Refined large N duality for knots

Abstract

We formulate large [Formula: see text] duality of [Formula: see text] refined Chern–Simons theory with a torus knot/link in [Formula: see text]. 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 [Formula: see text]-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in [Formula: see text] to refined BPS invariants in the resolved conifold. Assuming that the extra [Formula: see text] 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 also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.