Knot Categorification from Mirror Symmetry, Part II: Lagrangians

Abstract

I provide two solutions to the problem of categorifying quantum link invariants, which work uniformly for all gauge groups and originate in geometry and string theory. The first is based on a category of equivariant B-type branes on ${\cal X}$ which is a moduli space of singular $G$-monopoles on ${\mathbb R}^3$. In this paper, I give the second approach, which is based on a category of equivariant A-type branes on $Y$ with potential $W$. The first and the second approaches are related by equivariant homological mirror symmetry: $Y$ is homological mirror to $X$, a core locus of ${\cal X}$ preserved by an equivariant action related to $\mathfrak{q}$. The theory of equivariant A-branes on $Y$ is the same as the derived category of modules of an algebra $A$, which is a cousin of the algebra considered by Khovanov, Lauda, Rouquier and Webster, but simpler. The result is a new, geometric formulation of Khovanov homology, which generalizes to all groups. In part III, I will explain the string theory origin of the two approaches, and the relation to an approach being developed by Witten. The three parts may be read independently.