Vortices and Vermas

TL;DR

This paper investigates the algebraic and geometric structures of vortices and monopole operators in 3d N=4 gauge theories, revealing new mathematical frameworks and connections to quantum algebra, moduli spaces, and partition functions.

Contribution

It introduces a novel mathematical definition of the Coulomb-branch algebra and constructs vortex partition functions via coherent states, extending previous work on the AGT correspondence.

Findings

Monopole operators generate a non-commutative algebra quantizing the Coulomb-branch chiral ring.

Vortex moduli spaces are realized as handsaw quiver varieties, with monopole operators as interfaces.

Vortex partition functions are constructed as overlaps of Whittaker vectors for Coulomb-branch algebras.

Abstract

In three-dimensional gauge theories, monopole operators create and destroy vortices. We explore this idea in the context of 3d N=4 gauge theories in the presence of an Omega-background. In this case, monopole operators generate a non-commutative algebra that quantizes the Coulomb-branch chiral ring. The monopole operators act naturally on a Hilbert space, which is realized concretely as the equivariant cohomology of a moduli space of vortices. The action furnishes the space with the structure of a Verma module for the Coulomb-branch algebra. This leads to a new mathematical definition of the Coulomb-branch algebra itself, related to that of Braverman-Finkelberg-Nakajima. By introducing additional boundary conditions, we find a construction of vortex partition functions of 2d N=(2,2) theories as overlaps of coherent states (Whittaker vectors) for Coulomb-branch algebras, generalizing work of Braverman-Feigin-Finkelberg-Rybnikov on a finite version of the AGT correspondence. In the case of 3d linear quiver gauge theories, we use brane constructions to exhibit vortex moduli spaces as handsaw quiver varieties, and realize monopole operators as interfaces between handsaw-quiver quantum mechanics, generalizing work of Nakajima.