Vertex operators and 2-representations of quantum affine algebras

TL;DR

This paper develops a categorical framework to construct 2-representations of quantum affine algebras using vertex operators, leading to new categorifications of their basic representations and actions on derived categories.

Contribution

It introduces categorical vertex operators to build 2-representations of quantum affine algebras from Heisenberg 2-representations, advancing categorification methods.

Findings

Categorifies the Frenkel-Kac-Segal realization of quantum affine algebra representations.

Establishes categorical actions on derived categories of coherent sheaves on Hilbert schemes.

Provides a new approach to categorify quantum affine and toroidal algebras.

Abstract

We construct 2-representations of quantum affine algebras from 2-representations of quantum Heisenberg algebras. The main tool in this construction are categorical vertex operators, which are certain complexes in a Heisenberg 2-representation that recover vertex operators after passing to the Grothendieck group. As an application we categorify the Frenkel-Kac-Segal homogeneous realization of the basic representation of (simply laced) quantum affine algebras. This gives rise to categorical actions of quantum affine (and toroidal) algebras on derived categories of coherent sheaves on Hilbert schemes of points of ALE spaces.