Holomorphic curves, skeins and branes

The project concerns geometry at the borderline of topological theories in physics, low-dimensional topology, and symplectic geometry. This general area developed over the last three decades with both groundbreaking and famous results.

However, the theory that mediates the between the three subjects, counts of string instantons or holomorphic curves in Calabi-Yau threefolds with Lagrangian boundary condition was obstructed by wall-crossing phenomena that results in non-invariance under deformation.

This obstruction was removed by the PI and Shende by counting holomorphic curves by certain knot invariants of their boundary circles and several key results were thereby established.

This is however only the tip of an iceberg and the purpose of the project is to extend the method of skein valued curve counts and to apply it to derive central results with relevance both in mathematics and in topological physics.

Concrete goals include to develop derived or homological skein valued holomorphic curve theory, with skein-valued Symplectic Field Theory playing a key role and to find the skein-valued open curve analogue of the Gopakumar-Vafa formula where all curves come clustered as multiple covers of certain distinguished basic curves.

Applications include the, long sought after, higher genus extension of Lagrangian Floer cohomology and new recursive structures for colored HOMFLYPT invariants of knots and links.