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 $\mathcal{X}$ 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 $\mathcal{X}$ 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.

210 pages, 40 figures, minor corrections, downstairs algebra corrected, added Floer theory description of Khovanov's original complexes

References in corpus (14)

• Non-Abelian Anyons and Topological Quantum Computation
• Floer homology and knot complements
• 2-Kac-Moody algebras
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Spaces of stability conditions
• Noncommutative Counterparts of the Springer Resolution
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• Algebra of the Infrared: String Field Theoretic Structures in Massive $\mathcal{N}=(2,2)$ Field Theory In Two Dimensions
• ADE Little String Theory on a Riemann Surface (and Triality)
• D-Branes on Toric Calabi-Yau Varieties
• Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology
• Mirror Symmetry And Quantum Geometry
• Derived equivalences by quantization
• Monodromy of trigonometric KZ equations

Cited by in corpus (2)

• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Microsheaves from Hitchin fibers via Floer theory