Khovanov Homology from Mirror Symmetry

Khovanov showed, more than 20 years ago, that there is a deeper theory underlying the Jones polynomial. The``knot categorification problem” is to find a uniform description of this theory, for all gauge groups, which originates from physics. I found two solutions to this problem, related by a version of two dimensional (homological) mirror symmetry. They are based on two descriptions of the theory that lives on defects of the six dimensional (0,2) CFT, which are supported on a link times ``time". 

In this talk, I will focus on the description in terms of A-branes, which is new and surprising. (While the B-brane description is new as well, it shares flavors of theories mathematicians discovered earlier.) The theory turns out to be solvable explicitly. In the Khovanov homology case, the A-model gives a way to compute it which is significantly more efficient than the algebraic descriptions mathematicians have found.