Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology

Abstract

We describe rules for computing a homology theory of knots and links in ${\mathbb{R}}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of $SU(2)$ magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.

arXiv comment77 pages, 12 figures

References in corpus (15)

• Phases of $N=2$ Theories In Two Dimensions
• D-Branes And Mirror Symmetry
• A New Supersymmetric Index
• Khovanov-Rozansky Homology and Topological Strings
• Knot Invariants from Four-Dimensional Gauge Theory
• Knot Homology from Refined Chern-Simons Theory
• Fivebranes and 3-manifold homology
• Fukaya-Seidel category and gauge theory
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Algebra of the Infrared: String Field Theoretic Structures in Massive $\mathcal{N}=(2,2)$ Field Theory In Two Dimensions
• Quantum Holonomies from Spectral Networks and Framed BPS States
• Khovanov Homology And Gauge Theory
• Refined Chern-Simons Theory in Genus Two
• A homological definition of the Jones polynomial

Cited by in corpus (7)

• Gauge/Bethe correspondence from quiver BPS algebras
• Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?
• Rectangular superpolynomials for the figure-eight knot
• Checks of integrality properties in topological strings
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians