Abstract

We discuss relations between open topological strings and Chern-Simons theory and contact homology in the context of knot invariants. The starting point on the physics side is the relation between the HOMFLY polynomial and open topological strings (open Gromov-Witten invariants) in the cotangent bundle of the three-sphere, and (after large N transition) in the total space of the sum of two (-1)-line bundles over the projective line. The starting point on the geometry side is to apply contact homology, a theory of Floer homological nature for contact rather than symplectic manifolds, to the co-normal lift of a knot, which is a Legendrian torus in the unit cotangent bundle of the three-sphere with contact form the action form. The physics setup leads to a polynomial knot invariant (a Q-deformation of the A-polynomial) and the geometry setup leads to another polynomial knot invariant, the so called augmentation polynomial. It was recently observed that these two polynomial knot invariants seem to agree and we will discuss the underlying reason. This polynomial, conjecturally, also encodes data for the B-model mirror of the A-model theory mentioned above. If time permits we will also discuss the case of many component links where the corresponding mirror theory is a more involved higher dimensional theory involving a co-isotroppic brane. The talk is based on joint work with Aganagic, Ng, and Vafa.