Crane-Yetter(-Kauffman) have constructed 4-dimensional TQFT in such a way that Reshetikhin-Turaev theory lives on the boundary $\partial M$ of a 4-manifold $M$. Therefore, Crane-Yetter TQFT can be thought of a categorification of Chern-Simons theory. If we add surface defect (or foam) in Crane-Yetter theory that is an extension of a knot in Chern-Simons theory, then does it provide a categorification of quantum invariants of the knot which is Khovanov homology?

On the other hand, Witten has proposed in the paper that TQFT obtained by topologically twisting 4d N=4 SYM provides a framework for a categorification of Chern-Simons theory as well as Khovanov homology, which looks different from Crane-Yetter TQFT. I wonder how Crane-Yetter TQFT is related to Khovanov homology.

This post imported from StackExchange MathOverflow at 2017-07-02 10:59 (UTC), posted by SE-user Satoshi Nawata