Khovanov homology is a skew Howe 2-representation of categorified quantum slm

Joint work with Aaron Lauda and David Rose (University of Southern California).

Khovanov homology, which categorifies the Jones polynomial, was first introduced in a mostly combinatorial fashion. However, the Jones polynomial has deep roots in the theory of representations of the quantum group slm. We show that similar links appear at the categorified level as well: the original combinatorics of cobordisms (or rather a foamy version of them due to Blanchet) can be understood in terms of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of Khovanov homology arises as a family of 2-representations of categorified quantum slm. The heart of the process is given by categorified skew-Howe duality, due to Cautis-Kamnitzer-Licata, which drives to a very rich presentation of the Reshetikhin-Turaev invariants of knots.

The first talk will be devoted to review webs and skew-Howe duality, as well as the way the braiding can be understood in terms of a quantum Weyl group action, following Cautis, Kamnitzer, Licata and Morrison. The second lecture will focus on categorification of this phenomenon.