Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)

TL;DR

This paper connects Khovanov homology and foam constructions to higher representation theory, showing they arise from 2-representations of categorified quantum sl(m) via skew Howe duality, unifying foam-based categorifications.

Contribution

It demonstrates that foam constructions of Khovanov homology are 2-representations of categorified quantum sl(m), providing a new higher representation theoretic perspective.

Findings

Foam constructions arise from 2-representations of categorified quantum sl(m).

Unified construction of foam-based categorifications of Jones-Wenzl projectors.

Reveals the significance of modified foams in sl(2) and suggests similar modifications for sl(3).

Abstract

We show that Khovanov homology (and its sl(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 sl(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 sl(3) analogs purely from the higher representation theory of categorified quantum groups. In the sl(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 sl(3) foam category introduced here.