Knot Categorification from Mirror Symmetry, Part II: Lagrangians

Understand

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 $\mathfrak{q}$.