Action of the Witt algebra on categorified quantum groups

TL;DR

This paper constructs an action of the positive Witt algebra on categorified quantum groups related to simply-laced Lie algebras, connecting to foam and current algebra actions.

Contribution

It introduces a new Witt algebra action on categorified quantum groups and demonstrates its compatibility with trace decategorification and foam representations.

Findings

Witt algebra action on categorified quantum groups is established.

In type A, the action induces one on $rak{gl}_n$-foams, matching previous work.

The construction aligns with the action on the current algebra via trace decategorification.

Abstract

We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on $\mathfrak{gl}_n$-foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.