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

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.