Computations in monotone Floer theory

Abstract

Floer theory is a rich collection of tools for studying symplectic manifolds and their Lagrangian submanifolds with the help of holomorphic curves. Its origins lie in estimating the numbers of equilibria in Hamiltonian dynamics, and more recently it has become a major component of the Homological Mirror Symmetry conjecture. This work presents several new computations in Floer theory which combine the use of geometric symmetries, naturally arising in various contexts, with advanced algebraic structures related to Floer theory, like the string maps and the Fukaya category. The three main directions of our study are: the Floer cohomology for a pair of commuting symplectomorphisms; the Fukaya algebra of a Lagrangian submanifold invariant under a circle action; and rigidity properties of non-monotone Lagrangian submanifolds based on the use of low-area versions of the string maps. In each of the three mentioned setups we provide concrete applications of our general results to the study of symplectic manifolds. For example, we prove that Dehn twists in most projective hypersurfaces have infinite order in the symplectic mapping class group; prove that the real projective space split-generates the Fukaya category of the complex projective space and therefore must intersect any other Lagrangian submanifold that is nontrivial in that Fukaya category; and we exhibit a continuous family of Lagrangian tori in the complex projective plane that cannot be made disjoint from the standard Clifford torus by a Hamiltonian isotopy.