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 $\mathcal{X}$, the moduli space of singular $G$-monopoles, where $G$ is related to ${}^{L}G$ 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.

107 pages, 18 figures. v2: references added, minor corrections

References in corpus (5)

• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Spaces of stability conditions
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• Lectures on D-branes and Sheaves
• Algebra of the Infrared: String Field Theoretic Structures in Massive $\mathcal{N}=(2,2)$ Field Theory In Two Dimensions

Cited by in corpus (3)

• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs