Physics and Geometry Seminar

Seidel and Smith constructed a knot invariant from the Fukaya category of nilpotent slices, which they called symplectic Khovanov homology, and conjectured that it is isomorphic to Khovanov homology. One approach to proving Seidel and Smith┬ conjecture relies on showing that a given subcategory of the Fukaya category of such nilpotent slices is equivalent, as an A-infinity algebra, to Khovanov's arc algebra; in particular one must show that such a category is formal.

Using Manolescu's observation that the symplectic manifolds used to define symplectic Khovanov homology are Hilbert schemes of points on Milnor fibres of A_n singularities, I will construct a partial compactification of these spaces. Then, I will explain how this compactification gives rise to a Hochschild cohomology class of degree one, which induces a second grading on Lagrangian Floer homology groups, and implies formality of the subcategory by applying an abstract criterion due to Seidel. Time permitting, I will then explain how one can further use Wehrheim-Woodward and M'au's theory of A-infinity functors induced by Lagrangian correspondences to prove the formality of the cup and cap functors in symplectic Khovanov homology. This is joint work with Ivan Smith.