Worldsheet Interpretation of the Level-Rank Duality

TL;DR

This paper explores how level-rank duality in Chern-Simons theories leads to identities between open Gromov-Witten invariants of geometries related to knots and their mirrors, connecting topological string theories in A- and B-models.

Contribution

It demonstrates the implications of level-rank duality for topological string theory, establishing identities between invariants of mirror knot geometries.

Findings

Identities between open Gromov-Witten invariants of knot and mirror geometries

Dualities manifest in both A-model and B-model topological string theories

Connections between Chern-Simons duality and topological string invariants

Abstract

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.