Algebraic Geometry and Mirror Symmetry Seminar

Products on Fixed point Floer cohomology and closed string mirror symmetry for nodal curves Abstract: The fixed point Floer cohomology is a cochain complex generated by the fixed points of a symplectomorphism. It has been computed for all symplectomorphisms on (two real-dimensional) surfaces, but it enjoys additional algebraic structure coming from a pair of pants "product". We explain the computation of this "product" in the case of iterations of a Dehn twists on a surface. We find surprisingly, the resulting structure forms a polynomial ring. Via the lens of mirror symmetry, we relate this ring to sections of tensor powers of a degree 1 line bundle on a nodal elliptic curve. Finally, using the product structure, we explain how to define a version of genus 0 Gromov Witten invariants for nodal Riemann surfaces and compute it. All of this joint work with Maxim Jeffs and Ziwen Zhao.