Morse Theory of Loop Spaces and Hecke Algebras

One can associate an HDHF (symmetric) wrapped Fukaya category to a Liouville domain by counting higher genus curves, which are required to be branched covers. For the cotangent bundle of an orientable surface with genus at least one Honda, Tian, and Yuan showed that the A∞-algebra associated with k disjoint cotangent fibers is quasi-equivalent to the HOMFLY-PT braid skein algebra of the surface. In this talk, I will present a Morse-theoretic model that computes the HDHF A∞-algebra of k fibers of the cotangent bundle of an orientable smooth manifold. We use this model to describe the A∞-algebra of k cotangent fibers of the two-dimensional sphere, and show that it is quasi-equivalent to a certain dga. This talk is based on a joint work with Honda, Tian, and Yuan.