Link Homology from Homological Mirror Symmetry

TL;DR

This paper connects link homology to homological mirror symmetry by using Fukaya categories and A-branes, providing an explicit algorithm for computing Khovanov homology and extending it to all Lie algebras.

Contribution

It introduces a novel method to compute link homology via Fukaya categories and A-branes, offering an explicit algorithm for Khovanov homology and its extension to all Lie algebras.

Findings

Explicit algorithm for computing Khovanov homology.

Extension of the method to all Lie algebras.

Construction of projective resolutions using thimbles.

Abstract

We explain how to calculate link homology for a Lie algebra $\mathfrak{g}$ using the Fukaya category associated to a 2d A-model. Links are represented as configurations of particular A-branes and link homology is given by Homs between these A-branes. In the case of $\mathfrak{g}=\mathfrak{su}_2$, we explain how to explicitly construct projective resolutions of the relevant A-branes in terms of thimbles, whose algebra is known. This gives an explicit algorithm for computing Khovanov homology. This algorithm can be extended to all Lie algebras.