Products on Fixed point Floer homology and closed string mirror symmetry for nodal curves

For a symplectomorphism on a surface, its fixed point Floer cohomology is a cochain complex generated by its fixed points and differential defined by counts of J-holomorphic cylinders. The fixed point Floer cohomology has been computed for all symplectomorphisms on surfaces, but it enjoys additional algebraic structure. It has a “product” coming from counts of pairs of pants. We explain how to explicitly compute this “product” in the case the symplectomorphism is an iteration of a Dehn twists. As an application, we use these computations to calculate the Hochschild cohomology of the Fukaya category of a nodal Riemann surface, which we think of as a “quantum cohomology” for nodal curves . We illustrate how fixed point Floer homology can be used to describe closed string mirror symmetry for nodal curves, and show our results fit in with predictions from mirror symmetry. This is based on joint work with Ziwen Zhao, and work with Maxim Jeffs and Ziwen Zhao.