BPS states, knots and quivers

Abstract

We argue how to identify supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot we associate a quiver, so that various types of knot invariants are expressed in terms of characteristics of a moduli space of representations of the corresponding quiver. This statement can be regarded as a novel type of categorification of knot invariants, and among its various consequences we find that Labastida-Mariño-Ooguri-Vafa (LMOV) invariants of a knot can be expressed in terms of motivic Donaldson-Thomas invariants of the corresponding quiver; this proves integrality of LMOV invariants, conjectured originally based on string theory and M-theory arguments.

arXiv comment6 pages, 1 figure

References in corpus (15)

• Knot Invariants and Topological Strings
• Quantum Quivers and Hall/Hole Halos
• Khovanov-Rozansky Homology and Topological Strings
• Matrix factorizations and link homology II
• Knots, links and branes at large N
• On the algebras of BPS states
• Polynomial invariants for torus knots and topological strings
• On Link Invariants and Topological String Amplitudes
• Cohomological Hall algebra of a symmetric quiver
• Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants
• 3d analogs of Argyres-Douglas theories and knot homologies
• Sequencing BPS Spectra
• Knots, BPS states, and algebraic curves
• BPS counting for knots and combinatorics on words
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Cited by in corpus (26)

• Knots-quivers correspondence
• The quantum tropical vertex
• Blowup Equations for Refined Topological Strings
• Topological strings, strips and quivers
• Donaldson-Thomas invariants, torus knots, and lattice paths
• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories
• Differential expansion for link polynomials
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Branches, quivers, and ideals for knot complements
• Rational links and DT invariants of quivers
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Tangle addition and the knots-quivers correspondence
• Branes, quivers and wave-functions
• Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d $\mathcal{N}=2$ theories
• Nahm sums, quiver A-polynomials and topological recursion
• On explicit formulae of LMOV invariants
• Quiver diagonalization and open BPS states
• Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again
• 3d $\mathcal{N}=2$ Brane Webs and Quiver Matrices
• Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition
• Refined open topological strings revisited
• Open/Closed BPS Correspondence and Integrality
• The colored Jones polynomials as vortex partition functions
• Inverted state sums, inverted Habiro series, and indefinite theta functions
• Permutohedra for knots and quivers