Title: Commutative algebra and the categorification of the SU(N) HOMFLY
Speaker: Professor Lev Rozhansky (UNC at Chapel Hill)
I will review our joint work with M. Khovanov on the categorification of the HOMFLY polynomial.Date: Monday, March 28, 2005
The main goal of Khovanov's categorification program is to construct a functor from the cobordism category of manifolds into an appropriate category of complexes of graded vector spaces in such a way that the graded Euler characteristic of those complexes would match the quantum invariants of knots and links such as the Jones and HOMFLY polynomials. The starting point is a formula for a quantum polynomial, which presents it as an alternating sum over (non-invariant) polynomials with positive coefficients.
Khovanov has implemented this idea for the SU(2) Jones polynomial by using the Kauffman bracket. A similar formula for the SU(N) HOMFLY polynomial was provided in a paper by Murakami, Ohtsuki and Yamada. However, that latter formula was based on `resolving' a link into a set of special graphs rather than simple circles, and this complicated the guessing of an appropriate differential for the categorifying comples.
This difficulty was resolved through the use of a category of matrix factorizations. This category was introduced by D. Eisenbud in order to describe hypersurface singularities and it appeared recently in quantum field theory for the description of boundary conditions in 2-dimensional topological Landau-Ginzburg models. We associate special matrix factorizations to open graphs and then use their morphisms in order to construct the graded vector spaces and the differentials of our categorification.