Worldsheet Interpretation of the Level-Rank Duality

Level-rank duality relates the observables of two different Chern-Simons theories in which the roles of the Chern-Simons level and the rank of the gauge group are exchanged. In this note, we explore the consequences of this duality in the realm of topological string theory. We show that this duality induces a number of identities between the open Gromov-Witten invariants of the geometries associated with a knot ${\cal K}$ and its mirror image $\tilde{\cal K}$. We show how these identities arise both in the A-model and in the dual B-model.