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.

arXiv comment210 pages, 40 figures, minor corrections, downstairs algebra corrected, added Floer theory description of Khovanov's original complexes

References in corpus (60)

• Non-Abelian Anyons and Topological Quantum Computation
• S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory
• Mirror Symmetry in Three Dimensional Gauge Theories
• A diagrammatic approach to categorification of quantum groups I
• Floer homology and knot complements
• 2-Kac-Moody algebras
• D-Branes And Mirror Symmetry
• D-Branes on Calabi-Yau Spaces and Their Mirrors
• Analytic Continuation of Liouville Theory
• Gaudin Model, Bethe Ansatz and Critical Level
• Analytic Continuation Of Chern-Simons Theory
• On Mirror Symmetry in Three Dimensional Abelian Gauge Theories
• A functor-valued invariant of tangles
• Disk Instantons, Mirror Symmetry and the Duality Web
• Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories
• Perverse sheaves on a Loop group and Langlands' duality
• Khovanov-Rozansky Homology and Topological Strings
• The structure of 2D semi-simple field theories
• Bordered Heegaard Floer homology: Invariance and pairing
• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Quantum Groups and Quantum Cohomology
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Knot invariants and higher representation theory
• Knot Invariants from Four-Dimensional Gauge Theory
• Covariantly functorial wrapped Floer theory on Liouville sectors
• Quasiclassical asymptotics of solutions to the KZ equations
• Derived Categories and Zero-Brane Stability
• D-Brane Stability and Monodromy
• Gaudin models with irregular singularities
• Opers with irregular singularity and spectra of the shift of argument subalgebra
• Quiver Schur algebras and q-Fock space
• Spaces of stability conditions
• Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants
• Rational curves on hypersurfaces [after A. Givental]
• Noncommutative Counterparts of the Springer Resolution
• Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)
• Lectures on K-theoretic computations in enumerative geometry
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• Microlocal Morse theory of wrapped Fukaya categories
• Symplectic cohomology and q-intersection numbers
• Algebra of the Infrared: String Field Theoretic Structures in Massive $\mathcal{N}=(2,2)$ Field Theory In Two Dimensions
• Erratum to "The Homogeneous Coordinate Ring of a Toric Variety", along with the original paper
• ADE Little String Theory on a Riemann Surface (and Triality)
• D-Branes on Toric Calabi-Yau Varieties
• The symplectic arc algebra is formal
• Geometric Langlands duality and representations of algebraic groups over commutative rings
• T-Duality and Homological Mirror Symmetry of Toric Varieties
• Braid group actions on derived categories of coherent sheaves
• Khovanov Homology And Gauge Theory
• Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A
• Quantum q-Langlands Correspondence
• Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces
• Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems
• Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology
• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• Mirror Symmetry And Quantum Geometry
• Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
• Derived equivalences by quantization
• Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves
• 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