Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves

Abstract

We derive two geometric approaches to categorification of quantum invariants of links associated to an arbitrary compact simple Lie group $^L{G}$. In part I, we describe the first approach, based on an equivariant derived category of coherent sheaves on ${\cal X}$, the moduli space of singular $G$-monopoles, where $G$ is related to $^LG$ by Langlands duality. In part II, we describe the second approach, based on the derived category of a Fukaya-Seidel category of a Calabi-Yau $Y$ with potential $W$. The two approaches are related by a version of mirror symmetry, which plays a crucial role in the story. In part III, we explain the string theory origin of these results, and the relation to an approach due to Witten.