# Refined large N duality for knots

## TL;DR

This paper formulates a refined large N duality connecting refined Chern-Simons theory for torus knots in S^3 with refined BPS invariants in the resolved conifold, providing new insights into knot invariants and string dualities.

## Contribution

It explicitly relates refined Chern-Simons invariants of torus knots to refined BPS invariants via a duality framework, including predictions about cohomology of moduli spaces.

## Key findings

- Derived explicit form of low-energy effective actions in Type IIA string theory with D4-branes.
- Established a duality predicting graded dimensions of cohomology groups for moduli spaces of M2-M5 bound states.
- Proposed a positivity conjecture for refined Chern-Simons invariants of torus knots and discussed extensions to non-torus 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.