Algebraic sheaves of Floer homology groups via algebraic torus actions on the Fukaya category

Abstract: Let $(M,\omega_M)$ be a monotone or negatively monotone symplectic manifold, or a Weinstein manifold. One can construct an "action" of $H^1(M,\mathbb{G}_m)$ on the Fukaya category (wrapped Fukaya category in the exact case) that reflects the action of $Symp^0(M,\omega_M)$ on the set of Lagrangian branes. A priori this action is only analytic. The purpose of this work is to show the algebraicity of this action under some assumptions. We use this to prove a tameness result for the sheaf of Lagrangian Floer homology groups obtained by moving one of the Lagrangians via global symplectic isotopies. We also show the algebraicity of the locus of $z\in H^1(M,\mathbb{G}_m)$ that fix a Lagrangian brane in the Fukaya category. The latter has applications to Lagrangian flux. Finally, we prove a statement in mirror symmetry: in the Weinstein case, assume that $M$ is mirror to an affine or projective variety $X$, that there exists an exact Lagrangian torus $L\subset M$ such that $H^1(M)\to H^1(L)$ is surjective, and that $L$ is sent to a smooth point of $x\in X$ under the mirror equivalence. Then we construct a Zariski chart of $X$ containing $x$, that is isomorphic to $H^1(L,\mathbb{G}_m)$, and such that other points of this chart correspond to non-exact deformations of $L$ (possibly equipped with unitary local systems). In particular, this implies rationality of the irreducible component containing $x$; however, it is stronger. Under our assumptions, one can construct an algebraic action of $H^1(M,\mathbb{G}_m)$, namely the action by non-unitary local systems. By combining techniques from family Floer homology and non-commutative geometry, we prove that this action coincides with the geometric action mentioned in the first paragraph. We use this to deduce the theorems above.