Floer Homology and Khovanov Homology Reading Group

This talk is an introduction to the work of Witten on physical approaches to knots, especially based on hep-th 1101.3216. Classical approaches to the Jones polynomial are based on various combinatorial constructions which have the downside of not providing manifestly topological invariants. Witten's 1989 paper gives an interpretation via quantum Chern-Simons theory which is manifestly topological but which a priori gives a discrete set of invariants for which which one would not expect any sort of analytic continuation - let alone to a polynomial. The analytic continuation arises upon considering gradient flow for the Chern-Simons functional on the space of complexified gauge fields and replacing the integration cycle which was previously the real gauge fields with a (finite sum of) Lefschetz thimble(s). Miraculously, the flow equations match precisely with the equations studied previously by Kapustin and Witten in the context of the geometric Langlands program which describe the supersymmetry-invariant states on which the path integral localizes for a certain four dimensional topological field theory, and this is dual to a similar theory with somewhat peculiar boundary conditions modified by the presence of knots which force the path integral to localize on a FINITE set of points, thus reducing a computation in quantum gauge theory to one in classical gauge theory corresponding to counting solutions of differential equations with boundary conditions. Time permitting, we will discuss the five dimensional categorification from mathematical and physical viewpoints.