The purpose of this talk is to explore how Lagrangian Floer homology groups change under (non-Hamiltonian) symplectic isotopies on a (negatively) monotone symplectic manifold (M,ω) satisfying a strong non-degeneracy condition. More precisely, given two Lagrangian branes L,L′, consider family of Floer homology groups HF(ϕv(L),L′), where v ∈ H1(M,ℝ) and ϕv is the time-1 map of a symplectic isotopy with flux v. We show how to fit this collection into an algebraic sheaf over the algebraic torus H1(M,𝔾m). The main tool is the construction of an “algebraic action” of H1(M,𝔾m) on the Fukaya category. As an application, we deduce the change in Floer homology groups satisfy various tameness properties, for instance, the dimension is constant outside an algebraic subset of H1(M,𝔾m). Similarly, given closed 1-form α, which generates a symplectic isotopy denoted by ϕtα, the Floer homology groups HF(ϕtα(L),L′) have rank that is constant in t, with finitely many possible exceptions.
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
