Some recent work by Aganagic on knot categorification discusses two categorical approaches to categorification of quantum link invariants and suggests that the two approaches are connected by equivariant homological mirror symmetry, arriving at a geometric formulation of Khovanov homology.

The article states that both the categorical approaches ''come from'' string theory in some sense and that they can be related to a more physical approach which is being developed by Witten.

I think the type of article being referred to when she talks about Witten's approach is this one by Gaiotto and Witten, which is much more physically motivated. Can someone who knows this work explain (roughly speaking) how the physics approach is connected to what is described in the preprint of Aganagic, or where are the places where they connect? Are there parts of Witten's approach which are made more mathematically rigorous in the other two approaches mentioned by Aganagic?