Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra

TL;DR

This paper applies algebraic techniques to 3D $ ext{N}=2$ gauge theories, deriving vortex partition functions and qq-characters from quantum affine algebra representations, linking brane systems to algebraic structures.

Contribution

It introduces a novel network of intertwiners involving prefundamental and Fock representations to engineer 3D $ ext{N}=2$ theories using quantum affine algebra.

Findings

Derived vortex partition functions and qq-characters from algebraic networks.

Identified brane configurations corresponding to algebraic modules.

Highlighted the role of shifted quantum algebras in Higgsing.

Abstract

The algebraic engineering technique is applied to a class of 3D $\mathcal{N}=2$ gauge theories on the omega-deformed background $\mathbb{R}_\epsilon^2\times S^1$. The vortex partition function and the fundamental qq-character are obtained from a network of intertwiners between representations of the shifted (or asymptotic) quantum affine $\mathfrak{sl}(2)$ algebra. This network involves two types of representations, the prefundamental representation of Hernandez-Jimbo, and a new vertex representation acting on a bosonic Fock space. The brane system associated to this network is identified: D3 branes carry the prefundamental module while NS5-branes (+D5) support the Fock module. In the process, we highlight the role of shifted quantum algebras in implementing the Higgsing procedure.