How Does Gauge Theory Connect to Khovanov Homology of Knots?

[Kevin_Axion]

http://arxiv4.library.cornell.edu/abs/1101.3216
Fivebranes and Knots
Edward Witten
146 pages
(Submitted on 17 Jan 2011)
Abstract
We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approches involved other descriptions of the relevant spaces of BPS states). The starting point is a system of D3-branes ending on an NS5-brane with a nonzero theta-angle. On the one hand, this system can be related to a Chern-Simons gauge theory on the boundary of the D3-brane worldvolume; on the other hand, it can be studied by standard techniques of $S$-duality and $T$-duality. Combining the two approaches leads to a new and manifestly invariant description of the Jones polynomial of knots, and its generalizations, and to a manifestly invariant description of Khovanov homology, in terms of certain elliptic partial differential equations in four and five dimensions

Here is a link for more information about his research and what it is attempting to achieve: http://ldtopology.wordpress.com/2011/01/20/newsflash-wittens-new-preprint/

 

[AndyW]

Dear fellow scientist,

I am excited to see that you are exploring the connection between Khovanov homology and gauge theory in your recent paper, "Fivebranes and Knots". Your approach using D3-branes and an NS5-brane with a nonzero theta-angle is a novel and interesting one.

I believe that your work has the potential to provide a new and more comprehensive understanding of the Jones polynomial and its generalizations, as well as Khovanov homology. Your use of $S$-duality and $T$-duality in studying this system is a clever and powerful tool, and I am eager to see the results of combining these techniques.

Your description of Khovanov homology in terms of elliptic partial differential equations in four and five dimensions is intriguing, and I am curious to see how this approach will shed light on the topological properties of knots.

I am also interested in how your research may have implications for other areas of physics, such as string theory and quantum field theory. Perhaps your work could lead to new insights and connections between these fields.

Overall, I am impressed by the depth and creativity of your approach and I look forward to following your progress in this area. Your paper has already generated a lot of excitement and I am sure it will continue to do so in the scientific community.