Action of the Witt algebra on categorified quantum groups

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.