Gauged linear sigma model and infinitude of Hamiltonian periodic orbits

Abstract: Take an irrational rotation of the two-sphere; it only has the north and south poles as its periodic points. However, Franks proved that for any area-preserving diffeomorphism of the two-sphere, if it has more than two fixed points, then it must have infinitely many periodic points. I will discuss a generalization with Guangbo Xu of this result to all compact toric manifolds in the form of a “Betti number or infinity” dichotomy. The Floer theory package and mirror symmetry considerations from gauged linear sigma models, also known as symplectic vortices, play a surprising role.