Refined large N duality for knots

Abstract

We formulate large $N$ duality of $\mathrm{U}(N)$ refined Chern-Simons theory with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the $\Omega$-background. This form enables us to relate refined Chern-Simons invariants of a torus knot/link in $S^3$ to refined BPS invariants in the resolved conifold. Assuming that the extra $\mathrm{U}(1)$ global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2-M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be interpreted as a positivity conjecture of refined Chern-Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

Two Mathematica notebooks are attached as ancillary files; in v2, the title is changed, an extension to non-torus knots is included in section 6, typos are corrected, and references are added; v3, published version

References in corpus (4)

• Generalized Superconformal Index for Three Dimensional Field Theories
• Refined BPS invariants of 6d SCFTs from anomalies and modularity
• Refined Chern-Simons Theory and Topological String
• BPS states, torus links and wild character varieties

Cited by in corpus (8)

• Knots-quivers correspondence
• Refined geometric transition and $qq$-characters
• Checks of integrality properties in topological strings
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Gaussian distribution of LMOV numbers
• Refined Topological Branes
• Hopf superpolynomial from topological vertices
• Refined open topological strings revisited