Link Homologies and the Refined Topological Vertex

Abstract

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorifying quantum group invariants associated with other representations (R_1, R_2). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.

38 pages, 5 figures

References in corpus (6)

• Floer homology and knot complements
• The Refined Topological Vertex
• Matrix Factorizations and Kauffman Homology
• Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial
• On a proof of the Labastida-Marino-Ooguri-Vafa conjecture
• Proof of the Labastida-Marino-Ooguri-Vafa Conjecture

Cited by in corpus (56)

• The Refined Topological Vertex
• Refined BPS state counting from Nekrasov's formula and Macdonald functions
• Refined, Motivic, and Quantum
• Knot Homology from Refined Chern-Simons Theory
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Fiber-Base Duality and Global Symmetry Enhancement
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• Surface Operators and Knot Homologies
• Knots-quivers correspondence
• 5d/6d DE instantons from trivalent gluing of web diagrams
• Refined Chern-Simons Theory and Topological String
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Super-A-polynomials for Twist Knots
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• On 3d extensions of AGT relation
• Changing the preferred direction of the refined topological vertex
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• 3d-3d Correspondence Revisited
• Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials
• Computing Khovanov-Rozansky homology and defect fusion
• Orientifolds and the Refined Topological String
• Flop Invariance of Refined Topological Vertex and Link Homologies
• Homological algebra of knots and BPS states
• Macdonald operators and homological invariants of the colored Hopf link
• Refined Chern-Simons Theory and Knot Homology
• The MacMahon R-matrix
• Hopf link invariants and integrable hierarchies
• Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials
• Tangle blocks in the theory of link invariants
• Refined Hopf Link Revisited
• On Hopf-induced deformation of topological locus
• Challenges of beta-deformation
• Kerov functions for composite representations and Macdonald ideal
• Refined Topological Strings on Local ${\mathbb{P}}^2$
• A non-torus link from topological vertex
• Can tangle calculus be applicable to hyperpolynomials?
• q,t-Catalan numbers and knot homology
• Selfduality and Chern-Simons Theory
• Condensates and instanton - torus knot duality. Hidden Physics at UV scale
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• The Condensate from Torus Knots
• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• BPS states, torus links and wild character varieties
• Topological String on Toric CY3s in Large Complex Structure Limit
• M-theory interpretation of the real topological string
• Exact results and Schur expansions in quiver Chern-Simons-matter theories
• $S$-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants
• A hitchhiker's guide to Khovanov homology
• Hopf superpolynomial from topological vertices
• Towards a gauge theory interpretation of the real topological string
• Matrix model and dimensions at hypercube vertices
• Topological Vertex, String Amplitudes and Spectral Functions of Hyperbolic Geometry
• A Proof of the Full Marino-Vafa Conjecture
• Khovanov--Rozansky matrix factorization reduction for bipartite links
• The 1,2-coloured HOMFLY-PT link homology