Khovanov homology is a skew Howe $2$–representation of categorified quantum $\mathfrak{sl}_m$

Abstract

We show that Khovanov homology (and its ${\mathfrak{s}\mathfrak{l}}_3$ variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of $2$–representations of categorified quantum ${\mathfrak{s}\mathfrak{l}}_m$ via categorical skew Howe duality. Utilizing Cautis–Rozansky categorified clasps we also obtain a unified construction of foam-based categorifications of Jones–Wenzl projectors and their ${\mathfrak{s}\mathfrak{l}}_3$ analogs purely from the higher representation theory of categorified quantum groups. In the ${\mathfrak{s}\mathfrak{l}}_2$ case, this work reveals the importance of a modified class of foams introduced by Christian Blanchet which in turn suggest a similar modified version of the ${\mathfrak{s}\mathfrak{l}}_3$ foam category introduced here.