;DR

This paper constructs stable envelopes for vortex moduli spaces, linking enumerative geometry, quantum difference equations, and representation theory, with applications to quantum q-Langlands correspondence in a novel geometric setting.

Contribution

It explicitly constructs K-theoretic and elliptic stable envelopes for vortex moduli spaces and relates quantum difference equations to quantum KZ equations, revealing new geometric and representation-theoretic insights.

Findings

Identified quantum difference equations with quantum KZ equations.

Expressed monodromy in terms of geometric elliptic R-matrices.

Connected vortex moduli spaces to quantum q-Langlands correspondence.

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.