Algebraic torus actions on Fukaya categories and tameness of change in Floer homology under symplectic isotopies

Algebraic torus actions on Fukaya categories and tameness of change in Floer homology under symplectic isotopies

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Yusuf Barış Kartal, Princeton University

Zoom link:  :  https://umontreal.zoom.us/j/94366166514?pwd=OHBWcGluUmJwMFJyd2IwS1ROZ0FJdz0

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 $(M,\omega)$ satisfying a strong non-degeneracy condition. More precisely, given two Lagrangian branes $L,L'$, consider family of Floer homology groups $HF(\phi_v(L),L')$, where $v\in H^1(M,\mathbb R)$ and $\phi_v$ 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 $H^1(M,\mathbb G_m)$. The main tool is the construction of an "algebraic action" of $H^1(M,\mathbb G_m)$ 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 $H^1(M,\mathbb G_m)$. Similarly, given closed $1$-form $\alpha$, which generates a symplectic isotopy denoted by $\phi_\alpha^t$, the Floer homology groups $HF(\phi_\alpha^t(L),L')$ have rank that is constant in $t$, with finitely many possible exceptions.