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

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.

75 pages, tikz and xypic figures

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• sl(N)-Web categories
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• A hitchhiker's guide to Khovanov homology
• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• Trivalent vertices and bordered knot Floer homology in the standard basis
• gl(2) foams and the Khovanov homotopy type
• Categorification of infinite-dimensional ${\mathfrak{sl}}_2$-modules and braid group 2-actions I: tensor products
• A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$