Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving representations of the loop group $LG$ at level $k$, and another involving representations of the quantum group $U_q(\mathfrak{g})$ where $q$ is some function of $k$. This modular tensor category is in particular a braided monoidal category with duals, so an object in it gives a knot invariant. In the case that $G = \text{SU}(2)$ and the object is the defining representation this knot invariant is the Jones polynomial evaluated at $q$.

In the same way that the Jones polynomial comes from 3d Chern-Simons, it's expected that Khovanov homology comes from a 4d TQFT which gives 3d Chern-Simons upon dimensional reduction. I think this 4d TQFT should in particular assign something like a braided monoidal $2$-category with duals to the circle, the objects of which should have something to do with representations of $G$, and this braided monoidal $2$-category with duals should determine all of the knot invariants assigned by this 4d TQFT (Khovanov homology for various representations of $G$). I don't know how $k$ should figure into this story, since from the perspective of Khovanov homology $q$ is a formal variable.

Is a description of this braided monoidal $2$-category with duals known in terms analogous to either the loop group or the quantum group description of the corresponding braided monoidal category in the case of Chern-Simons?

I understand that various authors have been busy categorifying the representation theory of quantum groups, but I don't have a good sense of the extent to which this work lets you write down braided monoidal $2$-categories with duals. In the ones I'm aware of it looks like the objects categorifying particular representations of the quantum group are written down "by hand" rather than being described as "the such-and-such representations of this categorified quantum group"; in particular I don't think these constructions exhaust all of the possible objects. I have no idea if anything's been done on the loop group side.

Answers regarding the entire 4d TQFT giving Khovanov homology would also be more than welcome! (I don't have a good sense of what's known or expected here.)

  • $\begingroup$ I have no idea if this is in any way related to Khovanov homology, but Dennis Gaitsgory has done some interesting work in the direction of local geometric Langlands duality, where one would like to understand $\infty$-categories equipped with an action of the "algebraic" loop group $G((t))$. See, for example, math.harvard.edu/~gaitsgde/GL/shvsofcat.pdf. If this gives the "right notion" of a categorified loop group representation, it might also lead to an interesting TFT. $\endgroup$ – Yonatan Harpaz Feb 29 '16 at 20:51

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