The subject of topological strings has been important in both physics and mathematics. From the physical point of view it plays a prominent role in the understanding of string dualities. From the mathematical perspective, Gromov-Witten invariants play a central role in symplectic geometry and related subjects. In many cases amplitudes are directly computed using both physical and mathematical techniques.
The topological string grew out of attempts to extend computations which occurred in the physical string theory. It has turned out that the topological string has many physical applications far beyond those that motivated its original construction. In a sense, the topological string is a natural locus where mathematics and physics meet.
When asked about the use of topological string theory, different string theorists may give very different answers. Some will point at the many interesting mathematical results. Others believe that topological string theory is a very useful toy model to understand more about properties of ordinary string theory in a simplified setting. So to speak, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry. Topological string theory is obtained by a topological twist of the worldsheet description of ordinary string theory: the operators are given different spins. An interesting application of topological strings involves the study of knot and link invariants. In particular it is known that HOMFLY polynomials for links can be reformulated in terms of topological strings on the cotangent bundle of the 3-sphere.
The paper under review, studies the connection between topological strings and contact homology in the context of knot invariants. In particular, the authors establish the relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. They also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau.
Connecting to contact geometry, the paper studies intersection properties of branches of the augmentation variety guided by the relation to \(D\)-modules. This study leads to propose concrete geometric constructions of Lagrangian fillings for links. It also relates the augmentation variety with the large \(N\) limit of the colored HOMFLY, and conjecture to be related to a \(Q\)-deformation of the extension of polynomials associated with the link complement.