Refined large N duality for knots

Understand

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.