Tobias Ekholm: Knot contact homology, Chern-Simons, and topological strings
We discuss recent developments relating contact geometry in dimension 5 to physically inspired topology in dimension 3. A brief description is as follows:
In 1989 Witten explained the Jones and HOMFLY polynomials of knot theory in terms of Chern-Simons theory gauge theory in dimension 3. In 1992 he further showed that Chern-Simons theory of a 3-manifold is equivalent to an open topological string theory in its cotangent bundle. In 1999 Ooguri-Vafa reinterpreted this open string theory for the 3-sphere as a closed string theory in the resolved conifold X, or, in mathematical language, the Gromov-Witten theory of X that counts holomorphic curves. They similarly gave curve counting interpretations of the HOMFLY polynomial. In 2014 Aganagic, Ekholm, Ng, and Vafa related the counts of the simplest curves, the disks, for the HOMFLY to knot contact homology, which is a Floer homological theory in the unit cotangent bundle of the 3-sphere and which can be combinatorially computed. This gives a new effective way of approaching the theories discussed above from infinity. Work in progress gives similar descriptions of curve counts at arbitrary genus through a more elaborate theory at infinity.
The talk will survey these developments and will be accessible to a general mathematical audience.